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135 changed files (+30/-24638)
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.vscode/settings.json (new)
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@@ -0,0 +1,25 @@{ "files.exclude": { "**/.git": true, "**/.DS_Store": true, "**/*.olean": true, "**/.DS_yml": true, "**/.gitpod.yml": true, "**/.vscode": true, }, "editor.minimap.enabled": false, "editor.acceptSuggestionOnEnter": "off", "editor.quickSuggestions.comments": true, "scm.diffDecorationsGutterVisibility": "hover", "editor.semanticHighlighting.enabled": true, "editor.semanticTokenColorCustomizations": { "[Default Light+]": {"rules": {"variable": "#1a31da"}} }, "editor.unicodeHighlight.nonBasicASCII": false, "editor.unicodeHighlight.ambiguousCharacters": false, "editor.fontLigatures": true, "files.trimTrailingWhitespace": true, "files.trimFinalNewlines": true, "lean4.infoview.emphasizeFirstGoal": true, "lean4.infoview.showExpectedType": false }
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MIL (new)
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@@ -0,0 +1,1 @@../src
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@@ -1,2 +0,0 @@#eval "Hello, World!"
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MIL/C01_Introduction/S02_Overview.lean (deleted)
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@@ -1,57 +0,0 @@import Mathlib.Data.Nat.Basic import Mathlib.Data.Nat.Parity import Mathlib.Tactic open Nat -- These are pieces of data. #check 2 + 2 def f (x : ℕ) := x + 3 #check f -- These are propositions, of type `Prop`. #check 2 + 2 = 4 def FermatLastTheorem := ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n #check FermatLastTheorem -- These are proofs of propositions. theorem easy : 2 + 2 = 4 := rfl #check easy theorem hard : FermatLastTheorem := sorry #check hard -- Here are some proofs. example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ => have hmn : m * n = m * k + m * k := by rw [hk, mul_add] show ∃ l, m * n = l + l from ⟨_, hmn⟩ example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, hk⟩ => ⟨m * k, by rw [hk, mul_add]⟩ example : ∀ m n : Nat, Even n → Even (m * n) := by -- say m and n are natural numbers, and assume n=2*k rintro m n ⟨k, hk⟩ -- We need to prove m*n is twice a natural number. Let's show it's twice m*k. use m * k -- substitute in for n rw [hk] -- and now it's obvious ring example : ∀ m n : Nat, Even n → Even (m * n) := by rintro m n ⟨k, hk⟩; use m * k; rw [hk]; ring example : ∀ m n : Nat, Even n → Even (m * n) := by intros; simp [*, parity_simps]
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@@ -1,7 +0,0 @@import Mathlib.Data.Nat.Basic import Mathlib.Data.Nat.Parity import Mathlib.Tactic open Nat -- There are no exercises in this section.
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MIL/C02_Basics/S01_Calculating.lean (deleted)
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@@ -1,147 +0,0 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Real.Basic -- An example. example (a b c : ℝ) : a * b * c = b * (a * c) := by rw [mul_comm a b] rw [mul_assoc b a c] -- Try these. example (a b c : ℝ) : c * b * a = b * (a * c) := by sorry example (a b c : ℝ) : a * (b * c) = b * (a * c) := by sorry -- An example. example (a b c : ℝ) : a * b * c = b * c * a := by rw [mul_assoc] rw [mul_comm] /- Try doing the first of these without providing any arguments at all, and the second with only one argument. -/ example (a b c : ℝ) : a * (b * c) = b * (c * a) := by sorry example (a b c : ℝ) : a * (b * c) = b * (a * c) := by sorry -- Using facts from the local context. example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h'] rw [← mul_assoc] rw [h] rw [mul_assoc] example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by sorry example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by sorry -- Examples. example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ← mul_assoc, h, mul_assoc] section variable (a b c d e f g : ℝ) example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ← mul_assoc, h, mul_assoc] end section variable (a b c : ℝ) #check a #check a + b #check (a : ℝ) #check mul_comm a b #check (mul_comm a b : a * b = b * a) #check mul_assoc c a b #check mul_comm a #check mul_comm #check @mul_comm end section variable (a b : ℝ) example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by rw [mul_add, add_mul, add_mul] rw [← add_assoc, add_assoc (a * a)] rw [mul_comm b a, ← two_mul] example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := calc (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by rw [mul_add, add_mul, add_mul] _ = a * a + (b * a + a * b) + b * b := by rw [← add_assoc, add_assoc (a * a)] _ = a * a + 2 * (a * b) + b * b := by rw [mul_comm b a, ← two_mul] example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := calc (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by sorry _ = a * a + (b * a + a * b) + b * b := by sorry _ = a * a + 2 * (a * b) + b * b := by sorry end -- Try these. For the second, use the theorems listed underneath. section variable (a b c d : ℝ) example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := by sorry example (a b : ℝ) : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by sorry #check pow_two a #check mul_sub a b c #check add_mul a b c #check add_sub a b c #check sub_sub a b c #check add_zero a end -- Examples. section variable (a b c d : ℝ) example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by rw [hyp'] at hyp rw [mul_comm d a] at hyp rw [← two_mul (a * d)] at hyp rw [← mul_assoc 2 a d] at hyp exact hyp example : c * b * a = b * (a * c) := by ring example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by rw [hyp, hyp'] ring end example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by nth_rw 2 [h] rw [add_mul]
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@@ -1,146 +0,0 @@import Mathlib.Algebra.Ring.Defs import Mathlib.Data.Real.Basic import Mathlib.Tactic section variable (R : Type _) [Ring R] #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) #check (add_comm : ∀ a b : R, a + b = b + a) #check (zero_add : ∀ a : R, 0 + a = a) #check (add_left_neg : ∀ a : R, -a + a = 0) #check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c)) #check (mul_one : ∀ a : R, a * 1 = a) #check (one_mul : ∀ a : R, 1 * a = a) #check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c) #check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c) end section variable (R : Type _) [CommRing R] variable (a b c d : R) example : c * b * a = b * (a * c) := by ring example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by rw [hyp, hyp'] ring end namespace MyRing variable {R : Type _} [Ring R] theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add] theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] #check @MyRing.add_zero #check @add_zero end MyRing namespace MyRing variable {R : Type _} [Ring R] theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by rw [← add_assoc, add_left_neg, zero_add] -- Prove these: theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by sorry theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by sorry theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by sorry theorem mul_zero (a : R) : a * 0 = 0 := by have h : a * 0 + a * 0 = a * 0 + 0 := by rw [← mul_add, add_zero, add_zero] rw [add_left_cancel h] theorem zero_mul (a : R) : 0 * a = 0 := by sorry theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by sorry theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by sorry theorem neg_zero : (-0 : R) = 0 := by apply neg_eq_of_add_eq_zero rw [add_zero] theorem neg_neg (a : R) : - -a = a := by sorry end MyRing -- Examples. section variable {R : Type _} [Ring R] example (a b : R) : a - b = a + -b := sub_eq_add_neg a b end example (a b : ℝ) : a - b = a + -b := rfl example (a b : ℝ) : a - b = a + -b := by rfl namespace MyRing variable {R : Type _} [Ring R] theorem self_sub (a : R) : a - a = 0 := sorry theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by norm_num theorem two_mul (a : R) : 2 * a = a + a := sorry end MyRing section variable (A : Type _) [AddGroup A] #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) #check (zero_add : ∀ a : A, 0 + a = a) #check (add_left_neg : ∀ a : A, -a + a = 0) end section variable {G : Type _} [Group G] #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) #check (one_mul : ∀ a : G, 1 * a = a) #check (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) namespace MyGroup theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by sorry theorem mul_one (a : G) : a * 1 = a := by sorry theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by sorry end MyGroup end
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@@ -1,130 +0,0 @@import Mathlib.Analysis.SpecialFunctions.Log.Basic import Mathlib.Tactic variable (a b c d e : ℝ) open Real #check (le_refl : ∀ a : ℝ, a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) section variable (h : a ≤ b) (h' : b ≤ c) #check (le_refl : ∀ a : Real, a ≤ a) #check (le_refl a : a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) #check (le_trans h : b ≤ c → a ≤ c) #check (le_trans h h' : a ≤ c) end example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by apply le_trans · apply h₀ . apply h₁ example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by apply le_trans h₀ apply h₁ example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := le_trans h₀ h₁ example (x : ℝ) : x ≤ x := by apply le_refl example (x : ℝ) : x ≤ x := le_refl x #check (le_refl : ∀ a, a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) #check (lt_of_le_of_lt : a ≤ b → b < c → a < c) #check (lt_of_lt_of_le : a < b → b ≤ c → a < c) #check (lt_trans : a < b → b < c → a < c) -- Try this. example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by sorry example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by linarith section example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : d + a ≤ 5 * b := by linarith end example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by linarith [exp_le_exp.mpr h'] #check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) #check (exp_lt_exp : exp a < exp b ↔ a < b) #check (log_le_log : 0 < a → 0 < b → (log a ≤ log b ↔ a ≤ b)) #check (log_lt_log : 0 < a → a < b → log a < log b) #check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d) #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) #check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c) #check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d) #check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d) #check (add_lt_add_left : a < b → ∀ c, c + a < c + b) #check (add_lt_add_right : a < b → ∀ c, a + c < b + c) #check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b) #check (add_pos : 0 < a → 0 < b → 0 < a + b) #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) #check (exp_pos : ∀ a, 0 < exp a) #check @add_le_add_left example (h : a ≤ b) : exp a ≤ exp b := by rw [exp_le_exp] exact h example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := by apply add_lt_add_of_lt_of_le · apply add_lt_add_of_le_of_lt h₀ apply exp_lt_exp.mpr h₁ apply le_refl example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by sorry example : (0 : ℝ) < 1 := by norm_num example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by have h₀ : 0 < 1 + exp a := by sorry have h₁ : 0 < 1 + exp b := by sorry apply (log_le_log h₀ h₁).mpr sorry example : 0 ≤ a ^ 2 := by -- library_search exact sq_nonneg a example (h : a ≤ b) : c - exp b ≤ c - exp a := by sorry example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 calc a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring _ ≥ 0 := by apply pow_two_nonneg calc 2 * a * b = 2 * a * b + 0 := by ring _ ≤ 2 * a * b + (a ^ 2 - 2 * a * b + b ^ 2) := add_le_add (le_refl _) h _ = a ^ 2 + b ^ 2 := by ring example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 calc a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by sorry #check abs_le'.mpr
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@@ -1,90 +0,0 @@import Mathlib.Data.Real.Basic namespace C02S04 section variable (a b c d : ℝ) #check (min_le_left a b : min a b ≤ a) #check (min_le_right a b : min a b ≤ b) #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) example : min a b = min b a := by apply le_antisymm · show min a b ≤ min b a apply le_min · apply min_le_right apply min_le_left · show min b a ≤ min a b apply le_min · apply min_le_right apply min_le_left example : min a b = min b a := by have h : ∀ x y : ℝ, min x y ≤ min y x := by intro x y apply le_min apply min_le_right apply min_le_left apply le_antisymm apply h apply h example : min a b = min b a := by apply le_antisymm repeat apply le_min apply min_le_right apply min_le_left example : max a b = max b a := by sorry example : min (min a b) c = min a (min b c) := by sorry theorem aux : min a b + c ≤ min (a + c) (b + c) := by sorry example : min a b + c = min (a + c) (b + c) := by sorry #check (abs_add : ∀ a b : ℝ, abs (a + b) ≤ abs a + abs b) example : abs a - abs b ≤ abs (a - b) := sorry end section variable (w x y z : ℕ) example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := dvd_trans h₀ h₁ example : x ∣ y * x * z := by apply dvd_mul_of_dvd_left apply dvd_mul_left example : x ∣ x ^ 2 := by apply dvd_mul_left example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by sorry end section variable (m n : ℕ) open Nat #check (gcd_zero_right n : gcd n 0 = n) #check (gcd_zero_left n : gcd 0 n = n) #check (lcm_zero_right n : lcm n 0 = 0) #check (lcm_zero_left n : lcm 0 n = 0) example : gcd m n = gcd n m := by sorry end
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@@ -1,109 +0,0 @@import Mathlib.Topology.MetricSpace.Basic section variable {α : Type _} [PartialOrder α] variable (x y z : α) #check x ≤ y #check (le_refl x : x ≤ x) #check (le_trans : x ≤ y → y ≤ z → x ≤ z) #check x < y #check (lt_irrefl x : ¬x < x) #check (lt_trans : x < y → y < z → x < z) #check (lt_of_le_of_lt : x ≤ y → y < z → x < z) #check (lt_of_lt_of_le : x < y → y ≤ z → x < z) example : x < y ↔ x ≤ y ∧ x ≠ y := lt_iff_le_and_ne end section variable {α : Type _} [Lattice α] variable (x y z : α) #check x ⊓ y #check (inf_le_left : x ⊓ y ≤ x) #check (inf_le_right : x ⊓ y ≤ y) #check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y) #check x ⊔ y #check (le_sup_left : x ≤ x ⊔ y) #check (le_sup_right : y ≤ x ⊔ y) #check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z) example : x ⊓ y = y ⊓ x := by sorry example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by sorry example : x ⊔ y = y ⊔ x := by sorry example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by sorry theorem absorb1 : x ⊓ (x ⊔ y) = x := by sorry theorem absorb2 : x ⊔ x ⊓ y = x := by sorry end section variable {α : Type _} [DistribLattice α] variable (x y z : α) #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) end section variable {α : Type _} [Lattice α] variable (a b c : α) example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by sorry example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by sorry end section variable {R : Type _} [StrictOrderedRing R] variable (a b c : R) #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) #check (mul_pos : 0 < a → 0 < b → 0 < a * b) #check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) example : a ≤ b → 0 ≤ b - a := by sorry example : 0 ≤ b - a → a ≤ b := by sorry example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by sorry end section variable {X : Type _} [MetricSpace X] variable (x y z : X) #check (dist_self x : dist x x = 0) #check (dist_comm x y : dist x y = dist y x) #check (dist_triangle x y z : dist x z ≤ dist x y + dist y z) example (x y : X) : 0 ≤ dist x y := by sorry end
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@@ -1,33 +0,0 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Real.Basic example (a b c : ℝ) : c * b * a = b * (a * c) := by rw [mul_comm c b] rw [mul_assoc b c a] rw [mul_comm c a] example (a b c : ℝ) : a * (b * c) = b * (a * c) := by rw [← mul_assoc a b c] rw [mul_comm a b] rw [mul_assoc b a c] example (a b c : ℝ) : a * (b * c) = b * (c * a) := by rw [mul_comm] rw [mul_assoc] example (a b c : ℝ) : a * (b * c) = b * (a * c) := by rw [← mul_assoc] rw [mul_comm a] rw [mul_assoc] example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by rw [mul_assoc a] rw [h] rw [← mul_assoc] example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by rw [hyp] rw [hyp'] rw [mul_comm] rw [sub_self]
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@@ -1,73 +0,0 @@import Mathlib.Algebra.Ring.Defs import Mathlib.Data.Real.Basic import Mathlib.Tactic namespace MyRing variable {R : Type _} [Ring R] theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by rw [add_assoc, add_right_neg, add_zero] theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by rw [← neg_add_cancel_left a b, h, neg_add_cancel_left] theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by rw [← add_neg_cancel_right a b, h, add_neg_cancel_right] theorem zero_mul (a : R) : 0 * a = 0 := by have h : 0 * a + 0 * a = 0 * a + 0 := by rw [← add_mul, add_zero, add_zero] rw [add_left_cancel h] theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by rw [← neg_add_cancel_left a b, h, add_zero] theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by symm apply neg_eq_of_add_eq_zero rw [add_comm, h] theorem neg_zero : (-0 : R) = 0 := by apply neg_eq_of_add_eq_zero rw [add_zero] theorem neg_neg (a : R) : - -a = a := by apply neg_eq_of_add_eq_zero rw [add_left_neg] end MyRing namespace MyRing variable {R : Type _} [Ring R] theorem self_sub (a : R) : a - a = 0 := by rw [sub_eq_add_neg, add_right_neg] theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by norm_num theorem two_mul (a : R) : 2 * a = a + a := by rw [← one_add_one_eq_two, add_mul, one_mul] end MyRing section variable {G : Type _} [Group G] namespace MyGroup theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by have h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 := by rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] rw [← h, ← mul_assoc, mul_left_inv, one_mul] theorem mul_one (a : G) : a * 1 = a := by rw [← mul_left_inv a, ← mul_assoc, mul_right_inv, one_mul] theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by rw [← one_mul (b⁻¹ * a⁻¹), ← mul_left_inv (a * b), mul_assoc, mul_assoc, ← mul_assoc b b⁻¹, mul_right_inv, one_mul, mul_right_inv, mul_one] end MyGroup end
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@@ -1,61 +0,0 @@import Mathlib.Analysis.SpecialFunctions.Log.Basic import Mathlib.Tactic variable (a b c d e : ℝ) open Real example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by apply lt_of_le_of_lt h₀ apply lt_trans h₁ exact lt_of_le_of_lt h₂ h₃ example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by apply add_le_add_left rw [exp_le_exp] apply add_le_add_left h₀ -- an alternative using `linarith`. example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by have : exp (a + d) ≤ exp (a + e) := by rw [exp_le_exp] linarith linarith [this] example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by have h₀ : 0 < 1 + exp a := by linarith [exp_pos a] have h₁ : 0 < 1 + exp b := by linarith [exp_pos b] apply (log_le_log h₀ h₁).mpr apply add_le_add_left (exp_le_exp.mpr h) -- SOLUTION. example (h : a ≤ b) : c - exp b ≤ c - exp a := by apply sub_le_sub_left exact exp_le_exp.mpr h -- alternatively: example (h : a ≤ b) : c - exp b ≤ c - exp a := by linarith [exp_le_exp.mpr h] theorem fact1 : a * b * 2 ≤ a ^ 2 + b ^ 2 := by have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 calc a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith theorem fact2 : -(a * b) * 2 ≤ a ^ 2 + b ^ 2 := by have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 calc a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by have h : (0 : ℝ) < 2 := by norm_num apply abs_le'.mpr constructor · rw [le_div_iff h] apply fact1 rw [le_div_iff h] apply fact2
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@@ -1,116 +0,0 @@import Mathlib.Data.Real.Basic namespace C02S04 section variable (a b c d : ℝ) #check (min_le_left a b : min a b ≤ a) #check (min_le_right a b : min a b ≤ b) #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) example : max a b = max b a := by apply le_antisymm repeat' apply max_le apply le_max_right apply le_max_left example : min (min a b) c = min a (min b c) := by apply le_antisymm · apply le_min · apply le_trans apply min_le_left apply min_le_left apply le_min · apply le_trans apply min_le_left apply min_le_right apply min_le_right apply le_min · apply le_min · apply min_le_left apply le_trans apply min_le_right apply min_le_left apply le_trans apply min_le_right apply min_le_right theorem aux : min a b + c ≤ min (a + c) (b + c) := by apply le_min · apply add_le_add_right apply min_le_left apply add_le_add_right apply min_le_right example : min a b + c = min (a + c) (b + c) := by apply le_antisymm · apply aux have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel] rw [h] apply add_le_add_right rw [sub_eq_add_neg] apply le_trans apply aux rw [add_neg_cancel_right, add_neg_cancel_right] example : abs a - abs b ≤ abs (a - b) := calc abs a - abs b = abs (a - b + b) - abs b := by rw [sub_add_cancel] _ ≤ abs (a - b) + abs b - abs b := by apply sub_le_sub_right apply abs_add _ ≤ abs (a - b) := by rw [add_sub_cancel] -- alternatively example : abs a - abs b ≤ abs (a - b) := by have h := abs_add (a - b) b rw [sub_add_cancel] at h linarith end section variable (w x y z : ℕ) example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := dvd_trans h₀ h₁ example : x ∣ y * x * z := by apply dvd_mul_of_dvd_left apply dvd_mul_left example : x ∣ x ^ 2 := by apply dvd_mul_left example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by apply dvd_add · apply dvd_add · apply dvd_mul_of_dvd_right apply dvd_mul_right apply dvd_mul_left rw [pow_two] apply dvd_mul_of_dvd_right exact h end section variable (m n : ℕ) open Nat #check (gcd_zero_right n : gcd n 0 = n) #check (gcd_zero_left n : gcd 0 n = n) #check (lcm_zero_right n : lcm n 0 = 0) #check (lcm_zero_left n : lcm 0 n = 0) example : gcd m n = gcd n m := by apply _root_.dvd_antisymm repeat' apply dvd_gcd apply gcd_dvd_right apply gcd_dvd_left end
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@@ -1,135 +0,0 @@import Mathlib.Topology.MetricSpace.Basic section variable {α : Type _} [Lattice α] variable (x y z : α) example : x ⊓ y = y ⊓ x := by apply le_antisymm repeat' apply le_inf · apply inf_le_right apply inf_le_left example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by apply le_antisymm · apply le_inf · apply le_trans apply inf_le_left apply inf_le_left apply le_inf · apply le_trans apply inf_le_left apply inf_le_right apply inf_le_right apply le_inf · apply le_inf · apply inf_le_left apply le_trans apply inf_le_right apply inf_le_left apply le_trans apply inf_le_right apply inf_le_right example : x ⊔ y = y ⊔ x := by apply le_antisymm repeat' apply sup_le · apply le_sup_right apply le_sup_left example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by apply le_antisymm · apply sup_le · apply sup_le apply le_sup_left · apply le_trans apply @le_sup_left _ _ y z apply le_sup_right apply le_trans apply @le_sup_right _ _ y z apply le_sup_right apply sup_le · apply le_trans apply @le_sup_left _ _ x y apply le_sup_left apply sup_le · apply le_trans apply @le_sup_right _ _ x y apply le_sup_left apply le_sup_right theorem absorb1 : x ⊓ (x ⊔ y) = x := by apply le_antisymm · apply inf_le_left apply le_inf · apply le_refl apply le_sup_left theorem absorb2 : x ⊔ x ⊓ y = x := by apply le_antisymm · apply sup_le · apply le_refl apply inf_le_left apply le_sup_left end section variable {α : Type _} [DistribLattice α] variable (x y z : α) #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) end section variable {α : Type _} [Lattice α] variable (a b c : α) example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by rw [h, @inf_comm _ _ (a ⊔ b), absorb1, @inf_comm _ _ (a ⊔ b), h, ← sup_assoc, @inf_comm _ _ c a, absorb2, inf_comm] example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by rw [h, @sup_comm _ _ (a ⊓ b), absorb2, @sup_comm _ _ (a ⊓ b), h, ← inf_assoc, @sup_comm _ _ c a, absorb1, sup_comm] end section variable {R : Type _} [StrictOrderedRing R] variable (a b c : R) theorem aux1 : a ≤ b → 0 ≤ b - a := by intro h rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b] apply add_le_add_left h theorem aux2 : 0 ≤ b - a → a ≤ b := by intro h rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)] apply add_le_add_left h example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by have h1 : 0 ≤ (b - a) * c := mul_nonneg (aux1 _ _ h) h' rw [sub_mul] at h1 exact aux2 _ _ h1 end section variable {X : Type _} [MetricSpace X] variable (x y z : X) example (x y : X) : 0 ≤ dist x y :=by have : 0 ≤ dist x y + dist y x := by rw [← dist_self x] apply dist_triangle linarith [dist_comm x y] end
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@@ -1,178 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S01 #check ∀ x : ℝ, 0 ≤ x → abs x = x #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := sorry section variable (a b δ : ℝ) variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) variable (ha : abs a < δ) (hb : abs b < δ) #check my_lemma a b δ #check my_lemma a b δ h₀ h₁ #check my_lemma a b δ h₀ h₁ ha hb end theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := sorry section variable (a b δ : ℝ) variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) variable (ha : abs a < δ) (hb : abs b < δ) #check my_lemma2 h₀ h₁ ha hb end theorem my_lemma3 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by intro x y ε epos ele1 xlt ylt sorry theorem my_lemma4 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by intro x y ε epos ele1 xlt ylt calc abs (x * y) = abs x * abs y := sorry _ ≤ abs x * ε := sorry _ < 1 * ε := sorry _ = ε := sorry def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x section variable (f g : ℝ → ℝ) (a b : ℝ) example (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x ↦ f x + g x) (a + b) := by intro x dsimp apply add_le_add apply hfa apply hgb example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := sorry example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := sorry example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : FnUb (fun x => f x * g x) (a * b) := sorry end section variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] #check @add_le_add def FnUb' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a theorem fnUb_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) end example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b := @h section variable (f g : ℝ → ℝ) example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := by intro a b aleb apply add_le_add apply mf aleb apply mg aleb example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := fun a b aleb => add_le_add (mf aleb) (mg aleb) example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := sorry example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := sorry def FnEven (f : ℝ → ℝ) : Prop := ∀ x, f x = f (-x) def FnOdd (f : ℝ → ℝ) : Prop := ∀ x, f x = -f (-x) example (ef : FnEven f) (eg : FnEven g) : FnEven fun x => f x + g x := by intro x calc (fun x => f x + g x) x = f x + g x := rfl _ = f (-x) + g (-x) := by rw [ef, eg] example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by sorry example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by sorry example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by sorry end section variable {α : Type _} (r s t : Set α) example : s ⊆ s := by intro x xs exact xs theorem Subset.refl : s ⊆ s := fun x xs => xs theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by sorry end section variable {α : Type _} [PartialOrder α] variable (s : Set α) (a b : α) def SetUb (s : Set α) (a : α) := ∀ x, x ∈ s → x ≤ a example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := sorry end section open Function example (c : ℝ) : Injective fun x => x + c := by intro x₁ x₂ h' exact (add_left_inj c).mp h' example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by sorry variable {α : Type _} {β : Type _} {γ : Type _} variable {g : β → γ} {f : α → β} example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by sorry end
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@@ -1,131 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S02 example : ∃ x : ℝ, 2 < x ∧ x < 3 := by use 5 / 2 norm_num example : ∃ x : ℝ, 2 < x ∧ x < 3 := have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num ⟨5 / 2, h⟩ example : ∃ x : ℝ, 2 < x ∧ x < 3 := ⟨5 / 2, by norm_num⟩ def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def FnHasUb (f : ℝ → ℝ) := ∃ a, FnUb f a def FnHasLb (f : ℝ → ℝ) := ∃ a, FnLb f a theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) section variable {f g : ℝ → ℝ} example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by cases' ubf with a ubfa cases' ubg with b ubgb use a + b apply fnUb_add ubfa ubgb example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by sorry example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by sorry example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by rcases ubf with ⟨a, ubfa⟩ rcases ubg with ⟨b, ubgb⟩ exact ⟨a + b, fnUb_add ubfa ubgb⟩ example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by rintro ⟨a, ubfa⟩ ⟨b, ubgb⟩ exact ⟨a + b, fnUb_add ubfa ubgb⟩ example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := fun ⟨a, ubfa⟩ ⟨b, ubgb⟩ => ⟨a + b, fnUb_add ubfa ubgb⟩ end section variable {α : Type _} [CommRing α] def SumOfSquares (x : α) := ∃ a b, x = a ^ 2 + b ^ 2 theorem sumOfSquares_mul {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : SumOfSquares (x * y) := by rcases sosx with ⟨a, b, xeq⟩ rcases sosy with ⟨c, d, yeq⟩ rw [xeq, yeq] use a * c - b * d, a * d + b * c ring theorem sumOfSquares_mul' {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : SumOfSquares (x * y) := by rcases sosx with ⟨a, b, rfl⟩ rcases sosy with ⟨c, d, rfl⟩ use a * c - b * d, a * d + b * c ring end section variable {a b c : ℕ} example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by cases' divab with d beq cases' divbc with e ceq rw [ceq, beq] use d * e; ring example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by sorry end section open Function example {c : ℝ} : Surjective fun x => x + c := by intro x use x - c dsimp; ring example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by sorry example (x y : ℝ) (h : x - y ≠ 0) : (x ^ 2 - y ^ 2) / (x - y) = x + y := by field_simp [h] ring example {f : ℝ → ℝ} (h : Surjective f) : ∃ x, f x ^ 2 = 4 := by cases' h 2 with x hx use x rw [hx] norm_num end section open Function variable {α : Type _} {β : Type _} {γ : Type _} variable {g : β → γ} {f : α → β} example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by sorry end
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MIL/C03_Logic/S03_Negation.lean (deleted)
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@@ -1,137 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S03 section variable (a b : ℝ) example (h : a < b) : ¬b < a := by intro h' have : a < a := lt_trans h h' apply lt_irrefl a this def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def FnHasUb (f : ℝ → ℝ) := ∃ a, FnUb f a def FnHasLb (f : ℝ → ℝ) := ∃ a, FnLb f a variable (f : ℝ → ℝ) example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by intro fnub cases' fnub with a fnuba cases' h a with x hx have : f x ≤ a := fnuba x linarith example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := sorry example : ¬FnHasUb fun x => x := sorry #check (not_le_of_gt : a > b → ¬a ≤ b) #check (not_lt_of_ge : a ≥ b → ¬a < b) #check (lt_of_not_ge : ¬a ≥ b → a < b) #check (le_of_not_gt : ¬a > b → a ≤ b) example (h : Monotone f) (h' : f a < f b) : a < b := by sorry example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by sorry example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by intro h let f := fun x : ℝ => (0 : ℝ) have monof : Monotone f := by sorry have h' : f 1 ≤ f 0 := le_refl _ sorry example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by sorry end section variable {α : Type _} (P : α → Prop) (Q : Prop) example (h : ¬∃ x, P x) : ∀ x, ¬P x := by sorry example (h : ∀ x, ¬P x) : ¬∃ x, P x := by sorry example (h : ¬∀ x, P x) : ∃ x, ¬P x := by sorry example (h : ∃ x, ¬P x) : ¬∀ x, P x := by sorry example (h : ¬∀ x, P x) : ∃ x, ¬P x := by by_contra h' apply h intro x show P x by_contra h'' exact h' ⟨x, h''⟩ example (h : ¬¬Q) : Q := by sorry example (h : Q) : ¬¬Q := by sorry end section variable (f : ℝ → ℝ) example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by sorry example (h : ¬∀ a, ∃ x, f x > a) : FnHasUb f := by push_neg at h exact h example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by simp only [FnHasUb, FnUb] at h push_neg at h exact h example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by sorry example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by contrapose! h exact h example (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := by contrapose! h use x / 2 constructor <;> linarith end section variable (a : ℕ) example (h : 0 < 0) : a > 37 := by exfalso apply lt_irrefl 0 h example (h : 0 < 0) : a > 37 := absurd h (lt_irrefl 0) example (h : 0 < 0) : a > 37 := by have h' : ¬0 < 0 := lt_irrefl 0 contradiction end
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@@ -1,135 +0,0 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum namespace C03S04 example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by constructor · assumption intro h apply h₁ rw [h] example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := ⟨h₀, fun h => h₁ (by rw [h])⟩ example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := have h : x ≠ y := by contrapose! h₁ rw [h₁] ⟨h₀, h⟩ example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by cases' h with h₀ h₁ contrapose! h₁ exact le_antisymm h₀ h₁ example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := by rintro ⟨h₀, h₁⟩ h' exact h₁ (le_antisymm h₀ h') example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := fun ⟨h₀, h₁⟩ h' => h₁ (le_antisymm h₀ h') example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by intro h' apply h.right exact le_antisymm h.left h' example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := fun h' => h.right (le_antisymm h.left h') example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := sorry example : ∃ x : ℝ, 2 < x ∧ x < 4 := ⟨5 / 2, by norm_num, by norm_num⟩ example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := by rintro ⟨z, xltz, zlty⟩ exact lt_trans xltz zlty example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := fun ⟨z, xltz, zlty⟩ => lt_trans xltz zlty example : ∃ x : ℝ, 2 < x ∧ x < 4 := by use 5 / 2 constructor <;> norm_num example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prime n := by use 5 use 7 norm_num sorry example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬y ≤ x := by rintro ⟨h₀, h₁⟩ use h₀ exact fun h' => h₁ (le_antisymm h₀ h') example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := by constructor · contrapose! rintro rfl rfl contrapose! exact le_antisymm h example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := ⟨fun h₀ h₁ => h₀ (by rw [h₁]), fun h₀ h₁ => h₀ (le_antisymm h h₁)⟩ example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := sorry theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := have h' : x ^ 2 = 0 := by sorry pow_eq_zero h' example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := sorry section example (x : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := by rw [abs_lt] intro h constructor <;> linarith example : 3 ∣ Nat.gcd 6 15 := by rw [Nat.dvd_gcd_iff] constructor <;> norm_num end theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by rw [Monotone] push_neg rfl example : ¬Monotone fun x : ℝ => -x := by sorry section variable {α : Type _} [PartialOrder α] variable (a b : α) example : a < b ↔ a ≤ b ∧ a ≠ b := by rw [lt_iff_le_not_le] sorry end section variable {α : Type _} [Preorder α] variable (a b c : α) example : ¬a < a := by rw [lt_iff_le_not_le] sorry example : a < b → b < c → a < c := by simp only [lt_iff_le_not_le] sorry end
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MIL/C03_Logic/S05_Disjunction.lean (deleted)
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@@ -1,101 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S05 section variable {x y : ℝ} example (h : y > x ^ 2) : y > 0 ∨ y < -1 := by left linarith [pow_two_nonneg x] example (h : -y > x ^ 2 + 1) : y > 0 ∨ y < -1 := by right linarith [pow_two_nonneg x] example (h : y > 0) : y > 0 ∨ y < -1 := Or.inl h example (h : y < -1) : y > 0 ∨ y < -1 := Or.inr h example : x < abs y → x < y ∨ x < -y := by cases' le_or_gt 0 y with h h · rw [abs_of_nonneg h] intro h left exact h rw [abs_of_neg h] intro h; right; exact h namespace MyAbs theorem le_abs_self (x : ℝ) : x ≤ abs x := by sorry theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by sorry theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by sorry theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by sorry theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by sorry end MyAbs end example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := by rcases lt_trichotomy x 0 with (xlt | xeq | xgt) · left exact xlt · contradiction right; exact xgt example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := by rcases h with (⟨a, rfl⟩ | ⟨b, rfl⟩) · rw [mul_assoc] apply dvd_mul_right rw [mul_comm, mul_assoc] apply dvd_mul_right example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by sorry example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by sorry example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by sorry section variable {R : Type _} [CommRing R] [IsDomain R] variable (x y : R) example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by sorry example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by sorry end example (P : Prop) : ¬¬P → P := by intro h cases em P · assumption contradiction example (P : Prop) : ¬¬P → P := by intro h by_cases h' : P · assumption contradiction example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by sorry
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@@ -1,99 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S06 def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε example : (fun x y : ℝ => (x + y) ^ 2) = fun x y : ℝ => x ^ 2 + 2 * x * y + y ^ 2 := by ext ring example (a b : ℝ) : abs a = abs (a - b + b) := by congr ring example {a : ℝ} (h : 1 < a) : a < a * a := by convert(mul_lt_mul_right _).2 h · rw [one_mul] exact lt_trans zero_lt_one h theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by intro ε εpos use 0 intro n nge; dsimp rw [sub_self, abs_zero] apply εpos theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : ConvergesTo (fun n => s n + t n) (a + b) := by intro ε εpos dsimp have ε2pos : 0 < ε / 2 := by linarith cases' cs (ε / 2) ε2pos with Ns hs cases' ct (ε / 2) ε2pos with Nt ht use max Ns Nt sorry theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : ConvergesTo (fun n => c * s n) (c * a) := by by_cases h : c = 0 · convert convergesTo_const 0 · rw [h, MulZeroClass.zero_mul] rw [h, MulZeroClass.zero_mul] have acpos : 0 < abs c := abs_pos.mpr h sorry theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : ∃ N b, ∀ n, N ≤ n → abs (s n) < b := by cases' cs 1 zero_lt_one with N h use N, abs a + 1 sorry theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) : ConvergesTo (fun n => s n * t n) 0 := by intro ε εpos dsimp rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)) have pos₀ : ε / B > 0 := div_pos εpos Bpos cases' ct _ pos₀ with N₁ h₁ sorry theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : ConvergesTo (fun n => s n * t n) (a * b) := by have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by apply aux cs convert convergesTo_add ct (convergesTo_const (-b)) ring have := convergesTo_add h₁ (convergesTo_mul_const b cs) convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 · ext; ring ring theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : a = b := by by_contra abne have : abs (a - b) > 0 := by sorry let ε := abs (a - b) / 2 have εpos : ε > 0 := by change abs (a - b) / 2 > 0 linarith cases' sa ε εpos with Na hNa cases' sb ε εpos with Nb hNb let N := max Na Nb have absa : abs (s N - a) < ε := by sorry have absb : abs (s N - b) < ε := by sorry have : abs (a - b) < abs (a - b) := by sorry exact lt_irrefl _ this section variable {α : Type _} [LinearOrder α] def ConvergesTo' (s : α → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε end
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@@ -1,129 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S01 def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x section variable (f g : ℝ → ℝ) (a b : ℝ) example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := by intro x apply add_le_add apply hfa apply hgb example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := by intro x apply mul_nonneg apply nnf apply nng example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : FnUb (fun x => f x * g x) (a * b) := by intro x apply mul_le_mul apply hfa apply hfb apply nng apply nna end section variable (f g : ℝ → ℝ) example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := by intro a b aleb apply mul_le_mul_of_nonneg_left _ nnc apply mf aleb example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := fun a b aleb => mul_le_mul_of_nonneg_left (mf aleb) nnc example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := by intro a b aleb apply mf apply mg apply aleb example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := fun a b aleb => mf (mg aleb) def FnEven (f : ℝ → ℝ) : Prop := ∀ x, f x = f (-x) def FnOdd (f : ℝ → ℝ) : Prop := ∀ x, f x = -f (-x) example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by intro x calc (fun x => f x * g x) x = f x * g x := rfl _ = f (-x) * g (-x) := by rw [of, og, neg_mul_neg] example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by intro x dsimp rw [ef, og, neg_mul_eq_mul_neg] example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by intro x dsimp rw [og, ← ef] end section variable {α : Type _} (r s t : Set α) example : r ⊆ s → s ⊆ t → r ⊆ t := by intro rsubs ssubt x xr apply ssubt apply rsubs apply xr theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := fun rsubs ssubt x xr => ssubt (rsubs xr) end section variable {α : Type _} [PartialOrder α] variable (s : Set α) (a b : α) def SetUb (s : Set α) (a : α) := ∀ x, x ∈ s → x ≤ a example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := by intro x xs apply le_trans (h x xs) h' example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := fun x xs => le_trans (h x xs) h' end section open Function example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by intro x₁ x₂ h' apply (mul_right_inj' h).mp h' variable {α : Type _} {β : Type _} {γ : Type _} variable {g : β → γ} {f : α → β} example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by intro x₁ x₂ h apply injf apply injg apply h end
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@@ -1,81 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S02 def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def FnHasUb (f : ℝ → ℝ) := ∃ a, FnUb f a def FnHasLb (f : ℝ → ℝ) := ∃ a, FnLb f a theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) section variable {f g : ℝ → ℝ} example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by cases' lbf with a lbfa cases' lbg with b lbgb use a + b intro x exact add_le_add (lbfa x) (lbgb x) example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by cases' ubf with a lbfa use c * a intro x exact mul_le_mul_of_nonneg_left (lbfa x) h end section variable {a b c : ℕ} example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by rcases divab with ⟨d, rfl⟩ rcases divbc with ⟨e, rfl⟩ use d * e; ring example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by rcases divab with ⟨d, rfl⟩ rcases divac with ⟨e, rfl⟩ use d + e; ring end section open Function example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by intro x use x / c dsimp; rw [mul_div_cancel' _ h] example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by intro x use x / c field_simp [h] ; ring end section open Function variable {α : Type _} {β : Type _} {γ : Type _} variable {g : β → γ} {f : α → β} example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by intro z rcases surjg z with ⟨y, rfl⟩ rcases surjf y with ⟨x, rfl⟩ use x end
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@@ -1,111 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S03 section variable (a b : ℝ) def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def FnHasUb (f : ℝ → ℝ) := ∃ a, FnUb f a def FnHasLb (f : ℝ → ℝ) := ∃ a, FnLb f a variable (f : ℝ → ℝ) example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := by rintro ⟨a, ha⟩ rcases h a with ⟨x, hx⟩ have := ha x linarith example : ¬FnHasUb fun x => x := by rintro ⟨a, ha⟩ have : a + 1 ≤ a := ha (a + 1) linarith example (h : Monotone f) (h' : f a < f b) : a < b := by apply lt_of_not_ge intro h'' apply absurd h' apply not_lt_of_ge (h h'') example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by intro h'' apply absurd h' apply not_lt_of_ge apply h'' h example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by intro h let f := fun x : ℝ => (0 : ℝ) have monof : Monotone f := by intro a b leab rfl have h' : f 1 ≤ f 0 := le_refl _ have : (1 : ℝ) ≤ 0 := h monof h' linarith example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by apply le_of_not_gt intro h' linarith [h _ h'] end section variable {α : Type _} (P : α → Prop) (Q : Prop) example (h : ¬∃ x, P x) : ∀ x, ¬P x := by intro x Px apply h use x exact Px example (h : ∀ x, ¬P x) : ¬∃ x, P x := by rintro ⟨x, Px⟩ exact h x Px example (h : ∃ x, ¬P x) : ¬∀ x, P x := by intro h' rcases h with ⟨x, nPx⟩ apply nPx apply h' example (h : ¬¬Q) : Q := by by_contra h' exact h h' example (h : Q) : ¬¬Q := by intro h' exact h' h end section variable (f : ℝ → ℝ) example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by intro a by_contra h' apply h use a intro x apply le_of_not_gt intro h'' apply h' use x exact h'' example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by rw [Monotone] at h push_neg at h exact h end
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@@ -1,94 +0,0 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum namespace C03S04 example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by cases' h with h0 h1 constructor · exact h0 intro h2 apply h1 apply Nat.dvd_antisymm h0 h2 example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := by constructor · rintro ⟨h0, h1⟩ constructor · exact h0 intro h2 apply h1 rw [h2] rintro ⟨h0, h1⟩ constructor · exact h0 intro h2 apply h1 apply le_antisymm h0 h2 theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := have h' : x ^ 2 = 0 := by linarith [pow_two_nonneg x, pow_two_nonneg y] pow_eq_zero h' example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by constructor · intro h constructor · exact aux h rw [add_comm] at h exact aux h rintro ⟨rfl, rfl⟩ norm_num theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by rw [Monotone] push_neg rfl example : ¬Monotone fun x : ℝ => -x := by rw [not_monotone_iff] use 0, 1 norm_num section variable {α : Type _} [PartialOrder α] variable (a b : α) example : a < b ↔ a ≤ b ∧ a ≠ b := by rw [lt_iff_le_not_le] constructor · rintro ⟨h0, h1⟩ constructor · exact h0 intro h2 apply h1 rw [h2] rintro ⟨h0, h1⟩ constructor · exact h0 intro h2 apply h1 apply le_antisymm h0 h2 end section variable {α : Type _} [Preorder α] variable (a b c : α) example : ¬a < a := by rw [lt_iff_le_not_le] rintro ⟨h0, h1⟩ exact h1 h0 example : a < b → b < c → a < c := by simp only [lt_iff_le_not_le] rintro ⟨h0, h1⟩ ⟨h2, h3⟩ constructor · apply le_trans h0 h2 intro h4 apply h1 apply le_trans h2 h4 end
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@@ -1,141 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S05 section variable {x y : ℝ} namespace MyAbs theorem le_abs_self (x : ℝ) : x ≤ abs x := by cases' le_or_gt 0 x with h h · rw [abs_of_nonneg h] rw [abs_of_neg h] linarith theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by cases' le_or_gt 0 x with h h · rw [abs_of_nonneg h] linarith rw [abs_of_neg h] theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by cases' le_or_gt 0 (x + y) with h h · rw [abs_of_nonneg h] linarith [le_abs_self x, le_abs_self y] rw [abs_of_neg h] linarith [neg_le_abs_self x, neg_le_abs_self y] theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by cases' le_or_gt 0 y with h h · rw [abs_of_nonneg h] constructor · intro h' left exact h' intro h' cases' h' with h' h' · exact h' linarith rw [abs_of_neg h] constructor · intro h' right exact h' intro h' cases' h' with h' h' · linarith exact h' theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by cases' le_or_gt 0 x with h h · rw [abs_of_nonneg h] constructor · intro h' constructor · linarith exact h' intro h' cases' h' with h1 h2 exact h2 rw [abs_of_neg h] constructor · intro h' constructor · linarith linarith intro h' linarith end MyAbs end example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by rcases h with ⟨x, y, rfl | rfl⟩ <;> linarith [sq_nonneg x, sq_nonneg y] example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] have h'' : (x + 1) * (x - 1) = 0 := by rw [← h'] ring cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 · right exact eq_neg_iff_add_eq_zero.mpr h1 left exact eq_of_sub_eq_zero h1 example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] have h'' : (x + y) * (x - y) = 0 := by rw [← h'] ring cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 · right exact eq_neg_iff_add_eq_zero.mpr h1 left exact eq_of_sub_eq_zero h1 section variable {R : Type _} [CommRing R] [IsDomain R] variable (x y : R) example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] have h'' : (x + 1) * (x - 1) = 0 := by rw [← h'] ring cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 · right exact eq_neg_iff_add_eq_zero.mpr h1 left exact eq_of_sub_eq_zero h1 example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] have h'' : (x + y) * (x - y) = 0 := by rw [← h'] ring cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 · right exact eq_neg_iff_add_eq_zero.mpr h1 left exact eq_of_sub_eq_zero h1 end example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by constructor · intro h by_cases h' : P · right exact h h' left exact h' rintro (h | h) · intro h' exact absurd h' h intro exact h
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@@ -1,127 +0,0 @@import Mathlib.Data.Real.Basic namespace C03S06 def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by intro ε εpos use 0 intro n nge; dsimp rw [sub_self, abs_zero] apply εpos theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : ConvergesTo (fun n => s n + t n) (a + b) := by intro ε εpos dsimp have ε2pos : 0 < ε / 2 := by linarith cases' cs (ε / 2) ε2pos with Ns hs cases' ct (ε / 2) ε2pos with Nt ht use max Ns Nt intro n hn have ngeNs : n ≥ Ns := le_of_max_le_left hn have ngeNt : n ≥ Nt := le_of_max_le_right hn calc |s n + t n - (a + b)| = |s n - a + (t n - b)| := by congr ring _ ≤ |s n - a| + |t n - b| := (abs_add _ _) _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt)) _ = ε := by norm_num theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : ConvergesTo (fun n => c * s n) (c * a) := by by_cases h : c = 0 · convert convergesTo_const 0 · rw [h, MulZeroClass.zero_mul] rw [h, MulZeroClass.zero_mul] have acpos : 0 < abs c := abs_pos.mpr h intro ε εpos dsimp have εcpos : 0 < ε / abs c := by apply div_pos εpos acpos cases' cs (ε / abs c) εcpos with Ns hs use Ns intro n ngt calc |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub] _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos) _ = ε := mul_div_cancel' _ (ne_of_lt acpos).symm theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : ∃ N b, ∀ n, N ≤ n → abs (s n) < b := by cases' cs 1 zero_lt_one with N h use N, abs a + 1 intro n ngt calc |s n| = |s n - a + a| := by congr abel _ ≤ |s n - a| + |a| := (abs_add _ _) _ < |a| + 1 := by linarith [h n ngt] theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) : ConvergesTo (fun n => s n * t n) 0 := by intro ε εpos dsimp rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)) have pos₀ : ε / B > 0 := div_pos εpos Bpos cases' ct _ pos₀ with N₁ h₁ use max N₀ N₁ intro n ngt have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt calc |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero] _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _)) _ = ε := mul_div_cancel' _ (ne_of_lt Bpos).symm theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : ConvergesTo (fun n => s n * t n) (a * b) := by have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by apply aux cs convert convergesTo_add ct (convergesTo_const (-b)) ring have := convergesTo_add h₁ (convergesTo_mul_const b cs) convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 · ext; ring ring theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : a = b := by by_contra abne have : abs (a - b) > 0 := by apply lt_of_le_of_ne · apply abs_nonneg intro h'' apply abne apply eq_of_abs_sub_eq_zero h''.symm let ε := abs (a - b) / 2 have εpos : ε > 0 := by change abs (a - b) / 2 > 0 linarith cases' sa ε εpos with Na hNa cases' sb ε εpos with Nb hNb let N := max Na Nb have absa : abs (s N - a) < ε := by apply hNa apply le_max_left have absb : abs (s N - b) < ε := by apply hNb apply le_max_right have : abs (a - b) < abs (a - b) calc abs (a - b) = abs (-(s N - a) + (s N - b)) := by congr ring _ ≤ abs (-(s N - a)) + abs (s N - b) := (abs_add _ _) _ = abs (s N - a) + abs (s N - b) := by rw [abs_neg] _ < ε + ε := (add_lt_add absa absb) _ = abs (a - b) := by norm_num exact lt_irrefl _ this
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MIL/C04_Sets_and_Functions/S01_Sets.lean (deleted)
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@@ -1,247 +0,0 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Nat.Prime import Mathlib.Data.Nat.Parity import Mathlib.Tactic section variable {α : Type _} variable (s t u : Set α) open Set example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by rw [subset_def, inter_def, inter_def] rw [subset_def] at h dsimp rintro x ⟨xs, xu⟩ exact ⟨h _ xs, xu⟩ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by simp only [subset_def, mem_inter_iff] at * rintro x ⟨xs, xu⟩ exact ⟨h _ xs, xu⟩ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by intro x xsu exact ⟨h xsu.1, xsu.2⟩ theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by intro x hx have xs : x ∈ s := hx.1 have xtu : x ∈ t ∪ u := hx.2 cases' xtu with xt xu · left show x ∈ s ∩ t exact ⟨xs, xt⟩ right show x ∈ s ∩ u exact ⟨xs, xu⟩ example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by rintro x ⟨xs, xt | xu⟩ · left exact ⟨xs, xt⟩ right; exact ⟨xs, xu⟩ example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by sorry example : (s \ t) \ u ⊆ s \ (t ∪ u) := by intro x xstu have xs : x ∈ s := xstu.1.1 have xnt : x ∉ t := xstu.1.2 have xnu : x ∉ u := xstu.2 constructor · exact xs intro xtu -- x ∈ t ∨ x ∈ u cases' xtu with xt xu · show False exact xnt xt show False; exact xnu xu example : (s \ t) \ u ⊆ s \ (t ∪ u) := by rintro x ⟨⟨xs, xnt⟩, xnu⟩ use xs rintro (xt | xu) <;> contradiction example : s \ (t ∪ u) ⊆ (s \ t) \ u := by sorry example : s ∩ t = t ∩ s := by ext x simp only [mem_inter_iff] constructor · rintro ⟨xs, xt⟩ exact ⟨xt, xs⟩ rintro ⟨xt, xs⟩; exact ⟨xs, xt⟩ example : s ∩ t = t ∩ s := Set.ext fun x => ⟨fun ⟨xs, xt⟩ => ⟨xt, xs⟩, fun ⟨xt, xs⟩ => ⟨xs, xt⟩⟩ example : s ∩ t = t ∩ s := by ext x; simp [and_comm] example : s ∩ t = t ∩ s := by apply Subset.antisymm · rintro x ⟨xs, xt⟩ exact ⟨xt, xs⟩ rintro x ⟨xt, xs⟩; exact ⟨xs, xt⟩ example : s ∩ t = t ∩ s := Subset.antisymm sorry sorry example : s ∩ (s ∪ t) = s := by sorry example : s ∪ s ∩ t = s := by sorry example : s \ t ∪ t = s ∪ t := by sorry example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by sorry def evens : Set ℕ := { n | Even n } def odds : Set ℕ := { n | ¬Even n } example : evens ∪ odds = univ := by rw [evens, odds] ext n simp apply Classical.em example (x : ℕ) (h : x ∈ (∅ : Set ℕ)) : False := h example (x : ℕ) : x ∈ (univ : Set ℕ) := trivial example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by sorry #print Prime #print Nat.Prime example (n : ℕ) : Prime n ↔ Nat.Prime n := Nat.prime_iff.symm example (n : ℕ) (h : Prime n) : Nat.Prime n := by rw [Nat.prime_iff] exact h example (n : ℕ) (h : Prime n) : Nat.Prime n := by rwa [Nat.prime_iff] end section variable (s t : Set ℕ) example (h₀ : ∀ x ∈ s, ¬Even x) (h₁ : ∀ x ∈ s, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by intro x xs constructor · apply h₀ x xs apply h₁ x xs example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by rcases h with ⟨x, xs, _, prime_x⟩ use x, xs exact prime_x section variable (ssubt : s ⊆ t) example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by sorry example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by sorry end end section variable {α I : Type _} variable (A B : I → Set α) variable (s : Set α) open Set example : (s ∩ ⋃ i, A i) = ⋃ i, A i ∩ s := by ext x simp only [mem_inter_iff, mem_iUnion] constructor · rintro ⟨xs, ⟨i, xAi⟩⟩ exact ⟨i, xAi, xs⟩ rintro ⟨i, xAi, xs⟩ exact ⟨xs, ⟨i, xAi⟩⟩ example : (⋂ i, A i ∩ B i) = (⋂ i, A i) ∩ ⋂ i, B i := by ext x simp only [mem_inter_iff, mem_iInter] constructor · intro h constructor · intro i exact (h i).1 intro i exact (h i).2 rintro ⟨h1, h2⟩ i constructor · exact h1 i exact h2 i example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by sorry def primes : Set ℕ := { x | Nat.Prime x } example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } :=by ext rw [mem_iUnion₂] simp example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } := by ext simp example : (⋂ p ∈ primes, { x | ¬p ∣ x }) ⊆ { x | x = 1 } := by intro x contrapose! simp apply Nat.exists_prime_and_dvd example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by sorry end section open Set variable {α : Type _} (s : Set (Set α)) example : ⋃₀ s = ⋃ t ∈ s, t := by ext x rw [mem_iUnion₂] simp example : ⋂₀ s = ⋂ t ∈ s, t := by ext x rw [mem_iInter₂] rfl end
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@@ -1,213 +0,0 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Analysis.SpecialFunctions.Log.Basic section variable {α β : Type _} variable (f : α → β) variable (s t : Set α) variable (u v : Set β) open Function open Set example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := by ext rfl example : f '' (s ∪ t) = f '' s ∪ f '' t := by ext y; constructor · rintro ⟨x, xs | xt, rfl⟩ · left use x, xs right use x, xt rintro (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩) · use x, Or.inl xs use x, Or.inr xt example : s ⊆ f ⁻¹' (f '' s) := by intro x xs show f x ∈ f '' s use x, xs example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by sorry example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by sorry example : f '' (f ⁻¹' u) ⊆ u := by sorry example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by sorry example (h : s ⊆ t) : f '' s ⊆ f '' t := by sorry example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by sorry example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by sorry example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by sorry example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by sorry example : f '' s \ f '' t ⊆ f '' (s \ t) := by sorry example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := by sorry example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by sorry example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u := by sorry example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by sorry example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by sorry variable {I : Type _} (A : I → Set α) (B : I → Set β) example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by ext y; simp constructor · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ use i, x exact ⟨xAi, fxeq⟩ rintro ⟨i, x, xAi, fxeq⟩ exact ⟨x, ⟨i, xAi⟩, fxeq⟩ example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by intro y; simp intro x h fxeq i use x exact ⟨h i, fxeq⟩ example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by intro y; simp intro h rcases h i with ⟨x, xAi, fxeq⟩ use x; constructor · intro i' rcases h i' with ⟨x', x'Ai, fx'eq⟩ have : f x = f x' := by rw [fxeq, fx'eq] have : x = x' := injf this rw [this] exact x'Ai exact fxeq example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by ext x simp example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by ext x simp example : InjOn f s ↔ ∀ x₁ ∈ s, ∀ x₂ ∈ s, f x₁ = f x₂ → x₁ = x₂ := Iff.refl _ end section open Set Real example : InjOn log { x | x > 0 } := by intro x xpos y ypos intro e -- log x = log y calc x = exp (log x) := by rw [exp_log xpos] _ = exp (log y) := by rw [e] _ = y := by rw [exp_log ypos] example : range exp = { y | y > 0 } := by ext y; constructor · rintro ⟨x, rfl⟩ apply exp_pos intro ypos use log y rw [exp_log ypos] example : InjOn sqrt { x | x ≥ 0 } := by sorry example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by sorry example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by sorry example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by sorry end section variable {α β : Type _} [Inhabited α] #check (default : α) variable (P : α → Prop) (h : ∃ x, P x) #check Classical.choose h example : P (Classical.choose h) := Classical.choose_spec h noncomputable section open Classical def inverse (f : α → β) : β → α := fun y : β => if h : ∃ x, f x = y then Classical.choose h else default theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by rw [inverse]; dsimp; rw [dif_pos h] exact Classical.choose_spec h variable (f : α → β) open Function example : Injective f ↔ LeftInverse (inverse f) f := sorry example : Surjective f ↔ RightInverse (inverse f) f := sorry end section variable {α : Type _} open Function theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by intro f surjf let S := { i | i ∉ f i } rcases surjf S with ⟨j, h⟩ have h₁ : j ∉ f j := by intro h' have : j ∉ f j := by rwa [h] at h' contradiction have h₂ : j ∈ S sorry have h₃ : j ∉ S sorry contradiction -- COMMENTS: TODO: improve this end
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@@ -1,98 +0,0 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Tactic open Set open Function noncomputable section open Classical variable {α β : Type _} [Nonempty β] section variable (f : α → β) (g : β → α) def sbAux : ℕ → Set α | 0 => univ \ g '' univ | n + 1 => g '' (f '' sbAux n) def sbSet := ⋃ n, sbAux f g n def sbFun (x : α) : β := if x ∈ sbSet f g then f x else invFun g x theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by have : x ∈ g '' univ := by contrapose! hx rw [sbSet, mem_iUnion] use 0 rw [sbAux, mem_diff] sorry have : ∃ y, g y = x := by sorry sorry theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by set A := sbSet f g with A_def set h := sbFun f g with h_def intro x₁ x₂ intro (hxeq : h x₁ = h x₂) show x₁ = x₂ simp only [h_def, sbFun, ← A_def] at hxeq by_cases xA : x₁ ∈ A ∨ x₂ ∈ A · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA · symm apply this hxeq.symm xA.symm (xA.resolve_left x₁A) have x₂A : x₂ ∈ A := by apply not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A have x₂eq : x₂ = g (f x₁) := by sorry rcases x₁A with ⟨n, hn⟩ rw [A_def, sbSet, mem_iUnion] use n + 1 simp [sbAux] exact ⟨x₁, hn, x₂eq.symm⟩ sorry push_neg at xA sorry theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by set A := sbSet f g with A_def set h := sbFun f g with h_def intro y by_cases gyA : g y ∈ A · rw [A_def, sbSet, mem_iUnion] at gyA rcases gyA with ⟨n, hn⟩ cases' n with n · simp [sbAux] at hn simp [sbAux] at hn rcases hn with ⟨x, xmem, hx⟩ use x have : x ∈ A := by rw [A_def, sbSet, mem_iUnion] exact ⟨n, xmem⟩ simp only [h_def, sbFun, if_pos this] exact hg hx sorry end theorem schroeder_bernstein {f : α → β} {g : β → α} (hf : Injective f) (hg : Injective g) : ∃ h : α → β, Bijective h := ⟨sbFun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ -- Auxiliary information section variable (g : β → α) (x : α) #check (invFun g : α → β) #check (leftInverse_invFun : Injective g → LeftInverse (invFun g) g) #check (leftInverse_invFun : Injective g → ∀ y, invFun g (g y) = y) #check (invFun_eq : (∃ y, g y = x) → g (invFun g x) = x) end
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@@ -1,158 +0,0 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Nat.Prime import Mathlib.Data.Nat.Parity import Mathlib.Tactic section variable {α : Type _} variable (s t u : Set α) open Set example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by rintro x (⟨xs, xt⟩ | ⟨xs, xu⟩) · use xs left exact xt use xs; right; exact xu example : s \ (t ∪ u) ⊆ (s \ t) \ u := by rintro x ⟨xs, xntu⟩ constructor use xs · intro xt exact xntu (Or.inl xt) intro xu apply xntu (Or.inr xu) example : s ∩ t = t ∩ s := Subset.antisymm (fun x ⟨xs, xt⟩ => ⟨xt, xs⟩) fun x ⟨xt, xs⟩ => ⟨xs, xt⟩ example : s ∩ (s ∪ t) = s := by ext x; constructor · rintro ⟨xs, _⟩ exact xs intro xs use xs; left; exact xs example : s ∪ s ∩ t = s := by ext x; constructor · rintro (xs | ⟨xs, xt⟩) <;> exact xs intro xs; left; exact xs example : s \ t ∪ t = s ∪ t := by ext x; constructor · rintro (⟨xs, nxt⟩ | xt) · left exact xs right exact xt by_cases h : x ∈ t · intro right exact h rintro (xs | xt) · left use xs exact h right; exact xt example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by ext x; constructor · rintro (⟨xs, xnt⟩ | ⟨xt, xns⟩) · constructor left exact xs rintro ⟨_, xt⟩ contradiction constructor right exact xt rintro ⟨xs, _⟩ contradiction rintro ⟨xs | xt, nxst⟩ · left use xs intro xt apply nxst constructor <;> assumption right; use xt; intro xs apply nxst constructor <;> assumption example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by intro n simp intro nprime cases' Nat.Prime.eq_two_or_odd nprime with h h · rw [h] intro linarith rw [Nat.even_iff, h] norm_num end section variable (s t : Set ℕ) section variable (ssubt : s ⊆ t) example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by intro x xs constructor · apply h₀ x (ssubt xs) apply h₁ x (ssubt xs) example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by rcases h with ⟨x, xs, _, px⟩ use x, ssubt xs exact px end end section variable {α I : Type _} variable (A B : I → Set α) variable (s : Set α) open Set example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by ext x simp only [mem_union, mem_iInter] constructor · rintro (xs | xI) · intro i right exact xs intro i left exact xI i intro h by_cases xs : x ∈ s · left exact xs right intro i cases h i · assumption contradiction def primes : Set ℕ := { x | Nat.Prime x } example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by apply eq_univ_of_forall intro x simp rcases Nat.exists_infinite_primes x with ⟨p, primep, pge⟩ use p, pge exact primep end
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@@ -1,252 +0,0 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Analysis.SpecialFunctions.Log.Basic section variable {α β : Type _} variable (f : α → β) variable (s t : Set α) variable (u v : Set β) open Function open Set example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by constructor · intro h x xs have : f x ∈ f '' s := mem_image_of_mem _ xs exact h this intro h y ymem rcases ymem with ⟨x, xs, fxeq⟩ rw [← fxeq] apply h xs example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by rintro x ⟨y, ys, fxeq⟩ rw [← h fxeq] exact ys example : f '' (f ⁻¹' u) ⊆ u := by rintro y ⟨x, xmem, rfl⟩ exact xmem example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by intro y yu rcases h y with ⟨x, fxeq⟩ use x constructor · show f x ∈ u rw [fxeq] exact yu exact fxeq example (h : s ⊆ t) : f '' s ⊆ f '' t := by rintro y ⟨x, xs, fxeq⟩ use x, h xs exact fxeq example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by intro x; apply h example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by ext x; rfl example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by rintro y ⟨x, ⟨xs, xt⟩, rfl⟩ constructor . use x, xs . use x, xt example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by rintro y ⟨⟨x₁, x₁s, rfl⟩, ⟨x₂, x₂t, fx₂eq⟩⟩ use x₁ constructor . use x₁s rw [← h fx₂eq] exact x₂t . rfl example : f '' s \ f '' t ⊆ f '' (s \ t) := by rintro y ⟨⟨x₁, x₁s, rfl⟩, h⟩ use x₁ constructor . constructor . exact x₁s . intro h' apply h use x₁, h' . rfl example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := fun x => id example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by ext y; constructor · rintro ⟨⟨x, xs, rfl⟩, fxv⟩ use x, ⟨xs, fxv⟩ rintro ⟨x, ⟨⟨xs, fxv⟩, rfl⟩⟩ exact ⟨⟨x, xs, rfl⟩, fxv⟩ example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∩ u := by rintro y ⟨x, ⟨xs, fxu⟩, rfl⟩ exact ⟨⟨x, xs, rfl⟩, fxu⟩ example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by rintro x ⟨xs, fxu⟩ exact ⟨⟨x, xs, rfl⟩, fxu⟩ example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by rintro x (xs | fxu) · left exact ⟨x, xs, rfl⟩ right; exact fxu variable {I : Type _} (A : I → Set α) (B : I → Set β) example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by ext y; simp constructor · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ use i, x exact ⟨xAi, fxeq⟩ rintro ⟨i, x, xAi, fxeq⟩ exact ⟨x, ⟨i, xAi⟩, fxeq⟩ example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by intro y; simp intro x h fxeq i use x exact ⟨h i, fxeq⟩ example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by intro y; simp intro h rcases h i with ⟨x, xAi, fxeq⟩ use x; constructor · intro i' rcases h i' with ⟨x', x'Ai, fx'eq⟩ have : f x = f x' := by rw [fxeq, fx'eq] have : x = x' := injf this rw [this] exact x'Ai exact fxeq example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by ext x simp example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by ext x simp end section open Set Real example : InjOn sqrt { x | x ≥ 0 } := by intro x xnonneg y ynonneg intro e calc x = sqrt x ^ 2 := by rw [sq_sqrt xnonneg] _ = sqrt y ^ 2 := by rw [e] _ = y := by rw [sq_sqrt ynonneg] example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by intro x xnonneg y ynonneg intro e dsimp at * calc x = sqrt (x ^ 2) := by rw [sqrt_sq xnonneg] _ = sqrt (y ^ 2) := by rw [e] _ = y := by rw [sqrt_sq ynonneg] example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by ext y; constructor · rintro ⟨x, ⟨xnonneg, rfl⟩⟩ apply sqrt_nonneg intro ynonneg use y ^ 2 dsimp at * constructor apply pow_nonneg ynonneg apply sqrt_sq assumption example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by ext y constructor · rintro ⟨x, rfl⟩ dsimp at * apply pow_two_nonneg intro ynonneg use sqrt y exact sq_sqrt ynonneg end section variable {α β : Type _} [Inhabited α] noncomputable section open Classical def inverse (f : α → β) : β → α := fun y : β => if h : ∃ x, f x = y then Classical.choose h else default theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by rw [inverse]; dsimp; rw [dif_pos h] exact Classical.choose_spec h variable (f : α → β) open Function example : Injective f ↔ LeftInverse (inverse f) f := by constructor · intro h y apply h apply inverse_spec use y intro h x1 x2 e rw [← h x1, ← h x2, e] example : Injective f ↔ LeftInverse (inverse f) f := ⟨fun h y => h (inverse_spec _ ⟨y, rfl⟩), fun h x1 x2 e => by rw [← h x1, ← h x2, e]⟩ example : Surjective f ↔ RightInverse (inverse f) f := by constructor · intro h y apply inverse_spec apply h intro h y use inverse f y apply h example : Surjective f ↔ RightInverse (inverse f) f := ⟨fun h y => inverse_spec _ (h _), fun h y => ⟨inverse f y, h _⟩⟩ end section variable {α : Type _} open Function theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by intro f surjf let S := { i | i ∉ f i } rcases surjf S with ⟨j, h⟩ have h₁ : j ∉ f j := by intro h' have : j ∉ f j := by rwa [h] at h' contradiction have h₂ : j ∈ S := h₁ have h₃ : j ∉ S := by rwa [h] at h₁ contradiction end
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@@ -1,88 +0,0 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Tactic open Set open Function noncomputable section open Classical variable {α β : Type _} [Nonempty β] section variable (f : α → β) (g : β → α) def sbAux : ℕ → Set α | 0 => univ \ g '' univ | n + 1 => g '' (f '' sbAux n) def sbSet := ⋃ n, sbAux f g n def sbFun (x : α) : β := if x ∈ sbSet f g then f x else invFun g x theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by have : x ∈ g '' univ := by contrapose! hx rw [sbSet, mem_iUnion] use 0 rw [sbAux, mem_diff] exact ⟨mem_univ _, hx⟩ have : ∃ y, g y = x := by simp at this assumption exact invFun_eq this theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by set A := sbSet f g with A_def set h := sbFun f g with h_def intro x₁ x₂ intro (hxeq : h x₁ = h x₂) show x₁ = x₂ simp only [h_def, sbFun, ← A_def] at hxeq by_cases xA : x₁ ∈ A ∨ x₂ ∈ A · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA · symm apply this hxeq.symm xA.symm (xA.resolve_left x₁A) have x₂A : x₂ ∈ A := by apply not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A have x₂eq : x₂ = g (f x₁) := by rw [hxeq, sb_right_inv f g x₂nA] rcases x₁A with ⟨n, hn⟩ rw [A_def, sbSet, mem_iUnion] use n + 1 simp [sbAux] exact ⟨x₁, hn, x₂eq.symm⟩ rw [if_pos x₁A, if_pos x₂A] at hxeq exact hf hxeq push_neg at xA rw [if_neg xA.1, if_neg xA.2] at hxeq rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by set A := sbSet f g with A_def set h := sbFun f g with h_def intro y by_cases gyA : g y ∈ A · rw [A_def, sbSet, mem_iUnion] at gyA rcases gyA with ⟨n, hn⟩ cases' n with n · simp [sbAux] at hn simp [sbAux] at hn rcases hn with ⟨x, xmem, hx⟩ use x have : x ∈ A := by rw [A_def, sbSet, mem_iUnion] exact ⟨n, xmem⟩ simp only [h_def, sbFun, if_pos this] exact hg hx use g y simp only [h_def, sbFun, if_neg gyA] apply leftInverse_invFun hg end
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@@ -1,121 +0,0 @@import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum.GCD import Mathlib.Tactic.NormNum.Prime #print Nat.coprime example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := h example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := by rw [Nat.coprime] at h exact h example : Nat.coprime 12 7 := by norm_num example : Nat.gcd 12 8 = 4 := by norm_num #check @Nat.prime_def_lt example (p : ℕ) (prime_p : Nat.Prime p) : 2 ≤ p ∧ ∀ m : ℕ, m < p → m ∣ p → m = 1 := by rwa [Nat.prime_def_lt] at prime_p #check Nat.Prime.eq_one_or_self_of_dvd example (p : ℕ) (prime_p : Nat.Prime p) : ∀ m : ℕ, m ∣ p → m = 1 ∨ m = p := prime_p.eq_one_or_self_of_dvd example : Nat.Prime 17 := by norm_num -- commonly used example : Nat.Prime 2 := Nat.prime_two example : Nat.Prime 3 := Nat.prime_three #check @Nat.Prime.dvd_mul #check Nat.Prime.dvd_mul Nat.prime_two #check Nat.prime_two.dvd_mul theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by rw [pow_two, Nat.prime_two.dvd_mul] at h cases h <;> assumption example {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := Nat.Prime.dvd_of_dvd_pow Nat.prime_two h example (a b c : Nat) (h : a * b = a * c) (h' : a ≠ 0) : b = c := -- library_search suggests the following: (mul_right_inj' h').mp h example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by intro sqr_eq have : 2 ∣ m := by sorry obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by rw [← sqr_eq, meq] ring have : 2 * k ^ 2 = n ^ 2 := sorry have : 2 ∣ n := by sorry have : 2 ∣ m.gcd n := by sorry have : 2 ∣ 1 := by sorry norm_num at this example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by sorry #check Nat.factors #check Nat.prime_of_mem_factors #check Nat.prod_factors #check Nat.factors_unique theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : (m * n).factorization p = m.factorization p + n.factorization p := by rw [Nat.factorization_mul mnez nnez] rfl theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by rw [Nat.factorization_pow] rfl theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by rw [prime_p.factorization] simp example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by intro sqr_eq have nsqr_nez : n ^ 2 ≠ 0 := by simpa have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by sorry have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by sorry have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this norm_num at this example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : k ∣ r.factorization p := by cases' r with r · simp have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) have eq1 : (m ^ k).factorization p = k * m.factorization p := by sorry have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by sorry have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] rw [this] sorry #check multiplicity
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@@ -1,148 +0,0 @@import Mathlib.Data.Nat.GCD.Basic import Mathlib.Algebra.BigOperators.Basic import Mathlib.Tactic example (n : Nat) : n.succ ≠ Nat.zero := Nat.succ_ne_zero n example (m n : Nat) (h : m.succ = n.succ) : m = n := Nat.succ.inj h def fac : ℕ → ℕ | 0 => 1 | n + 1 => (n + 1) * fac n example : fac 0 = 1 := rfl example : fac 0 = 1 := by rw [fac] example : fac 0 = 1 := by simp [fac] example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := rfl example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by rw [fac] example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by simp [fac] theorem fac_pos (n : ℕ) : 0 < fac n := by induction' n with n ih · rw [fac] exact zero_lt_one rw [fac] exact mul_pos n.succ_pos ih theorem dvd_fac {i n : ℕ} (ipos : 0 < i) (ile : i ≤ n) : i ∣ fac n := by induction' n with n ih · exact absurd ipos (not_lt_of_ge ile) rw [fac] cases' Nat.of_le_succ ile with h h · apply dvd_mul_of_dvd_right (ih h) rw [h] apply dvd_mul_right theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by cases' n with n · simp [fac] sorry section variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) #check Finset.sum s f #check Finset.prod s f open BigOperators open Finset example : s.sum f = ∑ x in s, f x := rfl example : s.prod f = ∏ x in s, f x := rfl example : (range n).sum f = ∑ x in range n, f x := rfl example : (range n).prod f = ∏ x in range n, f x := rfl example (f : ℕ → ℕ) : (∑ x in range 0, f x) = 0 := Finset.sum_range_zero f example (f : ℕ → ℕ) (n : ℕ) : (∑ x in range n.succ, f x) = (∑ x in range n, f x) + f n := Finset.sum_range_succ f n example (f : ℕ → ℕ) : (∏ x in range 0, f x) = 1 := Finset.prod_range_zero f example (f : ℕ → ℕ) (n : ℕ) : (∏ x in range n.succ, f x) = (∏ x in range n, f x) * f n := Finset.prod_range_succ f n example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := by induction' n with n ih · rw [fac, prod_range_zero] rw [fac, ih, prod_range_succ, mul_comm] example (a b c d e f : ℕ) : a * (b * c * f * (d * e)) = d * (a * f * e) * (c * b) := by simp [mul_assoc, mul_comm, mul_left_comm] theorem sum_id (n : ℕ) : (∑ i in range (n + 1), i) = n * (n + 1) / 2 := by symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) induction' n with n ih · simp rw [Finset.sum_range_succ, mul_add 2, ← ih, Nat.succ_eq_add_one] ring theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by sorry end inductive MyNat | zero : MyNat | succ : MyNat → MyNat namespace MyNat def add : MyNat → MyNat → MyNat | x, zero => x | x, succ y => succ (add x y) def mul : MyNat → MyNat → MyNat | x, zero => zero | x, succ y => add (mul x y) x theorem zero_add (n : MyNat) : add zero n = n := by induction' n with n ih · rfl rw [add, ih] theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by induction' n with n ih · rfl rw [add, ih] rfl theorem add_comm (m n : MyNat) : add m n = add n m := by induction' n with n ih · rw [zero_add] rfl rw [add, succ_add, ih] theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by sorry theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by sorry theorem zero_mul (n : MyNat) : mul zero n = zero := by sorry theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by sorry theorem mul_comm (m n : MyNat) : mul m n = mul n m := by sorry end MyNat
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@@ -1,233 +0,0 @@import Mathlib.Data.Nat.Prime import Mathlib.Algebra.BigOperators.Order import Mathlib.Tactic import Mathlib.Tactic.IntervalCases open BigOperators namespace C05S03 theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by cases m; contradiction case succ m => cases m; contradiction repeat' apply Nat.succ_le_succ apply zero_le example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by by_contra h push_neg at h interval_cases m <;> contradiction example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by by_contra h push_neg at h revert h0 h1 revert h m decide theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by by_cases np : n.Prime · use n, np induction' n using Nat.strong_induction_on with n ih dsimp at ih rw [Nat.prime_def_lt] at np push_neg at np rcases np h with ⟨m, mltn, mdvdn, mne1⟩ have : m ≠ 0 := by intro mz rw [mz, zero_dvd_iff] at mdvdn linarith have mgt2 : 2 ≤ m := two_le this mne1 by_cases mp : m.Prime · use m, mp exact mdvdn . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ use p, pp apply pdvd.trans mdvdn theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by intro n have : 2 ≤ Nat.factorial (n + 1) + 1 := by sorry rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ refine' ⟨p, _, pp⟩ show p > n by_contra ple push_neg at ple have : p ∣ Nat.factorial (n + 1) := by sorry have : p ∣ 1 := by sorry show False sorry open Finset section variable {α : Type _} [DecidableEq α] (r s t : Finset α) example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by rw [subset_iff] intro x rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter] tauto example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by simp [subset_iff] intro x tauto example : r ∩ s ∪ r ∩ t ⊆ r ∩ (s ∪ t) := by simp [subset_iff] intro x tauto example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by ext x simp tauto end section variable {α : Type _} [DecidableEq α] (r s t : Finset α) example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by sorry example : (r \ s) \ t = r \ (s ∪ t) := by sorry end example (s : Finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ ∏ i in s, i := Finset.dvd_prod_of_mem _ h theorem _root_.Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) (h : p ∣ q) : p = q := by sorry theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by intro h₀ h₁ induction' s using Finset.induction_on with a s ans ih · simp at h₁ linarith [prime_p.two_le] simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ rw [mem_insert] sorry example (s : Finset ℕ) (x : ℕ) : x ∈ s.filter Nat.Prime ↔ x ∈ s ∧ x.Prime := mem_filter theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by intro s by_contra h push_neg at h set s' := s.filter Nat.Prime with s'_def have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by intro n simp [s'_def] apply h have : 2 ≤ (∏ i in s', i) + 1 := by sorry rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ have : p ∣ ∏ i in s', i := by sorry have : p ∣ 1 := by convert Nat.dvd_sub' pdvd this simp show False sorry theorem bounded_of_ex_finset (Q : ℕ → Prop) : (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by rintro ⟨s, hs⟩ use s.sup id + 1 intro k Qk apply Nat.lt_succ_of_le show id k ≤ s.sup id apply le_sup (hs k Qk) theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by rintro ⟨n, hn⟩ use (range (n + 1)).filter Q intro k simp [Nat.lt_succ_iff] exact hn k example : 27 % 4 = 3 := by norm_num example (n : ℕ) : (4 * n + 3) % 4 = 3 := by rw [add_comm, Nat.add_mul_mod_self_left] norm_num theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by revert h rw [Nat.mul_mod] have : m % 4 < 4 := Nat.mod_lt m (by norm_num) interval_cases hm : m % 4 <;> simp [hm] have : n % 4 < 4 := Nat.mod_lt n (by norm_num) interval_cases hn : n % 4 <;> simp [hn] theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le <;> · intro neq rw [neq] at h norm_num at h theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by sorry theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by by_cases np : n.Prime · use n exact ⟨np, dvd_rfl, h⟩ induction' n using Nat.strong_induction_on with n ih dsimp at ih rw [Nat.prime_def_lt] at np push_neg at np rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ have mge2 : 2 ≤ m := by apply two_le _ mne1 intro mz rw [mz, zero_dvd_iff] at mdvdn linarith have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn have : m % 4 = 3 ∨ n / m % 4 = 3 := by apply mod_4_eq_3_or_mod_4_eq_3 rw [neq, h] cases' this with h1 h1 . sorry . sorry example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by rwa [mem_erase] at h example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by simp at h assumption theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by by_contra h push_neg at h cases' h with n hn have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by apply ex_finset_of_bounded use n contrapose! hn rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ exact ⟨p, pltn, pp, p4⟩ cases' this with s hs have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by sorry rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ have ps : p ∈ s := by sorry have pne3 : p ≠ 3 := by sorry have : p ∣ 4 * ∏ i in erase s 3, i := by sorry have : p ∣ 3 := by sorry have : p = 3 := by sorry contradiction
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@@ -1,105 +0,0 @@import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum.GCD import Mathlib.Tactic.NormNum.Prime theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by rw [pow_two, Nat.prime_two.dvd_mul] at h cases h <;> assumption example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by intro sqr_eq have : 2 ∣ m := by apply even_of_even_sqr rw [sqr_eq] apply dvd_mul_right obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by rw [← sqr_eq, meq] ring have : 2 * k ^ 2 = n ^ 2 := (mul_right_inj' (by norm_num)).mp this have : 2 ∣ n := by apply even_of_even_sqr rw [← this] apply dvd_mul_right have : 2 ∣ m.gcd n := by apply Nat.dvd_gcd <;> assumption have : 2 ∣ 1 := by convert this symm exact coprime_mn norm_num at this example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by intro sqr_eq have : p ∣ m := by apply prime_p.dvd_of_dvd_pow rw [sqr_eq] apply dvd_mul_right obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this have : p * (p * k ^ 2) = p * n ^ 2 := by rw [← sqr_eq, meq] ring have : p * k ^ 2 = n ^ 2 := by apply (mul_right_inj' _).mp this exact prime_p.ne_zero have : p ∣ n := by apply prime_p.dvd_of_dvd_pow rw [← this] apply dvd_mul_right have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption have : p ∣ 1 := by convert this symm exact coprime_mn have : 2 ≤ 1 := by apply prime_p.two_le.trans exact Nat.le_of_dvd zero_lt_one this norm_num at this theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : (m * n).factorization p = m.factorization p + n.factorization p := by rw [Nat.factorization_mul mnez nnez] rfl theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by rw [Nat.factorization_pow] rfl theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by rw [prime_p.factorization] simp example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by intro sqr_eq have nsqr_nez : n ^ 2 ≠ 0 := by simpa have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by rw [factorization_pow'] have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow', add_comm] have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this norm_num at this example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : k ∣ r.factorization p := by cases' r with r · simp have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) have eq1 : (m ^ k).factorization p = k * m.factorization p := by rw [factorization_pow'] have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm] have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] rw [this] apply Nat.dvd_sub' <;> apply Nat.dvd_mul_right
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@@ -1,98 +0,0 @@import Mathlib.Data.Nat.GCD.Basic import Mathlib.Algebra.BigOperators.Basic import Mathlib.Tactic def fac : ℕ → ℕ | 0 => 1 | n + 1 => (n + 1) * fac n theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by cases' n with n · simp [fac] induction' n with n ih · simp [fac] simp at * rw [pow_succ, fac] apply Nat.mul_le_mul _ ih repeat' apply Nat.succ_le_succ apply zero_le section variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) open BigOperators open Finset theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6) induction' n with n ih · simp rw [Finset.sum_range_succ, mul_add 6, ← ih, Nat.succ_eq_add_one] ring end inductive MyNat | zero : MyNat | succ : MyNat → MyNat namespace MyNat def add : MyNat → MyNat → MyNat | x, zero => x | x, succ y => succ (add x y) def mul : MyNat → MyNat → MyNat | x, zero => zero | x, succ y => add (mul x y) x theorem zero_add (n : MyNat) : add zero n = n := by induction' n with n ih · rfl rw [add, ih] theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by induction' n with n ih · rfl rw [add, ih] rfl theorem add_comm (m n : MyNat) : add m n = add n m := by induction' n with n ih · rw [zero_add] rfl rw [add, succ_add, ih] theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by induction' k with k ih · rfl rw [add, ih] rfl theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by induction' k with k ih · rfl rw [add, mul, mul, ih, add_assoc] theorem zero_mul (n : MyNat) : mul zero n = zero := by induction' n with n ih · rfl rw [mul, ih] rfl theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by induction' n with n ih · rfl rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add] rfl theorem mul_comm (m n : MyNat) : mul m n = mul n m := by induction' n with n ih · rw [zero_mul] rfl rw [mul, ih, succ_mul] end MyNat
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@@ -1,242 +0,0 @@import Mathlib.Data.Nat.Prime import Mathlib.Algebra.BigOperators.Order import Mathlib.Tactic import Mathlib.Tactic.IntervalCases open BigOperators namespace C05S03 theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by cases m; contradiction case succ m => cases m; contradiction repeat' apply Nat.succ_le_succ apply zero_le theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by by_cases np : n.Prime · use n, np induction' n using Nat.strong_induction_on with n ih dsimp at ih rw [Nat.prime_def_lt] at np push_neg at np rcases np h with ⟨m, mltn, mdvdn, mne1⟩ have : m ≠ 0 := by intro mz rw [mz, zero_dvd_iff] at mdvdn linarith have mgt2 : 2 ≤ m := two_le this mne1 by_cases mp : m.Prime · use m, mp exact mdvdn . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ use p, pp apply pdvd.trans mdvdn theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by intro n have : 2 ≤ Nat.factorial (n + 1) + 1 := by apply Nat.succ_le_succ exact Nat.succ_le_of_lt (Nat.factorial_pos _) rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ refine' ⟨p, _, pp⟩ show p > n by_contra ple push_neg at ple have : p ∣ Nat.factorial (n + 1) := by apply Nat.dvd_factorial apply pp.pos linarith have : p ∣ 1 := by convert Nat.dvd_sub' pdvd this simp show False have := Nat.le_of_dvd zero_lt_one this linarith [pp.two_le] open Finset section variable {α : Type _} [DecidableEq α] (r s t : Finset α) example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by ext x rw [mem_inter, mem_union, mem_union, mem_union, mem_inter] tauto example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by ext x simp tauto example : (r \ s) \ t = r \ (s ∪ t) := by ext x rw [mem_sdiff, mem_sdiff, mem_sdiff, mem_union] tauto example : (r \ s) \ t = r \ (s ∪ t) := by ext x simp tauto end theorem _root_.Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) (h : p ∣ q) : p = q := by cases prime_q.eq_one_or_self_of_dvd _ h · linarith [prime_p.two_le] assumption theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by intro h₀ h₁ induction' s using Finset.induction_on with a s ans ih · simp at h₁ linarith [prime_p.two_le] simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ rw [mem_insert] cases' h₁ with h₁ h₁ · left exact prime_p.eq_of_dvd_of_prime h₀.1 h₁ right exact ih h₀.2 h₁ theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by intro s by_contra h push_neg at h set s' := s.filter Nat.Prime with s'_def have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by intro n simp [s'_def] apply h have : 2 ≤ (∏ i in s', i) + 1 := by apply Nat.succ_le_succ apply Nat.succ_le_of_lt apply Finset.prod_pos intro n ns' apply (mem_s'.mp ns').pos rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ have : p ∣ ∏ i in s', i := by apply dvd_prod_of_mem rw [mem_s'] apply pp have : p ∣ 1 := by convert Nat.dvd_sub' pdvd this simp show False have := Nat.le_of_dvd zero_lt_one this linarith [pp.two_le] theorem bounded_of_ex_finset (Q : ℕ → Prop) : (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by rintro ⟨s, hs⟩ use s.sup id + 1 intro k Qk apply Nat.lt_succ_of_le show id k ≤ s.sup id apply le_sup (hs k Qk) theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by rintro ⟨n, hn⟩ use (range (n + 1)).filter Q intro k simp [Nat.lt_succ_iff] exact hn k theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by revert h rw [Nat.mul_mod] have : m % 4 < 4 := Nat.mod_lt m (by norm_num) interval_cases hm : m % 4 <;> simp [hm] have : n % 4 < 4 := Nat.mod_lt n (by norm_num) interval_cases hn : n % 4 <;> simp [hn] theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le <;> · intro neq rw [neq] at h norm_num at h theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by constructor · exact Nat.div_dvd_of_dvd h₀ exact Nat.div_lt_self (lt_of_le_of_lt (zero_le _) h₂) h₁ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by by_cases np : n.Prime · use n exact ⟨np, dvd_rfl, h⟩ induction' n using Nat.strong_induction_on with n ih dsimp at ih rw [Nat.prime_def_lt] at np push_neg at np rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ have mge2 : 2 ≤ m := by apply two_le _ mne1 intro mz rw [mz, zero_dvd_iff] at mdvdn linarith have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn have : m % 4 = 3 ∨ n / m % 4 = 3 := by apply mod_4_eq_3_or_mod_4_eq_3 rw [neq, h] cases' this with h1 h1 · by_cases mp : m.Prime · use m exact ⟨mp, mdvdn, h1⟩ rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩ use p exact ⟨pp, pdvd.trans mdvdn, p4eq⟩ obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn by_cases nmp : (n / m).Prime · use n / m exact ⟨nmp, nmdvdn, h1⟩ rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩ use p exact ⟨pp, pdvd.trans nmdvdn, p4eq⟩ theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by by_contra h push_neg at h cases' h with n hn have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by apply ex_finset_of_bounded use n contrapose! hn rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ exact ⟨p, pltn, pp, p4⟩ cases' this with s hs have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by rw [add_comm, Nat.add_mul_mod_self_left] norm_num rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ have ps : p ∈ s := by rw [← hs p] exact ⟨pp, p4eq⟩ have pne3 : p ≠ 3 := by intro peq rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd rw [Nat.prime_three.dvd_mul] at pdvd norm_num at pdvd have : 3 ∈ s.erase 3 := by apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd intro n simp [← hs n] tauto simp at this have : p ∣ 4 * ∏ i in erase s 3, i := by apply dvd_trans _ (dvd_mul_left _ _) apply dvd_prod_of_mem simp constructor <;> assumption have : p ∣ 3 := by convert Nat.dvd_sub' pdvd this simp have : p = 3 := by apply pp.eq_of_dvd_of_prime Nat.prime_three this contradiction
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MIL/C06_Structures/S01_Structures.lean (deleted)
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@@ -1,224 +0,0 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic namespace C06S01 noncomputable section @[ext] structure Point where x : ℝ y : ℝ z : ℝ #check Point.ext example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by ext repeat' assumption def myPoint1 : Point where x := 2 y := -1 z := 4 def myPoint2 : Point := ⟨2, -1, 4⟩ def myPoint3 := Point.mk 2 (-1) 4 structure Point' where build :: x : ℝ y : ℝ z : ℝ #check Point'.build 2 (-1) 4 namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def add' (a b : Point) : Point where x := a.x + b.x y := a.y + b.y z := a.z + b.z #check add myPoint1 myPoint2 #check myPoint1.add myPoint2 end Point #check Point.add myPoint1 myPoint2 #check myPoint1.add myPoint2 namespace Point protected theorem add_comm (a b : Point) : add a b = add b a := by rw [add, add] ext <;> dsimp repeat' apply add_comm example (a b : Point) : add a b = add b a := by simp [add, add_comm] theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := rfl def addAlt : Point → Point → Point | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ def addAlt' : Point → Point → Point | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by cases a cases b rfl theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by rcases a with ⟨xa, ya, za⟩ rcases b with ⟨xb, yb, zb⟩ rw [addAlt, addAlt] ext <;> dsimp apply add_comm repeat' apply add_comm example (a b : Point) : addAlt a b = addAlt b a := by rcases a with ⟨xa, ya, za⟩ rcases b with ⟨xb, yb, zb⟩ simp [addAlt, add_comm] example : ∀ a b : Point, addAlt a b = addAlt b a := by rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ simp [addAlt, add_comm] example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by simp [add, add_comm] protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by sorry def smul (r : ℝ) (a : Point) : Point := sorry theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by sorry end Point structure StandardTwoSimplex where x : ℝ y : ℝ z : ℝ x_nonneg : 0 ≤ x y_nonneg : 0 ≤ y z_nonneg : 0 ≤ z sum_eq : x + y + z = 1 namespace StandardTwoSimplex def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex where x := a.y y := a.x z := a.z x_nonneg := a.y_nonneg y_nonneg := a.x_nonneg z_nonneg := a.z_nonneg sum_eq := by rw [add_comm a.y a.x, a.sum_eq] noncomputable section def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex where x := (a.x + b.x) / 2 y := (a.y + b.y) / 2 z := (a.z + b.z) / 2 x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardTwoSimplex) : StandardTwoSimplex := sorry end end StandardTwoSimplex open BigOperators structure StandardSimplex (n : ℕ) where V : Fin n → ℝ NonNeg : ∀ i : Fin n, 0 ≤ V i sum_eq_one : (∑ i, V i) = 1 namespace StandardSimplex def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n where V i := (a.V i + b.V i) / 2 NonNeg := by intro i apply div_nonneg · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex structure IsLinear (f : ℝ → ℝ) where is_additive : ∀ x y, f (x + y) = f x + f y preserves_mul : ∀ x c, f (c * x) = c * f x section variable (f : ℝ → ℝ) (linf : IsLinear f) #check linf.is_additive #check linf.preserves_mul end def Point'' := ℝ × ℝ × ℝ def IsLinear' (f : ℝ → ℝ) := (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x def PReal := { y : ℝ // 0 < y } section variable (x : PReal) #check x.val #check x.property #check x.1 #check x.2 end def StandardTwoSimplex' := { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } def StandardSimplex' (n : ℕ) := { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } def StdSimplex := Σ n : ℕ, StandardSimplex n section variable (s : StdSimplex) #check s.fst #check s.snd #check s.1 #check s.2 end
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@@ -1,171 +0,0 @@import Mathlib.Data.Real.Basic namespace C06S02 structure Group₁ (α : Type _) where mul : α → α → α one : α inv : α → α mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) mul_one : ∀ x : α, mul x one = x one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one structure Group₁Cat where α : Type _ str : Group₁ α section variable (α β γ : Type _) variable (f : α ≃ β) (g : β ≃ γ) #check Equiv α β #check (f.toFun : α → β) #check (f.invFun : β → α) #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) #check (Equiv.refl α : α ≃ α) #check (f.symm : β ≃ α) #check (f.trans g : α ≃ γ) example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := rfl example (x : α) : (f.trans g) x = g (f x) := rfl example : (f.trans g : α → γ) = g ∘ f := rfl end example (α : Type _) : Equiv.Perm α = (α ≃ α) := rfl def permGroup {α : Type _} : Group₁ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm one_mul := Equiv.trans_refl mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm structure AddGroup₁ (α : Type _) where (add : α → α → α) -- fill in the rest @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : Point) : Point := sorry def zero : Point := sorry def add_group_point : AddGroup₁ point := sorry end Point section variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) #check f * g #check mul_assoc f g g⁻¹ -- group power, defined for any group #check g ^ n example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] example : f * g * g⁻¹ = f := mul_inv_cancel_right f g example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := mul_inv_cancel_right f g end class Group₂ (α : Type _) where mul : α → α → α one : α inv : α → α mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) mul_one : ∀ x : α, mul x one = x one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one instance {α : Type _} : Group₂ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm one_mul := Equiv.trans_refl mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm #check @Group₂.mul def mySquare {α : Type _} [Group₂ α] (x : α) := Group₂.mul x x #check @mySquare section variable {β : Type _} (f g : Equiv.Perm β) example : Group₂.mul f g = g.trans f := rfl example : mySquare f = f.trans f := rfl end instance : Inhabited Point where default := ⟨0, 0, 0⟩ #check (default : Point) example : ([] : List Point).headI = default := rfl instance : Add Point where add := Point.add section variable (x y : Point) #check x + y example : x + y = Point.add x y := rfl end instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := ⟨Group₂.mul⟩ instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := ⟨Group₂.one⟩ instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := ⟨Group₂.inv⟩ section variable {α : Type _} (f g : Equiv.Perm α) #check f * 1 * g⁻¹ def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := rfl end class AddGroup₂ (α : Type _) where add : α → α → α -- fill in the rest
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@@ -1,273 +0,0 @@import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Tactic @[ext] structure gaussInt where re : ℤ im : ℤ namespace gaussInt instance : Zero gaussInt := ⟨⟨0, 0⟩⟩ instance : One gaussInt := ⟨⟨1, 0⟩⟩ instance : Add gaussInt := ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := ⟨fun x => ⟨-x.re, -x.im⟩⟩ instance : Mul gaussInt := ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := rfl theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := rfl theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp] theorem zero_re : (0 : gaussInt).re = 0 := rfl @[simp] theorem zero_im : (0 : gaussInt).im = 0 := rfl @[simp] theorem one_re : (1 : gaussInt).re = 1 := rfl @[simp] theorem one_im : (1 : gaussInt).im = 0 := rfl @[simp] theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := rfl @[simp] theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := rfl @[simp] theorem neg_re (x : gaussInt) : (-x).re = -x.re := rfl @[simp] theorem neg_im (x : gaussInt) : (-x).im = -x.im := rfl @[simp] theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl @[simp] theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) add_assoc := by intros ext <;> simp <;> ring zero_add := by intro ext <;> simp add_zero := by intro ext <;> simp add_left_neg := by intro ext <;> simp add_comm := by intros ext <;> simp <;> ring mul_assoc := by intros ext <;> simp <;> ring one_mul := by intro ext <;> simp mul_one := by intro ext <;> simp left_distrib := by intros ext <;> simp <;> ring right_distrib := by intros ext <;> simp <;> ring mul_comm := by intros ext <;> simp <;> ring zero_mul := sorry mul_zero := sorry @[simp] theorem sub_re (x y : gaussInt) : (x - y).re = x.re - y.re := rfl @[simp] theorem sub_im (x y : gaussInt) : (x - y).im = x.im - y.im := rfl instance : Nontrivial gaussInt := by use 0, 1 rw [Ne, gaussInt.ext_iff] simp end gaussInt example (a b : ℤ) : a = b * (a / b) + a % b := Eq.symm <| Int.ediv_add_emod a b example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := Int.emod_nonneg a example (a b : ℤ) : b ≠ 0 → a % b < abs b := Int.emod_lt a namespace Int def div' (a b : ℤ) := (a + b / 2) / b def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by rw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne'] have := Int.emod_lt_of_pos (a + b / 2) h have := Int.ediv_add_emod b 2 have := Int.emod_lt_of_pos b zero_lt_two revert this; intro this -- FIXME, this should not be needed linarith theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] end Int theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by sorry namespace gaussInt def norm (x : gaussInt) := x.re ^ 2 + x.im ^ 2 @[simp] theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by sorry theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by sorry theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by sorry theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by sorry def conj (x : gaussInt) : gaussInt := ⟨x.re, -x.im⟩ @[simp] theorem conj_re (x : gaussInt) : (conj x).re = x.re := rfl @[simp] theorem conj_im (x : gaussInt) : (conj x).im = -x.im := rfl theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] instance : Div gaussInt := ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ instance : Mod gaussInt := ⟨fun x y => x - y * (x / y)⟩ theorem div_def (x y : gaussInt) : x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := rfl theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y · calc norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] _ = abs (Int.mod' (x.re * y.re + x.im * y.im) (norm y)) ^ 2 + abs (Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)) ^ 2 := by simp [H1, norm, sq_abs] _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos _ = norm y / 2 * (norm y / 2 * 2) := by ring _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos _ < norm y := by apply Int.ediv_lt_of_lt_mul · norm_num · linarith theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : ¬(norm (x * y)).natAbs < (norm x).natAbs := by apply not_lt_of_ge rw [norm_mul, Int.natAbs_mul] apply le_mul_of_one_le_right (Nat.zero_le _) apply Int.ofNat_le.1 rw [coe_natAbs_norm] exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) instance : EuclideanDomain gaussInt := { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl r := Measure (Int.natAbs ∘ norm) r_wellFounded := (measure (Int.natAbs ∘ norm)).2 remainder_lt := natAbs_norm_mod_lt mul_left_not_lt := not_norm_mul_left_lt_norm } example (x : gaussInt) : Irreducible x ↔ Prime x := PrincipalIdealRing.irreducible_iff_prime end gaussInt
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@@ -1,99 +0,0 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic namespace C06S01 noncomputable section @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by simp [add, add_assoc] def smul (r : ℝ) (a : Point) : Point := ⟨r * a.x, r * a.y, r * a.z⟩ theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by simp [add, smul, mul_add] end Point structure StandardTwoSimplex where x : ℝ y : ℝ z : ℝ x_nonneg : 0 ≤ x y_nonneg : 0 ≤ y z_nonneg : 0 ≤ z sum_eq : x + y + z = 1 namespace StandardTwoSimplex noncomputable section def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardTwoSimplex) : StandardTwoSimplex where x := lambda * a.x + (1 - lambda) * b.x y := lambda * a.y + (1 - lambda) * b.y z := lambda * a.z + (1 - lambda) * b.z x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) sum_eq := by trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) · ring simp [a.sum_eq, b.sum_eq] end end StandardTwoSimplex open BigOperators structure StandardSimplex (n : ℕ) where V : Fin n → ℝ NonNeg : ∀ i : Fin n, 0 ≤ V i sum_eq_one : (∑ i, V i) = 1 namespace StandardSimplex def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n where V i := (a.V i + b.V i) / 2 NonNeg := by intro i apply div_nonneg · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex namespace StandardSimplex def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardSimplex n) : StandardSimplex n where V i := lambda * a.V i + (1 - lambda) * b.V i NonNeg i := add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) sum_eq_one := by trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] simp [a.sum_eq_one, b.sum_eq_one] end StandardSimplex
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@@ -1,74 +0,0 @@import Mathlib.Data.Real.Basic namespace C06S02 structure AddGroup₁ (α : Type _) where add : α → α → α zero : α neg : α → α add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) add_zero : ∀ x : α, add x zero = x zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : Point) : Point := ⟨-a.x, -a.y, -a.z⟩ def zero : Point := ⟨0, 0, 0⟩ def addGroupPoint : AddGroup₁ Point where add := Point.add zero := Point.zero neg := Point.neg add_assoc := by simp [Point.add, add_assoc] add_zero := by simp [Point.add, Point.zero] zero_add := by simp [Point.add, Point.zero] add_left_neg := by simp [Point.add, Point.neg, Point.zero] end Point class AddGroup₂ (α : Type _) where add : α → α → α zero : α neg : α → α add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) add_zero : ∀ x : α, add x zero = x zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := ⟨AddGroup₂.add⟩ instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := ⟨AddGroup₂.zero⟩ instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := ⟨AddGroup₂.neg⟩ instance : AddGroup₂ Point where add := Point.add zero := Point.zero neg := Point.neg add_assoc := by simp [Point.add, add_assoc] add_zero := by simp [Point.add, Point.zero] zero_add := by simp [Point.add, Point.zero] add_left_neg := by simp [Point.add, Point.neg, Point.zero] section variable (x y : Point) #check x + -y + 0 end
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@@ -1,287 +0,0 @@import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Tactic @[ext] structure gaussInt where re : ℤ im : ℤ namespace gaussInt instance : Zero gaussInt := ⟨⟨0, 0⟩⟩ instance : One gaussInt := ⟨⟨1, 0⟩⟩ instance : Add gaussInt := ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := ⟨fun x => ⟨-x.re, -x.im⟩⟩ instance : Mul gaussInt := ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := rfl theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := rfl theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp] theorem zero_re : (0 : gaussInt).re = 0 := rfl @[simp] theorem zero_im : (0 : gaussInt).im = 0 := rfl @[simp] theorem one_re : (1 : gaussInt).re = 1 := rfl @[simp] theorem one_im : (1 : gaussInt).im = 0 := rfl @[simp] theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := rfl @[simp] theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := rfl @[simp] theorem neg_re (x : gaussInt) : (-x).re = -x.re := rfl @[simp] theorem neg_im (x : gaussInt) : (-x).im = -x.im := rfl @[simp] theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl @[simp] theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) add_assoc := by intros ext <;> simp <;> ring zero_add := by intro ext <;> simp add_zero := by intro ext <;> simp add_left_neg := by intro ext <;> simp add_comm := by intros ext <;> simp <;> ring mul_assoc := by intros ext <;> simp <;> ring one_mul := by intro ext <;> simp mul_one := by intro ext <;> simp left_distrib := by intros ext <;> simp <;> ring right_distrib := by intros ext <;> simp <;> ring mul_comm := by intros ext <;> simp <;> ring zero_mul := sorry mul_zero := sorry @[simp] theorem sub_re (x y : gaussInt) : (x - y).re = x.re - y.re := rfl @[simp] theorem sub_im (x y : gaussInt) : (x - y).im = x.im - y.im := rfl instance : Nontrivial gaussInt := by use 0, 1 rw [Ne, gaussInt.ext_iff] simp end gaussInt namespace Int def div' (a b : ℤ) := (a + b / 2) / b def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by rw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne'] have := Int.emod_lt_of_pos (a + b / 2) h have := Int.ediv_add_emod b 2 have := Int.emod_lt_of_pos b zero_lt_two revert this; intro this -- FIXME, this should not be needed linarith theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] end Int private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := haveI h' : x ^ 2 = 0 := by apply le_antisymm _ (sq_nonneg x) rw [← h] apply le_add_of_nonneg_right (sq_nonneg y) pow_eq_zero h' theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by constructor · intro h constructor · exact aux h rw [add_comm] at h exact aux h rintro ⟨rfl, rfl⟩ norm_num namespace gaussInt def norm (x : gaussInt) := x.re ^ 2 + x.im ^ 2 @[simp] theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by apply add_nonneg <;> apply sq_nonneg theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] rfl theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] simp [norm_nonneg] theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by simp [norm] ring def conj (x : gaussInt) : gaussInt := ⟨x.re, -x.im⟩ @[simp] theorem conj_re (x : gaussInt) : (conj x).re = x.re := rfl @[simp] theorem conj_im (x : gaussInt) : (conj x).im = -x.im := rfl theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] instance : Div gaussInt := ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ instance : Mod gaussInt := ⟨fun x y => x - y * (x / y)⟩ theorem div_def (x y : gaussInt) : x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := rfl theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y · calc norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] _ = abs (Int.mod' (x.re * y.re + x.im * y.im) (norm y)) ^ 2 + abs (Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)) ^ 2 := by simp [H1, norm, sq_abs] _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos _ = norm y / 2 * (norm y / 2 * 2) := by ring _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos _ < norm y := by apply Int.ediv_lt_of_lt_mul · norm_num · linarith theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : ¬(norm (x * y)).natAbs < (norm x).natAbs := by apply not_lt_of_ge rw [norm_mul, Int.natAbs_mul] apply le_mul_of_one_le_right (Nat.zero_le _) apply Int.ofNat_le.1 rw [coe_natAbs_norm] exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) instance : EuclideanDomain gaussInt := { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl r := Measure (Int.natAbs ∘ norm) r_wellFounded := (measure (Int.natAbs ∘ norm)).2 remainder_lt := natAbs_norm_mod_lt mul_left_not_lt := not_norm_mul_left_lt_norm } example (x : gaussInt) : Irreducible x ↔ Prime x := PrincipalIdealRing.irreducible_iff_prime end gaussInt
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MIL/C07_Hierarchies/S01_Basics.lean (deleted)
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@@ -1,308 +0,0 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic class One₁ (α : Type) where /-- The element one -/ one : α #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α @[class] structure One₂ (α : Type) where /-- The element one -/ one : α #check One₂.one example (α : Type) [One₁ α] : α := One₁.one example (α : Type) [One₁ α] := (One₁.one : α) @[inherit_doc] notation "𝟙" => One₁.one example {α : Type} [One₁ α] : α := 𝟙 example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl class Dia₁ (α : Type) where dia : α → α → α infixl:70 " ⋄ " => Dia₁.dia class Semigroup₁ (α : Type) where toDia₁ : Dia₁ α /-- Diamond is associative -/ dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) attribute [instance] Semigroup₁.toDia₁ example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b class Semigroup₂ (α : Type) extends Dia₁ α where /-- Diamond is associative -/ dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where /-- One is a left neutral element for diamond. -/ one_dia : ∀ a : α, 𝟙 ⋄ a = a /-- One is a right neutral element for diamond -/ dia_one : ∀ a : α, a ⋄ 𝟙 = a set_option trace.Meta.synthInstance true in example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α class Monoid₂ (α : Type) where toSemigroup₁ : Semigroup₁ α toDiaOneClass₁ : DiaOneClass₁ α example {α : Type} [Monoid₁ α] : (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ #check Monoid₂.mk /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ #check Monoid₁.mk #check Monoid₁.toSemigroup₁ #check Monoid₁.toDiaOneClass₁ class Inv₁ (α : Type) where /-- The inversion function -/ inv : α → α @[inherit_doc] postfix:max "⁻¹" => Inv₁.inv class Group₁ (G : Type) extends Monoid₁ G, Inv G where inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] export DiaOneClass₁ (one_dia dia_one) export Semigroup₁ (dia_assoc) export Group₁ (inv_dia) example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := sorry lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := sorry class AddSemigroup₃ (α : Type) extends Add α where /-- Multiplication is associative -/ add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) @[to_additive AddSemigroup₃] class Semigroup₃ (α : Type) extends Mul α where /-- Multiplication is associative -/ mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α @[to_additive AddMonoid₃] class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α attribute [to_additive existing] Monoid₃.toMulOneClass export Semigroup₃ (mul_assoc₃) export AddSemigroup₃ (add_assoc₃) whatsnew in @[to_additive] lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] #check left_neg_eq_right_neg' class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where add_comm : ∀ a b : α, a + b = b + a @[to_additive AddCommSemigroup₃] class CommSemigroup₃ (α : Type) extends Semigroup₃ α where mul_comm : ∀ a b : α, a * b = b * a class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α @[to_additive AddCommMonoid₃] class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where neg_add : ∀ a : G, -a + a = 0 @[to_additive AddGroup₃] class Group₃ (G : Type) extends Monoid₃ G, Inv G where inv_mul : ∀ a : G, a⁻¹ * a = 1 attribute [simp] Group₃.inv_mul AddGroup₃.neg_add @[to_additive] lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := sorry @[to_additive (attr := simp)] lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by sorry @[to_additive] lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by sorry @[to_additive] lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by sorry class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G @[to_additive AddCommGroup₃] class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where /-- Multiplication is left distributive over addition -/ left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c /-- Multiplication is right distributive over addition -/ right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := { Ring₃.toAddGroup₃ with add_comm := by sorry } instance : Ring₃ ℤ where add := (· + ·) add_assoc₃ := add_assoc zero := 0 zero_add := by simp add_zero := by simp neg := (- ·) neg_add := by simp mul := (· * ·) mul_assoc₃ := mul_assoc one := 1 one_mul := by simp mul_one := by simp zero_mul := by simp mul_zero := by simp left_distrib := Int.mul_add right_distrib := Int.add_mul class LE₁ (α : Type) where /-- The Less-or-Equal relation. -/ le : α → α → Prop @[inherit_doc] infix:50 " ≤₁ " => LE₁.le class Preorder₁ (α : Type) class PartialOrder₁ (α : Type) class OrderedCommMonoid₁ (α : Type) instance : OrderedCommMonoid₁ ℕ where class SMul₃ (α : Type) (β : Type) where /-- Scalar multiplication -/ smul : α → β → β infixr:73 " • " => SMul₃.smul class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where zero_smul : ∀ m : M, (0 : R) • m = 0 one_smul : ∀ m : M, (1 : R) • m = m mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where smul := fun r s ↦ r*s zero_smul := zero_mul one_smul := one_mul mul_smul := mul_assoc₃ add_smul := Ring₃.right_distrib smul_add := Ring₃.left_distrib def nsmul₁ [Zero M] [Add M] : ℕ → M → M | 0, _ => 0 | n + 1, a => a + nsmul₁ n a def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M | Int.ofNat n, a => nsmul₁ n a | Int.negSucc n, a => -nsmul₁ n.succ a instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where smul := zsmul₁ zero_smul := sorry one_smul := sorry mul_smul := sorry add_smul := sorry smul_add := sorry #synth Module₁ ℤ ℤ -- abGrpModule ℤ class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where /-- Multiplication by a natural number. -/ nsmul : ℕ → M → M := nsmul₁ /-- Multiplication by `(0 : ℕ)` gives `0`. -/ nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where add := fun p q ↦ (p.1 + q.1, p.2 + q.2) add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ zero := (0, 0) zero_add := fun a ↦ by ext <;> apply zero_add add_zero := fun a ↦ by ext <;> apply add_zero instance : AddMonoid₄ ℤ where add := (· + ·) add_assoc₃ := Int.add_assoc zero := 0 zero_add := Int.zero_add add_zero := Int.add_zero nsmul := fun n m ↦ (n : ℤ) * m nsmul_zero := Int.zero_mul nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m by rw [Int.add_mul, Int.add_comm, Int.one_mul] example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl
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MIL/C07_Hierarchies/S02_Morphisms.lean (deleted)
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@@ -1,114 +0,0 @@import Mathlib.Topology.Instances.Real def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where map_one : f 1 = 1 map_mul : ∀ g g', f (g * g') = f g * f g' example : Continuous (id : ℝ → ℝ) := continuous_id @[ext] structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where toFun : G → H map_one : toFun 1 = 1 map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where coe := MonoidHom₁.toFun attribute [coe] MonoidHom₁.toFun example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one @[ext] structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where toFun : G → H map_zero : toFun 0 = 0 map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where coe := AddMonoidHom₁.toFun attribute [coe] AddMonoidHom₁.toFun @[ext] structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where toFun : F → M → N map_one : ∀ f : F, toFun f 1 = 1 map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where coe := MonoidHomClass₁.toFun class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where toFun : F → M → N map_one : ∀ f : F, toFun f 1 = 1 map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where coe := MonoidHomClass₂.toFun attribute [coe] MonoidHomClass₂.toFun instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where toFun := MonoidHom₁.toFun map_one := fun f ↦ f.map_one map_mul := fun f ↦ f.map_mul instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where toFun := fun f ↦ f.toMonoidHom₁.toFun map_one := fun f ↦ f.toMonoidHom₁.map_one map_mul := fun f ↦ f.toMonoidHom₁.map_mul lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := by rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := map_inv_of_inv f h example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := map_inv_of_inv f h class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends FunLike F M (fun _ ↦ N) where map_one : ∀ f : F, f 1 = 1 map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where coe := MonoidHom₁.toFun coe_injective' := MonoidHom₁.ext map_one := MonoidHom₁.map_one map_mul := MonoidHom₁.map_mul @[ext] structure OrderPresHom (α β : Type) [LE α] [LE β] where toFun : α → β le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' @[ext] structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends MonoidHom₁ M N, OrderPresHom M N class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : OrderPresHomClass (OrderPresMonoidHom α β) α β where instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : MonoidHomClass₃ (OrderPresMonoidHom α β) α β := sorry
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MIL/C07_Hierarchies/S03_Subobjects.lean (deleted)
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@@ -1,88 +0,0 @@import Mathlib.GroupTheory.QuotientGroup @[ext] structure Submonoid₁ (M : Type) [Monoid M] where /-- The carrier of a submonoid. -/ carrier : Set M /-- The product of two elements of a submonoid belongs to the submonoid. -/ mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier /-- The unit element belongs to the submonoid. -/ one_mem : 1 ∈ carrier /-- Submonoids in `M` can be seen as sets in `M`. -/ instance [Monoid M] : SetLike (Submonoid₁ M) M where coe := Submonoid₁.carrier coe_injective' := Submonoid₁.ext example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) one := ⟨1, N.one_mem⟩ one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) example [Monoid M] (N : Submonoid₁ M) : Monoid N where mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) one := ⟨1, N.one_mem⟩ one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s one_mem : ∀ s : S, 1 ∈ s instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where mul_mem := Submonoid₁.mul_mem one_mem := Submonoid₁.one_mem instance [Monoid M] : Inf (Submonoid₁ M) := ⟨fun S₁ S₂ => { carrier := S₁ ∩ S₂ one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z iseqv := { refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ trans := by sorry } instance [CommMonoid M] : HasQuotient M (Submonoid M) where quotient' := fun N ↦ Quotient N.Setoid def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where mul := Quotient.map₂' (· * ·) (by sorry ) mul_assoc := by sorry one := QuotientMonoid.mk N 1 one_mul := by sorry mul_one := by sorry
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@@ -1,348 +0,0 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic class One₁ (α : Type) where /-- The element one -/ one : α #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α @[class] structure One₂ (α : Type) where /-- The element one -/ one : α #check One₂.one example (α : Type) [One₁ α] : α := One₁.one example (α : Type) [One₁ α] := (One₁.one : α) @[inherit_doc] notation "𝟙" => One₁.one example {α : Type} [One₁ α] : α := 𝟙 example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl class Dia₁ (α : Type) where dia : α → α → α infixl:70 " ⋄ " => Dia₁.dia class Semigroup₁ (α : Type) where toDia₁ : Dia₁ α /-- Diamond is associative -/ dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) attribute [instance] Semigroup₁.toDia₁ example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b class Semigroup₂ (α : Type) extends Dia₁ α where /-- Diamond is associative -/ dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where /-- One is a left neutral element for diamond. -/ one_dia : ∀ a : α, 𝟙 ⋄ a = a /-- One is a right neutral element for diamond -/ dia_one : ∀ a : α, a ⋄ 𝟙 = a set_option trace.Meta.synthInstance true in example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α class Monoid₂ (α : Type) where toSemigroup₁ : Semigroup₁ α toDiaOneClass₁ : DiaOneClass₁ α example {α : Type} [Monoid₁ α] : (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ #check Monoid₂.mk /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ #check Monoid₁.mk #check Monoid₁.toSemigroup₁ #check Monoid₁.toDiaOneClass₁ class Inv₁ (α : Type) where /-- The inversion function -/ inv : α → α @[inherit_doc] postfix:max "⁻¹" => Inv₁.inv class Group₁ (G : Type) extends Monoid₁ G, Inv G where inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] export DiaOneClass₁ (one_dia dia_one) export Semigroup₁ (dia_assoc) export Group₁ (inv_dia) example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := left_inv_eq_right_inv₁ (inv_dia a) h lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := by rw [← inv_dia a⁻¹, inv_eq_of_dia (inv_dia a)] class AddSemigroup₃ (α : Type) extends Add α where /-- Multiplication is associative -/ add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) @[to_additive AddSemigroup₃] class Semigroup₃ (α : Type) extends Mul α where /-- Multiplication is associative -/ mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α @[to_additive AddMonoid₃] class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α attribute [to_additive existing] Monoid₃.toMulOneClass export Semigroup₃ (mul_assoc₃) export AddSemigroup₃ (add_assoc₃) whatsnew in @[to_additive] lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] #check left_neg_eq_right_neg' class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where add_comm : ∀ a b : α, a + b = b + a @[to_additive AddCommSemigroup₃] class CommSemigroup₃ (α : Type) extends Semigroup₃ α where mul_comm : ∀ a b : α, a * b = b * a class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α @[to_additive AddCommMonoid₃] class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where neg_add : ∀ a : G, -a + a = 0 @[to_additive AddGroup₃] class Group₃ (G : Type) extends Monoid₃ G, Inv G where inv_mul : ∀ a : G, a⁻¹ * a = 1 attribute [simp] Group₃.inv_mul AddGroup₃.neg_add @[to_additive] lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := left_inv_eq_right_inv' (Group₃.inv_mul a) h @[to_additive (attr := simp)] lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by rw [← inv_mul a⁻¹, inv_eq_of_mul (inv_mul a)] @[to_additive] lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by simpa [← mul_assoc₃] using congr_arg (a⁻¹ * ·) h @[to_additive] lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by simpa [mul_assoc₃] using congr_arg (· * a⁻¹) h class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G @[to_additive AddCommGroup₃] class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where /-- Multiplication is left distributive over addition -/ left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c /-- Multiplication is right distributive over addition -/ right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := { Ring₃.toAddGroup₃ with add_comm := by intro a b have : a + (a + b + b) = a + (b + a + b) := calc a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃] _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib] _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib] _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib] _ = (a + b) + (a + b) := by simp _ = a + (b + a + b) := by simp [add_assoc₃] exact add_right_cancel₃ (add_left_cancel₃ this) } instance : Ring₃ ℤ where add := (· + ·) add_assoc₃ := add_assoc zero := 0 zero_add := by simp add_zero := by simp neg := (- ·) neg_add := by simp mul := (· * ·) mul_assoc₃ := mul_assoc one := 1 one_mul := by simp mul_one := by simp zero_mul := by simp mul_zero := by simp left_distrib := Int.mul_add right_distrib := Int.add_mul class LE₁ (α : Type) where /-- The Less-or-Equal relation. -/ le : α → α → Prop @[inherit_doc] infix:50 " ≤₁ " => LE₁.le class Preorder₁ (α : Type) extends LE₁ α where le_refl : ∀ a : α, a ≤₁ a le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c class PartialOrder₁ (α : Type) extends Preorder₁ α where le_antisymm : ∀ a b : α, a ≤₁ b → b ≤₁ a → a = b class OrderedCommMonoid₁ (α : Type) extends PartialOrder₁ α, CommMonoid₃ α where mul_of_le : ∀ a b : α, a ≤₁ b → ∀ c : α, c * a ≤₁ c * b instance : OrderedCommMonoid₁ ℕ where le := (· ≤ ·) le_refl := fun _ ↦ le_rfl le_trans := fun _ _ _ ↦ le_trans le_antisymm := fun _ _ ↦ le_antisymm mul := (· * ·) mul_assoc₃ := mul_assoc one := 1 one_mul := one_mul mul_one := mul_one mul_comm := mul_comm mul_of_le := fun _ _ h c ↦ Nat.mul_le_mul_left c h class SMul₃ (α : Type) (β : Type) where /-- Scalar multiplication -/ smul : α → β → β infixr:73 " • " => SMul₃.smul class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where zero_smul : ∀ m : M, (0 : R) • m = 0 one_smul : ∀ m : M, (1 : R) • m = m mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where smul := fun r s ↦ r*s zero_smul := zero_mul one_smul := one_mul mul_smul := mul_assoc₃ add_smul := Ring₃.right_distrib smul_add := Ring₃.left_distrib def nsmul₁ [Zero M] [Add M] : ℕ → M → M | 0, _ => 0 | n + 1, a => a + nsmul₁ n a def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M | Int.ofNat n, a => nsmul₁ n a | Int.negSucc n, a => -nsmul₁ n.succ a instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where smul := zsmul₁ zero_smul := sorry one_smul := sorry mul_smul := sorry add_smul := sorry smul_add := sorry #synth Module₁ ℤ ℤ -- abGrpModule ℤ class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where /-- Multiplication by a natural number. -/ nsmul : ℕ → M → M := nsmul₁ /-- Multiplication by `(0 : ℕ)` gives `0`. -/ nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where add := fun p q ↦ (p.1 + q.1, p.2 + q.2) add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ zero := (0, 0) zero_add := fun a ↦ by ext <;> apply zero_add add_zero := fun a ↦ by ext <;> apply add_zero instance : AddMonoid₄ ℤ where add := (· + ·) add_assoc₃ := Int.add_assoc zero := 0 zero_add := Int.zero_add add_zero := Int.add_zero nsmul := fun n m ↦ (n : ℤ) * m nsmul_zero := Int.zero_mul nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m by rw [Int.add_mul, Int.add_comm, Int.one_mul] example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl class LT₁ (α : Type) where /-- The Less-Than relation -/ lt : α → α → Prop @[inherit_doc] infix:50 " <₁ " => LT₁.lt class PreOrder₂ (α : Type) extends LE₁ α, LT₁ α where le_refl : ∀ a : α, a ≤₁ a le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c lt := fun a b => a ≤₁ b ∧ ¬b ≤₁ a lt_iff_le_not_le : ∀ a b : α, a <₁ b ↔ a ≤₁ b ∧ ¬b ≤₁ a := by intros; rfl
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@@ -1,126 +0,0 @@import Mathlib.Topology.Instances.Real def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where map_one : f 1 = 1 map_mul : ∀ g g', f (g * g') = f g * f g' example : Continuous (id : ℝ → ℝ) := continuous_id @[ext] structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where toFun : G → H map_one : toFun 1 = 1 map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where coe := MonoidHom₁.toFun attribute [coe] MonoidHom₁.toFun example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one @[ext] structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where toFun : G → H map_zero : toFun 0 = 0 map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where coe := AddMonoidHom₁.toFun attribute [coe] AddMonoidHom₁.toFun @[ext] structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where toFun : F → M → N map_one : ∀ f : F, toFun f 1 = 1 map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where coe := MonoidHomClass₁.toFun class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where toFun : F → M → N map_one : ∀ f : F, toFun f 1 = 1 map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where coe := MonoidHomClass₂.toFun attribute [coe] MonoidHomClass₂.toFun instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where toFun := MonoidHom₁.toFun map_one := fun f ↦ f.map_one map_mul := fun f ↦ f.map_mul instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where toFun := fun f ↦ f.toMonoidHom₁.toFun map_one := fun f ↦ f.toMonoidHom₁.map_one map_mul := fun f ↦ f.toMonoidHom₁.map_mul lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := by rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := map_inv_of_inv f h example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := map_inv_of_inv f h class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends FunLike F M (fun _ ↦ N) where map_one : ∀ f : F, f 1 = 1 map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where coe := MonoidHom₁.toFun coe_injective' := MonoidHom₁.ext map_one := MonoidHom₁.map_one map_mul := MonoidHom₁.map_mul @[ext] structure OrderPresHom (α β : Type) [LE α] [LE β] where toFun : α → β le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' @[ext] structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends MonoidHom₁ M N, OrderPresHom M N class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] extends FunLike F α (fun _ ↦ β) where le_of_le : ∀ (f : F) a a', a ≤ a' → f a ≤ f a' instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where coe := OrderPresHom.toFun coe_injective' := OrderPresHom.ext le_of_le := OrderPresHom.le_of_le instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : OrderPresHomClass (OrderPresMonoidHom α β) α β where coe := fun f ↦ f.toOrderPresHom.toFun coe_injective' := OrderPresMonoidHom.ext le_of_le := fun f ↦ f.toOrderPresHom.le_of_le instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : MonoidHomClass₃ (OrderPresMonoidHom α β) α β where coe := fun f ↦ f.toOrderPresHom.toFun coe_injective' := OrderPresMonoidHom.ext map_one := fun f ↦ f.toMonoidHom₁.map_one map_mul := fun f ↦ f.toMonoidHom₁.map_mul
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@@ -1,126 +0,0 @@import Mathlib.GroupTheory.QuotientGroup @[ext] structure Submonoid₁ (M : Type) [Monoid M] where /-- The carrier of a submonoid. -/ carrier : Set M /-- The product of two elements of a submonoid belongs to the submonoid. -/ mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier /-- The unit element belongs to the submonoid. -/ one_mem : 1 ∈ carrier /-- Submonoids in `M` can be seen as sets in `M`. -/ instance [Monoid M] : SetLike (Submonoid₁ M) M where coe := Submonoid₁.carrier coe_injective' := Submonoid₁.ext example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) one := ⟨1, N.one_mem⟩ one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) example [Monoid M] (N : Submonoid₁ M) : Monoid N where mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) one := ⟨1, N.one_mem⟩ one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s one_mem : ∀ s : S, 1 ∈ s instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where mul_mem := Submonoid₁.mul_mem one_mem := Submonoid₁.one_mem @[ext] structure Subgroup₁ (G : Type) [Group G] extends Submonoid₁ G where /-- The inverse of an element of a subgroup belongs to the subgroup. -/ inv_mem {a} : a ∈ carrier → a⁻¹ ∈ carrier /-- Subgroups in `M` can be seen as sets in `M`. -/ instance [Group G] : SetLike (Subgroup₁ G) G where coe := fun H ↦ H.toSubmonoid₁.carrier coe_injective' := Subgroup₁.ext instance [Group G] (H : Subgroup₁ G) : Group H := { SubMonoid₁Monoid H.toSubmonoid₁ with inv := fun x ↦ ⟨x⁻¹, H.inv_mem x.property⟩ mul_left_inv := fun x ↦ SetCoe.ext (mul_left_inv (x : G)) } class SubgroupClass₁ (S : Type _) (G : Type) [Group G] [SetLike S G] extends SubmonoidClass₁ S G : Prop where inv_mem : ∀ (s : S) {a : G}, a ∈ s → a⁻¹ ∈ s instance [Group G] : SubmonoidClass₁ (Subgroup₁ G) G where mul_mem := fun H ↦ H.toSubmonoid₁.mul_mem one_mem := fun H ↦ H.toSubmonoid₁.one_mem instance [Group G] : SubgroupClass₁ (Subgroup₁ G) G := { (inferInstance : SubmonoidClass₁ (Subgroup₁ G) G) with inv_mem := Subgroup₁.inv_mem } instance [Monoid M] : Inf (Submonoid₁ M) := ⟨fun S₁ S₂ => { carrier := S₁ ∩ S₂ one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z iseqv := { refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ trans := by rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩ refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc] } instance [CommMonoid M] : HasQuotient M (Submonoid M) where quotient' := fun N ↦ Quotient N.Setoid def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where mul := Quotient.map₂' (· * ·) (by rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩ refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha, mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc] ) mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ apply Quotient.sound dsimp only rw [mul_assoc] apply @Setoid.refl M N.Setoid one := QuotientMonoid.mk N 1 one_mul := by rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul] ; apply @Setoid.refl M N.Setoid mul_one := by rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one] ; apply @Setoid.refl M N.Setoid
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MIL/C08_Topology/S01_Filters.lean (deleted)
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@@ -1,105 +0,0 @@import Mathlib.Topology.Instances.Real open Set Filter Topology def principal {α : Type _} (s : Set α) : Filter α where sets := { t | s ⊆ t } univ_sets := sorry sets_of_superset := sorry inter_sets := sorry example : Filter ℕ := { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } univ_sets := sorry sets_of_superset := sorry inter_sets := sorry } def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := ∀ V ∈ G, f ⁻¹' V ∈ F def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := map f F ≤ G example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) : Tendsto₂ f F G ↔ Tendsto₁ f F G := Iff.rfl #check (@Filter.map_mono : ∀ {α β} {m : α → β}, Monotone (map m)) #check (@Filter.map_map : ∀ {α β γ} {f : Filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := sorry variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) section variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) end example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ˢ 𝓝 y₀ := nhds_prod_eq #check le_inf_iff example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : Tendsto f atTop (𝓝 (x₀, y₀)) ↔ Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := sorry example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Ioo (x₀ - ε) (x₀ + ε) := nhds_basis_Ioo_pos x₀ example (u : ℕ → ℝ) (x₀ : ℝ) : Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] simp example (P Q : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) : ∀ᶠ n in atTop, P n ∧ Q n := hP.and hQ example (u v : ℕ → ℝ) (h : ∀ᶠ n in atTop, u n = v n) (x₀ : ℝ) : Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := tendsto_congr' h example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) : Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := tendsto_congr' h #check @eventually_of_forall #check @Eventually.mono #check @Eventually.and example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by apply (hP.and (hQ.and hR)).mono rintro n ⟨h, h', h''⟩ exact h'' ⟨h, h'⟩ example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by filter_upwards [hP, hQ, hR] intro n h h' h'' exact h'' ⟨h, h'⟩ #check mem_closure_iff_clusterPt #check le_principal_iff #check neBot_of_le example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := sorry
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MIL/C08_Topology/S02_Metric_Spaces.lean (deleted)
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@@ -1,206 +0,0 @@import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter open Topology Filter variable {X : Type _} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) #check (dist_nonneg : 0 ≤ dist a b) #check (dist_eq_zero : dist a b = 0 ↔ a = b) #check (dist_comm a b : dist a b = dist b a) #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. #check EMetricSpace #check PseudoMetricSpace #check PseudoEMetricSpace example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := Metric.tendsto_atTop example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := Metric.continuous_iff example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := by apply Continuous.dist exact hf.comp continuous_fst exact hf.comp continuous_snd example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := (hf.comp continuous_fst).dist (hf.comp continuous_snd) example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := hf.fst'.dist hf.snd' example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := sorry example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := Metric.continuousAt_iff variable (r : ℝ) example : Metric.ball a r = { b | dist b a < r } := rfl example : Metric.closedBall a r = { b | dist b a ≤ r } := rfl example (hr : 0 < r) : a ∈ Metric.ball a r := Metric.mem_ball_self hr example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := Metric.mem_closedBall_self hr example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.isOpen_iff example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := isOpen_compl_iff.symm example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) (hus : ∀ n, u n ∈ s) : a ∈ s := hs.mem_of_tendsto hu (eventually_of_forall hus) example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := Metric.mem_closure_iff example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := sorry example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.nhds_basis_ball.mem_iff example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := Metric.nhds_basis_closedBall.mem_iff example : IsCompact (Set.Icc 0 1 : Set ℝ) := isCompact_Icc example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := hs.exists_forall_le hs' hfs example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := hs.exists_forall_ge hs' hfs example {s : Set X} (hs : IsCompact s) : IsClosed s := hs.isClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ #check IsCompact.isClosed example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := Metric.uniformContinuous_iff example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := Metric.cauchySeq_iff example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := Metric.cauchySeq_iff' example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete hu open BigOperators open Finset theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by rw [Metric.cauchySeq_iff'] intro ε ε_pos obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn calc dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry _ ≤ 1 / 2 ^ N * 2 := sorry _ < ε := sorry open Metric example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) := by let B : ℕ → ℝ := fun n => (1 / 2) ^ n have Bpos : ∀ n, 0 < B n sorry /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by sorry choose! center radius Hpos HB Hball using this intro x rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] intro ε εpos /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n => Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) let c : ℕ → X := fun n => (F n).1 let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by sorry have rB : ∀ n, r n ≤ B n := by sorry have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry sorry
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@@ -1,155 +0,0 @@import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter Topology section variable {X : Type _} [TopologicalSpace X] example : IsOpen (univ : Set X) := isOpen_univ example : IsOpen (∅ : Set X) := isOpen_empty example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := isOpen_iUnion hs example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := isOpen_iInter hs variable {Y : Type _} [TopologicalSpace Y] example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := continuous_def example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := Iff.rfl example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := Iff.rfl example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := mem_nhds_iff example (x : X) : pure x ≤ 𝓝 x := pure_le_nhds x example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := pure_le_nhds x h example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := eventually_eventually_nhds.mpr h #check TopologicalSpace.mkOfNhds #check TopologicalSpace.nhds_mkOfNhds example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := sorry end -- BOTH. variable {X Y : Type _} example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := TopologicalSpace.coinduced f example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := TopologicalSpace.induced f example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := coinduced_le_iff_le_induced #check coinduced_compose #check induced_compose example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := Iff.rfl example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := continuous_iff_coinduced_le example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := rfl example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) (hb : Tendsto u atTop (𝓝 b)) : a = b := tendsto_nhds_unique ha hb example [TopologicalSpace X] [RegularSpace X] (a : X) : (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := closed_nhds_basis a example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := nhds_basis_opens' x theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := sorry example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry #check @HasBasis.tendsto_right_iff example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := mem_closure_iff_seq_limit variable [TopologicalSpace X] example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := Iff.rfl example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := Iff.rfl example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu variable [TopologicalSpace Y] example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := ClusterPt.map H hfx hf example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : IsCompact (f '' s) := by intro F F_ne F_le have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left sorry example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := hs.elim_finite_subcover U hUo hsU example [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
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@@ -1,71 +0,0 @@import Mathlib.Topology.Instances.Real open Set Filter Topology -- In the next example we could use `tauto` in each proof instead of knowing the lemmas example {α : Type _} (s : Set α) : Filter α := { sets := { t | s ⊆ t } univ_sets := subset_univ s sets_of_superset := fun hU hUV => Subset.trans hU hUV inter_sets := fun hU hV => subset_inter hU hV } example : Filter ℕ := { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } univ_sets := by use 42 simp sets_of_superset := by rintro U V ⟨N, hN⟩ hUV use N tauto inter_sets := by rintro U V ⟨N, hN⟩ ⟨N', hN'⟩ use max N N' intro b hb rw [max_le_iff] at hb constructor <;> tauto } def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := ∀ V ∈ G, f ⁻¹' V ∈ F example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := calc map (g ∘ f) F = map g (map f F) := by rw [map_map] _ ≤ map g G := (map_mono hf) _ ≤ H := hg example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by intro V hV rw [preimage_comp] apply hf apply hg exact hV example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : Tendsto f atTop (𝓝 (x₀, y₀)) ↔ Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := calc Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl _ ↔ map f atTop ≤ 𝓝 x₀ ×ˢ 𝓝 y₀ := by rw [nhds_prod_eq] _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by rw [map_map, map_map] -- an alternative solution example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : Tendsto f atTop (𝓝 (x₀, y₀)) ↔ Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by rw [nhds_prod_eq] unfold Tendsto SProd.sprod Filter.instSProd Filter.prod erw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)
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@@ -1,371 +0,0 @@import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter open Topology Filter variable {X : Type _} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) #check (dist_nonneg : 0 ≤ dist a b) #check (dist_eq_zero : dist a b = 0 ↔ a = b) #check (dist_comm a b : dist a b = dist b a) #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. #check EMetricSpace #check PseudoMetricSpace #check PseudoEMetricSpace example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := Metric.tendsto_atTop example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := Metric.continuous_iff example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := by apply Continuous.dist exact hf.comp continuous_fst exact hf.comp continuous_snd example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := (hf.comp continuous_fst).dist (hf.comp continuous_snd) example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := hf.fst'.dist hf.snd' example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := sorry example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := hf.comp <| (continuous_pow 2).add continuous_id example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := Metric.continuousAt_iff variable (r : ℝ) example : Metric.ball a r = { b | dist b a < r } := rfl example : Metric.closedBall a r = { b | dist b a ≤ r } := rfl example (hr : 0 < r) : a ∈ Metric.ball a r := Metric.mem_ball_self hr example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := Metric.mem_closedBall_self hr example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.isOpen_iff example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := isOpen_compl_iff.symm example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) (hus : ∀ n, u n ∈ s) : a ∈ s := hs.mem_of_tendsto hu (eventually_of_forall hus) example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := Metric.mem_closure_iff example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := sorry example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := by rw [Metric.tendsto_atTop] at hu rw [Metric.mem_closure_iff] intro ε ε_pos rcases hu ε ε_pos with ⟨N, hN⟩ refine' ⟨u N, hs _, _⟩ rw [dist_comm] exact hN N le_rfl example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.nhds_basis_ball.mem_iff example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := Metric.nhds_basis_closedBall.mem_iff example : IsCompact (Set.Icc 0 1 : Set ℝ) := isCompact_Icc example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := hs.exists_forall_le hs' hfs example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := hs.exists_forall_ge hs' hfs example {s : Set X} (hs : IsCompact s) : IsClosed s := hs.isClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ #check IsCompact.isClosed example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := Metric.uniformContinuous_iff example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := by rw [Metric.uniformContinuous_iff] intro ε ε_pos let φ : X × X → ℝ := fun p => dist (f p.1) (f p.2) have φ_cont : Continuous φ := hf.fst'.dist hf.snd' let K := { p : X × X | ε ≤ φ p } have K_closed : IsClosed K := isClosed_le continuous_const φ_cont have K_cpct : IsCompact K := K_closed.isCompact cases' eq_empty_or_nonempty K with hK hK · use 1, by norm_num intro x y _ have : (x, y) ∉ K := by simp [hK] simpa using this · rcases K_cpct.exists_forall_le hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩ use dist x₀ x₁ constructor · change _ < _ rw [dist_pos] intro h have : ε ≤ 0 := by simpa [*] using xx_in linarith · intro x x' contrapose! intro hxx' exact H (x, x') hxx' example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := Metric.cauchySeq_iff example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := Metric.cauchySeq_iff' example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete hu open BigOperators open Finset theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by rw [Metric.cauchySeq_iff'] intro ε ε_pos obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn calc dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry _ ≤ 1 / 2 ^ N * 2 := sorry _ < ε := sorry example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by rw [Metric.cauchySeq_iff'] intro ε ε_pos obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by have : Tendsto (fun N : ℕ => (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by rw [← MulZeroClass.zero_mul (2 : ℝ)] apply Tendsto.mul simp_rw [← one_div_pow (2 : ℝ)] apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith exact tendsto_const_nhds rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn calc dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := (dist_le_range_sum_dist (fun i => u (N + i)) k) _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] _ ≤ 1 / 2 ^ N * 2 := (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _))) _ < ε := hN open Metric example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) := by let B : ℕ → ℝ := fun n => (1 / 2) ^ n have Bpos : ∀ n, 0 < B n sorry /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by sorry choose! center radius Hpos HB Hball using this intro x rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] intro ε εpos /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n => Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) let c : ℕ → X := fun n => (F n).1 let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by sorry have rB : ∀ n, r n ≤ B n := by sorry have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry sorry example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) := by let B : ℕ → ℝ := fun n => (1 / 2) ^ n have Bpos : ∀ n, 0 < B n := fun n => pow_pos sorry n /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by intro n x δ δpos have : x ∈ closure (f n) := hd n x rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ rw [dist_comm] at xy obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ show 0 < min (min (δ / 2) r) (B (n + 1)) exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) exact min_le_right _ _ show z ∈ closedBall x δ exact calc dist z x ≤ dist z y + dist y x := dist_triangle _ _ _ _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le) _ ≤ δ / 2 + δ / 2 := (add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _) _ = δ := add_halves δ show z ∈ f n exact hr (calc dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz _ ≤ r := (min_le_left _ _).trans (min_le_right _ _) ) choose! center radius Hpos HB Hball using this refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n => Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) let c : ℕ → X := fun n => (F n).1 let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by intro n induction' n with n hn exact lt_min εpos (Bpos 0) exact Hpos n (c n) (r n) hn have rB : ∀ n, r n ≤ B n := by intro n induction' n with n hn exact min_le_right _ _ exact HB n (c n) (r n) (rpos n) have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => Hball n (c n) (r n) (rpos n) have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by intro n rw [dist_comm] have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := mem_closedBall_self (rpos <| n + 1).le have I := calc closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := (incl n).trans (inter_subset_left _ _) _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) exact I A have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by intro n refine' Nat.le_induction _ fun m hnm h => _ · exact Subset.rfl · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) have yball : ∀ n, y ∈ closedBall (c n) (r n) := by intro n refine' isClosed_ball.mem_of_tendsto ylim _ refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ exact I n m hm (mem_closedBall_self (rpos _).le) constructor · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] intro n have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := Subset.trans (incl n) (inter_subset_right _ _) exact this (yball (n + 1)) calc dist y x ≤ r 0 := yball 0 _ ≤ ε := min_le_left _ _
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@@ -1,207 +0,0 @@import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter Topology section variable {X : Type _} [TopologicalSpace X] example : IsOpen (univ : Set X) := isOpen_univ example : IsOpen (∅ : Set X) := isOpen_empty example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := isOpen_iUnion hs example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := isOpen_iInter hs variable {Y : Type _} [TopologicalSpace Y] example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := continuous_def example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := Iff.rfl example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := Iff.rfl example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := mem_nhds_iff example (x : X) : pure x ≤ 𝓝 x := pure_le_nhds x example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := pure_le_nhds x h example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := eventually_eventually_nhds.mpr h #check TopologicalSpace.mkOfNhds #check TopologicalSpace.nhds_mkOfNhds example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := sorry example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := by intro a s s_in refine' ⟨{ y | s ∈ n y }, H a (fun x => x ∈ s) s_in, _, by tauto⟩ rintro y (hy : s ∈ n y) exact H₀ y hy end -- BOTH. variable {X Y : Type _} example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := TopologicalSpace.coinduced f example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := TopologicalSpace.induced f example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := coinduced_le_iff_le_induced #check coinduced_compose #check induced_compose example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := Iff.rfl example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := continuous_iff_coinduced_le example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := rfl example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) (hb : Tendsto u atTop (𝓝 b)) : a = b := tendsto_nhds_unique ha hb example [TopologicalSpace X] [RegularSpace X] (a : X) : (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := closed_nhds_basis a example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := nhds_basis_opens' x theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := sorry example {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry #check @HasBasis.tendsto_right_iff example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by choose φ hφ using hf use φ constructor · rw [continuous_iff_continuousAt] intro x suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] intro V' V'_in V'_closed obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in suffices : ∀ y ∈ V, φ y ∈ V' exact mem_of_superset V_in this intro y y_in have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y apply V'_closed.mem_of_tendsto (hφ y) exact mem_of_superset (preimage_mem_comap hVx) hV · intro a have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a exact tendsto_nhds_unique lim f_cont.continuousAt example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := mem_closure_iff_seq_limit variable [TopologicalSpace X] example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := Iff.rfl example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := Iff.rfl example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu variable [TopologicalSpace Y] example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := ClusterPt.map H hfx hf example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : IsCompact (f '' s) := by intro F F_ne F_le have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left sorry example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : IsCompact (f '' s) := by intro F F_ne F_le have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal] have Hne : (𝓟 s ⊓ comap f F).NeBot := by apply NeBot.of_map rwa [map_eq, inf_of_le_right F_le] have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left rcases hs Hle with ⟨x, x_in, hx⟩ refine' ⟨f x, mem_image_of_mem f x_in, _⟩ apply hx.map hf.continuousAt rw [Tendsto, map_eq] exact inf_le_right example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := hs.elim_finite_subcover U hUo hsU example [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
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@@ -1,40 +0,0 @@import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv import Mathlib.Analysis.Calculus.MeanValue open Set Filter open Topology Filter Classical Real noncomputable section open Real /-- The sin function has derivative 1 at 0. -/ example : HasDerivAt sin 1 0 := by simpa using hasDerivAt_sin 0 example (x : ℝ) : DifferentiableAt ℝ sin x := (hasDerivAt_sin x).differentiableAt example {f : ℝ → ℝ} {x a : ℝ} (h : HasDerivAt f a x) : deriv f x = a := h.deriv example {f : ℝ → ℝ} {x : ℝ} (h : ¬DifferentiableAt ℝ f x) : deriv f x = 0 := deriv_zero_of_not_differentiableAt h example {f g : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) : deriv (f + g) x = deriv f x + deriv g x := deriv_add hf hg example {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMin f a) : deriv f a = 0 := h.deriv_eq_zero example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : ∃ c ∈ Ioo a b, deriv f c = 0 := exists_deriv_eq_zero f hab hfc hfI example (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Icc a b)) (hf' : DifferentiableOn ℝ f (Ioo a b)) : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) := exists_deriv_eq_slope f hab hf hf' example : deriv (fun x : ℝ ↦ x ^ 5) 6 = 5 * 6 ^ 4 := by simp example : deriv sin π = -1 := by simp
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@@ -1,185 +0,0 @@import Mathlib.Analysis.NormedSpace.BanachSteinhaus import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Calculus.Inverse import Mathlib.Analysis.Calculus.ContDiff import Mathlib.Analysis.Calculus.FDeriv.Prod open Set Filter open Topology Filter noncomputable section section variable {E : Type _} [NormedAddCommGroup E] example (x : E) : 0 ≤ ‖x‖ := norm_nonneg x example {x : E} : ‖x‖ = 0 ↔ x = 0 := norm_eq_zero example (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ := norm_add_le x y example : MetricSpace E := by infer_instance example {X : Type _} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : Continuous fun x => ‖f x‖ := hf.norm variable [NormedSpace ℝ E] example (a : ℝ) (x : E) : ‖a • x‖ = |a| * ‖x‖ := norm_smul a x example [FiniteDimensional ℝ E] : CompleteSpace E := by infer_instance example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := norm_mul x y example (𝕜 : Type _) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := NormedField.exists_one_lt_norm 𝕜 example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (E : Type _) [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] : CompleteSpace E := FiniteDimensional.complete 𝕜 E end section variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] example : E →L[𝕜] E := ContinuousLinearMap.id 𝕜 E example (f : E →L[𝕜] F) : E → F := f example (f : E →L[𝕜] F) : Continuous f := f.cont example (f : E →L[𝕜] F) (x y : E) : f (x + y) = f x + f y := f.map_add x y example (f : E →L[𝕜] F) (a : 𝕜) (x : E) : f (a • x) = a • f x := f.map_smul a x variable (f : E →L[𝕜] F) example (x : E) : ‖f x‖ ≤ ‖f‖ * ‖x‖ := f.le_op_norm x example {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M := f.op_norm_le_bound hMp hM end section variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] open Metric example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C' := by -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } -- each of these sets is closed have hc : ∀ n : ℕ, IsClosed (e n) sorry -- the union is the entire space; this is where we use `h` have hU : (⋃ n : ℕ, e n) = univ sorry /- apply the Baire category theorem to conclude that for some `m : ℕ`, `e m` contains some `x` -/ obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ‖k‖ := sorry -- show all elements in the ball have norm bounded by `m` after applying any `g i` have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m sorry have εk_pos : 0 < ε / ‖k‖ := sorry refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ sorry sorry end open Asymptotics open Asymptotics example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := isBigOWith_iff example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (l : Filter α) (f : α → E) (g : α → F) : f =O[l] g ↔ ∃ C, IsBigOWith C l f g := isBigO_iff_isBigOWith example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (l : Filter α) (f : α → E) (g : α → F) : f =o[l] g ↔ ∀ C > 0, IsBigOWith C l f g := isLittleO_iff_forall_isBigOWith example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : f ~[l] g ↔ (f - g) =o[l] g := Iff.rfl section variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : HasFDerivAt f f' x₀ ↔ (fun x => f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x => x - x₀ := Iff.rfl example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) (hff' : HasFDerivAt f f' x₀) : fderiv 𝕜 f x₀ = f' := hff'.fderiv example (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F := iteratedFDeriv 𝕜 n f example (n : WithTop ℕ) {f : E → F} : ContDiff 𝕜 n f ↔ (∀ m : ℕ, (m : WithTop ℕ) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧ ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x := contDiff_iff_continuous_differentiable example {𝕂 : Type _} [IsROrC 𝕂] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := hf.hasStrictFDerivAt hn section LocalInverse variable [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E := HasStrictFDerivAt.localInverse f f' a hf example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x := hf.eventually_left_inverse example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : ∀ᶠ x in 𝓝 (f a), f (hf.localInverse f f' a x) = x := hf.eventually_right_inverse example [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (f'.symm : F →L[𝕜] E) (f a) := HasStrictFDerivAt.to_localInverse hf end LocalInverse #check HasFDerivWithinAt #check HasFDerivAtFilter end
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MIL/C09_Differential_Calculus/solutions/Solutions_S01_Elementary_Differential_Calculus.lean (deleted)
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@@ -1,8 +0,0 @@import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv import Mathlib.Analysis.Calculus.MeanValue open Set Filter open Topology Filter Classical Real noncomputable section
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MIL/C09_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean (deleted)
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@@ -1,62 +0,0 @@import Mathlib.Analysis.NormedSpace.BanachSteinhaus import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Calculus.Inverse import Mathlib.Analysis.Calculus.ContDiff import Mathlib.Analysis.Calculus.FDeriv.Prod open Set Filter open Topology Filter noncomputable section section variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] open Metric example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C' := by -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } -- each of these sets is closed have hc : ∀ n : ℕ, IsClosed (e n) := fun i => isClosed_iInter fun i => isClosed_le (g i).cont.norm continuous_const -- the union is the entire space; this is where we use `h` have hU : (⋃ n : ℕ, e n) = univ := by refine' eq_univ_of_forall fun x => _ cases' h x with C hC obtain ⟨m, hm⟩ := exists_nat_ge C exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i => le_trans (hC i) hm⟩ /- apply the Baire category theorem to conclude that for some `m : ℕ`, `e m` contains some `x` -/ obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_iUnion_of_closed hc hU obtain ⟨ε, ε_pos, hε : ball x ε ⊆ interior (e m)⟩ := isOpen_iff.mp isOpen_interior x hx obtain ⟨k : 𝕜, hk : 1 < ‖k‖⟩ := NormedField.exists_one_lt_norm 𝕜 -- show all elements in the ball have norm bounded by `m` after applying any `g i` have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m := by intro z hz i replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i apply interior_subset hz have εk_pos : 0 < ε / ‖k‖ := div_pos ε_pos (zero_lt_one.trans hk) refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ · exact div_nonneg (Nat.cast_nonneg _) εk_pos.le intro y le_y y_lt calc ‖g i y‖ = ‖g i (y + x) - g i x‖ := by rw [(g i).map_add, add_sub_cancel] _ ≤ ‖g i (y + x)‖ + ‖g i x‖ := (norm_sub_le _ _) _ ≤ m + m := (add_le_add (real_norm_le (y + x) (by rwa [add_comm, add_mem_ball_iff_norm]) i) (real_norm_le x (mem_ball_self ε_pos) i)) _ = (m + m : ℕ) := by norm_cast _ ≤ (m + m : ℕ) * (‖y‖ / (ε / ‖k‖)) := (le_mul_of_one_le_right (Nat.cast_nonneg _) ((one_le_div <| div_pos ε_pos (zero_lt_one.trans hk)).2 le_y)) _ = (m + m : ℕ) / (ε / ‖k‖) * ‖y‖ := (mul_comm_div _ _ _).symm end
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@@ -1,370 +0,0 @@<!DOCTYPE html> <html class="writer-html5" lang="en" > <head> <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>1. Introduction — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> <link rel="shortcut icon" href="_static/favicon.ico"/> <!--[if lt IE 9]> <script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" /> <link rel="next" title="2. Basics" href="C02_Basics.html" /> <link rel="prev" title="Mathematics in Lean" href="index.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul class="current"> <li class="toctree-l1 current"><a class="current reference internal" href="#">1. Introduction</a><ul> <li class="toctree-l2"><a class="reference internal" href="#getting-started">1.1. Getting Started</a></li> <li class="toctree-l2"><a class="reference internal" href="#overview">1.2. Overview</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">1. </span>Introduction</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C01_Introduction.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="introduction"> <span id="id1"></span><h1><span class="section-number">1. </span>Introduction<a class="headerlink" href="#introduction" title="Permalink to this heading"></a></h1> <section id="getting-started"> <h2><span class="section-number">1.1. </span>Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this heading"></a></h2> <p>The goal of this book is to teach you to formalize mathematics using the Lean 4 interactive proof assistant. It assumes that you know some mathematics, but it does not require much. Although we will cover examples ranging from number theory to measure theory and analysis, we will focus on elementary aspects of those fields, in the hopes that if they are not familiar to you, you can pick them up as you go. We also don’t presuppose any background in formalization. Formalization can be seen as a kind of computer programming: we will write mathematical definitions, theorems, and proofs in a regimented language, like a programming language, that Lean can understand. In return, Lean provides feedback and information, interprets expressions and guarantees that they are well-formed, and ultimately certifies the correctness of our proofs.</p> <p>You can learn more about Lean from the <a class="reference external" href="https://leanprover.github.io">Lean project page</a> and the <a class="reference external" href="https://leanprover-community.github.io/">Lean community web pages</a>. This tutorial is based on Lean’s large and ever-growing library, <em>mathlib</em>. We also strongly recommend taking a look at the <a class="reference external" href="https://leanprover.zulipchat.com/">Lean Zulip online chat group</a> if you haven’t already. You’ll find a lively and welcoming community of Lean enthusiasts there, happy to answer questions and offer moral support.</p> <p>Although you can read a pdf or html version of this book online, it designed to be read interactively, running Lean from inside the VS Code editor. To get started:</p> <ol class="arabic simple"> <li><p>Install Lean 4 and VS Code following these <a class="reference external" href="https://github.com/leanprover/lean4/blob/master/doc/quickstart.md">instructions</a>.</p></li> <li><p>In a terminal, navigate to the folder where you want to put a copy of the repository, and type <code class="docutils literal notranslate"><span class="pre">git</span> <span class="pre">clone</span> <span class="pre">git@github.com:leanprover-community/mathematics_in_lean.git</span></code> to fetch it from github.</p></li> <li><p>Navigate to <code class="docutils literal notranslate"><span class="pre">mathematics_in_lean</span></code>, and execute <code class="docutils literal notranslate"><span class="pre">lake</span> <span class="pre">exe</span> <span class="pre">cache</span> <span class="pre">get</span></code> to fetch a compiled version of the library, <code class="docutils literal notranslate"><span class="pre">mathlib</span></code>.</p></li> <li><p>Type <code class="docutils literal notranslate"><span class="pre">code</span> <span class="pre">.</span></code> to open the folder in <code class="docutils literal notranslate"><span class="pre">VS</span> <span class="pre">Code</span></code>. Alternatively, you can run <code class="docutils literal notranslate"><span class="pre">VS</span> <span class="pre">Code</span></code> and choose <code class="docutils literal notranslate"><span class="pre">Open</span> <span class="pre">Folder</span></code> from the <code class="docutils literal notranslate"><span class="pre">File</span></code> menu. Be sure to open the folder <code class="docutils literal notranslate"><span class="pre">mathematics_in_lean</span></code>, not any other folder.</p></li> </ol> <p>Opening any Lean file will simultaneously open this book in a VS Code window. You can update to a newer version by tying <code class="docutils literal notranslate"><span class="pre">git</span> <span class="pre">pull</span></code> followed by <code class="docutils literal notranslate"><span class="pre">lake</span> <span class="pre">exe</span> <span class="pre">cache</span> <span class="pre">get</span></code> inside the <code class="docutils literal notranslate"><span class="pre">mathematics_in_lean</span></code> folder.</p> <p>Alternatively, you can run Lean and VS Code in the cloud, using <a class="reference external" href="https://gitpod.io/">Gitpod</a>. You can find instructions as to how to do that on the Mathematics in Lean <a class="reference external" href="https://github.com/leanprover-community/mathematics_in_lean">project page</a> on Github.</p> <p>Each section in this book has an associated Lean file with examples and exercises. You can find them in the folder <cite>src</cite>, organized by chapter. We recommend making a copy of that folder so that you can experiment with the files as you go, while leaving the originals intact. The text will often include examples, like this one:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#eval</span> <span class="s2">"Hello, World!"</span> </pre></div> </div> <p>You should be able to find the corresponding example in the associated Lean file. If you click on the line, VS Code will show you Lean’s feedback in the <code class="docutils literal notranslate"><span class="pre">Lean</span> <span class="pre">Goal</span></code> window, and if you hover your cursor over the <code class="docutils literal notranslate"><span class="pre">#eval</span></code> command VS Code will show you Lean’s response to this command in a pop-up window. You are encouraged to edit the file and try examples of your own.</p> <p>This book moreover provides lots of challenging exercises for you to try. Don’t rush past these! Lean is about <em>doing</em> mathematics interactively, not just reading about it. Working through the exercises is central to the experience. You can always compare your solutions to the ones in the <code class="docutils literal notranslate"><span class="pre">solutions</span></code> folder associated with each section.</p> </section> <section id="overview"> <h2><span class="section-number">1.2. </span>Overview<a class="headerlink" href="#overview" title="Permalink to this heading"></a></h2> <p>Put simply, Lean is a tool for building complex expressions in a formal language known as <em>dependent type theory</em>.</p> <p id="index-0">Every expression has a <em>type</em>, and you can use the <cite>#check</cite> command to print it. Some expressions have types like <cite>ℕ</cite> or <cite>ℕ → ℕ</cite>. These are mathematical objects.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="mi">2</span> <span class="bp">+</span> <span class="mi">2</span> <span class="kd">def</span> <span class="n">f</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:=</span> <span class="n">x</span> <span class="bp">+</span> <span class="mi">3</span> <span class="k">#check</span> <span class="n">f</span> </pre></div> </div> <p>Some expressions have type <cite>Prop</cite>. These are mathematical statements.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="mi">2</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">4</span> <span class="kd">def</span> <span class="n">FermatLastTheorem</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">n</span> <span class="bp">></span> <span class="mi">2</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">y</span> <span class="bp">*</span> <span class="n">z</span> <span class="bp">≠</span> <span class="mi">0</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp">^</span> <span class="n">n</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="n">n</span> <span class="bp">≠</span> <span class="n">z</span> <span class="bp">^</span> <span class="n">n</span> <span class="k">#check</span> <span class="n">FermatLastTheorem</span> </pre></div> </div> <p>Some expressions have a type, <cite>P</cite>, where <cite>P</cite> itself has type <cite>Prop</cite>. Such an expression is a proof of the proposition <cite>P</cite>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">easy</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">4</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="k">#check</span> <span class="n">easy</span> <span class="kd">theorem</span> <span class="n">hard</span> <span class="o">:</span> <span class="n">FermatLastTheorem</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="k">#check</span> <span class="n">hard</span> </pre></div> </div> <p>If you manage to construct an expression of type <cite>fermat_last_theorem</cite> and Lean accepts it as a term of that type, you have done something very impressive. (Using <code class="docutils literal notranslate"><span class="pre">sorry</span></code> is cheating, and Lean knows it.) So now you know the game. All that is left to learn are the rules.</p> <p>This book is complementary to a companion tutorial, <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, which provides a more thorough introduction to the underlying logical framework and core syntax of Lean. <em>Theorem Proving in Lean</em> is for people who prefer to read a user manual cover to cover before using a new dishwasher. If you are the kind of person who prefers to hit the <em>start</em> button and figure out how to activate the potscrubber feature later, it makes more sense to start here and refer back to <em>Theorem Proving in Lean</em> as necessary.</p> <p>Another thing that distinguishes <em>Mathematics in Lean</em> from <em>Theorem Proving in Lean</em> is that here we place a much greater emphasis on the use of <em>tactics</em>. Given that we are trying to build complex expressions, Lean offers two ways of going about it: we can write down the expressions themselves (that is, suitable text descriptions thereof), or we can provide Lean with <em>instructions</em> as to how to construct them. For example, the following expression represents a proof of the fact that if <code class="docutils literal notranslate"><span class="pre">n</span></code> is even then so is <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">m</span> <span class="n">n</span> <span class="o">⟨</span><span class="n">k</span><span class="o">,</span> <span class="o">(</span><span class="n">hk</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">k</span> <span class="bp">+</span> <span class="n">k</span><span class="o">)⟩</span> <span class="bp">=></span> <span class="k">have</span> <span class="n">hmn</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">k</span> <span class="bp">+</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">k</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hk</span><span class="o">,</span> <span class="n">mul_add</span><span class="o">]</span> <span class="k">show</span> <span class="bp">∃</span> <span class="n">l</span><span class="o">,</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">l</span> <span class="bp">+</span> <span class="n">l</span> <span class="k">from</span> <span class="o">⟨</span><span class="n">_</span><span class="o">,</span> <span class="n">hmn</span><span class="o">⟩</span> </pre></div> </div> <p>The <em>proof term</em> can be compressed to a single line:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">m</span> <span class="n">n</span> <span class="o">⟨</span><span class="n">k</span><span class="o">,</span> <span class="n">hk</span><span class="o">⟩</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">m</span> <span class="bp">*</span> <span class="n">k</span><span class="o">,</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hk</span><span class="o">,</span> <span class="n">mul_add</span><span class="o">]⟩</span> </pre></div> </div> <p>The following is, instead, a <em>tactic-style</em> proof of the same theorem:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="c1">-- say m and n are natural numbers, and assume n=2*k</span> <span class="n">rintro</span> <span class="n">m</span> <span class="n">n</span> <span class="o">⟨</span><span class="n">k</span><span class="o">,</span> <span class="n">hk</span><span class="o">⟩</span> <span class="c1">-- We need to prove m*n is twice a natural number. Let's show it's twice m*k.</span> <span class="n">use</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">k</span> <span class="c1">-- substitute in for n</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hk</span><span class="o">]</span> <span class="c1">-- and now it's obvious</span> <span class="n">ring</span> </pre></div> </div> <p>As you enter each line of such a proof in VS Code, Lean displays the <em>proof state</em> in a separate window, telling you what facts you have already established and what tasks remain to prove your theorem. You can replay the proof by stepping through the lines, since Lean will continue to show you the state of the proof at the point where the cursor is. In this example, you will then see that the first line of the proof introduces <code class="docutils literal notranslate"><span class="pre">m</span></code> and <code class="docutils literal notranslate"><span class="pre">n</span></code> (we could have renamed them at that point, if we wanted to), and also decomposes the hypothesis <code class="docutils literal notranslate"><span class="pre">Even</span> <span class="pre">n</span></code> to a <code class="docutils literal notranslate"><span class="pre">k</span></code> and the assumption that <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">=</span> <span class="pre">2</span> <span class="pre">*</span> <span class="pre">k</span></code>. The second line, <code class="docutils literal notranslate"><span class="pre">use</span> <span class="pre">m</span> <span class="pre">*</span> <span class="pre">k</span></code>, declares that we are going to show that <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span></code> is even by showing <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">=</span> <span class="pre">2</span> <span class="pre">*</span> <span class="pre">(m</span> <span class="pre">*</span> <span class="pre">k)</span></code>. The next line uses the <code class="docutils literal notranslate"><span class="pre">rewrite</span></code> tactic to replace <code class="docutils literal notranslate"><span class="pre">n</span></code> by <code class="docutils literal notranslate"><span class="pre">2</span> <span class="pre">*</span> <span class="pre">k</span></code> in the goal, and the <cite>ring</cite> tactic solves the resulting goal <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">(2</span> <span class="pre">*</span> <span class="pre">k)</span> <span class="pre">=</span> <span class="pre">2</span> <span class="pre">*</span> <span class="pre">(m</span> <span class="pre">*</span> <span class="pre">k)</span></code>.</p> <p>The ability to build a proof in small steps with incremental feedback is extremely powerful. For that reason, tactic proofs are often easier and quicker to write than proof terms. There isn’t a sharp distinction between the two: tactic proofs can be inserted in proof terms, as we did with the phrase <code class="docutils literal notranslate"><span class="pre">by</span> <span class="pre">rw</span> <span class="pre">[hk,</span> <span class="pre">mul_left_comm]</span></code> in the example above. We will also see that, conversely, it is often useful to insert a short proof term in the middle of a tactic proof. That said, in this book, our emphasis will be on the use of tactics.</p> <p>In our example, the tactic proof can also be reduced to a one-liner:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="n">m</span> <span class="n">n</span> <span class="o">⟨</span><span class="n">k</span><span class="o">,</span> <span class="n">hk</span><span class="o">⟩</span><span class="bp">;</span> <span class="n">use</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">k</span><span class="bp">;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hk</span><span class="o">]</span><span class="bp">;</span> <span class="n">ring</span> </pre></div> </div> <p>Here we have used tactics to carry out small proof steps. But they can also provide substantial automation, and justify longer calculations and bigger inferential steps. For example, we can invoke Lean’s simplifier with specific rules for simplifying statements about parity to prove our theorem automatically.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span><span class="bp">;</span> <span class="n">simp</span> <span class="o">[</span><span class="bp">*</span><span class="o">,</span> <span class="n">parity_simps</span><span class="o">]</span> </pre></div> </div> <p>Another big difference between the two introductions is that <em>Theorem Proving in Lean</em> depends only on core Lean and its built-in tactics, whereas <em>Mathematics in Lean</em> is built on top of Lean’s powerful and ever-growing library, <em>mathlib</em>. As a result, we can show you how to use some of the mathematical objects and theorems in the library, and some of the very useful tactics. This book is not meant to be used as an overview of the library; the <a class="reference external" href="https://leanprover-community.github.io/">community</a> web pages contain extensive documentation. Rather, our goal is to introduce you to the style of thinking that underlies that formalization, so that you are comfortable browsing the library and finding things on your own.</p> <p>Interactive theorem proving can be frustrating, and the learning curve is steep. But the Lean community is very welcoming to newcomers, and people are available on the <a class="reference external" href="https://leanprover.zulipchat.com/">Lean Zulip chat group</a> round the clock to answer questions. We hope to see you there, and have no doubt that soon enough you, too, will be able to answer such questions and contribute to the development of <em>mathlib</em>.</p> <p>So here is your mission, should you choose to accept it: dive in, try the exercises, come to Zulip with questions, and have fun. But be forewarned: interactive theorem proving will challenge you to think about mathematics and mathematical reasoning in fundamentally new ways. Your life may never be the same.</p> <p><em>Acknowledgments.</em> We are grateful to Gabriel Ebner for setting up the infrastructure for running this tutorial in VS Code, and to Scott Morrison and Mario Carneiro for help porting it from Lean 3. We are also grateful for help and corrections from Bryan Gin-ge Chen, Johan Commelin, Mathieu Guay-Paquet, Julian Külshammer, Giovanni Mascellani, Hunter Monroe, Pietro Monticone, Bartosz Piotrowski, and Guilherme Silva. Our work has been partially supported by the Hoskinson Center for Formal Mathematics.</p> </section> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="index.html" class="btn btn-neutral float-left" title="Mathematics in Lean" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> <a href="C02_Basics.html" class="btn btn-neutral float-right" title="2. Basics" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> </div> <hr/> <div role="contentinfo"> <p>© Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. 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Logic" href="C03_Logic.html" /> <link rel="prev" title="1. Introduction" href="C01_Introduction.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul class="current"> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">2. Basics</a><ul> <li class="toctree-l2"><a class="reference internal" href="#calculating">2.1. Calculating</a></li> <li class="toctree-l2"><a class="reference internal" href="#proving-identities-in-algebraic-structures">2.2. Proving Identities in Algebraic Structures</a></li> <li class="toctree-l2"><a class="reference internal" href="#using-theorems-and-lemmas">2.3. Using Theorems and Lemmas</a></li> <li class="toctree-l2"><a class="reference internal" href="#more-on-order-and-divisibility">2.4. More on Order and Divisibility</a></li> <li class="toctree-l2"><a class="reference internal" href="#proving-facts-about-algebraic-structures">2.5. Proving Facts about Algebraic Structures</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">2. </span>Basics</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C02_Basics.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="basics"> <span id="id1"></span><h1><span class="section-number">2. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h1> <p>This chapter is designed to introduce you to the nuts and bolts of mathematical reasoning in Lean: calculating, applying lemmas and theorems, and reasoning about generic structures.</p> <section id="calculating"> <h2><span class="section-number">2.1. </span>Calculating<a class="headerlink" href="#calculating" title="Permalink to this heading"></a></h2> <p>We generally learn to carry out mathematical calculations without thinking of them as proofs. But when we justify each step in a calculation, as Lean requires us to do, the net result is a proof that the left-hand side of the calculation is equal to the right-hand side.</p> <p id="index-0">In Lean, stating a theorem is tantamount to stating a goal, namely, the goal of proving the theorem. Lean provides the rewriting tactic <code class="docutils literal notranslate"><span class="pre">rw</span></code>, to replace the left-hand side of an identity by the right-hand side in the goal. If <code class="docutils literal notranslate"><span class="pre">a</span></code>, <code class="docutils literal notranslate"><span class="pre">b</span></code>, and <code class="docutils literal notranslate"><span class="pre">c</span></code> are real numbers, <code class="docutils literal notranslate"><span class="pre">mul_assoc</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">c</span></code> is the identity <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">b</span> <span class="pre">*</span> <span class="pre">c</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">(b</span> <span class="pre">*</span> <span class="pre">c)</span></code> and <code class="docutils literal notranslate"><span class="pre">mul_comm</span> <span class="pre">a</span> <span class="pre">b</span></code> is the identity <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">b</span> <span class="pre">=</span> <span class="pre">b</span> <span class="pre">*</span> <span class="pre">a</span></code>. Lean provides automation that generally eliminates the need to refer the facts like these explicitly, but they are useful for the purposes of illustration. In Lean, multiplication associates to the left, so the left-hand side of <code class="docutils literal notranslate"><span class="pre">mul_assoc</span></code> could also be written <code class="docutils literal notranslate"><span class="pre">(a</span> <span class="pre">*</span> <span class="pre">b)</span> <span class="pre">*</span> <span class="pre">c</span></code>. However, it is generally good style to be mindful of Lean’s notational conventions and leave out parentheses when Lean does as well.</p> <p>Let’s try out <code class="docutils literal notranslate"><span class="pre">rw</span></code>.</p> <div class="highlight-lean notranslate" id="index-1"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_comm</span> <span class="n">a</span> <span class="n">b</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_assoc</span> <span class="n">b</span> <span class="n">a</span> <span class="n">c</span><span class="o">]</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">import</span></code> line at the beginning of the example imports the theory of the real numbers from <code class="docutils literal notranslate"><span class="pre">mathlib</span></code>. For the sake of brevity, we generally suppress information like this when it is repeated from example to example.</p> <p>You are welcome to make changes to see what happens. You can type the <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> character as <code class="docutils literal notranslate"><span class="pre">\R</span></code> or <code class="docutils literal notranslate"><span class="pre">\real</span></code> in VS Code. The symbol doesn’t appear until you hit space or the tab key. If you hover over a symbol when reading a Lean file, VS Code will show you the syntax that can be used to enter it. If you are curious to see all available abreviations, you can hit Ctrl-Shift-p and then type abbreviations to get access to the <code class="docutils literal notranslate"><span class="pre">Lean</span> <span class="pre">4:</span> <span class="pre">Show</span> <span class="pre">all</span> <span class="pre">abbreviations</span></code> command. If your keyboard does not have an easily accessible backslash, you can change the leading character by changing the <code class="docutils literal notranslate"><span class="pre">lean.input.leader</span></code> setting.</p> <p id="index-2">When a cursor is in the middle of a tactic proof, Lean reports on the current <em>proof state</em> in the <em>Lean infoview</em> window. As you move your cursor past each step of the proof, you can see the state change. A typical proof state in Lean might look as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="n">goal</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">h₁</span> <span class="o">:</span> <span class="n">Prime</span> <span class="n">x</span><span class="o">,</span> <span class="n">h₂</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">x</span><span class="o">,</span> <span class="n">h₃</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">></span> <span class="n">x</span> <span class="bp">⊢</span> <span class="n">y</span> <span class="bp">≥</span> <span class="mi">4</span> </pre></div> </div> <p>The lines before the one that begins with <code class="docutils literal notranslate"><span class="pre">⊢</span></code> denote the <em>context</em>: they are the objects and assumptions currently at play. In this example, these include two objects, <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code>, each a natural number. They also include three assumptions, labelled <code class="docutils literal notranslate"><span class="pre">h₁</span></code>, <code class="docutils literal notranslate"><span class="pre">h₂</span></code>, and <code class="docutils literal notranslate"><span class="pre">h₃</span></code>. In Lean, everything in a context is labelled with an identifier. You can type these subscripted labels as <code class="docutils literal notranslate"><span class="pre">h\1</span></code>, <code class="docutils literal notranslate"><span class="pre">h\2</span></code>, and <code class="docutils literal notranslate"><span class="pre">h\3</span></code>, but any legal identifiers would do: you can use <code class="docutils literal notranslate"><span class="pre">h1</span></code>, <code class="docutils literal notranslate"><span class="pre">h2</span></code>, <code class="docutils literal notranslate"><span class="pre">h3</span></code> instead, or <code class="docutils literal notranslate"><span class="pre">foo</span></code>, <code class="docutils literal notranslate"><span class="pre">bar</span></code>, and <code class="docutils literal notranslate"><span class="pre">baz</span></code>. The last line represents the <em>goal</em>, that is, the fact to be proved. Sometimes people use <em>target</em> for the fact to be proved, and <em>goal</em> for the combination of the context and the target. In practice, the intended meaning is usually clear.</p> <p>Try proving these identities, in each case replacing <code class="docutils literal notranslate"><span class="pre">sorry</span></code> by a tactic proof. With the <code class="docutils literal notranslate"><span class="pre">rw</span></code> tactic, you can use a left arrow (<code class="docutils literal notranslate"><span class="pre">\l</span></code>) to reverse an identity. For example, <code class="docutils literal notranslate"><span class="pre">rw</span> <span class="pre">←</span> <span class="pre">mul_assoc</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">c</span></code> replaces <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">(b</span> <span class="pre">*</span> <span class="pre">c)</span></code> by <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">b</span> <span class="pre">*</span> <span class="pre">c</span></code> in the current goal. Note that the left-pointing arrow refers to going from right to left in the identity provided by <code class="docutils literal notranslate"><span class="pre">mul_assoc</span></code>, it has nothing to do with the left or right side of the goal.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>You can also use identities like <code class="docutils literal notranslate"><span class="pre">mul_assoc</span></code> and <code class="docutils literal notranslate"><span class="pre">mul_comm</span></code> without arguments. In this case, the rewrite tactic tries to match the left-hand side with an expression in the goal, using the first pattern it finds.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_assoc</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_comm</span><span class="o">]</span> </pre></div> </div> <p>You can also provide <em>partial</em> information. For example, <code class="docutils literal notranslate"><span class="pre">mul_comm</span> <span class="pre">a</span></code> matches any pattern of the form <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">?</span></code> and rewrites it to <code class="docutils literal notranslate"><span class="pre">?</span> <span class="pre">*</span> <span class="pre">a</span></code>. Try doing the first of these examples without providing any arguments at all, and the second with only one argument.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">c</span> <span class="bp">*</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>You an also use <code class="docutils literal notranslate"><span class="pre">rw</span></code> with facts from the local context.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="n">e</span> <span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">d</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">e</span> <span class="bp">=</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">e</span><span class="o">)</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="o">(</span><span class="n">d</span> <span class="bp">*</span> <span class="n">f</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h'</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">mul_assoc</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_assoc</span><span class="o">]</span> </pre></div> </div> <p>Try these, using the theorem <cite>sub_self</cite> for the second one:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="n">e</span> <span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">e</span> <span class="bp">*</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">d</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">e</span> <span class="bp">*</span> <span class="n">f</span> <span class="bp">*</span> <span class="n">d</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">hyp</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">-</span> <span class="n">d</span><span class="o">)</span> <span class="o">(</span><span class="n">hyp'</span> <span class="o">:</span> <span class="n">d</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Multiple rewrite commands can be carried out with a single command, by listing the relevant identities separated by commas inside the square brackets.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="n">e</span> <span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">d</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">e</span> <span class="bp">=</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">e</span><span class="o">)</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="o">(</span><span class="n">d</span> <span class="bp">*</span> <span class="n">f</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h'</span><span class="o">,</span> <span class="bp">←</span> <span class="n">mul_assoc</span><span class="o">,</span> <span class="n">h</span><span class="o">,</span> <span class="n">mul_assoc</span><span class="o">]</span> </pre></div> </div> <p>You still see the incremental progress by placing the cursor after a comma in any list of rewrites.</p> <p>Another trick is that we can declare variables once and for all outside an example or theorem. Lean then includes them automatically.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="n">e</span> <span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">d</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">e</span> <span class="bp">=</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">e</span><span class="o">)</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="o">(</span><span class="n">d</span> <span class="bp">*</span> <span class="n">f</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h'</span><span class="o">,</span> <span class="bp">←</span> <span class="n">mul_assoc</span><span class="o">,</span> <span class="n">h</span><span class="o">,</span> <span class="n">mul_assoc</span><span class="o">]</span> </pre></div> </div> <p>Inspection of the tactic state at the beginning of the above proof reveals that Lean indeed included all variables. We can delimit the scope of the declaration by putting it in a <code class="docutils literal notranslate"><span class="pre">section</span> <span class="pre">...</span> <span class="pre">end</span></code> block. Finally, recall from the introduction that Lean provides us with a command to determine the type of an expression:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="k">#check</span> <span class="n">a</span> <span class="k">#check</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="k">#check</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="k">#check</span> <span class="n">mul_comm</span> <span class="n">a</span> <span class="n">b</span> <span class="k">#check</span> <span class="o">(</span><span class="n">mul_comm</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="n">mul_assoc</span> <span class="n">c</span> <span class="n">a</span> <span class="n">b</span> <span class="k">#check</span> <span class="n">mul_comm</span> <span class="n">a</span> <span class="k">#check</span> <span class="n">mul_comm</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">mul_comm</span> <span class="kd">end</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">#check</span></code> command works for both objects and facts. In response to the command <code class="docutils literal notranslate"><span class="pre">#check</span> <span class="pre">a</span></code>, Lean reports that <code class="docutils literal notranslate"><span class="pre">a</span></code> has type <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>. In response to the command <code class="docutils literal notranslate"><span class="pre">#check</span> <span class="pre">mul_comm</span> <span class="pre">a</span> <span class="pre">b</span></code>, Lean reports that <code class="docutils literal notranslate"><span class="pre">mul_comm</span> <span class="pre">a</span> <span class="pre">b</span></code> is a proof of the fact <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">b</span> <span class="pre">=</span> <span class="pre">b</span> <span class="pre">*</span> <span class="pre">a</span></code>. The command <code class="docutils literal notranslate"><span class="pre">#check</span> <span class="pre">(a</span> <span class="pre">:</span> <span class="pre">ℝ)</span></code> states our expectation that the type of <code class="docutils literal notranslate"><span class="pre">a</span></code> is <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>, and Lean will raise an error if that is not the case. We will explain the output of the last three <code class="docutils literal notranslate"><span class="pre">#check</span></code> commands later, but in the meanwhile, you can take a look at them, and experiment with some <code class="docutils literal notranslate"><span class="pre">#check</span></code> commands of your own.</p> <p>Let’s try some more examples. The theorem <code class="docutils literal notranslate"><span class="pre">two_mul</span> <span class="pre">a</span></code> says that <code class="docutils literal notranslate"><span class="pre">2</span> <span class="pre">*</span> <span class="pre">a</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">a</span></code>. The theorems <code class="docutils literal notranslate"><span class="pre">add_mul</span></code> and <code class="docutils literal notranslate"><span class="pre">mul_add</span></code> express the distributivity of multiplication over addition, and the theorem <code class="docutils literal notranslate"><span class="pre">add_assoc</span></code> expresses the associativity of addition. Use the <code class="docutils literal notranslate"><span class="pre">#check</span></code> command to see the precise statements.</p> <div class="highlight-lean notranslate" id="index-3"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_add</span><span class="o">,</span> <span class="n">add_mul</span><span class="o">,</span> <span class="n">add_mul</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">add_assoc</span><span class="o">,</span> <span class="n">add_assoc</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">a</span><span class="o">)]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_comm</span> <span class="n">b</span> <span class="n">a</span><span class="o">,</span> <span class="bp">←</span> <span class="n">two_mul</span><span class="o">]</span> </pre></div> </div> <p>Whereas it is possible to figure out what it going on in this proof by stepping through it in the editor, it is hard to read on its own. Lean provides a more structured way of writing proofs like this using the <code class="docutils literal notranslate"><span class="pre">calc</span></code> keyword.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="k">calc</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_add</span><span class="o">,</span> <span class="n">add_mul</span><span class="o">,</span> <span class="n">add_mul</span><span class="o">]</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">add_assoc</span><span class="o">,</span> <span class="n">add_assoc</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">a</span><span class="o">)]</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_comm</span> <span class="n">b</span> <span class="n">a</span><span class="o">,</span> <span class="bp">←</span> <span class="n">two_mul</span><span class="o">]</span> </pre></div> </div> <p>Notice that the proof does <em>not</em> begin with <code class="docutils literal notranslate"><span class="pre">by</span></code>: an expression that begins with <code class="docutils literal notranslate"><span class="pre">calc</span></code> is a <em>proof term</em>. A <code class="docutils literal notranslate"><span class="pre">calc</span></code> expression can also be used inside a tactic proof, but Lean interprets it as the instruction to use the resulting proof term to solve the goal. The <code class="docutils literal notranslate"><span class="pre">calc</span></code> syntax is finicky: the dots and underscores and justification have to be in the format indicated above. Lean uses indentation to determine things like where a block of tactics or a <code class="docutils literal notranslate"><span class="pre">calc</span></code> block begins and ends; try changing the indentation in the proof above to see what happens.</p> <p>One way to write a <code class="docutils literal notranslate"><span class="pre">calc</span></code> proof is to outline it first using the <code class="docutils literal notranslate"><span class="pre">sorry</span></code> tactic for justification, make sure Lean accepts the expression modulo these, and then justify the individual steps using tactics.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="k">calc</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Try proving the following identity using both a pure <code class="docutils literal notranslate"><span class="pre">rw</span></code> proof and a more structured <code class="docutils literal notranslate"><span class="pre">calc</span></code> proof:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">c</span> <span class="bp">+</span> <span class="n">d</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">d</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>The following exercise is a little more challenging. You can use the theorems listed underneath.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">#check</span> <span class="n">pow_two</span> <span class="n">a</span> <span class="k">#check</span> <span class="n">mul_sub</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="k">#check</span> <span class="n">add_mul</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="k">#check</span> <span class="n">add_sub</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="k">#check</span> <span class="n">sub_sub</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="k">#check</span> <span class="n">add_zero</span> <span class="n">a</span> </pre></div> </div> <p id="index-4">We can also perform rewriting in an assumption in the context. For example, <code class="docutils literal notranslate"><span class="pre">rw</span> <span class="pre">[mul_comm</span> <span class="pre">a</span> <span class="pre">b]</span> <span class="pre">at</span> <span class="pre">hyp</span></code> replaces <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">b</span></code> by <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">*</span> <span class="pre">a</span></code> in the assumption <code class="docutils literal notranslate"><span class="pre">hyp</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">hyp</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">d</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">hyp'</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span><span class="o">)</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hyp'</span><span class="o">]</span> <span class="n">at</span> <span class="n">hyp</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_comm</span> <span class="n">d</span> <span class="n">a</span><span class="o">]</span> <span class="n">at</span> <span class="n">hyp</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">two_mul</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">d</span><span class="o">)]</span> <span class="n">at</span> <span class="n">hyp</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">mul_assoc</span> <span class="mi">2</span> <span class="n">a</span> <span class="n">d</span><span class="o">]</span> <span class="n">at</span> <span class="n">hyp</span> <span class="n">exact</span> <span class="n">hyp</span> </pre></div> </div> <p id="index-5">In the last step, the <code class="docutils literal notranslate"><span class="pre">exact</span></code> tactic can use <code class="docutils literal notranslate"><span class="pre">hyp</span></code> to solve the goal because at that point <code class="docutils literal notranslate"><span class="pre">hyp</span></code> matches the goal exactly.</p> <p id="index-6">We close this section by noting that <code class="docutils literal notranslate"><span class="pre">mathlib</span></code> provides a useful bit of automation with a <code class="docutils literal notranslate"><span class="pre">ring</span></code> tactic, which is designed to prove identities in any commutative ring as long as they follow purely from the ring axioms, without using any local assumption.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="kd">example</span> <span class="o">(</span><span class="n">hyp</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">d</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">hyp'</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span><span class="o">)</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hyp</span><span class="o">,</span> <span class="n">hyp'</span><span class="o">]</span> <span class="n">ring</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">ring</span></code> tactic is imported indirectly when we import <code class="docutils literal notranslate"><span class="pre">Mathlib.Data.Real.Basic</span></code>, but we will see in the next section that it can be used for calculations on structures other than the real numbers. It can be imported explicitly with the command <code class="docutils literal notranslate"><span class="pre">import</span> <span class="pre">Mathlib.Tactic</span></code>. We will see there are similar tactics for other common kind of algebraic structures.</p> <p>There is a variation of <code class="docutils literal notranslate"><span class="pre">rw</span></code> called <code class="docutils literal notranslate"><span class="pre">nth_rewrite</span></code> that allows you to replace only particular instances of an expression in the goal. Possible matches are enumerated starting with 1, so in the following example, <code class="docutils literal notranslate"><span class="pre">nth_rewrite</span> <span class="pre">2</span> <span class="pre">h</span></code> replaces the second occurrence of <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code> with <code class="docutils literal notranslate"><span class="pre">c</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">nth_rw</span> <span class="mi">2</span> <span class="o">[</span><span class="n">h</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_mul</span><span class="o">]</span> </pre></div> </div> </section> <section id="proving-identities-in-algebraic-structures"> <span id="id2"></span><h2><span class="section-number">2.2. </span>Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this heading"></a></h2> <p id="index-7">Mathematically, a ring consists of a collection of objects, <span class="math notranslate nohighlight">\(R\)</span>, operations <span class="math notranslate nohighlight">\(+\)</span> <span class="math notranslate nohighlight">\(\times\)</span>, and constants <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>, and an operation <span class="math notranslate nohighlight">\(x \mapsto -x\)</span> such that:</p> <ul class="simple"> <li><p><span class="math notranslate nohighlight">\(R\)</span> with <span class="math notranslate nohighlight">\(+\)</span> is an <em>abelian group</em>, with <span class="math notranslate nohighlight">\(0\)</span> as the additive identity and negation as inverse.</p></li> <li><p>Multiplication is associative with identity <span class="math notranslate nohighlight">\(1\)</span>, and multiplication distributes over addition.</p></li> </ul> <p>In Lean, the collection of objects is represented as a <em>type</em>, <code class="docutils literal notranslate"><span class="pre">R</span></code>. The ring axioms are as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">[</span><span class="n">Ring</span> <span class="n">R</span><span class="o">]</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_assoc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">+</span> <span class="o">(</span><span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">))</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_comm</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">zero_add</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_left_neg</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="bp">-</span><span class="n">a</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">0</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">mul_assoc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">c</span><span class="o">))</span> <span class="k">#check</span> <span class="o">(</span><span class="n">mul_one</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="mi">1</span> <span class="bp">=</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">one_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="mi">1</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">mul_add</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> </pre></div> </div> <p>You will learn more about the square brackets in the first line later, but for the time being, suffice it to say that the declaration gives us a type, <code class="docutils literal notranslate"><span class="pre">R</span></code>, and a ring structure on <code class="docutils literal notranslate"><span class="pre">R</span></code>. Lean then allows us to use generic ring notation with elements of <code class="docutils literal notranslate"><span class="pre">R</span></code>, and to make use of a library of theorems about rings.</p> <p>The names of some of the theorems should look familiar: they are exactly the ones we used to calculate with the real numbers in the last section. Lean is good not only for proving things about concrete mathematical structures like the natural numbers and the integers, but also for proving things about abstract structures, characterized axiomatically, like rings. Moreover, Lean supports <em>generic reasoning</em> about both abstract and concrete structures, and can be trained to recognized appropriate instances. So any theorem about rings can be applied to concrete rings like the integers, <code class="docutils literal notranslate"><span class="pre">ℤ</span></code>, the rational numbers, <code class="docutils literal notranslate"><span class="pre">ℚ</span></code>, and the complex numbers <code class="docutils literal notranslate"><span class="pre">ℂ</span></code>. It can also be applied to any instance of an abstract structure that extends rings, such as any <em>ordered ring</em> or any <em>field</em>.</p> <p id="index-8">Not all important properties of the real numbers hold in an arbitrary ring, however. For example, multiplication on the real numbers is commutative, but that does not hold in general. If you have taken a course in linear algebra, you will recognize that, for every <span class="math notranslate nohighlight">\(n\)</span>, the <span class="math notranslate nohighlight">\(n\)</span> by <span class="math notranslate nohighlight">\(n\)</span> matrices of real numbers form a ring in which commutativity usually fails. If we declare <code class="docutils literal notranslate"><span class="pre">R</span></code> to be a <em>commutative</em> ring, in fact, all the theorems in the last section continue to hold when we replace <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> by <code class="docutils literal notranslate"><span class="pre">R</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">[</span><span class="n">CommRing</span> <span class="n">R</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="kd">example</span> <span class="o">(</span><span class="n">hyp</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">d</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">hyp'</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span><span class="o">)</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hyp</span><span class="o">,</span> <span class="n">hyp'</span><span class="o">]</span> <span class="n">ring</span> </pre></div> </div> <p>We leave it to you to check that all the other proofs go through unchanged. Notice that when a proof is short, like <code class="docutils literal notranslate"><span class="pre">by</span> <span class="pre">ring</span></code> or <code class="docutils literal notranslate"><span class="pre">by</span> <span class="pre">linarith</span></code> or <code class="docutils literal notranslate"><span class="pre">by</span> <span class="pre">sorry</span></code>, it is common (and permissible) to put it on the same line as the <code class="docutils literal notranslate"><span class="pre">by</span></code>. Good proof-writing style should strike a balance between concision and readability.</p> <p>The goal of this section is to strengthen the skills you have developed in the last section and apply them to reasoning axiomatically about rings. We will start with the axioms listed above, and use them to derive other facts. Most of the facts we prove are already in <code class="docutils literal notranslate"><span class="pre">mathlib</span></code>. We will give the versions we prove the same names to help you learn the contents of the library as well as the naming conventions.</p> <p id="index-9">Lean provides an organizational mechanism similar to those used in programming languages: when a definition or theorem <code class="docutils literal notranslate"><span class="pre">foo</span></code> is introduced in a <em>namespace</em> <code class="docutils literal notranslate"><span class="pre">bar</span></code>, its full name is <code class="docutils literal notranslate"><span class="pre">bar.foo</span></code>. The command <code class="docutils literal notranslate"><span class="pre">open</span> <span class="pre">bar</span></code> later <em>opens</em> the namespace, which allows us to use the shorter name <code class="docutils literal notranslate"><span class="pre">foo</span></code>. To avoid errors due to name clashes, in the next example we put our versions of the library theorems in a new namespace called <code class="docutils literal notranslate"><span class="pre">MyRing.</span></code></p> <p>The next example shows that we do not need <code class="docutils literal notranslate"><span class="pre">add_zero</span></code> or <code class="docutils literal notranslate"><span class="pre">add_right_neg</span></code> as ring axioms, because they follow from the other axioms.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">namespace</span> <span class="n">MyRing</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Ring</span> <span class="n">R</span><span class="o">]</span> <span class="kd">theorem</span> <span class="n">add_zero</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">0</span> <span class="bp">=</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_comm</span><span class="o">,</span> <span class="n">zero_add</span><span class="o">]</span> <span class="kd">theorem</span> <span class="n">add_right_neg</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">a</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_comm</span><span class="o">,</span> <span class="n">add_left_neg</span><span class="o">]</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">MyRing.add_zero</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">add_zero</span> <span class="kd">end</span> <span class="n">MyRing</span> </pre></div> </div> <p>The net effect is that we can temporarily reprove a theorem in the library, and then go on using the library version after that. But don’t cheat! In the exercises that follow, take care to use only the general facts about rings that we have proved earlier in this section.</p> <p>(If you are paying careful attention, you may have noticed that we changed the round brackets in <code class="docutils literal notranslate"><span class="pre">(R</span> <span class="pre">:</span> <span class="pre">Type</span> <span class="pre">_)</span></code> for curly brackets in <code class="docutils literal notranslate"><span class="pre">{R</span> <span class="pre">:</span> <span class="pre">Type</span> <span class="pre">_}</span></code>. This declares <code class="docutils literal notranslate"><span class="pre">R</span></code> to be an <em>implicit argument</em>. We will explain what this means in a moment, but don’t worry about it in the meanwhile.)</p> <p>Here is a useful theorem:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">neg_add_cancel_left</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="bp">-</span><span class="n">a</span> <span class="bp">+</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">add_assoc</span><span class="o">,</span> <span class="n">add_left_neg</span><span class="o">,</span> <span class="n">zero_add</span><span class="o">]</span> </pre></div> </div> <p>Prove the companion version:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">add_neg_cancel_right</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Use these to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">add_left_cancel</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">add_right_cancel</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>With enough planning, you can do each of them with three rewrites.</p> <p id="index-10">We can now explain the use of the curly braces. Imagine you are in a situation where you have <code class="docutils literal notranslate"><span class="pre">a</span></code>, <code class="docutils literal notranslate"><span class="pre">b</span></code>, and <code class="docutils literal notranslate"><span class="pre">c</span></code> in your context, as well as a hypothesis <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">c</span></code>, and you would like to draw the conclusion <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">=</span> <span class="pre">c</span></code>. In Lean, you can apply a theorem to hypotheses and facts just the same way that you can apply them to objects, so you might think that <code class="docutils literal notranslate"><span class="pre">add_left_cancel</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">c</span> <span class="pre">h</span></code> is a proof of the fact <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">=</span> <span class="pre">c</span></code>. But notice that explicitly writing <code class="docutils literal notranslate"><span class="pre">a</span></code>, <code class="docutils literal notranslate"><span class="pre">b</span></code>, and <code class="docutils literal notranslate"><span class="pre">c</span></code> is redundant, because the hypothesis <code class="docutils literal notranslate"><span class="pre">h</span></code> makes it clear that those are the objects we have in mind. In this case, typing a few extra characters is not onerous, but if we wanted to apply <code class="docutils literal notranslate"><span class="pre">add_left_cancel</span></code> to more complicated expressions, writing them would be tedious. In cases like these, Lean allows us to mark arguments as <em>implicit</em>, meaning that they are supposed to be left out and inferred by other means, such as later arguments and hypotheses. The curly brackets in <code class="docutils literal notranslate"><span class="pre">{a</span> <span class="pre">b</span> <span class="pre">c</span> <span class="pre">:</span> <span class="pre">R}</span></code> do exactly that. So, given the statement of the theorem above, the correct expression is simply <code class="docutils literal notranslate"><span class="pre">add_left_cancel</span> <span class="pre">h</span></code>.</p> <p>To illustrate, let us show that <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">0</span> <span class="pre">=</span> <span class="pre">0</span></code> follows from the ring axioms.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">mul_zero</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="mi">0</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="mi">0</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">*</span> <span class="mi">0</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="mi">0</span> <span class="bp">+</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">mul_add</span><span class="o">,</span> <span class="n">add_zero</span><span class="o">,</span> <span class="n">add_zero</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_left_cancel</span> <span class="n">h</span><span class="o">]</span> </pre></div> </div> <p id="index-11">We have used a new trick! If you step through the proof, you can see what is going on. The <code class="docutils literal notranslate"><span class="pre">have</span></code> tactic introduces a new goal, <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">0</span> <span class="pre">+</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">0</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">0</span> <span class="pre">+</span> <span class="pre">0</span></code>, with the same context as the original goal. The fact that the next line is indented indicates that Lean is expecting a block of tactics that serves to prove this new goal. The indentation therefore promotes a modular style of proof: the indented subproof establishes the goal that was introduced by the <code class="docutils literal notranslate"><span class="pre">have</span></code>. After that, we are back to proving the original goal, except a new hypothesis <code class="docutils literal notranslate"><span class="pre">h</span></code> has been added: having proved it, we are now free to use it. At this point, the goal is exactly the result of <code class="docutils literal notranslate"><span class="pre">add_left_cancel</span> <span class="pre">h</span></code>. We could equally well have closed the proof with <code class="docutils literal notranslate"><span class="pre">apply</span> <span class="pre">add_left_cancel</span> <span class="pre">h</span></code> or <code class="docutils literal notranslate"><span class="pre">exact</span> <span class="pre">add_left_cancel</span> <span class="pre">h</span></code>.</p> <p>Remember that multiplication is not assumed to be commutative, so the following theorem also requires some work.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">zero_mul</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>By now, you should also be able replace each <code class="docutils literal notranslate"><span class="pre">sorry</span></code> in the next exercise with a proof, still using only facts about rings that we have established in this section.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">neg_eq_of_add_eq_zero</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">=</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="bp">-</span><span class="n">a</span> <span class="bp">=</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">eq_neg_of_add_eq_zero</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">=</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">=</span> <span class="bp">-</span><span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">neg_zero</span> <span class="o">:</span> <span class="o">(</span><span class="bp">-</span><span class="mi">0</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">neg_eq_of_add_eq_zero</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_zero</span><span class="o">]</span> <span class="kd">theorem</span> <span class="n">neg_neg</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="bp">-</span> <span class="bp">-</span><span class="n">a</span> <span class="bp">=</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>We had to use the annotation <code class="docutils literal notranslate"><span class="pre">(-0</span> <span class="pre">:</span> <span class="pre">R)</span></code> instead of <code class="docutils literal notranslate"><span class="pre">0</span></code> in the third theorem because without specifying <code class="docutils literal notranslate"><span class="pre">R</span></code> it is impossible for Lean to infer which <code class="docutils literal notranslate"><span class="pre">0</span></code> we have in mind, and by default it would be interpreted as a natural number.</p> <p>In Lean, subtraction in a ring is provably equal to addition of the additive inverse.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">b</span> <span class="o">:=</span> <span class="n">sub_eq_add_neg</span> <span class="n">a</span> <span class="n">b</span> </pre></div> </div> <p>On the real numbers, it is <em>defined</em> that way:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">b</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rfl</span> </pre></div> </div> <p id="index-12">The proof term <code class="docutils literal notranslate"><span class="pre">rfl</span></code> is short for <code class="docutils literal notranslate"><span class="pre">reflexivity</span></code>. Presenting it as a proof of <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">-</span> <span class="pre">b</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">-b</span></code> forces Lean to unfold the definition and recognize both sides as being the same. The <code class="docutils literal notranslate"><span class="pre">reflexivity</span></code> tactic, which can be abbreviated as <code class="docutils literal notranslate"><span class="pre">rfl</span></code>, does the same. This is an instance of what is known as a <em>definitional equality</em> in Lean’s underlying logic. This means that not only can one rewrite with <code class="docutils literal notranslate"><span class="pre">sub_eq_add_neg</span></code> to replace <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">-</span> <span class="pre">b</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">-b</span></code>, but in some contexts, when dealing with the real numbers, you can use the two sides of the equation interchangeably. For example, you now have enough information to prove the theorem <code class="docutils literal notranslate"><span class="pre">self_sub</span></code> from the last section:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">self_sub</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">-</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Show that you can prove this using <code class="docutils literal notranslate"><span class="pre">rw</span></code>, but if you replace the arbitrary ring <code class="docutils literal notranslate"><span class="pre">R</span></code> by the real numbers, you can also prove it using either <code class="docutils literal notranslate"><span class="pre">apply</span></code> or <code class="docutils literal notranslate"><span class="pre">exact</span></code>.</p> <p>Lean knows that <code class="docutils literal notranslate"><span class="pre">1</span> <span class="pre">+</span> <span class="pre">1</span> <span class="pre">=</span> <span class="pre">2</span></code> holds in any ring. With a bit of effort, you can use that to prove the theorem <code class="docutils literal notranslate"><span class="pre">two_mul</span></code> from the last section:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">one_add_one_eq_two</span> <span class="o">:</span> <span class="mi">1</span> <span class="bp">+</span> <span class="mi">1</span> <span class="bp">=</span> <span class="o">(</span><span class="mi">2</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">norm_num</span> <span class="kd">theorem</span> <span class="n">two_mul</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">a</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-13">We close this section by noting that some of the facts about addition and negation that we established above do not need the full strength of the ring axioms, or even commutativity of addition. The weaker notion of a <em>group</em> can be axiomatized as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">A</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">[</span><span class="n">AddGroup</span> <span class="n">A</span><span class="o">]</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_assoc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">A</span><span class="o">,</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">+</span> <span class="o">(</span><span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">))</span> <span class="k">#check</span> <span class="o">(</span><span class="n">zero_add</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">A</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_left_neg</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">A</span><span class="o">,</span> <span class="bp">-</span><span class="n">a</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">0</span><span class="o">)</span> </pre></div> </div> <p>It is conventional to use additive notation when the group operation is commutative, and multiplicative notation otherwise. So Lean defines a multiplicative version as well as the additive version (and also their abelian variants, <code class="docutils literal notranslate"><span class="pre">AddCommGroup</span></code> and <code class="docutils literal notranslate"><span class="pre">CommGroup</span></code>).</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Group</span> <span class="n">G</span><span class="o">]</span> <span class="k">#check</span> <span class="o">(</span><span class="n">mul_assoc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">G</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">c</span><span class="o">))</span> <span class="k">#check</span> <span class="o">(</span><span class="n">one_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">,</span> <span class="mi">1</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">mul_left_inv</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">,</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> </pre></div> </div> <p>If you are feeling cocky, try proving the following facts about groups, using only these axioms. You will need to prove a number of helper lemmas along the way. The proofs we have carried out in this section provide some hints.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">mul_right_inv</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">mul_one</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="mi">1</span> <span class="bp">=</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">mul_inv_rev</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">G</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="n">b</span><span class="bp">⁻¹</span> <span class="bp">*</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-14">Explicitly invoking those lemmas is tedious, so mathlib provides tactics similar to <cite>ring</cite> in order to cover most uses: <cite>group</cite> is for non-commutative multiplicative groups, <cite>abel</cite> for abelian additive groups, and <cite>noncomm_ring</cite> for non-commutative groups. It may seem odd that the algebraic structures are called <cite>Ring</cite> and <cite>CommRing</cite> while the tactics are named <cite>noncomm_ring</cite> and <cite>ring</cite>. This is partly for historical reasons, but also for the convenience of using a shorter name for the tactic that deals with commutative rings, since it is used more often.</p> </section> <section id="using-theorems-and-lemmas"> <span id="id3"></span><h2><span class="section-number">2.3. </span>Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this heading"></a></h2> <p id="index-15">Rewriting is great for proving equations, but what about other sorts of theorems? For example, how can we prove an inequality, like the fact that <span class="math notranslate nohighlight">\(a + e^b \le a + e^c\)</span> holds whenever <span class="math notranslate nohighlight">\(b \le c\)</span>? We have already seen that theorems can be applied to arguments and hypotheses, and that the <code class="docutils literal notranslate"><span class="pre">apply</span></code> and <code class="docutils literal notranslate"><span class="pre">exact</span></code> tactics can be used to solve goals. In this section, we will make good use of these tools.</p> <p>Consider the library theorems <code class="docutils literal notranslate"><span class="pre">le_refl</span></code> and <code class="docutils literal notranslate"><span class="pre">le_trans</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">le_refl</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_trans</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> </pre></div> </div> <p>As we explain in more detail in <a class="reference internal" href="C03_Logic.html#implication-and-the-universal-quantifier"><span class="std std-numref">Section 3.1</span></a>, the implicit parentheses in the statement of <code class="docutils literal notranslate"><span class="pre">le_trans</span></code> associate to the right, so it should be interpreted as <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">≤</span> <span class="pre">b</span> <span class="pre">→</span> <span class="pre">(b</span> <span class="pre">≤</span> <span class="pre">c</span> <span class="pre">→</span> <span class="pre">a</span> <span class="pre">≤</span> <span class="pre">c)</span></code>. The library designers have set the arguments to <code class="docutils literal notranslate"><span class="pre">le_trans</span></code> implicit, so that Lean will <em>not</em> let you provide them explicitly (unless you really insist, as we will discuss later). Rather, it expects to infer them from the context in which they are used. For example, when hypotheses <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">a</span> <span class="pre">≤</span> <span class="pre">b</span></code> and <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">b</span> <span class="pre">≤</span> <span class="pre">c</span></code> are in the context, all the following work:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_refl</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">Real</span><span class="o">,</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_refl</span> <span class="n">a</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_trans</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_trans</span> <span class="n">h</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_trans</span> <span class="n">h</span> <span class="n">h'</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> </pre></div> </div> <p id="index-16">The <code class="docutils literal notranslate"><span class="pre">apply</span></code> tactic takes a proof of a general statement or implication, tries to match the conclusion with the current goal, and leaves the hypotheses, if any, as new goals. If the given proof matches the goal exactly (modulo <em>definitional</em> equality), you can use the <code class="docutils literal notranslate"><span class="pre">exact</span></code> tactic instead of <code class="docutils literal notranslate"><span class="pre">apply</span></code>. So, all of these work:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">z</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">z</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">le_trans</span> <span class="bp">·</span> <span class="n">apply</span> <span class="n">h₀</span> <span class="bp">.</span> <span class="n">apply</span> <span class="n">h₁</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">z</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">z</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">le_trans</span> <span class="n">h₀</span> <span class="n">apply</span> <span class="n">h₁</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">z</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">z</span> <span class="o">:=</span> <span class="n">le_trans</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">le_refl</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">le_refl</span> <span class="n">x</span> </pre></div> </div> <p>In the first example, applying <code class="docutils literal notranslate"><span class="pre">le_trans</span></code> creates two goals, and we use the dots to indicate where the proof of each begins. The dots are optional, but they serve to <em>focus</em> the goal: within the block introduced by the dot, only one goal is visible, and it must be completed before the end of the block. Here we end the first block by starting a new one with another dot. We could just as well have decreased the indentation. In the fourth example and in the last example, we avoid going into tactic mode entirely: <code class="docutils literal notranslate"><span class="pre">le_trans</span> <span class="pre">h₀</span> <span class="pre">h₁</span></code> and <code class="docutils literal notranslate"><span class="pre">le_refl</span> <span class="pre">x</span></code> are the proof terms we need.</p> <p>Here are a few more library theorems:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">le_refl</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_trans</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_of_le_of_lt</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">c</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_of_lt_of_le</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_trans</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">c</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">c</span><span class="o">)</span> </pre></div> </div> <p>Use them together with <code class="docutils literal notranslate"><span class="pre">apply</span></code> and <code class="docutils literal notranslate"><span class="pre">exact</span></code> to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">c</span><span class="o">)</span> <span class="o">(</span><span class="n">h₂</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">d</span><span class="o">)</span> <span class="o">(</span><span class="n">h₃</span> <span class="o">:</span> <span class="n">d</span> <span class="bp"><</span> <span class="n">e</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">e</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-17">In fact, Lean has a tactic that does this sort of thing automatically:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">c</span><span class="o">)</span> <span class="o">(</span><span class="n">h₂</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">d</span><span class="o">)</span> <span class="o">(</span><span class="n">h₃</span> <span class="o">:</span> <span class="n">d</span> <span class="bp"><</span> <span class="n">e</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">e</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">linarith</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">linarith</span></code> tactic is designed to handle <em>linear arithmetic</em>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">≤</span> <span class="mi">3</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="mi">1</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">h''</span> <span class="o">:</span> <span class="n">d</span> <span class="bp">=</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:</span> <span class="n">d</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">≤</span> <span class="mi">5</span> <span class="bp">*</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">linarith</span> </pre></div> </div> <p>In addition to equations and inequalities in the context, <code class="docutils literal notranslate"><span class="pre">linarith</span></code> will use additional inequalities that you pass as arguments. In the next example, <code class="docutils literal notranslate"><span class="pre">exp_le_exp.mpr</span> <span class="pre">h'</span></code> is a proof of <code class="docutils literal notranslate"><span class="pre">exp</span> <span class="pre">b</span> <span class="pre">≤</span> <span class="pre">exp</span> <span class="pre">c</span></code>, as we will explain in a moment. Notice that, in Lean, we write <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> to denote the application of a function <code class="docutils literal notranslate"><span class="pre">f</span></code> to the argument <code class="docutils literal notranslate"><span class="pre">x</span></code>, exactly the same way we write <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">x</span></code> to denote the result of applying a fact or theorem <code class="docutils literal notranslate"><span class="pre">h</span></code> to the argument <code class="docutils literal notranslate"><span class="pre">x</span></code>. Parentheses are only needed for compound arguments, as in <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">(x</span> <span class="pre">+</span> <span class="pre">y)</span></code>. Without the parentheses, <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">y</span></code> would be parsed as <code class="docutils literal notranslate"><span class="pre">(f</span> <span class="pre">x)</span> <span class="pre">+</span> <span class="pre">y</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">1</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">exp</span> <span class="n">b</span> <span class="bp">≤</span> <span class="mi">3</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">exp</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">exp_le_exp.mpr</span> <span class="n">h'</span><span class="o">]</span> </pre></div> </div> <p id="index-18">Here are some more theorems in the library that can be used to establish inequalities on the real numbers.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">exp_le_exp</span> <span class="o">:</span> <span class="n">exp</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">exp</span> <span class="n">b</span> <span class="bp">↔</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">exp_lt_exp</span> <span class="o">:</span> <span class="n">exp</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">exp</span> <span class="n">b</span> <span class="bp">↔</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">log_le_log</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="o">(</span><span class="n">log</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">log</span> <span class="n">b</span> <span class="bp">↔</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">))</span> <span class="k">#check</span> <span class="o">(</span><span class="n">log_lt_log</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">log</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">log</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_le_add</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">d</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">d</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_le_add_left</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="bp">∀</span> <span class="n">c</span><span class="o">,</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_le_add_right</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="bp">∀</span> <span class="n">c</span><span class="o">,</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_lt_add_of_le_of_lt</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">c</span> <span class="bp"><</span> <span class="n">d</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">d</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_lt_add_of_lt_of_le</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">d</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">d</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_lt_add_left</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="bp">∀</span> <span class="n">c</span><span class="o">,</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_lt_add_right</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="bp">∀</span> <span class="n">c</span><span class="o">,</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_pos_of_pos_of_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">exp_pos</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">exp</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">add_le_add_left</span> </pre></div> </div> <p>Some of the theorems, <code class="docutils literal notranslate"><span class="pre">exp_le_exp</span></code>, <code class="docutils literal notranslate"><span class="pre">exp_lt_exp</span></code>, and <code class="docutils literal notranslate"><span class="pre">log_le_log</span></code> use a <em>bi-implication</em>, which represents the phrase “if and only if.” (You can type it in VS Code with <code class="docutils literal notranslate"><span class="pre">\lr</span></code> of <code class="docutils literal notranslate"><span class="pre">\iff</span></code>). We will discuss this connective in greater detail in the next chapter. Such a theorem can be used with <code class="docutils literal notranslate"><span class="pre">rw</span></code> to rewrite a goal to an equivalent one:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">exp</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">exp</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">exp_le_exp</span><span class="o">]</span> <span class="n">exact</span> <span class="n">h</span> </pre></div> </div> <p>In this section, however, we will use the fact that if <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">A</span> <span class="pre">↔</span> <span class="pre">B</span></code> is such an equivalence, then <code class="docutils literal notranslate"><span class="pre">h.mp</span></code> establishes the forward direction, <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">→</span> <span class="pre">B</span></code>, and <code class="docutils literal notranslate"><span class="pre">h.mpr</span></code> establishes the reverse direction, <code class="docutils literal notranslate"><span class="pre">B</span> <span class="pre">→</span> <span class="pre">A</span></code>. Here, <code class="docutils literal notranslate"><span class="pre">mp</span></code> stands for “modus ponens” and <code class="docutils literal notranslate"><span class="pre">mpr</span></code> stands for “modus ponens reverse.” You can also use <code class="docutils literal notranslate"><span class="pre">h.1</span></code> and <code class="docutils literal notranslate"><span class="pre">h.2</span></code> for <code class="docutils literal notranslate"><span class="pre">h.mp</span></code> and <code class="docutils literal notranslate"><span class="pre">h.mpr</span></code>, respectively, if you prefer. Thus the following proof works:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">c</span> <span class="bp"><</span> <span class="n">d</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">exp</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">e</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">exp</span> <span class="n">d</span> <span class="bp">+</span> <span class="n">e</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">add_lt_add_of_lt_of_le</span> <span class="bp">·</span> <span class="n">apply</span> <span class="n">add_lt_add_of_le_of_lt</span> <span class="n">h₀</span> <span class="n">apply</span> <span class="n">exp_lt_exp.mpr</span> <span class="n">h₁</span> <span class="n">apply</span> <span class="n">le_refl</span> </pre></div> </div> <p>The first line, <code class="docutils literal notranslate"><span class="pre">apply</span> <span class="pre">add_lt_add_of_lt_of_le</span></code>, creates two goals, and once again we use a dot to separate the proof of the first from the proof of the second.</p> <p id="index-19">Try the following examples on your own. The example in the middle shows you that the <code class="docutils literal notranslate"><span class="pre">norm_num</span></code> tactic can be used to solve concrete numeric goals.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">d</span> <span class="bp">≤</span> <span class="n">e</span><span class="o">)</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">exp</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">d</span><span class="o">)</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">exp</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">e</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="mi">0</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="bp"><</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">norm_num</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">log</span> <span class="o">(</span><span class="mi">1</span> <span class="bp">+</span> <span class="n">exp</span> <span class="n">a</span><span class="o">)</span> <span class="bp">≤</span> <span class="n">log</span> <span class="o">(</span><span class="mi">1</span> <span class="bp">+</span> <span class="n">exp</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="n">h₀</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="mi">1</span> <span class="bp">+</span> <span class="n">exp</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">h₁</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="mi">1</span> <span class="bp">+</span> <span class="n">exp</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">apply</span> <span class="o">(</span><span class="n">log_le_log</span> <span class="n">h₀</span> <span class="n">h₁</span><span class="o">)</span><span class="bp">.</span><span class="n">mpr</span> <span class="gr">sorry</span> </pre></div> </div> <p>From these examples, it should be clear that being able to find the library theorems you need constitutes an important part of formalization. There are a number of strategies you can use:</p> <ul class="simple"> <li><p>You can browse mathlib in its <a class="reference external" href="https://github.com/leanprover-community/mathlib">GitHub repository</a>.</p></li> <li><p>You can use the API documentation on the mathlib <a class="reference external" href="https://leanprover-community.github.io/mathlib_docs/">web pages</a>.</p></li> <li><p>You can rely on mathlib naming conventions and tab completion in the editor to guess a theorem name. In Lean, a theorem named <code class="docutils literal notranslate"><span class="pre">A_of_B_of_C</span></code> establishes something of the form <code class="docutils literal notranslate"><span class="pre">A</span></code> from hypotheses of the form <code class="docutils literal notranslate"><span class="pre">B</span></code> and <code class="docutils literal notranslate"><span class="pre">C</span></code>, where <code class="docutils literal notranslate"><span class="pre">A</span></code>, <code class="docutils literal notranslate"><span class="pre">B</span></code>, and <code class="docutils literal notranslate"><span class="pre">C</span></code> approximate the way we might read the goals out loud. So a theorem establishing something like <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">+</span> <span class="pre">y</span> <span class="pre">≤</span> <span class="pre">...</span></code> will probably start with <code class="docutils literal notranslate"><span class="pre">add_le</span></code>. Typing <code class="docutils literal notranslate"><span class="pre">add_le</span></code> and hitting tab will give you some helpful choices.</p></li> <li><p>If you right-click on an existing theorem name in VS Code, the editor will show a menu with the option to jump to the file where the theorem is defined, and you can find similar theorems nearby.</p></li> <li><p>You can use the <code class="docutils literal notranslate"><span class="pre">library_search</span></code> tactic, which tries to find the relevant theorem in the library.</p></li> </ul> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="c1">-- library_search</span> <span class="n">exact</span> <span class="n">sq_nonneg</span> <span class="n">a</span> </pre></div> </div> <p>To try out <code class="docutils literal notranslate"><span class="pre">library_search</span></code> in this example, delete the <code class="docutils literal notranslate"><span class="pre">exact</span></code> command and uncomment the previous line. Using these tricks, see if you can find what you need to do the next example:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">-</span> <span class="n">exp</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">-</span> <span class="n">exp</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Using the same tricks, confirm that <code class="docutils literal notranslate"><span class="pre">linarith</span></code> instead of <code class="docutils literal notranslate"><span class="pre">library_search</span></code> can also finish the job.</p> <p>Here is another example of an inequality:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="n">h</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="k">calc</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="n">_</span> <span class="bp">≥</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">pow_two_nonneg</span> <span class="k">calc</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="n">_</span> <span class="bp">≤</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="o">(</span><span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:=</span> <span class="n">add_le_add</span> <span class="o">(</span><span class="n">le_refl</span> <span class="n">_</span><span class="o">)</span> <span class="n">h</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> </pre></div> </div> <p>Mathlib tends to put spaces around binary operations like <code class="docutils literal notranslate"><span class="pre">*</span></code> and <code class="docutils literal notranslate"><span class="pre">^</span></code>, but in this example, the more compressed format increases readability. There are a number of things worth noticing. First, an expression <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">≥</span> <span class="pre">t</span></code> is definitionally equivalent to <code class="docutils literal notranslate"><span class="pre">t</span> <span class="pre">≤</span> <span class="pre">s</span></code>. In principle, this means one should be able to use them interchangeably. But some of Lean’s automation does not recognize the equivalence, so mathlib tends to favor <code class="docutils literal notranslate"><span class="pre">≤</span></code> over <code class="docutils literal notranslate"><span class="pre">≥</span></code>. Second, we have used the <code class="docutils literal notranslate"><span class="pre">ring</span></code> tactic extensively. It is a real timesaver! Finally, notice that in the second line of the second <code class="docutils literal notranslate"><span class="pre">calc</span></code> proof, instead of writing <code class="docutils literal notranslate"><span class="pre">by</span> <span class="pre">exact</span> <span class="pre">add_le_add</span> <span class="pre">(le_refl</span> <span class="pre">_)</span> <span class="pre">h</span></code>, we can simply write the proof term <code class="docutils literal notranslate"><span class="pre">add_le_add</span> <span class="pre">(le_refl</span> <span class="pre">_)</span> <span class="pre">h</span></code>.</p> <p>In fact, the only cleverness in the proof above is figuring out the hypothesis <code class="docutils literal notranslate"><span class="pre">h</span></code>. Once we have it, the second calculation involves only linear arithmetic, and <code class="docutils literal notranslate"><span class="pre">linarith</span></code> can handle it:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="n">h</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="k">calc</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="n">_</span> <span class="bp">≥</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">pow_two_nonneg</span> <span class="n">linarith</span> </pre></div> </div> <p>How nice! We challenge you to use these ideas to prove the following theorem. You can use the theorem <code class="docutils literal notranslate"><span class="pre">abs_le'.mpr</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">≤</span> <span class="o">(</span><span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">#check</span> <span class="n">abs_le'.mpr</span> </pre></div> </div> <p>If you managed to solve this, congratulations! You are well on your way to becoming a master formalizer.</p> </section> <section id="more-on-order-and-divisibility"> <span id="id4"></span><h2><span class="section-number">2.4. </span>More on Order and Divisibility<a class="headerlink" href="#more-on-order-and-divisibility" title="Permalink to this heading"></a></h2> <p id="index-20">The <code class="docutils literal notranslate"><span class="pre">min</span></code> function on the real numbers is uniquely characterized by the following three facts:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">min_le_left</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">min_le_right</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_min</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">→</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span><span class="o">)</span> </pre></div> </div> <p>Can you guess the names of the theorems that characterize <code class="docutils literal notranslate"><span class="pre">max</span></code> in a similar way?</p> <p>Notice that we have to apply <code class="docutils literal notranslate"><span class="pre">min</span></code> to a pair of arguments <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code> by writing <code class="docutils literal notranslate"><span class="pre">min</span> <span class="pre">a</span> <span class="pre">b</span></code> rather than <code class="docutils literal notranslate"><span class="pre">min</span> <span class="pre">(a,</span> <span class="pre">b)</span></code>. Formally, <code class="docutils literal notranslate"><span class="pre">min</span></code> is a function of type <code class="docutils literal notranslate"><span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code>. When we write a type like this with multiple arrows, the convention is that the implicit parentheses associate to the right, so the type is interpreted as <code class="docutils literal notranslate"><span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">(ℝ</span> <span class="pre">→</span> <span class="pre">ℝ)</span></code>. The net effect is that if <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code> have type <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> then <code class="docutils literal notranslate"><span class="pre">min</span> <span class="pre">a</span></code> has type <code class="docutils literal notranslate"><span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> and <code class="docutils literal notranslate"><span class="pre">min</span> <span class="pre">a</span> <span class="pre">b</span></code> has type <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>, so <code class="docutils literal notranslate"><span class="pre">min</span></code> acts like a function of two arguments, as we expect. Handling multiple arguments in this way is known as <em>currying</em>, after the logician Haskell Curry.</p> <p>The order of operations in Lean can also take some getting used to. Function application binds tighter than infix operations, so the expression <code class="docutils literal notranslate"><span class="pre">min</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">+</span> <span class="pre">c</span></code> is interpreted as <code class="docutils literal notranslate"><span class="pre">(min</span> <span class="pre">a</span> <span class="pre">b)</span> <span class="pre">+</span> <span class="pre">c</span></code>. With time, these conventions will become second nature.</p> <p>Using the theorem <code class="docutils literal notranslate"><span class="pre">le_antisymm</span></code>, we can show that two real numbers are equal if each is less than or equal to the other. Using this and the facts above, we can show that <code class="docutils literal notranslate"><span class="pre">min</span></code> is commutative:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">min</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">le_antisymm</span> <span class="bp">·</span> <span class="k">show</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">min</span> <span class="n">b</span> <span class="n">a</span> <span class="n">apply</span> <span class="n">le_min</span> <span class="bp">·</span> <span class="n">apply</span> <span class="n">min_le_right</span> <span class="n">apply</span> <span class="n">min_le_left</span> <span class="bp">·</span> <span class="k">show</span> <span class="n">min</span> <span class="n">b</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="n">apply</span> <span class="n">le_min</span> <span class="bp">·</span> <span class="n">apply</span> <span class="n">min_le_right</span> <span class="n">apply</span> <span class="n">min_le_left</span> </pre></div> </div> <p id="index-21">Here we have used dots to separate proofs of different goals. Our usage is inconsistent: at the outer level, we use dots and indentation for both goals, whereas for the nested proofs, we use dots only until a single goal remains. Both conventions are reasonable and useful. We also use the <code class="docutils literal notranslate"><span class="pre">show</span></code> tactic to structure the proof and indicate what is being proved in each block. The proof still works without the <code class="docutils literal notranslate"><span class="pre">show</span></code> commands, but using them makes the proof easier to read and maintain.</p> <p>It may bother you that the the proof is repetitive. To foreshadow skills you will learn later on, we note that one way to avoid the repetition is to state a local lemma and then use it:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">min</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="n">min</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">min</span> <span class="n">y</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">y</span> <span class="n">apply</span> <span class="n">le_min</span> <span class="n">apply</span> <span class="n">min_le_right</span> <span class="n">apply</span> <span class="n">min_le_left</span> <span class="n">apply</span> <span class="n">le_antisymm</span> <span class="n">apply</span> <span class="n">h</span> <span class="n">apply</span> <span class="n">h</span> </pre></div> </div> <p>We will say more about the universal quantifier in <a class="reference internal" href="C03_Logic.html#implication-and-the-universal-quantifier"><span class="std std-numref">Section 3.1</span></a>, but suffice it to say here that the hypothesis <code class="docutils literal notranslate"><span class="pre">h</span></code> says that the desired inequality holds for any <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code>, and the <code class="docutils literal notranslate"><span class="pre">intro</span></code> tactic introduces an arbitrary <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code> to establish the conclusion. The first <code class="docutils literal notranslate"><span class="pre">apply</span></code> after <code class="docutils literal notranslate"><span class="pre">le_antisymm</span></code> implicitly uses <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">a</span> <span class="pre">b</span></code>, whereas the second one uses <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">b</span> <span class="pre">a</span></code>.</p> <p id="index-22">Another solution is to use the <code class="docutils literal notranslate"><span class="pre">repeat</span></code> tactic, which applies a tactic (or a block) as many times as it can.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">min</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">le_antisymm</span> <span class="n">repeat</span> <span class="n">apply</span> <span class="n">le_min</span> <span class="n">apply</span> <span class="n">min_le_right</span> <span class="n">apply</span> <span class="n">min_le_left</span> </pre></div> </div> <p>In any case, whether or not you use these tricks, we encourage you to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">max</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">max</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">min</span> <span class="o">(</span><span class="n">min</span> <span class="n">a</span> <span class="n">b</span><span class="o">)</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">min</span> <span class="n">a</span> <span class="o">(</span><span class="n">min</span> <span class="n">b</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Of course, you are welcome to prove the associativity of <code class="docutils literal notranslate"><span class="pre">max</span></code> as well.</p> <p>It is an interesting fact that <code class="docutils literal notranslate"><span class="pre">min</span></code> distributes over <code class="docutils literal notranslate"><span class="pre">max</span></code> the way that multiplication distributes over addition, and vice-versa. In other words, on the real numbers, we have the identity <code class="docutils literal notranslate"><span class="pre">min</span> <span class="pre">a</span> <span class="pre">(max</span> <span class="pre">b</span> <span class="pre">c)</span> <span class="pre">≤</span> <span class="pre">max</span> <span class="pre">(min</span> <span class="pre">a</span> <span class="pre">b)</span> <span class="pre">(min</span> <span class="pre">a</span> <span class="pre">c)</span></code> as well as the corresponding version with <code class="docutils literal notranslate"><span class="pre">max</span></code> and <code class="docutils literal notranslate"><span class="pre">min</span></code> switched. But in the next section we will see that this does <em>not</em> follow from the transitivity and reflexivity of <code class="docutils literal notranslate"><span class="pre">≤</span></code> and the characterizing properties of <code class="docutils literal notranslate"><span class="pre">min</span></code> and <code class="docutils literal notranslate"><span class="pre">max</span></code> enumerated above. We need to use the fact that <code class="docutils literal notranslate"><span class="pre">≤</span></code> on the real numbers is a <em>total order</em>, which is to say, it satisfies <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">x</span> <span class="pre">y,</span> <span class="pre">x</span> <span class="pre">≤</span> <span class="pre">y</span> <span class="pre">∨</span> <span class="pre">y</span> <span class="pre">≤</span> <span class="pre">x</span></code>. Here the disjunction symbol, <code class="docutils literal notranslate"><span class="pre">∨</span></code>, represents “or”. In the first case, we have <code class="docutils literal notranslate"><span class="pre">min</span> <span class="pre">x</span> <span class="pre">y</span> <span class="pre">=</span> <span class="pre">x</span></code>, and in the second case, we have <code class="docutils literal notranslate"><span class="pre">min</span> <span class="pre">x</span> <span class="pre">y</span> <span class="pre">=</span> <span class="pre">y</span></code>. We will learn how to reason by cases in <a class="reference internal" href="C03_Logic.html#disjunction"><span class="std std-numref">Section 3.5</span></a>, but for now we will stick to examples that don’t require the case split.</p> <p>Here is one such example:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">aux</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">min</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="o">(</span><span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">min</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="o">(</span><span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>It is clear that <code class="docutils literal notranslate"><span class="pre">aux</span></code> provides one of the two inequalities needed to prove the equality, but applying it to suitable values yields the other direction as well. As a hint, you can use the theorem <code class="docutils literal notranslate"><span class="pre">add_neg_cancel_right</span></code> and the <code class="docutils literal notranslate"><span class="pre">linarith</span></code> tactic.</p> <p id="index-23">Lean’s naming convention is made manifest in the library’s name for the triangle inequality:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">abs_add</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">≤</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">abs</span> <span class="n">b</span><span class="o">)</span> </pre></div> </div> <p>Use it to prove the following variant:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">-</span> <span class="n">abs</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>See if you can do this in three lines or less. You can use the theorem <code class="docutils literal notranslate"><span class="pre">sub_add_cancel</span></code>.</p> <p id="index-24">Another important relation that we will make use of in the sections to come is the divisibility relation on the natural numbers, <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∣</span> <span class="pre">y</span></code>. Be careful: the divisibility symbol is <em>not</em> the ordinary bar on your keyboard. Rather, it is a unicode character obtained by typing <code class="docutils literal notranslate"><span class="pre">\|</span></code> in VS Code. By convention, mathlib uses <code class="docutils literal notranslate"><span class="pre">dvd</span></code> to refer to it in theorem names.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∣</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">∣</span> <span class="n">z</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∣</span> <span class="n">z</span> <span class="o">:=</span> <span class="n">dvd_trans</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∣</span> <span class="n">y</span> <span class="bp">*</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">z</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">dvd_mul_of_dvd_left</span> <span class="n">apply</span> <span class="n">dvd_mul_left</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∣</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">dvd_mul_left</span> </pre></div> </div> <p>In the last example, the exponent is a natural number, and applying <code class="docutils literal notranslate"><span class="pre">dvd_mul_left</span></code> forces Lean to expand the definition of <code class="docutils literal notranslate"><span class="pre">x^2</span></code> to <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">x^1</span></code>. See if you can guess the names of the theorems you need to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∣</span> <span class="n">w</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∣</span> <span class="n">y</span> <span class="bp">*</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">z</span><span class="o">)</span> <span class="bp">+</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">w</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-25">With respect to divisibility, the <em>greatest common divisor</em>, <code class="docutils literal notranslate"><span class="pre">gcd</span></code>, and least common multiple, <code class="docutils literal notranslate"><span class="pre">lcm</span></code>, are analogous to <code class="docutils literal notranslate"><span class="pre">min</span></code> and <code class="docutils literal notranslate"><span class="pre">max</span></code>. Since every number divides <code class="docutils literal notranslate"><span class="pre">0</span></code>, <code class="docutils literal notranslate"><span class="pre">0</span></code> is really the greatest element with respect to divisibility:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="kn">open</span> <span class="n">Nat</span> <span class="k">#check</span> <span class="o">(</span><span class="n">gcd_zero_right</span> <span class="n">n</span> <span class="o">:</span> <span class="n">gcd</span> <span class="n">n</span> <span class="mi">0</span> <span class="bp">=</span> <span class="n">n</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">gcd_zero_left</span> <span class="n">n</span> <span class="o">:</span> <span class="n">gcd</span> <span class="mi">0</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">n</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lcm_zero_right</span> <span class="n">n</span> <span class="o">:</span> <span class="n">lcm</span> <span class="n">n</span> <span class="mi">0</span> <span class="bp">=</span> <span class="mi">0</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lcm_zero_left</span> <span class="n">n</span> <span class="o">:</span> <span class="n">lcm</span> <span class="mi">0</span> <span class="n">n</span> <span class="bp">=</span> <span class="mi">0</span><span class="o">)</span> </pre></div> </div> <p>The functions <code class="docutils literal notranslate"><span class="pre">gcd</span></code> and <code class="docutils literal notranslate"><span class="pre">lcm</span></code> for natural numbers are in the <code class="docutils literal notranslate"><span class="pre">Nat</span></code> namespace, which means that the full identifiers are <code class="docutils literal notranslate"><span class="pre">Nat.gcd</span></code> and <code class="docutils literal notranslate"><span class="pre">Nat.lcm</span></code>. Similarly, the names of the theorems listed are prefixed by <code class="docutils literal notranslate"><span class="pre">Nat</span></code>. The command <code class="docutils literal notranslate"><span class="pre">open</span> <span class="pre">Nat</span></code> opens the namespace, allowing us to use the shorter names.</p> <p>See if you can guess the names of the theorems you will need to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">gcd</span> <span class="n">m</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">gcd</span> <span class="n">n</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Hint: you can use <code class="docutils literal notranslate"><span class="pre">dvd_antisymm</span></code>, but if you do, Lean will complain that the expression is ambiguous between the generic theorem and the version <code class="docutils literal notranslate"><span class="pre">Nat.dvd_antisymm</span></code>, the one specifically for the natural numbers. You can use <code class="docutils literal notranslate"><span class="pre">_root_.dvd_antisymm</span></code> to specify the generic one; either one will work.</p> </section> <section id="proving-facts-about-algebraic-structures"> <span id="id5"></span><h2><span class="section-number">2.5. </span>Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this heading"></a></h2> <p id="index-26">In <a class="reference internal" href="#proving-identities-in-algebraic-structures"><span class="std std-numref">Section 2.2</span></a>, we saw that many common identities governing the real numbers hold in more general classes of algebraic structures, such as commutative rings. We can use any axioms we want to describe an algebraic structure, not just equations. For example, a <em>partial order</em> consists of a set with a binary relation that is reflexive and transitive, like <code class="docutils literal notranslate"><span class="pre">≤</span></code> on the real numbers. Lean knows about partial orders:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">PartialOrder</span> <span class="n">α</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="k">#check</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_refl</span> <span class="n">x</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">x</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_trans</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">→</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">z</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">z</span><span class="o">)</span> </pre></div> </div> <p>Here we are adopting the mathlib convention of using letters like <code class="docutils literal notranslate"><span class="pre">α</span></code>, <code class="docutils literal notranslate"><span class="pre">β</span></code>, and <code class="docutils literal notranslate"><span class="pre">γ</span></code> (entered as <code class="docutils literal notranslate"><span class="pre">\a</span></code>, <code class="docutils literal notranslate"><span class="pre">\b</span></code>, and <code class="docutils literal notranslate"><span class="pre">\g</span></code>) for arbitrary types. The library often uses letters like <code class="docutils literal notranslate"><span class="pre">R</span></code> and <code class="docutils literal notranslate"><span class="pre">G</span></code> for the carries of algebraic structures like rings and groups, respectively, but in general Greek letters are used for types, especially when there is little or no structure associated with them.</p> <p>Associated to any partial order, <code class="docutils literal notranslate"><span class="pre">≤</span></code>, there is also a <em>strict partial order</em>, <code class="docutils literal notranslate"><span class="pre"><</span></code>, which acts somewhat like <code class="docutils literal notranslate"><span class="pre"><</span></code> on the real numbers. Saying that <code class="docutils literal notranslate"><span class="pre">x</span></code> is less than <code class="docutils literal notranslate"><span class="pre">y</span></code> in this order is equivalent to saying that it is less-than-or-equal to <code class="docutils literal notranslate"><span class="pre">y</span></code> and not equal to <code class="docutils literal notranslate"><span class="pre">y</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_irrefl</span> <span class="n">x</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">x</span> <span class="bp"><</span> <span class="n">x</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_trans</span> <span class="o">:</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="bp">→</span> <span class="n">y</span> <span class="bp"><</span> <span class="n">z</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">z</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_of_le_of_lt</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">→</span> <span class="n">y</span> <span class="bp"><</span> <span class="n">z</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">z</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_of_lt_of_le</span> <span class="o">:</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="bp">→</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">z</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">z</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="o">:=</span> <span class="n">lt_iff_le_and_ne</span> </pre></div> </div> <p>In this example, the symbol <code class="docutils literal notranslate"><span class="pre">∧</span></code> stands for “and,” the symbol <code class="docutils literal notranslate"><span class="pre">¬</span></code> stands for “not,” and <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">≠</span> <span class="pre">y</span></code> abbreviates <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">(x</span> <span class="pre">=</span> <span class="pre">y)</span></code>. In <a class="reference internal" href="C03_Logic.html#logic"><span class="std std-numref">Chapter 3</span></a>, you will learn how to use these logical connectives to <em>prove</em> that <code class="docutils literal notranslate"><span class="pre"><</span></code> has the properties indicated.</p> <p id="index-27">A <em>lattice</em> is a structure that extends a partial order with operations <code class="docutils literal notranslate"><span class="pre">⊓</span></code> and <code class="docutils literal notranslate"><span class="pre">⊔</span></code> that are analogous to <code class="docutils literal notranslate"><span class="pre">min</span></code> and <code class="docutils literal notranslate"><span class="pre">max</span></code> on the real numbers:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Lattice</span> <span class="n">α</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="k">#check</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="k">#check</span> <span class="o">(</span><span class="n">inf_le_left</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">inf_le_right</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_inf</span> <span class="o">:</span> <span class="n">z</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">→</span> <span class="n">z</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">→</span> <span class="n">z</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span><span class="o">)</span> <span class="k">#check</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_sup_left</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_sup_right</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">sup_le</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">z</span> <span class="bp">→</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">z</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">z</span><span class="o">)</span> </pre></div> </div> <p>The characterizations of <code class="docutils literal notranslate"><span class="pre">⊓</span></code> and <code class="docutils literal notranslate"><span class="pre">⊔</span></code> justify calling them the <em>greatest lower bound</em> and <em>least upper bound</em>, respectively. You can type them in VS code using <code class="docutils literal notranslate"><span class="pre">\glb</span></code> and <code class="docutils literal notranslate"><span class="pre">\lub</span></code>. The symbols are also often called then <em>infimum</em> and the <em>supremum</em>, and mathlib refers to them as <code class="docutils literal notranslate"><span class="pre">inf</span></code> and <code class="docutils literal notranslate"><span class="pre">sup</span></code> in theorem names. To further complicate matters, they are also often called <em>meet</em> and <em>join</em>. Therefore, if you work with lattices, you have to keep the following dictionary in mind:</p> <ul class="simple"> <li><p><code class="docutils literal notranslate"><span class="pre">⊓</span></code> is the <em>greatest lower bound</em>, <em>infimum</em>, or <em>meet</em>.</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">⊔</span></code> is the <em>least upper bound</em>, <em>supremum</em>, or <em>join</em>.</p></li> </ul> <p>Some instances of lattices include:</p> <ul class="simple"> <li><p><code class="docutils literal notranslate"><span class="pre">min</span></code> and <code class="docutils literal notranslate"><span class="pre">max</span></code> on any total order, such as the integers or real numbers with <code class="docutils literal notranslate"><span class="pre">≤</span></code></p></li> <li><p><code class="docutils literal notranslate"><span class="pre">∩</span></code> and <code class="docutils literal notranslate"><span class="pre">∪</span></code> on the collection of subsets of some domain, with the ordering <code class="docutils literal notranslate"><span class="pre">⊆</span></code></p></li> <li><p><code class="docutils literal notranslate"><span class="pre">∧</span></code> and <code class="docutils literal notranslate"><span class="pre">∨</span></code> on boolean truth values, with ordering <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">≤</span> <span class="pre">y</span></code> if either <code class="docutils literal notranslate"><span class="pre">x</span></code> is false or <code class="docutils literal notranslate"><span class="pre">y</span></code> is true</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">gcd</span></code> and <code class="docutils literal notranslate"><span class="pre">lcm</span></code> on the natural numbers (or positive natural numbers), with the divisibility ordering, <code class="docutils literal notranslate"><span class="pre">∣</span></code></p></li> <li><p>the collection of linear subspaces of a vector space, where the greatest lower bound is given by the intersection, the least upper bound is given by the sum of the two spaces, and the ordering is inclusion</p></li> <li><p>the collection of topologies on a set (or, in Lean, a type), where the greatest lower bound of two topologies consists of the topology that is generated by their union, the least upper bound is their intersection, and the ordering is reverse inclusion</p></li> </ul> <p>You can check that, as with <code class="docutils literal notranslate"><span class="pre">min</span></code> / <code class="docutils literal notranslate"><span class="pre">max</span></code> and <code class="docutils literal notranslate"><span class="pre">gcd</span></code> / <code class="docutils literal notranslate"><span class="pre">lcm</span></code>, you can prove the commutativity and associativity of the infimum and supremum using only their characterizing axioms, together with <code class="docutils literal notranslate"><span class="pre">le_refl</span></code> and <code class="docutils literal notranslate"><span class="pre">le_trans</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">y</span> <span class="bp">⊓</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="bp">⊓</span> <span class="n">z</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">y</span> <span class="bp">⊓</span> <span class="n">z</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">y</span> <span class="bp">⊔</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span> <span class="bp">⊔</span> <span class="n">z</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="o">(</span><span class="n">y</span> <span class="bp">⊔</span> <span class="n">z</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>You can find these theorems in the mathlib as <code class="docutils literal notranslate"><span class="pre">inf_comm</span></code>, <code class="docutils literal notranslate"><span class="pre">inf_assoc</span></code>, <code class="docutils literal notranslate"><span class="pre">sup_comm</span></code>, and <code class="docutils literal notranslate"><span class="pre">sup_assoc</span></code>, respectively.</p> <p>Another good exercise is to prove the <em>absorption laws</em> using only those axioms:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">absorb1</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">absorb2</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>These can be found in mathlib with the names <code class="docutils literal notranslate"><span class="pre">inf_sup_self</span></code> and <code class="docutils literal notranslate"><span class="pre">sup_inf_self</span></code>.</p> <p>A lattice that satisfies the additional identities <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">⊓</span> <span class="pre">(y</span> <span class="pre">⊔</span> <span class="pre">z)</span> <span class="pre">=</span> <span class="pre">(x</span> <span class="pre">⊓</span> <span class="pre">y)</span> <span class="pre">⊔</span> <span class="pre">(x</span> <span class="pre">⊓</span> <span class="pre">z)</span></code> and <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">⊔</span> <span class="pre">(y</span> <span class="pre">⊓</span> <span class="pre">z)</span> <span class="pre">=</span> <span class="pre">(x</span> <span class="pre">⊔</span> <span class="pre">y)</span> <span class="pre">⊓</span> <span class="pre">(x</span> <span class="pre">⊔</span> <span class="pre">z)</span></code> is called a <em>distributive lattice</em>. Lean knows about these too:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">DistribLattice</span> <span class="n">α</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">inf_sup_left</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">y</span> <span class="bp">⊔</span> <span class="n">z</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="bp">⊔</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">z</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">inf_sup_right</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span><span class="o">)</span> <span class="bp">⊓</span> <span class="n">z</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">z</span> <span class="bp">⊔</span> <span class="n">y</span> <span class="bp">⊓</span> <span class="n">z</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">sup_inf_left</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span> <span class="bp">⊓</span> <span class="n">z</span> <span class="bp">=</span> <span class="o">(</span><span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span><span class="o">)</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">x</span> <span class="bp">⊔</span> <span class="n">z</span><span class="o">))</span> <span class="k">#check</span> <span class="o">(</span><span class="n">sup_inf_right</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="bp">⊔</span> <span class="n">z</span> <span class="bp">=</span> <span class="o">(</span><span class="n">x</span> <span class="bp">⊔</span> <span class="n">z</span><span class="o">)</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">y</span> <span class="bp">⊔</span> <span class="n">z</span><span class="o">))</span> </pre></div> </div> <p>The left and right versions are easily shown to be equivalent, given the commutativity of <code class="docutils literal notranslate"><span class="pre">⊓</span></code> and <code class="docutils literal notranslate"><span class="pre">⊔</span></code>. It is a good exercise to show that not every lattice is distributive by providing an explicit description of a nondistributive lattice with finitely many elements. It is also a good exercise to show that in any lattice, either distributivity law implies the other:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Lattice</span> <span class="n">α</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">y</span> <span class="bp">⊔</span> <span class="n">z</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">y</span> <span class="bp">⊔</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">z</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">⊔</span> <span class="n">b</span> <span class="bp">⊓</span> <span class="n">c</span> <span class="bp">=</span> <span class="o">(</span><span class="n">a</span> <span class="bp">⊔</span> <span class="n">b</span><span class="o">)</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">a</span> <span class="bp">⊔</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span> <span class="bp">⊓</span> <span class="n">z</span> <span class="bp">=</span> <span class="o">(</span><span class="n">x</span> <span class="bp">⊔</span> <span class="n">y</span><span class="o">)</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">x</span> <span class="bp">⊔</span> <span class="n">z</span><span class="o">))</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">⊓</span> <span class="o">(</span><span class="n">b</span> <span class="bp">⊔</span> <span class="n">c</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">⊓</span> <span class="n">b</span> <span class="bp">⊔</span> <span class="n">a</span> <span class="bp">⊓</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>It is possible to combine axiomatic structures into larger ones. For example, a <em>strict ordered ring</em> consists of a commutative ring together with a partial order on the carrier satisfying additional axioms that say that the ring operations are compatible with the order:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">StrictOrderedRing</span> <span class="n">R</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_le_add_left</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="bp">∀</span> <span class="n">c</span><span class="o">,</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">mul_pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> </pre></div> </div> <p><a class="reference internal" href="C03_Logic.html#logic"><span class="std std-numref">Chapter 3</span></a> will provide the means to derive the following from <code class="docutils literal notranslate"><span class="pre">mul_pos</span></code> and the definition of <code class="docutils literal notranslate"><span class="pre"><</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">mul_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> </pre></div> </div> <p>It is then an extended exercise to show that many common facts used to reason about arithmetic and the ordering on the real numbers hold generically for any ordered ring. Here are a couple of examples you can try, using only properties of rings, partial orders, and the facts enumerated in the last two examples:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">-</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">-</span> <span class="n">a</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-28">Finally, here is one last example. A <em>metric space</em> consists of a set equipped with a notion of distance, <code class="docutils literal notranslate"><span class="pre">dist</span> <span class="pre">x</span> <span class="pre">y</span></code>, mapping any pair of elements to a real number. The distance function is assumed to satisfy the following axioms:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">X</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">X</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">dist_self</span> <span class="n">x</span> <span class="o">:</span> <span class="n">dist</span> <span class="n">x</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">dist_comm</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">dist</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">dist</span> <span class="n">y</span> <span class="n">x</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">dist_triangle</span> <span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">dist</span> <span class="n">x</span> <span class="n">z</span> <span class="bp">≤</span> <span class="n">dist</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">+</span> <span class="n">dist</span> <span class="n">y</span> <span class="n">z</span><span class="o">)</span> </pre></div> </div> <p>Having mastered this section, you can show that it follows from these axioms that distances are always nonnegative:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">X</span><span class="o">)</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">dist</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>We recommend making use of the theorem <code class="docutils literal notranslate"><span class="pre">nonneg_of_mul_nonneg_left</span></code>. As you may have guessed, this theorem is called <code class="docutils literal notranslate"><span class="pre">dist_nonneg</span></code> in mathlib.</p> </section> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C01_Introduction.html" class="btn btn-neutral float-left" title="1. Introduction" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> <a href="C03_Logic.html" class="btn btn-neutral float-right" title="3. Logic" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> </div> <hr/> <div role="contentinfo"> <p>© Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. 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Logic — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> <link rel="shortcut icon" href="_static/favicon.ico"/> <!--[if lt IE 9]> <script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" /> <link rel="next" title="4. Sets and Functions" href="C04_Sets_and_Functions.html" /> <link rel="prev" title="2. Basics" href="C02_Basics.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul class="current"> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">3. Logic</a><ul> <li class="toctree-l2"><a class="reference internal" href="#implication-and-the-universal-quantifier">3.1. Implication and the Universal Quantifier</a></li> <li class="toctree-l2"><a class="reference internal" href="#the-existential-quantifier">3.2. The Existential Quantifier</a></li> <li class="toctree-l2"><a class="reference internal" href="#negation">3.3. Negation</a></li> <li class="toctree-l2"><a class="reference internal" href="#conjunction-and-bi-implication">3.4. Conjunction and Bi-implication</a></li> <li class="toctree-l2"><a class="reference internal" href="#disjunction">3.5. Disjunction</a></li> <li class="toctree-l2"><a class="reference internal" href="#sequences-and-convergence">3.6. Sequences and Convergence</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">3. </span>Logic</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C03_Logic.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="logic"> <span id="id1"></span><h1><span class="section-number">3. </span>Logic<a class="headerlink" href="#logic" title="Permalink to this heading"></a></h1> <p>In the last chapter, we dealt with equations, inequalities, and basic mathematical statements like “<span class="math notranslate nohighlight">\(x\)</span> divides <span class="math notranslate nohighlight">\(y\)</span>.” Complex mathematical statements are built up from simple ones like these using logical terms like “and,” “or,” “not,” and “if … then,” “every,” and “some.” In this chapter, we show you how to work with statements that are built up in this way.</p> <section id="implication-and-the-universal-quantifier"> <span id="id2"></span><h2><span class="section-number">3.1. </span>Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this heading"></a></h2> <p>Consider the statement after the <code class="docutils literal notranslate"><span class="pre">#check</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">x</span> </pre></div> </div> <p>In words, we would say “for every real number <code class="docutils literal notranslate"><span class="pre">x</span></code>, if <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre">≤</span> <span class="pre">x</span></code> then the absolute value of <code class="docutils literal notranslate"><span class="pre">x</span></code> equals <code class="docutils literal notranslate"><span class="pre">x</span></code>”. We can also have more complicated statements like:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span> <span class="n">ε</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">ε</span> <span class="bp">≤</span> <span class="mi">1</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> </pre></div> </div> <p>In words, we would say “for every <code class="docutils literal notranslate"><span class="pre">x</span></code>, <code class="docutils literal notranslate"><span class="pre">y</span></code>, and <code class="docutils literal notranslate"><span class="pre">ε</span></code>, if <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">ε</span> <span class="pre">≤</span> <span class="pre">1</span></code>, the absolute value of <code class="docutils literal notranslate"><span class="pre">x</span></code> is less than <code class="docutils literal notranslate"><span class="pre">ε</span></code>, and the absolute value of <code class="docutils literal notranslate"><span class="pre">y</span></code> is less than <code class="docutils literal notranslate"><span class="pre">ε</span></code>, then the absolute value of <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">y</span></code> is less than <code class="docutils literal notranslate"><span class="pre">ε</span></code>.” In Lean, in a sequence of implications there are implicit parentheses grouped to the right. So the expression above means “if <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">ε</span></code> then if <code class="docutils literal notranslate"><span class="pre">ε</span> <span class="pre">≤</span> <span class="pre">1</span></code> then if <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">ε</span></code> …” As a result, the expression says that all the assumptions together imply the conclusion.</p> <p>You have already seen that even though the universal quantifier in this statement ranges over objects and the implication arrows introduce hypotheses, Lean treats the two in very similar ways. In particular, if you have proved a theorem of that form, you can apply it to objects and hypotheses in the same way:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span> <span class="n">ε</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">ε</span> <span class="bp">≤</span> <span class="mi">1</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">δ</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">δ</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">δ</span> <span class="bp">≤</span> <span class="mi">1</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">ha</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">δ</span><span class="o">)</span> <span class="o">(</span><span class="n">hb</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">δ</span><span class="o">)</span> <span class="k">#check</span> <span class="n">my_lemma</span> <span class="n">a</span> <span class="n">b</span> <span class="n">δ</span> <span class="k">#check</span> <span class="n">my_lemma</span> <span class="n">a</span> <span class="n">b</span> <span class="n">δ</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="k">#check</span> <span class="n">my_lemma</span> <span class="n">a</span> <span class="n">b</span> <span class="n">δ</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="n">ha</span> <span class="n">hb</span> <span class="kd">end</span> </pre></div> </div> <p>You have also already seen that it is common in Lean to use curly brackets to make quantified variables implicit when they can be inferred from subsequent hypotheses. When we do that, we can just apply a lemma to the hypotheses without mentioning the objects.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma2</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">ε</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">},</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">ε</span> <span class="bp">≤</span> <span class="mi">1</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">δ</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">δ</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="n">δ</span> <span class="bp">≤</span> <span class="mi">1</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">ha</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">δ</span><span class="o">)</span> <span class="o">(</span><span class="n">hb</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">δ</span><span class="o">)</span> <span class="k">#check</span> <span class="n">my_lemma2</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="n">ha</span> <span class="n">hb</span> <span class="kd">end</span> </pre></div> </div> <p>At this stage, you also know that if you use the <code class="docutils literal notranslate"><span class="pre">apply</span></code> tactic to apply <code class="docutils literal notranslate"><span class="pre">my_lemma</span></code> to a goal of the form <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">(a</span> <span class="pre">*</span> <span class="pre">b)</span> <span class="pre"><</span> <span class="pre">δ</span></code>, you are left with new goals that require you to prove each of the hypotheses.</p> <p id="index-0">To prove a statement like this, use the <code class="docutils literal notranslate"><span class="pre">intro</span></code> tactic. Take a look at what it does in this example:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma3</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">ε</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">},</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">ε</span> <span class="bp">≤</span> <span class="mi">1</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">y</span> <span class="n">ε</span> <span class="n">epos</span> <span class="n">ele1</span> <span class="n">xlt</span> <span class="n">ylt</span> <span class="gr">sorry</span> </pre></div> </div> <p>We can use any names we want for the universally quantified variables; they do not have to be <code class="docutils literal notranslate"><span class="pre">x</span></code>, <code class="docutils literal notranslate"><span class="pre">y</span></code>, and <code class="docutils literal notranslate"><span class="pre">ε</span></code>. Notice that we have to introduce the variables even though they are marked implicit: making them implicit means that we leave them out when we write an expression <em>using</em> <code class="docutils literal notranslate"><span class="pre">my_lemma</span></code>, but they are still an essential part of the statement that we are proving. After the <code class="docutils literal notranslate"><span class="pre">intro</span></code> command, the goal is what it would have been at the start if we listed all the variables and hypotheses <em>before</em> the colon, as we did in the last section. In a moment, we will see why it is sometimes necessary to introduce variables and hypotheses after the proof begins.</p> <p>To help you prove the lemma, we will start you off:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma4</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">ε</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">},</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">ε</span> <span class="bp">≤</span> <span class="mi">1</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">→</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">y</span> <span class="n">ε</span> <span class="n">epos</span> <span class="n">ele1</span> <span class="n">xlt</span> <span class="n">ylt</span> <span class="k">calc</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">abs</span> <span class="n">y</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">≤</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">ε</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp"><</span> <span class="mi">1</span> <span class="bp">*</span> <span class="n">ε</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">ε</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Finish the proof using the theorems <code class="docutils literal notranslate"><span class="pre">abs_mul</span></code>, <code class="docutils literal notranslate"><span class="pre">mul_le_mul</span></code>, <code class="docutils literal notranslate"><span class="pre">abs_nonneg</span></code>, <code class="docutils literal notranslate"><span class="pre">mul_lt_mul_right</span></code>, and <code class="docutils literal notranslate"><span class="pre">one_mul</span></code>. Remember that you can find theorems like these using tab completion. Remember also that you can use <code class="docutils literal notranslate"><span class="pre">.mp</span></code> and <code class="docutils literal notranslate"><span class="pre">.mpr</span></code> or <code class="docutils literal notranslate"><span class="pre">.1</span></code> and <code class="docutils literal notranslate"><span class="pre">.2</span></code> to extract the two directions of an if-and-only-if statement.</p> <p>Universal quantifiers are often hidden in definitions, and Lean will unfold definitions to expose them when necessary. For example, let’s define two predicates, <code class="docutils literal notranslate"><span class="pre">FnUb</span> <span class="pre">f</span> <span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">FnLb</span> <span class="pre">f</span> <span class="pre">a</span></code>, where <code class="docutils literal notranslate"><span class="pre">f</span></code> is a function from the real numbers to the real numbers and <code class="docutils literal notranslate"><span class="pre">a</span></code> is a real number. The first says that <code class="docutils literal notranslate"><span class="pre">a</span></code> is an upper bound on the values of <code class="docutils literal notranslate"><span class="pre">f</span></code>, and the second says that <code class="docutils literal notranslate"><span class="pre">a</span></code> is a lower bound on the values of <code class="docutils literal notranslate"><span class="pre">f</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">FnUb</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">a</span> <span class="kd">def</span> <span class="n">FnLb</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">f</span> <span class="n">x</span> </pre></div> </div> <p id="index-1">In the next example, <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">↦</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code> is a name for the function that maps <code class="docutils literal notranslate"><span class="pre">x</span></code> to <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">hfa</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">↦</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">dsimp</span> <span class="n">apply</span> <span class="n">add_le_add</span> <span class="n">apply</span> <span class="n">hfa</span> <span class="n">apply</span> <span class="n">hgb</span> </pre></div> </div> <p id="index-2">Applying <code class="docutils literal notranslate"><span class="pre">intro</span></code> to the goal <code class="docutils literal notranslate"><span class="pre">FnUb</span> <span class="pre">(fun</span> <span class="pre">x</span> <span class="pre">↦</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x)</span> <span class="pre">(a</span> <span class="pre">+</span> <span class="pre">b)</span></code> forces Lean to unfold the definition of <code class="docutils literal notranslate"><span class="pre">FnUb</span></code> and introduce <code class="docutils literal notranslate"><span class="pre">x</span></code> for the universal quantifier. The goal is then <code class="docutils literal notranslate"><span class="pre">(fun</span> <span class="pre">(x</span> <span class="pre">:</span> <span class="pre">ℝ)</span> <span class="pre">↦</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x)</span> <span class="pre">x</span> <span class="pre">≤</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code>. But applying <code class="docutils literal notranslate"><span class="pre">(fun</span> <span class="pre">x</span> <span class="pre">↦</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x)</span></code> to <code class="docutils literal notranslate"><span class="pre">x</span></code> should result in <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code>, and the <code class="docutils literal notranslate"><span class="pre">dsimp</span></code> command performs that simplification. (The “d” stands for “definitional.”) You can delete that command and the proof still works; Lean would have to perform that contraction anyhow to make sense of the next <code class="docutils literal notranslate"><span class="pre">apply</span></code>. The <code class="docutils literal notranslate"><span class="pre">dsimp</span></code> command simply makes the goal more readable and helps us figure out what to do next. Another option is to use the <code class="docutils literal notranslate"><span class="pre">change</span></code> tactic by writing <code class="docutils literal notranslate"><span class="pre">change</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span> <span class="pre">≤</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code>. This helps make the proof more readable, and gives you more control over how the goal is transformed.</p> <p>The rest of the proof is routine. The last two <code class="docutils literal notranslate"><span class="pre">apply</span></code> commands force Lean to unfold the definitions of <code class="docutils literal notranslate"><span class="pre">FnUb</span></code> in the hypotheses. Try carrying out similar proofs of these:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">hfa</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">nnf</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">f</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">nng</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">g</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="mi">0</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">hfa</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hfb</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">nng</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">g</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">nna</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Even though we have defined <code class="docutils literal notranslate"><span class="pre">FnUb</span></code> and <code class="docutils literal notranslate"><span class="pre">FnLb</span></code> for functions from the reals to the reals, you should recognize that the definitions and proofs are much more general. The definitions make sense for functions between any two types for which there is a notion of order on the codomain. Checking the type of the theorem <code class="docutils literal notranslate"><span class="pre">add_le_add</span></code> shows that it holds of any structure that is an “ordered additive commutative monoid”; the details of what that means don’t matter now, but it is worth knowing that the natural numbers, integers, rationals, and real numbers are all instances. So if we prove the theorem <code class="docutils literal notranslate"><span class="pre">fnUb_add</span></code> at that level of generality, it will apply in all these instances.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">OrderedCancelAddCommMonoid</span> <span class="n">R</span><span class="o">]</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">add_le_add</span> <span class="kd">def</span> <span class="n">FnUb'</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">R</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">a</span> <span class="kd">theorem</span> <span class="n">fnUb_add</span> <span class="o">{</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">R</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">}</span> <span class="o">(</span><span class="n">hfa</span> <span class="o">:</span> <span class="n">FnUb'</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="o">:</span> <span class="n">FnUb'</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnUb'</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">add_le_add</span> <span class="o">(</span><span class="n">hfa</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="n">x</span><span class="o">)</span> </pre></div> </div> <p>You have already seen square brackets like these in Section <a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures"><span class="std std-numref">Section 2.2</span></a>, though we still haven’t explained what they mean. For concreteness, we will stick to the real numbers for most of our examples, but it is worth knowing that mathlib contains definitions and theorems that work at a high level of generality.</p> <p id="index-3">For another example of a hidden universal quantifier, mathlib defines a predicate <code class="docutils literal notranslate"><span class="pre">Monotone</span></code>, which says that a function is nondecreasing in its arguments:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span><span class="o">},</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">f</span> <span class="n">b</span> <span class="o">:=</span> <span class="bp">@</span><span class="n">h</span> </pre></div> </div> <p>The property <code class="docutils literal notranslate"><span class="pre">Monotone</span> <span class="pre">f</span></code> is defined to be exactly the expression after the colon. We need to put the <code class="docutils literal notranslate"><span class="pre">@</span></code> symbol before <code class="docutils literal notranslate"><span class="pre">h</span></code> because if we don’t, Lean expands the implicit arguments to <code class="docutils literal notranslate"><span class="pre">h</span></code> and inserts placeholders.</p> <p>Proving statements about monotonicity involves using <code class="docutils literal notranslate"><span class="pre">intro</span></code> to introduce two variables, say, <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code>, and the hypothesis <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">≤</span> <span class="pre">b</span></code>. To <em>use</em> a monotonicity hypothesis, you can apply it to suitable arguments and hypotheses, and then apply the resulting expression to the goal. Or you can apply it to the goal and let Lean help you work backwards by displaying the remaining hypotheses as new subgoals.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">a</span> <span class="n">b</span> <span class="n">aleb</span> <span class="n">apply</span> <span class="n">add_le_add</span> <span class="n">apply</span> <span class="n">mf</span> <span class="n">aleb</span> <span class="n">apply</span> <span class="n">mg</span> <span class="n">aleb</span> </pre></div> </div> <p>When a proof is this short, it is often convenient to give a proof term instead. To describe a proof that temporarily introduces objects <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code> and a hypothesis <code class="docutils literal notranslate"><span class="pre">aleb</span></code>, Lean uses the notation <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">aleb</span> <span class="pre">=></span> <span class="pre">...</span></code>. This is analogous to the way that an expression like <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">=></span> <span class="pre">x^2</span></code> describes a function by temporarily naming an object, <code class="docutils literal notranslate"><span class="pre">x</span></code>, and then using it to describe a value. So the <code class="docutils literal notranslate"><span class="pre">intro</span></code> command in the previous proof corresponds to the lambda abstraction in the next proof term. The <code class="docutils literal notranslate"><span class="pre">apply</span></code> commands then correspond to building the application of the theorem to its arguments.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">a</span> <span class="n">b</span> <span class="n">aleb</span> <span class="bp">=></span> <span class="n">add_le_add</span> <span class="o">(</span><span class="n">mf</span> <span class="n">aleb</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="n">aleb</span><span class="o">)</span> </pre></div> </div> <p>Here is a useful trick: if you start writing the proof term <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">aleb</span> <span class="pre">=></span> <span class="pre">_</span></code> using an underscore where the rest of the expression should go, Lean will flag an error, indicating that it can’t guess the value of that expression. If you check the Lean Goal window in VS Code or hover over the squiggly error marker, Lean will show you the goal that the remaining expression has to solve.</p> <p>Try proving these, with either tactics or proof terms:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">nnc</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Here are some more examples. A function <span class="math notranslate nohighlight">\(f\)</span> from <span class="math notranslate nohighlight">\(\Bbb R\)</span> to <span class="math notranslate nohighlight">\(\Bbb R\)</span> is said to be <em>even</em> if <span class="math notranslate nohighlight">\(f(-x) = f(x)\)</span> for every <span class="math notranslate nohighlight">\(x\)</span>, and <em>odd</em> if <span class="math notranslate nohighlight">\(f(-x) = -f(x)\)</span> for every <span class="math notranslate nohighlight">\(x\)</span>. The following example defines these two notions formally and establishes one fact about them. You can complete the proofs of the others.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">FnEven</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">f</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</span><span class="o">)</span> <span class="kd">def</span> <span class="n">FnOdd</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">=</span> <span class="bp">-</span><span class="n">f</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">eg</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="k">calc</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">f</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</span><span class="o">)</span> <span class="bp">+</span> <span class="n">g</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">ef</span><span class="o">,</span> <span class="n">eg</span><span class="o">]</span> <span class="kd">example</span> <span class="o">(</span><span class="n">of</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-4">The first proof can be shortened using <code class="docutils literal notranslate"><span class="pre">dsimp</span></code> or <code class="docutils literal notranslate"><span class="pre">change</span></code> to get rid of the lambda. But you can check that the subsequent <code class="docutils literal notranslate"><span class="pre">rw</span></code> won’t work unless we get rid of the lambda explicitly, because otherwise it cannot find the patterns <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">x</span></code> in the expression. Contrary to some other tactics, <code class="docutils literal notranslate"><span class="pre">rw</span></code> operates on the syntactic level, it won’t unfold definitions or apply reductions for you (it has a variant called <code class="docutils literal notranslate"><span class="pre">erw</span></code> that tries a little harder in this direction, but not much harder).</p> <p>You can find implicit universal quantifiers all over the place, once you know how to spot them. Mathlib includes a good library for rudimentary set theory. Lean’s logical foundation imposes the restriction that when we talk about sets, we are always talking about sets of elements of some type. If <code class="docutils literal notranslate"><span class="pre">x</span></code> has type <code class="docutils literal notranslate"><span class="pre">α</span></code> and <code class="docutils literal notranslate"><span class="pre">s</span></code> has type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">α</span></code>, then <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span></code> is a proposition that asserts that <code class="docutils literal notranslate"><span class="pre">x</span></code> is an element of <code class="docutils literal notranslate"><span class="pre">s</span></code>. If <code class="docutils literal notranslate"><span class="pre">s</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span></code> are of type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">α</span></code>, then the subset relation <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">⊆</span> <span class="pre">t</span></code> is defined to mean <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">{x</span> <span class="pre">:</span> <span class="pre">α},</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">→</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">t</span></code>. The variable in the quantifier is marked implicit so that given <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">s</span> <span class="pre">⊆</span> <span class="pre">t</span></code> and <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span></code>, we can write <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">h'</span></code> as justification for <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">t</span></code>. The following example provides a tactic proof and a proof term justifying the reflexivity of the subset relation, and asks you to do the same for transitivity.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">r</span> <span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">xs</span> <span class="n">exact</span> <span class="n">xs</span> <span class="kd">theorem</span> <span class="n">Subset.refl</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">xs</span> <span class="bp">=></span> <span class="n">xs</span> <span class="kd">theorem</span> <span class="n">Subset.trans</span> <span class="o">:</span> <span class="n">r</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="bp">→</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span> <span class="bp">→</span> <span class="n">r</span> <span class="bp">⊆</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Just as we defined <code class="docutils literal notranslate"><span class="pre">FnUb</span></code> for functions, we can define <code class="docutils literal notranslate"><span class="pre">SetUb</span> <span class="pre">s</span> <span class="pre">a</span></code> to mean that <code class="docutils literal notranslate"><span class="pre">a</span></code> is an upper bound on the set <code class="docutils literal notranslate"><span class="pre">s</span></code>, assuming <code class="docutils literal notranslate"><span class="pre">s</span></code> is a set of elements of some type that has an order associated with it. In the next example, we ask you to prove that if <code class="docutils literal notranslate"><span class="pre">a</span></code> is a bound on <code class="docutils literal notranslate"><span class="pre">s</span></code> and <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">≤</span> <span class="pre">b</span></code>, then <code class="docutils literal notranslate"><span class="pre">b</span></code> is a bound on <code class="docutils literal notranslate"><span class="pre">s</span></code> as well.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">PartialOrder</span> <span class="n">α</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="kd">def</span> <span class="n">SetUb</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">a</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">SetUb</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">SetUb</span> <span class="n">s</span> <span class="n">b</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-5">We close this section with one last important example. A function <span class="math notranslate nohighlight">\(f\)</span> is said to be <em>injective</em> if for every <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>, if <span class="math notranslate nohighlight">\(f(x_1) = f(x_2)\)</span> then <span class="math notranslate nohighlight">\(x_1 = x_2\)</span>. Mathlib defines <code class="docutils literal notranslate"><span class="pre">Function.Injective</span> <span class="pre">f</span></code> with <code class="docutils literal notranslate"><span class="pre">x₁</span></code> and <code class="docutils literal notranslate"><span class="pre">x₂</span></code> implicit. The next example shows that, on the real numbers, any function that adds a constant is injective. We then ask you to show that multiplication by a nonzero constant is also injective.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Function</span> <span class="kd">example</span> <span class="o">(</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">x</span> <span class="bp">+</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x₁</span> <span class="n">x₂</span> <span class="n">h'</span> <span class="n">exact</span> <span class="o">(</span><span class="n">add_left_inj</span> <span class="n">c</span><span class="o">)</span><span class="bp">.</span><span class="n">mp</span> <span class="n">h'</span> <span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">c</span> <span class="bp">*</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Finally, show that the composition of two injective functions is injective:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">γ</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">g</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">γ</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">}</span> <span class="kd">example</span> <span class="o">(</span><span class="n">injg</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">g</span><span class="o">)</span> <span class="o">(</span><span class="n">injf</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">g</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="the-existential-quantifier"> <span id="id3"></span><h2><span class="section-number">3.2. </span>The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this heading"></a></h2> <p>The existential quantifier, which can be entered as <code class="docutils literal notranslate"><span class="pre">\ex</span></code> in VS Code, is used to represent the phrase “there exists.” The formal expression <code class="docutils literal notranslate"><span class="pre">∃</span> <span class="pre">x</span> <span class="pre">:</span> <span class="pre">ℝ,</span> <span class="pre">2</span> <span class="pre"><</span> <span class="pre">x</span> <span class="pre">∧</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">3</span></code> in Lean says that there is a real number between 2 and 3. (We will discuss the conjunction symbol, <code class="docutils literal notranslate"><span class="pre">∧</span></code>, in <a class="reference internal" href="#conjunction-and-biimplication"><span class="std std-numref">Section 3.4</span></a>.) The canonical way to prove such a statement is to exhibit a real number and show that it has the stated property. The number 2.5, which we can enter as <code class="docutils literal notranslate"><span class="pre">5</span> <span class="pre">/</span> <span class="pre">2</span></code> or <code class="docutils literal notranslate"><span class="pre">(5</span> <span class="pre">:</span> <span class="pre">ℝ)</span> <span class="pre">/</span> <span class="pre">2</span></code> when Lean cannot infer from context that we have the real numbers in mind, has the required property, and the <code class="docutils literal notranslate"><span class="pre">norm_num</span></code> tactic can prove that it meets the description.</p> <p id="index-6">There are a few ways we can put the information together. Given a goal that begins with an existential quantifier, the <code class="docutils literal notranslate"><span class="pre">use</span></code> tactic is used to provide the object, leaving the goal of proving the property.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">2</span> <span class="bp"><</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp"><</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">use</span> <span class="mi">5</span> <span class="bp">/</span> <span class="mi">2</span> <span class="n">norm_num</span> </pre></div> </div> <p id="index-7">Alternatively, we can use Lean’s <em>anonymous constructor</em> notation to construct the proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">2</span> <span class="bp"><</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp"><</span> <span class="mi">3</span> <span class="o">:=</span> <span class="k">have</span> <span class="n">h</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp"><</span> <span class="o">(</span><span class="mi">5</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">∧</span> <span class="o">(</span><span class="mi">5</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp"><</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">norm_num</span> <span class="o">⟨</span><span class="mi">5</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">,</span> <span class="n">h</span><span class="o">⟩</span> </pre></div> </div> <p>The left and right angle brackets, which can be entered as <code class="docutils literal notranslate"><span class="pre">\<</span></code> and <code class="docutils literal notranslate"><span class="pre">\></span></code> respectively, tell Lean to put together the given data using whatever construction is appropriate for the current goal. We can use the notation without going first into tactic mode:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">2</span> <span class="bp"><</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp"><</span> <span class="mi">3</span> <span class="o">:=</span> <span class="o">⟨</span><span class="mi">5</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">,</span> <span class="kd">by</span> <span class="n">norm_num</span><span class="o">⟩</span> </pre></div> </div> <p>So now we know how to <em>prove</em> an exists statement. But how do we <em>use</em> one? If we know that there exists an object with a certain property, we should be able to give a name to an arbitrary one and reason about it. For example, remember the predicates <code class="docutils literal notranslate"><span class="pre">FnUb</span> <span class="pre">f</span> <span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">FnLb</span> <span class="pre">f</span> <span class="pre">a</span></code> from the last section, which say that <code class="docutils literal notranslate"><span class="pre">a</span></code> is an upper bound or lower bound on <code class="docutils literal notranslate"><span class="pre">f</span></code>, respectively. We can use the existential quantifier to say that “<code class="docutils literal notranslate"><span class="pre">f</span></code> is bounded” without specifying the bound:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">FnUb</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">a</span> <span class="kd">def</span> <span class="n">FnLb</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">f</span> <span class="n">x</span> <span class="kd">def</span> <span class="n">FnHasUb</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:=</span> <span class="bp">∃</span> <span class="n">a</span><span class="o">,</span> <span class="n">FnUb</span> <span class="n">f</span> <span class="n">a</span> <span class="kd">def</span> <span class="n">FnHasLb</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:=</span> <span class="bp">∃</span> <span class="n">a</span><span class="o">,</span> <span class="n">FnLb</span> <span class="n">f</span> <span class="n">a</span> </pre></div> </div> <p>We can use the theorem <code class="docutils literal notranslate"><span class="pre">FnUb_add</span></code> from the last section to prove that if <code class="docutils literal notranslate"><span class="pre">f</span></code> and <code class="docutils literal notranslate"><span class="pre">g</span></code> have upper bounds, then so does <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">=></span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="kd">example</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">ubg</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases'</span> <span class="n">ubf</span> <span class="k">with</span> <span class="n">a</span> <span class="n">ubfa</span> <span class="n">cases'</span> <span class="n">ubg</span> <span class="k">with</span> <span class="n">b</span> <span class="n">ubgb</span> <span class="n">use</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="n">apply</span> <span class="n">fnUb_add</span> <span class="n">ubfa</span> <span class="n">ubgb</span> </pre></div> </div> <p id="index-8">The <code class="docutils literal notranslate"><span class="pre">cases'</span></code> tactic unpacks the information in the existential quantifier. Given the hypothesis <code class="docutils literal notranslate"><span class="pre">ubf</span></code> that there is an upper bound for <code class="docutils literal notranslate"><span class="pre">f</span></code>, <code class="docutils literal notranslate"><span class="pre">cases'</span></code> adds a new variable for an upper bound to the context, together with the hypothesis that it has the given property. The <code class="docutils literal notranslate"><span class="pre">with</span></code> clause allows us to specify the names we want Lean to use. The goal is left unchanged; what <em>has</em> changed is that we can now use the new object and the new hypothesis to prove the goal. This is a common pattern in mathematics: we unpack objects whose existence is asserted or implied by some hypothesis, and then use it to establish the existence of something else.</p> <p>Try using this pattern to establish the following. You might find it useful to turn some of the examples from the last section into named theorems, as we did with <code class="docutils literal notranslate"><span class="pre">fn_ub_add</span></code>, or you can insert the arguments directly into the proofs.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">lbf</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">lbg</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">≥</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-9">The task of unpacking information in a hypothesis is so important that Lean and mathlib provide a number of ways to do it. A cousin of the <code class="docutils literal notranslate"><span class="pre">cases'</span></code> tactic, <code class="docutils literal notranslate"><span class="pre">rcases</span></code>, is more flexible in that it allows us to unpack nested data. (The “r” stands for “recursive.”) In the <code class="docutils literal notranslate"><span class="pre">with</span></code> clause for unpacking an existential quantifier, we name the object and the hypothesis by presenting them as a pattern <code class="docutils literal notranslate"><span class="pre">⟨a,</span> <span class="pre">h⟩</span></code> that <code class="docutils literal notranslate"><span class="pre">rcases</span></code> then tries to match. The <code class="docutils literal notranslate"><span class="pre">rintro</span></code> tactic is a combination of <code class="docutils literal notranslate"><span class="pre">intro</span></code> and <code class="docutils literal notranslate"><span class="pre">rcases</span></code>. These examples illustrate their use:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">ubg</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">ubf</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">⟩</span> <span class="n">rcases</span> <span class="n">ubg</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">,</span> <span class="n">fnUb_add</span> <span class="n">ubfa</span> <span class="n">ubgb</span><span class="o">⟩</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span> <span class="bp">→</span> <span class="n">FnHasUb</span> <span class="n">g</span> <span class="bp">→</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">⟩</span> <span class="o">⟨</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">,</span> <span class="n">fnUb_add</span> <span class="n">ubfa</span> <span class="n">ubgb</span><span class="o">⟩</span> </pre></div> </div> <p>In fact, Lean also supports a pattern-matching lambda in expressions and proof terms:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span> <span class="bp">→</span> <span class="n">FnHasUb</span> <span class="n">g</span> <span class="bp">→</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">⟩</span> <span class="o">⟨</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">⟩</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">,</span> <span class="n">fnUb_add</span> <span class="n">ubfa</span> <span class="n">ubgb</span><span class="o">⟩</span> </pre></div> </div> <p>These are power-user moves, and there is no harm in favoring the use of <code class="docutils literal notranslate"><span class="pre">cases'</span></code> until you are more comfortable with the existential quantifier. But we will come to learn that all of these tools, including <code class="docutils literal notranslate"><span class="pre">cases'</span></code>, <code class="docutils literal notranslate"><span class="pre">use</span></code>, and the anonymous constructors, are like Swiss army knives when it comes to theorem proving. They can be used for a wide range of purposes, not just for unpacking exists statements.</p> <p>To illustrate one way that <code class="docutils literal notranslate"><span class="pre">rcases</span></code> can be used, we prove an old mathematical chestnut: if two integers <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code> can each be written as a sum of two squares, then so can their product, <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">y</span></code>. In fact, the statement is true for any commutative ring, not just the integers. In the next example, <code class="docutils literal notranslate"><span class="pre">rcases</span></code> unpacks two existential quantifiers at once. We then provide the magic values needed to express <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">y</span></code> as a sum of squares as a list to the <code class="docutils literal notranslate"><span class="pre">use</span></code> statement, and we use <code class="docutils literal notranslate"><span class="pre">ring</span></code> to verify that they work.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">CommRing</span> <span class="n">α</span><span class="o">]</span> <span class="kd">def</span> <span class="n">SumOfSquares</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:=</span> <span class="bp">∃</span> <span class="n">a</span> <span class="n">b</span><span class="o">,</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="kd">theorem</span> <span class="n">sumOfSquares_mul</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">α</span><span class="o">}</span> <span class="o">(</span><span class="n">sosx</span> <span class="o">:</span> <span class="n">SumOfSquares</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">sosy</span> <span class="o">:</span> <span class="n">SumOfSquares</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="n">SumOfSquares</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">sosx</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">a</span><span class="o">,</span> <span class="n">b</span><span class="o">,</span> <span class="n">xeq</span><span class="o">⟩</span> <span class="n">rcases</span> <span class="n">sosy</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">c</span><span class="o">,</span> <span class="n">d</span><span class="o">,</span> <span class="n">yeq</span><span class="o">⟩</span> <span class="n">rw</span> <span class="o">[</span><span class="n">xeq</span><span class="o">,</span> <span class="n">yeq</span><span class="o">]</span> <span class="n">use</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">d</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="n">ring</span> </pre></div> </div> <p>This proof doesn’t provide much insight, but here is one way to motivate it. A <em>Gaussian integer</em> is a number of the form <span class="math notranslate nohighlight">\(a + bi\)</span> where <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are integers and <span class="math notranslate nohighlight">\(i = \sqrt{-1}\)</span>. The <em>norm</em> of the Gaussian integer <span class="math notranslate nohighlight">\(a + bi\)</span> is, by definition, <span class="math notranslate nohighlight">\(a^2 + b^2\)</span>. So the norm of a Gaussian integer is a sum of squares, and any sum of squares can be expressed in this way. The theorem above reflects the fact that norm of a product of Gaussian integers is the product of their norms: if <span class="math notranslate nohighlight">\(x\)</span> is the norm of <span class="math notranslate nohighlight">\(a + bi\)</span> and <span class="math notranslate nohighlight">\(y\)</span> in the norm of <span class="math notranslate nohighlight">\(c + di\)</span>, then <span class="math notranslate nohighlight">\(xy\)</span> is the norm of <span class="math notranslate nohighlight">\((a + bi) (c + di)\)</span>. Our cryptic proof illustrates the fact that the proof that is easiest to formalize isn’t always the most perspicuous one. In the chapters to come, we will provide you with the means to define the Gaussian integers and use them to provide an alternative proof.</p> <p>The pattern of unpacking an equation inside an existential quantifier and then using it to rewrite an expression in the goal comes up often, so much so that the <code class="docutils literal notranslate"><span class="pre">rcases</span></code> tactic provides an abbreviation: if you use the keyword <code class="docutils literal notranslate"><span class="pre">rfl</span></code> in place of a new identifier, <code class="docutils literal notranslate"><span class="pre">rcases</span></code> does the rewriting automatically (this trick doesn’t work with pattern-matching lambdas).</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">sumOfSquares_mul'</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">α</span><span class="o">}</span> <span class="o">(</span><span class="n">sosx</span> <span class="o">:</span> <span class="n">SumOfSquares</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">sosy</span> <span class="o">:</span> <span class="n">SumOfSquares</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="n">SumOfSquares</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">sosx</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">a</span><span class="o">,</span> <span class="n">b</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩</span> <span class="n">rcases</span> <span class="n">sosy</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">c</span><span class="o">,</span> <span class="n">d</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">d</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">d</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="n">ring</span> </pre></div> </div> <p>As with the universal quantifier, you can find existential quantifiers hidden all over if you know how to spot them. For example, divisibility is implicitly an “exists” statement.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">divab</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∣</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">divbc</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">∣</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∣</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases'</span> <span class="n">divab</span> <span class="k">with</span> <span class="n">d</span> <span class="n">beq</span> <span class="n">cases'</span> <span class="n">divbc</span> <span class="k">with</span> <span class="n">e</span> <span class="n">ceq</span> <span class="n">rw</span> <span class="o">[</span><span class="n">ceq</span><span class="o">,</span> <span class="n">beq</span><span class="o">]</span> <span class="n">use</span> <span class="n">d</span> <span class="bp">*</span> <span class="n">e</span><span class="bp">;</span> <span class="n">ring</span> </pre></div> </div> <p>And once again, this provides a nice setting for using <code class="docutils literal notranslate"><span class="pre">rcases</span></code> with <code class="docutils literal notranslate"><span class="pre">rfl</span></code>. Try it out in the proof above. It feels pretty good!</p> <p>Then try proving the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">divab</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∣</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">divac</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∣</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∣</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-10">For another important example, a function <span class="math notranslate nohighlight">\(f : \alpha \to \beta\)</span> is said to be <em>surjective</em> if for every <span class="math notranslate nohighlight">\(y\)</span> in the codomain, <span class="math notranslate nohighlight">\(\beta\)</span>, there is an <span class="math notranslate nohighlight">\(x\)</span> in the domain, <span class="math notranslate nohighlight">\(\alpha\)</span>, such that <span class="math notranslate nohighlight">\(f(x) = y\)</span>. Notice that this statement includes both a universal and an existential quantifier, which explains why the next example makes use of both <code class="docutils literal notranslate"><span class="pre">intro</span></code> and <code class="docutils literal notranslate"><span class="pre">use</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">x</span> <span class="bp">+</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">use</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">c</span> <span class="n">dsimp</span><span class="bp">;</span> <span class="n">ring</span> </pre></div> </div> <p>Try this example yourself:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">c</span> <span class="bp">*</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-11">At this point, it is worth mentioning that there is a tactic, <cite>field_simp</cite>, that will often clear denominators in a useful way. It can be used in conjunction with the <code class="docutils literal notranslate"><span class="pre">ring</span></code> tactic.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">y</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">-</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">/</span> <span class="o">(</span><span class="n">x</span> <span class="bp">-</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">field_simp</span> <span class="o">[</span><span class="n">h</span><span class="o">]</span> <span class="n">ring</span> </pre></div> </div> <p>You can use the theorem <code class="docutils literal notranslate"><span class="pre">div_mul_cancel</span></code>. The next example uses a surjectivity hypothesis by applying it to a suitable value. Note that you can use <code class="docutils literal notranslate"><span class="pre">cases'</span></code> with any expression, not just a hypothesis.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">4</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases'</span> <span class="n">h</span> <span class="mi">2</span> <span class="k">with</span> <span class="n">x</span> <span class="n">hx</span> <span class="n">use</span> <span class="n">x</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hx</span><span class="o">]</span> <span class="n">norm_num</span> </pre></div> </div> <p>See if you can use these methods to show that the composition of surjective functions is surjective.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">γ</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">g</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">γ</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">}</span> <span class="kd">example</span> <span class="o">(</span><span class="n">surjg</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">g</span><span class="o">)</span> <span class="o">(</span><span class="n">surjf</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">g</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="negation"> <span id="id4"></span><h2><span class="section-number">3.3. </span>Negation<a class="headerlink" href="#negation" title="Permalink to this heading"></a></h2> <p>The symbol <code class="docutils literal notranslate"><span class="pre">¬</span></code> is meant to express negation, so <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">y</span></code> says that <code class="docutils literal notranslate"><span class="pre">x</span></code> is not less than <code class="docutils literal notranslate"><span class="pre">y</span></code>, <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">y</span></code> (or, equivalently, <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">≠</span> <span class="pre">y</span></code>) says that <code class="docutils literal notranslate"><span class="pre">x</span></code> is not equal to <code class="docutils literal notranslate"><span class="pre">y</span></code>, and <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">∃</span> <span class="pre">z,</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">z</span> <span class="pre">∧</span> <span class="pre">z</span> <span class="pre"><</span> <span class="pre">y</span></code> says that there does not exist a <code class="docutils literal notranslate"><span class="pre">z</span></code> strictly between <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code>. In Lean, the notation <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">A</span></code> abbreviates <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">→</span> <span class="pre">False</span></code>, which you can think of as saying that <code class="docutils literal notranslate"><span class="pre">A</span></code> implies a contradiction. Practically speaking, this means that you already know something about how to work with negations: you can prove <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">A</span></code> by introducing a hypothesis <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">A</span></code> and proving <code class="docutils literal notranslate"><span class="pre">False</span></code>, and if you have <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">¬</span> <span class="pre">A</span></code> and <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">A</span></code>, then applying <code class="docutils literal notranslate"><span class="pre">h</span></code> to <code class="docutils literal notranslate"><span class="pre">h'</span></code> yields <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p> <p>To illustrate, consider the irreflexivity principle <code class="docutils literal notranslate"><span class="pre">lt_irrefl</span></code> for a strict order, which says that we have <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">a</span> <span class="pre"><</span> <span class="pre">a</span></code> for every <code class="docutils literal notranslate"><span class="pre">a</span></code>. The asymmetry principle <code class="docutils literal notranslate"><span class="pre">lt_asymm</span></code> says that we have <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre"><</span> <span class="pre">b</span> <span class="pre">→</span> <span class="pre">¬</span> <span class="pre">b</span> <span class="pre"><</span> <span class="pre">a</span></code>. Let’s show that <code class="docutils literal notranslate"><span class="pre">lt_asymm</span></code> follows from <code class="docutils literal notranslate"><span class="pre">lt_irrefl</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">b</span> <span class="bp"><</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">h'</span> <span class="k">have</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">a</span> <span class="o">:=</span> <span class="n">lt_trans</span> <span class="n">h</span> <span class="n">h'</span> <span class="n">apply</span> <span class="n">lt_irrefl</span> <span class="n">a</span> <span class="n">this</span> </pre></div> </div> <p id="index-12">This example introduces a couple of new tricks. First, when you use <code class="docutils literal notranslate"><span class="pre">have</span></code> without providing a label, Lean uses the name <code class="docutils literal notranslate"><span class="pre">this</span></code>, providing a convenient way to refer back to it. Also, the <code class="docutils literal notranslate"><span class="pre">from</span></code> tactic is syntactic sugar for <code class="docutils literal notranslate"><span class="pre">exact</span></code>, providing a nice way to justify a <code class="docutils literal notranslate"><span class="pre">have</span></code> with an explicit proof term. But what you should really be paying attention to in this proof is the result of the <code class="docutils literal notranslate"><span class="pre">intro</span></code> tactic, which leaves a goal of <code class="docutils literal notranslate"><span class="pre">False</span></code>, and the fact that we eventually prove <code class="docutils literal notranslate"><span class="pre">False</span></code> by applying <code class="docutils literal notranslate"><span class="pre">lt_irrefl</span></code> to a proof of <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre"><</span> <span class="pre">a</span></code>.</p> <p>Here is another example, which uses the predicate <code class="docutils literal notranslate"><span class="pre">FnHasUb</span></code> defined in the last section, which says that a function has an upper bound.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">></span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">FnHasUb</span> <span class="n">f</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">fnub</span> <span class="n">cases'</span> <span class="n">fnub</span> <span class="k">with</span> <span class="n">a</span> <span class="n">fnuba</span> <span class="n">cases'</span> <span class="n">h</span> <span class="n">a</span> <span class="k">with</span> <span class="n">x</span> <span class="n">hx</span> <span class="k">have</span> <span class="o">:</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">a</span> <span class="o">:=</span> <span class="n">fnuba</span> <span class="n">x</span> <span class="n">linarith</span> </pre></div> </div> <p>See if you can prove these in a similar way:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">FnHasLb</span> <span class="n">f</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">x</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Mathlib offers a number of useful theorems for relating orders and negations:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">not_le_of_gt</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">></span> <span class="n">b</span> <span class="bp">→</span> <span class="bp">¬</span><span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">not_lt_of_ge</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≥</span> <span class="n">b</span> <span class="bp">→</span> <span class="bp">¬</span><span class="n">a</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_of_not_ge</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">a</span> <span class="bp">≥</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">le_of_not_gt</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">a</span> <span class="bp">></span> <span class="n">b</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> </pre></div> </div> <p>Recall the predicate <code class="docutils literal notranslate"><span class="pre">Monotone</span> <span class="pre">f</span></code>, which says that <code class="docutils literal notranslate"><span class="pre">f</span></code> is nondecreasing. Use some of the theorems just enumerated to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">f</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">f</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">f</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">Monotone</span> <span class="n">f</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Remember that it is often convenient to use <code class="docutils literal notranslate"><span class="pre">linarith</span></code> when a goal follows from linear equations and inequalities that in the context.</p> <p>We can show that the first example in the last snippet cannot be proved if we replace <code class="docutils literal notranslate"><span class="pre"><</span></code> by <code class="docutils literal notranslate"><span class="pre">≤</span></code>. Notice that we can prove the negation of a universally quantified statement by giving a counterexample. Complete the proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">¬∀</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">},</span> <span class="n">Monotone</span> <span class="n">f</span> <span class="bp">→</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span><span class="o">},</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">f</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">h</span> <span class="k">let</span> <span class="n">f</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">=></span> <span class="o">(</span><span class="mi">0</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="k">have</span> <span class="n">monof</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">h'</span> <span class="o">:</span> <span class="n">f</span> <span class="mi">1</span> <span class="bp">≤</span> <span class="n">f</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">le_refl</span> <span class="n">_</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-13">This example introduces the <code class="docutils literal notranslate"><span class="pre">let</span></code> tactic, which adds a <em>local definition</em> to the context. If you put the cursor after the <code class="docutils literal notranslate"><span class="pre">let</span></code> command, in the goal window you will see that the definition <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span> <span class="pre">:=</span> <span class="pre">fun</span> <span class="pre">x</span> <span class="pre">=></span> <span class="pre">0</span></code> has been added to the context. Lean will unfold the definition of <code class="docutils literal notranslate"><span class="pre">f</span></code> when it has to. In particular, when we prove <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">1</span> <span class="pre">≤</span> <span class="pre">f</span> <span class="pre">0</span></code> with <code class="docutils literal notranslate"><span class="pre">le_refl</span></code>, Lean reduces <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">1</span></code> and <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">0</span></code> to <code class="docutils literal notranslate"><span class="pre">0</span></code>.</p> <p>Use <code class="docutils literal notranslate"><span class="pre">le_of_not_gt</span></code> to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">ε</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Implicit in many of the proofs we have just done is the fact that if <code class="docutils literal notranslate"><span class="pre">P</span></code> is any property, saying that there is nothing with property <code class="docutils literal notranslate"><span class="pre">P</span></code> is the same as saying that everything fails to have property <code class="docutils literal notranslate"><span class="pre">P</span></code>, and saying that not everything has property <code class="docutils literal notranslate"><span class="pre">P</span></code> is equivalent to saying that something fails to have property <code class="docutils literal notranslate"><span class="pre">P</span></code>. In other words, all four of the following implications are valid (but one of them cannot be proved with what we explained so far):</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">P</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">(</span><span class="n">Q</span> <span class="o">:</span> <span class="kt">Prop</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">P</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="bp">¬</span><span class="n">P</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="bp">¬</span><span class="n">P</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">P</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">P</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="bp">¬</span><span class="n">P</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="bp">¬</span><span class="n">P</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">P</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>The first, second, and fourth are straightforward to prove using the methods you have already seen. We encourage you to try it. The third is more difficult, however, because it concludes that an object exists from the fact that its nonexistence is contradictory. This is an instance of <em>classical</em> mathematical reasoning. We can use proof by contradiction to prove the third implication as follows.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">P</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="bp">¬</span><span class="n">P</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_contra</span> <span class="n">h'</span> <span class="n">apply</span> <span class="n">h</span> <span class="n">intro</span> <span class="n">x</span> <span class="k">show</span> <span class="n">P</span> <span class="n">x</span> <span class="n">by_contra</span> <span class="n">h''</span> <span class="n">exact</span> <span class="n">h'</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">h''</span><span class="o">⟩</span> </pre></div> </div> <p id="index-14">Make sure you understand how this works. The <code class="docutils literal notranslate"><span class="pre">by_contra</span></code> tactic allows us to prove a goal <code class="docutils literal notranslate"><span class="pre">Q</span></code> by assuming <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">Q</span></code> and deriving a contradiction. In fact, it is equivalent to using the equivalence <code class="docutils literal notranslate"><span class="pre">not_not</span> <span class="pre">:</span> <span class="pre">¬</span> <span class="pre">¬</span> <span class="pre">Q</span> <span class="pre">↔</span> <span class="pre">Q</span></code>. Confirm that you can prove the forward direction of this equivalence using <code class="docutils literal notranslate"><span class="pre">by_contra</span></code>, while the reverse direction follows from the ordinary rules for negation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬¬</span><span class="n">Q</span><span class="o">)</span> <span class="o">:</span> <span class="n">Q</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Q</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬¬</span><span class="n">Q</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Use proof by contradiction to establish the following, which is the converse of one of the implications we proved above. (Hint: use <code class="docutils literal notranslate"><span class="pre">intro</span></code> first.)</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">></span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-15">It is often tedious to work with compound statements with a negation in front, and it is a common mathematical pattern to replace such statements with equivalent forms in which the negation has been pushed inward. To facilitate this, mathlib offers a <code class="docutils literal notranslate"><span class="pre">push_neg</span></code> tactic, which restates the goal in this way. The command <code class="docutils literal notranslate"><span class="pre">push_neg</span> <span class="pre">at</span> <span class="pre">h</span></code> restates the hypothesis <code class="docutils literal notranslate"><span class="pre">h</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬∀</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">></span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">h</span> <span class="n">exact</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">></span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">FnHasUb</span><span class="o">,</span> <span class="n">FnUb</span><span class="o">]</span> <span class="n">at</span> <span class="n">h</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">h</span> <span class="n">exact</span> <span class="n">h</span> </pre></div> </div> <p>In the second example, we use Lean’s simplifier to expand the definitions of <code class="docutils literal notranslate"><span class="pre">FnHasUb</span></code> and <code class="docutils literal notranslate"><span class="pre">FnUb</span></code>. (We need to use <code class="docutils literal notranslate"><span class="pre">simp</span></code> rather than <code class="docutils literal notranslate"><span class="pre">rw</span></code> to expand <code class="docutils literal notranslate"><span class="pre">FnUb</span></code>, because it appears in the scope of a quantifier.) You can verify that in the examples above with <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">∃</span> <span class="pre">x,</span> <span class="pre">P</span> <span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">∀</span> <span class="pre">x,</span> <span class="pre">P</span> <span class="pre">x</span></code>, the <code class="docutils literal notranslate"><span class="pre">push_neg</span></code> tactic does the expected thing. Without even knowing how to use the conjunction symbol, you should be able to use <code class="docutils literal notranslate"><span class="pre">push_neg</span></code> to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="n">y</span><span class="o">,</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">f</span> <span class="n">y</span> <span class="bp"><</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-16">Mathlib also has a tactic, <code class="docutils literal notranslate"><span class="pre">contrapose</span></code>, which transforms a goal <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">→</span> <span class="pre">B</span></code> to <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">B</span> <span class="pre">→</span> <span class="pre">¬</span> <span class="pre">A</span></code>. Similarly, given a goal of proving <code class="docutils literal notranslate"><span class="pre">B</span></code> from hypothesis <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">A</span></code>, <code class="docutils literal notranslate"><span class="pre">contrapose</span> <span class="pre">h</span></code> leaves you with a goal of proving <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">A</span></code> from hypothesis <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">B</span></code>. Using <code class="docutils literal notranslate"><span class="pre">contrapose!</span></code> instead of <code class="docutils literal notranslate"><span class="pre">contrapose</span></code> applies <code class="docutils literal notranslate"><span class="pre">push_neg</span></code> to the goal and the relevant hypothesis as well.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">></span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h</span> <span class="n">exact</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">ε</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h</span> <span class="n">use</span> <span class="n">x</span> <span class="bp">/</span> <span class="mi">2</span> <span class="n">constructor</span> <span class="bp"><;></span> <span class="n">linarith</span> </pre></div> </div> <p>We have not yet explained the <code class="docutils literal notranslate"><span class="pre">constructor</span></code> command or the use of the semicolon after it, but we will do that in the next section.</p> <p>We close this section with the principle of <em>ex falso</em>, which says that anything follows from a contradiction. In Lean, this is represented by <code class="docutils literal notranslate"><span class="pre">False.elim</span></code>, which establishes <code class="docutils literal notranslate"><span class="pre">False</span> <span class="pre">→</span> <span class="pre">P</span></code> for any proposition <code class="docutils literal notranslate"><span class="pre">P</span></code>. This may seem like a strange principle, but it comes up fairly often. We often prove a theorem by splitting on cases, and sometimes we can show that one of the cases is contradictory. In that case, we need to assert that the contradiction establishes the goal so we can move on to the next one. (We will see instances of reasoning by cases in <a class="reference internal" href="#disjunction"><span class="std std-numref">Section 3.5</span></a>.)</p> <p id="index-17">Lean provides a number of ways of closing a goal once a contradiction has been reached.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">></span> <span class="mi">37</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">exfalso</span> <span class="n">apply</span> <span class="n">lt_irrefl</span> <span class="mi">0</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">></span> <span class="mi">37</span> <span class="o">:=</span> <span class="n">absurd</span> <span class="n">h</span> <span class="o">(</span><span class="n">lt_irrefl</span> <span class="mi">0</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">></span> <span class="mi">37</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="n">h'</span> <span class="o">:</span> <span class="bp">¬</span><span class="mi">0</span> <span class="bp"><</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">lt_irrefl</span> <span class="mi">0</span> <span class="n">contradiction</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">exfalso</span></code> tactic replaces the current goal with the goal of proving <code class="docutils literal notranslate"><span class="pre">False</span></code>. Given <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">P</span></code> and <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">¬</span> <span class="pre">P</span></code>, the term <code class="docutils literal notranslate"><span class="pre">absurd</span> <span class="pre">h</span> <span class="pre">h'</span></code> establishes any proposition. Finally, the <code class="docutils literal notranslate"><span class="pre">contradiction</span></code> tactic tries to close a goal by finding a contradiction in the hypotheses, such as a pair of the form <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">P</span></code> and <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">¬</span> <span class="pre">P</span></code>. Of course, in this example, <code class="docutils literal notranslate"><span class="pre">linarith</span></code> also works.</p> </section> <section id="conjunction-and-bi-implication"> <span id="conjunction-and-biimplication"></span><h2><span class="section-number">3.4. </span>Conjunction and Bi-implication<a class="headerlink" href="#conjunction-and-bi-implication" title="Permalink to this heading"></a></h2> <p id="index-18">You have already seen that the conjunction symbol, <code class="docutils literal notranslate"><span class="pre">∧</span></code>, is used to express “and.” The <code class="docutils literal notranslate"><span class="pre">constructor</span></code> tactic allows you to prove a statement of the form <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">∧</span> <span class="pre">B</span></code> by proving <code class="docutils literal notranslate"><span class="pre">A</span></code> and then proving <code class="docutils literal notranslate"><span class="pre">B</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">assumption</span> <span class="n">intro</span> <span class="n">h</span> <span class="n">apply</span> <span class="n">h₁</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">]</span> </pre></div> </div> <p id="index-19">In this example, the <code class="docutils literal notranslate"><span class="pre">assumption</span></code> tactic tells Lean to find an assumption that will solve the goal. Notice that the final <code class="docutils literal notranslate"><span class="pre">rw</span></code> finishes the goal by applying the reflexivity of <code class="docutils literal notranslate"><span class="pre">≤</span></code>. The following are alternative ways of carrying out the previous examples using the anonymous constructor angle brackets. The first is a slick proof-term version of the previous proof, which drops into tactic mode at the keyword <code class="docutils literal notranslate"><span class="pre">by</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">h₀</span><span class="o">,</span> <span class="k">fun</span> <span class="n">h</span> <span class="bp">=></span> <span class="n">h₁</span> <span class="o">(</span><span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">])⟩</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="o">:=</span> <span class="k">have</span> <span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h₁</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h₁</span><span class="o">]</span> <span class="o">⟨</span><span class="n">h₀</span><span class="o">,</span> <span class="n">h</span><span class="o">⟩</span> </pre></div> </div> <p><em>Using</em> a conjunction instead of proving one involves unpacking the proofs of the two parts. You can use the <code class="docutils literal notranslate"><span class="pre">cases</span></code> tactic for that, as well as <code class="docutils literal notranslate"><span class="pre">rcases</span></code>, <code class="docutils literal notranslate"><span class="pre">rintro</span></code>, or a pattern-matching lambda, all in a manner similar to the way they are used with the existential quantifier.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases'</span> <span class="n">h</span> <span class="k">with</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h₁</span> <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="bp">→</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">h₀</span><span class="o">,</span> <span class="n">h₁</span><span class="o">⟩</span> <span class="n">h'</span> <span class="n">exact</span> <span class="n">h₁</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h₀</span> <span class="n">h'</span><span class="o">)</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="bp">→</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">h₀</span><span class="o">,</span> <span class="n">h₁</span><span class="o">⟩</span> <span class="n">h'</span> <span class="bp">=></span> <span class="n">h₁</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h₀</span> <span class="n">h'</span><span class="o">)</span> </pre></div> </div> <p>In contrast to using an existential quantifier, you can also extract proofs of the two components of a hypothesis <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">A</span> <span class="pre">∧</span> <span class="pre">B</span></code> by writing <code class="docutils literal notranslate"><span class="pre">h.left</span></code> and <code class="docutils literal notranslate"><span class="pre">h.right</span></code>, or, equivalently, <code class="docutils literal notranslate"><span class="pre">h.1</span></code> and <code class="docutils literal notranslate"><span class="pre">h.2</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">h'</span> <span class="n">apply</span> <span class="n">h.right</span> <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h.left</span> <span class="n">h'</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">h'</span> <span class="bp">=></span> <span class="n">h.right</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h.left</span> <span class="n">h'</span><span class="o">)</span> </pre></div> </div> <p>Try using these techniques to come up with various ways of proving of the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">m</span> <span class="bp">≠</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">n</span> <span class="bp">∧</span> <span class="bp">¬</span><span class="n">n</span> <span class="bp">∣</span> <span class="n">m</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>You can nest uses of <code class="docutils literal notranslate"><span class="pre">∃</span></code> and <code class="docutils literal notranslate"><span class="pre">∧</span></code> with anonymous constructors, <code class="docutils literal notranslate"><span class="pre">rintro</span></code>, and <code class="docutils literal notranslate"><span class="pre">rcases</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">2</span> <span class="bp"><</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp"><</span> <span class="mi">4</span> <span class="o">:=</span> <span class="o">⟨</span><span class="mi">5</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">,</span> <span class="kd">by</span> <span class="n">norm_num</span><span class="o">,</span> <span class="kd">by</span> <span class="n">norm_num</span><span class="o">⟩</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∃</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">z</span> <span class="bp">∧</span> <span class="n">z</span> <span class="bp"><</span> <span class="n">y</span><span class="o">)</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">z</span><span class="o">,</span> <span class="n">xltz</span><span class="o">,</span> <span class="n">zlty</span><span class="o">⟩</span> <span class="n">exact</span> <span class="n">lt_trans</span> <span class="n">xltz</span> <span class="n">zlty</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∃</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">z</span> <span class="bp">∧</span> <span class="n">z</span> <span class="bp"><</span> <span class="n">y</span><span class="o">)</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">z</span><span class="o">,</span> <span class="n">xltz</span><span class="o">,</span> <span class="n">zlty</span><span class="o">⟩</span> <span class="bp">=></span> <span class="n">lt_trans</span> <span class="n">xltz</span> <span class="n">zlty</span> </pre></div> </div> <p>You can also use the <code class="docutils literal notranslate"><span class="pre">use</span></code> tactic:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">2</span> <span class="bp"><</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp"><</span> <span class="mi">4</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">use</span> <span class="mi">5</span> <span class="bp">/</span> <span class="mi">2</span> <span class="n">constructor</span> <span class="bp"><;></span> <span class="n">norm_num</span> <span class="kd">example</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="mi">4</span> <span class="bp"><</span> <span class="n">m</span> <span class="bp">∧</span> <span class="n">m</span> <span class="bp"><</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">n</span> <span class="bp"><</span> <span class="mi">10</span> <span class="bp">∧</span> <span class="n">Nat.Prime</span> <span class="n">m</span> <span class="bp">∧</span> <span class="n">Nat.Prime</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">use</span> <span class="mi">5</span> <span class="n">use</span> <span class="mi">7</span> <span class="n">norm_num</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">h₀</span><span class="o">,</span> <span class="n">h₁</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">h₀</span> <span class="n">exact</span> <span class="k">fun</span> <span class="n">h'</span> <span class="bp">=></span> <span class="n">h₁</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h₀</span> <span class="n">h'</span><span class="o">)</span> </pre></div> </div> <p>In the first example, the semicolon after the <code class="docutils literal notranslate"><span class="pre">constructor</span></code> command tells Lean to use the <code class="docutils literal notranslate"><span class="pre">norm_num</span></code> tactic on both of the goals that result.</p> <p>In Lean, <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">↔</span> <span class="pre">B</span></code> is <em>not</em> defined to be <code class="docutils literal notranslate"><span class="pre">(A</span> <span class="pre">→</span> <span class="pre">B)</span> <span class="pre">∧</span> <span class="pre">(B</span> <span class="pre">→</span> <span class="pre">A)</span></code>, but it could have been, and it behaves roughly the same way. You have already seen that you can write <code class="docutils literal notranslate"><span class="pre">h.mp</span></code> and <code class="docutils literal notranslate"><span class="pre">h.mpr</span></code> or <code class="docutils literal notranslate"><span class="pre">h.1</span></code> and <code class="docutils literal notranslate"><span class="pre">h.2</span></code> for the two directions of <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">A</span> <span class="pre">↔</span> <span class="pre">B</span></code>. You can also use <code class="docutils literal notranslate"><span class="pre">cases</span></code> and friends. To prove an if-and-only-if statement, you can uses <code class="docutils literal notranslate"><span class="pre">constructor</span></code> or angle brackets, just as you would if you were proving a conjunction.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">rintro</span> <span class="n">rfl</span> <span class="n">rfl</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="o">:=</span> <span class="o">⟨</span><span class="k">fun</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="bp">=></span> <span class="n">h₀</span> <span class="o">(</span><span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h₁</span><span class="o">]),</span> <span class="k">fun</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="bp">=></span> <span class="n">h₀</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h</span> <span class="n">h₁</span><span class="o">)⟩</span> </pre></div> </div> <p>The last proof term is inscrutable. Remember that you can use underscores while writing an expression like that to see what Lean expects.</p> <p>Try out the various techniques and gadgets you have just seen in order to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="bp">¬</span><span class="n">y</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">≠</span> <span class="n">y</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>For a more interesting exercise, show that for any two real numbers <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code>, <code class="docutils literal notranslate"><span class="pre">x^2</span> <span class="pre">+</span> <span class="pre">y^2</span> <span class="pre">=</span> <span class="pre">0</span></code> if and only if <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">=</span> <span class="pre">0</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span> <span class="pre">=</span> <span class="pre">0</span></code>. We suggest proving an auxiliary lemma using <code class="docutils literal notranslate"><span class="pre">linarith</span></code>, <code class="docutils literal notranslate"><span class="pre">pow_two_nonneg</span></code>, and <code class="docutils literal notranslate"><span class="pre">pow_eq_zero</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">aux</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="k">have</span> <span class="n">h'</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">pow_eq_zero</span> <span class="n">h'</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">∧</span> <span class="n">y</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>In Lean, bi-implication leads a double-life. You can treat it like a conjunction and use its two parts separately. But Lean also knows that it is a reflexive, symmetric, and transitive relation between propositions, and you can also use it with <code class="docutils literal notranslate"><span class="pre">calc</span></code> and <code class="docutils literal notranslate"><span class="pre">rw</span></code>. It is often convenient to rewrite a statement to an equivalent one. In the next example, we use <code class="docutils literal notranslate"><span class="pre">abs_lt</span></code> to replace an expression of the form <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">y</span></code> by the equivalent expression <code class="docutils literal notranslate"><span class="pre">-</span> <span class="pre">y</span> <span class="pre"><</span> <span class="pre">x</span> <span class="pre">∧</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">y</span></code>, and in the one after that we use <code class="docutils literal notranslate"><span class="pre">Nat.dvd_gcd_iff</span></code> to replace an expression of the form <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">∣</span> <span class="pre">Nat.gcd</span> <span class="pre">n</span> <span class="pre">k</span></code> by the equivalent expression <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">∣</span> <span class="pre">n</span> <span class="pre">∧</span> <span class="pre">m</span> <span class="pre">∣</span> <span class="pre">k</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="mi">3</span><span class="o">)</span> <span class="bp"><</span> <span class="mi">5</span> <span class="bp">→</span> <span class="bp">-</span><span class="mi">8</span> <span class="bp"><</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp"><</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">abs_lt</span><span class="o">]</span> <span class="n">intro</span> <span class="n">h</span> <span class="n">constructor</span> <span class="bp"><;></span> <span class="n">linarith</span> <span class="kd">example</span> <span class="o">:</span> <span class="mi">3</span> <span class="bp">∣</span> <span class="n">Nat.gcd</span> <span class="mi">6</span> <span class="mi">15</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Nat.dvd_gcd_iff</span><span class="o">]</span> <span class="n">constructor</span> <span class="bp"><;></span> <span class="n">norm_num</span> </pre></div> </div> <p>See if you can use <code class="docutils literal notranslate"><span class="pre">rw</span></code> with the theorem below to provide a short proof that negation is not a nondecreasing function. (Note that <code class="docutils literal notranslate"><span class="pre">push_neg</span></code> won’t unfold definitions for you, so the <code class="docutils literal notranslate"><span class="pre">rw</span> <span class="pre">monotone</span></code> in the proof of the theorem is needed.)</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">not_monotone_iff</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">Monotone</span> <span class="n">f</span> <span class="bp">↔</span> <span class="bp">∃</span> <span class="n">x</span> <span class="n">y</span><span class="o">,</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">y</span> <span class="bp">∧</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">></span> <span class="n">f</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Monotone</span><span class="o">]</span> <span class="n">push_neg</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">=></span> <span class="bp">-</span><span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>The remaining exercises in this section are designed to give you some more practice with conjunction and bi-implication. Remember that a <em>partial order</em> is a binary relation that is transitive, reflexive, and antisymmetric. An even weaker notion sometimes arises: a <em>preorder</em> is just a reflexive, transitive relation. For any pre-order <code class="docutils literal notranslate"><span class="pre">≤</span></code>, Lean axiomatizes the associated strict pre-order by <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre"><</span> <span class="pre">b</span> <span class="pre">↔</span> <span class="pre">a</span> <span class="pre">≤</span> <span class="pre">b</span> <span class="pre">∧</span> <span class="pre">¬</span> <span class="pre">b</span> <span class="pre">≤</span> <span class="pre">a</span></code>. Show that if <code class="docutils literal notranslate"><span class="pre">≤</span></code> is a partial order, then <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre"><</span> <span class="pre">b</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">≤</span> <span class="pre">b</span> <span class="pre">∧</span> <span class="pre">a</span> <span class="pre">≠</span> <span class="pre">b</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">PartialOrder</span> <span class="n">α</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">↔</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">∧</span> <span class="n">a</span> <span class="bp">≠</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">lt_iff_le_not_le</span><span class="o">]</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-20">Beyond logical operations, you should not need anything more than <code class="docutils literal notranslate"><span class="pre">le_refl</span></code> and <code class="docutils literal notranslate"><span class="pre">le_antisymm</span></code>. Then show that even in the case where <code class="docutils literal notranslate"><span class="pre">≤</span></code> is only assumed to be a preorder, we can prove that the strict order is irreflexive and transitive. You do not need anything more than <code class="docutils literal notranslate"><span class="pre">le_refl</span></code> and <code class="docutils literal notranslate"><span class="pre">le_trans</span></code>. In the second example, for convenience, we use the simplifier rather than <code class="docutils literal notranslate"><span class="pre">rw</span></code> to express <code class="docutils literal notranslate"><span class="pre"><</span></code> in terms of <code class="docutils literal notranslate"><span class="pre">≤</span></code> and <code class="docutils literal notranslate"><span class="pre">¬</span></code>. We will come back to the simplifier later, but here we are only relying on the fact that it will use the indicated lemma repeatedly, even if it needs to be instantiated to different values.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Preorder</span> <span class="n">α</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">a</span> <span class="bp"><</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">lt_iff_le_not_le</span><span class="o">]</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">c</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">lt_iff_le_not_le</span><span class="o">]</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="disjunction"> <span id="id5"></span><h2><span class="section-number">3.5. </span>Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this heading"></a></h2> <p id="index-21">The canonical way to prove a disjunction <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">∨</span> <span class="pre">B</span></code> is to prove <code class="docutils literal notranslate"><span class="pre">A</span></code> or to prove <code class="docutils literal notranslate"><span class="pre">B</span></code>. The <code class="docutils literal notranslate"><span class="pre">left</span></code> tactic chooses <code class="docutils literal notranslate"><span class="pre">A</span></code>, and the <code class="docutils literal notranslate"><span class="pre">right</span></code> tactic chooses <code class="docutils literal notranslate"><span class="pre">B</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">></span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">></span> <span class="mi">0</span> <span class="bp">∨</span> <span class="n">y</span> <span class="bp"><</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">left</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">pow_two_nonneg</span> <span class="n">x</span><span class="o">]</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">-</span><span class="n">y</span> <span class="bp">></span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">></span> <span class="mi">0</span> <span class="bp">∨</span> <span class="n">y</span> <span class="bp"><</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">right</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">pow_two_nonneg</span> <span class="n">x</span><span class="o">]</span> </pre></div> </div> <p>We cannot use an anonymous constructor to construct a proof of an “or” because Lean would have to guess which disjunct we are trying to prove. When we write proof terms we can use <code class="docutils literal notranslate"><span class="pre">Or.inl</span></code> and <code class="docutils literal notranslate"><span class="pre">Or.inr</span></code> instead to make the choice explicitly. Here, <code class="docutils literal notranslate"><span class="pre">inl</span></code> is short for “introduction left” and <code class="docutils literal notranslate"><span class="pre">inr</span></code> is short for “introduction right.”</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">></span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">></span> <span class="mi">0</span> <span class="bp">∨</span> <span class="n">y</span> <span class="bp"><</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="n">Or.inl</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">y</span> <span class="bp"><</span> <span class="bp">-</span><span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">></span> <span class="mi">0</span> <span class="bp">∨</span> <span class="n">y</span> <span class="bp"><</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="n">Or.inr</span> <span class="n">h</span> </pre></div> </div> <p>It may seem strange to prove a disjunction by proving one side or the other. In practice, which case holds usually depends a case distinction that is implicit or explicit in the assumptions and the data. The <code class="docutils literal notranslate"><span class="pre">cases</span></code> tactic allows us to make use of a hypothesis of the form <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">∨</span> <span class="pre">B</span></code>. In contrast to the use of <code class="docutils literal notranslate"><span class="pre">cases</span></code> with conjunction or an existential quantifier, here the <code class="docutils literal notranslate"><span class="pre">cases</span></code> tactic produces <em>two</em> goals. Both have the same conclusion, but in the first case, <code class="docutils literal notranslate"><span class="pre">A</span></code> is assumed to be true, and in the second case, <code class="docutils literal notranslate"><span class="pre">B</span></code> is assumed to be true. In other words, as the name suggests, the <code class="docutils literal notranslate"><span class="pre">cases</span></code> tactic carries out a proof by cases. As usual, we can tell Lean what names to use for the hypotheses. In the next example, we tell Lean to use the name <code class="docutils literal notranslate"><span class="pre">h</span></code> on each branch.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">→</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="bp">∨</span> <span class="n">x</span> <span class="bp"><</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases'</span> <span class="n">le_or_gt</span> <span class="mi">0</span> <span class="n">y</span> <span class="k">with</span> <span class="n">h</span> <span class="n">h</span> <span class="bp">·</span> <span class="n">rw</span> <span class="o">[</span><span class="n">abs_of_nonneg</span> <span class="n">h</span><span class="o">]</span> <span class="n">intro</span> <span class="n">h</span> <span class="n">left</span> <span class="n">exact</span> <span class="n">h</span> <span class="n">rw</span> <span class="o">[</span><span class="n">abs_of_neg</span> <span class="n">h</span><span class="o">]</span> <span class="n">intro</span> <span class="n">h</span><span class="bp">;</span> <span class="n">right</span><span class="bp">;</span> <span class="n">exact</span> <span class="n">h</span> </pre></div> </div> <p>The absolute value function is defined in such a way that we can immediately prove that <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">≥</span> <span class="pre">0</span></code> implies <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">x</span></code> (this is the theorem <code class="docutils literal notranslate"><span class="pre">abs_of_nonneg</span></code>) and <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre"><</span> <span class="pre">0</span></code> implies <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">-x</span></code> (this is <code class="docutils literal notranslate"><span class="pre">abs_of_neg</span></code>). The expression <code class="docutils literal notranslate"><span class="pre">le_or_gt</span> <span class="pre">0</span> <span class="pre">x</span></code> establishes <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre">≤</span> <span class="pre">x</span> <span class="pre">∨</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">0</span></code>, allowing us to split on those two cases. Try proving the triangle inequality using the two first two theorems in the next snippet. They are given the same names they have in mathlib.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">namespace</span> <span class="n">MyAbs</span> <span class="kd">theorem</span> <span class="n">le_abs_self</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">abs</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">neg_le_abs_self</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="bp">-</span><span class="n">x</span> <span class="bp">≤</span> <span class="n">abs</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">abs_add</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="o">)</span> <span class="bp">≤</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">abs</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>In case you enjoyed these (pun intended) and you want more practice with disjunction, try these.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">lt_abs</span> <span class="o">:</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="bp">∨</span> <span class="n">x</span> <span class="bp"><</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">abs_lt</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="bp">↔</span> <span class="bp">-</span><span class="n">y</span> <span class="bp"><</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>You can also use <code class="docutils literal notranslate"><span class="pre">rcases</span></code> and <code class="docutils literal notranslate"><span class="pre">rintro</span></code> with disjunctions. When these result in a genuine case split with multiple goals, the patterns for each new goal are separated by a vertical bar.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp"><</span> <span class="mi">0</span> <span class="bp">∨</span> <span class="n">x</span> <span class="bp">></span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">lt_trichotomy</span> <span class="n">x</span> <span class="mi">0</span> <span class="k">with</span> <span class="o">(</span><span class="n">xlt</span> <span class="bp">|</span> <span class="n">xeq</span> <span class="bp">|</span> <span class="n">xgt</span><span class="o">)</span> <span class="bp">·</span> <span class="n">left</span> <span class="n">exact</span> <span class="n">xlt</span> <span class="bp">·</span> <span class="n">contradiction</span> <span class="n">right</span><span class="bp">;</span> <span class="n">exact</span> <span class="n">xgt</span> </pre></div> </div> <p>You can still nest patterns and use the <code class="docutils literal notranslate"><span class="pre">rfl</span></code> keyword to substitute equations:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="n">k</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">n</span> <span class="bp">∨</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">k</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">k</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">h</span> <span class="k">with</span> <span class="o">(⟨</span><span class="n">a</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩</span> <span class="bp">|</span> <span class="o">⟨</span><span class="n">b</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩)</span> <span class="bp">·</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_assoc</span><span class="o">]</span> <span class="n">apply</span> <span class="n">dvd_mul_right</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_comm</span><span class="o">,</span> <span class="n">mul_assoc</span><span class="o">]</span> <span class="n">apply</span> <span class="n">dvd_mul_right</span> </pre></div> </div> <p>See if you can prove the following with a single (long) line. Use <code class="docutils literal notranslate"><span class="pre">rcases</span></code> to unpack the hypotheses and split on cases, and use a semicolon and <code class="docutils literal notranslate"><span class="pre">linarith</span></code> to solve each branch.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="n">y</span><span class="o">,</span> <span class="n">z</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">∨</span> <span class="n">z</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">z</span> <span class="bp">≥</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>On the real numbers, an equation <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">y</span> <span class="pre">=</span> <span class="pre">0</span></code> tells us that <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">=</span> <span class="pre">0</span></code> or <code class="docutils literal notranslate"><span class="pre">y</span> <span class="pre">=</span> <span class="pre">0</span></code>. In mathlib, this fact is known as <code class="docutils literal notranslate"><span class="pre">eq_zero_or_eq_zero_of_mul_eq_zero</span></code>, and it is another nice example of how a disjunction can arise. See if you can use it to prove the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">1</span> <span class="bp">∨</span> <span class="n">x</span> <span class="bp">=</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">y</span> <span class="bp">∨</span> <span class="n">x</span> <span class="bp">=</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Remember that you can use the <code class="docutils literal notranslate"><span class="pre">ring</span></code> tactic to help with calculations.</p> <p>In an arbitrary ring <span class="math notranslate nohighlight">\(R\)</span>, an element <span class="math notranslate nohighlight">\(x\)</span> such that <span class="math notranslate nohighlight">\(x y = 0\)</span> for some nonzero <span class="math notranslate nohighlight">\(y\)</span> is called a <em>left zero divisor</em>, an element <span class="math notranslate nohighlight">\(x\)</span> such that <span class="math notranslate nohighlight">\(y x = 0\)</span> for some nonzero <span class="math notranslate nohighlight">\(y\)</span> is called a <em>right zero divisor</em>, and an element that is either a left or right zero divisor is called simply a <em>zero divisor</em>. The theorem <code class="docutils literal notranslate"><span class="pre">eq_zero_or_eq_zero_of_mul_eq_zero</span></code> says that the real numbers have no nontrivial zero divisors. A commutative ring with this property is called an <em>integral domain</em>. Your proofs of the two theorems above should work equally well in any integral domain:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">CommRing</span> <span class="n">R</span><span class="o">]</span> <span class="o">[</span><span class="n">IsDomain</span> <span class="n">R</span><span class="o">]</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">1</span> <span class="bp">∨</span> <span class="n">x</span> <span class="bp">=</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">y</span> <span class="bp">∨</span> <span class="n">x</span> <span class="bp">=</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>In fact, if you are careful, you can prove the first theorem without using commutativity of multiplication. In that case, it suffices to assume that <code class="docutils literal notranslate"><span class="pre">R</span></code> is a <code class="docutils literal notranslate"><span class="pre">Ring</span></code> instead of an <code class="docutils literal notranslate"><span class="pre">CommRing</span></code>.</p> <p id="index-22">Sometimes in a proof we want to split on cases depending on whether some statement is true or not. For any proposition <code class="docutils literal notranslate"><span class="pre">P</span></code>, we can use <code class="docutils literal notranslate"><span class="pre">em</span> <span class="pre">P</span> <span class="pre">:</span> <span class="pre">P</span> <span class="pre">∨</span> <span class="pre">¬</span> <span class="pre">P</span></code>. The name <code class="docutils literal notranslate"><span class="pre">em</span></code> is short for “excluded middle.”</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">P</span> <span class="o">:</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬¬</span><span class="n">P</span> <span class="bp">→</span> <span class="n">P</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">h</span> <span class="n">cases</span> <span class="n">em</span> <span class="n">P</span> <span class="bp">·</span> <span class="n">assumption</span> <span class="n">contradiction</span> </pre></div> </div> <p id="index-23">Alternatively, you can use the <code class="docutils literal notranslate"><span class="pre">by_cases</span></code> tactic.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">P</span> <span class="o">:</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬¬</span><span class="n">P</span> <span class="bp">→</span> <span class="n">P</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">h</span> <span class="n">by_cases</span> <span class="n">h'</span> <span class="o">:</span> <span class="n">P</span> <span class="bp">·</span> <span class="n">assumption</span> <span class="n">contradiction</span> </pre></div> </div> <p>Notice that the <code class="docutils literal notranslate"><span class="pre">by_cases</span></code> tactic lets you specify a label for the hypothesis that is introduced in each branch, in this case, <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">P</span></code> in one and <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">¬</span> <span class="pre">P</span></code> in the other. If you leave out the label, Lean uses <code class="docutils literal notranslate"><span class="pre">h</span></code> by default. Try proving the following equivalence, using <code class="docutils literal notranslate"><span class="pre">by_cases</span></code> to establish one direction.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">P</span> <span class="n">Q</span> <span class="o">:</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">:</span> <span class="n">P</span> <span class="bp">→</span> <span class="n">Q</span> <span class="bp">↔</span> <span class="bp">¬</span><span class="n">P</span> <span class="bp">∨</span> <span class="n">Q</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="sequences-and-convergence"> <span id="id6"></span><h2><span class="section-number">3.6. </span>Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this heading"></a></h2> <p>We now have enough skills at our disposal to do some real mathematics. In Lean, we can represent a sequence <span class="math notranslate nohighlight">\(s_0, s_1, s_2, \ldots\)</span> of real numbers as a function <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">ℕ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code>. Such a sequence is said to <em>converge</em> to a number <span class="math notranslate nohighlight">\(a\)</span> if for every <span class="math notranslate nohighlight">\(\varepsilon > 0\)</span> there is a point beyond which the sequence remains within <span class="math notranslate nohighlight">\(\varepsilon\)</span> of <span class="math notranslate nohighlight">\(a\)</span>, that is, there is a number <span class="math notranslate nohighlight">\(N\)</span> such that for every <span class="math notranslate nohighlight">\(n \ge N\)</span>, <span class="math notranslate nohighlight">\(| s_n - a | < \varepsilon\)</span>. In Lean, we can render this as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">N</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">n</span> <span class="bp">≥</span> <span class="n">N</span><span class="o">,</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">n</span> <span class="bp">-</span> <span class="n">a</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> </pre></div> </div> <p>The notation <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">ε</span> <span class="pre">></span> <span class="pre">0,</span> <span class="pre">...</span></code> is a convenient abbreviation for <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">ε,</span> <span class="pre">ε</span> <span class="pre">></span> <span class="pre">0</span> <span class="pre">→</span> <span class="pre">...</span></code>, and, similarly, <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">n</span> <span class="pre">≥</span> <span class="pre">N,</span> <span class="pre">...</span></code> abbreviates <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">n,</span> <span class="pre">n</span> <span class="pre">≥</span> <span class="pre">N</span> <span class="pre">→</span>  <span class="pre">...</span></code>. And remember that <code class="docutils literal notranslate"><span class="pre">ε</span> <span class="pre">></span> <span class="pre">0</span></code>, in turn, is defined as <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">ε</span></code>, and <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">≥</span> <span class="pre">N</span></code> is defined as <code class="docutils literal notranslate"><span class="pre">N</span> <span class="pre">≤</span> <span class="pre">n</span></code>.</p> <p id="index-24">In this section, we’ll establish some properties of convergence. But first, we will discuss three tactics for working with equality that will prove useful. The first, the <code class="docutils literal notranslate"><span class="pre">ext</span></code> tactic, gives us a way of proving that two functions are equal. Let <span class="math notranslate nohighlight">\(f(x) = x + 1\)</span> and <span class="math notranslate nohighlight">\(g(x) = 1 + x\)</span> be functions from reals to reals. Then, of course, <span class="math notranslate nohighlight">\(f = g\)</span>, because they return the same value for every <span class="math notranslate nohighlight">\(x\)</span>. The <code class="docutils literal notranslate"><span class="pre">ext</span></code> tactic enables us to prove an equation between functions by proving that their values are the same at all the values of their arguments.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">=></span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="o">)</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">=></span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">y</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">ring</span> </pre></div> </div> <p id="index-25">We’ll see later that <code class="docutils literal notranslate"><span class="pre">ext</span></code> is actually more general, and also one can specify the name of the variables that appear. For instance you can try to replace <code class="docutils literal notranslate"><span class="pre">ext</span></code> with <code class="docutils literal notranslate"><span class="pre">ext</span> <span class="pre">u</span> <span class="pre">v</span></code> in the above proof. The second tactic, the <code class="docutils literal notranslate"><span class="pre">congr</span></code> tactic, allows us to prove an equation between two expressions by reconciling the parts that are different:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">congr</span> <span class="n">ring</span> </pre></div> </div> <p>Here the <code class="docutils literal notranslate"><span class="pre">congr</span></code> tactic peels off the <code class="docutils literal notranslate"><span class="pre">abs</span></code> on each side, leaving us to prove <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">-</span> <span class="pre">b</span> <span class="pre">+</span> <span class="pre">b</span></code>.</p> <p id="index-26">Finally, the <code class="docutils literal notranslate"><span class="pre">convert</span></code> tactic is used to apply a theorem to a goal when the conclusion of the theorem doesn’t quite match. For example, suppose we want to prove <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre"><</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">a</span></code> from <code class="docutils literal notranslate"><span class="pre">1</span> <span class="pre"><</span> <span class="pre">a</span></code>. A theorem in the library, <code class="docutils literal notranslate"><span class="pre">mul_lt_mul_right</span></code>, will let us prove <code class="docutils literal notranslate"><span class="pre">1</span> <span class="pre">*</span> <span class="pre">a</span> <span class="pre"><</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">a</span></code>. One possibility is to work backwards and rewrite the goal so that it has that form. Instead, the <code class="docutils literal notranslate"><span class="pre">convert</span></code> tactic lets us apply the theorem as it is, and leaves us with the task of proving the equations that are needed to make the goal match.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">1</span> <span class="bp"><</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">convert</span><span class="o">(</span><span class="n">mul_lt_mul_right</span> <span class="n">_</span><span class="o">)</span><span class="bp">.</span><span class="mi">2</span> <span class="n">h</span> <span class="bp">·</span> <span class="n">rw</span> <span class="o">[</span><span class="n">one_mul</span><span class="o">]</span> <span class="n">exact</span> <span class="n">lt_trans</span> <span class="n">zero_lt_one</span> <span class="n">h</span> </pre></div> </div> <p>This example illustrates another useful trick: when we apply an expression with an underscore and Lean can’t fill it in for us automatically, it simply leaves it for us as another goal.</p> <p>The following shows that any constant sequence <span class="math notranslate nohighlight">\(a, a, a, \ldots\)</span> converges.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_const</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">=></span> <span class="n">a</span><span class="o">)</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">ε</span> <span class="n">εpos</span> <span class="n">use</span> <span class="mi">0</span> <span class="n">intro</span> <span class="n">n</span> <span class="n">nge</span><span class="bp">;</span> <span class="n">dsimp</span> <span class="n">rw</span> <span class="o">[</span><span class="n">sub_self</span><span class="o">,</span> <span class="n">abs_zero</span><span class="o">]</span> <span class="n">apply</span> <span class="n">εpos</span> </pre></div> </div> <p>Lean has a tactic, <code class="docutils literal notranslate"><span class="pre">simp</span></code>, which can often save you the trouble of carrying out steps like <code class="docutils literal notranslate"><span class="pre">rw</span> <span class="pre">[sub_self,</span> <span class="pre">abs_zero]</span></code> by hand. We will tell you more about it soon.</p> <p>For a more interesting theorem, let’s show that if <code class="docutils literal notranslate"><span class="pre">s</span></code> converges to <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span></code> converges to <code class="docutils literal notranslate"><span class="pre">b</span></code>, then <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">n</span> <span class="pre">↦</span> <span class="pre">s</span> <span class="pre">n</span> <span class="pre">+</span> <span class="pre">t</span> <span class="pre">n</span></code> converges to <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code>. It is helpful to have a clear pen-and-paper proof in mind before you start writing a formal one. Given <code class="docutils literal notranslate"><span class="pre">ε</span></code> greater than <code class="docutils literal notranslate"><span class="pre">0</span></code>, the idea is to use the hypotheses to obtain an <code class="docutils literal notranslate"><span class="pre">Ns</span></code> such that beyond that point, <code class="docutils literal notranslate"><span class="pre">s</span></code> is within <code class="docutils literal notranslate"><span class="pre">ε</span> <span class="pre">/</span> <span class="pre">2</span></code> of <code class="docutils literal notranslate"><span class="pre">a</span></code>, and an <code class="docutils literal notranslate"><span class="pre">Nt</span></code> such that beyond that point, <code class="docutils literal notranslate"><span class="pre">t</span></code> is within <code class="docutils literal notranslate"><span class="pre">ε</span> <span class="pre">/</span> <span class="pre">2</span></code> of <code class="docutils literal notranslate"><span class="pre">b</span></code>. Then, whenever <code class="docutils literal notranslate"><span class="pre">n</span></code> is greater than or equal to the maximum of <code class="docutils literal notranslate"><span class="pre">Ns</span></code> and <code class="docutils literal notranslate"><span class="pre">Nt</span></code>, the sequence <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">n</span> <span class="pre">↦</span> <span class="pre">s</span> <span class="pre">n</span> <span class="pre">+</span> <span class="pre">t</span> <span class="pre">n</span></code> should be within <code class="docutils literal notranslate"><span class="pre">ε</span></code> of <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code>. The following example begins to implement this strategy. See if you can finish it off.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_add</span> <span class="o">{</span><span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">ct</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">t</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="n">s</span> <span class="n">n</span> <span class="bp">+</span> <span class="n">t</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">ε</span> <span class="n">εpos</span> <span class="n">dsimp</span> <span class="k">have</span> <span class="n">ε2pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">linarith</span> <span class="n">cases'</span> <span class="n">cs</span> <span class="o">(</span><span class="n">ε</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">ε2pos</span> <span class="k">with</span> <span class="n">Ns</span> <span class="n">hs</span> <span class="n">cases'</span> <span class="n">ct</span> <span class="o">(</span><span class="n">ε</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">ε2pos</span> <span class="k">with</span> <span class="n">Nt</span> <span class="n">ht</span> <span class="n">use</span> <span class="n">max</span> <span class="n">Ns</span> <span class="n">Nt</span> <span class="gr">sorry</span> </pre></div> </div> <p>As hints, you can use <code class="docutils literal notranslate"><span class="pre">le_of_max_le_left</span></code> and <code class="docutils literal notranslate"><span class="pre">le_of_max_le_right</span></code>, and <code class="docutils literal notranslate"><span class="pre">norm_num</span></code> can prove <code class="docutils literal notranslate"><span class="pre">ε</span> <span class="pre">/</span> <span class="pre">2</span> <span class="pre">+</span> <span class="pre">ε</span> <span class="pre">/</span> <span class="pre">2</span> <span class="pre">=</span> <span class="pre">ε</span></code>. Also, it is helpful to use the <code class="docutils literal notranslate"><span class="pre">congr</span></code> tactic to show that <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">(s</span> <span class="pre">n</span> <span class="pre">+</span> <span class="pre">t</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">(a</span> <span class="pre">+</span> <span class="pre">b))</span></code> is equal to <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">((s</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">a)</span> <span class="pre">+</span> <span class="pre">(t</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">b)),</span></code> since then you can use the triangle inequality. Notice that we marked all the variables <code class="docutils literal notranslate"><span class="pre">s</span></code>, <code class="docutils literal notranslate"><span class="pre">t</span></code>, <code class="docutils literal notranslate"><span class="pre">a</span></code>, and <code class="docutils literal notranslate"><span class="pre">b</span></code> implicit because they can be inferred from the hypotheses.</p> <p>Proving the same theorem with multiplication in place of addition is tricky. We will get there by proving some auxiliary statements first. See if you can also finish off the next proof, which shows that if <code class="docutils literal notranslate"><span class="pre">s</span></code> converges to <code class="docutils literal notranslate"><span class="pre">a</span></code>, then <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">n</span> <span class="pre">=></span> <span class="pre">c</span> <span class="pre">*</span> <span class="pre">s</span> <span class="pre">n</span></code> converges to <code class="docutils literal notranslate"><span class="pre">c</span> <span class="pre">*</span> <span class="pre">a</span></code>. It is helpful to split into cases depending on whether <code class="docutils literal notranslate"><span class="pre">c</span></code> is equal to zero or not. We have taken care of the zero case, and we have left you to prove the result with the extra assumption that <code class="docutils literal notranslate"><span class="pre">c</span></code> is nonzero.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_mul_const</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="n">c</span> <span class="bp">*</span> <span class="n">s</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">c</span> <span class="bp">*</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_cases</span> <span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">·</span> <span class="n">convert</span> <span class="n">convergesTo_const</span> <span class="mi">0</span> <span class="bp">·</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">,</span> <span class="n">MulZeroClass.zero_mul</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">,</span> <span class="n">MulZeroClass.zero_mul</span><span class="o">]</span> <span class="k">have</span> <span class="n">acpos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">abs</span> <span class="n">c</span> <span class="o">:=</span> <span class="n">abs_pos.mpr</span> <span class="n">h</span> <span class="gr">sorry</span> </pre></div> </div> <p>The next theorem is also independently interesting: it shows that a convergent sequence is eventually bounded in absolute value. We have started you off; see if you can finish it.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">exists_abs_le_of_convergesTo</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">N</span> <span class="n">b</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">N</span> <span class="bp">≤</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">n</span><span class="o">)</span> <span class="bp"><</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases'</span> <span class="n">cs</span> <span class="mi">1</span> <span class="n">zero_lt_one</span> <span class="k">with</span> <span class="n">N</span> <span class="n">h</span> <span class="n">use</span> <span class="n">N</span><span class="o">,</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">1</span> <span class="gr">sorry</span> </pre></div> </div> <p>In fact, the theorem could be strengthened to assert that there is a bound <code class="docutils literal notranslate"><span class="pre">b</span></code> that holds for all values of <code class="docutils literal notranslate"><span class="pre">n</span></code>. But this version is strong enough for our purposes, and we will see at the end of this section that it holds more generally.</p> <p>The next lemma is auxiliary: we prove that if <code class="docutils literal notranslate"><span class="pre">s</span></code> converges to <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span></code> converges to <code class="docutils literal notranslate"><span class="pre">0</span></code>, then <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">n</span> <span class="pre">=></span> <span class="pre">s</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">t</span> <span class="pre">n</span></code> converges to <code class="docutils literal notranslate"><span class="pre">0</span></code>. To do so, we use the previous theorem to find a <code class="docutils literal notranslate"><span class="pre">B</span></code> that bounds <code class="docutils literal notranslate"><span class="pre">s</span></code> beyond some point <code class="docutils literal notranslate"><span class="pre">N₀</span></code>. See if you can understand the strategy we have outlined and finish the proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">aux</span> <span class="o">{</span><span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">ct</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">t</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">t</span> <span class="n">n</span><span class="o">)</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">ε</span> <span class="n">εpos</span> <span class="n">dsimp</span> <span class="n">rcases</span> <span class="n">exists_abs_le_of_convergesTo</span> <span class="n">cs</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">N₀</span><span class="o">,</span> <span class="n">B</span><span class="o">,</span> <span class="n">h₀</span><span class="o">⟩</span> <span class="k">have</span> <span class="n">Bpos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">B</span> <span class="o">:=</span> <span class="n">lt_of_le_of_lt</span> <span class="o">(</span><span class="n">abs_nonneg</span> <span class="n">_</span><span class="o">)</span> <span class="o">(</span><span class="n">h₀</span> <span class="n">N₀</span> <span class="o">(</span><span class="n">le_refl</span> <span class="n">_</span><span class="o">))</span> <span class="k">have</span> <span class="n">pos₀</span> <span class="o">:</span> <span class="n">ε</span> <span class="bp">/</span> <span class="n">B</span> <span class="bp">></span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">div_pos</span> <span class="n">εpos</span> <span class="n">Bpos</span> <span class="n">cases'</span> <span class="n">ct</span> <span class="n">_</span> <span class="n">pos₀</span> <span class="k">with</span> <span class="n">N₁</span> <span class="n">h₁</span> <span class="gr">sorry</span> </pre></div> </div> <p>If you have made it this far, congratulations! We are now within striking distance of our theorem. The following proof finishes it off.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_mul</span> <span class="o">{</span><span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">ct</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">t</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">t</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="n">h₁</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="o">(</span><span class="n">t</span> <span class="n">n</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">b</span><span class="o">))</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">aux</span> <span class="n">cs</span> <span class="n">convert</span> <span class="n">convergesTo_add</span> <span class="n">ct</span> <span class="o">(</span><span class="n">convergesTo_const</span> <span class="o">(</span><span class="bp">-</span><span class="n">b</span><span class="o">))</span> <span class="n">ring</span> <span class="k">have</span> <span class="o">:=</span> <span class="n">convergesTo_add</span> <span class="n">h₁</span> <span class="o">(</span><span class="n">convergesTo_mul_const</span> <span class="n">b</span> <span class="n">cs</span><span class="o">)</span> <span class="n">convert</span> <span class="n">convergesTo_add</span> <span class="n">h₁</span> <span class="o">(</span><span class="n">convergesTo_mul_const</span> <span class="n">b</span> <span class="n">cs</span><span class="o">)</span> <span class="n">using</span> <span class="mi">1</span> <span class="bp">·</span> <span class="n">ext</span><span class="bp">;</span> <span class="n">ring</span> <span class="n">ring</span> </pre></div> </div> <p>For another challenging exercise, try filling out the following sketch of a proof that limits are unique. (If you are feeling bold, you can delete the proof sketch and try proving it from scratch.)</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_unique</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">sa</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">sb</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_contra</span> <span class="n">abne</span> <span class="k">have</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">></span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">let</span> <span class="n">ε</span> <span class="o">:=</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="k">have</span> <span class="n">εpos</span> <span class="o">:</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">change</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">></span> <span class="mi">0</span> <span class="n">linarith</span> <span class="n">cases'</span> <span class="n">sa</span> <span class="n">ε</span> <span class="n">εpos</span> <span class="k">with</span> <span class="n">Na</span> <span class="n">hNa</span> <span class="n">cases'</span> <span class="n">sb</span> <span class="n">ε</span> <span class="n">εpos</span> <span class="k">with</span> <span class="n">Nb</span> <span class="n">hNb</span> <span class="k">let</span> <span class="n">N</span> <span class="o">:=</span> <span class="n">max</span> <span class="n">Na</span> <span class="n">Nb</span> <span class="k">have</span> <span class="n">absa</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">N</span> <span class="bp">-</span> <span class="n">a</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">absb</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">N</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp"><</span> <span class="n">abs</span> <span class="o">(</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">exact</span> <span class="n">lt_irrefl</span> <span class="n">_</span> <span class="n">this</span> </pre></div> </div> <p>We close the section with the observation that our proofs can be generalized. For example, the only properties that we have used of the natural numbers is that their structure carries a partial order with <code class="docutils literal notranslate"><span class="pre">min</span></code> and <code class="docutils literal notranslate"><span class="pre">max</span></code>. You can check that everything still works if you replace <code class="docutils literal notranslate"><span class="pre">ℕ</span></code> everywhere by any linear order <code class="docutils literal notranslate"><span class="pre">α</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">LinearOrder</span> <span class="n">α</span><span class="o">]</span> <span class="kd">def</span> <span class="n">ConvergesTo'</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">N</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">n</span> <span class="bp">≥</span> <span class="n">N</span><span class="o">,</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">n</span> <span class="bp">-</span> <span class="n">a</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> </pre></div> </div> <p>In <a class="reference internal" href="C08_Topology.html#filters"><span class="std std-numref">Section 8.1</span></a>, we will see that mathlib has mechanisms for dealing with convergence in vastly more general terms, not only abstracting away particular features of the domain and codomain, but also abstracting over different types of convergence.</p> </section> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C02_Basics.html" class="btn btn-neutral float-left" title="2. 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Sets and Functions — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> <link rel="shortcut icon" href="_static/favicon.ico"/> <!--[if lt IE 9]> <script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" /> <link rel="next" title="5. Number Theory" href="C05_Number_Theory.html" /> <link rel="prev" title="3. Logic" href="C03_Logic.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul class="current"> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">4. Sets and Functions</a><ul> <li class="toctree-l2"><a class="reference internal" href="#sets">4.1. Sets</a></li> <li class="toctree-l2"><a class="reference internal" href="#functions">4.2. Functions</a></li> <li class="toctree-l2"><a class="reference internal" href="#the-schroder-bernstein-theorem">4.3. The Schröder-Bernstein Theorem</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">4. </span>Sets and Functions</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C04_Sets_and_Functions.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="sets-and-functions"> <span id="id1"></span><h1><span class="section-number">4. </span>Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this heading"></a></h1> <p>The vocabulary of sets, relations, and functions provides a uniform language for carrying out constructions in all the branches of mathematics. Since functions and relations can be defined in terms of sets, axiomatic set theory can be used as a foundation for mathematics.</p> <p>Lean’s foundation is based instead on the primitive notion of a <em>type</em>, and it includes ways of defining functions between types. Every expression in Lean has a type: there are natural numbers, real numbers, functions from reals to reals, groups, vector spaces, and so on. Some expressions <em>are</em> types, which is to say, their type is <code class="docutils literal notranslate"><span class="pre">Type</span></code>. Lean and mathlib provide ways of defining new types, and ways of defining objects of those types.</p> <p>Conceptually, you can think of a type as just a set of objects. Requiring every object to have a type has some advantages. For example, it makes it possible to overload notation like <code class="docutils literal notranslate"><span class="pre">+</span></code>, and it sometimes makes input less verbose because Lean can infer a lot of information from an object’s type. The type system also enables Lean to flag errors when you apply a function to the wrong number of arguments, or apply a function to arguments of the wrong type.</p> <p>Lean’s library does define elementary set-theoretic notions. In contrast to set theory, in Lean a set is always a set of objects of some type, such as a set natural numbers or a set of functions from real numbers to real numbers. The distinction between types and set takes some getting used to, but this chapter will take you through the essentials.</p> <section id="sets"> <span id="id2"></span><h2><span class="section-number">4.1. </span>Sets<a class="headerlink" href="#sets" title="Permalink to this heading"></a></h2> <p id="index-0">If <code class="docutils literal notranslate"><span class="pre">α</span></code> is any type, the type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">α</span></code> consists of sets of elements of <code class="docutils literal notranslate"><span class="pre">α</span></code>. This type supports the usual set-theoretic operations and relations. For example, <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">⊆</span> <span class="pre">t</span></code> says that <code class="docutils literal notranslate"><span class="pre">s</span></code> is a subset of <code class="docutils literal notranslate"><span class="pre">t</span></code>, <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">∩</span> <span class="pre">t</span></code> denotes the intersection of <code class="docutils literal notranslate"><span class="pre">s</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span></code>, and <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">∪</span> <span class="pre">t</span></code> denotes their union. The subset relation can be typed with <code class="docutils literal notranslate"><span class="pre">\ss</span></code> or <code class="docutils literal notranslate"><span class="pre">\sub</span></code>, intersection can be typed with <code class="docutils literal notranslate"><span class="pre">\i</span></code> or <code class="docutils literal notranslate"><span class="pre">\cap</span></code>, and union can be typed with <code class="docutils literal notranslate"><span class="pre">\un</span></code> or <code class="docutils literal notranslate"><span class="pre">\cup</span></code>. The library also defines the set <code class="docutils literal notranslate"><span class="pre">univ</span></code>, which consists of all the elements of type <code class="docutils literal notranslate"><span class="pre">α</span></code>, and the empty set, <code class="docutils literal notranslate"><span class="pre">∅</span></code>, which can be typed as <code class="docutils literal notranslate"><span class="pre">\empty</span></code>. Given <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">α</span></code> and <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">α</span></code>, the expression <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span></code> says that <code class="docutils literal notranslate"><span class="pre">x</span></code> is a member of <code class="docutils literal notranslate"><span class="pre">s</span></code>. Theorems that mention set membership often include <code class="docutils literal notranslate"><span class="pre">mem</span></code> in their name. The expression <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∉</span> <span class="pre">s</span></code> abbreviates <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span></code>. You can type <code class="docutils literal notranslate"><span class="pre">∈</span></code> as <code class="docutils literal notranslate"><span class="pre">\in</span></code> or <code class="docutils literal notranslate"><span class="pre">\mem</span></code> and <code class="docutils literal notranslate"><span class="pre">∉</span></code> as <code class="docutils literal notranslate"><span class="pre">\notin</span></code>.</p> <p id="index-1">One way to prove things about sets is to use <code class="docutils literal notranslate"><span class="pre">rw</span></code> or the simplifier to expand the definitions. In the second example below, we use <code class="docutils literal notranslate"><span class="pre">simp</span> <span class="pre">only</span></code> to tell the simplifier to use only the list of identities we give it, and not its full database of identities. Unlike <code class="docutils literal notranslate"><span class="pre">rw</span></code>, <code class="docutils literal notranslate"><span class="pre">simp</span></code> can perform simplifications inside a universal or existential quantifier. If you step through the proof, you can see the effects of these commands.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">s</span> <span class="n">t</span> <span class="n">u</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="kn">open</span> <span class="n">Set</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">subset_def</span><span class="o">,</span> <span class="n">inter_def</span><span class="o">,</span> <span class="n">inter_def</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">subset_def</span><span class="o">]</span> <span class="n">at</span> <span class="n">h</span> <span class="n">dsimp</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">h</span> <span class="n">_</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">subset_def</span><span class="o">,</span> <span class="n">mem_inter_iff</span><span class="o">]</span> <span class="n">at</span> <span class="bp">*</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">h</span> <span class="n">_</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> </pre></div> </div> <p>In this example, we open the <code class="docutils literal notranslate"><span class="pre">set</span></code> namespace to have access to the shorter names for the theorems. But, in fact, we can delete the calls to <code class="docutils literal notranslate"><span class="pre">rw</span></code> and <code class="docutils literal notranslate"><span class="pre">simp</span></code> entirely:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">xsu</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">h</span> <span class="n">xsu.1</span><span class="o">,</span> <span class="n">xsu.2</span><span class="o">⟩</span> </pre></div> </div> <p>What is going on here is known as <em>definitional reduction</em>: to make sense of the <code class="docutils literal notranslate"><span class="pre">intro</span></code> command and the anonymous constructors Lean is forced to expand the definitions. The following examples also illustrate the phenomenon:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">foo</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">u</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">h</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">u</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">h</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> </pre></div> </div> <p>Due to a quirk of how Lean processes its input, the first example fails if we replace <code class="docutils literal notranslate"><span class="pre">theorem</span> <span class="pre">foo</span></code> with <code class="docutils literal notranslate"><span class="pre">example</span></code>. This illustrates the pitfalls of relying on definitional reduction too heavily. It is often convenient, but sometimes we have to fall back on unfolding definitions manually.</p> <p>To deal with unions, we can use <code class="docutils literal notranslate"><span class="pre">Set.union_def</span></code> and <code class="docutils literal notranslate"><span class="pre">Set.mem_union</span></code>. Since <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">∪</span> <span class="pre">t</span></code> unfolds to <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">∨</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">t</span></code>, we can also use the <code class="docutils literal notranslate"><span class="pre">cases</span></code> tactic to force a definitional reduction.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">t</span> <span class="bp">∪</span> <span class="n">u</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">∪</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">hx</span> <span class="k">have</span> <span class="n">xs</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">hx.1</span> <span class="k">have</span> <span class="n">xtu</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">t</span> <span class="bp">∪</span> <span class="n">u</span> <span class="o">:=</span> <span class="n">hx.2</span> <span class="n">cases'</span> <span class="n">xtu</span> <span class="k">with</span> <span class="n">xt</span> <span class="n">xu</span> <span class="bp">·</span> <span class="n">left</span> <span class="k">show</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">⟩</span> <span class="n">right</span> <span class="k">show</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> </pre></div> </div> <p>Since intersection binds tighter than union, the use of parentheses in the expression <code class="docutils literal notranslate"><span class="pre">(s</span> <span class="pre">∩</span> <span class="pre">t)</span> <span class="pre">∪</span> <span class="pre">(s</span> <span class="pre">∩</span> <span class="pre">u)</span></code> is unnecessary, but they make the meaning of the expression clearer. The following is a shorter proof of the same fact:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">t</span> <span class="bp">∪</span> <span class="n">u</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">∪</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span> <span class="bp">|</span> <span class="n">xu</span><span class="o">⟩</span> <span class="bp">·</span> <span class="n">left</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">⟩</span> <span class="n">right</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">⟩</span> </pre></div> </div> <p>As an exercise, try proving the other inclusion:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">∪</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">t</span> <span class="bp">∪</span> <span class="n">u</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>It might help to know that when using <code class="docutils literal notranslate"><span class="pre">rintro</span></code>, sometimes we need to use parentheses around a disjunctive pattern <code class="docutils literal notranslate"><span class="pre">h1</span> <span class="pre">|</span> <span class="pre">h2</span></code> to get Lean to parse it correctly.</p> <p>The library also defines set difference, <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">\</span> <span class="pre">t</span></code>, where the backslash is a special unicode character entered as <code class="docutils literal notranslate"><span class="pre">\\</span></code>. The expression <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">\</span> <span class="pre">t</span></code> expands to <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">∧</span> <span class="pre">x</span> <span class="pre">∉</span> <span class="pre">t</span></code>. (The <code class="docutils literal notranslate"><span class="pre">∉</span></code> can be entered as <code class="docutils literal notranslate"><span class="pre">\notin</span></code>.) It can be rewritten manually using <code class="docutils literal notranslate"><span class="pre">Set.diff_eq</span></code> and <code class="docutils literal notranslate"><span class="pre">dsimp</span></code> or <code class="docutils literal notranslate"><span class="pre">Set.mem_diff</span></code>, but the following two proofs of the same inclusion show how to avoid using them.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">s</span> <span class="bp">\</span> <span class="n">t</span><span class="o">)</span> <span class="bp">\</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="bp">\</span> <span class="o">(</span><span class="n">t</span> <span class="bp">∪</span> <span class="n">u</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">xstu</span> <span class="k">have</span> <span class="n">xs</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">xstu.1.1</span> <span class="k">have</span> <span class="n">xnt</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∉</span> <span class="n">t</span> <span class="o">:=</span> <span class="n">xstu.1.2</span> <span class="k">have</span> <span class="n">xnu</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∉</span> <span class="n">u</span> <span class="o">:=</span> <span class="n">xstu.2</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">exact</span> <span class="n">xs</span> <span class="n">intro</span> <span class="n">xtu</span> <span class="c1">-- x ∈ t ∨ x ∈ u</span> <span class="n">cases'</span> <span class="n">xtu</span> <span class="k">with</span> <span class="n">xt</span> <span class="n">xu</span> <span class="bp">·</span> <span class="k">show</span> <span class="n">False</span> <span class="n">exact</span> <span class="n">xnt</span> <span class="n">xt</span> <span class="k">show</span> <span class="n">False</span><span class="bp">;</span> <span class="n">exact</span> <span class="n">xnu</span> <span class="n">xu</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">s</span> <span class="bp">\</span> <span class="n">t</span><span class="o">)</span> <span class="bp">\</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="bp">\</span> <span class="o">(</span><span class="n">t</span> <span class="bp">∪</span> <span class="n">u</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">⟨⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xnt</span><span class="o">⟩,</span> <span class="n">xnu</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">xs</span> <span class="n">rintro</span> <span class="o">(</span><span class="n">xt</span> <span class="bp">|</span> <span class="n">xu</span><span class="o">)</span> <span class="bp"><;></span> <span class="n">contradiction</span> </pre></div> </div> <p>As an exercise, prove the reverse inclusion:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">\</span> <span class="o">(</span><span class="n">t</span> <span class="bp">∪</span> <span class="n">u</span><span class="o">)</span> <span class="bp">⊆</span> <span class="o">(</span><span class="n">s</span> <span class="bp">\</span> <span class="n">t</span><span class="o">)</span> <span class="bp">\</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>To prove that two sets are equal, it suffices to show that every element of one is an element of the other. This principle is known as “extensionality,” and, unsurprisingly, the <code class="docutils literal notranslate"><span class="pre">ext</span></code> tactic is equipped to handle it.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">mem_inter_iff</span><span class="o">]</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">⟩</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">⟩</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">⟩</span> </pre></div> </div> <p>Once again, deleting the line <code class="docutils literal notranslate"><span class="pre">simp</span> <span class="pre">only</span> <span class="pre">[mem_inter_iff]</span></code> does not harm the proof. In fact, if you like inscrutable proof terms, the following one-line proof is for you:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">Set.ext</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="o">⟨</span><span class="k">fun</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">⟩</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">⟩,</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">⟩</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">⟩⟩</span> </pre></div> </div> <p>The dollar sign is a useful syntax: writing <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">$</span> <span class="pre">...</span></code> is essentially the same as writing <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">(...)</span></code>, but it saves us the trouble of having to close a set of parentheses at the end of a long expression. Here is an even shorter proof, using the simplifier:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span><span class="bp">;</span> <span class="n">simp</span> <span class="o">[</span><span class="n">and_comm</span><span class="o">]</span> </pre></div> </div> <p>An alternative to using <code class="docutils literal notranslate"><span class="pre">ext</span></code> is to use the theorem <code class="docutils literal notranslate"><span class="pre">Subset.antisymm</span></code> which allows us to prove an equation <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">=</span> <span class="pre">t</span></code> between sets by proving <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">⊆</span> <span class="pre">t</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span> <span class="pre">⊆</span> <span class="pre">s</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">Subset.antisymm</span> <span class="bp">·</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">⟩</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">⟨</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">⟩</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">⟩</span> </pre></div> </div> <p>Try finishing this proof term:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">t</span> <span class="bp">∩</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">Subset.antisymm</span> <span class="gr">sorry</span> <span class="gr">sorry</span> </pre></div> </div> <p>Remember that you can replace <cite>sorry</cite> by an underscore, and when you hover over it, Lean will show you what it expects at that point.</p> <p>Here are some set-theoretic identities you might enjoy proving:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="bp">=</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">\</span> <span class="n">t</span> <span class="bp">∪</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">\</span> <span class="n">t</span> <span class="bp">∪</span> <span class="n">t</span> <span class="bp">\</span> <span class="n">s</span> <span class="bp">=</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="bp">\</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>When it comes to representing sets, here is what is going on underneath the hood. In type theory, a <em>property</em> or <em>predicate</em> on a type <code class="docutils literal notranslate"><span class="pre">α</span></code> is just a function <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">:</span> <span class="pre">α</span> <span class="pre">→</span> <span class="pre">Prop</span></code>. This makes sense: given <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">:</span> <span class="pre">α</span></code>, <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">a</span></code> is just the proposition that <code class="docutils literal notranslate"><span class="pre">P</span></code> holds of <code class="docutils literal notranslate"><span class="pre">a</span></code>. In the library, <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">α</span></code> is defined to be <code class="docutils literal notranslate"><span class="pre">α</span> <span class="pre">→</span> <span class="pre">Prop</span></code> and <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span></code> is defined to be <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">x</span></code>. In other words, sets are really properties, treated as objects.</p> <p>The library also defines set-builder notation. The expression <code class="docutils literal notranslate"><span class="pre">{</span> <span class="pre">y</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">y</span> <span class="pre">}</span></code> unfolds to <code class="docutils literal notranslate"><span class="pre">(fun</span> <span class="pre">y</span> <span class="pre">↦</span> <span class="pre">P</span> <span class="pre">y)</span></code>, so <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">{</span> <span class="pre">y</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">y</span> <span class="pre">}</span></code> reduces to <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">x</span></code>. So we can turn the property of being even into the set of even numbers:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">evens</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">ℕ</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">n</span> <span class="bp">|</span> <span class="n">Even</span> <span class="n">n</span> <span class="o">}</span> <span class="kd">def</span> <span class="n">odds</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">ℕ</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">n</span> <span class="bp">|</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">n</span> <span class="o">}</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">evens</span> <span class="bp">∪</span> <span class="n">odds</span> <span class="bp">=</span> <span class="n">univ</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">evens</span><span class="o">,</span> <span class="n">odds</span><span class="o">]</span> <span class="n">ext</span> <span class="n">n</span> <span class="n">simp</span> <span class="n">apply</span> <span class="n">Classical.em</span> </pre></div> </div> <p>You should step through this proof and make sure you understand what is going on. Try deleting the line <code class="docutils literal notranslate"><span class="pre">rw</span> <span class="pre">[evens,</span> <span class="pre">odds]</span></code> and confirm that the proof still works.</p> <p>In fact, set-builder notation is used to define</p> <ul class="simple"> <li><p><code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">∩</span> <span class="pre">t</span></code> as <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">∧</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">t}</span></code>,</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">∪</span> <span class="pre">t</span></code> as <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">∨</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">t}</span></code>,</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">∅</span></code> as <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">False}</span></code>, and</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">univ</span></code> as <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">True}</span></code>.</p></li> </ul> <p>We often need to indicate the type of <code class="docutils literal notranslate"><span class="pre">∅</span></code> and <code class="docutils literal notranslate"><span class="pre">univ</span></code> explicitly, because Lean has trouble guessing which ones we mean. The following examples show how Lean unfolds the last two definitions when needed. In the second one, <code class="docutils literal notranslate"><span class="pre">trivial</span></code> is the canonical proof of <code class="docutils literal notranslate"><span class="pre">True</span></code> in the library.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="o">(</span><span class="bp">∅</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">ℕ</span><span class="o">))</span> <span class="o">:</span> <span class="n">False</span> <span class="o">:=</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="o">(</span><span class="n">univ</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:=</span> <span class="n">trivial</span> </pre></div> </div> <p>As an exercise, prove the following inclusion. Use <code class="docutils literal notranslate"><span class="pre">intro</span> <span class="pre">n</span></code> to unfold the definition of subset, and use the simplifier to reduce the set-theoretic constructions to logic. We also recommend using the theorems <code class="docutils literal notranslate"><span class="pre">Nat.Prime.eq_two_or_odd</span></code> and <code class="docutils literal notranslate"><span class="pre">Nat.even_iff</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">{</span> <span class="n">n</span> <span class="bp">|</span> <span class="n">Nat.Prime</span> <span class="n">n</span> <span class="o">}</span> <span class="bp">∩</span> <span class="o">{</span> <span class="n">n</span> <span class="bp">|</span> <span class="n">n</span> <span class="bp">></span> <span class="mi">2</span> <span class="o">}</span> <span class="bp">⊆</span> <span class="o">{</span> <span class="n">n</span> <span class="bp">|</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">n</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Be careful: it is somewhat confusing that the library has multiple versions of the predicate <code class="docutils literal notranslate"><span class="pre">Prime</span></code>. The most general one makes sense in any commutative monoid with a zero element. The predicate <code class="docutils literal notranslate"><span class="pre">Nat.Prime</span></code> is specific to the natural numbers. Fortunately, there is a theorem that says that in the specific case, the two notions agree, so you can always rewrite one to the other.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#print</span> <span class="n">Prime</span> <span class="k">#print</span> <span class="n">Nat.Prime</span> <span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="n">Prime</span> <span class="n">n</span> <span class="bp">↔</span> <span class="n">Nat.Prime</span> <span class="n">n</span> <span class="o">:=</span> <span class="n">Nat.prime_iff.symm</span> <span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Prime</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Nat.prime_iff</span><span class="o">]</span> <span class="n">exact</span> <span class="n">h</span> </pre></div> </div> <p id="index-2">The <cite>rwa</cite> tactic follows a rewrite with the assumption tactic.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Prime</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rwa</span> <span class="o">[</span><span class="n">Nat.prime_iff</span><span class="o">]</span> </pre></div> </div> <p id="index-3">Lean introduces the notation <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s,</span> <span class="pre">...</span></code>, “for every <code class="docutils literal notranslate"><span class="pre">x</span></code> in <code class="docutils literal notranslate"><span class="pre">s</span></code> .,” as an abbreviation for <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">x,</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">→</span> <span class="pre">...</span></code>. It also introduces the notation <code class="docutils literal notranslate"><span class="pre">∃</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s,</span> <span class="pre">...,</span></code> “there exists an <code class="docutils literal notranslate"><span class="pre">x</span></code> in <code class="docutils literal notranslate"><span class="pre">s</span></code> such that ..” These are sometimes known as <em>bounded quantifiers</em>, because the construction serves to restrict their significance to the set <code class="docutils literal notranslate"><span class="pre">s</span></code>. As a result, theorems in the library that make use of them often contain <code class="docutils literal notranslate"><span class="pre">ball</span></code> or <code class="docutils literal notranslate"><span class="pre">bex</span></code> in the name. The theorem <code class="docutils literal notranslate"><span class="pre">bex_def</span></code> asserts that <code class="docutils literal notranslate"><span class="pre">∃</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s,</span> <span class="pre">...</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">∃</span> <span class="pre">x,</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">∧</span> <span class="pre">...,</span></code> but when they are used with <code class="docutils literal notranslate"><span class="pre">rintro</span></code>, <code class="docutils literal notranslate"><span class="pre">use</span></code>, and anonymous constructors, these two expressions behave roughly the same. As a result, we usually don’t need to use <code class="docutils literal notranslate"><span class="pre">bex_def</span></code> to transform them explicitly. Here is are some examples of how they are used:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">ℕ</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">Prime</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">Prime</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">xs</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">apply</span> <span class="n">h₀</span> <span class="n">x</span> <span class="n">xs</span> <span class="n">apply</span> <span class="n">h₁</span> <span class="n">x</span> <span class="n">xs</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">Prime</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">Prime</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">h</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">xs</span><span class="o">,</span> <span class="n">_</span><span class="o">,</span> <span class="n">prime_x</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">x</span><span class="o">,</span> <span class="n">xs</span> <span class="n">exact</span> <span class="n">prime_x</span> </pre></div> </div> <p>See if you can prove these slight variations:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">ssubt</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">t</span><span class="o">,</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">t</span><span class="o">,</span> <span class="n">Prime</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">Prime</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">¬</span><span class="n">Even</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">Prime</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">t</span><span class="o">,</span> <span class="n">Prime</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">end</span> </pre></div> </div> <p>Indexed unions and intersections are another important set-theoretic construction. We can model a sequence <span class="math notranslate nohighlight">\(A_0, A_1, A_2, \ldots\)</span> of sets of elements of <code class="docutils literal notranslate"><span class="pre">α</span></code> as a function <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">:</span> <span class="pre">ℕ</span> <span class="pre">→</span> <span class="pre">Set</span> <span class="pre">α</span></code>, in which case <code class="docutils literal notranslate"><span class="pre">⋃</span> <span class="pre">i,</span> <span class="pre">A</span> <span class="pre">i</span></code> denotes their union, and <code class="docutils literal notranslate"><span class="pre">⋂</span> <span class="pre">i,</span> <span class="pre">A</span> <span class="pre">i</span></code> denotes their intersection. There is nothing special about the natural numbers here, so <code class="docutils literal notranslate"><span class="pre">ℕ</span></code> can be replaced by any type <code class="docutils literal notranslate"><span class="pre">I</span></code> used to index the sets. The following illustrates their use.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="n">I</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">A</span> <span class="n">B</span> <span class="o">:</span> <span class="n">I</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="kn">open</span> <span class="n">Set</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∩</span> <span class="bp">⋃</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="bp">⋃</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span> <span class="bp">∩</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">mem_inter_iff</span><span class="o">,</span> <span class="n">mem_iUnion</span><span class="o">]</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="o">⟨</span><span class="n">i</span><span class="o">,</span> <span class="n">xAi</span><span class="o">⟩⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">i</span><span class="o">,</span> <span class="n">xAi</span><span class="o">,</span> <span class="n">xs</span><span class="o">⟩</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">i</span><span class="o">,</span> <span class="n">xAi</span><span class="o">,</span> <span class="n">xs</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xs</span><span class="o">,</span> <span class="o">⟨</span><span class="n">i</span><span class="o">,</span> <span class="n">xAi</span><span class="o">⟩⟩</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span> <span class="bp">∩</span> <span class="n">B</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="o">(</span><span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span><span class="o">)</span> <span class="bp">∩</span> <span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">B</span> <span class="n">i</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">mem_inter_iff</span><span class="o">,</span> <span class="n">mem_iInter</span><span class="o">]</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">intro</span> <span class="n">h</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">intro</span> <span class="n">i</span> <span class="n">exact</span> <span class="o">(</span><span class="n">h</span> <span class="n">i</span><span class="o">)</span><span class="bp">.</span><span class="mi">1</span> <span class="n">intro</span> <span class="n">i</span> <span class="n">exact</span> <span class="o">(</span><span class="n">h</span> <span class="n">i</span><span class="o">)</span><span class="bp">.</span><span class="mi">2</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">h1</span><span class="o">,</span> <span class="n">h2</span><span class="o">⟩</span> <span class="n">i</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">exact</span> <span class="n">h1</span> <span class="n">i</span> <span class="n">exact</span> <span class="n">h2</span> <span class="n">i</span> </pre></div> </div> <p>Parentheses are often needed with an indexed union or intersection because, as with the quantifiers, the scope of the bound variable extends as far as it can.</p> <p>Try proving the following identity. One direction requires classical logic! We recommend using <code class="docutils literal notranslate"><span class="pre">by_cases</span> <span class="pre">xs</span> <span class="pre">:</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span></code> at an appropriate point in the proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span> <span class="bp">∪</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Mathlib also has bounded unions and intersections, which are analogous to the bounded quantifiers. You can unpack their meaning with <code class="docutils literal notranslate"><span class="pre">mem_Union₂</span></code> and <code class="docutils literal notranslate"><span class="pre">mem_Inter₂</span></code>. As the following examples show, Lean’s simplifier carries out these replacements as well.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">primes</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">ℕ</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">Nat.Prime</span> <span class="n">x</span> <span class="o">}</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="bp">⋃</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">primes</span><span class="o">,</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">p</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">x</span> <span class="o">})</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="bp">∃</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">primes</span><span class="o">,</span> <span class="n">p</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">x</span> <span class="o">}</span> <span class="o">:=</span><span class="kd">by</span> <span class="n">ext</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mem_iUnion₂</span><span class="o">]</span> <span class="n">simp</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="bp">⋃</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">primes</span><span class="o">,</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">p</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">x</span> <span class="o">})</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="bp">∃</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">primes</span><span class="o">,</span> <span class="n">p</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">x</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">simp</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="bp">⋂</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">primes</span><span class="o">,</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="bp">¬</span><span class="n">p</span> <span class="bp">∣</span> <span class="n">x</span> <span class="o">})</span> <span class="bp">⊆</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">simp</span> <span class="n">apply</span> <span class="n">Nat.exists_prime_and_dvd</span> </pre></div> </div> <p>Try solving the following example, which is similar. If you start typing <code class="docutils literal notranslate"><span class="pre">eq_univ</span></code>, tab completion will tell you that <code class="docutils literal notranslate"><span class="pre">apply</span> <span class="pre">eq_univ_of_forall</span></code> is a good way to start the proof. We also recommend using the theorem <code class="docutils literal notranslate"><span class="pre">Nat.exists_infinite_primes</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="bp">⋃</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">primes</span><span class="o">,</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">p</span> <span class="o">})</span> <span class="bp">=</span> <span class="n">univ</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Give a collection of sets, <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">(Set</span> <span class="pre">α)</span></code>, their union, <code class="docutils literal notranslate"><span class="pre">⋃₀</span> <span class="pre">s</span></code>, has type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">α</span></code> and is defined as <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">∃</span> <span class="pre">t</span> <span class="pre">∈</span> <span class="pre">s,</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">t}</span></code>. Similarly, their intersection, <code class="docutils literal notranslate"><span class="pre">⋂₀</span> <span class="pre">s</span></code>, is defined as <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">∀</span> <span class="pre">t</span> <span class="pre">∈</span> <span class="pre">s,</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">t}</span></code>. These operations are called <code class="docutils literal notranslate"><span class="pre">sUnion</span></code> and <code class="docutils literal notranslate"><span class="pre">sInter</span></code>, respectively. The following examples show their relationship to bounded union and intersection.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="o">(</span><span class="n">Set</span> <span class="n">α</span><span class="o">))</span> <span class="kd">example</span> <span class="o">:</span> <span class="bp">⋃₀</span> <span class="n">s</span> <span class="bp">=</span> <span class="bp">⋃</span> <span class="n">t</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mem_iUnion₂</span><span class="o">]</span> <span class="n">simp</span> <span class="kd">example</span> <span class="o">:</span> <span class="bp">⋂₀</span> <span class="n">s</span> <span class="bp">=</span> <span class="bp">⋂</span> <span class="n">t</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mem_iInter₂</span><span class="o">]</span> <span class="n">rfl</span> </pre></div> </div> <p>In the library, these identities are called <code class="docutils literal notranslate"><span class="pre">sUnion_eq_biUnion</span></code> and <code class="docutils literal notranslate"><span class="pre">sInter_eq_biInter</span></code>.</p> </section> <section id="functions"> <span id="id3"></span><h2><span class="section-number">4.2. </span>Functions<a class="headerlink" href="#functions" title="Permalink to this heading"></a></h2> <p>If <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">α</span> <span class="pre">→</span> <span class="pre">β</span></code> is a function and <code class="docutils literal notranslate"><span class="pre">p</span></code> is a set of elements of type <code class="docutils literal notranslate"><span class="pre">β</span></code>, the library defines <code class="docutils literal notranslate"><span class="pre">preimage</span> <span class="pre">f</span> <span class="pre">p</span></code>, written <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">⁻¹'</span> <span class="pre">p</span></code>, to be <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">p}</span></code>. The expression <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">f</span> <span class="pre">⁻¹'</span> <span class="pre">p</span></code> reduces to <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">p</span></code>. This is often convenient, as in the following example:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">u</span> <span class="n">v</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">β</span><span class="o">)</span> <span class="kn">open</span> <span class="n">Function</span> <span class="kn">open</span> <span class="n">Set</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="o">(</span><span class="n">u</span> <span class="bp">∩</span> <span class="n">v</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span> <span class="bp">∩</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">v</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">rfl</span> </pre></div> </div> <p>If <code class="docutils literal notranslate"><span class="pre">s</span></code> is a set of elements of type <code class="docutils literal notranslate"><span class="pre">α</span></code>, the library also defines <code class="docutils literal notranslate"><span class="pre">image</span> <span class="pre">f</span> <span class="pre">s</span></code>, written <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">''</span> <span class="pre">s</span></code>, to be <code class="docutils literal notranslate"><span class="pre">{y</span> <span class="pre">|</span> <span class="pre">∃</span> <span class="pre">x,</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">∧</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">y}</span></code>. So a hypothesis <code class="docutils literal notranslate"><span class="pre">y</span> <span class="pre">∈</span> <span class="pre">f</span> <span class="pre">''</span> <span class="pre">s</span></code> decomposes to a triple <code class="docutils literal notranslate"><span class="pre">⟨x,</span> <span class="pre">xs,</span> <span class="pre">xeq⟩</span></code> with <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">α</span></code> satisfying the hypotheses <code class="docutils literal notranslate"><span class="pre">xs</span> <span class="pre">:</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span></code> and <code class="docutils literal notranslate"><span class="pre">xeq</span> <span class="pre">:</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">y</span></code>. The <code class="docutils literal notranslate"><span class="pre">rfl</span></code> tag in the <code class="docutils literal notranslate"><span class="pre">rintro</span></code> tactic (see <a class="reference internal" href="C03_Logic.html#the-existential-quantifier"><span class="std std-numref">Section 3.2</span></a>) was made precisely for this sort of situation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">y</span><span class="bp">;</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">xs</span> <span class="bp">|</span> <span class="n">xt</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩</span> <span class="bp">·</span> <span class="n">left</span> <span class="n">use</span> <span class="n">x</span><span class="o">,</span> <span class="n">xs</span> <span class="n">right</span> <span class="n">use</span> <span class="n">x</span><span class="o">,</span> <span class="n">xt</span> <span class="n">rintro</span> <span class="o">(⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">xs</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩</span> <span class="bp">|</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">xt</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩)</span> <span class="bp">·</span> <span class="n">use</span> <span class="n">x</span><span class="o">,</span> <span class="n">Or.inl</span> <span class="n">xs</span> <span class="n">use</span> <span class="n">x</span><span class="o">,</span> <span class="n">Or.inr</span> <span class="n">xt</span> </pre></div> </div> <p>Notice also that the <code class="docutils literal notranslate"><span class="pre">use</span></code> tactic applies <code class="docutils literal notranslate"><span class="pre">rfl</span></code> to close goals when it can.</p> <p>Here is another example:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="n">s</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">xs</span> <span class="k">show</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="n">use</span> <span class="n">x</span><span class="o">,</span> <span class="n">xs</span> </pre></div> </div> <p>We can replace the line <code class="docutils literal notranslate"><span class="pre">use</span> <span class="pre">x,</span> <span class="pre">xs</span></code> by <code class="docutils literal notranslate"><span class="pre">apply</span> <span class="pre">mem_image_of_mem</span> <span class="pre">f</span> <span class="pre">xs</span></code> if we want to use a theorem specifically designed for that purpose. But knowing that the image is defined in terms of an existential quantifier is often convenient.</p> <p>The following equivalence is a good exercise:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">v</span> <span class="bp">↔</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">v</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>It shows that <code class="docutils literal notranslate"><span class="pre">image</span> <span class="pre">f</span></code> and <code class="docutils literal notranslate"><span class="pre">preimage</span> <span class="pre">f</span></code> are an instance of what is known as a <em>Galois connection</em> between <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">α</span></code> and <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">β</span></code>, each partially ordered by the subset relation. In the library, this equivalence is named <code class="docutils literal notranslate"><span class="pre">image_subset_iff</span></code>. In practice, the right-hand side is often the more useful representation, because <code class="docutils literal notranslate"><span class="pre">y</span> <span class="pre">∈</span> <span class="pre">f</span> <span class="pre">⁻¹'</span> <span class="pre">t</span></code> unfolds to <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">y</span> <span class="pre">∈</span> <span class="pre">t</span></code> whereas working with <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">∈</span> <span class="pre">f</span> <span class="pre">''</span> <span class="pre">s</span></code> requires decomposing an existential quantifier.</p> <p>Here is a long list of set-theoretic identities for you to enjoy. You don’t have to do all of them at once; do a few of them, and set the rest aside for a rainy day.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="n">s</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="o">(</span><span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">''</span> <span class="o">(</span><span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">v</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">v</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="o">(</span><span class="n">u</span> <span class="bp">∪</span> <span class="n">v</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span> <span class="bp">∪</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">v</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">t</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">''</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">\</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">t</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">''</span> <span class="o">(</span><span class="n">s</span> <span class="bp">\</span> <span class="n">t</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span> <span class="bp">\</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">v</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="o">(</span><span class="n">u</span> <span class="bp">\</span> <span class="n">v</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">v</span> <span class="bp">=</span> <span class="n">f</span> <span class="bp">''</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∩</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">v</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">''</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∩</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">u</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">u</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">u</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>You can also try your hand at the next group of exercises, which characterize the behavior of images and preimages with respect to indexed unions and intersections. In the third exercise, the argument <code class="docutils literal notranslate"><span class="pre">i</span> <span class="pre">:</span> <span class="pre">I</span></code> is needed to guarantee that the index set is nonempty. To prove any of these, we recommend using <code class="docutils literal notranslate"><span class="pre">ext</span></code> or <code class="docutils literal notranslate"><span class="pre">intro</span></code> to unfold the meaning of an equation or inclusion between sets, and then calling <code class="docutils literal notranslate"><span class="pre">simp</span></code> to unpack the conditions for membership.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">I</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">A</span> <span class="o">:</span> <span class="n">I</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">B</span> <span class="o">:</span> <span class="n">I</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">β</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="bp">⋃</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="bp">⋃</span> <span class="n">i</span><span class="o">,</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">A</span> <span class="n">i</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">y</span><span class="bp">;</span> <span class="n">simp</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="o">⟨</span><span class="n">i</span><span class="o">,</span> <span class="n">xAi</span><span class="o">⟩,</span> <span class="n">fxeq</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">i</span><span class="o">,</span> <span class="n">x</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">xAi</span><span class="o">,</span> <span class="n">fxeq</span><span class="o">⟩</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">i</span><span class="o">,</span> <span class="n">x</span><span class="o">,</span> <span class="n">xAi</span><span class="o">,</span> <span class="n">fxeq</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="o">⟨</span><span class="n">i</span><span class="o">,</span> <span class="n">xAi</span><span class="o">⟩,</span> <span class="n">fxeq</span><span class="o">⟩</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span><span class="o">)</span> <span class="bp">⊆</span> <span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">A</span> <span class="n">i</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">y</span><span class="bp">;</span> <span class="n">simp</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">h</span> <span class="n">fxeq</span> <span class="n">i</span> <span class="n">use</span> <span class="n">x</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">h</span> <span class="n">i</span><span class="o">,</span> <span class="n">fxeq</span><span class="o">⟩</span> <span class="kd">example</span> <span class="o">(</span><span class="n">i</span> <span class="o">:</span> <span class="n">I</span><span class="o">)</span> <span class="o">(</span><span class="n">injf</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">A</span> <span class="n">i</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">''</span> <span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">A</span> <span class="n">i</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">y</span><span class="bp">;</span> <span class="n">simp</span> <span class="n">intro</span> <span class="n">h</span> <span class="n">rcases</span> <span class="n">h</span> <span class="n">i</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">xAi</span><span class="o">,</span> <span class="n">fxeq</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">x</span><span class="bp">;</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">intro</span> <span class="n">i'</span> <span class="n">rcases</span> <span class="n">h</span> <span class="n">i'</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">x'</span><span class="o">,</span> <span class="n">x'Ai</span><span class="o">,</span> <span class="n">fx'eq</span><span class="o">⟩</span> <span class="k">have</span> <span class="o">:</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">x'</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">fxeq</span><span class="o">,</span> <span class="n">fx'eq</span><span class="o">]</span> <span class="k">have</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">x'</span> <span class="o">:=</span> <span class="n">injf</span> <span class="n">this</span> <span class="n">rw</span> <span class="o">[</span><span class="n">this</span><span class="o">]</span> <span class="n">exact</span> <span class="n">x'Ai</span> <span class="n">exact</span> <span class="n">fxeq</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">f</span> <span class="bp">⁻¹'</span> <span class="bp">⋃</span> <span class="n">i</span><span class="o">,</span> <span class="n">B</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="bp">⋃</span> <span class="n">i</span><span class="o">,</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">B</span> <span class="n">i</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span> <span class="n">simp</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">f</span> <span class="bp">⁻¹'</span> <span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">B</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">B</span> <span class="n">i</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span> <span class="n">simp</span> </pre></div> </div> <p>The library defines a predicate <code class="docutils literal notranslate"><span class="pre">InjOn</span> <span class="pre">f</span> <span class="pre">s</span></code> to say that <code class="docutils literal notranslate"><span class="pre">f</span></code> is injective on <code class="docutils literal notranslate"><span class="pre">s</span></code>. It is defined as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">InjOn</span> <span class="n">f</span> <span class="n">s</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">x₁</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">x₂</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">f</span> <span class="n">x₁</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">x₂</span> <span class="bp">→</span> <span class="n">x₁</span> <span class="bp">=</span> <span class="n">x₂</span> <span class="o">:=</span> <span class="n">Iff.refl</span> <span class="n">_</span> </pre></div> </div> <p>The statement <code class="docutils literal notranslate"><span class="pre">Injective</span> <span class="pre">f</span></code> is provably equivalent to <code class="docutils literal notranslate"><span class="pre">InjOn</span> <span class="pre">f</span> <span class="pre">univ</span></code>. Similarly, the library defines <code class="docutils literal notranslate"><span class="pre">range</span> <span class="pre">f</span></code> to be <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">∃y,</span> <span class="pre">f</span> <span class="pre">y</span> <span class="pre">=</span> <span class="pre">x}</span></code>, so <code class="docutils literal notranslate"><span class="pre">range</span> <span class="pre">f</span></code> is provably equal to <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">''</span> <span class="pre">univ</span></code>. This is a common theme in mathlib: although many properties of functions are defined relative to their full domain, there are often relativized versions that restrict the statements to a subset of the domain type.</p> <p>Here is are some examples of <code class="docutils literal notranslate"><span class="pre">InjOn</span></code> and <code class="docutils literal notranslate"><span class="pre">range</span></code> in use:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Set</span> <span class="n">Real</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">InjOn</span> <span class="n">log</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">></span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">xpos</span> <span class="n">y</span> <span class="n">ypos</span> <span class="n">intro</span> <span class="n">e</span> <span class="c1">-- log x = log y</span> <span class="k">calc</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">exp</span> <span class="o">(</span><span class="n">log</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">exp_log</span> <span class="n">xpos</span><span class="o">]</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">exp</span> <span class="o">(</span><span class="n">log</span> <span class="n">y</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">e</span><span class="o">]</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">exp_log</span> <span class="n">ypos</span><span class="o">]</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">range</span> <span class="n">exp</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">y</span> <span class="bp">|</span> <span class="n">y</span> <span class="bp">></span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">y</span><span class="bp">;</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩</span> <span class="n">apply</span> <span class="n">exp_pos</span> <span class="n">intro</span> <span class="n">ypos</span> <span class="n">use</span> <span class="n">log</span> <span class="n">y</span> <span class="n">rw</span> <span class="o">[</span><span class="n">exp_log</span> <span class="n">ypos</span><span class="o">]</span> </pre></div> </div> <p>Try proving these:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">InjOn</span> <span class="n">sqrt</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">≥</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">InjOn</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">{</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">≥</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">sqrt</span> <span class="bp">''</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">≥</span> <span class="mi">0</span> <span class="o">}</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">y</span> <span class="bp">|</span> <span class="n">y</span> <span class="bp">≥</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">range</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">|</span> <span class="n">y</span> <span class="bp">≥</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>To define the inverse of a function <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">α</span> <span class="pre">→</span> <span class="pre">β</span></code>, we will use two new ingredients. First, we need to deal with the fact that an arbitrary type in Lean may be empty. To define the inverse to <code class="docutils literal notranslate"><span class="pre">f</span></code> at <code class="docutils literal notranslate"><span class="pre">y</span></code> when there is no <code class="docutils literal notranslate"><span class="pre">x</span></code> satisfying <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">y</span></code>, we want to assign a default value in <code class="docutils literal notranslate"><span class="pre">α</span></code>. Adding the annotation <code class="docutils literal notranslate"><span class="pre">[Inhabited</span> <span class="pre">α]</span></code> as a variable is tantamount to assuming that <code class="docutils literal notranslate"><span class="pre">α</span></code> has a preferred element, which is denoted <code class="docutils literal notranslate"><span class="pre">default</span></code>. Second, in the case where there is more than one <code class="docutils literal notranslate"><span class="pre">x</span></code> such that <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">y</span></code>, the inverse function needs to <em>choose</em> one of them. This requires an appeal to the <em>axiom of choice</em>. Lean allows various ways of accessing it; one convenient method is to use the classical <code class="docutils literal notranslate"><span class="pre">some</span></code> operator, illustrated below.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Inhabited</span> <span class="n">α</span><span class="o">]</span> <span class="k">#check</span> <span class="o">(</span><span class="n">default</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">P</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">P</span> <span class="n">x</span><span class="o">)</span> <span class="k">#check</span> <span class="n">Classical.choose</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">P</span> <span class="o">(</span><span class="n">Classical.choose</span> <span class="n">h</span><span class="o">)</span> <span class="o">:=</span> <span class="n">Classical.choose_spec</span> <span class="n">h</span> </pre></div> </div> <p>Given <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">∃</span> <span class="pre">x,</span> <span class="pre">P</span> <span class="pre">x</span></code>, the value of <code class="docutils literal notranslate"><span class="pre">Classical.some</span> <span class="pre">h</span></code> is some <code class="docutils literal notranslate"><span class="pre">x</span></code> satisfying <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">x</span></code>. The theorem <code class="docutils literal notranslate"><span class="pre">Classical.some_spec</span> <span class="pre">h</span></code> says that <code class="docutils literal notranslate"><span class="pre">Classical.some</span> <span class="pre">h</span></code> meets this specification.</p> <p>With these in hand, we can define the inverse function as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">noncomputable</span> <span class="kn">section</span> <span class="kn">open</span> <span class="n">Classical</span> <span class="kd">def</span> <span class="n">inverse</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">)</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">α</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">y</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">=></span> <span class="k">if</span> <span class="n">h</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">y</span> <span class="k">then</span> <span class="n">Classical.choose</span> <span class="n">h</span> <span class="k">else</span> <span class="n">default</span> <span class="kd">theorem</span> <span class="n">inverse_spec</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">}</span> <span class="o">(</span><span class="n">y</span> <span class="o">:</span> <span class="n">β</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="o">(</span><span class="n">inverse</span> <span class="n">f</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">inverse</span><span class="o">]</span><span class="bp">;</span> <span class="n">dsimp</span><span class="bp">;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">dif_pos</span> <span class="n">h</span><span class="o">]</span> <span class="n">exact</span> <span class="n">Classical.choose_spec</span> <span class="n">h</span> </pre></div> </div> <p>The lines <code class="docutils literal notranslate"><span class="pre">noncomputable</span> <span class="pre">theory</span></code> and <code class="docutils literal notranslate"><span class="pre">open</span> <span class="pre">Classical</span></code> are needed because we are using classical logic in an essential way. On input <code class="docutils literal notranslate"><span class="pre">y</span></code>, the function <code class="docutils literal notranslate"><span class="pre">inverse</span> <span class="pre">f</span></code> returns some value of <code class="docutils literal notranslate"><span class="pre">x</span></code> satisfying <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">y</span></code> if there is one, and a default element of <code class="docutils literal notranslate"><span class="pre">α</span></code> otherwise. This is an instance of a <em>dependent if</em> construction, since in the positive case, the value returned, <code class="docutils literal notranslate"><span class="pre">Classical.choose</span> <span class="pre">h</span></code>, depends on the assumption <code class="docutils literal notranslate"><span class="pre">h</span></code>. The identity <code class="docutils literal notranslate"><span class="pre">dif_pos</span> <span class="pre">h</span></code> rewrites <code class="docutils literal notranslate"><span class="pre">if</span> <span class="pre">h</span> <span class="pre">:</span> <span class="pre">e</span> <span class="pre">then</span> <span class="pre">a</span> <span class="pre">else</span> <span class="pre">b</span></code> to <code class="docutils literal notranslate"><span class="pre">a</span></code> given <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">e</span></code>, and, similarly, <code class="docutils literal notranslate"><span class="pre">dif_neg</span> <span class="pre">h</span></code> rewrites it to <code class="docutils literal notranslate"><span class="pre">b</span></code> given <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">¬</span> <span class="pre">e</span></code>. The theorem <code class="docutils literal notranslate"><span class="pre">inverse_spec</span></code> says that <code class="docutils literal notranslate"><span class="pre">inverse</span> <span class="pre">f</span></code> meets the first part of this specification.</p> <p>Don’t worry if you do not fully understand how these work. The theorem <code class="docutils literal notranslate"><span class="pre">inverse_spec</span></code> alone should be enough to show that <code class="docutils literal notranslate"><span class="pre">inverse</span> <span class="pre">f</span></code> is a left inverse if and only if <code class="docutils literal notranslate"><span class="pre">f</span></code> is injective and a right inverse if and only if <code class="docutils literal notranslate"><span class="pre">f</span></code> is surjective. Look up the definition of <code class="docutils literal notranslate"><span class="pre">LeftInverse</span></code> and <code class="docutils literal notranslate"><span class="pre">RightInverse</span></code> by double-clicking or right-clicking on them in VS Code, or using the commands <code class="docutils literal notranslate"><span class="pre">#print</span> <span class="pre">LeftInverse</span></code> and <code class="docutils literal notranslate"><span class="pre">#print</span> <span class="pre">RightInverse</span></code>. Then try to prove the two theorems. They are tricky! It helps to do the proofs on paper before you start hacking through the details. You should be able to prove each of them with about a half-dozen short lines. If you are looking for an extra challenge, try to condense each proof to a single-line proof term.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">)</span> <span class="kn">open</span> <span class="n">Function</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span> <span class="bp">↔</span> <span class="n">LeftInverse</span> <span class="o">(</span><span class="n">inverse</span> <span class="n">f</span><span class="o">)</span> <span class="n">f</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">f</span> <span class="bp">↔</span> <span class="n">RightInverse</span> <span class="o">(</span><span class="n">inverse</span> <span class="n">f</span><span class="o">)</span> <span class="n">f</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>We close this section with a type-theoretic statement of Cantor’s famous theorem that there is no surjective function from a set to its power set. See if you can understand the proof, and then fill in the two lines that are missing.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">Cantor</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">α</span><span class="o">,</span> <span class="bp">¬</span><span class="n">Surjective</span> <span class="n">f</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">f</span> <span class="n">surjf</span> <span class="k">let</span> <span class="n">S</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">i</span> <span class="bp">|</span> <span class="n">i</span> <span class="bp">∉</span> <span class="n">f</span> <span class="n">i</span> <span class="o">}</span> <span class="n">rcases</span> <span class="n">surjf</span> <span class="n">S</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">j</span><span class="o">,</span> <span class="n">h</span><span class="o">⟩</span> <span class="k">have</span> <span class="n">h₁</span> <span class="o">:</span> <span class="n">j</span> <span class="bp">∉</span> <span class="n">f</span> <span class="n">j</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">h'</span> <span class="k">have</span> <span class="o">:</span> <span class="n">j</span> <span class="bp">∉</span> <span class="n">f</span> <span class="n">j</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rwa</span> <span class="o">[</span><span class="n">h</span><span class="o">]</span> <span class="n">at</span> <span class="n">h'</span> <span class="n">contradiction</span> <span class="k">have</span> <span class="n">h₂</span> <span class="o">:</span> <span class="n">j</span> <span class="bp">∈</span> <span class="n">S</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">h₃</span> <span class="o">:</span> <span class="n">j</span> <span class="bp">∉</span> <span class="n">S</span> <span class="gr">sorry</span> <span class="n">contradiction</span> </pre></div> </div> </section> <section id="the-schroder-bernstein-theorem"> <span id="the-schroeder-bernstein-theorem"></span><h2><span class="section-number">4.3. </span>The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this heading"></a></h2> <p>We close this chapter with an elementary but nontrivial theorem of set theory. Let <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span> be sets. (In our formalization, they will actually be types.) Suppose <span class="math notranslate nohighlight">\(f : \alpha → \beta\)</span> and <span class="math notranslate nohighlight">\(g : \beta → \alpha\)</span> are both injective. Intuitively, this means that <span class="math notranslate nohighlight">\(\alpha\)</span> is no bigger than <span class="math notranslate nohighlight">\(\beta\)</span> and vice-versa. If <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span> are finite, this implies that they have the same cardinality, which is equivalent to saying that there is a bijection between them. In the nineteenth century, Cantor stated that same result holds even in the case where <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span> are infinite. This was eventually established by Dedekind, Schröder, and Bernstein independently.</p> <p>Our formalization will introduce some new methods that we will explain in greater detail in chapters to come. Don’t worry if they go by too quickly here. Our goal is to show you that you already have the skills to contribute to the formal proof of a real mathematical result.</p> <p>To understand the idea behind the proof, consider the image of the map <span class="math notranslate nohighlight">\(g\)</span> in <span class="math notranslate nohighlight">\(\alpha\)</span>. On that image, the inverse of <span class="math notranslate nohighlight">\(g\)</span> is defined and is a bijection with <span class="math notranslate nohighlight">\(\beta\)</span>.</p> <a class="reference internal image-reference" href="_images/schroeder_bernstein1.png"><img alt="the Schröder Bernstein theorem" class="align-center" src="_images/schroeder_bernstein1.png" style="height: 150px;" /></a> <p>The problem is that the bijection does not include the shaded region in the diagram, which is nonempty if <span class="math notranslate nohighlight">\(g\)</span> is not surjective. Alternatively, we can use <span class="math notranslate nohighlight">\(f\)</span> to map all of <span class="math notranslate nohighlight">\(\alpha\)</span> to <span class="math notranslate nohighlight">\(\beta\)</span>, but in that case the problem is that if <span class="math notranslate nohighlight">\(f\)</span> is not surjective, it will miss some elements of <span class="math notranslate nohighlight">\(\beta\)</span>.</p> <a class="reference internal image-reference" href="_images/schroeder_bernstein2.png"><img alt="the Schröder Bernstein theorem" class="align-center" src="_images/schroeder_bernstein2.png" style="height: 150px;" /></a> <p>But now consider the composition <span class="math notranslate nohighlight">\(g \circ f\)</span> from <span class="math notranslate nohighlight">\(\alpha\)</span> to itself. Because the composition is injective, it forms a bijection between <span class="math notranslate nohighlight">\(\alpha\)</span> and its image, yielding a scaled-down copy of <span class="math notranslate nohighlight">\(\alpha\)</span> inside itself.</p> <a class="reference internal image-reference" href="_images/schroeder_bernstein3.png"><img alt="the Schröder Bernstein theorem" class="align-center" src="_images/schroeder_bernstein3.png" style="height: 150px;" /></a> <p>This composition maps the inner shaded ring to yet another such set, which we can think of as an even smaller concentric shaded ring, and so on. This yields a concentric sequence of shaded rings, each of which is in bijective correspondence with the next. If we map each ring to the next and leave the unshaded parts of <span class="math notranslate nohighlight">\(\alpha\)</span> alone, we have a bijection of <span class="math notranslate nohighlight">\(\alpha\)</span> with the image of <span class="math notranslate nohighlight">\(g\)</span>. Composing with <span class="math notranslate nohighlight">\(g^{-1}\)</span>, this yields the desired bijection between <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span>.</p> <p>We can describe this bijection more simply. Let <span class="math notranslate nohighlight">\(A\)</span> be the union of the sequence of shaded regions, and define <span class="math notranslate nohighlight">\(h : \alpha \to \beta\)</span> as follows:</p> <div class="math notranslate nohighlight"> \[\begin{split}h(x) = \begin{cases} f(x) & \text{if $x \in A$} \\ g^{-1}(x) & \text{otherwise.} \end{cases}\end{split}\]</div> <p>In other words, we use <span class="math notranslate nohighlight">\(f\)</span> on the shaded parts, and we use the inverse of <span class="math notranslate nohighlight">\(g\)</span> everywhere else. The resulting map <span class="math notranslate nohighlight">\(h\)</span> is injective because each component is injective and the images of the two components are disjoint. To see that it is surjective, suppose we are given a <span class="math notranslate nohighlight">\(y\)</span> in <span class="math notranslate nohighlight">\(\beta\)</span>, and consider <span class="math notranslate nohighlight">\(g(y)\)</span>. If <span class="math notranslate nohighlight">\(g(y)\)</span> is in one of the shaded regions, it cannot be in the first ring, so we have <span class="math notranslate nohighlight">\(g(y) = g(f(x))\)</span> for some <span class="math notranslate nohighlight">\(x\)</span> is in the previous ring. By the injectivity of <span class="math notranslate nohighlight">\(g\)</span>, we have <span class="math notranslate nohighlight">\(h(x) = f(x) = y\)</span>. If <span class="math notranslate nohighlight">\(g(y)\)</span> is not in the shaded region, then by the definition of <span class="math notranslate nohighlight">\(h\)</span>, we have <span class="math notranslate nohighlight">\(h(g(y))= y\)</span>. Either way, <span class="math notranslate nohighlight">\(y\)</span> is in the image of <span class="math notranslate nohighlight">\(h\)</span>.</p> <p>This argument should sound plausible, but the details are delicate. Formalizing the proof will not only improve our confidence in the result, but also help us understand it better. Because the proof uses classical logic, we tell Lean that our definitions will generally not be computable.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">noncomputable</span> <span class="kn">section</span> <span class="kn">open</span> <span class="n">Classical</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Nonempty</span> <span class="n">β</span><span class="o">]</span> </pre></div> </div> <p>The annotation <code class="docutils literal notranslate"><span class="pre">[Nonempty</span> <span class="pre">β]</span></code> specifies that <code class="docutils literal notranslate"><span class="pre">β</span></code> is nonempty. We use it because the mathlib primitive that we will use to construct <span class="math notranslate nohighlight">\(g^{-1}\)</span> requires it. The case of the theorem where <span class="math notranslate nohighlight">\(\beta\)</span> is empty is trivial, and even though it would not be hard to generalize the formalization to cover that case as well, we will not bother. Specifically, we need the hypothesis <code class="docutils literal notranslate"><span class="pre">[Nonempty</span> <span class="pre">β]</span></code> for the operation <code class="docutils literal notranslate"><span class="pre">invFun</span></code> that is defined in mathlib. Given <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">α</span></code>, <code class="docutils literal notranslate"><span class="pre">invFun</span> <span class="pre">g</span> <span class="pre">x</span></code> chooses a preimage of <code class="docutils literal notranslate"><span class="pre">x</span></code> in <code class="docutils literal notranslate"><span class="pre">β</span></code> if there is one, and returns an arbitrary element of <code class="docutils literal notranslate"><span class="pre">β</span></code> otherwise. The function <code class="docutils literal notranslate"><span class="pre">invFun</span> <span class="pre">g</span></code> is always a left inverse if <code class="docutils literal notranslate"><span class="pre">g</span></code> is injective and a right inverse if <code class="docutils literal notranslate"><span class="pre">g</span></code> is surjective.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">invFun</span> <span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">leftInverse_invFun</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">g</span> <span class="bp">→</span> <span class="n">LeftInverse</span> <span class="o">(</span><span class="n">invFun</span> <span class="n">g</span><span class="o">)</span> <span class="n">g</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">leftInverse_invFun</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">g</span> <span class="bp">→</span> <span class="bp">∀</span> <span class="n">y</span><span class="o">,</span> <span class="n">invFun</span> <span class="n">g</span> <span class="o">(</span><span class="n">g</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">y</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">invFun_eq</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∃</span> <span class="n">y</span><span class="o">,</span> <span class="n">g</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">x</span><span class="o">)</span> <span class="bp">→</span> <span class="n">g</span> <span class="o">(</span><span class="n">invFun</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span><span class="o">)</span> </pre></div> </div> <p>We define the set corresponding to the union of the shaded regions as follows.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">α</span><span class="o">)</span> <span class="kd">def</span> <span class="n">sbAux</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">α</span> <span class="bp">|</span> <span class="mi">0</span> <span class="bp">=></span> <span class="n">univ</span> <span class="bp">\</span> <span class="n">g</span> <span class="bp">''</span> <span class="n">univ</span> <span class="bp">|</span> <span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span> <span class="bp">=></span> <span class="n">g</span> <span class="bp">''</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="n">sbAux</span> <span class="n">n</span><span class="o">)</span> <span class="kd">def</span> <span class="n">sbSet</span> <span class="o">:=</span> <span class="bp">⋃</span> <span class="n">n</span><span class="o">,</span> <span class="n">sbAux</span> <span class="n">f</span> <span class="n">g</span> <span class="n">n</span> </pre></div> </div> <p>The definition <code class="docutils literal notranslate"><span class="pre">sb_aux</span></code> is an example of a <em>recursive definition</em>, which we will explain in the next chapter. It defines a sequence of sets</p> <div class="math notranslate nohighlight"> \[\begin{split}S_0 &= \alpha ∖ g(\beta) \\ S_{n+1} &= g(f(S_n)).\end{split}\]</div> <p>The definition <code class="docutils literal notranslate"><span class="pre">sb_set</span></code> corresponds to the set <span class="math notranslate nohighlight">\(A = \bigcup_{n \in \mathbb{N}} S_n\)</span> in our proof sketch. The function <span class="math notranslate nohighlight">\(h\)</span> described above is now defined as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">sbFun</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="n">β</span> <span class="o">:=</span> <span class="k">if</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">sbSet</span> <span class="n">f</span> <span class="n">g</span> <span class="k">then</span> <span class="n">f</span> <span class="n">x</span> <span class="k">else</span> <span class="n">invFun</span> <span class="n">g</span> <span class="n">x</span> </pre></div> </div> <p>We will need the fact that our definition of <span class="math notranslate nohighlight">\(g^{-1}\)</span> is a right inverse on the complement of <span class="math notranslate nohighlight">\(A\)</span>, which is to say, on the non-shaded regions of <span class="math notranslate nohighlight">\(\alpha\)</span>. This is so because the outermost ring, <span class="math notranslate nohighlight">\(S_0\)</span>, is equal to <span class="math notranslate nohighlight">\(\alpha \setminus g(\beta)\)</span>, so the complement of <span class="math notranslate nohighlight">\(A\)</span> is contained in <span class="math notranslate nohighlight">\(g(\beta)\)</span>. As a result, for every <span class="math notranslate nohighlight">\(x\)</span> in the complement of <span class="math notranslate nohighlight">\(A\)</span>, there is a <span class="math notranslate nohighlight">\(y\)</span> such that <span class="math notranslate nohighlight">\(g(y) = x\)</span>. (By the injectivity of <span class="math notranslate nohighlight">\(g\)</span>, this <span class="math notranslate nohighlight">\(y\)</span> is unique, but next theorem says only that <code class="docutils literal notranslate"><span class="pre">inv_fun</span> <span class="pre">g</span> <span class="pre">x</span></code> returns some <code class="docutils literal notranslate"><span class="pre">y</span></code> such that <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">y</span> <span class="pre">=</span> <span class="pre">x</span></code>.)</p> <p>Step through the proof below, make sure you understand what is going on, and fill in the remaining parts. You will need to use <code class="docutils literal notranslate"><span class="pre">inv_fun_eq</span></code> at the end. Notice that rewriting with <code class="docutils literal notranslate"><span class="pre">sb_aux</span></code> here replaces <code class="docutils literal notranslate"><span class="pre">sb_aux</span> <span class="pre">f</span> <span class="pre">g</span> <span class="pre">0</span></code> with the right-hand side of the corresponding defining equation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">sb_right_inv</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">}</span> <span class="o">(</span><span class="n">hx</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∉</span> <span class="n">sbSet</span> <span class="n">f</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">g</span> <span class="o">(</span><span class="n">invFun</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">g</span> <span class="bp">''</span> <span class="n">univ</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">hx</span> <span class="n">rw</span> <span class="o">[</span><span class="n">sbSet</span><span class="o">,</span> <span class="n">mem_iUnion</span><span class="o">]</span> <span class="n">use</span> <span class="mi">0</span> <span class="n">rw</span> <span class="o">[</span><span class="n">sbAux</span><span class="o">,</span> <span class="n">mem_diff</span><span class="o">]</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">y</span><span class="o">,</span> <span class="n">g</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="gr">sorry</span> </pre></div> </div> <p>We now turn to the proof that <span class="math notranslate nohighlight">\(h\)</span> is injective. Informally, the proof goes as follows. First, suppose <span class="math notranslate nohighlight">\(h(x_1) = h(x_2)\)</span>. If <span class="math notranslate nohighlight">\(x_1\)</span> is in <span class="math notranslate nohighlight">\(A\)</span>, then <span class="math notranslate nohighlight">\(h(x_1) = f(x_1)\)</span>, and we can show that <span class="math notranslate nohighlight">\(x_2\)</span> is in <span class="math notranslate nohighlight">\(A\)</span> as follows. If it isn’t, then we have <span class="math notranslate nohighlight">\(h(x_2) = g^{-1}(x_2)\)</span>. From <span class="math notranslate nohighlight">\(f(x_1) = h(x_1) = h(x_2)\)</span> we have <span class="math notranslate nohighlight">\(g(f(x_1)) = x_2\)</span>. From the definition of <span class="math notranslate nohighlight">\(A\)</span>, since <span class="math notranslate nohighlight">\(x_1\)</span> is in <span class="math notranslate nohighlight">\(A\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span> is in <span class="math notranslate nohighlight">\(A\)</span> as well, a contradiction. Hence, if <span class="math notranslate nohighlight">\(x_1\)</span> is in <span class="math notranslate nohighlight">\(A\)</span>, so is <span class="math notranslate nohighlight">\(x_2\)</span>, in which case we have <span class="math notranslate nohighlight">\(f(x_1) = h(x_1) = h(x_2) = f(x_2)\)</span>. The injectivity of <span class="math notranslate nohighlight">\(f\)</span> then implies <span class="math notranslate nohighlight">\(x_1 = x_2\)</span>. The symmetric argument shows that if <span class="math notranslate nohighlight">\(x_2\)</span> is in <span class="math notranslate nohighlight">\(A\)</span>, then so is <span class="math notranslate nohighlight">\(x_1\)</span>, which again implies <span class="math notranslate nohighlight">\(x_1 = x_2\)</span>.</p> <p>The only remaining possibility is that neither <span class="math notranslate nohighlight">\(x_1\)</span> nor <span class="math notranslate nohighlight">\(x_2\)</span> is in <span class="math notranslate nohighlight">\(A\)</span>. In that case, we have <span class="math notranslate nohighlight">\(g^{-1}(x_1) = h(x_1) = h(x_2) = g^{-1}(x_2)\)</span>. Applying <span class="math notranslate nohighlight">\(g\)</span> to both sides yields <span class="math notranslate nohighlight">\(x_1 = x_2\)</span>.</p> <p>Once again, we encourage you to step through the following proof to see how the argument plays out in Lean. See if you can finish off the proof using <code class="docutils literal notranslate"><span class="pre">sb_right_inv</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">sb_injective</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">hg</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="o">(</span><span class="n">sbFun</span> <span class="n">f</span> <span class="n">g</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">set</span> <span class="n">A</span> <span class="o">:=</span> <span class="n">sbSet</span> <span class="n">f</span> <span class="n">g</span> <span class="k">with</span> <span class="n">A_def</span> <span class="n">set</span> <span class="n">h</span> <span class="o">:=</span> <span class="n">sbFun</span> <span class="n">f</span> <span class="n">g</span> <span class="k">with</span> <span class="n">h_def</span> <span class="n">intro</span> <span class="n">x₁</span> <span class="n">x₂</span> <span class="n">intro</span> <span class="o">(</span><span class="n">hxeq</span> <span class="o">:</span> <span class="n">h</span> <span class="n">x₁</span> <span class="bp">=</span> <span class="n">h</span> <span class="n">x₂</span><span class="o">)</span> <span class="k">show</span> <span class="n">x₁</span> <span class="bp">=</span> <span class="n">x₂</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">h_def</span><span class="o">,</span> <span class="n">sbFun</span><span class="o">,</span> <span class="bp">←</span> <span class="n">A_def</span><span class="o">]</span> <span class="n">at</span> <span class="n">hxeq</span> <span class="n">by_cases</span> <span class="n">xA</span> <span class="o">:</span> <span class="n">x₁</span> <span class="bp">∈</span> <span class="n">A</span> <span class="bp">∨</span> <span class="n">x₂</span> <span class="bp">∈</span> <span class="n">A</span> <span class="bp">·</span> <span class="n">wlog</span> <span class="n">x₁A</span> <span class="o">:</span> <span class="n">x₁</span> <span class="bp">∈</span> <span class="n">A</span> <span class="n">generalizing</span> <span class="n">x₁</span> <span class="n">x₂</span> <span class="n">hxeq</span> <span class="n">xA</span> <span class="bp">·</span> <span class="n">symm</span> <span class="n">apply</span> <span class="n">this</span> <span class="n">hxeq.symm</span> <span class="n">xA.symm</span> <span class="o">(</span><span class="n">xA.resolve_left</span> <span class="n">x₁A</span><span class="o">)</span> <span class="k">have</span> <span class="n">x₂A</span> <span class="o">:</span> <span class="n">x₂</span> <span class="bp">∈</span> <span class="n">A</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">not_imp_self.mp</span> <span class="n">intro</span> <span class="o">(</span><span class="n">x₂nA</span> <span class="o">:</span> <span class="n">x₂</span> <span class="bp">∉</span> <span class="n">A</span><span class="o">)</span> <span class="n">rw</span> <span class="o">[</span><span class="n">if_pos</span> <span class="n">x₁A</span><span class="o">,</span> <span class="n">if_neg</span> <span class="n">x₂nA</span><span class="o">]</span> <span class="n">at</span> <span class="n">hxeq</span> <span class="n">rw</span> <span class="o">[</span><span class="n">A_def</span><span class="o">,</span> <span class="n">sbSet</span><span class="o">,</span> <span class="n">mem_iUnion</span><span class="o">]</span> <span class="n">at</span> <span class="n">x₁A</span> <span class="k">have</span> <span class="n">x₂eq</span> <span class="o">:</span> <span class="n">x₂</span> <span class="bp">=</span> <span class="n">g</span> <span class="o">(</span><span class="n">f</span> <span class="n">x₁</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">rcases</span> <span class="n">x₁A</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">n</span><span class="o">,</span> <span class="n">hn</span><span class="o">⟩</span> <span class="n">rw</span> <span class="o">[</span><span class="n">A_def</span><span class="o">,</span> <span class="n">sbSet</span><span class="o">,</span> <span class="n">mem_iUnion</span><span class="o">]</span> <span class="n">use</span> <span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span> <span class="n">simp</span> <span class="o">[</span><span class="n">sbAux</span><span class="o">]</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">x₁</span><span class="o">,</span> <span class="n">hn</span><span class="o">,</span> <span class="n">x₂eq.symm</span><span class="o">⟩</span> <span class="gr">sorry</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">xA</span> <span class="gr">sorry</span> </pre></div> </div> <p>The proof introduces some new tactics. To start with, notice the <code class="docutils literal notranslate"><span class="pre">set</span></code> tactic, which introduces abbreviations <code class="docutils literal notranslate"><span class="pre">A</span></code> and <code class="docutils literal notranslate"><span class="pre">h</span></code> for <code class="docutils literal notranslate"><span class="pre">sb_set</span> <span class="pre">f</span> <span class="pre">g</span></code> and <code class="docutils literal notranslate"><span class="pre">sb_fun</span> <span class="pre">f</span> <span class="pre">g</span></code> respectively. We name the corresponding defining equations <code class="docutils literal notranslate"><span class="pre">A_def</span></code> and <code class="docutils literal notranslate"><span class="pre">h_def</span></code>. The abbreviations are definitional, which is to say, Lean will sometimes unfold them automatically when needed. But not always; for example, when using <code class="docutils literal notranslate"><span class="pre">rw</span></code>, we generally need to use <code class="docutils literal notranslate"><span class="pre">A_def</span></code> and <code class="docutils literal notranslate"><span class="pre">h_def</span></code> explicitly. So the definitions bring a tradeoff: they can make expressions shorter and more readable, but they sometimes require us to do more work.</p> <p>A more interesting tactic is the <code class="docutils literal notranslate"><span class="pre">wlog</span></code> tactic, which encapsulates the symmetry argument in the informal proof above. We will not dwell on it now, but notice that it does exactly what we want. If you hover over the tactic you can take a look at its documentation.</p> <p>The argument for surjectivity is even easier. Given <span class="math notranslate nohighlight">\(y\)</span> in <span class="math notranslate nohighlight">\(\beta\)</span>, we consider two cases, depending on whether <span class="math notranslate nohighlight">\(g(y)\)</span> is in <span class="math notranslate nohighlight">\(A\)</span>. If it is, it can’t be in <span class="math notranslate nohighlight">\(S_0\)</span>, the outermost ring, because by definition that is disjoint from the image of <span class="math notranslate nohighlight">\(g\)</span>. Thus it is an element of <span class="math notranslate nohighlight">\(S_{n+1}\)</span> for some <span class="math notranslate nohighlight">\(n\)</span>. This means that it is of the form <span class="math notranslate nohighlight">\(g(f(x))\)</span> for some <span class="math notranslate nohighlight">\(x\)</span> in <span class="math notranslate nohighlight">\(S_n\)</span>. By the injectivity of <span class="math notranslate nohighlight">\(g\)</span>, we have <span class="math notranslate nohighlight">\(f(x) = y\)</span>. In the case where <span class="math notranslate nohighlight">\(g(y)\)</span> is in the complement of <span class="math notranslate nohighlight">\(A\)</span>, we immediately have <span class="math notranslate nohighlight">\(h(g(y))= y\)</span>, and we are done.</p> <p>Once again, we encourage you to step through the proof and fill in the missing parts. The tactic <code class="docutils literal notranslate"><span class="pre">cases</span> <span class="pre">n</span> <span class="pre">with</span> <span class="pre">n</span></code> splits on the cases <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">y</span> <span class="pre">∈</span> <span class="pre">sb_aux</span> <span class="pre">f</span> <span class="pre">g</span> <span class="pre">0</span></code> and <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">y</span> <span class="pre">∈</span> <span class="pre">sb_aux</span> <span class="pre">f</span> <span class="pre">g</span> <span class="pre">n.succ</span></code>. In both cases, calling the simplifier with <code class="docutils literal notranslate"><span class="pre">simp</span> <span class="pre">[sb_aux]</span></code> applies the corresponding defining equation of <code class="docutils literal notranslate"><span class="pre">sb_aux</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">sb_surjective</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">hg</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="o">(</span><span class="n">sbFun</span> <span class="n">f</span> <span class="n">g</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">set</span> <span class="n">A</span> <span class="o">:=</span> <span class="n">sbSet</span> <span class="n">f</span> <span class="n">g</span> <span class="k">with</span> <span class="n">A_def</span> <span class="n">set</span> <span class="n">h</span> <span class="o">:=</span> <span class="n">sbFun</span> <span class="n">f</span> <span class="n">g</span> <span class="k">with</span> <span class="n">h_def</span> <span class="n">intro</span> <span class="n">y</span> <span class="n">by_cases</span> <span class="n">gyA</span> <span class="o">:</span> <span class="n">g</span> <span class="n">y</span> <span class="bp">∈</span> <span class="n">A</span> <span class="bp">·</span> <span class="n">rw</span> <span class="o">[</span><span class="n">A_def</span><span class="o">,</span> <span class="n">sbSet</span><span class="o">,</span> <span class="n">mem_iUnion</span><span class="o">]</span> <span class="n">at</span> <span class="n">gyA</span> <span class="n">rcases</span> <span class="n">gyA</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">n</span><span class="o">,</span> <span class="n">hn</span><span class="o">⟩</span> <span class="n">cases'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="bp">·</span> <span class="n">simp</span> <span class="o">[</span><span class="n">sbAux</span><span class="o">]</span> <span class="n">at</span> <span class="n">hn</span> <span class="n">simp</span> <span class="o">[</span><span class="n">sbAux</span><span class="o">]</span> <span class="n">at</span> <span class="n">hn</span> <span class="n">rcases</span> <span class="n">hn</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">xmem</span><span class="o">,</span> <span class="n">hx</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">x</span> <span class="k">have</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">A</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">A_def</span><span class="o">,</span> <span class="n">sbSet</span><span class="o">,</span> <span class="n">mem_iUnion</span><span class="o">]</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">n</span><span class="o">,</span> <span class="n">xmem</span><span class="o">⟩</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">h_def</span><span class="o">,</span> <span class="n">sbFun</span><span class="o">,</span> <span class="n">if_pos</span> <span class="n">this</span><span class="o">]</span> <span class="n">exact</span> <span class="n">hg</span> <span class="n">hx</span> <span class="gr">sorry</span> </pre></div> </div> <p>We can now put it all together. The final statement is short and sweet, and the proof uses the fact that <code class="docutils literal notranslate"><span class="pre">Bijective</span> <span class="pre">h</span></code> unfolds to <code class="docutils literal notranslate"><span class="pre">Injective</span> <span class="pre">h</span> <span class="pre">∧</span> <span class="pre">Surjective</span> <span class="pre">h</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">schroeder_bernstein</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">}</span> <span class="o">{</span><span class="n">g</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">α</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">hg</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">h</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">,</span> <span class="n">Bijective</span> <span class="n">h</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">sbFun</span> <span class="n">f</span> <span class="n">g</span><span class="o">,</span> <span class="n">sb_injective</span> <span class="n">f</span> <span class="n">g</span> <span class="n">hf</span> <span class="n">hg</span><span class="o">,</span> <span class="n">sb_surjective</span> <span class="n">f</span> <span class="n">g</span> <span class="n">hf</span> <span class="n">hg</span><span class="o">⟩</span> </pre></div> </div> </section> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C03_Logic.html" class="btn btn-neutral float-left" title="3. Logic" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> <a href="C05_Number_Theory.html" class="btn btn-neutral float-right" title="5. Number Theory" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> </div> <hr/> <div role="contentinfo"> <p>© Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot.</p> </div> Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>. </footer> </div> </div> </section> </div> <script> jQuery(function () { SphinxRtdTheme.Navigation.enable(true); }); </script> </body> </html>
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Number Theory — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> <link rel="shortcut icon" href="_static/favicon.ico"/> <!--[if lt IE 9]> <script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" /> <link rel="next" title="6. Structures" href="C06_Structures.html" /> <link rel="prev" title="4. Sets and Functions" href="C04_Sets_and_Functions.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul class="current"> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">5. Number Theory</a><ul> <li class="toctree-l2"><a class="reference internal" href="#irrational-roots">5.1. Irrational Roots</a></li> <li class="toctree-l2"><a class="reference internal" href="#induction-and-recursion">5.2. Induction and Recursion</a></li> <li class="toctree-l2"><a class="reference internal" href="#id2">5.3. Induction and Recursion</a></li> <li class="toctree-l2"><a class="reference internal" href="#infinitely-many-primes">5.4. Infinitely Many Primes</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">5. </span>Number Theory</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C05_Number_Theory.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="number-theory"> <span id="id1"></span><h1><span class="section-number">5. </span>Number Theory<a class="headerlink" href="#number-theory" title="Permalink to this heading"></a></h1> <p>In this chapter, we show you how to formalize some elementary results in number theory. As we deal with more substantive mathematical content, the proofs will get longer and more involved, building on the skills you have already mastered.</p> <section id="irrational-roots"> <span id="section-irrational-roots"></span><h2><span class="section-number">5.1. </span>Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this heading"></a></h2> <p>Let’s start with a fact known to the ancient greeks, namely, that the square root of 2 is irrational. If we suppose otherwise, we can write <span class="math notranslate nohighlight">\(\sqrt{2} = a / b\)</span> as a fraction in lowest terms. Squaring both sides yields <span class="math notranslate nohighlight">\(a^2 = 2 b^2\)</span>, which implies that <span class="math notranslate nohighlight">\(a\)</span> is even. If we write <span class="math notranslate nohighlight">\(a = 2c\)</span>, then we get <span class="math notranslate nohighlight">\(4c^2 = 2 b^2\)</span> and hence <span class="math notranslate nohighlight">\(b^2 = 2 c^2\)</span>. This implies that <span class="math notranslate nohighlight">\(b\)</span> is also even, contradicting the fact that we have assumed that <span class="math notranslate nohighlight">\(a / b\)</span> has been reduced to lowest terms.</p> <p>Saying that <span class="math notranslate nohighlight">\(a / b\)</span> is a fraction in lowest terms means that <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> do not have any factors in common, which is to say, they are <em>coprime</em>. Mathlib defines the predicate <code class="docutils literal notranslate"><span class="pre">Nat.coprime</span> <span class="pre">m</span> <span class="pre">n</span></code> to be <code class="docutils literal notranslate"><span class="pre">Nat.gcd</span> <span class="pre">m</span> <span class="pre">n</span> <span class="pre">=</span> <span class="pre">1</span></code>. Using Lean’s anonymous projection notation, if <code class="docutils literal notranslate"><span class="pre">s</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span></code> are expressions of type <code class="docutils literal notranslate"><span class="pre">Nat</span></code>, we can write <code class="docutils literal notranslate"><span class="pre">s.coprime</span> <span class="pre">t</span></code> instead of <code class="docutils literal notranslate"><span class="pre">Nat.coprime</span> <span class="pre">s</span> <span class="pre">t</span></code>, and similarly for <code class="docutils literal notranslate"><span class="pre">Nat.gcd</span></code>. As usual, Lean will often unfold the definition of <code class="docutils literal notranslate"><span class="pre">Nat.coprime</span></code> automatically when necessary, but we can also do it manually by rewriting or simplifying with the identifier <code class="docutils literal notranslate"><span class="pre">Nat.coprime</span></code>. The <code class="docutils literal notranslate"><span class="pre">norm_num</span></code> tactic is smart enough to compute concrete values.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#print</span> <span class="n">Nat.coprime</span> <span class="kd">example</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m.coprime</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">m.gcd</span> <span class="n">n</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m.coprime</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">m.gcd</span> <span class="n">n</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Nat.coprime</span><span class="o">]</span> <span class="n">at</span> <span class="n">h</span> <span class="n">exact</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Nat.coprime</span> <span class="mi">12</span> <span class="mi">7</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">norm_num</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Nat.gcd</span> <span class="mi">12</span> <span class="mi">8</span> <span class="bp">=</span> <span class="mi">4</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">norm_num</span> </pre></div> </div> <p>We have already encountered the <code class="docutils literal notranslate"><span class="pre">gcd</span></code> function in <a class="reference internal" href="C02_Basics.html#more-on-order-and-divisibility"><span class="std std-numref">Section 2.4</span></a>. There is also a version of <code class="docutils literal notranslate"><span class="pre">gcd</span></code> for the integers; we will return to a discussion of the relationship between different number systems below. There are even a generic <code class="docutils literal notranslate"><span class="pre">gcd</span></code> function and generic notions of <code class="docutils literal notranslate"><span class="pre">Prime</span></code> and <code class="docutils literal notranslate"><span class="pre">coprime</span></code> that make sense in general classes of algebraic structures. We will come to understand how Lean manages this generality in the next chapter. In the meanwhile, in this section, we will restrict attention to the natural numbers.</p> <p>We also need the notion of a prime number, <code class="docutils literal notranslate"><span class="pre">Nat.Prime</span></code>. The theorem <code class="docutils literal notranslate"><span class="pre">Nat.prime_def_lt</span></code> provides one familiar characterization, and <code class="docutils literal notranslate"><span class="pre">Nat.Prime.eq_one_or_self_of_dvd</span></code> provides another.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">@</span><span class="n">Nat.prime_def_lt</span> <span class="kd">example</span> <span class="o">(</span><span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="n">p</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">p</span> <span class="bp">∧</span> <span class="bp">∀</span> <span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">m</span> <span class="bp"><</span> <span class="n">p</span> <span class="bp">→</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">p</span> <span class="bp">→</span> <span class="n">m</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rwa</span> <span class="o">[</span><span class="n">Nat.prime_def_lt</span><span class="o">]</span> <span class="n">at</span> <span class="n">prime_p</span> <span class="k">#check</span> <span class="n">Nat.Prime.eq_one_or_self_of_dvd</span> <span class="kd">example</span> <span class="o">(</span><span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="n">p</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">p</span> <span class="bp">→</span> <span class="n">m</span> <span class="bp">=</span> <span class="mi">1</span> <span class="bp">∨</span> <span class="n">m</span> <span class="bp">=</span> <span class="n">p</span> <span class="o">:=</span> <span class="n">prime_p.eq_one_or_self_of_dvd</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="mi">17</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">norm_num</span> <span class="c1">-- commonly used</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="mi">2</span> <span class="o">:=</span> <span class="n">Nat.prime_two</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="mi">3</span> <span class="o">:=</span> <span class="n">Nat.prime_three</span> </pre></div> </div> <p>In the natural numbers, a prime number has the property that it cannot be written as a product of nontrivial factors. In a broader mathematical context, an element of a ring that has this property is said to be <em>irreducible</em>. An element of a ring is said to be <em>prime</em> if whenever it divides a product, it divides one of the factors. It is an important property of the natural numbers that in that setting the two notions coincide, giving rise to the theorem <code class="docutils literal notranslate"><span class="pre">Nat.Prime.dvd_mul</span></code>.</p> <p>We can use this fact to establish a key property in the argument above: if the square of a number is even, then that number is even as well. Mathlib defines the predicate <code class="docutils literal notranslate"><span class="pre">Even</span></code> in <code class="docutils literal notranslate"><span class="pre">Data.Nat.Parity</span></code>, but for reasons that will become clear below, we will simply use <code class="docutils literal notranslate"><span class="pre">2</span> <span class="pre">∣</span> <span class="pre">m</span></code> to express that <code class="docutils literal notranslate"><span class="pre">m</span></code> is even.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">@</span><span class="n">Nat.Prime.dvd_mul</span> <span class="k">#check</span> <span class="n">Nat.Prime.dvd_mul</span> <span class="n">Nat.prime_two</span> <span class="k">#check</span> <span class="n">Nat.prime_two.dvd_mul</span> <span class="kd">theorem</span> <span class="n">even_of_even_sqr</span> <span class="o">{</span><span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">m</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">pow_two</span><span class="o">,</span> <span class="n">Nat.prime_two.dvd_mul</span><span class="o">]</span> <span class="n">at</span> <span class="n">h</span> <span class="n">cases</span> <span class="n">h</span> <span class="bp"><;></span> <span class="n">assumption</span> <span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">m</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">m</span> <span class="o">:=</span> <span class="n">Nat.Prime.dvd_of_dvd_pow</span> <span class="n">Nat.prime_two</span> <span class="n">h</span> </pre></div> </div> <p>As we proceed, you will need to become proficient at finding the facts you need. Remember that if you can guess the prefix of the name and you have imported the relevant library, you can use tab completion (sometimes with <code class="docutils literal notranslate"><span class="pre">ctrl-tab</span></code>) to find what you are looking for. You can use <code class="docutils literal notranslate"><span class="pre">ctrl-click</span></code> on any identifier to jump to the file where it is defined, which enables you to browse definitions and theorems nearby. You can also use the search engine on the <a class="reference external" href="https://leanprover-community.github.io/">Lean community web pages</a>, and if all else fails, don’t hesitate to ask on <a class="reference external" href="https://leanprover.zulipchat.com/">Zulip</a>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> <span class="c1">-- library_search suggests the following:</span> <span class="o">(</span><span class="n">mul_right_inj'</span> <span class="n">h'</span><span class="o">)</span><span class="bp">.</span><span class="n">mp</span> <span class="n">h</span> </pre></div> </div> <p>The heart of our proof of the irrationality of the square root of two is contained in the following theorem. See if you can fill out the proof sketch, using <code class="docutils literal notranslate"><span class="pre">even_of_even_sqr</span></code> and the theorem <code class="docutils literal notranslate"><span class="pre">Nat.dvd_gcd</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">coprime_mn</span> <span class="o">:</span> <span class="n">m.coprime</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">≠</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">sqr_eq</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">obtain</span> <span class="o">⟨</span><span class="n">k</span><span class="o">,</span> <span class="n">meq</span><span class="o">⟩</span> <span class="o">:=</span> <span class="n">dvd_iff_exists_eq_mul_left.mp</span> <span class="n">this</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="mi">2</span> <span class="bp">*</span> <span class="n">k</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">=</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">sqr_eq</span><span class="o">,</span> <span class="n">meq</span><span class="o">]</span> <span class="n">ring</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">k</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="n">n</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="n">m.gcd</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">∣</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">norm_num</span> <span class="n">at</span> <span class="n">this</span> </pre></div> </div> <p>In fact, with very few changes, we can replace <code class="docutils literal notranslate"><span class="pre">2</span></code> by an arbitrary prime. Give it a try in the next example. At the end of the proof, you’ll need to derive a contradiction from <code class="docutils literal notranslate"><span class="pre">p</span> <span class="pre">∣</span> <span class="pre">1</span></code>. You can use <code class="docutils literal notranslate"><span class="pre">Nat.Prime.two_le</span></code>, which says that any prime number is greater than or equal to two, and <code class="docutils literal notranslate"><span class="pre">Nat.le_of_dvd</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">coprime_mn</span> <span class="o">:</span> <span class="n">m.coprime</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">p.Prime</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">≠</span> <span class="n">p</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Let us consider another approach. Here is a quick proof that if <span class="math notranslate nohighlight">\(p\)</span> is prime, then <span class="math notranslate nohighlight">\(m^2 \ne p n^2\)</span>: if we assume <span class="math notranslate nohighlight">\(m^2 = p n^2\)</span> and consider the factorization of <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(n\)</span> into primes, then <span class="math notranslate nohighlight">\(p\)</span> occurs an even number of times on the left side of the equation and an odd number of times on the right, a contradiction. Note that this argument requires that <span class="math notranslate nohighlight">\(n\)</span> and hence <span class="math notranslate nohighlight">\(m\)</span> are not equal to zero. The formalization below confirms that this assumption is sufficient.</p> <p>The unique factorization theorem says that any natural number other than zero can be written as the product of primes in a unique way. Mathlib contains a formal version of this, expressed in terms of a function <code class="docutils literal notranslate"><span class="pre">Nat.factors</span></code>, which returns the list of prime factors of a number in nondecreasing order. The library proves that all the elements of <code class="docutils literal notranslate"><span class="pre">Nat.factors</span> <span class="pre">n</span></code> are prime, that any <code class="docutils literal notranslate"><span class="pre">n</span></code> greater than zero is equal to the product of its factors, and that if <code class="docutils literal notranslate"><span class="pre">n</span></code> is equal to the product of another list of prime numbers, then that list is a permutation of <code class="docutils literal notranslate"><span class="pre">Nat.factors</span> <span class="pre">n</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">Nat.factors</span> <span class="k">#check</span> <span class="n">Nat.prime_of_mem_factors</span> <span class="k">#check</span> <span class="n">Nat.prod_factors</span> <span class="k">#check</span> <span class="n">Nat.factors_unique</span> </pre></div> </div> <p>You can browse these theorems and others nearby, even though we have not talked about list membership, products, or permutations yet. We won’t need any of that for the task at hand. We will instead use the fact that Mathlib has a function <code class="docutils literal notranslate"><span class="pre">Nat.factorization</span></code>, that represents the same data as a function. Specifically, <code class="docutils literal notranslate"><span class="pre">Nat.factorization</span> <span class="pre">n</span> <span class="pre">p</span></code>, which we can also write <code class="docutils literal notranslate"><span class="pre">n.factorization</span> <span class="pre">p</span></code>, returns the multiplicity of <code class="docutils literal notranslate"><span class="pre">p</span></code> in the prime factorization of <code class="docutils literal notranslate"><span class="pre">n</span></code>. We will use the following three facts.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">factorization_mul'</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">mnez</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">nnez</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="n">factorization</span> <span class="n">p</span> <span class="bp">=</span> <span class="n">m.factorization</span> <span class="n">p</span> <span class="bp">+</span> <span class="n">n.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Nat.factorization_mul</span> <span class="n">mnez</span> <span class="n">nnez</span><span class="o">]</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">factorization_pow'</span> <span class="o">(</span><span class="n">n</span> <span class="n">k</span> <span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">n</span> <span class="bp">^</span> <span class="n">k</span><span class="o">)</span><span class="bp">.</span><span class="n">factorization</span> <span class="n">p</span> <span class="bp">=</span> <span class="n">k</span> <span class="bp">*</span> <span class="n">n.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Nat.factorization_pow</span><span class="o">]</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">Nat.Prime.factorization'</span> <span class="o">{</span><span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">p.Prime</span><span class="o">)</span> <span class="o">:</span> <span class="n">p.factorization</span> <span class="n">p</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">prime_p.factorization</span><span class="o">]</span> <span class="n">simp</span> </pre></div> </div> <p>In fact, <code class="docutils literal notranslate"><span class="pre">n.factorization</span></code> is defined in Lean as a function of finite support, which explains the strange notation you will see as you step through the proofs above. Don’t worry about this now. For our purposes here, we can use the three theorems above as a black box.</p> <p>The next example shows that the simplifier is smart enough to replace <code class="docutils literal notranslate"><span class="pre">n^2</span> <span class="pre">≠</span> <span class="pre">0</span></code> by <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">≠</span> <span class="pre">0</span></code>. The tactic <code class="docutils literal notranslate"><span class="pre">simpa</span></code> just calls <code class="docutils literal notranslate"><span class="pre">simp</span></code> followed by <code class="docutils literal notranslate"><span class="pre">assumption</span></code>.</p> <p>See if you can use the identities above to fill in the missing parts of the proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">nnz</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">p.Prime</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">≠</span> <span class="n">p</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">sqr_eq</span> <span class="k">have</span> <span class="n">nsqr_nez</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">≠</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simpa</span> <span class="k">have</span> <span class="n">eq1</span> <span class="o">:</span> <span class="n">Nat.factorization</span> <span class="o">(</span><span class="n">m</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="n">p</span> <span class="bp">=</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">m.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">eq2</span> <span class="o">:</span> <span class="o">(</span><span class="n">p</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span><span class="bp">.</span><span class="n">factorization</span> <span class="n">p</span> <span class="bp">=</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">n.factorization</span> <span class="n">p</span> <span class="bp">+</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">m.factorization</span> <span class="n">p</span> <span class="bp">%</span> <span class="mi">2</span> <span class="bp">=</span> <span class="o">(</span><span class="mi">2</span> <span class="bp">*</span> <span class="n">n.factorization</span> <span class="n">p</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">%</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">eq1</span><span class="o">,</span> <span class="n">sqr_eq</span><span class="o">,</span> <span class="n">eq2</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_comm</span><span class="o">,</span> <span class="n">Nat.add_mul_mod_self_left</span><span class="o">,</span> <span class="n">Nat.mul_mod_right</span><span class="o">]</span> <span class="n">at</span> <span class="n">this</span> <span class="n">norm_num</span> <span class="n">at</span> <span class="n">this</span> </pre></div> </div> <p>A nice thing about this proof is that it also generalizes. There is nothing special about <code class="docutils literal notranslate"><span class="pre">2</span></code>; with small changes, the proof shows that whenever we write <code class="docutils literal notranslate"><span class="pre">m^k</span> <span class="pre">=</span> <span class="pre">r</span> <span class="pre">*</span> <span class="pre">n^k</span></code>, the multiplicity of any prime <code class="docutils literal notranslate"><span class="pre">p</span></code> in <code class="docutils literal notranslate"><span class="pre">r</span></code> has to be a multiple of <code class="docutils literal notranslate"><span class="pre">k</span></code>.</p> <p>To use <code class="docutils literal notranslate"><span class="pre">Nat.count_factors_mul_of_pos</span></code> with <code class="docutils literal notranslate"><span class="pre">r</span> <span class="pre">*</span> <span class="pre">n^k</span></code>, we need to know that <code class="docutils literal notranslate"><span class="pre">r</span></code> is positive. But when <code class="docutils literal notranslate"><span class="pre">r</span></code> is zero, the theorem below is trivial, and easily proved by the simplifier. So the proof is carried out in cases. The line <code class="docutils literal notranslate"><span class="pre">cases</span> <span class="pre">r</span> <span class="pre">with</span> <span class="pre">r</span></code> replaces the goal with two versions: one in which <code class="docutils literal notranslate"><span class="pre">r</span></code> is replaced by <code class="docutils literal notranslate"><span class="pre">0</span></code>, and the other in which <code class="docutils literal notranslate"><span class="pre">r</span></code> is replaces by <code class="docutils literal notranslate"><span class="pre">r.succ</span></code>, the successor of <code class="docutils literal notranslate"><span class="pre">r</span></code>. In the second case, we can use the theorem <code class="docutils literal notranslate"><span class="pre">r.succ_ne_zero</span></code>, which establishes <code class="docutils literal notranslate"><span class="pre">r.succ</span> <span class="pre">≠</span> <span class="pre">0</span></code>.</p> <p>Notice also that the line that begins <code class="docutils literal notranslate"><span class="pre">have</span> <span class="pre">:</span> <span class="pre">npow_nz</span></code> provides a short proof-term proof of <code class="docutils literal notranslate"><span class="pre">n^k</span> <span class="pre">≠</span> <span class="pre">0</span></code>. To understand how it works, try replacing it with a tactic proof, and then think about how the tactics describe the proof term.</p> <p>See if you can fill in the missing parts of the proof below. At the very end, you can use <code class="docutils literal notranslate"><span class="pre">Nat.dvd_sub'</span></code> and <code class="docutils literal notranslate"><span class="pre">Nat.dvd_mul_right</span></code> to finish it off.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="n">k</span> <span class="n">r</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">nnz</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">pow_eq</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">^</span> <span class="n">k</span> <span class="bp">=</span> <span class="n">r</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">^</span> <span class="n">k</span><span class="o">)</span> <span class="o">{</span><span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">p.Prime</span><span class="o">)</span> <span class="o">:</span> <span class="n">k</span> <span class="bp">∣</span> <span class="n">r.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases'</span> <span class="n">r</span> <span class="k">with</span> <span class="n">r</span> <span class="bp">·</span> <span class="n">simp</span> <span class="k">have</span> <span class="n">npow_nz</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">^</span> <span class="n">k</span> <span class="bp">≠</span> <span class="mi">0</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">npowz</span> <span class="bp">=></span> <span class="n">nnz</span> <span class="o">(</span><span class="n">pow_eq_zero</span> <span class="n">npowz</span><span class="o">)</span> <span class="k">have</span> <span class="n">eq1</span> <span class="o">:</span> <span class="o">(</span><span class="n">m</span> <span class="bp">^</span> <span class="n">k</span><span class="o">)</span><span class="bp">.</span><span class="n">factorization</span> <span class="n">p</span> <span class="bp">=</span> <span class="n">k</span> <span class="bp">*</span> <span class="n">m.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">eq2</span> <span class="o">:</span> <span class="o">(</span><span class="n">r.succ</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">^</span> <span class="n">k</span><span class="o">)</span><span class="bp">.</span><span class="n">factorization</span> <span class="n">p</span> <span class="bp">=</span> <span class="n">k</span> <span class="bp">*</span> <span class="n">n.factorization</span> <span class="n">p</span> <span class="bp">+</span> <span class="n">r.succ.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="n">r.succ.factorization</span> <span class="n">p</span> <span class="bp">=</span> <span class="n">k</span> <span class="bp">*</span> <span class="n">m.factorization</span> <span class="n">p</span> <span class="bp">-</span> <span class="n">k</span> <span class="bp">*</span> <span class="n">n.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">eq1</span><span class="o">,</span> <span class="n">pow_eq</span><span class="o">,</span> <span class="n">eq2</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">,</span> <span class="n">Nat.add_sub_cancel</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">this</span><span class="o">]</span> <span class="gr">sorry</span> </pre></div> </div> <p>There are a number of ways in which we might want to improve on these results. To start with, a proof that the square root of two is irrational should say something about the square root of two, which can be understood as an element of the real or complex numbers. And stating that it is irrational should say something about the rational numbers, namely, that no rational number is equal to it. Moreover, we should extend the theorems in this section to the integers. Although it is mathematically obvious that if we could write the square root of two as a quotient of two integers then we could write it as a quotient of two natural numbers, proving this formally requires some effort.</p> <p>In Mathlib, the natural numbers, the integers, the rationals, the reals, and the complex numbers are represented by separate data types. Restricting attention to the separate domains is often helpful: we will see that it is easy to do induction on the natural numbers, and it is easiest to reason about divisibility of integers when the real numbers are not part of the picture. But having to mediate between the different domains is a headache, one we will have to contend with. We will return to this issue later in this chapter.</p> <p>We should also expect to be able to strengthen the conclusion of the last theorem to say that the number <code class="docutils literal notranslate"><span class="pre">r</span></code> is a <code class="docutils literal notranslate"><span class="pre">k</span></code>-th power, since its <code class="docutils literal notranslate"><span class="pre">k</span></code>-th root is just the product of each prime dividing <code class="docutils literal notranslate"><span class="pre">r</span></code> raised to its multiplicity in <code class="docutils literal notranslate"><span class="pre">r</span></code> divided by <code class="docutils literal notranslate"><span class="pre">k</span></code>. To be able to do that we will need better means for reasoning about products and sums over a finite set, which is also a topic we will return to.</p> <p>In fact, the results in this section are all established in much greater generality in mathlib, in <code class="docutils literal notranslate"><span class="pre">Data.Real.Irrational</span></code>. The notion of <code class="docutils literal notranslate"><span class="pre">multiplicity</span></code> is defined for an arbitrary commutative monoid, and that it takes values in the extended natural numbers <code class="docutils literal notranslate"><span class="pre">enat</span></code>, which adds the value infinity to the natural numbers. In the next chapter, we will begin to develop the means to appreciate the way that Lean supports this sort of generality.</p> </section> <section id="induction-and-recursion"> <span id="section-induction-and-recursion"></span><h2><span class="section-number">5.2. </span>Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this heading"></a></h2> <p>The set of natural numbers <span class="math notranslate nohighlight">\(\mathbb{N} = \{ 0, 1, 2, \ldots \}\)</span> is not only fundamentally important in its own right, but also a plays a central role in the construction of new mathematical objects. Lean’s foundation allows us to declare <em>inductive types</em>, which are types generated inductively by a given list of <em>constructors</em>. In Lean, the natural numbers are declared as follows. .. _section_induction_and_recursion:</p> </section> <section id="id2"> <h2><span class="section-number">5.3. </span>Induction and Recursion<a class="headerlink" href="#id2" title="Permalink to this heading"></a></h2> <p>The set of natural numbers <span class="math notranslate nohighlight">\(\mathbb{N} = \{ 0, 1, 2, \ldots \}\)</span> is not only fundamentally important in its own right, but also a plays a central role in the construction of new mathematical objects. Lean’s foundation allows us to declare <em>inductive types</em>, which are types generated inductively by a given list of <em>constructors</em>. In Lean, the natural numbers are declared as follows.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">inductive</span> <span class="n">Nat</span> <span class="bp">|</span> <span class="n">zero</span> <span class="o">:</span> <span class="n">Nat</span> <span class="bp">|</span> <span class="n">succ</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">)</span> <span class="o">:</span> <span class="n">Nat</span> </pre></div> </div> <p>You can find this in the library by writting <code class="docutils literal notranslate"><span class="pre">#check</span> <span class="pre">Nat</span></code> and then using <code class="docutils literal notranslate"><span class="pre">ctrl-click</span></code> on the identifier <code class="docutils literal notranslate"><span class="pre">Nat</span></code>. The command specifies that <code class="docutils literal notranslate"><span class="pre">Nat</span></code> is the datatype generated freely and inductively by the two constructors <code class="docutils literal notranslate"><span class="pre">zero</span> <span class="pre">:</span> <span class="pre">Nat</span></code> and <code class="docutils literal notranslate"><span class="pre">succ</span> <span class="pre">:</span> <span class="pre">Nat</span> <span class="pre">→</span> <span class="pre">Nat</span></code>. Of course, the library introduces notation <code class="docutils literal notranslate"><span class="pre">ℕ</span></code> and <code class="docutils literal notranslate"><span class="pre">0</span></code> for <code class="docutils literal notranslate"><span class="pre">nat</span></code> and <code class="docutils literal notranslate"><span class="pre">zero</span></code> respectively. (Numerals are translated to binary representations, but we don’t have to worry about the details of that now.)</p> <p>What “freely” means for the working mathematician is that the type <code class="docutils literal notranslate"><span class="pre">Nat</span></code> has an element <code class="docutils literal notranslate"><span class="pre">zero</span></code> and an injective successor function <code class="docutils literal notranslate"><span class="pre">succ</span></code> whose image does not include <code class="docutils literal notranslate"><span class="pre">zero</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">)</span> <span class="o">:</span> <span class="n">n.succ</span> <span class="bp">≠</span> <span class="n">Nat.zero</span> <span class="o">:=</span> <span class="n">Nat.succ_ne_zero</span> <span class="n">n</span> <span class="kd">example</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m.succ</span> <span class="bp">=</span> <span class="n">n.succ</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">=</span> <span class="n">n</span> <span class="o">:=</span> <span class="n">Nat.succ.inj</span> <span class="n">h</span> </pre></div> </div> <p>What the word “inductively” means for the working mathematician is that the natural numbers comes with a principle of proof by induction and a principle of definition by recursion. This section will show you how to use these.</p> <p>Here is an example of a recursive definition of the factorial function.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">fac</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℕ</span> <span class="bp">|</span> <span class="mi">0</span> <span class="bp">=></span> <span class="mi">1</span> <span class="bp">|</span> <span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span> <span class="bp">=></span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">*</span> <span class="n">fac</span> <span class="n">n</span> </pre></div> </div> <p>The syntax takes some getting used to. Notice that there is no <code class="docutils literal notranslate"><span class="pre">:=</span></code> on the first line. The next two lines provide the base case and inductive step for a recursive definition. These equations hold definitionally, but they can also be used manually by giving the name <code class="docutils literal notranslate"><span class="pre">fac</span></code> to <code class="docutils literal notranslate"><span class="pre">simp</span></code> or <code class="docutils literal notranslate"><span class="pre">rw</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">fac</span> <span class="mi">0</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">fac</span> <span class="mi">0</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">fac</span><span class="o">]</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">fac</span> <span class="mi">0</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">fac</span><span class="o">]</span> <span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="n">fac</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">=</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">*</span> <span class="n">fac</span> <span class="n">n</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="n">fac</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">=</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">*</span> <span class="n">fac</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">fac</span><span class="o">]</span> <span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="n">fac</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">=</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">*</span> <span class="n">fac</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">fac</span><span class="o">]</span> </pre></div> </div> <p>The factorial function is actually already defined in mathlib as <code class="docutils literal notranslate"><span class="pre">Nat.factorial</span></code>. Once again, you can jump to it by typing <code class="docutils literal notranslate"><span class="pre">#check</span> <span class="pre">Nat.factorial</span></code> and using <code class="docutils literal notranslate"><span class="pre">ctrl-click.</span></code> For illustrative purposes, we will continue using <code class="docutils literal notranslate"><span class="pre">fac</span></code> in the examples. The annotation <code class="docutils literal notranslate"><span class="pre">@[simp]</span></code> before the definition of <code class="docutils literal notranslate"><span class="pre">Nat.factorial</span></code> specifies that the defining equation should be added to the database of identities that the simplifier uses by default.</p> <p>The principle of induction says that we can prove a general statement about the natural numbers by proving that the statement holds of 0 and that whenever it holds of a natural number <span class="math notranslate nohighlight">\(n\)</span>, it also holds of <span class="math notranslate nohighlight">\(n + 1\)</span>. The line <code class="docutils literal notranslate"><span class="pre">induction</span> <span class="pre">n</span> <span class="pre">with</span> <span class="pre">n</span> <span class="pre">ih</span></code> in the proof below therefore results in two goals: in the first we need to prove <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">fac</span> <span class="pre">0</span></code>, and in the second we have the added assumption <code class="docutils literal notranslate"><span class="pre">ih</span> <span class="pre">:</span> <span class="pre">0</span> <span class="pre"><</span> <span class="pre">fac</span> <span class="pre">n</span></code> and a required to prove <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">fac</span> <span class="pre">(n</span> <span class="pre">+</span> <span class="pre">1)</span></code>. The phrase <code class="docutils literal notranslate"><span class="pre">with</span> <span class="pre">n</span> <span class="pre">ih</span></code> serves to name the variable and the assumption for the inductive hypothesis, and you can choose whatever names you want for them.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">fac_pos</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">fac</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">induction'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="bp">·</span> <span class="n">rw</span> <span class="o">[</span><span class="n">fac</span><span class="o">]</span> <span class="n">exact</span> <span class="n">zero_lt_one</span> <span class="n">rw</span> <span class="o">[</span><span class="n">fac</span><span class="o">]</span> <span class="n">exact</span> <span class="n">mul_pos</span> <span class="n">n.succ_pos</span> <span class="n">ih</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">induction</span></code> tactic is smart enough to include hypotheses that depend on the induction variable as part of the induction hypothesis. Step through the next example to see what is going on.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">dvd_fac</span> <span class="o">{</span><span class="n">i</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">ipos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">i</span><span class="o">)</span> <span class="o">(</span><span class="n">ile</span> <span class="o">:</span> <span class="n">i</span> <span class="bp">≤</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">i</span> <span class="bp">∣</span> <span class="n">fac</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">induction'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="bp">·</span> <span class="n">exact</span> <span class="n">absurd</span> <span class="n">ipos</span> <span class="o">(</span><span class="n">not_lt_of_ge</span> <span class="n">ile</span><span class="o">)</span> <span class="n">rw</span> <span class="o">[</span><span class="n">fac</span><span class="o">]</span> <span class="n">cases'</span> <span class="n">Nat.of_le_succ</span> <span class="n">ile</span> <span class="k">with</span> <span class="n">h</span> <span class="n">h</span> <span class="bp">·</span> <span class="n">apply</span> <span class="n">dvd_mul_of_dvd_right</span> <span class="o">(</span><span class="n">ih</span> <span class="n">h</span><span class="o">)</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">]</span> <span class="n">apply</span> <span class="n">dvd_mul_right</span> </pre></div> </div> <p>The following example provides a crude lower bound for the factorial function. It turns out to be easier to start with a proof by cases, so that the remainder of the proof starts with the case <span class="math notranslate nohighlight">\(n = 1\)</span>. See if you can complete the argument with a proof by induction.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">pow_two_le_fac</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">^</span> <span class="o">(</span><span class="n">n</span> <span class="bp">-</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">≤</span> <span class="n">fac</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="bp">·</span> <span class="n">simp</span> <span class="o">[</span><span class="n">fac</span><span class="o">]</span> <span class="gr">sorry</span> </pre></div> </div> <p>Induction is often used to prove identities involving finite sums and products. Mathlib defines the expressions <code class="docutils literal notranslate"><span class="pre">Finset.sum</span> <span class="pre">s</span> <span class="pre">f</span></code> where <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Finset</span> <span class="pre">α</span></code> if a finite set of elements of the type <code class="docutils literal notranslate"><span class="pre">α</span></code> and <code class="docutils literal notranslate"><span class="pre">f</span></code> is a function defined on <code class="docutils literal notranslate"><span class="pre">α</span></code>. The codomain of <code class="docutils literal notranslate"><span class="pre">f</span></code> can be any type that supports a commutative, associative addition operation with a zero element. If you import <code class="docutils literal notranslate"><span class="pre">Algebra.BigOperators.Basic</span></code> and issue the command <code class="docutils literal notranslate"><span class="pre">open</span> <span class="pre">BigOperators</span></code>, you can use the more suggestive notation <code class="docutils literal notranslate"><span class="pre">∑</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">s,</span> <span class="pre">f</span> <span class="pre">x</span></code>. Of course, there is are an analogous operation and notation for finite products.</p> <p>We will talk about the <code class="docutils literal notranslate"><span class="pre">Finset</span></code> type and the operations it supports in the next section, and again in a later chapter. For now, we will only make use of <code class="docutils literal notranslate"><span class="pre">Finset.range</span> <span class="pre">n</span></code>, which is the finite set of natural numbers less than <code class="docutils literal notranslate"><span class="pre">n</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="k">#check</span> <span class="n">Finset.sum</span> <span class="n">s</span> <span class="n">f</span> <span class="k">#check</span> <span class="n">Finset.prod</span> <span class="n">s</span> <span class="n">f</span> <span class="kn">open</span> <span class="n">BigOperators</span> <span class="kn">open</span> <span class="n">Finset</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">s.sum</span> <span class="n">f</span> <span class="bp">=</span> <span class="bp">∑</span> <span class="n">x</span> <span class="k">in</span> <span class="n">s</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">s.prod</span> <span class="n">f</span> <span class="bp">=</span> <span class="bp">∏</span> <span class="n">x</span> <span class="k">in</span> <span class="n">s</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">range</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="n">sum</span> <span class="n">f</span> <span class="bp">=</span> <span class="bp">∑</span> <span class="n">x</span> <span class="k">in</span> <span class="n">range</span> <span class="n">n</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">range</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="n">prod</span> <span class="n">f</span> <span class="bp">=</span> <span class="bp">∏</span> <span class="n">x</span> <span class="k">in</span> <span class="n">range</span> <span class="n">n</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>The facts <code class="docutils literal notranslate"><span class="pre">Finset.sum_range_zero</span></code> and <code class="docutils literal notranslate"><span class="pre">Finset.sum_range_succ</span></code> provide a recursive description summation up to <span class="math notranslate nohighlight">\(n\)</span>, and similarly for products.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∑</span> <span class="n">x</span> <span class="k">in</span> <span class="n">range</span> <span class="mi">0</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">Finset.sum_range_zero</span> <span class="n">f</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∑</span> <span class="n">x</span> <span class="k">in</span> <span class="n">range</span> <span class="n">n.succ</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="o">(</span><span class="bp">∑</span> <span class="n">x</span> <span class="k">in</span> <span class="n">range</span> <span class="n">n</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">+</span> <span class="n">f</span> <span class="n">n</span> <span class="o">:=</span> <span class="n">Finset.sum_range_succ</span> <span class="n">f</span> <span class="n">n</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∏</span> <span class="n">x</span> <span class="k">in</span> <span class="n">range</span> <span class="mi">0</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="n">Finset.prod_range_zero</span> <span class="n">f</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∏</span> <span class="n">x</span> <span class="k">in</span> <span class="n">range</span> <span class="n">n.succ</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="o">(</span><span class="bp">∏</span> <span class="n">x</span> <span class="k">in</span> <span class="n">range</span> <span class="n">n</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">n</span> <span class="o">:=</span> <span class="n">Finset.prod_range_succ</span> <span class="n">f</span> <span class="n">n</span> </pre></div> </div> <p>The first identity in each pair holds definitionally, which is to say, you can replace the proofs by <code class="docutils literal notranslate"><span class="pre">rfl</span></code>.</p> <p>The following expresses the factorial function that we defined as a product.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="n">fac</span> <span class="n">n</span> <span class="bp">=</span> <span class="bp">∏</span> <span class="n">i</span> <span class="k">in</span> <span class="n">range</span> <span class="n">n</span><span class="o">,</span> <span class="o">(</span><span class="n">i</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">induction'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="bp">·</span> <span class="n">rw</span> <span class="o">[</span><span class="n">fac</span><span class="o">,</span> <span class="n">prod_range_zero</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">fac</span><span class="o">,</span> <span class="n">ih</span><span class="o">,</span> <span class="n">prod_range_succ</span><span class="o">,</span> <span class="n">mul_comm</span><span class="o">]</span> </pre></div> </div> <p>The fact that we include <code class="docutils literal notranslate"><span class="pre">mul_comm</span></code> as a simplification rule deserves comment. It should seem dangerous to simplify with the identity <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">y</span> <span class="pre">=</span> <span class="pre">y</span> <span class="pre">*</span> <span class="pre">x</span></code>, which would ordinarily loop indefinitely. Lean’s simplifier is smart enough to recognize that, and applies the rule only in the case where the resulting term has a smaller value in some fixed but arbitrary ordering of the terms. The following example shows that simplifying using the three rules <code class="docutils literal notranslate"><span class="pre">mul_assoc</span></code>, <code class="docutils literal notranslate"><span class="pre">mul_comm</span></code>, and <code class="docutils literal notranslate"><span class="pre">mul_left_comm</span></code> manages to identify products that are the same up to the placement of parentheses and ordering of variables.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="n">e</span> <span class="n">f</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="bp">*</span> <span class="o">(</span><span class="n">d</span> <span class="bp">*</span> <span class="n">e</span><span class="o">))</span> <span class="bp">=</span> <span class="n">d</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">f</span> <span class="bp">*</span> <span class="n">e</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="n">c</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">mul_assoc</span><span class="o">,</span> <span class="n">mul_comm</span><span class="o">,</span> <span class="n">mul_left_comm</span><span class="o">]</span> </pre></div> </div> <p>Roughly, the rules work by pushing parentheses to the right and then re-ordering the expressions on both sides until they both follow the same canonical order. Simplifying with these rules, and the corresponding rules for addition, is a handy trick.</p> <p>Returning to summation identities, we suggest stepping through the following proof that the sum of the natural numbers up to an including <span class="math notranslate nohighlight">\(n\)</span> is <span class="math notranslate nohighlight">\(n (n + 1) / 2\)</span>. The first step of the proof clears the denominator. This is generally useful when formalizing identities, because calculations with division generally have side conditions. (It is similarly useful to avoid using subtraction on the natural numbers when possible.)</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">sum_id</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∑</span> <span class="n">i</span> <span class="k">in</span> <span class="n">range</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">),</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="n">n</span> <span class="bp">*</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">symm</span><span class="bp">;</span> <span class="n">apply</span> <span class="n">Nat.div_eq_of_eq_mul_right</span> <span class="o">(</span><span class="kd">by</span> <span class="n">norm_num</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="mi">2</span><span class="o">)</span> <span class="n">induction'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="bp">·</span> <span class="n">simp</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Finset.sum_range_succ</span><span class="o">,</span> <span class="n">mul_add</span> <span class="mi">2</span><span class="o">,</span> <span class="bp">←</span> <span class="n">ih</span><span class="o">,</span> <span class="n">Nat.succ_eq_add_one</span><span class="o">]</span> <span class="n">ring</span> </pre></div> </div> <p>We encourage you to prove the analogous identity for sums of squares, and other identities you can find on the web.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">sum_sqr</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∑</span> <span class="n">i</span> <span class="k">in</span> <span class="n">range</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">),</span> <span class="n">i</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">=</span> <span class="n">n</span> <span class="bp">*</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">*</span> <span class="o">(</span><span class="mi">2</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">6</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>In Lean’s core library, addition and multiplication are themselves defined using recursive definitions, and their fundamental properties are established using induction. If you like thinking about foundational topics like that, you might enjoy working through proofs of the commutativity and associativity of multiplication and addition and the distributivity of multiplication over addition. You can do this on a copy of the natural numbers following the outline below. Notice that we can use the <code class="docutils literal notranslate"><span class="pre">induction</span></code> tactic with <code class="docutils literal notranslate"><span class="pre">MyNat</span></code>; Lean is smart enough to know to use the relevant induction principle (which is, of course, the same as that for <code class="docutils literal notranslate"><span class="pre">Nat</span></code>).</p> <p>We start you off with the commutativity of addition. A good rule of thumb is that because addition and multiplication are defined by recursion on the second argument, it is generally advantageous to do proofs by induction on a variable that occurs in that position. It is a bit tricky to decide which variable to use in the proof of associativity.</p> <p>It can be confusing to write things without the usual notation for zero, one, addition, and multiplication. We will learn how to define such notation later. Working in the namespace <code class="docutils literal notranslate"><span class="pre">MyNat</span></code> means that we can write <code class="docutils literal notranslate"><span class="pre">zero</span></code> and <code class="docutils literal notranslate"><span class="pre">succ</span></code> rather than <code class="docutils literal notranslate"><span class="pre">MyNat.zero</span></code> and <code class="docutils literal notranslate"><span class="pre">MyNat.succ</span></code>, and that these interpretations of the names take precedence over others. Outside the namespace, the full name of the <code class="docutils literal notranslate"><span class="pre">add</span></code> defined below, for example, is <code class="docutils literal notranslate"><span class="pre">MyNat.add</span></code>.</p> <p>If you find that you <em>really</em> enjoy this sort of thing, try defining truncated subtraction and exponentiation and proving some of their properties as well. Remember that truncated subtraction cuts off at zero. To define that, it is useful to define a predecessor function, <code class="docutils literal notranslate"><span class="pre">pred</span></code>, that subtracts one from any nonzero number and fixes zero. The function <code class="docutils literal notranslate"><span class="pre">pred</span></code> can be defined by a simple instance of recursion.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">inductive</span> <span class="n">MyNat</span> <span class="bp">|</span> <span class="n">zero</span> <span class="o">:</span> <span class="n">MyNat</span> <span class="bp">|</span> <span class="n">succ</span> <span class="o">:</span> <span class="n">MyNat</span> <span class="bp">→</span> <span class="n">MyNat</span> <span class="kn">namespace</span> <span class="n">MyNat</span> <span class="kd">def</span> <span class="n">add</span> <span class="o">:</span> <span class="n">MyNat</span> <span class="bp">→</span> <span class="n">MyNat</span> <span class="bp">→</span> <span class="n">MyNat</span> <span class="bp">|</span> <span class="n">x</span><span class="o">,</span> <span class="n">zero</span> <span class="bp">=></span> <span class="n">x</span> <span class="bp">|</span> <span class="n">x</span><span class="o">,</span> <span class="n">succ</span> <span class="n">y</span> <span class="bp">=></span> <span class="n">succ</span> <span class="o">(</span><span class="n">add</span> <span class="n">x</span> <span class="n">y</span><span class="o">)</span> <span class="kd">def</span> <span class="n">mul</span> <span class="o">:</span> <span class="n">MyNat</span> <span class="bp">→</span> <span class="n">MyNat</span> <span class="bp">→</span> <span class="n">MyNat</span> <span class="bp">|</span> <span class="n">x</span><span class="o">,</span> <span class="n">zero</span> <span class="bp">=></span> <span class="n">zero</span> <span class="bp">|</span> <span class="n">x</span><span class="o">,</span> <span class="n">succ</span> <span class="n">y</span> <span class="bp">=></span> <span class="n">add</span> <span class="o">(</span><span class="n">mul</span> <span class="n">x</span> <span class="n">y</span><span class="o">)</span> <span class="n">x</span> <span class="kd">theorem</span> <span class="n">zero_add</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">MyNat</span><span class="o">)</span> <span class="o">:</span> <span class="n">add</span> <span class="n">zero</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">induction'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="bp">·</span> <span class="n">rfl</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add</span><span class="o">,</span> <span class="n">ih</span><span class="o">]</span> <span class="kd">theorem</span> <span class="n">succ_add</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">MyNat</span><span class="o">)</span> <span class="o">:</span> <span class="n">add</span> <span class="o">(</span><span class="n">succ</span> <span class="n">m</span><span class="o">)</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">succ</span> <span class="o">(</span><span class="n">add</span> <span class="n">m</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">induction'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="bp">·</span> <span class="n">rfl</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add</span><span class="o">,</span> <span class="n">ih</span><span class="o">]</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">add_comm</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">MyNat</span><span class="o">)</span> <span class="o">:</span> <span class="n">add</span> <span class="n">m</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">add</span> <span class="n">n</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">induction'</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="bp">·</span> <span class="n">rw</span> <span class="o">[</span><span class="n">zero_add</span><span class="o">]</span> <span class="n">rfl</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add</span><span class="o">,</span> <span class="n">succ_add</span><span class="o">,</span> <span class="n">ih</span><span class="o">]</span> <span class="kd">theorem</span> <span class="n">add_assoc</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="n">k</span> <span class="o">:</span> <span class="n">MyNat</span><span class="o">)</span> <span class="o">:</span> <span class="n">add</span> <span class="o">(</span><span class="n">add</span> <span class="n">m</span> <span class="n">n</span><span class="o">)</span> <span class="n">k</span> <span class="bp">=</span> <span class="n">add</span> <span class="n">m</span> <span class="o">(</span><span class="n">add</span> <span class="n">n</span> <span class="n">k</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">mul_add</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="n">k</span> <span class="o">:</span> <span class="n">MyNat</span><span class="o">)</span> <span class="o">:</span> <span class="n">mul</span> <span class="n">m</span> <span class="o">(</span><span class="n">add</span> <span class="n">n</span> <span class="n">k</span><span class="o">)</span> <span class="bp">=</span> <span class="n">add</span> <span class="o">(</span><span class="n">mul</span> <span class="n">m</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">mul</span> <span class="n">m</span> <span class="n">k</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">zero_mul</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">MyNat</span><span class="o">)</span> <span class="o">:</span> <span class="n">mul</span> <span class="n">zero</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">zero</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">succ_mul</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">MyNat</span><span class="o">)</span> <span class="o">:</span> <span class="n">mul</span> <span class="o">(</span><span class="n">succ</span> <span class="n">m</span><span class="o">)</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">add</span> <span class="o">(</span><span class="n">mul</span> <span class="n">m</span> <span class="n">n</span><span class="o">)</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">mul_comm</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">MyNat</span><span class="o">)</span> <span class="o">:</span> <span class="n">mul</span> <span class="n">m</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">mul</span> <span class="n">n</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">end</span> <span class="n">MyNat</span> </pre></div> </div> </section> <section id="infinitely-many-primes"> <span id="section-infinitely-many-primes"></span><h2><span class="section-number">5.4. </span>Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this heading"></a></h2> <p>Let us continue our exploration of induction and recursion with another mathematical standard: a proof that there are infinitely many primes. One way to formulate this is as the statement that for every natural number <span class="math notranslate nohighlight">\(n\)</span>, there is a prime number greater than <span class="math notranslate nohighlight">\(n\)</span>. To prove this, let <span class="math notranslate nohighlight">\(p\)</span> be any prime factor of <span class="math notranslate nohighlight">\(n! + 1\)</span>. If <span class="math notranslate nohighlight">\(p\)</span> is less than <span class="math notranslate nohighlight">\(n\)</span>, it divides <span class="math notranslate nohighlight">\(n!\)</span>. Since it also divides <span class="math notranslate nohighlight">\(n! + 1\)</span>, it divides 1, a contradiction. Hence <span class="math notranslate nohighlight">\(p\)</span> is greater than <span class="math notranslate nohighlight">\(n\)</span>.</p> <p>To formalize that proof, we need to show that any number greater than or equal to 2 has a prime factor. To do that, we will need to show that any natural number that is not equal to 0 or 1 is greater-than or equal to 2. And this brings us to a quirky feature of formalization: it is often trivial statements like this that are among the most annoying to formalize. Here we consider a few ways to do it.</p> <p>To start with, we can use the <code class="docutils literal notranslate"><span class="pre">cases</span></code> tactic and the fact that the successor function respects the ordering on the natural numbers.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">two_le</span> <span class="o">{</span><span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h0</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">h1</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases</span> <span class="n">m</span><span class="bp">;</span> <span class="n">contradiction</span> <span class="n">case</span> <span class="n">succ</span> <span class="n">m</span> <span class="bp">=></span> <span class="n">cases</span> <span class="n">m</span><span class="bp">;</span> <span class="n">contradiction</span> <span class="n">repeat'</span> <span class="n">apply</span> <span class="n">Nat.succ_le_succ</span> <span class="n">apply</span> <span class="n">zero_le</span> </pre></div> </div> <p>Another strategy is to use the tactic <code class="docutils literal notranslate"><span class="pre">interval_cases</span></code>, which automatically splits the goal into cases when the variable in question is contained in an interval of natural numbers or integers. Remember that you can hover over it to see its documentation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h0</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">h1</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_contra</span> <span class="n">h</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">h</span> <span class="n">interval_cases</span> <span class="n">m</span> <span class="bp"><;></span> <span class="n">contradiction</span> </pre></div> </div> <p>Recall that the semicolon after <code class="docutils literal notranslate"><span class="pre">interval_cases</span> <span class="pre">m</span></code> means that the next tactic is applied to each of the cases that it generates. Yet another option is to use the tactic, <code class="docutils literal notranslate"><span class="pre">decide</span></code>, which tries to find a decision procedure to solve the problem. Lean knows that you can decide the truth value of a statement that begins with a bounded quantifier <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">x,</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">n</span> <span class="pre">→</span> <span class="pre">...</span></code> or <code class="docutils literal notranslate"><span class="pre">∃</span> <span class="pre">x,</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">n</span> <span class="pre">∧</span> <span class="pre">...</span></code> by deciding each of the finitely many instances.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h0</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">h1</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_contra</span> <span class="n">h</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">h</span> <span class="n">revert</span> <span class="n">h0</span> <span class="n">h1</span> <span class="n">revert</span> <span class="n">h</span> <span class="n">m</span> <span class="n">decide</span> </pre></div> </div> <p>With the theorem <code class="docutils literal notranslate"><span class="pre">two_le</span></code> in hand, let’s start by showing that every natural number greater than two has a prime divisor. Mathlib contains a function <code class="docutils literal notranslate"><span class="pre">Nat.minFac</span></code> that returns the smallest prime divisor, but for the sake of learning new parts of the library, we’ll avoid using it and prove the theorem directly.</p> <p>Here, ordinary induction isn’t enough. We want to use <em>strong induction</em>, which allows us to prove that every natural number <span class="math notranslate nohighlight">\(n\)</span> has a property <span class="math notranslate nohighlight">\(P\)</span> by showing that for every number <span class="math notranslate nohighlight">\(n\)</span>, if <span class="math notranslate nohighlight">\(P\)</span> holds of all values less than <span class="math notranslate nohighlight">\(n\)</span>, it holds at <span class="math notranslate nohighlight">\(n\)</span> as well. In Lean, this principle is called <code class="docutils literal notranslate"><span class="pre">Nat.strong_induction_on</span></code>, and we can use the <code class="docutils literal notranslate"><span class="pre">with</span></code> keyword to tell the induction tactic to use it. Notice that when we do that, there is no base case; it is subsumed by the general induction step.</p> <p>The argument is simply as follows. Assuming <span class="math notranslate nohighlight">\(n ≥ 2\)</span>, if <span class="math notranslate nohighlight">\(n\)</span> is prime, we’re done. If it isn’t, then by one of the characterizations of what it means to be a prime number, it has a nontrivial factor, <span class="math notranslate nohighlight">\(m\)</span>, and we can apply the inductive hypothesis to that. Step through the next proof to see how that plays out. The line <code class="docutils literal notranslate"><span class="pre">dsimp</span> <span class="pre">at</span> <span class="pre">ih</span></code> simplifies the expression of the inductive hypothesis to make it more readable. The proof still works if you delete that line.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">exists_prime_factor</span> <span class="o">{</span><span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">p</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">p.Prime</span> <span class="bp">∧</span> <span class="n">p</span> <span class="bp">∣</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_cases</span> <span class="n">np</span> <span class="o">:</span> <span class="n">n.Prime</span> <span class="bp">·</span> <span class="n">use</span> <span class="n">n</span><span class="o">,</span> <span class="n">np</span> <span class="n">induction'</span> <span class="n">n</span> <span class="n">using</span> <span class="n">Nat.strong_induction_on</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="n">dsimp</span> <span class="n">at</span> <span class="n">ih</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Nat.prime_def_lt</span><span class="o">]</span> <span class="n">at</span> <span class="n">np</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">np</span> <span class="n">rcases</span> <span class="n">np</span> <span class="n">h</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">m</span><span class="o">,</span> <span class="n">mltn</span><span class="o">,</span> <span class="n">mdvdn</span><span class="o">,</span> <span class="n">mne1</span><span class="o">⟩</span> <span class="k">have</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">mz</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mz</span><span class="o">,</span> <span class="n">zero_dvd_iff</span><span class="o">]</span> <span class="n">at</span> <span class="n">mdvdn</span> <span class="n">linarith</span> <span class="k">have</span> <span class="n">mgt2</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">m</span> <span class="o">:=</span> <span class="n">two_le</span> <span class="n">this</span> <span class="n">mne1</span> <span class="n">by_cases</span> <span class="n">mp</span> <span class="o">:</span> <span class="n">m.Prime</span> <span class="bp">·</span> <span class="n">use</span> <span class="n">m</span><span class="o">,</span> <span class="n">mp</span> <span class="n">exact</span> <span class="n">mdvdn</span> <span class="bp">.</span> <span class="n">rcases</span> <span class="n">ih</span> <span class="n">m</span> <span class="n">mltn</span> <span class="n">mgt2</span> <span class="n">mp</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">p</span><span class="o">,</span> <span class="n">pp</span><span class="o">,</span> <span class="n">pdvd</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">p</span><span class="o">,</span> <span class="n">pp</span> <span class="n">apply</span> <span class="n">pdvd.trans</span> <span class="n">mdvdn</span> </pre></div> </div> <p>We can now prove the following formulation of our theorem. See if you can fill out the sketch. You can use <code class="docutils literal notranslate"><span class="pre">Nat.factorial_pos</span></code>, <code class="docutils literal notranslate"><span class="pre">Nat.dvd_factorial</span></code>, and <code class="docutils literal notranslate"><span class="pre">Nat.dvd_sub</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">primes_infinite</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">p</span> <span class="bp">></span> <span class="n">n</span><span class="o">,</span> <span class="n">Nat.Prime</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">n</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">Nat.factorial</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">+</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">rcases</span> <span class="n">exists_prime_factor</span> <span class="n">this</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">p</span><span class="o">,</span> <span class="n">pp</span><span class="o">,</span> <span class="n">pdvd</span><span class="o">⟩</span> <span class="n">refine'</span> <span class="o">⟨</span><span class="n">p</span><span class="o">,</span> <span class="n">_</span><span class="o">,</span> <span class="n">pp</span><span class="o">⟩</span> <span class="k">show</span> <span class="n">p</span> <span class="bp">></span> <span class="n">n</span> <span class="n">by_contra</span> <span class="n">ple</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">ple</span> <span class="k">have</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">∣</span> <span class="n">Nat.factorial</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">∣</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">show</span> <span class="n">False</span> <span class="gr">sorry</span> </pre></div> </div> <p>Let’s consider a variation of the proof above, where instead of using the factorial function, we suppose that we are given by a finite set <span class="math notranslate nohighlight">\(\{ p_1, \ldots, p_n \}\)</span> and we consider a prime factor of <span class="math notranslate nohighlight">\(\prod_{i = 1}^n p_i + 1\)</span>. That prime factor has to be distinct from each <span class="math notranslate nohighlight">\(p_i\)</span>, showing that there is no finite set that contains all the prime numbers.</p> <p>Formalizing this argument requires us to reason about finite sets. In Lean, for any type <code class="docutils literal notranslate"><span class="pre">α</span></code>, the type <code class="docutils literal notranslate"><span class="pre">Finset</span> <span class="pre">α</span></code> represents finite sets of elements of type <code class="docutils literal notranslate"><span class="pre">α</span></code>. Reasoning about finite sets computationally requires having a procedure to test equality on <code class="docutils literal notranslate"><span class="pre">α</span></code>, which is why the snippet below includes the assumption <code class="docutils literal notranslate"><span class="pre">[DecidableEq</span> <span class="pre">α]</span></code>. For concrete data types like <code class="docutils literal notranslate"><span class="pre">ℕ</span></code>, <code class="docutils literal notranslate"><span class="pre">ℤ</span></code>, and <code class="docutils literal notranslate"><span class="pre">ℚ</span></code>, the assumption is satisfied automatically. When reasoning about the real numbers, it can be satisfied using classical logic and abandoning the computational interpretation.</p> <p>We use the command <code class="docutils literal notranslate"><span class="pre">open</span> <span class="pre">Finset</span></code> to avail ourselves of shorter names for the relevant theorems. Unlike the case with sets, most equivalences involving finsets do not hold definitionally, so they need to be expanded manually using equivalances like <code class="docutils literal notranslate"><span class="pre">Finset.subset_iff</span></code>, <code class="docutils literal notranslate"><span class="pre">Finset.mem_union</span></code>, <code class="docutils literal notranslate"><span class="pre">Finset.mem_inter</span></code>, and <code class="docutils literal notranslate"><span class="pre">Finset.mem_sdiff</span></code>. The <code class="docutils literal notranslate"><span class="pre">ext</span></code> tactic can still be used to reduce show that two finite sets are equal by showing that every element of one is an element of the other.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Finset</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">DecidableEq</span> <span class="n">α</span><span class="o">]</span> <span class="o">(</span><span class="n">r</span> <span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">α</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">r</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">r</span> <span class="bp">∩</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">r</span> <span class="bp">∩</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">subset_iff</span><span class="o">]</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mem_inter</span><span class="o">,</span> <span class="n">mem_union</span><span class="o">,</span> <span class="n">mem_union</span><span class="o">,</span> <span class="n">mem_inter</span><span class="o">,</span> <span class="n">mem_inter</span><span class="o">]</span> <span class="n">tauto</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">r</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">r</span> <span class="bp">∩</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">r</span> <span class="bp">∩</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">subset_iff</span><span class="o">]</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">tauto</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">r</span> <span class="bp">∩</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">r</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">⊆</span> <span class="n">r</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">subset_iff</span><span class="o">]</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">tauto</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">r</span> <span class="bp">∩</span> <span class="n">s</span> <span class="bp">∪</span> <span class="n">r</span> <span class="bp">∩</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">r</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span> <span class="n">simp</span> <span class="n">tauto</span> <span class="kd">end</span> </pre></div> </div> <p>We have used a new trick: the <code class="docutils literal notranslate"><span class="pre">tauto</span></code> tactic (and a strengthened version, <code class="docutils literal notranslate"><span class="pre">tauto!</span></code>, which uses classical logic) can be used to dispense with propositional tautologies. See if you can use these methods to prove the two examples below.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">r</span> <span class="bp">∪</span> <span class="n">s</span><span class="o">)</span> <span class="bp">∩</span> <span class="o">(</span><span class="n">r</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="bp">=</span> <span class="n">r</span> <span class="bp">∪</span> <span class="n">s</span> <span class="bp">∩</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">r</span> <span class="bp">\</span> <span class="n">s</span><span class="o">)</span> <span class="bp">\</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">r</span> <span class="bp">\</span> <span class="o">(</span><span class="n">s</span> <span class="bp">∪</span> <span class="n">t</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>The theorem <code class="docutils literal notranslate"><span class="pre">Finset.dvd_prod_of_mem</span></code> tells us that if an <code class="docutils literal notranslate"><span class="pre">n</span></code> is an element of a finite set <code class="docutils literal notranslate"><span class="pre">s</span></code>, then <code class="docutils literal notranslate"><span class="pre">n</span></code> divides <code class="docutils literal notranslate"><span class="pre">∏</span> <span class="pre">i</span> <span class="pre">in</span> <span class="pre">s,</span> <span class="pre">i</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">∣</span> <span class="bp">∏</span> <span class="n">i</span> <span class="k">in</span> <span class="n">s</span><span class="o">,</span> <span class="n">i</span> <span class="o">:=</span> <span class="n">Finset.dvd_prod_of_mem</span> <span class="n">_</span> <span class="n">h</span> </pre></div> </div> <p>We also need to know that the converse holds in the case where <code class="docutils literal notranslate"><span class="pre">n</span></code> is prime and <code class="docutils literal notranslate"><span class="pre">s</span></code> is a set of primes. To show that, we need the following lemma, which you should be able to prove using the theorem <code class="docutils literal notranslate"><span class="pre">Nat.Prime.eq_one_or_self_of_dvd</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">_root_.Nat.Prime.eq_of_dvd_of_prime</span> <span class="o">{</span><span class="n">p</span> <span class="n">q</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="n">p</span><span class="o">)</span> <span class="o">(</span><span class="n">prime_q</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="n">q</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">∣</span> <span class="n">q</span><span class="o">)</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">=</span> <span class="n">q</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>We can use this lemma to show that if a prime <code class="docutils literal notranslate"><span class="pre">p</span></code> divides a product of a finite set of primes, then it divides one of them. Mathlib provides a useful principle of induction on finite sets: to show that a property holds of an arbitrary finite set <code class="docutils literal notranslate"><span class="pre">s</span></code>, show that it holds of the empty set, and show that it is preserved when we add a single new element <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">∉</span> <span class="pre">s</span></code>. The principle is known as <code class="docutils literal notranslate"><span class="pre">Finset.induction_on</span></code>. When we tell the induction tactic to use it, we can also specify the names <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">s</span></code>, the name for the assumption <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">∉</span> <span class="pre">s</span></code> in the inductive step, and the name of the inductive hypothesis. The expression <code class="docutils literal notranslate"><span class="pre">Finset.insert</span> <span class="pre">a</span> <span class="pre">s</span></code> denotes the union of <code class="docutils literal notranslate"><span class="pre">s</span></code> with the singleton <code class="docutils literal notranslate"><span class="pre">a</span></code>. The identities <code class="docutils literal notranslate"><span class="pre">Finset.prod_empty</span></code> and <code class="docutils literal notranslate"><span class="pre">Finset.prod_insert</span></code> then provide the relevant rewrite rules for the product. In the proof below, the first <code class="docutils literal notranslate"><span class="pre">simp</span></code> applies <code class="docutils literal notranslate"><span class="pre">Finset.prod_empty</span></code>. Step through the beginning of the proof to see the induction unfold, and then finish it off.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">mem_of_dvd_prod_primes</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">{</span><span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">p.Prime</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∀</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">Nat.Prime</span> <span class="n">n</span><span class="o">)</span> <span class="bp">→</span> <span class="o">(</span><span class="n">p</span> <span class="bp">∣</span> <span class="bp">∏</span> <span class="n">n</span> <span class="k">in</span> <span class="n">s</span><span class="o">,</span> <span class="n">n</span><span class="o">)</span> <span class="bp">→</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="n">induction'</span> <span class="n">s</span> <span class="n">using</span> <span class="n">Finset.induction_on</span> <span class="k">with</span> <span class="n">a</span> <span class="n">s</span> <span class="n">ans</span> <span class="n">ih</span> <span class="bp">·</span> <span class="n">simp</span> <span class="n">at</span> <span class="n">h₁</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">prime_p.two_le</span><span class="o">]</span> <span class="n">simp</span> <span class="o">[</span><span class="n">Finset.prod_insert</span> <span class="n">ans</span><span class="o">,</span> <span class="n">prime_p.dvd_mul</span><span class="o">]</span> <span class="n">at</span> <span class="n">h₀</span> <span class="n">h₁</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mem_insert</span><span class="o">]</span> <span class="gr">sorry</span> </pre></div> </div> <p>We need one last property of finite sets. Given an element <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">α</span></code> and a predicate <code class="docutils literal notranslate"><span class="pre">P</span></code> on <code class="docutils literal notranslate"><span class="pre">α</span></code>, in <a class="reference internal" href="C04_Sets_and_Functions.html#sets-and-functions"><span class="std std-numref">Chapter 4</span></a> we wrote <code class="docutils literal notranslate"><span class="pre">{</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">s</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">x</span> <span class="pre">}</span></code> for the set of elements of <code class="docutils literal notranslate"><span class="pre">s</span></code> that satisfy <code class="docutils literal notranslate"><span class="pre">P</span></code>. Given <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Finset</span> <span class="pre">α</span></code>, the analogous notion is written <code class="docutils literal notranslate"><span class="pre">s.filter</span> <span class="pre">P</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s.filter</span> <span class="n">Nat.Prime</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span> <span class="bp">∧</span> <span class="n">x.Prime</span> <span class="o">:=</span> <span class="n">mem_filter</span> </pre></div> </div> <p>We now prove an alternative formulation of the statement that there are infinitely many primes, namely, that given any <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Finset</span> <span class="pre">ℕ</span></code>, there is a prime <code class="docutils literal notranslate"><span class="pre">p</span></code> that is not an element of <code class="docutils literal notranslate"><span class="pre">s</span></code>. Aiming for a contradiction, we assume that all the primes are in <code class="docutils literal notranslate"><span class="pre">s</span></code>, and then cut down to a set <code class="docutils literal notranslate"><span class="pre">s'</span></code> that contains all and only the primes. Taking the product of that set, adding one, and finding a prime factor of the result leads to the contradiction we are looking for. See if you can complete the sketch below. You can use <code class="docutils literal notranslate"><span class="pre">Finset.prod_pos</span></code> in the proof of the first <code class="docutils literal notranslate"><span class="pre">have</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">primes_infinite'</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">Nat</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">p</span><span class="o">,</span> <span class="n">Nat.Prime</span> <span class="n">p</span> <span class="bp">∧</span> <span class="n">p</span> <span class="bp">∉</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">s</span> <span class="n">by_contra</span> <span class="n">h</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">h</span> <span class="n">set</span> <span class="n">s'</span> <span class="o">:=</span> <span class="n">s.filter</span> <span class="n">Nat.Prime</span> <span class="k">with</span> <span class="n">s'_def</span> <span class="k">have</span> <span class="n">mem_s'</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">},</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">s'</span> <span class="bp">↔</span> <span class="n">n.Prime</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">n</span> <span class="n">simp</span> <span class="o">[</span><span class="n">s'_def</span><span class="o">]</span> <span class="n">apply</span> <span class="n">h</span> <span class="k">have</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="o">(</span><span class="bp">∏</span> <span class="n">i</span> <span class="k">in</span> <span class="n">s'</span><span class="o">,</span> <span class="n">i</span><span class="o">)</span> <span class="bp">+</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">rcases</span> <span class="n">exists_prime_factor</span> <span class="n">this</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">p</span><span class="o">,</span> <span class="n">pp</span><span class="o">,</span> <span class="n">pdvd</span><span class="o">⟩</span> <span class="k">have</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">∣</span> <span class="bp">∏</span> <span class="n">i</span> <span class="k">in</span> <span class="n">s'</span><span class="o">,</span> <span class="n">i</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">∣</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">convert</span> <span class="n">Nat.dvd_sub'</span> <span class="n">pdvd</span> <span class="n">this</span> <span class="n">simp</span> <span class="k">show</span> <span class="n">False</span> <span class="gr">sorry</span> </pre></div> </div> <p>We have thus seen two ways of saying that there are infinitely many primes: saying that they are not bounded by any <code class="docutils literal notranslate"><span class="pre">n</span></code>, and saying that they are not contained in any finite set <code class="docutils literal notranslate"><span class="pre">s</span></code>. The two proofs below show that these formulations are equivalent. In the second, in order to form <code class="docutils literal notranslate"><span class="pre">s.filter</span> <span class="pre">Q</span></code>, we have to assume that there is a procedure for deciding whether or not <code class="docutils literal notranslate"><span class="pre">Q</span></code> holds. Lean knows that there is a procedure for <code class="docutils literal notranslate"><span class="pre">Nat.Prime</span></code>. In general, if we use classical logic by writing <code class="docutils literal notranslate"><span class="pre">open</span> <span class="pre">Classical</span></code>, we can dispense with the assumption.</p> <p>In mathlib, <code class="docutils literal notranslate"><span class="pre">Finset.sup</span> <span class="pre">s</span> <span class="pre">f</span></code> denotes the supremum of the values of <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> as <code class="docutils literal notranslate"><span class="pre">x</span></code> ranges over <code class="docutils literal notranslate"><span class="pre">s</span></code>, returning <code class="docutils literal notranslate"><span class="pre">0</span></code> in the case where <code class="docutils literal notranslate"><span class="pre">s</span></code> is empty and the codomain of <code class="docutils literal notranslate"><span class="pre">f</span></code> is <code class="docutils literal notranslate"><span class="pre">ℕ</span></code>. In the first proof, we use <code class="docutils literal notranslate"><span class="pre">s.sup</span> <span class="pre">id</span></code>, where <code class="docutils literal notranslate"><span class="pre">id</span></code> is the identity function, to refer to the maximum value in <code class="docutils literal notranslate"><span class="pre">s</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">bounded_of_ex_finset</span> <span class="o">(</span><span class="n">Q</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∃</span> <span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ℕ</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">k</span><span class="o">,</span> <span class="n">Q</span> <span class="n">k</span> <span class="bp">→</span> <span class="n">k</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">)</span> <span class="bp">→</span> <span class="bp">∃</span> <span class="n">n</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">k</span><span class="o">,</span> <span class="n">Q</span> <span class="n">k</span> <span class="bp">→</span> <span class="n">k</span> <span class="bp"><</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">s</span><span class="o">,</span> <span class="n">hs</span><span class="o">⟩</span> <span class="n">use</span> <span class="n">s.sup</span> <span class="n">id</span> <span class="bp">+</span> <span class="mi">1</span> <span class="n">intro</span> <span class="n">k</span> <span class="n">Qk</span> <span class="n">apply</span> <span class="n">Nat.lt_succ_of_le</span> <span class="k">show</span> <span class="n">id</span> <span class="n">k</span> <span class="bp">≤</span> <span class="n">s.sup</span> <span class="n">id</span> <span class="n">apply</span> <span class="n">le_sup</span> <span class="o">(</span><span class="n">hs</span> <span class="n">k</span> <span class="n">Qk</span><span class="o">)</span> <span class="kd">theorem</span> <span class="n">ex_finset_of_bounded</span> <span class="o">(</span><span class="n">Q</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">[</span><span class="n">DecidablePred</span> <span class="n">Q</span><span class="o">]</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∃</span> <span class="n">n</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">k</span><span class="o">,</span> <span class="n">Q</span> <span class="n">k</span> <span class="bp">→</span> <span class="n">k</span> <span class="bp">≤</span> <span class="n">n</span><span class="o">)</span> <span class="bp">→</span> <span class="bp">∃</span> <span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ℕ</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">k</span><span class="o">,</span> <span class="n">Q</span> <span class="n">k</span> <span class="bp">↔</span> <span class="n">k</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">n</span><span class="o">,</span> <span class="n">hn</span><span class="o">⟩</span> <span class="n">use</span> <span class="o">(</span><span class="n">range</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">))</span><span class="bp">.</span><span class="n">filter</span> <span class="n">Q</span> <span class="n">intro</span> <span class="n">k</span> <span class="n">simp</span> <span class="o">[</span><span class="n">Nat.lt_succ_iff</span><span class="o">]</span> <span class="n">exact</span> <span class="n">hn</span> <span class="n">k</span> </pre></div> </div> <p>A small variation on our second proof that there are infinitely many primes shows that there are infinitely many primes congruent to 3 modulo 4. The argument goes as follows. First, notice that if the product of two numbers <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(n\)</span> is equal to 3 modulo 4, then one of the two numbers is congruent to three modulo 4. After all, both have to be odd, and if they are both congruent to 1 modulo 4, so is their product. We can use this observation to show that if some number greater than 2 is congruent to 3 modulo 4, then that number has a prime divisor that is also congruent to 3 modulo 4.</p> <p>Now suppose there are only finitely many prime numbers congruent to 3 modulo 4, say, <span class="math notranslate nohighlight">\(p_1, \ldots, p_k\)</span>. Without loss of generality, we can assume that <span class="math notranslate nohighlight">\(p_1 = 3\)</span>. Consider the product <span class="math notranslate nohighlight">\(4 \prod_{i = 2}^k p_i + 3\)</span>. It is easy to see that this is congruent to 3 modulo 4, so it has a prime factor <span class="math notranslate nohighlight">\(p\)</span> congruent to 3 modulo 4. It can’t be the case that <span class="math notranslate nohighlight">\(p = 3\)</span>; since <span class="math notranslate nohighlight">\(p\)</span> divides <span class="math notranslate nohighlight">\(4 \prod_{i = 2}^k p_i + 3\)</span>, if <span class="math notranslate nohighlight">\(p\)</span> were equal to 3 then it would also divide <span class="math notranslate nohighlight">\(\prod_{i = 2}^k p_i\)</span>, which implies that <span class="math notranslate nohighlight">\(p\)</span> is equal to one of the <span class="math notranslate nohighlight">\(p_i\)</span> for <span class="math notranslate nohighlight">\(i = 2, \ldots, k\)</span>; and we have excluded 3 from this list. So <span class="math notranslate nohighlight">\(p\)</span> has to be one of the other elements <span class="math notranslate nohighlight">\(p_i\)</span>. But in that case, <span class="math notranslate nohighlight">\(p\)</span> divides <span class="math notranslate nohighlight">\(4 \prod_{i = 2}^k p_i\)</span> and hence 3, which contradicts the fact that it is not 3.</p> <p>In Lean, the notation <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">%</span> <span class="pre">m</span></code>, read “<code class="docutils literal notranslate"><span class="pre">n</span></code> modulo <code class="docutils literal notranslate"><span class="pre">m</span></code>,” denotes the remainder of the division of <code class="docutils literal notranslate"><span class="pre">n</span></code> by <code class="docutils literal notranslate"><span class="pre">m</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="mi">27</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">norm_num</span> </pre></div> </div> <p>We can then render the statement “<code class="docutils literal notranslate"><span class="pre">n</span></code> is congruent to 3 modulo 4” as <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">%</span> <span class="pre">4</span> <span class="pre">=</span> <span class="pre">3</span></code>. The following example and theorems sum up the facts about this function that we will need to use below. The first named theorem is another illustration of reasoning by a small number of cases. In the second named theorem, remember that the semicolon means that the subsequent tactic block is applied to both of the goals that result from the application of <code class="docutils literal notranslate"><span class="pre">two_le</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="mi">4</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">+</span> <span class="mi">3</span><span class="o">)</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_comm</span><span class="o">,</span> <span class="n">Nat.add_mul_mod_self_left</span><span class="o">]</span> <span class="n">norm_num</span> <span class="kd">theorem</span> <span class="n">mod_4_eq_3_or_mod_4_eq_3</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="bp">∨</span> <span class="n">n</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">revert</span> <span class="n">h</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Nat.mul_mod</span><span class="o">]</span> <span class="k">have</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp"><</span> <span class="mi">4</span> <span class="o">:=</span> <span class="n">Nat.mod_lt</span> <span class="n">m</span> <span class="o">(</span><span class="kd">by</span> <span class="n">norm_num</span><span class="o">)</span> <span class="n">interval_cases</span> <span class="n">hm</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="o">[</span><span class="n">hm</span><span class="o">]</span> <span class="k">have</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp"><</span> <span class="mi">4</span> <span class="o">:=</span> <span class="n">Nat.mod_lt</span> <span class="n">n</span> <span class="o">(</span><span class="kd">by</span> <span class="n">norm_num</span><span class="o">)</span> <span class="n">interval_cases</span> <span class="n">hn</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="o">[</span><span class="n">hn</span><span class="o">]</span> <span class="kd">theorem</span> <span class="n">two_le_of_mod_4_eq_3</span> <span class="o">{</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">two_le</span> <span class="bp"><;></span> <span class="bp">·</span> <span class="n">intro</span> <span class="n">neq</span> <span class="n">rw</span> <span class="o">[</span><span class="n">neq</span><span class="o">]</span> <span class="n">at</span> <span class="n">h</span> <span class="n">norm_num</span> <span class="n">at</span> <span class="n">h</span> </pre></div> </div> <p>We will also need the following fact, which says that if <code class="docutils literal notranslate"><span class="pre">m</span></code> is a nontrivial divisor of <code class="docutils literal notranslate"><span class="pre">n</span></code>, then so is <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">/</span> <span class="pre">m</span></code>. See if you can complete the proof using <code class="docutils literal notranslate"><span class="pre">Nat.div_dvd_of_dvd</span></code> and <code class="docutils literal notranslate"><span class="pre">Nat.div_lt_self</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">aux</span> <span class="o">{</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">h₀</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">h₁</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">m</span><span class="o">)</span> <span class="o">(</span><span class="n">h₂</span> <span class="o">:</span> <span class="n">m</span> <span class="bp"><</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">/</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">n</span> <span class="bp">/</span> <span class="n">m</span> <span class="bp"><</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Now put all the pieces together to prove that any number congruent to 3 modulo 4 has a prime divisor with that same property.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">exists_prime_factor_mod_4_eq_3</span> <span class="o">{</span><span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">p</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">p.Prime</span> <span class="bp">∧</span> <span class="n">p</span> <span class="bp">∣</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">p</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_cases</span> <span class="n">np</span> <span class="o">:</span> <span class="n">n.Prime</span> <span class="bp">·</span> <span class="n">use</span> <span class="n">n</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">np</span><span class="o">,</span> <span class="n">dvd_rfl</span><span class="o">,</span> <span class="n">h</span><span class="o">⟩</span> <span class="n">induction'</span> <span class="n">n</span> <span class="n">using</span> <span class="n">Nat.strong_induction_on</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="n">dsimp</span> <span class="n">at</span> <span class="n">ih</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Nat.prime_def_lt</span><span class="o">]</span> <span class="n">at</span> <span class="n">np</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">np</span> <span class="n">rcases</span> <span class="n">np</span> <span class="o">(</span><span class="n">two_le_of_mod_4_eq_3</span> <span class="n">h</span><span class="o">)</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">m</span><span class="o">,</span> <span class="n">mltn</span><span class="o">,</span> <span class="n">mdvdn</span><span class="o">,</span> <span class="n">mne1</span><span class="o">⟩</span> <span class="k">have</span> <span class="n">mge2</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">m</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">two_le</span> <span class="n">_</span> <span class="n">mne1</span> <span class="n">intro</span> <span class="n">mz</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mz</span><span class="o">,</span> <span class="n">zero_dvd_iff</span><span class="o">]</span> <span class="n">at</span> <span class="n">mdvdn</span> <span class="n">linarith</span> <span class="k">have</span> <span class="n">neq</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">*</span> <span class="o">(</span><span class="n">n</span> <span class="bp">/</span> <span class="n">m</span><span class="o">)</span> <span class="bp">=</span> <span class="n">n</span> <span class="o">:=</span> <span class="n">Nat.mul_div_cancel'</span> <span class="n">mdvdn</span> <span class="k">have</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="bp">∨</span> <span class="n">n</span> <span class="bp">/</span> <span class="n">m</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">mod_4_eq_3_or_mod_4_eq_3</span> <span class="n">rw</span> <span class="o">[</span><span class="n">neq</span><span class="o">,</span> <span class="n">h</span><span class="o">]</span> <span class="n">cases'</span> <span class="n">this</span> <span class="k">with</span> <span class="n">h1</span> <span class="n">h1</span> <span class="bp">.</span> <span class="gr">sorry</span> <span class="bp">.</span> <span class="gr">sorry</span> </pre></div> </div> <p>We are in the home stretch. Given a set <code class="docutils literal notranslate"><span class="pre">s</span></code> of prime numbers, we need to talk about the result of removing 3 from that set, if it is present. The function <code class="docutils literal notranslate"><span class="pre">Finset.erase</span></code> handles that.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">∈</span> <span class="n">erase</span> <span class="n">s</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">m</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rwa</span> <span class="o">[</span><span class="n">mem_erase</span><span class="o">]</span> <span class="n">at</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">∈</span> <span class="n">erase</span> <span class="n">s</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">≠</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">m</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">at</span> <span class="n">h</span> <span class="n">assumption</span> </pre></div> </div> <p>We are now ready to prove that there are infinitely many primes congruent to 3 modulo 4. Fill in the missing parts below. Our solution uses <code class="docutils literal notranslate"><span class="pre">Nat.dvd_add_iff_left</span></code> and <code class="docutils literal notranslate"><span class="pre">Nat.dvd_sub'</span></code> along the way.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">primes_mod_4_eq_3_infinite</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">p</span> <span class="bp">></span> <span class="n">n</span><span class="o">,</span> <span class="n">Nat.Prime</span> <span class="n">p</span> <span class="bp">∧</span> <span class="n">p</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_contra</span> <span class="n">h</span> <span class="n">push_neg</span> <span class="n">at</span> <span class="n">h</span> <span class="n">cases'</span> <span class="n">h</span> <span class="k">with</span> <span class="n">n</span> <span class="n">hn</span> <span class="k">have</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">s</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">Nat</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">p.Prime</span> <span class="bp">∧</span> <span class="n">p</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="bp">↔</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">ex_finset_of_bounded</span> <span class="n">use</span> <span class="n">n</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">hn</span> <span class="n">rcases</span> <span class="n">hn</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">p</span><span class="o">,</span> <span class="o">⟨</span><span class="n">pp</span><span class="o">,</span> <span class="n">p4</span><span class="o">⟩,</span> <span class="n">pltn</span><span class="o">⟩</span> <span class="n">exact</span> <span class="o">⟨</span><span class="n">p</span><span class="o">,</span> <span class="n">pltn</span><span class="o">,</span> <span class="n">pp</span><span class="o">,</span> <span class="n">p4</span><span class="o">⟩</span> <span class="n">cases'</span> <span class="n">this</span> <span class="k">with</span> <span class="n">s</span> <span class="n">hs</span> <span class="k">have</span> <span class="n">h₁</span> <span class="o">:</span> <span class="o">((</span><span class="mi">4</span> <span class="bp">*</span> <span class="bp">∏</span> <span class="n">i</span> <span class="k">in</span> <span class="n">erase</span> <span class="n">s</span> <span class="mi">3</span><span class="o">,</span> <span class="n">i</span><span class="o">)</span> <span class="bp">+</span> <span class="mi">3</span><span class="o">)</span> <span class="bp">%</span> <span class="mi">4</span> <span class="bp">=</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">rcases</span> <span class="n">exists_prime_factor_mod_4_eq_3</span> <span class="n">h₁</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">p</span><span class="o">,</span> <span class="n">pp</span><span class="o">,</span> <span class="n">pdvd</span><span class="o">,</span> <span class="n">p4eq</span><span class="o">⟩</span> <span class="k">have</span> <span class="n">ps</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">pne3</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">≠</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">∣</span> <span class="mi">4</span> <span class="bp">*</span> <span class="bp">∏</span> <span class="n">i</span> <span class="k">in</span> <span class="n">erase</span> <span class="n">s</span> <span class="mi">3</span><span class="o">,</span> <span class="n">i</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">∣</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="n">p</span> <span class="bp">=</span> <span class="mi">3</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">contradiction</span> </pre></div> </div> <p>If you managed to complete the proof, congratulations! This has been a serious feat of formalization.</p> </section> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C04_Sets_and_Functions.html" class="btn btn-neutral float-left" title="4. Sets and Functions" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> <a href="C06_Structures.html" class="btn btn-neutral float-right" title="6. Structures" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> </div> <hr/> <div role="contentinfo"> <p>© Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. 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Structures — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> <link rel="shortcut icon" href="_static/favicon.ico"/> <!--[if lt IE 9]> <script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" /> <link rel="next" title="7. Hierarchies" href="C07_Hierarchies.html" /> <link rel="prev" title="5. Number Theory" href="C05_Number_Theory.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul class="current"> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">6. Structures</a><ul> <li class="toctree-l2"><a class="reference internal" href="#defining-structures">6.1. Defining structures</a></li> <li class="toctree-l2"><a class="reference internal" href="#algebraic-structures">6.2. Algebraic Structures</a></li> <li class="toctree-l2"><a class="reference internal" href="#building-the-gaussian-integers">6.3. Building the Gaussian Integers</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">6. </span>Structures</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C06_Structures.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="structures"> <span id="id1"></span><h1><span class="section-number">6. </span>Structures<a class="headerlink" href="#structures" title="Permalink to this heading"></a></h1> <p>Modern mathematics makes essential use of algebraic structures, which encapsulate patterns that can be instantiated in multiple settings. The subject provides various ways of defining such structures and constructing particular instances.</p> <p>Lean therefore provides corresponding ways of defining structures formally and working with them. You have already seen examples of algebraic structures in Lean, such as rings and lattices, which were discussed in <a class="reference internal" href="C02_Basics.html#basics"><span class="std std-numref">Chapter 2</span></a>. This chapter will explain the mysterious square bracket annotations that you saw there, <code class="docutils literal notranslate"><span class="pre">[Ring</span> <span class="pre">α]</span></code> and <code class="docutils literal notranslate"><span class="pre">[Lattice</span> <span class="pre">α]</span></code>. It will also show you how to define and use algebraic structures on your own.</p> <p>For more technical detail, you can consult <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, and a paper by Anne Baanen, <a class="reference external" href="https://arxiv.org/abs/2202.01629">Use and abuse of instance parameters in the Lean mathematical library</a>.</p> <section id="defining-structures"> <span id="section-structures"></span><h2><span class="section-number">6.1. </span>Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this heading"></a></h2> <p>In the broadest sense of the term, a <em>structure</em> is a specification of a collection of data, possibly with constraints that the data is required to satisfy. An <em>instance</em> of the structure is a particular bundle of data satisfying the constraints. For example, we can specify that a point is a tuple of three real numbers:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">Point</span> <span class="n">where</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">@[ext]</span></code> annotation tells Lean to automatically generate theorems that can be used to prove that two instances of a structure are equal when their components are equal, a property known as <em>extensionality</em>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">Point.ext</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">(</span><span class="n">hx</span> <span class="o">:</span> <span class="n">a.x</span> <span class="bp">=</span> <span class="n">b.x</span><span class="o">)</span> <span class="o">(</span><span class="n">hy</span> <span class="o">:</span> <span class="n">a.y</span> <span class="bp">=</span> <span class="n">b.y</span><span class="o">)</span> <span class="o">(</span><span class="n">hz</span> <span class="o">:</span> <span class="n">a.z</span> <span class="bp">=</span> <span class="n">b.z</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">repeat'</span> <span class="n">assumption</span> </pre></div> </div> <p>We can then define particular instances of the <code class="docutils literal notranslate"><span class="pre">Point</span></code> structure. Lean provides multiple ways of doing that.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">myPoint1</span> <span class="o">:</span> <span class="n">Point</span> <span class="n">where</span> <span class="n">x</span> <span class="o">:=</span> <span class="mi">2</span> <span class="n">y</span> <span class="o">:=</span> <span class="bp">-</span><span class="mi">1</span> <span class="n">z</span> <span class="o">:=</span> <span class="mi">4</span> <span class="kd">def</span> <span class="n">myPoint2</span> <span class="o">:</span> <span class="n">Point</span> <span class="o">:=</span> <span class="o">⟨</span><span class="mi">2</span><span class="o">,</span> <span class="bp">-</span><span class="mi">1</span><span class="o">,</span> <span class="mi">4</span><span class="o">⟩</span> <span class="kd">def</span> <span class="n">myPoint3</span> <span class="o">:=</span> <span class="n">Point.mk</span> <span class="mi">2</span> <span class="o">(</span><span class="bp">-</span><span class="mi">1</span><span class="o">)</span> <span class="mi">4</span> </pre></div> </div> <p>In the first example, the fields of the structure are named explicitly. The function <code class="docutils literal notranslate"><span class="pre">Point.mk</span></code> referred to in the definition of <code class="docutils literal notranslate"><span class="pre">myPoint3</span></code> is known as the <em>constructor</em> for the <code class="docutils literal notranslate"><span class="pre">Point</span></code> structure, because it serves to construct elements. You can specify a different name if you want, like <code class="docutils literal notranslate"><span class="pre">build</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">structure</span> <span class="n">Point'</span> <span class="n">where</span> <span class="n">build</span> <span class="o">::</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="k">#check</span> <span class="n">Point'.build</span> <span class="mi">2</span> <span class="o">(</span><span class="bp">-</span><span class="mi">1</span><span class="o">)</span> <span class="mi">4</span> </pre></div> </div> <p>The next two examples show how to define functions on structures. Whereas the second example makes the <code class="docutils literal notranslate"><span class="pre">Point.mk</span></code> constructor explicit, the first example uses an anonymous constructor for brevity. Lean can infer the relevant constructor from the indicated type of <code class="docutils literal notranslate"><span class="pre">add</span></code>. It is conventional to put definitions and theorems associated with a structure like <code class="docutils literal notranslate"><span class="pre">Point</span></code> in a namespace with the same name. In the example below, because we have opened the <code class="docutils literal notranslate"><span class="pre">Point</span></code> namespace, the full name of <code class="docutils literal notranslate"><span class="pre">add</span></code> is <code class="docutils literal notranslate"><span class="pre">Point.add</span></code>. When the namespace is not open, we have to use the full name. But remember that it is often convenient to use anonymous projection notation, which allows us to write <code class="docutils literal notranslate"><span class="pre">a.add</span> <span class="pre">b</span></code> instead of <code class="docutils literal notranslate"><span class="pre">Point.add</span> <span class="pre">a</span> <span class="pre">b</span></code>. Lean interprets the former as the latter because <code class="docutils literal notranslate"><span class="pre">a</span></code> has type <code class="docutils literal notranslate"><span class="pre">Point</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">namespace</span> <span class="n">Point</span> <span class="kd">def</span> <span class="n">add</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">Point</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">a.x</span> <span class="bp">+</span> <span class="n">b.x</span><span class="o">,</span> <span class="n">a.y</span> <span class="bp">+</span> <span class="n">b.y</span><span class="o">,</span> <span class="n">a.z</span> <span class="bp">+</span> <span class="n">b.z</span><span class="o">⟩</span> <span class="kd">def</span> <span class="n">add'</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">Point</span> <span class="n">where</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">a.x</span> <span class="bp">+</span> <span class="n">b.x</span> <span class="n">y</span> <span class="o">:=</span> <span class="n">a.y</span> <span class="bp">+</span> <span class="n">b.y</span> <span class="n">z</span> <span class="o">:=</span> <span class="n">a.z</span> <span class="bp">+</span> <span class="n">b.z</span> <span class="k">#check</span> <span class="n">add</span> <span class="n">myPoint1</span> <span class="n">myPoint2</span> <span class="k">#check</span> <span class="n">myPoint1.add</span> <span class="n">myPoint2</span> <span class="kd">end</span> <span class="n">Point</span> <span class="k">#check</span> <span class="n">Point.add</span> <span class="n">myPoint1</span> <span class="n">myPoint2</span> <span class="k">#check</span> <span class="n">myPoint1.add</span> <span class="n">myPoint2</span> </pre></div> </div> <p>Below we will continue to put definitions in the relevant namespace, but we will leave the namespacing commands out of the quoted snippets. To prove properties of the addition function, we can use <code class="docutils literal notranslate"><span class="pre">rw</span></code> to expand the definition and <code class="docutils literal notranslate"><span class="pre">ext</span></code> to reduce an equation between two elements of the structure to equations between the components. Below we use the <code class="docutils literal notranslate"><span class="pre">protected</span></code> keyword so that the name of the theorem is <code class="docutils literal notranslate"><span class="pre">Point.add_comm</span></code>, even when the namespace is open. This is helpful when we want to avoid ambiguity with a generic theorem like <code class="docutils literal notranslate"><span class="pre">add_comm</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">protected</span> <span class="kd">theorem</span> <span class="n">add_comm</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">add</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">add</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add</span><span class="o">,</span> <span class="n">add</span><span class="o">]</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">dsimp</span> <span class="n">repeat'</span> <span class="n">apply</span> <span class="n">add_comm</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">add</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">add</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">add</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> </pre></div> </div> <p>Because Lean can unfold definitions and simplify projections internally, sometimes the equations we want hold definitionally.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">add_x</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">a.add</span> <span class="n">b</span><span class="o">)</span><span class="bp">.</span><span class="n">x</span> <span class="bp">=</span> <span class="n">a.x</span> <span class="bp">+</span> <span class="n">b.x</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>It is also possible to define functions on structures using pattern matching, in a manner similar to the way we defined recursive functions in <a class="reference internal" href="C05_Number_Theory.html#section-induction-and-recursion"><span class="std std-numref">Section 5.2</span></a>. The definitions <code class="docutils literal notranslate"><span class="pre">addAlt</span></code> and <code class="docutils literal notranslate"><span class="pre">addAlt'</span></code> below are essentially the same; the only difference is that we use anonymous constructor notation in the second. Although it is sometimes convenient to define functions this way, the definitional properties are not as convenient. For example, the expressions <code class="docutils literal notranslate"><span class="pre">addAlt</span> <span class="pre">a</span> <span class="pre">b</span></code> and <code class="docutils literal notranslate"><span class="pre">addAlt'</span> <span class="pre">a</span> <span class="pre">b</span></code> cannot be simplified until we decompose <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code> into components, which we can do with <code class="docutils literal notranslate"><span class="pre">cases</span></code>, <code class="docutils literal notranslate"><span class="pre">rcases</span></code>, etc.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">addAlt</span> <span class="o">:</span> <span class="n">Point</span> <span class="bp">→</span> <span class="n">Point</span> <span class="bp">→</span> <span class="n">Point</span> <span class="bp">|</span> <span class="n">Point.mk</span> <span class="n">x₁</span> <span class="n">y₁</span> <span class="n">z₁</span><span class="o">,</span> <span class="n">Point.mk</span> <span class="n">x₂</span> <span class="n">y₂</span> <span class="n">z₂</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">x₁</span> <span class="bp">+</span> <span class="n">x₂</span><span class="o">,</span> <span class="n">y₁</span> <span class="bp">+</span> <span class="n">y₂</span><span class="o">,</span> <span class="n">z₁</span> <span class="bp">+</span> <span class="n">z₂</span><span class="o">⟩</span> <span class="kd">def</span> <span class="n">addAlt'</span> <span class="o">:</span> <span class="n">Point</span> <span class="bp">→</span> <span class="n">Point</span> <span class="bp">→</span> <span class="n">Point</span> <span class="bp">|</span> <span class="o">⟨</span><span class="n">x₁</span><span class="o">,</span> <span class="n">y₁</span><span class="o">,</span> <span class="n">z₁</span><span class="o">⟩,</span> <span class="o">⟨</span><span class="n">x₂</span><span class="o">,</span> <span class="n">y₂</span><span class="o">,</span> <span class="n">z₂</span><span class="o">⟩</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">x₁</span> <span class="bp">+</span> <span class="n">x₂</span><span class="o">,</span> <span class="n">y₁</span> <span class="bp">+</span> <span class="n">y₂</span><span class="o">,</span> <span class="n">z₁</span> <span class="bp">+</span> <span class="n">z₂</span><span class="o">⟩</span> <span class="kd">theorem</span> <span class="n">addAlt_x</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">a.addAlt</span> <span class="n">b</span><span class="o">)</span><span class="bp">.</span><span class="n">x</span> <span class="bp">=</span> <span class="n">a.x</span> <span class="bp">+</span> <span class="n">b.x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases</span> <span class="n">a</span> <span class="n">cases</span> <span class="n">b</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">addAlt_comm</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">addAlt</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">addAlt</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">a</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">⟩</span> <span class="n">rcases</span> <span class="n">b</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">⟩</span> <span class="n">rw</span> <span class="o">[</span><span class="n">addAlt</span><span class="o">,</span> <span class="n">addAlt</span><span class="o">]</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">dsimp</span> <span class="n">apply</span> <span class="n">add_comm</span> <span class="n">repeat'</span> <span class="n">apply</span> <span class="n">add_comm</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">addAlt</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">addAlt</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">a</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">⟩</span> <span class="n">rcases</span> <span class="n">b</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">⟩</span> <span class="n">simp</span> <span class="o">[</span><span class="n">addAlt</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> <span class="kd">example</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">,</span> <span class="n">addAlt</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">addAlt</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">⟨</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">⟩</span> <span class="o">⟨</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">⟩</span> <span class="n">simp</span> <span class="o">[</span><span class="n">addAlt</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> <span class="kd">example</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">,</span> <span class="n">add</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">add</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">⟩</span> <span class="o">⟨</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">⟩</span> <span class="bp">=></span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">add</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> </pre></div> </div> <p>Mathematical constructions often involve taking apart bundled information and putting it together again in different ways. It therefore makes sense that Lean and mathlib offer so many ways of doing this efficiently. As an exercise, try proving that <code class="docutils literal notranslate"><span class="pre">Point.add</span></code> is associative. Then define scalar multiplication for a point and show that it distributes over addition.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">protected</span> <span class="kd">theorem</span> <span class="n">add_assoc</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">a.add</span> <span class="n">b</span><span class="o">)</span><span class="bp">.</span><span class="n">add</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a.add</span> <span class="o">(</span><span class="n">b.add</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">def</span> <span class="n">smul</span> <span class="o">(</span><span class="n">r</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">Point</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">smul_distrib</span> <span class="o">(</span><span class="n">r</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">smul</span> <span class="n">r</span> <span class="n">a</span><span class="o">)</span><span class="bp">.</span><span class="n">add</span> <span class="o">(</span><span class="n">smul</span> <span class="n">r</span> <span class="n">b</span><span class="o">)</span> <span class="bp">=</span> <span class="n">smul</span> <span class="n">r</span> <span class="o">(</span><span class="n">a.add</span> <span class="n">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Using structures is only the first step on the road to algebraic abstraction. We don’t yet have a way to link <code class="docutils literal notranslate"><span class="pre">Point.add</span></code> to the generic <code class="docutils literal notranslate"><span class="pre">+</span></code> symbol, or to connect <code class="docutils literal notranslate"><span class="pre">Point.add_comm</span></code> and <code class="docutils literal notranslate"><span class="pre">Point.add_assoc</span></code> to the generic <code class="docutils literal notranslate"><span class="pre">add_comm</span></code> and <code class="docutils literal notranslate"><span class="pre">add_assoc</span></code> theorems. These tasks belong to the <em>algebraic</em> aspect of using structures, and we will explain how to carry them out in the next section. For now, just think of a structure as a way of bundling together objects and information.</p> <p>It is especially useful that a structure can specify not only data types but also constraints that the data must satisfy. In Lean, the latter are represented as fields of type <code class="docutils literal notranslate"><span class="pre">Prop</span></code>. For example, the <em>standard 2-simplex</em> is defined to be the set of points <span class="math notranslate nohighlight">\((x, y, z)\)</span> satisfying <span class="math notranslate nohighlight">\(x ≥ 0\)</span>, <span class="math notranslate nohighlight">\(y ≥ 0\)</span>, <span class="math notranslate nohighlight">\(z ≥ 0\)</span>, and <span class="math notranslate nohighlight">\(x + y + z = 1\)</span>. If you are not familiar with the notion, you should draw a picture, and convince yourself that this set is the equilateral triangle in three-space with vertices <span class="math notranslate nohighlight">\((1, 0, 0)\)</span>, <span class="math notranslate nohighlight">\((0, 1, 0)\)</span>, and <span class="math notranslate nohighlight">\((0, 0, 1)\)</span>, together with its interior. We can represent it in Lean as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">structure</span> <span class="n">StandardTwoSimplex</span> <span class="n">where</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">x_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">x</span> <span class="n">y_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">y</span> <span class="n">z_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">z</span> <span class="n">sum_eq</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">+</span> <span class="n">z</span> <span class="bp">=</span> <span class="mi">1</span> </pre></div> </div> <p>Notice that the last four fields refer to <code class="docutils literal notranslate"><span class="pre">x</span></code>, <code class="docutils literal notranslate"><span class="pre">y</span></code>, and <code class="docutils literal notranslate"><span class="pre">z</span></code>, that is, the first three fields. We can define a map from the two-simplex to itself that swaps <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">swapXy</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">StandardTwoSimplex</span><span class="o">)</span> <span class="o">:</span> <span class="n">StandardTwoSimplex</span> <span class="n">where</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">a.y</span> <span class="n">y</span> <span class="o">:=</span> <span class="n">a.x</span> <span class="n">z</span> <span class="o">:=</span> <span class="n">a.z</span> <span class="n">x_nonneg</span> <span class="o">:=</span> <span class="n">a.y_nonneg</span> <span class="n">y_nonneg</span> <span class="o">:=</span> <span class="n">a.x_nonneg</span> <span class="n">z_nonneg</span> <span class="o">:=</span> <span class="n">a.z_nonneg</span> <span class="n">sum_eq</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_comm</span> <span class="n">a.y</span> <span class="n">a.x</span><span class="o">,</span> <span class="n">a.sum_eq</span><span class="o">]</span> </pre></div> </div> <p>More interestingly, we can compute the midpoint of two points on the simplex. We have added the phrase <code class="docutils literal notranslate"><span class="pre">noncomputable</span> <span class="pre">section</span></code> at the beginning of this file in order to use division on the real numbers.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">noncomputable</span> <span class="kn">section</span> <span class="kd">def</span> <span class="n">midpoint</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">StandardTwoSimplex</span><span class="o">)</span> <span class="o">:</span> <span class="n">StandardTwoSimplex</span> <span class="n">where</span> <span class="n">x</span> <span class="o">:=</span> <span class="o">(</span><span class="n">a.x</span> <span class="bp">+</span> <span class="n">b.x</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="n">y</span> <span class="o">:=</span> <span class="o">(</span><span class="n">a.y</span> <span class="bp">+</span> <span class="n">b.y</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="n">z</span> <span class="o">:=</span> <span class="o">(</span><span class="n">a.z</span> <span class="bp">+</span> <span class="n">b.z</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="n">x_nonneg</span> <span class="o">:=</span> <span class="n">div_nonneg</span> <span class="o">(</span><span class="n">add_nonneg</span> <span class="n">a.x_nonneg</span> <span class="n">b.x_nonneg</span><span class="o">)</span> <span class="o">(</span><span class="kd">by</span> <span class="n">norm_num</span><span class="o">)</span> <span class="n">y_nonneg</span> <span class="o">:=</span> <span class="n">div_nonneg</span> <span class="o">(</span><span class="n">add_nonneg</span> <span class="n">a.y_nonneg</span> <span class="n">b.y_nonneg</span><span class="o">)</span> <span class="o">(</span><span class="kd">by</span> <span class="n">norm_num</span><span class="o">)</span> <span class="n">z_nonneg</span> <span class="o">:=</span> <span class="n">div_nonneg</span> <span class="o">(</span><span class="n">add_nonneg</span> <span class="n">a.z_nonneg</span> <span class="n">b.z_nonneg</span><span class="o">)</span> <span class="o">(</span><span class="kd">by</span> <span class="n">norm_num</span><span class="o">)</span> <span class="n">sum_eq</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">field_simp</span><span class="bp">;</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">a.sum_eq</span><span class="o">,</span> <span class="n">b.sum_eq</span><span class="o">]</span> </pre></div> </div> <p>Here we have established <code class="docutils literal notranslate"><span class="pre">x_nonneg</span></code>, <code class="docutils literal notranslate"><span class="pre">y_nonneg</span></code>, and <code class="docutils literal notranslate"><span class="pre">z_nonneg</span></code> with concise proof terms, but establish <code class="docutils literal notranslate"><span class="pre">sum_eq</span></code> in tactic mode, using <code class="docutils literal notranslate"><span class="pre">by</span></code>.</p> <p>Given a parameter <span class="math notranslate nohighlight">\(\lambda\)</span> satisfying <span class="math notranslate nohighlight">\(0 \le \lambda \le 1\)</span>, we can take the weighted average <span class="math notranslate nohighlight">\(\lambda a + (1 - \lambda) b\)</span> of two points <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> in the standard 2-simplex. We challenge you to define that function, in analogy to the <code class="docutils literal notranslate"><span class="pre">midpoint</span></code> function above.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">weightedAverage</span> <span class="o">(</span><span class="n">lambda</span> <span class="o">:</span> <span class="n">Real</span><span class="o">)</span> <span class="o">(</span><span class="n">lambda_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">lambda</span><span class="o">)</span> <span class="o">(</span><span class="n">lambda_le</span> <span class="o">:</span> <span class="n">lambda</span> <span class="bp">≤</span> <span class="mi">1</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">StandardTwoSimplex</span><span class="o">)</span> <span class="o">:</span> <span class="n">StandardTwoSimplex</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Structures can depend on parameters. For example, we can generalize the standard 2-simplex to the standard <span class="math notranslate nohighlight">\(n\)</span>-simplex for any <span class="math notranslate nohighlight">\(n\)</span>. At this stage, you don’t have to know anything about the type <cite>Fin n</cite> except that it has <span class="math notranslate nohighlight">\(n\)</span> elements, and that Lean knows how to sum over it.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">BigOperators</span> <span class="kd">structure</span> <span class="n">StandardSimplex</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="n">where</span> <span class="n">V</span> <span class="o">:</span> <span class="n">Fin</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">ℝ</span> <span class="n">NonNeg</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">i</span> <span class="o">:</span> <span class="n">Fin</span> <span class="n">n</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">V</span> <span class="n">i</span> <span class="n">sum_eq_one</span> <span class="o">:</span> <span class="o">(</span><span class="bp">∑</span> <span class="n">i</span><span class="o">,</span> <span class="n">V</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="mi">1</span> <span class="kn">namespace</span> <span class="n">StandardSimplex</span> <span class="kd">def</span> <span class="n">midpoint</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">StandardSimplex</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">StandardSimplex</span> <span class="n">n</span> <span class="n">where</span> <span class="n">V</span> <span class="n">i</span> <span class="o">:=</span> <span class="o">(</span><span class="n">a.V</span> <span class="n">i</span> <span class="bp">+</span> <span class="n">b.V</span> <span class="n">i</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="n">NonNeg</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">i</span> <span class="n">apply</span> <span class="n">div_nonneg</span> <span class="bp">·</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">a.NonNeg</span> <span class="n">i</span><span class="o">,</span> <span class="n">b.NonNeg</span> <span class="n">i</span><span class="o">]</span> <span class="n">norm_num</span> <span class="n">sum_eq_one</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">div_eq_mul_inv</span><span class="o">,</span> <span class="bp">←</span> <span class="n">Finset.sum_mul</span><span class="o">,</span> <span class="n">Finset.sum_add_distrib</span><span class="o">,</span> <span class="n">a.sum_eq_one</span><span class="o">,</span> <span class="n">b.sum_eq_one</span><span class="o">]</span> <span class="n">field_simp</span> <span class="kd">end</span> <span class="n">StandardSimplex</span> </pre></div> </div> <p>As an exercise, see if you can define the weighted average of two points in the standard <span class="math notranslate nohighlight">\(n\)</span>-simplex. You can use <code class="docutils literal notranslate"><span class="pre">Finset.sum_add_distrib</span></code> and <code class="docutils literal notranslate"><span class="pre">Finset.mul_sum</span></code> to manipulate the relevant sums. We have seen that structures can be used to bundle together data and properties. Interestingly, they can also be used to bundle together properties without the data. For example, the next structure, <code class="docutils literal notranslate"><span class="pre">IsLinear</span></code>, bundles together the two components of linearity.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">structure</span> <span class="n">IsLinear</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="n">where</span> <span class="n">is_additive</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span><span class="o">,</span> <span class="n">f</span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">f</span> <span class="n">y</span> <span class="n">preserves_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">c</span><span class="o">,</span> <span class="n">f</span> <span class="o">(</span><span class="n">c</span> <span class="bp">*</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">x</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">linf</span> <span class="o">:</span> <span class="n">IsLinear</span> <span class="n">f</span><span class="o">)</span> <span class="k">#check</span> <span class="n">linf.is_additive</span> <span class="k">#check</span> <span class="n">linf.preserves_mul</span> <span class="kd">end</span> </pre></div> </div> <p>It is worth pointing out that structures are not the only way to bundle together data. The <code class="docutils literal notranslate"><span class="pre">Point</span></code> data structure can be defined using the generic type product, and <code class="docutils literal notranslate"><span class="pre">IsLinear</span></code> can be defined with a simple <code class="docutils literal notranslate"><span class="pre">and</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">Point''</span> <span class="o">:=</span> <span class="n">ℝ</span> <span class="bp">×</span> <span class="n">ℝ</span> <span class="bp">×</span> <span class="n">ℝ</span> <span class="kd">def</span> <span class="n">IsLinear'</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span><span class="o">,</span> <span class="n">f</span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">f</span> <span class="n">y</span><span class="o">)</span> <span class="bp">∧</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">c</span><span class="o">,</span> <span class="n">f</span> <span class="o">(</span><span class="n">c</span> <span class="bp">*</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">x</span> </pre></div> </div> <p>Generic type constructions can even be used in place of structures with dependencies between their components. For example, the <em>subtype</em> construction combines a piece of data with a property. You can think of the type <code class="docutils literal notranslate"><span class="pre">PReal</span></code> in the next example as being the type of positive real numbers. Any <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">PReal</span></code> has two components: the value, and the property of being positive. You can access these components as <code class="docutils literal notranslate"><span class="pre">x.val</span></code>, which has type <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>, and <code class="docutils literal notranslate"><span class="pre">x.property</span></code>, which represents the fact <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">x.val</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">PReal</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">//</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">y</span> <span class="o">}</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">PReal</span><span class="o">)</span> <span class="k">#check</span> <span class="n">x.val</span> <span class="k">#check</span> <span class="n">x.property</span> <span class="k">#check</span> <span class="n">x.1</span> <span class="k">#check</span> <span class="n">x.2</span> <span class="kd">end</span> </pre></div> </div> <p>We could have used subtypes to define the standard 2-simplex, as well as the standard <span class="math notranslate nohighlight">\(n\)</span>-simplex for an arbitrary <span class="math notranslate nohighlight">\(n\)</span>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">StandardTwoSimplex'</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">p</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">×</span> <span class="n">ℝ</span> <span class="bp">×</span> <span class="n">ℝ</span> <span class="bp">//</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">p.1</span> <span class="bp">∧</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">p.2.1</span> <span class="bp">∧</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">p.2.2</span> <span class="bp">∧</span> <span class="n">p.1</span> <span class="bp">+</span> <span class="n">p.2.1</span> <span class="bp">+</span> <span class="n">p.2.2</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">}</span> <span class="kd">def</span> <span class="n">StandardSimplex'</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">v</span> <span class="o">:</span> <span class="n">Fin</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">ℝ</span> <span class="bp">//</span> <span class="o">(</span><span class="bp">∀</span> <span class="n">i</span> <span class="o">:</span> <span class="n">Fin</span> <span class="n">n</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">v</span> <span class="n">i</span><span class="o">)</span> <span class="bp">∧</span> <span class="o">(</span><span class="bp">∑</span> <span class="n">i</span><span class="o">,</span> <span class="n">v</span> <span class="n">i</span><span class="o">)</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">}</span> </pre></div> </div> <p>Similarly, <em>Sigma types</em> are generalizations of ordered pairs, whereby the type of the second component depends on the type of the first.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">StdSimplex</span> <span class="o">:=</span> <span class="bp">Σ</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">StandardSimplex</span> <span class="n">n</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">StdSimplex</span><span class="o">)</span> <span class="k">#check</span> <span class="n">s.fst</span> <span class="k">#check</span> <span class="n">s.snd</span> <span class="k">#check</span> <span class="n">s.1</span> <span class="k">#check</span> <span class="n">s.2</span> <span class="kd">end</span> </pre></div> </div> <p>Given <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">StdSimplex</span></code>, the first component <code class="docutils literal notranslate"><span class="pre">s.fst</span></code> is a natural number, and the second component is an element of the corresponding simplex <code class="docutils literal notranslate"><span class="pre">StandardSimplex</span> <span class="pre">s.fst</span></code>. The difference between a Sigma type and a subtype is that the second component of a Sigma type is data rather than a proposition.</p> <p>But even though we can use products, subtypes, and Sigma types instead of structures, using structures has a number of advantages. Defining a structure abstracts away the underlying representation and provides custom names for the functions that access the components. This makes proofs more robust: proofs that rely only on the interface to a structure will generally continue to work when we change the definition, as long as we redefine the old accessors in terms of the new definition. Moreover, as we are about to see, Lean provides support for weaving structures together into a rich, interconnected hierarchy, and for managing the interactions between them.</p> </section> <section id="algebraic-structures"> <span id="section-algebraic-structures"></span><h2><span class="section-number">6.2. </span>Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this heading"></a></h2> <p>To clarify what we mean by the phrase <em>algebraic structure</em>, it will help to consider some examples.</p> <ol class="arabic simple"> <li><p>A <em>partially ordered set</em> consists of a set <span class="math notranslate nohighlight">\(P\)</span> and a binary relation <span class="math notranslate nohighlight">\(\le\)</span> on <span class="math notranslate nohighlight">\(P\)</span> that is transitive and antireflexive.</p></li> <li><p>A <em>group</em> consists of a set <span class="math notranslate nohighlight">\(G\)</span> with an associative binary operation, an identity element <span class="math notranslate nohighlight">\(1\)</span>, and a function <span class="math notranslate nohighlight">\(g \mapsto g^{-1}\)</span> that returns an inverse for each <span class="math notranslate nohighlight">\(g\)</span> in <span class="math notranslate nohighlight">\(G\)</span>. A group is <em>abelian</em> or <em>commutative</em> if the operation is commutative.</p></li> <li><p>A <em>lattice</em> is a partially ordered set with meets and joins.</p></li> <li><p>A <em>ring</em> consists of an (additively written) abelian group <span class="math notranslate nohighlight">\((R, +, 0, x \mapsto -x)\)</span> together with an associative multiplication operation <span class="math notranslate nohighlight">\(\cdot\)</span> and an identity <span class="math notranslate nohighlight">\(1\)</span>, such that multiplication distributes over addition. A ring is <em>commutative</em> if the multiplication is commutative.</p></li> <li><p>An <em>ordered ring</em> <span class="math notranslate nohighlight">\((R, +, 0, -, \cdot, 1, \le)\)</span> consists of a ring together with a partial order on its elements, such that <span class="math notranslate nohighlight">\(a \le b\)</span> implies <span class="math notranslate nohighlight">\(a + c \le b + c\)</span> for every <span class="math notranslate nohighlight">\(a\)</span>, <span class="math notranslate nohighlight">\(b\)</span>, and <span class="math notranslate nohighlight">\(c\)</span> in <span class="math notranslate nohighlight">\(R\)</span>, and <span class="math notranslate nohighlight">\(0 \le a\)</span> and <span class="math notranslate nohighlight">\(0 \le b\)</span> implies <span class="math notranslate nohighlight">\(0 \le a b\)</span> for every <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> in <span class="math notranslate nohighlight">\(R\)</span>.</p></li> <li><p>A <em>metric space</em> consists of a set <span class="math notranslate nohighlight">\(X\)</span> and a function <span class="math notranslate nohighlight">\(d : X \times X \to \mathbb{R}\)</span> such that the following hold:</p> <ul class="simple"> <li><p><span class="math notranslate nohighlight">\(d(x, y) \ge 0\)</span> for every <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> in <span class="math notranslate nohighlight">\(X\)</span>.</p></li> <li><p><span class="math notranslate nohighlight">\(d(x, y) = 0\)</span> if and only if <span class="math notranslate nohighlight">\(x = y\)</span>.</p></li> <li><p><span class="math notranslate nohighlight">\(d(x, y) = d(y, x)\)</span> for every <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> in <span class="math notranslate nohighlight">\(X\)</span>.</p></li> <li><p><span class="math notranslate nohighlight">\(d(x, z) \le d(x, y) + d(y, z)\)</span> for every <span class="math notranslate nohighlight">\(x\)</span>, <span class="math notranslate nohighlight">\(y\)</span>, and <span class="math notranslate nohighlight">\(z\)</span> in <span class="math notranslate nohighlight">\(X\)</span>.</p></li> </ul> </li> <li><p>A <em>topological space</em> consists of a set <span class="math notranslate nohighlight">\(X\)</span> and a collection <span class="math notranslate nohighlight">\(\mathcal T\)</span> of subsets of <span class="math notranslate nohighlight">\(X\)</span>, called the <em>open subsets of</em> <span class="math notranslate nohighlight">\(X\)</span>, such that the following hold:</p> <ul class="simple"> <li><p>The empty set and <span class="math notranslate nohighlight">\(X\)</span> are open.</p></li> <li><p>The intersection of two open sets is open.</p></li> <li><p>An arbitrary union of open sets is open.</p></li> </ul> </li> </ol> <p>In each of these examples, the elements of the structure belong to a set, the <em>carrier set</em>, that sometimes stands proxy for the entire structure. For example, when we say “let <span class="math notranslate nohighlight">\(G\)</span> be a group” and then “let <span class="math notranslate nohighlight">\(g \in G\)</span>,” we are using <span class="math notranslate nohighlight">\(G\)</span> to stand for both the structure and its carrier. Not every algebraic structure is associated with a single carrier set in this way. For example, a <em>bipartite graph</em> involves a relation between two sets, as does a <em>Galois connection</em>, A <em>category</em> also involves two sets of interest, commonly called the <em>objects</em> and the <em>morphisms</em>.</p> <p>The examples indicate some of the things that a proof assistant has to do in order to support algebraic reasoning. First, it needs to recognize concrete instances of structures. The number systems <span class="math notranslate nohighlight">\(\mathbb{Z}\)</span>, <span class="math notranslate nohighlight">\(\mathbb{Q}\)</span>, and <span class="math notranslate nohighlight">\(\mathbb{R}\)</span> are all ordered rings, and we should be able to apply a generic theorem about ordered rings in any of these instances. Sometimes a concrete set may be an instance of a structure in more than one way. For example, in addition to the usual topology on <span class="math notranslate nohighlight">\(\mathbb{R}\)</span>, which forms the basis for real analysis, we can also consider the <em>discrete</em> topology on <span class="math notranslate nohighlight">\(\mathbb{R}\)</span>, in which every set is open.</p> <p>Second, a proof assistant needs to support generic notation on structures. In Lean, the notation <code class="docutils literal notranslate"><span class="pre">*</span></code> is used for multiplication in all the usual number systems, as well as for multiplication in generic groups and rings. When we use an expression like <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">*</span> <span class="pre">y</span></code>, Lean has to use information about the types of <code class="docutils literal notranslate"><span class="pre">f</span></code>, <code class="docutils literal notranslate"><span class="pre">x</span></code>, and <code class="docutils literal notranslate"><span class="pre">y</span></code> to determine which multiplication we have in mind.</p> <p>Third, it needs to deal with the fact that structures can inherit definitions, theorems, and notation from other structures in various ways. Some structures extend others by adding more axioms. A commutative ring is still a ring, so any definition that makes sense in a ring also makes sense in a commutative ring, and any theorem that holds in a ring also holds in a commutative ring. Some structures extend others by adding more data. For example, the additive part of any ring is an additive group. The ring structure adds a multiplication and an identity, as well as axioms that govern them and relate them to the additive part. Sometimes we can define one structure in terms of another. Any metric space has a canonical topology associated with it, the <em>metric space topology</em>, and there are various topologies that can be associated with any linear ordering.</p> <p>Finally, it is important to keep in mind that mathematics allows us to use functions and operations to define structures in the same way we use functions and operations to define numbers. Products and powers of groups are again groups. For every <span class="math notranslate nohighlight">\(n\)</span>, the integers modulo <span class="math notranslate nohighlight">\(n\)</span> form a ring, and for every <span class="math notranslate nohighlight">\(k > 0\)</span>, the <span class="math notranslate nohighlight">\(k \times k\)</span> matrices of polynomials with coefficients in that ring again form a ring. Thus we can calculate with structures just as easily as we can calculate with their elements. This means that algebraic structures lead dual lives in mathematics, as containers for collections of objects and as objects in their own right. A proof assistant has to accommodate this dual role.</p> <p>When dealing with elements of a type that has an algebraic structure associated with it, a proof assistant needs to recognize the structure and find the relevant definitions, theorems, and notation. All this should sound like a lot of work, and it is. But Lean uses a small collection of fundamental mechanisms to carry out these tasks. The goal of this section is to explain these mechanisms and show you how to use them.</p> <p>The first ingredient is almost too obvious to mention: formally speaking, algebraic structures are structures in the sense of <a class="reference internal" href="#section-structures"><span class="std std-numref">Section 6.1</span></a>. An algebraic structure is a specification of a bundle of data satisfying some axiomatic hypotheses, and we saw in <a class="reference internal" href="#section-structures"><span class="std std-numref">Section 6.1</span></a> that this is exactly what the <code class="docutils literal notranslate"><span class="pre">structure</span></code> command is designed to accommodate. It’s a marriage made in heaven!</p> <p>Given a data type <code class="docutils literal notranslate"><span class="pre">α</span></code>, we can define the group structure on <code class="docutils literal notranslate"><span class="pre">α</span></code> as follows.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">structure</span> <span class="n">Group₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="n">where</span> <span class="n">mul</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="n">one</span> <span class="o">:</span> <span class="n">α</span> <span class="n">inv</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="n">mul_assoc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">mul</span> <span class="o">(</span><span class="n">mul</span> <span class="n">x</span> <span class="n">y</span><span class="o">)</span> <span class="n">z</span> <span class="bp">=</span> <span class="n">mul</span> <span class="n">x</span> <span class="o">(</span><span class="n">mul</span> <span class="n">y</span> <span class="n">z</span><span class="o">)</span> <span class="n">mul_one</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">mul</span> <span class="n">x</span> <span class="n">one</span> <span class="bp">=</span> <span class="n">x</span> <span class="n">one_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">mul</span> <span class="n">one</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">x</span> <span class="n">mul_left_inv</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">mul</span> <span class="o">(</span><span class="n">inv</span> <span class="n">x</span><span class="o">)</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">one</span> </pre></div> </div> <p>Notice that the type <code class="docutils literal notranslate"><span class="pre">α</span></code> is a <em>parameter</em> in the definition of <code class="docutils literal notranslate"><span class="pre">group₁</span></code>. So you should think of an object <code class="docutils literal notranslate"><span class="pre">struc</span> <span class="pre">:</span> <span class="pre">Group₁</span> <span class="pre">α</span></code> as being a group structure on <code class="docutils literal notranslate"><span class="pre">α</span></code>. We saw in <a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures"><span class="std std-numref">Section 2.2</span></a> that the counterpart <code class="docutils literal notranslate"><span class="pre">mul_right_inv</span></code> to <code class="docutils literal notranslate"><span class="pre">mul_left_inv</span></code> follows from the other group axioms, so there is no need to add it to the definition.</p> <p>This definition of a group is similar to the definition of <code class="docutils literal notranslate"><span class="pre">Group</span></code> in mathlib, and we have chosen the name <code class="docutils literal notranslate"><span class="pre">Group₁</span></code> to distinguish our version. If you write <code class="docutils literal notranslate"><span class="pre">#check</span> <span class="pre">Group</span></code> and ctrl-click on the definition, you will see that the mathlib version of <code class="docutils literal notranslate"><span class="pre">Group</span></code> is defined to extend another structure; we will explain how to do that later. If you type <code class="docutils literal notranslate"><span class="pre">#print</span> <span class="pre">Group</span></code> you will also see that the mathlib version of <code class="docutils literal notranslate"><span class="pre">Group</span></code> has a number of extra fields. For reasons we will explain later, sometimes it is useful to add redundant information to a structure, so that there are additional fields for objects and functions that can be defined from the core data. Don’t worry about that for now. Rest assured that our simplified version <code class="docutils literal notranslate"><span class="pre">Group₁</span></code> is morally the same as the definition of a group that mathlib uses.</p> <p>It is sometimes useful to bundle the type together with the structure, and mathlib also contains a definition of a <code class="docutils literal notranslate"><span class="pre">GroupCat</span></code> structure that is equivalent to the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">structure</span> <span class="n">Group₁Cat</span> <span class="n">where</span> <span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span> <span class="n">str</span> <span class="o">:</span> <span class="n">Group₁</span> <span class="n">α</span> </pre></div> </div> <p>The mathlib version is found in <code class="docutils literal notranslate"><span class="pre">Algebra.Category.Group.Basic</span></code>, and you can <code class="docutils literal notranslate"><span class="pre">#check</span></code> it if you add this to the imports at the beginning of the examples file.</p> <p>For reasons that will become clearer below, it is more often useful to keep the type <code class="docutils literal notranslate"><span class="pre">α</span></code> separate from the structure <code class="docutils literal notranslate"><span class="pre">Group</span> <span class="pre">α</span></code>. We refer to the two objects together as a <em>partially bundled structure</em>, since the representation combines most, but not all, of the components into one structure. It is common in mathlib to use capital roman letters like <code class="docutils literal notranslate"><span class="pre">G</span></code> for a type when it is used as the carrier type for a group.</p> <p>Let’s construct a group, which is to say, an element of the <code class="docutils literal notranslate"><span class="pre">Group₁</span></code> type. For any pair of types <code class="docutils literal notranslate"><span class="pre">α</span></code> and <code class="docutils literal notranslate"><span class="pre">β</span></code>, Mathlib defines the type <code class="docutils literal notranslate"><span class="pre">Equiv</span> <span class="pre">α</span> <span class="pre">β</span></code> of <em>equivalences</em> between <code class="docutils literal notranslate"><span class="pre">α</span></code> and <code class="docutils literal notranslate"><span class="pre">β</span></code>. Mathlib also defines the suggestive notation <code class="docutils literal notranslate"><span class="pre">α</span> <span class="pre">≃</span> <span class="pre">β</span></code> for this type. An element <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">α</span> <span class="pre">≃</span> <span class="pre">β</span></code> is a bijection between <code class="docutils literal notranslate"><span class="pre">α</span></code> and <code class="docutils literal notranslate"><span class="pre">β</span></code> represented by four components: a function <code class="docutils literal notranslate"><span class="pre">f.toFun</span></code> from <code class="docutils literal notranslate"><span class="pre">α</span></code> to <code class="docutils literal notranslate"><span class="pre">β</span></code>, the inverse function <code class="docutils literal notranslate"><span class="pre">f.invFun</span></code> from <code class="docutils literal notranslate"><span class="pre">β</span></code> to <code class="docutils literal notranslate"><span class="pre">α</span></code>, and two properties that specify these functions are indeed inverse to one another.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">α</span> <span class="n">β</span> <span class="n">γ</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">≃</span> <span class="n">β</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">≃</span> <span class="n">γ</span><span class="o">)</span> <span class="k">#check</span> <span class="n">Equiv</span> <span class="n">α</span> <span class="n">β</span> <span class="k">#check</span> <span class="o">(</span><span class="n">f.toFun</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">f.invFun</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">α</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">f.right_inv</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">β</span><span class="o">,</span> <span class="n">f</span> <span class="o">(</span><span class="n">f.invFun</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">f.left_inv</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">f.invFun</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">Equiv.refl</span> <span class="n">α</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">≃</span> <span class="n">α</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">f.symm</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">≃</span> <span class="n">α</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">f.trans</span> <span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">≃</span> <span class="n">γ</span><span class="o">)</span> </pre></div> </div> <p>Notice the creative naming of the last three constructions. We think of the identity function <code class="docutils literal notranslate"><span class="pre">Equiv.refl</span></code>, the inverse operation <code class="docutils literal notranslate"><span class="pre">Equiv.symm</span></code>, and the composition operation <code class="docutils literal notranslate"><span class="pre">Equiv.trans</span></code> as explicit evidence that the property of being in bijective correspondence is an equivalence relation.</p> <p>Notice also that <code class="docutils literal notranslate"><span class="pre">f.trans</span> <span class="pre">g</span></code> requires composing the forward functions in reverse order. Mathlib has declared a <em>coercion</em> from <code class="docutils literal notranslate"><span class="pre">Equiv</span> <span class="pre">α</span> <span class="pre">β</span></code> to the function type <code class="docutils literal notranslate"><span class="pre">α</span> <span class="pre">→</span> <span class="pre">β</span></code>, so we can omit writing <code class="docutils literal notranslate"><span class="pre">.toFun</span></code> and have Lean insert it for us.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">f.trans</span> <span class="n">g</span><span class="o">)</span><span class="bp">.</span><span class="n">toFun</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">g.toFun</span> <span class="o">(</span><span class="n">f.toFun</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">f.trans</span> <span class="n">g</span><span class="o">)</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">g</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">f.trans</span> <span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">γ</span><span class="o">)</span> <span class="bp">=</span> <span class="n">g</span> <span class="bp">∘</span> <span class="n">f</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>Mathlib also defines the type <code class="docutils literal notranslate"><span class="pre">perm</span> <span class="pre">α</span></code> of equivalences between <code class="docutils literal notranslate"><span class="pre">α</span></code> and itself.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">:</span> <span class="n">Equiv.Perm</span> <span class="n">α</span> <span class="bp">=</span> <span class="o">(</span><span class="n">α</span> <span class="bp">≃</span> <span class="n">α</span><span class="o">)</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>It should be clear that <code class="docutils literal notranslate"><span class="pre">Equiv.Perm</span> <span class="pre">α</span></code> forms a group under composition of equivalences. We orient things so that <code class="docutils literal notranslate"><span class="pre">mul</span> <span class="pre">f</span> <span class="pre">g</span></code> is equal to <code class="docutils literal notranslate"><span class="pre">g.trans</span> <span class="pre">f</span></code>, whose forward function is <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">∘</span> <span class="pre">g</span></code>. In other words, multiplication is what we ordinarily think of as composition of the bijections. Here we define this group:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">permGroup</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">:</span> <span class="n">Group₁</span> <span class="o">(</span><span class="n">Equiv.Perm</span> <span class="n">α</span><span class="o">)</span> <span class="n">where</span> <span class="n">mul</span> <span class="n">f</span> <span class="n">g</span> <span class="o">:=</span> <span class="n">Equiv.trans</span> <span class="n">g</span> <span class="n">f</span> <span class="n">one</span> <span class="o">:=</span> <span class="n">Equiv.refl</span> <span class="n">α</span> <span class="n">inv</span> <span class="o">:=</span> <span class="n">Equiv.symm</span> <span class="n">mul_assoc</span> <span class="n">f</span> <span class="n">g</span> <span class="n">h</span> <span class="o">:=</span> <span class="o">(</span><span class="n">Equiv.trans_assoc</span> <span class="n">_</span> <span class="n">_</span> <span class="n">_</span><span class="o">)</span><span class="bp">.</span><span class="n">symm</span> <span class="n">one_mul</span> <span class="o">:=</span> <span class="n">Equiv.trans_refl</span> <span class="n">mul_one</span> <span class="o">:=</span> <span class="n">Equiv.refl_trans</span> <span class="n">mul_left_inv</span> <span class="o">:=</span> <span class="n">Equiv.self_trans_symm</span> </pre></div> </div> <p>In fact, mathlib defines exactly this <code class="docutils literal notranslate"><span class="pre">Group</span></code> structure on <code class="docutils literal notranslate"><span class="pre">Equiv.Perm</span> <span class="pre">α</span></code> in the file <code class="docutils literal notranslate"><span class="pre">GroupTheory.Perm.Basic</span></code>. As always, you can hover over the theorems used in the definition of <code class="docutils literal notranslate"><span class="pre">permGroup</span></code> to see their statements, and you can jump to their definitions in the original file to learn more about how they are implemented.</p> <p>In ordinary mathematics, we generally think of notation as independent of structure. For example, we can consider groups <span class="math notranslate nohighlight">\((G_1, \cdot, 1, \cdot^{-1})\)</span>, <span class="math notranslate nohighlight">\((G_2, \circ, e, i(\cdot))\)</span>, and <span class="math notranslate nohighlight">\((G_3, +, 0, -)\)</span>. In the first case, we write the binary operation as <span class="math notranslate nohighlight">\(\cdot\)</span>, the identity at <span class="math notranslate nohighlight">\(1\)</span>, and the inverse function as <span class="math notranslate nohighlight">\(x \mapsto x^{-1}\)</span>. In the second and third cases, we use the notational alternatives shown. When we formalize the notion of a group in Lean, however, the notation is more tightly linked to the structure. In Lean, the components of any <code class="docutils literal notranslate"><span class="pre">Group</span></code> are named <code class="docutils literal notranslate"><span class="pre">mul</span></code>, <code class="docutils literal notranslate"><span class="pre">one</span></code>, and <code class="docutils literal notranslate"><span class="pre">inv</span></code>, and in a moment we will see how multiplicative notation is set up to refer to them. If we want to use additive notation, we instead use an isomorphic structure <code class="docutils literal notranslate"><span class="pre">AdditiveGroup</span></code>. Its components are named <code class="docutils literal notranslate"><span class="pre">add</span></code>, <code class="docutils literal notranslate"><span class="pre">zero</span></code>, and <code class="docutils literal notranslate"><span class="pre">neg</span></code>, and the associated notation is what you would expect it to be.</p> <p>Recall the type <code class="docutils literal notranslate"><span class="pre">Point</span></code> that we defined in <a class="reference internal" href="#section-structures"><span class="std std-numref">Section 6.1</span></a>, and the addition function that we defined there. These definitions are reproduced in the examples file that accompanies this section. As an exercise, define an <code class="docutils literal notranslate"><span class="pre">AddGroup₁</span></code> structure that is similar to the <code class="docutils literal notranslate"><span class="pre">Group₁</span></code> structure we defined above, except that it uses the additive naming scheme just described. Define negation and a zero on the <code class="docutils literal notranslate"><span class="pre">Point</span></code> data type, and define the <code class="docutils literal notranslate"><span class="pre">AddGroup₁</span></code> structure on <code class="docutils literal notranslate"><span class="pre">Point</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">structure</span> <span class="n">AddGroup₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="n">where</span> <span class="o">(</span><span class="n">add</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span><span class="o">)</span> <span class="c1">-- fill in the rest</span> <span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">Point</span> <span class="n">where</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">y</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="n">z</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="kn">namespace</span> <span class="n">Point</span> <span class="kd">def</span> <span class="n">add</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">Point</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">a.x</span> <span class="bp">+</span> <span class="n">b.x</span><span class="o">,</span> <span class="n">a.y</span> <span class="bp">+</span> <span class="n">b.y</span><span class="o">,</span> <span class="n">a.z</span> <span class="bp">+</span> <span class="n">b.z</span><span class="o">⟩</span> <span class="kd">def</span> <span class="n">neg</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">Point</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">def</span> <span class="n">zero</span> <span class="o">:</span> <span class="n">Point</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">def</span> <span class="n">add_group_point</span> <span class="o">:</span> <span class="n">AddGroup₁</span> <span class="n">point</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">end</span> <span class="n">Point</span> </pre></div> </div> <p>We are making progress. Now we know how to define algebraic structures in Lean, and we know how to define instances of those structures. But we also want to associate notation with structures so that we can use it with each instance. Moreover, we want to arrange it so that we can define an operation on a structure and use it with any particular instance, and we want to arrange it so that we can prove a theorem about a structure and use it with any instance.</p> <p>In fact, mathlib is already set up to use generic group notation, definitions, and theorems for <code class="docutils literal notranslate"><span class="pre">Equiv.Perm</span> <span class="pre">α</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">Equiv.Perm</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="k">#check</span> <span class="n">f</span> <span class="bp">*</span> <span class="n">g</span> <span class="k">#check</span> <span class="n">mul_assoc</span> <span class="n">f</span> <span class="n">g</span> <span class="n">g</span><span class="bp">⁻¹</span> <span class="c1">-- group power, defined for any group</span> <span class="k">#check</span> <span class="n">g</span> <span class="bp">^</span> <span class="n">n</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">*</span> <span class="n">g</span> <span class="bp">*</span> <span class="n">g</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="n">f</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_assoc</span><span class="o">,</span> <span class="n">mul_right_inv</span><span class="o">,</span> <span class="n">mul_one</span><span class="o">]</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">*</span> <span class="n">g</span> <span class="bp">*</span> <span class="n">g</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="n">f</span> <span class="o">:=</span> <span class="n">mul_inv_cancel_right</span> <span class="n">f</span> <span class="n">g</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">Equiv.Perm</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="n">g.symm.trans</span> <span class="o">(</span><span class="n">g.trans</span> <span class="n">f</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="o">:=</span> <span class="n">mul_inv_cancel_right</span> <span class="n">f</span> <span class="n">g</span> </pre></div> </div> <p>You can check that this is not the case for the additive group structure on <code class="docutils literal notranslate"><span class="pre">Point</span></code> that we asked you to define above. Our task now is to understand that magic that goes on under the hood in order to make the examples for <code class="docutils literal notranslate"><span class="pre">Equiv.Perm</span> <span class="pre">α</span></code> work the way they do.</p> <p>The issue is that Lean needs to be able to <em>find</em> the relevant notation and the implicit group structure, using the information that is found in the expressions that we type. Similarly, when we write <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">+</span> <span class="pre">y</span></code> with expressions <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code> that have type <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>, Lean needs to interpret the <code class="docutils literal notranslate"><span class="pre">+</span></code> symbol as the relevant addition function on the reals. It also has to recognize the type <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> as an instance of a commutative ring, so that all the definitions and theorems for a commutative ring are available. For another example, continuity is defined in Lean relative to any two topological spaces. When we have <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℂ</span></code> and we write <code class="docutils literal notranslate"><span class="pre">Continuous</span> <span class="pre">f</span></code>, Lean has to find the relevant topologies on <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> and <code class="docutils literal notranslate"><span class="pre">ℂ</span></code>.</p> <p>The magic is achieved with a combination of three things.</p> <ol class="arabic simple"> <li><p><em>Logic.</em> A definition that should be interpreted in any group takes, as arguments, the type of the group and the group structure as arguments. Similarly, a theorem about the elements of an arbitrary group begins with universal quantifiers over the type of the group and the group structure.</p></li> <li><p><em>Implicit arguments.</em> The arguments for the type and the structure are generally left implicit, so that we do not have to write them or see them in the Lean information window. Lean fills the information in for us silently.</p></li> <li><p><em>Type class inference.</em> Also known as <em>class inference</em>, this is a simple but powerful mechanism that enables us to register information for Lean to use later on. When Lean is called on to fill in implicit arguments to a definition, theorem, or piece of notation, it can make use of information that has been registered.</p></li> </ol> <p>Whereas an annotation <code class="docutils literal notranslate"><span class="pre">(grp</span> <span class="pre">:</span> <span class="pre">Group</span> <span class="pre">G)</span></code> tells Lean that it should expect to be given that argument explicitly and the annotation <code class="docutils literal notranslate"><span class="pre">{grp</span> <span class="pre">:</span> <span class="pre">Group</span> <span class="pre">G}</span></code> tells Lean that it should try to figure it out from contextual cues in the expression, the annotation <code class="docutils literal notranslate"><span class="pre">[grp</span> <span class="pre">:</span> <span class="pre">Group</span> <span class="pre">G]</span></code> tells Lean that the corresponding argument should be synthesized using type class inference. Since the whole point to the use of such arguments is that we generally do not need to refer to them explicitly, Lean allows us to write <code class="docutils literal notranslate"><span class="pre">[Group</span> <span class="pre">G]</span></code> and leave the name anonymous. You have probably already noticed that Lean chooses names like <code class="docutils literal notranslate"><span class="pre">_inst_1</span></code> automatically. When we use the anonymous square-bracket annotation with the <code class="docutils literal notranslate"><span class="pre">variables</span></code> command, then as long as the variables are still in scope, Lean automatically adds the argument <code class="docutils literal notranslate"><span class="pre">[Group</span> <span class="pre">G]</span></code> to any definition or theorem that mentions <code class="docutils literal notranslate"><span class="pre">G</span></code>.</p> <p>How do we register the information that Lean needs to use to carry out the search? Returning to our group example, we need only make two changes. First, instead of using the <code class="docutils literal notranslate"><span class="pre">structure</span></code> command to define the group structure, we use the keyword <code class="docutils literal notranslate"><span class="pre">class</span></code> to indicate that it is a candidate for class inference. Second, instead of defining particular instances with <code class="docutils literal notranslate"><span class="pre">def</span></code>, we use the keyword <code class="docutils literal notranslate"><span class="pre">instance</span></code> to register the particular instance with Lean. As with the names of class variables, we are allowed to leave the name of an instance definition anonymous, since in general we intend Lean to find it and put it to use without troubling us with the details.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Group₂</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="n">where</span> <span class="n">mul</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="n">one</span> <span class="o">:</span> <span class="n">α</span> <span class="n">inv</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="n">mul_assoc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">mul</span> <span class="o">(</span><span class="n">mul</span> <span class="n">x</span> <span class="n">y</span><span class="o">)</span> <span class="n">z</span> <span class="bp">=</span> <span class="n">mul</span> <span class="n">x</span> <span class="o">(</span><span class="n">mul</span> <span class="n">y</span> <span class="n">z</span><span class="o">)</span> <span class="n">mul_one</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">mul</span> <span class="n">x</span> <span class="n">one</span> <span class="bp">=</span> <span class="n">x</span> <span class="n">one_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">mul</span> <span class="n">one</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">x</span> <span class="n">mul_left_inv</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">mul</span> <span class="o">(</span><span class="n">inv</span> <span class="n">x</span><span class="o">)</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">one</span> <span class="kd">instance</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">:</span> <span class="n">Group₂</span> <span class="o">(</span><span class="n">Equiv.Perm</span> <span class="n">α</span><span class="o">)</span> <span class="n">where</span> <span class="n">mul</span> <span class="n">f</span> <span class="n">g</span> <span class="o">:=</span> <span class="n">Equiv.trans</span> <span class="n">g</span> <span class="n">f</span> <span class="n">one</span> <span class="o">:=</span> <span class="n">Equiv.refl</span> <span class="n">α</span> <span class="n">inv</span> <span class="o">:=</span> <span class="n">Equiv.symm</span> <span class="n">mul_assoc</span> <span class="n">f</span> <span class="n">g</span> <span class="n">h</span> <span class="o">:=</span> <span class="o">(</span><span class="n">Equiv.trans_assoc</span> <span class="n">_</span> <span class="n">_</span> <span class="n">_</span><span class="o">)</span><span class="bp">.</span><span class="n">symm</span> <span class="n">one_mul</span> <span class="o">:=</span> <span class="n">Equiv.trans_refl</span> <span class="n">mul_one</span> <span class="o">:=</span> <span class="n">Equiv.refl_trans</span> <span class="n">mul_left_inv</span> <span class="o">:=</span> <span class="n">Equiv.self_trans_symm</span> </pre></div> </div> <p>The following illustrates their use.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">@</span><span class="n">Group₂.mul</span> <span class="kd">def</span> <span class="n">mySquare</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Group₂</span> <span class="n">α</span><span class="o">]</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:=</span> <span class="n">Group₂.mul</span> <span class="n">x</span> <span class="n">x</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">mySquare</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">Equiv.Perm</span> <span class="n">β</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Group₂.mul</span> <span class="n">f</span> <span class="n">g</span> <span class="bp">=</span> <span class="n">g.trans</span> <span class="n">f</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">mySquare</span> <span class="n">f</span> <span class="bp">=</span> <span class="n">f.trans</span> <span class="n">f</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">end</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">#check</span></code> command shows that <code class="docutils literal notranslate"><span class="pre">Group₂.mul</span></code> has an implicit argument <code class="docutils literal notranslate"><span class="pre">[Group₂</span> <span class="pre">α]</span></code> that we expect to be found by class inference, where <code class="docutils literal notranslate"><span class="pre">α</span></code> is the type of the arguments to <code class="docutils literal notranslate"><span class="pre">Group₂.mul</span></code>. In other words, <code class="docutils literal notranslate"><span class="pre">{α</span> <span class="pre">:</span> <span class="pre">Type*}</span></code> is the implicit argument for the type of the group elements and <code class="docutils literal notranslate"><span class="pre">[Group₂</span> <span class="pre">α]</span></code> is the implicit argument for the group structure on <code class="docutils literal notranslate"><span class="pre">α</span></code>. Similarly, when we define a generic squaring function <code class="docutils literal notranslate"><span class="pre">my_square</span></code> for <code class="docutils literal notranslate"><span class="pre">Group₂</span></code>, we use an implicit argument <code class="docutils literal notranslate"><span class="pre">{α</span> <span class="pre">:</span> <span class="pre">Type*}</span></code> for the type of the elements and an implicit argument <code class="docutils literal notranslate"><span class="pre">[Group₂</span> <span class="pre">α]</span></code> for the <code class="docutils literal notranslate"><span class="pre">Group₂</span></code> structure.</p> <p>In the first example, when we write <code class="docutils literal notranslate"><span class="pre">Group₂.mul</span> <span class="pre">f</span> <span class="pre">g</span></code>, the type of <code class="docutils literal notranslate"><span class="pre">f</span></code> and <code class="docutils literal notranslate"><span class="pre">g</span></code> tells Lean that in the argument <code class="docutils literal notranslate"><span class="pre">α</span></code> to <code class="docutils literal notranslate"><span class="pre">Group₂.mul</span></code> has to be instantiated to <code class="docutils literal notranslate"><span class="pre">Equiv.Perm</span> <span class="pre">β</span></code>. That means that Lean has to find an element of <code class="docutils literal notranslate"><span class="pre">Group₂</span> <span class="pre">(Equiv.Perm</span> <span class="pre">β)</span></code>. The previous <code class="docutils literal notranslate"><span class="pre">instance</span></code> declaration tells Lean exactly how to do that. Problem solved!</p> <p>This simple mechanism for registering information so that Lean can find it when it needs it is remarkably useful. Here is one way it comes up. In Lean’s foundation, a data type <code class="docutils literal notranslate"><span class="pre">α</span></code> may be empty. In a number of applications, however, it is useful to know that a type has at least one element. For example, the function <code class="docutils literal notranslate"><span class="pre">List.head</span></code>, which returns the first element of a list, can return the default value when the list is empty. To make that work, the Lean library defines a class <code class="docutils literal notranslate"><span class="pre">Inhabited</span> <span class="pre">α</span></code>, which does nothing more than store a default value. We can show that the <code class="docutils literal notranslate"><span class="pre">Point</span></code> type is an instance:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Inhabited</span> <span class="n">Point</span> <span class="n">where</span> <span class="n">default</span> <span class="o">:=</span> <span class="o">⟨</span><span class="mi">0</span><span class="o">,</span> <span class="mi">0</span><span class="o">,</span> <span class="mi">0</span><span class="o">⟩</span> <span class="k">#check</span> <span class="o">(</span><span class="n">default</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="o">([]</span> <span class="o">:</span> <span class="n">List</span> <span class="n">Point</span><span class="o">)</span><span class="bp">.</span><span class="n">headI</span> <span class="bp">=</span> <span class="n">default</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>The class inference mechanism is also used for generic notation. The expression <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">+</span> <span class="pre">y</span></code> is an abbreviation for <code class="docutils literal notranslate"><span class="pre">Add.add</span> <span class="pre">x</span> <span class="pre">y</span></code> where—you guessed it—<code class="docutils literal notranslate"><span class="pre">Add</span> <span class="pre">α</span></code> is a class that stores a binary function on <code class="docutils literal notranslate"><span class="pre">α</span></code>. Writing <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">+</span> <span class="pre">y</span></code> tells Lean to find a registered instance of <code class="docutils literal notranslate"><span class="pre">[Add.add</span> <span class="pre">α]</span></code> and use the corresponding function. Below, we register the addition function for <code class="docutils literal notranslate"><span class="pre">Point</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Add</span> <span class="n">Point</span> <span class="n">where</span> <span class="n">add</span> <span class="o">:=</span> <span class="n">Point.add</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="k">#check</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">y</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">Point.add</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">end</span> </pre></div> </div> <p>In this way, we can assign the notation <code class="docutils literal notranslate"><span class="pre">+</span></code> to binary operations on other types as well.</p> <p>But we can do even better. We have seen that <code class="docutils literal notranslate"><span class="pre">*</span></code> can be used in any group, <code class="docutils literal notranslate"><span class="pre">+</span></code> can be used in any additive group, and both can be used in any ring. When we define a new instance of a ring in Lean, we don’t have to define <code class="docutils literal notranslate"><span class="pre">+</span></code> and <code class="docutils literal notranslate"><span class="pre">*</span></code> for that instance, because Lean knows that these are defined for every ring. We can use this method to specify notation for our <code class="docutils literal notranslate"><span class="pre">Group₂</span></code> class:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="n">hasMulGroup₂</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Group₂</span> <span class="n">α</span><span class="o">]</span> <span class="o">:</span> <span class="n">Mul</span> <span class="n">α</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">Group₂.mul</span><span class="o">⟩</span> <span class="kd">instance</span> <span class="n">hasOneGroup₂</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Group₂</span> <span class="n">α</span><span class="o">]</span> <span class="o">:</span> <span class="n">One</span> <span class="n">α</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">Group₂.one</span><span class="o">⟩</span> <span class="kd">instance</span> <span class="n">hasInvGroup₂</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Group₂</span> <span class="n">α</span><span class="o">]</span> <span class="o">:</span> <span class="n">Inv</span> <span class="n">α</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">Group₂.inv</span><span class="o">⟩</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">Equiv.Perm</span> <span class="n">α</span><span class="o">)</span> <span class="k">#check</span> <span class="n">f</span> <span class="bp">*</span> <span class="mi">1</span> <span class="bp">*</span> <span class="n">g</span><span class="bp">⁻¹</span> <span class="kd">def</span> <span class="n">foo</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">*</span> <span class="mi">1</span> <span class="bp">*</span> <span class="n">g</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="n">g.symm.trans</span> <span class="o">((</span><span class="n">Equiv.refl</span> <span class="n">α</span><span class="o">)</span><span class="bp">.</span><span class="n">trans</span> <span class="n">f</span><span class="o">)</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">end</span> </pre></div> </div> <p>In this case, we have to supply names for the instances, because Lean has a hard time coming up with good defaults. What makes this approach work is that Lean carries out a recursive search. According to the instances we have declared, Lean can find an instance of <code class="docutils literal notranslate"><span class="pre">Mul</span> <span class="pre">(Equiv.Perm</span> <span class="pre">α)</span></code> by finding an instance of <code class="docutils literal notranslate"><span class="pre">Group₂</span> <span class="pre">(Equiv.Perm</span> <span class="pre">α)</span></code>, and it can find an instance of <code class="docutils literal notranslate"><span class="pre">Group₂</span> <span class="pre">(Equiv.Perm</span> <span class="pre">α)</span></code> because we have provided one. Lean is capable of finding these two facts and chaining them together.</p> <p>The example we have just given is dangerous, because Lean’s library also has an instance of <code class="docutils literal notranslate"><span class="pre">Group</span> <span class="pre">(Equiv.Perm</span> <span class="pre">α)</span></code>, and multiplication is defined on any group. So it is ambiguous as to which instance is found. In fact, Lean favors more recent declarations unless you explicitly specify a different priority. Also, there is another way to tell Lean that one structure is an instance of another, using the <code class="docutils literal notranslate"><span class="pre">extends</span></code> keyword. This is how <code class="docutils literal notranslate"><span class="pre">mathlib</span></code> specifies that, for example, every commutative ring is a ring. You can find more information in a <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean4/type_classes.html#managing-type-class-inference">section on class inference</a> in <em>Theorem Proving in Lean</em>.</p> <p>In general, it is a bad idea to specify a value of <code class="docutils literal notranslate"><span class="pre">*</span></code> for an instance of an algebraic structure that already has the notation defined. Redefining the notion of <code class="docutils literal notranslate"><span class="pre">Group</span></code> in Lean is an artificial example. In this case, however, both interpretations of the group notation unfold to <code class="docutils literal notranslate"><span class="pre">Equiv.trans</span></code>, <code class="docutils literal notranslate"><span class="pre">Equiv.refl</span></code>, and <code class="docutils literal notranslate"><span class="pre">Equiv.symm</span></code>, in the same way.</p> <p>As a similarly artificial exercise, define a class <code class="docutils literal notranslate"><span class="pre">AddGroup₂</span></code> in analogy to <code class="docutils literal notranslate"><span class="pre">Group₂</span></code>. Define the usual notation for addition, negation, and zero on any <code class="docutils literal notranslate"><span class="pre">AddGroup₂</span></code> using the classes <code class="docutils literal notranslate"><span class="pre">Add</span></code>, <code class="docutils literal notranslate"><span class="pre">Neg</span></code>, and <code class="docutils literal notranslate"><span class="pre">Zero</span></code>. Then show <code class="docutils literal notranslate"><span class="pre">Point</span></code> is an instance of <code class="docutils literal notranslate"><span class="pre">AddGroup₂</span></code>. Try it out and make sure that the additive group notation works for elements of <code class="docutils literal notranslate"><span class="pre">Point</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">AddGroup₂</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="n">where</span> <span class="n">add</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="c1">-- fill in the rest</span> </pre></div> </div> <p>It is not a big problem that we have already declared instances <code class="docutils literal notranslate"><span class="pre">Add</span></code>, <code class="docutils literal notranslate"><span class="pre">Neg</span></code>, and <code class="docutils literal notranslate"><span class="pre">Zero</span></code> for <code class="docutils literal notranslate"><span class="pre">Point</span></code> above. Once again, the two ways of synthesizing the notation should come up with the same answer.</p> <p>Class inference is subtle, and you have to be careful when using it, because it configures automation that invisibly governs the interpretation of the expressions we type. When used wisely, however, class inference is a powerful tool. It is what makes algebraic reasoning possible in Lean.</p> </section> <section id="building-the-gaussian-integers"> <span id="section-building-the-gaussian-integers"></span><h2><span class="section-number">6.3. </span>Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this heading"></a></h2> <p>We will now illustrate the use of the algebraic hierarchy in Lean by building an important mathematical object, the <em>Gaussian integers</em>, and showing that it is a Euclidean domain. In other words, according to the terminology we have been using, we will define the Gaussian integers and show that they are an instance of the Euclidean domain structure.</p> <p>In ordinary mathematical terms, the set of Gaussian integers <span class="math notranslate nohighlight">\(\Bbb{Z}[i]\)</span> is the set of complex numbers <span class="math notranslate nohighlight">\(\{ a + b i \mid a, b \in \Bbb{Z}\}\)</span>. But rather than define them as a subset of the complex numbers, our goal here is to define them as a data type in their own right. We do this by representing a Gaussian integer as a pair of integers, which we think of as the <em>real</em> and <em>imaginary</em> parts.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">gaussInt</span> <span class="n">where</span> <span class="n">re</span> <span class="o">:</span> <span class="n">ℤ</span> <span class="n">im</span> <span class="o">:</span> <span class="n">ℤ</span> </pre></div> </div> <p>We first show that the Gaussian integers have the structure of a ring, with <code class="docutils literal notranslate"><span class="pre">0</span></code> defined to be <code class="docutils literal notranslate"><span class="pre">⟨0,</span> <span class="pre">0⟩</span></code>, <code class="docutils literal notranslate"><span class="pre">1</span></code> defined to be <code class="docutils literal notranslate"><span class="pre">⟨1,</span> <span class="pre">0⟩</span></code>, and addition defined pointwise. To work out the definition of multiplication, remember that we want the element <span class="math notranslate nohighlight">\(i\)</span>, represented by <code class="docutils literal notranslate"><span class="pre">⟨0,</span> <span class="pre">1⟩</span></code>, to be a square root of <span class="math notranslate nohighlight">\(-1\)</span>. Thus we want</p> <div class="math notranslate nohighlight"> \[\begin{split}(a + bi) (c + di) & = ac + bci + adi + bd i^2 \\ & = (ac - bd) + (bc + ad)i.\end{split}\]</div> <p>This explains the definition of <code class="docutils literal notranslate"><span class="pre">hasMul</span></code> below.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Zero</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">⟨⟨</span><span class="mi">0</span><span class="o">,</span> <span class="mi">0</span><span class="o">⟩⟩</span> <span class="kd">instance</span> <span class="o">:</span> <span class="n">One</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">⟨⟨</span><span class="mi">1</span><span class="o">,</span> <span class="mi">0</span><span class="o">⟩⟩</span> <span class="kd">instance</span> <span class="o">:</span> <span class="n">Add</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">⟨</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">x.re</span> <span class="bp">+</span> <span class="n">y.re</span><span class="o">,</span> <span class="n">x.im</span> <span class="bp">+</span> <span class="n">y.im</span><span class="o">⟩⟩</span> <span class="kd">instance</span> <span class="o">:</span> <span class="n">Neg</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">⟨</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="o">⟨</span><span class="bp">-</span><span class="n">x.re</span><span class="o">,</span> <span class="bp">-</span><span class="n">x.im</span><span class="o">⟩⟩</span> <span class="kd">instance</span> <span class="o">:</span> <span class="n">Mul</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">⟨</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="bp">-</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">,</span> <span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.im</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.re</span><span class="o">⟩⟩</span> </pre></div> </div> <p>As noted in <a class="reference internal" href="#section-structures"><span class="std std-numref">Section 6.1</span></a>, it is a good idea to put all the definitions related to a data type in a namespace with the same name. Thus in the Lean files associated with this chapter, these definitions are made in the <code class="docutils literal notranslate"><span class="pre">GaussInt</span></code> namespace.</p> <p>Notice that here we are defining the interpretations of the notation <code class="docutils literal notranslate"><span class="pre">0</span></code>, <code class="docutils literal notranslate"><span class="pre">1</span></code>, <code class="docutils literal notranslate"><span class="pre">+</span></code>, <code class="docutils literal notranslate"><span class="pre">-</span></code>, and <code class="docutils literal notranslate"><span class="pre">*</span></code> directly, rather than naming them <code class="docutils literal notranslate"><span class="pre">GaussInt.zero</span></code> and the like and assigning the notation to those. It is often useful to have an explicit name for the definitions, for example, to use with <code class="docutils literal notranslate"><span class="pre">simp</span></code> and <code class="docutils literal notranslate"><span class="pre">rewrite</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">zero_def</span> <span class="o">:</span> <span class="o">(</span><span class="mi">0</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="bp">=</span> <span class="o">⟨</span><span class="mi">0</span><span class="o">,</span> <span class="mi">0</span><span class="o">⟩</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">one_def</span> <span class="o">:</span> <span class="o">(</span><span class="mi">1</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="bp">=</span> <span class="o">⟨</span><span class="mi">1</span><span class="o">,</span> <span class="mi">0</span><span class="o">⟩</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">add_def</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">=</span> <span class="o">⟨</span><span class="n">x.re</span> <span class="bp">+</span> <span class="n">y.re</span><span class="o">,</span> <span class="n">x.im</span> <span class="bp">+</span> <span class="n">y.im</span><span class="o">⟩</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">neg_def</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="bp">-</span><span class="n">x</span> <span class="bp">=</span> <span class="o">⟨</span><span class="bp">-</span><span class="n">x.re</span><span class="o">,</span> <span class="bp">-</span><span class="n">x.im</span><span class="o">⟩</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">mul_def</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">y</span> <span class="bp">=</span> <span class="o">⟨</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="bp">-</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">,</span> <span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.im</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.re</span><span class="o">⟩</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>It is also useful to name the rules that compute the real and imaginary parts, and to declare them to the simplifier.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">zero_re</span> <span class="o">:</span> <span class="o">(</span><span class="mi">0</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">zero_im</span> <span class="o">:</span> <span class="o">(</span><span class="mi">0</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">one_re</span> <span class="o">:</span> <span class="o">(</span><span class="mi">1</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">one_im</span> <span class="o">:</span> <span class="o">(</span><span class="mi">1</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">add_re</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="bp">=</span> <span class="n">x.re</span> <span class="bp">+</span> <span class="n">y.re</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">add_im</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="bp">=</span> <span class="n">x.im</span> <span class="bp">+</span> <span class="n">y.im</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">neg_re</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="bp">=</span> <span class="bp">-</span><span class="n">x.re</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">neg_im</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="bp">=</span> <span class="bp">-</span><span class="n">x.im</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">mul_re</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="bp">=</span> <span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="bp">-</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.im</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">mul_im</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="bp">=</span> <span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.im</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>It is now surprisingly easy to show that the Gaussian integers are an instance of a commutative ring. We are putting the structure concept to good use. Each particular Gaussian integer is an instance of the <code class="docutils literal notranslate"><span class="pre">gaussInt</span></code> structure, whereas the type <code class="docutils literal notranslate"><span class="pre">gaussInt</span></code> itself, together with the relevant operations, is an instance of the <code class="docutils literal notranslate"><span class="pre">CommRing</span></code> structure. The <code class="docutils literal notranslate"><span class="pre">CommRing</span></code> structure, in turn, extends the notational structures <code class="docutils literal notranslate"><span class="pre">Zero</span></code>, <code class="docutils literal notranslate"><span class="pre">One</span></code>, <code class="docutils literal notranslate"><span class="pre">Add</span></code>, <code class="docutils literal notranslate"><span class="pre">Neg</span></code>, and <code class="docutils literal notranslate"><span class="pre">Mul</span></code>.</p> <p>If you type <code class="docutils literal notranslate"><span class="pre">instance</span> <span class="pre">:</span> <span class="pre">CommRing</span> <span class="pre">gaussInt</span> <span class="pre">:=</span> <span class="pre">_</span></code>, click on the light bulb that appears in VS Code, and then ask Lean to fill in a skeleton for the structure definition, you will see a scary number of entries. Jumping to the definition of the structure, however, shows that many of the fields have default definitions that Lean will fill in for you automatically. The essential ones appear in the definition below. In each case, the relevant identity is proved by unfolding definitions, using the <code class="docutils literal notranslate"><span class="pre">ext</span></code> tactic to reduce the identities to their real and imaginary components, simplifying, and, if necessary, carrying out the relevant ring calculation in the integers.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="n">instCommRing</span> <span class="o">:</span> <span class="n">CommRing</span> <span class="n">gaussInt</span> <span class="n">where</span> <span class="n">zero</span> <span class="o">:=</span> <span class="mi">0</span> <span class="n">one</span> <span class="o">:=</span> <span class="mi">1</span> <span class="n">add</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">·</span> <span class="bp">+</span> <span class="bp">·</span><span class="o">)</span> <span class="n">neg</span> <span class="n">x</span> <span class="o">:=</span> <span class="bp">-</span><span class="n">x</span> <span class="n">mul</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">·</span> <span class="bp">*</span> <span class="bp">·</span><span class="o">)</span> <span class="n">add_assoc</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="bp"><;></span> <span class="n">ring</span> <span class="n">zero_add</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="n">add_zero</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="n">add_left_neg</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="n">add_comm</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="bp"><;></span> <span class="n">ring</span> <span class="n">mul_assoc</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="bp"><;></span> <span class="n">ring</span> <span class="n">one_mul</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="n">mul_one</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="n">left_distrib</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="bp"><;></span> <span class="n">ring</span> <span class="n">right_distrib</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="bp"><;></span> <span class="n">ring</span> <span class="n">mul_comm</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="bp"><;></span> <span class="n">ring</span> <span class="n">zero_mul</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">mul_zero</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Lean’s library defines the class of <em>nontrivial</em> types to be types with at least two distinct elements. In the context of a ring, this is equivalent to saying the the zero is not equal to the one. Since some common theorems depend on that fact, we may as well establish it now.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Nontrivial</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">use</span> <span class="mi">0</span><span class="o">,</span> <span class="mi">1</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Ne</span><span class="o">,</span> <span class="n">gaussInt.ext_iff</span><span class="o">]</span> <span class="n">simp</span> </pre></div> </div> <p>We will now show that the Gaussian integers have an important additional property. A <em>Euclidean domain</em> is a ring <span class="math notranslate nohighlight">\(R\)</span> equipped with a <em>norm</em> function <span class="math notranslate nohighlight">\(N : R \to \mathbb{N}\)</span> with the following two properties:</p> <ul class="simple"> <li><p>For every <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b \ne 0\)</span> in <span class="math notranslate nohighlight">\(R\)</span>, there are <span class="math notranslate nohighlight">\(q\)</span> and <span class="math notranslate nohighlight">\(r\)</span> in <span class="math notranslate nohighlight">\(R\)</span> such that <span class="math notranslate nohighlight">\(a = bq + r\)</span> and either <span class="math notranslate nohighlight">\(r = 0\)</span> or <cite>N(r) < N(b)</cite>.</p></li> <li><p>For every <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b \ne 0\)</span>, <span class="math notranslate nohighlight">\(N(a) \le N(ab)\)</span>.</p></li> </ul> <p>The ring of integers <span class="math notranslate nohighlight">\(\Bbb{Z}\)</span> with <span class="math notranslate nohighlight">\(N(a) = |a|\)</span> is an archetypal example of a Euclidean domain. In that case, we can take <span class="math notranslate nohighlight">\(q\)</span> to be the result of integer division of <span class="math notranslate nohighlight">\(a\)</span> by <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(r\)</span> to be the remainder. These functions are defined in Lean so that the satisfy the following:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">/</span> <span class="n">b</span><span class="o">)</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">%</span> <span class="n">b</span> <span class="o">:=</span> <span class="n">Eq.symm</span> <span class="bp"><|</span> <span class="n">Int.ediv_add_emod</span> <span class="n">a</span> <span class="n">b</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">≠</span> <span class="mi">0</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">a</span> <span class="bp">%</span> <span class="n">b</span> <span class="o">:=</span> <span class="n">Int.emod_nonneg</span> <span class="n">a</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">≠</span> <span class="mi">0</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">%</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">abs</span> <span class="n">b</span> <span class="o">:=</span> <span class="n">Int.emod_lt</span> <span class="n">a</span> </pre></div> </div> <p>In an arbitrary ring, an element <span class="math notranslate nohighlight">\(a\)</span> is said to be a <em>unit</em> if it divides <span class="math notranslate nohighlight">\(1\)</span>. A nonzero element <span class="math notranslate nohighlight">\(a\)</span> is said to be <em>irreducible</em> if it cannot be written in the form <span class="math notranslate nohighlight">\(a = bc\)</span> where neither <span class="math notranslate nohighlight">\(b\)</span> nor <span class="math notranslate nohighlight">\(c\)</span> is a unit. In the integers, every irreducible element <span class="math notranslate nohighlight">\(a\)</span> is <em>prime</em>, which is to say, whenever <span class="math notranslate nohighlight">\(a\)</span> divides a product <span class="math notranslate nohighlight">\(bc\)</span>, it divides either <span class="math notranslate nohighlight">\(b\)</span> or <span class="math notranslate nohighlight">\(c\)</span>. But in other rings this property can fail. In the ring <span class="math notranslate nohighlight">\(\Bbb{Z}[\sqrt{-5}]\)</span>, we have</p> <div class="math notranslate nohighlight"> \[6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}),\]</div> <p>and the elements <span class="math notranslate nohighlight">\(2\)</span>, <span class="math notranslate nohighlight">\(3\)</span>, <span class="math notranslate nohighlight">\(1 + \sqrt{-5}\)</span>, and <span class="math notranslate nohighlight">\(1 - \sqrt{-5}\)</span> are all irreducible, but they are not prime. For example, <span class="math notranslate nohighlight">\(2\)</span> divides the product <span class="math notranslate nohighlight">\((1 + \sqrt{-5})(1 - \sqrt{-5})\)</span>, but it does not divide either factor. In particular, we no longer have unique factorization: the number <span class="math notranslate nohighlight">\(6\)</span> can be factored into irreducible elements in more than one way.</p> <p>In contrast, every Euclidean domain is a unique factorization domain, which implies that every irreducible element is prime. The axioms for a Euclidean domain imply that one can write any nonzero element as a finite product of irreducible elements. They also imply that one can use the Euclidean algorithm to find a greatest common divisor of any two nonzero elements <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code>, i.e.~an element that is divisible by any other common divisor. This, in turn, implies that factorization into irreducible elements is unique up to multiplication by units.</p> <p>We now show that the Gaussian integers are a Euclidean domain with the norm defined by <span class="math notranslate nohighlight">\(N(a + bi) = (a + bi)(a - bi) = a^2 + b^2\)</span>. The Gaussian integer <span class="math notranslate nohighlight">\(a - bi\)</span> is called the <em>conjugate</em> of <span class="math notranslate nohighlight">\(a + bi\)</span>. It is not hard to check that for any complex numbers <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>, we have <span class="math notranslate nohighlight">\(N(xy) = N(x)N(y)\)</span>.</p> <p>To see that this definition of the norm makes the complex numbers a Euclidean domain, only the first property is challenging. Suppose we want to write <span class="math notranslate nohighlight">\(a + bi = (c + di) q + r\)</span> for suitable <span class="math notranslate nohighlight">\(q\)</span> and <span class="math notranslate nohighlight">\(r\)</span>. Treating <span class="math notranslate nohighlight">\(a + bi\)</span> and <span class="math notranslate nohighlight">\(c + di\)</span> are complex numbers, carry out the division</p> <div class="math notranslate nohighlight"> \[\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c-di)} = \frac{ac + bd}{c^2 + d^2} + \frac{bc -ad}{c^2+d^2} i.\]</div> <p>The real and imaginary parts might not be integers, but we can round them to the nearest integers <span class="math notranslate nohighlight">\(u\)</span> and <span class="math notranslate nohighlight">\(v\)</span>. We can then express the right-hand size as <span class="math notranslate nohighlight">\((u + vi) + (u' + v'i)\)</span>, where <span class="math notranslate nohighlight">\(u' + v'i\)</span> is the part left over. Note that we have <span class="math notranslate nohighlight">\(|u'| \le 1/2\)</span> and <span class="math notranslate nohighlight">\(|v'| \le 1/2\)</span>, and hence</p> <div class="math notranslate nohighlight"> \[N(u' + v' i) = (u')^2 + (v')^2 \le 1/4 + 1/4 \le 1/2.\]</div> <p>Multiplying through by <span class="math notranslate nohighlight">\(c + di\)</span>, we have</p> <div class="math notranslate nohighlight"> \[a + bi = (c + di) (u + vi) + (c + di) (u' + v'i).\]</div> <p>Setting <span class="math notranslate nohighlight">\(q = u + vi\)</span> and <span class="math notranslate nohighlight">\(r = (c + di) (u' + v'i)\)</span>, we have <span class="math notranslate nohighlight">\(a + bi = (c + di) q + r\)</span>, and we only need to bound <span class="math notranslate nohighlight">\(N(r)\)</span>:</p> <div class="math notranslate nohighlight"> \[N(r) = N(c + di)N(u' + v'i) \le N(c + di) \cdot 1/2 < N(c + di).\]</div> <p>The argument we just carried out requires viewing the Gaussian integers as a subset of the complex numbers. One option for formalizing it in Lean is therefore to embed the Gaussian integers in the complex numbers, embed the integers in the Gaussian integers, define the rounding function from the real numbers to the integers, and take great care to pass back and forth between these number systems appropriately. In fact, this is exactly the approach that is followed in mathlib, where the Gaussian integers themselves are constructed as a special case of a ring of <em>quadratic integers</em>. See the file <a class="reference external" href="https://github.com/leanprover-community/mathlib/blob/master/src/number_theory/zsqrtd/gaussian_int.lean">gaussian_int.lean</a>.</p> <p>Here we will instead carry out an argument that stays in the integers. This illustrates an choice one commonly faces when formalizing mathematics. Given an argument that requires concepts or machinery that is not already in the library, one has two choices: either formalizes the concepts or machinery needed, or adapt the argument to make use of concepts and machinery you already have. The first choice is generally a good investment of time when the results can be used in other contexts. Pragmatically speaking, however, sometimes seeking a more elementary proof is more efficient.</p> <p>The usual quotient-remainder theorem for the integers says that for every <span class="math notranslate nohighlight">\(a\)</span> and nonzero <span class="math notranslate nohighlight">\(b\)</span>, there are <span class="math notranslate nohighlight">\(q\)</span> and <span class="math notranslate nohighlight">\(r\)</span> such that <span class="math notranslate nohighlight">\(a = b q + r\)</span> and <span class="math notranslate nohighlight">\(0 \le r < b\)</span>. Here we will make use of the following variation, which says that there are <span class="math notranslate nohighlight">\(q'\)</span> and <span class="math notranslate nohighlight">\(r'\)</span> such that <span class="math notranslate nohighlight">\(a = b q' + r'\)</span> and <span class="math notranslate nohighlight">\(|r'| \le b/2\)</span>. You can check that if the value of <span class="math notranslate nohighlight">\(r\)</span> in the first statement satisfies <span class="math notranslate nohighlight">\(r \le b/2\)</span>, we can take <span class="math notranslate nohighlight">\(q' = q\)</span> and <span class="math notranslate nohighlight">\(r' = r\)</span>, and otherwise we can take <span class="math notranslate nohighlight">\(q' = q + 1\)</span> and <span class="math notranslate nohighlight">\(r' = r - b\)</span>. We are grateful to Heather Macbeth for suggesting the following more elegant approach, which avoids definition by cases. We simply add <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">/</span> <span class="pre">2</span></code> to <code class="docutils literal notranslate"><span class="pre">a</span></code> before dividing and then subtract it from the remainder.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">div'</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">:=</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">/</span> <span class="n">b</span> <span class="kd">def</span> <span class="n">mod'</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">:=</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">%</span> <span class="n">b</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span> <span class="kd">theorem</span> <span class="n">div'_add_mod'</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">div'</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">mod'</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">div'</span><span class="o">,</span> <span class="n">mod'</span><span class="o">]</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">Int.ediv_add_emod</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">b</span><span class="o">]</span> <span class="kd">theorem</span> <span class="n">abs_mod'_le</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">mod'</span> <span class="n">a</span> <span class="n">b</span><span class="o">)</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mod'</span><span class="o">,</span> <span class="n">abs_le</span><span class="o">]</span> <span class="n">constructor</span> <span class="bp">·</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">Int.emod_nonneg</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">h.ne'</span><span class="o">]</span> <span class="k">have</span> <span class="o">:=</span> <span class="n">Int.emod_lt_of_pos</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">h</span> <span class="k">have</span> <span class="o">:=</span> <span class="n">Int.ediv_add_emod</span> <span class="n">b</span> <span class="mi">2</span> <span class="k">have</span> <span class="o">:=</span> <span class="n">Int.emod_lt_of_pos</span> <span class="n">b</span> <span class="n">zero_lt_two</span> <span class="n">revert</span> <span class="n">this</span><span class="bp">;</span> <span class="n">intro</span> <span class="n">this</span> <span class="c1">-- FIXME, this should not be needed</span> <span class="n">linarith</span> </pre></div> </div> <p>Note the use of our old friend, <code class="docutils literal notranslate"><span class="pre">linarith</span></code>. We will also need to express <code class="docutils literal notranslate"><span class="pre">mod'</span></code> in terms of <code class="docutils literal notranslate"><span class="pre">div'</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">mod'_eq</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">:</span> <span class="n">mod'</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">div'</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">div'_add_mod'</span> <span class="n">a</span> <span class="n">b</span><span class="o">]</span> </pre></div> </div> <p>We will use the fact that <span class="math notranslate nohighlight">\(x^2 + y^2\)</span> is equal to zero if and only if <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> are both zero. As an exercise, we ask you to prove that this holds in any ordered ring.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">sq_add_sq_eq_zero</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">LinearOrderedRing</span> <span class="n">α</span><span class="o">]</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">∧</span> <span class="n">y</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>We will put all the remaining definitions and theorems in this section in the <code class="docutils literal notranslate"><span class="pre">gaussInt</span></code> namespace. First, we define the <code class="docutils literal notranslate"><span class="pre">norm</span></code> function and ask you to establish some of its properties. The proofs are all short.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">norm</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:=</span> <span class="n">x.re</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">^</span> <span class="mi">2</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">norm_nonneg</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">norm</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">norm_eq_zero</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">norm</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">norm_pos</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">norm</span> <span class="n">x</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">≠</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">theorem</span> <span class="n">norm_mul</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">norm</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">norm</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>Next we define the conjugate function:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">conj</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">x.re</span><span class="o">,</span> <span class="bp">-</span><span class="n">x.im</span><span class="o">⟩</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">conj_re</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">conj</span> <span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="bp">=</span> <span class="n">x.re</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">@[simp]</span> <span class="kd">theorem</span> <span class="n">conj_im</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">conj</span> <span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="bp">=</span> <span class="bp">-</span><span class="n">x.im</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">norm_conj</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">norm</span> <span class="o">(</span><span class="n">conj</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">norm</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">norm</span><span class="o">]</span> </pre></div> </div> <p>Finally, we define division for the Gaussian integers with the notation <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">/</span> <span class="pre">y</span></code>, that rounds the complex quotient to the nearest Gaussian integer. We use our bespoke <code class="docutils literal notranslate"><span class="pre">Int.div'</span></code> for that purpose. As we calculated above, if <code class="docutils literal notranslate"><span class="pre">x</span></code> is <span class="math notranslate nohighlight">\(a + bi\)</span> and <code class="docutils literal notranslate"><span class="pre">y</span></code> is <span class="math notranslate nohighlight">\(c + di\)</span>, then the real and imaginary parts of <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">/</span> <span class="pre">y</span></code> are the nearest integers to</p> <div class="math notranslate nohighlight"> \[\frac{ac + bd}{c^2 + d^2} \quad \text{and} \quad \frac{bc -ad}{c^2+d^2},\]</div> <p>respectively. Here the numerators are the real and imaginary parts of <span class="math notranslate nohighlight">\((a + bi) (c - di)\)</span>, and the denominators are both equal to the norm of <span class="math notranslate nohighlight">\(c + di\)</span>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Div</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">⟨</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=></span> <span class="o">⟨</span><span class="n">Int.div'</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">),</span> <span class="n">Int.div'</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">)⟩⟩</span> </pre></div> </div> <p>Having defined <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">/</span> <span class="pre">y</span></code>, We define <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code> to be the remainder, <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">-</span> <span class="pre">(x</span> <span class="pre">/</span> <span class="pre">y)</span> <span class="pre">*</span> <span class="pre">y</span></code>. As above, we record the definitions in the theorems <code class="docutils literal notranslate"><span class="pre">div_def</span></code> and <code class="docutils literal notranslate"><span class="pre">mod_def</span></code> so that we can use them with <code class="docutils literal notranslate"><span class="pre">simp</span></code> and <code class="docutils literal notranslate"><span class="pre">rewrite</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Mod</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">⟨</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=></span> <span class="n">x</span> <span class="bp">-</span> <span class="n">y</span> <span class="bp">*</span> <span class="o">(</span><span class="n">x</span> <span class="bp">/</span> <span class="n">y</span><span class="o">)⟩</span> <span class="kd">theorem</span> <span class="n">div_def</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">/</span> <span class="n">y</span> <span class="bp">=</span> <span class="o">⟨</span><span class="n">Int.div'</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">),</span> <span class="n">Int.div'</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">)⟩</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">mod_def</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">%</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">y</span> <span class="bp">*</span> <span class="o">(</span><span class="n">x</span> <span class="bp">/</span> <span class="n">y</span><span class="o">)</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>These definitions immediately yield <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">=</span> <span class="pre">y</span> <span class="pre">*</span> <span class="pre">(x</span> <span class="pre">/</span> <span class="pre">y)</span> <span class="pre">+</span> <span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code> for every <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code>, so all we need to do is show that the norm of <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code> is less than the norm of <code class="docutils literal notranslate"><span class="pre">y</span></code> when <code class="docutils literal notranslate"><span class="pre">y</span></code> is not zero.</p> <p>We just defined the real and imaginary parts of <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">/</span> <span class="pre">y</span></code> to be <code class="docutils literal notranslate"><span class="pre">div'</span> <span class="pre">(x</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y).re</span> <span class="pre">(norm</span> <span class="pre">y)</span></code> and <code class="docutils literal notranslate"><span class="pre">div'</span> <span class="pre">(x</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y).im</span> <span class="pre">(norm</span> <span class="pre">y)</span></code>, respectively. Calculating, we have</p> <blockquote> <div><p><code class="docutils literal notranslate"><span class="pre">(x</span> <span class="pre">%</span> <span class="pre">y)</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y</span> <span class="pre">=</span> <span class="pre">(x</span> <span class="pre">-</span> <span class="pre">x</span> <span class="pre">/</span> <span class="pre">y</span> <span class="pre">*</span> <span class="pre">y)</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y</span> <span class="pre">=</span> <span class="pre">x</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y</span> <span class="pre">-</span> <span class="pre">x</span> <span class="pre">*</span> <span class="pre">(y</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">j)</span></code></p> </div></blockquote> <p>The real and imaginary parts of the right-hand side are exactly <code class="docutils literal notranslate"><span class="pre">mod'</span> <span class="pre">(x</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y).re</span> <span class="pre">(norm</span> <span class="pre">y)</span></code> and <code class="docutils literal notranslate"><span class="pre">mod'</span> <span class="pre">(x</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y).im</span> <span class="pre">(norm</span> <span class="pre">y)</span></code>. By the properties of <code class="docutils literal notranslate"><span class="pre">div'</span></code> and <code class="docutils literal notranslate"><span class="pre">mod'</span></code>, these are guaranteed to be less than or equal to <code class="docutils literal notranslate"><span class="pre">norm</span> <span class="pre">y</span> <span class="pre">/</span> <span class="pre">2</span></code>. So we have</p> <blockquote> <div><p><code class="docutils literal notranslate"><span class="pre">norm</span> <span class="pre">((x</span> <span class="pre">%</span> <span class="pre">y)</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y)</span> <span class="pre">≤</span> <span class="pre">(norm</span> <span class="pre">y</span> <span class="pre">/</span> <span class="pre">2)^2</span> <span class="pre">+</span> <span class="pre">(norm</span> <span class="pre">y</span> <span class="pre">/</span> <span class="pre">2)^2</span> <span class="pre">≤</span> <span class="pre">(norm</span> <span class="pre">y</span> <span class="pre">/</span> <span class="pre">2)</span> <span class="pre">*</span> <span class="pre">norm</span> <span class="pre">y</span></code>.</p> </div></blockquote> <p>On the other hand, we have</p> <blockquote> <div><p><code class="docutils literal notranslate"><span class="pre">norm</span> <span class="pre">((x</span> <span class="pre">%</span> <span class="pre">y)</span> <span class="pre">*</span> <span class="pre">conj</span> <span class="pre">y)</span> <span class="pre">=</span> <span class="pre">norm</span> <span class="pre">(x</span> <span class="pre">%</span> <span class="pre">y)</span> <span class="pre">*</span> <span class="pre">norm</span> <span class="pre">(conj</span> <span class="pre">y)</span> <span class="pre">=</span> <span class="pre">norm</span> <span class="pre">(x</span> <span class="pre">%</span> <span class="pre">y)</span> <span class="pre">*</span> <span class="pre">norm</span> <span class="pre">y</span></code>.</p> </div></blockquote> <p>Dividing through by <code class="docutils literal notranslate"><span class="pre">norm</span> <span class="pre">y</span></code> we have <code class="docutils literal notranslate"><span class="pre">norm</span> <span class="pre">(x</span> <span class="pre">%</span> <span class="pre">y)</span> <span class="pre">≤</span> <span class="pre">(norm</span> <span class="pre">y)</span> <span class="pre">/</span> <span class="pre">2</span> <span class="pre"><</span> <span class="pre">norm</span> <span class="pre">y</span></code>, as required.</p> <p>This messy calculation is carried out in the next proof. We encourage you to step through the details and see if you can find a nicer argument.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">norm_mod_lt</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">{</span><span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">}</span> <span class="o">(</span><span class="n">hy</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="bp">%</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">norm</span> <span class="bp"><</span> <span class="n">y.norm</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="n">norm_y_pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">norm</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rwa</span> <span class="o">[</span><span class="n">norm_pos</span><span class="o">]</span> <span class="k">have</span> <span class="n">H1</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">%</span> <span class="n">y</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span> <span class="bp">=</span> <span class="o">⟨</span><span class="n">Int.mod'</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">),</span> <span class="n">Int.mod'</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">)⟩</span> <span class="bp">·</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">simp</span> <span class="o">[</span><span class="n">Int.mod'_eq</span><span class="o">,</span> <span class="n">mod_def</span><span class="o">,</span> <span class="n">div_def</span><span class="o">,</span> <span class="n">norm</span><span class="o">]</span> <span class="bp"><;></span> <span class="n">ring</span> <span class="k">have</span> <span class="n">H2</span> <span class="o">:</span> <span class="n">norm</span> <span class="o">(</span><span class="n">x</span> <span class="bp">%</span> <span class="n">y</span><span class="o">)</span> <span class="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">·</span> <span class="k">calc</span> <span class="n">norm</span> <span class="o">(</span><span class="n">x</span> <span class="bp">%</span> <span class="n">y</span><span class="o">)</span> <span class="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">norm</span> <span class="o">(</span><span class="n">x</span> <span class="bp">%</span> <span class="n">y</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">norm_mul</span><span class="o">,</span> <span class="n">norm_conj</span><span class="o">]</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">abs</span> <span class="o">(</span><span class="n">Int.mod'</span> <span class="o">(</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">)</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">))</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">abs</span> <span class="o">(</span><span class="n">Int.mod'</span> <span class="o">(</span><span class="bp">-</span><span class="o">(</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">)</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.re</span><span class="o">)</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">))</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">H1</span><span class="o">,</span> <span class="n">norm</span><span class="o">,</span> <span class="n">sq_abs</span><span class="o">]</span> <span class="n">_</span> <span class="bp">≤</span> <span class="o">(</span><span class="n">y.norm</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="o">(</span><span class="n">y.norm</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">gcongr</span> <span class="bp"><;></span> <span class="n">apply</span> <span class="n">Int.abs_mod'_le</span> <span class="n">_</span> <span class="n">_</span> <span class="n">norm_y_pos</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">*</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="n">_</span> <span class="bp">≤</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">gcongr</span><span class="bp">;</span> <span class="n">apply</span> <span class="n">Int.ediv_mul_le</span><span class="bp">;</span> <span class="n">norm_num</span> <span class="k">calc</span> <span class="n">norm</span> <span class="o">(</span><span class="n">x</span> <span class="bp">%</span> <span class="n">y</span><span class="o">)</span> <span class="bp">≤</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="n">le_of_mul_le_mul_right</span> <span class="n">H2</span> <span class="n">norm_y_pos</span> <span class="n">_</span> <span class="bp"><</span> <span class="n">norm</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">Int.ediv_lt_of_lt_mul</span> <span class="bp">·</span> <span class="n">norm_num</span> <span class="bp">·</span> <span class="n">linarith</span> </pre></div> </div> <p>We are in the home stretch. Our <code class="docutils literal notranslate"><span class="pre">norm</span></code> function maps Gaussian integers to nonnegative integers. We need a function that maps Gaussian integers to natural numbers, and we obtain that by composing <code class="docutils literal notranslate"><span class="pre">norm</span></code> with the function <code class="docutils literal notranslate"><span class="pre">Int.natAbs</span></code>, which maps integers to the natural numbers. The first of the next two lemmas establishes that mapping the norm to the natural numbers and back to the integers does not change the value. The second one re-expresses the fact that the norm is decreasing.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">coe_natAbs_norm</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x.norm.natAbs</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x.norm</span> <span class="o">:=</span> <span class="n">Int.natAbs_of_nonneg</span> <span class="o">(</span><span class="n">norm_nonneg</span> <span class="n">_</span><span class="o">)</span> <span class="kd">theorem</span> <span class="n">natAbs_norm_mod_lt</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">(</span><span class="n">hy</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="bp">%</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">norm.natAbs</span> <span class="bp"><</span> <span class="n">y.norm.natAbs</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">Int.ofNat_lt.1</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">Int.coe_natAbs</span><span class="o">,</span> <span class="n">abs_of_nonneg</span><span class="o">,</span> <span class="n">norm_nonneg</span><span class="o">]</span> <span class="n">apply</span> <span class="n">norm_mod_lt</span> <span class="n">x</span> <span class="n">hy</span> </pre></div> </div> <p>We also need to establish the second key property of the norm function on a Euclidean domain.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">not_norm_mul_left_lt_norm</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">{</span><span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">}</span> <span class="o">(</span><span class="n">hy</span> <span class="o">:</span> <span class="n">y</span> <span class="bp">≠</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="bp">¬</span><span class="o">(</span><span class="n">norm</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="o">))</span><span class="bp">.</span><span class="n">natAbs</span> <span class="bp"><</span> <span class="o">(</span><span class="n">norm</span> <span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">natAbs</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">not_lt_of_ge</span> <span class="n">rw</span> <span class="o">[</span><span class="n">norm_mul</span><span class="o">,</span> <span class="n">Int.natAbs_mul</span><span class="o">]</span> <span class="n">apply</span> <span class="n">le_mul_of_one_le_right</span> <span class="o">(</span><span class="n">Nat.zero_le</span> <span class="n">_</span><span class="o">)</span> <span class="n">apply</span> <span class="n">Int.ofNat_le.1</span> <span class="n">rw</span> <span class="o">[</span><span class="n">coe_natAbs_norm</span><span class="o">]</span> <span class="n">exact</span> <span class="n">Int.add_one_le_of_lt</span> <span class="o">((</span><span class="n">norm_pos</span> <span class="n">_</span><span class="o">)</span><span class="bp">.</span><span class="n">mpr</span> <span class="n">hy</span><span class="o">)</span> </pre></div> </div> <p>We can now put it together to show that the Gaussian integers are an instance of a Euclidean domain. We use the quotient and remainder function we have defined. The mathlib definition of a Euclidean domain is more general than the one above in that it allows us to show that remainder decreases with respect to any well-founded measure. Comparing the values of a norm function that returns natural numbers is just one instance of such a measure, and in that case, the required properties are the theorems <code class="docutils literal notranslate"><span class="pre">natAbs_norm_mod_lt</span></code> and <code class="docutils literal notranslate"><span class="pre">not_norm_mul_left_lt_norm</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">EuclideanDomain</span> <span class="n">gaussInt</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">gaussInt.instCommRing</span> <span class="k">with</span> <span class="n">quotient</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">·</span> <span class="bp">/</span> <span class="bp">·</span><span class="o">)</span> <span class="n">remainder</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">·</span> <span class="bp">%</span> <span class="bp">·</span><span class="o">)</span> <span class="n">quotient_mul_add_remainder_eq</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=></span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">only</span><span class="bp">;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mod_def</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">,</span> <span class="n">sub_add_cancel</span><span class="o">]</span> <span class="n">quotient_zero</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">div_def</span><span class="o">,</span> <span class="n">norm</span><span class="o">,</span> <span class="n">Int.div'</span><span class="o">]</span> <span class="n">rfl</span> <span class="n">r</span> <span class="o">:=</span> <span class="n">Measure</span> <span class="o">(</span><span class="n">Int.natAbs</span> <span class="bp">∘</span> <span class="n">norm</span><span class="o">)</span> <span class="n">r_wellFounded</span> <span class="o">:=</span> <span class="o">(</span><span class="n">measure</span> <span class="o">(</span><span class="n">Int.natAbs</span> <span class="bp">∘</span> <span class="n">norm</span><span class="o">))</span><span class="bp">.</span><span class="mi">2</span> <span class="n">remainder_lt</span> <span class="o">:=</span> <span class="n">natAbs_norm_mod_lt</span> <span class="n">mul_left_not_lt</span> <span class="o">:=</span> <span class="n">not_norm_mul_left_lt_norm</span> <span class="o">}</span> </pre></div> </div> <p>An immediate payoff is that we now know that, in the Gaussian integers, the notions of being prime and being irreducible coincide.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">Irreducible</span> <span class="n">x</span> <span class="bp">↔</span> <span class="n">Prime</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">PrincipalIdealRing.irreducible_iff_prime</span> </pre></div> </div> </section> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C05_Number_Theory.html" class="btn btn-neutral float-left" title="5. Number Theory" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> <a href="C07_Hierarchies.html" class="btn btn-neutral float-right" title="7. Hierarchies" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> </div> <hr/> <div role="contentinfo"> <p>© Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot.</p> </div> Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>. </footer> </div> </div> </section> </div> <script> jQuery(function () { SphinxRtdTheme.Navigation.enable(true); }); </script> </body> </html>
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Hierarchies — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> <link rel="shortcut icon" href="_static/favicon.ico"/> <!--[if lt IE 9]> <script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" /> <link rel="next" title="8. Topology" href="C08_Topology.html" /> <link rel="prev" title="6. Structures" href="C06_Structures.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul class="current"> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">7. Hierarchies</a><ul> <li class="toctree-l2"><a class="reference internal" href="#basics">7.1. Basics</a></li> <li class="toctree-l2"><a class="reference internal" href="#morphisms">7.2. Morphisms</a></li> <li class="toctree-l2"><a class="reference internal" href="#sub-objects">7.3. Sub-objects</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">7. </span>Hierarchies</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C07_Hierarchies.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="hierarchies"> <span id="id1"></span><h1><span class="section-number">7. </span>Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this heading"></a></h1> <p>We have seen in <a class="reference internal" href="C06_Structures.html#structures"><span class="std std-numref">Chapter 6</span></a> how to define the class of groups and build instances of this class, and then how to build an instance of the commutative ring class. But of course there is a hierarchy here: a commutative ring is in particular an additive group. In this chapter we will study how to build such hierarchies. They appear in all branches of mathematics but in this chapter the emphasis will be on algebraic examples.</p> <p>It may seem premature to discuss how to build hierarchies before more discussions about using existing hierarchies. But some understanding of the technology underlying hierarchies is required to use them. So you should probably still read this chapter, but without trying too hard to remember everything on your first read, then read the following chapters and come back here for a second reading.</p> <p>In this chapter, we will redefine (simpler versions of) many things that appear in Mathlib so we will used indices to distinguish our version. For instance we will have <code class="docutils literal notranslate"><span class="pre">Ring₁</span></code> as our version of <code class="docutils literal notranslate"><span class="pre">Ring</span></code>. Since we will gradually explain more powerful ways of formalizing structures, those indices will sometimes grow beyond one.</p> <section id="basics"> <span id="section-hierarchies-basics"></span><h2><span class="section-number">7.1. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h2> <p>At the very bottom of all hierarchies in Lean, we find data-carrying classes. The following class records that the given type <code class="docutils literal notranslate"><span class="pre">α</span></code> is endowed with a distinguished element called <code class="docutils literal notranslate"><span class="pre">one</span></code>. At this stage, it has no property at all.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">One₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="n">where</span> <span class="sd">/-- The element one -/</span> <span class="n">one</span> <span class="o">:</span> <span class="n">α</span> </pre></div> </div> <p>Since we’ll make a much heavier use of classes in this chapter, we need to understand some more details about what the <code class="docutils literal notranslate"><span class="pre">class</span></code> command is doing. First, the <code class="docutils literal notranslate"><span class="pre">class</span></code> command above defines a structure <code class="docutils literal notranslate"><span class="pre">One₁</span></code> with parameter <code class="docutils literal notranslate"><span class="pre">α</span> <span class="pre">:</span> <span class="pre">Type</span></code> and a single field <code class="docutils literal notranslate"><span class="pre">one</span></code>. It also mark this structure as a class so that arguments of type <code class="docutils literal notranslate"><span class="pre">One₁</span> <span class="pre">α</span></code> for some type <code class="docutils literal notranslate"><span class="pre">α</span></code> will be inferrable using the instance resolution procedure, as long as they are marked as instance-implicit, ie appear between square brackets. Those two effects could also have been achieved using the <code class="docutils literal notranslate"><span class="pre">structure</span></code> command with <code class="docutils literal notranslate"><span class="pre">class</span></code> attribute, ie writing <code class="docutils literal notranslate"><span class="pre">@[class]</span> <span class="pre">structure</span></code> instance of <code class="docutils literal notranslate"><span class="pre">class</span></code>. But the class command also ensures that <code class="docutils literal notranslate"><span class="pre">One₁</span> <span class="pre">α</span></code> appears as an instance-implicit argument in its own fields. Compare:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">One₁.one</span> <span class="c1">-- One₁.one {α : Type} [self : One₁ α] : α</span> <span class="kd">@[class]</span> <span class="kd">structure</span> <span class="n">One₂</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="n">where</span> <span class="sd">/-- The element one -/</span> <span class="n">one</span> <span class="o">:</span> <span class="n">α</span> <span class="k">#check</span> <span class="n">One₂.one</span> </pre></div> </div> <p>In the second check, we can see that <code class="docutils literal notranslate"><span class="pre">self</span> <span class="pre">:</span> <span class="pre">One₂</span> <span class="pre">α</span></code> is an explicit argument. Let us make sure the first version is indeed usable without any explicit argument.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">One₁</span> <span class="n">α</span><span class="o">]</span> <span class="o">:</span> <span class="n">α</span> <span class="o">:=</span> <span class="n">One₁.one</span> </pre></div> </div> <p>Remark: in the above example, the argument <code class="docutils literal notranslate"><span class="pre">One₁</span> <span class="pre">α</span></code> is marked as instance-implicit, which is a bit silly since this affects only <em>uses</em> of the declaration and declaration created by the <code class="docutils literal notranslate"><span class="pre">example</span></code> command cannot be used. However it allows to avoid giving a name to that argument and, more importantly, it starts installing the good habit of marking <code class="docutils literal notranslate"><span class="pre">One₁</span> <span class="pre">α</span></code> arguments as instance-implicit.</p> <p>Another remark is that all this will work only when Lean knows what is <code class="docutils literal notranslate"><span class="pre">α</span></code>. In the above example, leaving out the type ascription <code class="docutils literal notranslate"><span class="pre">:</span> <span class="pre">α</span></code> would generate an error message like: <code class="docutils literal notranslate"><span class="pre">typeclass</span> <span class="pre">instance</span> <span class="pre">problem</span> <span class="pre">is</span> <span class="pre">stuck,</span> <span class="pre">it</span> <span class="pre">is</span> <span class="pre">often</span> <span class="pre">due</span> <span class="pre">to</span> <span class="pre">metavariables</span> <span class="pre">One₁</span> <span class="pre">(?m.263</span> <span class="pre">α)</span></code> where <code class="docutils literal notranslate"><span class="pre">?m.263</span> <span class="pre">α</span></code> means “some type depending on <code class="docutils literal notranslate"><span class="pre">α</span></code>” (and 263 is simply an auto-generated index that would be useful to distinguish between several unknown things). Another way to avoid this issue would be to use a type annotation, as in:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">One₁</span> <span class="n">α</span><span class="o">]</span> <span class="o">:=</span> <span class="o">(</span><span class="n">One₁.one</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> </pre></div> </div> <p>You may have already encountered that issue when playing with limits of sequences in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence"><span class="std std-numref">Section 3.6</span></a> if you tried to state for instance that <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">1</span></code> without telling Lean whether you meant this inequality to be about natural numbers or real numbers.</p> <p>Our next task is to assign a notation to <code class="docutils literal notranslate"><span class="pre">One₁.one</span></code>. This we don’t want collisions with the builtin notation for <code class="docutils literal notranslate"><span class="pre">1</span></code>, we will use <code class="docutils literal notranslate"><span class="pre">𝟙</span></code>. This is achieved by the following command where the first line tells Lean to use the documentation of <code class="docutils literal notranslate"><span class="pre">One₁.one</span></code> as documentation for the symbol <code class="docutils literal notranslate"><span class="pre">𝟙</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[inherit_doc]</span> <span class="kd">notation</span> <span class="s2">"𝟙"</span> <span class="bp">=></span> <span class="n">One₁.one</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">One₁</span> <span class="n">α</span><span class="o">]</span> <span class="o">:</span> <span class="n">α</span> <span class="o">:=</span> <span class="mi">𝟙</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">One₁</span> <span class="n">α</span><span class="o">]</span> <span class="o">:</span> <span class="o">(</span><span class="mi">𝟙</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="bp">=</span> <span class="mi">𝟙</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>We now want a data-carrying class recording a binary operation. We don’t want to choose between addition and multiplication for now so we’ll use diamond.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Dia₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="n">where</span> <span class="n">dia</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="kd">infixl</span><span class="o">:</span><span class="mi">70</span> <span class="s2">" ⋄ "</span> <span class="bp">=></span> <span class="n">Dia₁.dia</span> </pre></div> </div> <p>As in the <code class="docutils literal notranslate"><span class="pre">One₁</span></code> example, the operation has no property at all at this stage. Let us now define the class of semigroup structures where the operation is denoted by <code class="docutils literal notranslate"><span class="pre">⋄</span></code>. For now, we define it by hand as a structure with two fields, a <code class="docutils literal notranslate"><span class="pre">Dia₁</span></code> instance and some <code class="docutils literal notranslate"><span class="pre">Prop</span></code>-valued field <code class="docutils literal notranslate"><span class="pre">dia_assoc</span></code> asserting associativity of <code class="docutils literal notranslate"><span class="pre">⋄</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Semigroup₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="n">where</span> <span class="n">toDia₁</span> <span class="o">:</span> <span class="n">Dia₁</span> <span class="n">α</span> <span class="sd">/-- Diamond is associative -/</span> <span class="n">dia_assoc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">b</span> <span class="bp">⋄</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="o">(</span><span class="n">b</span> <span class="bp">⋄</span> <span class="n">c</span><span class="o">)</span> </pre></div> </div> <p>Note that while stating <cite>dia_assoc</cite>, the previously defined field <cite>toDia₁</cite> is in the local context hence can be used when Lean searches for an instance of <cite>Dia₁ α</cite> to make sense of <cite>a ⋄ b</cite>. However this <cite>toDia₁</cite> field does not become part of the type class instances database. Hence doing <code class="docutils literal notranslate"><span class="pre">example</span> <span class="pre">{α</span> <span class="pre">:</span> <span class="pre">Type}</span> <span class="pre">[Semigroup₁</span> <span class="pre">α]</span> <span class="pre">(a</span> <span class="pre">b</span> <span class="pre">:</span> <span class="pre">α)</span> <span class="pre">:</span> <span class="pre">α</span> <span class="pre">:=</span> <span class="pre">a</span> <span class="pre">⋄</span> <span class="pre">b</span></code> would fail with error message <code class="docutils literal notranslate"><span class="pre">failed</span> <span class="pre">to</span> <span class="pre">synthesize</span> <span class="pre">instance</span> <span class="pre">Dia₁</span> <span class="pre">α</span></code>.</p> <p>We can fix this by adding the <code class="docutils literal notranslate"><span class="pre">instance</span></code> attribute later.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">attribute</span> <span class="o">[</span><span class="kd">instance</span><span class="o">]</span> <span class="n">Semigroup₁.toDia₁</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Semigroup₁</span> <span class="n">α</span><span class="o">]</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="n">α</span> <span class="o">:=</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">b</span> </pre></div> </div> <p>Before building up, we need a more convenient way to extend structures than explicitly writing fields like <cite>toDia₁</cite> and adding the instance attribute by hand. The <code class="docutils literal notranslate"><span class="pre">class</span></code> supports this using the <code class="docutils literal notranslate"><span class="pre">extends</span></code> syntax as in:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Semigroup₂</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Dia₁</span> <span class="n">α</span> <span class="n">where</span> <span class="sd">/-- Diamond is associative -/</span> <span class="n">dia_assoc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">b</span> <span class="bp">⋄</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="o">(</span><span class="n">b</span> <span class="bp">⋄</span> <span class="n">c</span><span class="o">)</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Semigroup₂</span> <span class="n">α</span><span class="o">]</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="n">α</span> <span class="o">:=</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">b</span> </pre></div> </div> <p>Note this syntax is also available in the <code class="docutils literal notranslate"><span class="pre">structure</span></code> command, although it that case it fixes only the hurdle of writing fields such as <cite>toDia₁</cite> since there is no instance to define in that case.</p> <p>Let us now try to combine a diamond operation and a distinguished one with axioms saying this element is neutral on both sides.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">DiaOneClass₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">One₁</span> <span class="n">α</span><span class="o">,</span> <span class="n">Dia₁</span> <span class="n">α</span> <span class="n">where</span> <span class="sd">/-- One is a left neutral element for diamond. -/</span> <span class="n">one_dia</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="mi">𝟙</span> <span class="bp">⋄</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">a</span> <span class="sd">/-- One is a right neutral element for diamond -/</span> <span class="n">dia_one</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="mi">𝟙</span> <span class="bp">=</span> <span class="n">a</span> </pre></div> </div> <p>In the next example, we tell Lean that <code class="docutils literal notranslate"><span class="pre">α</span></code> has a <code class="docutils literal notranslate"><span class="pre">DiaOneClass₁</span></code> structure and state a property that uses both a <cite>Dia₁</cite> instance and a <cite>One₁</cite> instance. In order to see how Lean finds those instances we set a tracing option whose result can be seen in the info view. This result is rather terse by default but can be expended by clicking one lines ending with black arrows. It includes failed attempts where Lean tried to find instances before having enough type information to succceed. The successful attempts do involve the instances generated by the <code class="docutils literal notranslate"><span class="pre">extends</span></code> syntax.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">set_option</span> <span class="n">trace.Meta.synthInstance</span> <span class="n">true</span> <span class="k">in</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">DiaOneClass₁</span> <span class="n">α</span><span class="o">]</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">b</span> <span class="bp">=</span> <span class="mi">𝟙</span> </pre></div> </div> <p>Note that we don’t need to include extra fields where combining existing classes. Hence we can define monoids as:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Monoid₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Semigroup₁</span> <span class="n">α</span><span class="o">,</span> <span class="n">DiaOneClass₁</span> <span class="n">α</span> </pre></div> </div> <p>While the above definition seems straightforward, it hides an important subtlety. Both <code class="docutils literal notranslate"><span class="pre">Semigroup₁</span> <span class="pre">α</span></code> and <code class="docutils literal notranslate"><span class="pre">DiaOneClass₁</span> <span class="pre">α</span></code> extend <code class="docutils literal notranslate"><span class="pre">Dia₁</span> <span class="pre">α</span></code>, so one could fear that having a <code class="docutils literal notranslate"><span class="pre">Monoid₁</span> <span class="pre">α</span></code> instance gives two unrelated diamond operations on <code class="docutils literal notranslate"><span class="pre">α</span></code>, one coming from a field <code class="docutils literal notranslate"><span class="pre">Monoid₁.toSemigroup₁</span></code> and one coming from a field <code class="docutils literal notranslate"><span class="pre">Monoid₁.toDiaOneClass₁</span></code>.</p> <p>Indeed if we try to build a monoid class by hand using:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Monoid₂</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="n">where</span> <span class="n">toSemigroup₁</span> <span class="o">:</span> <span class="n">Semigroup₁</span> <span class="n">α</span> <span class="n">toDiaOneClass₁</span> <span class="o">:</span> <span class="n">DiaOneClass₁</span> <span class="n">α</span> </pre></div> </div> <p>then we get two completely unrelated diamond operations <code class="docutils literal notranslate"><span class="pre">Monoid₂.toSemigroup₁.toDia₁.dia</span></code> and <code class="docutils literal notranslate"><span class="pre">Monoid₂.toDiaOneClass₁.toDia₁.dia</span></code>.</p> <p>The version generated using the <code class="docutils literal notranslate"><span class="pre">extends</span></code> syntax does not have this defect.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Monoid₁</span> <span class="n">α</span><span class="o">]</span> <span class="o">:</span> <span class="o">(</span><span class="n">Monoid₁.toSemigroup₁.toDia₁.dia</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span><span class="o">)</span> <span class="bp">=</span> <span class="n">Monoid₁.toDiaOneClass₁.toDia₁.dia</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>So the <code class="docutils literal notranslate"><span class="pre">class</span></code> command did some magic for us (and the <code class="docutils literal notranslate"><span class="pre">structure</span></code> command would have done it too). An easy way to see what are the fields of our classes is to check their constructor. Compare:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="c">/-</span><span class="cm"> Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/</span> <span class="k">#check</span> <span class="n">Monoid₂.mk</span> <span class="c">/-</span><span class="cm"> Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/</span> <span class="k">#check</span> <span class="n">Monoid₁.mk</span> </pre></div> </div> <p>So we see that <code class="docutils literal notranslate"><span class="pre">Monoid₁</span></code> takes <code class="docutils literal notranslate"><span class="pre">Semigroup₁</span> <span class="pre">α</span></code> argument as expected but then it won’t take a would-be overlapping <code class="docutils literal notranslate"><span class="pre">DiaOneClass₁</span> <span class="pre">α</span></code> argument but instead tears it appart and includes only the non-overlapping parts. And it also auto-generated an instance <code class="docutils literal notranslate"><span class="pre">Monoid₁.toDiaOneClass₁</span></code> which is <em>not</em> a field but has the expected signature which, from the end-user point of view, restores the symmetry between the two extended classes <code class="docutils literal notranslate"><span class="pre">Semigroup₁</span></code> and <code class="docutils literal notranslate"><span class="pre">DiaOneClass₁</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">Monoid₁.toSemigroup₁</span> <span class="k">#check</span> <span class="n">Monoid₁.toDiaOneClass₁</span> </pre></div> </div> <p>We are now very close to defining groups. We could add to the monoid structure a field asserting the existence of an inverse for every element. But then we would need to work to access these inverses. In practice it is more convenient to add it as data. To optimize reusability, we define a new data-carrying class, and then give it some notation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Inv₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="n">where</span> <span class="sd">/-- The inversion function -/</span> <span class="n">inv</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="kd">@[inherit_doc]</span> <span class="kd">postfix</span><span class="o">:</span><span class="n">max</span> <span class="s2">"⁻¹"</span> <span class="bp">=></span> <span class="n">Inv₁.inv</span> <span class="kd">class</span> <span class="n">Group₁</span> <span class="o">(</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Monoid₁</span> <span class="n">G</span><span class="o">,</span> <span class="n">Inv</span> <span class="n">G</span> <span class="n">where</span> <span class="n">inv_dia</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">,</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="bp">⋄</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">𝟙</span> </pre></div> </div> <p>The above definition may seem too weak, we only ask that <code class="docutils literal notranslate"><span class="pre">a⁻¹</span></code> is a left-inverse of <code class="docutils literal notranslate"><span class="pre">a</span></code>. But the other side is automatic. In order to prove that, we need a preliminary lemma.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">lemma</span> <span class="n">left_inv_eq_right_inv₁</span> <span class="o">{</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Monoid₁</span> <span class="n">M</span><span class="o">]</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">M</span><span class="o">}</span> <span class="o">(</span><span class="n">hba</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">⋄</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">𝟙</span><span class="o">)</span> <span class="o">(</span><span class="n">hac</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">𝟙</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">DiaOneClass₁.one_dia</span> <span class="n">c</span><span class="o">,</span> <span class="bp">←</span> <span class="n">hba</span><span class="o">,</span> <span class="n">Semigroup₁.dia_assoc</span><span class="o">,</span> <span class="n">hac</span><span class="o">,</span> <span class="n">DiaOneClass₁.dia_one</span> <span class="n">b</span><span class="o">]</span> </pre></div> </div> <p>In this lemma, it is pretty annoying to give full names, especially since it requires knowing which part of the hierarchy provides those facts. One way to fix this is to use the <code class="docutils literal notranslate"><span class="pre">export</span></code> command to copy those facts as lemmas in the root name space.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">export</span> <span class="n">DiaOneClass₁</span> <span class="o">(</span><span class="n">one_dia</span> <span class="n">dia_one</span><span class="o">)</span> <span class="kn">export</span> <span class="n">Semigroup₁</span> <span class="o">(</span><span class="n">dia_assoc</span><span class="o">)</span> <span class="kn">export</span> <span class="n">Group₁</span> <span class="o">(</span><span class="n">inv_dia</span><span class="o">)</span> </pre></div> </div> <p>We can then rewrite the above proof as:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Monoid₁</span> <span class="n">M</span><span class="o">]</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">M</span><span class="o">}</span> <span class="o">(</span><span class="n">hba</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">⋄</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">𝟙</span><span class="o">)</span> <span class="o">(</span><span class="n">hac</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">𝟙</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">one_dia</span> <span class="n">c</span><span class="o">,</span> <span class="bp">←</span> <span class="n">hba</span><span class="o">,</span> <span class="n">dia_assoc</span><span class="o">,</span> <span class="n">hac</span><span class="o">,</span> <span class="n">dia_one</span> <span class="n">b</span><span class="o">]</span> </pre></div> </div> <p>It is now your turn to prove things about our algebraic structures.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">lemma</span> <span class="n">inv_eq_of_dia</span> <span class="o">[</span><span class="n">Group₁</span> <span class="n">G</span><span class="o">]</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">G</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">b</span> <span class="bp">=</span> <span class="mi">𝟙</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="n">b</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">lemma</span> <span class="n">dia_inv</span> <span class="o">[</span><span class="n">Group₁</span> <span class="n">G</span><span class="o">]</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">⋄</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="mi">𝟙</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>At this stage we would like to move on to define rings, but there is a serious issue. A ring structure on a type contains both an additive group structure and a multiplicative monoid structure, and some properties about their interaction. But so far we hard-coded a notation <code class="docutils literal notranslate"><span class="pre">⋄</span></code> for all our operations. More fundamentally, the type class system assumes every type has only one instance of each type class. There are various ways to solve this issue. Surprisingly mathlib uses the naive idea to duplicate everything for additive and multiplicative theories with the help of some code-generating attribute. Structures and classes are defined in both additive and multiplicative notation with an attibute <code class="docutils literal notranslate"><span class="pre">to_additive</span></code> linking them. In case of multiple inheritance like for semi-groups, the auto-generated “symmetry-restoring” instances need also to be marked. This is a bit technical you don’t need to understand details. The important point is that lemmas are then only stated in multiplicative notation and marked with the attribute <code class="docutils literal notranslate"><span class="pre">to_additive</span></code> to generate the additive version as <code class="docutils literal notranslate"><span class="pre">left_inv_eq_right_inv'</span></code> with it’s auto-generated additive version <code class="docutils literal notranslate"><span class="pre">left_neg_eq_right_neg'</span></code>. In order to check the name of this additive version we used the <code class="docutils literal notranslate"><span class="pre">whatsnew</span> <span class="pre">in</span></code> command on top of <code class="docutils literal notranslate"><span class="pre">left_inv_eq_right_inv'</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">AddSemigroup₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Add</span> <span class="n">α</span> <span class="n">where</span> <span class="sd">/-- Multiplication is associative -/</span> <span class="n">add_assoc₃</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">+</span> <span class="o">(</span><span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="kd">@[to_additive AddSemigroup₃]</span> <span class="kd">class</span> <span class="n">Semigroup₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Mul</span> <span class="n">α</span> <span class="n">where</span> <span class="sd">/-- Multiplication is associative -/</span> <span class="n">mul_assoc₃</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="kd">class</span> <span class="n">AddMonoid₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">AddSemigroup₃</span> <span class="n">α</span><span class="o">,</span> <span class="n">AddZeroClass</span> <span class="n">α</span> <span class="kd">@[to_additive AddMonoid₃]</span> <span class="kd">class</span> <span class="n">Monoid₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Semigroup₃</span> <span class="n">α</span><span class="o">,</span> <span class="n">MulOneClass</span> <span class="n">α</span> <span class="kn">attribute</span> <span class="o">[</span><span class="n">to_additive</span> <span class="n">existing</span><span class="o">]</span> <span class="n">Monoid₃.toMulOneClass</span> <span class="kn">export</span> <span class="n">Semigroup₃</span> <span class="o">(</span><span class="n">mul_assoc₃</span><span class="o">)</span> <span class="kn">export</span> <span class="n">AddSemigroup₃</span> <span class="o">(</span><span class="n">add_assoc₃</span><span class="o">)</span> <span class="n">whatsnew</span> <span class="k">in</span> <span class="kd">@[to_additive]</span> <span class="kd">lemma</span> <span class="n">left_inv_eq_right_inv'</span> <span class="o">{</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Monoid₃</span> <span class="n">M</span><span class="o">]</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">M</span><span class="o">}</span> <span class="o">(</span><span class="n">hba</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> <span class="o">(</span><span class="n">hac</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">one_mul</span> <span class="n">c</span><span class="o">,</span> <span class="bp">←</span> <span class="n">hba</span><span class="o">,</span> <span class="n">mul_assoc₃</span><span class="o">,</span> <span class="n">hac</span><span class="o">,</span> <span class="n">mul_one</span> <span class="n">b</span><span class="o">]</span> <span class="k">#check</span> <span class="n">left_neg_eq_right_neg'</span> </pre></div> </div> <p>Equipped with this technology, we can easily define also commutative semigroups, monoids and groups, and then define rings.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">AddCommSemigroup₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">AddSemigroup₃</span> <span class="n">α</span> <span class="n">where</span> <span class="n">add_comm</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">a</span> <span class="kd">@[to_additive AddCommSemigroup₃]</span> <span class="kd">class</span> <span class="n">CommSemigroup₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Semigroup₃</span> <span class="n">α</span> <span class="n">where</span> <span class="n">mul_comm</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">a</span> <span class="kd">class</span> <span class="n">AddCommMonoid₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">AddMonoid₃</span> <span class="n">α</span><span class="o">,</span> <span class="n">AddCommSemigroup₃</span> <span class="n">α</span> <span class="kd">@[to_additive AddCommMonoid₃]</span> <span class="kd">class</span> <span class="n">CommMonoid₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Monoid₃</span> <span class="n">α</span><span class="o">,</span> <span class="n">CommSemigroup₃</span> <span class="n">α</span> <span class="kd">class</span> <span class="n">AddGroup₃</span> <span class="o">(</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">AddMonoid₃</span> <span class="n">G</span><span class="o">,</span> <span class="n">Neg</span> <span class="n">G</span> <span class="n">where</span> <span class="n">neg_add</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">,</span> <span class="bp">-</span><span class="n">a</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">0</span> <span class="kd">@[to_additive AddGroup₃]</span> <span class="kd">class</span> <span class="n">Group₃</span> <span class="o">(</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Monoid₃</span> <span class="n">G</span><span class="o">,</span> <span class="n">Inv</span> <span class="n">G</span> <span class="n">where</span> <span class="n">inv_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">,</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">1</span> </pre></div> </div> <p>We should remember to tagged lemmas with <code class="docutils literal notranslate"><span class="pre">simp</span></code> when approriate.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">attribute</span> <span class="o">[</span><span class="n">simp</span><span class="o">]</span> <span class="n">Group₃.inv_mul</span> <span class="n">AddGroup₃.neg_add</span> </pre></div> </div> <p>Then we need to repeat ourselves a bit since we switch to standard notations, but at least <code class="docutils literal notranslate"><span class="pre">to_additive</span></code> does the work of translating from the multiplicative notation to the additive one.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[to_additive]</span> <span class="kd">lemma</span> <span class="n">inv_eq_of_mul</span> <span class="o">[</span><span class="n">Group₃</span> <span class="n">G</span><span class="o">]</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">G</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="n">b</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Note that <code class="docutils literal notranslate"><span class="pre">to_additive</span></code> can be ask to tag a lemma with <code class="docutils literal notranslate"><span class="pre">simp</span></code> and propagate that attribute to the additive version as follows.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[to_additive (attr := simp)]</span> <span class="kd">lemma</span> <span class="n">Group₃.mul_inv</span> <span class="o">{</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Group₃</span> <span class="n">G</span><span class="o">]</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">G</span><span class="o">}</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">a</span><span class="bp">⁻¹</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">@[to_additive]</span> <span class="kd">lemma</span> <span class="n">mul_left_cancel₃</span> <span class="o">{</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Group₃</span> <span class="n">G</span><span class="o">]</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">G</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">@[to_additive]</span> <span class="kd">lemma</span> <span class="n">mul_right_cancel₃</span> <span class="o">{</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Group₃</span> <span class="n">G</span><span class="o">]</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">G</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">b</span><span class="bp">*</span><span class="n">a</span> <span class="bp">=</span> <span class="n">c</span><span class="bp">*</span><span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">class</span> <span class="n">AddCommGroup₃</span> <span class="o">(</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">AddGroup₃</span> <span class="n">G</span><span class="o">,</span> <span class="n">AddCommMonoid₃</span> <span class="n">G</span> <span class="kd">@[to_additive AddCommGroup₃]</span> <span class="kd">class</span> <span class="n">CommGroup₃</span> <span class="o">(</span><span class="n">G</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">Group₃</span> <span class="n">G</span><span class="o">,</span> <span class="n">CommMonoid₃</span> <span class="n">G</span> </pre></div> </div> <p>We are now ready for rings. For demonstration puprposes we won’t assume that addition is commutative, and then immediately provide an instance of <code class="docutils literal notranslate"><span class="pre">AddCommGroup₃</span></code>. Mathlib does not play this game, first because in practice this does not make any ring instance easier and also because Mathlib’s algebraic hierarchy goes through semi-rings which are like rings but without opposites so that the proof below does not work for them. What we gain here, besides a nice exercise if you have never seen it, is an example of building an instance using the syntax that allows to provide a parent structure and some extra fields.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Ring₃</span> <span class="o">(</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">AddGroup₃</span> <span class="n">R</span><span class="o">,</span> <span class="n">Monoid₃</span> <span class="n">R</span><span class="o">,</span> <span class="n">MulZeroClass</span> <span class="n">R</span> <span class="n">where</span> <span class="sd">/-- Multiplication is left distributive over addition -/</span> <span class="n">left_distrib</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">+</span> <span class="n">c</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="sd">/-- Multiplication is right distributive over addition -/</span> <span class="n">right_distrib</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">R</span><span class="o">,</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="kd">instance</span> <span class="o">{</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">Ring₃</span> <span class="n">R</span><span class="o">]</span> <span class="o">:</span> <span class="n">AddCommGroup₃</span> <span class="n">R</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">Ring₃.toAddGroup₃</span> <span class="k">with</span> <span class="n">add_comm</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="o">}</span> </pre></div> </div> <p>Of course we can also build concrete instances, such as a ring structure on integers (of course the instance below uses that all the work is already done in Mathlib).</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Ring₃</span> <span class="n">ℤ</span> <span class="n">where</span> <span class="n">add</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">·</span> <span class="bp">+</span> <span class="bp">·</span><span class="o">)</span> <span class="n">add_assoc₃</span> <span class="o">:=</span> <span class="n">add_assoc</span> <span class="n">zero</span> <span class="o">:=</span> <span class="mi">0</span> <span class="n">zero_add</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">add_zero</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">neg</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">-</span> <span class="bp">·</span><span class="o">)</span> <span class="n">neg_add</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">mul</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">·</span> <span class="bp">*</span> <span class="bp">·</span><span class="o">)</span> <span class="n">mul_assoc₃</span> <span class="o">:=</span> <span class="n">mul_assoc</span> <span class="n">one</span> <span class="o">:=</span> <span class="mi">1</span> <span class="n">one_mul</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">mul_one</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">zero_mul</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">mul_zero</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">left_distrib</span> <span class="o">:=</span> <span class="n">Int.mul_add</span> <span class="n">right_distrib</span> <span class="o">:=</span> <span class="n">Int.add_mul</span> </pre></div> </div> <p>As an exercise you can now set up a simple hierarchy for order relations, including a class for ordered commutative monoids, which have both a partial order and a commutative monoid structure such that <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">:</span> <span class="pre">α,</span> <span class="pre">a</span> <span class="pre">≤</span> <span class="pre">b</span> <span class="pre">→</span> <span class="pre">∀</span> <span class="pre">c</span> <span class="pre">:</span> <span class="pre">α,</span> <span class="pre">c</span> <span class="pre">*</span> <span class="pre">a</span> <span class="pre">≤</span> <span class="pre">c</span> <span class="pre">*</span> <span class="pre">b</span></code>. Of course you need to add fields and maybe <code class="docutils literal notranslate"><span class="pre">extends</span></code> clauses to the following classes.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">LE₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="n">where</span> <span class="sd">/-- The Less-or-Equal relation. -/</span> <span class="n">le</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="kt">Prop</span> <span class="kd">@[inherit_doc]</span> <span class="kd">infix</span><span class="o">:</span><span class="mi">50</span> <span class="s2">" ≤₁ "</span> <span class="bp">=></span> <span class="n">LE₁.le</span> <span class="kd">class</span> <span class="n">Preorder₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">class</span> <span class="n">PartialOrder₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">class</span> <span class="n">OrderedCommMonoid₁</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">instance</span> <span class="o">:</span> <span class="n">OrderedCommMonoid₁</span> <span class="n">ℕ</span> <span class="n">where</span> </pre></div> </div> <p>We now want to discuss algebraic structures involving several types. The prime example is modules over rings. If you don’t know what is a module, you can pretend it means vector space and think that all our rings are fields. Those structures are commutative additive groups equipped with a scalar multiplication by elements of some ring.</p> <p>We first define the data-carrying type class of scalar multiplication by some type <code class="docutils literal notranslate"><span class="pre">α</span></code> on some type <code class="docutils literal notranslate"><span class="pre">β</span></code>, and give it a right associative notation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">SMul₃</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">(</span><span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="n">where</span> <span class="sd">/-- Scalar multiplication -/</span> <span class="n">smul</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">β</span> <span class="kd">infixr</span><span class="o">:</span><span class="mi">73</span> <span class="s2">" • "</span> <span class="bp">=></span> <span class="n">SMul₃.smul</span> </pre></div> </div> <p>Then we can define modules (again think about vector spaces if you don’t know what is a module).</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">Module₁</span> <span class="o">(</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Ring₃</span> <span class="n">R</span><span class="o">]</span> <span class="o">(</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">AddCommGroup₃</span> <span class="n">M</span><span class="o">]</span> <span class="kd">extends</span> <span class="n">SMul₃</span> <span class="n">R</span> <span class="n">M</span> <span class="n">where</span> <span class="n">zero_smul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">m</span> <span class="o">:</span> <span class="n">M</span><span class="o">,</span> <span class="o">(</span><span class="mi">0</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="bp">•</span> <span class="n">m</span> <span class="bp">=</span> <span class="mi">0</span> <span class="n">one_smul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">m</span> <span class="o">:</span> <span class="n">M</span><span class="o">,</span> <span class="o">(</span><span class="mi">1</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="bp">•</span> <span class="n">m</span> <span class="bp">=</span> <span class="n">m</span> <span class="n">mul_smul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">M</span><span class="o">),</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="o">)</span> <span class="bp">•</span> <span class="n">m</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">•</span> <span class="n">b</span> <span class="bp">•</span> <span class="n">m</span> <span class="n">add_smul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">M</span><span class="o">),</span> <span class="o">(</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="bp">•</span> <span class="n">m</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">•</span> <span class="n">m</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">•</span> <span class="n">m</span> <span class="n">smul_add</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">(</span><span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">M</span><span class="o">),</span> <span class="n">a</span> <span class="bp">•</span> <span class="o">(</span><span class="n">m</span> <span class="bp">+</span> <span class="n">n</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">•</span> <span class="n">m</span> <span class="bp">+</span> <span class="n">a</span> <span class="bp">•</span> <span class="n">n</span> </pre></div> </div> <p>There is something interesting going on here. While it isn’t too surprising that the ring structure on <code class="docutils literal notranslate"><span class="pre">R</span></code> is a parameter in this definition, you probably expected <code class="docutils literal notranslate"><span class="pre">AddCommGroup3</span> <span class="pre">M</span></code> to be part of the <code class="docutils literal notranslate"><span class="pre">extends</span></code> clause just as <code class="docutils literal notranslate"><span class="pre">SMul₃</span> <span class="pre">R</span> <span class="pre">M</span></code> is. Trying to do that would lead to a mysterious sounding error message: <code class="docutils literal notranslate"><span class="pre">cannot</span> <span class="pre">find</span> <span class="pre">synthesization</span> <span class="pre">order</span> <span class="pre">for</span> <span class="pre">instance</span> <span class="pre">Module₁.toAddCommGroup₃</span> <span class="pre">with</span> <span class="pre">type</span> <span class="pre">(R</span> <span class="pre">:</span> <span class="pre">Type)</span> <span class="pre">→</span> <span class="pre">[inst</span> <span class="pre">:</span> <span class="pre">Ring₃</span> <span class="pre">R]</span> <span class="pre">→</span> <span class="pre">{M</span> <span class="pre">:</span> <span class="pre">Type}</span> <span class="pre">→</span> <span class="pre">[self</span> <span class="pre">:</span> <span class="pre">Module₁</span> <span class="pre">R</span> <span class="pre">M]</span> <span class="pre">→</span> <span class="pre">AddCommGroup₃</span> <span class="pre">M</span> <span class="pre">all</span> <span class="pre">remaining</span> <span class="pre">arguments</span> <span class="pre">have</span> <span class="pre">metavariables:</span> <span class="pre">Ring₃</span> <span class="pre">?R</span> <span class="pre">@Module₁</span> <span class="pre">?R</span> <span class="pre">?inst✝</span> <span class="pre">M</span></code>. In order to understand this message, you need to remember that such an <code class="docutils literal notranslate"><span class="pre">extends</span></code> clause would lead to a field <code class="docutils literal notranslate"><span class="pre">Module₃.toAddCommGroup₃</span></code> marked as an instance. This instance would have the signature appearing in the error message: <code class="docutils literal notranslate"><span class="pre">(R</span> <span class="pre">:</span> <span class="pre">Type)</span> <span class="pre">→</span> <span class="pre">[inst</span> <span class="pre">:</span> <span class="pre">Ring₃</span> <span class="pre">R]</span> <span class="pre">→</span> <span class="pre">{M</span> <span class="pre">:</span> <span class="pre">Type}</span> <span class="pre">→</span> <span class="pre">[self</span> <span class="pre">:</span> <span class="pre">Module₁</span> <span class="pre">R</span> <span class="pre">M]</span> <span class="pre">→</span> <span class="pre">AddCommGroup₃</span> <span class="pre">M</span></code>. With such an instance in the type class database, each time Lean would look for a <code class="docutils literal notranslate"><span class="pre">AddCommGroup₃</span> <span class="pre">M</span></code> instance for some <code class="docutils literal notranslate"><span class="pre">M</span></code>, it would need to go hunting for a completely unspecified type <code class="docutils literal notranslate"><span class="pre">R``and</span> <span class="pre">a</span> <span class="pre">``Ring₃</span> <span class="pre">R</span></code> instance before embarking on the main quest of finding a <code class="docutils literal notranslate"><span class="pre">Module₁</span> <span class="pre">R</span> <span class="pre">M</span></code> instance. Those two side-quests are represented by the meta-variables mentionned in the error message and denoted by <code class="docutils literal notranslate"><span class="pre">?R</span></code> and <code class="docutils literal notranslate"><span class="pre">?inst✝</span></code> there. Such a <code class="docutils literal notranslate"><span class="pre">Module₃.toAddCommGroup₃</span></code> instance would then be a huge trap for the instance resolution procedure and then <code class="docutils literal notranslate"><span class="pre">class</span></code> command refuses to set it up.</p> <p>What about <code class="docutils literal notranslate"><span class="pre">extends</span> <span class="pre">SMul₃</span> <span class="pre">R</span> <span class="pre">M</span></code> then? That one creates a field <code class="docutils literal notranslate"><span class="pre">Module₁.toSMul₃</span> <span class="pre">:</span> <span class="pre">{R</span> <span class="pre">:</span> <span class="pre">Type}</span> <span class="pre">→</span>  <span class="pre">[inst</span> <span class="pre">:</span> <span class="pre">Ring₃</span> <span class="pre">R]</span> <span class="pre">→</span> <span class="pre">{M</span> <span class="pre">:</span> <span class="pre">Type}</span> <span class="pre">→</span> <span class="pre">[inst_1</span> <span class="pre">:</span> <span class="pre">AddCommGroup₃</span> <span class="pre">M]</span> <span class="pre">→</span> <span class="pre">[self</span> <span class="pre">:</span> <span class="pre">Module₁</span> <span class="pre">R</span> <span class="pre">M]</span> <span class="pre">→</span> <span class="pre">SMul₃</span> <span class="pre">R</span> <span class="pre">M</span></code> whose end result <code class="docutils literal notranslate"><span class="pre">SMul₃</span> <span class="pre">R</span> <span class="pre">M</span></code> mentions both <code class="docutils literal notranslate"><span class="pre">R</span></code> and <code class="docutils literal notranslate"><span class="pre">M</span></code> so this field can safely be used as an instance. The rule is easy to remember: each class appearing in the <code class="docutils literal notranslate"><span class="pre">extends</span></code> clause should mention every type appearing in the parameters.</p> <p>Let us create our first module instance: a ring is a module over itself using its multiplication as a scalar multiplication.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="n">selfModule</span> <span class="o">(</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Ring₃</span> <span class="n">R</span><span class="o">]</span> <span class="o">:</span> <span class="n">Module₁</span> <span class="n">R</span> <span class="n">R</span> <span class="n">where</span> <span class="n">smul</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">r</span> <span class="n">s</span> <span class="bp">↦</span> <span class="n">r</span><span class="bp">*</span><span class="n">s</span> <span class="n">zero_smul</span> <span class="o">:=</span> <span class="n">zero_mul</span> <span class="n">one_smul</span> <span class="o">:=</span> <span class="n">one_mul</span> <span class="n">mul_smul</span> <span class="o">:=</span> <span class="n">mul_assoc₃</span> <span class="n">add_smul</span> <span class="o">:=</span> <span class="n">Ring₃.right_distrib</span> <span class="n">smul_add</span> <span class="o">:=</span> <span class="n">Ring₃.left_distrib</span> </pre></div> </div> <p>As a second example, every abelian group is a module over <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> (this is one of the reason to generalize the theory of vector spaces by allowing non-invertible scalars). First one can define scalar multiplication by a natural number for any type equipped with a zero and an addition: <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">•</span> <span class="pre">a</span></code> is defined as <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">⋯</span> <span class="pre">+</span> <span class="pre">a</span></code> where <code class="docutils literal notranslate"><span class="pre">a</span></code> appears <code class="docutils literal notranslate"><span class="pre">n</span></code> times. Then this is extended to scalar multiplication by an integer by ensuring <code class="docutils literal notranslate"><span class="pre">(-1)</span> <span class="pre">•</span> <span class="pre">a</span> <span class="pre">=</span> <span class="pre">-a</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">nsmul₁</span> <span class="o">[</span><span class="n">Zero</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Add</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">M</span> <span class="bp">|</span> <span class="mi">0</span><span class="o">,</span> <span class="n">_</span> <span class="bp">=></span> <span class="mi">0</span> <span class="bp">|</span> <span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">,</span> <span class="n">a</span> <span class="bp">=></span> <span class="n">a</span> <span class="bp">+</span> <span class="n">nsmul₁</span> <span class="n">n</span> <span class="n">a</span> <span class="kd">def</span> <span class="n">zsmul₁</span> <span class="o">{</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Zero</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Add</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Neg</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="n">ℤ</span> <span class="bp">→</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">M</span> <span class="bp">|</span> <span class="n">Int.ofNat</span> <span class="n">n</span><span class="o">,</span> <span class="n">a</span> <span class="bp">=></span> <span class="n">nsmul₁</span> <span class="n">n</span> <span class="n">a</span> <span class="bp">|</span> <span class="n">Int.negSucc</span> <span class="n">n</span><span class="o">,</span> <span class="n">a</span> <span class="bp">=></span> <span class="bp">-</span><span class="n">nsmul₁</span> <span class="n">n.succ</span> <span class="n">a</span> </pre></div> </div> <p>Proving this gives rise to a module structure is a bit tedious and not interesting for the current discussion, so we will sorry all axioms. You are <em>not</em> asked to replace those sorries with proofs. If you insist on doing it then you will probably want to state and prove several intermediate lemmas about <code class="docutils literal notranslate"><span class="pre">nsmul₁</span></code> and <code class="docutils literal notranslate"><span class="pre">zsmul₁</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="n">abGrpModule</span> <span class="o">(</span><span class="n">A</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">AddCommGroup₃</span> <span class="n">A</span><span class="o">]</span> <span class="o">:</span> <span class="n">Module₁</span> <span class="n">ℤ</span> <span class="n">A</span> <span class="n">where</span> <span class="n">smul</span> <span class="o">:=</span> <span class="n">zsmul₁</span> <span class="n">zero_smul</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">one_smul</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">mul_smul</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">add_smul</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">smul_add</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>A much more important issue is that we now have two module structures over the ring <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> for <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> itself: <code class="docutils literal notranslate"><span class="pre">abGrpModule</span> <span class="pre">ℤ</span></code> since <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> is a abelian group, and <code class="docutils literal notranslate"><span class="pre">selfModule</span> <span class="pre">ℤ</span></code> since <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> is a ring. Those two module structure correspond to the same abelian group structure, but it is not obvious that they have the same scalar multiplication. They actually do, but this isn’t true by definition, it requires a proof. This is very bad news for the type class instance resolution procedure and will lead to very frustating failures for users of this hierarchy. When directly asked to find an instance, Lean will pick one, and we can see which one using:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="bp">#</span><span class="n">synth</span> <span class="n">Module₁</span> <span class="n">ℤ</span> <span class="n">ℤ</span> <span class="c1">-- abGrpModule ℤ</span> </pre></div> </div> <p>But in a more indirect context it can happen that Lean infers the one and then gets confused. This situation is known as a bad diamond. This has nothing to do with the diamond operation we used above, it refers to the way one can draw the paths from <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> to its <code class="docutils literal notranslate"><span class="pre">Module₁</span> <span class="pre">ℤ</span></code> going through either <code class="docutils literal notranslate"><span class="pre">AddCommGroup₃</span> <span class="pre">ℤ</span></code> or <code class="docutils literal notranslate"><span class="pre">Ring₃</span> <span class="pre">ℤ</span></code>.</p> <p>It is important to understand that not all diamonds are bad. In fact there are diamonds everywhere in mathlib, and also in this chapter. Already at the very beginning we saw one can go from <code class="docutils literal notranslate"><span class="pre">Monoid₁</span> <span class="pre">α</span></code> to <code class="docutils literal notranslate"><span class="pre">Dia₁</span> <span class="pre">α</span></code> through either <code class="docutils literal notranslate"><span class="pre">Semigroup₁</span> <span class="pre">α</span></code> or <code class="docutils literal notranslate"><span class="pre">DiaOneClass₁</span> <span class="pre">α</span></code> and thanks to the work done by the <code class="docutils literal notranslate"><span class="pre">class</span></code> command, the resulting two <code class="docutils literal notranslate"><span class="pre">Dia₁</span> <span class="pre">α</span></code> instances are definitionnaly equal. In particular a diamond having a <code class="docutils literal notranslate"><span class="pre">Prop</span></code>-valued class at the bottom cannot be bad since any too proofs of the same statement are definitionnaly equal.</p> <p>But the diamond we created with modules is definitely bad. The offending piece is the <code class="docutils literal notranslate"><span class="pre">smul</span></code> field which is data, not a proof, and we have two constructions that are not definitionnaly equal. The robust way of fixing this issue is to make sure that going from a rich structure to a poor structure is always done by forgetting data, not by defining data. This well-known pattern as been named “forgetful inheritance” and extensively discussed in <a class="reference external" href="https://inria.hal.science/hal-02463336">https://inria.hal.science/hal-02463336</a>.</p> <p>In our concrete case, we can modify the definition of <code class="docutils literal notranslate"><span class="pre">AddMonoid₃</span></code> to include a <code class="docutils literal notranslate"><span class="pre">nsmul</span></code> data field and some <code class="docutils literal notranslate"><span class="pre">Prop</span></code>-valued fields ensuring this operation is provably the one we constructed above. Those fields are given default values using <code class="docutils literal notranslate"><span class="pre">:=</span></code> after their type in the definition below. Thanks to these default values, most instances would be constructed exactly as with our previous definitions. But in the special case of <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> we will be able to provide specific values.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">AddMonoid₄</span> <span class="o">(</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="kd">extends</span> <span class="n">AddSemigroup₃</span> <span class="n">M</span><span class="o">,</span> <span class="n">AddZeroClass</span> <span class="n">M</span> <span class="n">where</span> <span class="sd">/-- Multiplication by a natural number. -/</span> <span class="n">nsmul</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">M</span> <span class="o">:=</span> <span class="n">nsmul₁</span> <span class="sd">/-- Multiplication by `(0 : ℕ)` gives `0`. -/</span> <span class="n">nsmul_zero</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">nsmul</span> <span class="mi">0</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span><span class="bp">;</span> <span class="n">rfl</span> <span class="sd">/-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/</span> <span class="n">nsmul_succ</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">x</span><span class="o">),</span> <span class="n">nsmul</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">nsmul</span> <span class="n">n</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intros</span><span class="bp">;</span> <span class="n">rfl</span> <span class="kd">instance</span> <span class="n">mySMul</span> <span class="o">{</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">}</span> <span class="o">[</span><span class="n">AddMonoid₄</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="n">SMul</span> <span class="n">ℕ</span> <span class="n">M</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">AddMonoid₄.nsmul</span><span class="o">⟩</span> </pre></div> </div> <p>Let us check we can still construct a product monoid instance without providing the <code class="docutils literal notranslate"><span class="pre">nsmul</span></code> related fields.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">(</span><span class="n">M</span> <span class="n">N</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">AddMonoid₄</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">AddMonoid₄</span> <span class="n">N</span><span class="o">]</span> <span class="o">:</span> <span class="n">AddMonoid₄</span> <span class="o">(</span><span class="n">M</span> <span class="bp">×</span> <span class="n">N</span><span class="o">)</span> <span class="n">where</span> <span class="n">add</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">p</span> <span class="n">q</span> <span class="bp">↦</span> <span class="o">(</span><span class="n">p.1</span> <span class="bp">+</span> <span class="n">q.1</span><span class="o">,</span> <span class="n">p.2</span> <span class="bp">+</span> <span class="n">q.2</span><span class="o">)</span> <span class="n">add_assoc₃</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="bp">↦</span> <span class="kd">by</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">apply</span> <span class="n">add_assoc₃</span> <span class="n">zero</span> <span class="o">:=</span> <span class="o">(</span><span class="mi">0</span><span class="o">,</span> <span class="mi">0</span><span class="o">)</span> <span class="n">zero_add</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">a</span> <span class="bp">↦</span> <span class="kd">by</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">apply</span> <span class="n">zero_add</span> <span class="n">add_zero</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">a</span> <span class="bp">↦</span> <span class="kd">by</span> <span class="n">ext</span> <span class="bp"><;></span> <span class="n">apply</span> <span class="n">add_zero</span> </pre></div> </div> <p>And now let us handle the special case of <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> where we want to build <code class="docutils literal notranslate"><span class="pre">nsmul</span></code> using the coercion of <code class="docutils literal notranslate"><span class="pre">ℕ</span></code> to <code class="docutils literal notranslate"><span class="pre">ℤ</span></code> and the multiplication on <code class="docutils literal notranslate"><span class="pre">ℤ</span></code>. Note in particular how the proof fields contain more work than in the default value above.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">AddMonoid₄</span> <span class="n">ℤ</span> <span class="n">where</span> <span class="n">add</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">·</span> <span class="bp">+</span> <span class="bp">·</span><span class="o">)</span> <span class="n">add_assoc₃</span> <span class="o">:=</span> <span class="n">Int.add_assoc</span> <span class="n">zero</span> <span class="o">:=</span> <span class="mi">0</span> <span class="n">zero_add</span> <span class="o">:=</span> <span class="n">Int.zero_add</span> <span class="n">add_zero</span> <span class="o">:=</span> <span class="n">Int.add_zero</span> <span class="n">nsmul</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="n">m</span> <span class="bp">↦</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="bp">*</span> <span class="n">m</span> <span class="n">nsmul_zero</span> <span class="o">:=</span> <span class="n">Int.zero_mul</span> <span class="n">nsmul_succ</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="n">m</span> <span class="bp">↦</span> <span class="k">show</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="bp">*</span> <span class="n">m</span> <span class="bp">=</span> <span class="n">m</span> <span class="bp">+</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">m</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Int.add_mul</span><span class="o">,</span> <span class="n">Int.add_comm</span><span class="o">,</span> <span class="n">Int.one_mul</span><span class="o">]</span> </pre></div> </div> <p>Let us check we solved our issue. Because Lean already has a definition of scalar multiplication of a natural number and an integer, and we want to make sure our instance is used, we won’t use the <code class="docutils literal notranslate"><span class="pre">•</span></code> notation but call <code class="docutils literal notranslate"><span class="pre">SMul.mul</span></code> and explicitly provide our instance defined above.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">ℤ</span><span class="o">)</span> <span class="o">:</span> <span class="n">SMul.smul</span> <span class="o">(</span><span class="n">self</span> <span class="o">:=</span> <span class="n">mySMul</span><span class="o">)</span> <span class="n">n</span> <span class="n">m</span> <span class="bp">=</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">m</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>This story then continues with incorporating a <code class="docutils literal notranslate"><span class="pre">zsmul</span></code> field into the definition of groups and similar tricks. You are now ready to read the definition of monoids, groups, rings and modules in mathlib. There are more complicated than what we have seen here, because they are part of a huge hierarchy, but all principles have been explained above.</p> <p>As an exercise, you can come back to the order relation hierarchy you built above and try to incorportate a type class <code class="docutils literal notranslate"><span class="pre">LT₁</span></code> carrying the Less-Than notation <code class="docutils literal notranslate"><span class="pre"><₁</span></code> and make sure that every preorder comes with a <code class="docutils literal notranslate"><span class="pre"><₁</span></code> which has a default value built from <code class="docutils literal notranslate"><span class="pre">≤₁</span></code> and a <code class="docutils literal notranslate"><span class="pre">Prop</span></code>-valued field asserting the natural relation between those two comparison operators. TEXT. -/</p> </section> <section id="morphisms"> <span id="section-hierarchies-morphisms"></span><h2><span class="section-number">7.2. </span>Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this heading"></a></h2> <p>So far in this chapter, we discussed how to create a hierarchy of mathematical structures. But defining structures is not really completed until we have morphisms. There are two main approaches here. The most obvious one is to define a predicate on functions.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">isMonoidHom₁</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">G</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">H</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">G</span> <span class="bp">→</span> <span class="n">H</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="n">f</span> <span class="mi">1</span> <span class="bp">=</span> <span class="mi">1</span> <span class="bp">∧</span> <span class="bp">∀</span> <span class="n">g</span> <span class="n">g'</span><span class="o">,</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="bp">*</span> <span class="n">g'</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">g</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">g'</span> </pre></div> </div> <p>In this definition, it is a bit unpleasant to use a conjunction. In particular users will need to remember the ordering we chose when they want to access the two conditions. So we could use a structure instead.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">structure</span> <span class="n">isMonoidHom₂</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">G</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">H</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">G</span> <span class="bp">→</span> <span class="n">H</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="n">where</span> <span class="n">map_one</span> <span class="o">:</span> <span class="n">f</span> <span class="mi">1</span> <span class="bp">=</span> <span class="mi">1</span> <span class="n">map_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">g</span> <span class="n">g'</span><span class="o">,</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="bp">*</span> <span class="n">g'</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">g</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">g'</span> </pre></div> </div> <p>Once we are here, it is even tempting to make it a class and use the type class instance resolution procedure to automatically infer <code class="docutils literal notranslate"><span class="pre">isMonoidHom₂</span></code> for complicated functions out of instances for simpler functions. For instance a composition of monoid morphisms is a monoid morphism and this seems like a useful instance. However such an instance would be very tricky for the resolution procedure since it would need to hunt down <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">∘</span> <span class="pre">f</span></code> everywhere. Seeing it failing in <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">(f</span> <span class="pre">x)</span></code> would be very frustrating. More generally one must always keep in mind that recognizing which function is applied in a given expression is a very difficult problem, called the “higher-order unification problem”. So Mathlib does not use this class approach.</p> <p>A more fundamental question is whether we use predicates as above (using either a <code class="docutils literal notranslate"><span class="pre">def</span></code> or a <code class="docutils literal notranslate"><span class="pre">structure</span></code>) or use structures bundling a function and predicates. This is partly a psychological issue. It is extremely rare to consider a function between monoids that is not a morphism. It really feels like “monoid morphism” is not an adjective you can assign to a bare function, it is a noun. On the other hand one can argue that a continuous function between topological spaces is really a function that happens to be continuous. This is one reason why Mathlib has a <code class="docutils literal notranslate"><span class="pre">Continuous</span></code> predicate. For instance you can write:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="o">(</span><span class="n">id</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:=</span> <span class="n">continuous_id</span> </pre></div> </div> <p>We still have bundles continuous functions, which are convenient for instance to put a topology on a space of continuous functions, but they are not the primary tool to work with continuity.</p> <p>By constrast, morphisms between monoids (or other algebraic structures) are bundled as in:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">MonoidHom₁</span> <span class="o">(</span><span class="n">G</span> <span class="n">H</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">G</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">H</span><span class="o">]</span> <span class="n">where</span> <span class="n">toFun</span> <span class="o">:</span> <span class="n">G</span> <span class="bp">→</span> <span class="n">H</span> <span class="n">map_one</span> <span class="o">:</span> <span class="n">toFun</span> <span class="mi">1</span> <span class="bp">=</span> <span class="mi">1</span> <span class="n">map_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">g</span> <span class="n">g'</span><span class="o">,</span> <span class="n">toFun</span> <span class="o">(</span><span class="n">g</span> <span class="bp">*</span> <span class="n">g'</span><span class="o">)</span> <span class="bp">=</span> <span class="n">toFun</span> <span class="n">g</span> <span class="bp">*</span> <span class="n">toFun</span> <span class="n">g'</span> </pre></div> </div> <p>Of course we don’t want to type <code class="docutils literal notranslate"><span class="pre">toFun</span></code> everywhere so we register a coercion using the <code class="docutils literal notranslate"><span class="pre">CoeFun</span></code> type class. Its first argument is the type we want to coerce to a function. The second argument describes the target function type. In our case it is always <code class="docutils literal notranslate"><span class="pre">G</span> <span class="pre">→</span> <span class="pre">H</span></code> for every <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">MonoidHom₁</span> <span class="pre">G</span> <span class="pre">H</span></code>. We also tag <code class="docutils literal notranslate"><span class="pre">MonoidHom₁.toFun</span></code> with the <code class="docutils literal notranslate"><span class="pre">coe</span></code> attribute to make sure it is displayed almost invisibly in the tactic state, simply by a <code class="docutils literal notranslate"><span class="pre">↑</span></code> prefix.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">G</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">H</span><span class="o">]</span> <span class="o">:</span> <span class="n">CoeFun</span> <span class="o">(</span><span class="n">MonoidHom₁</span> <span class="n">G</span> <span class="n">H</span><span class="o">)</span> <span class="o">(</span><span class="k">fun</span> <span class="n">_</span> <span class="bp">↦</span> <span class="n">G</span> <span class="bp">→</span> <span class="n">H</span><span class="o">)</span> <span class="n">where</span> <span class="n">coe</span> <span class="o">:=</span> <span class="n">MonoidHom₁.toFun</span> <span class="kn">attribute</span> <span class="o">[</span><span class="n">coe</span><span class="o">]</span> <span class="n">MonoidHom₁.toFun</span> </pre></div> </div> <p>Let us check we can indeed apply a bundled monoid morphism to an element.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">G</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">H</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">MonoidHom₁</span> <span class="n">G</span> <span class="n">H</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="mi">1</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="n">f.map_one</span> </pre></div> </div> <p>We can do the same with other kind of morphisms until we reach ring morphisms.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">AddMonoidHom₁</span> <span class="o">(</span><span class="n">G</span> <span class="n">H</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">AddMonoid</span> <span class="n">G</span><span class="o">]</span> <span class="o">[</span><span class="n">AddMonoid</span> <span class="n">H</span><span class="o">]</span> <span class="n">where</span> <span class="n">toFun</span> <span class="o">:</span> <span class="n">G</span> <span class="bp">→</span> <span class="n">H</span> <span class="n">map_zero</span> <span class="o">:</span> <span class="n">toFun</span> <span class="mi">0</span> <span class="bp">=</span> <span class="mi">0</span> <span class="n">map_add</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">g</span> <span class="n">g'</span><span class="o">,</span> <span class="n">toFun</span> <span class="o">(</span><span class="n">g</span> <span class="bp">+</span> <span class="n">g'</span><span class="o">)</span> <span class="bp">=</span> <span class="n">toFun</span> <span class="n">g</span> <span class="bp">+</span> <span class="n">toFun</span> <span class="n">g'</span> <span class="kd">instance</span> <span class="o">[</span><span class="n">AddMonoid</span> <span class="n">G</span><span class="o">]</span> <span class="o">[</span><span class="n">AddMonoid</span> <span class="n">H</span><span class="o">]</span> <span class="o">:</span> <span class="n">CoeFun</span> <span class="o">(</span><span class="n">AddMonoidHom₁</span> <span class="n">G</span> <span class="n">H</span><span class="o">)</span> <span class="o">(</span><span class="k">fun</span> <span class="n">_</span> <span class="bp">↦</span> <span class="n">G</span> <span class="bp">→</span> <span class="n">H</span><span class="o">)</span> <span class="n">where</span> <span class="n">coe</span> <span class="o">:=</span> <span class="n">AddMonoidHom₁.toFun</span> <span class="kn">attribute</span> <span class="o">[</span><span class="n">coe</span><span class="o">]</span> <span class="n">AddMonoidHom₁.toFun</span> <span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">RingHom₁</span> <span class="o">(</span><span class="n">R</span> <span class="n">S</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Ring</span> <span class="n">R</span><span class="o">]</span> <span class="o">[</span><span class="n">Ring</span> <span class="n">S</span><span class="o">]</span> <span class="kd">extends</span> <span class="n">MonoidHom₁</span> <span class="n">R</span> <span class="n">S</span><span class="o">,</span> <span class="n">AddMonoidHom₁</span> <span class="n">R</span> <span class="n">S</span> </pre></div> </div> <p>There are a couple of issues about this approach. A minor one is we don’t quite know where to put the <code class="docutils literal notranslate"><span class="pre">coe</span></code> attribute since the <code class="docutils literal notranslate"><span class="pre">RingHom₁.toFun</span></code> does not exist, the relevant function is <code class="docutils literal notranslate"><span class="pre">MonoidHom₁.toFun</span> <span class="pre">∘</span> <span class="pre">RingHom₁.toMonoidHom₁</span></code> which is not a declaration that can be tagged with an attribute (but we could still define a <code class="docutils literal notranslate"><span class="pre">CoeFun</span>  <span class="pre">(RingHom₁</span> <span class="pre">R</span> <span class="pre">S)</span> <span class="pre">(fun</span> <span class="pre">_</span> <span class="pre">↦</span> <span class="pre">R</span> <span class="pre">→</span> <span class="pre">S)</span></code> instance). A much more important one is that lemmas about monoid morphisms won’t directly apply to ring morphisms. This leaves the alternative of either juggling with <code class="docutils literal notranslate"><span class="pre">RingHom₁.toMonoidHom₁</span></code> each time we want to apply a monoid morphism lemma or restate every such lemmas for ring morphisms. Neither option is appealing so Mathlib uses a new hierarchy trick here. The idea is to define a type class for objects that are at least monoid morphisms, instantiate that class with both monoid morphisms and ring morphisms and use it to state every lemma. In the definition below, <code class="docutils literal notranslate"><span class="pre">F</span></code> could be <code class="docutils literal notranslate"><span class="pre">MonoidHom₁</span> <span class="pre">M</span> <span class="pre">N</span></code>, or <code class="docutils literal notranslate"><span class="pre">RingHom₁</span> <span class="pre">M</span> <span class="pre">N</span></code> if <code class="docutils literal notranslate"><span class="pre">M</span></code> and <code class="docutils literal notranslate"><span class="pre">N</span></code> have a ring structure.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">MonoidHomClass₁</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">(</span><span class="n">M</span> <span class="n">N</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="n">where</span> <span class="n">toFun</span> <span class="o">:</span> <span class="n">F</span> <span class="bp">→</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">N</span> <span class="n">map_one</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">f</span> <span class="o">:</span> <span class="n">F</span><span class="o">,</span> <span class="n">toFun</span> <span class="n">f</span> <span class="mi">1</span> <span class="bp">=</span> <span class="mi">1</span> <span class="n">map_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">f</span> <span class="n">g</span> <span class="n">g'</span><span class="o">,</span> <span class="n">toFun</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="bp">*</span> <span class="n">g'</span><span class="o">)</span> <span class="bp">=</span> <span class="n">toFun</span> <span class="n">f</span> <span class="n">g</span> <span class="bp">*</span> <span class="n">toFun</span> <span class="n">f</span> <span class="n">g'</span> </pre></div> </div> <p>However there is a problem with the above implementation. We haven’t registered a coercion to function instance yet. Let us try to do it now.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">badInst</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="o">[</span><span class="n">MonoidHomClass₁</span> <span class="n">F</span> <span class="n">M</span> <span class="n">N</span><span class="o">]</span> <span class="o">:</span> <span class="n">CoeFun</span> <span class="n">F</span> <span class="o">(</span><span class="k">fun</span> <span class="n">_</span> <span class="bp">↦</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">N</span><span class="o">)</span> <span class="n">where</span> <span class="n">coe</span> <span class="o">:=</span> <span class="n">MonoidHomClass₁.toFun</span> </pre></div> </div> <p>Making the an instance would be bad. When faced with something like <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> where the type of <code class="docutils literal notranslate"><span class="pre">f</span></code> is not a function type, Lean will try to find a <code class="docutils literal notranslate"><span class="pre">CoeFun</span></code> instance to coerce <code class="docutils literal notranslate"><span class="pre">f</span></code> into a function. The above function has type: <code class="docutils literal notranslate"><span class="pre">{M</span> <span class="pre">N</span> <span class="pre">F</span> <span class="pre">:</span> <span class="pre">Type}</span> <span class="pre">→</span> <span class="pre">[Monoid</span> <span class="pre">M]</span> <span class="pre">→</span> <span class="pre">[Monoid</span> <span class="pre">N]</span> <span class="pre">→</span> <span class="pre">[MonoidHomClass₁</span> <span class="pre">F</span> <span class="pre">M</span> <span class="pre">N]</span> <span class="pre">→</span> <span class="pre">CoeFun</span> <span class="pre">F</span> <span class="pre">(fun</span> <span class="pre">x</span> <span class="pre">↦</span> <span class="pre">M</span> <span class="pre">→</span> <span class="pre">N)</span></code> so, when it trying to apply it, it wouldn’t be a priori clear to Lean in which order the unknown types <code class="docutils literal notranslate"><span class="pre">M</span></code>, <code class="docutils literal notranslate"><span class="pre">N</span></code> and <code class="docutils literal notranslate"><span class="pre">F</span></code> should be inferred. This is a kind of bad instance that is slightly different from the one we saw already, but it boils down to the same issue: without knowing <code class="docutils literal notranslate"><span class="pre">M</span></code>, Lean would have to search for a monoid instance on an unknown type, hence hopelessly try <em>every</em> monoid instance in the database. If you are curious to see the effect of such an instance you can type <code class="docutils literal notranslate"><span class="pre">set_option</span> <span class="pre">synthInstance.checkSynthOrder</span> <span class="pre">false</span> <span class="pre">in</span></code> on top of the above declaration, replace <code class="docutils literal notranslate"><span class="pre">def</span> <span class="pre">badInst</span></code> with <code class="docutils literal notranslate"><span class="pre">instance</span></code>, and look for random failures in this file.</p> <p>Here the solution is easy, we need to tell Lean to first search what is <code class="docutils literal notranslate"><span class="pre">F</span></code> and then deduce <code class="docutils literal notranslate"><span class="pre">M</span></code> and <code class="docutils literal notranslate"><span class="pre">N</span></code>. This is done using the <code class="docutils literal notranslate"><span class="pre">outParam</span></code> function. This function is defined as the identity function, but is still recognized by the type class machinery and triggers the desired behavior. Hence we can retry defining our class, paying attention to the <code class="docutils literal notranslate"><span class="pre">outParam</span></code> function:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">MonoidHomClass₂</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">(</span><span class="n">M</span> <span class="n">N</span> <span class="o">:</span> <span class="n">outParam</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="n">where</span> <span class="n">toFun</span> <span class="o">:</span> <span class="n">F</span> <span class="bp">→</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">N</span> <span class="n">map_one</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">f</span> <span class="o">:</span> <span class="n">F</span><span class="o">,</span> <span class="n">toFun</span> <span class="n">f</span> <span class="mi">1</span> <span class="bp">=</span> <span class="mi">1</span> <span class="n">map_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">f</span> <span class="n">g</span> <span class="n">g'</span><span class="o">,</span> <span class="n">toFun</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="bp">*</span> <span class="n">g'</span><span class="o">)</span> <span class="bp">=</span> <span class="n">toFun</span> <span class="n">f</span> <span class="n">g</span> <span class="bp">*</span> <span class="n">toFun</span> <span class="n">f</span> <span class="n">g'</span> <span class="kd">instance</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="o">[</span><span class="n">MonoidHomClass₂</span> <span class="n">F</span> <span class="n">M</span> <span class="n">N</span><span class="o">]</span> <span class="o">:</span> <span class="n">CoeFun</span> <span class="n">F</span> <span class="o">(</span><span class="k">fun</span> <span class="n">_</span> <span class="bp">↦</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">N</span><span class="o">)</span> <span class="n">where</span> <span class="n">coe</span> <span class="o">:=</span> <span class="n">MonoidHomClass₂.toFun</span> <span class="kn">attribute</span> <span class="o">[</span><span class="n">coe</span><span class="o">]</span> <span class="n">MonoidHomClass₂.toFun</span> </pre></div> </div> <p>Now we can proceed with our plan to instantiate this class.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">(</span><span class="n">M</span> <span class="n">N</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="o">:</span> <span class="n">MonoidHomClass₂</span> <span class="o">(</span><span class="n">MonoidHom₁</span> <span class="n">M</span> <span class="n">N</span><span class="o">)</span> <span class="n">M</span> <span class="n">N</span> <span class="n">where</span> <span class="n">toFun</span> <span class="o">:=</span> <span class="n">MonoidHom₁.toFun</span> <span class="n">map_one</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">f</span> <span class="bp">↦</span> <span class="n">f.map_one</span> <span class="n">map_mul</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">f</span> <span class="bp">↦</span> <span class="n">f.map_mul</span> <span class="kd">instance</span> <span class="o">(</span><span class="n">R</span> <span class="n">S</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Ring</span> <span class="n">R</span><span class="o">]</span> <span class="o">[</span><span class="n">Ring</span> <span class="n">S</span><span class="o">]</span> <span class="o">:</span> <span class="n">MonoidHomClass₂</span> <span class="o">(</span><span class="n">RingHom₁</span> <span class="n">R</span> <span class="n">S</span><span class="o">)</span> <span class="n">R</span> <span class="n">S</span> <span class="n">where</span> <span class="n">toFun</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">f</span> <span class="bp">↦</span> <span class="n">f.toMonoidHom₁.toFun</span> <span class="n">map_one</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">f</span> <span class="bp">↦</span> <span class="n">f.toMonoidHom₁.map_one</span> <span class="n">map_mul</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">f</span> <span class="bp">↦</span> <span class="n">f.toMonoidHom₁.map_mul</span> </pre></div> </div> <p>As promised every lemma we prove about <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">F</span></code> assuming an instance of <code class="docutils literal notranslate"><span class="pre">MonoidHomClass₁</span> <span class="pre">F</span></code> will apply both to monoid morphims and ring morphisms. Let us see an example lemma and check it applies to both situations.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">lemma</span> <span class="n">map_inv_of_inv</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="o">[</span><span class="n">MonoidHomClass₂</span> <span class="n">F</span> <span class="n">M</span> <span class="n">N</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">F</span><span class="o">)</span> <span class="o">{</span><span class="n">m</span> <span class="n">m'</span> <span class="o">:</span> <span class="n">M</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m</span><span class="bp">*</span><span class="n">m'</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">m'</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="bp">←</span> <span class="n">MonoidHomClass₂.map_mul</span><span class="o">,</span> <span class="n">h</span><span class="o">,</span> <span class="n">MonoidHomClass₂.map_one</span><span class="o">]</span> <span class="kd">example</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">MonoidHom₁</span> <span class="n">M</span> <span class="n">N</span><span class="o">)</span> <span class="o">{</span><span class="n">m</span> <span class="n">m'</span> <span class="o">:</span> <span class="n">M</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">m</span><span class="bp">*</span><span class="n">m'</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">m'</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="n">map_inv_of_inv</span> <span class="n">f</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">[</span><span class="n">Ring</span> <span class="n">R</span><span class="o">]</span> <span class="o">[</span><span class="n">Ring</span> <span class="n">S</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">RingHom₁</span> <span class="n">R</span> <span class="n">S</span><span class="o">)</span> <span class="o">{</span><span class="n">r</span> <span class="n">r'</span> <span class="o">:</span> <span class="n">R</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">r</span><span class="bp">*</span><span class="n">r'</span> <span class="bp">=</span> <span class="mi">1</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="n">r</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">r'</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="n">map_inv_of_inv</span> <span class="n">f</span> <span class="n">h</span> </pre></div> </div> <p>At first sight, it may look like we got back to our old bad idea of making <code class="docutils literal notranslate"><span class="pre">MonoidHom₁</span></code> a class. But we haven’t. Everything is shifted one level of abstraction up. The type class resolution procedure won’t be looking for functions, it will be looking for either <code class="docutils literal notranslate"><span class="pre">MonoidHom₁</span></code> or <code class="docutils literal notranslate"><span class="pre">RingHom₁</span></code>.</p> <p>One remaining issue with our approach is the presence of repeatitive code around the <code class="docutils literal notranslate"><span class="pre">toFun</span></code> field and the corresponding <code class="docutils literal notranslate"><span class="pre">CoeFun</span></code> instance and <code class="docutils literal notranslate"><span class="pre">coe</span></code> attribute. It would also be better to record that this pattern is used only for function with extra properties, meaning that the coercion to functions should be injective. So Mathlib adds one more layer of abstraction with the base class <code class="docutils literal notranslate"><span class="pre">FunLike</span></code>. Let us redefine our <code class="docutils literal notranslate"><span class="pre">MonoidHomClass</span></code> on top of this base layer.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">MonoidHomClass₃</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">(</span><span class="n">M</span> <span class="n">N</span> <span class="o">:</span> <span class="n">outParam</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="kd">extends</span> <span class="n">FunLike</span> <span class="n">F</span> <span class="n">M</span> <span class="o">(</span><span class="k">fun</span> <span class="n">_</span> <span class="bp">↦</span> <span class="n">N</span><span class="o">)</span> <span class="n">where</span> <span class="n">map_one</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">f</span> <span class="o">:</span> <span class="n">F</span><span class="o">,</span> <span class="n">f</span> <span class="mi">1</span> <span class="bp">=</span> <span class="mi">1</span> <span class="n">map_mul</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">F</span><span class="o">)</span> <span class="n">g</span> <span class="n">g'</span><span class="o">,</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="bp">*</span> <span class="n">g'</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">g</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">g'</span> <span class="kd">instance</span> <span class="o">(</span><span class="n">M</span> <span class="n">N</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="o">:</span> <span class="n">MonoidHomClass₃</span> <span class="o">(</span><span class="n">MonoidHom₁</span> <span class="n">M</span> <span class="n">N</span><span class="o">)</span> <span class="n">M</span> <span class="n">N</span> <span class="n">where</span> <span class="n">coe</span> <span class="o">:=</span> <span class="n">MonoidHom₁.toFun</span> <span class="n">coe_injective'</span> <span class="o">:=</span> <span class="n">MonoidHom₁.ext</span> <span class="n">map_one</span> <span class="o">:=</span> <span class="n">MonoidHom₁.map_one</span> <span class="n">map_mul</span> <span class="o">:=</span> <span class="n">MonoidHom₁.map_mul</span> </pre></div> </div> <p>Of course the hierarchy of morphisms does not stop here. We could go on and define a class <code class="docutils literal notranslate"><span class="pre">RingHomClass₃</span></code> extending <code class="docutils literal notranslate"><span class="pre">MonoidHomClass₃</span></code> and instantiate it on <code class="docutils literal notranslate"><span class="pre">RingHom</span></code> and then later on <code class="docutils literal notranslate"><span class="pre">AlgebraHom</span></code> (algebras are rings with some extra structure). But we’ve covered the main formalization ideas used in Mathlib for morphisms and you should be ready to understand how morphisms are defined in Mathlib.</p> <p>As an exercise, you should try to define your class of bundled order-preserving function between ordered types, and then order preserving monoid morphisms. This is for training purposes only. Like continuous functions, order preserving functions are primarily unbundled in Mathlib where they are defined by the <code class="docutils literal notranslate"><span class="pre">Monotone</span></code> predicate. Of course you need to complete the class definitions below.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">OrderPresHom</span> <span class="o">(</span><span class="n">α</span> <span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">LE</span> <span class="n">α</span><span class="o">]</span> <span class="o">[</span><span class="n">LE</span> <span class="n">β</span><span class="o">]</span> <span class="n">where</span> <span class="n">toFun</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span> <span class="n">le_of_le</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="n">a'</span><span class="o">,</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">a'</span> <span class="bp">→</span> <span class="n">toFun</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">toFun</span> <span class="n">a'</span> <span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">OrderPresMonoidHom</span> <span class="o">(</span><span class="n">M</span> <span class="n">N</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">LE</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">N</span><span class="o">]</span> <span class="o">[</span><span class="n">LE</span> <span class="n">N</span><span class="o">]</span> <span class="kd">extends</span> <span class="n">MonoidHom₁</span> <span class="n">M</span> <span class="n">N</span><span class="o">,</span> <span class="n">OrderPresHom</span> <span class="n">M</span> <span class="n">N</span> <span class="kd">class</span> <span class="n">OrderPresHomClass</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">(</span><span class="n">α</span> <span class="n">β</span> <span class="o">:</span> <span class="n">outParam</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">LE</span> <span class="n">α</span><span class="o">]</span> <span class="o">[</span><span class="n">LE</span> <span class="n">β</span><span class="o">]</span> <span class="kd">instance</span> <span class="o">(</span><span class="n">α</span> <span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">LE</span> <span class="n">α</span><span class="o">]</span> <span class="o">[</span><span class="n">LE</span> <span class="n">β</span><span class="o">]</span> <span class="o">:</span> <span class="n">OrderPresHomClass</span> <span class="o">(</span><span class="n">OrderPresHom</span> <span class="n">α</span> <span class="n">β</span><span class="o">)</span> <span class="n">α</span> <span class="n">β</span> <span class="n">where</span> <span class="kd">instance</span> <span class="o">(</span><span class="n">α</span> <span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">LE</span> <span class="n">α</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">α</span><span class="o">]</span> <span class="o">[</span><span class="n">LE</span> <span class="n">β</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">β</span><span class="o">]</span> <span class="o">:</span> <span class="n">OrderPresHomClass</span> <span class="o">(</span><span class="n">OrderPresMonoidHom</span> <span class="n">α</span> <span class="n">β</span><span class="o">)</span> <span class="n">α</span> <span class="n">β</span> <span class="n">where</span> <span class="kd">instance</span> <span class="o">(</span><span class="n">α</span> <span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">LE</span> <span class="n">α</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">α</span><span class="o">]</span> <span class="o">[</span><span class="n">LE</span> <span class="n">β</span><span class="o">]</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">β</span><span class="o">]</span> <span class="o">:</span> <span class="n">MonoidHomClass₃</span> <span class="o">(</span><span class="n">OrderPresMonoidHom</span> <span class="n">α</span> <span class="n">β</span><span class="o">)</span> <span class="n">α</span> <span class="n">β</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="sub-objects"> <span id="section-hierarchies-subobjects"></span><h2><span class="section-number">7.3. </span>Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this heading"></a></h2> <p>After defining some algebraic structure and its morphisms, the next step is to consider sets that inherit this algebraic structure, for instance subgroups or subrings. This largely overlaps our previous topic. Indeed a set in <code class="docutils literal notranslate"><span class="pre">X</span></code> is implemented as a function from <code class="docutils literal notranslate"><span class="pre">X</span></code> to <code class="docutils literal notranslate"><span class="pre">Prop</span></code> so sub-objects are function satisfying a certain predicate. Hence we can reuse of lot of the ideas that led to the <code class="docutils literal notranslate"><span class="pre">FunLike</span></code> class and its descendants. We won’t reuse <code class="docutils literal notranslate"><span class="pre">FunLike</span></code> itself because this would break the abstraction barrier from <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">X</span></code> to <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">→</span> <span class="pre">Prop</span></code>. Instead there is a <code class="docutils literal notranslate"><span class="pre">SetLike</span></code> class. Instead of wrapping an injection into a function type, that class wraps an injection into a <code class="docutils literal notranslate"><span class="pre">Set</span></code> type and defines the corresponding coercion and <code class="docutils literal notranslate"><span class="pre">Membership</span></code> instance.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">@[ext]</span> <span class="kd">structure</span> <span class="n">Submonoid₁</span> <span class="o">(</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="n">where</span> <span class="sd">/-- The carrier of a submonoid. -/</span> <span class="n">carrier</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">M</span> <span class="sd">/-- The product of two elements of a submonoid belongs to the submonoid. -/</span> <span class="n">mul_mem</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span><span class="o">}</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">carrier</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp">∈</span> <span class="n">carrier</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">∈</span> <span class="n">carrier</span> <span class="sd">/-- The unit element belongs to the submonoid. -/</span> <span class="n">one_mem</span> <span class="o">:</span> <span class="mi">1</span> <span class="bp">∈</span> <span class="n">carrier</span> <span class="sd">/-- Submonoids in `M` can be seen as sets in `M`. -/</span> <span class="kd">instance</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="n">SetLike</span> <span class="o">(</span><span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="n">M</span> <span class="n">where</span> <span class="n">coe</span> <span class="o">:=</span> <span class="n">Submonoid₁.carrier</span> <span class="n">coe_injective'</span> <span class="o">:=</span> <span class="n">Submonoid₁.ext</span> </pre></div> </div> <p>Equipped with the above <code class="docutils literal notranslate"><span class="pre">SetLike</span></code> instance, we can already state naturally that a submonoid <code class="docutils literal notranslate"><span class="pre">N</span></code> contains <code class="docutils literal notranslate"><span class="pre">1</span></code> without using <code class="docutils literal notranslate"><span class="pre">N.carrier</span></code>. We can also silently treat <code class="docutils literal notranslate"><span class="pre">N</span></code> as a set in <code class="docutils literal notranslate"><span class="pre">M</span></code> as take its direct image under a map.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="mi">1</span> <span class="bp">∈</span> <span class="n">N</span> <span class="o">:=</span> <span class="n">N.one_mem</span> <span class="kd">example</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="o">(</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">α</span><span class="o">)</span> <span class="o">:=</span> <span class="n">f</span> <span class="bp">''</span> <span class="n">N</span> </pre></div> </div> <p>We also have a coercion to <code class="docutils literal notranslate"><span class="pre">Type</span></code> which uses <code class="docutils literal notranslate"><span class="pre">Subtype</span></code> so, given a submonoid <code class="docutils literal notranslate"><span class="pre">N</span></code> we can write a parameter <code class="docutils literal notranslate"><span class="pre">(x</span> <span class="pre">:</span> <span class="pre">N)</span></code> which can be coerced to an element of <code class="docutils literal notranslate"><span class="pre">M</span></code> belonging to <code class="docutils literal notranslate"><span class="pre">N</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">N</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">M</span><span class="o">)</span> <span class="bp">∈</span> <span class="n">N</span> <span class="o">:=</span> <span class="n">x.property</span> </pre></div> </div> <p>Using this coercion to <code class="docutils literal notranslate"><span class="pre">Type</span></code> we can also tackle the task of equipping a submonoid with a monoid structure. We will use the coercion from the type associated to <code class="docutils literal notranslate"><span class="pre">N</span></code> as above, and the lemma <code class="docutils literal notranslate"><span class="pre">SetCoe.ext</span></code> asserting this coercion is injective. Both are provided by the <code class="docutils literal notranslate"><span class="pre">SetLike</span></code> instance.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="n">SubMonoid₁Monoid</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monoid</span> <span class="n">N</span> <span class="n">where</span> <span class="n">mul</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">↦</span> <span class="o">⟨</span><span class="n">x</span><span class="bp">*</span><span class="n">y</span><span class="o">,</span> <span class="n">N.mul_mem</span> <span class="n">x.property</span> <span class="n">y.property</span><span class="o">⟩</span> <span class="n">mul_assoc</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="n">z</span> <span class="bp">↦</span> <span class="n">SetCoe.ext</span> <span class="o">(</span><span class="n">mul_assoc</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">M</span><span class="o">)</span> <span class="n">y</span> <span class="n">z</span><span class="o">)</span> <span class="n">one</span> <span class="o">:=</span> <span class="o">⟨</span><span class="mi">1</span><span class="o">,</span> <span class="n">N.one_mem</span><span class="o">⟩</span> <span class="n">one_mul</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">↦</span> <span class="n">SetCoe.ext</span> <span class="o">(</span><span class="n">one_mul</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">M</span><span class="o">))</span> <span class="n">mul_one</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">↦</span> <span class="n">SetCoe.ext</span> <span class="o">(</span><span class="n">mul_one</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">M</span><span class="o">))</span> </pre></div> </div> <p>Note that, in the above instance, instead of using the coercion to <code class="docutils literal notranslate"><span class="pre">M</span></code> and calling the <code class="docutils literal notranslate"><span class="pre">property</span></code> field, we could have used destructuring binders as follows.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monoid</span> <span class="n">N</span> <span class="n">where</span> <span class="n">mul</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">hx</span><span class="o">⟩</span> <span class="o">⟨</span><span class="n">y</span><span class="o">,</span> <span class="n">hy</span><span class="o">⟩</span> <span class="bp">↦</span> <span class="o">⟨</span><span class="n">x</span><span class="bp">*</span><span class="n">y</span><span class="o">,</span> <span class="n">N.mul_mem</span> <span class="n">hx</span> <span class="n">hy</span><span class="o">⟩</span> <span class="n">mul_assoc</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">_</span><span class="o">⟩</span> <span class="o">⟨</span><span class="n">y</span><span class="o">,</span> <span class="n">_</span><span class="o">⟩</span> <span class="o">⟨</span><span class="n">z</span><span class="o">,</span> <span class="n">_</span><span class="o">⟩</span> <span class="bp">↦</span> <span class="n">SetCoe.ext</span> <span class="o">(</span><span class="n">mul_assoc</span> <span class="n">x</span> <span class="n">y</span> <span class="n">z</span><span class="o">)</span> <span class="n">one</span> <span class="o">:=</span> <span class="o">⟨</span><span class="mi">1</span><span class="o">,</span> <span class="n">N.one_mem</span><span class="o">⟩</span> <span class="n">one_mul</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">_</span><span class="o">⟩</span> <span class="bp">↦</span> <span class="n">SetCoe.ext</span> <span class="o">(</span><span class="n">one_mul</span> <span class="n">x</span><span class="o">)</span> <span class="n">mul_one</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">x</span><span class="o">,</span> <span class="n">_</span><span class="o">⟩</span> <span class="bp">↦</span> <span class="n">SetCoe.ext</span> <span class="o">(</span><span class="n">mul_one</span> <span class="n">x</span><span class="o">)</span> </pre></div> </div> <p>In order to apply lemmas about submonoids to subgroups or subrings, we need a class, just like for morphisms. Note this class take a <code class="docutils literal notranslate"><span class="pre">SetLike</span></code> instance as a parameter so it does not need a carrier field and can use the membership notation in its fields.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">class</span> <span class="n">SubmonoidClass₁</span> <span class="o">(</span><span class="n">S</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">(</span><span class="n">M</span> <span class="o">:</span> <span class="kt">Type</span><span class="o">)</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">[</span><span class="n">SetLike</span> <span class="n">S</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="n">where</span> <span class="n">mul_mem</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">S</span><span class="o">)</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">M</span><span class="o">},</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">s</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp">∈</span> <span class="n">s</span> <span class="bp">→</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">∈</span> <span class="n">s</span> <span class="n">one_mem</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">s</span> <span class="o">:</span> <span class="n">S</span><span class="o">,</span> <span class="mi">1</span> <span class="bp">∈</span> <span class="n">s</span> <span class="kd">instance</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="n">SubmonoidClass₁</span> <span class="o">(</span><span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="n">M</span> <span class="n">where</span> <span class="n">mul_mem</span> <span class="o">:=</span> <span class="n">Submonoid₁.mul_mem</span> <span class="n">one_mem</span> <span class="o">:=</span> <span class="n">Submonoid₁.one_mem</span> </pre></div> </div> <p>As an exercise you should define a <code class="docutils literal notranslate"><span class="pre">Subgroup₁</span></code> structure, endow it with a <code class="docutils literal notranslate"><span class="pre">SetLike</span></code> instance and a <code class="docutils literal notranslate"><span class="pre">SubmonoidClass₁</span></code> instance, put a <code class="docutils literal notranslate"><span class="pre">Group</span></code> instance on the subtype associated to a <code class="docutils literal notranslate"><span class="pre">Subgroup₁</span></code> and define a <code class="docutils literal notranslate"><span class="pre">SubgroupClass₁</span></code> class.</p> <p>Another very important thing to know about subobjects of a given algebraic object in Mathlib always form a complete lattice, and this structure is used a lot. For instance you may look for the lemma saying that an intersection of submonoids is a submonoid. But this won’t be a lemma, this will be an infimum construction. Let us do the case of two submonoids.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="n">Inf</span> <span class="o">(</span><span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="o">:=</span> <span class="o">⟨</span><span class="k">fun</span> <span class="n">S₁</span> <span class="n">S₂</span> <span class="bp">=></span> <span class="o">{</span> <span class="n">carrier</span> <span class="o">:=</span> <span class="n">S₁</span> <span class="bp">∩</span> <span class="n">S₂</span> <span class="n">one_mem</span> <span class="o">:=</span> <span class="o">⟨</span><span class="n">S₁.one_mem</span><span class="o">,</span> <span class="n">S₂.one_mem</span><span class="o">⟩</span> <span class="n">mul_mem</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">hx</span><span class="o">,</span> <span class="n">hx'</span><span class="o">⟩</span> <span class="o">⟨</span><span class="n">hy</span><span class="o">,</span> <span class="n">hy'</span><span class="o">⟩</span> <span class="bp">↦</span> <span class="o">⟨</span><span class="n">S₁.mul_mem</span> <span class="n">hx</span> <span class="n">hy</span><span class="o">,</span> <span class="n">S₂.mul_mem</span> <span class="n">hx'</span> <span class="n">hy'</span><span class="o">⟩</span> <span class="o">}⟩</span> </pre></div> </div> <p>This allows to get the intersections of two submonoids as a submonoid.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="n">P</span> <span class="o">:</span> <span class="n">Submonoid₁</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="n">Submonoid₁</span> <span class="n">M</span> <span class="o">:=</span> <span class="n">N</span> <span class="bp">⊓</span> <span class="n">P</span> </pre></div> </div> <p>You may think it’s a shame that we had to use the inf symbol <code class="docutils literal notranslate"><span class="pre">⊓</span></code> in the above example instead of the intersection symbol <code class="docutils literal notranslate"><span class="pre">∩</span></code>. But think about the supremum. The union of two submonoids is not a submonoid. However submonoids still form a lattice (even a complete one). Actually <code class="docutils literal notranslate"><span class="pre">N</span> <span class="pre">⊔</span> <span class="pre">P</span></code> is the submonoid generated by the union of <code class="docutils literal notranslate"><span class="pre">N</span></code> and <code class="docutils literal notranslate"><span class="pre">P</span></code> and of course it would be very confusing to denote it by <code class="docutils literal notranslate"><span class="pre">N</span> <span class="pre">∪</span> <span class="pre">P</span></code>. So you can see the use of <code class="docutils literal notranslate"><span class="pre">N</span> <span class="pre">⊓</span> <span class="pre">P</span></code> as much more consistent. It is also a lot more consistent across various kind of algebraic structures. It may look a bit weird at first to see the sum of two vector subspace <code class="docutils literal notranslate"><span class="pre">E</span></code> and <code class="docutils literal notranslate"><span class="pre">F</span></code> denoted by <code class="docutils literal notranslate"><span class="pre">E</span> <span class="pre">⊔</span> <span class="pre">F</span></code> instead of <code class="docutils literal notranslate"><span class="pre">E</span> <span class="pre">+</span> <span class="pre">F</span></code>. But you will get used to it. And soon you will consider the <code class="docutils literal notranslate"><span class="pre">E</span> <span class="pre">+</span> <span class="pre">F</span></code> notation as a distraction emphasizing the anecdotal fact that elements of <code class="docutils literal notranslate"><span class="pre">E</span> <span class="pre">⊔</span> <span class="pre">F</span></code> can be written as a sum of an element of <code class="docutils literal notranslate"><span class="pre">E</span></code> and an element of <code class="docutils literal notranslate"><span class="pre">F</span></code> instead of emphasizing the fundamental fact that <code class="docutils literal notranslate"><span class="pre">E</span> <span class="pre">⊔</span> <span class="pre">F</span></code> is the smallest vector subspace containing both <code class="docutils literal notranslate"><span class="pre">E</span></code> and <code class="docutils literal notranslate"><span class="pre">F</span></code>.</p> <p>Our last topic for this chapter is that of quotients. Again we want to explain how convenient notation are built and code duplication is avoided in Mathlib. Here the main device is the <code class="docutils literal notranslate"><span class="pre">HasQuotient</span></code> class which allows notations like <code class="docutils literal notranslate"><span class="pre">M</span> <span class="pre">⧸</span> <span class="pre">N</span></code>. Beware the quotient symbol <code class="docutils literal notranslate"><span class="pre">⧸</span></code> is a special unicode character, not a regular ASCII division symbol.</p> <p>As an example, we will build the quotient of a commutative monoid by a submonoid, leave proofs to you.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">Submonoid.Setoid</span> <span class="o">[</span><span class="n">CommMonoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="n">Setoid</span> <span class="n">M</span> <span class="n">where</span> <span class="n">r</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">↦</span> <span class="bp">∃</span> <span class="n">w</span> <span class="bp">∈</span> <span class="n">N</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">z</span> <span class="bp">∈</span> <span class="n">N</span><span class="o">,</span> <span class="n">x</span><span class="bp">*</span><span class="n">w</span> <span class="bp">=</span> <span class="n">y</span><span class="bp">*</span><span class="n">z</span> <span class="n">iseqv</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">refl</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">↦</span> <span class="o">⟨</span><span class="mi">1</span><span class="o">,</span> <span class="n">N.one_mem</span><span class="o">,</span> <span class="mi">1</span><span class="o">,</span> <span class="n">N.one_mem</span><span class="o">,</span> <span class="n">rfl</span><span class="o">⟩</span> <span class="n">symm</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">⟨</span><span class="n">w</span><span class="o">,</span> <span class="n">hw</span><span class="o">,</span> <span class="n">z</span><span class="o">,</span> <span class="n">hz</span><span class="o">,</span> <span class="n">h</span><span class="o">⟩</span> <span class="bp">↦</span> <span class="o">⟨</span><span class="n">z</span><span class="o">,</span> <span class="n">hz</span><span class="o">,</span> <span class="n">w</span><span class="o">,</span> <span class="n">hw</span><span class="o">,</span> <span class="n">h.symm</span><span class="o">⟩</span> <span class="n">trans</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="o">}</span> <span class="kd">instance</span> <span class="o">[</span><span class="n">CommMonoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="n">HasQuotient</span> <span class="n">M</span> <span class="o">(</span><span class="n">Submonoid</span> <span class="n">M</span><span class="o">)</span> <span class="n">where</span> <span class="n">quotient'</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">N</span> <span class="bp">↦</span> <span class="n">Quotient</span> <span class="n">N.Setoid</span> <span class="kd">def</span> <span class="n">QuotientMonoid.mk</span> <span class="o">[</span><span class="n">CommMonoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="n">M</span> <span class="bp">→</span> <span class="n">M</span> <span class="bp">⧸</span> <span class="n">N</span> <span class="o">:=</span> <span class="n">Quotient.mk</span> <span class="n">N.Setoid</span> <span class="kd">instance</span> <span class="o">[</span><span class="n">CommMonoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monoid</span> <span class="o">(</span><span class="n">M</span> <span class="bp">⧸</span> <span class="n">N</span><span class="o">)</span> <span class="n">where</span> <span class="n">mul</span> <span class="o">:=</span> <span class="n">Quotient.map₂'</span> <span class="o">(</span><span class="bp">·</span> <span class="bp">*</span> <span class="bp">·</span><span class="o">)</span> <span class="o">(</span><span class="kd">by</span> <span class="gr">sorry</span> <span class="o">)</span> <span class="n">mul_assoc</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">one</span> <span class="o">:=</span> <span class="n">QuotientMonoid.mk</span> <span class="n">N</span> <span class="mi">1</span> <span class="n">one_mul</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">mul_one</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>-/</p> </section> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C06_Structures.html" class="btn btn-neutral float-left" title="6. 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Topology — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> <link rel="shortcut icon" href="_static/favicon.ico"/> <!--[if lt IE 9]> <script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" /> <link rel="next" title="9. Differential Calculus" href="C09_Differential_Calculus.html" /> <link rel="prev" title="7. Hierarchies" href="C07_Hierarchies.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul class="current"> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">8. Topology</a><ul> <li class="toctree-l2"><a class="reference internal" href="#filters">8.1. Filters</a></li> <li class="toctree-l2"><a class="reference internal" href="#metric-spaces">8.2. Metric spaces</a><ul> <li class="toctree-l3"><a class="reference internal" href="#convergence-and-continuity">8.2.1. Convergence and continuity</a></li> <li class="toctree-l3"><a class="reference internal" href="#balls-open-sets-and-closed-sets">8.2.2. Balls, open sets and closed sets</a></li> <li class="toctree-l3"><a class="reference internal" href="#compactness">8.2.3. Compactness</a></li> <li class="toctree-l3"><a class="reference internal" href="#uniformly-continuous-functions">8.2.4. Uniformly continuous functions</a></li> <li class="toctree-l3"><a class="reference internal" href="#completeness">8.2.5. Completeness</a></li> </ul> </li> <li class="toctree-l2"><a class="reference internal" href="#topological-spaces">8.3. Topological spaces</a><ul> <li class="toctree-l3"><a class="reference internal" href="#fundamentals">8.3.1. Fundamentals</a></li> <li class="toctree-l3"><a class="reference internal" href="#separation-and-countability">8.3.2. Separation and countability</a></li> <li class="toctree-l3"><a class="reference internal" href="#id5">8.3.3. Compactness</a></li> </ul> </li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">8. </span>Topology</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C08_Topology.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <span class="target" id="topology"></span><section id="index-0"> <span id="id1"></span><h1><span class="section-number">8. </span>Topology<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <p>Calculus is based on the concept of a function, which is used to model quantities that depend on one another. For example, it is common to study quantities that change over time. The notion of a <em>limit</em> is also fundamental. We may say that the limit of a function <span class="math notranslate nohighlight">\(f(x)\)</span> is a value <span class="math notranslate nohighlight">\(b\)</span> as <span class="math notranslate nohighlight">\(x\)</span> approaches a value <span class="math notranslate nohighlight">\(a\)</span>, or that <span class="math notranslate nohighlight">\(f(x)\)</span> <em>converges to</em> <span class="math notranslate nohighlight">\(b\)</span> as <span class="math notranslate nohighlight">\(x\)</span> approaches <span class="math notranslate nohighlight">\(a\)</span>. Equivalently, we may say that a <span class="math notranslate nohighlight">\(f(x)\)</span> approaches <span class="math notranslate nohighlight">\(a\)</span> as <span class="math notranslate nohighlight">\(x\)</span> approaches a value <span class="math notranslate nohighlight">\(b\)</span>, or that it <em>tends to</em> <span class="math notranslate nohighlight">\(b\)</span> as <span class="math notranslate nohighlight">\(x\)</span> tends to <span class="math notranslate nohighlight">\(a\)</span>. We have already begun to consider such notions in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence"><span class="std std-numref">Section 3.6</span></a>.</p> <p><em>Topology</em> is the abstract study of limits and continuity. Having covered the essentials of formalization in Chapters <a class="reference internal" href="C02_Basics.html#basics"><span class="std std-numref">2</span></a> to <a class="reference internal" href="C06_Structures.html#structures"><span class="std std-numref">6</span></a>, in this chapter, we will explain how topological notions are formalized in mathlib. Not only do topological abstractions apply in much greater generality, but that also, somewhat paradoxically, make it easier to reason about limits and continuity in concrete instances.</p> <p>Topological notions build on quite a few layers of mathematical structure. The first layer is naive set theory, as described in <a class="reference internal" href="C04_Sets_and_Functions.html#sets-and-functions"><span class="std std-numref">Chapter 4</span></a>. The next layer is the theory of <em>filters</em>, which we will describe in <a class="reference internal" href="#filters"><span class="std std-numref">Section 8.1</span></a>. On top of that, we layer the theories of <em>topological spaces</em>, <em>metric spaces</em>, and a slightly more exotic intermediate notion called a <em>uniform space</em>.</p> <p>Whereas previous chapters relied on mathematical notions that were likely familiar to you, the notion of a filter less well known, even to many working mathematicians. The notion is essential, however, for formalizing mathematics effectively. Let us explain why. Let <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> be any function. We can consider the limit of <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> as <code class="docutils literal notranslate"><span class="pre">x</span></code> approaches some value <code class="docutils literal notranslate"><span class="pre">x₀</span></code>, but we can also consider the limit of <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> as <code class="docutils literal notranslate"><span class="pre">x</span></code> approaches infinity or negative infinity. We can moreover consider the limit of <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> as <code class="docutils literal notranslate"><span class="pre">x</span></code> approaches <code class="docutils literal notranslate"><span class="pre">x₀</span></code> from the right, conventionally written <code class="docutils literal notranslate"><span class="pre">x₀⁺</span></code>, or from the left, written <code class="docutils literal notranslate"><span class="pre">x₀⁻</span></code>. There are variations where <code class="docutils literal notranslate"><span class="pre">x</span></code> approaches <code class="docutils literal notranslate"><span class="pre">x₀</span></code> or <code class="docutils literal notranslate"><span class="pre">x₀⁺</span></code> or <code class="docutils literal notranslate"><span class="pre">x₀⁻</span></code> but is not allowed to take on the value <code class="docutils literal notranslate"><span class="pre">x₀</span></code> itself. This results in at least eight ways that <code class="docutils literal notranslate"><span class="pre">x</span></code> can approach something. We can also restrict to rational values of <code class="docutils literal notranslate"><span class="pre">x</span></code> or place other constraints on the domain, but let’s stick to those 8 cases.</p> <p>We have a similar variety of options on the codomain: we can specify that <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> approaches a value from the left or right, or that it approaches positive or negative infinity, and so on. For example, we may wish to say that <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> tends to <code class="docutils literal notranslate"><span class="pre">+∞</span></code> when <code class="docutils literal notranslate"><span class="pre">x</span></code> tends to <code class="docutils literal notranslate"><span class="pre">x₀</span></code> from the right without being equal to <code class="docutils literal notranslate"><span class="pre">x₀</span></code>. This results in 64 different kinds of limit statements, and we haven’t even begun to deal with limits of sequences, as we did in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence"><span class="std std-numref">Section 3.6</span></a>.</p> <p>The problem is compounded even further when it comes to the supporting lemmas. For instance, limits compose: if <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> tends to <code class="docutils literal notranslate"><span class="pre">y₀</span></code> when <code class="docutils literal notranslate"><span class="pre">x</span></code> tends to <code class="docutils literal notranslate"><span class="pre">x₀</span></code> and <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">y</span></code> tends to <code class="docutils literal notranslate"><span class="pre">z₀</span></code> when <code class="docutils literal notranslate"><span class="pre">y</span></code> tends to <code class="docutils literal notranslate"><span class="pre">y₀</span></code> then <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">∘</span> <span class="pre">f</span> <span class="pre">x</span></code> tends to <code class="docutils literal notranslate"><span class="pre">z₀</span></code> when <code class="docutils literal notranslate"><span class="pre">x</span></code> tends to <code class="docutils literal notranslate"><span class="pre">x₀</span></code>. There are three notions of “tends to” at play here, each of which can be instantiated in any of the eight ways described in the previous paragraph. This results in 512 lemmas, a lot to have to add to a library! Informally, mathematicians generally prove two or three of these and simply note that the rest can be proved “in the same way.” Formalizing mathematics requires making the relevant notion of “sameness” fully explicit, and that is exactly what Bourbaki’s theory of filters manages to do.</p> <section id="filters"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">8.1. </span>Filters<a class="headerlink" href="#filters" title="Permalink to this heading"></a></h2> <p>A <em>filter</em> on a type <code class="docutils literal notranslate"><span class="pre">X</span></code> is a collection of sets of <code class="docutils literal notranslate"><span class="pre">X</span></code> that satisfies three conditions that we will spell out below. The notion supports two related ideas:</p> <ul class="simple"> <li><p><em>limits</em>, including all the kinds of limits discussed above: finite and infinite limits of sequences, finite and infinite limits of functions at a point or at infinity, and so on.</p></li> <li><p><em>things happening eventually</em>, including things happening for large enough <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">:</span> <span class="pre">ℕ</span></code>, or sufficiently near a point <code class="docutils literal notranslate"><span class="pre">x</span></code>, or for sufficiently close pairs of points, or almost everywhere in the sense of measure theory. Dually, filters can also express the idea of <em>things happening often</em>: for arbitrarily large <code class="docutils literal notranslate"><span class="pre">n</span></code>, at a point in any neighborhood of given a point, etc.</p></li> </ul> <p>The filters that correspond to these descriptions will be defined later in this section, but we can already name them:</p> <ul class="simple"> <li><p><code class="docutils literal notranslate"><span class="pre">(at_top</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">ℕ)</span></code>, made of sets of <code class="docutils literal notranslate"><span class="pre">ℕ</span></code> containing <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">n</span> <span class="pre">≥</span> <span class="pre">N}</span></code> for some <code class="docutils literal notranslate"><span class="pre">N</span></code></p></li> <li><p><code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x</span></code>, made of neighborhoods of <code class="docutils literal notranslate"><span class="pre">x</span></code> in a topological space</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">𝓤</span> <span class="pre">X</span></code>, made of entourages of a uniform space (uniform spaces generalize metric spaces and topological groups)</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">μ.a_e</span></code> , made of sets whose complement has zero measure with respect to a measure <code class="docutils literal notranslate"><span class="pre">μ</span></code>.</p></li> </ul> <p>The general definition is as follows: a filter <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">X</span></code> is a collection of sets <code class="docutils literal notranslate"><span class="pre">F.sets</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">(Set</span> <span class="pre">X)</span></code> satisfying the following:</p> <ul class="simple"> <li><p><code class="docutils literal notranslate"><span class="pre">F.univ_sets</span> <span class="pre">:</span> <span class="pre">univ</span> <span class="pre">∈</span> <span class="pre">F.sets</span></code></p></li> <li><p><code class="docutils literal notranslate"><span class="pre">F.sets_of_superset</span> <span class="pre">:</span> <span class="pre">∀</span> <span class="pre">{U</span> <span class="pre">V},</span> <span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F.sets</span> <span class="pre">→</span> <span class="pre">U</span> <span class="pre">⊆</span> <span class="pre">V</span> <span class="pre">→</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F.sets</span></code></p></li> <li><p><code class="docutils literal notranslate"><span class="pre">F.inter_sets</span> <span class="pre">:</span> <span class="pre">∀</span> <span class="pre">{U</span> <span class="pre">V},</span> <span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F.sets</span> <span class="pre">→</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F.sets</span> <span class="pre">→</span> <span class="pre">U</span> <span class="pre">∩</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F.sets</span></code>.</p></li> </ul> <p>The first condition says that the set of all elements of <code class="docutils literal notranslate"><span class="pre">X</span></code> belongs to <code class="docutils literal notranslate"><span class="pre">F.sets</span></code>. The second condition says that if <code class="docutils literal notranslate"><span class="pre">U</span></code> belongs to <code class="docutils literal notranslate"><span class="pre">F.sets</span></code> then anything containing <code class="docutils literal notranslate"><span class="pre">U</span></code> also belongs to <code class="docutils literal notranslate"><span class="pre">F.sets</span></code>. The third condition says that <code class="docutils literal notranslate"><span class="pre">F.sets</span></code> is closed under finite intersections. In mathlib, a filter <code class="docutils literal notranslate"><span class="pre">F</span></code> is defined to be a structure bundling <code class="docutils literal notranslate"><span class="pre">F.sets</span></code> and its three properties, but the properties carry no additional data, and it is convenient to blur the distinction between <code class="docutils literal notranslate"><span class="pre">F</span></code> and <code class="docutils literal notranslate"><span class="pre">F.sets</span></code>. We therefore define <code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span></code> to mean <code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F.sets</span></code>. This explains why the word <code class="docutils literal notranslate"><span class="pre">sets</span></code> appears in the names of some lemmas that that mention <code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span></code>.</p> <p>It may help to think of a filter as defining a notion of a “sufficiently large” set. The first condition then says that <code class="docutils literal notranslate"><span class="pre">univ</span></code> is sufficiently large, the second one says that a set containing a sufficiently large set is sufficiently large and the third one says that the intersection of two sufficiently large sets is sufficiently large.</p> <p>It may be even more useful to think of a filter on a type <code class="docutils literal notranslate"><span class="pre">X</span></code> as a generalized element of <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">X</span></code>. For instance, <code class="docutils literal notranslate"><span class="pre">at_top</span></code> is the “set of very large numbers” and <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x₀</span></code> is the “set of points very close to <code class="docutils literal notranslate"><span class="pre">x₀</span></code>.” One manifestation of this view is that we can associate to any <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X</span></code> the so-called <em>principal filter</em> consisting of all sets that contain <code class="docutils literal notranslate"><span class="pre">s</span></code>. This definition is already in mathlib and has a notation <code class="docutils literal notranslate"><span class="pre">𝓟</span></code> (localized in the <code class="docutils literal notranslate"><span class="pre">Filter</span></code> namespace). For the purpose of demonstration, we ask you to take this opportunity to work out the definition here.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">principal</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">α</span> <span class="n">where</span> <span class="n">sets</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">t</span> <span class="bp">|</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="n">t</span> <span class="o">}</span> <span class="n">univ_sets</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">sets_of_superset</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">inter_sets</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>For our second example, we ask you to define the filter <code class="docutils literal notranslate"><span class="pre">at_top</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">ℕ</span></code>. (We could use any type with a preorder instead of <code class="docutils literal notranslate"><span class="pre">ℕ</span></code>.)</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">ℕ</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">sets</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">s</span> <span class="bp">|</span> <span class="bp">∃</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">b</span><span class="o">,</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="n">b</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">}</span> <span class="n">univ_sets</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">sets_of_superset</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">inter_sets</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="o">}</span> </pre></div> </div> <p>We can also directly define the filter <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x</span></code> of neighborhoods of any <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">ℝ</span></code>. In the real numbers, a neighborhood of <code class="docutils literal notranslate"><span class="pre">x</span></code> is a set containing an open interval <span class="math notranslate nohighlight">\((x_0 - \varepsilon, x_0 + \varepsilon)\)</span>, defined in mathlib as <code class="docutils literal notranslate"><span class="pre">Ioo</span> <span class="pre">(x₀</span> <span class="pre">-</span> <span class="pre">ε)</span> <span class="pre">(x₀</span> <span class="pre">+</span> <span class="pre">ε)</span></code>. (This is notion of a neighborhood is only a special case of a more general construction in mathlib.)</p> <p>With these examples, we can already define what is means for a function <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Y</span></code> to converge to some <code class="docutils literal notranslate"><span class="pre">G</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">Y</span></code> along some <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">X</span></code>, as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">Tendsto₁</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">G</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">Y</span><span class="o">)</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">V</span> <span class="bp">∈</span> <span class="n">G</span><span class="o">,</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">V</span> <span class="bp">∈</span> <span class="n">F</span> </pre></div> </div> <p>When <code class="docutils literal notranslate"><span class="pre">X</span></code> is <code class="docutils literal notranslate"><span class="pre">ℕ</span></code> and <code class="docutils literal notranslate"><span class="pre">Y</span></code> is <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>, <code class="docutils literal notranslate"><span class="pre">Tendsto₁</span> <span class="pre">u</span> <span class="pre">at_top</span> <span class="pre">(𝓝</span> <span class="pre">x)</span></code> is equivalent to saying that the sequence <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">:</span> <span class="pre">ℕ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> converges to the real number <code class="docutils literal notranslate"><span class="pre">x</span></code>. When both <code class="docutils literal notranslate"><span class="pre">X</span></code> and <code class="docutils literal notranslate"><span class="pre">Y</span></code> are <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>, <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">f</span> <span class="pre">(𝓝</span> <span class="pre">x₀)</span> <span class="pre">(𝓝</span> <span class="pre">y₀)</span></code> is equivalent to the familiar notion <span class="math notranslate nohighlight">\(\lim_{x \to x₀} f(x) = y₀\)</span>. All of the other kinds of limits mentioned in the introduction are also equivalent to instances of <code class="docutils literal notranslate"><span class="pre">Tendsto₁</span></code> for suitable choices of filters on the source and target.</p> <p>The notion <code class="docutils literal notranslate"><span class="pre">Tendsto₁</span></code> above is definitionally equivalent to the notion <code class="docutils literal notranslate"><span class="pre">Tendsto</span></code> that is defined in mathlib, but the latter is defined more abstractly. The problem with the definition of <code class="docutils literal notranslate"><span class="pre">Tendsto₁</span></code> is that it exposes a quantifier and elements of <code class="docutils literal notranslate"><span class="pre">G</span></code>, and it hides the intuition that we get by viewing filters as generalized sets. We can hide the quantifier <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">V</span></code> and make the intuition more salient by using more algebraic and set-theoretic machinery. The first ingredient is the <em>pushforward</em> operation <span class="math notranslate nohighlight">\(f_*\)</span> associated to any map <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Y</span></code>, denoted <code class="docutils literal notranslate"><span class="pre">Filter.map</span> <span class="pre">f</span></code> in mathlib. Given a filter <code class="docutils literal notranslate"><span class="pre">F</span></code> on <code class="docutils literal notranslate"><span class="pre">X</span></code>, <code class="docutils literal notranslate"><span class="pre">Filter.map</span> <span class="pre">f</span> <span class="pre">F</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">Y</span></code> is defined so that <code class="docutils literal notranslate"><span class="pre">V</span> <span class="pre">∈</span> <span class="pre">Filter.map</span> <span class="pre">f</span> <span class="pre">F</span> <span class="pre">↔</span> <span class="pre">f</span> <span class="pre">⁻¹'</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F</span></code> holds definitionally. In this examples file we’ve opened the <code class="docutils literal notranslate"><span class="pre">Filter</span></code> namespace so that <code class="docutils literal notranslate"><span class="pre">Filter.map</span></code> can be written as <code class="docutils literal notranslate"><span class="pre">map</span></code>. This means that we can rewrite the definition of <code class="docutils literal notranslate"><span class="pre">Tendsto</span></code> using the order relation on <code class="docutils literal notranslate"><span class="pre">Filter</span> <span class="pre">Y</span></code>, which is reversed inclusion of the set of members. In other words, given <code class="docutils literal notranslate"><span class="pre">G</span> <span class="pre">H</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">Y</span></code>, we have <code class="docutils literal notranslate"><span class="pre">G</span> <span class="pre">≤</span> <span class="pre">H</span> <span class="pre">↔</span> <span class="pre">∀</span> <span class="pre">V</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">Y,</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">H</span> <span class="pre">→</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">G</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">Tendsto₂</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">G</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">Y</span><span class="o">)</span> <span class="o">:=</span> <span class="n">map</span> <span class="n">f</span> <span class="n">F</span> <span class="bp">≤</span> <span class="n">G</span> <span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">G</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">Y</span><span class="o">)</span> <span class="o">:</span> <span class="n">Tendsto₂</span> <span class="n">f</span> <span class="n">F</span> <span class="n">G</span> <span class="bp">↔</span> <span class="n">Tendsto₁</span> <span class="n">f</span> <span class="n">F</span> <span class="n">G</span> <span class="o">:=</span> <span class="n">Iff.rfl</span> </pre></div> </div> <p>It may seem that the order relation on filters is backward. But recall that we can view filters on <code class="docutils literal notranslate"><span class="pre">X</span></code> as generalized elements of <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">X</span></code>, via the inclusion of <code class="docutils literal notranslate"><span class="pre">𝓟</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">X</span></code> which maps any set <code class="docutils literal notranslate"><span class="pre">s</span></code> to the corresponding principal filter. This inclusion is order preserving, so the order relation on <code class="docutils literal notranslate"><span class="pre">Filter</span></code> can indeed be seen as the natural inclusion relation between generalized sets. In this analogy, pushforward is analogous to the direct image. And, indeed, <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">f</span> <span class="pre">(𝓟</span> <span class="pre">s)</span> <span class="pre">=</span> <span class="pre">𝓟</span> <span class="pre">(f</span> <span class="pre">''</span> <span class="pre">s)</span></code>.</p> <p>We can now understand intuitively why a sequence <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">:</span> <span class="pre">ℕ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> converges to a point <code class="docutils literal notranslate"><span class="pre">x₀</span></code> if and only if we have <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">u</span> <span class="pre">at_top</span> <span class="pre">≤</span> <span class="pre">𝓝</span> <span class="pre">x₀</span></code>. The inequality means the “direct image under <code class="docutils literal notranslate"><span class="pre">u</span></code>” of “the set of very big natural numbers” is “included” in “the set of points very close to <code class="docutils literal notranslate"><span class="pre">x₀</span></code>.”</p> <p>As promised, the definition of <code class="docutils literal notranslate"><span class="pre">Tendsto₂</span></code> does not exhibit any quantifiers or sets. It also leverages the algebraic properties of the pushforward operation. First, each <code class="docutils literal notranslate"><span class="pre">Filter.map</span> <span class="pre">f</span></code> is monotone. And, second, <code class="docutils literal notranslate"><span class="pre">Filter.map</span></code> is compatible with composition.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="bp">@</span><span class="n">Filter.map_mono</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">α</span> <span class="n">β</span><span class="o">}</span> <span class="o">{</span><span class="n">m</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">},</span> <span class="n">Monotone</span> <span class="o">(</span><span class="n">map</span> <span class="n">m</span><span class="o">))</span> <span class="k">#check</span> <span class="o">(</span><span class="bp">@</span><span class="n">Filter.map_map</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">α</span> <span class="n">β</span> <span class="n">γ</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">α</span><span class="o">}</span> <span class="o">{</span><span class="n">m</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">β</span><span class="o">}</span> <span class="o">{</span><span class="n">m'</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">γ</span><span class="o">},</span> <span class="n">map</span> <span class="n">m'</span> <span class="o">(</span><span class="n">map</span> <span class="n">m</span> <span class="n">f</span><span class="o">)</span> <span class="bp">=</span> <span class="n">map</span> <span class="o">(</span><span class="n">m'</span> <span class="bp">∘</span> <span class="n">m</span><span class="o">)</span> <span class="n">f</span><span class="o">)</span> </pre></div> </div> <p>Together these two properties allow us to prove that limits compose, yielding in one shot all 256 variants of the composition lemma described in the introduction, and lots more. You can practice proving the following statement using either the definition of <code class="docutils literal notranslate"><span class="pre">Tendsto₁</span></code> in terms of the universal quantifier or the algebraic definition, together with the two lemmas above.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="n">Z</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">G</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">Y</span><span class="o">}</span> <span class="o">{</span><span class="n">H</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">Z</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">{</span><span class="n">g</span> <span class="o">:</span> <span class="n">Y</span> <span class="bp">→</span> <span class="n">Z</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Tendsto₁</span> <span class="n">f</span> <span class="n">F</span> <span class="n">G</span><span class="o">)</span> <span class="o">(</span><span class="n">hg</span> <span class="o">:</span> <span class="n">Tendsto₁</span> <span class="n">g</span> <span class="n">G</span> <span class="n">H</span><span class="o">)</span> <span class="o">:</span> <span class="n">Tendsto₁</span> <span class="o">(</span><span class="n">g</span> <span class="bp">∘</span> <span class="n">f</span><span class="o">)</span> <span class="n">F</span> <span class="n">H</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>The pushforward construction uses a map to push filters from the map source to the map target. There also a <em>pullback</em> operation, <code class="docutils literal notranslate"><span class="pre">Filter.comap</span></code>, going in the other direction. This generalizes the preimage operation on sets. For any map <code class="docutils literal notranslate"><span class="pre">f</span></code>, <code class="docutils literal notranslate"><span class="pre">Filter.map</span> <span class="pre">f</span></code> and <code class="docutils literal notranslate"><span class="pre">Filter.comap</span> <span class="pre">f</span></code> form what is known as a <em>Galois connection</em>, which is to say, they satisfy</p> <blockquote> <div><p><code class="docutils literal notranslate"><span class="pre">Filter.map_le_iff_le_comap</span> <span class="pre">:</span> <span class="pre">Filter.map</span> <span class="pre">f</span> <span class="pre">F</span> <span class="pre">≤</span> <span class="pre">G</span> <span class="pre">↔</span> <span class="pre">F</span> <span class="pre">≤</span> <span class="pre">Filter.comap</span> <span class="pre">f</span> <span class="pre">G</span></code></p> </div></blockquote> <p>for every <code class="docutils literal notranslate"><span class="pre">F</span></code> and <code class="docutils literal notranslate"><span class="pre">G</span></code>. This operation could be used to provided another formulation of <code class="docutils literal notranslate"><span class="pre">Tendsto</span></code> that would be provably (but not definitionally) equivalent to the one in mathlib.</p> <p>The <code class="docutils literal notranslate"><span class="pre">comap</span></code> operation can be used to restrict filters to a subtype. For instance, suppose we have <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code>, <code class="docutils literal notranslate"><span class="pre">x₀</span> <span class="pre">:</span> <span class="pre">ℝ</span></code> and <code class="docutils literal notranslate"><span class="pre">y₀</span> <span class="pre">:</span> <span class="pre">ℝ</span></code>, and suppose we want to state that <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> approaches <code class="docutils literal notranslate"><span class="pre">y₀</span></code> when <code class="docutils literal notranslate"><span class="pre">x</span></code> approaches <code class="docutils literal notranslate"><span class="pre">x₀</span></code> within the rational numbers. We can pull the filter <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x₀</span></code> back to <code class="docutils literal notranslate"><span class="pre">ℚ</span></code> using the coercion map <code class="docutils literal notranslate"><span class="pre">(↑)</span> <span class="pre">:</span> <span class="pre">ℚ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> and state <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">(f</span> <span class="pre">∘</span> <span class="pre">(↑)</span> <span class="pre">:</span> <span class="pre">ℚ</span> <span class="pre">→</span> <span class="pre">ℝ)</span> <span class="pre">(comap</span> <span class="pre">(↑)</span> <span class="pre">(𝓝</span> <span class="pre">x₀))</span> <span class="pre">(𝓝</span> <span class="pre">y₀)</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="n">y₀</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="k">#check</span> <span class="n">comap</span> <span class="o">((</span><span class="bp">↑</span><span class="o">)</span> <span class="o">:</span> <span class="n">ℚ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">)</span> <span class="k">#check</span> <span class="n">Tendsto</span> <span class="o">(</span><span class="n">f</span> <span class="bp">∘</span> <span class="o">(</span><span class="bp">↑</span><span class="o">))</span> <span class="o">(</span><span class="n">comap</span> <span class="o">((</span><span class="bp">↑</span><span class="o">)</span> <span class="o">:</span> <span class="n">ℚ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">))</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">y₀</span><span class="o">)</span> </pre></div> </div> <p>The pullback operation is also compatible with composition, but it <em>contravariant</em>, which is to say, it reverses the order of the arguments.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">section</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="n">β</span> <span class="n">γ</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">α</span><span class="o">)</span> <span class="o">{</span><span class="n">m</span> <span class="o">:</span> <span class="n">γ</span> <span class="bp">→</span> <span class="n">β</span><span class="o">}</span> <span class="o">{</span><span class="n">n</span> <span class="o">:</span> <span class="n">β</span> <span class="bp">→</span> <span class="n">α</span><span class="o">}</span> <span class="k">#check</span> <span class="o">(</span><span class="n">comap_comap</span> <span class="o">:</span> <span class="n">comap</span> <span class="n">m</span> <span class="o">(</span><span class="n">comap</span> <span class="n">n</span> <span class="n">F</span><span class="o">)</span> <span class="bp">=</span> <span class="n">comap</span> <span class="o">(</span><span class="n">n</span> <span class="bp">∘</span> <span class="n">m</span><span class="o">)</span> <span class="n">F</span><span class="o">)</span> <span class="kd">end</span> </pre></div> </div> <p>Let’s now shift attention to the plane <code class="docutils literal notranslate"><span class="pre">ℝ</span> <span class="pre">×</span> <span class="pre">ℝ</span></code> and try to understand how the neighborhoods of a point <code class="docutils literal notranslate"><span class="pre">(x₀,</span> <span class="pre">y₀)</span></code> are related to <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x₀</span></code> and <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">y₀</span></code>. There is a product operation <code class="docutils literal notranslate"><span class="pre">Filter.prod</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">Y</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">(X</span> <span class="pre">×</span> <span class="pre">Y)</span></code>, denoted by <code class="docutils literal notranslate"><span class="pre">×ᶠ</span></code>, which answers this question:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">𝓝</span> <span class="o">(</span><span class="n">x₀</span><span class="o">,</span> <span class="n">y₀</span><span class="o">)</span> <span class="bp">=</span> <span class="bp">𝓝</span> <span class="n">x₀</span> <span class="bp">×ˢ</span> <span class="bp">𝓝</span> <span class="n">y₀</span> <span class="o">:=</span> <span class="n">nhds_prod_eq</span> </pre></div> </div> <p>The product operation is defined in terms of the pullback operation and the <code class="docutils literal notranslate"><span class="pre">inf</span></code> operation:</p> <blockquote> <div><p><code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">×ᶠ</span> <span class="pre">G</span> <span class="pre">=</span> <span class="pre">(comap</span> <span class="pre">prod.fst</span> <span class="pre">F)</span> <span class="pre">⊓</span> <span class="pre">(comap</span> <span class="pre">prod.snd</span> <span class="pre">G)</span></code>.</p> </div></blockquote> <p>Here the <code class="docutils literal notranslate"><span class="pre">inf</span></code> operation refers to the lattice structure on <code class="docutils literal notranslate"><span class="pre">filter</span> <span class="pre">X</span></code> for any type <code class="docutils literal notranslate"><span class="pre">X</span></code>, whereby <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">⊓</span> <span class="pre">G</span></code> is the greatest filter that is smaller than both <code class="docutils literal notranslate"><span class="pre">F</span></code> and <code class="docutils literal notranslate"><span class="pre">G</span></code>. Thus the <code class="docutils literal notranslate"><span class="pre">inf</span></code> operation generalizes the notion of the intersection of sets.</p> <p>A lot of proofs in mathlib use all of the aforementioned structure (<code class="docutils literal notranslate"><span class="pre">map</span></code>, <code class="docutils literal notranslate"><span class="pre">comap</span></code>, <code class="docutils literal notranslate"><span class="pre">inf</span></code>, <code class="docutils literal notranslate"><span class="pre">sup</span></code>, and <code class="docutils literal notranslate"><span class="pre">prod</span></code>) to give algebraic proofs about convergence without ever referring to members of filters. You can practice doing this in a proof of the following lemma, unfolding the definition of <code class="docutils literal notranslate"><span class="pre">Tendsto</span></code> and <code class="docutils literal notranslate"><span class="pre">Filter.prod</span></code> if needed.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">le_inf_iff</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span> <span class="bp">×</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="n">y₀</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">f</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="o">(</span><span class="n">x₀</span><span class="o">,</span> <span class="n">y₀</span><span class="o">))</span> <span class="bp">↔</span> <span class="n">Tendsto</span> <span class="o">(</span><span class="n">Prod.fst</span> <span class="bp">∘</span> <span class="n">f</span><span class="o">)</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">)</span> <span class="bp">∧</span> <span class="n">Tendsto</span> <span class="o">(</span><span class="n">Prod.snd</span> <span class="bp">∘</span> <span class="n">f</span><span class="o">)</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">y₀</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>The ordered type <code class="docutils literal notranslate"><span class="pre">filter</span> <span class="pre">X</span></code> is actually a <em>complete</em> lattice, which is to say, there is a bottom element, there is a top element, and every set of filters on <code class="docutils literal notranslate"><span class="pre">X</span></code> has an <code class="docutils literal notranslate"><span class="pre">Inf</span></code> and a <code class="docutils literal notranslate"><span class="pre">Sup</span></code>.</p> <p>Note that given the second property in the definition of a filter (if <code class="docutils literal notranslate"><span class="pre">U</span></code> belongs to <code class="docutils literal notranslate"><span class="pre">F</span></code> then anything larger than <code class="docutils literal notranslate"><span class="pre">U</span></code> also belongs to <code class="docutils literal notranslate"><span class="pre">F</span></code>), the first property (the set of all inhabitants of <code class="docutils literal notranslate"><span class="pre">X</span></code> belongs to <code class="docutils literal notranslate"><span class="pre">F</span></code>) is equivalent to the property that <code class="docutils literal notranslate"><span class="pre">F</span></code> is not the empty collection of sets. This shouldn’t be confused with the more subtle question as to whether the empty set is an <em>element</em> of <code class="docutils literal notranslate"><span class="pre">F</span></code>. The definition of a filter does not prohibit <code class="docutils literal notranslate"><span class="pre">∅</span> <span class="pre">∈</span> <span class="pre">F</span></code>, but if the empty set is in <code class="docutils literal notranslate"><span class="pre">F</span></code> then every set is in <code class="docutils literal notranslate"><span class="pre">F</span></code>, which is to say, <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">U</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X,</span> <span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span></code>. In this case, <code class="docutils literal notranslate"><span class="pre">F</span></code> is a rather trivial filter, which is precisely the bottom element of the complete lattice <code class="docutils literal notranslate"><span class="pre">filter</span> <span class="pre">X</span></code>. This contrasts with the definition of filters in Bourbaki, which doesn’t allow filters containing the empty set.</p> <p>Because we include the trivial filter in our definition, we sometimes need to explicitly assume nontriviality in some lemmas. In return, however, the theory has nicer global properties. We have already seen that including the trivial filter gives us a bottom element. It also allows us to define <code class="docutils literal notranslate"><span class="pre">principal</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">X</span></code>, which maps <code class="docutils literal notranslate"><span class="pre">∅</span></code> to <code class="docutils literal notranslate"><span class="pre">⊥</span></code>, without adding a precondition to rule out the empty set. And it allows us to define the pullback operation without a precondition as well. Indeed, it can happen that <code class="docutils literal notranslate"><span class="pre">comap</span> <span class="pre">f</span> <span class="pre">F</span> <span class="pre">=</span> <span class="pre">⊥</span></code> although <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">≠</span> <span class="pre">⊥</span></code>. For instance, given <code class="docutils literal notranslate"><span class="pre">x₀</span> <span class="pre">:</span> <span class="pre">ℝ</span></code> and <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">ℝ</span></code>, the pullback of <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x₀</span></code> under the coercion from the subtype corresponding to <code class="docutils literal notranslate"><span class="pre">s</span></code> is nontrivial if and only if <code class="docutils literal notranslate"><span class="pre">x₀</span></code> belongs to the closure of <code class="docutils literal notranslate"><span class="pre">s</span></code>.</p> <p>In order to manage lemmas that do need to assume some filter is nontrivial, mathlib has a type class <code class="docutils literal notranslate"><span class="pre">Filter.ne_bot</span></code>, and the library has lemmas that assume <code class="docutils literal notranslate"><span class="pre">(F</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">X)</span> <span class="pre">[F.ne_bot]</span></code>. The instance database knows, for example, that <code class="docutils literal notranslate"><span class="pre">(at_top</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">ℕ).ne_bot</span></code>, and it knows that pushing forward a nontrivial filter gives a nontrivial filter. As a result, a lemma assuming <code class="docutils literal notranslate"><span class="pre">[F.ne_bot]</span></code> will automatically apply to <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">u</span> <span class="pre">at_top</span></code> for any sequence <code class="docutils literal notranslate"><span class="pre">u</span></code>.</p> <p>Our tour of the algebraic properties of filters and their relation to limits is essentially done, but we have not yet justified our claim to have recaptured the usual limit notions. Superficially, it may seem that <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">u</span> <span class="pre">at_top</span> <span class="pre">(𝓝</span> <span class="pre">x₀)</span></code> is stronger than the notion of convergence defined in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence"><span class="std std-numref">Section 3.6</span></a> because we ask that <em>every</em> neighborhood of <code class="docutils literal notranslate"><span class="pre">x₀</span></code> has a preimage belonging to <code class="docutils literal notranslate"><span class="pre">at_top</span></code>, whereas the usual definition only requires this for the standard neighborhoods <code class="docutils literal notranslate"><span class="pre">Ioo</span> <span class="pre">(x₀</span> <span class="pre">-</span> <span class="pre">ε)</span> <span class="pre">(x₀</span> <span class="pre">+</span> <span class="pre">ε)</span></code>. The key is that, by definition, every neighborhood contains such a standard one. This observation leads to the notion of a <em>filter basis</em>.</p> <p>Given <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">X</span></code>, a family of sets <cite>s : ι → Set X</cite> is a basis for <code class="docutils literal notranslate"><span class="pre">F</span></code> if for every set <code class="docutils literal notranslate"><span class="pre">U</span></code>, we have <code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span></code> if and only if it contains some <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">i</span></code>. In other words, formally speaking, <code class="docutils literal notranslate"><span class="pre">s</span></code> is a basis if it satisfies <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">U</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X,</span> <span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span> <span class="pre">↔</span> <span class="pre">∃</span> <span class="pre">i,</span> <span class="pre">s</span> <span class="pre">i</span> <span class="pre">⊆</span> <span class="pre">U</span></code>. It is even more flexible to consider a predicate on <code class="docutils literal notranslate"><span class="pre">ι</span></code> that selects only some of the values <code class="docutils literal notranslate"><span class="pre">i</span></code> in the indexing type. In the case of <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x₀</span></code>, we want <code class="docutils literal notranslate"><span class="pre">ι</span></code> to be <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>, we write <code class="docutils literal notranslate"><span class="pre">ε</span></code> for <code class="docutils literal notranslate"><span class="pre">i</span></code>, and the predicate should select the positive values of <code class="docutils literal notranslate"><span class="pre">ε</span></code>. So the fact that the sets <code class="docutils literal notranslate"><span class="pre">Ioo</span>  <span class="pre">(x₀</span> <span class="pre">-</span> <span class="pre">ε)</span> <span class="pre">(x₀</span> <span class="pre">+</span> <span class="pre">ε)</span></code> form a basis for the neighborhood topology on <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> is stated as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x₀</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">HasBasis</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">)</span> <span class="o">(</span><span class="k">fun</span> <span class="n">ε</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">=></span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">ε</span><span class="o">)</span> <span class="k">fun</span> <span class="n">ε</span> <span class="bp">=></span> <span class="n">Ioo</span> <span class="o">(</span><span class="n">x₀</span> <span class="bp">-</span> <span class="n">ε</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="bp">+</span> <span class="n">ε</span><span class="o">)</span> <span class="o">:=</span> <span class="n">nhds_basis_Ioo_pos</span> <span class="n">x₀</span> </pre></div> </div> <p>There is also a nice basis for the filter <code class="docutils literal notranslate"><span class="pre">at_top</span></code>. The lemma <code class="docutils literal notranslate"><span class="pre">Filter.has_basis.tendsto_iff</span></code> allows us to reformulate a statement of the form <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">f</span> <span class="pre">F</span> <span class="pre">G</span></code> given bases for <code class="docutils literal notranslate"><span class="pre">F</span></code> and <code class="docutils literal notranslate"><span class="pre">G</span></code>. Putting these pieces together gives us essentially the notion of convergence that we used in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence"><span class="std std-numref">Section 3.6</span></a>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">)</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">N</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">n</span> <span class="bp">≥</span> <span class="n">N</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">Ioo</span> <span class="o">(</span><span class="n">x₀</span> <span class="bp">-</span> <span class="n">ε</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="bp">+</span> <span class="n">ε</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">have</span> <span class="o">:</span> <span class="n">atTop.HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">_</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">=></span> <span class="n">True</span><span class="o">)</span> <span class="n">Ici</span> <span class="o">:=</span> <span class="n">atTop_basis</span> <span class="n">rw</span> <span class="o">[</span><span class="n">this.tendsto_iff</span> <span class="o">(</span><span class="n">nhds_basis_Ioo_pos</span> <span class="n">x₀</span><span class="o">)]</span> <span class="n">simp</span> </pre></div> </div> <p>We now show how filters facilitate working with properties that hold for sufficiently large numbers or for points that are sufficiently close to a given point. In <a class="reference internal" href="C03_Logic.html#sequences-and-convergence"><span class="std std-numref">Section 3.6</span></a>, we were often faced with the situation where we knew that some property <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">n</span></code> holds for sufficiently large <code class="docutils literal notranslate"><span class="pre">n</span></code> and that some other property <code class="docutils literal notranslate"><span class="pre">Q</span> <span class="pre">n</span></code> holds for sufficiently large <code class="docutils literal notranslate"><span class="pre">n</span></code>. Using <code class="docutils literal notranslate"><span class="pre">cases</span></code> twice gave us <code class="docutils literal notranslate"><span class="pre">N_P</span></code> and <code class="docutils literal notranslate"><span class="pre">N_Q</span></code> satisfying <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">n</span> <span class="pre">≥</span> <span class="pre">N_P,</span> <span class="pre">P</span> <span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">n</span> <span class="pre">≥</span> <span class="pre">N_Q,</span> <span class="pre">Q</span> <span class="pre">n</span></code>. Using <code class="docutils literal notranslate"><span class="pre">set</span> <span class="pre">N</span> <span class="pre">:=</span> <span class="pre">max</span> <span class="pre">N_P</span> <span class="pre">N_Q</span></code>, we could eventually prove <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">n</span> <span class="pre">≥</span> <span class="pre">N,</span> <span class="pre">P</span> <span class="pre">n</span> <span class="pre">∧</span> <span class="pre">Q</span> <span class="pre">n</span></code>. Doing this repeatedly becomes tiresome.</p> <p>We can do better by noting that the statement “<code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">Q</span> <span class="pre">n</span></code> hold for large enough <code class="docutils literal notranslate"><span class="pre">n</span></code>” means that we have <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n}</span> <span class="pre">∈</span> <span class="pre">at_top</span></code> and <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">Q</span> <span class="pre">n}</span> <span class="pre">∈</span> <span class="pre">at_top</span></code>. The fact that <code class="docutils literal notranslate"><span class="pre">at_top</span></code> is a filter implies that the intersection of two elements of <code class="docutils literal notranslate"><span class="pre">at_top</span></code> is again in <code class="docutils literal notranslate"><span class="pre">at_top</span></code>, so we have <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n</span> <span class="pre">∧</span> <span class="pre">Q</span> <span class="pre">n}</span> <span class="pre">∈</span> <span class="pre">at_top</span></code>. Writing <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n}</span> <span class="pre">∈</span> <span class="pre">at_top</span></code> is unpleasant, but we can use the more suggestive notation <code class="docutils literal notranslate"><span class="pre">∀ᶠ</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">at_top,</span> <span class="pre">P</span> <span class="pre">n</span></code>. Here the superscripted <code class="docutils literal notranslate"><span class="pre">f</span></code> stands for “Filter.” You can think of the notation as saying that for all <code class="docutils literal notranslate"><span class="pre">n</span></code> in the “set of very large numbers,” <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">n</span></code> holds. The <code class="docutils literal notranslate"><span class="pre">∀ᶠ</span></code> notation stands for <code class="docutils literal notranslate"><span class="pre">Filter.Eventually</span></code>, and the lemma <code class="docutils literal notranslate"><span class="pre">Filter.Eventually.and</span></code> uses the intersection property of filters to do what we just described:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">P</span> <span class="n">Q</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">(</span><span class="n">hP</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">P</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">hQ</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">Q</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">P</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">Q</span> <span class="n">n</span> <span class="o">:=</span> <span class="n">hP.and</span> <span class="n">hQ</span> </pre></div> </div> <p>This notation is so convenient and intuitive that we also have specializations when <code class="docutils literal notranslate"><span class="pre">P</span></code> is an equality or inequality statement. For example, let <code class="docutils literal notranslate"><span class="pre">u</span></code> and <code class="docutils literal notranslate"><span class="pre">v</span></code> be two sequences of real numbers, and let us show that if <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">v</span> <span class="pre">n</span></code> coincide for sufficiently large <code class="docutils literal notranslate"><span class="pre">n</span></code> then <code class="docutils literal notranslate"><span class="pre">u</span></code> tends to <code class="docutils literal notranslate"><span class="pre">x₀</span></code> if and only if <code class="docutils literal notranslate"><span class="pre">v</span></code> tends to <code class="docutils literal notranslate"><span class="pre">x₀</span></code>. First we’ll use the generic <code class="docutils literal notranslate"><span class="pre">Eventually</span></code> and then the one specialized for the equality predicate, <code class="docutils literal notranslate"><span class="pre">Eventually_eq</span></code>. The two statements are definitionally equivalent so the same proof work in both cases.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">u</span> <span class="n">v</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">v</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">)</span> <span class="bp">↔</span> <span class="n">Tendsto</span> <span class="n">v</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">)</span> <span class="o">:=</span> <span class="n">tendsto_congr'</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">(</span><span class="n">u</span> <span class="n">v</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">u</span> <span class="bp">=ᶠ</span><span class="o">[</span><span class="n">atTop</span><span class="o">]</span> <span class="n">v</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">)</span> <span class="bp">↔</span> <span class="n">Tendsto</span> <span class="n">v</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">)</span> <span class="o">:=</span> <span class="n">tendsto_congr'</span> <span class="n">h</span> </pre></div> </div> <p>It is instructive to review the definition of filters in terms of <code class="docutils literal notranslate"><span class="pre">Eventually</span></code>. Given <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">X</span></code>, for any predicates <code class="docutils literal notranslate"><span class="pre">P</span></code> and <code class="docutils literal notranslate"><span class="pre">Q</span></code> on <code class="docutils literal notranslate"><span class="pre">X</span></code>,</p> <ul class="simple"> <li><p>the condition <code class="docutils literal notranslate"><span class="pre">univ</span> <span class="pre">∈</span> <span class="pre">F</span></code> ensures <code class="docutils literal notranslate"><span class="pre">(∀</span> <span class="pre">x,</span> <span class="pre">P</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x</span></code>,</p></li> <li><p>the condition <code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span> <span class="pre">→</span> <span class="pre">U</span> <span class="pre">⊆</span> <span class="pre">V</span> <span class="pre">→</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F</span></code> ensures <code class="docutils literal notranslate"><span class="pre">(∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">(∀</span> <span class="pre">x,</span> <span class="pre">P</span> <span class="pre">x</span> <span class="pre">→</span> <span class="pre">Q</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">Q</span> <span class="pre">x</span></code>, and</p></li> <li><p>the condition <code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span> <span class="pre">→</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F</span> <span class="pre">→</span> <span class="pre">U</span> <span class="pre">∩</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F</span></code> ensures <code class="docutils literal notranslate"><span class="pre">(∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">(∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">Q</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x</span> <span class="pre">∧</span> <span class="pre">Q</span> <span class="pre">x</span></code>.</p></li> </ul> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">@</span><span class="n">eventually_of_forall</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">Eventually.mono</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">Eventually.and</span> </pre></div> </div> <p>The second item, corresponding to <code class="docutils literal notranslate"><span class="pre">eventually.mono</span></code>, supports nice ways of using filters, especially when combined with <code class="docutils literal notranslate"><span class="pre">eventually.and</span></code>. The <code class="docutils literal notranslate"><span class="pre">filter_upwards</span></code> tactic allows us to combine them. Compare:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">P</span> <span class="n">Q</span> <span class="n">R</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">(</span><span class="n">hP</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">P</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">hQ</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">Q</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">hR</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">P</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">Q</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">R</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">R</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="o">(</span><span class="n">hP.and</span> <span class="o">(</span><span class="n">hQ.and</span> <span class="n">hR</span><span class="o">))</span><span class="bp">.</span><span class="n">mono</span> <span class="n">rintro</span> <span class="n">n</span> <span class="o">⟨</span><span class="n">h</span><span class="o">,</span> <span class="n">h'</span><span class="o">,</span> <span class="n">h''</span><span class="o">⟩</span> <span class="n">exact</span> <span class="n">h''</span> <span class="o">⟨</span><span class="n">h</span><span class="o">,</span> <span class="n">h'</span><span class="o">⟩</span> <span class="kd">example</span> <span class="o">(</span><span class="n">P</span> <span class="n">Q</span> <span class="n">R</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">(</span><span class="n">hP</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">P</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">hQ</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">Q</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">hR</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">P</span> <span class="n">n</span> <span class="bp">∧</span> <span class="n">Q</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">R</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">R</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">filter_upwards</span> <span class="o">[</span><span class="n">hP</span><span class="o">,</span> <span class="n">hQ</span><span class="o">,</span> <span class="n">hR</span><span class="o">]</span> <span class="n">intro</span> <span class="n">n</span> <span class="n">h</span> <span class="n">h'</span> <span class="n">h''</span> <span class="n">exact</span> <span class="n">h''</span> <span class="o">⟨</span><span class="n">h</span><span class="o">,</span> <span class="n">h'</span><span class="o">⟩</span> </pre></div> </div> <p>Readers who know about measure theory will note that the filter <code class="docutils literal notranslate"><span class="pre">μ.ae</span></code> of sets whose complement has measure zero (aka “the set consisting of almost every point”) is not very useful as the source or target of <code class="docutils literal notranslate"><span class="pre">Tendsto</span></code>, but it can be conveniently used with <code class="docutils literal notranslate"><span class="pre">eventually</span></code> to say that a property holds for almost every point.</p> <p>There is a dual version of <code class="docutils literal notranslate"><span class="pre">∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x</span></code>, which is occasionally useful: <code class="docutils literal notranslate"><span class="pre">∃ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x</span></code> means <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">¬P</span> <span class="pre">x}</span> <span class="pre">∉</span> <span class="pre">F</span></code>. For example, <code class="docutils literal notranslate"><span class="pre">∃ᶠ</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">at_top,</span> <span class="pre">P</span> <span class="pre">n</span></code> means there are arbitrarily large <code class="docutils literal notranslate"><span class="pre">n</span></code> such that <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">n</span></code> holds. The <code class="docutils literal notranslate"><span class="pre">∃ᶠ</span></code> notation stands for <code class="docutils literal notranslate"><span class="pre">Filter.frequently</span></code>.</p> <p>For a more sophisticated example, consider the following statement about a sequence <code class="docutils literal notranslate"><span class="pre">u</span></code>, a set <code class="docutils literal notranslate"><span class="pre">M</span></code>, and a value <code class="docutils literal notranslate"><span class="pre">x</span></code>:</p> <blockquote> <div><p>If <code class="docutils literal notranslate"><span class="pre">u</span></code> converges to <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">n</span></code> belongs to <code class="docutils literal notranslate"><span class="pre">M</span></code> for sufficiently large <code class="docutils literal notranslate"><span class="pre">n</span></code> then <code class="docutils literal notranslate"><span class="pre">x</span></code> is in the closure of <code class="docutils literal notranslate"><span class="pre">M</span></code>.</p> </div></blockquote> <p>This can be formalized as follows:</p> <blockquote> <div><p><code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">u</span> <span class="pre">at_top</span> <span class="pre">(𝓝</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">(∀ᶠ</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">at_top,</span> <span class="pre">u</span> <span class="pre">n</span> <span class="pre">∈</span> <span class="pre">M)</span> <span class="pre">→</span> <span class="pre">x</span> <span class="pre">∈</span> <span class="pre">closure</span> <span class="pre">M</span></code>.</p> </div></blockquote> <p>This is a special case of the theorem <code class="docutils literal notranslate"><span class="pre">mem_closure_of_tendsto</span></code> from the topology library. See if you can prove it using the quoted lemmas, using the fact that <code class="docutils literal notranslate"><span class="pre">cluster_pt</span> <span class="pre">x</span> <span class="pre">F</span></code> means <code class="docutils literal notranslate"><span class="pre">(𝓝</span> <span class="pre">x</span> <span class="pre">⊓</span> <span class="pre">F).ne_bot</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">mem_closure_iff_clusterPt</span> <span class="k">#check</span> <span class="n">le_principal_iff</span> <span class="k">#check</span> <span class="n">neBot_of_le</span> <span class="kd">example</span> <span class="o">(</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">M</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">hux</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x</span><span class="o">))</span> <span class="o">(</span><span class="n">huM</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">n</span> <span class="k">in</span> <span class="n">atTop</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">closure</span> <span class="n">M</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="metric-spaces"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">8.2. </span>Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this heading"></a></h2> <p>Examples in the previous section focus on sequences of real numbers. In this section we will go up a bit in generality and focus on metric spaces. A metric space is a type <code class="docutils literal notranslate"><span class="pre">X</span></code> equipped with a distance function <code class="docutils literal notranslate"><span class="pre">dist</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> which is a generalization of the function <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">y</span> <span class="pre">↦</span> <span class="pre">|x</span> <span class="pre">-</span> <span class="pre">y|</span></code> from the case where <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">=</span> <span class="pre">ℝ</span></code>.</p> <p>Introducing such a space is easy and we will check all properties required from the distance function.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">X</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">X</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">dist</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">dist_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">dist</span> <span class="n">a</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">dist_eq_zero</span> <span class="o">:</span> <span class="n">dist</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">↔</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">dist_comm</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">dist</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">dist</span> <span class="n">b</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">dist_triangle</span> <span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">dist</span> <span class="n">a</span> <span class="n">c</span> <span class="bp">≤</span> <span class="n">dist</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">dist</span> <span class="n">b</span> <span class="n">c</span><span class="o">)</span> </pre></div> </div> <p>Note we also have variants where the distance can be infinite or where <code class="docutils literal notranslate"><span class="pre">dist</span> <span class="pre">a</span> <span class="pre">b</span></code> can be zero without having <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">b</span></code> or both. They are called <code class="docutils literal notranslate"><span class="pre">EMetricSpace</span></code>, <code class="docutils literal notranslate"><span class="pre">PseudoMetricSpace</span></code> and <code class="docutils literal notranslate"><span class="pre">PseudoEMetricSpace</span></code> respectively (here “e” stands for “extended”).</p> <p>Note that our journey from <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> to metric spaces jumped over the special case of normed spaces that also require linear algebra and will be explained as part of the calculus chapter.</p> <section id="convergence-and-continuity"> <h3><span class="section-number">8.2.1. </span>Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this heading"></a></h3> <p>Using distance functions, we can already define convergent sequences and continuous functions between metric spaces. They are actually defined in a more general setting covered in the next section, but we have lemmas recasting the definition is terms of distances.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">a</span><span class="o">)</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">N</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">n</span> <span class="bp">≥</span> <span class="n">N</span><span class="o">,</span> <span class="n">dist</span> <span class="o">(</span><span class="n">u</span> <span class="n">n</span><span class="o">)</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="n">Metric.tendsto_atTop</span> <span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">δ</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">x'</span><span class="o">,</span> <span class="n">dist</span> <span class="n">x'</span> <span class="n">x</span> <span class="bp"><</span> <span class="n">δ</span> <span class="bp">→</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">x'</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="n">Metric.continuous_iff</span> </pre></div> </div> <p id="index-3">A <em>lot</em> of lemmas have some continuity assumptions, no we end up proving a lot of continuity results and there is a <code class="docutils literal notranslate"><span class="pre">continuity</span></code> tactic devoted to this task. Let’s prove a continuity statement that will be needed in an exercise below. Notice that Lean knows how to treat a product of two metric spaces as a metric space, so it makes sense to consider continuous functions from <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">×</span> <span class="pre">X</span></code> to <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>. In particular the (uncurried version of the) distance function is such a function.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">×</span> <span class="n">X</span> <span class="bp">=></span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">continuity</span> </pre></div> </div> <p>This tactic is a bit slow, so it is also useful to know how to do it by hand. We first need to use that <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">p</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">×</span> <span class="pre">X</span> <span class="pre">↦</span> <span class="pre">f</span> <span class="pre">p.1</span></code> is continuous because it is the composition of <code class="docutils literal notranslate"><span class="pre">f</span></code>, which is continuous by assumption <code class="docutils literal notranslate"><span class="pre">hf</span></code>, and the projection <code class="docutils literal notranslate"><span class="pre">prod.fst</span></code> whose continuity is the content of the lemma <code class="docutils literal notranslate"><span class="pre">continuous_fst</span></code>. The composition property is <code class="docutils literal notranslate"><span class="pre">Continuous.comp</span></code> which is in the <code class="docutils literal notranslate"><span class="pre">Continuous</span></code> namespace so we can use dot notation to compress <code class="docutils literal notranslate"><span class="pre">Continuous.comp</span> <span class="pre">hf</span> <span class="pre">continuous_fst</span></code> into <code class="docutils literal notranslate"><span class="pre">hf.comp</span> <span class="pre">continuous_fst</span></code> which is actually more readable since it really reads as composing our assumption and our lemma. We can do the same for the second component to get continuity of <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">p</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">×</span> <span class="pre">X</span> <span class="pre">↦</span> <span class="pre">f</span> <span class="pre">p.2</span></code>. We then assemble those two continuities using <code class="docutils literal notranslate"><span class="pre">Continuous.prod_mk</span></code> to get <code class="docutils literal notranslate"><span class="pre">(hf.comp</span> <span class="pre">continuous_fst).prod_mk</span> <span class="pre">(hf.comp</span> <span class="pre">continuous_snd)</span> <span class="pre">:</span> <span class="pre">Continuous</span> <span class="pre">(fun</span> <span class="pre">p</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">×</span> <span class="pre">X</span> <span class="pre">↦</span> <span class="pre">(f</span> <span class="pre">p.1,</span> <span class="pre">f</span> <span class="pre">p.2))</span></code> and compose once more to get our full proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">×</span> <span class="n">X</span> <span class="bp">=></span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="n">continuous_dist.comp</span> <span class="o">((</span><span class="n">hf.comp</span> <span class="n">continuous_fst</span><span class="o">)</span><span class="bp">.</span><span class="n">prod_mk</span> <span class="o">(</span><span class="n">hf.comp</span> <span class="n">continuous_snd</span><span class="o">))</span> </pre></div> </div> <p>The combination of <code class="docutils literal notranslate"><span class="pre">Continuous.prod_mk</span></code> and <code class="docutils literal notranslate"><span class="pre">continuous_dist</span></code> via <code class="docutils literal notranslate"><span class="pre">Continuous.comp</span></code> feels clunky, even when heavily using dot notation as above. A more serious issue is that this nice proof requires a lot of planning. Lean accepts the above proof term because it is a full term proving a statement which is definitionally equivalent to our goal, the crucial definition to unfold being that of a composition of functions. Indeed our target function <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">p</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">×</span> <span class="pre">X</span> <span class="pre">↦</span> <span class="pre">dist</span> <span class="pre">(f</span> <span class="pre">p.1)</span> <span class="pre">(f</span> <span class="pre">p.2)</span></code> is not presented as a composition. The proof term we provided proves continuity of <code class="docutils literal notranslate"><span class="pre">dist</span> <span class="pre">∘</span> <span class="pre">(fun</span> <span class="pre">p</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">×</span> <span class="pre">X</span> <span class="pre">↦</span> <span class="pre">(f</span> <span class="pre">p.1,</span> <span class="pre">f</span> <span class="pre">p.2))</span></code> which happens to be definitionally equal to our target function. But if we try to build this proof gradually using tactics starting with <code class="docutils literal notranslate"><span class="pre">apply</span> <span class="pre">continuous_dist.comp</span></code> then Lean’s elaborator will fail to recognize a composition and refuse to apply this lemma. It is especially bad at this when products of types are involved.</p> <p>A better lemma to apply here is <code class="docutils literal notranslate"><span class="pre">Continuous.dist</span> <span class="pre">{f</span> <span class="pre">g</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Y}</span> <span class="pre">:</span> <span class="pre">Continuous</span> <span class="pre">f</span> <span class="pre">→</span> <span class="pre">Continuous</span> <span class="pre">g</span> <span class="pre">→</span> <span class="pre">Continuous</span> <span class="pre">(fun</span> <span class="pre">x</span> <span class="pre">↦</span> <span class="pre">dist</span> <span class="pre">(f</span> <span class="pre">x)</span> <span class="pre">(g</span> <span class="pre">x))</span></code> which is nicer to Lean’s elaborator and also provides a shorter proof when directly providing a full proof term, as can be seen from the following two new proofs of the above statement:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">×</span> <span class="n">X</span> <span class="bp">=></span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">Continuous.dist</span> <span class="n">exact</span> <span class="n">hf.comp</span> <span class="n">continuous_fst</span> <span class="n">exact</span> <span class="n">hf.comp</span> <span class="n">continuous_snd</span> <span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">×</span> <span class="n">X</span> <span class="bp">=></span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="o">(</span><span class="n">hf.comp</span> <span class="n">continuous_fst</span><span class="o">)</span><span class="bp">.</span><span class="n">dist</span> <span class="o">(</span><span class="n">hf.comp</span> <span class="n">continuous_snd</span><span class="o">)</span> </pre></div> </div> <p>Note that, without the elaboration issue coming from composition, another way to compress our proof would be to use <code class="docutils literal notranslate"><span class="pre">Continuous.prod_map</span></code> which is sometimes useful and gives as an alternate proof term <code class="docutils literal notranslate"><span class="pre">continuous_dist.comp</span> <span class="pre">(hf.prod_map</span> <span class="pre">hf)</span></code> which even shorter to type.</p> <p>Since it is sad to decide between a version which is better for elaboration and a version which is shorter to type, let us wrap this discussion with a last bit of compression offered by <code class="docutils literal notranslate"><span class="pre">Continuous.fst'</span></code> which allows to compress <code class="docutils literal notranslate"><span class="pre">hf.comp</span> <span class="pre">continuous_fst</span></code> to <code class="docutils literal notranslate"><span class="pre">hf.fst'</span></code> (and the same with <code class="docutils literal notranslate"><span class="pre">snd</span></code>) and get our final proof, now bordering obfuscation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">×</span> <span class="n">X</span> <span class="bp">=></span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="n">hf.fst'.dist</span> <span class="n">hf.snd'</span> </pre></div> </div> <p>It’s your turn now to prove some continuity lemma. After trying the continuity tactic, you will need <code class="docutils literal notranslate"><span class="pre">Continuous.add</span></code>, <code class="docutils literal notranslate"><span class="pre">continuous_pow</span></code> and <code class="docutils literal notranslate"><span class="pre">continuous_id</span></code> to do it by hand.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">=></span> <span class="n">f</span> <span class="o">(</span><span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>So far we saw continuity as a global notion, but one can also define continuity at a point.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">X</span><span class="o">)</span> <span class="o">:</span> <span class="n">ContinuousAt</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">δ</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">x</span><span class="o">},</span> <span class="n">dist</span> <span class="n">x</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">δ</span> <span class="bp">→</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="n">Metric.continuousAt_iff</span> </pre></div> </div> </section> <section id="balls-open-sets-and-closed-sets"> <h3><span class="section-number">8.2.2. </span>Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this heading"></a></h3> <p>Once we have a distance function, the most important geometric definitions are (open) balls and closed balls.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">r</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Metric.ball</span> <span class="n">a</span> <span class="n">r</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">b</span> <span class="bp">|</span> <span class="n">dist</span> <span class="n">b</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">r</span> <span class="o">}</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">Metric.closedBall</span> <span class="n">a</span> <span class="n">r</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">b</span> <span class="bp">|</span> <span class="n">dist</span> <span class="n">b</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">r</span> <span class="o">}</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>Note that <cite>r</cite> is any real number here, there is no sign restriction. Of course some statements do require a radius condition.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">hr</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">r</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">Metric.ball</span> <span class="n">a</span> <span class="n">r</span> <span class="o">:=</span> <span class="n">Metric.mem_ball_self</span> <span class="n">hr</span> <span class="kd">example</span> <span class="o">(</span><span class="n">hr</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">r</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">Metric.closedBall</span> <span class="n">a</span> <span class="n">r</span> <span class="o">:=</span> <span class="n">Metric.mem_closedBall_self</span> <span class="n">hr</span> </pre></div> </div> <p>Once we have balls, we can define open sets. They are actually defined in a more general setting covered in the next section, but we have lemmas recasting the definition is terms of balls.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">)</span> <span class="o">:</span> <span class="n">IsOpen</span> <span class="n">s</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="n">Metric.ball</span> <span class="n">x</span> <span class="n">ε</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">Metric.isOpen_iff</span> </pre></div> </div> <p>Then closed sets are sets whose complement is open. Their important property is they are closed under limits. The closure of a set is the smallest subset containing it.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">IsClosed</span> <span class="n">s</span> <span class="bp">↔</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="n">s</span><span class="bp">ᶜ</span><span class="o">)</span> <span class="o">:=</span> <span class="n">isOpen_compl_iff.symm</span> <span class="kd">example</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsClosed</span> <span class="n">s</span><span class="o">)</span> <span class="o">{</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hu</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">a</span><span class="o">))</span> <span class="o">(</span><span class="n">hus</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">hs.mem_of_tendsto</span> <span class="n">hu</span> <span class="o">(</span><span class="n">eventually_of_forall</span> <span class="n">hus</span><span class="o">)</span> <span class="kd">example</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">closure</span> <span class="n">s</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">b</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">Metric.ball</span> <span class="n">b</span> <span class="n">ε</span> <span class="o">:=</span> <span class="n">Metric.mem_closure_iff</span> </pre></div> </div> <p>Do the next exercise without using <cite>mem_closure_iff_seq_limit</cite></p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hu</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">a</span><span class="o">))</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">closure</span> <span class="n">s</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Remember from the filters sections that neighborhood filters play a big role in mathlib. In the metric space context, the crucial point is that balls provide bases for those filters. The main lemmas here are <code class="docutils literal notranslate"><span class="pre">Metric.nhds_basis_ball</span></code> and <code class="docutils literal notranslate"><span class="pre">Metric.nhds_basis_closedBall</span></code> that claim this for open and closed balls with positive radius. The center point is an implicit argument so we can invoke <code class="docutils literal notranslate"><span class="pre">Filter.HasBasis.mem_iff</span></code> as in the following example.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∈</span> <span class="bp">𝓝</span> <span class="n">x</span> <span class="bp">↔</span> <span class="bp">∃</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="n">Metric.ball</span> <span class="n">x</span> <span class="n">ε</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">Metric.nhds_basis_ball.mem_iff</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∈</span> <span class="bp">𝓝</span> <span class="n">x</span> <span class="bp">↔</span> <span class="bp">∃</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="n">Metric.closedBall</span> <span class="n">x</span> <span class="n">ε</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">Metric.nhds_basis_closedBall.mem_iff</span> </pre></div> </div> </section> <section id="compactness"> <h3><span class="section-number">8.2.3. </span>Compactness<a class="headerlink" href="#compactness" title="Permalink to this heading"></a></h3> <p>Compactness is an important topological notion. It distinguishes subsets of a metric space that enjoy the same kind of properties as segments in reals compared to other intervals:</p> <ul class="simple"> <li><p>Any sequence taking value in a compact set has a subsequence that converges in this set</p></li> <li><p>Any continuous function on a nonempty compact set with values in real numbers is bounded and achieves its bounds somewhere (this is called the extreme values theorem).</p></li> <li><p>Compact sets are closed sets.</p></li> </ul> <p>Let us first check that the unit interval in reals is indeed a compact set, and then check the above claims for compact sets in general metric spaces. In the second statement we only need continuity on the given set so we will use <code class="docutils literal notranslate"><span class="pre">ContinuousOn</span></code> instead of <code class="docutils literal notranslate"><span class="pre">Continuous</span></code>, and we will give separate statements for the minimum and the maximum. Of course all these results are deduced from more general versions, some of which will be discussed in later sections.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="o">(</span><span class="n">Set.Icc</span> <span class="mi">0</span> <span class="mi">1</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:=</span> <span class="n">isCompact_Icc</span> <span class="kd">example</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span><span class="o">)</span> <span class="o">{</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hu</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">φ</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">StrictMono</span> <span class="n">φ</span> <span class="bp">∧</span> <span class="n">Tendsto</span> <span class="o">(</span><span class="n">u</span> <span class="bp">∘</span> <span class="n">φ</span><span class="o">)</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">hs.tendsto_subseq</span> <span class="n">hu</span> <span class="kd">example</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span><span class="o">)</span> <span class="o">(</span><span class="n">hs'</span> <span class="o">:</span> <span class="n">s.Nonempty</span><span class="o">)</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">hfs</span> <span class="o">:</span> <span class="n">ContinuousOn</span> <span class="n">f</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">y</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">f</span> <span class="n">y</span> <span class="o">:=</span> <span class="n">hs.exists_forall_le</span> <span class="n">hs'</span> <span class="n">hfs</span> <span class="kd">example</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span><span class="o">)</span> <span class="o">(</span><span class="n">hs'</span> <span class="o">:</span> <span class="n">s.Nonempty</span><span class="o">)</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">hfs</span> <span class="o">:</span> <span class="n">ContinuousOn</span> <span class="n">f</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">y</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">f</span> <span class="n">y</span> <span class="bp">≤</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">hs.exists_forall_ge</span> <span class="n">hs'</span> <span class="n">hfs</span> <span class="kd">example</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="n">IsClosed</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">hs.isClosed</span> </pre></div> </div> <p>We can also metric spaces which are globally compact, using an extra <code class="docutils literal notranslate"><span class="pre">Prop</span></code>-valued type class:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">CompactSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="o">(</span><span class="n">univ</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">)</span> <span class="o">:=</span> <span class="n">isCompact_univ</span> </pre></div> </div> <p>In a compact metric space any closed set is compact, this is <code class="docutils literal notranslate"><span class="pre">IsCompact.isClosed</span></code>.</p> </section> <section id="uniformly-continuous-functions"> <h3><span class="section-number">8.2.4. </span>Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this heading"></a></h3> <p>We now turn to uniformity notions on metric spaces : uniformly continuous functions, Cauchy sequences and completeness. Again those are defined in a more general context but we have lemmas in the metric name space to access their elementary definitions. We start with uniform continuity.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">:</span> <span class="n">UniformContinuous</span> <span class="n">f</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">δ</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∀</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">X</span><span class="o">},</span> <span class="n">dist</span> <span class="n">a</span> <span class="n">b</span> <span class="bp"><</span> <span class="n">δ</span> <span class="bp">→</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">b</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="n">Metric.uniformContinuous_iff</span> </pre></div> </div> <p>In order to practice manipulating all those definitions, we will prove that continuous functions from a compact metric space to a metric space are uniformly continuous (we will see a more general version in a later section).</p> <p>We will first give an informal sketch. Let <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Y</span></code> be a continuous function from a compact metric space to a metric space. We fix <code class="docutils literal notranslate"><span class="pre">ε</span> <span class="pre">></span> <span class="pre">0</span></code> and start looking for some <code class="docutils literal notranslate"><span class="pre">δ</span></code>.</p> <p>Let <code class="docutils literal notranslate"><span class="pre">φ</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">×</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">ℝ</span> <span class="pre">:=</span> <span class="pre">fun</span> <span class="pre">p</span> <span class="pre">↦</span> <span class="pre">dist</span> <span class="pre">(f</span> <span class="pre">p.1)</span> <span class="pre">(f</span> <span class="pre">p.2)</span></code> and let <code class="docutils literal notranslate"><span class="pre">K</span> <span class="pre">:=</span> <span class="pre">{</span> <span class="pre">p</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">×</span> <span class="pre">X</span> <span class="pre">|</span> <span class="pre">ε</span> <span class="pre">≤</span> <span class="pre">φ</span> <span class="pre">p</span> <span class="pre">}</span></code>. Observe <code class="docutils literal notranslate"><span class="pre">φ</span></code> is continuous since <code class="docutils literal notranslate"><span class="pre">f</span></code> and distance are continuous. And <code class="docutils literal notranslate"><span class="pre">K</span></code> is clearly closed (use <code class="docutils literal notranslate"><span class="pre">isClosed_le</span></code>) hence compact since <code class="docutils literal notranslate"><span class="pre">X</span></code> is compact.</p> <p>Then we discuss two possibilities using <code class="docutils literal notranslate"><span class="pre">eq_empty_or_nonempty</span></code>. If <code class="docutils literal notranslate"><span class="pre">K</span></code> is empty then we are clearly done (we can set <code class="docutils literal notranslate"><span class="pre">δ</span> <span class="pre">=</span> <span class="pre">1</span></code> for instance). So let’s assume <code class="docutils literal notranslate"><span class="pre">K</span></code> is not empty, and use the extreme value theorem to choose <code class="docutils literal notranslate"><span class="pre">(x₀,</span> <span class="pre">x₁)</span></code> attaining the infimum of the distance function on <code class="docutils literal notranslate"><span class="pre">K</span></code>. We can then set <code class="docutils literal notranslate"><span class="pre">δ</span> <span class="pre">=</span> <span class="pre">dist</span> <span class="pre">x₀</span> <span class="pre">x₁</span></code> and check everything works.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">CompactSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">UniformContinuous</span> <span class="n">f</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="completeness"> <h3><span class="section-number">8.2.5. </span>Completeness<a class="headerlink" href="#completeness" title="Permalink to this heading"></a></h3> <p>A Cauchy sequence in a metric space is a sequence whose terms get closer and closer to each other. There are a couple of equivalent ways to state that idea. In particular converging sequences are Cauchy. The converse is true only in so-called <em>complete</em> spaces.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">)</span> <span class="o">:</span> <span class="n">CauchySeq</span> <span class="n">u</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">N</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">m</span> <span class="bp">≥</span> <span class="n">N</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">n</span> <span class="bp">≥</span> <span class="n">N</span><span class="o">,</span> <span class="n">dist</span> <span class="o">(</span><span class="n">u</span> <span class="n">m</span><span class="o">)</span> <span class="o">(</span><span class="n">u</span> <span class="n">n</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="n">Metric.cauchySeq_iff</span> <span class="kd">example</span> <span class="o">(</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">)</span> <span class="o">:</span> <span class="n">CauchySeq</span> <span class="n">u</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">N</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">n</span> <span class="bp">≥</span> <span class="n">N</span><span class="o">,</span> <span class="n">dist</span> <span class="o">(</span><span class="n">u</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">u</span> <span class="n">N</span><span class="o">)</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="n">Metric.cauchySeq_iff'</span> <span class="kd">example</span> <span class="o">[</span><span class="n">CompleteSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">(</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">hu</span> <span class="o">:</span> <span class="n">CauchySeq</span> <span class="n">u</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="n">cauchySeq_tendsto_of_complete</span> <span class="n">hu</span> </pre></div> </div> <p>We’ll practice using this definition by proving a convenient criterion which is a special case of a criterion appearing in mathlib. This is also a good opportunity to practice using big sums in a geometric context. In addition to the explanations from the filters section, you will probably need <code class="docutils literal notranslate"><span class="pre">tendsto_pow_atTop_nhds_0_of_lt_1</span></code>, <code class="docutils literal notranslate"><span class="pre">Tendsto.mul</span></code> and <code class="docutils literal notranslate"><span class="pre">dist_le_range_sum_dist</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">cauchySeq_of_le_geometric_two'</span> <span class="o">{</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hu</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">dist</span> <span class="o">(</span><span class="n">u</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">u</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">))</span> <span class="bp">≤</span> <span class="o">(</span><span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">CauchySeq</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">Metric.cauchySeq_iff'</span><span class="o">]</span> <span class="n">intro</span> <span class="n">ε</span> <span class="n">ε_pos</span> <span class="n">obtain</span> <span class="o">⟨</span><span class="n">N</span><span class="o">,</span> <span class="n">hN</span><span class="o">⟩</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">N</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">^</span> <span class="n">N</span> <span class="bp">*</span> <span class="mi">2</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">use</span> <span class="n">N</span> <span class="n">intro</span> <span class="n">n</span> <span class="n">hn</span> <span class="n">obtain</span> <span class="o">⟨</span><span class="n">k</span><span class="o">,</span> <span class="n">rfl</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">N</span> <span class="bp">+</span> <span class="n">k</span><span class="o">⟩</span> <span class="o">:=</span> <span class="n">le_iff_exists_add.mp</span> <span class="n">hn</span> <span class="k">calc</span> <span class="n">dist</span> <span class="o">(</span><span class="n">u</span> <span class="o">(</span><span class="n">N</span> <span class="bp">+</span> <span class="n">k</span><span class="o">))</span> <span class="o">(</span><span class="n">u</span> <span class="n">N</span><span class="o">)</span> <span class="bp">=</span> <span class="n">dist</span> <span class="o">(</span><span class="n">u</span> <span class="o">(</span><span class="n">N</span> <span class="bp">+</span> <span class="mi">0</span><span class="o">))</span> <span class="o">(</span><span class="n">u</span> <span class="o">(</span><span class="n">N</span> <span class="bp">+</span> <span class="n">k</span><span class="o">))</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">≤</span> <span class="bp">∑</span> <span class="n">i</span> <span class="k">in</span> <span class="n">range</span> <span class="n">k</span><span class="o">,</span> <span class="n">dist</span> <span class="o">(</span><span class="n">u</span> <span class="o">(</span><span class="n">N</span> <span class="bp">+</span> <span class="n">i</span><span class="o">))</span> <span class="o">(</span><span class="n">u</span> <span class="o">(</span><span class="n">N</span> <span class="bp">+</span> <span class="o">(</span><span class="n">i</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)))</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">≤</span> <span class="bp">∑</span> <span class="n">i</span> <span class="k">in</span> <span class="n">range</span> <span class="n">k</span><span class="o">,</span> <span class="o">(</span><span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="bp">^</span> <span class="o">(</span><span class="n">N</span> <span class="bp">+</span> <span class="n">i</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">=</span> <span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">^</span> <span class="n">N</span> <span class="bp">*</span> <span class="bp">∑</span> <span class="n">i</span> <span class="k">in</span> <span class="n">range</span> <span class="n">k</span><span class="o">,</span> <span class="o">(</span><span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="n">i</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">≤</span> <span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">^</span> <span class="n">N</span> <span class="bp">*</span> <span class="mi">2</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp"><</span> <span class="n">ε</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>We are ready for the final boss of this section: Baire’s theorem for complete metric spaces! The proof skeleton below shows interesting techniques. It uses the <code class="docutils literal notranslate"><span class="pre">choose</span></code> tactic in its exclamation mark variant (you should experiment with removing this exclamation mark) and it shows how to define something inductively in the middle of a proof using <code class="docutils literal notranslate"><span class="pre">Nat.rec_on</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Metric</span> <span class="kd">example</span> <span class="o">[</span><span class="n">CompleteSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">ho</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="n">f</span> <span class="n">n</span><span class="o">))</span> <span class="o">(</span><span class="n">hd</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">Dense</span> <span class="o">(</span><span class="n">f</span> <span class="n">n</span><span class="o">))</span> <span class="o">:</span> <span class="n">Dense</span> <span class="o">(</span><span class="bp">⋂</span> <span class="n">n</span><span class="o">,</span> <span class="n">f</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="k">let</span> <span class="n">B</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="o">(</span><span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="n">n</span> <span class="k">have</span> <span class="n">Bpos</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">B</span> <span class="n">n</span> <span class="gr">sorry</span> <span class="c">/-</span><span class="cm"> Translate the density assumption into two functions `center` and `radius` associating</span> <span class="cm"> to any n, x, δ, δpos a center and a positive radius such that</span> <span class="cm"> `closedBall center radius` is included both in `f n` and in `closedBall x δ`.</span> <span class="cm"> We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/</span> <span class="k">have</span> <span class="o">:</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">),</span> <span class="bp">∀</span> <span class="n">δ</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">y</span> <span class="o">:</span> <span class="n">X</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">r</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="n">r</span> <span class="bp">≤</span> <span class="n">B</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">)</span> <span class="bp">∧</span> <span class="n">closedBall</span> <span class="n">y</span> <span class="n">r</span> <span class="bp">⊆</span> <span class="n">closedBall</span> <span class="n">x</span> <span class="n">δ</span> <span class="bp">∩</span> <span class="n">f</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">choose</span><span class="bp">!</span> <span class="n">center</span> <span class="n">radius</span> <span class="n">Hpos</span> <span class="n">HB</span> <span class="n">Hball</span> <span class="n">using</span> <span class="n">this</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mem_closure_iff_nhds_basis</span> <span class="n">nhds_basis_closedBall</span><span class="o">]</span> <span class="n">intro</span> <span class="n">ε</span> <span class="n">εpos</span> <span class="c">/-</span><span class="cm"> `ε` is positive. We have to find a point in the ball of radius `ε` around `x`</span> <span class="cm"> belonging to all `f n`. For this, we construct inductively a sequence</span> <span class="cm"> `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included</span> <span class="cm"> in the previous ball and in `f n`, and such that `r n` is small enough to ensure</span> <span class="cm"> that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs</span> <span class="cm"> to all the `f n`. -/</span> <span class="k">let</span> <span class="n">F</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span> <span class="bp">×</span> <span class="n">ℝ</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="n">Nat.recOn</span> <span class="n">n</span> <span class="o">(</span><span class="n">Prod.mk</span> <span class="n">x</span> <span class="o">(</span><span class="n">min</span> <span class="n">ε</span> <span class="o">(</span><span class="n">B</span> <span class="mi">0</span><span class="o">)))</span> <span class="k">fun</span> <span class="n">n</span> <span class="n">p</span> <span class="bp">=></span> <span class="n">Prod.mk</span> <span class="o">(</span><span class="n">center</span> <span class="n">n</span> <span class="n">p.1</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">(</span><span class="n">radius</span> <span class="n">n</span> <span class="n">p.1</span> <span class="n">p.2</span><span class="o">)</span> <span class="k">let</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="o">(</span><span class="n">F</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="mi">1</span> <span class="k">let</span> <span class="n">r</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℝ</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="o">(</span><span class="n">F</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="mi">2</span> <span class="k">have</span> <span class="n">rpos</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">r</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">rB</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">r</span> <span class="n">n</span> <span class="bp">≤</span> <span class="n">B</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">incl</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">closedBall</span> <span class="o">(</span><span class="n">c</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">))</span> <span class="o">(</span><span class="n">r</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">))</span> <span class="bp">⊆</span> <span class="n">closedBall</span> <span class="o">(</span><span class="n">c</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">r</span> <span class="n">n</span><span class="o">)</span> <span class="bp">∩</span> <span class="n">f</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">cdist</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">dist</span> <span class="o">(</span><span class="n">c</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">c</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">))</span> <span class="bp">≤</span> <span class="n">B</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="o">:</span> <span class="n">CauchySeq</span> <span class="n">c</span> <span class="o">:=</span> <span class="n">cauchySeq_of_le_geometric_two'</span> <span class="n">cdist</span> <span class="c1">-- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`.</span> <span class="n">rcases</span> <span class="n">cauchySeq_tendsto_of_complete</span> <span class="n">this</span> <span class="k">with</span> <span class="o">⟨</span><span class="n">y</span><span class="o">,</span> <span class="n">ylim</span><span class="o">⟩</span> <span class="c1">-- this point `y` will be the desired point. We will check that it belongs to all</span> <span class="c1">-- `f n` and to `ball x ε`.</span> <span class="n">use</span> <span class="n">y</span> <span class="k">have</span> <span class="n">I</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">m</span> <span class="bp">≥</span> <span class="n">n</span><span class="o">,</span> <span class="n">closedBall</span> <span class="o">(</span><span class="n">c</span> <span class="n">m</span><span class="o">)</span> <span class="o">(</span><span class="n">r</span> <span class="n">m</span><span class="o">)</span> <span class="bp">⊆</span> <span class="n">closedBall</span> <span class="o">(</span><span class="n">c</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">r</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">yball</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">y</span> <span class="bp">∈</span> <span class="n">closedBall</span> <span class="o">(</span><span class="n">c</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">r</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="gr">sorry</span> </pre></div> </div> </section> </section> <section id="topological-spaces"> <span id="index-4"></span><span id="id4"></span><h2><span class="section-number">8.3. </span>Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> <section id="fundamentals"> <h3><span class="section-number">8.3.1. </span>Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> <p>We now go up in generality and introduce topological spaces. We will review the two main ways to define topological spaces and then explain how the category of topological spaces is much better behaved than the category of metric spaces. Note that we won’t be using mathlib category theory here, only having a somewhat categorical point of view.</p> <p>The first way to think about the transition from metric spaces to topological spaces is that we only remember the notion of open sets (or equivalently the notion of closed sets). From this point of view, a topological space is a type equipped with a collection of sets that are called open sets. This collection has to satisfy a number of axioms presented below (this collection is slightly redundant but we will ignore that).</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">section</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">X</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="n">univ</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">)</span> <span class="o">:=</span> <span class="n">isOpen_univ</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="bp">∅</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">)</span> <span class="o">:=</span> <span class="n">isOpen_empty</span> <span class="kd">example</span> <span class="o">{</span><span class="n">ι</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">ι</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">i</span><span class="o">,</span> <span class="n">IsOpen</span> <span class="bp"><|</span> <span class="n">s</span> <span class="n">i</span><span class="o">)</span> <span class="o">:</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="bp">⋃</span> <span class="n">i</span><span class="o">,</span> <span class="n">s</span> <span class="n">i</span><span class="o">)</span> <span class="o">:=</span> <span class="n">isOpen_iUnion</span> <span class="n">hs</span> <span class="kd">example</span> <span class="o">{</span><span class="n">ι</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Fintype</span> <span class="n">ι</span><span class="o">]</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">ι</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">i</span><span class="o">,</span> <span class="n">IsOpen</span> <span class="bp"><|</span> <span class="n">s</span> <span class="n">i</span><span class="o">)</span> <span class="o">:</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="bp">⋂</span> <span class="n">i</span><span class="o">,</span> <span class="n">s</span> <span class="n">i</span><span class="o">)</span> <span class="o">:=</span> <span class="n">isOpen_iInter</span> <span class="n">hs</span> </pre></div> </div> <p>Closed sets are then defined as sets whose complement is open. A function between topological spaces is (globally) continuous if all preimages of open sets are open.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">s</span><span class="o">,</span> <span class="n">IsOpen</span> <span class="n">s</span> <span class="bp">→</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">s</span><span class="o">)</span> <span class="o">:=</span> <span class="n">continuous_def</span> </pre></div> </div> <p>With this definition we already see that, compared to metric spaces, topological spaces only remember enough information to talk about continuous functions: two topological structures on a type are the same if and only if they have the same continuous functions (indeed the identity function will be continuous in both direction if and only if the two structures have the same open sets).</p> <p>However as soon as we move on to continuity at a point we see the limitations of the approach based on open sets. In mathlib it is much more frequent to think of topological spaces as types equipped with a neighborhood filter <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x</span></code> attached to each point <code class="docutils literal notranslate"><span class="pre">x</span></code> (the corresponding function <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">X</span></code> satisfies certain conditions explained further down). Remember from the filters section that these gadget play two related roles. First <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x</span></code> is seen as the generalized set of points of <code class="docutils literal notranslate"><span class="pre">X</span></code> that are close to <code class="docutils literal notranslate"><span class="pre">x</span></code>. And then it is seen as giving a way to say, for any predicate <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Prop</span></code>, that this predicates holds for points that are close enough to <code class="docutils literal notranslate"><span class="pre">x</span></code>. Let us state that <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Y</span></code> is continuous at <code class="docutils literal notranslate"><span class="pre">x</span></code>. The purely filtery way is to say that the direct image under <code class="docutils literal notranslate"><span class="pre">f</span></code> of the generalized set of points that are close to <code class="docutils literal notranslate"><span class="pre">x</span></code> is contained in the generalized set of points that are close to <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code>. Recall this spelled either <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">f</span> <span class="pre">(𝓝</span> <span class="pre">x)</span> <span class="pre">≤</span> <span class="pre">𝓝</span> <span class="pre">(f</span> <span class="pre">x)</span></code> or <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">f</span> <span class="pre">(𝓝</span> <span class="pre">x)</span> <span class="pre">(𝓝</span> <span class="pre">(f</span> <span class="pre">x))</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">ContinuousAt</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">↔</span> <span class="n">map</span> <span class="n">f</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x</span><span class="o">)</span> <span class="bp">≤</span> <span class="bp">𝓝</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="n">Iff.rfl</span> </pre></div> </div> <p>One can also spell it using both neighborhoods seen as ordinary sets and a neighborhood filter seen as a generalized set: “for any neighborhood <code class="docutils literal notranslate"><span class="pre">U</span></code> of <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code>, all points close to <code class="docutils literal notranslate"><span class="pre">x</span></code> are sent to <code class="docutils literal notranslate"><span class="pre">U</span></code>”. Note that the proof is again <code class="docutils literal notranslate"><span class="pre">iff.rfl</span></code>, this point of view is definitionally equivalent to the previous one.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">ContinuousAt</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">U</span> <span class="bp">∈</span> <span class="bp">𝓝</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">),</span> <span class="bp">∀ᶠ</span> <span class="n">x</span> <span class="k">in</span> <span class="bp">𝓝</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">U</span> <span class="o">:=</span> <span class="n">Iff.rfl</span> </pre></div> </div> <p>We now explain how to go from one point of view to the other. In terms of open sets, we can simply define members of <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x</span></code> as sets that contain an open set containing <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">∈</span> <span class="bp">𝓝</span> <span class="n">x</span> <span class="bp">↔</span> <span class="bp">∃</span> <span class="n">t</span><span class="o">,</span> <span class="n">t</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="bp">∧</span> <span class="n">IsOpen</span> <span class="n">t</span> <span class="bp">∧</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">t</span> <span class="o">:=</span> <span class="n">mem_nhds_iff</span> </pre></div> </div> <p>To go in the other direction we need to discuss the condition that <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">X</span></code> must satisfy in order to be the neighborhood function of a topology.</p> <p>The first constraint is that <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x</span></code>, seen as a generalized set, contains the set <code class="docutils literal notranslate"><span class="pre">{x}</span></code> seen as the generalized set <code class="docutils literal notranslate"><span class="pre">pure</span> <span class="pre">x</span></code> (explaining this weird name would be too much of a digression, so we simply accept it for now). Another way to say it is that if a predicate holds for points close to <code class="docutils literal notranslate"><span class="pre">x</span></code> then it holds at <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">)</span> <span class="o">:</span> <span class="n">pure</span> <span class="n">x</span> <span class="bp">≤</span> <span class="bp">𝓝</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">pure_le_nhds</span> <span class="n">x</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">P</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">y</span> <span class="k">in</span> <span class="bp">𝓝</span> <span class="n">x</span><span class="o">,</span> <span class="n">P</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="n">P</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">pure_le_nhds</span> <span class="n">x</span> <span class="n">h</span> </pre></div> </div> <p>Then a more subtle requirement is that, for any predicate <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Prop</span></code> and any <code class="docutils literal notranslate"><span class="pre">x</span></code>, if <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">y</span></code> holds for <code class="docutils literal notranslate"><span class="pre">y</span></code> close to <code class="docutils literal notranslate"><span class="pre">x</span></code> then for <code class="docutils literal notranslate"><span class="pre">y</span></code> close to <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">z</span></code> close to <code class="docutils literal notranslate"><span class="pre">y</span></code>, <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">z</span></code> holds. More precisely we have:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">P</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">y</span> <span class="k">in</span> <span class="bp">𝓝</span> <span class="n">x</span><span class="o">,</span> <span class="n">P</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">y</span> <span class="k">in</span> <span class="bp">𝓝</span> <span class="n">x</span><span class="o">,</span> <span class="bp">∀ᶠ</span> <span class="n">z</span> <span class="k">in</span> <span class="bp">𝓝</span> <span class="n">y</span><span class="o">,</span> <span class="n">P</span> <span class="n">z</span> <span class="o">:=</span> <span class="n">eventually_eventually_nhds.mpr</span> <span class="n">h</span> </pre></div> </div> <p>Those two results characterize the functions <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">X</span></code> that are neighborhood functions for a topological space structure on <code class="docutils literal notranslate"><span class="pre">X</span></code>. There is a still a function <code class="docutils literal notranslate"><span class="pre">topological_space.mk_of_nhds</span> <span class="pre">:</span> <span class="pre">(X</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">X)</span> <span class="pre">→</span> <span class="pre">topological_space</span> <span class="pre">X</span></code> but it will give back its input as a neighborhood function only if it satisfies the above two constraints. More precisely we have a lemma <code class="docutils literal notranslate"><span class="pre">topological_space.nhds_mk_of_nhds</span></code> saying that in a different way and our next exercise deduces this different way from how we stated it above.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">Filter</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">H₀</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="n">pure</span> <span class="n">a</span> <span class="bp">≤</span> <span class="n">n</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">H</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">α</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">p</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="kt">Prop</span><span class="o">,</span> <span class="o">(</span><span class="bp">∀ᶠ</span> <span class="n">x</span> <span class="k">in</span> <span class="n">n</span> <span class="n">a</span><span class="o">,</span> <span class="n">p</span> <span class="n">x</span><span class="o">)</span> <span class="bp">→</span> <span class="bp">∀ᶠ</span> <span class="n">y</span> <span class="k">in</span> <span class="n">n</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∀ᶠ</span> <span class="n">x</span> <span class="k">in</span> <span class="n">n</span> <span class="n">y</span><span class="o">,</span> <span class="n">p</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">s</span> <span class="bp">∈</span> <span class="n">n</span> <span class="n">a</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">t</span> <span class="bp">∈</span> <span class="n">n</span> <span class="n">a</span><span class="o">,</span> <span class="n">t</span> <span class="bp">⊆</span> <span class="n">s</span> <span class="bp">∧</span> <span class="bp">∀</span> <span class="n">a'</span> <span class="bp">∈</span> <span class="n">t</span><span class="o">,</span> <span class="n">s</span> <span class="bp">∈</span> <span class="n">n</span> <span class="n">a'</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Note that <code class="docutils literal notranslate"><span class="pre">topological_space.mk_of_nhds</span></code> is not so frequently used, but it still good to know in what precise sense the neighborhood filters is all there is in a topological space structure.</p> <p>The next thing to know in order to efficiently use topological spaces in mathlib is that we use a lot of formal properties of <code class="docutils literal notranslate"><span class="pre">topological_space</span> <span class="pre">:</span> <span class="pre">Type</span> <span class="pre">u</span> <span class="pre">→</span> <span class="pre">Type</span> <span class="pre">u</span></code>. From a purely mathematical point of view, those formal properties are a very clean way to explain how topological spaces solve issues that metric spaces have. From this point of view, the issues solved by topological spaces is that metric spaces enjoy very little fonctoriality, and have very bad categorical properties in general. This comes on top of the fact already discussed that metric spaces contain a lot of geometrical information that is not topologically relevant.</p> <p>Let us focus on fonctoriality first. A metric space structure can be induced on a subset or, equivalently, it can be pulled back by an injective map. But that’s pretty much everything. They cannot be pulled back by general map or pushed forward, even by surjective maps.</p> <p>In particular there is no sensible distance to put on a quotient of a metric space or on an uncountable products of metric spaces. Consider for instance the type <code class="docutils literal notranslate"><span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code>, seen as a product of copies of <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> indexed by <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>. We would like to say that pointwise convergence of sequences of functions is a respectable notion of convergence. But there is no distance on <code class="docutils literal notranslate"><span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> that gives this notion of convergence. Relatedly, there is no distance ensuring that a map <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">(ℝ</span> <span class="pre">→</span> <span class="pre">ℝ)</span></code> is continuous if and only <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">↦</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">t</span></code> is continuous for every <code class="docutils literal notranslate"><span class="pre">t</span> <span class="pre">:</span> <span class="pre">ℝ</span></code>.</p> <p>We now review the data used to solve all those issues. First we can use any map <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Y</span></code> to push or pull topologies from one side to the other. Those two operations form a Galois connection.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">TopologicalSpace</span> <span class="n">Y</span> <span class="o">:=</span> <span class="n">TopologicalSpace.coinduced</span> <span class="n">f</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">Y</span> <span class="bp">→</span> <span class="n">TopologicalSpace</span> <span class="n">X</span> <span class="o">:=</span> <span class="n">TopologicalSpace.induced</span> <span class="n">f</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">(</span><span class="n">T_X</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">T_Y</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">Y</span><span class="o">)</span> <span class="o">:</span> <span class="n">TopologicalSpace.coinduced</span> <span class="n">f</span> <span class="n">T_X</span> <span class="bp">≤</span> <span class="n">T_Y</span> <span class="bp">↔</span> <span class="n">T_X</span> <span class="bp">≤</span> <span class="n">TopologicalSpace.induced</span> <span class="n">f</span> <span class="n">T_Y</span> <span class="o">:=</span> <span class="n">coinduced_le_iff_le_induced</span> </pre></div> </div> <p>Those operations are compactible with composition of functions. As usual, pushing forward is covariant and pulling back is contravariant, see <code class="docutils literal notranslate"><span class="pre">coinduced_compose</span></code> and <code class="docutils literal notranslate"><span class="pre">induced_compose</span></code>. On paper we will use notations <span class="math notranslate nohighlight">\(f_*T\)</span> for <code class="docutils literal notranslate"><span class="pre">topological_space.coinduced</span> <span class="pre">f</span> <span class="pre">T</span></code> and <span class="math notranslate nohighlight">\(f^*T\)</span> for <code class="docutils literal notranslate"><span class="pre">topological_space.induced</span> <span class="pre">f</span> <span class="pre">T</span></code>.</p> <p>Then the next big piece is a complete lattice structure on <code class="docutils literal notranslate"><span class="pre">topological_structure</span> <span class="pre">X</span></code> for any given structure. If you think of topologies are being primarily the data of open sets then you expect the order relation on <code class="docutils literal notranslate"><span class="pre">topological_structure</span> <span class="pre">X</span></code> to come from <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">(Set</span> <span class="pre">X)</span></code>, ie you expect <code class="docutils literal notranslate"><span class="pre">t</span> <span class="pre">≤</span> <span class="pre">t'</span></code> if a set <code class="docutils literal notranslate"><span class="pre">u</span></code> is open for <code class="docutils literal notranslate"><span class="pre">t'</span></code> as soon as it is open for <code class="docutils literal notranslate"><span class="pre">t</span></code>. However we already know that mathlib focuses on neighborhoods more than open sets so, for any <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">X</span></code> we want <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">T</span> <span class="pre">:</span> <span class="pre">topological_space</span> <span class="pre">X</span> <span class="pre">↦</span> <span class="pre">@nhds</span> <span class="pre">X</span> <span class="pre">T</span> <span class="pre">x</span></code> to be order preserving. And we know the order relation on <code class="docutils literal notranslate"><span class="pre">filter</span> <span class="pre">X</span></code> is designed to ensure an order preserving <code class="docutils literal notranslate"><span class="pre">principal</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">Filter</span> <span class="pre">X</span></code>, allowing to see filters as generalized sets. So the order relation we do use on <code class="docutils literal notranslate"><span class="pre">topological_structure</span> <span class="pre">X</span></code> is opposite to the one coming from <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">(Set</span> <span class="pre">X)</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">T</span> <span class="n">T'</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">T</span> <span class="bp">≤</span> <span class="n">T'</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">s</span><span class="o">,</span> <span class="n">T'.IsOpen</span> <span class="n">s</span> <span class="bp">→</span> <span class="n">T.IsOpen</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">Iff.rfl</span> </pre></div> </div> <p>Now we can recover continuity by combining the push-foward (or pull-back) operation with the order relation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">T_X</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">T_Y</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">Y</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span> <span class="bp">↔</span> <span class="n">TopologicalSpace.coinduced</span> <span class="n">f</span> <span class="n">T_X</span> <span class="bp">≤</span> <span class="n">T_Y</span> <span class="o">:=</span> <span class="n">continuous_iff_coinduced_le</span> </pre></div> </div> <p>With this definition and the compatibility of push-forward and composition, we get for free the universal property that, for any topological space <span class="math notranslate nohighlight">\(Z\)</span>, a function <span class="math notranslate nohighlight">\(g : Y → Z\)</span> is continuous for the topology <span class="math notranslate nohighlight">\(f_*T_X\)</span> if and only if <span class="math notranslate nohighlight">\(g ∘ f\)</span> is continuous.</p> <div class="math notranslate nohighlight"> \[\begin{split}g \text{ continuous } &⇔ g_*(f_*T_X) ≤ T_Z \\ &⇔ (g ∘ f)_* T_X ≤ T_Z \\ &⇔ g ∘ f \text{ continuous}\end{split}\]</div> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">Z</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">)</span> <span class="o">(</span><span class="n">T_X</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">T_Z</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="n">Z</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">Y</span> <span class="bp">→</span> <span class="n">Z</span><span class="o">)</span> <span class="o">:</span> <span class="bp">@</span><span class="n">Continuous</span> <span class="n">Y</span> <span class="n">Z</span> <span class="o">(</span><span class="n">TopologicalSpace.coinduced</span> <span class="n">f</span> <span class="n">T_X</span><span class="o">)</span> <span class="n">T_Z</span> <span class="n">g</span> <span class="bp">↔</span> <span class="bp">@</span><span class="n">Continuous</span> <span class="n">X</span> <span class="n">Z</span> <span class="n">T_X</span> <span class="n">T_Z</span> <span class="o">(</span><span class="n">g</span> <span class="bp">∘</span> <span class="n">f</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">continuous_iff_coinduced_le</span><span class="o">,</span> <span class="n">coinduced_compose</span><span class="o">,</span> <span class="n">continuous_iff_coinduced_le</span><span class="o">]</span> </pre></div> </div> <p>So we already get quotient topologies (using the projection map as <code class="docutils literal notranslate"><span class="pre">f</span></code>). This wasn’t using that <code class="docutils literal notranslate"><span class="pre">topological_space</span> <span class="pre">X</span></code> is a complete lattice for all <code class="docutils literal notranslate"><span class="pre">X</span></code>. Let’s now see how all this structure proves the existence of the product topology by abstract non-sense. We considered the case of <code class="docutils literal notranslate"><span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> above, but let’s now consider the general case of <code class="docutils literal notranslate"><span class="pre">Π</span> <span class="pre">i,</span> <span class="pre">X</span> <span class="pre">i</span></code> for some <code class="docutils literal notranslate"><span class="pre">ι</span> <span class="pre">:</span> <span class="pre">Type*</span></code> and <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">:</span> <span class="pre">ι</span> <span class="pre">→</span> <span class="pre">Type*</span></code>. We want, for any topological space <code class="docutils literal notranslate"><span class="pre">Z</span></code> and any function <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">Z</span> <span class="pre">→</span> <span class="pre">Π</span> <span class="pre">i,</span> <span class="pre">X</span> <span class="pre">i</span></code>, that <code class="docutils literal notranslate"><span class="pre">f</span></code> is continuous if and only if <code class="docutils literal notranslate"><span class="pre">(fun</span> <span class="pre">x</span> <span class="pre">↦</span> <span class="pre">x</span> <span class="pre">i)</span> <span class="pre">∘</span> <span class="pre">f</span></code> is continuous. Let us explore that constraint “on papar” using notation <span class="math notranslate nohighlight">\(p_i\)</span> for the projection <code class="docutils literal notranslate"><span class="pre">(fun</span> <span class="pre">(x</span> <span class="pre">:</span> <span class="pre">Π</span> <span class="pre">i,</span> <span class="pre">X</span> <span class="pre">i)</span> <span class="pre">↦</span> <span class="pre">x</span> <span class="pre">i)</span></code>:</p> <div class="math notranslate nohighlight"> \[\begin{split}(∀ i, p_i ∘ f \text{ continuous}) &⇔ ∀ i, (p_i ∘ f)_* T_Z ≤ T_{X_i} \\ &⇔ ∀ i, (p_i)_* f_* T_Z ≤ T_{X_i}\\ &⇔ ∀ i, f_* T_Z ≤ (p_i)^*T_{X_i}\\ &⇔ f_* T_Z ≤ \inf \left[(p_i)^*T_{X_i}\right]\end{split}\]</div> <p>So we see that what is the topology we want on <code class="docutils literal notranslate"><span class="pre">Π</span> <span class="pre">i,</span> <span class="pre">X</span> <span class="pre">i</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">ι</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">(</span><span class="n">X</span> <span class="o">:</span> <span class="n">ι</span> <span class="bp">→</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">(</span><span class="n">T_X</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">i</span><span class="o">,</span> <span class="n">TopologicalSpace</span> <span class="bp"><|</span> <span class="n">X</span> <span class="n">i</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="k">Pi</span><span class="bp">.</span><span class="n">topologicalSpace</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="o">(</span><span class="bp">∀</span> <span class="n">i</span><span class="o">,</span> <span class="n">X</span> <span class="n">i</span><span class="o">))</span> <span class="bp">=</span> <span class="bp">⨅</span> <span class="n">i</span><span class="o">,</span> <span class="n">TopologicalSpace.induced</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">x</span> <span class="n">i</span><span class="o">)</span> <span class="o">(</span><span class="n">T_X</span> <span class="n">i</span><span class="o">)</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div> <p>This ends our tour of how mathlib thinks that topological spaces fix defects of the theory of metric spaces by being a more functorial theory and having a complete lattice structure for any fixed type.</p> </section> <section id="separation-and-countability"> <h3><span class="section-number">8.3.2. </span>Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this heading"></a></h3> <p>We saw that the category of topological spaces have very nice properties. The price to pay for this is existence of rather pathological topological spaces. There are a number of assumptions you can make on a topological space to ensure its behavior is closer to what metric spaces do. The most important is <code class="docutils literal notranslate"><span class="pre">t2_space</span></code>, also called “Hausdorff”, that will ensure that limits are unique. A stronger separation property is regularity that ensure that each point has a basis of closed neighborhood.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">T2Space</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">ha</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">a</span><span class="o">))</span> <span class="o">(</span><span class="n">hb</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">b</span><span class="o">))</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">b</span> <span class="o">:=</span> <span class="n">tendsto_nhds_unique</span> <span class="n">ha</span> <span class="n">hb</span> <span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">RegularSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">X</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">a</span><span class="o">)</span><span class="bp">.</span><span class="n">HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span> <span class="bp">=></span> <span class="n">s</span> <span class="bp">∈</span> <span class="bp">𝓝</span> <span class="n">a</span> <span class="bp">∧</span> <span class="n">IsClosed</span> <span class="n">s</span><span class="o">)</span> <span class="n">id</span> <span class="o">:=</span> <span class="n">closed_nhds_basis</span> <span class="n">a</span> </pre></div> </div> <p>Note that, in every topological space, each point has a basis of open neighborhood, by definition.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">t</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span> <span class="bp">=></span> <span class="n">t</span> <span class="bp">∈</span> <span class="bp">𝓝</span> <span class="n">x</span> <span class="bp">∧</span> <span class="n">IsOpen</span> <span class="n">t</span><span class="o">)</span> <span class="n">id</span> <span class="o">:=</span> <span class="n">nhds_basis_opens'</span> <span class="n">x</span> </pre></div> </div> <p>Our main goal is now to prove the basic theorem which allows extension by continuity. From Bourbaki’s general topology book, I.8.5, Theorem 1 (taking only the non-trivial implication):</p> <p>Let <span class="math notranslate nohighlight">\(X\)</span> be a topological space, <span class="math notranslate nohighlight">\(A\)</span> a dense subset of <span class="math notranslate nohighlight">\(X\)</span>, <span class="math notranslate nohighlight">\(f : A → Y\)</span> a continuous mapping of <span class="math notranslate nohighlight">\(A\)</span> into a regular space <span class="math notranslate nohighlight">\(Y\)</span>. If, for each <span class="math notranslate nohighlight">\(x\)</span> in <span class="math notranslate nohighlight">\(X\)</span>, <span class="math notranslate nohighlight">\(f(y)\)</span> tends to a limit in <span class="math notranslate nohighlight">\(Y\)</span> when <span class="math notranslate nohighlight">\(y\)</span> tends to <span class="math notranslate nohighlight">\(x\)</span> while remaining in <span class="math notranslate nohighlight">\(A\)</span> then there exists a continuous extension <span class="math notranslate nohighlight">\(φ\)</span> of <span class="math notranslate nohighlight">\(f\)</span> to <span class="math notranslate nohighlight">\(X\)</span>.</p> <p>Actually <code class="docutils literal notranslate"><span class="pre">mathlib</span></code> contains a more general version of the above lemma, <code class="docutils literal notranslate"><span class="pre">DenseInducing.continuousAt_extend</span></code>, but we’ll stick to Bourbaki’s version here.</p> <p>Remember that, given <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X</span></code>, <code class="docutils literal notranslate"><span class="pre">↥A</span></code> is the subtype associated to <code class="docutils literal notranslate"><span class="pre">A</span></code>, and Lean will automatically insert that funny up arrow when needed. And the (inclusion) coercion map is <code class="docutils literal notranslate"><span class="pre">(↑)</span> <span class="pre">:</span> <span class="pre">A</span> <span class="pre">→</span> <span class="pre">X</span></code>. The assumption “tends to <span class="math notranslate nohighlight">\(x\)</span> while remaining in <span class="math notranslate nohighlight">\(A\)</span>” corresponds to the pull-back filter <code class="docutils literal notranslate"><span class="pre">comap</span> <span class="pre">(↑)</span> <span class="pre">(𝓝</span> <span class="pre">x)</span></code>.</p> <p>Let’s prove first an auxiliary lemma, extracted to simplify the context (in particular we don’t need Y to be a topological space here).</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">aux</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</span> <span class="n">A</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">A</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">A</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">f</span> <span class="o">(</span><span class="n">comap</span> <span class="n">c</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x</span><span class="o">))</span> <span class="n">F</span><span class="o">)</span> <span class="o">{</span><span class="n">V'</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">V'_in</span> <span class="o">:</span> <span class="n">V'</span> <span class="bp">∈</span> <span class="n">F</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">V</span> <span class="bp">∈</span> <span class="bp">𝓝</span> <span class="n">x</span><span class="o">,</span> <span class="n">IsOpen</span> <span class="n">V</span> <span class="bp">∧</span> <span class="n">c</span> <span class="bp">⁻¹'</span> <span class="n">V</span> <span class="bp">⊆</span> <span class="n">f</span> <span class="bp">⁻¹'</span> <span class="n">V'</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> <p>Let’s now turn to the main proof of the extension by continuity theorem.</p> <p>When Lean needs a topology on <code class="docutils literal notranslate"><span class="pre">↥A</span></code> it will use the induced topology, thanks to the instance <code class="docutils literal notranslate"><span class="pre">subtype.topological_space</span></code>. This all happens automatically. The only relevant lemma is <code class="docutils literal notranslate"><span class="pre">nhds_induced</span> <span class="pre">(↑)</span> <span class="pre">:</span> <span class="pre">∀</span> <span class="pre">a</span> <span class="pre">:</span> <span class="pre">↥A,</span> <span class="pre">𝓝</span> <span class="pre">a</span> <span class="pre">=</span> <span class="pre">comap</span> <span class="pre">(↑)</span> <span class="pre">(𝓝</span> <span class="pre">↑a)</span></code> (this is actually a general lemma about induced topologies).</p> <p>The proof outline is:</p> <p>The main assumption and the axiom of choice give a function <code class="docutils literal notranslate"><span class="pre">φ</span></code> such that <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">x,</span> <span class="pre">Tendsto</span> <span class="pre">f</span> <span class="pre">(comap</span> <span class="pre">(↑)</span> <span class="pre">$</span> <span class="pre">𝓝</span> <span class="pre">x)</span> <span class="pre">(𝓝</span> <span class="pre">(φ</span> <span class="pre">x))</span></code> (because <code class="docutils literal notranslate"><span class="pre">Y</span></code> is Hausdorff, <code class="docutils literal notranslate"><span class="pre">φ</span></code> is entirely determined, but we won’t need that until we try to prove that <code class="docutils literal notranslate"><span class="pre">φ</span></code> indeed extends <code class="docutils literal notranslate"><span class="pre">f</span></code>).</p> <p>Let’s first prove <code class="docutils literal notranslate"><span class="pre">φ</span></code> is continuous. Fix any <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">X</span></code>. Since <code class="docutils literal notranslate"><span class="pre">Y</span></code> is regular, it suffices to check that for every <em>closed</em> neighborhood <code class="docutils literal notranslate"><span class="pre">V'</span></code> of <code class="docutils literal notranslate"><span class="pre">φ</span> <span class="pre">x</span></code>, <code class="docutils literal notranslate"><span class="pre">φ</span> <span class="pre">⁻¹'</span> <span class="pre">V'</span> <span class="pre">∈</span> <span class="pre">𝓝</span> <span class="pre">x</span></code>. The limit assumption gives (through the auxiliary lemma above) some <code class="docutils literal notranslate"><span class="pre">V</span> <span class="pre">∈</span> <span class="pre">𝓝</span> <span class="pre">x</span></code> such <code class="docutils literal notranslate"><span class="pre">is_open</span> <span class="pre">V</span> <span class="pre">∧</span> <span class="pre">(↑)</span> <span class="pre">⁻¹'</span> <span class="pre">V</span> <span class="pre">⊆</span> <span class="pre">f</span> <span class="pre">⁻¹'</span> <span class="pre">V'</span></code>. Since <code class="docutils literal notranslate"><span class="pre">V</span> <span class="pre">∈</span> <span class="pre">𝓝</span> <span class="pre">x</span></code>, it suffices to prove <code class="docutils literal notranslate"><span class="pre">V</span> <span class="pre">⊆</span> <span class="pre">φ</span> <span class="pre">⁻¹'</span> <span class="pre">V'</span></code>, ie <code class="docutils literal notranslate"><span class="pre">∀</span> <span class="pre">y</span> <span class="pre">∈</span> <span class="pre">V,</span> <span class="pre">φ</span> <span class="pre">y</span> <span class="pre">∈</span> <span class="pre">V'</span></code>. Let’s fix <code class="docutils literal notranslate"><span class="pre">y</span></code> in <code class="docutils literal notranslate"><span class="pre">V</span></code>. Because <code class="docutils literal notranslate"><span class="pre">V</span></code> is <em>open</em>, it is a neighborhood of <code class="docutils literal notranslate"><span class="pre">y</span></code>. In particular <code class="docutils literal notranslate"><span class="pre">(↑)</span> <span class="pre">⁻¹'</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">comap</span> <span class="pre">(↑)</span> <span class="pre">(𝓝</span> <span class="pre">y)</span></code> and a fortiori <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">⁻¹'</span> <span class="pre">V'</span> <span class="pre">∈</span> <span class="pre">comap</span> <span class="pre">(↑)</span> <span class="pre">(𝓝</span> <span class="pre">y)</span></code>. In addition <code class="docutils literal notranslate"><span class="pre">comap</span> <span class="pre">(↑)$</span> <span class="pre">𝓝</span> <span class="pre">y</span> <span class="pre">≠</span> <span class="pre">⊥</span></code> because <code class="docutils literal notranslate"><span class="pre">A</span></code> is dense. Because we know <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">f</span> <span class="pre">(comap</span> <span class="pre">(↑)</span> <span class="pre">$</span> <span class="pre">𝓝</span> <span class="pre">y)</span> <span class="pre">(𝓝</span> <span class="pre">(φ</span> <span class="pre">y))</span></code> this implies <code class="docutils literal notranslate"><span class="pre">φ</span> <span class="pre">y</span> <span class="pre">∈</span> <span class="pre">closure</span> <span class="pre">V'</span></code> and, since <code class="docutils literal notranslate"><span class="pre">V'</span></code> is closed, we have proved <code class="docutils literal notranslate"><span class="pre">φ</span> <span class="pre">y</span> <span class="pre">∈</span> <span class="pre">V'</span></code>.</p> <p>It remains to prove that <code class="docutils literal notranslate"><span class="pre">φ</span></code> extends <code class="docutils literal notranslate"><span class="pre">f</span></code>. This is were continuity of <code class="docutils literal notranslate"><span class="pre">f</span></code> enters the discussion, together with the fact that <code class="docutils literal notranslate"><span class="pre">Y</span></code> is Hausdorff.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">[</span><span class="n">RegularSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">A</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hA</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">closure</span> <span class="n">A</span><span class="o">)</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">A</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">f_cont</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">c</span> <span class="o">:</span> <span class="n">Y</span><span class="o">,</span> <span class="n">Tendsto</span> <span class="n">f</span> <span class="o">(</span><span class="n">comap</span> <span class="o">(</span><span class="bp">↑</span><span class="o">)</span> <span class="bp"><|</span> <span class="bp">𝓝</span> <span class="n">x</span><span class="o">)</span> <span class="bp"><|</span> <span class="bp">𝓝</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">φ</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">,</span> <span class="n">Continuous</span> <span class="n">φ</span> <span class="bp">∧</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">A</span><span class="o">,</span> <span class="n">φ</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">a</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">HasBasis.tendsto_right_iff</span> </pre></div> </div> <p>In addition to separation property, the main kind of assumption you can make on a topological space to bring it closer to metric spaces is countability assumption. The main one is first countability asking that every point has a countable neighborhood basic. In particular this ensures that closure of sets can be understood using sequences.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">TopologicalSpace.FirstCountableTopology</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">closure</span> <span class="n">s</span> <span class="bp">↔</span> <span class="bp">∃</span> <span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">,</span> <span class="o">(</span><span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">)</span> <span class="bp">∧</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">mem_closure_iff_seq_limit</span> </pre></div> </div> </section> <section id="id5"> <h3><span class="section-number">8.3.3. </span>Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> <p>Let us now discuss how compactness is defined for topological spaces. As usual there are several ways to think about it and mathlib goes for the filter version.</p> <p>We first need to define cluster points of filters. Given a filter <code class="docutils literal notranslate"><span class="pre">F</span></code> on a topological space <code class="docutils literal notranslate"><span class="pre">X</span></code>, a point <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">X</span></code> is a cluster point of <code class="docutils literal notranslate"><span class="pre">F</span></code> if <code class="docutils literal notranslate"><span class="pre">F</span></code>, seen as a generalized set, has non-empty intersection with the generalized set of points that are close to <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p> <p>Then we can say that a set <code class="docutils literal notranslate"><span class="pre">s</span></code> is compact if every nonempty generalized set <code class="docutils literal notranslate"><span class="pre">F</span></code> contained in <code class="docutils literal notranslate"><span class="pre">s</span></code>, ie such that <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">≤</span> <span class="pre">𝓟</span> <span class="pre">s</span></code>, has a cluster point in <code class="docutils literal notranslate"><span class="pre">s</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="kd">example</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">ClusterPt</span> <span class="n">x</span> <span class="n">F</span> <span class="bp">↔</span> <span class="n">NeBot</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">x</span> <span class="bp">⊓</span> <span class="n">F</span><span class="o">)</span> <span class="o">:=</span> <span class="n">Iff.rfl</span> <span class="kd">example</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">X</span><span class="o">)</span> <span class="o">[</span><span class="n">NeBot</span> <span class="n">F</span><span class="o">],</span> <span class="n">F</span> <span class="bp">≤</span> <span class="bp">𝓟</span> <span class="n">s</span> <span class="bp">→</span> <span class="bp">∃</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="n">ClusterPt</span> <span class="n">a</span> <span class="n">F</span> <span class="o">:=</span> <span class="n">Iff.rfl</span> </pre></div> </div> <p>For instance if <code class="docutils literal notranslate"><span class="pre">F</span></code> is <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">u</span> <span class="pre">at_top</span></code>, the image under <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">:</span> <span class="pre">ℕ</span> <span class="pre">→</span> <span class="pre">X</span></code> of <code class="docutils literal notranslate"><span class="pre">at_top</span></code>, the generalized set of very large natural numbers, then the assumption <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">≤</span> <span class="pre">𝓟</span> <span class="pre">s</span></code> means that <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">n</span></code> belongs to <code class="docutils literal notranslate"><span class="pre">s</span></code> for <code class="docutils literal notranslate"><span class="pre">n</span></code> large enough. Saying that <code class="docutils literal notranslate"><span class="pre">x</span></code> is a cluster point of <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">u</span> <span class="pre">at_top</span></code> says the image of very large numbers intersects the set of points that are close to <code class="docutils literal notranslate"><span class="pre">x</span></code>. In case <code class="docutils literal notranslate"><span class="pre">𝓝</span> <span class="pre">x</span></code> has a countable basis, we can interpret this as saying that <code class="docutils literal notranslate"><span class="pre">u</span></code> has a subsequence converging to <code class="docutils literal notranslate"><span class="pre">x</span></code>, and we get back what compactness looks like in metric spaces.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace.FirstCountableTopology</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">u</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span><span class="o">)</span> <span class="o">(</span><span class="n">hu</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">a</span> <span class="bp">∈</span> <span class="n">s</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">φ</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">StrictMono</span> <span class="n">φ</span> <span class="bp">∧</span> <span class="n">Tendsto</span> <span class="o">(</span><span class="n">u</span> <span class="bp">∘</span> <span class="n">φ</span><span class="o">)</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">hs.tendsto_subseq</span> <span class="n">hu</span> </pre></div> </div> <p>Cluster points behave nicely with continuous functions.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">X</span><span class="o">}</span> <span class="o">{</span><span class="n">G</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">H</span> <span class="o">:</span> <span class="n">ClusterPt</span> <span class="n">x</span> <span class="n">F</span><span class="o">)</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hfx</span> <span class="o">:</span> <span class="n">ContinuousAt</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">f</span> <span class="n">F</span> <span class="n">G</span><span class="o">)</span> <span class="o">:</span> <span class="n">ClusterPt</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="n">G</span> <span class="o">:=</span> <span class="n">ClusterPt.map</span> <span class="n">H</span> <span class="n">hfx</span> <span class="n">hf</span> </pre></div> </div> <p>As an exercise, we will prove that the image of a compact set under a continuous map is compact. In addition to what we saw already, you should use <code class="docutils literal notranslate"><span class="pre">Filter.push_pull</span></code> and <code class="docutils literal notranslate"><span class="pre">ne_bot.of_map</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="n">s</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">F</span> <span class="n">F_ne</span> <span class="n">F_le</span> <span class="k">have</span> <span class="n">map_eq</span> <span class="o">:</span> <span class="n">map</span> <span class="n">f</span> <span class="o">(</span><span class="bp">𝓟</span> <span class="n">s</span> <span class="bp">⊓</span> <span class="n">comap</span> <span class="n">f</span> <span class="n">F</span><span class="o">)</span> <span class="bp">=</span> <span class="bp">𝓟</span> <span class="o">(</span><span class="n">f</span> <span class="bp">''</span> <span class="n">s</span><span class="o">)</span> <span class="bp">⊓</span> <span class="n">F</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">Hne</span> <span class="o">:</span> <span class="o">(</span><span class="bp">𝓟</span> <span class="n">s</span> <span class="bp">⊓</span> <span class="n">comap</span> <span class="n">f</span> <span class="n">F</span><span class="o">)</span><span class="bp">.</span><span class="n">NeBot</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">Hle</span> <span class="o">:</span> <span class="bp">𝓟</span> <span class="n">s</span> <span class="bp">⊓</span> <span class="n">comap</span> <span class="n">f</span> <span class="n">F</span> <span class="bp">≤</span> <span class="bp">𝓟</span> <span class="n">s</span> <span class="o">:=</span> <span class="n">inf_le_left</span> <span class="gr">sorry</span> </pre></div> </div> <p>One can also express compactness in terms of open covers: <code class="docutils literal notranslate"><span class="pre">s</span></code> is compact if every family of open sets that cover <code class="docutils literal notranslate"><span class="pre">s</span></code> has a finite covering sub-family.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">ι</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span><span class="o">)</span> <span class="o">(</span><span class="n">U</span> <span class="o">:</span> <span class="n">ι</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">hUo</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">i</span><span class="o">,</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="n">U</span> <span class="n">i</span><span class="o">))</span> <span class="o">(</span><span class="n">hsU</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="bp">⋃</span> <span class="n">i</span><span class="o">,</span> <span class="n">U</span> <span class="n">i</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">t</span> <span class="o">:</span> <span class="n">Finset</span> <span class="n">ι</span><span class="o">,</span> <span class="n">s</span> <span class="bp">⊆</span> <span class="bp">⋃</span> <span class="n">i</span> <span class="bp">∈</span> <span class="n">t</span><span class="o">,</span> <span class="n">U</span> <span class="n">i</span> <span class="o">:=</span> <span class="n">hs.elim_finite_subcover</span> <span class="n">U</span> <span class="n">hUo</span> <span class="n">hsU</span> </pre></div> </div> <p>A topological space <code class="docutils literal notranslate"><span class="pre">X</span></code> is compact if <code class="docutils literal notranslate"><span class="pre">(univ</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X)</span></code> is compact.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">CompactSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="o">(</span><span class="n">univ</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">)</span> <span class="o">:=</span> <span class="n">isCompact_univ</span> </pre></div> 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Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">9. Differential Calculus</a><ul> <li class="toctree-l2"><a class="reference internal" href="#elementary-differential-calculus">9.1. Elementary Differential Calculus</a></li> <li class="toctree-l2"><a class="reference internal" href="#differential-calculus-in-normed-spaces">9.2. Differential Calculus in Normed Spaces</a><ul> <li class="toctree-l3"><a class="reference internal" href="#id3">9.2.1. Normed spaces</a></li> <li class="toctree-l3"><a class="reference internal" href="#continuous-linear-maps">9.2.2. Continuous linear maps</a></li> <li class="toctree-l3"><a class="reference internal" href="#asymptotic-comparisons">9.2.3. Asymptotic comparisons</a></li> <li class="toctree-l3"><a class="reference internal" href="#differentiability">9.2.4. Differentiability</a></li> </ul> </li> </ul> </li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">9. </span>Differential Calculus</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C09_Differential_Calculus.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <span class="target" id="differential-calculus"></span><section id="index-0"> <span id="id1"></span><h1><span class="section-number">9. </span>Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <p>We now consider the formalization of notions from <em>analysis</em>, starting with differentiation in this chapter and turning integration and measure theory in the next. In <a class="reference internal" href="#elementary-differential-calculus"><span class="std std-numref">Section 9.1</span></a>, we stick with the setting of functions from the real numbers to the real numbers, which is familiar from any introductory calculus class. In <a class="reference internal" href="#normed-spaces"><span class="std std-numref">Section 9.2</span></a>, we then consider the notion of a derivative in a much broader setting.</p> <section id="elementary-differential-calculus"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">9.1. </span>Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this heading"></a></h2> <p>Let <code class="docutils literal notranslate"><span class="pre">f</span></code> be a function from the reals to the reals. There is a difference between talking about the derivative of <code class="docutils literal notranslate"><span class="pre">f</span></code> at a single point and talking about the derivative function. In mathlib, the first notion is represented as follows.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Real</span> <span class="sd">/-- The sin function has derivative 1 at 0. -/</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">HasDerivAt</span> <span class="n">sin</span> <span class="mi">1</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simpa</span> <span class="n">using</span> <span class="n">hasDerivAt_sin</span> <span class="mi">0</span> </pre></div> </div> <p>We can also express that <code class="docutils literal notranslate"><span class="pre">f</span></code> is differentiable at a point without specifying its derivative there by writing <code class="docutils literal notranslate"><span class="pre">differentiable_at</span> <span class="pre">ℝ</span></code>. We specify <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> explicitly because in a slightly more general context, when talking about functions from <code class="docutils literal notranslate"><span class="pre">ℂ</span></code> to <code class="docutils literal notranslate"><span class="pre">ℂ</span></code>, we want to be able to distinguish between being differentiable in the real sense and being differentiable in the sense of the complex derivative.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">DifferentiableAt</span> <span class="n">ℝ</span> <span class="n">sin</span> <span class="n">x</span> <span class="o">:=</span> <span class="o">(</span><span class="n">hasDerivAt_sin</span> <span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">differentiableAt</span> </pre></div> </div> <p>It would be inconvenient to have to provide a proof of differentiability every time we want to refer to a derivative. So mathlib provides a function <code class="docutils literal notranslate"><span class="pre">deriv</span> <span class="pre">f</span> <span class="pre">:</span> <span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> that is defined for any function <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">ℝ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> but is defined to take the value <code class="docutils literal notranslate"><span class="pre">0</span></code> at any point where <code class="docutils literal notranslate"><span class="pre">f</span></code> is not differentiable.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">HasDerivAt</span> <span class="n">f</span> <span class="n">a</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="n">deriv</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">a</span> <span class="o">:=</span> <span class="n">h.deriv</span> <span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">¬</span><span class="n">DifferentiableAt</span> <span class="n">ℝ</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="n">deriv</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">deriv_zero_of_not_differentiableAt</span> <span class="n">h</span> </pre></div> </div> <p>Of course there are many lemmas about <code class="docutils literal notranslate"><span class="pre">deriv</span></code> that do require differentiability assumptions. For instance, you should think about a counterexample to the next lemma without the differentiability assumptions.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">DifferentiableAt</span> <span class="n">ℝ</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">hg</span> <span class="o">:</span> <span class="n">DifferentiableAt</span> <span class="n">ℝ</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="n">deriv</span> <span class="o">(</span><span class="n">f</span> <span class="bp">+</span> <span class="n">g</span><span class="o">)</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">deriv</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">deriv</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">deriv_add</span> <span class="n">hf</span> <span class="n">hg</span> </pre></div> </div> <p>Interestingly, however, there are statements that can avoid differentiability assumptions by taking advantage of the fact that the value of <code class="docutils literal notranslate"><span class="pre">deriv</span></code> defaults to zero when the function is not differentiable. So making sense of the following statement requires knowing the precise definition of <code class="docutils literal notranslate"><span class="pre">deriv</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">IsLocalMin</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">deriv</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">h.deriv_eq_zero</span> </pre></div> </div> <p>We can eve state Rolle’s theorem without any differentiability assumptions, which seems even weirder.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">hab</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">hfc</span> <span class="o">:</span> <span class="n">ContinuousOn</span> <span class="n">f</span> <span class="o">(</span><span class="n">Icc</span> <span class="n">a</span> <span class="n">b</span><span class="o">))</span> <span class="o">(</span><span class="n">hfI</span> <span class="o">:</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">c</span> <span class="bp">∈</span> <span class="n">Ioo</span> <span class="n">a</span> <span class="n">b</span><span class="o">,</span> <span class="n">deriv</span> <span class="n">f</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">exists_deriv_eq_zero</span> <span class="n">f</span> <span class="n">hab</span> <span class="n">hfc</span> <span class="n">hfI</span> </pre></div> </div> <p>Of course, this trick does not work for the general mean value theorem.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">hab</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">ContinuousOn</span> <span class="n">f</span> <span class="o">(</span><span class="n">Icc</span> <span class="n">a</span> <span class="n">b</span><span class="o">))</span> <span class="o">(</span><span class="n">hf'</span> <span class="o">:</span> <span class="n">DifferentiableOn</span> <span class="n">ℝ</span> <span class="n">f</span> <span class="o">(</span><span class="n">Ioo</span> <span class="n">a</span> <span class="n">b</span><span class="o">))</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">c</span> <span class="bp">∈</span> <span class="n">Ioo</span> <span class="n">a</span> <span class="n">b</span><span class="o">,</span> <span class="n">deriv</span> <span class="n">f</span> <span class="n">c</span> <span class="bp">=</span> <span class="o">(</span><span class="n">f</span> <span class="n">b</span> <span class="bp">-</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="bp">/</span> <span class="o">(</span><span class="n">b</span> <span class="bp">-</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">exists_deriv_eq_slope</span> <span class="n">f</span> <span class="n">hab</span> <span class="n">hf</span> <span class="n">hf'</span> </pre></div> </div> <p>Lean can automatically compute some simple derivatives using the <code class="docutils literal notranslate"><span class="pre">simp</span></code> tactic.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">deriv</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">↦</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">5</span><span class="o">)</span> <span class="mi">6</span> <span class="bp">=</span> <span class="mi">5</span> <span class="bp">*</span> <span class="mi">6</span> <span class="bp">^</span> <span class="mi">4</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">deriv</span> <span class="n">sin</span> <span class="n">π</span> <span class="bp">=</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> </pre></div> </div> </section> <section id="differential-calculus-in-normed-spaces"> <span id="normed-spaces"></span><span id="index-2"></span><h2><span class="section-number">9.2. </span>Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this heading"></a></h2> <section id="id3"> <h3><span class="section-number">9.2.1. </span>Normed spaces<a class="headerlink" href="#id3" title="Permalink to this heading"></a></h3> <p>Differentiation can be generalized beyond <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> using the notion of a <em>normed vector space</em>, which encapsulates both direction and distance. We start with the notion of a <em>normed group</em>, which as an additive commutative group equipped with a real-valued norm function satisfying the following conditions.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="bp">‖</span><span class="n">x</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">norm_nonneg</span> <span class="n">x</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">E</span><span class="o">}</span> <span class="o">:</span> <span class="bp">‖</span><span class="n">x</span><span class="bp">‖</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">↔</span> <span class="n">x</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">norm_eq_zero</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> <span class="bp">‖</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="bp">‖</span><span class="n">x</span><span class="bp">‖</span> <span class="bp">+</span> <span class="bp">‖</span><span class="n">y</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">norm_add_le</span> <span class="n">x</span> <span class="n">y</span> </pre></div> </div> <p>Every normed space is a metric space with distance function <span class="math notranslate nohighlight">\(d(x, y) = \| x - y \|\)</span>, and hence it is also a topological space. Lean and mathlib know this.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">MetricSpace</span> <span class="n">E</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">infer_instance</span> <span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">E</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="bp">‖</span><span class="n">f</span> <span class="n">x</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">hf.norm</span> </pre></div> </div> <p>In order to use the notion of a norm with concepts from linear algebra, we add the assumption <code class="docutils literal notranslate"><span class="pre">normed_space</span> <span class="pre">ℝ</span> <span class="pre">E</span></code> on top of <code class="docutils literal notranslate"><span class="pre">normed_add_group</span> <span class="pre">E</span></code>. This stipulates that <code class="docutils literal notranslate"><span class="pre">E</span></code> is a vector space over <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> and that scalar multiplication satisfies the following condition.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="n">ℝ</span> <span class="n">E</span><span class="o">]</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> <span class="bp">‖</span><span class="n">a</span> <span class="bp">•</span> <span class="n">x</span><span class="bp">‖</span> <span class="bp">=</span> <span class="bp">|</span><span class="n">a</span><span class="bp">|</span> <span class="bp">*</span> <span class="bp">‖</span><span class="n">x</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">norm_smul</span> <span class="n">a</span> <span class="n">x</span> </pre></div> </div> <p>A complete normed space is known as a <em>Banach space</em>. Every finite-dimensional vector space is complete.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">FiniteDimensional</span> <span class="n">ℝ</span> <span class="n">E</span><span class="o">]</span> <span class="o">:</span> <span class="n">CompleteSpace</span> <span class="n">E</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">infer_instance</span> </pre></div> </div> <p>In all the previous examples, we used the real numbers as the base field. More generally, we can make sense of calculus with a vector space over any <em>non-discrete normed field</em>. These are fields that are equipped with a real-valued norm that is multiplicative and has the property that not every element has norm zero or one (equivalently, there is an element whose norm is bigger than one).</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="bp">𝕜</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">[</span><span class="n">NontriviallyNormedField</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="bp">𝕜</span><span class="o">)</span> <span class="o">:</span> <span class="bp">‖</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="bp">‖</span> <span class="bp">=</span> <span class="bp">‖</span><span class="n">x</span><span class="bp">‖</span> <span class="bp">*</span> <span class="bp">‖</span><span class="n">y</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">norm_mul</span> <span class="n">x</span> <span class="n">y</span> <span class="kd">example</span> <span class="o">(</span><span class="bp">𝕜</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">[</span><span class="n">NontriviallyNormedField</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">x</span> <span class="o">:</span> <span class="bp">𝕜</span><span class="o">,</span> <span class="mi">1</span> <span class="bp"><</span> <span class="bp">‖</span><span class="n">x</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">NormedField.exists_one_lt_norm</span> <span class="bp">𝕜</span> </pre></div> </div> <p>A finite-dimensional vector space over a nondiscrete normed field is complete as long as the field itself is complete.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="bp">𝕜</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">[</span><span class="n">NontriviallyNormedField</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">(</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">E</span><span class="o">]</span> <span class="o">[</span><span class="n">CompleteSpace</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">[</span><span class="n">FiniteDimensional</span> <span class="bp">𝕜</span> <span class="n">E</span><span class="o">]</span> <span class="o">:</span> <span class="n">CompleteSpace</span> <span class="n">E</span> <span class="o">:=</span> <span class="n">FiniteDimensional.complete</span> <span class="bp">𝕜</span> <span class="n">E</span> </pre></div> </div> </section> <section id="continuous-linear-maps"> <h3><span class="section-number">9.2.2. </span>Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this heading"></a></h3> <p>We now turn to the morphisms in the category of normed spaces, namely, continuous linear maps. In mathlib, the type of <code class="docutils literal notranslate"><span class="pre">𝕜</span></code>-linear continuous maps between normed spaces <code class="docutils literal notranslate"><span class="pre">E</span></code> and <code class="docutils literal notranslate"><span class="pre">F</span></code> is written <code class="docutils literal notranslate"><span class="pre">E</span> <span class="pre">→L[𝕜]</span> <span class="pre">F</span></code>. They are implemented as <em>bundled maps</em>, which means that an element of this type a structure that that includes the function itself and the properties of being linear and continuous. Lean will insert a coercion so that a continuous linear map can be treated as a function.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="bp">𝕜</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NontriviallyNormedField</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">F</span><span class="o">]</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">E</span> <span class="o">:=</span> <span class="n">ContinuousLinearMap.id</span> <span class="bp">𝕜</span> <span class="n">E</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span> <span class="o">:=</span> <span class="n">f</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span> <span class="o">:=</span> <span class="n">f.cont</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">f</span> <span class="n">y</span> <span class="o">:=</span> <span class="n">f.map_add</span> <span class="n">x</span> <span class="n">y</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="bp">𝕜</span><span class="o">)</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="o">(</span><span class="n">a</span> <span class="bp">•</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">a</span> <span class="bp">•</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">f.map_smul</span> <span class="n">a</span> <span class="n">x</span> </pre></div> </div> <p>Continuous linear maps have an operator norm that is characterized by the following properties.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> <span class="bp">‖</span><span class="n">f</span> <span class="n">x</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="bp">‖</span><span class="n">f</span><span class="bp">‖</span> <span class="bp">*</span> <span class="bp">‖</span><span class="n">x</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">f.le_op_norm</span> <span class="n">x</span> <span class="kd">example</span> <span class="o">{</span><span class="n">M</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">hMp</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">M</span><span class="o">)</span> <span class="o">(</span><span class="n">hM</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="bp">‖</span><span class="n">f</span> <span class="n">x</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">M</span> <span class="bp">*</span> <span class="bp">‖</span><span class="n">x</span><span class="bp">‖</span><span class="o">)</span> <span class="o">:</span> <span class="bp">‖</span><span class="n">f</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">M</span> <span class="o">:=</span> <span class="n">f.op_norm_le_bound</span> <span class="n">hMp</span> <span class="n">hM</span> </pre></div> </div> <p>There is also a notion of bundled continuous linear <em>isomorphism</em>. Their type of such isomorphisms is <code class="docutils literal notranslate"><span class="pre">E</span> <span class="pre">≃L[𝕜]</span> <span class="pre">F</span></code>.</p> <p>As a challenging exercise, you can prove the Banach-Steinhaus theorem, also known as the Uniform Boundedness Principle. The principle states that a family of continuous linear maps from a Banach space into a normed space is pointwise bounded, then the norms of these linear maps are uniformly bounded. The main ingredient is Baire’s theorem <code class="docutils literal notranslate"><span class="pre">nonempty_interior_of_Union_of_closed.</span></code> (You proved a version of this in the topology chapter.) Minor ingredients include <code class="docutils literal notranslate"><span class="pre">continuous_linear_map.op_norm_le_of_shell</span></code>, <code class="docutils literal notranslate"><span class="pre">interior_subset</span></code> and <code class="docutils literal notranslate"><span class="pre">interior_Inter_subset</span></code> and <code class="docutils literal notranslate"><span class="pre">is_closed_le</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="bp">𝕜</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NontriviallyNormedField</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">F</span><span class="o">]</span> <span class="kn">open</span> <span class="n">Metric</span> <span class="kd">example</span> <span class="o">{</span><span class="n">ι</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">CompleteSpace</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">g</span> <span class="o">:</span> <span class="n">ι</span> <span class="bp">→</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">C</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">i</span><span class="o">,</span> <span class="bp">‖</span><span class="n">g</span> <span class="n">i</span> <span class="n">x</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">C</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">C'</span><span class="o">,</span> <span class="bp">∀</span> <span class="n">i</span><span class="o">,</span> <span class="bp">‖</span><span class="n">g</span> <span class="n">i</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">C'</span> <span class="o">:=</span> <span class="kd">by</span> <span class="c1">-- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n`</span> <span class="k">let</span> <span class="n">e</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">Set</span> <span class="n">E</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=></span> <span class="bp">⋂</span> <span class="n">i</span> <span class="o">:</span> <span class="n">ι</span><span class="o">,</span> <span class="o">{</span> <span class="n">x</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">|</span> <span class="bp">‖</span><span class="n">g</span> <span class="n">i</span> <span class="n">x</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">n</span> <span class="o">}</span> <span class="c1">-- each of these sets is closed</span> <span class="k">have</span> <span class="n">hc</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">IsClosed</span> <span class="o">(</span><span class="n">e</span> <span class="n">n</span><span class="o">)</span> <span class="gr">sorry</span> <span class="c1">-- the union is the entire space; this is where we use `h`</span> <span class="k">have</span> <span class="n">hU</span> <span class="o">:</span> <span class="o">(</span><span class="bp">⋃</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">e</span> <span class="n">n</span><span class="o">)</span> <span class="bp">=</span> <span class="n">univ</span> <span class="gr">sorry</span> <span class="c">/-</span><span class="cm"> apply the Baire category theorem to conclude that for some `m : ℕ`,</span> <span class="cm"> `e m` contains some `x` -/</span> <span class="n">obtain</span> <span class="o">⟨</span><span class="n">m</span><span class="o">,</span> <span class="n">x</span><span class="o">,</span> <span class="n">hx</span><span class="o">⟩</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">m</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">x</span><span class="o">,</span> <span class="n">x</span> <span class="bp">∈</span> <span class="n">interior</span> <span class="o">(</span><span class="n">e</span> <span class="n">m</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">obtain</span> <span class="o">⟨</span><span class="n">ε</span><span class="o">,</span> <span class="n">ε_pos</span><span class="o">,</span> <span class="n">hε</span><span class="o">⟩</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">ε</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="n">ball</span> <span class="n">x</span> <span class="n">ε</span> <span class="bp">⊆</span> <span class="n">interior</span> <span class="o">(</span><span class="n">e</span> <span class="n">m</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">obtain</span> <span class="o">⟨</span><span class="n">k</span><span class="o">,</span> <span class="n">hk</span><span class="o">⟩</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">k</span> <span class="o">:</span> <span class="bp">𝕜</span><span class="o">,</span> <span class="mi">1</span> <span class="bp"><</span> <span class="bp">‖</span><span class="n">k</span><span class="bp">‖</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="c1">-- show all elements in the ball have norm bounded by `m` after applying any `g i`</span> <span class="k">have</span> <span class="n">real_norm_le</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">z</span> <span class="bp">∈</span> <span class="n">ball</span> <span class="n">x</span> <span class="n">ε</span><span class="o">,</span> <span class="bp">∀</span> <span class="o">(</span><span class="n">i</span> <span class="o">:</span> <span class="n">ι</span><span class="o">),</span> <span class="bp">‖</span><span class="n">g</span> <span class="n">i</span> <span class="n">z</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">m</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">εk_pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">ε</span> <span class="bp">/</span> <span class="bp">‖</span><span class="n">k</span><span class="bp">‖</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">refine'</span> <span class="o">⟨(</span><span class="n">m</span> <span class="bp">+</span> <span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="bp">/</span> <span class="o">(</span><span class="n">ε</span> <span class="bp">/</span> <span class="bp">‖</span><span class="n">k</span><span class="bp">‖</span><span class="o">),</span> <span class="k">fun</span> <span class="n">i</span> <span class="bp">=></span> <span class="n">ContinuousLinearMap.op_norm_le_of_shell</span> <span class="n">ε_pos</span> <span class="n">_</span> <span class="n">hk</span> <span class="n">_</span><span class="o">⟩</span> <span class="gr">sorry</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="asymptotic-comparisons"> <h3><span class="section-number">9.2.3. </span>Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this heading"></a></h3> <p>Defining differentiability also requires asymptotic comparisons. Mathlib has an extensive library covering the big O and little o relations, whose definitions are shown below. Opening the <code class="docutils literal notranslate"><span class="pre">asymptotics</span></code> locale allows us to use the corresponding notation. Here we will only use little o to define differentiability.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Asymptotics</span> <span class="kn">open</span> <span class="n">Asymptotics</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">(</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">l</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">F</span><span class="o">)</span> <span class="o">:</span> <span class="n">IsBigOWith</span> <span class="n">c</span> <span class="n">l</span> <span class="n">f</span> <span class="n">g</span> <span class="bp">↔</span> <span class="bp">∀ᶠ</span> <span class="n">x</span> <span class="k">in</span> <span class="n">l</span><span class="o">,</span> <span class="bp">‖</span><span class="n">f</span> <span class="n">x</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">*</span> <span class="bp">‖</span><span class="n">g</span> <span class="n">x</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">isBigOWith_iff</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">(</span><span class="n">l</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">F</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">=</span><span class="n">O</span><span class="o">[</span><span class="n">l</span><span class="o">]</span> <span class="n">g</span> <span class="bp">↔</span> <span class="bp">∃</span> <span class="n">C</span><span class="o">,</span> <span class="n">IsBigOWith</span> <span class="n">C</span> <span class="n">l</span> <span class="n">f</span> <span class="n">g</span> <span class="o">:=</span> <span class="n">isBigO_iff_isBigOWith</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">(</span><span class="n">l</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">F</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">=</span><span class="n">o</span><span class="o">[</span><span class="n">l</span><span class="o">]</span> <span class="n">g</span> <span class="bp">↔</span> <span class="bp">∀</span> <span class="n">C</span> <span class="bp">></span> <span class="mi">0</span><span class="o">,</span> <span class="n">IsBigOWith</span> <span class="n">C</span> <span class="n">l</span> <span class="n">f</span> <span class="n">g</span> <span class="o">:=</span> <span class="n">isLittleO_iff_forall_isBigOWith</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">(</span><span class="n">l</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">~</span><span class="o">[</span><span class="n">l</span><span class="o">]</span> <span class="n">g</span> <span class="bp">↔</span> <span class="o">(</span><span class="n">f</span> <span class="bp">-</span> <span class="n">g</span><span class="o">)</span> <span class="bp">=</span><span class="n">o</span><span class="o">[</span><span class="n">l</span><span class="o">]</span> <span class="n">g</span> <span class="o">:=</span> <span class="n">Iff.rfl</span> </pre></div> </div> </section> <section id="differentiability"> <h3><span class="section-number">9.2.4. </span>Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> <p>We are now ready to discuss differentiable functions between normed spaces. In analogy the elementary one-dimensional, mathlib defines a predicate <code class="docutils literal notranslate"><span class="pre">has_fderiv_at</span></code> and a function <code class="docutils literal notranslate"><span class="pre">fderiv</span></code>. Here the letter “f” stands for <em>Fréchet</em>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="bp">𝕜</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NontriviallyNormedField</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">F</span><span class="o">]</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> <span class="n">HasFDerivAt</span> <span class="n">f</span> <span class="n">f'</span> <span class="n">x₀</span> <span class="bp">↔</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">f</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">f</span> <span class="n">x₀</span> <span class="bp">-</span> <span class="n">f'</span> <span class="o">(</span><span class="n">x</span> <span class="bp">-</span> <span class="n">x₀</span><span class="o">))</span> <span class="bp">=</span><span class="n">o</span><span class="o">[</span><span class="bp">𝓝</span> <span class="n">x₀</span><span class="o">]</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">x</span> <span class="bp">-</span> <span class="n">x₀</span> <span class="o">:=</span> <span class="n">Iff.rfl</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">(</span><span class="n">hff'</span> <span class="o">:</span> <span class="n">HasFDerivAt</span> <span class="n">f</span> <span class="n">f'</span> <span class="n">x₀</span><span class="o">)</span> <span class="o">:</span> <span class="n">fderiv</span> <span class="bp">𝕜</span> <span class="n">f</span> <span class="n">x₀</span> <span class="bp">=</span> <span class="n">f'</span> <span class="o">:=</span> <span class="n">hff'.fderiv</span> </pre></div> </div> <p>We also have iterated derivatives that take values in the type of multilinear maps <code class="docutils literal notranslate"><span class="pre">E</span> <span class="pre">[×n]→L[𝕜]</span> <span class="pre">F</span></code>, and we have continuously differential functions. The type <code class="docutils literal notranslate"><span class="pre">with_top</span> <span class="pre">ℕ</span></code> is <code class="docutils literal notranslate"><span class="pre">ℕ</span></code> with an additional element <code class="docutils literal notranslate"><span class="pre">⊤</span></code> that is bigger than every natural number. So <span class="math notranslate nohighlight">\(\mathcal{C}^\infty\)</span> functions are functions <code class="docutils literal notranslate"><span class="pre">f</span></code> that satisfy <code class="docutils literal notranslate"><span class="pre">cont_diff</span> <span class="pre">𝕜</span> <span class="pre">⊤</span> <span class="pre">f</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">)</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">E</span><span class="o">[</span><span class="bp">×</span><span class="n">n</span><span class="o">]</span><span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span> <span class="o">:=</span> <span class="n">iteratedFDeriv</span> <span class="bp">𝕜</span> <span class="n">n</span> <span class="n">f</span> <span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">}</span> <span class="o">:</span> <span class="n">ContDiff</span> <span class="bp">𝕜</span> <span class="n">n</span> <span class="n">f</span> <span class="bp">↔</span> <span class="o">(</span><span class="bp">∀</span> <span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">ℕ</span><span class="o">)</span> <span class="bp">≤</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">iteratedFDeriv</span> <span class="bp">𝕜</span> <span class="n">m</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">∧</span> <span class="bp">∀</span> <span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">ℕ</span><span class="o">)</span> <span class="bp"><</span> <span class="n">n</span> <span class="bp">→</span> <span class="n">Differentiable</span> <span class="bp">𝕜</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=></span> <span class="n">iteratedFDeriv</span> <span class="bp">𝕜</span> <span class="n">m</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">contDiff_iff_continuous_differentiable</span> </pre></div> </div> <p>There is a stricter notion of differentiability called <code class="docutils literal notranslate"><span class="pre">has_strict_fderiv_at</span></code>, which is used in the statement of the inverse function theorem and the statement of the implicit function theorem, both of which are in mathlib. Over <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> or <code class="docutils literal notranslate"><span class="pre">ℂ</span></code>, continuously differentiable functions are strictly differentiable.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="bp">𝕂</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">IsROrC</span> <span class="bp">𝕂</span><span class="o">]</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕂</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕂</span> <span class="n">F</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">E</span><span class="o">}</span> <span class="o">{</span><span class="n">n</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">ℕ</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">ContDiffAt</span> <span class="bp">𝕂</span> <span class="n">n</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">hn</span> <span class="o">:</span> <span class="mi">1</span> <span class="bp">≤</span> <span class="n">n</span><span class="o">)</span> <span class="o">:</span> <span class="n">HasStrictFDerivAt</span> <span class="n">f</span> <span class="o">(</span><span class="n">fderiv</span> <span class="bp">𝕂</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">hf.hasStrictFDerivAt</span> <span class="n">hn</span> </pre></div> </div> <p>The local inverse theorem is stated using an operation that produces an inverse function from a function and the assumptions that the function is strictly differentiable at a point <code class="docutils literal notranslate"><span class="pre">a</span></code> and that its derivative is an isomorphism.</p> <p>The first example below gets this local inverse. The next one states that it is indeed a local inverse from the left and from the right, and that it is strictly differentiable.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">section</span> <span class="n">LocalInverse</span> <span class="kd">variable</span> <span class="o">[</span><span class="n">CompleteSpace</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">≃</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">E</span><span class="o">}</span> <span class="kd">example</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">HasStrictFDerivAt</span> <span class="n">f</span> <span class="o">(</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">F</span> <span class="bp">→</span> <span class="n">E</span> <span class="o">:=</span> <span class="n">HasStrictFDerivAt.localInverse</span> <span class="n">f</span> <span class="n">f'</span> <span class="n">a</span> <span class="n">hf</span> <span class="kd">example</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">HasStrictFDerivAt</span> <span class="n">f</span> <span class="o">(</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">x</span> <span class="k">in</span> <span class="bp">𝓝</span> <span class="n">a</span><span class="o">,</span> <span class="n">hf.localInverse</span> <span class="n">f</span> <span class="n">f'</span> <span class="n">a</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">hf.eventually_left_inverse</span> <span class="kd">example</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">HasStrictFDerivAt</span> <span class="n">f</span> <span class="o">(</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∀ᶠ</span> <span class="n">x</span> <span class="k">in</span> <span class="bp">𝓝</span> <span class="o">(</span><span class="n">f</span> <span class="n">a</span><span class="o">),</span> <span class="n">f</span> <span class="o">(</span><span class="n">hf.localInverse</span> <span class="n">f</span> <span class="n">f'</span> <span class="n">a</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">hf.eventually_right_inverse</span> <span class="kd">example</span> <span class="o">[</span><span class="n">CompleteSpace</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">≃</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">E</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">HasStrictFDerivAt</span> <span class="n">f</span> <span class="o">(</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">HasStrictFDerivAt</span> <span class="o">(</span><span class="n">HasStrictFDerivAt.localInverse</span> <span class="n">f</span> <span class="n">f'</span> <span class="n">a</span> <span class="n">hf</span><span class="o">)</span> <span class="o">(</span><span class="n">f'.symm</span> <span class="o">:</span> <span class="n">F</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">E</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">HasStrictFDerivAt.to_localInverse</span> <span class="n">hf</span> <span class="kd">end</span> <span class="n">LocalInverse</span> </pre></div> </div> <p>This has been only a quick tour of the differential calculus in mathlib. The library contains many variations that we have not discussed. For example, you may want to use one-sided derivatives in the one-dimensional setting. The means to do so are found in mathlib in a more general context; see <code class="docutils literal notranslate"><span class="pre">HasFDerivWithinAt</span></code> or the even more general <code class="docutils literal notranslate"><span class="pre">HasFDerivAtFilter</span></code>.</p> </section> </section> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C08_Topology.html" class="btn btn-neutral float-left" title="8. 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@@ -1,7 +0,0 @@.. _introduction: Introduction ============ .. include:: C01_Introduction/S01_Getting_Started.inc .. include:: C01_Introduction/S02_Overview.inc
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@@ -1,19 +0,0 @@.. _basics: Basics ====== This chapter is designed to introduce you to the nuts and bolts of mathematical reasoning in Lean: calculating, applying lemmas and theorems, and reasoning about generic structures. .. include:: C02_Basics/S01_Calculating.inc .. include:: C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.inc .. include:: C02_Basics/S03_Using_Theorems_and_Lemmas.inc .. include:: C02_Basics/S04_More_on_Order_and_Divisibility.inc .. include:: C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.inc
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@@ -1,37 +0,0 @@.. _logic: Logic ===== In the last chapter, we dealt with equations, inequalities, and basic mathematical statements like ":math:`x` divides :math:`y`." Complex mathematical statements are built up from simple ones like these using logical terms like "and," "or," "not," and "if ... then," "every," and "some." In this chapter, we show you how to work with statements that are built up in this way. .. include:: C03_Logic/S01_Implication_and_the_Universal_Quantifier.inc .. include:: C03_Logic/S02_The_Existential_Quantifier.inc .. include:: C03_Logic/S03_Negation.inc .. include:: C03_Logic/S04_Conjunction_and_Bi-implication.inc .. include:: C03_Logic/S05_Disjunction.inc .. include:: C03_Logic/S06_Sequences_and_Convergence.inc
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@@ -1,45 +0,0 @@.. _sets_and_functions: Sets and Functions ================== The vocabulary of sets, relations, and functions provides a uniform language for carrying out constructions in all the branches of mathematics. Since functions and relations can be defined in terms of sets, axiomatic set theory can be used as a foundation for mathematics. Lean's foundation is based instead on the primitive notion of a *type*, and it includes ways of defining functions between types. Every expression in Lean has a type: there are natural numbers, real numbers, functions from reals to reals, groups, vector spaces, and so on. Some expressions *are* types, which is to say, their type is ``Type``. Lean and mathlib provide ways of defining new types, and ways of defining objects of those types. Conceptually, you can think of a type as just a set of objects. Requiring every object to have a type has some advantages. For example, it makes it possible to overload notation like ``+``, and it sometimes makes input less verbose because Lean can infer a lot of information from an object's type. The type system also enables Lean to flag errors when you apply a function to the wrong number of arguments, or apply a function to arguments of the wrong type. Lean's library does define elementary set-theoretic notions. In contrast to set theory, in Lean a set is always a set of objects of some type, such as a set natural numbers or a set of functions from real numbers to real numbers. The distinction between types and set takes some getting used to, but this chapter will take you through the essentials. .. include:: C04_Sets_and_Functions/S01_Sets.inc .. include:: C04_Sets_and_Functions/S02_Functions.inc .. include:: C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.inc
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@@ -1,14 +0,0 @@.. _number_theory: Number Theory ============= In this chapter, we show you how to formalize some elementary results in number theory. As we deal with more substantive mathematical content, the proofs will get longer and more involved, building on the skills you have already mastered. .. include:: C05_Number_Theory/S01_Irrational_Roots.inc .. include:: C05_Number_Theory/S02_Induction_and_Recursion.inc .. include:: C05_Number_Theory/S03_Infinitely_Many_Primes.inc
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@@ -1,29 +0,0 @@.. _structures: Structures ========== Modern mathematics makes essential use of algebraic structures, which encapsulate patterns that can be instantiated in multiple settings. The subject provides various ways of defining such structures and constructing particular instances. Lean therefore provides corresponding ways of defining structures formally and working with them. You have already seen examples of algebraic structures in Lean, such as rings and lattices, which were discussed in :numref:`Chapter %s <basics>`. This chapter will explain the mysterious square bracket annotations that you saw there, ``[Ring α]`` and ``[Lattice α]``. It will also show you how to define and use algebraic structures on your own. For more technical detail, you can consult `Theorem Proving in Lean <https://leanprover.github.io/theorem_proving_in_lean/>`_, and a paper by Anne Baanen, `Use and abuse of instance parameters in the Lean mathematical library <https://arxiv.org/abs/2202.01629>`_. .. include:: C06_Structures/S01_Structures.inc .. include:: C06_Structures/S02_Algebraic_Structures.inc .. include:: C06_Structures/S03_Building_the_Gaussian_Integers.inc
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@@ -1,26 +0,0 @@.. _hierarchies: Hierarchies =========== We have seen in :numref:`Chapter %s <structures>` how to define the class of groups and build instances of this class, and then how to build an instance of the commutative ring class. But of course there is a hierarchy here: a commutative ring is in particular an additive group. In this chapter we will study how to build such hierarchies. They appear in all branches of mathematics but in this chapter the emphasis will be on algebraic examples. It may seem premature to discuss how to build hierarchies before more discussions about using existing hierarchies. But some understanding of the technology underlying hierarchies is required to use them. So you should probably still read this chapter, but without trying too hard to remember everything on your first read, then read the following chapters and come back here for a second reading. In this chapter, we will redefine (simpler versions of) many things that appear in Mathlib so we will used indices to distinguish our version. For instance we will have ``Ring₁`` as our version of ``Ring``. Since we will gradually explain more powerful ways of formalizing structures, those indices will sometimes grow beyond one. .. include:: C07_Hierarchies/S01_Basics.inc .. include:: C07_Hierarchies/S02_Morphisms.inc .. include:: C07_Hierarchies/S03_Subobjects.inc
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@@ -1,84 +0,0 @@.. _topology: .. index:: topology Topology ======== Calculus is based on the concept of a function, which is used to model quantities that depend on one another. For example, it is common to study quantities that change over time. The notion of a *limit* is also fundamental. We may say that the limit of a function :math:`f(x)` is a value :math:`b` as :math:`x` approaches a value :math:`a`, or that :math:`f(x)` *converges to* :math:`b` as :math:`x` approaches :math:`a`. Equivalently, we may say that a :math:`f(x)` approaches :math:`a` as :math:`x` approaches a value :math:`b`, or that it *tends to* :math:`b` as :math:`x` tends to :math:`a`. We have already begun to consider such notions in :numref:`sequences_and_convergence`. *Topology* is the abstract study of limits and continuity. Having covered the essentials of formalization in Chapters :numref:`%s <basics>` to :numref:`%s <structures>`, in this chapter, we will explain how topological notions are formalized in mathlib. Not only do topological abstractions apply in much greater generality, but that also, somewhat paradoxically, make it easier to reason about limits and continuity in concrete instances. Topological notions build on quite a few layers of mathematical structure. The first layer is naive set theory, as described in :numref:`Chapter %s <sets_and_functions>`. The next layer is the theory of *filters*, which we will describe in :numref:`filters`. On top of that, we layer the theories of *topological spaces*, *metric spaces*, and a slightly more exotic intermediate notion called a *uniform space*. Whereas previous chapters relied on mathematical notions that were likely familiar to you, the notion of a filter less well known, even to many working mathematicians. The notion is essential, however, for formalizing mathematics effectively. Let us explain why. Let ``f : ℝ → ℝ`` be any function. We can consider the limit of ``f x`` as ``x`` approaches some value ``x₀``, but we can also consider the limit of ``f x`` as ``x`` approaches infinity or negative infinity. We can moreover consider the limit of ``f x`` as ``x`` approaches ``x₀`` from the right, conventionally written ``x₀⁺``, or from the left, written ``x₀⁻``. There are variations where ``x`` approaches ``x₀`` or ``x₀⁺`` or ``x₀⁻`` but is not allowed to take on the value ``x₀`` itself. This results in at least eight ways that ``x`` can approach something. We can also restrict to rational values of ``x`` or place other constraints on the domain, but let's stick to those 8 cases. We have a similar variety of options on the codomain: we can specify that ``f x`` approaches a value from the left or right, or that it approaches positive or negative infinity, and so on. For example, we may wish to say that ``f x`` tends to ``+∞`` when ``x`` tends to ``x₀`` from the right without being equal to ``x₀``. This results in 64 different kinds of limit statements, and we haven't even begun to deal with limits of sequences, as we did in :numref:`sequences_and_convergence`. The problem is compounded even further when it comes to the supporting lemmas. For instance, limits compose: if ``f x`` tends to ``y₀`` when ``x`` tends to ``x₀`` and ``g y`` tends to ``z₀`` when ``y`` tends to ``y₀`` then ``g ∘ f x`` tends to ``z₀`` when ``x`` tends to ``x₀``. There are three notions of "tends to" at play here, each of which can be instantiated in any of the eight ways described in the previous paragraph. This results in 512 lemmas, a lot to have to add to a library! Informally, mathematicians generally prove two or three of these and simply note that the rest can be proved "in the same way." Formalizing mathematics requires making the relevant notion of "sameness" fully explicit, and that is exactly what Bourbaki's theory of filters manages to do. .. include:: C08_Topology/S01_Filters.inc .. include:: C08_Topology/S02_Metric_Spaces.inc .. include:: C08_Topology/S03_Topological_Spaces.inc
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@@ -1,18 +0,0 @@.. _differential_calculus: .. index:: differential calculus Differential Calculus ===================== We now consider the formalization of notions from *analysis*, starting with differentiation in this chapter and turning integration and measure theory in the next. In :numref:`elementary_differential_calculus`, we stick with the setting of functions from the real numbers to the real numbers, which is familiar from any introductory calculus class. In :numref:`normed_spaces`, we then consider the notion of a derivative in a much broader setting. .. include:: C09_Differential_Calculus/S01_Elementary_Differential_Calculus.inc .. include:: C09_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.inc
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@@ -1,2 +0,0 @@Index =====
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@@ -1,23 +0,0 @@Mathematics in Lean =================== .. toctree:: :numbered: :maxdepth: 2 C01_Introduction C02_Basics C03_Logic C04_Sets_and_Functions C05_Number_Theory C06_Structures C07_Hierarchies C08_Topology C09_Differential_Calculus .. toctree:: :hidden: genindex
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html/_static/language_data.js (deleted)
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@@ -1,199 +0,0 @@/* * language_data.js * ~~~~~~~~~~~~~~~~ * * This script contains the language-specific data used by searchtools.js, * namely the list of stopwords, stemmer, scorer and splitter. * * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"]; /* Non-minified version is copied as a separate JS file, is available */ /** * Porter Stemmer */ var Stemmer = function() { var step2list = { ational: 'ate', tional: 'tion', enci: 'ence', anci: 'ance', izer: 'ize', bli: 'ble', alli: 'al', entli: 'ent', eli: 'e', ousli: 'ous', ization: 'ize', ation: 'ate', ator: 'ate', alism: 'al', iveness: 'ive', fulness: 'ful', ousness: 'ous', aliti: 'al', iviti: 'ive', biliti: 'ble', logi: 'log' }; var step3list = { icate: 'ic', ative: '', alize: 'al', iciti: 'ic', ical: 'ic', ful: '', ness: '' }; var c = "[^aeiou]"; // consonant var v = "[aeiouy]"; // vowel var C = c + "[^aeiouy]*"; // consonant sequence var V = v + "[aeiou]*"; // vowel sequence var mgr0 = "^(" + C + ")?" + V + C; // [C]VC... is m>0 var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$"; // [C]VC[V] is m=1 var mgr1 = "^(" + C + ")?" + V + C + V + C; // [C]VCVC... is m>1 var s_v = "^(" + C + ")?" + v; // vowel in stem this.stemWord = function (w) { var stem; var suffix; var firstch; var origword = w; if (w.length < 3) return w; var re; var re2; var re3; var re4; firstch = w.substr(0,1); if (firstch == "y") w = firstch.toUpperCase() + w.substr(1); // Step 1a re = /^(.+?)(ss|i)es$/; re2 = /^(.+?)([^s])s$/; if (re.test(w)) w = w.replace(re,"$1$2"); else if (re2.test(w)) w = w.replace(re2,"$1$2"); // Step 1b re = /^(.+?)eed$/; re2 = /^(.+?)(ed|ing)$/; if (re.test(w)) { var fp = re.exec(w); re = new RegExp(mgr0); if (re.test(fp[1])) { re = /.$/; w = w.replace(re,""); } } else if (re2.test(w)) { var fp = re2.exec(w); stem = fp[1]; re2 = new RegExp(s_v); if (re2.test(stem)) { w = stem; re2 = /(at|bl|iz)$/; re3 = new RegExp("([^aeiouylsz])\\1$"); re4 = new RegExp("^" + C + v + "[^aeiouwxy]$"); if (re2.test(w)) w = w + "e"; else if (re3.test(w)) { re = /.$/; w = w.replace(re,""); } else if (re4.test(w)) w = w + "e"; } } // Step 1c re = /^(.+?)y$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; re = new RegExp(s_v); if (re.test(stem)) w = stem + "i"; } // Step 2 re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; suffix = fp[2]; re = new RegExp(mgr0); if (re.test(stem)) w = stem + step2list[suffix]; } // Step 3 re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; suffix = fp[2]; re = new RegExp(mgr0); if (re.test(stem)) w = stem + step3list[suffix]; } // Step 4 re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/; re2 = /^(.+?)(s|t)(ion)$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; re = new RegExp(mgr1); if (re.test(stem)) w = stem; } else if (re2.test(w)) { var fp = re2.exec(w); stem = fp[1] + fp[2]; re2 = new RegExp(mgr1); if (re2.test(stem)) w = stem; } // Step 5 re = /^(.+?)e$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; re = new RegExp(mgr1); re2 = new RegExp(meq1); re3 = new RegExp("^" + C + v + "[^aeiouwxy]$"); if (re.test(stem) || (re2.test(stem) && !(re3.test(stem)))) w = stem; } re = /ll$/; re2 = new RegExp(mgr1); if (re.test(w) && re2.test(w)) { re = /.$/; w = w.replace(re,""); } // and turn initial Y back to y if (firstch == "y") w = firstch.toLowerCase() + w.substr(1); return w; } }
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html/_static/minus.png (deleted)
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html/_static/plus.png (deleted)
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html/_static/pygments.css (deleted)
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@@ -1,74 +0,0 @@pre { line-height: 125%; } td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; } span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; } td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; } span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; } .highlight .hll { background-color: #ffffcc } .highlight { background: #f8f8f8; } .highlight .c { color: #3D7B7B; font-style: italic } /* Comment */ .highlight .err { border: 1px solid #FF0000 } /* Error */ .highlight .k { color: #008000; font-weight: bold } /* Keyword */ .highlight .o { color: #666666 } /* Operator */ .highlight .ch { color: #3D7B7B; font-style: italic } /* Comment.Hashbang */ .highlight .cm { color: #3D7B7B; font-style: italic } /* Comment.Multiline */ .highlight .cp { color: #9C6500 } /* Comment.Preproc */ .highlight .cpf { color: #3D7B7B; font-style: italic } /* Comment.PreprocFile */ .highlight .c1 { color: #3D7B7B; font-style: italic } /* Comment.Single */ .highlight .cs { color: #3D7B7B; font-style: italic } /* Comment.Special */ .highlight .gd { color: #A00000 } /* Generic.Deleted */ .highlight .ge { font-style: italic } /* Generic.Emph */ .highlight .gr { color: #E40000 } /* Generic.Error */ .highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */ .highlight .gi { color: #008400 } /* Generic.Inserted */ .highlight .go { color: #717171 } /* Generic.Output */ .highlight .gp { color: #000080; font-weight: bold } /* Generic.Prompt */ .highlight .gs { font-weight: bold } /* Generic.Strong */ .highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */ .highlight .gt { color: #0044DD } /* Generic.Traceback */ .highlight .kc { color: #008000; font-weight: bold } /* Keyword.Constant */ .highlight .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */ .highlight .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */ .highlight .kp { color: #008000 } /* Keyword.Pseudo */ .highlight .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */ .highlight .kt { color: #B00040 } /* Keyword.Type */ .highlight .m { color: #666666 } /* Literal.Number */ .highlight .s { color: #BA2121 } /* Literal.String */ .highlight .na { color: #687822 } /* Name.Attribute */ .highlight .nb { color: #008000 } /* Name.Builtin */ .highlight .nc { color: #0000FF; font-weight: bold } /* Name.Class */ .highlight .no { color: #880000 } /* Name.Constant */ .highlight .nd { color: #AA22FF } /* Name.Decorator */ .highlight .ni { color: #717171; font-weight: bold } /* Name.Entity */ .highlight .ne { color: #CB3F38; font-weight: bold } /* Name.Exception */ .highlight .nf { color: #0000FF } /* Name.Function */ .highlight .nl { color: #767600 } /* Name.Label */ .highlight .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */ .highlight .nt { color: #008000; font-weight: bold } /* Name.Tag */ .highlight .nv { color: #19177C } /* Name.Variable */ .highlight .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */ .highlight .w { color: #bbbbbb } /* Text.Whitespace */ .highlight .mb { color: #666666 } /* Literal.Number.Bin */ .highlight .mf { color: #666666 } /* Literal.Number.Float */ .highlight .mh { color: #666666 } /* Literal.Number.Hex */ .highlight .mi { color: #666666 } /* Literal.Number.Integer */ .highlight .mo { color: #666666 } /* Literal.Number.Oct */ .highlight .sa { color: #BA2121 } /* Literal.String.Affix */ .highlight .sb { color: #BA2121 } /* Literal.String.Backtick */ .highlight .sc { color: #BA2121 } /* Literal.String.Char */ .highlight .dl { color: #BA2121 } /* Literal.String.Delimiter */ .highlight .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */ .highlight .s2 { color: #BA2121 } /* Literal.String.Double */ .highlight .se { color: #AA5D1F; font-weight: bold } /* Literal.String.Escape */ .highlight .sh { color: #BA2121 } /* Literal.String.Heredoc */ .highlight .si { color: #A45A77; font-weight: bold } /* Literal.String.Interpol */ .highlight .sx { color: #008000 } /* Literal.String.Other */ .highlight .sr { color: #A45A77 } /* Literal.String.Regex */ .highlight .s1 { color: #BA2121 } /* Literal.String.Single */ .highlight .ss { color: #19177C } /* Literal.String.Symbol */ .highlight .bp { color: #008000 } /* Name.Builtin.Pseudo */ .highlight .fm { color: #0000FF } /* Name.Function.Magic */ .highlight .vc { color: #19177C } /* Name.Variable.Class */ .highlight .vg { color: #19177C } /* Name.Variable.Global */ .highlight .vi { color: #19177C } /* Name.Variable.Instance */ .highlight .vm { color: #19177C } /* Name.Variable.Magic */ .highlight .il { color: #666666 } /* Literal.Number.Integer.Long */
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html/_static/searchtools.js (deleted)
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@@ -1,566 +0,0 @@/* * searchtools.js * ~~~~~~~~~~~~~~~~ * * Sphinx JavaScript utilities for the full-text search. * * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ "use strict"; /** * Simple result scoring code. */ if (typeof Scorer === "undefined") { var Scorer = { // Implement the following function to further tweak the score for each result // The function takes a result array [docname, title, anchor, descr, score, filename] // and returns the new score. /* score: result => { const [docname, title, anchor, descr, score, filename] = result return score }, */ // query matches the full name of an object objNameMatch: 11, // or matches in the last dotted part of the object name objPartialMatch: 6, // Additive scores depending on the priority of the object objPrio: { 0: 15, // used to be importantResults 1: 5, // used to be objectResults 2: -5, // used to be unimportantResults }, // Used when the priority is not in the mapping. objPrioDefault: 0, // query found in title title: 15, partialTitle: 7, // query found in terms term: 5, partialTerm: 2, }; } const _removeChildren = (element) => { while (element && element.lastChild) element.removeChild(element.lastChild); }; /** * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping */ const _escapeRegExp = (string) => string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string const _displayItem = (item, searchTerms) => { const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; const [docName, title, anchor, descr, score, _filename] = item; let listItem = document.createElement("li"); let requestUrl; let linkUrl; if (docBuilder === "dirhtml") { // dirhtml builder let dirname = docName + "/"; if (dirname.match(/\/index\/$/)) dirname = dirname.substring(0, dirname.length - 6); else if (dirname === "index/") dirname = ""; requestUrl = docUrlRoot + dirname; linkUrl = requestUrl; } else { // normal html builders requestUrl = docUrlRoot + docName + docFileSuffix; linkUrl = docName + docLinkSuffix; } let linkEl = listItem.appendChild(document.createElement("a")); linkEl.href = linkUrl + anchor; linkEl.dataset.score = score; linkEl.innerHTML = title; if (descr) listItem.appendChild(document.createElement("span")).innerHTML = " (" + descr + ")"; else if (showSearchSummary) fetch(requestUrl) .then((responseData) => responseData.text()) .then((data) => { if (data) listItem.appendChild( Search.makeSearchSummary(data, searchTerms) ); }); Search.output.appendChild(listItem); }; const _finishSearch = (resultCount) => { Search.stopPulse(); Search.title.innerText = _("Search Results"); if (!resultCount) Search.status.innerText = Documentation.gettext( "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories." ); else Search.status.innerText = _( `Search finished, found ${resultCount} page(s) matching the search query.` ); }; const _displayNextItem = ( results, resultCount, searchTerms ) => { // results left, load the summary and display it // this is intended to be dynamic (don't sub resultsCount) if (results.length) { _displayItem(results.pop(), searchTerms); setTimeout( () => _displayNextItem(results, resultCount, searchTerms), 5 ); } // search finished, update title and status message else _finishSearch(resultCount); }; /** * Default splitQuery function. Can be overridden in ``sphinx.search`` with a * custom function per language. * * The regular expression works by splitting the string on consecutive characters * that are not Unicode letters, numbers, underscores, or emoji characters. * This is the same as ``\W+`` in Python, preserving the surrogate pair area. */ if (typeof splitQuery === "undefined") { var splitQuery = (query) => query .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu) .filter(term => term) // remove remaining empty strings } /** * Search Module */ const Search = { _index: null, _queued_query: null, _pulse_status: -1, htmlToText: (htmlString) => { const htmlElement = new DOMParser().parseFromString(htmlString, 'text/html'); htmlElement.querySelectorAll(".headerlink").forEach((el) => { el.remove() }); const docContent = htmlElement.querySelector('[role="main"]'); if (docContent !== undefined) return docContent.textContent; console.warn( "Content block not found. Sphinx search tries to obtain it via '[role=main]'. Could you check your theme or template." ); return ""; }, init: () => { const query = new URLSearchParams(window.location.search).get("q"); document .querySelectorAll('input[name="q"]') .forEach((el) => (el.value = query)); if (query) Search.performSearch(query); }, loadIndex: (url) => (document.body.appendChild(document.createElement("script")).src = url), setIndex: (index) => { Search._index = index; if (Search._queued_query !== null) { const query = Search._queued_query; Search._queued_query = null; Search.query(query); } }, hasIndex: () => Search._index !== null, deferQuery: (query) => (Search._queued_query = query), stopPulse: () => (Search._pulse_status = -1), startPulse: () => { if (Search._pulse_status >= 0) return; const pulse = () => { Search._pulse_status = (Search._pulse_status + 1) % 4; Search.dots.innerText = ".".repeat(Search._pulse_status); if (Search._pulse_status >= 0) window.setTimeout(pulse, 500); }; pulse(); }, /** * perform a search for something (or wait until index is loaded) */ performSearch: (query) => { // create the required interface elements const searchText = document.createElement("h2"); searchText.textContent = _("Searching"); const searchSummary = document.createElement("p"); searchSummary.classList.add("search-summary"); searchSummary.innerText = ""; const searchList = document.createElement("ul"); searchList.classList.add("search"); const out = document.getElementById("search-results"); Search.title = out.appendChild(searchText); Search.dots = Search.title.appendChild(document.createElement("span")); Search.status = out.appendChild(searchSummary); Search.output = out.appendChild(searchList); const searchProgress = document.getElementById("search-progress"); // Some themes don't use the search progress node if (searchProgress) { searchProgress.innerText = _("Preparing search..."); } Search.startPulse(); // index already loaded, the browser was quick! if (Search.hasIndex()) Search.query(query); else Search.deferQuery(query); }, /** * execute search (requires search index to be loaded) */ query: (query) => { const filenames = Search._index.filenames; const docNames = Search._index.docnames; const titles = Search._index.titles; const allTitles = Search._index.alltitles; const indexEntries = Search._index.indexentries; // stem the search terms and add them to the correct list const stemmer = new Stemmer(); const searchTerms = new Set(); const excludedTerms = new Set(); const highlightTerms = new Set(); const objectTerms = new Set(splitQuery(query.toLowerCase().trim())); splitQuery(query.trim()).forEach((queryTerm) => { const queryTermLower = queryTerm.toLowerCase(); // maybe skip this "word" // stopwords array is from language_data.js if ( stopwords.indexOf(queryTermLower) !== -1 || queryTerm.match(/^\d+$/) ) return; // stem the word let word = stemmer.stemWord(queryTermLower); // select the correct list if (word[0] === "-") excludedTerms.add(word.substr(1)); else { searchTerms.add(word); highlightTerms.add(queryTermLower); } }); if (SPHINX_HIGHLIGHT_ENABLED) { // set in sphinx_highlight.js localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" ")) } // console.debug("SEARCH: searching for:"); // console.info("required: ", [...searchTerms]); // console.info("excluded: ", [...excludedTerms]); // array of [docname, title, anchor, descr, score, filename] let results = []; _removeChildren(document.getElementById("search-progress")); const queryLower = query.toLowerCase(); for (const [title, foundTitles] of Object.entries(allTitles)) { if (title.toLowerCase().includes(queryLower) && (queryLower.length >= title.length/2)) { for (const [file, id] of foundTitles) { let score = Math.round(100 * queryLower.length / title.length) results.push([ docNames[file], titles[file] !== title ? `${titles[file]} > ${title}` : title, id !== null ? "#" + id : "", null, score, filenames[file], ]); } } } // search for explicit entries in index directives for (const [entry, foundEntries] of Object.entries(indexEntries)) { if (entry.includes(queryLower) && (queryLower.length >= entry.length/2)) { for (const [file, id] of foundEntries) { let score = Math.round(100 * queryLower.length / entry.length) results.push([ docNames[file], titles[file], id ? "#" + id : "", null, score, filenames[file], ]); } } } // lookup as object objectTerms.forEach((term) => results.push(...Search.performObjectSearch(term, objectTerms)) ); // lookup as search terms in fulltext results.push(...Search.performTermsSearch(searchTerms, excludedTerms)); // let the scorer override scores with a custom scoring function if (Scorer.score) results.forEach((item) => (item[4] = Scorer.score(item))); // now sort the results by score (in opposite order of appearance, since the // display function below uses pop() to retrieve items) and then // alphabetically results.sort((a, b) => { const leftScore = a[4]; const rightScore = b[4]; if (leftScore === rightScore) { // same score: sort alphabetically const leftTitle = a[1].toLowerCase(); const rightTitle = b[1].toLowerCase(); if (leftTitle === rightTitle) return 0; return leftTitle > rightTitle ? -1 : 1; // inverted is intentional } return leftScore > rightScore ? 1 : -1; }); // remove duplicate search results // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept let seen = new Set(); results = results.reverse().reduce((acc, result) => { let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(','); if (!seen.has(resultStr)) { acc.push(result); seen.add(resultStr); } return acc; }, []); results = results.reverse(); // for debugging //Search.lastresults = results.slice(); // a copy // console.info("search results:", Search.lastresults); // print the results _displayNextItem(results, results.length, searchTerms); }, /** * search for object names */ performObjectSearch: (object, objectTerms) => { const filenames = Search._index.filenames; const docNames = Search._index.docnames; const objects = Search._index.objects; const objNames = Search._index.objnames; const titles = Search._index.titles; const results = []; const objectSearchCallback = (prefix, match) => { const name = match[4] const fullname = (prefix ? prefix + "." : "") + name; const fullnameLower = fullname.toLowerCase(); if (fullnameLower.indexOf(object) < 0) return; let score = 0; const parts = fullnameLower.split("."); // check for different match types: exact matches of full name or // "last name" (i.e. last dotted part) if (fullnameLower === object || parts.slice(-1)[0] === object) score += Scorer.objNameMatch; else if (parts.slice(-1)[0].indexOf(object) > -1) score += Scorer.objPartialMatch; // matches in last name const objName = objNames[match[1]][2]; const title = titles[match[0]]; // If more than one term searched for, we require other words to be // found in the name/title/description const otherTerms = new Set(objectTerms); otherTerms.delete(object); if (otherTerms.size > 0) { const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase(); if ( [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0) ) return; } let anchor = match[3]; if (anchor === "") anchor = fullname; else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname; const descr = objName + _(", in ") + title; // add custom score for some objects according to scorer if (Scorer.objPrio.hasOwnProperty(match[2])) score += Scorer.objPrio[match[2]]; else score += Scorer.objPrioDefault; results.push([ docNames[match[0]], fullname, "#" + anchor, descr, score, filenames[match[0]], ]); }; Object.keys(objects).forEach((prefix) => objects[prefix].forEach((array) => objectSearchCallback(prefix, array) ) ); return results; }, /** * search for full-text terms in the index */ performTermsSearch: (searchTerms, excludedTerms) => { // prepare search const terms = Search._index.terms; const titleTerms = Search._index.titleterms; const filenames = Search._index.filenames; const docNames = Search._index.docnames; const titles = Search._index.titles; const scoreMap = new Map(); const fileMap = new Map(); // perform the search on the required terms searchTerms.forEach((word) => { const files = []; const arr = [ { files: terms[word], score: Scorer.term }, { files: titleTerms[word], score: Scorer.title }, ]; // add support for partial matches if (word.length > 2) { const escapedWord = _escapeRegExp(word); Object.keys(terms).forEach((term) => { if (term.match(escapedWord) && !terms[word]) arr.push({ files: terms[term], score: Scorer.partialTerm }); }); Object.keys(titleTerms).forEach((term) => { if (term.match(escapedWord) && !titleTerms[word]) arr.push({ files: titleTerms[word], score: Scorer.partialTitle }); }); } // no match but word was a required one if (arr.every((record) => record.files === undefined)) return; // found search word in contents arr.forEach((record) => { if (record.files === undefined) return; let recordFiles = record.files; if (recordFiles.length === undefined) recordFiles = [recordFiles]; files.push(...recordFiles); // set score for the word in each file recordFiles.forEach((file) => { if (!scoreMap.has(file)) scoreMap.set(file, {}); scoreMap.get(file)[word] = record.score; }); }); // create the mapping files.forEach((file) => { if (fileMap.has(file) && fileMap.get(file).indexOf(word) === -1) fileMap.get(file).push(word); else fileMap.set(file, [word]); }); }); // now check if the files don't contain excluded terms const results = []; for (const [file, wordList] of fileMap) { // check if all requirements are matched // as search terms with length < 3 are discarded const filteredTermCount = [...searchTerms].filter( (term) => term.length > 2 ).length; if ( wordList.length !== searchTerms.size && wordList.length !== filteredTermCount ) continue; // ensure that none of the excluded terms is in the search result if ( [...excludedTerms].some( (term) => terms[term] === file || titleTerms[term] === file || (terms[term] || []).includes(file) || (titleTerms[term] || []).includes(file) ) ) break; // select one (max) score for the file. const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w])); // add result to the result list results.push([ docNames[file], titles[file], "", null, score, filenames[file], ]); } return results; }, /** * helper function to return a node containing the * search summary for a given text. keywords is a list * of stemmed words. */ makeSearchSummary: (htmlText, keywords) => { const text = Search.htmlToText(htmlText); if (text === "") return null; const textLower = text.toLowerCase(); const actualStartPosition = [...keywords] .map((k) => textLower.indexOf(k.toLowerCase())) .filter((i) => i > -1) .slice(-1)[0]; const startWithContext = Math.max(actualStartPosition - 120, 0); const top = startWithContext === 0 ? "" : "..."; const tail = startWithContext + 240 < text.length ? "..." : ""; let summary = document.createElement("p"); summary.classList.add("context"); summary.textContent = top + text.substr(startWithContext, 240).trim() + tail; return summary; }, }; _ready(Search.init);
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@@ -1,553 +0,0 @@<!DOCTYPE html> <html class="writer-html5" lang="en" > <head> <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>Index — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> <link rel="shortcut icon" href="_static/favicon.ico"/> <!--[if lt IE 9]> <script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="#" /> <link rel="search" title="Search" href="search.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul class="current"> <li class="toctree-l1 current"><a class="current reference internal" href="#">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="index.html">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active">Index</li> <li class="wy-breadcrumbs-aside"> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <h1 id="index">Index</h1> <div class="genindex-jumpbox"> <a href="#A"><strong>A</strong></a> | <a href="#B"><strong>B</strong></a> | <a href="#C"><strong>C</strong></a> | <a href="#D"><strong>D</strong></a> | <a href="#E"><strong>E</strong></a> | <a href="#F"><strong>F</strong></a> | <a href="#G"><strong>G</strong></a> | <a href="#H"><strong>H</strong></a> | <a href="#I"><strong>I</strong></a> | <a href="#L"><strong>L</strong></a> | <a href="#M"><strong>M</strong></a> | <a href="#N"><strong>N</strong></a> | <a href="#O"><strong>O</strong></a> | <a href="#P"><strong>P</strong></a> | <a href="#R"><strong>R</strong></a> | <a href="#S"><strong>S</strong></a> | <a href="#T"><strong>T</strong></a> | <a href="#U"><strong>U</strong></a> </div> <h2 id="A">A</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-23">absolute value</a> </li> <li><a href="C03_Logic.html#index-17">absurd</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-7">anonymous constructor</a> </li> <li><a href="C02_Basics.html#index-16">apply</a> </li> <li><a href="C03_Logic.html#index-19">assumption</a> </li> </ul></td> </tr></table> <h2 id="B">B</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C04_Sets_and_Functions.html#index-3">bounded quantifiers</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-23">by_cases</a> </li> <li><a href="C03_Logic.html#index-14">by_contra</a> </li> </ul></td> </tr></table> <h2 id="C">C</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-3">calc</a> </li> <li><a href="C03_Logic.html#index-8">cases</a> </li> <li><a href="C03_Logic.html#index-2">change</a> </li> <li><a href="C01_Introduction.html#index-0">check</a> </li> <li> command <ul> <li><a href="C02_Basics.html#index-9">open</a> </li> </ul></li> <li> commands <ul> <li><a href="C01_Introduction.html#index-0">check</a> </li> </ul></li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-8">commutative ring</a> </li> <li><a href="C03_Logic.html#index-25">congr</a> </li> <li><a href="C03_Logic.html#index-18">constructor</a> </li> <li><a href="C08_Topology.html#index-3">continuity</a> </li> <li><a href="C03_Logic.html#index-17">contradiction</a> </li> <li><a href="C03_Logic.html#index-16">contrapose</a> </li> <li><a href="C03_Logic.html#index-26">convert</a> </li> </ul></td> </tr></table> <h2 id="D">D</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-12">definitional equality</a> </li> <li><a href="C09_Differential_Calculus.html#index-0">differential calculus</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-24">divisibility</a> </li> <li><a href="C03_Logic.html#index-2">dsimp</a> </li> </ul></td> </tr></table> <h2 id="E">E</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C09_Differential_Calculus.html#index-1">elementary calculus</a> </li> <li><a href="C03_Logic.html#index-4">erw</a> </li> <li><a href="C02_Basics.html#index-5">exact</a> </li> <li><a href="C03_Logic.html#index-22">excluded middle</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-17">exfalso</a> </li> <li><a href="C02_Basics.html#index-18">exponential</a> </li> <li><a href="C03_Logic.html#index-24">ext</a> </li> <li><a href="C03_Logic.html#index-24">extensionality</a> </li> </ul></td> </tr></table> <h2 id="F">F</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-11">field_simp</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C08_Topology.html#index-1">Filter</a> </li> <li><a href="C03_Logic.html#index-12">from</a> </li> </ul></td> </tr></table> <h2 id="G">G</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-25">gcd</a> </li> <li><a href="C02_Basics.html#index-2">goal</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-13">group (algebraic structure)</a> <ul> <li><a href="C02_Basics.html#index-14">(tactic)</a> </li> </ul></li> </ul></td> </tr></table> <h2 id="H">H</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-11">have</a>, <a href="C03_Logic.html#index-12">[1]</a> </li> </ul></td> </tr></table> <h2 id="I">I</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-10">implicit argument</a> </li> <li><a href="C02_Basics.html#index-15">inequalities</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-5">injective function</a> </li> <li><a href="C10_Integration_and_Measure_Theory.html#index-0">integration</a> </li> <li><a href="C03_Logic.html#index-0">intro</a> </li> </ul></td> </tr></table> <h2 id="L">L</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-1">lambda abstraction</a> </li> <li><a href="C02_Basics.html#index-27">lattice</a> </li> <li><a href="C02_Basics.html#index-25">lcm</a> </li> <li><a href="C03_Logic.html#index-21">left</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-13">let</a> </li> <li><a href="C02_Basics.html#index-17">linarith</a> </li> <li><a href="C02_Basics.html#index-2">local context</a> </li> <li><a href="C02_Basics.html#index-18">logarithm</a> </li> </ul></td> </tr></table> <h2 id="M">M</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-20">max</a> </li> <li><a href="C02_Basics.html#index-28">metric space</a>, <a href="C08_Topology.html#index-2">[1]</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-20">min</a> </li> <li><a href="C03_Logic.html#index-3">monotone function</a> </li> </ul></td> </tr></table> <h2 id="N">N</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-9">namespace</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-19">norm_num</a> </li> <li><a href="C09_Differential_Calculus.html#index-2">normed space</a> </li> </ul></td> </tr></table> <h2 id="O">O</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-9">open</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-26">order relation</a> </li> </ul></td> </tr></table> <h2 id="P">P</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-26">partial order</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-2">proof state</a> </li> <li><a href="C03_Logic.html#index-15">push_neg</a> </li> </ul></td> </tr></table> <h2 id="R">R</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-9">rcases</a> </li> <li><a href="C02_Basics.html#index-1">real numbers</a> </li> <li><a href="C02_Basics.html#index-12">reflexivity</a> </li> <li><a href="C02_Basics.html#index-22">repeat</a> </li> <li><a href="C02_Basics.html#index-0">rewrite</a> </li> <li><a href="C02_Basics.html#index-12">rfl</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-21">right</a> </li> <li><a href="C02_Basics.html#index-7">ring (algebraic structure)</a> <ul> <li><a href="C02_Basics.html#index-6">(tactic)</a> </li> </ul></li> <li><a href="C03_Logic.html#index-9">rintro</a> </li> <li><a href="C02_Basics.html#index-0">rw</a>, <a href="C02_Basics.html#index-4">[1]</a> </li> <li><a href="C04_Sets_and_Functions.html#index-2">rwa</a> </li> </ul></td> </tr></table> <h2 id="S">S</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C04_Sets_and_Functions.html#index-0">set operations</a> </li> <li><a href="C02_Basics.html#index-21">show</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-20">simp</a>, <a href="C04_Sets_and_Functions.html#index-1">[1]</a> </li> <li><a href="C03_Logic.html#index-10">surjective function</a> </li> </ul></td> </tr></table> <h2 id="T">T</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li> tactic <ul> <li><a href="C03_Logic.html#index-11">field_simp</a> </li> </ul></li> <li> tactics <ul> <li><a href="C02_Basics.html#index-14">abel</a> </li> <li><a href="C02_Basics.html#index-16">apply</a> </li> <li><a href="C03_Logic.html#index-19">assumption</a> </li> <li><a href="C03_Logic.html#index-23">by_cases</a> </li> <li><a href="C03_Logic.html#index-14">by_contra and by_contradiction</a> </li> <li><a href="C02_Basics.html#index-3">calc</a> </li> <li><a href="C03_Logic.html#index-8">cases</a> </li> <li><a href="C03_Logic.html#index-2">change</a> </li> <li><a href="C03_Logic.html#index-25">congr</a> </li> <li><a href="C03_Logic.html#index-18">constructor</a> </li> <li><a href="C08_Topology.html#index-3">continuity</a> </li> <li><a href="C03_Logic.html#index-17">contradiction</a> </li> <li><a href="C03_Logic.html#index-16">contrapose</a> </li> <li><a href="C03_Logic.html#index-26">convert</a> </li> <li><a href="C03_Logic.html#index-2">dsimp</a> </li> <li><a href="C03_Logic.html#index-4">erw</a> </li> <li><a href="C02_Basics.html#index-5">exact</a> </li> <li><a href="C03_Logic.html#index-17">exfalso</a> </li> <li><a href="C03_Logic.html#index-24">ext</a> </li> <li><a href="C03_Logic.html#index-12">from</a> </li> <li><a href="C02_Basics.html#index-14">group</a> </li> <li><a href="C02_Basics.html#index-11">have</a>, <a href="C03_Logic.html#index-12">[1]</a> </li> <li><a href="C03_Logic.html#index-0">intro</a> </li> <li><a href="C03_Logic.html#index-21">left</a> </li> <li><a href="C03_Logic.html#index-13">let</a> </li> <li><a href="C02_Basics.html#index-17">linarith</a> </li> <li><a href="C02_Basics.html#index-14">noncomm_ring</a> </li> <li><a href="C02_Basics.html#index-19">norm_num</a> </li> <li><a href="C03_Logic.html#index-15">push_neg</a> </li> <li><a href="C03_Logic.html#index-9">rcases</a> </li> <li><a href="C02_Basics.html#index-12">refl and reflexivity</a> </li> <li><a href="C02_Basics.html#index-22">repeat</a> </li> <li><a href="C03_Logic.html#index-21">right</a> </li> <li><a href="C02_Basics.html#index-6">ring</a> </li> <li><a href="C03_Logic.html#index-9">rintro</a> </li> <li><a href="C02_Basics.html#index-0">rw and rewrite</a>, <a href="C02_Basics.html#index-4">[1]</a> </li> <li><a href="C04_Sets_and_Functions.html#index-2">rwa</a> </li> <li><a href="C02_Basics.html#index-21">show</a> </li> <li><a href="C03_Logic.html#index-20">simp</a>, <a href="C04_Sets_and_Functions.html#index-1">[1]</a> </li> <li><a href="C03_Logic.html#index-6">use</a> </li> </ul></li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-12">this</a> </li> <li><a href="C08_Topology.html#index-4">topological space</a> </li> <li><a href="C08_Topology.html#index-0">topology</a> </li> </ul></td> </tr></table> <h2 id="U">U</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-6">use</a> </li> </ul></td> </tr></table> </div> </div> <footer> <hr/> <div role="contentinfo"> <p>© Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. 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Introduction" href="C01_Introduction.html" /> </head> <body class="wy-body-for-nav"> <div class="wy-grid-for-nav"> <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="#" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> <ul> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> </ul> </div> </div> </nav> <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > <i data-toggle="wy-nav-top" class="fa fa-bars"></i> <a href="#">Mathematics in Lean</a> </nav> <div class="wy-nav-content"> <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="#" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active">Mathematics in Lean</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/index.rst.txt" rel="nofollow"> View page source</a> </li> </ul> <hr/> </div> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="mathematics-in-lean"> <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this heading"></a></h1> <div class="toctree-wrapper compound"> <ul> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a><ul> <li class="toctree-l2"><a class="reference internal" href="C01_Introduction.html#getting-started">1.1. Getting Started</a></li> <li class="toctree-l2"><a class="reference internal" href="C01_Introduction.html#overview">1.2. Overview</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a><ul> <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#calculating">2.1. Calculating</a></li> <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures">2.2. Proving Identities in Algebraic Structures</a></li> <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#using-theorems-and-lemmas">2.3. Using Theorems and Lemmas</a></li> <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#more-on-order-and-divisibility">2.4. More on Order and Divisibility</a></li> <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#proving-facts-about-algebraic-structures">2.5. Proving Facts about Algebraic Structures</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a><ul> <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#implication-and-the-universal-quantifier">3.1. Implication and the Universal Quantifier</a></li> <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#the-existential-quantifier">3.2. The Existential Quantifier</a></li> <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#negation">3.3. Negation</a></li> <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#conjunction-and-bi-implication">3.4. Conjunction and Bi-implication</a></li> <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#disjunction">3.5. Disjunction</a></li> <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#sequences-and-convergence">3.6. Sequences and Convergence</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a><ul> <li class="toctree-l2"><a class="reference internal" href="C04_Sets_and_Functions.html#sets">4.1. Sets</a></li> <li class="toctree-l2"><a class="reference internal" href="C04_Sets_and_Functions.html#functions">4.2. Functions</a></li> <li class="toctree-l2"><a class="reference internal" href="C04_Sets_and_Functions.html#the-schroder-bernstein-theorem">4.3. The Schröder-Bernstein Theorem</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a><ul> <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#irrational-roots">5.1. Irrational Roots</a></li> <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#induction-and-recursion">5.2. Induction and Recursion</a></li> <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#id2">5.3. Induction and Recursion</a></li> <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#infinitely-many-primes">5.4. Infinitely Many Primes</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a><ul> <li class="toctree-l2"><a class="reference internal" href="C06_Structures.html#defining-structures">6.1. Defining structures</a></li> <li class="toctree-l2"><a class="reference internal" href="C06_Structures.html#algebraic-structures">6.2. Algebraic Structures</a></li> <li class="toctree-l2"><a class="reference internal" href="C06_Structures.html#building-the-gaussian-integers">6.3. Building the Gaussian Integers</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a><ul> <li class="toctree-l2"><a class="reference internal" href="C07_Hierarchies.html#basics">7.1. Basics</a></li> <li class="toctree-l2"><a class="reference internal" href="C07_Hierarchies.html#morphisms">7.2. Morphisms</a></li> <li class="toctree-l2"><a class="reference internal" href="C07_Hierarchies.html#sub-objects">7.3. Sub-objects</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a><ul> <li class="toctree-l2"><a class="reference internal" href="C08_Topology.html#filters">8.1. Filters</a></li> <li class="toctree-l2"><a class="reference internal" href="C08_Topology.html#metric-spaces">8.2. Metric spaces</a></li> <li class="toctree-l2"><a class="reference internal" href="C08_Topology.html#topological-spaces">8.3. Topological spaces</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a><ul> <li class="toctree-l2"><a class="reference internal" href="C09_Differential_Calculus.html#elementary-differential-calculus">9.1. Elementary Differential Calculus</a></li> <li class="toctree-l2"><a class="reference internal" href="C09_Differential_Calculus.html#differential-calculus-in-normed-spaces">9.2. Differential Calculus in Normed Spaces</a></li> </ul> </li> </ul> </div> <div class="toctree-wrapper compound"> </div> </section> </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C01_Introduction.html" class="btn btn-neutral float-right" title="1. Introduction" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> </div> <hr/> <div role="contentinfo"> <p>© Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. 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@@ -1,27 +1,1 @@{"version": 4, "packagesDir": "lake-packages", "packages": [{"git": {"url": "https://github.com/leanprover-community/mathlib4", "subDir?": null, "rev": "2db7650f98b1ea3543e48b6b02aa9112e33bb68b", "name": "mathlib", "inputRev?": "master"}}, {"git": {"url": "https://github.com/gebner/quote4", "subDir?": null, "rev": "c71f94e34c1cda52eef5c93dc9da409ab2727420", "name": "Qq", "inputRev?": "master"}}, {"git": {"url": "https://github.com/JLimperg/aesop", "subDir?": null, "rev": "ca73109cc40837bc61df8024c9016da4b4f99d4c", "name": "aesop", "inputRev?": "master"}}, {"git": {"url": "https://github.com/leanprover/std4", "subDir?": null, "rev": "6932c4ea52914dc6b0488944e367459ddc4d01a6", "name": "std", "inputRev?": "main"}}]} ../lake-manifest.json
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@@ -1,13 +1,1 @@import Lake open Lake DSL package mil { -- add package configuration options here } @[default_target] lean_lib MIL { -- add library configuration options here } require mathlib from git "https://github.com/leanprover-community/mathlib4"@"master" ../lakefile.lean
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@@ -1,1 +1,1 @@leanprover/lean4:nightly-2023-05-31 ../lean-toolchain
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