mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A K' H' : C
eK : K' ≅ h.K
eH : H' ≅ h.H
⊢ ((Cocone.precompose (parallelPair.ext (Iso.refl S.X₁) eK.symm ⋯ ⋯).inv).obj (CokernelCofork.ofπ h.π ⋯)).ι.app
        WalkingParallelPair.zero ≫
      eH.symm.hom =
    (CokernelCofork.ofπ (eK.hom ≫ h.π ≫ eH.inv) ⋯).ι.app WalkingParallelPair.zero

pp: case one
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A K' H' : C
eK : K' ≅ h.K
eH : H' ≅ h.H
⊢ ((Cocone.precompose (parallelPair.ext (Iso.refl S.X₁) eK.symm ⋯ ⋯).inv).obj (CokernelCofork.ofπ h.π ⋯)).ι.app
        WalkingParallelPair.one ≫
      eH.symm.hom =
    (CokernelCofork.ofπ (eK.hom ≫ h.π ≫ eH.inv) ⋯).ι.app WalkingParallelPair.one

Tactic:
<;>

Goal state after:
case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A K' H' : C
eK : K' ≅ h.K
eH : H' ≅ h.H
⊢ ((Cocone.precompose (parallelPair.ext (Iso.refl S.X₁) eK.symm ⋯ ⋯).inv).obj (CokernelCofork.ofπ h.π ⋯)).ι.app
        WalkingParallelPair.zero ≫
      eH.symm.hom =
    (CokernelCofork.ofπ (eK.hom ≫ h.π ≫ eH.inv) ⋯).ι.app WalkingParallelPair.zero

case one
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A K' H' : C
eK : K' ≅ h.K
eH : H' ≅ h.H
⊢ ((Cocone.precompose (parallelPair.ext (Iso.refl S.X₁) eK.symm ⋯ ⋯).inv).obj (CokernelCofork.ofπ h.π ⋯)).ι.app
        WalkingParallelPair.one ≫
      eH.symm.hom =
    (CokernelCofork.ofπ (eK.hom ≫ h.π ≫ eH.inv) ⋯).ι.app WalkingParallelPair.one
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A Q' H' : C
eQ : Q' ≅ h.Q
eH : H' ≅ h.H
⊢ ((Cone.postcompose (parallelPair.ext eQ.symm (Iso.refl S.X₃) ⋯ ⋯).hom).obj (KernelFork.ofι h.ι ⋯)).π.app
      WalkingParallelPair.zero =
    eH.symm.hom ≫ (KernelFork.ofι (eH.hom ≫ h.ι ≫ eQ.inv) ⋯).π.app WalkingParallelPair.zero

pp: case one
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A Q' H' : C
eQ : Q' ≅ h.Q
eH : H' ≅ h.H
⊢ ((Cone.postcompose (parallelPair.ext eQ.symm (Iso.refl S.X₃) ⋯ ⋯).hom).obj (KernelFork.ofι h.ι ⋯)).π.app
      WalkingParallelPair.one =
    eH.symm.hom ≫ (KernelFork.ofι (eH.hom ≫ h.ι ≫ eQ.inv) ⋯).π.app WalkingParallelPair.one

Tactic:
<;>

Goal state after:
case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A Q' H' : C
eQ : Q' ≅ h.Q
eH : H' ≅ h.H
⊢ ((Cone.postcompose (parallelPair.ext eQ.symm (Iso.refl S.X₃) ⋯ ⋯).hom).obj (KernelFork.ofι h.ι ⋯)).π.app
      WalkingParallelPair.zero =
    eH.symm.hom ≫ (KernelFork.ofι (eH.hom ≫ h.ι ≫ eQ.inv) ⋯).π.app WalkingParallelPair.zero

case one
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A Q' H' : C
eQ : Q' ≅ h.Q
eH : H' ≅ h.H
⊢ ((Cone.postcompose (parallelPair.ext eQ.symm (Iso.refl S.X₃) ⋯ ⋯).hom).obj (KernelFork.ofι h.ι ⋯)).π.app
      WalkingParallelPair.one =
    eH.symm.hom ≫ (KernelFork.ofι (eH.hom ≫ h.ι ≫ eQ.inv) ⋯).π.app WalkingParallelPair.one
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasRightHomology
inst✝ : S₂.HasRightHomology
⊢ (rightHomologyMap φ).op = S₂.leftHomologyOpIso.inv ≫ leftHomologyMap (opMap φ) ≫ S₁.leftHomologyOpIso.hom

Tactic:
dsimp [leftHomologyOpIso, LeftHomologyData.leftHomologyIso, leftHomologyMap,
    rightHomologyMap]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasRightHomology
inst✝ : S₂.HasRightHomology
⊢ (rightHomologyMap' φ S₁.rightHomologyData S₂.rightHomologyData).op =
    leftHomologyMap' (𝟙 S₂.op) S₂.rightHomologyData.op S₂.op.leftHomologyData ≫
      leftHomologyMap' (opMap φ) S₂.op.leftHomologyData S₁.op.leftHomologyData ≫
        leftHomologyMap' (𝟙 S₁.op) S₁.op.leftHomologyData S₁.rightHomologyData.op
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
h : Mono (opcyclesMap φ)
⊢ Mono (homologyMap φ)

Tactic:
have : Mono (homologyMap φ ≫ S₂.homologyι) := by
    rw [homologyι_naturality φ]
    apply mono_comp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
h : Mono (opcyclesMap φ)
this : Mono (homologyMap φ ≫ S₂.homologyι)
⊢ Mono (homologyMap φ)
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 : Cone F
h₁ : IsLimit (π₁.mapCone c)
h₂ : IsLimit (π₂.mapCone c)
h₃ : IsLimit (π₃.mapCone c)
s : Cone F
j : J
⊢ ({ τ₁ := h₁.lift (π₁.mapCone s), τ₂ := h₂.lift (π₂.mapCone s), τ₃ := h₃.lift (π₃.mapCone s), comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        c.π.app j).τ₁ =
    (s.π.app j).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 : Cone F
h₁ : IsLimit (π₁.mapCone c)
h₂ : IsLimit (π₂.mapCone c)
h₃ : IsLimit (π₃.mapCone c)
s : Cone F
j : J
⊢ ({ τ₁ := h₁.lift (π₁.mapCone s), τ₂ := h₂.lift (π₂.mapCone s), τ₃ := h₃.lift (π₃.mapCone s), comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        c.π.app j).τ₂ =
    (s.π.app j).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 : Cone F
h₁ : IsLimit (π₁.mapCone c)
h₂ : IsLimit (π₂.mapCone c)
h₃ : IsLimit (π₃.mapCone c)
s : Cone F
j : J
⊢ ({ τ₁ := h₁.lift (π₁.mapCone s), τ₂ := h₂.lift (π₂.mapCone s), τ₃ := h₃.lift (π₃.mapCone s), comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        c.π.app j).τ₃ =
    (s.π.app j).τ₃

Tactic:
<;>

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

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 : Cone F
h₁ : IsLimit (π₁.mapCone c)
h₂ : IsLimit (π₂.mapCone c)
h₃ : IsLimit (π₃.mapCone c)
s : Cone F
j : J
⊢ ({ τ₁ := h₁.lift (π₁.mapCone s), τ₂ := h₂.lift (π₂.mapCone s), τ₃ := h₃.lift (π₃.mapCone s), comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        c.π.app j).τ₂ =
    (s.π.app j).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 : Cone F
h₁ : IsLimit (π₁.mapCone c)
h₂ : IsLimit (π₂.mapCone c)
h₃ : IsLimit (π₃.mapCone c)
s : Cone F
j : J
⊢ ({ τ₁ := h₁.lift (π₁.mapCone s), τ₂ := h₂.lift (π₂.mapCone s), τ₃ := h₃.lift (π₃.mapCone s), comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        c.π.app j).τ₃ =
    (s.π.app j).τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b c : S₁ ⟶ S₂
⊢ (a + b + c).τ₁ = (a + (b + c)).τ₁

pp: case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b c : S₁ ⟶ S₂
⊢ (a + b + c).τ₂ = (a + (b + c)).τ₂

pp: case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b c : S₁ ⟶ S₂
⊢ (a + b + c).τ₃ = (a + (b + c)).τ₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b c : S₁ ⟶ S₂
⊢ (a + b + c).τ₁ = (a + (b + c)).τ₁

case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b c : S₁ ⟶ S₂
⊢ (a + b + c).τ₂ = (a + (b + c)).τ₂

case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b c : S₁ ⟶ S₂
⊢ (a + b + c).τ₃ = (a + (b + c)).τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (0 + a).τ₁ = a.τ₁

pp: case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (0 + a).τ₂ = a.τ₂

pp: case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (0 + a).τ₃ = a.τ₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (0 + a).τ₁ = a.τ₁

case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (0 + a).τ₂ = a.τ₂

case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (0 + a).τ₃ = a.τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (a + 0).τ₁ = a.τ₁

pp: case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (a + 0).τ₂ = a.τ₂

pp: case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (a + 0).τ₃ = a.τ₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (a + 0).τ₁ = a.τ₁

case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (a + 0).τ₂ = a.τ₂

case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (a + 0).τ₃ = a.τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a - b).τ₁ = (a + -b).τ₁

pp: case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a - b).τ₂ = (a + -b).τ₂

pp: case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a - b).τ₃ = (a + -b).τ₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a - b).τ₁ = (a + -b).τ₁

case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a - b).τ₂ = (a + -b).τ₂

case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a - b).τ₃ = (a + -b).τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (-a + a).τ₁ = Hom.τ₁ 0

pp: case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (-a + a).τ₂ = Hom.τ₂ 0

pp: case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (-a + a).τ₃ = Hom.τ₃ 0

Tactic:
<;>

Goal state after:
case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (-a + a).τ₁ = Hom.τ₁ 0

case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (-a + a).τ₂ = Hom.τ₂ 0

case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a : S₁ ⟶ S₂
⊢ (-a + a).τ₃ = Hom.τ₃ 0
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a + b).τ₁ = (b + a).τ₁

pp: case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a + b).τ₂ = (b + a).τ₂

pp: case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a + b).τ₃ = (b + a).τ₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a + b).τ₁ = (b + a).τ₁

case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a + b).τ₂ = (b + a).τ₂

case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
a b : S₁ ⟶ S₂
⊢ (a + b).τ₃ = (b + a).τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

Goal state before:
pp: case h₃
J : Type u_1
C : Type u_2
inst✝⁵ : Category.{v_1, u_1} J
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
inst✝² : HasColimit (F ⋙ π₁)
inst✝¹ : HasColimit (F ⋙ π₂)
inst✝ : HasColimit (F ⋙ π₃)
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ (F.map f ≫
        { τ₁ := colimit.ι (F ⋙ π₁) x✝, τ₂ := colimit.ι (F ⋙ π₂) x✝, τ₃ := colimit.ι (F ⋙ π₃) x✝, comm₁₂ := ⋯,
          comm₂₃ := ⋯ }).τ₃ =
    ({ τ₁ := colimit.ι (F ⋙ π₁) x✝¹, τ₂ := colimit.ι (F ⋙ π₂) x✝¹, τ₃ := colimit.ι (F ⋙ π₃) x✝¹, comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        ((const J).obj
              { X₁ := colimit (F ⋙ π₁), X₂ := colimit (F ⋙ π₂), X₃ := colimit (F ⋙ π₃),
                f := colimMap (F.whiskerLeft π₁Toπ₂), g := colimMap (F.whiskerLeft π₂Toπ₃), zero := ⋯ }).map
          f).τ₃

Tactic:
simp [← colimit.w (F ⋙ π₃) f]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : HasZeroMorphisms C
inst✝⁷ : HasZeroMorphisms D
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
F : C ⥤ D
inst✝⁶ : F.PreservesZeroMorphisms
inst✝⁵ : S₁.HasHomology
inst✝⁴ : S₂.HasHomology
inst✝³ : (S₁.map F).HasHomology
inst✝² : (S₂.map F).HasHomology
inst✝¹ : F.PreservesLeftHomologyOf S₁
inst✝ : F.PreservesLeftHomologyOf S₂
⊢ homologyMap (F.mapShortComplex.map φ) ≫ (S₂.mapHomologyIso F).hom = (S₁.mapHomologyIso F).hom ≫ F.map (homologyMap φ)

Tactic:
dsimp only [homologyMap, homologyMap', mapHomologyIso, LeftHomologyData.homologyIso,
    LeftHomologyData.leftHomologyIso, leftHomologyMapIso', leftHomologyIso,
    Iso.symm, Iso.trans, Iso.refl]

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : HasZeroMorphisms C
inst✝⁷ : HasZeroMorphisms D
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
F : C ⥤ D
inst✝⁶ : F.PreservesZeroMorphisms
inst✝⁵ : S₁.HasHomology
inst✝⁴ : S₂.HasHomology
inst✝³ : (S₁.map F).HasHomology
inst✝² : (S₂.map F).HasHomology
inst✝¹ : F.PreservesLeftHomologyOf S₁
inst✝ : F.PreservesLeftHomologyOf S₂
⊢ leftHomologyMap' (F.mapShortComplex.map φ) (S₁.map F).homologyData.left (S₂.map F).homologyData.left ≫
      leftHomologyMap' (𝟙 (S₂.map F)) (S₂.map F).homologyData.left (S₂.map F).leftHomologyData ≫
        leftHomologyMap' (𝟙 (S₂.map F)) (S₂.map F).leftHomologyData (S₂.homologyData.left.map F) =
    (leftHomologyMap' (𝟙 (S₁.map F)) (S₁.map F).homologyData.left (S₁.map F).leftHomologyData ≫
        leftHomologyMap' (𝟙 (S₁.map F)) (S₁.map F).leftHomologyData (S₁.homologyData.left.map F)) ≫
      F.map (leftHomologyMap' φ S₁.homologyData.left S₂.homologyData.left)
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h✝ : (ConcreteCategory.hom f) y ∈ Set.range ⇑(ConcreteCategory.hom f)
⊢ (ConcreteCategory.hom g) (Classical.choose h✝) = (ConcreteCategory.hom g).toFun y

pp: case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h✝ : (ConcreteCategory.hom f) y ∉ Set.range ⇑(ConcreteCategory.hom f)
⊢ ⋯.some = (ConcreteCategory.hom g).toFun y

Tactic:
<;>

Goal state after:
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h✝ : (ConcreteCategory.hom f) y ∈ Set.range ⇑(ConcreteCategory.hom f)
⊢ (ConcreteCategory.hom g) (Classical.choose h✝) = (ConcreteCategory.hom g).toFun y

case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h✝ : (ConcreteCategory.hom f) y ∉ Set.range ⇑(ConcreteCategory.hom f)
⊢ ⋯.some = (ConcreteCategory.hom g).toFun y
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
P Q : C
inst✝² : HasBinaryProduct P Q
inst✝¹ : Injective P
inst✝ : Injective Q
X✝ Y✝ : C
g : X✝ ⟶ P ⨯ Q
f : X✝ ⟶ Y✝
mono : Mono f
⊢ ∃ h, f ≫ h = g

Tactic:
by
    use Limits.prod.lift (factorThru (g ≫ Limits.prod.fst) f) (factorThru (g ≫ Limits.prod.snd) f)
    simp only [prod.comp_lift, comp_factorThru]
    ext
    · simp only [prod.lift_fst]
    · simp only [prod.lift_snd]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S : ShortComplex C
hS : S.ShortExact
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : PreservesFiniteLimits F
inst✝ : PreservesFiniteColimits F
⊢ (S.map F).ShortExact

Tactic:
have := hS.mono_f

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S : ShortComplex C
hS : S.ShortExact
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : PreservesFiniteLimits F
inst✝ : PreservesFiniteColimits F
this : Mono S.f
⊢ (S.map F).ShortExact
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
hS : S.ShortExact
inst✝ : Balanced C
this : HasZeroObject C
⊢ IsIso S.f ↔ IsZero S.X₃

Tactic:
have := hS.mono_f

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
hS : S.ShortExact
inst✝ : Balanced C
this✝ : HasZeroObject C
this : Mono S.f
⊢ IsIso S.f ↔ IsZero S.X₃
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
hS : S.ShortExact
inst✝ : Balanced C
this : HasZeroObject C
⊢ IsIso S.g ↔ IsZero S.X₁

Tactic:
have := hS.mono_f

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
hS : S.ShortExact
inst✝ : Balanced C
this✝ : HasZeroObject C
this : Mono S.f
⊢ IsIso S.g ↔ IsZero S.X₁
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Preadditive C
inst✝ : Balanced C
S : ShortComplex C
hS : S.ShortExact
⊢ IsLimit (KernelFork.ofι S.f ⋯)

Tactic:
have := hS.mono_f

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Preadditive C
inst✝ : Balanced C
S : ShortComplex C
hS : S.ShortExact
this : Mono S.f
⊢ IsLimit (KernelFork.ofι S.f ⋯)
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: case mp
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
h₁ : S₁.HomologyData
z₁ : IsZero h₁.left.H
⊢ S₂.Exact

Tactic:
exact ⟨HomologyData.ofEpiOfIsIsoOfMono φ h₁, z₁⟩

Goal state after:
Module: Mathlib.CategoryTheory.Adjunction.Restrict

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
D' : Type u₄
inst✝ : Category.{v₄, u₄} D'
iC : C ⥤ C'
iD : D ⥤ D'
L' : C' ⥤ D'
R' : D' ⥤ C'
adj : L' ⊣ R'
hiC : iC.FullyFaithful
hiD : iD.FullyFaithful
L : C ⥤ D
R : D ⥤ C
comm1 : iC ⋙ L' ≅ L ⋙ iD
comm2 : iD ⋙ R' ≅ R ⋙ iC
X : D
⊢ iD.map ((adj.restrictFullyFaithful hiC hiD comm1 comm2).counit.app X) =
    comm1.inv.app (R.obj X) ≫ L'.map (comm2.inv.app X) ≫ adj.counit.app (iD.obj X)

Tactic:
by
  dsimp [restrictFullyFaithful]
  simp

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
s : S.Splitting
F : C ⥤ D
inst✝ : F.Additive
⊢ F.map (𝟙 S.X₁) = 𝟙 (F.obj S.X₁)

Tactic:
F.map_id

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
s : S.Splitting
F : C ⥤ D
inst✝ : F.Additive
⊢ 𝟙 (F.obj S.X₁) = 𝟙 (F.obj S.X₁)
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: case e'_3
C : Type u₁
inst✝ : Category.{v₁, u₁} C
B : C
X Y : Subobject B
f : underlying.obj X ⟶ underlying.obj Y
w : f ≫ Y.arrow = X.arrow
⊢ X = mk X.arrow

pp: case e'_4
C : Type u₁
inst✝ : Category.{v₁, u₁} C
B : C
X Y : Subobject B
f : underlying.obj X ⟶ underlying.obj Y
w : f ≫ Y.arrow = X.arrow
⊢ Y = mk Y.arrow

Tactic:
<;>

Goal state after:
case e'_3
C : Type u₁
inst✝ : Category.{v₁, u₁} C
B : C
X Y : Subobject B
f : underlying.obj X ⟶ underlying.obj Y
w : f ≫ Y.arrow = X.arrow
⊢ X = mk X.arrow

case e'_4
C : Type u₁
inst✝ : Category.{v₁, u₁} C
B : C
X Y : Subobject B
f : underlying.obj X ⟶ underlying.obj Y
w : f ≫ Y.arrow = X.arrow
⊢ Y = mk Y.arrow
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

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 : Preadditive C
inst✝² : Preadditive D
S : ShortComplex C
inst✝¹ : Balanced C
hS : S.Exact
inst✝ : Mono S.f
⊢ IsLimit (KernelFork.ofι S.f ⋯)

Tactic:
have := hS.hasHomology
  have := hS.isIso_toCycles
  exact IsLimit.ofIsoLimit S.cyclesIsKernel
    (Fork.ext (asIso S.toCycles).symm (by simp))

Goal state after: