mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f ↦ K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f ↦ limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f ↦ limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
f : (p : J × J) × (p.1 ⟶ p.2)
⊢ G.map
      ((Fan.mk (∏ᶜ K.obj) fun f ↦ limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd).π.app { as := f }) =
    G.map (Pi.π K.obj f.fst.1 ≫ K.map f.snd)

Tactic:
congr

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: case e'_2
R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Surjective ⇑f.range.subtype.dualMap
rr_surj : Function.Surjective ⇑f.rangeRestrict
this : f.rangeRestrict.dualMap.range = f.rangeRestrict.ker.dualAnnihilator
⊢ f.dualMap.range = f.rangeRestrict.dualMap.range

Tactic:
calc
      _ = range ((range f).subtype.comp f.rangeRestrict).dualMap := by simp
      _ = _ := ?_
    rw [← dualMap_comp_dualMap, range_comp_of_range_eq_top]
    rwa [range_eq_top]

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_3
V₁ : Type u_4
V₂ : Type u_5
inst✝⁴ : Field K
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
f : V₁ →ₗ[K] V₂
⊢ f.ker = ⊥ ↔ Function.Injective ⇑f

Tactic:
LinearMap.ker_eq_bot

Goal state after:
K : Type u_3
V₁ : Type u_4
V₂ : Type u_5
inst✝⁴ : Field K
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
f : V₁ →ₗ[K] V₂
⊢ Function.Injective ⇑f ↔ Function.Injective ⇑f
Module: Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasColimitsOfShape (Discrete J) C
inst✝⁴ : HasColimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasCoequalizers C
G : C ⥤ D
inst✝² : PreservesColimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesColimitsOfShape (Discrete J) G
inst✝ : PreservesColimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∐ K.obj
Q : C := ∐ fun f ↦ K.obj f.fst.1
s : Q ⟶ P := Sigma.desc fun f ↦ K.map f.snd ≫ colimit.ι (Discrete.functor K.obj) { as := f.fst.2 }
t : Q ⟶ P := Sigma.desc fun f ↦ colimit.ι (Discrete.functor K.obj) { as := f.fst.1 }
I : C := coequalizer s t
i : P ⟶ I := coequalizer.π s t
f : (p : J × J) × (p.1 ⟶ p.2)
⊢ G.map
      ((Cofan.mk (∐ K.obj) fun f ↦ K.map f.snd ≫ colimit.ι (Discrete.functor K.obj) { as := f.fst.2 }).ι.app
        { as := f }) =
    G.map (K.map f.snd ≫ Sigma.ι K.obj f.fst.2)

Tactic:
congr

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ : C
inst✝ : HasZeroMorphisms C
X Y : C
f : X ⟶ Y
hf : NormalMono f
X' Y' : C
f' : X' ⟶ Y'
e : Arrow.mk f ≅ Arrow.mk f'
this : Arrow.Hom.left e.inv ≫ f = f' ≫ Arrow.Hom.right e.inv
⊢ Arrow.Hom.left e.inv ≫ f ≫ g = 0

Tactic:
hf.w,

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ : C
inst✝ : HasZeroMorphisms C
X Y : C
f : X ⟶ Y
hf : NormalMono f
X' Y' : C
f' : X' ⟶ Y'
e : Arrow.mk f ≅ Arrow.mk f'
this : Arrow.Hom.left e.inv ≫ f = f' ≫ Arrow.Hom.right e.inv
⊢ Arrow.Hom.left e.inv ≫ 0 = 0
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)
b' : kernel (prod.lift f g) ⟶ (KernelFork.ofι b hbg).pt
hb' : b' ≫ Fork.ι (KernelFork.ofι b hbg) = kernel.ι (prod.lift f g)
s : PullbackCone a b
m : s.pt ⟶ kernel (prod.lift f g)
h₁ : m ≫ a' = s.fst
x✝ : m ≫ b' = s.snd
⊢ m ≫ kernel.ι (prod.lift f g) = m ≫ a' ≫ a

Tactic:
congr

Goal state after:
case e_a
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)
b' : kernel (prod.lift f g) ⟶ (KernelFork.ofι b hbg).pt
hb' : b' ≫ Fork.ι (KernelFork.ofι b hbg) = kernel.ι (prod.lift f g)
s : PullbackCone a b
m : s.pt ⟶ kernel (prod.lift f g)
h₁ : m ≫ a' = s.fst
x✝ : m ≫ b' = s.snd
⊢ kernel.ι (prod.lift f g) = a' ≫ a
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 : HasFiniteCoproducts C
inst✝³ : HasCokernels C
inst✝² : IsNormalEpiCategory C
X Y Z : C
a : X ⟶ Y
b : X ⟶ Z
inst✝¹ : Epi a
inst✝ : Epi b
P : C
f : P ⟶ X
hfa : f ≫ a = 0
i : IsColimit (CokernelCofork.ofπ a hfa)
Q : C
g : Q ⟶ X
hgb : g ≫ b = 0
i' : IsColimit (CokernelCofork.ofπ b hgb)
a' : (CokernelCofork.ofπ a hfa).pt ⟶ cokernel (coprod.desc f g)
ha' : Cofork.π (CokernelCofork.ofπ a hfa) ≫ a' = cokernel.π (coprod.desc f g)
b' : (CokernelCofork.ofπ b hgb).pt ⟶ cokernel (coprod.desc f g)
hb' : Cofork.π (CokernelCofork.ofπ b hgb) ≫ b' = cokernel.π (coprod.desc f g)
s : PushoutCocone a b
m : cokernel (coprod.desc f g) ⟶ s.pt
h₁ : a' ≫ m = s.inl
x✝ : b' ≫ m = s.inr
⊢ cokernel.π (coprod.desc f g) ≫ m = (a ≫ a') ≫ m

Tactic:
congr

Goal state after:
case e_a
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasFiniteCoproducts C
inst✝³ : HasCokernels C
inst✝² : IsNormalEpiCategory C
X Y Z : C
a : X ⟶ Y
b : X ⟶ Z
inst✝¹ : Epi a
inst✝ : Epi b
P : C
f : P ⟶ X
hfa : f ≫ a = 0
i : IsColimit (CokernelCofork.ofπ a hfa)
Q : C
g : Q ⟶ X
hgb : g ≫ b = 0
i' : IsColimit (CokernelCofork.ofπ b hgb)
a' : (CokernelCofork.ofπ a hfa).pt ⟶ cokernel (coprod.desc f g)
ha' : Cofork.π (CokernelCofork.ofπ a hfa) ≫ a' = cokernel.π (coprod.desc f g)
b' : (CokernelCofork.ofπ b hgb).pt ⟶ cokernel (coprod.desc f g)
hb' : Cofork.π (CokernelCofork.ofπ b hgb) ≫ b' = cokernel.π (coprod.desc f g)
s : PushoutCocone a b
m : cokernel (coprod.desc f g) ⟶ s.pt
h₁ : a' ≫ m = s.inl
x✝ : b' ≫ m = s.inr
⊢ cokernel.π (coprod.desc f g) = a ≫ a'
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| prod.lift a 0 ≫ σ

Tactic:
congr

Goal state after:
case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| prod.lift a 0

case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| σ
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| prod.lift a 0

pp: case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| σ

Tactic:
congr

Goal state after:
case a.f
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| a

case a.g
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| 0

case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| σ
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| 0 - (0 - a)

Tactic:
congr

Goal state after:
case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| 0

case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
| 0 - a
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
| 0 - 0 - (0 - a - b)

Tactic:
congr

Goal state after:
case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
| 0 - 0

case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
| 0 - a - b
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c : X ⟶ Y
| a + b + c

Tactic:
congr

Goal state after:
case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c : X ⟶ Y
| a + b

case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c : X ⟶ Y
| c
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A B : GrpCat
f : A ⟶ B
b : ↑B
hb : b ∈ (Hom.hom f).range
⊢ fromCoset ⟨b • ↑(Hom.hom f).range, ⋯⟩ = fromCoset ⟨↑(Hom.hom f).range, ⋯⟩

Tactic:
congr

Goal state after:
A B : GrpCat
f : A ⟶ B
b : ↑B
hb : b ∈ (Hom.hom f).range
⊢ b • ↑(Hom.hom f).range = ↑(Hom.hom f).range
Module: Mathlib.Algebra.Homology.ShortComplex.Basic

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
inst✝ : HasZeroMorphisms E
S✝ S₁ S₂ S₃ : ShortComplex C
S : ShortComplex Cᵒᵖ
⊢ S.g.unop ≫ S.f.unop = 0

Tactic:
simp only [← unop_comp, S.zero]

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
inst✝ : HasZeroMorphisms E
S✝ S₁ S₂ S₃ : ShortComplex C
S : ShortComplex Cᵒᵖ
⊢ Quiver.Hom.unop 0 = 0
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hf : S.f = 0
hg : S.g = 0
⊢ (ofZeros S hf hg).f' = 0

Tactic:
← cancel_mono ((ofZeros S hf hg).i),

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hf : S.f = 0
hg : S.g = 0
⊢ (ofZeros S hf hg).f' ≫ (ofZeros S hf hg).i = 0 ≫ (ofZeros S hf hg).i
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: case mk.mk
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.LeftHomologyData
h₂ : S₂.LeftHomologyData
φK✝¹ : h₁.K ⟶ h₂.K
φH✝¹ : h₁.H ⟶ h₂.H
commi✝¹ : φK✝¹ ≫ h₂.i = h₁.i ≫ φ.τ₂
commf'✝¹ : h₁.f' ≫ φK✝¹ = φ.τ₁ ≫ h₂.f'
commπ✝¹ : h₁.π ≫ φH✝¹ = φK✝¹ ≫ h₂.π
φK✝ : h₁.K ⟶ h₂.K
φH✝ : h₁.H ⟶ h₂.H
commi✝ : φK✝ ≫ h₂.i = h₁.i ≫ φ.τ₂
commf'✝ : h₁.f' ≫ φK✝ = φ.τ₁ ≫ h₂.f'
commπ✝ : h₁.π ≫ φH✝ = φK✝ ≫ h₂.π
hK :
  { φK := φK✝¹, φH := φH✝¹, commi := commi✝¹, commf' := commf'✝¹, commπ := commπ✝¹ }.φK =
    { φK := φK✝, φH := φH✝, commi := commi✝, commf' := commf'✝, commπ := commπ✝ }.φK
hH :
  { φK := φK✝¹, φH := φH✝¹, commi := commi✝¹, commf' := commf'✝¹, commπ := commπ✝¹ }.φH =
    { φK := φK✝, φH := φH✝, commi := commi✝, commf' := commf'✝, commπ := commπ✝ }.φH
⊢ { φK := φK✝¹, φH := φH✝¹, commi := commi✝¹, commf' := commf'✝¹, commπ := commπ✝¹ } =
    { φK := φK✝, φH := φH✝, commi := commi✝, commf' := commf'✝, commπ := commπ✝ }

Tactic:
congr

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
W'✝ : C
x : W'✝ ⟶ S₂.X₂
hx : x ≫ S₂.g = 0
b : W'✝ ⟶ h.K
hb : b ≫ i = x
⊢ b ≫ h.i = h.liftK (x ≫ inv φ.τ₂) ⋯ ≫ h.i

Tactic:
← cancel_mono φ.τ₂,

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
W'✝ : C
x : W'✝ ⟶ S₂.X₂
hx : x ≫ S₂.g = 0
b : W'✝ ⟶ h.K
hb : b ≫ i = x
⊢ (b ≫ h.i) ≫ φ.τ₂ = (h.liftK (x ≫ inv φ.τ₂) ⋯ ≫ h.i) ≫ φ.τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
inst✝ : S.HasHomology
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ IsIso (leftRightHomologyComparison' h₁ h₂)

Tactic:
leftRightHomologyComparison'_compatibility h₁ S.homologyData.left h₂
    S.homologyData.right

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
inst✝ : S.HasHomology
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ IsIso
    (leftHomologyMap' (𝟙 S) h₁ S.homologyData.left ≫
      leftRightHomologyComparison' S.homologyData.left S.homologyData.right ≫
        rightHomologyMap' (𝟙 S) S.homologyData.right h₂)
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₂.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₂.f ⋯)
eq : hi.lift (KernelFork.ofι S₂.f ⋯) ≫ i = S₂.f
⊢ (φ.τ₁ ≫ f') ≫ h.i = h.f' ≫ h.i

Tactic:
← cancel_mono φ.τ₂,

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₂.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₂.f ⋯)
eq : hi.lift (KernelFork.ofι S₂.f ⋯) ≫ i = S₂.f
⊢ ((φ.τ₁ ≫ f') ≫ h.i) ≫ φ.τ₂ = (h.f' ≫ h.i) ≫ φ.τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: case mk.mk
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✝¹ : h₁.Q ⟶ h₂.Q
φH✝¹ : h₁.H ⟶ h₂.H
commp✝¹ : h₁.p ≫ φQ✝¹ = φ.τ₂ ≫ h₂.p
commg'✝¹ : φQ✝¹ ≫ h₂.g' = h₁.g' ≫ φ.τ₃
commι✝¹ : φH✝¹ ≫ h₂.ι = h₁.ι ≫ φQ✝¹
φQ✝ : h₁.Q ⟶ h₂.Q
φH✝ : h₁.H ⟶ h₂.H
commp✝ : h₁.p ≫ φQ✝ = φ.τ₂ ≫ h₂.p
commg'✝ : φQ✝ ≫ h₂.g' = h₁.g' ≫ φ.τ₃
commι✝ : φH✝ ≫ h₂.ι = h₁.ι ≫ φQ✝
hQ :
  { φQ := φQ✝¹, φH := φH✝¹, commp := commp✝¹, commg' := commg'✝¹, commι := commι✝¹ }.φQ =
    { φQ := φQ✝, φH := φH✝, commp := commp✝, commg' := commg'✝, commι := commι✝ }.φQ
hH :
  { φQ := φQ✝¹, φH := φH✝¹, commp := commp✝¹, commg' := commg'✝¹, commι := commι✝¹ }.φH =
    { φQ := φQ✝, φH := φH✝, commp := commp✝, commg' := commg'✝, commι := commι✝ }.φH
⊢ { φQ := φQ✝¹, φH := φH✝¹, commp := commp✝¹, commg' := commg'✝¹, commι := commι✝¹ } =
    { φQ := φQ✝, φH := φH✝, commp := commp✝, commg' := commg'✝, commι := commι✝ }

Tactic:
congr

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
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
c₁ : KernelFork S₁.g
hc₁ : IsLimit c₁
hf₂ : S₂.f = 0
c₂ : KernelFork S₂.g
hc₂ : IsLimit c₂
f : c₁.pt ⟶ c₂.pt
comm : Fork.ι c₁ ≫ φ.τ₂ = f ≫ Fork.ι c₂
⊢ φ.τ₂ ≫ (RightHomologyData.ofIsLimitKernelFork S₂ hf₂ c₂ hc₂).g' =
    (RightHomologyData.ofIsLimitKernelFork S₁ hf₁ c₁ hc₁).g' ≫ φ.τ₃

Tactic:
by simp only [RightHomologyData.ofIsLimitKernelFork_g', φ.comm₂₃]

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
S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
γ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂
γ₂ : RightHomologyMapData φ₂ h₂ h₃ := rightHomologyMapData φ₂ h₂ h₃
⊢ rightHomologyMap' (φ₁ ≫ φ₂) h₁ h₃ = γ₁.φH ≫ rightHomologyMap' φ₂ h₂ h₃

Tactic:
γ₂.rightHomologyMap'_eq,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
γ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂
γ₂ : RightHomologyMapData φ₂ h₂ h₃ := rightHomologyMapData φ₂ h₂ h₃
⊢ rightHomologyMap' (φ₁ ≫ φ₂) h₁ h₃ = γ₁.φH ≫ γ₂.φH
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

Goal state before:
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 : Cone F
h₁ : IsLimit (π₁.mapCone c)
h₂ : IsLimit (π₂.mapCone c)
h₃ : IsLimit (π₃.mapCone c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
⊢ m.τ₃ =
    { τ₁ := h₁.lift (π₁.mapCone s), τ₂ := h₂.lift (π₂.mapCone s), τ₃ := h₃.lift (π₃.mapCone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₃

Tactic:
· exact h₃.uniq (π₃.mapCone s) _ (fun j => π₃.congr_map (hm j))

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
f' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯
hf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ
wπ : f' ≫ cokernel.π (kernel.ι γ) = 0
e : Abelian.image S.f ≅ kernel γ :=
  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))
he : e.hom ≫ kernel.ι γ = S.abelianImageToKernel
fac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ
Z'✝ : C
x : kernel S.g ⟶ Z'✝
hx : f' ≫ x = 0
b : cokernel (kernel.ι γ) ⟶ Z'✝
hb : cokernel.π (kernel.ι γ) ≫ b = x
⊢ coequalizer.π (kernel.ι γ) 0 ≫ b = coequalizer.π (kernel.ι γ) 0 ≫ cokernel.desc (kernel.ι γ) x ⋯

Tactic:
by simp only [hb, cokernel.π_desc]

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 : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ rightHomologyMap' φ₂ h₁ h₂ + rightHomologyMap' (S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯) h₁ h₂ =
    rightHomologyMap' φ₂ h₁ h₂

Tactic:
rightHomologyMap'_nullHomotopic,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ rightHomologyMap' φ₂ h₁ h₂ + 0 = rightHomologyMap' φ₂ h₁ h₂
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
inst✝² : Balanced C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
inst✝¹ : IsIso φ.τ₁
inst✝ : IsIso φ.τ₃
this✝³ : Mono S₁.f
this✝² : Mono S₂.f
this✝¹ : Epi S₁.g
this✝ : Epi S₂.g
this : Mono φ.τ₂
⊢ IsIso φ.τ₂

Tactic:
have := epi_τ₂_of_exact_of_epi φ h₂.exact

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
inst✝² : Balanced C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
inst✝¹ : IsIso φ.τ₁
inst✝ : IsIso φ.τ₃
this✝⁴ : Mono S₁.f
this✝³ : Mono S₂.f
this✝² : Epi S₁.g
this✝¹ : Epi S₂.g
this✝ : Mono φ.τ₂
this : Epi φ.τ₂
⊢ IsIso φ.τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: case mk
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
s' : S.Splitting
this : Epi S.g
r✝ : S.X₂ ⟶ S.X₁
s✝ : S.X₃ ⟶ S.X₂
f_r✝ : S.f ≫ r✝ = 𝟙 S.X₁
s_g✝ : s✝ ≫ S.g = 𝟙 S.X₃
id✝ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂
h : { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.r = s'.r
eq : { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.s = s'.s
⊢ { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ } = s'

Tactic:
congr

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

Goal state before:
pp: case mk
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
s' : S.Splitting
this : Mono S.f
r✝ : S.X₂ ⟶ S.X₁
s✝ : S.X₃ ⟶ S.X₂
f_r✝ : S.f ≫ r✝ = 𝟙 S.X₁
s_g✝ : s✝ ≫ S.g = 𝟙 S.X₃
id✝ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂
h : { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.s = s'.s
eq : { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ }.r = s'.r
⊢ { r := r✝, s := s✝, f_r := f_r✝, s_g := s_g✝, id := id✝ } = s'

Tactic:
congr

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
B A₁ A₂ A₃ : C
f : A₁ ⟶ B
inst✝² : Mono f
g : A₂ ⟶ B
inst✝¹ : Mono g
h : A₃ ⟶ B
inst✝ : Mono h
h₁ : mk f ≤ mk g
h₂ : mk g ≤ mk h
⊢ ofMkLEMk f g h₁ ≫ ofMkLEMk g h h₂ = ofMkLEMk f h ⋯

Tactic:
by
  simp only [ofLE, ofMkLEMk, ← Functor.map_comp_assoc underlying, assoc,
    Iso.hom_inv_id_assoc]
  congr 1

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
B A₁ : C
f : A₁ ⟶ B
inst✝ : Mono f
⊢ ofMkLEMk f f ⋯ = 𝟙 A₁

Tactic:
apply (cancel_mono f).mp
  simp

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

Goal state before:
pp: case mpr.refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
w : φ.τ₂ ≫ S₂.g = 0
h₁ : { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact
h₂ : Mono φ.τ₂
⊢ S₂.liftCycles φ.τ₂ ⋯ ≫ h₁.lift S₂.iCycles ⋯ = 𝟙 S₁.X₂

Tactic:
← cancel_mono φ.τ₂,

Goal state after:
case mpr.refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
w : φ.τ₂ ≫ S₂.g = 0
h₁ : { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact
h₂ : Mono φ.τ₂
⊢ (S₂.liftCycles φ.τ₂ ⋯ ≫ h₁.lift S₂.iCycles ⋯) ≫ φ.τ₂ = 𝟙 S₁.X₂ ≫ φ.τ₂
Module: Mathlib.CategoryTheory.Subobject.Limits

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
f g : X ⟶ Y
inst✝¹ : HasEqualizer f g
W : C
h : W ⟶ X
inst✝ : HasPullbacks C
⊢ (Subobject.pullback h).obj (equalizerSubobject f g) = equalizerSubobject (h ≫ f) (h ≫ g)

Tactic:
refine skeletal _ ⟨iso_of_both_ways (homOfFactors ?_) (homOfFactors ?_)⟩

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
X Y : C
f g : X ⟶ Y
inst✝¹ : HasEqualizer f g
W : C
h : W ⟶ X
inst✝ : HasPullbacks C
⊢ (equalizerSubobject (h ≫ f) (h ≫ g)).Factors ((Subobject.pullback h).obj (equalizerSubobject f g)).arrow

case refine_2
C : Type u
inst✝² : Category.{v, u} C
X Y : C
f g : X ⟶ Y
inst✝¹ : HasEqualizer f g
W : C
h : W ⟶ X
inst✝ : HasPullbacks C
⊢ ((Subobject.pullback h).obj (equalizerSubobject f g)).Factors (equalizerSubobject (h ≫ f) (h ≫ g)).arrow