mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : CartesianMonoidalCategory C
inst✝ : BraidedCategory C
X Y : C
⊢ (β_ X Y).hom ≫ fst Y X = snd X Y

Tactic:
simp [fst_def, snd_def, ← BraidedCategory.braiding_naturality_left_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : CartesianMonoidalCategory C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : CartesianMonoidalCategory D
F : C ⥤ D
inst✝ : F.OplaxMonoidal
X Y : C
⊢ δ F X Y ≫ snd (F.obj X) (F.obj Y) = F.map (toUnit X ▷ Y) ≫ F.map (λ_ Y).hom

Tactic:
← whiskerRight_snd (F.map (toUnit X)),

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : CartesianMonoidalCategory C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : CartesianMonoidalCategory D
F : C ⥤ D
inst✝ : F.OplaxMonoidal
X Y : C
⊢ δ F X Y ≫ F.map (toUnit X) ▷ F.obj Y ≫ snd (F.obj (𝟙_ C)) (F.obj Y) = F.map (toUnit X ▷ Y) ≫ F.map (λ_ Y).hom
Module: Mathlib.RingTheory.HopfAlgebra.Convolution

Goal state before:
pp: R : Type u_1
C : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring C
inst✝ : HopfAlgebra R C
⊢ δ ∘ₗ mul' R C ∘ₗ (id ⊗ₘ antipode R) ∘ₗ δ = δ ∘ₗ (toConv id * toConv (antipode R)).ofConv

Tactic:
simp [LinearMap.convMul_def]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Closed.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : MonoidalCategory C
A B : C
inst✝¹ : Closed A
inst✝ : Closed B
f : B ⟶ A
X Y : C
g : A ⊗ Y ⟶ X
⊢ uncurry (curry g ≫ (pre f).app X) = uncurry (curry (f ▷ Y ≫ g))

Tactic:
by
  rw [uncurry_curry, uncurry_eq, MonoidalCategory.whiskerLeft_comp, assoc,
    id_tensor_pre_app_comp_ev, whisker_exchange_assoc, whiskerLeft_curry_ihom_ev_app]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Closed.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
A : C
inst✝ : Closed A
⊢ pre (𝟙 A) = 𝟙 (ihom A)

Tactic:
rw [pre, Functor.map_id]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
A : C
inst✝ : Closed A
⊢ (conjugateEquiv (ihom.adjunction A) (ihom.adjunction A)) (𝟙 ((tensoringLeft C).obj A)) = 𝟙 (ihom A)
Module: Mathlib.Data.W.Basic

Goal state before:
pp: α : Type u_1
β : α → Type u_2
inst✝² : (a : α) → Fintype (β a)
inst✝¹ : (a : α) → Encodable (β a)
inst✝ : Encodable α
h' : (n : ℕ) → Encodable (WType' β n)
f : WType β → (n : ℕ) × WType' β n := fun t ↦ ⟨t.depth, ⟨t, ⋯⟩⟩
⊢ Encodable (WType β)

Tactic:
let finv : (Σ n, WType' β n) → WType β := fun p => p.2.1

Goal state after:
α : Type u_1
β : α → Type u_2
inst✝² : (a : α) → Fintype (β a)
inst✝¹ : (a : α) → Encodable (β a)
inst✝ : Encodable α
h' : (n : ℕ) → Encodable (WType' β n)
f : WType β → (n : ℕ) × WType' β n := fun t ↦ ⟨t.depth, ⟨t, ⋯⟩⟩
finv : (n : ℕ) × WType' β n → WType β := fun p ↦ ↑p.snd
⊢ Encodable (WType β)
Module: Mathlib.LinearAlgebra.Dimension.ErdosKaplansky

Goal state before:
pp: case inr
K : Type u
inst✝ : DivisionRing K
aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)
card_K : ℵ₀ < #K
this✝ : Module.rank K (ℕ → K) < #K
ιK : Type u
bK : Module.Basis ιK K (ℕ → K)
L : Subfield K := Subfield.closure (Set.range fun i ↦ bK i.1 i.2)
hLK : #↥L < #K
this : Module (↥L)ᵐᵒᵖ K := Module.compHom K (RingHom.op L.subtype)
ιL : Type u
bL : Module.Basis ιL (↥L)ᵐᵒᵖ K
card_ιL : ℵ₀ ≤ #ιL
e : ℕ ↪ ιL
rep_e : ((bK.repr (⇑bL ∘ ⇑e)).sum fun i a ↦ a • bK i) = ⇑bL ∘ ⇑e
⊢ False

Tactic:
Finsupp.sum

Goal state after:
case inr
K : Type u
inst✝ : DivisionRing K
aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)
card_K : ℵ₀ < #K
this✝ : Module.rank K (ℕ → K) < #K
ιK : Type u
bK : Module.Basis ιK K (ℕ → K)
L : Subfield K := Subfield.closure (Set.range fun i ↦ bK i.1 i.2)
hLK : #↥L < #K
this : Module (↥L)ᵐᵒᵖ K := Module.compHom K (RingHom.op L.subtype)
ιL : Type u
bL : Module.Basis ιL (↥L)ᵐᵒᵖ K
card_ιL : ℵ₀ ≤ #ιL
e : ℕ ↪ ιL
rep_e : ∑ a ∈ (bK.repr (⇑bL ∘ ⇑e)).support, (bK.repr (⇑bL ∘ ⇑e)) a • bK a = ⇑bL ∘ ⇑e
⊢ False
Module: Mathlib.LinearAlgebra.Dual.Basis

Goal state before:
pp: R : Type u_1
M : Type u_2
ι : Type u_3
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
e : ι → M
ε : ι → Dual R M
h : DualBases e ε
⊢ ⇑h.basis = e

Tactic:
by
  ext i
  rw [Basis.apply_eq_iff]
  ext j
  rcases eq_or_ne i j with rfl | hne
  · simp [h.eval_same]
  · simp [hne, h.eval_of_ne hne.symm]

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.RankNullity

Goal state before:
pp: R : Type u_1
M : Type u
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : HasRankNullity.{u, u_1} R
inst✝¹ : StrongRankCondition R
inst✝ : Module.Finite R M
N : Submodule R M
h : finrank R ↥N < finrank R M
s : Finset (M ⧸ N)
hs : s.card = finrank R (M ⧸ N)
hs' : LinearIndependent R Subtype.val
⊢ ¬finrank R (M ⧸ N) = 0

Tactic:
finrank_quotient,

Goal state after:
R : Type u_1
M : Type u
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : HasRankNullity.{u, u_1} R
inst✝¹ : StrongRankCondition R
inst✝ : Module.Finite R M
N : Submodule R M
h : finrank R ↥N < finrank R M
s : Finset (M ⧸ N)
hs : s.card = finrank R (M ⧸ N)
hs' : LinearIndependent R Subtype.val
⊢ ¬finrank R M - finrank R ↥N = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring R₁
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : Module R₁ M₁
inst✝⁴ : CommSemiring R₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module R₂ M₂
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
x : M₂
⊢ 0 x = 0

Tactic:
zero_apply

Goal state after:
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring R₁
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : Module R₁ M₁
inst✝⁴ : CommSemiring R₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module R₂ M₂
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
x : M₂
⊢ 0 = 0
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: case e'_4
K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
this : Nontrivial K
⊢ lift.{v, u_2} (Module.rank K M) = lift.{max u_2 v, max u_2 v} (Module.rank L (M →ₗ[K] L))

Tactic:
lift_id,

Goal state after:
case e'_4
K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
this : Nontrivial K
⊢ lift.{v, u_2} (Module.rank K M) = Module.rank L (M →ₗ[K] L)
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: K : Type u_13
K₁ : Type u_14
V : Type u_16
V₁ : Type u_17
n : Type u_19
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
inst✝² : Field K₁
inst✝¹ : AddCommGroup V₁
inst✝ : Module K₁ V₁
I₁ I₁' : K₁ →+* K
B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V
v : n → V₁
hv₁ : B.IsOrthoᵢ v
hv₂ : ∀ (i : n), (B (v i)) (v i) ≠ 0
s : Finset n
w : n → K₁
hs : ∑ i ∈ s, w i • v i = 0
i : n
hi : i ∈ s
⊢ 0 (v i) = 0

Tactic:
zero_apply

Goal state after:
K : Type u_13
K₁ : Type u_14
V : Type u_16
V₁ : Type u_17
n : Type u_19
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
inst✝² : Field K₁
inst✝¹ : AddCommGroup V₁
inst✝ : Module K₁ V₁
I₁ I₁' : K₁ →+* K
B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V
v : n → V₁
hv₁ : B.IsOrthoᵢ v
hv₂ : ∀ (i : n), (B (v i)) (v i) ≠ 0
s : Finset n
w : n → K₁
hs : ∑ i ∈ s, w i • v i = 0
i : n
hi : i ∈ s
⊢ 0 = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : CommSemiring R₁
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R₁ M₁
inst✝² : CommSemiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
h : ∀ (x : M₁), B x = 0 → x = 0
x : M₁
hB : ∀ (y : M₂), (B x) y = 0
x✝ : M₂
⊢ (B x) x✝ = 0 x✝

Tactic:
zero_apply

Goal state after:
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : CommSemiring R₁
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R₁ M₁
inst✝² : CommSemiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
h : ∀ (x : M₁), B x = 0 → x = 0
x : M₁
hB : ∀ (y : M₂), (B x) y = 0
x✝ : M₂
⊢ (B x) x✝ = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
hB : B.IsSymm
W : Submodule R M
hW : IsCompl W B.ker
hB' : (B.domRestrict₁₂ W W).IsRefl
⊢ (B.domRestrict₁₂ W W).Nondegenerate

Tactic:
LinearMap.IsRefl.nondegenerate_iff_separatingLeft hB'

Goal state after:
R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
hB : B.IsSymm
W : Submodule R M
hW : IsCompl W B.ker
hB' : (B.domRestrict₁₂ W W).IsRefl
⊢ (B.domRestrict₁₂ W W).SeparatingLeft
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
⊢ ∀ {ι : Type u_4} [Finite ι] (W : ι → Subspace K V₁), dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)

Tactic:
apply Finite.induction_empty_option

Goal state after:
case of_equiv
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
⊢ ∀ {α β : Type u_4} (a : α ≃ β),
    (∀ (W : α → Subspace K V₁), dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)) →
      ∀ (W : β → Subspace K V₁), dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)

case h_empty
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
⊢ ∀ (W : PEmpty.{u_4 + 1} → Subspace K V₁), dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)

case h_option
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
⊢ ∀ {α : Type u_4} [Fintype α],
    (∀ (W : α → Subspace K V₁), dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)) →
      ∀ (W : Option α → Subspace K V₁), dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)
Module: Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasKernels C
inst✝² : IsNormalMonoCategory C
X Y Z : C
a : X ⟶ Z
b : Y ⟶ Z
inst✝¹ : Mono a
inst✝ : Mono b
P : C
f : Z ⟶ P
haf : a ≫ f = 0
i : IsLimit (KernelFork.ofι a haf)
Q : C
g : Z ⟶ Q
hbg : b ≫ g = 0
i' : IsLimit (KernelFork.ofι b hbg)
⊢ kernel.ι (prod.lift f g) ≫ prod.lift f g ≫ prod.fst = 0 ≫ prod.fst

Tactic:
kernel.condition_assoc

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasKernels C
inst✝² : IsNormalMonoCategory C
X Y Z : C
a : X ⟶ Z
b : Y ⟶ Z
inst✝¹ : Mono a
inst✝ : Mono b
P : C
f : Z ⟶ P
haf : a ≫ f = 0
i : IsLimit (KernelFork.ofι a haf)
Q : C
g : Z ⟶ Q
hbg : b ≫ g = 0
i' : IsLimit (KernelFork.ofι b hbg)
⊢ 0 ≫ prod.fst = 0 ≫ prod.fst
Module: Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasKernels C
inst✝² : IsNormalMonoCategory C
X Y Z : C
a : X ⟶ Z
b : Y ⟶ Z
inst✝¹ : Mono a
inst✝ : Mono b
P : C
f : Z ⟶ P
haf : a ≫ f = 0
i : IsLimit (KernelFork.ofι a haf)
Q : C
g : Z ⟶ Q
hbg : b ≫ g = 0
i' : IsLimit (KernelFork.ofι b hbg)
a' : kernel (prod.lift f g) ⟶ (KernelFork.ofι a haf).pt
ha' : a' ≫ Fork.ι (KernelFork.ofι a haf) = kernel.ι (prod.lift f g)
⊢ kernel.ι (prod.lift f g) ≫ prod.lift f g ≫ prod.snd = 0 ≫ prod.snd

Tactic:
kernel.condition_assoc

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasKernels C
inst✝² : IsNormalMonoCategory C
X Y Z : C
a : X ⟶ Z
b : Y ⟶ Z
inst✝¹ : Mono a
inst✝ : Mono b
P : C
f : Z ⟶ P
haf : a ≫ f = 0
i : IsLimit (KernelFork.ofι a haf)
Q : C
g : Z ⟶ Q
hbg : b ≫ g = 0
i' : IsLimit (KernelFork.ofι b hbg)
a' : kernel (prod.lift f g) ⟶ (KernelFork.ofι a haf).pt
ha' : a' ≫ Fork.ι (KernelFork.ofι a haf) = kernel.ι (prod.lift f g)
⊢ 0 ≫ prod.snd = 0 ≫ prod.snd
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
A : C
hlp : prod.lift (𝟙 A) 0 ≫ prod.snd = 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) (fun s ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
⊢ z = 0

Tactic:
obtain ⟨t, ht⟩ := CokernelCofork.IsColimit.desc' hp2 _ h

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
A : C
hlp : prod.lift (𝟙 A) 0 ≫ prod.snd = 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) (fun s ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : Cofork.π (CokernelCofork.ofπ prod.snd hlp) ≫ t = cokernel.π (diag A) ≫ z
⊢ z = 0
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
f : X ⟶ Y
g : (CokernelCofork.ofπ σ ⋯).pt ⟶ Y
hg : Cofork.π (CokernelCofork.ofπ σ ⋯) ≫ g = prod.map f f ≫ σ
⊢ f = f ≫ prod.lift (𝟙 Y) 0 ≫ σ

Tactic:
rw [lift_σ, Category.comp_id]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
S₁ S₂ S₃ S₄ : ShortComplex C
inst✝³ : S₁.HasHomology
inst✝² : S₂.HasHomology
inst✝¹ : S₃.HasHomology
inst✝ : S₄.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ : QuasiIso φ
hφ' : QuasiIso φ'
⊢ QuasiIso (φ ≫ φ')

Tactic:
quasiIso_iff

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
S₁ S₂ S₃ S₄ : ShortComplex C
inst✝³ : S₁.HasHomology
inst✝² : S₂.HasHomology
inst✝¹ : S₃.HasHomology
inst✝ : S₄.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφ' : IsIso (homologyMap φ')
⊢ IsIso (homologyMap (φ ≫ φ'))
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ : QuasiIso φ
hφφ' : QuasiIso (φ ≫ φ')
⊢ QuasiIso φ'

Tactic:
quasiIso_iff

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφφ' : IsIso (homologyMap (φ ≫ φ'))
⊢ IsIso (homologyMap φ')
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ' : QuasiIso φ'
hφφ' : QuasiIso (φ ≫ φ')
⊢ QuasiIso φ

Tactic:
quasiIso_iff

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ' : IsIso (homologyMap φ')
hφφ' : IsIso (homologyMap (φ ≫ φ'))
⊢ IsIso (homologyMap φ)
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
⊢ QuasiIso φ ↔ IsIso (S₂.liftCycles φ.τ₂ ⋯)

Tactic:
by
  let H : LeftHomologyMapData φ (LeftHomologyData.ofZeros S₁ hf₁ hg₁)
      (LeftHomologyData.ofIsLimitKernelFork S₂ hf₂ _ S₂.cyclesIsKernel) :=
    { φK := S₂.liftCycles φ.τ₂ (by rw [φ.comm₂₃, hg₁, zero_comp])
      φH := S₂.liftCycles φ.τ₂ (by rw [φ.comm₂₃, hg₁, zero_comp]) }
  exact H.quasiIso_iff

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
e : S₁ ≅ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
⊢ homologyMap' e.hom h₁ h₂ ≫ homologyMap' e.inv h₂ h₁ = 𝟙 h₁.left.H

Tactic:
by rw [← homologyMap'_comp, e.hom_inv_id, homologyMap'_id]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
φQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯
commg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃
⊢ (h₁.ι ≫ φQ) ≫ h₂.g' = 0

Tactic:
rw [assoc, commg', RightHomologyData.ι_g'_assoc, zero_comp]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₂.τ₂ = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + φ₁.τ₂

Tactic:
rw [h.comm₂, comp_neg, neg_comp]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₂.τ₂ = -S₁.g ≫ h.h₂ + -h.h₁ ≫ S₂.f + (S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂)
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

Goal state before:
pp: case h₁
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m.τ₁ =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₁

pp: case h₂
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m.τ₂ =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₂

pp: case h₃
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m.τ₃ =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₃

Tactic:
· exact h₁.uniq (π₁.mapCocone s) _ (fun j => π₁.congr_map (hm j))

Goal state after:
case h₂
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m.τ₂ =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₂

case h₃
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m.τ₃ =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ (φ₁ + φ₃).τ₃ = h.h₃ + h'.h₃ + (h.h₂ + h'.h₂) ≫ S₂.g + (φ₂ + φ₄).τ₃

Tactic:
add_τ₃,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ φ₁.τ₃ + φ₃.τ₃ = h.h₃ + h'.h₃ + (h.h₂ + h'.h₂) ≫ S₂.g + (φ₂ + φ₄).τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ φ₁.τ₃ + φ₃.τ₃ = h.h₃ + h'.h₃ + (h.h₂ + h'.h₂) ≫ S₂.g + (φ₂ + φ₄).τ₃

Tactic:
add_τ₃,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ φ₁.τ₃ + φ₃.τ₃ = h.h₃ + h'.h₃ + (h.h₂ + h'.h₂) ≫ S₂.g + (φ₂.τ₃ + φ₄.τ₃)
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₂ ⟶ S₃
⊢ (φ₁ ≫ ψ).τ₁ = S₁.f ≫ h.h₁ ≫ ψ.τ₁ + h.h₀ ≫ ψ.τ₁ + (φ₂ ≫ ψ).τ₁

Tactic:
rw [comp_τ₁, comp_τ₁, h.comm₁, add_comp, add_comp, assoc]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
S : ShortComplex C
hg : S.g = 0
inst✝ : PreservesColimit (parallelPair S.f 0) F
h : S.LeftHomologyData
⊢ PreservesColimit (parallelPair h.f' 0) F

Tactic:
by
      have := h.isIso_i hg
      let e : parallelPair h.f' 0 ≅ parallelPair S.f 0 :=
        parallelPair.ext (Iso.refl _) (asIso h.i) (by simp) (by simp)
      exact Limits.preservesColimit_of_iso_diagram F e.symm

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Ab

Goal state before:
pp: S : ShortComplex Ab
x : ↑S.X₁
⊢ (ConcreteCategory.hom (S.f ≫ S.g)) x = 0

Tactic:
S.zero

Goal state after:
S : ShortComplex Ab
x : ↑S.X₁
⊢ (ConcreteCategory.hom 0) x = 0
Module: Mathlib.Algebra.Homology.ShortComplex.Ab

Goal state before:
pp: case mp
S : ShortComplex Ab
h : (AddCommGrpCat.Hom.hom S.g).ker ≤ (AddCommGrpCat.Hom.hom S.f).range
x₁ : ↑S.X₁
⊢ (ConcreteCategory.hom (S.f ≫ S.g)) x₁ = 0

Tactic:
S.zero

Goal state after:
case mp
S : ShortComplex Ab
h : (AddCommGrpCat.Hom.hom S.g).ker ≤ (AddCommGrpCat.Hom.hom S.f).range
x₁ : ↑S.X₁
⊢ (ConcreteCategory.hom 0) x₁ = 0
Module: Mathlib.Algebra.Homology.ShortComplex.Ab

Goal state before:
pp: S : ShortComplex Ab
hS : S.Exact
inst✝¹ : Finite ↑S.X₁
inst✝ : Finite ↑S.X₃
this : Finite ↥(AddCommGrpCat.Hom.hom S.f).range
⊢ Finite (↑S.X₂ ⧸ (AddCommGrpCat.Hom.hom S.g).ker)

Tactic:
exact .of_equiv _ (QuotientAddGroup.quotientKerEquivRange _).toEquiv.symm

Goal state after: