Changes

46 changed files (+438/-2299)

• MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C02S04 + section variable (a b c d : ℝ)
• MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C02S04 + section variable (a b c d : ℝ)
• MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S01 + #check ∀ x : ℝ, 0 ≤ x → |x| = x #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε
• MIL/C03_Logic/S02_The_Existential_Quantifier.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S02 + example : ∃ x : ℝ, 2 < x ∧ x < 3 := by use 5 / 2 norm_num
  • @@ -34,8 +35,8 @@ 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 + rcases ubf with ⟨a, ubfa⟩ + rcases ubg with ⟨b, ubgb⟩ use a + b apply fnUb_add ubfa ubgb
  • @@ -45,11 +46,6 @@ example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x ↦ f x + g x := by
    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⟩
  • @@ -59,6 +55,35 @@ example : FnHasUb f → FnHasUb g → FnHasUb fun x ↦ f x + g x :=
    end + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by + obtain ⟨a, ubfa⟩ := ubf + obtain ⟨b, ubgb⟩ := ubg + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by + cases ubf + case intro a ubfa => + cases ubg + case intro b ubgb => + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by + cases ubf + next a ubfa => + cases ubg + next b ubgb => + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by + match ubf, ubg with + | ⟨a, ubfa⟩, ⟨b, ubgb⟩ => + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := + match ubf, ubg with + | ⟨a, ubfa⟩, ⟨b, ubgb⟩ => + ⟨a + b, fnUb_add ubfa ubgb⟩ + section variable {α : Type _} [CommRing α]
  • @@ -87,8 +112,8 @@ 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 + rcases divab with ⟨d, beq⟩ + rcases divbc with ⟨e, ceq⟩ rw [ceq, beq] use d * e; ring
  • @@ -114,7 +139,7 @@ example (x y : ℝ) (h : x - y ≠ 0) : (x ^ 2 - y ^ 2) / (x - y) = x + y := by
    ring example {f : ℝ → ℝ} (h : Surjective f) : ∃ x, f x ^ 2 = 4 := by - cases' h 2 with x hx + rcases h 2 with ⟨x, hx⟩ use x rw [hx] norm_num
• MIL/C03_Logic/S03_Negation.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S03 + section variable (a b : ℝ)
  • @@ -26,8 +27,8 @@ 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 + rcases fnub with ⟨a, fnuba⟩ + rcases h a with ⟨x, hx⟩ have : f x ≤ a := fnuba x linarith
  • @@ -102,7 +103,7 @@ example (h : ¬∀ a, ∃ x, f x > a) : FnHasUb f := by
    exact h example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by - simp only [FnHasUb, FnUb] at h + dsimp only [FnHasUb, FnUb] at h push_neg at h exact h
• MIL/C03_Logic/S04_Conjunction_and_Bi-implication.lean
  • @@ -3,6 +3,7 @@ import Mathlib.Data.Real.Basic
    import Mathlib.Data.Nat.Prime namespace C03S04 + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by constructor · assumption
  • @@ -20,7 +21,7 @@ example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y :=
    ⟨h₀, h⟩ example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by - cases' h with h₀ h₁ + rcases h with ⟨h₀, h₁⟩ contrapose! h₁ exact le_antisymm h₀ h₁
  • @@ -31,6 +32,29 @@ example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := by
    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 + have ⟨h₀, h₁⟩ := h + contrapose! h₁ + exact le_antisymm h₀ h₁ + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + cases h + case intro h₀ h₁ => + contrapose! h₁ + exact le_antisymm h₀ h₁ + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + cases h + next h₀ h₁ => + contrapose! h₁ + exact le_antisymm h₀ h₁ + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + match h with + | ⟨h₀, h₁⟩ => + contrapose! h₁ + exact le_antisymm h₀ h₁ + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by intro h' apply h.right
• MIL/C03_Logic/S05_Disjunction.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S05 + section variable {x y : ℝ}
  • @@ -21,13 +22,39 @@ example (h : y < -1) : y > 0 ∨ y < -1 :=
    Or.inr h example : x < |y| → x < y ∨ x < -y := by - cases' le_or_gt 0 y with h h + rcases 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 + intro h; left; exact h + . rw [abs_of_neg h] + intro h; right; exact h + + example : x < |y| → x < y ∨ x < -y := by + cases le_or_gt 0 y + case inl h => + rw [abs_of_nonneg h] + intro h; left; exact h + case inr h => + rw [abs_of_neg h] + intro h; right; exact h + + example : x < |y| → x < y ∨ x < -y := by + cases le_or_gt 0 y + next h => + rw [abs_of_nonneg h] + intro h; left; exact h + next h => + rw [abs_of_neg h] + intro h; right; exact h + + + example : x < |y| → x < y ∨ x < -y := by + match le_or_gt 0 y with + | Or.inl h => + rw [abs_of_nonneg h] + intro h; left; exact h + | Or.inr h => + rw [abs_of_neg h] + intro h; right; exact h namespace MyAbs
  • @@ -51,18 +78,18 @@ end MyAbs
    end example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := by - rcases lt_trichotomy x 0 with (xlt | xeq | xgt) + rcases lt_trichotomy x 0 with xlt | xeq | xgt · left exact xlt · contradiction - right; exact xgt + . right; exact xgt example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := by - rcases h with (⟨a, rfl⟩ | ⟨b, rfl⟩) + rcases h with ⟨a, rfl⟩ | ⟨b, rfl⟩ · rw [mul_assoc] apply dvd_mul_right - rw [mul_comm, 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
  • @@ -89,7 +116,7 @@ example (P : Prop) : ¬¬P → P := by
    intro h cases em P · assumption - contradiction + . contradiction example (P : Prop) : ¬¬P → P := by intro h
• MIL/C03_Logic/S06_Sequences_and_Convergence.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S06 + def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε
  • @@ -31,8 +32,8 @@ theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ}
    intro ε εpos dsimp -- this line is not needed but cleans up the goal a bit. have ε2pos : 0 < ε / 2 := by linarith - cases' cs (ε / 2) ε2pos with Ns hs - cases' ct (ε / 2) ε2pos with Nt ht + rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩ + rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩ use max Ns Nt sorry
  • @@ -49,7 +50,7 @@ theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : Conver
    theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : ∃ N b, ∀ n, N ≤ n → |s n| < b := by - cases' cs 1 zero_lt_one with N h + rcases cs 1 zero_lt_one with ⟨N, h⟩ use N, |a| + 1 sorry
  • @@ -60,7 +61,7 @@ theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : Converges
    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 + rcases ct _ pos₀ with ⟨N₁, h₁⟩ sorry theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ}
  • @@ -84,8 +85,8 @@ theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ}
    have εpos : ε > 0 := by change |a - b| / 2 > 0 linarith - cases' sa ε εpos with Na hNa - cases' sb ε εpos with Nb hNb + rcases sa ε εpos with ⟨Na, hNa⟩ + rcases sb ε εpos with ⟨Nb, hNb⟩ let N := max Na Nb have absa : |s N - a| < ε := by sorry have absb : |s N - b| < ε := by sorry
• MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S01 + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a
• MIL/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S02 + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a
  • @@ -23,14 +24,14 @@ 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 + rcases lbf with ⟨a, lbfa⟩ + rcases 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 + rcases ubf with ⟨a, lbfa⟩ use c * a intro x exact mul_le_mul_of_nonneg_left (lbfa x) h
• MIL/C03_Logic/solutions/Solutions_S03_Negation.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S03 + section variable (a b : ℝ)
• MIL/C03_Logic/solutions/Solutions_S04_Conjunction_and_Bi-implication.lean
  • @@ -3,8 +3,9 @@ import Mathlib.Data.Real.Basic
    import Mathlib.Data.Nat.Prime namespace C03S04 + example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by - cases' h with h0 h1 + rcases h with ⟨h0, h1⟩ constructor · exact h0 intro h2
• MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S05 + section variable {x y : ℝ}
  • @@ -9,64 +10,64 @@ variable {x y : ℝ}
    namespace MyAbs theorem le_abs_self (x : ℝ) : x ≤ |x| := by - cases' le_or_gt 0 x with h h + rcases le_or_gt 0 x with h | h · rw [abs_of_nonneg h] - rw [abs_of_neg h] - linarith + . rw [abs_of_neg h] + linarith theorem neg_le_abs_self (x : ℝ) : -x ≤ |x| := by - cases' le_or_gt 0 x with h h + rcases le_or_gt 0 x with h | h · rw [abs_of_nonneg h] linarith - rw [abs_of_neg h] + . rw [abs_of_neg h] theorem abs_add (x y : ℝ) : |x + y| ≤ |x| + |y| := by - cases' le_or_gt 0 (x + y) with h h + rcases 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] + . rw [abs_of_neg h] + linarith [neg_le_abs_self x, neg_le_abs_self y] theorem lt_abs : x < |y| ↔ x < y ∨ x < -y := by - cases' le_or_gt 0 y with h h + rcases 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 + . intro h' + rcases 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' + . intro h' + rcases h' with h' | h' + · linarith + . exact h' theorem abs_lt : |x| < y ↔ -y < x ∧ x < y := by - cases' le_or_gt 0 x with h h + rcases 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' + . intro h' + rcases h' with ⟨h1, h2⟩ + exact h2 + . rw [abs_of_neg h] constructor - · linarith - linarith - intro h' - linarith + · intro h' + constructor + · linarith + . linarith + . intro h' + linarith end MyAbs
  • @@ -80,22 +81,22 @@ example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
    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 + rcases 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 + . 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 + rcases 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 + . left + exact eq_of_sub_eq_zero h1 section variable {R : Type _} [CommRing R] [IsDomain R]
  • @@ -106,22 +107,22 @@ example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
    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 + rcases 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 + . 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 + rcases 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 + . left + exact eq_of_sub_eq_zero h1 end
  • @@ -131,10 +132,10 @@ example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by
    by_cases h' : P · right exact h h' - left - exact h' + . left + exact h' rintro (h | h) · intro h' exact absurd h' h - intro - exact h + . intro + exact h
• MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C03S06 + def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε
  • @@ -18,8 +19,8 @@ theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ}
    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 + rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩ + rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩ use max Ns Nt intro n hn have ngeNs : n ≥ Ns := le_of_max_le_left hn
  • @@ -44,7 +45,7 @@ theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : Conver
    intro ε εpos dsimp have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos - cases' cs (ε / |c|) εcpos with Ns hs + rcases cs (ε / |c|) εcpos with ⟨Ns, hs⟩ use Ns intro n ngt calc
  • @@ -54,7 +55,7 @@ theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : Conver
    theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : ∃ N b, ∀ n, N ≤ n → |s n| < b := by - cases' cs 1 zero_lt_one with N h + rcases cs 1 zero_lt_one with ⟨N, h⟩ use N, |a| + 1 intro n ngt calc
  • @@ -71,7 +72,7 @@ theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : Converges
    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 + rcases ct _ pos₀ with ⟨N₁, h₁⟩ use max N₀ N₁ intro n ngt have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt
  • @@ -107,8 +108,8 @@ theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ}
    have εpos : ε > 0 := by change |a - b| / 2 > 0 linarith - cases' sa ε εpos with Na hNa - cases' sb ε εpos with Nb hNb + rcases sa ε εpos with ⟨Na, hNa⟩ + rcases sb ε εpos with ⟨Nb, hNb⟩ let N := max Na Nb have absa : |s N - a| < ε := by apply hNa
• MIL/C04_Sets_and_Functions/S01_Sets.lean
  • @@ -34,19 +34,18 @@ 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 + rcases xtu with xt | xu · left show x ∈ s ∩ t exact ⟨xs, xt⟩ - right - show x ∈ s ∩ u - exact ⟨xs, xu⟩ + . 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⟩ + · left; exact ⟨xs, xt⟩ + . right; exact ⟨xs, xu⟩ example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by sorry
  • @@ -59,10 +58,9 @@ example : (s \ t) \ u ⊆ s \ (t ∪ u) := by
    · exact xs intro xtu -- x ∈ t ∨ x ∈ u - cases' xtu with xt xu - · show False - exact xnt xt - show False; exact xnu xu + rcases 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⟩
  • @@ -75,9 +73,8 @@ 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⟩ + · 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⟩⟩
  • @@ -86,9 +83,8 @@ 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⟩ + · rintro x ⟨xs, xt⟩; exact ⟨xt, xs⟩ + . rintro x ⟨xt, xs⟩; exact ⟨xs, xt⟩ example : s ∩ t = t ∩ s := Subset.antisymm sorry sorry
• MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean
  • @@ -67,7 +67,7 @@ theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun
    by_cases gyA : g y ∈ A · rw [A_def, sbSet, mem_iUnion] at gyA rcases gyA with ⟨n, hn⟩ - cases' n with n + rcases n with _ | n · simp [sbAux] at hn simp [sbAux] at hn rcases hn with ⟨x, xmem, hx⟩
  • @@ -95,4 +95,3 @@ variable (g : β → α) (x : α)
    #check (invFun_eq : (∃ y, g y = x) → g (invFun g x) = x) end -
• MIL/C04_Sets_and_Functions/solutions/Solutions_S01_Sets.lean
  • @@ -10,10 +10,8 @@ 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 + · use xs; left; exact xt + . use xs; right; exact xu example : s \ (t ∪ u) ⊆ (s \ t) \ u := by rintro x ⟨xs, xntu⟩
  • @@ -32,21 +30,21 @@ example : s ∩ (s ∪ t) = s := by
    ext x; constructor · rintro ⟨xs, _⟩ exact xs - intro xs - use xs; left; 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 + . 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 + . right + exact xt by_cases h : x ∈ t · intro right
  • @@ -65,26 +63,26 @@ example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by
    exact xs rintro ⟨_, xt⟩ contradiction - constructor - right - exact xt - rintro ⟨xs, _⟩ - 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 + . 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 + rcases Nat.Prime.eq_two_or_odd nprime with h | h · rw [h] intro linarith
• MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
  • @@ -70,7 +70,7 @@ theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun
    by_cases gyA : g y ∈ A · rw [A_def, sbSet, mem_iUnion] at gyA rcases gyA with ⟨n, hn⟩ - cases' n with n + rcases n with _ | n · simp [sbAux] at hn simp [sbAux] at hn rcases hn with ⟨x, xmem, hx⟩
• MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean
  • @@ -103,7 +103,7 @@ example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2
    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 + rcases 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
• MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean
  • @@ -41,13 +41,13 @@ 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 + rcases 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 + rcases n with _ | n · simp [fac] sorry section
• MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
  • @@ -5,6 +5,7 @@ import Mathlib.Tactic
    open BigOperators namespace C05S03 + theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by cases m; contradiction case succ m =>
  • @@ -194,7 +195,7 @@ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) :
    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 + rcases this with h1 | h1 . sorry . sorry example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by
  • @@ -207,14 +208,14 @@ example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s
    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 + rcases 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 + rcases 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⟩
• MIL/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean
  • @@ -88,7 +88,7 @@ example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2
    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 + rcases 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
• MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean
  • @@ -7,7 +7,7 @@ def fac : ℕ → ℕ
    | n + 1 => (n + 1) * fac n theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by - cases' n with n + rcases n with _ | n · simp [fac] induction' n with n ih · simp [fac]
• MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
  • @@ -5,6 +5,7 @@ import Mathlib.Tactic
    open BigOperators namespace C05S03 + theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by cases m; contradiction case succ m =>
  • @@ -95,7 +96,7 @@ theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) :
    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 + rcases h₁ with h₁ | h₁ · left exact prime_p.eq_of_dvd_of_prime h₀.1 h₁ right
  • @@ -183,7 +184,7 @@ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) :
    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 + rcases this with h1 | h1 · by_cases mp : m.Prime · use m exact ⟨mp, mdvdn, h1⟩
  • @@ -201,14 +202,14 @@ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) :
    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 + rcases 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 + rcases 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
• MIL/C05_Number_Theory/S01_Irrational_Roots.lean (deleted)
  • @@ -1,120 +0,0 @@
    - import Mathlib.Tactic - import Mathlib.Data.Nat.Factorization.Basic - import Mathlib.Data.Nat.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 := - -- apply? 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 -
• MIL/C05_Number_Theory/S02_Induction_and_Recursion.lean (deleted)
  • @@ -1,147 +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
• MIL/C05_Number_Theory/S03_Infinitely_Many_Primes.lean (deleted)
  • @@ -1,232 +0,0 @@
    - import Mathlib.Data.Nat.Prime - import Mathlib.Algebra.BigOperators.Order - import Mathlib.Tactic - - 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 -
• MIL/C05_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean (deleted)
  • @@ -1,104 +0,0 @@
    - import Mathlib.Tactic - import Mathlib.Data.Nat.Factorization.Basic - import Mathlib.Data.Nat.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 -
• MIL/C05_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean (deleted)
  • @@ -1,97 +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
• MIL/C05_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean (deleted)
  • @@ -1,241 +0,0 @@
    - import Mathlib.Data.Nat.Prime - import Mathlib.Algebra.BigOperators.Order - import Mathlib.Tactic - - 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 -
• MIL/C06_Structures/S02_Algebraic_Structures.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C06S02 + structure Group₁ (α : Type _) where mul : α → α → α one : α
• MIL/C06_Structures/solutions/Solutions_S02_Algebraic_Structures.lean
  • @@ -2,6 +2,7 @@ import Mathlib.Tactic
    import Mathlib.Data.Real.Basic namespace C06S02 + structure AddGroup₁ (α : Type _) where add : α → α → α zero : α
• MIL/C07_Topology/S01_Filters.lean (deleted)
  • @@ -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 ×ᶠ 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 -
• MIL/C07_Topology/S02_Metric_Spaces.lean (deleted)
  • @@ -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 -
• MIL/C07_Topology/S03_Topological_Spaces.lean (deleted)
  • @@ -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 -
• MIL/C07_Topology/solutions/Solutions_S01_Filters.lean (deleted)
  • @@ -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 ×ᶠ 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 Filter.prod - rw [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)
• MIL/C07_Topology/solutions/Solutions_S02_Metric_Spaces.lean (deleted)
  • @@ -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 _ _ - -
• MIL/C07_Topology/solutions/Solutions_S03_Topological_Spaces.lean (deleted)
  • @@ -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 -
• MIL/C08_Topology/solutions/Solutions_S02_Metric_Spaces.lean
  • @@ -148,7 +148,7 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace
    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 + rcases eq_empty_or_nonempty K with hK | hK · use 1, by norm_num intro x y _ have : (x, y) ∉ K := by simp [hK]
• MIL/C09_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean
  • @@ -29,7 +29,7 @@ example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x,
    -- 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 + rcases 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`,
• html/C01_Introduction.html
  • @@ -156,9 +156,9 @@ as described above.</p>
    <p>This textbook and the associated repository are still a work in progress. You can update the repository by typing <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. - (This assumes that you have not changed the contents of the <cite>MIL</cite> folder, + (This assumes that you have not changed the contents of the <code class="docutils literal notranslate"><span class="pre">MIL</span></code> folder, which is why we suggested making a copy.)</p> - <p>We intend for you to work on the exercises in the <cite>MIL</cite> folder while reading the + <p>We intend for you to work on the exercises in the <code class="docutils literal notranslate"><span class="pre">MIL</span></code> folder while reading the textbook, which contains explanations, instructions, and hints. 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">&quot;Hello, World!&quot;</span>
• html/C03_Logic.html
  • @@ -559,30 +559,32 @@ then so does <code class="docutils literal notranslate"><span class="pre">fun</s
    <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">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</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">&#8614;</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&#39;</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&#39;</span> <span class="n">ubg</span> <span class="k">with</span> <span class="n">b</span> <span class="n">ubgb</span> + <span class="n">rcases</span> <span class="n">ubf</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> + <span class="n">rcases</span> <span class="n">ubg</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</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 + <p id="index-8">The <code class="docutils literal notranslate"><span class="pre">rcases</span></code> tactic unpacks the information in the existential quantifier. + The annotations like <code class="docutils literal notranslate"><span class="pre">&#10216;a,</span> <span class="pre">ubfa&#10217;</span></code>, written with the + same angle brackets as the anonymous constructors, + are known as <em>patterns</em>, and they describe the information + that we expect to find when we unpack the main argument. 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. + <code class="docutils literal notranslate"><span class="pre">rcases</span> <span class="pre">ubf</span> <span class="pre">with</span> <span class="pre">&#10216;a,</span> <span class="pre">ubfa&#10217;</span></code> adds a new variable <code class="docutils literal notranslate"><span class="pre">a</span></code> + for an upper bound to the context, + together with the hypothesis <code class="docutils literal notranslate"><span class="pre">ubfa</span></code> that it has the given property. 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: + This is a common method of reasoning 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. + <p>Try using this method 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>,
  • @@ -595,24 +597,11 @@ into the proofs.</p>
    <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 &#8220;r&#8221; stands for &#8220;recursive.&#8221;) - 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">&#10216;a,</span> <span class="pre">h&#10217;</span></code> that <code class="docutils literal notranslate"><span class="pre">rcases</span></code> then tries to match. + <p id="index-9">The &#8220;r&#8221; in <code class="docutils literal notranslate"><span class="pre">rcases</span></code> stands for &#8220;recursive,&#8221; because it allows + us to use arbitrarily complex patterns to unpack nested data. 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">&#8614;</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">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> - <span class="n">rcases</span> <span class="n">ubg</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> - <span class="n">exact</span> <span class="o">&#10216;</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">&#10217;</span> - - <span class="kd">example</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="n">g</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</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> + 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>:</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">&#8594;</span> <span class="n">FnHasUb</span> <span class="n">g</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</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">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="n">exact</span> <span class="o">&#10216;</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">&#10217;</span> </pre></div>
  • @@ -623,14 +612,62 @@ in expressions and proof terms:</p>
    <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</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">&#10217;</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>The task of unpacking information in a hypothesis is + so important that Lean and mathlib provide a number of + ways to do it. For example, the <code class="docutils literal notranslate"><span class="pre">obtain</span></code> tactic provides suggestive syntax:</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">&#8614;</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">obtain</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> <span class="o">:=</span> <span class="n">ubf</span> + <span class="n">obtain</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="o">:=</span> <span class="n">ubg</span> + <span class="n">exact</span> <span class="o">&#10216;</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">&#10217;</span> + </pre></div> + </div> + <p>Think of the first <code class="docutils literal notranslate"><span class="pre">obtain</span></code> instruction as matching the &#8220;contents&#8221; of <code class="docutils literal notranslate"><span class="pre">ubf</span></code> + with the given pattern and assigning the components to the named variables. + <code class="docutils literal notranslate"><span class="pre">rcases</span></code> and <code class="docutils literal notranslate"><span class="pre">obtain</span></code> are said to <code class="docutils literal notranslate"><span class="pre">destruct</span></code> their arguments, though + there is a small difference in that <code class="docutils literal notranslate"><span class="pre">rcases</span></code> clears <code class="docutils literal notranslate"><span class="pre">ubf</span></code> from the context + when it is done, whereas it is still present after <code class="docutils literal notranslate"><span class="pre">obtain</span></code>.</p> + <p>Lean also supports syntax that is similar to that used in other functional programming + languages:</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">&#8614;</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="n">case</span> <span class="n">intro</span> <span class="n">a</span> <span class="n">ubfa</span> <span class="bp">=&gt;</span> + <span class="n">cases</span> <span class="n">ubg</span> + <span class="n">case</span> <span class="n">intro</span> <span class="n">b</span> <span class="n">ubgb</span> <span class="bp">=&gt;</span> + <span class="n">exact</span> <span class="o">&#10216;</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">&#10217;</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">&#8614;</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="n">next</span> <span class="n">a</span> <span class="n">ubfa</span> <span class="bp">=&gt;</span> + <span class="n">cases</span> <span class="n">ubg</span> + <span class="n">next</span> <span class="n">b</span> <span class="n">ubgb</span> <span class="bp">=&gt;</span> + <span class="n">exact</span> <span class="o">&#10216;</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">&#10217;</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">&#8614;</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="k">match</span> <span class="n">ubf</span><span class="o">,</span> <span class="n">ubg</span> <span class="k">with</span> + <span class="bp">|</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;,</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> + <span class="n">exact</span> <span class="o">&#10216;</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">&#10217;</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">&#8614;</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">match</span> <span class="n">ubf</span><span class="o">,</span> <span class="n">ubg</span> <span class="k">with</span> + <span class="bp">|</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;,</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> + <span class="o">&#10216;</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">&#10217;</span> + </pre></div> + </div> + <p>In the first example, if you put your cursor after <code class="docutils literal notranslate"><span class="pre">cases</span> <span class="pre">ubf</span></code>, + you will see that the tactic produces a single goal, which Lean has tagged + <code class="docutils literal notranslate"><span class="pre">intro</span></code>. (The particular name chosen comes from the internal name for + the axiomatic primitive that bulids a proof of an existential statement.) + The <code class="docutils literal notranslate"><span class="pre">case</span></code> tactic then names the components. The second example is similar, + except using <code class="docutils literal notranslate"><span class="pre">next</span></code> instead of <code class="docutils literal notranslate"><span class="pre">case</span></code> means that you can avoid mentioning + <code class="docutils literal notranslate"><span class="pre">intro</span></code>. The word <code class="docutils literal notranslate"><span class="pre">match</span></code> in the last two examples highlights that + what we are doing here is what computer scientists call &#8220;pattern matching.&#8221; + Notice that the third proof begins by <code class="docutils literal notranslate"><span class="pre">by</span></code>, after which the tactic version + of <code class="docutils literal notranslate"><span class="pre">match</span></code> expects a tactic proof on the right side of the arrow. + The last example is a proof term: there are no tactics in sight.</p> + <p>For the rest of this book, we will stick to <code class="docutils literal notranslate"><span class="pre">rcases</span></code>, <code class="docutils literal notranslate"><span class="pre">rintros</span></code>, and <code class="docutils literal notranslate"><span class="pre">obtain</span></code>, + as the preferred ways of using an existential quantifier. + But it can&#8217;t hurt to see the alternative syntax, especially if there is + a chance you will find yourself in the company of computer scientists.</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
  • @@ -697,8 +734,8 @@ you can find existential quantifiers hidden all over
    if you know how to spot them. For example, divisibility is implicitly an &#8220;exists&#8221; 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">&#8739;</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">&#8739;</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">&#8739;</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">cases&#39;</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&#39;</span> <span class="n">divbc</span> <span class="k">with</span> <span class="n">e</span> <span class="n">ceq</span> + <span class="n">rcases</span> <span class="n">divab</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">d</span><span class="o">,</span> <span class="n">beq</span><span class="o">&#10217;</span> + <span class="n">rcases</span> <span class="n">divbc</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">e</span><span class="o">,</span> <span class="n">ceq</span><span class="o">&#10217;</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>
  • @@ -741,10 +778,10 @@ It can be used in conjunction with the <code class="docutils literal notranslate
    </div> <p>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, + Note that you can use <code class="docutils literal notranslate"><span class="pre">rcases</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">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</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">&#8707;</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&#39;</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">rcases</span> <span class="n">h</span> <span class="mi">2</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">x</span><span class="o">,</span> <span class="n">hx</span><span class="o">&#10217;</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>
  • @@ -806,8 +843,8 @@ predicate <code class="docutils literal notranslate"><span class="pre">FnHasUb</
    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">&#8704;</span> <span class="n">a</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">&gt;</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</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&#39;</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&#39;</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="n">rcases</span> <span class="n">fnub</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">fnuba</span><span class="o">&#10217;</span> + <span class="n">rcases</span> <span class="n">h</span> <span class="n">a</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">x</span><span class="o">,</span> <span class="n">hx</span><span class="o">&#10217;</span> <span class="k">have</span> <span class="o">:</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">&#8804;</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>
  • @@ -947,18 +984,18 @@ The command <code class="docutils literal notranslate"><span class="pre">push_ne
    <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">&#172;</span><span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">a</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">&gt;</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">dsimp</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&#8217;s simplifier to + <p>In the second example, we use dsimp 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> + (We need to use <code class="docutils literal notranslate"><span class="pre">dsimp</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">&#172;</span> <span class="pre">&#8707;</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">&#172;</span> <span class="pre">&#8704;</span> <span class="pre">x,</span> <span class="pre">P</span> <span class="pre">x</span></code>, + with <code class="docutils literal notranslate"><span class="pre">&#172;&#8707;</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">&#172;&#8704;</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,
  • @@ -969,11 +1006,11 @@ to prove the following:</p>
    </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">&#8594;</span> <span class="pre">B</span></code> to <code class="docutils literal notranslate"><span class="pre">&#172;</span> <span class="pre">B</span> <span class="pre">&#8594;</span> <span class="pre">&#172;</span> <span class="pre">A</span></code>. + which transforms a goal <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">&#8594;</span> <span class="pre">B</span></code> to <code class="docutils literal notranslate"><span class="pre">&#172;B</span> <span class="pre">&#8594;</span> <span class="pre">&#172;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">&#172;</span> <span class="pre">A</span></code> from hypothesis <code class="docutils literal notranslate"><span class="pre">&#172;</span> <span class="pre">B</span></code>. + <code class="docutils literal notranslate"><span class="pre">&#172;A</span></code> from hypothesis <code class="docutils literal notranslate"><span class="pre">&#172;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>
  • @@ -1064,12 +1101,12 @@ which drops into tactic mode at the keyword <code class="docutils literal notran
    </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 <code class="docutils literal notranslate"><span class="pre">fun</span></code>, + You can use the <code class="docutils literal notranslate"><span class="pre">rcases</span></code> tactic for that, + as well as <code class="docutils literal notranslate"><span class="pre">rintro</span></code> or a pattern-matching <code class="docutils literal notranslate"><span class="pre">fun</span></code>, 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">&#8477;</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">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">cases&#39;</span> <span class="n">h</span> <span class="k">with</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> + <span class="n">rcases</span> <span class="n">h</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="n">h&#8321;</span><span class="o">&#10217;</span> <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h&#8321;</span> <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span>
  • @@ -1081,6 +1118,36 @@ the existential quantifier.</p>
    <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="n">h&#8321;</span><span class="o">&#10217;</span> <span class="n">h&#39;</span> <span class="bp">&#8614;</span> <span class="n">h&#8321;</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#39;</span><span class="o">)</span> </pre></div> </div> + <p>In analogy to the <code class="docutils literal notranslate"><span class="pre">obtain</span></code> tactic, which we used with the existential + quantifier, there is also a pattern-matching <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">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</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">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="k">have</span> <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="n">h&#8321;</span><span class="o">&#10217;</span> <span class="o">:=</span> <span class="n">h</span> + <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h&#8321;</span> + <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> + </pre></div> + </div> + <p>In contrast to <code class="docutils literal notranslate"><span class="pre">rcases</span></code>, here the <code class="docutils literal notranslate"><span class="pre">have</span></code> tactic leaves <code class="docutils literal notranslate"><span class="pre">h</span></code> in the context. + And even though we won&#8217;t use them, once again we have the computer scientists&#8217; + pattern-matching syntax:</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">&#8477;</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">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</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="n">case</span> <span class="n">intro</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> <span class="bp">=&gt;</span> + <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h&#8321;</span> + <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</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">&#8477;</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">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</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="n">next</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> <span class="bp">=&gt;</span> + <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h&#8321;</span> + <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</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">&#8477;</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">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="k">match</span> <span class="n">h</span> <span class="k">with</span> + <span class="bp">|</span> <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="n">h&#8321;</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> + <span class="n">contrapose</span><span class="bp">!</span> <span class="n">h&#8321;</span> + <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</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">&#8743;</span> <span class="pre">B</span></code>
  • @@ -1294,38 +1361,86 @@ Here, <code class="docutils literal notranslate"><span class="pre">inl</span></c
    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 + The <code class="docutils literal notranslate"><span class="pre">rcases</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">&#8744;</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 + In contrast to the use of <code class="docutils literal notranslate"><span class="pre">rcases</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. + here the <code class="docutils literal notranslate"><span class="pre">rcases</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. + the <code class="docutils literal notranslate"><span class="pre">rcases</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">&lt;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">cases&#39;</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="n">rcases</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="bp">|</span> <span class="n">h</span> <span class="bp">&#183;</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> + <span class="n">intro</span> <span class="n">h</span><span class="bp">;</span> <span class="n">left</span><span class="bp">;</span> <span class="n">exact</span> <span class="n">h</span> + <span class="bp">.</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>Notice that the pattern changes from <code class="docutils literal notranslate"><span class="pre">&#10216;h&#8320;,</span> <span class="pre">h&#8321;&#10217;</span></code> in the case of + a conjunction to <code class="docutils literal notranslate"><span class="pre">h&#8320;</span> <span class="pre">|</span> <span class="pre">h&#8321;</span></code> in the case of a disjunction. + Think of the first pattern as matching against data the contains + <em>both</em> an <code class="docutils literal notranslate"><span class="pre">h&#8320;</span></code> and a <code class="docutils literal notranslate"><span class="pre">h&#8321;</span></code>, whereas second pattern, with the bar, + matches against data that contains <em>either</em> an <code class="docutils literal notranslate"><span class="pre">h&#8320;</span></code> or <code class="docutils literal notranslate"><span class="pre">h&#8321;</span></code>. + In this case, because the two goals are separate, we have chosen + to use the same name, <code class="docutils literal notranslate"><span class="pre">h</span></code>, in each case.</p> <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">&#8805;</span> <span class="pre">0</span></code> implies <code class="docutils literal notranslate"><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">&lt;</span> <span class="pre">0</span></code> implies <code class="docutils literal notranslate"><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">&#8804;</span> <span class="pre">x</span> <span class="pre">&#8744;</span> <span class="pre">x</span> <span class="pre">&lt;</span> <span class="pre">0</span></code>, - allowing us to split on those two cases. - Try proving the triangle inequality using the two + allowing us to split on those two cases.</p> + <p>Lean also supports the computer scientists&#8217; pattern-matching + syntax for disjunction. Now the <code class="docutils literal notranslate"><span class="pre">cases</span></code> tactic is more attractive, + because it allows us to name each <code class="docutils literal notranslate"><span class="pre">case</span></code>, and name the hypothesis + that is introduced closer to where it is used.</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">&lt;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&lt;</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="n">case</span> <span class="n">inl</span> <span class="n">h</span> <span class="bp">=&gt;</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="bp">;</span> <span class="n">left</span><span class="bp">;</span> <span class="n">exact</span> <span class="n">h</span> + <span class="n">case</span> <span class="n">inr</span> <span class="n">h</span> <span class="bp">=&gt;</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 names <code class="docutils literal notranslate"><span class="pre">inl</span></code> and <code class="docutils literal notranslate"><span class="pre">inr</span></code> are short for &#8220;intro left&#8221; and &#8220;intro right,&#8221; + respectively. Using <code class="docutils literal notranslate"><span class="pre">case</span></code> has the advantage is that you can prove the + cases in either order; Lean uses the tag to find the relevant goal. + If you don&#8217;t care about that, you can use <code class="docutils literal notranslate"><span class="pre">next</span></code>, or <code class="docutils literal notranslate"><span class="pre">match</span></code>, + or even a pattern-matching <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">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&lt;</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="n">next</span> <span class="n">h</span> <span class="bp">=&gt;</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="bp">;</span> <span class="n">left</span><span class="bp">;</span> <span class="n">exact</span> <span class="n">h</span> + <span class="n">next</span> <span class="n">h</span> <span class="bp">=&gt;</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> + + + <span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="k">match</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="bp">|</span> <span class="n">Or.inl</span> <span class="n">h</span> <span class="bp">=&gt;</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="bp">;</span> <span class="n">left</span><span class="bp">;</span> <span class="n">exact</span> <span class="n">h</span> + <span class="bp">|</span> <span class="n">Or.inr</span> <span class="n">h</span> <span class="bp">=&gt;</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>In the case of the <code class="docutils literal notranslate"><span class="pre">match</span></code>, we need to use the full names + <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> of the canonical ways to prove a disjunction. + In this textbook, we will generally use <code class="docutils literal notranslate"><span class="pre">rcases</span></code> to split on the + caess of a disjunction.</p> + <p>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>
  • @@ -1350,25 +1465,25 @@ try these.</p>
    <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. + <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 nested 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">&#8477;</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">&#8800;</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="mi">0</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&gt;</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="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="n">xlt</span> <span class="bp">|</span> <span class="n">xeq</span> <span class="bp">|</span> <span class="n">xgt</span> <span class="bp">&#183;</span> <span class="n">left</span> <span class="n">exact</span> <span class="n">xlt</span> <span class="bp">&#183;</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> + <span class="bp">.</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">&#8469;</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">&#8739;</span> <span class="n">n</span> <span class="bp">&#8744;</span> <span class="n">m</span> <span class="bp">&#8739;</span> <span class="n">k</span><span class="o">)</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">&#8739;</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">(&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">rfl</span><span class="o">&#10217;</span> <span class="bp">|</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">rfl</span><span class="o">&#10217;)</span> + <span class="n">rcases</span> <span class="n">h</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">rfl</span><span class="o">&#10217;</span> <span class="bp">|</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">rfl</span><span class="o">&#10217;</span> <span class="bp">&#183;</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> + <span class="bp">.</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.
  • @@ -1428,7 +1543,7 @@ The name <code class="docutils literal notranslate"><span class="pre">em</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">&#183;</span> <span class="n">assumption</span> - <span class="n">contradiction</span> + <span class="bp">.</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>
  • @@ -1562,8 +1677,8 @@ See if you can finish it off.</p>
    <span class="n">intro</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="n">dsimp</span> <span class="c1">-- this line is not needed but cleans up the goal a bit.</span> <span class="k">have</span> <span class="n">&#949;2pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</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&#39;</span> <span class="n">cs</span> <span class="o">(</span><span class="n">&#949;</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">&#949;2pos</span> <span class="k">with</span> <span class="n">Ns</span> <span class="n">hs</span> - <span class="n">cases&#39;</span> <span class="n">ct</span> <span class="o">(</span><span class="n">&#949;</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">&#949;2pos</span> <span class="k">with</span> <span class="n">Nt</span> <span class="n">ht</span> + <span class="n">rcases</span> <span class="n">cs</span> <span class="o">(</span><span class="n">&#949;</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">&#949;2pos</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">Ns</span><span class="o">,</span> <span class="n">hs</span><span class="o">&#10217;</span> + <span class="n">rcases</span> <span class="n">ct</span> <span class="o">(</span><span class="n">&#949;</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">&#949;2pos</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">Nt</span><span class="o">,</span> <span class="n">ht</span><span class="o">&#10217;</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>
  • @@ -1605,7 +1720,7 @@ 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">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">&#8477;</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">&#8707;</span> <span class="n">N</span> <span class="n">b</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="n">N</span> <span class="bp">&#8804;</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">s</span> <span class="n">n</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">cases&#39;</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">rcases</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="o">&#10216;</span><span class="n">N</span><span class="o">,</span> <span class="n">h</span><span class="o">&#10217;</span> <span class="n">use</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="bp">+</span> <span class="mi">1</span> <span class="gr">sorry</span> </pre></div>
  • @@ -1629,7 +1744,7 @@ and finish the proof.</p>
    <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">&#10216;</span><span class="n">N&#8320;</span><span class="o">,</span> <span class="n">B</span><span class="o">,</span> <span class="n">h&#8320;</span><span class="o">&#10217;</span> <span class="k">have</span> <span class="n">Bpos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</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">h&#8320;</span> <span class="n">N&#8320;</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">pos&#8320;</span> <span class="o">:</span> <span class="n">&#949;</span> <span class="bp">/</span> <span class="n">B</span> <span class="bp">&gt;</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">div_pos</span> <span class="n">&#949;pos</span> <span class="n">Bpos</span> - <span class="n">cases&#39;</span> <span class="n">ct</span> <span class="n">_</span> <span class="n">pos&#8320;</span> <span class="k">with</span> <span class="n">N&#8321;</span> <span class="n">h&#8321;</span> + <span class="n">rcases</span> <span class="n">ct</span> <span class="n">_</span> <span class="n">pos&#8320;</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">N&#8321;</span><span class="o">,</span> <span class="n">h&#8321;</span><span class="o">&#10217;</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -1663,8 +1778,8 @@ you can delete the proof sketch and try proving it from scratch.)</p>
    <span class="k">have</span> <span class="n">&#949;pos</span> <span class="o">:</span> <span class="n">&#949;</span> <span class="bp">&gt;</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">change</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="bp">/</span> <span class="mi">2</span> <span class="bp">&gt;</span> <span class="mi">0</span> <span class="n">linarith</span> - <span class="n">cases&#39;</span> <span class="n">sa</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="k">with</span> <span class="n">Na</span> <span class="n">hNa</span> - <span class="n">cases&#39;</span> <span class="n">sb</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="k">with</span> <span class="n">Nb</span> <span class="n">hNb</span> + <span class="n">rcases</span> <span class="n">sa</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">Na</span><span class="o">,</span> <span class="n">hNa</span><span class="o">&#10217;</span> + <span class="n">rcases</span> <span class="n">sb</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">Nb</span><span class="o">,</span> <span class="n">hNb</span><span class="o">&#10217;</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="bp">|</span><span class="n">s</span> <span class="n">N</span> <span class="bp">-</span> <span class="n">a</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</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="bp">|</span><span class="n">s</span> <span class="n">N</span> <span class="bp">-</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span>
• html/C04_Sets_and_Functions.html
  • @@ -199,13 +199,13 @@ we can also use the <code class="docutils literal notranslate"><span class="pre"
    <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">&#8712;</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">&#8712;</span> <span class="n">t</span> <span class="bp">&#8746;</span> <span class="n">u</span> <span class="o">:=</span> <span class="n">hx.2</span> - <span class="n">cases&#39;</span> <span class="n">xtu</span> <span class="k">with</span> <span class="n">xt</span> <span class="n">xu</span> + <span class="n">rcases</span> <span class="n">xtu</span> <span class="k">with</span> <span class="n">xt</span> <span class="bp">|</span> <span class="n">xu</span> <span class="bp">&#183;</span> <span class="n">left</span> <span class="k">show</span> <span class="n">x</span> <span class="bp">&#8712;</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">t</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> - <span class="n">right</span> - <span class="k">show</span> <span class="n">x</span> <span class="bp">&#8712;</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">u</span> - <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> + <span class="bp">.</span> <span class="n">right</span> + <span class="k">show</span> <span class="n">x</span> <span class="bp">&#8712;</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">u</span> + <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> </pre></div> </div> <p>Since intersection binds tighter than union,
  • @@ -214,9 +214,8 @@ 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">&#8745;</span> <span class="o">(</span><span class="n">t</span> <span class="bp">&#8746;</span> <span class="n">u</span><span class="o">)</span> <span class="bp">&#8838;</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">t</span> <span class="bp">&#8746;</span> <span class="n">s</span> <span class="bp">&#8745;</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">&#10216;</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">&#10217;</span> - <span class="bp">&#183;</span> <span class="n">left</span> - <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> - <span class="n">right</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> + <span class="bp">&#183;</span> <span class="n">left</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> + <span class="bp">.</span> <span class="n">right</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> </pre></div> </div> <p>As an exercise, try proving the other inclusion:</p>
  • @@ -245,10 +244,9 @@ show how to avoid using them.</p>
    <span class="bp">&#183;</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 &#8712; t &#8744; x &#8712; u</span> - <span class="n">cases&#39;</span> <span class="n">xtu</span> <span class="k">with</span> <span class="n">xt</span> <span class="n">xu</span> - <span class="bp">&#183;</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="n">rcases</span> <span class="n">xtu</span> <span class="k">with</span> <span class="n">xt</span> <span class="bp">|</span> <span class="n">xu</span> + <span class="bp">&#183;</span> <span class="k">show</span> <span class="n">False</span><span class="bp">;</span> <span class="n">exact</span> <span class="n">xnt</span> <span class="n">xt</span> + <span class="bp">.</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">&#8838;</span> <span class="n">s</span> <span class="bp">\</span> <span class="o">(</span><span class="n">t</span> <span class="bp">&#8746;</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">&#10216;&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xnt</span><span class="o">&#10217;,</span> <span class="n">xnu</span><span class="o">&#10217;</span>
  • @@ -271,9 +269,8 @@ the <code class="docutils literal notranslate"><span class="pre">ext</span></cod
    <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">&#183;</span> <span class="n">rintro</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> - <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span> - <span class="n">rintro</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> + <span class="bp">&#183;</span> <span class="n">rintro</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span> + <span class="bp">.</span> <span class="n">rintro</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</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>
  • @@ -295,9 +292,8 @@ which allows us to prove an equation <code class="docutils literal notranslate">
    between sets by proving <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">&#8838;</span> <span class="pre">t</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span> <span class="pre">&#8838;</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">&#8745;</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">t</span> <span class="bp">&#8745;</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">&#183;</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> - <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span> - <span class="n">rintro</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> + <span class="bp">&#183;</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span> + <span class="bp">.</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span><span class="bp">;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> </pre></div> </div> <p>Try finishing this proof term:</p>
  • @@ -1128,7 +1124,7 @@ applies the corresponding defining equation of <code class="docutils literal not
    <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">&#8712;</span> <span class="n">A</span> <span class="bp">&#183;</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">&#10216;</span><span class="n">n</span><span class="o">,</span> <span class="n">hn</span><span class="o">&#10217;</span> - <span class="n">cases&#39;</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> + <span class="n">rcases</span> <span class="n">n</span> <span class="k">with</span> <span class="n">_</span> <span class="bp">|</span> <span class="n">n</span> <span class="bp">&#183;</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">&#10216;</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">&#10217;</span>
• html/C05_Elementary_Number_Theory.html
  • @@ -342,7 +342,7 @@ At the very end, you can use <code class="docutils literal notranslate"><span cl
    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">&#8469;</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">&#8800;</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">&#8469;</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">&#8739;</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&#39;</span> <span class="n">r</span> <span class="k">with</span> <span class="n">r</span> + <span class="n">rcases</span> <span class="n">r</span> <span class="k">with</span> <span class="n">_</span> <span class="bp">|</span> <span class="n">r</span> <span class="bp">&#183;</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">&#8800;</span> <span class="mi">0</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">npowz</span> <span class="bp">&#8614;</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>
  • @@ -498,7 +498,7 @@ Step through the next example to see what is going on.</p>
    <span class="n">induction&#39;</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> <span class="n">ih</span> <span class="bp">&#183;</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&#39;</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="n">rcases</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="bp">|</span> <span class="n">h</span> <span class="bp">&#183;</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>
  • @@ -511,7 +511,7 @@ 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">&#8469;</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">&#8804;</span> <span class="n">fac</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">cases&#39;</span> <span class="n">n</span> <span class="k">with</span> <span class="n">n</span> + <span class="n">rcases</span> <span class="n">n</span> <span class="k">with</span> <span class="n">_</span> <span class="bp">|</span> <span class="n">n</span> <span class="bp">&#183;</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>
  • @@ -1090,7 +1090,7 @@ same property.</p>
    <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">&#8744;</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&#39;</span> <span class="n">this</span> <span class="k">with</span> <span class="n">h1</span> <span class="n">h1</span> + <span class="n">rcases</span> <span class="n">this</span> <span class="k">with</span> <span class="n">h1</span> <span class="bp">|</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>
  • @@ -1114,14 +1114,14 @@ 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">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">p</span> <span class="bp">&gt;</span> <span class="n">n</span><span class="o">,</span> <span class="n">Nat.Prime</span> <span class="n">p</span> <span class="bp">&#8743;</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&#39;</span> <span class="n">h</span> <span class="k">with</span> <span class="n">n</span> <span class="n">hn</span> + <span class="n">rcases</span> <span class="n">h</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">n</span><span class="o">,</span> <span class="n">hn</span><span class="o">&#10217;</span> <span class="k">have</span> <span class="o">:</span> <span class="bp">&#8707;</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">&#8704;</span> <span class="n">p</span> <span class="o">:</span> <span class="n">&#8469;</span><span class="o">,</span> <span class="n">p.Prime</span> <span class="bp">&#8743;</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">&#8596;</span> <span class="n">p</span> <span class="bp">&#8712;</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">&#10216;</span><span class="n">p</span><span class="o">,</span> <span class="o">&#10216;</span><span class="n">pp</span><span class="o">,</span> <span class="n">p4</span><span class="o">&#10217;,</span> <span class="n">pltn</span><span class="o">&#10217;</span> <span class="n">exact</span> <span class="o">&#10216;</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">&#10217;</span> - <span class="n">cases&#39;</span> <span class="n">this</span> <span class="k">with</span> <span class="n">s</span> <span class="n">hs</span> + <span class="n">rcases</span> <span class="n">this</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">s</span><span class="o">,</span> <span class="n">hs</span><span class="o">&#10217;</span> <span class="k">have</span> <span class="n">h&#8321;</span> <span class="o">:</span> <span class="o">((</span><span class="mi">4</span> <span class="bp">*</span> <span class="bp">&#8719;</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">h&#8321;</span> <span class="k">with</span> <span class="o">&#10216;</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">&#10217;</span>
• html/searchindex.js
  • @@ -1,1 +1,1 @@
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</span>Hierarchies", "<span class=\"section-number\">8. </span>Topology", "<span class=\"section-number\">9. </span>Differential Calculus", "<span class=\"section-number\">10. </span>Integration and Measure Theory", "Index", "Mathematics in Lean"], "terms": {"The": [0, 1, 4, 5, 6, 7, 8, 9, 11], "goal": [0, 1, 2, 3, 4, 5, 7], "thi": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "book": [0, 2, 7], "teach": 0, "you": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "formal": [0, 1, 2, 3, 4, 5, 6, 7, 8], "mathemat": [0, 1, 2, 3, 4, 5, 6, 7], "us": [0, 2, 3, 4, 5, 6, 7, 8, 9, 11], "lean": [0, 1, 2, 3, 4, 5, 6, 7, 8], "4": [0, 1, 2, 4, 5, 7, 8], "interact": [0, 5, 6, 9], "proof": [0, 1, 2, 3, 4, 5, 6, 7, 8], "assist": [0, 2, 5], "It": [0, 1, 2, 3, 4, 5, 6, 7, 8], "assum": [0, 1, 2, 3, 4, 6, 7], "know": [0, 1, 2, 3, 4, 5, 6, 7, 8], "some": [0, 1, 2, 3, 4, 5, 6, 7, 8], "doe": [0, 1, 2, 3, 4, 5, 6, 7, 8], "requir": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "much": [0, 2, 4, 6, 7, 8], "although": [0, 3, 4, 5, 6, 7], "we": [0, 1, 2, 3, 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• mathematics_in_lean.pdf