dc6139bd - mathematics_in

Authored by: Patrick Massot
Authored at: 2023-12-04 09:48:18 -0500
Committed by: Patrick Massot
Committed at: 2023-12-04 09:48:18 -0500
Object name: dc6139bdc9bbea02a5482108fde1e0e3e0f01ca5
Parent: ff0d782b

Update lun. 04 déc. 2023 09:48:18 EST

Changes

32 changed files (+1265/-1345)

MIL/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean
- @@ -33,10 +33,10 @@ example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x ↦ f x + g x := by
  exact add_le_add (lbfa x) (lbgb x) example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x ↦ c * f x := by - rcases ubf with ⟨a, lbfa⟩ + rcases ubf with ⟨a, ubfa⟩ use c * a intro x - exact mul_le_mul_of_nonneg_left (lbfa x) h + exact mul_le_mul_of_nonneg_left (ubfa x) h end
MIL/C04_Sets_and_Functions/S02_Functions.lean
- @@ -81,39 +81,19 @@ example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by
  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 - rintro ⟨i, x, xAi, fxeq⟩ - exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + sorry example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by - intro y; simp - intro x h fxeq i - use x - exact ⟨h i, fxeq⟩ + sorry example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by - intro y; simp - intro h - rcases h i with ⟨x, xAi, fxeq⟩ - use x; constructor - · intro i' - rcases h i' with ⟨x', x'Ai, fx'eq⟩ - have : f x = f x' := by rw [fxeq, fx'eq] - have : x = x' := injf this - rw [this] - exact x'Ai - exact fxeq + sorry example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by - ext x - simp + sorry example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by - ext x - simp + sorry example : InjOn f s ↔ ∀ x₁ ∈ s, ∀ x₂ ∈ s, f x₁ = f x₂ → x₁ = x₂ := Iff.refl _
MIL/C08_Groups_and_Rings/S01_Groups.lean
- @@ -5,53 +5,53 @@ import Mathlib.GroupTheory.PresentedGroup
  import MIL.Common - example {M : Type*} [Monoid M] (x : M) : x*1 = x := mul_one x + example {M : Type*} [Monoid M] (x : M) : x * 1 = x := mul_one x example {M : Type*} [AddCommMonoid M] (x y : M) : x + y = y + x := add_comm x y example {M N : Type*} [Monoid M] [Monoid N] (x y : M) (f : M →* N) : f (x * y) = f x * f y := - f.map_mul x y + f.map_mul x y example {M N : Type*} [AddMonoid M] [AddMonoid N] (f : M →+ N) : f 0 = 0 := - f.map_zero + f.map_zero example {M N P : Type*} [AddMonoid M] [AddMonoid N] [AddMonoid P] - (f : M →+ N) (g : N →+ P) : M →+ P := g.comp f + (f : M →+ N) (g : N →+ P) : M →+ P := g.comp f example {G : Type*} [Group G] (x : G) : x * x⁻¹ = 1 := mul_inv_self x - example {G : Type*} [Group G] (x y z : G) : x * (y * z) * (x*z)⁻¹ * (x * y * x⁻¹)⁻¹ = 1 := by + example {G : Type*} [Group G] (x y z : G) : x * (y * z) * (x * z)⁻¹ * (x * y * x⁻¹)⁻¹ = 1 := by group example {G : Type*} [AddCommGroup G] (x y z : G) : z + x + (y - z - x) = y := by abel example {G H : Type*} [Group G] [Group H] (x y : G) (f : G →* H) : f (x * y) = f x * f y := - f.map_mul x y + f.map_mul x y example {G H : Type*} [Group G] [Group H] (x : G) (f : G →* H) : f (x⁻¹) = (f x)⁻¹ := - f.map_inv x + f.map_inv x example {G H : Type*} [Group G] [Group H] (f : G → H) (h : ∀ x y, f (x * y) = f x * f y) : - G →* H := - MonoidHom.mk' f h + G →* H := + MonoidHom.mk' f h example {G H : Type*} [Group G] [Group H] (f : G ≃* H) : - f.trans f.symm = MulEquiv.refl G := - f.self_trans_symm + f.trans f.symm = MulEquiv.refl G := + f.self_trans_symm noncomputable example {G H : Type*} [Group G] [Group H] - (f : G →* H) (h : Function.Bijective f) : + (f : G →* H) (h : Function.Bijective f) : G ≃* H := MulEquiv.ofBijective f h example {G : Type*} [Group G] (H : Subgroup G) {x y : G} (hx : x ∈ H) (hy : y ∈ H) : - x * y ∈ H := - H.mul_mem hx hy + x * y ∈ H := + H.mul_mem hx hy example {G : Type*} [Group G] (H : Subgroup G) {x : G} (hx : x ∈ H) : - x⁻¹ ∈ H := - H.inv_mem hx + x⁻¹ ∈ H := + H.inv_mem hx example : AddSubgroup ℚ where carrier := Set.range ((↑) : ℤ → ℚ)
- @@ -72,7 +72,7 @@ example {G : Type*} [Group G] (H : Subgroup G) : Group H := inferInstance
  example {G : Type*} [Group G] (H : Subgroup G) : Group {x : G // x ∈ H} := inferInstance example {G : Type*} [Group G] (H H' : Subgroup G) : - ((H ⊓ H' : Subgroup G) : Set G) = (H : Set G) ∩ (H' : Set G) := rfl + ((H ⊓ H' : Subgroup G) : Set G) = (H : Set G) ∩ (H' : Set G) := rfl example {G : Type*} [Group G] (H H' : Subgroup G) : ((H ⊔ H' : Subgroup G) : Set G) = Subgroup.closure ((H : Set G) ∪ (H' : Set G)) := by
- @@ -116,28 +116,28 @@ variable {G H : Type*} [Group G] [Group H]
  open Subgroup - example (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T :=by + example (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T := by sorry - example (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T :=by + example (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T := by sorry variable {K : Type*} [Group K] -- Remember you can use the `ext` tactic to prove an equality of subgroups. example (φ : G →* H) (ψ : H →* K) (U : Subgroup K) : - comap (ψ.comp φ) U = comap φ (comap ψ U) := by + comap (ψ.comp φ) U = comap φ (comap ψ U) := by sorry -- Pushing a subgroup along one homomorphism and then another is equal to - -- pushing it forward along the composite of the homomorphisms. + -- pushing it forward along the composite of the homomorphisms. example (φ : G →* H) (ψ : H →* K) (S : Subgroup G) : - map (ψ.comp φ) S = map ψ (S.map φ) := by + map (ψ.comp φ) S = map ψ (S.map φ) := by sorry end exercises - attribute [local instance 10] setFintype Classical.propDecidable + open scoped Classical open Fintype
- @@ -159,12 +159,12 @@ lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} [Fintype H] :
  #check card_dvd_of_le lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) [Fintype H] [Fintype K] - (h : (card H).Coprime (card K)) : H ⊓ K = ⊥ := by - sorry + (h : (card H).Coprime (card K)) : H ⊓ K = ⊥ := by + sorry open Equiv example {X : Type*} [Finite X] : Subgroup.closure {σ : Perm X | Perm.IsCycle σ} = ⊤ := - Perm.closure_isCycle + Perm.closure_isCycle #simp [mul_assoc] c[1, 2, 3] * c[2, 3, 4]
- @@ -199,20 +199,20 @@ end FreeGroup
  noncomputable section GroupActions example {G X : Type*} [Group G] [MulAction G X] (g g': G) (x : X) : - g • (g' • x) = (g * g') • x := - (mul_smul g g' x).symm + g • (g' • x) = (g * g') • x := + (mul_smul g g' x).symm example {G X : Type*} [AddGroup G] [AddAction G X] (g g' : G) (x : X) : - g +ᵥ (g' +ᵥ x) = (g + g') +ᵥ x := - (add_vadd g g' x).symm + g +ᵥ (g' +ᵥ x) = (g + g') +ᵥ x := + (add_vadd g g' x).symm open MulAction example {G X : Type*} [Group G] [MulAction G X] : G →* Equiv.Perm X := - toPermHom G X + toPermHom G X def CayleyIsoMorphism (G : Type*) [Group G] : G ≃* (toPermHom G G).range := - Equiv.Perm.subgroupOfMulAction G G + Equiv.Perm.subgroupOfMulAction G G example {G X : Type*} [Group G] [MulAction G X] : X ≃ (ω : orbitRel.Quotient G X) × (orbit G (Quotient.out' ω)) :=
- @@ -228,7 +228,7 @@ example {G : Type*} [Group G] (H : Subgroup G) : G ≃ (G ⧸ H) × H :=
  variable {G : Type*} [Group G] lemma conjugate_one (H : Subgroup G) : conjugate 1 H = H := by - sorry + sorry instance : MulAction G (Subgroup G) where smul := conjugate
- @@ -244,11 +244,11 @@ noncomputable section QuotientGroup
  example {G : Type*} [Group G] (H : Subgroup G) [H.Normal] : Group (G ⧸ H) := inferInstance example {G : Type*} [Group G] (H : Subgroup G) [H.Normal] : G →* G ⧸ H := - QuotientGroup.mk' H + QuotientGroup.mk' H example {G : Type*} [Group G] (N : Subgroup G) [N.Normal] {M : Type*} - [Group M] (φ : G →* M) (h : N ≤ MonoidHom.ker φ) : G ⧸ N →* M := - QuotientGroup.lift N φ h + [Group M] (φ : G →* M) (h : N ≤ MonoidHom.ker φ) : G ⧸ N →* M := + QuotientGroup.lift N φ h example {G : Type*} [Group G] {M : Type*} [Group M] (φ : G →* M) : G ⧸ MonoidHom.ker φ →* MonoidHom.range φ :=
- @@ -260,10 +260,10 @@ example {G G': Type*} [Group G] [Group G']
  QuotientGroup.map N N' φ h example {G : Type*} [Group G] {M N : Subgroup G} [M.Normal] - [N.Normal] (h : M = N) : G ⧸ M ≃* G ⧸ N := QuotientGroup.quotientMulEquivOfEq h + [N.Normal] (h : M = N) : G ⧸ M ≃* G ⧸ N := QuotientGroup.quotientMulEquivOfEq h section - variable {G : Type*} [Group G] {H K : Subgroup G} + variable {G : Type*} [Group G] {H K : Subgroup G} open MonoidHom
- @@ -272,7 +272,7 @@ open MonoidHom
  #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right - lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G⧸H) = card K := by + lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G ⧸ H) = card K := by sorry variable [H.Normal] [K.Normal] [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K)
- @@ -281,9 +281,9 @@ variable [H.Normal] [K.Normal] [Fintype G] (h : Disjoint H K) (h' : card G = car
  #check restrict #check ker_restrict - def iso₁ [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K) : K ≃* G⧸H := by + def iso₁ [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K) : K ≃* G ⧸ H := by sorry - def iso₂ : G ≃* (G⧸K) × (G⧸H) := by + def iso₂ : G ≃* (G ⧸ K) × (G ⧸ H) := by sorry #check MulEquiv.prodCongr
MIL/C08_Groups_and_Rings/S02_Rings.lean
- @@ -7,13 +7,13 @@ import MIL.Common
  noncomputable section - example {R : Type*} [CommRing R] (x y : R) : (x + y)^2 = x^2 + y^2 + 2*x*y := by ring + example {R : Type*} [CommRing R] (x y : R) : (x + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y := by ring - example (x y : ℕ) : (x + y)^2 = x^2 + y^2 + 2*x*y := by ring + example (x y : ℕ) : (x + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y := by ring example (x : ℤˣ) : x = 1 ∨ x = -1 := Int.units_eq_one_or x - example {M : Type*} [Monoid M] (x : Mˣ) : (x : M)*x⁻¹ = 1 := Units.mul_inv x + example {M : Type*} [Monoid M] (x : Mˣ) : (x : M) * x⁻¹ = 1 := Units.mul_inv x example {M : Type*} [Monoid M] : Group Mˣ := inferInstance
- @@ -25,7 +25,7 @@ example {R S : Type*} [Ring R] [Ring S] (f : R →+* S) : Rˣ →* Sˣ :=
  example {R : Type*} [Ring R] (S : Subring R) : Ring S := inferInstance - example {R : Type*} [CommRing R] (I : Ideal R) : R →+* R⧸I := + example {R : Type*} [CommRing R] (I : Ideal R) : R →+* R ⧸ I := Ideal.Quotient.mk I example {R : Type*} [CommRing R] {a : R} {I : Ideal R} :
- @@ -118,10 +118,10 @@ theorem isCoprime_Inf {I : Ideal R} {J : ι → Ideal R} {s : Finset ι}
  rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc - 1 = I + K := sorry - _ = I + K*(I + J i) := sorry - _ = (1+K)*I + K*J i := sorry - _ ≤ I + K ⊓ J i := sorry + 1 = I + K := sorry + _ = I + K * (I + J i) := sorry + _ = (1 + K) * I + K * J i := sorry + _ ≤ I + K ⊓ J i := sorry lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R} (hI : ∀ i j, i ≠ j → IsCoprime (I i) (I j)) : Surjective (chineseMap I) := by classical
- @@ -133,11 +133,11 @@ lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R}
  sorry sorry choose e he using key - use mk _ (∑ i, f i*e i) + use mk _ (∑ i, f i * e i) sorry noncomputable def chineseIso [Fintype ι] (f : ι → Ideal R) - (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* ∀ i, R ⧸ f i := + (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i := { Equiv.ofBijective _ ⟨chineseMap_inj f, chineseMap_surj hf⟩, chineseMap f with }
- @@ -158,13 +158,13 @@ example {R : Type*} [CommRing R] : R[X] := X
  example {R : Type*} [CommRing R] (r : R) := X - C r - example {R : Type*} [CommRing R] (r : R) : (X + C r) * (X - C r) = X^2 - C (r ^ 2) := by + example {R : Type*} [CommRing R] (r : R) : (X + C r) * (X - C r) = X ^ 2 - C (r ^ 2) := by rw [C.map_pow] ring - example {R : Type*} [CommRing R](r:R) : (C r).coeff 0 = r := by simp + example {R : Type*} [CommRing R] (r:R) : (C r).coeff 0 = r := by simp - example {R : Type*} [CommRing R] : (X^2 + 2*X + C 3 : R[X]).coeff 1 = 2 := by simp + example {R : Type*} [CommRing R] : (X ^ 2 + 2 * X + C 3 : R[X]).coeff 1 = 2 := by simp example {R : Type*} [Semiring R] [NoZeroDivisors R] {p q : R[X]} : degree (p * q) = degree p + degree q :=
- @@ -188,14 +188,14 @@ example {R : Type*} [CommRing R] [IsDomain R] (r : R) : (X - C r).roots = {r} :=
  roots_X_sub_C r example {R : Type*} [CommRing R] [IsDomain R] (r : R) (n : ℕ): - ((X - C r)^n).roots = n • {r} := + ((X - C r) ^ n).roots = n • {r} := by simp - example : aeval Complex.I (X^2 + 1 : ℝ[X]) = 0 := by simp + example : aeval Complex.I (X ^ 2 + 1 : ℝ[X]) = 0 := by simp open Complex Polynomial - example : aroots (X^2 + 1 : ℝ[X]) ℂ = {Complex.I, -I} := by + example : aroots (X ^ 2 + 1 : ℝ[X]) ℂ = {Complex.I, -I} := by suffices roots (X ^ 2 + 1 : ℂ[X]) = {I, -I} by simpa [aroots_def] have factored : (X ^ 2 + 1 : ℂ[X]) = (X - C I) * (X - C (-I)) := by rw [C_neg]
- @@ -212,7 +212,7 @@ example : IsAlgClosed ℂ := inferInstance
  #check (Complex.ofReal : ℝ →+* ℂ) - example : (X^2 + 1 : ℝ[X]).eval₂ Complex.ofReal Complex.I = 0 := by simp + example : (X ^ 2 + 1 : ℝ[X]).eval₂ Complex.ofReal Complex.I = 0 := by simp open MvPolynomial
MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean
- @@ -29,12 +29,12 @@ variable {G H : Type*} [Group G] [Group H]
 open Subgroup - example (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T :=by + example (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T := by intro x hx rw [mem_comap] at * -- Lean does not need this line exact hST hx - example (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T :=by + example (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T := by intro x hx rw [mem_map] at * -- Lean does not need this line rcases hx with ⟨y, hy, rfl⟩
- @@ -44,16 +44,16 @@ variable {K : Type*} [Group K]
 -- Remember you can use the `ext` tactic to prove an equality of subgroups. example (φ : G →* H) (ψ : H →* K) (U : Subgroup K) : - comap (ψ.comp φ) U = comap φ (comap ψ U) := by + comap (ψ.comp φ) U = comap φ (comap ψ U) := by -- The whole proof could be ``rfl``, but let's decompose it a bit. ext x simp only [mem_comap] rfl -- Pushing a subgroup along one homomorphism and then another is equal to - -- pushing it forward along the composite of the homomorphisms. + -- pushing it forward along the composite of the homomorphisms. example (φ : G →* H) (ψ : H →* K) (S : Subgroup G) : - map (ψ.comp φ) S = map ψ (S.map φ) := by + map (ψ.comp φ) S = map ψ (S.map φ) := by ext x simp only [mem_map] constructor
- @@ -67,7 +67,7 @@ example (φ : G →* H) (ψ : H →* K) (S : Subgroup G) :
 end exercises - attribute [local instance 10] setFintype Classical.propDecidable + open scoped Classical open Fintype open Subgroup
- @@ -84,7 +84,7 @@ lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} [Fintype H] :
 _ = 1 := (hy' 1 H.one_mem).symm lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) [Fintype H] [Fintype K] - (h : (card H).Coprime (card K)) : H ⊓ K = ⊥ := by + (h : (card H).Coprime (card K)) : H ⊓ K = ⊥ := by have D₁ : card (H ⊓ K : Subgroup G) ∣ card H := card_dvd_of_le inf_le_left have D₂ : card (H ⊓ K : Subgroup G) ∣ card K := card_dvd_of_le inf_le_right exact eq_bot_iff_card.2 (Nat.eq_one_of_dvd_coprimes h D₁ D₂)
- @@ -119,7 +119,7 @@ end GroupActions
 noncomputable section QuotientGroup section - variable {G : Type*} [Group G] {H K : Subgroup G} + variable {G : Type*} [Group G] {H K : Subgroup G} open MonoidHom
- @@ -128,7 +128,7 @@ open MonoidHom
 #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right - lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G⧸H) = card K := by + lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G ⧸ H) = card K := by have := calc card (G ⧸ H) * card H = card G := by rw [← H.index_eq_card, H.index_mul_card] _ = card K * card H := by rw [h', mul_comm]
- @@ -142,7 +142,7 @@ variable [H.Normal] [K.Normal] [Fintype G] (h : Disjoint H K) (h' : card G = car
 #check restrict #check ker_restrict - def iso₁ [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K) : K ≃* G⧸H := by + def iso₁ [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K) : K ≃* G ⧸ H := by apply MulEquiv.ofBijective ((QuotientGroup.mk' H).restrict K) rw [bijective_iff_injective_and_card] constructor
- @@ -151,8 +151,8 @@ def iso₁ [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K) : K
 · symm exact aux_card_eq h' - def iso₂ : G ≃* (G⧸K) × (G⧸H) := by - apply MulEquiv.ofBijective <| (QuotientGroup.mk' K).prod (QuotientGroup.mk' H) + def iso₂ : G ≃* (G ⧸ K) × (G ⧸ H) := by + apply MulEquiv.ofBijective <| (QuotientGroup.mk' K).prod (QuotientGroup.mk' H) rw [bijective_iff_injective_and_card] constructor · rw [← ker_eq_bot_iff, ker_prod]
MIL/C08_Groups_and_Rings/solutions/Solutions_S02_Rings.lean
- @@ -44,10 +44,10 @@ theorem isCoprime_Inf {I : Ideal R} {J : ι → Ideal R} {s : Finset ι}
  rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc - 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm - _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] - _ = (1+K)*I + K*J i := by ring - _ ≤ I + K ⊓ J i := by gcongr ; apply mul_le_left ; apply mul_le_inf + 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm + _ = I + K * (I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] + _ = (1 + K) * I + K * J i := by ring + _ ≤ I + K ⊓ J i := by gcongr ; apply mul_le_left ; apply mul_le_inf lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R}
- @@ -67,7 +67,7 @@ lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R}
  rfl · exact fun j hj ↦ eq_zero_iff_mem.mpr (he j hj) choose e he using key - use mk _ (∑ i, f i*e i) + use mk _ (∑ i, f i * e i) ext i rw [chineseMap_mk', map_sum, Fintype.sum_eq_single i] · simp [(he i).1, hf]
- @@ -75,7 +75,7 @@ lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R}
  simp [(he j).2 i hj.symm] noncomputable def chineseIso [Fintype ι] (f : ι → Ideal R) - (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* ∀ i, R ⧸ f i := + (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i := { Equiv.ofBijective _ ⟨chineseMap_inj f, chineseMap_surj hf⟩, chineseMap f with }
README.md
- @@ -40,7 +40,7 @@ At that point, you can open the textbook in a side panel in VS Code as follows:
  1. Type `ctrl-shift-P` (`command-shift-P` in macOS). - 2. Type `Lean 4: Open Documentation View` in the bar that appears, and then + 2. Type `Lean 4: Docview: Open Docview` in the bar that appears, and then press return. (You can press return to select it as soon as it is highlighted in the menu.)
html/.buildinfo
- @@ -1,4 +1,4 @@
  # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. - config: 9821478498fa33697c987b601aae688f + config: 803b947a09f3db5a52c6c529e3761a53 tags: 645f666f9bcd5a90fca523b33c5a78b7
html/C01_Introduction.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>1. Introduction — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -85,10 +83,10 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="introduction"> - <h1>1. Introduction<a class="headerlink" href="#introduction" title="Permalink to this heading"></a></h1> - <section id="getting-started"> - <h2>1.1. Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this heading"></a></h2> + <div class="section" id="introduction"> + <h1>1. Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline"></a></h1> + <div class="section" id="getting-started"> + <h2>1.1. Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this headline"></a></h2> The goal of this book is to teach you to formalize mathematics using the Lean 4 interactive proof assistant. It assumes that you know some mathematics, but it does not require much.
- @@ -173,9 +171,9 @@ You don’t have to do all of them; when you feel comfortable that you have
 the relevant skills, feel free to move on. You can always compare your solutions to the ones in the <code class="docutils literal notranslate">solutions</code> folder associated with each section. - </section> - <section id="overview"> - <h2>1.2. Overview<a class="headerlink" href="#overview" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="overview"> + <h2>1.2. Overview<a class="headerlink" href="#overview" title="Permalink to this headline"></a></h2> Put simply, Lean is a tool for building complex expressions in a formal language known as dependent type theory. Every expression has a type, and you can use the <cite>#check</cite> command to
- @@ -347,8 +345,8 @@ Giovanni Mascellani, Isaiah Mindich, Hunter Monroe, Pietro Monticone, Oliver Nas
 Bartosz Piotrowski, Nicolas Rolland, Guilherme Silva, Floris van Doorn, and Eric Wieser. Our work has been partially supported by the Hoskinson Center for Formal Mathematics. - </section> - </section> + </div> + </div> </div>
html/C02_Basics.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>2. Basics — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -89,14 +87,14 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="basics"> - <h1>2. Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h1> + <div class="section" id="basics"> + <h1>2. Basics<a class="headerlink" href="#basics" title="Permalink to this headline"></a></h1> This chapter is designed to introduce you to the nuts and bolts of mathematical reasoning in Lean: calculating, applying lemmas and theorems, and reasoning about generic structures. - <section id="calculating"> - <h2>2.1. Calculating<a class="headerlink" href="#calculating" title="Permalink to this heading"></a></h2> + <div class="section" id="calculating"> + <h2>2.1. Calculating<a class="headerlink" href="#calculating" title="Permalink to this headline"></a></h2> We generally learn to carry out mathematical calculations without thinking of them as proofs. But when we justify each step in a calculation,
- @@ -385,9 +383,9 @@ occurrence of <code class="docutils literal notranslate">a</sp
 rw [add_mul] </pre></div> </div> - </section> - <section id="proving-identities-in-algebraic-structures"> - <h2>2.2. Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="proving-identities-in-algebraic-structures"> + <h2>2.2. Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this headline"></a></h2> Mathematically, a ring consists of a collection of objects, $R$, operations $+$ $\times$, and constants $0$ and $1$, and an operation $x \mapsto -x$ such that:
- @@ -705,9 +703,9 @@ It may seem odd that the algebraic structures are called
 <cite>noncomm_ring</cite> and <cite>ring</cite>. This is partly for historical reasons, but also for the convenience of using a shorter name for the tactic that deals with commutative rings, since it is used more often. - </section> - <section id="using-theorems-and-lemmas"> - <h2>2.3. Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="using-theorems-and-lemmas"> + <h2>2.3. Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this headline"></a></h2> Rewriting is great for proving equations, but what about other sorts of theorems? For example, how can we prove an inequality,
- @@ -974,9 +972,9 @@ to two goals; see <a class="reference internal" href="C03_Logic.html#conjunction
 </div> If you managed to solve this, congratulations! You are well on your way to becoming a master formalizer. - </section> - <section id="more-examples-using-apply-and-rw"> - <h2>2.4. More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="more-examples-using-apply-and-rw"> + <h2>2.4. More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Permalink to this headline"></a></h2> The <code class="docutils literal notranslate">min</code> function on the real numbers is uniquely characterized by the following three facts: <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (min_le_left a b : min a b ≤ a)
- @@ -1176,9 +1174,9 @@ theorem and the version <code class="docutils literal notranslate">_root_.dvd_antisymm</code> to specify the generic one; either one will work. - </section> - <section id="proving-facts-about-algebraic-structures"> - <h2>2.5. Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="proving-facts-about-algebraic-structures"> + <h2>2.5. Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this headline"></a></h2> In <a class="reference internal" href="#proving-identities-in-algebraic-structures">Section 2.2</a>, we saw that many common identities governing the real numbers hold in more general classes of algebraic structures,
- @@ -1391,8 +1389,8 @@ always nonnegative:
 </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. - </section> - </section> + </div> + </div> </div>
html/C03_Logic.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>3. Logic — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -90,8 +88,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="logic"> - <h1>3. Logic<a class="headerlink" href="#logic" title="Permalink to this heading"></a></h1> + <div class="section" id="logic"> + <h1>3. Logic<a class="headerlink" href="#logic" title="Permalink to this headline"></a></h1> In the last chapter, we dealt with equations, inequalities, and basic mathematical statements like “$x$ divides $y$.”
- @@ -101,8 +99,8 @@ using logical terms like “and,” “or,” “not,”
 “if … then,” “every,” and “some.” In this chapter, we show you how to work with statements that are built up in this way. - <section id="implication-and-the-universal-quantifier"> - <h2>3.1. Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this heading"></a></h2> + <div class="section" id="implication-and-the-universal-quantifier"> + <h2>3.1. Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this headline"></a></h2> Consider the statement after the <code class="docutils literal notranslate">#check</code>: <div class="highlight-lean notranslate"><div class="highlight"><pre>#check ∀ x : ℝ, 0 ≤ x → |x| = x </pre></div>
- @@ -490,9 +488,9 @@ a lemma name.
 sorry </pre></div> </div> - </section> - <section id="the-existential-quantifier"> - <h2>3.2. The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="the-existential-quantifier"> + <h2>3.2. The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this headline"></a></h2> The existential quantifier, which can be entered as <code class="docutils literal notranslate">\ex</code> in VS Code, is used to represent the phrase “there exists.” The formal expression <code class="docutils literal notranslate">∃ x : ℝ, 2 < x ∧ x < 3</code> in Lean says
- @@ -811,9 +809,9 @@ the composition of surjective functions is surjective.
 sorry </pre></div> </div> - </section> - <section id="negation"> - <h2>3.3. Negation<a class="headerlink" href="#negation" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="negation"> + <h2>3.3. Negation<a class="headerlink" href="#negation" title="Permalink to this headline"></a></h2> The symbol <code class="docutils literal notranslate">¬</code> is meant to express negation, so <code class="docutils literal notranslate">¬ x < y</code> says that <code class="docutils literal notranslate">x</code> is not less than <code class="docutils literal notranslate">y</code>, <code class="docutils literal notranslate">¬ x = y</code> (or, equivalently, <code class="docutils literal notranslate">x ≠ y</code>) says that
- @@ -1076,9 +1074,9 @@ Finally, the <code class="docutils literal notranslate">contra
 by finding a contradiction in the hypotheses, such as a pair of the form <code class="docutils literal notranslate">h : P</code> and <code class="docutils literal notranslate">h' : ¬ P</code>. Of course, in this example, <code class="docutils literal notranslate">linarith</code> also works. - </section> - <section id="conjunction-and-iff"> - <h2>3.4. Conjunction and Iff<a class="headerlink" href="#conjunction-and-iff" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="conjunction-and-iff"> + <h2>3.4. Conjunction and Iff<a class="headerlink" href="#conjunction-and-iff" title="Permalink to this headline"></a></h2> You have already seen that the conjunction symbol, <code class="docutils literal notranslate">∧</code>, is used to express “and.” The <code class="docutils literal notranslate">constructor</code> tactic allows you to prove a statement of
- @@ -1131,8 +1129,7 @@ the existential quantifier.
 fun ⟨h₀, h₁⟩ h' ↦ h₁ (le_antisymm h₀ h') </pre></div> </div> - In analogy to the <code class="docutils literal notranslate">obtain</code> tactic, which we used with the existential - quantifier, there is also a pattern-matching <code class="docutils literal notranslate">have</code>: + In analogy to the <code class="docutils literal notranslate">obtain</code> tactic, there is also a pattern-matching <code class="docutils literal notranslate">have</code>: <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by have ⟨h₀, h₁⟩ := h contrapose! h₁
- @@ -1336,9 +1333,9 @@ to be instantiated to different values.
 sorry </pre></div> </div> - </section> - <section id="disjunction"> - <h2>3.5. Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="disjunction"> + <h2>3.5. Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this headline"></a></h2> The canonical way to prove a disjunction <code class="docutils literal notranslate">A ∨ B</code> is to prove <code class="docutils literal notranslate">A</code> or to prove <code class="docutils literal notranslate">B</code>. The <code class="docutils literal notranslate">left</code> tactic chooses <code class="docutils literal notranslate">A</code>,
- @@ -1579,9 +1576,9 @@ using <code class="docutils literal notranslate">by_cases</spa
 sorry </pre></div> </div> - </section> - <section id="sequences-and-convergence"> - <h2>3.6. Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="sequences-and-convergence"> + <h2>3.6. Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this headline"></a></h2> We now have enough skills at our disposal to do some real mathematics. In Lean, we can represent a sequence $s_0, s_1, s_2, \ldots$ of real numbers as a function <code class="docutils literal notranslate">s : ℕ → ℝ</code>.
- @@ -1816,8 +1813,8 @@ for dealing with convergence in vastly more general terms,
 not only abstracting away particular features of the domain and codomain, but also abstracting over different types of convergence. - </section> - </section> + </div> + </div> </div>
html/C04_Sets_and_Functions.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>4. Sets and Functions — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -87,8 +85,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="sets-and-functions"> - <h1>4. Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this heading"></a></h1> + <div class="section" id="sets-and-functions"> + <h1>4. Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this headline"></a></h1> The vocabulary of sets, relations, and functions provides a uniform language for carrying out constructions in all the branches of mathematics.
- @@ -120,8 +118,8 @@ such as a set of natural numbers or a set of functions
 from real numbers to real numbers. The distinction between types and sets takes some getting used to, but this chapter will take you through the essentials. - <section id="sets"> - <h2>4.1. Sets<a class="headerlink" href="#sets" title="Permalink to this heading"></a></h2> + <div class="section" id="sets"> + <h2>4.1. Sets<a class="headerlink" href="#sets" title="Permalink to this headline"></a></h2> If <code class="docutils literal notranslate">α</code> is any type, the type <code class="docutils literal notranslate">Set α</code> consists of sets of elements of <code class="docutils literal notranslate">α</code>. This type supports the usual set-theoretic operations and relations.
- @@ -557,9 +555,9 @@ and intersection.
 </div> In the library, these identities are called <code class="docutils literal notranslate">sUnion_eq_biUnion</code> and <code class="docutils literal notranslate">sInter_eq_biInter</code>. - </section> - <section id="functions"> - <h2>4.2. Functions<a class="headerlink" href="#functions" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="functions"> + <h2>4.2. Functions<a class="headerlink" href="#functions" title="Permalink to this headline"></a></h2> If <code class="docutils literal notranslate">f : α → β</code> is a function and <code class="docutils literal notranslate">p</code> is a set of elements of type <code class="docutils literal notranslate">β</code>, the library defines <code class="docutils literal notranslate">preimage f p</code>, written <code class="docutils literal notranslate">f ⁻¹' p</code>,
- @@ -689,39 +687,19 @@ and then calling <code class="docutils literal notranslate">si
 <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 - rintro ⟨i, x, xAi, fxeq⟩ - exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + sorry example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by - intro y; simp - intro x h fxeq i - use x - exact ⟨h i, fxeq⟩ + sorry example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by - intro y; simp - intro h - rcases h i with ⟨x, xAi, fxeq⟩ - use x; constructor - · intro i' - rcases h i' with ⟨x', x'Ai, fx'eq⟩ - have : f x = f x' := by rw [fxeq, fx'eq] - have : x = x' := injf this - rw [this] - exact x'Ai - exact fxeq + sorry example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by - ext x - simp + sorry example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by - ext x - simp + sorry </pre></div> </div> The library defines a predicate <code class="docutils literal notranslate">InjOn f s</code> to say that
- @@ -824,7 +802,7 @@ as follows:
 exact Classical.choose_spec h </pre></div> </div> - The lines <code class="docutils literal notranslate">noncomputable theory</code> and <code class="docutils literal notranslate">open Classical</code> + The lines <code class="docutils literal notranslate">noncomputable section</code> and <code class="docutils literal notranslate">open Classical</code> are needed because we are using classical logic in an essential way. On input <code class="docutils literal notranslate">y</code>, the function <code class="docutils literal notranslate">inverse f</code> returns some value of <code class="docutils literal notranslate">x</code> satisfying <code class="docutils literal notranslate">f x = y</code> if there is one,
- @@ -883,9 +861,9 @@ and then fill in the two lines that are missing.
 contradiction </pre></div> </div> - </section> - <section id="the-schroder-bernstein-theorem"> - <h2>4.3. The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="the-schroder-bernstein-theorem"> + <h2>4.3. The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this headline"></a></h2> We close this chapter with an elementary but nontrivial theorem of set theory. Let $\alpha$ and $\beta$ be sets. (In our formalization, they will actually be types.)
- @@ -1017,11 +995,11 @@ contained in $g(\beta)$.
 As a result, for every $x$ in the complement of $A$, there is a $y$ such that $g(y) = x$. (By the injectivity of $g$, this $y$ is unique, - but next theorem says only that <code class="docutils literal notranslate">inv_fun g x</code> returns some <code class="docutils literal notranslate">y</code> + but next theorem says only that <code class="docutils literal notranslate">invFun g x</code> returns some <code class="docutils literal notranslate">y</code> such that <code class="docutils literal notranslate">g y = x</code>.) Step through the proof below, make sure you understand what is going on, and fill in the remaining parts. - You will need to use <code class="docutils literal notranslate">inv_fun_eq</code> at the end. + You will need to use <code class="docutils literal notranslate">invFun_eq</code> at the end. Notice that rewriting with <code class="docutils literal notranslate">sb_aux</code> here replaces <code class="docutils literal notranslate">sb_aux f g 0</code> with the right-hand side of the corresponding defining equation. <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by
- @@ -1144,8 +1122,8 @@ and the proof uses the fact that <code class="docutils literal notranslate"><spa
 ⟨sbFun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ </pre></div> </div> - </section> - </section> + </div> + </div> </div>
html/C05_Elementary_Number_Theory.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>5. Elementary Number Theory — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -87,16 +85,16 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="elementary-number-theory"> - <h1>5. Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Permalink to this heading"></a></h1> + <div class="section" id="elementary-number-theory"> + <h1>5. Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Permalink to this headline"></a></h1> In this chapter, we show you how to formalize some elementary results in number theory. As we deal with more substantive mathematical content, the proofs will get longer and more involved, building on the skills you have already mastered. - <section id="irrational-roots"> - <h2>5.1. Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this heading"></a></h2> - Let’s start with a fact known to the ancient greeks, namely, + <div class="section" id="irrational-roots"> + <h2>5.1. Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this headline"></a></h2> + Let’s start with a fact known to the ancient Greeks, namely, that the square root of 2 is irrational. If we suppose otherwise, we can write $\sqrt{2} = a / b$ as a fraction
- @@ -394,9 +392,9 @@ and that it takes values in the extended natural numbers <code class="docutils l
 which adds the value infinity to the natural numbers. In the next chapter, we will begin to develop the means to appreciate the way that Lean supports this sort of generality. - </section> - <section id="induction-and-recursion"> - <h2>5.2. Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="induction-and-recursion"> + <h2>5.2. Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this headline"></a></h2> The set of natural numbers $\mathbb{N} = \{ 0, 1, 2, \ldots \}$ is not only fundamentally important in its own right, but also a plays a central role in the construction of new mathematical objects.
- @@ -699,9 +697,9 @@ The function <code class="docutils literal notranslate">pred</
 end MyNat </pre></div> </div> - </section> - <section id="infinitely-many-primes"> - <h2>5.3. Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="infinitely-many-primes"> + <h2>5.3. Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this headline"></a></h2> Let us continue our exploration of induction and recursion with another mathematical standard: a proof that there are infinitely many primes. One way to formulate this is as the statement that
- @@ -1137,8 +1135,8 @@ along the way.
 </div> If you managed to complete the proof, congratulations! This has been a serious feat of formalization. - </section> - </section> + </div> + </div> </div>
html/C06_Structures.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>6. Structures — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -87,8 +85,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="structures"> - <h1>6. Structures<a class="headerlink" href="#structures" title="Permalink to this heading"></a></h1> + <div class="section" id="structures"> + <h1>6. Structures<a class="headerlink" href="#structures" title="Permalink to this headline"></a></h1> Modern mathematics makes essential use of algebraic structures, which encapsulate patterns that can be instantiated in
- @@ -107,8 +105,8 @@ It will also show you how to define and use
 algebraic structures on your own. For more technical detail, you can consult <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, and a paper by Anne Baanen, <a class="reference external" href="https://arxiv.org/abs/2202.01629">Use and abuse of instance parameters in the Lean mathematical library</a>. - <section id="defining-structures"> - <h2>6.1. Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this heading"></a></h2> + <div class="section" id="defining-structures"> + <h2>6.1. Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this headline"></a></h2> In the broadest sense of the term, a structure is a specification of a collection of data, possibly with constraints that the data is required to satisfy.
- @@ -467,9 +465,9 @@ as long as we redefine the old accessors in terms of the new definition.
 Moreover, as we are about to see, Lean provides support for weaving structures together into a rich, interconnected hierarchy, and for managing the interactions between them. - </section> - <section id="algebraic-structures"> - <h2>6.2. Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="algebraic-structures"> + <h2>6.2. Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this headline"></a></h2> To clarify what we mean by the phrase algebraic structure, it will help to consider some examples. <ol class="arabic simple">
- @@ -1016,9 +1014,9 @@ because it configures automation that invisibly governs the interpretation of
 the expressions we type. When used wisely, however, class inference is a powerful tool. It is what makes algebraic reasoning possible in Lean. - </section> - <section id="building-the-gaussian-integers"> - <h2>6.3. Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="building-the-gaussian-integers"> + <h2>6.3. Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this headline"></a></h2> We will now illustrate the use of the algebraic hierarchy in Lean by building an important mathematical object, the Gaussian integers, and showing that it is a Euclidean domain. In other words, according to
- @@ -1044,7 +1042,7 @@ be a square root of $-1$. Th
 <div class="math notranslate nohighlight"> \[\begin{split}(a + bi) (c + di) & = ac + bci + adi + bd i^2 \\ & = (ac - bd) + (bc + ad)i.\end{split}\]</div> - This explains the definition of <code class="docutils literal notranslate">hasMul</code> below. + This explains the definition of <code class="docutils literal notranslate">Mul</code> below. <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Zero gaussInt := ⟨⟨0, 0⟩⟩
- @@ -1512,8 +1510,8 @@ the notions of being prime and being irreducible coincide.
 PrincipalIdealRing.irreducible_iff_prime </pre></div> </div> - </section> - </section> + </div> + </div> </div>
html/C07_Hierarchies.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>7. Hierarchies — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -86,8 +84,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="hierarchies"> - <h1>7. Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this heading"></a></h1> + <div class="section" id="hierarchies"> + <h1>7. Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this headline"></a></h1> We have seen in <a class="reference internal" href="C06_Structures.html#structures">Chapter 6</a> how to define the class of groups and build instances of this class, and then how to build an instance of the commutative ring class. But of course there is a hierarchy here: a
- @@ -103,8 +101,8 @@ the following chapters and come back here for a second reading.
 so we will used indices to distinguish our version. For instance we will have <code class="docutils literal notranslate">Ring₁</code> as our version of <code class="docutils literal notranslate">Ring</code>. Since we will gradually explain more powerful ways of formalizing structures, those indices will sometimes grow beyond one. - <section id="basics"> - <h2>7.1. Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h2> + <div class="section" id="basics"> + <h2>7.1. Basics<a class="headerlink" href="#basics" title="Permalink to this headline"></a></h2> At the very bottom of all hierarchies in Lean, we find data-carrying classes. The following class records that the given type <code class="docutils literal notranslate">α</code> is endowed with a distinguished element called <code class="docutils literal notranslate">one</code>. At this stage, it has no property at all.
- @@ -320,9 +318,9 @@ everything for additive and multiplicative theories with the help of some code-g
 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 semi-groups, the auto-generated “symmetry-restoring” instances need also to be marked. - This is a bit technical you don’t need to understand details. The important point is that + This is a bit technical; you don’t need to understand details. The important point is that lemmas are then only stated in multiplicative notation and marked with the attribute <code class="docutils literal notranslate">to_additive</code> - to generate the additive version as <code class="docutils literal notranslate">left_inv_eq_right_inv'</code> with it’s auto-generated additive + to generate the additive version as <code class="docutils literal notranslate">left_inv_eq_right_inv'</code> with its auto-generated additive version <code class="docutils literal notranslate">left_neg_eq_right_neg'</code>. In order to check the name of this additive version we used the <code class="docutils literal notranslate">whatsnew in</code> command on top of <code class="docutils literal notranslate">left_inv_eq_right_inv'</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddSemigroup₃ (α : Type) extends Add α where
- @@ -385,7 +383,7 @@ groups, and then define rings.
 sorry </pre></div> </div> - Note that <code class="docutils literal notranslate">to_additive</code> can be ask to tag a lemma with <code class="docutils literal notranslate">simp</code> and propagate that attribute + Note that <code class="docutils literal notranslate">to_additive</code> can be asked to tag a lemma with <code class="docutils literal notranslate">simp</code> and propagate that attribute to the additive version as follows. <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive (attr := simp)] lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by
- @@ -408,7 +406,7 @@ to the additive version as follows.
 We are now ready for rings. For demonstration purposes we won’t assume that addition is commutative, and then immediately provide an instance of <code class="docutils literal notranslate">AddCommGroup₃</code>. Mathlib does not play this game, first because in practice this does not make any ring instance easier and - also because Mathlib’s algebraic hierarchy goes through semi-rings which are like rings but without + also because Mathlib’s algebraic hierarchy goes through semirings which are like rings but without opposites so that the proof below does not work for them. What we gain here, besides a nice exercise if you have never seen it, is an example of building an instance using the syntax that allows to provide a parent structure and some extra fields.
- @@ -629,9 +627,9 @@ to incorporate a type class <code class="docutils literal notranslate"><₁</code> which has a default value built from <code class="docutils literal notranslate">≤₁</code> and a <code class="docutils literal notranslate">Prop</code>-valued field asserting the natural relation between those two comparison operators. -/ - </section> - <section id="morphisms"> - <h2>7.2. Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="morphisms"> + <h2>7.2. Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this headline"></a></h2> So far in this chapter, we discussed how to create a hierarchy of mathematical structures. But defining structures is not really completed until we have morphisms. There are two main approaches here. The most obvious one is to define a predicate on functions.
- @@ -833,9 +831,9 @@ definitions below.
 := sorry </pre></div> </div> - </section> - <section id="sub-objects"> - <h2>7.3. Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="sub-objects"> + <h2>7.3. Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this headline"></a></h2> After defining some algebraic structure and its morphisms, the next step is to consider sets that inherit this algebraic structure, for instance subgroups or subrings. This largely overlaps our previous topic. Indeed a set in <code class="docutils literal notranslate">X</code> is implemented as a function from
- @@ -971,8 +969,8 @@ the <code class="docutils literal notranslate">@</c
 sorry </pre></div> </div> - </section> - </section> + </div> + </div> </div>
html/C08_Groups_and_Rings.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>8. Groups and Rings — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -99,8 +97,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="groups-and-rings"> - <h1>8. Groups and Rings<a class="headerlink" href="#groups-and-rings" title="Permalink to this heading"></a></h1> + <div class="section" id="groups-and-rings"> + <h1>8. Groups and Rings<a class="headerlink" href="#groups-and-rings" title="Permalink to this headline"></a></h1> We saw in <a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures">Section 2.2</a> how to reason about operations in groups and rings. Later, in <a class="reference internal" href="C06_Structures.html#section-algebraic-structures">Section 6.2</a>, we saw how to define abstract algebraic structures, such as group structures, as well as concrete instances
- @@ -114,10 +112,10 @@ There is some overlap with the discussion of
 decisions behind the way the topics are treated. So making sense of some of the examples may require reviewing the background from <a class="reference internal" href="C07_Hierarchies.html#hierarchies">Chapter 7</a>. - <section id="monoids-and-groups"> - <h2>8.1. Monoids and Groups<a class="headerlink" href="#monoids-and-groups" title="Permalink to this heading"></a></h2> - <section id="monoids-and-their-morphisms"> - <h3>8.1.1. Monoids and their morphisms<a class="headerlink" href="#monoids-and-their-morphisms" title="Permalink to this heading"></a></h3> + <div class="section" id="monoids-and-groups"> + <h2>8.1. Monoids and Groups<a class="headerlink" href="#monoids-and-groups" title="Permalink to this headline"></a></h2> + <div class="section" id="monoids-and-their-morphisms"> + <h3>8.1.1. Monoids and their morphisms<a class="headerlink" href="#monoids-and-their-morphisms" title="Permalink to this headline"></a></h3> Courses in abstract algebra often start with groups and then progress to rings, fields, and vector spaces. This involves some contortions when discussing multiplication on rings since the multiplication operation does not come from a group structure
- @@ -126,7 +124,7 @@ The most common fix, when doing mathematics with pen and paper,
 is to leave those proofs as exercises. A less efficient but safer and more formalization-friendly way of proceeding is to use monoids. A monoid structure on a type <cite>M</cite> is an internal composition law that is associative and has a neutral element. - Monoids are used primarily to accommodate both groups and the multiplicative structure + Monoids are used primarily to accommodate both groups and the multiplicative structure of rings. But there are also a number of natural examples; for instance, the set of natural numbers equipped with addition forms a monoid. From a practical point of view, you can mostly ignore monoids when using Mathlib. But you need
- @@ -139,7 +137,7 @@ argument (in other words, in square brackets).
 By default, <code class="docutils literal notranslate">Monoid</code> uses multiplicative notation for the operation; for additive notation use <code class="docutils literal notranslate">AddMonoid</code> instead. The commutative versions of these structures add the prefix <code class="docutils literal notranslate">Comm</code> before <code class="docutils literal notranslate">Monoid</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M : Type*} [Monoid M] (x : M) : x*1 = x := mul_one x + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M : Type*} [Monoid M] (x : M) : x * 1 = x := mul_one x example {M : Type*} [AddCommMonoid M] (x y : M) : x + y = y + x := add_comm x y </pre></div>
- @@ -151,10 +149,10 @@ it is generally confusing to use additive notation with a non-commutative operat
 we apply it to elements of <code class="docutils literal notranslate">M</code>. The additive version is called <code class="docutils literal notranslate">AddMonoidHom</code> and written <code class="docutils literal notranslate">M →+ N</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M N : Type*} [Monoid M] [Monoid N] (x y : M) (f : M →* N) : f (x * y) = f x * f y := - f.map_mul x y + f.map_mul x y example {M N : Type*} [AddMonoid M] [AddMonoid N] (f : M →+ N) : f 0 = 0 := - f.map_zero + f.map_zero </pre></div> </div> These morphisms are bundled maps, i.e. they package together a map and some of its properties.
- @@ -162,12 +160,12 @@ Remember that <a class="reference internal" href="C07_Hierarchies.html#section-h
 here we simply note the slightly unfortunate consequence that we cannot use ordinary function composition to compose maps. Instead, we need to use <code class="docutils literal notranslate">MonoidHom.comp</code> and <code class="docutils literal notranslate">AddMonoidHom.comp</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M N P : Type*} [AddMonoid M] [AddMonoid N] [AddMonoid P] - (f : M →+ N) (g : N →+ P) : M →+ P := g.comp f + (f : M →+ N) (g : N →+ P) : M →+ P := g.comp f </pre></div> </div> - </section> - <section id="groups-and-their-morphisms"> - <h3>8.1.2. Groups and their morphisms<a class="headerlink" href="#groups-and-their-morphisms" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="groups-and-their-morphisms"> + <h3>8.1.2. Groups and their morphisms<a class="headerlink" href="#groups-and-their-morphisms" title="Permalink to this headline"></a></h3> We will have much more to say about groups, which are monoids with the extra property that every element has an inverse. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x : G) : x * x⁻¹ = 1 := mul_inv_self x
- @@ -176,7 +174,7 @@ property that every element has an inverse.
 Similar to the <code class="docutils literal notranslate">ring</code> tactic that we saw earlier, there is a <code class="docutils literal notranslate">group</code> tactic that proves any identity that holds in any group. (Equivalently, it proves the identities that hold in free groups.) - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x y z : G) : x * (y * z) * (x*z)⁻¹ * (x * y * x⁻¹)⁻¹ = 1 := by + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x y z : G) : x * (y * z) * (x * z)⁻¹ * (x * y * x⁻¹)⁻¹ = 1 := by group </pre></div> </div>
- @@ -189,12 +187,12 @@ free groups.)
 morphism is nothing more than a monoid morphism between groups. So we can copy and paste one of our earlier examples, replacing <code class="docutils literal notranslate">Monoid</code> with <code class="docutils literal notranslate">Group</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (x y : G) (f : G →* H) : f (x * y) = f x * f y := - f.map_mul x y + f.map_mul x y </pre></div> </div> Of course we do get some new properties, such as this one: <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (x : G) (f : G →* H) : f (x⁻¹) = (f x)⁻¹ := - f.map_inv x + f.map_inv x </pre></div> </div> You may be worried that constructing group morphisms will require us to do unnecessary work since
- @@ -203,8 +201,8 @@ while this is automatic in the case of group morphisms. In practice the extra wo
 but, to avoid it, there is a function building a group morphism from a function between groups that is compatible with the composition laws. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (f : G → H) (h : ∀ x y, f (x * y) = f x * f y) : - G →* H := - MonoidHom.mk' f h + G →* H := + MonoidHom.mk' f h </pre></div> </div> There is also a type <code class="docutils literal notranslate">MulEquiv</code> of group (or monoid) isomorphisms denoted by <code class="docutils literal notranslate">≃*</code> (and
- @@ -216,30 +214,30 @@ Using anonymous projector notation, the first two can be written <code class="do
 <code class="docutils literal notranslate">f.trans g</code> respectively. Elements of this type are automatically coerced to morphisms and functions when necessary. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (f : G ≃* H) : - f.trans f.symm = MulEquiv.refl G := - f.self_trans_symm + f.trans f.symm = MulEquiv.refl G := + f.self_trans_symm </pre></div> </div> One can use <code class="docutils literal notranslate">MulEquiv.ofBijective</code> to build an isomorphism from a bijective morphism. Doing so makes the inverse function noncomputable. <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example {G H : Type*} [Group G] [Group H] - (f : G →* H) (h : Function.Bijective f) : + (f : G →* H) (h : Function.Bijective f) : G ≃* H := MulEquiv.ofBijective f h </pre></div> </div> - </section> - <section id="subgroups"> - <h3>8.1.3. Subgroups<a class="headerlink" href="#subgroups" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="subgroups"> + <h3>8.1.3. Subgroups<a class="headerlink" href="#subgroups" title="Permalink to this headline"></a></h3> Just as group morphisms are bundled, a subgroup of <code class="docutils literal notranslate">G</code> is also a bundled structure consisting of a set in <code class="docutils literal notranslate">G</code> with the relevant closure properties. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) {x y : G} (hx : x ∈ H) (hy : y ∈ H) : - x * y ∈ H := - H.mul_mem hx hy + x * y ∈ H := + H.mul_mem hx hy example {G : Type*} [Group G] (H : Subgroup G) {x : G} (hx : x ∈ H) : - x⁻¹ ∈ H := - H.inv_mem hx + x⁻¹ ∈ H := + H.inv_mem hx </pre></div> </div> In the example above, it is important to understand that <code class="docutils literal notranslate">Subgroup G</code> is the type of subgroups
- @@ -281,18 +279,18 @@ So the above example can be restated more explicitly as:
 <code class="docutils literal notranslate">IsSubgroup : Set G → Prop</code> is that one can easily endow <code class="docutils literal notranslate">Subgroup G</code> with additional structure. Importantly, it has the structure of a complete lattice structure with respect to inclusion. For instance, instead of having a lemma stating that an intersection of - two subgroups of <code class="docutils literal notranslate">G</code> if again a subgroup, we - have use the lattice operation <code class="docutils literal notranslate">⊓</code> to construct the intersection. We can then apply arbitrary + two subgroups of <code class="docutils literal notranslate">G</code> is again a subgroup, we + have used the lattice operation <code class="docutils literal notranslate">⊓</code> to construct the intersection. We can then apply arbitrary lemmas about lattices to the construction. Let us check that the set underlying the infimum of two subgroups is indeed, by definition, their intersection. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H H' : Subgroup G) : - ((H ⊓ H' : Subgroup G) : Set G) = (H : Set G) ∩ (H' : Set G) := rfl + ((H ⊓ H' : Subgroup G) : Set G) = (H : Set G) ∩ (H' : Set G) := rfl </pre></div> </div> It may look strange to have a different notation for what amounts to the intersection of the underlying sets, but the correspondence does not carry over to the supremum operation and set - union, since a union of subgroup is not a subgroup. + union, since a union of subgroups is not, in general, a subgroup. Instead one needs to use the subgroup generated by the union, which is done using <code class="docutils literal notranslate">Subgroup.closure</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H H' : Subgroup G) :
- @@ -331,7 +329,7 @@ by an element of the ambient group.
 group morphisms. The naming convention in Mathlib is to call those operations <code class="docutils literal notranslate">map</code> and <code class="docutils literal notranslate">comap</code>. These are not the common mathematical terms, but they have the advantage of being - shorter than “pushforward” and “direct image.”” + shorter than “pushforward” and “direct image.” <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (G' : Subgroup G) (f : G →* H) : Subgroup H := Subgroup.map f G'
- @@ -361,23 +359,23 @@ from these exercises.
 open Subgroup - example (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T :=by + example (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T := by sorry - example (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T :=by + example (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T := by sorry variable {K : Type*} [Group K] -- Remember you can use the `ext` tactic to prove an equality of subgroups. example (φ : G →* H) (ψ : H →* K) (U : Subgroup K) : - comap (ψ.comp φ) U = comap φ (comap ψ U) := by + comap (ψ.comp φ) U = comap φ (comap ψ U) := by sorry -- Pushing a subgroup along one homomorphism and then another is equal to - -- pushing it forward along the composite of the homomorphisms. + -- pushing it forward along the composite of the homomorphisms. example (φ : G →* H) (ψ : H →* K) (S : Subgroup G) : - map (ψ.comp φ) S = map ψ (S.map φ) := by + map (ψ.comp φ) S = map ψ (S.map φ) := by sorry end exercises
- @@ -386,9 +384,9 @@ from these exercises.
 Let us finish this introduction to subgroups in Mathlib with two very classical results. Lagrange theorem states the cardinality of a subgroup of a finite group divides the cardinality of the group. Sylow’s first theorem is a famous partial converse to Lagrange’s theorem. - Since this corner of Mathlib is partly set up to allow computation, we need to tell - Lean to use nonconstructive logic, using the following <code class="docutils literal notranslate">attribute</code> command. - <div class="highlight-lean notranslate"><div class="highlight"><pre>attribute [local instance 10] setFintype Classical.propDecidable + While this corner of Mathlib is partly set up to allow computation, we can tell + Lean to use nonconstructive logic anyway using the following <code class="docutils literal notranslate">open scoped</code> command. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open scoped Classical open Fintype
- @@ -403,7 +401,7 @@ Lean to use nonconstructive logic, using the following <code class="docutils lit
 </pre></div> </div> The next two exercises derive a corollary of Lagrange’s lemma. (This is also already in Mathlib, - so do not use <cite>exact?</cite> too quickly.) + so do not use <code class="docutils literal notranslate">exact?</code> too quickly.) <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} [Fintype H] : H = ⊥ ↔ card H = 1 := by suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by
- @@ -413,13 +411,13 @@ so do not use <cite>exact?</cite> too quickly.)
 #check card_dvd_of_le lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) [Fintype H] [Fintype K] - (h : (card H).Coprime (card K)) : H ⊓ K = ⊥ := by - sorry + (h : (card H).Coprime (card K)) : H ⊓ K = ⊥ := by + sorry </pre></div> </div> - </section> - <section id="concrete-groups"> - <h3>8.1.4. Concrete groups<a class="headerlink" href="#concrete-groups" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="concrete-groups"> + <h3>8.1.4. Concrete groups<a class="headerlink" href="#concrete-groups" title="Permalink to this headline"></a></h3> One can also manipulate concrete groups in Mathlib, although this is typically more complicated than working with the abstract theory. For instance, given any type <code class="docutils literal notranslate">X</code>, the group of permutations of <code class="docutils literal notranslate">X</code> is <code class="docutils literal notranslate">Equiv.Perm X</code>.
- @@ -429,7 +427,7 @@ generated by cycles if <code class="docutils literal notranslate"><div class="highlight"><pre>open Equiv example {X : Type*} [Finite X] : Subgroup.closure {σ : Perm X | Perm.IsCycle σ} = ⊤ := - Perm.closure_isCycle + Perm.closure_isCycle </pre></div> </div> One can be fully concrete and compute actual products of cycles. Below we use the <code class="docutils literal notranslate">#simp</code> command,
- @@ -493,9 +491,9 @@ to <code class="docutils literal notranslate">PresentedGroup.t
 end FreeGroup </pre></div> </div> - </section> - <section id="group-actions"> - <h3>8.1.5. Group actions<a class="headerlink" href="#group-actions" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="group-actions"> + <h3>8.1.5. Group actions<a class="headerlink" href="#group-actions" title="Permalink to this headline"></a></h3> One important way that group theory interacts with the rest of mathematics is through the use of group actions. An action of a group <code class="docutils literal notranslate">G</code> on some type <code class="docutils literal notranslate">X</code> is nothing more than a morphism from <code class="docutils literal notranslate">G</code> to
- @@ -510,28 +508,28 @@ a point <code class="docutils literal notranslate">x</c
 <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section GroupActions example {G X : Type*} [Group G] [MulAction G X] (g g': G) (x : X) : - g • (g' • x) = (g * g') • x := - (mul_smul g g' x).symm + g • (g' • x) = (g * g') • x := + (mul_smul g g' x).symm </pre></div> </div> There is also a version for additive group called <code class="docutils literal notranslate">AddAction</code>, where the action is denoted by <code class="docutils literal notranslate">+ᵥ</code>. This is used for instance in the definition of affine spaces. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [AddGroup G] [AddAction G X] (g g' : G) (x : X) : - g +ᵥ (g' +ᵥ x) = (g + g') +ᵥ x := - (add_vadd g g' x).symm + g +ᵥ (g' +ᵥ x) = (g + g') +ᵥ x := + (add_vadd g g' x).symm </pre></div> </div> The underlying group morphism is called <code class="docutils literal notranslate">MulAction.toPermHom</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>open MulAction example {G X : Type*} [Group G] [MulAction G X] : G →* Equiv.Perm X := - toPermHom G X + toPermHom G X </pre></div> </div> As an illustration let us see how to define the Cayley isomorphism embedding of any group <code class="docutils literal notranslate">G</code> into a permutation group, namely <code class="docutils literal notranslate">Perm G</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>def CayleyIsoMorphism (G : Type*) [Group G] : G ≃* (toPermHom G G).range := - Equiv.Perm.subgroupOfMulAction G G + Equiv.Perm.subgroupOfMulAction G G </pre></div> </div> Note that nothing before the above definition required having a group rather than a monoid (or any
- @@ -579,7 +577,7 @@ conjugation, using our definition of <code class="docutils literal notranslate">
 <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {G : Type*} [Group G] lemma conjugate_one (H : Subgroup G) : conjugate 1 H = H := by - sorry + sorry instance : MulAction G (Subgroup G) where smul := conjugate
- @@ -591,33 +589,33 @@ conjugation, using our definition of <code class="docutils literal notranslate">
 end GroupActions </pre></div> </div> - </section> - <section id="quotient-groups"> - <h3>8.1.6. Quotient groups<a class="headerlink" href="#quotient-groups" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="quotient-groups"> + <h3>8.1.6. Quotient groups<a class="headerlink" href="#quotient-groups" title="Permalink to this headline"></a></h3> In the above discussion of subgroups acting on groups, we saw the quotient <code class="docutils literal notranslate">G ⧸ H</code> appear. In general this is only a type. It can be endowed with a group structure such that the quotient map is a group morphism if and only if <code class="docutils literal notranslate">H</code> is a normal subgroup (and this group structure is then unique). - The normality assumption is a type class <code class="docutils literal notranslate">Subgroup.Normal</code> so that type class inference can use to - derive the group structure on the quotient. + The normality assumption is a type class <code class="docutils literal notranslate">Subgroup.Normal</code> so that type class inference can use it + to derive the group structure on the quotient. <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section QuotientGroup example {G : Type*} [Group G] (H : Subgroup G) [H.Normal] : Group (G ⧸ H) := inferInstance example {G : Type*} [Group G] (H : Subgroup G) [H.Normal] : G →* G ⧸ H := - QuotientGroup.mk' H + QuotientGroup.mk' H </pre></div> </div> The universal property of quotient groups is accessed through <code class="docutils literal notranslate">QuotientGroup.lift</code>: a group morphism <code class="docutils literal notranslate">φ</code> descends to <code class="docutils literal notranslate">G ⧸ N</code> as soon as its kernel contains <code class="docutils literal notranslate">N</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (N : Subgroup G) [N.Normal] {M : Type*} - [Group M] (φ : G →* M) (h : N ≤ MonoidHom.ker φ) : G ⧸ N →* M := - QuotientGroup.lift N φ h + [Group M] (φ : G →* M) (h : N ≤ MonoidHom.ker φ) : G ⧸ N →* M := + QuotientGroup.lift N φ h </pre></div> </div> The fact that the target group is called <code class="docutils literal notranslate">M</code> is the above snippet is a clue that having a monoid structure on <code class="docutils literal notranslate">M</code> would be enough. - An important special case is when <code class="docutils literal notranslate">N = ker φ</code> In that case the descended morphism is + An important special case is when <code class="docutils literal notranslate">N = ker φ</code>. In that case the descended morphism is injective and we get a group isomorphism onto its image. This result is often called the first isomorphism theorem. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] {M : Type*} [Group M] (φ : G →* M) :
- @@ -643,7 +641,7 @@ involve an existential quantifier.
 is not enough to make the corresponding quotients equal. However the universal properties does give an isomorphism in this case. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] {M N : Subgroup G} [M.Normal] - [N.Normal] (h : M = N) : G ⧸ M ≃* G ⧸ N := QuotientGroup.quotientMulEquivOfEq h + [N.Normal] (h : M = N) : G ⧸ M ≃* G ⧸ N := QuotientGroup.quotientMulEquivOfEq h </pre></div> </div> As a final series of exercises for this section, we will prove that if <code class="docutils literal notranslate">H</code> and <code class="docutils literal notranslate">K</code> are disjoint
- @@ -653,7 +651,7 @@ then <code class="docutils literal notranslate">G</code
 We start with playing a bit with Lagrange’s lemma, without assuming the subgroups are normal or disjoint. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable {G : Type*} [Group G] {H K : Subgroup G} + variable {G : Type*} [Group G] {H K : Subgroup G} open MonoidHom
- @@ -662,7 +660,7 @@ or disjoint.
 #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right - lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G⧸H) = card K := by + lemma aux_card_eq [Fintype G] (h' : card G = card H * card K) : card (G ⧸ H) = card K := by sorry </pre></div> </div>
- @@ -675,14 +673,14 @@ condition. Now we construct the first building block of the desired isomorphism.
 #check restrict #check ker_restrict - def iso₁ [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K) : K ≃* G⧸H := by + def iso₁ [Fintype G] (h : Disjoint H K) (h' : card G = card H * card K) : K ≃* G ⧸ H := by sorry </pre></div> </div> Now we can define our second building block. We will need <code class="docutils literal notranslate">MonoidHom.prod</code>, which builds a morphism from <code class="docutils literal notranslate">G₀</code> to <code class="docutils literal notranslate">G₁ × G₂</code> out of morphisms from <code class="docutils literal notranslate">G₀</code> to <code class="docutils literal notranslate">G₁</code> and <code class="docutils literal notranslate">G₂</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def iso₂ : G ≃* (G⧸K) × (G⧸H) := by + <div class="highlight-lean notranslate"><div class="highlight"><pre>def iso₂ : G ≃* (G ⧸ K) × (G ⧸ H) := by sorry </pre></div> </div>
- @@ -693,32 +691,32 @@ morphisms from <code class="docutils literal notranslate">G&#8
 sorry </pre></div> </div> - </section> - </section> - <section id="rings"> - <h2>8.2. Rings<a class="headerlink" href="#rings" title="Permalink to this heading"></a></h2> - <section id="rings-their-units-morphisms-and-subrings"> - <h3>8.2.1. Rings, their units, morphisms and subrings<a class="headerlink" href="#rings-their-units-morphisms-and-subrings" title="Permalink to this heading"></a></h3> + </div> + </div> + <div class="section" id="rings"> + <h2>8.2. Rings<a class="headerlink" href="#rings" title="Permalink to this headline"></a></h2> + <div class="section" id="rings-their-units-morphisms-and-subrings"> + <h3>8.2.1. Rings, their units, morphisms and subrings<a class="headerlink" href="#rings-their-units-morphisms-and-subrings" title="Permalink to this headline"></a></h3> The type of ring structures on a type <code class="docutils literal notranslate">R</code> is <code class="docutils literal notranslate">Ring R</code>. The variant where multiplication is assumed to be commutative is <code class="docutils literal notranslate">CommRing R</code>. We have already seen that the <code class="docutils literal notranslate">ring</code> tactic will prove any equality that follows from the axioms of a commutative ring. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (x y : R) : (x + y)^2 = x^2 + y^2 + 2*x*y := by ring + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (x y : R) : (x + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y := by ring </pre></div> </div> - More exotic variants do not require that the addition on <code class="docutils literal notranslate">R</code> forms a group but only an additive + More exotic variants do not require that the addition on <code class="docutils literal notranslate">R</code> forms a group but only an additive monoid. The corresponding type classes are <code class="docutils literal notranslate">Semiring R</code> and <code class="docutils literal notranslate">CommSemiring R</code>. The type of natural numbers is an important instance of <code class="docutils literal notranslate">CommSemiring R</code>, as is any type of functions taking values in the natural numbers. Another important example is the type of ideals in a ring, which will be discussed below. The name of the <code class="docutils literal notranslate">ring</code> tactic is doubly misleading, since it assumes commutativity but works in semirings as well. In other words, it applies to any <code class="docutils literal notranslate">CommSemiring</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x y : ℕ) : (x + y)^2 = x^2 + y^2 + 2*x*y := by ring + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x y : ℕ) : (x + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y := by ring </pre></div> </div> There are also versions of the ring and semiring classes that do not assume the existence of a multiplicative unit or the associativity of multiplication. We will not discuss those here. - Some concepts that are traditionally taught in an introduction to ring theory are actually about + Some concepts that are traditionally taught in an introduction to ring theory are actually about the underlying multiplicative monoid. A prominent example is the definition of the units of a ring. Every (multiplicative) monoid <code class="docutils literal notranslate">M</code> has a predicate <code class="docutils literal notranslate">IsUnit : M → Prop</code> asserting existence of a two-sided inverse, a
- @@ -731,7 +729,7 @@ In the commutative case, one also has <code class="docutils literal notranslate"
 which builds <code class="docutils literal notranslate">x</code> seen as unit. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℤˣ) : x = 1 ∨ x = -1 := Int.units_eq_one_or x - example {M : Type*} [Monoid M] (x : Mˣ) : (x : M)*x⁻¹ = 1 := Units.mul_inv x + example {M : Type*} [Monoid M] (x : Mˣ) : (x : M) * x⁻¹ = 1 := Units.mul_inv x example {M : Type*} [Monoid M] : Group Mˣ := inferInstance </pre></div>
- @@ -745,7 +743,7 @@ with notation <code class="docutils literal notranslate">R</sp
 Units.map f </pre></div> </div> - The isormophism variant is <code class="docutils literal notranslate">RingEquiv</code>, with notation <code class="docutils literal notranslate">≃+*</code>. + The isomorphism variant is <code class="docutils literal notranslate">RingEquiv</code>, with notation <code class="docutils literal notranslate">≃+*</code>. As with submonoids and subgroups, there is a <code class="docutils literal notranslate">Subring R</code> type for subrings of a ring <code class="docutils literal notranslate">R</code>, but this type is a lot less useful than the type of subgroups since one cannot quotient a ring by a subring.
- @@ -753,9 +751,9 @@ a subring.
 </pre></div> </div> Also notice that <code class="docutils literal notranslate">RingHom.range</code> produces a subring. - </section> - <section id="ideals-and-quotients"> - <h3>8.2.2. Ideals and quotients<a class="headerlink" href="#ideals-and-quotients" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="ideals-and-quotients"> + <h3>8.2.2. Ideals and quotients<a class="headerlink" href="#ideals-and-quotients" title="Permalink to this headline"></a></h3> For historical reasons, Mathlib only has a theory of ideals for commutative rings. (The ring library was originally developed to make quick progress toward the foundations of modern algebraic geometry.) So in this section we will work with commutative (semi)rings.
- @@ -765,7 +763,7 @@ safely ignored since most (but not all) relevant lemmas are restated in the spec
 ideals. But anonymous projection notation won’t always work as expected. For instance, one cannot replace <code class="docutils literal notranslate">Ideal.Quotient.mk I</code> by <code class="docutils literal notranslate">I.Quotient.mk</code> in the snippet below because the parser immediately replaces <code class="docutils literal notranslate">Ideal R</code> with <code class="docutils literal notranslate">Submodule R R</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (I : Ideal R) : R →+* R⧸I := + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (I : Ideal R) : R →+* R ⧸ I := Ideal.Quotient.mk I example {R : Type*} [CommRing R] {a : R} {I : Ideal R} :
- @@ -827,7 +825,7 @@ your font, those can be pretty hard to distinguish.
 Ideal.quotientInfRingEquivPiQuotient f hf </pre></div> </div> - The elementary version of the Chinese remainder theorem, a statement about <code class="docutils literal notranslate">Zmod</code>, can be easily + The elementary version of the Chinese remainder theorem, a statement about <code class="docutils literal notranslate">ZMod</code>, can be easily deduced from the previous one: <div class="highlight-lean notranslate"><div class="highlight"><pre>open BigOperators PiNotation
- @@ -881,7 +879,7 @@ of our <code class="docutils literal notranslate">chineseMap</
 </div> We take the opportunity to use induction on <code class="docutils literal notranslate">Finset</code>. Relevant lemmas on <code class="docutils literal notranslate">Finset</code> are given below. - Remember that the <code class="docutils literal notranslate">ring</code> tactic work for semirings and that the ideals of a ring form a semiring. + Remember that the <code class="docutils literal notranslate">ring</code> tactic works for semirings and that the ideals of a ring form a semiring. <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Finset.mem_insert_of_mem #check Finset.mem_insert_self
- @@ -896,10 +894,10 @@ Remember that the <code class="docutils literal notranslate">r
 rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc - 1 = I + K := sorry - _ = I + K*(I + J i) := sorry - _ = (1+K)*I + K*J i := sorry - _ ≤ I + K ⊓ J i := sorry + 1 = I + K := sorry + _ = I + K * (I + J i) := sorry + _ = (1 + K) * I + K * J i := sorry + _ ≤ I + K ⊓ J i := sorry </pre></div> </div> We can now prove surjectivity of the map appearing in the Chinese remainder theorem.
- @@ -914,25 +912,25 @@ Remember that the <code class="docutils literal notranslate">r
 sorry sorry choose e he using key - use mk _ (∑ i, f i*e i) + use mk _ (∑ i, f i * e i) sorry </pre></div> </div> Now all the pieces come together in the following: <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable def chineseIso [Fintype ι] (f : ι → Ideal R) - (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* ∀ i, R ⧸ f i := + (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i := { Equiv.ofBijective _ ⟨chineseMap_inj f, chineseMap_surj hf⟩, chineseMap f with } </pre></div> </div> - </section> - <section id="algebras-and-polynomials"> - <h3>8.2.3. Algebras and polynomials<a class="headerlink" href="#algebras-and-polynomials" title="Permalink to this heading"></a></h3> - Given a commutative (semi)ring <code class="docutils literal notranslate">R</code>, an algebra over ``R`` is a semiring <code class="docutils literal notranslate">A</code> equipped + </div> + <div class="section" id="algebras-and-polynomials"> + <h3>8.2.3. Algebras and polynomials<a class="headerlink" href="#algebras-and-polynomials" title="Permalink to this headline"></a></h3> + Given a commutative (semi)ring <code class="docutils literal notranslate">R</code>, an algebra over <code class="docutils literal notranslate">R</code> is a semiring <code class="docutils literal notranslate">A</code> equipped with a ring morphism whose image commutes with every element of <code class="docutils literal notranslate">A</code>. This is encoded as a type class <code class="docutils literal notranslate">Algebra R A</code>. The morphism from <code class="docutils literal notranslate">R</code> to <code class="docutils literal notranslate">A</code> is called the structure map and is denoted - <code class="docutils literal notranslate">algebraMap R A : R →*+ A</code> in Lean. + <code class="docutils literal notranslate">algebraMap R A : R →+* A</code> in Lean. Multiplication of <code class="docutils literal notranslate">a : A</code> by <code class="docutils literal notranslate">algebraMap R A r</code> for some <code class="docutils literal notranslate">r : R</code> is called the scalar multiplication of <code class="docutils literal notranslate">a</code> by <code class="docutils literal notranslate">r</code> and denoted by <code class="docutils literal notranslate">r • a</code>. Note that this notion of algebra is sometimes called an associative unital algebra to emphasize the
- @@ -973,15 +971,15 @@ algebra can be inferred from our use of <code class="docutils literal notranslat
 Because <code class="docutils literal notranslate">C</code> is a ring morphism from <code class="docutils literal notranslate">R</code> to <code class="docutils literal notranslate">R[X]</code>, we can use all ring morphisms lemmas such as <code class="docutils literal notranslate">map_zero</code>, <code class="docutils literal notranslate">map_one</code>, <code class="docutils literal notranslate">map_mul</code>, and <code class="docutils literal notranslate">map_pow</code> before computing in the ring <code class="docutils literal notranslate">R[X]</code>. For example: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (r : R) : (X + C r) * (X - C r) = X^2 - C (r ^ 2) := by + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (r : R) : (X + C r) * (X - C r) = X ^ 2 - C (r ^ 2) := by rw [C.map_pow] ring </pre></div> </div> You can access coefficients using <code class="docutils literal notranslate">Polynomial.coeff</code> - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R](r:R) : (C r).coeff 0 = r := by simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (r:R) : (C r).coeff 0 = r := by simp - example {R : Type*} [CommRing R] : (X^2 + 2*X + C 3 : R[X]).coeff 1 = 2 := by simp + example {R : Type*} [CommRing R] : (X ^ 2 + 2 * X + C 3 : R[X]).coeff 1 = 2 := by simp </pre></div> </div> Defining the degree of a polynomial is always tricky because of the special case of the zero
- @@ -1020,7 +1018,7 @@ using <code class="docutils literal notranslate">Polynomial.ev
 example {R : Type*} [CommRing R] (r : R) : (X - C r).eval r = 0 := by simp </pre></div> </div> - In particular, there is a predicate, <code class="docutils literal notranslate">IsRoot</code>, that hold of elements <code class="docutils literal notranslate">r</code> in <code class="docutils literal notranslate">R</code> where a + In particular, there is a predicate, <code class="docutils literal notranslate">IsRoot</code>, that holds for elements <code class="docutils literal notranslate">r</code> in <code class="docutils literal notranslate">R</code> where a polynomial vanishes. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (P : R[X]) (r : R) : IsRoot P r ↔ P.eval r = 0 := Iff.rfl </pre></div>
- @@ -1036,19 +1034,19 @@ since otherwise the definition does not have good properties.
 roots_X_sub_C r example {R : Type*} [CommRing R] [IsDomain R] (r : R) (n : ℕ): - ((X - C r)^n).roots = n • {r} := + ((X - C r) ^ n).roots = n • {r} := by simp </pre></div> </div> - Both <cite>Polynomial.eval</cite> and <cite>Polynomial.roots</cite> consider only the coefficients ring. They do not - allow us to say that <code class="docutils literal notranslate">X^2 - 2 : ℚ[X]</code> has a root in <code class="docutils literal notranslate">ℝ</code> or that <code class="docutils literal notranslate">X^2 + 1 : ℝ[X]</code> has a root in + Both <code class="docutils literal notranslate">Polynomial.eval</code> and <code class="docutils literal notranslate">Polynomial.roots</code> consider only the coefficients ring. They do not + allow us to say that <code class="docutils literal notranslate">X ^ 2 - 2 : ℚ[X]</code> has a root in <code class="docutils literal notranslate">ℝ</code> or that <code class="docutils literal notranslate">X ^ 2 + 1 : ℝ[X]</code> has a root in <code class="docutils literal notranslate">ℂ</code>. For this, we need <code class="docutils literal notranslate">Polynomial.aeval</code>, which will evaluate <code class="docutils literal notranslate">P : R[X]</code> in any <code class="docutils literal notranslate">R</code>-algebra. More precisely, given a semiring <code class="docutils literal notranslate">A</code> and an instance of <code class="docutils literal notranslate">Algebra R A</code>, <code class="docutils literal notranslate">Polynomial.aeval</code> sends every element of <code class="docutils literal notranslate">a</code> along the <code class="docutils literal notranslate">R</code>-algebra morphism of evaluation at <code class="docutils literal notranslate">a</code>. Since <code class="docutils literal notranslate">AlgHom</code> has a coercion to functions, one can apply it to a polynomial. But <code class="docutils literal notranslate">aeval</code> does not have a polynomial as an argument, so one cannot use dot notation like in <code class="docutils literal notranslate">P.eval</code> above. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : aeval Complex.I (X^2 + 1 : ℝ[X]) = 0 := by simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : aeval Complex.I (X ^ 2 + 1 : ℝ[X]) = 0 := by simp </pre></div> </div> The function corresponding to <code class="docutils literal notranslate">roots</code> in this context is <code class="docutils literal notranslate">aroots</code> which takes a polynomial
- @@ -1056,7 +1054,7 @@ and then an algebra and outputs a multiset (with the same caveat about the zero
 for <code class="docutils literal notranslate">roots</code>). <div class="highlight-lean notranslate"><div class="highlight"><pre>open Complex Polynomial - example : aroots (X^2 + 1 : ℝ[X]) ℂ = {Complex.I, -I} := by + example : aroots (X ^ 2 + 1 : ℝ[X]) ℂ = {Complex.I, -I} := by suffices roots (X ^ 2 + 1 : ℂ[X]) = {I, -I} by simpa [aroots_def] have factored : (X ^ 2 + 1 : ℂ[X]) = (X - C I) * (X - C (-I)) := by rw [C_neg]
- @@ -1078,7 +1076,7 @@ since it does not assume the existence of a <code class="docutils literal notran
 you would expect. <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (Complex.ofReal : ℝ →+* ℂ) - example : (X^2 + 1 : ℝ[X]).eval₂ Complex.ofReal Complex.I = 0 := by simp + example : (X ^ 2 + 1 : ℝ[X]).eval₂ Complex.ofReal Complex.I = 0 := by simp </pre></div> </div> Let us end by mentioning multivariate polynomials briefly. Given a commutative semiring <code class="docutils literal notranslate">R</code>,
- @@ -1101,9 +1099,9 @@ determined by the number of arguments and some type <code class="docutils litera
 <div class="highlight-lean notranslate"><div class="highlight"><pre>example : MvPolynomial.eval ![0, 1] circleEquation = 0 := by simp [circleEquation] </pre></div> </div> - </section> - </section> - </section> + </div> + </div> + </div> </div>
html/C09_Topology.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>9. Topology — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -99,8 +97,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>9. Topology<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + <div class="section" id="index-0"> + <h1>9. Topology<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> Calculus is based on the concept of a function, which is used to model quantities that depend on one another. For example, it is common to study quantities that change over time.
- @@ -167,8 +165,8 @@ and simply note that the rest can be proved “in the same way.”
 Formalizing mathematics requires making the relevant notion of “sameness” fully explicit, and that is exactly what Bourbaki’s theory of filters manages to do. - <section id="filters"> - <h2>9.1. Filters<a class="headerlink" href="#filters" title="Permalink to this heading"></a></h2> + <div class="section" id="filters"> + <h2>9.1. Filters<a class="headerlink" href="#filters" title="Permalink to this headline"></a></h2> A filter on a type <code class="docutils literal notranslate">X</code> is a collection of sets of <code class="docutils literal notranslate">X</code> that satisfies three conditions that we will spell out below. The notion supports two related ideas:
- @@ -410,7 +408,7 @@ neighborhood topology on <code class="docutils literal notranslate"></div> </div> There is also a nice basis for the filter <code class="docutils literal notranslate">atTop</code>. The lemma - <code class="docutils literal notranslate">Filter.has_basis.tendsto_iff</code> allows + <code class="docutils literal notranslate">Filter.HasBasis.tendsto_iff</code> allows us to reformulate a statement of the form <code class="docutils literal notranslate">Tendsto f F G</code> given bases for <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">G</code>. Putting these pieces together gives us essentially the notion of convergence
- @@ -520,9 +518,9 @@ by definition, the assumption <code class="docutils literal notranslate">sorry </pre></div> </div> - </section> - <section id="metric-spaces"> - <h2>9.2. Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="metric-spaces"> + <h2>9.2. Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this headline"></a></h2> Examples in the previous section focus on sequences of real numbers. In this section we will go up a bit in generality and focus on metric spaces. A metric space is a type <code class="docutils literal notranslate">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>.
- @@ -540,8 +538,8 @@ the function <code class="docutils literal notranslate">fun</s
 They are called <code class="docutils literal notranslate">EMetricSpace</code>, <code class="docutils literal notranslate">PseudoMetricSpace</code> and <code class="docutils literal notranslate">PseudoEMetricSpace</code> respectively (here “e” stands for “extended”). Note that our journey from <code class="docutils literal notranslate">ℝ</code> to metric spaces jumped over the special case of normed spaces that also require linear algebra and will be explained as part of the calculus chapter. - <section id="convergence-and-continuity"> - <h3>9.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this heading"></a></h3> + <div class="section" id="convergence-and-continuity"> + <h3>9.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this headline"></a></h3> Using distance functions, we can already define convergent sequences and continuous functions between metric spaces. They are actually defined in a more general setting covered in the next section, but we have lemmas recasting the definition is terms of distances.
- @@ -628,9 +626,9 @@ and get our final proof, now bordering obfuscation.
 Metric.continuousAt_iff </pre></div> </div> - </section> - <section id="balls-open-sets-and-closed-sets"> - <h3>9.2.2. Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="balls-open-sets-and-closed-sets"> + <h3>9.2.2. Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this headline"></a></h3> Once we have a distance function, the most important geometric definitions are (open) balls and closed balls. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (r : ℝ)
- @@ -685,9 +683,9 @@ argument so we can invoke <code class="docutils literal notranslate">Metric.nhds_basis_closedBall.mem_iff </pre></div> </div> - </section> - <section id="compactness"> - <h3>9.2.3. Compactness<a class="headerlink" href="#compactness" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="compactness"> + <h3>9.2.3. Compactness<a class="headerlink" href="#compactness" title="Permalink to this headline"></a></h3> Compactness is an important topological notion. It distinguishes subsets of a metric space that enjoy the same kind of properties as segments in reals compared to other intervals: <ul class="simple">
- @@ -728,9 +726,9 @@ are deduced from more general versions, some of which will be discussed in later
 </pre></div> </div> In a compact metric space any closed set is compact, this is <code class="docutils literal notranslate">IsClosed.isCompact</code>. - </section> - <section id="uniformly-continuous-functions"> - <h3>9.2.4. Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="uniformly-continuous-functions"> + <h3>9.2.4. Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this headline"></a></h3> 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.
- @@ -759,9 +757,9 @@ of the distance function on <code class="docutils literal notranslate">sorry </pre></div> </div> - </section> - <section id="completeness"> - <h3>9.2.5. Completeness<a class="headerlink" href="#completeness" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="completeness"> + <h3>9.2.5. Completeness<a class="headerlink" href="#completeness" title="Permalink to this headline"></a></h3> A Cauchy sequence in a metric space is a sequence whose terms get closer and closer to each other. There are a couple of equivalent ways to state that idea. In particular converging sequences are Cauchy. The converse is true only in so-called complete
- @@ -850,12 +848,12 @@ define something inductively in the middle of a proof using <code class="docutil
 sorry </pre></div> </div> - </section> - </section> - <section id="topological-spaces"> - <h2>9.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> - <section id="fundamentals"> - <h3>9.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> + </div> + </div> + <div class="section" id="topological-spaces"> + <h2>9.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this headline"></a></h2> + <div class="section" id="fundamentals"> + <h3>9.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this headline"></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
- @@ -1043,9 +1041,9 @@ Let us explore that constraint “on paper” using notation 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"> - <h3>9.3.2. Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="separation-and-countability"> + <h3>9.3.2. Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this headline"></a></h3> We saw that the category of topological spaces have very nice properties. The price to pay for this is existence of rather pathological topological spaces. There are a number of assumptions you can make on a topological space to ensure its behavior
- @@ -1132,9 +1130,9 @@ of sets can be understood using sequences.
 mem_closure_iff_seq_limit </pre></div> </div> - </section> - <section id="id5"> - <h3>9.3.3. Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="id5"> + <h3>9.3.3. Compactness<a class="headerlink" href="#id5" title="Permalink to this headline"></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. 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>,
- @@ -1190,9 +1188,9 @@ cover <code class="docutils literal notranslate">s</cod
 hs.elim_finite_subcover U hUo hsU </pre></div> </div> - </section> - </section> - </section> + </div> + </div> + </div> </div>
html/C10_Differential_Calculus.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>10. Differential Calculus — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -92,8 +90,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>10. Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + <div class="section" id="index-0"> + <h1>10. Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> We now consider the formalization of notions from analysis, starting with differentiation in this chapter and turning integration and measure theory in the next.
- @@ -102,8 +100,8 @@ setting of functions from the real numbers to the real numbers,
 which is familiar from any introductory calculus class. In <a class="reference internal" href="#normed-spaces">Section 10.2</a>, we then consider the notion of a derivative in a much broader setting. - <section id="elementary-differential-calculus"> - <h2>10.1. Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this heading"></a></h2> + <div class="section" id="elementary-differential-calculus"> + <h2>10.1. Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this headline"></a></h2> 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.
- @@ -176,11 +174,11 @@ seems even weirder.
 example : deriv sin π = -1 := by simp </pre></div> </div> - </section> - <section id="differential-calculus-in-normed-spaces"> - <h2>10.2. Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this heading"></a></h2> - <section id="id3"> - <h3>10.2.1. Normed spaces<a class="headerlink" href="#id3" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="differential-calculus-in-normed-spaces"> + <h2>10.2. Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this headline"></a></h2> + <div class="section" id="id3"> + <h3>10.2.1. Normed spaces<a class="headerlink" href="#id3" title="Permalink to this headline"></a></h3> Differentiation can be generalized beyond <code class="docutils literal notranslate">ℝ</code> using the notion of a normed vector space, which encapsulates both direction and distance. We start with the notion of a normed group, which as an additive commutative
- @@ -243,9 +241,9 @@ complete as long as the field itself is complete.
 FiniteDimensional.complete 𝕜 E </pre></div> </div> - </section> - <section id="continuous-linear-maps"> - <h3>10.2.2. Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="continuous-linear-maps"> + <h3>10.2.2. Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this headline"></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
- @@ -325,9 +323,9 @@ Minor ingredients include <code class="docutils literal notranslate">sorry </pre></div> </div> - </section> - <section id="asymptotic-comparisons"> - <h3>10.2.3. Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="asymptotic-comparisons"> + <h3>10.2.3. Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this headline"></a></h3> Defining differentiability also requires asymptotic comparisons. Mathlib has an extensive library covering the big O and little o relations, whose definitions are shown below.
- @@ -353,9 +351,9 @@ Here we will only use little o to define differentiability.
 Iff.rfl </pre></div> </div> - </section> - <section id="differentiability"> - <h3>10.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> + </div> + <div class="section" id="differentiability"> + <h3>10.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this headline"></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>.
- @@ -438,9 +436,9 @@ For example, you may want to use one-sided derivatives in the
 one-dimensional setting. The means to do so are found in Mathlib in a more general context; see <code class="docutils literal notranslate">HasFDerivWithinAt</code> or the even more general <code class="docutils literal notranslate">HasFDerivAtFilter</code>. - </section> - </section> - </section> + </div> + </div> + </div> </div>
html/C11_Integration_and_Measure_Theory.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>11. Integration and Measure Theory — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script>
- @@ -87,10 +85,10 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>11. Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> - <section id="elementary-integration"> - <h2>11.1. Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this heading"></a></h2> + <div class="section" id="index-0"> + <h1>11. Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> + <div class="section" id="elementary-integration"> + <h2>11.1. Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this headline"></a></h2> We first focus on integration of functions on finite intervals in <code class="docutils literal notranslate">ℝ</code>. We can integrate elementary functions. <div class="highlight-lean notranslate"><div class="highlight"><pre>open MeasureTheory intervalIntegral
- @@ -127,9 +125,9 @@ which are not shown here, are not equivalent.)
 rfl </pre></div> </div> - </section> - <section id="measure-theory"> - <h2>11.2. Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="measure-theory"> + <h2>11.2. Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this headline"></a></h2> 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,
- @@ -204,9 +202,9 @@ almost everywhere.
 Iff.rfl </pre></div> </div> - </section> - <section id="integration"> - <h2>11.3. Integration<a class="headerlink" href="#integration" title="Permalink to this heading"></a></h2> + </div> + <div class="section" id="integration"> + <h2>11.3. Integration<a class="headerlink" href="#integration" title="Permalink to this headline"></a></h2> Now that we have measurable spaces and measures we can consider integrals. As explained above, Mathlib uses a very general notion of integration that allows any Banach space as the target.
- @@ -281,8 +279,8 @@ gives finite mass to compact sets, and give positive mass to open sets.
 integral_image_eq_integral_abs_det_fderiv_smul μ hs hf h_inj g </pre></div> </div> - </section> - </section> + </div> + </div> </div>
html/_static/_sphinx_javascript_frameworks_compat.js (deleted)
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html/_static/basic.css
- @@ -222,7 +222,7 @@ table.modindextable td {
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- @@ -335,13 +335,13 @@ p.sidebar-title {
  font-weight: bold; } - div.admonition, div.topic, aside.topic, blockquote { + div.admonition, div.topic, blockquote { clear: left; } /* -- topics ---------------------------------------------------------------- */ - div.topic, aside.topic { + div.topic { border: 1px solid #ccc; padding: 7px; margin: 10px 0 10px 0;
- @@ -380,7 +380,6 @@ div.body p.centered {
  div.sidebar > :last-child, aside.sidebar > :last-child, div.topic > :last-child, - aside.topic > :last-child, div.admonition > :last-child { margin-bottom: 0; }
- @@ -388,7 +387,6 @@ div.admonition > :last-child {
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- @@ -430,6 +428,10 @@ table.docutils td, table.docutils th {
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- @@ -613,7 +615,6 @@ ul.simple p {
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- @@ -631,33 +632,6 @@ dl.citation > dd:after {
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html/_static/doctools.js
- @@ -2,263 +2,325 @@
 * doctools.js * ~~~~~~~~~~~ * - * Base JavaScript utilities for all Sphinx HTML documentation. + * Sphinx JavaScript utilities for all documentation. * * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ - "use strict"; - const _ready = (callback) => { - if (document.readyState !== "loading") { - callback(); - } else { - document.addEventListener("DOMContentLoaded", callback); + /** + * select a different prefix for underscore + */ + $u = _.noConflict(); + + /** + * make the code below compatible with browsers without + * an installed firebug like debugger + if (!window.console || !console.firebug) { + var names = ["log", "debug", "info", "warn", "error", "assert", "dir", + "dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace", + "profile", "profileEnd"]; + window.console = {}; + for (var i = 0; i < names.length; ++i) + window.console[names[i]] = function() {}; + } + */ + + /** + * small helper function to urldecode strings + * + * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL + */ + jQuery.urldecode = function(x) { + if (!x) { + return x } + return decodeURIComponent(x.replace(/\+/g, ' ')); }; /** - * highlight a given string on a node by wrapping it in - * span elements with the given class name. + * small helper function to urlencode strings */ - const _highlight = (node, addItems, text, className) => { - if (node.nodeType === Node.TEXT_NODE) { - const val = node.nodeValue; - const parent = node.parentNode; - const pos = val.toLowerCase().indexOf(text); - if ( - pos >= 0 && - !parent.classList.contains(className) && - !parent.classList.contains("nohighlight") - ) { - let span; + jQuery.urlencode = encodeURIComponent; - const closestNode = parent.closest("body, svg, foreignObject"); - const isInSVG = closestNode && closestNode.matches("svg"); - if (isInSVG) { - span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); - } else { - span = document.createElement("span"); - span.classList.add(className); - } + /** + * This function returns the parsed url parameters of the + * current request. 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translated : translated[0]; }, - ngettext: (singular, plural, n) => { - const translated = Documentation.TRANSLATIONS[singular]; - if (typeof translated !== "undefined") - return translated[Documentation.PLURAL_EXPR(n)]; - return n === 1 ? singular : plural; + ngettext : function(singular, plural, n) { + var translated = Documentation.TRANSLATIONS[singular]; + if (typeof translated === 'undefined') + return (n == 1) ? singular : plural; + return translated[Documentation.PLURALEXPR(n)]; }, - addTranslations: (catalog) => { - Object.assign(Documentation.TRANSLATIONS, catalog.messages); - Documentation.PLURAL_EXPR = new Function( - "n", - `return (${catalog.plural_expr})` - ); - Documentation.LOCALE = catalog.locale; + addTranslations : function(catalog) { + for (var key in catalog.messages) + this.TRANSLATIONS[key] = catalog.messages[key]; + this.PLURAL_EXPR = new Function('n', 'return +(' + catalog.plural_expr + ')'); + this.LOCALE = catalog.locale; }, /** - * highlight the search words provided in the url in the text + * add context elements like header anchor links */ - highlightSearchWords: () => { - const highlight = - new URLSearchParams(window.location.search).get("highlight") || ""; - const terms = highlight.toLowerCase().split(/\s+/).filter(x => x); - if (terms.length === 0) return; // nothing to do - - // There should never be more than one element matching "div.body" - const divBody = document.querySelectorAll("div.body"); - const body = divBody.length ? divBody[0] : document.querySelector("body"); - window.setTimeout(() => { - terms.forEach((term) => _highlightText(body, term, "highlighted")); - }, 10); - - const searchBox = document.getElementById("searchbox"); - if (searchBox === null) return; - searchBox.appendChild( - document - .createRange() - .createContextualFragment( - '' + - '<a href="javascript:Documentation.hideSearchWords()">' + - Documentation.gettext("Hide Search Matches") + - "</a>" - ) - ); + addContextElements : function() { + $('div[id] > :header:first').each(function() { + $('<a class="headerlink">\u00B6</a>'). + attr('href', '#' + this.id). + attr('title', _('Permalink to this headline')). + appendTo(this); + }); + $('dt[id]').each(function() { + $('<a class="headerlink">\u00B6</a>'). + attr('href', '#' + this.id). + attr('title', _('Permalink to this definition')). + appendTo(this); + }); }, /** - * helper function to hide the search marks again + * workaround a firefox stupidity + * see: https://bugzilla.mozilla.org/show_bug.cgi?id=645075 */ - hideSearchWords: () => { - document - .querySelectorAll("#searchbox .highlight-link") - .forEach((el) => el.remove()); - document - .querySelectorAll("span.highlighted") - .forEach((el) => el.classList.remove("highlighted")); - const url = new URL(window.location); - url.searchParams.delete("highlight"); - window.history.replaceState({}, "", url); + fixFirefoxAnchorBug : function() { + if (document.location.hash && $.browser.mozilla) + window.setTimeout(function() { + document.location.href += ''; + }, 10); }, /** - * helper function to focus on search bar + * highlight the search words provided in the url in the text */ - focusSearchBar: () => { - document.querySelectorAll("input[name=q]")[0]?.focus(); + highlightSearchWords : function() { + var params = $.getQueryParameters(); + var terms = (params.highlight) ? params.highlight[0].split(/\s+/) : []; + if (terms.length) { + var body = $('div.body'); + if (!body.length) { + body = $('body'); + } + window.setTimeout(function() { + $.each(terms, function() { + body.highlightText(this.toLowerCase(), 'highlighted'); + }); + }, 10); + $('<a href="javascript:Documentation.' + + 'hideSearchWords()">' + _('Hide Search Matches') + '</a>') + .appendTo($('#searchbox')); + } }, /** - * Initialise the domain index toggle buttons + * init the domain index toggle buttons */ - initDomainIndexTable: () => { - const toggler = (el) => { - const idNumber = el.id.substr(7); - const toggledRows = document.querySelectorAll(`tr.cg-${idNumber}`); - if (el.src.substr(-9) === "minus.png") { - el.src = `${el.src.substr(0, el.src.length - 9)}plus.png`; - toggledRows.forEach((el) => (el.style.display = "none")); - } else { - el.src = `${el.src.substr(0, el.src.length - 8)}minus.png`; - toggledRows.forEach((el) => (el.style.display = "")); - } - }; - - const togglerElements = document.querySelectorAll("img.toggler"); - togglerElements.forEach((el) => - el.addEventListener("click", (event) => toggler(event.currentTarget)) - ); - togglerElements.forEach((el) => (el.style.display = "")); - if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) togglerElements.forEach(toggler); + initIndexTable : function() { + var togglers = $('img.toggler').click(function() { + var src = $(this).attr('src'); + var idnum = $(this).attr('id').substr(7); + $('tr.cg-' + idnum).toggle(); + if (src.substr(-9) === 'minus.png') + $(this).attr('src', src.substr(0, src.length-9) + 'plus.png'); + else + $(this).attr('src', src.substr(0, src.length-8) + 'minus.png'); + }).css('display', ''); + if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) { + togglers.click(); + } }, - initOnKeyListeners: () => { - // only install a listener if it is really needed - if ( - !DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS && - !DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS - ) - return; + /** + * helper function to hide the search marks again + */ + hideSearchWords : function() { + $('#searchbox .highlight-link').fadeOut(300); + $('span.highlighted').removeClass('highlighted'); + var url = new URL(window.location); + url.searchParams.delete('highlight'); + window.history.replaceState({}, '', url); + }, - const blacklistedElements = new Set([ - "TEXTAREA", - "INPUT", - "SELECT", - "BUTTON", - ]); - document.addEventListener("keydown", (event) => { - if (blacklistedElements.has(document.activeElement.tagName)) return; // bail for input elements - if (event.altKey || event.ctrlKey || event.metaKey) return; // bail with special keys + /** + * make the url absolute + */ + makeURL : function(relativeURL) { + return DOCUMENTATION_OPTIONS.URL_ROOT + '/' + relativeURL; + }, - if (!event.shiftKey) { - switch (event.key) { - case "ArrowLeft": - if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; + /** + * get the current relative url + */ + getCurrentURL : function() { + var path = document.location.pathname; + var parts = path.split(/\//); + $.each(DOCUMENTATION_OPTIONS.URL_ROOT.split(/\//), function() { + if (this === '..') + parts.pop(); + }); + var url = parts.join('/'); + return path.substring(url.lastIndexOf('/') + 1, path.length - 1); + }, - const prevLink = document.querySelector('link[rel="prev"]'); - if (prevLink && prevLink.href) { - window.location.href = prevLink.href; - event.preventDefault(); + initOnKeyListeners: function() { + $(document).keydown(function(event) { + var activeElementType = document.activeElement.tagName; + // don't navigate when in search box, textarea, dropdown or button + if (activeElementType !== 'TEXTAREA' && activeElementType !== 'INPUT' && activeElementType !== 'SELECT' + && activeElementType !== 'BUTTON' && !event.altKey && !event.ctrlKey && !event.metaKey + && !event.shiftKey) { + switch (event.keyCode) { + case 37: // left + var prevHref = $('link[rel="prev"]').prop('href'); + if (prevHref) { + window.location.href = prevHref; + return false; } break; - case "ArrowRight": - if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; - - const nextLink = document.querySelector('link[rel="next"]'); - if (nextLink && nextLink.href) { - window.location.href = nextLink.href; - event.preventDefault(); + case 39: // right + var nextHref = $('link[rel="next"]').prop('href'); + if (nextHref) { + window.location.href = nextHref; + return false; } break; - case "Escape": - if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; - Documentation.hideSearchWords(); - event.preventDefault(); } } - - // some keyboard layouts may need Shift to get / - switch (event.key) { - case "/": - if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; - Documentation.focusSearchBar(); - event.preventDefault(); - } }); - }, + } }; // quick alias for translations - const _ = Documentation.gettext; + _ = Documentation.gettext; - _ready(Documentation.init); + $(document).ready(function() { + Documentation.init(); + });
html/_static/documentation_options.js
- @@ -1,14 +1,12 @@
  var DOCUMENTATION_OPTIONS = { URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'), VERSION: '0.1', - LANGUAGE: 'en', + LANGUAGE: 'None', COLLAPSE_INDEX: false, BUILDER: 'html', FILE_SUFFIX: '.html', LINK_SUFFIX: '.html', HAS_SOURCE: true, SOURCELINK_SUFFIX: '.txt', - NAVIGATION_WITH_KEYS: false, - SHOW_SEARCH_SUMMARY: true, - ENABLE_SEARCH_SHORTCUTS: false, + NAVIGATION_WITH_KEYS: false };
html/_static/jquery-3.6.0.js > html/_static/jquery-3.5.1.js
- @@ -1,15 +1,15 @@
 /*! - * jQuery JavaScript Library v3.6.0 + * jQuery JavaScript Library v3.5.1 * https://jquery.com/ * * Includes Sizzle.js * https://sizzlejs.com/ * - * Copyright OpenJS Foundation and other contributors + * Copyright JS Foundation and other contributors * Released under the MIT license * https://jquery.org/license * - * Date: 2021-03-02T17:08Z + * Date: 2020-05-04T22:49Z */ ( function( global, factory ) {
- @@ -76,16 +76,12 @@ var support = {};
 var isFunction = function isFunction( obj ) { - // Support: Chrome <=57, Firefox <=52 - // In some browsers, typeof returns "function" for HTML <object> elements - // (i.e., `typeof document.createElement( "object" ) === "function"`). - // We don't want to classify *any* DOM node as a function. - // Support: QtWeb <=3.8.5, WebKit <=534.34, wkhtmltopdf tool <=0.12.5 - // Plus for old WebKit, typeof returns "function" for HTML collections - // (e.g., `typeof document.getElementsByTagName("div") === "function"`). (gh-4756) - return typeof obj === "function" && typeof obj.nodeType !== "number" && - typeof obj.item !== "function"; - }; + // Support: Chrome <=57, Firefox <=52 + // In some browsers, typeof returns "function" for HTML <object> elements + // (i.e., `typeof document.createElement( "object" ) === "function"`). + // We don't want to classify *any* DOM node as a function. + return typeof obj === "function" && typeof obj.nodeType !== "number"; + }; var isWindow = function isWindow( obj ) {
- @@ -151,7 +147,7 @@ function toType( obj ) {
 var - version = "3.6.0", + version = "3.5.1", // Define a local copy of jQuery jQuery = function( selector, context ) {
- @@ -405,7 +401,7 @@ jQuery.extend( {
 if ( isArrayLike( Object( arr ) ) ) { jQuery.merge( ret, typeof arr === "string" ? - [ arr ] : arr + [ arr ] : arr ); } else { push.call( ret, arr );
- @@ -500,9 +496,9 @@ if ( typeof Symbol === "function" ) {
 // Populate the class2type map jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ), - function( _i, name ) { - class2type[ "[object " + name + "]" ] = name.toLowerCase(); - } ); + function( _i, name ) { + class2type[ "[object " + name + "]" ] = name.toLowerCase(); + } ); function isArrayLike( obj ) {
- @@ -522,14 +518,14 @@ function isArrayLike( obj ) {
 } var Sizzle = /*! - * Sizzle CSS Selector Engine v2.3.6 + * Sizzle CSS Selector Engine v2.3.5 * https://sizzlejs.com/ * * Copyright JS Foundation and other contributors * Released under the MIT license * https://js.foundation/ * - * Date: 2021-02-16 + * Date: 2020-03-14 */ ( function( window ) { var i,
- @@ -1112,8 +1108,8 @@ support = Sizzle.support = {};
 * @returns {Boolean} True iff elem is a non-HTML XML node */ isXML = Sizzle.isXML = function( elem ) { - var namespace = elem && elem.namespaceURI, - docElem = elem && ( elem.ownerDocument || elem ).documentElement; + var namespace = elem.namespaceURI, + docElem = ( elem.ownerDocument || elem ).documentElement; // Support: IE <=8 // Assume HTML when documentElement doesn't yet exist, such as inside loading iframes
- @@ -3028,9 +3024,9 @@ var rneedsContext = jQuery.expr.match.needsContext;
 function nodeName( elem, name ) { - return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); + return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); - } + }; var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i );
- @@ -4001,8 +3997,8 @@ jQuery.extend( {
 resolveContexts = Array( i ), resolveValues = slice.call( arguments ), - // the primary Deferred - primary = jQuery.Deferred(), + // the master Deferred + master = jQuery.Deferred(), // subordinate callback factory updateFunc = function( i ) {
- @@ -4010,30 +4006,30 @@ jQuery.extend( {
 resolveContexts[ i ] = this; resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value; if ( !( --remaining ) ) { - primary.resolveWith( resolveContexts, resolveValues ); + master.resolveWith( resolveContexts, resolveValues ); } }; }; // Single- and empty arguments are adopted like Promise.resolve if ( remaining <= 1 ) { - adoptValue( singleValue, primary.done( updateFunc( i ) ).resolve, primary.reject, + adoptValue( singleValue, master.done( updateFunc( i ) ).resolve, master.reject, !remaining ); // Use .then() to unwrap secondary thenables (cf. gh-3000) - if ( primary.state() === "pending" || + if ( master.state() === "pending" || isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { - return primary.then(); + return master.then(); } } // Multiple arguments are aggregated like Promise.all array elements while ( i-- ) { - adoptValue( resolveValues[ i ], updateFunc( i ), primary.reject ); + adoptValue( resolveValues[ i ], updateFunc( i ), master.reject ); } - return primary.promise(); + return master.promise(); } } );
- @@ -4184,8 +4180,8 @@ var access = function( elems, fn, key, value, chainable, emptyGet, raw ) {
 for ( ; i < len; i++ ) { fn( elems[ i ], key, raw ? - value : - value.call( elems[ i ], i, fn( elems[ i ], key ) ) + value : + value.call( elems[ i ], i, fn( elems[ i ], key ) ) ); } }
- @@ -5093,7 +5089,10 @@ function buildFragment( elems, context, scripts, selection, ignored ) {
 } - var rtypenamespace = /^([^.]*)(?:\.(.+)|)/; + var + rkeyEvent = /^key/, + rmouseEvent = /^(?:mouse|pointer|contextmenu|drag|drop)|click/, + rtypenamespace = /^([^.]*)(?:\.(.+)|)/; function returnTrue() { return true;
- @@ -5388,8 +5387,8 @@ jQuery.event = {
 event = jQuery.event.fix( nativeEvent ), handlers = ( - dataPriv.get( this, "events" ) || Object.create( null ) - )[ event.type ] || [], + dataPriv.get( this, "events" ) || Object.create( null ) + )[ event.type ] || [], special = jQuery.event.special[ event.type ] || {}; // Use the fix-ed jQuery.Event rather than the (read-only) native event
- @@ -5513,12 +5512,12 @@ jQuery.event = {
 get: isFunction( hook ) ? function() { if ( this.originalEvent ) { - return hook( this.originalEvent ); + return hook( this.originalEvent ); } } : function() { if ( this.originalEvent ) { - return this.originalEvent[ name ]; + return this.originalEvent[ name ]; } },
- @@ -5657,13 +5656,7 @@ function leverageNative( el, type, expectSync ) {
 // Cancel the outer synthetic event event.stopImmediatePropagation(); event.preventDefault(); - - // Support: Chrome 86+ - // In Chrome, if an element having a focusout handler is blurred by - // clicking outside of it, it invokes the handler synchronously. If - // that handler calls `.remove()` on the element, the data is cleared, - // leaving `result` undefined. We need to guard against this. - return result && result.value; + return result.value; } // If this is an inner synthetic event for an event with a bubbling surrogate
- @@ -5828,7 +5821,34 @@ jQuery.each( {
 targetTouches: true, toElement: true, touches: true, - which: true + + which: function( event ) { + var button = event.button; + + // Add which for key events + if ( event.which == null && rkeyEvent.test( event.type ) ) { + return event.charCode != null ? event.charCode : event.keyCode; + } + + // Add which for click: 1 === left; 2 === middle; 3 === right + if ( !event.which && button !== undefined && rmouseEvent.test( event.type ) ) { + if ( button & 1 ) { + return 1; + } + + if ( button & 2 ) { + return 3; + } + + if ( button & 4 ) { + return 2; + } + + return 0; + } + + return event.which; + } }, jQuery.event.addProp ); jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) {
- @@ -5854,12 +5874,6 @@ jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateTyp
 return true; }, - // Suppress native focus or blur as it's already being fired - // in leverageNative. - _default: function() { - return true; - }, - delegateType: delegateType }; } );
- @@ -6527,10 +6541,6 @@ var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" );
 // set in CSS while `offset*` properties report correct values. // Behavior in IE 9 is more subtle than in newer versions & it passes // some versions of this test; make sure not to make it pass there! - // - // Support: Firefox 70+ - // Only Firefox includes border widths - // in computed dimensions. (gh-4529) reliableTrDimensions: function() { var table, tr, trChild, trStyle; if ( reliableTrDimensionsVal == null ) {
- @@ -6538,32 +6548,17 @@ var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" );
 tr = document.createElement( "tr" ); trChild = document.createElement( "div" ); - table.style.cssText = "position:absolute;left:-11111px;border-collapse:separate"; - tr.style.cssText = "border:1px solid"; - - // Support: Chrome 86+ - // Height set through cssText does not get applied. - // Computed height then comes back as 0. + table.style.cssText = "position:absolute;left:-11111px"; tr.style.height = "1px"; trChild.style.height = "9px"; - // Support: Android 8 Chrome 86+ - // In our bodyBackground.html iframe, - // display for all div elements is set to "inline", - // which causes a problem only in Android 8 Chrome 86. - // Ensuring the div is display: block - // gets around this issue. - trChild.style.display = "block"; - documentElement .appendChild( table ) .appendChild( tr ) .appendChild( trChild ); trStyle = window.getComputedStyle( tr ); - reliableTrDimensionsVal = ( parseInt( trStyle.height, 10 ) + - parseInt( trStyle.borderTopWidth, 10 ) + - parseInt( trStyle.borderBottomWidth, 10 ) ) === tr.offsetHeight; + reliableTrDimensionsVal = parseInt( trStyle.height ) > 3; documentElement.removeChild( table ); }
- @@ -7027,10 +7022,10 @@ jQuery.each( [ "height", "width" ], function( _i, dimension ) {
 // Running getBoundingClientRect on a disconnected node // in IE throws an error. ( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ? - swap( elem, cssShow, function() { - return getWidthOrHeight( elem, dimension, extra ); - } ) : - getWidthOrHeight( elem, dimension, extra ); + swap( elem, cssShow, function() { + return getWidthOrHeight( elem, dimension, extra ); + } ) : + getWidthOrHeight( elem, dimension, extra ); } },
- @@ -7089,7 +7084,7 @@ jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft,
 swap( elem, { marginLeft: 0 }, function() { return elem.getBoundingClientRect().left; } ) - ) + "px"; + ) + "px"; } } );
- @@ -7228,7 +7223,7 @@ Tween.propHooks = {
 if ( jQuery.fx.step[ tween.prop ] ) { jQuery.fx.step[ tween.prop ]( tween ); } else if ( tween.elem.nodeType === 1 && ( - jQuery.cssHooks[ tween.prop ] || + jQuery.cssHooks[ tween.prop ] || tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) { jQuery.style( tween.elem, tween.prop, tween.now + tween.unit ); } else {
- @@ -7473,7 +7468,7 @@ function defaultPrefilter( elem, props, opts ) {
 anim.done( function() { - /* eslint-enable no-loop-func */ + /* eslint-enable no-loop-func */ // The final step of a "hide" animation is actually hiding the element if ( !hidden ) {
- @@ -7593,7 +7588,7 @@ function Animation( elem, properties, options ) {
 tweens: [], createTween: function( prop, end ) { var tween = jQuery.Tween( elem, animation.opts, prop, end, - animation.opts.specialEasing[ prop ] || animation.opts.easing ); + animation.opts.specialEasing[ prop ] || animation.opts.easing ); animation.tweens.push( tween ); return tween; },
- @@ -7766,8 +7761,7 @@ jQuery.fn.extend( {
 anim.stop( true ); } }; - - doAnimation.finish = doAnimation; + doAnimation.finish = doAnimation; return empty || optall.queue === false ? this.each( doAnimation ) :
- @@ -8407,8 +8401,8 @@ jQuery.fn.extend( {
 if ( this.setAttribute ) { this.setAttribute( "class", className || value === false ? - "" : - dataPriv.get( this, "__className__" ) || "" + "" : + dataPriv.get( this, "__className__" ) || "" ); } }
- @@ -8423,7 +8417,7 @@ jQuery.fn.extend( {
 while ( ( elem = this[ i++ ] ) ) { if ( elem.nodeType === 1 && ( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) { - return true; + return true; } }
- @@ -8713,7 +8707,9 @@ jQuery.extend( jQuery.event, {
 special.bindType || type; // jQuery handler - handle = ( dataPriv.get( cur, "events" ) || Object.create( null ) )[ event.type ] && + handle = ( + dataPriv.get( cur, "events" ) || Object.create( null ) + )[ event.type ] && dataPriv.get( cur, "handle" ); if ( handle ) { handle.apply( cur, data );
- @@ -8860,7 +8856,7 @@ var rquery = ( /\?/ );
 // Cross-browser xml parsing jQuery.parseXML = function( data ) { - var xml, parserErrorElem; + var xml; if ( !data || typeof data !== "string" ) { return null; }
- @@ -8869,17 +8865,12 @@ jQuery.parseXML = function( data ) {
 // IE throws on parseFromString with invalid input. try { xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" ); - } catch ( e ) {} + } catch ( e ) { + xml = undefined; + } - parserErrorElem = xml && xml.getElementsByTagName( "parsererror" )[ 0 ]; - if ( !xml || parserErrorElem ) { - jQuery.error( "Invalid XML: " + ( - parserErrorElem ? - jQuery.map( parserErrorElem.childNodes, function( el ) { - return el.textContent; - } ).join( "\n" ) : - data - ) ); + if ( !xml || xml.getElementsByTagName( "parsererror" ).length ) { + jQuery.error( "Invalid XML: " + data ); } return xml; };
- @@ -8980,14 +8971,16 @@ jQuery.fn.extend( {
 // Can add propHook for "elements" to filter or add form elements var elements = jQuery.prop( this, "elements" ); return elements ? jQuery.makeArray( elements ) : this; - } ).filter( function() { + } ) + .filter( function() { var type = this.type; // Use .is( ":disabled" ) so that fieldset[disabled] works return this.name && !jQuery( this ).is( ":disabled" ) && rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) && ( this.checked || !rcheckableType.test( type ) ); - } ).map( function( _i, elem ) { + } ) + .map( function( _i, elem ) { var val = jQuery( this ).val(); if ( val == null ) {
- @@ -9040,8 +9033,7 @@ var
 // Anchor tag for parsing the document origin originAnchor = document.createElement( "a" ); - - originAnchor.href = location.href; + originAnchor.href = location.href; // Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport function addToPrefiltersOrTransports( structure ) {
- @@ -9422,8 +9414,8 @@ jQuery.extend( {
 // Context for global events is callbackContext if it is a DOM node or jQuery collection globalEventContext = s.context && ( callbackContext.nodeType || callbackContext.jquery ) ? - jQuery( callbackContext ) : - jQuery.event, + jQuery( callbackContext ) : + jQuery.event, // Deferreds deferred = jQuery.Deferred(),
- @@ -9735,10 +9727,8 @@ jQuery.extend( {
 response = ajaxHandleResponses( s, jqXHR, responses ); } - // Use a noop converter for missing script but not if jsonp - if ( !isSuccess && - jQuery.inArray( "script", s.dataTypes ) > -1 && - jQuery.inArray( "json", s.dataTypes ) < 0 ) { + // Use a noop converter for missing script + if ( !isSuccess && jQuery.inArray( "script", s.dataTypes ) > -1 ) { s.converters[ "text script" ] = function() {}; }
- @@ -10476,6 +10466,12 @@ jQuery.offset = {
 options.using.call( elem, props ); } else { + if ( typeof props.top === "number" ) { + props.top += "px"; + } + if ( typeof props.left === "number" ) { + props.left += "px"; + } curElem.css( props ); } }
- @@ -10644,11 +10640,8 @@ jQuery.each( [ "top", "left" ], function( _i, prop ) {
 // Create innerHeight, innerWidth, height, width, outerHeight and outerWidth methods jQuery.each( { Height: "height", Width: "width" }, function( name, type ) { - jQuery.each( { - padding: "inner" + name, - content: type, - "": "outer" + name - }, function( defaultExtra, funcName ) { + jQuery.each( { padding: "inner" + name, content: type, "": "outer" + name }, + function( defaultExtra, funcName ) { // Margin is only for outerHeight, outerWidth jQuery.fn[ funcName ] = function( margin, value ) {
- @@ -10733,8 +10726,7 @@ jQuery.fn.extend( {
 } } ); - jQuery.each( - ( "blur focus focusin focusout resize scroll click dblclick " + + jQuery.each( ( "blur focus focusin focusout resize scroll click dblclick " + "mousedown mouseup mousemove mouseover mouseout mouseenter mouseleave " + "change select submit keydown keypress keyup contextmenu" ).split( " " ), function( _i, name ) {
- @@ -10745,8 +10737,7 @@ jQuery.each(
 this.on( name, null, data, fn ) : this.trigger( name ); }; - } - ); + } );
html/_static/jquery.js
- @@ -1,2 +1,2 @@
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html/_static/language_data.js
- @@ -10,7 +10,7 @@
 * */ - var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"]; + var stopwords = ["a","and","are","as","at","be","but","by","for","if","in","into","is","it","near","no","not","of","on","or","such","that","the","their","then","there","these","they","this","to","was","will","with"]; /* Non-minified version is copied as a separate JS file, is available */
- @@ -197,3 +197,101 @@ var Stemmer = function() {
 } } + + + + var splitChars = (function() { + var result = {}; + var singles = [96, 180, 187, 191, 215, 247, 749, 885, 903, 907, 909, 930, 1014, 1648, + 1748, 1809, 2416, 2473, 2481, 2526, 2601, 2609, 2612, 2615, 2653, 2702, + 2706, 2729, 2737, 2740, 2857, 2865, 2868, 2910, 2928, 2948, 2961, 2971, + 2973, 3085, 3089, 3113, 3124, 3213, 3217, 3241, 3252, 3295, 3341, 3345, + 3369, 3506, 3516, 3633, 3715, 3721, 3736, 3744, 3748, 3750, 3756, 3761, + 3781, 3912, 4239, 4347, 4681, 4695, 4697, 4745, 4785, 4799, 4801, 4823, + 4881, 5760, 5901, 5997, 6313, 7405, 8024, 8026, 8028, 8030, 8117, 8125, + 8133, 8181, 8468, 8485, 8487, 8489, 8494, 8527, 11311, 11359, 11687, 11695, + 11703, 11711, 11719, 11727, 11735, 12448, 12539, 43010, 43014, 43019, 43587, + 43696, 43713, 64286, 64297, 64311, 64317, 64319, 64322, 64325, 65141]; + var i, j, start, end; + for (i = 0; i < singles.length; i++) { + result[singles[i]] = true; + } + var ranges = [[0, 47], [58, 64], [91, 94], [123, 169], [171, 177], [182, 184], [706, 709], + [722, 735], [741, 747], [751, 879], [888, 889], [894, 901], [1154, 1161], + [1318, 1328], [1367, 1368], [1370, 1376], [1416, 1487], [1515, 1519], [1523, 1568], + [1611, 1631], [1642, 1645], [1750, 1764], [1767, 1773], [1789, 1790], [1792, 1807], + [1840, 1868], [1958, 1968], [1970, 1983], [2027, 2035], [2038, 2041], [2043, 2047], + [2070, 2073], [2075, 2083], [2085, 2087], [2089, 2307], [2362, 2364], [2366, 2383], + [2385, 2391], [2402, 2405], [2419, 2424], [2432, 2436], [2445, 2446], [2449, 2450], + [2483, 2485], [2490, 2492], [2494, 2509], [2511, 2523], [2530, 2533], [2546, 2547], + [2554, 2564], [2571, 2574], [2577, 2578], [2618, 2648], [2655, 2661], [2672, 2673], + [2677, 2692], [2746, 2748], [2750, 2767], [2769, 2783], [2786, 2789], [2800, 2820], + [2829, 2830], [2833, 2834], [2874, 2876], [2878, 2907], [2914, 2917], [2930, 2946], + [2955, 2957], [2966, 2968], [2976, 2978], [2981, 2983], [2987, 2989], [3002, 3023], + [3025, 3045], [3059, 3076], [3130, 3132], 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6319], [6390, 6399], [6429, 6469], [6510, 6511], [6517, 6527], [6572, 6592], + [6600, 6607], [6619, 6655], [6679, 6687], [6741, 6783], [6794, 6799], [6810, 6822], + [6824, 6916], [6964, 6980], [6988, 6991], [7002, 7042], [7073, 7085], [7098, 7167], + [7204, 7231], [7242, 7244], [7294, 7400], [7410, 7423], [7616, 7679], [7958, 7959], + [7966, 7967], [8006, 8007], [8014, 8015], [8062, 8063], [8127, 8129], [8141, 8143], + [8148, 8149], [8156, 8159], [8173, 8177], [8189, 8303], [8306, 8307], [8314, 8318], + [8330, 8335], [8341, 8449], [8451, 8454], [8456, 8457], [8470, 8472], [8478, 8483], + [8506, 8507], [8512, 8516], [8522, 8525], [8586, 9311], [9372, 9449], [9472, 10101], + [10132, 11263], [11493, 11498], [11503, 11516], [11518, 11519], [11558, 11567], + [11622, 11630], [11632, 11647], [11671, 11679], [11743, 11822], [11824, 12292], + [12296, 12320], [12330, 12336], [12342, 12343], [12349, 12352], [12439, 12444], + [12544, 12548], [12590, 12592], [12687, 12689], [12694, 12703], [12728, 12783], + [12800, 12831], [12842, 12880], [12896, 12927], [12938, 12976], [12992, 13311], + [19894, 19967], [40908, 40959], [42125, 42191], [42238, 42239], [42509, 42511], + [42540, 42559], [42592, 42593], [42607, 42622], [42648, 42655], [42736, 42774], + [42784, 42785], [42889, 42890], [42893, 43002], [43043, 43055], [43062, 43071], + [43124, 43137], [43188, 43215], [43226, 43249], [43256, 43258], [43260, 43263], + [43302, 43311], [43335, 43359], [43389, 43395], [43443, 43470], [43482, 43519], + [43561, 43583], [43596, 43599], [43610, 43615], [43639, 43641], [43643, 43647], + [43698, 43700], [43703, 43704], [43710, 43711], [43715, 43738], [43742, 43967], + [44003, 44015], [44026, 44031], [55204, 55215], [55239, 55242], [55292, 55295], + [57344, 63743], [64046, 64047], [64110, 64111], [64218, 64255], [64263, 64274], + [64280, 64284], [64434, 64466], [64830, 64847], [64912, 64913], [64968, 65007], + [65020, 65135], [65277, 65295], [65306, 65312], [65339, 65344], [65371, 65381], + [65471, 65473], [65480, 65481], [65488, 65489], [65496, 65497]]; + for (i = 0; i < ranges.length; i++) { + start = ranges[i][0]; + end = ranges[i][1]; + for (j = start; j <= end; j++) { + result[j] = true; + } + } + return result; + })(); + + function splitQuery(query) { + var result = []; + var start = -1; + for (var i = 0; i < query.length; i++) { + if (splitChars[query.charCodeAt(i)]) { + if (start !== -1) { + result.push(query.slice(start, i)); + start = -1; + } + } else if (start === -1) { + start = i; + } + } + if (start !== -1) { + result.push(query.slice(start)); + } + return result; + } + +
html/_static/searchtools.js
- @@ -8,20 +8,18 @@
 * :license: BSD, see LICENSE for details. * */ - "use strict"; - /** - * Simple result scoring code. - */ - if (typeof Scorer === "undefined") { + if (!Scorer) { + /** + * Simple result scoring code. + */ var Scorer = { // Implement the following function to further tweak the score for each result - // The function takes a result array [docname, title, anchor, descr, score, filename] + // The function takes a result array [filename, title, anchor, descr, score] // and returns the new score. /* - score: result => { - const [docname, title, anchor, descr, score, filename] = result - return score + score: function(result) { + return result[4]; }, */
- @@ -30,11 +28,9 @@ if (typeof Scorer === "undefined") {
 // or matches in the last dotted part of the object name objPartialMatch: 6, // Additive scores depending on the priority of the object - objPrio: { - 0: 15, // used to be importantResults - 1: 5, // used to be objectResults - 2: -5, // used to be unimportantResults - }, + objPrio: {0: 15, // used to be importantResults + 1: 5, // used to be objectResults + 2: -5}, // used to be unimportantResults // Used when the priority is not in the mapping. objPrioDefault: 0,
- @@ -43,455 +39,456 @@ if (typeof Scorer === "undefined") {
 partialTitle: 7, // query found in terms term: 5, - partialTerm: 2, + partialTerm: 2 }; } - const _removeChildren = (element) => { - while (element && element.lastChild) element.removeChild(element.lastChild); - }; - - /** - * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping - */ - const _escapeRegExp = (string) => - string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string - - const _displayItem = (item, highlightTerms, searchTerms) => { - const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; - const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; - const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; - const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; - const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; - - const [docName, title, anchor, descr] = item; - - let listItem = document.createElement("li"); - let requestUrl; - let linkUrl; - if (docBuilder === "dirhtml") { - // dirhtml builder - let dirname = docName + "/"; - if (dirname.match(/\/index\/$/)) - dirname = dirname.substring(0, dirname.length - 6); - else if (dirname === "index/") dirname = ""; - requestUrl = docUrlRoot + dirname; - linkUrl = requestUrl; - } else { - // normal html builders - requestUrl = docUrlRoot + docName + docFileSuffix; - linkUrl = docName + docLinkSuffix; - } - const params = new URLSearchParams(); - params.set("highlight", [...highlightTerms].join(" ")); - let linkEl = listItem.appendChild(document.createElement("a")); - linkEl.href = linkUrl + "?" + params.toString() + anchor; - linkEl.innerHTML = title; - if (descr) - listItem.appendChild(document.createElement("span")).innerText = - " (" + descr + ")"; - else if (showSearchSummary) - fetch(requestUrl) - .then((responseData) => responseData.text()) - .then((data) => { - if (data) - listItem.appendChild( - Search.makeSearchSummary(data, searchTerms, highlightTerms) - ); - }); - Search.output.appendChild(listItem); - }; - const _finishSearch = (resultCount) => { - Search.stopPulse(); - Search.title.innerText = _("Search Results"); - if (!resultCount) - Search.status.innerText = Documentation.gettext( - "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories." - ); - else - Search.status.innerText = _( - `Search finished, found ${resultCount} page(s) matching the search query.` - ); - }; - const _displayNextItem = ( - results, - resultCount, - highlightTerms, - searchTerms - ) => { - // results left, load the summary and display it - // this is intended to be dynamic (don't sub resultsCount) - if (results.length) { - _displayItem(results.pop(), highlightTerms, searchTerms); - setTimeout( - () => _displayNextItem(results, resultCount, highlightTerms, searchTerms), - 5 - ); + if (!splitQuery) { + function splitQuery(query) { + return query.split(/\s+/); } - // search finished, update title and status message - else _finishSearch(resultCount); - }; - - /** - * Default splitQuery function. Can be overridden in ``sphinx.search`` with a - * custom function per language. - * - * The regular expression works by splitting the string on consecutive characters - * that are not Unicode letters, numbers, underscores, or emoji characters. - * This is the same as ``\W+`` in Python, preserving the surrogate pair area. - */ - if (typeof splitQuery === "undefined") { - var splitQuery = (query) => query - .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu) - .filter(term => term) // remove remaining empty strings } /** * Search Module */ - const Search = { - _index: null, - _queued_query: null, - _pulse_status: -1, - - htmlToText: (htmlString) => { - const htmlElement = document - .createRange() - .createContextualFragment(htmlString); - _removeChildren(htmlElement.querySelectorAll(".headerlink")); - const docContent = htmlElement.querySelector('[role="main"]'); - if (docContent !== undefined) return docContent.textContent; - console.warn( - "Content block not found. Sphinx search tries to obtain it via '[role=main]'. Could you check your theme or template." - ); - return ""; + var Search = { + + _index : null, + _queued_query : null, + _pulse_status : -1, + + htmlToText : function(htmlString) { + var virtualDocument = document.implementation.createHTMLDocument('virtual'); + var htmlElement = $(htmlString, virtualDocument); + htmlElement.find('.headerlink').remove(); + docContent = htmlElement.find('[role=main]')[0]; + if(docContent === undefined) { + console.warn("Content block not found. Sphinx search tries to obtain it " + + "via '[role=main]'. Could you check your theme or template."); + return ""; + } + return docContent.textContent || docContent.innerText; }, - init: () => { - const query = new URLSearchParams(window.location.search).get("q"); - document - .querySelectorAll('input[name="q"]') - .forEach((el) => (el.value = query)); - if (query) Search.performSearch(query); + init : function() { + var params = $.getQueryParameters(); + if (params.q) { + var query = params.q[0]; + $('input[name="q"]')[0].value = query; + this.performSearch(query); + } }, - loadIndex: (url) => - (document.body.appendChild(document.createElement("script")).src = url), + loadIndex : function(url) { + $.ajax({type: "GET", url: url, data: null, + dataType: "script", cache: true, + complete: function(jqxhr, textstatus) { + if (textstatus != "success") { + document.getElementById("searchindexloader").src = url; + } + }}); + }, - setIndex: (index) => { - Search._index = index; - if (Search._queued_query !== null) { - const query = Search._queued_query; - Search._queued_query = null; - Search.query(query); + setIndex : function(index) { + var q; + this._index = index; + if ((q = this._queued_query) !== null) { + this._queued_query = null; + Search.query(q); } }, - hasIndex: () => Search._index !== null, - - deferQuery: (query) => (Search._queued_query = query), + hasIndex : function() { + return this._index !== null; + }, - stopPulse: () => (Search._pulse_status = -1), + deferQuery : function(query) { + this._queued_query = query; + }, - startPulse: () => { - if (Search._pulse_status >= 0) return; + stopPulse : function() { + this._pulse_status = 0; + }, - const pulse = () => { + startPulse : function() { + if (this._pulse_status >= 0) + return; + function pulse() { + var i; Search._pulse_status = (Search._pulse_status + 1) % 4; - Search.dots.innerText = ".".repeat(Search._pulse_status); - if (Search._pulse_status >= 0) window.setTimeout(pulse, 500); - }; + var dotString = ''; + for (i = 0; i < Search._pulse_status; i++) + dotString += '.'; + Search.dots.text(dotString); + if (Search._pulse_status > -1) + window.setTimeout(pulse, 500); + } pulse(); }, /** * perform a search for something (or wait until index is loaded) */ - performSearch: (query) => { + performSearch : function(query) { // create the required interface elements - const searchText = document.createElement("h2"); - searchText.textContent = _("Searching"); - const searchSummary = document.createElement("p"); - searchSummary.classList.add("search-summary"); - searchSummary.innerText = ""; - const searchList = document.createElement("ul"); - searchList.classList.add("search"); - - const out = document.getElementById("search-results"); - Search.title = out.appendChild(searchText); - Search.dots = Search.title.appendChild(document.createElement("span")); - Search.status = out.appendChild(searchSummary); - Search.output = out.appendChild(searchList); - - const searchProgress = document.getElementById("search-progress"); - // Some themes don't use the search progress node - if (searchProgress) { - searchProgress.innerText = _("Preparing search..."); - } - Search.startPulse(); + this.out = $('#search-results'); + this.title = $('<h2>' + _('Searching') + '</h2>').appendTo(this.out); + this.dots = $('').appendTo(this.title); + this.status = $(' ').appendTo(this.out); + this.output = $('<ul class="search"/>').appendTo(this.out); + + $('#search-progress').text(_('Preparing search...')); + this.startPulse(); // index already loaded, the browser was quick! - if (Search.hasIndex()) Search.query(query); - else Search.deferQuery(query); + if (this.hasIndex()) + this.query(query); + else + this.deferQuery(query); }, /** * execute search (requires search index to be loaded) */ - query: (query) => { - // stem the search terms and add them to the correct list - const stemmer = new Stemmer(); - const searchTerms = new Set(); - const excludedTerms = new Set(); - const highlightTerms = new Set(); - const objectTerms = new Set(splitQuery(query.toLowerCase().trim())); - splitQuery(query.trim()).forEach((queryTerm) => { - const queryTermLower = queryTerm.toLowerCase(); - - // maybe skip this "word" - // stopwords array is from language_data.js - if ( - stopwords.indexOf(queryTermLower) !== -1 || - queryTerm.match(/^\d+$/) - ) - return; + query : function(query) { + var i; + + // stem the searchterms and add them to the correct list + var stemmer = new Stemmer(); + var searchterms = []; + var excluded = []; + var hlterms = []; + var tmp = splitQuery(query); + var objectterms = []; + for (i = 0; i < tmp.length; i++) { + if (tmp[i] !== "") { + objectterms.push(tmp[i].toLowerCase()); + } + if ($u.indexOf(stopwords, tmp[i].toLowerCase()) != -1 || tmp[i] === "") { + // skip this "word" + continue; + } // stem the word - let word = stemmer.stemWord(queryTermLower); + var word = stemmer.stemWord(tmp[i].toLowerCase()); + // prevent stemmer from cutting word smaller than two chars + if(word.length < 3 && tmp[i].length >= 3) { + word = tmp[i]; + } + var toAppend; // select the correct list - if (word[0] === "-") excludedTerms.add(word.substr(1)); + if (word[0] == '-') { + toAppend = excluded; + word = word.substr(1); + } else { - searchTerms.add(word); - highlightTerms.add(queryTermLower); + toAppend = searchterms; + hlterms.push(tmp[i].toLowerCase()); } - }); + // only add if not already in the list + if (!$u.contains(toAppend, word)) + toAppend.push(word); + } + var highlightstring = '?highlight=' + $.urlencode(hlterms.join(" ")); - // console.debug("SEARCH: searching for:"); - // console.info("required: ", [...searchTerms]); - // console.info("excluded: ", [...excludedTerms]); + // console.debug('SEARCH: searching for:'); + // console.info('required: ', searchterms); + // console.info('excluded: ', excluded); + + // prepare search + var terms = this._index.terms; + var titleterms = this._index.titleterms; - // array of [docname, title, anchor, descr, score, filename] - let results = []; - _removeChildren(document.getElementById("search-progress")); + // array of [filename, title, anchor, descr, score] + var results = []; + $('#search-progress').empty(); // lookup as object - objectTerms.forEach((term) => - results.push(...Search.performObjectSearch(term, objectTerms)) - ); + for (i = 0; i < objectterms.length; i++) { + var others = [].concat(objectterms.slice(0, i), + objectterms.slice(i+1, objectterms.length)); + results = results.concat(this.performObjectSearch(objectterms[i], others)); + } // lookup as search terms in fulltext - results.push(...Search.performTermsSearch(searchTerms, excludedTerms)); + results = results.concat(this.performTermsSearch(searchterms, excluded, terms, titleterms)); // let the scorer override scores with a custom scoring function - if (Scorer.score) results.forEach((item) => (item[4] = Scorer.score(item))); + if (Scorer.score) { + for (i = 0; i < results.length; i++) + results[i][4] = Scorer.score(results[i]); + } // now sort the results by score (in opposite order of appearance, since the // display function below uses pop() to retrieve items) and then // alphabetically - results.sort((a, b) => { - const leftScore = a[4]; - const rightScore = b[4]; - if (leftScore === rightScore) { + results.sort(function(a, b) { + var left = a[4]; + var right = b[4]; + if (left > right) { + return 1; + } else if (left < right) { + return -1; + } else { // same score: sort alphabetically - const leftTitle = a[1].toLowerCase(); - const rightTitle = b[1].toLowerCase(); - if (leftTitle === rightTitle) return 0; - return leftTitle > rightTitle ? -1 : 1; // inverted is intentional + left = a[1].toLowerCase(); + right = b[1].toLowerCase(); + return (left > right) ? -1 : ((left < right) ? 1 : 0); } - return leftScore > rightScore ? 1 : -1; }); - // remove duplicate search results - // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept - let seen = new Set(); - results = results.reverse().reduce((acc, result) => { - let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(','); - if (!seen.has(resultStr)) { - acc.push(result); - seen.add(resultStr); - } - return acc; - }, []); - - results = results.reverse(); - // for debugging //Search.lastresults = results.slice(); // a copy - // console.info("search results:", Search.lastresults); + //console.info('search results:', Search.lastresults); // print the results - _displayNextItem(results, results.length, highlightTerms, searchTerms); + var resultCount = results.length; + function displayNextItem() { + // results left, load the summary and display it + if (results.length) { + var item = results.pop(); + var listItem = $('<li></li>'); + var requestUrl = ""; + var linkUrl = ""; + if (DOCUMENTATION_OPTIONS.BUILDER === 'dirhtml') { + // dirhtml builder + var dirname = item[0] + '/'; + if (dirname.match(/\/index\/$/)) { + dirname = dirname.substring(0, dirname.length-6); + } else if (dirname == 'index/') { + dirname = ''; + } + requestUrl = DOCUMENTATION_OPTIONS.URL_ROOT + dirname; + linkUrl = requestUrl; + + } else { + // normal html builders + requestUrl = DOCUMENTATION_OPTIONS.URL_ROOT + item[0] + DOCUMENTATION_OPTIONS.FILE_SUFFIX; + linkUrl = item[0] + DOCUMENTATION_OPTIONS.LINK_SUFFIX; + } + listItem.append($('<a/>').attr('href', + linkUrl + + highlightstring + item[2]).html(item[1])); + if (item[3]) { + listItem.append($(' (' + item[3] + ')')); + Search.output.append(listItem); + setTimeout(function() { + displayNextItem(); + }, 5); + } else if (DOCUMENTATION_OPTIONS.HAS_SOURCE) { + $.ajax({url: requestUrl, + dataType: "text", + complete: function(jqxhr, textstatus) { + var data = jqxhr.responseText; + if (data !== '' && data !== undefined) { + var summary = Search.makeSearchSummary(data, searchterms, hlterms); + if (summary) { + listItem.append(summary); + } + } + Search.output.append(listItem); + setTimeout(function() { + displayNextItem(); + }, 5); + }}); + } else { + // no source available, just display title + Search.output.append(listItem); + setTimeout(function() { + displayNextItem(); + }, 5); + } + } + // search finished, update title and status message + else { + Search.stopPulse(); + Search.title.text(_('Search Results')); + if (!resultCount) + Search.status.text(_('Your search did not match any documents. Please make sure that all words are spelled correctly and that you\'ve selected enough categories.')); + else + Search.status.text(_('Search finished, found %s page(s) matching the search query.').replace('%s', resultCount)); + Search.status.fadeIn(500); + } + } + displayNextItem(); }, /** * search for object names */ - performObjectSearch: (object, objectTerms) => { - const filenames = Search._index.filenames; - const docNames = Search._index.docnames; - const objects = Search._index.objects; - const objNames = Search._index.objnames; - const titles = Search._index.titles; - - const results = []; - - const objectSearchCallback = (prefix, match) => { - const name = match[4] - const fullname = (prefix ? prefix + "." : "") + name; - const fullnameLower = fullname.toLowerCase(); - if (fullnameLower.indexOf(object) < 0) return; - - let score = 0; - const parts = fullnameLower.split("."); - - // check for different match types: exact matches of full name or - // "last name" (i.e. last dotted part) - if (fullnameLower === object || parts.slice(-1)[0] === object) - score += Scorer.objNameMatch; - else if (parts.slice(-1)[0].indexOf(object) > -1) - score += Scorer.objPartialMatch; // matches in last name - - const objName = objNames[match[1]][2]; - const title = titles[match[0]]; - - // If more than one term searched for, we require other words to be - // found in the name/title/description - const otherTerms = new Set(objectTerms); - otherTerms.delete(object); - if (otherTerms.size > 0) { - const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase(); - if ( - [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0) - ) - return; + performObjectSearch : function(object, otherterms) { + var filenames = this._index.filenames; + var docnames = this._index.docnames; + var objects = this._index.objects; + var objnames = this._index.objnames; + var titles = this._index.titles; + + var i; + var results = []; + + for (var prefix in objects) { + for (var iMatch = 0; iMatch != objects[prefix].length; ++iMatch) { + var match = objects[prefix][iMatch]; + var name = match[4]; + var fullname = (prefix ? prefix + '.' : '') + name; + var fullnameLower = fullname.toLowerCase() + if (fullnameLower.indexOf(object) > -1) { + var score = 0; + var parts = fullnameLower.split('.'); + // check for different match types: exact matches of full name or + // "last name" (i.e. last dotted part) + if (fullnameLower == object || parts[parts.length - 1] == object) { + score += Scorer.objNameMatch; + // matches in last name + } else if (parts[parts.length - 1].indexOf(object) > -1) { + score += Scorer.objPartialMatch; + } + var objname = objnames[match[1]][2]; + var title = titles[match[0]]; + // If more than one term searched for, we require other words to be + // found in the name/title/description + if (otherterms.length > 0) { + var haystack = (prefix + ' ' + name + ' ' + + objname + ' ' + title).toLowerCase(); + var allfound = true; + for (i = 0; i < otherterms.length; i++) { + if (haystack.indexOf(otherterms[i]) == -1) { + allfound = false; + break; + } + } + if (!allfound) { + continue; + } + } + var descr = objname + _(', in ') + title; + + var anchor = match[3]; + if (anchor === '') + anchor = fullname; + else if (anchor == '-') + anchor = objnames[match[1]][1] + '-' + fullname; + // add custom score for some objects according to scorer + if (Scorer.objPrio.hasOwnProperty(match[2])) { + score += Scorer.objPrio[match[2]]; + } else { + score += Scorer.objPrioDefault; + } + results.push([docnames[match[0]], fullname, '#'+anchor, descr, score, filenames[match[0]]]); + } } + } - let anchor = match[3]; - if (anchor === "") anchor = fullname; - else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname; - - const descr = objName + _(", in ") + title; - - // add custom score for some objects according to scorer - if (Scorer.objPrio.hasOwnProperty(match[2])) - score += Scorer.objPrio[match[2]]; - else score += Scorer.objPrioDefault; - - results.push([ - docNames[match[0]], - fullname, - "#" + anchor, - descr, - score, - filenames[match[0]], - ]); - }; - Object.keys(objects).forEach((prefix) => - objects[prefix].forEach((array) => - objectSearchCallback(prefix, array) - ) - ); return results; }, + /** + * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions + */ + escapeRegExp : function(string) { + return string.replace(/[.*+\-?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string + }, + /** * search for full-text terms in the index */ - performTermsSearch: (searchTerms, excludedTerms) => { - // prepare search - const terms = Search._index.terms; - const titleTerms = Search._index.titleterms; - const docNames = Search._index.docnames; - const filenames = Search._index.filenames; - const titles = Search._index.titles; + performTermsSearch : function(searchterms, excluded, terms, titleterms) { + var docnames = this._index.docnames; + var filenames = this._index.filenames; + var titles = this._index.titles; - const scoreMap = new Map(); - const fileMap = new Map(); + var i, j, file; + var fileMap = {}; + var scoreMap = {}; + var results = []; // perform the search on the required terms - searchTerms.forEach((word) => { - const files = []; - const arr = [ - { files: terms[word], score: Scorer.term }, - { files: titleTerms[word], score: Scorer.title }, + for (i = 0; i < searchterms.length; i++) { + var word = searchterms[i]; + var files = []; + var _o = [ + {files: terms[word], score: Scorer.term}, + {files: titleterms[word], score: Scorer.title} ]; // add support for partial matches if (word.length > 2) { - const escapedWord = _escapeRegExp(word); - Object.keys(terms).forEach((term) => { - if (term.match(escapedWord) && !terms[word]) - arr.push({ files: terms[term], score: Scorer.partialTerm }); - }); - Object.keys(titleTerms).forEach((term) => { - if (term.match(escapedWord) && !titleTerms[word]) - arr.push({ files: titleTerms[word], score: Scorer.partialTitle }); - }); + var word_regex = this.escapeRegExp(word); + for (var w in terms) { + if (w.match(word_regex) && !terms[word]) { + _o.push({files: terms[w], score: Scorer.partialTerm}) + } + } + for (var w in titleterms) { + if (w.match(word_regex) && !titleterms[word]) { + _o.push({files: titleterms[w], score: Scorer.partialTitle}) + } + } } // no match but word was a required one - if (arr.every((record) => record.files === undefined)) return; - + if ($u.every(_o, function(o){return o.files === undefined;})) { + break; + } // found search word in contents - arr.forEach((record) => { - if (record.files === undefined) return; - - let recordFiles = record.files; - if (recordFiles.length === undefined) recordFiles = [recordFiles]; - files.push(...recordFiles); - - // set score for the word in each file - recordFiles.forEach((file) => { - if (!scoreMap.has(file)) scoreMap.set(file, {}); - scoreMap.get(file)[word] = record.score; - }); + $u.each(_o, function(o) { + var _files = o.files; + if (_files === undefined) + return + + if (_files.length === undefined) + _files = [_files]; + files = files.concat(_files); + + // set score for the word in each file to Scorer.term + for (j = 0; j < _files.length; j++) { + file = _files[j]; + if (!(file in scoreMap)) + scoreMap[file] = {}; + scoreMap[file][word] = o.score; + } }); // create the mapping - files.forEach((file) => { - if (fileMap.has(file) && fileMap.get(file).indexOf(word) === -1) - fileMap.get(file).push(word); - else fileMap.set(file, [word]); - }); - }); + for (j = 0; j < files.length; j++) { + file = files[j]; + if (file in fileMap && fileMap[file].indexOf(word) === -1) + fileMap[file].push(word); + else + fileMap[file] = [word]; + } + } // now check if the files don't contain excluded terms - const results = []; - for (const [file, wordList] of fileMap) { - // check if all requirements are matched + for (file in fileMap) { + var valid = true; - // as search terms with length < 3 are discarded - const filteredTermCount = [...searchTerms].filter( - (term) => term.length > 2 - ).length; + // check if all requirements are matched + var filteredTermCount = // as search terms with length < 3 are discarded: ignore + searchterms.filter(function(term){return term.length > 2}).length if ( - wordList.length !== searchTerms.size && - wordList.length !== filteredTermCount - ) - continue; + fileMap[file].length != searchterms.length && + fileMap[file].length != filteredTermCount + ) continue; // ensure that none of the excluded terms is in the search result - if ( - [...excludedTerms].some( - (term) => - terms[term] === file || - titleTerms[term] === file || - (terms[term] || []).includes(file) || - (titleTerms[term] || []).includes(file) - ) - ) - break; + for (i = 0; i < excluded.length; i++) { + if (terms[excluded[i]] == file || + titleterms[excluded[i]] == file || + $u.contains(terms[excluded[i]] || [], file) || + $u.contains(titleterms[excluded[i]] || [], file)) { + valid = false; + break; + } + } - // select one (max) score for the file. - const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w])); - // add result to the result list - results.push([ - docNames[file], - titles[file], - "", - null, - score, - filenames[file], - ]); + // if we have still a valid result we can add it to the result list + if (valid) { + // select one (max) score for the file. + // for better ranking, we should calculate ranking by using words statistics like basic tf-idf... + var score = $u.max($u.map(fileMap[file], function(w){return scoreMap[file][w]})); + results.push([docnames[file], titles[file], '', null, score, filenames[file]]); + } } return results; },
- @@ -499,33 +496,34 @@ const Search = {
 /** * helper function to return a node containing the * search summary for a given text. keywords is a list - * of stemmed words, highlightWords is the list of normal, unstemmed + * of stemmed words, hlwords is the list of normal, unstemmed * words. the first one is used to find the occurrence, the * latter for highlighting it. */ - makeSearchSummary: (htmlText, keywords, highlightWords) => { - const text = Search.htmlToText(htmlText).toLowerCase(); - if (text === "") return null; - - const actualStartPosition = [...keywords] - .map((k) => text.indexOf(k.toLowerCase())) - .filter((i) => i > -1) - .slice(-1)[0]; - const startWithContext = Math.max(actualStartPosition - 120, 0); - - const top = startWithContext === 0 ? "" : "..."; - const tail = startWithContext + 240 < text.length ? "..." : ""; - - let summary = document.createElement("div"); - summary.classList.add("context"); - summary.innerText = top + text.substr(startWithContext, 240).trim() + tail; - - highlightWords.forEach((highlightWord) => - _highlightText(summary, highlightWord, "highlighted") - ); - - return summary; - }, + makeSearchSummary : function(htmlText, keywords, hlwords) { + var text = Search.htmlToText(htmlText); + if (text == "") { + return null; + } + var textLower = text.toLowerCase(); + var start = 0; + $.each(keywords, function() { + var i = textLower.indexOf(this.toLowerCase()); + if (i > -1) + start = i; + }); + start = Math.max(start - 120, 0); + var excerpt = ((start > 0) ? '...' : '') + + $.trim(text.substr(start, 240)) + + ((start + 240 - text.length) ? '...' : ''); + var rv = $('').text(excerpt); + $.each(hlwords, function() { + rv = rv.highlightText(this, 'highlighted'); + }); + return rv; + } }; - _ready(Search.init); + $(document).ready(function() { + Search.init(); + });
html/genindex.html
- @@ -15,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="#" />
html/index.html
- @@ -1,8 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> - + <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>Mathematics in Lean — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
- @@ -16,7 +15,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -80,8 +78,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="mathematics-in-lean"> - <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this heading"></a></h1> + <div class="section" id="mathematics-in-lean"> + <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this headline"></a></h1> <div class="toctree-wrapper compound"> <ul> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a><ul>
- @@ -156,7 +154,7 @@
 </div> <div class="toctree-wrapper compound"> </div> - </section> + </div> </div>
html/search.html
- @@ -16,7 +16,6 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script src="_static/doctools.js"></script> <script src="_static/js/theme.js"></script> <script src="_static/searchtools.js"></script>
html/searchindex.js
- @@ -1,1 +1,1 @@
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Introduction","2. Basics","3. Logic","4. Sets and Functions","5. Elementary Number Theory","6. Structures","7. Hierarchies","8. Groups and Rings","9. Topology","10. Differential Calculus","11. Integration and Measure Theory","Index","Mathematics in Lean"],titleterms:{"function":[3,8],"schr\u00f6der":3,The:[2,3],about:1,action:7,algebra:[1,5,7],appli:1,asymptot:9,ball:8,basic:[1,6],bernstein:3,build:5,calcul:1,calculu:9,close:8,compact:8,comparison:9,complet:8,concret:7,conjunct:2,continu:[8,9],converg:[2,8],countabl:8,defin:5,differenti:9,disjunct:2,elementari:[4,9,10],exampl:1,existenti:2,fact:1,filter:8,fundament:8,gaussian:5,get:0,group:7,hierarchi:6,ideal:7,ident:1,iff:2,implic:2,index:11,induct:4,infinit:4,integ:5,integr:10,introduct:0,irrat:4,lean:12,lemma:1,linear:9,logic:2,mani:4,map:9,mathemat:12,measur:10,metric:8,monoid:7,more:1,morphism:[6,7],negat:2,norm:9,number:4,object:6,open:8,overview:0,polynomi:7,prime:4,prove:1,quantifi:2,quotient:7,recurs:4,ring:7,root:4,rw:1,separ:8,sequenc:2,set:[3,8],space:[8,9],start:0,structur:[1,5],sub:6,subgroup:7,subr:7,theorem:[1,3],theori:[4,10],topolog:8,uniformli:8,unit:7,univers:2,us:1}})
mathematics_in_lean.pdf