mathlib-initiative/mathlib-tactics huggingface dataset:
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
S S₁ S₂ : ShortComplex C
h : S.RightHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesColimit (parallelPair S.f 0) F
this : PreservesLimit (parallelPair h.g' 0) F
wp : F.map S.f ≫ F.map h.p = 0
hp : IsColimit ((CokernelCofork.ofπ h.p ⋯).map F)
g' : F.obj h.Q ⟶ F.obj S.X₃ := hp.desc (CokernelCofork.ofπ (S.map F).g ⋯)
hg' : g' = F.map h.g'
wι : F.map h.ι ≫ g' = 0
⊢ IsLimit (KernelFork.ofι (F.map h.ι) wι)

Tactic:
by

Goal state after:
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
S : ShortComplex C
inst✝¹ : S.HasRightHomology
inst✝ : HasCokernel S.f
h : S.RightHomologyData := S.rightHomologyData
this : HasLimit (parallelPair h.g' 0)
⊢ (parallelPair (cokernel.desc S.f S.g ⋯) 0).map WalkingParallelPairHom.right ≫
      (Iso.refl ((parallelPair (cokernel.desc S.f S.g ⋯) 0).obj WalkingParallelPair.one)).hom =
    ((colimit.isColimit (parallelPair S.f 0)).coconePointUniqueUpToIso h.hp).hom ≫
      (parallelPair h.g' 0).map WalkingParallelPairHom.right

Tactic:
by

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

Goal state before:
pp: 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 =
    { τ₁ := h₁.lift (π₁.mapCone s), τ₂ := h₂.lift (π₂.mapCone s), τ₃ := h₃.lift (π₃.mapCone s), comm₁₂ := ⋯,
      comm₂₃ := ⋯ }

Tactic:
by

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
S S₁ S₂ : ShortComplex C
h : S.RightHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
⊢ (S.map F).RightHomologyData

Tactic:
by

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

Goal state before:
pp: 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 : HasLimit (F ⋙ π₁)
inst✝¹ : HasLimit (F ⋙ π₂)
inst✝ : HasLimit (F ⋙ π₃)
⊢ limMap (F.whiskerLeft π₁Toπ₂) ≫ limMap (F.whiskerLeft π₂Toπ₃) = 0

Tactic:
by

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

Goal state before:
pp: 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 : HasLimit (F ⋙ π₁)
inst✝¹ : HasLimit (F ⋙ π₂)
inst✝ : HasLimit (F ⋙ π₃)
j : J
⊢ limit.π (F ⋙ π₁) j ≫ (F.obj j).f =
    (((const J).obj
              { X₁ := limit (F ⋙ π₁), X₂ := limit (F ⋙ π₂), X₃ := limit (F ⋙ π₃), f := limMap (F.whiskerLeft π₁Toπ₂),
                g := limMap (F.whiskerLeft π₂Toπ₃), zero := ⋯ }).obj
          j).f ≫
      limit.π (F ⋙ π₂) j

Tactic:
by

Goal state after:
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
S : ShortComplex C
inst✝¹ : S.HasRightHomology
inst✝ : HasCokernel S.f
⊢ HasKernel (cokernel.desc S.f S.g ⋯)

Tactic:
by

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

Goal state before:
pp: 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 : HasLimit (F ⋙ π₁)
inst✝¹ : HasLimit (F ⋙ π₂)
inst✝ : HasLimit (F ⋙ π₃)
j : J
⊢ limit.π (F ⋙ π₂) j ≫ (F.obj j).g =
    (((const J).obj
              { X₁ := limit (F ⋙ π₁), X₂ := limit (F ⋙ π₂), X₃ := limit (F ⋙ π₃), f := limMap (F.whiskerLeft π₁Toπ₂),
                g := limMap (F.whiskerLeft π₂Toπ₃), zero := ⋯ }).obj
          j).g ≫
      limit.π (F ⋙ π₃) j

Tactic:
by

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
S : ShortComplex C
h : S.RightHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
⊢ (h.map F).g' = F.map h.g'

Tactic:
by

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

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 S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
ψ : RightHomologyMapData φ h₁ h₂
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : h₁.IsPreservedBy F
inst✝ : h₂.IsPreservedBy F
⊢ (h₁.map F).p ≫ F.map ψ.φQ = (F.mapShortComplex.map φ).τ₂ ≫ (h₂.map F).p

Tactic:
by

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

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 S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
ψ : RightHomologyMapData φ h₁ h₂
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : h₁.IsPreservedBy F
inst✝ : h₂.IsPreservedBy F
⊢ F.map ψ.φQ ≫ (h₂.map F).g' = (h₁.map F).g' ≫ (F.mapShortComplex.map φ).τ₃

Tactic:
by

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 : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ IsIso (opcyclesMap' φ h₁ h₂)

Tactic:
by

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

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 S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
ψ : RightHomologyMapData φ h₁ h₂
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : h₁.IsPreservedBy F
inst✝ : h₂.IsPreservedBy F
⊢ F.map ψ.φH ≫ (h₂.map F).ι = (h₁.map F).ι ≫ F.map ψ.φQ

Tactic:
by

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

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 S₁ S₂ : ShortComplex C
h : S.HomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : h.left.IsPreservedBy F
inst✝ : h.right.IsPreservedBy F
⊢ (h.left.map F).π ≫ (F.mapIso h.iso).hom ≫ (h.right.map F).ι = (h.left.map F).i ≫ (h.right.map F).p

Tactic:
by

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

Goal state before:
pp: 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 : HasLimit (F ⋙ π₁)
inst✝¹ : HasLimit (F ⋙ π₂)
inst✝ : HasLimit (F ⋙ π₃)
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ ((const J).obj
            { X₁ := limit (F ⋙ π₁), X₂ := limit (F ⋙ π₂), X₃ := limit (F ⋙ π₃), f := limMap (F.whiskerLeft π₁Toπ₂),
              g := limMap (F.whiskerLeft π₂Toπ₃), zero := ⋯ }).map
        f ≫
      { τ₁ := limit.π (F ⋙ π₁) x✝, τ₂ := limit.π (F ⋙ π₂) x✝, τ₃ := limit.π (F ⋙ π₃) x✝, comm₁₂ := ⋯, comm₂₃ := ⋯ } =
    { τ₁ := limit.π (F ⋙ π₁) x✝¹, τ₂ := limit.π (F ⋙ π₂) x✝¹, τ₃ := limit.π (F ⋙ π₃) x✝¹, comm₁₂ := ⋯, comm₂₃ := ⋯ } ≫
      F.map f

Tactic:
by

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
⊢ Abelian.image.ι S.f ≫ S.g = 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
⊢ S.abelianImageToKernel ≫ kernel.ι S.g ≫ cokernel.π S.f = 0

Tactic:
by

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
W'✝ : C
k : W'✝ ⟶ kernel S.g
hk : k ≫ kernel.ι S.g ≫ cokernel.π S.f = 0
⊢ (k ≫ kernel.ι S.g) ≫ cokernel.π S.f = 0

Tactic:
by

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
W'✝ : C
k : W'✝ ⟶ kernel S.g
hk : k ≫ kernel.ι S.g ≫ cokernel.π S.f = 0
⊢ kernel.lift (cokernel.π S.f) (k ≫ kernel.ι S.g) ⋯ ≫ S.abelianImageToKernel = k

Tactic:
by

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
W'✝ : C
k : W'✝ ⟶ kernel S.g
hk : k ≫ kernel.ι S.g ≫ cokernel.π S.f = 0
b : W'✝ ⟶ Abelian.image S.f
hb : b ≫ S.abelianImageToKernel = k
⊢ b = kernel.lift (cokernel.π S.f) (k ≫ kernel.ι S.g) ⋯

Tactic:
by

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

Goal state before:
pp: 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 : HasLimit (F ⋙ π₁)
inst✝¹ : HasLimit (F ⋙ π₂)
inst✝ : HasLimit (F ⋙ π₃)
⊢ ∀ (j : J),
    (limit.cone (F ⋙ π₁)).π.app j = (Iso.refl (limit.cone (F ⋙ π₁)).pt).hom ≫ (π₁.mapCone (limitCone F)).π.app j

Tactic:
by

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

Goal state before:
pp: 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 : HasLimit (F ⋙ π₁)
inst✝¹ : HasLimit (F ⋙ π₂)
inst✝ : HasLimit (F ⋙ π₃)
⊢ ∀ (j : J),
    (limit.cone (F ⋙ π₂)).π.app j = (Iso.refl (limit.cone (F ⋙ π₂)).pt).hom ≫ (π₂.mapCone (limitCone F)).π.app j

Tactic:
by

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 ⋯
⊢ f' ≫ γ = 0

Tactic:
by

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

Goal state before:
pp: 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 : HasLimit (F ⋙ π₁)
inst✝¹ : HasLimit (F ⋙ π₂)
inst✝ : HasLimit (F ⋙ π₃)
⊢ ∀ (j : J),
    (limit.cone (F ⋙ π₃)).π.app j = (Iso.refl (limit.cone (F ⋙ π₃)).pt).hom ≫ (π₃.mapCone (limitCone F)).π.app j

Tactic:
by

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 ⋯
⊢ f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ

Tactic:
by

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

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
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ F.map (leftRightHomologyComparison' hₗ hᵣ) = leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F)

Tactic:
by

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
h : S.LeftHomologyData
inst✝ : h.IsPreservedBy F
h' : S.LeftHomologyData
this : PreservesColimit (parallelPair h.f' 0) F
⊢ (parallelPair h.f' 0).map WalkingParallelPairHom.left ≫ (ShortComplex.cyclesMapIso' (Iso.refl S) h h').hom =
    (Iso.refl ((parallelPair h.f' 0).obj WalkingParallelPair.zero)).hom ≫
      (parallelPair h'.f' 0).map WalkingParallelPairHom.left

Tactic:
by

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
⊢ f' ≫ cokernel.π (kernel.ι γ) = 0

Tactic:
by

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
h : S.LeftHomologyData
inst✝ : h.IsPreservedBy F
h' : S.LeftHomologyData
this : PreservesColimit (parallelPair h.f' 0) F
⊢ (parallelPair h.f' 0).map WalkingParallelPairHom.right ≫ (ShortComplex.cyclesMapIso' (Iso.refl S) h h').hom =
    (Iso.refl ((parallelPair h.f' 0).obj WalkingParallelPair.zero)).hom ≫
      (parallelPair h'.f' 0).map WalkingParallelPairHom.right

Tactic:
by

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
h : S.LeftHomologyData
inst✝ : h.IsPreservedBy F
h' : S.LeftHomologyData
⊢ PreservesColimit (parallelPair h'.f' 0) F

Tactic:
by

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
a b c : S₁ ⟶ S₂
⊢ a + b + c = a + (b + c)

Tactic:
by

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
a : S₁ ⟶ S₂
⊢ 0 + a = a

Tactic:
by

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
a : S₁ ⟶ S₂
⊢ a + 0 = a

Tactic:
by

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
h : S.RightHomologyData
inst✝ : h.IsPreservedBy F
h' : S.RightHomologyData
this : PreservesLimit (parallelPair h.g' 0) F
⊢ (parallelPair h.g' 0).map WalkingParallelPairHom.left ≫
      (Iso.refl ((parallelPair h.g' 0).obj WalkingParallelPair.one)).hom =
    (ShortComplex.opcyclesMapIso' (Iso.refl S) h h').hom ≫ (parallelPair h'.g' 0).map WalkingParallelPairHom.left

Tactic:
by

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
a b : S₁ ⟶ S₂
⊢ a - b = a + -b

Tactic:
by

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
a : S₁ ⟶ S₂
⊢ -a + a = 0

Tactic:
by

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
h : S.RightHomologyData
inst✝ : h.IsPreservedBy F
h' : S.RightHomologyData
this : PreservesLimit (parallelPair h.g' 0) F
⊢ (parallelPair h.g' 0).map WalkingParallelPairHom.right ≫
      (Iso.refl ((parallelPair h.g' 0).obj WalkingParallelPair.one)).hom =
    (ShortComplex.opcyclesMapIso' (Iso.refl S) h h').hom ≫ (parallelPair h'.g' 0).map WalkingParallelPairHom.right

Tactic:
by

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
a b : S₁ ⟶ S₂
⊢ a + b = b + a

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ leftHomologyMap' (-φ) h₁ h₂ = -leftHomologyMap' φ h₁ h₂

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ cyclesMap' (-φ) h₁ h₂ = -cyclesMap' φ h₁ h₂

Tactic:
by

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
φ φ' : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ leftHomologyMap' (φ + φ') h₁ h₂ = leftHomologyMap' φ h₁ h₂ + leftHomologyMap' φ' h₁ h₂

Tactic:
by

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
h : S.RightHomologyData
inst✝ : h.IsPreservedBy F
h' : S.RightHomologyData
⊢ PreservesLimit (parallelPair h'.g' 0) F

Tactic:
by

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
φ φ' : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ cyclesMap' (φ + φ') h₁ h₂ = cyclesMap' φ h₁ h₂ + cyclesMap' φ' h₁ h₂

Tactic:
by

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
φ φ' : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ leftHomologyMap' (φ - φ') h₁ h₂ = leftHomologyMap' φ h₁ h₂ - leftHomologyMap' φ' h₁ h₂

Tactic:
by

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
⊢ f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ

Tactic:
by

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
φ φ' : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ cyclesMap' (φ - φ') h₁ h₂ = cyclesMap' φ h₁ h₂ - cyclesMap' φ' h₁ h₂

Tactic:
by

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

Goal state before:
pp: 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
j : J
⊢ (π₁.mapCocone c).ι.app j ≫ h₁.desc (π₁.mapCocone s) ≫ s.pt.f =
    (π₁.mapCocone c).ι.app j ≫ c.pt.f ≫ h₂.desc (π₂.mapCocone s)

Tactic:
by

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

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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasLeftHomology
inst✝ : F.PreservesLeftHomologyOf S
⊢ (F.mapShortComplex.obj S).HasLeftHomology

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ rightHomologyMap' (-φ) h₁ h₂ = -rightHomologyMap' φ h₁ h₂

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ opcyclesMap' (-φ) h₁ h₂ = -opcyclesMap' φ h₁ h₂

Tactic:
by

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

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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasRightHomology
inst✝ : F.PreservesRightHomologyOf S
⊢ (F.mapShortComplex.obj S).HasRightHomology

Tactic:
by

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
φ φ' : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ rightHomologyMap' (φ + φ') h₁ h₂ = rightHomologyMap' φ h₁ h₂ + rightHomologyMap' φ' h₁ h₂

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : F.PreservesRightHomologyOf S
⊢ (F.mapShortComplex.obj S).HasHomology

Tactic:
by

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
⊢ kernel.ι γ ≫ x = 0

Tactic:
by

Goal state after: