49c20a40 - mathematics_in

Authored by: Jeremy Avigad
Authored at: 2023-08-11 17:49:22 -0400
Committed by: Jeremy Avigad
Committed at: 2023-08-11 17:49:22 -0400
Object name: 49c20a405daacee9f9271155388f1be5f6ba8f39
Parent: f5252520

Update Fri 11 Aug 2023 05:49:22 PM EDT

Changes

61 changed files (+405/-401)

MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean
- @@ -3,7 +3,7 @@ import Mathlib.Data.Real.Basic
  import Mathlib.Tactic section - variable (R : Type _) [Ring R] + variable (R : Type*) [Ring R] #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) #check (add_comm : ∀ a b : R, a + b = b + a)
- @@ -18,7 +18,7 @@ variable (R : Type _) [Ring R]
  end section - variable (R : Type _) [CommRing R] + variable (R : Type*) [CommRing R] variable (a b c d : R) example : c * b * a = b * (a * c) := by ring
- @@ -34,7 +34,7 @@ example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by
  end namespace MyRing - variable {R : Type _} [Ring R] + variable {R : Type*} [Ring R] theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add]
- @@ -46,7 +46,7 @@ theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg]
  end MyRing namespace MyRing - variable {R : Type _} [Ring R] + variable {R : Type*} [Ring R] theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by rw [← add_assoc, add_left_neg, zero_add]
- @@ -86,7 +86,7 @@ end MyRing
  -- Examples. section - variable {R : Type _} [Ring R] + variable {R : Type*} [Ring R] example (a b : R) : a - b = a + -b := sub_eq_add_neg a b
- @@ -100,7 +100,7 @@ example (a b : ℝ) : a - b = a + -b := by
  rfl namespace MyRing - variable {R : Type _} [Ring R] + variable {R : Type*} [Ring R] theorem self_sub (a : R) : a - a = 0 := sorry
- @@ -114,7 +114,7 @@ theorem two_mul (a : R) : 2 * a = a + a :=
  end MyRing section - variable (A : Type _) [AddGroup A] + variable (A : Type*) [AddGroup A] #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) #check (zero_add : ∀ a : A, 0 + a = a)
- @@ -123,7 +123,7 @@ variable (A : Type _) [AddGroup A]
  end section - variable {G : Type _} [Group G] + variable {G : Type*} [Group G] #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) #check (one_mul : ∀ a : G, 1 * a = a)
MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean
- @@ -2,7 +2,7 @@ import Mathlib.Tactic
 import Mathlib.Topology.MetricSpace.Basic section - variable {α : Type _} [PartialOrder α] + variable {α : Type*} [PartialOrder α] variable (x y z : α) #check x ≤ y
- @@ -21,7 +21,7 @@ example : x < y ↔ x ≤ y ∧ x ≠ y :=
 end section - variable {α : Type _} [Lattice α] + variable {α : Type*} [Lattice α] variable (x y z : α) #check x ⊓ y
- @@ -54,7 +54,7 @@ theorem absorb2 : x ⊔ x ⊓ y = x := by
 end section - variable {α : Type _} [DistribLattice α] + variable {α : Type*} [DistribLattice α] variable (x y z : α) #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z)
- @@ -64,7 +64,7 @@ variable (x y z : α)
 end section - variable {α : Type _} [Lattice α] + variable {α : Type*} [Lattice α] variable (a b c : α) example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by
- @@ -76,7 +76,7 @@ example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b
 end section - variable {R : Type _} [StrictOrderedRing R] + variable {R : Type*} [StrictOrderedRing R] variable (a b c : R) #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b)
- @@ -96,7 +96,7 @@ example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by
 end section - variable {X : Type _} [MetricSpace X] + variable {X : Type*} [MetricSpace X] variable (x y z : X) #check (dist_self x : dist x x = 0)
MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
- @@ -3,7 +3,7 @@ import Mathlib.Data.Real.Basic
  import Mathlib.Tactic namespace MyRing - variable {R : Type _} [Ring R] + variable {R : Type*} [Ring R] theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by rw [add_assoc, add_right_neg, add_zero]
- @@ -37,7 +37,7 @@ theorem neg_neg (a : R) : - -a = a := by
  end MyRing namespace MyRing - variable {R : Type _} [Ring R] + variable {R : Type*} [Ring R] theorem self_sub (a : R) : a - a = 0 := by rw [sub_eq_add_neg, add_right_neg]
- @@ -51,7 +51,7 @@ theorem two_mul (a : R) : 2 * a = a + a := by
  end MyRing section - variable {G : Type _} [Group G] + variable {G : Type*} [Group G] namespace MyGroup
MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean
- @@ -2,7 +2,7 @@ import Mathlib.Tactic
  import Mathlib.Topology.MetricSpace.Basic section - variable {α : Type _} [Lattice α] + variable {α : Type*} [Lattice α] variable (x y z : α) example : x ⊓ y = y ⊓ x := by
- @@ -78,7 +78,7 @@ theorem absorb2 : x ⊔ x ⊓ y = x := by
  end section - variable {α : Type _} [DistribLattice α] + variable {α : Type*} [DistribLattice α] variable (x y z : α) #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z)
- @@ -88,7 +88,7 @@ variable (x y z : α)
  end section - variable {α : Type _} [Lattice α] + variable {α : Type*} [Lattice α] variable (a b c : α) example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by
- @@ -102,7 +102,7 @@ example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b
  end section - variable {R : Type _} [StrictOrderedRing R] + variable {R : Type*} [StrictOrderedRing R] variable (a b c : R) theorem aux1 : a ≤ b → 0 ≤ b - a := by
- @@ -123,7 +123,7 @@ example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by
  end section - variable {X : Type _} [MetricSpace X] + variable {X : Type*} [MetricSpace X] variable (x y z : X) example (x y : X) : 0 ≤ dist x y :=by
MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean
- @@ -76,7 +76,7 @@ example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) :
  end section - variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] + variable {α : Type*} {R : Type*} [OrderedCancelAddCommMonoid R] #check add_le_add
- @@ -135,7 +135,7 @@ end
  section - variable {α : Type _} (r s t : Set α) + variable {α : Type*} (r s t : Set α) example : s ⊆ s := by intro x xs
- @@ -149,7 +149,7 @@ theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by
  end section - variable {α : Type _} [PartialOrder α] + variable {α : Type*} [PartialOrder α] variable (s : Set α) (a b : α) def SetUb (s : Set α) (a : α) :=
- @@ -171,7 +171,7 @@ example (c : ℝ) : Injective fun x ↦ x + c := by
  example {c : ℝ} (h : c ≠ 0) : Injective fun x ↦ c * x := by sorry - variable {α : Type _} {β : Type _} {γ : Type _} + variable {α : Type*} {β : Type*} {γ : Type*} variable {g : β → γ} {f : α → β} example (injg : Injective g) (injf : Injective f) : Injective fun x ↦ g (f x) := by
MIL/C03_Logic/S02_The_Existential_Quantifier.lean
- @@ -7,6 +7,15 @@ example : ∃ x : ℝ, 2 < x ∧ x < 3 := by
 use 5 / 2 norm_num + example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + have h1 : 2 < (5 : ℝ) / 2 := by norm_num + have h2 : (5 : ℝ) / 2 < 3 := by norm_num + use 5 / 2, h1, h2 + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num + use 5 / 2 + example : ∃ x : ℝ, 2 < x ∧ x < 3 := have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num ⟨5 / 2, h⟩
- @@ -86,7 +95,7 @@ example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x :=
 section - variable {α : Type _} [CommRing α] + variable {α : Type*} [CommRing α] def SumOfSquares (x : α) := ∃ a b, x = a ^ 2 + b ^ 2
- @@ -148,7 +157,7 @@ end
 section open Function - variable {α : Type _} {β : Type _} {γ : Type _} + variable {α : Type*} {β : Type*} {γ : Type*} variable {g : β → γ} {f : α → β} example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x ↦ g (f x) := by
MIL/C03_Logic/S03_Negation.lean
- @@ -62,7 +62,7 @@ example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by
 end section - variable {α : Type _} (P : α → Prop) (Q : Prop) + variable {α : Type*} (P : α → Prop) (Q : Prop) example (h : ¬∃ x, P x) : ∀ x, ¬P x := by sorry
MIL/C03_Logic/S04_Conjunction_and_Iff.lean
- @@ -133,7 +133,7 @@ example : ¬Monotone fun x : ℝ ↦ -x := by
 sorry section - variable {α : Type _} [PartialOrder α] + variable {α : Type*} [PartialOrder α] variable (a b : α) example : a < b ↔ a ≤ b ∧ a ≠ b := by
- @@ -143,7 +143,7 @@ example : a < b ↔ a ≤ b ∧ a ≠ b := by
 end section - variable {α : Type _} [Preorder α] + variable {α : Type*} [Preorder α] variable (a b c : α) example : ¬a < a := by
MIL/C03_Logic/S05_Disjunction.lean
- @@ -101,7 +101,7 @@ example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by
  sorry section - variable {R : Type _} [CommRing R] [IsDomain R] + variable {R : Type*} [CommRing R] [IsDomain R] variable (x y : R) example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
- @@ -94,7 +94,7 @@ theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ}
 exact lt_irrefl _ this section - variable {α : Type _} [LinearOrder α] + variable {α : Type*} [LinearOrder α] def ConvergesTo' (s : α → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε
MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean
- @@ -91,7 +91,7 @@ end
  section - variable {α : Type _} (r s t : Set α) + variable {α : Type*} (r s t : Set α) example : r ⊆ s → s ⊆ t → r ⊆ t := by intro rsubs ssubt x xr
- @@ -105,7 +105,7 @@ theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t :=
  end section - variable {α : Type _} [PartialOrder α] + variable {α : Type*} [PartialOrder α] variable (s : Set α) (a b : α) def SetUb (s : Set α) (a : α) :=
- @@ -128,7 +128,7 @@ example {c : ℝ} (h : c ≠ 0) : Injective fun x ↦ c * x := by
  intro x₁ x₂ h' apply (mul_right_inj' h).mp h' - variable {α : Type _} {β : Type _} {γ : Type _} + variable {α : Type*} {β : Type*} {γ : Type*} variable {g : β → γ} {f : α → β} example (injg : Injective g) (injf : Injective f) : Injective fun x ↦ g (f x) := by
MIL/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean
- @@ -71,7 +71,7 @@ end
  section open Function - variable {α : Type _} {β : Type _} {γ : Type _} + variable {α : Type*} {β : Type*} {γ : Type*} variable {g : β → γ} {f : α → β} example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x ↦ g (f x) := by
MIL/C03_Logic/solutions/Solutions_S03_Negation.lean
- @@ -61,13 +61,12 @@ example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by
 end section - variable {α : Type _} (P : α → Prop) (Q : Prop) + variable {α : Type*} (P : α → Prop) (Q : Prop) example (h : ¬∃ x, P x) : ∀ x, ¬P x := by intro x Px apply h use x - exact Px example (h : ∀ x, ¬P x) : ¬∃ x, P x := by rintro ⟨x, Px⟩
- @@ -102,7 +101,6 @@ example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by
 intro h'' apply h' use x - exact h'' example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by rw [Monotone] at h
MIL/C03_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean
- @@ -52,7 +52,7 @@ example : ¬Monotone fun x : ℝ ↦ -x := by
 norm_num section - variable {α : Type _} [PartialOrder α] + variable {α : Type*} [PartialOrder α] variable (a b : α) example : a < b ↔ a ≤ b ∧ a ≠ b := by
- @@ -74,7 +74,7 @@ example : a < b ↔ a ≤ b ∧ a ≠ b := by
 end section - variable {α : Type _} [Preorder α] + variable {α : Type*} [Preorder α] variable (a b c : α) example : ¬a < a := by
MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean
- @@ -99,7 +99,7 @@ example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by
  exact eq_of_sub_eq_zero h1 section - variable {R : Type _} [CommRing R] [IsDomain R] + variable {R : Type*} [CommRing R] [IsDomain R] variable (x y : R) example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
MIL/C04_Sets_and_Functions/S01_Sets.lean
- @@ -4,7 +4,7 @@ import Mathlib.Data.Nat.Parity
  import Mathlib.Tactic section - variable {α : Type _} + variable {α : Type*} variable (s t u : Set α) open Set
- @@ -150,7 +150,6 @@ example (h₀ : ∀ x ∈ s, ¬Even x) (h₁ : ∀ x ∈ s, Prime x) : ∀ x ∈
  example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by rcases h with ⟨x, xs, _, prime_x⟩ use x, xs - exact prime_x section variable (ssubt : s ⊆ t)
- @@ -166,7 +165,7 @@ end
  end section - variable {α I : Type _} + variable {α I : Type*} variable (A B : I → Set α) variable (s : Set α)
- @@ -227,7 +226,7 @@ section
  open Set - variable {α : Type _} (s : Set (Set α)) + variable {α : Type*} (s : Set (Set α)) example : ⋃₀ s = ⋃ t ∈ s, t := by ext x
MIL/C04_Sets_and_Functions/S02_Functions.lean
- @@ -5,7 +5,7 @@ import Mathlib.Analysis.SpecialFunctions.Log.Basic
  section - variable {α β : Type _} + variable {α β : Type*} variable (f : α → β) variable (s t : Set α) variable (u v : Set β)
- @@ -78,14 +78,13 @@ example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by
  example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by sorry - variable {I : Type _} (A : I → Set α) (B : I → Set β) + variable {I : Type*} (A : I → Set α) (B : I → Set β) example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by ext y; simp constructor · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ use i, x - exact ⟨xAi, fxeq⟩ rintro ⟨i, x, xAi, fxeq⟩ exact ⟨x, ⟨i, xAi⟩, fxeq⟩
- @@ -158,7 +157,7 @@ example : (range fun x ↦ x ^ 2) = { y : ℝ | y ≥ 0 } := by
  end section - variable {α β : Type _} [Inhabited α] + variable {α β : Type*} [Inhabited α] #check (default : α)
- @@ -193,7 +192,7 @@ example : Surjective f ↔ RightInverse (inverse f) f :=
  end section - variable {α : Type _} + variable {α : Type*} open Function theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by
MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean
- @@ -7,7 +7,7 @@ open Function
  noncomputable section open Classical - variable {α β : Type _} [Nonempty β] + variable {α β : Type*} [Nonempty β] section variable (f : α → β) (g : β → α)
MIL/C04_Sets_and_Functions/solutions/Solutions_S01_Sets.lean
- @@ -4,7 +4,7 @@ import Mathlib.Data.Nat.Parity
  import Mathlib.Tactic section - variable {α : Type _} + variable {α : Type*} variable (s t u : Set α) open Set
- @@ -52,7 +52,6 @@ example : s \ t ∪ t = s ∪ t := by
  rintro (xs | xt) · left use xs - exact h right; exact xt example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by
- @@ -107,14 +106,13 @@ example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈
  example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by rcases h with ⟨x, xs, _, px⟩ use x, ssubt xs - exact px end end section - variable {α I : Type _} + variable {α I : Type*} variable (A B : I → Set α) variable (s : Set α)
- @@ -150,7 +148,6 @@ example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by
  simp rcases Nat.exists_infinite_primes x with ⟨p, primep, pge⟩ use p, pge - exact primep end
MIL/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
- @@ -5,7 +5,7 @@ import Mathlib.Analysis.SpecialFunctions.Log.Basic
  section - variable {α β : Type _} + variable {α β : Type*} variable (f : α → β) variable (s t : Set α) variable (u v : Set β)
- @@ -45,7 +45,6 @@ example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by
  example (h : s ⊆ t) : f '' s ⊆ f '' t := by rintro y ⟨x, xs, fxeq⟩ use x, h xs - exact fxeq example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by intro x; apply h
- @@ -103,14 +102,13 @@ example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by
  exact ⟨x, xs, rfl⟩ right; exact fxu - variable {I : Type _} (A : I → Set α) (B : I → Set β) + variable {I : Type*} (A : I → Set α) (B : I → Set β) example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by ext y; simp constructor · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ use i, x - exact ⟨xAi, fxeq⟩ rintro ⟨i, x, xAi, fxeq⟩ exact ⟨x, ⟨i, xAi⟩, fxeq⟩
- @@ -191,7 +189,7 @@ example : (range fun x ↦ x ^ 2) = { y : ℝ | y ≥ 0 } := by
  end section - variable {α β : Type _} [Inhabited α] + variable {α β : Type*} [Inhabited α] noncomputable section
- @@ -235,7 +233,7 @@ example : Surjective f ↔ RightInverse (inverse f) f :=
  end section - variable {α : Type _} + variable {α : Type*} open Function theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by
MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
- @@ -7,7 +7,7 @@ open Function
  noncomputable section open Classical - variable {α β : Type _} [Nonempty β] + variable {α β : Type*} [Nonempty β] section variable (f : α → β) (g : β → α)
MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean
- @@ -52,7 +52,7 @@ theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by
  sorry section - variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) + variable {α : Type*} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) #check Finset.sum s f #check Finset.prod s f
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
- @@ -40,7 +40,6 @@ theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p
  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
- @@ -63,7 +62,7 @@ theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by
  open Finset section - variable {α : Type _} [DecidableEq α] (r s t : Finset α) + variable {α : Type*} [DecidableEq α] (r s t : Finset α) example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by rw [subset_iff]
- @@ -89,7 +88,7 @@ example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by
  end section - variable {α : Type _} [DecidableEq α] (r s t : Finset α) + variable {α : Type*} [DecidableEq α] (r s t : Finset α) example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by sorry
- @@ -180,7 +179,6 @@ 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
MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean
- @@ -19,7 +19,7 @@ theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by
  section - variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) + variable {α : Type*} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) open BigOperators open Finset
MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
- @@ -28,7 +28,6 @@ theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p
  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
- @@ -57,7 +56,7 @@ theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by
  open Finset section - variable {α : Type _} [DecidableEq α] (r s t : Finset α) + variable {α : Type*} [DecidableEq α] (r s t : Finset α) example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by ext x
- @@ -169,7 +168,6 @@ 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
- @@ -187,14 +185,12 @@ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) :
  rcases 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⟩
MIL/C06_Structures/S02_Algebraic_Structures.lean
- @@ -3,7 +3,7 @@ import Mathlib.Data.Real.Basic
  namespace C06S02 - structure Group₁ (α : Type _) where + structure Group₁ (α : Type*) where mul : α → α → α one : α inv : α → α
- @@ -13,11 +13,11 @@ structure Group₁ (α : Type _) where
  mul_left_inv : ∀ x : α, mul (inv x) x = one structure Group₁Cat where - α : Type _ + α : Type* str : Group₁ α section - variable (α β γ : Type _) + variable (α β γ : Type*) variable (f : α ≃ β) (g : β ≃ γ) #check Equiv α β
- @@ -40,10 +40,10 @@ example : (f.trans g : α → γ) = g ∘ f :=
  end - example (α : Type _) : Equiv.Perm α = (α ≃ α) := + example (α : Type*) : Equiv.Perm α = (α ≃ α) := rfl - def permGroup {α : Type _} : Group₁ (Equiv.Perm α) + def permGroup {α : Type*} : Group₁ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α
- @@ -53,7 +53,7 @@ def permGroup {α : Type _} : Group₁ (Equiv.Perm α)
  mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm - structure AddGroup₁ (α : Type _) where + structure AddGroup₁ (α : Type*) where (add : α → α → α) -- fill in the rest @[ext]
- @@ -76,7 +76,7 @@ def addGroupPoint : AddGroup₁ Point := sorry
  end Point section - variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) + variable {α : Type*} (f g : Equiv.Perm α) (n : ℕ) #check f * g #check mul_assoc f g g⁻¹
- @@ -89,12 +89,12 @@ example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one]
  example : f * g * g⁻¹ = f := mul_inv_cancel_right f g - example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := + example {α : Type*} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := mul_inv_cancel_right f g end - class Group₂ (α : Type _) where + class Group₂ (α : Type*) where mul : α → α → α one : α inv : α → α
- @@ -103,7 +103,7 @@ class Group₂ (α : Type _) where
  one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one - instance {α : Type _} : Group₂ (Equiv.Perm α) where + instance {α : Type*} : Group₂ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm
- @@ -114,13 +114,13 @@ instance {α : Type _} : Group₂ (Equiv.Perm α) where
  #check Group₂.mul - def mySquare {α : Type _} [Group₂ α] (x : α) := + def mySquare {α : Type*} [Group₂ α] (x : α) := Group₂.mul x x #check mySquare section - variable {β : Type _} (f g : Equiv.Perm β) + variable {β : Type*} (f g : Equiv.Perm β) example : Group₂.mul f g = g.trans f := rfl
- @@ -149,17 +149,17 @@ example : x + y = Point.add x y :=
  end - instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := + instance hasMulGroup₂ {α : Type*} [Group₂ α] : Mul α := ⟨Group₂.mul⟩ - instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := + instance hasOneGroup₂ {α : Type*} [Group₂ α] : One α := ⟨Group₂.one⟩ - instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := + instance hasInvGroup₂ {α : Type*} [Group₂ α] : Inv α := ⟨Group₂.inv⟩ section - variable {α : Type _} (f g : Equiv.Perm α) + variable {α : Type*} (f g : Equiv.Perm α) #check f * 1 * g⁻¹
- @@ -168,6 +168,6 @@ def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) :=
  end - class AddGroup₂ (α : Type _) where + class AddGroup₂ (α : Type*) where add : α → α → α -- fill in the rest
MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean
- @@ -173,7 +173,7 @@ theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_a
  end Int - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + theorem sq_add_sq_eq_zero {α : Type*} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by sorry namespace gaussInt
MIL/C06_Structures/solutions/Solutions_S02_Algebraic_Structures.lean
- @@ -3,7 +3,7 @@ import Mathlib.Data.Real.Basic
  namespace C06S02 - structure AddGroup₁ (α : Type _) where + structure AddGroup₁ (α : Type*) where add : α → α → α zero : α neg : α → α
- @@ -40,7 +40,7 @@ def addGroupPoint : AddGroup₁ Point where
  end Point - class AddGroup₂ (α : Type _) where + class AddGroup₂ (α : Type*) where add : α → α → α zero : α neg : α → α
- @@ -49,13 +49,13 @@ class AddGroup₂ (α : Type _) where
  zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero - instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := + instance hasAddAddGroup₂ {α : Type*} [AddGroup₂ α] : Add α := ⟨AddGroup₂.add⟩ - instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := + instance hasZeroAddGroup₂ {α : Type*} [AddGroup₂ α] : Zero α := ⟨AddGroup₂.zero⟩ - instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := + instance hasNegAddGroup₂ {α : Type*} [AddGroup₂ α] : Neg α := ⟨AddGroup₂.neg⟩ instance : AddGroup₂ Point where
MIL/C06_Structures/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean
- @@ -164,14 +164,14 @@ theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_a
  end Int - private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + private theorem aux {α : Type*} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := haveI h' : x ^ 2 = 0 := by apply le_antisymm _ (sq_nonneg x) rw [← h] apply le_add_of_nonneg_right (sq_nonneg y) pow_eq_zero h' - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + theorem sq_add_sq_eq_zero {α : Type*} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by constructor · intro h
MIL/C07_Hierarchies/S01_Basics.lean
- @@ -263,7 +263,7 @@ def nsmul₁ [Zero M] [Add M] : ℕ → M → M
  | 0, _ => 0 | n + 1, a => a + nsmul₁ n a - def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + def zsmul₁ {M : Type*} [Zero M] [Add M] [Neg M] : ℤ → M → M | Int.ofNat n, a => nsmul₁ n a | Int.negSucc n, a => -nsmul₁ n.succ a
MIL/C07_Hierarchies/solutions/Solutions_S01_Basics.lean
- @@ -291,7 +291,7 @@ def nsmul₁ [Zero M] [Add M] : ℕ → M → M
  | 0, _ => 0 | n + 1, a => a + nsmul₁ n a - def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + def zsmul₁ {M : Type*} [Zero M] [Add M] [Neg M] : ℤ → M → M | Int.ofNat n, a => nsmul₁ n a | Int.negSucc n, a => -nsmul₁ n.succ a
MIL/C07_Hierarchies/solutions/Solutions_S03_Subobjects.lean
- @@ -67,7 +67,7 @@ instance [Group G] (H : Subgroup₁ G) : Group H :=
  inv := fun x ↦ ⟨x⁻¹, H.inv_mem x.property⟩ mul_left_inv := fun x ↦ SetCoe.ext (mul_left_inv (x : G)) } - class SubgroupClass₁ (S : Type _) (G : Type) [Group G] [SetLike S G] + class SubgroupClass₁ (S : Type*) (G : Type) [Group G] [SetLike S G] extends SubmonoidClass₁ S G : Prop where inv_mem : ∀ (s : S) {a : G}, a ∈ s → a⁻¹ ∈ s
MIL/C08_Topology/S01_Filters.lean
- @@ -3,7 +3,7 @@ import Mathlib.Topology.Instances.Real
  open Set Filter Topology - def principal {α : Type _} (s : Set α) : Filter α + def principal {α : Type*} (s : Set α) : Filter α where sets := { t | s ⊆ t } univ_sets := sorry
- @@ -16,13 +16,13 @@ example : Filter ℕ :=
  sets_of_superset := sorry inter_sets := sorry } - def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + 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) := + 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) : + example {X Y : Type*} (f : X → Y) (F : Filter X) (G : Filter Y) : Tendsto₂ f F G ↔ Tendsto₁ f F G := Iff.rfl
- @@ -32,7 +32,7 @@ example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) :
  (@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} + 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
- @@ -43,7 +43,7 @@ variable (f : ℝ → ℝ) (x₀ y₀ : ℝ)
  #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} + 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)
MIL/C08_Topology/S02_Metric_Spaces.lean
- @@ -5,7 +5,7 @@ import Mathlib.Analysis.NormedSpace.BanachSteinhaus
 open Set Filter open Topology Filter - variable {X : Type _} [MetricSpace X] (a b c : X) + variable {X : Type*} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) #check (dist_nonneg : 0 ≤ dist a b)
- @@ -22,36 +22,36 @@ 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} : + 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) : + 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) : + 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) : + 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) : + 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) : + 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) : + 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
- @@ -112,18 +112,18 @@ example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ}
 example {s : Set X} (hs : IsCompact s) : IsClosed s := hs.isClosed - example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + 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} : + 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} + example {X : Type*} [MetricSpace X] [CompactSpace X] + {Y : Type*} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry
MIL/C08_Topology/S03_Topological_Spaces.lean
- @@ -5,7 +5,7 @@ import Mathlib.Analysis.NormedSpace.BanachSteinhaus
  open Set Filter Topology section - variable {X : Type _} [TopologicalSpace X] + variable {X : Type*} [TopologicalSpace X] example : IsOpen (univ : Set X) := isOpen_univ
- @@ -13,14 +13,14 @@ example : IsOpen (univ : Set X) :=
  example : IsOpen (∅ : Set X) := isOpen_empty - example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen (s i)) : IsOpen (⋃ i, s i) := + 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)) : + 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] + variable {Y : Type*} [TopologicalSpace Y] example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := continuous_def
- @@ -47,7 +47,7 @@ example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in
  #check TopologicalSpace.nhds_mkOfNhds - example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + 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
- @@ -55,7 +55,7 @@ example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a)
  end -- BOTH. - variable {X Y : Type _} + variable {X Y : Type*} example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := TopologicalSpace.coinduced f
- @@ -78,13 +78,13 @@ 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) + 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)) : + 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
- @@ -101,7 +101,7 @@ 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} + 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' :=
- @@ -147,6 +147,6 @@ example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs :
  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)) + 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
MIL/C08_Topology/solutions/Solutions_S01_Filters.lean
- @@ -4,7 +4,7 @@ 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 := + 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
- @@ -26,10 +26,10 @@ example : Filter ℕ :=
 rw [max_le_iff] at hb constructor <;> tauto } - def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + 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} + 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]
- @@ -37,7 +37,7 @@ example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X →
 _ ≤ H := hg - example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + 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]
MIL/C08_Topology/solutions/Solutions_S02_Metric_Spaces.lean
- @@ -5,7 +5,7 @@ import Mathlib.Analysis.NormedSpace.BanachSteinhaus
 open Set Filter open Topology Filter - variable {X : Type _} [MetricSpace X] (a b c : X) + variable {X : Type*} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) #check (dist_nonneg : 0 ≤ dist a b)
- @@ -22,29 +22,29 @@ 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} : + 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) : + 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) : + 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) : + 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) : + 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) : + 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'
- @@ -54,7 +54,7 @@ example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ ↦ f (x ^
 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) : + 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
- @@ -124,22 +124,22 @@ example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ}
 example {s : Set X} (hs : IsCompact s) : IsClosed s := hs.isClosed - example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + 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} : + 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} + 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} + 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
MIL/C08_Topology/solutions/Solutions_S03_Topological_Spaces.lean
- @@ -5,7 +5,7 @@ import Mathlib.Analysis.NormedSpace.BanachSteinhaus
  open Set Filter Topology section - variable {X : Type _} [TopologicalSpace X] + variable {X : Type*} [TopologicalSpace X] example : IsOpen (univ : Set X) := isOpen_univ
- @@ -13,14 +13,14 @@ example : IsOpen (univ : Set X) :=
  example : IsOpen (∅ : Set X) := isOpen_empty - example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen (s i)) : IsOpen (⋃ i, s i) := + 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)) : + 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] + variable {Y : Type*} [TopologicalSpace Y] example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := continuous_def
- @@ -47,12 +47,12 @@ example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in
  #check TopologicalSpace.nhds_mkOfNhds - example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + 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) + 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
- @@ -63,7 +63,7 @@ example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a)
  end -- BOTH. - variable {X Y : Type _} + variable {X Y : Type*} example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := TopologicalSpace.coinduced f
- @@ -86,13 +86,13 @@ 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) + 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)) : + 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
- @@ -109,13 +109,13 @@ 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} + 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} + 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
- @@ -199,6 +199,6 @@ example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs :
  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)) + 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
MIL/C09_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean
- @@ -14,7 +14,7 @@ noncomputable section
 section - variable {E : Type _} [NormedAddCommGroup E] + variable {E : Type*} [NormedAddCommGroup E] example (x : E) : 0 ≤ ‖x‖ := norm_nonneg x
- @@ -27,7 +27,7 @@ example (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ :=
 example : MetricSpace E := by infer_instance - example {X : Type _} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : + example {X : Type*} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : Continuous fun x ↦ ‖f x‖ := hf.norm
- @@ -38,13 +38,13 @@ example (a : ℝ) (x : E) : ‖a • x‖ = |a| * ‖x‖ :=
 example [FiniteDimensional ℝ E] : CompleteSpace E := by infer_instance - example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := + example (𝕜 : Type*) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := norm_mul x y - example (𝕜 : Type _) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := + example (𝕜 : Type*) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := NormedField.exists_one_lt_norm 𝕜 - example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (E : Type _) [NormedAddCommGroup E] + example (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*) [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] : CompleteSpace E := FiniteDimensional.complete 𝕜 E
- @@ -52,8 +52,8 @@ end
 section - variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] - [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] example : E →L[𝕜] E := ContinuousLinearMap.id 𝕜 E
- @@ -82,12 +82,12 @@ end
 section - variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] - [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] open Metric - example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + example {ι : Type*} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C' := by -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` let e : ℕ → Set E := fun n ↦ ⋂ i : ι, { x : E | ‖g i x‖ ≤ n }
- @@ -114,19 +114,19 @@ end
 open Asymptotics - example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) + example {α : Type*} {E : Type*} [NormedGroup E] {F : Type*} [NormedGroup F] (c : ℝ) (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := isBigOWith_iff - example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + example {α : Type*} {E : Type*} [NormedGroup E] {F : Type*} [NormedGroup F] (l : Filter α) (f : α → E) (g : α → F) : f =O[l] g ↔ ∃ C, IsBigOWith C l f g := isBigO_iff_isBigOWith - example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + example {α : Type*} {E : Type*} [NormedGroup E] {F : Type*} [NormedGroup F] (l : Filter α) (f : α → E) (g : α → F) : f =o[l] g ↔ ∀ C > 0, IsBigOWith C l f g := isLittleO_iff_forall_isBigOWith - example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : + example {α : Type*} {E : Type*} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : f ~[l] g ↔ (f - g) =o[l] g := Iff.rfl
- @@ -134,8 +134,8 @@ section
 open Topology - variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] - [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : HasFDerivAt f f' x₀ ↔ (fun x ↦ f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x ↦ x - x₀ :=
- @@ -153,7 +153,7 @@ example (n : WithTop ℕ) {f : E → F} :
 ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x ↦ iteratedFDeriv 𝕜 m f x := contDiff_iff_continuous_differentiable - example {𝕂 : Type _} [IsROrC 𝕂] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type _} + example {𝕂 : Type*} [IsROrC 𝕂] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := hf.hasStrictFDerivAt hn
MIL/C09_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean
- @@ -14,12 +14,12 @@ noncomputable section
  section - variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] - [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] open Metric - example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + example {ι : Type*} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C' := by -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` let e : ℕ → Set E := fun n ↦ ⋂ i : ι, { x : E | ‖g i x‖ ≤ n }
MIL/C10_Integration_and_Measure_Theory/S02_Measure_Theory.lean
- @@ -9,7 +9,7 @@ open Set Filter
  noncomputable section - variable {α : Type _} [MeasurableSpace α] + variable {α : Type*} [MeasurableSpace α] example : MeasurableSet (∅ : Set α) := MeasurableSet.empty
- @@ -24,7 +24,7 @@ example : Encodable ℕ := by infer_instance
  example (n : ℕ) : Encodable (Fin n) := by infer_instance - variable {ι : Type _} [Encodable ι] + variable {ι : Type*} [Encodable ι] example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := MeasurableSet.iUnion h
MIL/C10_Integration_and_Measure_Theory/S03_Integration.lean
- @@ -12,11 +12,11 @@ open Topology Filter ENNReal
  open MeasureTheory noncomputable section - variable {α : Type _} [MeasurableSpace α] + variable {α : Type*} [MeasurableSpace α] variable {μ : Measure α} section - variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} + variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) : ∫ a, f a + g a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ :=
- @@ -33,7 +33,7 @@ example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀
  Tendsto (fun n ↦ ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) := tendsto_integral_of_dominated_convergence bound hmeas hint hbound hlim - example {α : Type _} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type _} + example {α : Type*} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type*} [MeasurableSpace β] {ν : Measure β} [SigmaFinite ν] (f : α × β → E) (hf : Integrable f (μ.prod ν)) : ∫ z, f z ∂ μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ := integral_prod f hf
- @@ -44,7 +44,7 @@ section
  open Convolution - variable {𝕜 : Type _} {G : Type _} {E : Type _} {E' : Type _} {F : Type _} [NormedAddCommGroup E] + variable {𝕜 : Type*} {G : Type*} {E : Type*} {E' : Type*} {F : Type*} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedSpace ℝ F] [CompleteSpace F] [Sub G]
- @@ -55,8 +55,8 @@ example (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : Me
  end - example {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] - [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type _} + example {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] + [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} (hs : MeasurableSet s) (hf : ∀ x : E, x ∈ s → HasFDerivWithinAt f (f' x) s x) (h_inj : InjOn f s) (g : E → F) :
MIL/C10_Integration_and_Measure_Theory/solutions/Solutions_S02_Measure_Theory.lean
- @@ -9,9 +9,9 @@ open Set Filter
  noncomputable section - variable {α : Type _} [MeasurableSpace α] + variable {α : Type*} [MeasurableSpace α] - variable {ι : Type _} [Encodable ι] + variable {ι : Type*} [Encodable ι] open MeasureTheory variable {μ : Measure α}
MIL/C10_Integration_and_Measure_Theory/solutions/Solutions_S03_Integration.lean
- @@ -12,6 +12,6 @@ open Topology Filter ENNReal
  open MeasureTheory noncomputable section - variable {α : Type _} [MeasurableSpace α] + variable {α : Type*} [MeasurableSpace α] variable {μ : Measure α}
README.md
- @@ -58,7 +58,7 @@ which is why we suggested making a copy.)
  If you have a Gitpod account or are willing to sign up for one, just point your browser to [https://gitpod.io/#/https://github.com/leanprover-community/mathematics_in_lean](https://gitpod.io/#/https://github.com/leanprover-community/mathematics_in_lean). This creates a virtual machine in the cloud, - and installs Lean and mathlib. + and installs Lean and Mathlib. It then presents you with a VS Code window, running in a virtual copy of the repository. We still suggest making a copy of the `MIL` directory, as described
html/C01_Introduction.html
- @@ -108,7 +108,7 @@ and ultimately certifies the correctness of our proofs.
 <a class="reference external" href="https://leanprover.github.io">Lean project page</a> and the <a class="reference external" href="https://leanprover-community.github.io/">Lean community web pages</a>. - This tutorial is based on Lean’s large and ever-growing library, mathlib. + This tutorial is based on Lean’s large and ever-growing library, Mathlib. We also strongly recommend joining the <a class="reference external" href="https://leanprover.zulipchat.com/">Lean Zulip online chat group</a> if you haven’t already.
- @@ -307,7 +307,7 @@ prove our theorem automatically.
 Another big difference between the two introductions is that Theorem Proving in Lean depends only on core Lean and its built-in tactics, whereas Mathematics in Lean is built on top of Lean’s - powerful and ever-growing library, mathlib. + powerful and ever-growing library, Mathlib. As a result, we can show you how to use some of the mathematical objects and theorems in the library, and some of the very useful tactics.
- @@ -326,7 +326,7 @@ and people are available on the
 to answer questions. We hope to see you there, and have no doubt that soon enough you, too, will be able to answer such questions - and contribute to the development of mathlib. + and contribute to the development of Mathlib. So here is your mission, should you choose to accept it: dive in, try the exercises, come to Zulip with questions, and have fun. But be forewarned:
- @@ -339,10 +339,11 @@ and to Scott Morrison and Mario Carneiro for help porting it from Lean 4.
 We are also grateful for help and corrections from Julian Berman, Alex Best, Bulwi Cha, Bryan Gin-ge Chen, Mauricio Collaris, Johan Commelin, - Winston de Greef, + Denis Gorbachev, Winston de Greef, Mathieu Guay-Paquet, Julian Külshammer, + Martin C. Martin, Giovanni Mascellani, Isaiah Mindich, Hunter Monroe, Pietro Monticone, Oliver Nash, - Bartosz Piotrowski, Guilherme Silva, and Floris van Doorn. + Bartosz Piotrowski, Guilherme Silva, Floris van Doorn, and Eric Wieser. Our work has been partially supported by the Hoskinson Center for Formal Mathematics. </section>
html/C02_Basics.html
- @@ -122,11 +122,10 @@ notational conventions and leave out parentheses when Lean does as well.
 rw [mul_assoc b a c] </pre></div> </div> - The <code class="docutils literal notranslate">import</code> line at the beginning of the example - imports the theory of the real numbers from <code class="docutils literal notranslate">mathlib</code>. + The <code class="docutils literal notranslate">import</code> lines at the beginning of the associated examples file + import the theory of the real numbers from Mathlib, as well as useful automation. For the sake of brevity, - we generally suppress information like this when it - is repeated from example to example. + we generally suppress information like this in the textbook. You are welcome to make changes to see what happens. You can type the <code class="docutils literal notranslate">ℝ</code> character as <code class="docutils literal notranslate">\R</code> or <code class="docutils literal notranslate">\real</code> in VS Code.
- @@ -350,7 +349,7 @@ in the assumption <code class="docutils literal notranslate">h
 </div> In the last step, the <code class="docutils literal notranslate">exact</code> tactic can use <code class="docutils literal notranslate">hyp</code> to solve the goal because at that point <code class="docutils literal notranslate">hyp</code> matches the goal exactly. - We close this section by noting that <code class="docutils literal notranslate">mathlib</code> provides a + We close this section by noting that Mathlib provides a useful bit of automation with a <code class="docutils literal notranslate">ring</code> tactic, which is designed to prove identities in any commutative ring as long as they follow purely from the ring axioms, without using any local assumption.
- @@ -399,7 +398,7 @@ and multiplication distributes over addition.</li>
 </ul> In Lean, the collection of objects is represented as a type, <code class="docutils literal notranslate">R</code>. The ring axioms are as follows: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (R : Type _) [Ring R] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (R : Type*) [Ring R] #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) #check (add_comm : ∀ a b : R, a + b = b + a)
- @@ -446,7 +445,7 @@ form a ring in which commutativity usually fails. If we declare <code class="doc
 commutative ring, in fact, all the theorems in the last section continue to hold when we replace <code class="docutils literal notranslate">ℝ</code> by <code class="docutils literal notranslate">R</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (R : Type _) [CommRing R] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (R : Type*) [CommRing R] variable (a b c d : R) example : c * b * a = b * (a * c) := by ring
- @@ -471,7 +470,7 @@ you have developed in the last section
 and apply them to reasoning axiomatically about rings. We will start with the axioms listed above, and use them to derive other facts. - Most of the facts we prove are already in <code class="docutils literal notranslate">mathlib</code>. + Most of the facts we prove are already in Mathlib. We will give the versions we prove the same names to help you learn the contents of the library as well as the naming conventions.
- @@ -487,7 +486,7 @@ theorems in a new namespace called <code class="docutils literal notranslate"><s
 The next example shows that we do not need <code class="docutils literal notranslate">add_zero</code> or <code class="docutils literal notranslate">add_right_neg</code> as ring axioms, because they follow from the other axioms. <div class="highlight-lean notranslate"><div class="highlight"><pre>namespace MyRing - variable {R : Type _} [Ring R] + variable {R : Type*} [Ring R] theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add]
- @@ -505,8 +504,8 @@ But don’t cheat!
 In the exercises that follow, take care to use only the general facts about rings that we have proved earlier in this section. (If you are paying careful attention, you may have noticed that we - changed the round brackets in <code class="docutils literal notranslate">(R : Type _)</code> for - curly brackets in <code class="docutils literal notranslate">{R : Type _}</code>. + changed the round brackets in <code class="docutils literal notranslate">(R : Type*)</code> for + curly brackets in <code class="docutils literal notranslate">{R : Type*}</code>. This declares <code class="docutils literal notranslate">R</code> to be an implicit argument. We will explain what this means in a moment, but don’t worry about it in the meanwhile.)
- @@ -662,7 +661,7 @@ addition and negation that we established above do not
 need the full strength of the ring axioms, or even commutativity of addition. The weaker notion of a group can be axiomatized as follows: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (A : Type _) [AddGroup A] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (A : Type*) [AddGroup A] #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) #check (zero_add : ∀ a : A, 0 + a = a)
- @@ -675,7 +674,7 @@ and multiplicative notation otherwise.
 So Lean defines a multiplicative version as well as the additive version (and also their abelian variants, <code class="docutils literal notranslate">AddCommGroup</code> and <code class="docutils literal notranslate">CommGroup</code>). - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {G : Type _} [Group G] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {G : Type*} [Group G] #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) #check (one_mul : ∀ a : G, 1 * a = a)
- @@ -696,7 +695,7 @@ The proofs we have carried out in this section provide some hints.
 sorry </pre></div> </div> - Explicitly invoking those lemmas is tedious, so mathlib provides + Explicitly invoking those lemmas is tedious, so Mathlib provides tactics similar to <cite>ring</cite> in order to cover most uses: <cite>group</cite> is for non-commutative multiplicative groups, <cite>abel</cite> for abelian additive groups, and <cite>noncomm_ring</cite> for non-commutative rings.
- @@ -885,11 +884,11 @@ find the library theorems you need constitutes an important
 part of formalization. There are a number of strategies you can use: <ul class="simple"> - <li>You can browse mathlib in its + <li>You can browse Mathlib in its <a class="reference external" href="https://github.com/leanprover-community/mathlib4">GitHub repository</a>.</li> - <li>You can use the API documentation on the mathlib + <li>You can use the API documentation on the Mathlib <a class="reference external" href="https://leanprover-community.github.io/mathlib4_docs/">web pages</a>.</li> - <li>You can rely on mathlib naming conventions and Ctrl-space completion in + <li>You can rely on Mathlib naming conventions and Ctrl-space completion in the editor to guess a theorem name (or Cmd-space on a Mac keyboard). In Lean, a theorem named <code class="docutils literal notranslate">A_of_B_of_C</code> establishes something of the form <code class="docutils literal notranslate">A</code> from hypotheses of the form <code class="docutils literal notranslate">B</code> and <code class="docutils literal notranslate">C</code>,
- @@ -943,7 +942,7 @@ There are a number of things worth noticing.
 First, an expression <code class="docutils literal notranslate">s ≥ t</code> is definitionally equivalent to <code class="docutils literal notranslate">t ≤ s</code>. In principle, this means one should be able to use them interchangeably. But some of Lean’s automation does not recognize the equivalence, - so mathlib tends to favor <code class="docutils literal notranslate">≤</code> over <code class="docutils literal notranslate">≥</code>. + so Mathlib tends to favor <code class="docutils literal notranslate">≤</code> over <code class="docutils literal notranslate">≥</code>. Second, we have used the <code class="docutils literal notranslate">ring</code> tactic extensively. It is a real timesaver! Finally, notice that in the second line of the
- @@ -1125,7 +1124,7 @@ Be careful: the divisibility symbol is not the
 ordinary bar on your keyboard. Rather, it is a unicode character obtained by typing <code class="docutils literal notranslate">\|</code> in VS Code. - By convention, mathlib uses <code class="docutils literal notranslate">dvd</code> + By convention, Mathlib uses <code class="docutils literal notranslate">dvd</code> to refer to it in theorem names. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := dvd_trans h₀ h₁
- @@ -1187,7 +1186,7 @@ For example, a partial order consists of a set with a
 binary relation that is reflexive and transitive, like <code class="docutils literal notranslate">≤</code> on the real numbers. Lean knows about partial orders: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [PartialOrder α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [PartialOrder α] variable (x y z : α) #check x ≤ y
- @@ -1195,7 +1194,7 @@ Lean knows about partial orders:
 #check (le_trans : x ≤ y → y ≤ z → x ≤ z) </pre></div> </div> - Here we are adopting the mathlib convention of using + Here we are adopting the Mathlib convention of using letters like <code class="docutils literal notranslate">α</code>, <code class="docutils literal notranslate">β</code>, and <code class="docutils literal notranslate">γ</code> (entered as <code class="docutils literal notranslate">\a</code>, <code class="docutils literal notranslate">\b</code>, and <code class="docutils literal notranslate">\g</code>) for arbitrary types.
- @@ -1230,7 +1229,7 @@ has the properties indicated.
 A lattice is a structure that extends a partial order with operations <code class="docutils literal notranslate">⊓</code> and <code class="docutils literal notranslate">⊔</code> that are analogous to <code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code> on the real numbers: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [Lattice α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [Lattice α] variable (x y z : α) #check x ⊓ y
- @@ -1248,7 +1247,7 @@ the greatest lower bound and least upper bound, respectively.
 You can type them in VS code using <code class="docutils literal notranslate">\glb</code> and <code class="docutils literal notranslate">\lub</code>. The symbols are also often called then infimum and the supremum, - and mathlib refers to them as <code class="docutils literal notranslate">inf</code> and <code class="docutils literal notranslate">sup</code> in + and Mathlib refers to them as <code class="docutils literal notranslate">inf</code> and <code class="docutils literal notranslate">sup</code> in theorem names. To further complicate matters, they are also often called meet and join.
- @@ -1291,7 +1290,7 @@ together with <code class="docutils literal notranslate">le_re
 sorry </pre></div> </div> - You can find these theorems in the mathlib as <code class="docutils literal notranslate">inf_comm</code>, <code class="docutils literal notranslate">inf_assoc</code>, + You can find these theorems in the Mathlib as <code class="docutils literal notranslate">inf_comm</code>, <code class="docutils literal notranslate">inf_assoc</code>, <code class="docutils literal notranslate">sup_comm</code>, and <code class="docutils literal notranslate">sup_assoc</code>, respectively. Another good exercise is to prove the absorption laws using only those axioms:
- @@ -1302,12 +1301,12 @@ using only those axioms:
 sorry </pre></div> </div> - These can be found in mathlib with the names <code class="docutils literal notranslate">inf_sup_self</code> and <code class="docutils literal notranslate">sup_inf_self</code>. + These can be found in Mathlib with the names <code class="docutils literal notranslate">inf_sup_self</code> and <code class="docutils literal notranslate">sup_inf_self</code>. A lattice that satisfies the additional identities <code class="docutils literal notranslate">x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)</code> and <code class="docutils literal notranslate">x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)</code> is called a distributive lattice. Lean knows about these too: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [DistribLattice α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [DistribLattice α] variable (x y z : α) #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z)
- @@ -1324,7 +1323,7 @@ by providing an explicit description of a
 nondistributive lattice with finitely many elements. It is also a good exercise to show that in any lattice, either distributivity law implies the other: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [Lattice α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [Lattice α] variable (a b c : α) example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by
- @@ -1339,7 +1338,7 @@ For example, a strict ordered ring consists of a commutative ring toget
 with a partial order on the carrier satisfying additional axioms that say that the ring operations are compatible with the order: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {R : Type _} [StrictOrderedRing R] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {R : Type*} [StrictOrderedRing R] variable (a b c : R) #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b)
- @@ -1372,7 +1371,7 @@ A metric space consists of a set equipped with a notion of
 distance, <code class="docutils literal notranslate">dist x y</code>, mapping any pair of elements to a real number. The distance function is assumed to satisfy the following axioms: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X : Type _} [MetricSpace X] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X : Type*} [MetricSpace X] variable (x y z : X) #check (dist_self x : dist x x = 0)
- @@ -1388,7 +1387,7 @@ always nonnegative:
 </pre></div> </div> We recommend making use of the theorem <code class="docutils literal notranslate">nonneg_of_mul_nonneg_left</code>. - As you may have guessed, this theorem is called <code class="docutils literal notranslate">dist_nonneg</code> in mathlib. + As you may have guessed, this theorem is called <code class="docutils literal notranslate">dist_nonneg</code> in Mathlib. </section> </section>
html/C03_Logic.html
- @@ -281,7 +281,7 @@ but it is worth knowing that the natural numbers, integers, rationals,
 and real numbers are all instances. So if we prove the theorem <code class="docutils literal notranslate">fnUb_add</code> at that level of generality, it will apply in all these instances. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} {R : Type*} [OrderedCancelAddCommMonoid R] #check add_le_add
- @@ -297,10 +297,10 @@ Section <a class="reference internal" href="C02_Basics.html#proving-identities-i
 though we still haven’t explained what they mean. For concreteness, we will stick to the real numbers for most of our examples, - but it is worth knowing that mathlib contains definitions and theorems + but it is worth knowing that Mathlib contains definitions and theorems that work at a high level of generality. For another example of a hidden universal quantifier, - mathlib defines a predicate <code class="docutils literal notranslate">Monotone</code>, + Mathlib defines a predicate <code class="docutils literal notranslate">Monotone</code>, which says that a function is nondecreasing in its arguments: <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b := @h
- @@ -429,7 +429,7 @@ we can write <code class="docutils literal notranslate">h</spa
 The following example provides a tactic proof and a proof term justifying the reflexivity of the subset relation, and asks you to do the same for transitivity. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (r s t : Set α) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} (r s t : Set α) example : s ⊆ s := by intro x xs
- @@ -449,7 +449,7 @@ has an order associated with it.
 In the next example, we ask you to prove that if <code class="docutils literal notranslate">a</code> is a bound on <code class="docutils literal notranslate">s</code> and <code class="docutils literal notranslate">a ≤ b</code>, then <code class="docutils literal notranslate">b</code> is a bound on <code class="docutils literal notranslate">s</code> as well. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [PartialOrder α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [PartialOrder α] variable (s : Set α) (a b : α) def SetUb (s : Set α) (a : α) :=
- @@ -482,7 +482,7 @@ a lemma name.
 </pre></div> </div> Finally, show that the composition of two injective functions is injective: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} {β : Type _} {γ : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} {β : Type*} {γ : Type*} variable {g : β → γ} {f : α → β} example (injg : Injective g) (injf : Injective f) : Injective fun x ↦ g (f x) := by
- @@ -512,14 +512,28 @@ leaving the goal of proving the property.
 norm_num </pre></div> </div> + You can give the <code class="docutils literal notranslate">use</code> tactic proofs as well as data: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + have h1 : 2 < (5 : ℝ) / 2 := by norm_num + have h2 : (5 : ℝ) / 2 < 3 := by norm_num + use 5 / 2, h1, h2 + </pre></div> + </div> + In fact, the <code class="docutils literal notranslate">use</code> tactic automatically tries to use available assumptions as well. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num + use 5 / 2 + </pre></div> + </div> Alternatively, we can use Lean’s anonymous constructor notation - to construct the proof. + to construct a proof of an existential quantifier. <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num ⟨5 / 2, h⟩ </pre></div> </div> - The left and right angle brackets, + Notice that there is no <code class="docutils literal notranslate">by</code>; here we are giving an explicit proof term. + The left and right angle brackets, which can be entered as <code class="docutils literal notranslate">\<</code> and <code class="docutils literal notranslate">\></code> respectively, tell Lean to put together the given data using whatever construction is appropriate
- @@ -613,7 +627,7 @@ in expressions and proof terms:
 </pre></div> </div> The task of unpacking information in a hypothesis is - so important that Lean and mathlib provide a number of + so important that Lean and Mathlib provide a number of ways to do it. For example, the <code class="docutils literal notranslate">obtain</code> tactic provides suggestive syntax: <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by obtain ⟨a, ubfa⟩ := ubf
- @@ -680,7 +694,7 @@ quantifiers at once.
 We then provide the magic values needed to express <code class="docutils literal notranslate">x * y</code> as a sum of squares as a list to the <code class="docutils literal notranslate">use</code> statement, and we use <code class="docutils literal notranslate">ring</code> to verify that they work. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [CommRing α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [CommRing α] def SumOfSquares (x : α) := ∃ a b, x = a ^ 2 + b ^ 2
- @@ -789,7 +803,7 @@ not just a hypothesis.
 </div> See if you can use these methods to show that the composition of surjective functions is surjective. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} {β : Type _} {γ : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} {β : Type*} {γ : Type*} variable {g : β → γ} {f : α → β} example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x ↦ g (f x) := by
- @@ -912,7 +926,7 @@ is equivalent to saying that something fails to have property <code class="docut
 In other words, all four of the following implications are valid (but one of them cannot be proved with what we explained so far): - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (P : α → Prop) (Q : Prop) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} (P : α → Prop) (Q : Prop) example (h : ¬∃ x, P x) : ∀ x, ¬P x := by sorry
- @@ -974,7 +988,7 @@ a negation in front,
 and it is a common mathematical pattern to replace such statements with equivalent forms in which the negation has been pushed inward. - To facilitate this, mathlib offers a <code class="docutils literal notranslate">push_neg</code> tactic, + To facilitate this, Mathlib offers a <code class="docutils literal notranslate">push_neg</code> tactic, which restates the goal in this way. The command <code class="docutils literal notranslate">push_neg at h</code> restates the hypothesis <code class="docutils literal notranslate">h</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ¬∀ a, ∃ x, f x > a) : FnHasUb f := by
- @@ -1288,7 +1302,7 @@ Lean axiomatizes the associated strict pre-order by
 <code class="docutils literal notranslate">a < b ↔ a ≤ b ∧ ¬ b ≤ a</code>. Show that if <code class="docutils literal notranslate">≤</code> is a partial order, then <code class="docutils literal notranslate">a < b</code> is equivalent to <code class="docutils literal notranslate">a ≤ b ∧ a ≠ b</code>: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [PartialOrder α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [PartialOrder α] variable (a b : α) example : a < b ↔ a ≤ b ∧ a ≠ b := by
- @@ -1310,7 +1324,7 @@ We will come back to the simplifier later,
 but here we are only relying on the fact that it will use the indicated lemma repeatedly, even if it needs to be instantiated to different values. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [Preorder α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [Preorder α] variable (a b c : α) example : ¬a < a := by
- @@ -1440,7 +1454,7 @@ In this textbook, we will generally use <code class="docutils literal notranslat
 cases of a disjunction. 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. + They are given the same names they have in Mathlib. <div class="highlight-lean notranslate"><div class="highlight"><pre>namespace MyAbs theorem le_abs_self (x : ℝ) : x ≤ |x| := by
- @@ -1493,7 +1507,7 @@ and use a semicolon and <code class="docutils literal notranslate"> On the real numbers, an equation <code class="docutils literal notranslate">x * y = 0</code> tells us that <code class="docutils literal notranslate">x = 0</code> or <code class="docutils literal notranslate">y = 0</code>. - In mathlib, this fact is known as <code class="docutils literal notranslate">eq_zero_or_eq_zero_of_mul_eq_zero</code>, + In Mathlib, this fact is known as <code class="docutils literal notranslate">eq_zero_or_eq_zero_of_mul_eq_zero</code>, and it is another nice example of how a disjunction can arise. See if you can use it to prove the following: <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
- @@ -1518,7 +1532,7 @@ says that the real numbers have no nontrivial zero divisors.
 A commutative ring with this property is called an integral domain. Your proofs of the two theorems above should work equally well in any integral domain: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {R : Type _} [CommRing R] [IsDomain R] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {R : Type*} [CommRing R] [IsDomain R] variable (x y : R) example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
- @@ -1791,13 +1805,13 @@ natural numbers is that their structure carries a partial order
 with <code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code>. You can check that everything still works if you replace <code class="docutils literal notranslate">ℕ</code> everywhere by any linear order <code class="docutils literal notranslate">α</code>: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [LinearOrder α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [LinearOrder α] def ConvergesTo' (s : α → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε </pre></div> </div> - In <a class="reference internal" href="C08_Topology.html#filters">Section 8.1</a>, we will see that mathlib has mechanisms + In <a class="reference internal" href="C08_Topology.html#filters">Section 8.1</a>, we will see that Mathlib has mechanisms for dealing with convergence in vastly more general terms, not only abstracting away particular features of the domain and codomain,
html/C04_Sets_and_Functions.html
- @@ -101,7 +101,7 @@ groups, vector spaces, and so on.
 Some expressions are types, which is to say, their type is <code class="docutils literal notranslate">Type</code>. - Lean and mathlib provide ways of defining new types, + Lean and Mathlib provide ways of defining new types, and ways of defining objects of those types. Conceptually, you can think of a type as just a set of objects. Requiring every object to have a type has some advantages.
- @@ -149,7 +149,7 @@ Unlike <code class="docutils literal notranslate">rw</c
 inside a universal or existential quantifier. If you step through the proof, you can see the effects of these commands. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} variable (s t u : Set α) open Set
- @@ -430,7 +430,6 @@ Here is are some examples of how they are used:
 example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by rcases h with ⟨x, xs, _, prime_x⟩ use x, xs - exact prime_x </pre></div> </div> See if you can prove these slight variations:
- @@ -457,7 +456,7 @@ There is nothing special about the natural numbers here,
 so <code class="docutils literal notranslate">ℕ</code> can be replaced by any type <code class="docutils literal notranslate">I</code> used to index the sets. The following illustrates their use. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α I : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α I : Type*} variable (A B : I → Set α) variable (s : Set α)
- @@ -542,7 +541,7 @@ Similarly, their intersection, <code class="docutils literal notranslate">sUnion</code> and <code class="docutils literal notranslate">sInter</code>, respectively. The following examples show their relationship to bounded union and intersection. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (s : Set (Set α)) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} (s : Set (Set α)) example : ⋃₀ s = ⋃ t ∈ s, t := by ext x
- @@ -566,7 +565,7 @@ the library defines <code class="docutils literal notranslate">{x | f x ∈ p}</code>. The expression <code class="docutils literal notranslate">x ∈ f ⁻¹' p</code> reduces to <code class="docutils literal notranslate">f x ∈ p</code>. This is often convenient, as in the following example: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α β : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α β : Type*} variable (f : α → β) variable (s t : Set α) variable (u v : Set β)
- @@ -686,14 +685,13 @@ to guarantee that the index set is nonempty.
 To prove any of these, we recommend using <code class="docutils literal notranslate">ext</code> or <code class="docutils literal notranslate">intro</code> to unfold the meaning of an equation or inclusion between sets, and then calling <code class="docutils literal notranslate">simp</code> to unpack the conditions for membership. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {I : Type _} (A : I → Set α) (B : I → Set β) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {I : Type*} (A : I → Set α) (B : I → Set β) example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by ext y; simp constructor · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ use i, x - exact ⟨xAi, fxeq⟩ rintro ⟨i, x, xAi, fxeq⟩ exact ⟨x, ⟨i, xAi⟩, fxeq⟩
- @@ -737,7 +735,7 @@ to <code class="docutils literal notranslate">InjOn <sp
 Similarly, the library defines <code class="docutils literal notranslate">range f</code> to be <code class="docutils literal notranslate">{x | ∃y, f y = x}</code>, so <code class="docutils literal notranslate">range f</code> is provably equal to <code class="docutils literal notranslate">f '' univ</code>. - This is a common theme in mathlib: + This is a common theme in Mathlib: although many properties of functions are defined relative to their full domain, there are often relativized versions that restrict
- @@ -795,7 +793,7 @@ This requires an appeal to the axiom of choice.
 Lean allows various ways of accessing it; one convenient method is to use the classical <code class="docutils literal notranslate">choose</code> operator, illustrated below. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α β : Type _} [Inhabited α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α β : Type*} [Inhabited α] #check (default : α)
- @@ -963,17 +961,17 @@ Because the proof uses classical logic, we tell Lean that our definitions
 will generally not be computable. <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section open Classical - variable {α β : Type _} [Nonempty β] + variable {α β : Type*} [Nonempty β] </pre></div> </div> The annotation <code class="docutils literal notranslate">[Nonempty β]</code> specifies that <code class="docutils literal notranslate">β</code> is nonempty. - We use it because the mathlib primitive that we will use to + We use it because the Mathlib primitive that we will use to construct \(g^{-1}\) requires it. The case of the theorem where \(\beta\) is empty is trivial, and even though it would not be hard to generalize the formalization to cover that case as well, we will not bother. Specifically, we need the hypothesis <code class="docutils literal notranslate">[Nonempty β]</code> for the operation - <code class="docutils literal notranslate">invFun</code> that is defined in mathlib. + <code class="docutils literal notranslate">invFun</code> that is defined in Mathlib. Given <code class="docutils literal notranslate">x : α</code>, <code class="docutils literal notranslate">invFun g x</code> chooses a preimage of <code class="docutils literal notranslate">x</code> in <code class="docutils literal notranslate">β</code> if there is one, and returns an arbitrary element of <code class="docutils literal notranslate">β</code> otherwise.
html/C05_Elementary_Number_Theory.html
- @@ -385,7 +385,7 @@ To be able to do that we will need better means for reasoning about
 products and sums over a finite set, which is also a topic we will return to. In fact, the results in this section are all established in much - greater generality in mathlib, + greater generality in Mathlib, in <code class="docutils literal notranslate">Data.Real.Irrational</code>. The notion of <code class="docutils literal notranslate">multiplicity</code> is defined for an arbitrary commutative monoid,
- @@ -462,7 +462,7 @@ be used manually by giving the name <code class="docutils literal notranslate"><
 simp [fac] </pre></div> </div> - The factorial function is actually already defined in mathlib as + The factorial function is actually already defined in Mathlib as <code class="docutils literal notranslate">Nat.factorial</code>. Once again, you can jump to it by typing <code class="docutils literal notranslate">#check Nat.factorial</code> and using <code class="docutils literal notranslate">ctrl-click.</code> For illustrative purposes, we will continue using <code class="docutils literal notranslate">fac</code> in the examples.
- @@ -532,7 +532,7 @@ it supports in the next section, and again in a later chapter.
 For now, we will only make use of <code class="docutils literal notranslate">Finset.range n</code>, which is the finite set of natural numbers less than <code class="docutils literal notranslate">n</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) #check Finset.sum s f #check Finset.prod s f
- @@ -794,7 +794,6 @@ The proof still works if you delete that line.
 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
- @@ -850,7 +849,7 @@ that every element of one is an element of the other.
 <div class="highlight-lean notranslate"><div class="highlight"><pre>open Finset section - variable {α : Type _} [DecidableEq α] (r s t : Finset α) + variable {α : Type*} [DecidableEq α] (r s t : Finset α) example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by rw [subset_iff]
- @@ -981,7 +980,7 @@ is a procedure for deciding whether or not <code class="docutils literal notrans
 is a procedure for <code class="docutils literal notranslate">Nat.Prime</code>. In general, if we use classical logic by writing <code class="docutils literal notranslate">open Classical</code>, we can dispense with the assumption. - In mathlib, <code class="docutils literal notranslate">Finset.sup s f</code> denotes the supremum of the values of <code class="docutils literal notranslate">f x</code> as <code class="docutils literal notranslate">x</code> + In Mathlib, <code class="docutils literal notranslate">Finset.sup s f</code> denotes the supremum of the values of <code class="docutils literal notranslate">f x</code> as <code class="docutils literal notranslate">x</code> ranges over <code class="docutils literal notranslate">s</code>, returning <code class="docutils literal notranslate">0</code> in the case where <code class="docutils literal notranslate">s</code> is empty and the codomain of <code class="docutils literal notranslate">f</code> is <code class="docutils literal notranslate">ℕ</code>. In the first proof, we use <code class="docutils literal notranslate">s.sup id</code>, where <code class="docutils literal notranslate">id</code> is the identity function, to refer to the maximum value in <code class="docutils literal notranslate">s</code>.
- @@ -1075,7 +1074,6 @@ same property.
 ∃ 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
html/C06_Structures.html
- @@ -263,7 +263,7 @@ components, which we can do with <code class="docutils literal notranslate"><spa
 </div> Mathematical constructions often involve taking apart bundled information and putting it together again in different ways. - It therefore makes sense that Lean and mathlib offer so many ways + It therefore makes sense that Lean and Mathlib offer so many ways of doing this efficiently. As an exercise, try proving that <code class="docutils literal notranslate">Point.add</code> is associative. Then define scalar multiplication for a point and show that it
- @@ -604,7 +604,7 @@ this is exactly what the <code class="docutils literal notranslate"> Given a data type <code class="docutils literal notranslate">α</code>, we can define the group structure on <code class="docutils literal notranslate">α</code> as follows. - <div class="highlight-lean notranslate"><div class="highlight"><pre>structure Group₁ (α : Type _) where + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure Group₁ (α : Type*) where mul : α → α → α one : α inv : α → α
- @@ -622,12 +622,12 @@ that the counterpart <code class="docutils literal notranslate"> This definition of a group is similar to the definition of <code class="docutils literal notranslate">Group</code> in - mathlib, + Mathlib, and we have chosen the name <code class="docutils literal notranslate">Group₁</code> to distinguish our version. If you write <code class="docutils literal notranslate">#check Group</code> and ctrl-click on the definition, - you will see that the mathlib version of <code class="docutils literal notranslate">Group</code> is defined to + you will see that the Mathlib version of <code class="docutils literal notranslate">Group</code> is defined to extend another structure; we will explain how to do that later. - If you type <code class="docutils literal notranslate">#print Group</code> you will also see that the mathlib + If you type <code class="docutils literal notranslate">#print Group</code> you will also see that the Mathlib version of <code class="docutils literal notranslate">Group</code> has a number of extra fields. For reasons we will explain later, sometimes it is useful to add redundant information to a structure,
- @@ -635,24 +635,24 @@ so that there are additional fields for objects and functions
 that can be defined from the core data. Don’t worry about that for now. Rest assured that our simplified version <code class="docutils literal notranslate">Group₁</code> is - morally the same as the definition of a group that mathlib uses. + morally the same as the definition of a group that Mathlib uses. It is sometimes useful to bundle - the type together with the structure, and mathlib also + the type together with the structure, and Mathlib also contains a definition of a <code class="docutils literal notranslate">GroupCat</code> structure that is equivalent to the following: <div class="highlight-lean notranslate"><div class="highlight"><pre>structure Group₁Cat where - α : Type _ + α : Type* str : Group₁ α </pre></div> </div> - The mathlib version is found in <code class="docutils literal notranslate">Algebra.Category.Group.Basic</code>, + The Mathlib version is found in <code class="docutils literal notranslate">Algebra.Category.Group.Basic</code>, and you can <code class="docutils literal notranslate">#check</code> it if you add this to the imports at the beginning of the examples file. For reasons that will become clearer below, it is more often useful to keep the type <code class="docutils literal notranslate">α</code> separate from the structure <code class="docutils literal notranslate">Group α</code>. We refer to the two objects together as a partially bundled structure, since the representation combines most, but not all, of the components - into one structure. It is common in mathlib + into one structure. It is common in Mathlib to use capital roman letters like <code class="docutils literal notranslate">G</code> for a type when it is used as the carrier type for a group. Let’s construct a group, which is to say, an element of the <code class="docutils literal notranslate">Group₁</code> type.
- @@ -665,7 +665,7 @@ a function <code class="docutils literal notranslate">f.toFun<
 the inverse function <code class="docutils literal notranslate">f.invFun</code> from <code class="docutils literal notranslate">β</code> to <code class="docutils literal notranslate">α</code>, and two properties that specify these functions are indeed inverse to one another. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (α β γ : Type _) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (α β γ : Type*) variable (f : α ≃ β) (g : β ≃ γ) #check Equiv α β
- @@ -698,7 +698,7 @@ and have Lean insert it for us.
 </div> Mathlib also defines the type <code class="docutils literal notranslate">perm α</code> of equivalences between <code class="docutils literal notranslate">α</code> and itself. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (α : Type _) : Equiv.Perm α = (α ≃ α) := + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (α : Type*) : Equiv.Perm α = (α ≃ α) := rfl </pre></div> </div>
- @@ -707,7 +707,7 @@ of equivalences. We orient things so that <code class="docutils literal notransl
 equal to <code class="docutils literal notranslate">g.trans f</code>, whose forward function is <code class="docutils literal notranslate">f ∘ g</code>. In other words, multiplication is what we ordinarily think of as composition of the bijections. Here we define this group: - <div class="highlight-lean notranslate"><div class="highlight"><pre>def permGroup {α : Type _} : Group₁ (Equiv.Perm α) + <div class="highlight-lean notranslate"><div class="highlight"><pre>def permGroup {α : Type*} : Group₁ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α
- @@ -718,7 +718,7 @@ composition of the bijections. Here we define this group:
 mul_left_inv := Equiv.self_trans_symm </pre></div> </div> - In fact, mathlib defines exactly this <code class="docutils literal notranslate">Group</code> structure on <code class="docutils literal notranslate">Equiv.Perm α</code> + In fact, Mathlib defines exactly this <code class="docutils literal notranslate">Group</code> structure on <code class="docutils literal notranslate">Equiv.Perm α</code> in the file <code class="docutils literal notranslate">GroupTheory.Perm.Basic</code>. As always, you can hover over the theorems used in the definition of <code class="docutils literal notranslate">permGroup</code> to see their statements,
- @@ -749,7 +749,7 @@ to the <code class="docutils literal notranslate">Group₁
 additive naming scheme just described. Define negation and a zero on the <code class="docutils literal notranslate">Point</code> data type, and define the <code class="docutils literal notranslate">AddGroup₁</code> structure on <code class="docutils literal notranslate">Point</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>structure AddGroup₁ (α : Type _) where + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure AddGroup₁ (α : Type*) where (add : α → α → α) -- fill in the rest @[ext]
- @@ -781,9 +781,9 @@ Moreover, we want to arrange it so that we can define an operation
 on a structure and use it with any particular instance, and we want to arrange it so that we can prove a theorem about a structure and use it with any instance. - In fact, mathlib is already set up to use generic group notation, + In fact, Mathlib is already set up to use generic group notation, definitions, and theorems for <code class="docutils literal notranslate">Equiv.Perm α</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} (f g : Equiv.Perm α) (n : ℕ) #check f * g #check mul_assoc f g g⁻¹
- @@ -796,7 +796,7 @@ definitions, and theorems for <code class="docutils literal notranslate">example : f * g * g⁻¹ = f := mul_inv_cancel_right f g - example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := + example {α : Type*} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := mul_inv_cancel_right f g </pre></div> </div>
- @@ -861,7 +861,7 @@ Lean. As with the names of class variables, we are allowed to leave the
 name of an instance definition anonymous, since in general we intend Lean to find it and put it to use without troubling us with the details. - <div class="highlight-lean notranslate"><div class="highlight"><pre>class Group₂ (α : Type _) where + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Group₂ (α : Type*) where mul : α → α → α one : α inv : α → α
- @@ -870,7 +870,7 @@ without troubling us with the details.
 one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one - instance {α : Type _} : Group₂ (Equiv.Perm α) where + instance {α : Type*} : Group₂ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm
- @@ -883,13 +883,13 @@ without troubling us with the details.
 The following illustrates their use. <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Group₂.mul - def mySquare {α : Type _} [Group₂ α] (x : α) := + def mySquare {α : Type*} [Group₂ α] (x : α) := Group₂.mul x x #check mySquare section - variable {β : Type _} (f g : Equiv.Perm β) + variable {β : Type*} (f g : Equiv.Perm β) example : Group₂.mul f g = g.trans f := rfl
- @@ -965,17 +965,17 @@ When we define a new instance of a ring in Lean,
 we don’t have to define <code class="docutils literal notranslate">+</code> and <code class="docutils literal notranslate">*</code> for that instance, because Lean knows that these are defined for every ring. We can use this method to specify notation for our <code class="docutils literal notranslate">Group₂</code> class: - <div class="highlight-lean notranslate"><div class="highlight"><pre>instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance hasMulGroup₂ {α : Type*} [Group₂ α] : Mul α := ⟨Group₂.mul⟩ - instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := + instance hasOneGroup₂ {α : Type*} [Group₂ α] : One α := ⟨Group₂.one⟩ - instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := + instance hasInvGroup₂ {α : Type*} [Group₂ α] : Inv α := ⟨Group₂.inv⟩ section - variable {α : Type _} (f g : Equiv.Perm α) + variable {α : Type*} (f g : Equiv.Perm α) #check f * 1 * g⁻¹
- @@ -1001,7 +1001,7 @@ In fact, Lean favors more recent declarations unless you explicitly
 specify a different priority. Also, there is another way to tell Lean that one structure is an instance of another, using the <code class="docutils literal notranslate">extends</code> keyword. - This is how <code class="docutils literal notranslate">mathlib</code> specifies that, for example, + This is how Mathlib specifies that, for example, every commutative ring is a ring. You can find more information in a <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean4/type_classes.html#managing-type-class-inference">section on class inference</a> in Theorem Proving in Lean.
- @@ -1019,7 +1019,7 @@ using the classes <code class="docutils literal notranslate">A
 Then show <code class="docutils literal notranslate">Point</code> is an instance of <code class="docutils literal notranslate">AddGroup₂</code>. Try it out and make sure that the additive group notation works for elements of <code class="docutils literal notranslate">Point</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddGroup₂ (α : Type _) where + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddGroup₂ (α : Type*) where add : α → α → α -- fill in the rest </pre></div>
- @@ -1300,7 +1300,7 @@ is therefore to embed the Gaussian integers in the complex numbers, embed
 the integers in the Gaussian integers, define the rounding function from the real numbers to the integers, and take great care to pass back and forth between these number systems appropriately. - In fact, this is exactly the approach that is followed in mathlib, + In fact, this is exactly the approach that is followed in Mathlib, where the Gaussian integers themselves are constructed as a special case of a ring of quadratic integers. See the file <a class="reference external" href="https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean">GaussianInt.lean</a>.
- @@ -1356,7 +1356,7 @@ from the remainder.
 We will use the fact that \(x^2 + y^2\) is equal to zero if and only if \(x\) and \(y\) are both zero. As an exercise, we ask you to prove that this holds in any ordered ring. - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sq_add_sq_eq_zero {α : Type*} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by sorry </pre></div>
- @@ -1501,7 +1501,7 @@ on a Euclidean domain.
 We can now put it together to show that the Gaussian integers are an instance of a Euclidean domain. We use the quotient and remainder function we have defined. - The mathlib definition of a Euclidean domain is more general than the one + The Mathlib definition of a Euclidean domain is more general than the one above in that it allows us to show that remainder decreases with respect to any well-founded measure. Comparing the values of a norm function that returns natural numbers is
html/C07_Hierarchies.html
- @@ -153,7 +153,7 @@ to avoid this issue would be to use a type annotation, as in:
 in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence">Section 3.6</a> if you tried to state for instance that <code class="docutils literal notranslate">0 < 1</code> without telling Lean whether you meant this inequality to be about natural numbers or real numbers. - Our next task is to assign a notation to <code class="docutils literal notranslate">One₁.one</code>. This we don’t want collisions + Our next task is to assign a notation to <code class="docutils literal notranslate">One₁.one</code>. Since we don’t want collisions with the builtin notation for <code class="docutils literal notranslate">1</code>, we will use <code class="docutils literal notranslate">𝟙</code>. This is achieved by the following command where the first line tells Lean to use the documentation of <code class="docutils literal notranslate">One₁.one</code> as documentation for the symbol <code class="docutils literal notranslate">𝟙</code>.
- @@ -314,7 +314,7 @@ A ring structure on a type contains both an additive group structure and a multi
 monoid structure, and some properties about their interaction. But so far we hard-coded a notation <code class="docutils literal notranslate">⋄</code> for all our operations. More fundamentally, the type class system assumes every type has only one instance of each type class. There are various - ways to solve this issue. Surprisingly mathlib uses the naive idea to duplicate + ways to solve this issue. Surprisingly Mathlib uses the naive idea to duplicate everything for additive and multiplicative theories with the help of some code-generating attribute. Structures and classes are defined in both additive and multiplicative notation with an attribute <code class="docutils literal notranslate">to_additive</code> linking them. In case of multiple inheritance like for
- @@ -527,7 +527,7 @@ to scalar multiplication by an integer by ensuring <code class="docutils literal
 | 0, _ => 0 | n + 1, a => a + nsmul₁ n a - def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + def zsmul₁ {M : Type*} [Zero M] [Add M] [Neg M] : ℤ → M → M | Int.ofNat n, a => nsmul₁ n a | Int.negSucc n, a => -nsmul₁ n.succ a </pre></div>
- @@ -561,7 +561,7 @@ This situation is known as a bad diamond. This has nothing to do with the diamon
 we used above, it refers to the way one can draw the paths from <code class="docutils literal notranslate">ℤ</code> to its <code class="docutils literal notranslate">Module₁ ℤ</code> going through either <code class="docutils literal notranslate">AddCommGroup₃ ℤ</code> or <code class="docutils literal notranslate">Ring₃ ℤ</code>. It is important to understand that not all diamonds are bad. In fact there are diamonds everywhere - in mathlib, and also in this chapter. Already at the very beginning we saw one can go + in Mathlib, and also in this chapter. Already at the very beginning we saw one can go from <code class="docutils literal notranslate">Monoid₁ α</code> to <code class="docutils literal notranslate">Dia₁ α</code> through either <code class="docutils literal notranslate">Semigroup₁ α</code> or <code class="docutils literal notranslate">DiaOneClass₁ α</code> and thanks to the work done by the <code class="docutils literal notranslate">class</code> command, the resulting two <code class="docutils literal notranslate">Dia₁ α</code> instances are definitionally equal. In particular a diamond having a <code class="docutils literal notranslate">Prop</code>-valued class at the bottom
- @@ -621,7 +621,7 @@ the <code class="docutils literal notranslate">•<
 </div> This story then continues with incorporating a <code class="docutils literal notranslate">zsmul</code> field into the definition of groups and similar tricks. You are now ready to read the definition of monoids, groups, rings and modules - in mathlib. There are more complicated than what we have seen here, because they are part of a huge + in Mathlib. There are more complicated than what we have seen here, because they are part of a huge hierarchy, but all principles have been explained above. As an exercise, you can come back to the order relation hierarchy you built above and try to incorporate a type class <code class="docutils literal notranslate">LT₁</code> carrying the Less-Than notation <code class="docutils literal notranslate"><₁</code> and make sure
html/C08_Topology.html
- @@ -114,7 +114,7 @@ We have already begun to consider such notions in <a class="reference internal"
 Topology is the abstract study of limits and continuity. Having covered the essentials of formalization in Chapters <a class="reference internal" href="C02_Basics.html#basics">2</a> to <a class="reference internal" href="C06_Structures.html#structures">6</a>, - in this chapter, we will explain how topological notions are formalized in mathlib. + in this chapter, we will explain how topological notions are formalized in Mathlib. Not only do topological abstractions apply in much greater generality, but that also, somewhat paradoxically, make it easier to reason about limits and continuity in concrete instances.
- @@ -193,7 +193,7 @@ collection of sets <code class="docutils literal notranslate">
 The second condition says that if <code class="docutils literal notranslate">U</code> belongs to <code class="docutils literal notranslate">F.sets</code> then anything containing <code class="docutils literal notranslate">U</code> also belongs to <code class="docutils literal notranslate">F.sets</code>. The third condition says that <code class="docutils literal notranslate">F.sets</code> is closed under finite intersections. - In mathlib, a filter <code class="docutils literal notranslate">F</code> is defined to be a structure bundling <code class="docutils literal notranslate">F.sets</code> and its + In Mathlib, a filter <code class="docutils literal notranslate">F</code> is defined to be a structure bundling <code class="docutils literal notranslate">F.sets</code> and its three properties, but the properties carry no additional data, and it is convenient to blur the distinction between <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">F.sets</code>. We therefore define <code class="docutils literal notranslate">U ∈ F</code> to mean <code class="docutils literal notranslate">U ∈ F.sets</code>.
- @@ -208,9 +208,9 @@ as a generalized element of <code class="docutils literal notranslate">𝓝 x₀</code> is the “set of points very close to <code class="docutils literal notranslate">x₀</code>.” One manifestation of this view is that we can associate to any <code class="docutils literal notranslate">s : Set X</code> the so-called principal filter consisting of all sets that contain <code class="docutils literal notranslate">s</code>. - This definition is already in mathlib and has a notation <code class="docutils literal notranslate">𝓟</code> (localized in the <code class="docutils literal notranslate">Filter</code> namespace). + This definition is already in Mathlib and has a notation <code class="docutils literal notranslate">𝓟</code> (localized in the <code class="docutils literal notranslate">Filter</code> namespace). For the purpose of demonstration, we ask you to take this opportunity to work out the definition here. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def principal {α : Type _} (s : Set α) : Filter α + <div class="highlight-lean notranslate"><div class="highlight"><pre>def principal {α : Type*} (s : Set α) : Filter α where sets := { t | s ⊆ t } univ_sets := sorry
- @@ -230,12 +230,12 @@ For the purpose of demonstration, we ask you to take this opportunity to work ou
 We can also directly define the filter <code class="docutils literal notranslate">𝓝 x</code> of neighborhoods of any <code class="docutils literal notranslate">x : ℝ</code>. In the real numbers, a neighborhood of <code class="docutils literal notranslate">x</code> is a set containing an open interval \((x_0 - \varepsilon, x_0 + \varepsilon)\), - defined in mathlib as <code class="docutils literal notranslate">Ioo (x₀ - ε) (x₀ + ε)</code>. - (This is notion of a neighborhood is only a special case of a more general construction in mathlib.) + defined in Mathlib as <code class="docutils literal notranslate">Ioo (x₀ - ε) (x₀ + ε)</code>. + (This is notion of a neighborhood is only a special case of a more general construction in Mathlib.) With these examples, we can already define what is means for a function <code class="docutils literal notranslate">f : X → Y</code> to converge to some <code class="docutils literal notranslate">G : Filter Y</code> along some <code class="docutils literal notranslate">F : Filter X</code>, as follows: - <div class="highlight-lean notranslate"><div class="highlight"><pre>def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + <div class="highlight-lean notranslate"><div class="highlight"><pre>def Tendsto₁ {X Y : Type*} (f : X → Y) (F : Filter X) (G : Filter Y) := ∀ V ∈ G, f ⁻¹' V ∈ F </pre></div> </div>
- @@ -244,22 +244,22 @@ converges to the real number <code class="docutils literal notranslate">\(\lim_{x \to x₀} f(x) = y₀\). All of the other kinds of limits mentioned in the introduction are also equivalent to instances of <code class="docutils literal notranslate">Tendsto₁</code> for suitable choices of filters on the source and target. - The notion <code class="docutils literal notranslate">Tendsto₁</code> above is definitionally equivalent to the notion <code class="docutils literal notranslate">Tendsto</code> that is defined in mathlib, + The notion <code class="docutils literal notranslate">Tendsto₁</code> above is definitionally equivalent to the notion <code class="docutils literal notranslate">Tendsto</code> that is defined in Mathlib, but the latter is defined more abstractly. The problem with the definition of <code class="docutils literal notranslate">Tendsto₁</code> is that it exposes a quantifier and elements of <code class="docutils literal notranslate">G</code>, and it hides the intuition that we get by viewing filters as generalized sets. We can hide the quantifier <code class="docutils literal notranslate">∀ V</code> and make the intuition more salient by using more algebraic and set-theoretic machinery. The first ingredient is the pushforward operation \(f_*\) associated to any map <code class="docutils literal notranslate">f : X → Y</code>, - denoted <code class="docutils literal notranslate">Filter.map f</code> in mathlib. Given a filter <code class="docutils literal notranslate">F</code> on <code class="docutils literal notranslate">X</code>, <code class="docutils literal notranslate">Filter.map f F : Filter Y</code> is defined so that + denoted <code class="docutils literal notranslate">Filter.map f</code> in Mathlib. Given a filter <code class="docutils literal notranslate">F</code> on <code class="docutils literal notranslate">X</code>, <code class="docutils literal notranslate">Filter.map f F : Filter Y</code> is defined so that <code class="docutils literal notranslate">V ∈ Filter.map f F ↔ f ⁻¹' V ∈ F</code> holds definitionally. In this examples file we’ve opened the <code class="docutils literal notranslate">Filter</code> namespace so that <code class="docutils literal notranslate">Filter.map</code> can be written as <code class="docutils literal notranslate">map</code>. This means that we can rewrite the definition of <code class="docutils literal notranslate">Tendsto</code> using the order relation on <code class="docutils literal notranslate">Filter Y</code>, which is reversed inclusion of the set of members. In other words, given <code class="docutils literal notranslate">G H : Filter Y</code>, we have <code class="docutils literal notranslate">G ≤ H ↔ ∀ V : Set Y, V ∈ H → V ∈ G</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + <div class="highlight-lean notranslate"><div class="highlight"><pre>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) : + example {X Y : Type*} (f : X → Y) (F : Filter X) (G : Filter Y) : Tendsto₂ f F G ↔ Tendsto₁ f F G := Iff.rfl </pre></div>
- @@ -290,7 +290,7 @@ You can practice proving the following statement using either the definition
 of <code class="docutils literal notranslate">Tendsto₁</code> in terms of the universal quantifier or the algebraic definition, together with the two lemmas above. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + <div class="highlight-lean notranslate"><div class="highlight"><pre>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 </pre></div>
- @@ -306,7 +306,7 @@ which is to say, they satisfy
 </div></blockquote> for every <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">G</code>. This operation could be used to provided another formulation of <code class="docutils literal notranslate">Tendsto</code> that would be provably - (but not definitionally) equivalent to the one in mathlib. + (but not definitionally) equivalent to the one in Mathlib. The <code class="docutils literal notranslate">comap</code> operation can be used to restrict filters to a subtype. For instance, suppose we have <code class="docutils literal notranslate">f : ℝ → ℝ</code>, <code class="docutils literal notranslate">x₀ : ℝ</code> and <code class="docutils literal notranslate">y₀ : ℝ</code>, and suppose we want to state that <code class="docutils literal notranslate">f x</code> approaches <code class="docutils literal notranslate">y₀</code> when <code class="docutils literal notranslate">x</code> approaches <code class="docutils literal notranslate">x₀</code> within the rational numbers. We can pull the filter <code class="docutils literal notranslate">𝓝 x₀</code> back to <code class="docutils literal notranslate">ℚ</code> using the coercion map
- @@ -321,7 +321,7 @@ We can pull the filter <code class="docutils literal notranslate">The pullback operation is also compatible with composition, but it is contravariant, which is to say, it reverses the order of the arguments. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} + 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)
- @@ -342,7 +342,7 @@ which is to say, it reverses the order of the arguments.
 Here the <code class="docutils literal notranslate">inf</code> operation refers to the lattice structure on <code class="docutils literal notranslate">Filter X</code> for any type <code class="docutils literal notranslate">X</code>, whereby <code class="docutils literal notranslate">F ⊓ G</code> is the greatest filter that is smaller than both <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">G</code>. Thus the <code class="docutils literal notranslate">inf</code> operation generalizes the notion of the intersection of sets. - A lot of proofs in mathlib use all of the aforementioned structure (<code class="docutils literal notranslate">map</code>, <code class="docutils literal notranslate">comap</code>, <code class="docutils literal notranslate">inf</code>, <code class="docutils literal notranslate">sup</code>, and <code class="docutils literal notranslate">prod</code>) + A lot of proofs in Mathlib use all of the aforementioned structure (<code class="docutils literal notranslate">map</code>, <code class="docutils literal notranslate">comap</code>, <code class="docutils literal notranslate">inf</code>, <code class="docutils literal notranslate">sup</code>, and <code class="docutils literal notranslate">prod</code>) to give algebraic proofs about convergence without ever referring to members of filters. You can practice doing this in a proof of the following lemma, unfolding the definition of <code class="docutils literal notranslate">Tendsto</code> and <code class="docutils literal notranslate">Filter.prod</code> if needed.
- @@ -382,7 +382,7 @@ Indeed, it can happen that <code class="docutils literal notranslate">x₀ : ℝ</code> and <code class="docutils literal notranslate">s : Set ℝ</code>, the pullback of <code class="docutils literal notranslate">𝓝 x₀</code> under the coercion from the subtype corresponding to <code class="docutils literal notranslate">s</code> is nontrivial if and only if <code class="docutils literal notranslate">x₀</code> belongs to the closure of <code class="docutils literal notranslate">s</code>. - In order to manage lemmas that do need to assume some filter is nontrivial, mathlib has + In order to manage lemmas that do need to assume some filter is nontrivial, Mathlib has a type class <code class="docutils literal notranslate">Filter.NeBot</code>, and the library has lemmas that assume <code class="docutils literal notranslate">(F : Filter X) [F.NeBot]</code>. The instance database knows, for example, that <code class="docutils literal notranslate">(atTop : Filter ℕ).NeBot</code>, and it knows that pushing forward a nontrivial filter gives a nontrivial filter.
- @@ -526,7 +526,7 @@ by definition, the assumption <code class="docutils literal notranslate">X</code> equipped with a distance function <code class="docutils literal notranslate">dist : X → X → ℝ</code> which is a generalization of the function <code class="docutils literal notranslate">fun x y ↦ |x - y|</code> from the case where <code class="docutils literal notranslate">X = ℝ</code>. Introducing such a space is easy and we will check all properties required from the distance function. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X : Type _} [MetricSpace X] (a b c : X) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X : Type*} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) #check (dist_nonneg : 0 ≤ dist a b)
- @@ -548,7 +548,7 @@ but we have lemmas recasting the definition is terms of distances.
 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} : + 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
- @@ -559,7 +559,7 @@ is a <code class="docutils literal notranslate">continuity</sp
 in an exercise below. Notice that Lean knows how to treat a product of two metric spaces as a metric space, so it makes sense to consider continuous functions from <code class="docutils literal notranslate">X × X</code> to <code class="docutils literal notranslate">ℝ</code>. In particular the (uncurried version of the) distance function is such a function. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + <div class="highlight-lean notranslate"><div class="highlight"><pre>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 </pre></div> </div>
- @@ -574,7 +574,7 @@ We can do the same for the second component to get continuity of <code class="do
 those two continuities using <code class="docutils literal notranslate">Continuous.prod_mk</code> to get <code class="docutils literal notranslate">(hf.comp continuous_fst).prod_mk (hf.comp continuous_snd) : Continuous (fun p : X × X ↦ (f p.1, f p.2))</code> and compose once more to get our full proof. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + <div class="highlight-lean notranslate"><div class="highlight"><pre>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)) </pre></div>
- @@ -592,13 +592,13 @@ composition and refuse to apply this lemma. It is especially bad at this when pr
 <code class="docutils literal notranslate">Continuous.dist {f g : X → Y} : Continuous f → Continuous g → Continuous (fun x ↦ dist (f x) (g x))</code> which is nicer to Lean’s elaborator and also provides a shorter proof when directly providing a full proof term, as can be seen from the following two new proofs of the above statement: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + <div class="highlight-lean notranslate"><div class="highlight"><pre>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) : + 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) </pre></div>
- @@ -610,7 +610,7 @@ as an alternate proof term <code class="docutils literal notranslate">Continuous.fst'</code> which allows to compress <code class="docutils literal notranslate">hf.comp continuous_fst</code> to <code class="docutils literal notranslate">hf.fst'</code> (and the same with <code class="docutils literal notranslate">snd</code>) and get our final proof, now bordering obfuscation. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + <div class="highlight-lean notranslate"><div class="highlight"><pre>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' </pre></div>
- @@ -622,7 +622,7 @@ and get our final proof, now bordering obfuscation.
 </pre></div> </div> So far we saw continuity as a global notion, but one can also define continuity at a point. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : + <div class="highlight-lean notranslate"><div class="highlight"><pre>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 </pre></div>
- @@ -672,7 +672,7 @@ but we have lemmas recasting the definition is terms of balls.
 sorry </pre></div> </div> - Remember from the filters sections that neighborhood filters play a big role in mathlib. + Remember from the filters sections that neighborhood filters play a big role in Mathlib. In the metric space context, the crucial point is that balls provide bases for those filters. The main lemmas here are <code class="docutils literal notranslate">Metric.nhds_basis_ball</code> and <code class="docutils literal notranslate">Metric.nhds_basis_closedBall</code> that claim this for open and closed balls with positive radius. The center point is an implicit
- @@ -722,7 +722,7 @@ are deduced from more general versions, some of which will be discussed in later
 </pre></div> </div> We can also specify that a metric spaces is globally compact, using an extra <code class="docutils literal notranslate">Prop</code>-valued type class: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X : Type*} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ </pre></div> </div>
- @@ -733,7 +733,7 @@ are deduced from more general versions, some of which will be discussed in later
 We now turn to uniformity notions on metric spaces : uniformly continuous functions, Cauchy sequences and completeness. Again those are defined in a more general context but we have lemmas in the metric name space to access their elementary definitions. We start with uniform continuity. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : + <div class="highlight-lean notranslate"><div class="highlight"><pre>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
- @@ -752,8 +752,8 @@ And <code class="docutils literal notranslate">K</code>
 If <code class="docutils literal notranslate">K</code> is empty then we are clearly done (we can set <code class="docutils literal notranslate">δ = 1</code> for instance). So let’s assume <code class="docutils literal notranslate">K</code> is not empty, and use the extreme value theorem to choose <code class="docutils literal notranslate">(x₀, x₁)</code> attaining the infimum of the distance function on <code class="docutils literal notranslate">K</code>. We can then set <code class="docutils literal notranslate">δ = dist x₀ x₁</code> and check everything works. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X : Type _} [MetricSpace X] [CompactSpace X] - {Y : Type _} [MetricSpace Y] {f : X → Y} + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X : Type*} [MetricSpace X] [CompactSpace X] + {Y : Type*} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry </pre></div>
- @@ -779,7 +779,7 @@ spaces.
 </pre></div> </div> We’ll practice using this definition by proving a convenient criterion which is a special case of a - criterion appearing in mathlib. This is also a good opportunity to practice using big sums in + criterion appearing in Mathlib. This is also a good opportunity to practice using big sums in a geometric context. In addition to the explanations from the filters section, you will probably need <code class="docutils literal notranslate">tendsto_pow_atTop_nhds_0_of_lt_1</code>, <code class="docutils literal notranslate">Tendsto.mul</code> and <code class="docutils literal notranslate">dist_le_range_sum_dist</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem cauchySeq_of_le_geometric_two' {u : ℕ → X}
- @@ -857,14 +857,14 @@ define something inductively in the middle of a proof using <code class="docutil
 <h3>8.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> We now go up in generality and introduce topological spaces. We will review the two main ways to define topological spaces and then explain how the category of topological spaces is much better behaved than - the category of metric spaces. Note that we won’t be using mathlib category theory here, only having + the category of metric spaces. Note that we won’t be using Mathlib category theory here, only having a somewhat categorical point of view. The first way to think about the transition from metric spaces to topological spaces is that we only remember the notion of open sets (or equivalently the notion of closed sets). From this point of view, a topological space is a type equipped with a collection of sets that are called open sets. This collection has to satisfy a number of axioms presented below (this collection is slightly redundant but we will ignore that). <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable {X : Type _} [TopologicalSpace X] + variable {X : Type*} [TopologicalSpace X] example : IsOpen (univ : Set X) := isOpen_univ
- @@ -872,17 +872,17 @@ has to satisfy a number of axioms presented below (this collection is slightly r
 example : IsOpen (∅ : Set X) := isOpen_empty - example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen (s i)) : IsOpen (⋃ i, s i) := + 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)) : + example {ι : Type*} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen (s i)) : IsOpen (⋂ i, s i) := isOpen_iInter hs </pre></div> </div> Closed sets are then defined as sets whose complement is open. A function between topological spaces is (globally) continuous if all preimages of open sets are open. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {Y : Type _} [TopologicalSpace Y] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {Y : Type*} [TopologicalSpace Y] example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := continuous_def
- @@ -893,7 +893,7 @@ enough information to talk about continuous functions: two topological structure
 the same if and only if they have the same continuous functions (indeed the identity function will be continuous in both direction if and only if the two structures have the same open sets). However as soon as we move on to continuity at a point we see the limitations of the approach based - on open sets. In mathlib we frequently think of topological spaces as types equipped + on open sets. In Mathlib we frequently think of topological spaces as types equipped with a neighborhood filter <code class="docutils literal notranslate">𝓝 x</code> attached to each point <code class="docutils literal notranslate">x</code> (the corresponding function <code class="docutils literal notranslate">X → Filter X</code> satisfies certain conditions explained further down). Remember from the filters section that these gadget play two related roles. First <code class="docutils literal notranslate">𝓝 x</code> is seen as the generalized set of points of <code class="docutils literal notranslate">X</code>
- @@ -944,7 +944,7 @@ structure on <code class="docutils literal notranslate">X</spa
 but it will give back its input as a neighborhood function only if it satisfies the above two constraints. More precisely we have a lemma <code class="docutils literal notranslate">TopologicalSpace.nhds_mkOfNhds</code> saying that in a different way and our next exercise deduces this different way from how we stated it above. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + <div class="highlight-lean notranslate"><div class="highlight"><pre>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
- @@ -952,7 +952,7 @@ next exercise deduces this different way from how we stated it above.
 </div> Note that <code class="docutils literal notranslate">TopologicalSpace.mkOfNhds</code> is not so frequently used, but it still good to know in what precise sense the neighborhood filters is all there is in a topological space structure. - The next thing to know in order to efficiently use topological spaces in mathlib is that we use a lot + The next thing to know in order to efficiently use topological spaces in Mathlib is that we use a lot of formal properties of <code class="docutils literal notranslate">TopologicalSpace : Type u → Type u</code>. From a purely mathematical point of view, those formal properties are a very clean way to explain how topological spaces solve issues that metric spaces have. From this point of view, the issues solved by topological spaces is that metric spaces enjoy very
- @@ -969,7 +969,7 @@ sequences of functions is a respectable notion of convergence. But there is no d
 a map <code class="docutils literal notranslate">f : X → (ℝ → ℝ)</code> is continuous if and only <code class="docutils literal notranslate">fun x ↦ f x t</code> is continuous for every <code class="docutils literal notranslate">t : ℝ</code>. We now review the data used to solve all those issues. First we can use any map <code class="docutils literal notranslate">f : X → Y</code> to push or pull topologies from one side to the other. Those two operations form a Galois connection. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X Y : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X Y : Type*} example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := TopologicalSpace.coinduced f
- @@ -989,7 +989,7 @@ On paper we will use notations \(f_*T
 Then the next big piece is a complete lattice structure on <code class="docutils literal notranslate">TopologicalSpace X</code> for any given structure. If you think of topologies are being primarily the data of open sets then you expect the order relation on <code class="docutils literal notranslate">TopologicalSpace X</code> to come from <code class="docutils literal notranslate">Set (Set X)</code>, ie you expect <code class="docutils literal notranslate">t ≤ t'</code> - if a set <code class="docutils literal notranslate">u</code> is open for <code class="docutils literal notranslate">t'</code> as soon as it is open for <code class="docutils literal notranslate">t</code>. However we already know that mathlib focuses + if a set <code class="docutils literal notranslate">u</code> is open for <code class="docutils literal notranslate">t'</code> as soon as it is open for <code class="docutils literal notranslate">t</code>. However we already know that Mathlib focuses on neighborhoods more than open sets so, for any <code class="docutils literal notranslate">x : X</code> we want the map from topological spaces to neighborhoods <code class="docutils literal notranslate">fun T : TopologicalSpace X ↦ @nhds X T x</code> to be order preserving. And we know the order relation on <code class="docutils literal notranslate">Filter X</code> is designed to ensure an order
- @@ -1013,7 +1013,7 @@ a function \(g : Y → Z\)
 \[\begin{split}g \text{ continuous } &⇔ g_*(f_*T_X) ≤ T_Z \\ &⇔ (g ∘ f)_* T_X ≤ T_Z \\ &⇔ g ∘ f \text{ continuous}\end{split}\]</div> - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) + <div class="highlight-lean notranslate"><div class="highlight"><pre>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
- @@ -1034,13 +1034,13 @@ Let us explore that constraint “on paper” using notation So we see that what is the topology we want on <code class="docutils literal notranslate">Π i, X i</code>: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace (X i)) : + <div class="highlight-lean notranslate"><div class="highlight"><pre>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 </pre></div> </div> - This ends our tour of how mathlib thinks that topological spaces fix defects of the theory of metric spaces + This ends our tour of how Mathlib thinks that topological spaces fix defects of the theory of metric spaces by being a more functorial theory and having a complete lattice structure for any fixed type. </section> <section id="separation-and-countability">
- @@ -1074,7 +1074,7 @@ a continuous mapping of \(A\)
 \(f(y)\) tends to a limit in \(Y\) when \(y\) tends to \(x\) while remaining in \(A\) then there exists a continuous extension \(φ\) of \(f\) to \(X\). - Actually <code class="docutils literal notranslate">mathlib</code> contains a more general version of the above lemma, <code class="docutils literal notranslate">DenseInducing.continuousAt_extend</code>, + Actually Mathlib contains a more general version of the above lemma, <code class="docutils literal notranslate">DenseInducing.continuousAt_extend</code>, but we’ll stick to Bourbaki’s version here. Remember that, given <code class="docutils literal notranslate">A : Set X</code>, <code class="docutils literal notranslate">↥A</code> is the subtype associated to <code class="docutils literal notranslate">A</code>, and Lean will automatically insert that funny up arrow when needed. And the (inclusion) coercion map is <code class="docutils literal notranslate">(↑) : A → X</code>.
- @@ -1082,7 +1082,7 @@ The assumption “tends to \(x\)<
 <code class="docutils literal notranslate">comap (↑) (𝓝 x)</code>. Let’s prove first an auxiliary lemma, extracted to simplify the context (in particular we don’t need Y to be a topological space here). - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} + <div class="highlight-lean notranslate"><div class="highlight"><pre>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' :=
- @@ -1135,7 +1135,7 @@ of sets can be understood using sequences.
 <section id="id5"> <h3>8.3.3. Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> Let us now discuss how compactness is defined for topological spaces. As usual there are several ways - to think about it and mathlib goes for the filter version. + to think about it and Mathlib goes for the filter version. We first need to define cluster points of filters. Given a filter <code class="docutils literal notranslate">F</code> on a topological space <code class="docutils literal notranslate">X</code>, a point <code class="docutils literal notranslate">x : X</code> is a cluster point of <code class="docutils literal notranslate">F</code> if <code class="docutils literal notranslate">F</code>, seen as a generalized set, has non-empty intersection with the generalized set of points that are close to <code class="docutils literal notranslate">x</code>.
- @@ -1184,7 +1184,7 @@ compact. In addition to what we saw already, you should use <code class="docutil
 </div> One can also express compactness in terms of open covers: <code class="docutils literal notranslate">s</code> is compact if every family of open sets that cover <code class="docutils literal notranslate">s</code> has a finite covering sub-family. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) + <div class="highlight-lean notranslate"><div class="highlight"><pre>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 </pre></div>
html/C09_Differential_Calculus.html
- @@ -106,7 +106,7 @@ a much broader setting.
 Let <code class="docutils literal notranslate">f</code> be a function from the reals to the reals. There is a difference between talking about the derivative of <code class="docutils literal notranslate">f</code> at a single point and talking about the derivative function. - In mathlib, the first notion is represented as follows. + In Mathlib, the first notion is represented as follows. <div class="highlight-lean notranslate"><div class="highlight"><pre>open Real /-- The sin function has derivative 1 at 0. -/
- @@ -126,7 +126,7 @@ and being differentiable in the sense of the complex derivative.
 </div> It would be inconvenient to have to provide a proof of differentiability every time we want to refer to a derivative. - So mathlib provides a function <code class="docutils literal notranslate">deriv f : ℝ → ℝ</code> that is defined for any + So Mathlib provides a function <code class="docutils literal notranslate">deriv f : ℝ → ℝ</code> that is defined for any function <code class="docutils literal notranslate">f : ℝ → ℝ</code> but is defined to take the value <code class="docutils literal notranslate">0</code> at any point where <code class="docutils literal notranslate">f</code> is not differentiable. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} {x a : ℝ} (h : HasDerivAt f a x) : deriv f x = a :=
- @@ -185,7 +185,7 @@ seems even weirder.
 We start with the notion of a normed group, which as an additive commutative group equipped with a real-valued norm function satisfying the following conditions. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {E : Type _} [NormedAddCommGroup E] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {E : Type*} [NormedAddCommGroup E] example (x : E) : 0 ≤ ‖x‖ := norm_nonneg x
- @@ -199,10 +199,10 @@ satisfying the following conditions.
 </div> Every normed space is a metric space with distance function \(d(x, y) = \| x - y \|\), and hence it is also a topological space. - Lean and mathlib know this. + Lean and Mathlib know this. <div class="highlight-lean notranslate"><div class="highlight"><pre>example : MetricSpace E := by infer_instance - example {X : Type _} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : + example {X : Type*} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : Continuous fun x ↦ ‖f x‖ := hf.norm </pre></div>
- @@ -228,16 +228,16 @@ More generally, we can make sense of calculus with a vector space over any
 real-valued norm that is multiplicative and has the property that not every element has norm zero or one (equivalently, there is an element whose norm is bigger than one). - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (𝕜 : Type*) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := norm_mul x y - example (𝕜 : Type _) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := + example (𝕜 : Type*) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := NormedField.exists_one_lt_norm 𝕜 </pre></div> </div> A finite-dimensional vector space over a nontrivially normed field is complete as long as the field itself is complete. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (E : Type _) [NormedAddCommGroup E] + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*) [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] : CompleteSpace E := FiniteDimensional.complete 𝕜 E </pre></div>
- @@ -247,15 +247,15 @@ complete as long as the field itself is complete.
 <h3>9.2.2. Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this heading"></a></h3> We now turn to the morphisms in the category of normed spaces, namely, continuous linear maps. - In mathlib, the type of <code class="docutils literal notranslate">𝕜</code>-linear continuous maps between normed spaces + In Mathlib, the type of <code class="docutils literal notranslate">𝕜</code>-linear continuous maps between normed spaces <code class="docutils literal notranslate">E</code> and <code class="docutils literal notranslate">F</code> is written <code class="docutils literal notranslate">E →L[𝕜] F</code>. They are implemented as bundled maps, which means that an element of this type a structure that that includes the function itself and the properties of being linear and continuous. Lean will insert a coercion so that a continuous linear map can be treated as a function. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] - [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] example : E →L[𝕜] E := ContinuousLinearMap.id 𝕜 E
- @@ -295,12 +295,12 @@ The main ingredient is Baire’s theorem
 <code class="docutils literal notranslate">nonempty_interior_of_iUnion_of_closed</code>. (You proved a version of this in the topology chapter.) Minor ingredients include <code class="docutils literal notranslate">continuous_linear_map.op_norm_le_of_shell</code>, <code class="docutils literal notranslate">interior_subset</code> and <code class="docutils literal notranslate">interior_iInter_subset</code> and <code class="docutils literal notranslate">is_closed_le</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] - [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] open Metric - example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + example {ι : Type*} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C' := by -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` let e : ℕ → Set E := fun n ↦ ⋂ i : ι, { x : E | ‖g i x‖ ≤ n }
- @@ -335,19 +335,19 @@ notation.
 Here we will only use little o to define differentiability. <div class="highlight-lean notranslate"><div class="highlight"><pre>open Asymptotics - example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) + example {α : Type*} {E : Type*} [NormedGroup E] {F : Type*} [NormedGroup F] (c : ℝ) (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := isBigOWith_iff - example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + example {α : Type*} {E : Type*} [NormedGroup E] {F : Type*} [NormedGroup F] (l : Filter α) (f : α → E) (g : α → F) : f =O[l] g ↔ ∃ C, IsBigOWith C l f g := isBigO_iff_isBigOWith - example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + example {α : Type*} {E : Type*} [NormedGroup E] {F : Type*} [NormedGroup F] (l : Filter α) (f : α → E) (g : α → F) : f =o[l] g ↔ ∀ C > 0, IsBigOWith C l f g := isLittleO_iff_forall_isBigOWith - example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : + example {α : Type*} {E : Type*} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : f ~[l] g ↔ (f - g) =o[l] g := Iff.rfl </pre></div>
- @@ -357,13 +357,13 @@ Here we will only use little o to define differentiability.
 <h3>9.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> We are now ready to discuss differentiable functions between normed spaces. In analogy the elementary one-dimensional, - mathlib defines a predicate <code class="docutils literal notranslate">HasFDerivAt</code> and a function <code class="docutils literal notranslate">fderiv</code>. + Mathlib defines a predicate <code class="docutils literal notranslate">HasFDerivAt</code> and a function <code class="docutils literal notranslate">fderiv</code>. Here the letter “f” stands for Fréchet. <div class="highlight-lean notranslate"><div class="highlight"><pre>open Topology - variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] - [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : HasFDerivAt f f' x₀ ↔ (fun x ↦ f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x ↦ x - x₀ :=
- @@ -393,10 +393,10 @@ So \(\mathcal{C}^\infty\) func
 There is a stricter notion of differentiability called <code class="docutils literal notranslate">HasStrictFDerivAt</code>, which is used in the statement of the inverse function theorem and the statement of the implicit function - theorem, both of which are in mathlib. + theorem, both of which are in Mathlib. Over <code class="docutils literal notranslate">ℝ</code> or <code class="docutils literal notranslate">ℂ</code>, continuously differentiable functions are strictly differentiable. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {𝕂 : Type _} [IsROrC 𝕂] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {𝕂 : Type*} [IsROrC 𝕂] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := hf.hasStrictFDerivAt hn
- @@ -431,10 +431,10 @@ from the left and from the right, and that it is strictly differentiable.
 end LocalInverse </pre></div> </div> - This has been only a quick tour of the differential calculus in mathlib. + This has been only a quick tour of the differential calculus in Mathlib. The library contains many variations that we have not discussed. For example, you may want to use one-sided derivatives in the - one-dimensional setting. The means to do so are found in mathlib in a more + one-dimensional setting. The means to do so are found in Mathlib in a more general context; see <code class="docutils literal notranslate">HasFDerivWithinAt</code> or the even more general <code class="docutils literal notranslate">HasFDerivAtFilter</code>. </section>
html/C10_Integration_and_Measure_Theory.html
- @@ -119,7 +119,7 @@ which are not shown here, are not equivalent.)
 integral_eq_sub_of_hasDerivAt h h' </pre></div> </div> - Convolution is also defined in mathlib and its basic properties are proved. + Convolution is also defined in Mathlib and its basic properties are proved. <div class="highlight-lean notranslate"><div class="highlight"><pre>open Convolution example (f : ℝ → ℝ) (g : ℝ → ℝ) : f ⋆ g = fun x ↦ ∫ t, f t * g (x - t) :=
- @@ -129,7 +129,7 @@ which are not shown here, are not equivalent.)
 </section> <section id="measure-theory"> <h2>10.2. Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this heading"></a></h2> - The general context for integration in mathlib is measure theory. Even the elementary + The general context for integration in Mathlib is measure theory. Even the elementary integrals of the previous section are in fact Bochner integrals. Bochner integration is a generalization of Lebesgue integration where the target space can be any Banach space, not necessarily finite dimensional.
- @@ -141,10 +141,10 @@ The sets <code class="docutils literal notranslate">empty</spa
 the complement of a measurable set is measurable, and a countable union or intersection of measurable sets is measurable. Note that these axioms are redundant; if you <code class="docutils literal notranslate">#print MeasurableSpace</code>, - you will see the ones that mathlib uses. + you will see the ones that Mathlib uses. As the examples below show, countability assumptions can be expressed using the <code class="docutils literal notranslate">Encodable</code> type class. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [MeasurableSpace α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [MeasurableSpace α] example : MeasurableSet (∅ : Set α) := MeasurableSet.empty
- @@ -159,7 +159,7 @@ As the examples below show, countability assumptions can be expressed using the
 example (n : ℕ) : Encodable (Fin n) := by infer_instance - variable {ι : Type _} [Encodable ι] + variable {ι : Type*} [Encodable ι] example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := MeasurableSet.iUnion h
- @@ -173,7 +173,7 @@ As the examples below show, countability assumptions can be expressed using the
 \(\sigma\)-algebra is a function from the measurable sets to the extended non-negative reals that is additive on countable disjoint unions. - In mathlib, we don’t want to carry around measurability assumptions + In Mathlib, we don’t want to carry around measurability assumptions every time we write an application of the measure to a set. So we extend the measure to any set <code class="docutils literal notranslate">s</code> as the infimum of measures of measurable sets containing <code class="docutils literal notranslate">s</code>.
- @@ -197,7 +197,7 @@ measurability assumptions, but not all.
 holds almost everywhere if the set of elements where the property fails has measure 0. The collection of properties that hold almost everywhere form a filter, - but mathlib introduces special notation for saying that a property holds + but Mathlib introduces special notation for saying that a property holds almost everywhere. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {P : α → Prop} : (∀ᵐ x ∂μ, P x) ↔ ∀ᶠ x in μ.ae, P x := Iff.rfl
- @@ -207,7 +207,7 @@ almost everywhere.
 <section id="integration"> <h2>10.3. Integration<a class="headerlink" href="#integration" title="Permalink to this heading"></a></h2> Now that we have measurable spaces and measures we can consider integrals. - As explained above, mathlib uses a very general notion of + As explained above, Mathlib uses a very general notion of integration that allows any Banach space as the target. As usual, we don’t want our notation to carry around assumptions, so we define integration in such a way
- @@ -215,7 +215,7 @@ that an integral is equal to zero if the function in question is
 not integrable. Most lemmas having to do with integrals have integrability assumptions. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} + variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) : ∫ a, f a + g a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ :=
- @@ -234,7 +234,7 @@ So in all cases we have the following lemma.
 </pre></div> </div> We now quickly explain how to access the most important theorems in integration theory, starting - with the dominated convergence theorem. There are several versions in mathlib, + with the dominated convergence theorem. There are several versions in Mathlib, and here we only show the most basic one. <div class="highlight-lean notranslate"><div class="highlight"><pre>open Filter
- @@ -246,7 +246,7 @@ and here we only show the most basic one.
 </pre></div> </div> Then we have Fubini’s theorem for integrals on product type. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type _} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type*} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type*} [MeasurableSpace β] {ν : Measure β} [SigmaFinite ν] (f : α × β → E) (hf : Integrable f (μ.prod ν)) : ∫ z, f z ∂ μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ := integral_prod f hf
- @@ -256,7 +256,7 @@ and here we only show the most basic one.
 continuous bilinear form. <div class="highlight-lean notranslate"><div class="highlight"><pre>open Convolution - variable {𝕜 : Type _} {G : Type _} {E : Type _} {E' : Type _} {F : Type _} [NormedAddCommGroup E] + variable {𝕜 : Type*} {G : Type*} {E : Type*} {E' : Type*} {F : Type*} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedSpace ℝ F] [CompleteSpace F] [Sub G]
- @@ -266,13 +266,13 @@ continuous bilinear form.
 rfl </pre></div> </div> - Finally, mathlib has a very general version of the change-of-variables formula. + Finally, Mathlib has a very general version of the change-of-variables formula. In the statement below, <code class="docutils literal notranslate">BorelSpace E</code> means the \(\sigma\)-algebra on <code class="docutils literal notranslate">E</code> is generated by the open sets of <code class="docutils literal notranslate">E</code>, and <code class="docutils literal notranslate">IsAddHaarMeasure μ</code> means that the measure <code class="docutils literal notranslate">μ</code> is left-invariant, gives finite mass to compact sets, and give positive mass to open sets. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] - [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type _} + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] + [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} (hs : MeasurableSet s) (hf : ∀ x : E, x ∈ s → HasFDerivWithinAt f (f' x) s x) (h_inj : InjOn f s) (g : E → F) :
html/_sources/C04_Sets_and_Functions.rst.txt
- @@ -17,7 +17,7 @@ groups, vector spaces, and so on.
  Some expressions *are* types, which is to say, their type is ``Type``. - Lean and mathlib provide ways of defining new types, + Lean and Mathlib provide ways of defining new types, and ways of defining objects of those types. Conceptually, you can think of a type as just a set of objects.
html/_sources/C08_Topology.rst.txt
- @@ -20,7 +20,7 @@ We have already begun to consider such notions in :numref:`sequences_and_converg
 *Topology* is the abstract study of limits and continuity. Having covered the essentials of formalization in Chapters :numref:`%s <basics>` to :numref:`%s <structures>`, - in this chapter, we will explain how topological notions are formalized in mathlib. + in this chapter, we will explain how topological notions are formalized in Mathlib. Not only do topological abstractions apply in much greater generality, but that also, somewhat paradoxically, make it easier to reason about limits and continuity in concrete instances.
html/searchindex.js
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lake-manifest.json
- @@ -4,9 +4,9 @@
  [{"git": {"url": "https://github.com/EdAyers/ProofWidgets4", "subDir?": null, - "rev": "c43db94a8f495dad37829e9d7ad65483d68c86b8", + "rev": "a0c2cd0ac3245a0dade4f925bcfa97e06dd84229", "name": "proofwidgets", - "inputRev?": "v0.0.11"}}, + "inputRev?": "v0.0.13"}}, {"git": {"url": "https://github.com/mhuisi/lean4-cli.git", "subDir?": null,
- @@ -16,24 +16,24 @@
  {"git": {"url": "https://github.com/leanprover-community/mathlib4", "subDir?": null, - "rev": "758b7d6b19adfc8a0d12be3eafb48507e2e859e5", + "rev": "d567f7cfc07d1a56eabb808ca3be77113520505b", "name": "mathlib", "inputRev?": "master"}}, {"git": {"url": "https://github.com/gebner/quote4", "subDir?": null, - "rev": "ae84bd82cca324dc958583d6f1ae08429877dcb0", + "rev": "81cc13c524a68d0072561dbac276cd61b65872a6", "name": "Qq", "inputRev?": "master"}}, {"git": {"url": "https://github.com/JLimperg/aesop", "subDir?": null, - "rev": "f04538ab6ad07642368cf11d2702acc0a9b4bcee", + "rev": "d13a9666e6f430b940ef8d092f1219e964b52a09", "name": "aesop", "inputRev?": "master"}}, {"git": {"url": "https://github.com/leanprover/std4", "subDir?": null, - "rev": "dff883c55395438ae2a5c65ad5ddba084b600feb", + "rev": "dbffa8cb31b0c51b151453c4ff8f00ede2a84ed8", "name": "std", "inputRev?": "main"}}]}
lean-toolchain
- @@ -1,1 +1,1 @@
  - leanprover/lean4:nightly-2023-07-12 + leanprover/lean4:nightly-2023-08-05
mathematics_in_lean.pdf