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✝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
hr : S.RightHomologyData
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : (S.map F).HasHomology
inst✝ : F.PreservesRightHomologyOf S
⊢ S.mapHomologyIso' F = (hr.map F).homologyIso ≪≫ F.mapIso hr.homologyIso.symm

Tactic:
ext

Goal state after:
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
hr : S.RightHomologyData
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : (S.map F).HasHomology
inst✝ : F.PreservesRightHomologyOf S
⊢ (S.mapHomologyIso' F).hom = ((hr.map F).homologyIso ≪≫ F.mapIso hr.homologyIso.symm).hom
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
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
      comm₂₃ := ⋯ }

Tactic:
ext

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 : 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

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
h₁₂ : Homotopy φ₁ φ₂
h₂₃ : Homotopy φ₂ φ₃
⊢ φ₁.τ₂ = S₁.g ≫ (h₁₂.h₂ + h₂₃.h₂) + (h₁₂.h₁ + h₂₃.h₁) ≫ S₂.f + φ₃.τ₂

Tactic:
by rw [h₁₂.comm₂, h₂₃.comm₂, comp_add, add_comp]; abel

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Opposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.desc f (kernel.ι f.op).unop ⋯).op ≫ kernel.lift f.op (cokernel.π f).op ⋯ = 𝟙 (kernel f.op)

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ ((cokernel.desc f (kernel.ι f.op).unop ⋯).op ≫ kernel.lift f.op (cokernel.π f).op ⋯) ≫ equalizer.ι f.op 0 =
    𝟙 (kernel f.op) ≫ equalizer.ι f.op 0
Module: Mathlib.CategoryTheory.Abelian.Opposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ cokernel.desc f (kernel.ι f.op).unop ⋯ ≫ (kernel.lift f.op (cokernel.π f).op ⋯).unop = 𝟙 (cokernel f)

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ coequalizer.π f 0 ≫ cokernel.desc f (kernel.ι f.op).unop ⋯ ≫ (kernel.lift f.op (cokernel.π f).op ⋯).unop =
    coequalizer.π f 0 ≫ 𝟙 (cokernel f)
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.PreservesRightHomologyOf S₁
inst✝ : F.PreservesRightHomologyOf S₂
⊢ F.map (homologyMap φ) ≫ (S₂.mapHomologyIso' F).inv =
    (S₁.mapHomologyIso' F).inv ≫ homologyMap (F.mapShortComplex.map φ)

Tactic:
by
  rw [← cancel_epi (S₁.mapHomologyIso' F).hom, ← mapHomologyIso'_hom_naturality_assoc,
    Iso.hom_inv_id, comp_id, Iso.hom_inv_id_assoc]

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
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : F.PreservesRightHomologyOf S
⊢ S.mapHomologyIso' F = S.mapHomologyIso F

Tactic:
ext

Goal state after:
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
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : F.PreservesRightHomologyOf S
⊢ (S.mapHomologyIso' F).hom = (S.mapHomologyIso F).hom
Module: Mathlib.CategoryTheory.Abelian.Opposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ cokernel.desc f.op (kernel.ι f).op ⋯ ≫ (kernel.lift f (cokernel.π f.op).unop ⋯).op = 𝟙 (cokernel f.op)

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ coequalizer.π f.op 0 ≫ cokernel.desc f.op (kernel.ι f).op ⋯ ≫ (kernel.lift f (cokernel.π f.op).unop ⋯).op =
    coequalizer.π f.op 0 ≫ 𝟙 (cokernel f.op)
Module: Mathlib.CategoryTheory.Abelian.Opposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.desc f.op (kernel.ι f).op ⋯).unop ≫ kernel.lift f (cokernel.π f.op).unop ⋯ = 𝟙 (kernel f)

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ ((cokernel.desc f.op (kernel.ι f).op ⋯).unop ≫ kernel.lift f (cokernel.π f.op).unop ⋯) ≫ equalizer.ι f 0 =
    𝟙 (kernel f) ≫ equalizer.ι f 0
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesLeftHomologyOf S₁
inst✝¹ : F.PreservesLeftHomologyOf S₂
inst✝ : QuasiIso φ
γ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData
⊢ IsIso γ.φH

Tactic:
rw [← γ.quasiIso_iff]

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesLeftHomologyOf S₁
inst✝¹ : F.PreservesLeftHomologyOf S₂
inst✝ : QuasiIso φ
γ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData
⊢ QuasiIso φ
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: 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✝
⊢ (ConcreteCategory.hom
          (f ≫
            ↾fun z ↦
              if h : z ∈ Set.range ⇑(ConcreteCategory.hom f) then (ConcreteCategory.hom g) (Classical.choose h)
              else ⋯.some)).toFun
      y =
    (ConcreteCategory.hom g).toFun y

Tactic:
change dite (f y ∈ Set.range f) (fun h => g (Classical.choose h)) _ = _
      split_ifs <;> rename_i h
      · rw [mono_iff_injective] at mono
        simp [mono (Classical.choose_spec h)]
      · exact False.elim (h ⟨y, rfl⟩)

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesLeftHomologyOf S₁
inst✝¹ : F.PreservesLeftHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData
⊢ IsIso (F.map γ.φH) ↔ IsIso γ.φH

Tactic:
constructor

Goal state after:
case mp
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesLeftHomologyOf S₁
inst✝¹ : F.PreservesLeftHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData
⊢ IsIso (F.map γ.φH) → IsIso γ.φH

case mpr
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesLeftHomologyOf S₁
inst✝¹ : F.PreservesLeftHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData
⊢ IsIso γ.φH → IsIso (F.map γ.φH)
Module: Mathlib.CategoryTheory.Preadditive.Projective.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
⊢ EnoughProjectives C ↔ EnoughProjectives D

Tactic:
constructor

Goal state after:
case mp
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
⊢ EnoughProjectives C → EnoughProjectives D

case mpr
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
⊢ EnoughProjectives D → EnoughProjectives C
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
⊢ (opMap φ₁).τ₃.unop = (h.h₀.op + h.h₁.op ≫ S₁.op.g + (opMap φ₂).τ₃).unop

Tactic:
dsimp; rw [h.comm₁]; abel

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Projective.Basic

Goal state before:
pp: case mp
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
H : EnoughProjectives C
⊢ EnoughProjectives D

Tactic:
constructor

Goal state after:
case mp
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
H : EnoughProjectives C
⊢ ∀ (X : D), Nonempty (ProjectivePresentation X)
Module: Mathlib.CategoryTheory.Preadditive.Projective.Basic

Goal state before:
pp: case mp
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
H : EnoughProjectives C
X : D
⊢ Nonempty (ProjectivePresentation X)

Tactic:
constructor

Goal state after:
case mp
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
H : EnoughProjectives C
X : D
⊢ ProjectivePresentation X
Module: Mathlib.CategoryTheory.Preadditive.Projective.Basic

Goal state before:
pp: case mpr
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
H : EnoughProjectives D
⊢ EnoughProjectives C

Tactic:
constructor

Goal state after:
case mpr
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
H : EnoughProjectives D
⊢ ∀ (X : C), Nonempty (ProjectivePresentation X)
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : QuasiIso φ
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso γ.φH

Tactic:
rw [← γ.quasiIso_iff]

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : QuasiIso φ
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ QuasiIso φ
Module: Mathlib.CategoryTheory.Preadditive.Projective.Basic

Goal state before:
pp: case mpr
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
H : EnoughProjectives D
X : C
⊢ Nonempty (ProjectivePresentation X)

Tactic:
constructor

Goal state after:
case mpr
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ≌ D
H : EnoughProjectives D
X : C
⊢ ProjectivePresentation X
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: case h
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
⊢ prod.lift (g ≫ prod.fst) (g ≫ prod.snd) = g

Tactic:
ext

Goal state after:
case h.h₁
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
⊢ prod.lift (g ≫ prod.fst) (g ≫ prod.snd) ≫ prod.fst = g ≫ prod.fst

case h₂
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
⊢ prod.lift (g ≫ prod.fst) (g ≫ prod.snd) ≫ prod.snd = g ≫ prod.snd
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
P Q : C
inst✝³ : HasZeroMorphisms C
inst✝² : HasBinaryBiproduct P Q
inst✝¹ : Injective P
inst✝ : Injective Q
X✝ Y✝ : C
g : X✝ ⟶ P ⊞ Q
f : X✝ ⟶ Y✝
mono : Mono f
⊢ f ≫ biprod.lift (factorThru (g ≫ biprod.fst) f) (factorThru (g ≫ biprod.snd) f) = g

Tactic:
ext

Goal state after:
case h₀
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
P Q : C
inst✝³ : HasZeroMorphisms C
inst✝² : HasBinaryBiproduct P Q
inst✝¹ : Injective P
inst✝ : Injective Q
X✝ Y✝ : C
g : X✝ ⟶ P ⊞ Q
f : X✝ ⟶ Y✝
mono : Mono f
⊢ (f ≫ biprod.lift (factorThru (g ≫ biprod.fst) f) (factorThru (g ≫ biprod.snd) f)) ≫ biprod.fst = g ≫ biprod.fst

case h₁
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
P Q : C
inst✝³ : HasZeroMorphisms C
inst✝² : HasBinaryBiproduct P Q
inst✝¹ : Injective P
inst✝ : Injective Q
X✝ Y✝ : C
g : X✝ ⟶ P ⊞ Q
f : X✝ ⟶ Y✝
mono : Mono f
⊢ (f ≫ biprod.lift (factorThru (g ≫ biprod.fst) f) (factorThru (g ≫ biprod.snd) f)) ≫ biprod.snd = g ≫ biprod.snd
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso (F.map γ.φH) ↔ IsIso γ.φH

Tactic:
constructor

Goal state after:
case mp
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso (F.map γ.φH) → IsIso γ.φH

case mpr
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso γ.φH → IsIso (F.map γ.φH)
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
β : Type v
c : β → C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBiproduct c
inst✝ : ∀ (b : β), Injective (c b)
X✝ Y✝ : C
g : X✝ ⟶ ⨁ c
f : X✝ ⟶ Y✝
mono : Mono f
⊢ (f ≫ biproduct.lift fun b ↦ factorThru (g ≫ biproduct.π c b) f) = g

Tactic:
ext

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
β : Type v
c : β → C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBiproduct c
inst✝ : ∀ (b : β), Injective (c b)
X✝ Y✝ : C
g : X✝ ⟶ ⨁ c
f : X✝ ⟶ Y✝
mono : Mono f
j✝ : β
⊢ (f ≫ biproduct.lift fun b ↦ factorThru (g ≫ biproduct.π c b) f) ≫ biproduct.π c j✝ = g ≫ biproduct.π c j✝
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✝
S₁ S₂ : ShortComplex Cᵒᵖ
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ (unopMap φ₁).τ₃.op = (h.h₀.unop + h.h₁.unop ≫ S₁.unop.g + (unopMap φ₂).τ₃).op

Tactic:
dsimp; rw [h.comm₁]; abel

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
⊢ φ₁ = φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯

Tactic:
ext

Goal state after:
case h₁
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
⊢ φ₁.τ₁ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₁

case h₂
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
⊢ φ₁.τ₂ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₂

case h₃
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
⊢ φ₁.τ₃ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₃
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.RightHomologyData
⊢ h.g' = 0

Tactic:
by rw [← cancel_epi h.p, h.p_g', comp_zero, hg]

Goal state after:
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 : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₁ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₁

Tactic:
dsimp; rw [h.comm₁]; abel

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
H₁ : S₁.LeftHomologyData
H₂ : S₂.LeftHomologyData
h₀ : S₁.X₁ ⟶ S₂.X₁
h₀_f : h₀ ≫ S₂.f = 0
h₁ : S₁.X₂ ⟶ S₂.X₁
h₂ : S₁.X₃ ⟶ S₂.X₂
h₃ : S₁.X₃ ⟶ S₂.X₃
g_h₃ : S₁.g ≫ h₃ = 0
⊢ H₁.π ≫ 0 = H₂.liftK (H₁.i ≫ h₁ ≫ S₂.f) ⋯ ≫ H₂.π

Tactic:
rw [H₂.liftK_π_eq_zero_of_boundary (H₁.i ≫ h₁ ≫ S₂.f) (H₁.i ≫ h₁) (by rw [assoc]), comp_zero]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
e : S₁ ≅ S₂
⊢ S₁.ShortExact ↔ S₂.ShortExact

Tactic:
constructor

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
e : S₁ ≅ S₂
⊢ S₁.ShortExact → S₂.ShortExact

case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
e : S₁ ≅ S₂
⊢ S₂.ShortExact → S₁.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✝¹ : HasZeroObject C
this✝ : Mono S.f
this : Epi S.g
⊢ IsIso S.f ↔ IsZero S.X₃

Tactic:
constructor

Goal state after:
case mp
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✝¹ : HasZeroObject C
this✝ : Mono S.f
this : Epi S.g
⊢ IsIso S.f → IsZero S.X₃

case mpr
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✝¹ : HasZeroObject C
this✝ : Mono S.f
this : Epi S.g
⊢ IsZero S.X₃ → IsIso S.f