mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveNondegenerate

Goal state before:
pp: case succ
X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n + 1⦌
i : Fin (n + 1)
⊢ s.obj i.castSucc = s.obj i.succ ↔ ¬s.obj i.castSucc < s.obj i.succ

Tactic:
have := s.monotone i.castSucc_le_succ

Goal state after:
case succ
X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n + 1⦌
i : Fin (n + 1)
this : s.obj i.castSucc ≤ s.obj i.succ
⊢ s.obj i.castSucc = s.obj i.succ ↔ ¬s.obj i.castSucc < s.obj i.succ
Module: Mathlib.Algebra.Homology.Augment

Goal state before:
pp: V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : ChainComplex V ℕ
i : ℕ
⊢ C.d (i + 2) 0 = 0

Tactic:
by
  rw [C.shape]
  exact i.succ_succ_ne_one.symm

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case mp
X : SSet
n : ℕ
x : X _⦋n⦌
⊢ IsIso (toOfSimplex x) → Mono (yonedaEquiv.symm x)

pp: case mpr
X : SSet
n : ℕ
x : X _⦋n⦌
⊢ Mono (yonedaEquiv.symm x) → IsIso (toOfSimplex x)

Tactic:
· intro
    rw [← toOfSimplex_ι]
    infer_instance

Goal state after:
case mpr
X : SSet
n : ℕ
x : X _⦋n⦌
⊢ Mono (yonedaEquiv.symm x) → IsIso (toOfSimplex x)
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: case a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
S : SplitEpi f.hom
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1 + 1)
⊢ (ExtraDegeneracy.s f S (n + 1) ≫
        WidePullback.lift (WidePullback.base fun x ↦ f.hom)
          (fun i_1 ↦ WidePullback.π (fun x ↦ f.hom) (i.succ.succAbove i_1)) ⋯) ≫
      WidePullback.π (fun x ↦ f.hom) j =
    (WidePullback.lift (WidePullback.base fun x ↦ f.hom) (fun i_1 ↦ WidePullback.π (fun x ↦ f.hom) (i.succAbove i_1))
          ⋯ ≫
        ExtraDegeneracy.s f S n) ≫
      WidePullback.π (fun x ↦ f.hom) j

Tactic:
induction j using Fin.cases <;> simp

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: case a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
S : SplitEpi f.hom
n : ℕ
i : Fin (n + 1)
j : Fin (n + 1 + 1 + 1)
⊢ (ExtraDegeneracy.s f S n ≫
        WidePullback.lift (WidePullback.base fun x ↦ f.hom)
          (fun i_1 ↦ WidePullback.π (fun x ↦ f.hom) (i.succ.predAbove i_1)) ⋯) ≫
      WidePullback.π (fun x ↦ f.hom) j =
    (WidePullback.lift (WidePullback.base fun x ↦ f.hom) (fun i_1 ↦ WidePullback.π (fun x ↦ f.hom) (i.predAbove i_1))
          ⋯ ≫
        ExtraDegeneracy.s f S (n + 1)) ≫
      WidePullback.π (fun x ↦ f.hom) j

Tactic:
induction j using Fin.cases <;> simp

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

Goal state before:
pp: case pos
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : c₂.Rel i₂ (c₂.next i₂)
⊢ c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ (c₂.next i₂) ≫ K.d₂ c₁₂ i₁ (c₂.next i₂) i₁₂'' = 0

Tactic:
by_cases h₄ : c₂.Rel (c₂.next i₂) (c₂.next (c₂.next i₂))

Goal state after:
case pos
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : c₂.Rel i₂ (c₂.next i₂)
h₄ : c₂.Rel (c₂.next i₂) (c₂.next (c₂.next i₂))
⊢ c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ (c₂.next i₂) ≫ K.d₂ c₁₂ i₁ (c₂.next i₂) i₁₂'' = 0

case neg
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : c₂.Rel i₂ (c₂.next i₂)
h₄ : ¬c₂.Rel (c₂.next i₂) (c₂.next (c₂.next i₂))
⊢ c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ (c₂.next i₂) ≫ K.d₂ c₁₂ i₁ (c₂.next i₂) i₁₂'' = 0
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
i₁ i₁' : I₁
i₂ : I₂
j j' : J
h : c₁.π c₂ c (i₁', i₂) = j
h' : c₁.prev i₁' = i₁
⊢ K₁.ιMapBifunctor K₂ F c i₁' i₂ j h ≫ hom₁ h₁ f₂ F c j j' =
    c₁.ε₁ c₂ c (i₁, i₂) •
      (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫ (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ L₁.ιMapBifunctorOrZero L₂ F c i₁ i₂ j'

Tactic:
by
  subst h'
  simp [hom₁]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : (up ℤ).π (up ℤ) (up ℤ) (i₁, i₂) = n
⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫
      (total.map ((shiftFunctor₁ C x).map f) (up ℤ) ≫ (L.totalShift₁Iso x).hom).f n =
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫
      ((K.totalShift₁Iso x).hom ≫ (shiftFunctor (HomologicalComplex C (up ℤ)) x).map (total.map f (up ℤ))).f n

Tactic:
dsimp at h ⊢

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫
      (total.map ((shiftFunctor₁ C x).map f) (up ℤ)).f n ≫ (L.totalShift₁Iso x).hom.f n =
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫ (K.totalShift₁Iso x).hom.f n ≫ (total.map f (up ℤ)).f (n + x)
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
y : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : (up ℤ).π (up ℤ) (up ℤ) (i₁, i₂) = n
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫
      (total.map ((shiftFunctor₂ C y).map f) (up ℤ) ≫ (L.totalShift₂Iso y).hom).f n =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫
      ((K.totalShift₂Iso y).hom ≫ (shiftFunctor (HomologicalComplex C (up ℤ)) y).map (total.map f (up ℤ))).f n

Tactic:
dsimp at h ⊢

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
y : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫
      (total.map ((shiftFunctor₂ C y).map f) (up ℤ)).f n ≫ (L.totalShift₂Iso y).hom.f n =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫ (K.totalShift₂Iso y).hom.f n ≫ (total.map f (up ℤ)).f (n + y)
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : (up ℤ).π (up ℤ) (up ℤ) (n₁, n₂) = n
⊢ ((shiftFunctor₁ C x).obj ((shiftFunctor₂ C y).obj K)).ιTotal (up ℤ) n₁ n₂ n h ≫
      (((shiftFunctor₂ C y).obj K).totalShift₁Iso x ≪≫
              (shiftFunctor (CochainComplex C ℤ) x).mapIso (K.totalShift₂Iso y)).hom.f
        n =
    ((shiftFunctor₁ C x).obj ((shiftFunctor₂ C y).obj K)).ιTotal (up ℤ) n₁ n₂ n h ≫
      ((x * y).negOnePow •
              total.mapIso ((shiftFunctor₁₂CommIso C x y).app K) (up ℤ) ≪≫
                ((shiftFunctor₁ C x).obj K).totalShift₂Iso y ≪≫
                  (shiftFunctor (HomologicalComplex C (up ℤ)) y).mapIso (K.totalShift₁Iso x) ≪≫
                    (shiftFunctorComm (CochainComplex C ℤ) x y).app (K.total (up ℤ))).hom.f
        n

Tactic:
dsimp at h ⊢

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ ((shiftFunctor₁ C x).obj ((shiftFunctor₂ C y).obj K)).ιTotal (up ℤ) n₁ n₂ n h ≫
      (((shiftFunctor₂ C y).obj K).totalShift₁Iso x).hom.f n ≫ (K.totalShift₂Iso y).hom.f (n + x) =
    ((shiftFunctor₁ C x).obj ((shiftFunctor₂ C y).obj K)).ιTotal (up ℤ) n₁ n₂ n h ≫
      ((x * y).negOnePow •
        (total.map ((shiftFunctor₁₂CommIso C x y).hom.app K) (up ℤ)).f n ≫
          (((shiftFunctor₁ C x).obj K).totalShift₂Iso y).hom.f n ≫
            (K.totalShift₁Iso x).hom.f (n + y) ≫
              ((shiftFunctorComm (CochainComplex C ℤ) x y).hom.app (K.total (up ℤ))).f n)
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
L : CochainComplex C ℤ
hL : (HomotopyCategory.subcategoryAcyclic C).rightOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomotopyCategory.subcategoryAcyclic C ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K)
⊢ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map f = (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map 0

Tactic:
rw [hL ((HomotopyCategory.quotient _ _).map f) hK, Functor.map_zero]

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
α : Type u_2
c : ComplexShape α
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : α), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : K.HasPathObject
i : α
h₁ : IsZero (K.X i)
h₂ : ∀ (j : α), c.Rel j i → IsZero (K.X j)
⊢ IsZero (K.pathObject.X i)

Tactic:
by
  apply IsZero.unop
  dsimp [pathObject]
  refine homotopyCofiber.isZero_X _ _ ?_ (fun j hj ↦ IsZero.op (h₂ _ hj))
  exact IsZero.of_iso (by simpa using h₁.op)
    ((eval Cᵒᵖ c.symm i).mapBiprod K.op K.op)

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.TStructure

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
hX : ∀ (i : ℤ), n < i → IsZero ((homologyFunctor C i).obj X)
⊢ (Q.objPreimage X).IsLE n

Tactic:
rw [CochainComplex.isLE_iff]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
hX : ∀ (i : ℤ), n < i → IsZero ((homologyFunctor C i).obj X)
⊢ ∀ (i : ℤ), n < i → HomologicalComplex.ExactAt (Q.objPreimage X) i
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case pos.e_a
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²³ : Category.{v_1, u_1} C₁
inst✝²² : Category.{v_2, u_2} C₂
inst✝²¹ : Category.{v_3, u_5} C₃
inst✝²⁰ : Category.{v_4, u_6} C₄
inst✝¹⁹ : Category.{v_6, u_4} C₂₃
inst✝¹⁸ : HasZeroMorphisms C₁
inst✝¹⁷ : HasZeroMorphisms C₂
inst✝¹⁶ : HasZeroMorphisms C₃
inst✝¹⁵ : Preadditive C₂₃
inst✝¹⁴ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹³ : G₂₃.PreservesZeroMorphisms
inst✝¹² : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹¹ : F.PreservesZeroMorphisms
inst✝¹⁰ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁹ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁸ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁷ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁶ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁵ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁴ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝³ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝² : DecidableEq ι₂₃
inst✝¹ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₂₃ : ι₂₃ := c₂.π c₃ c₂₃ (i₂, i₃)
h₁ : c₂₃.Rel i₂₃ (c₂₃.next i₂₃)
h₂ : c₁.π c₂₃ c₄ (i₁, c₂₃.next i₂₃) = j'
⊢ c₁.ε₂ c₂₃ c₄ (i₁, i₂₃) •
      (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ i₃ (c₂.π c₃ c₂₃ (i₂, i₃)) ⋯) ≫
        (F.obj (K₁.X i₁)).map (mapBifunctor.D₁ K₂ K₃ G₂₃ c₂₃ i₂₃ (c₂₃.next i₂₃)) ≫
          K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂₃.next i₂₃) j' h₂ =
    d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'

Tactic:
rw [← Functor.map_comp_assoc, mapBifunctor.ι_D₁]
        by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
        · rw [d₂_eq _ _ _ _ _ _ _ _ _ h₃,
            mapBifunctor.d₁_eq _ _ _ _ h₃ _ _ (ComplexShape.next_π₁ c₃ c₂₃ h₃ i₃).symm,
            Functor.map_units_smul, Functor.map_comp, Linear.units_smul_comp,
            assoc, smul_smul, smul_left_cancel_iff,
            ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ _ (by
              dsimp [ComplexShape.r]
              rw [← h₂, ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄,
                ComplexShape.next_π₁ c₃ c₂₃ h₃ i₃]), ι_eq]
        · rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₃,
            mapBifunctor.d₁_eq_zero _ _ _ _ _ _ _ h₃,
            Functor.map_zero, zero_comp, smul_zero]

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.Cylinder

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
A : C
inst✝⁴ : CategoryWithWeakEquivalences C
P : Cylinder A
inst✝³ : HasBinaryCoproducts C
inst✝² : CategoryWithCofibrations C
inst✝¹ : P.IsGood
inst✝ : (cofibrations C).RespectsIso
⊢ Cofibration P.symm.i

Tactic:
have hi : cofibrations C P.i := by rw [← cofibration_iff]; infer_instance
    rw [P.symm_i, cofibration_iff]
    refine ((cofibrations C).arrow_mk_iso_iff ?_).2 hi
    exact Arrow.isoMk (coprod.braiding A A) (Iso.refl _)

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h : LeftHomotopyRel f g
⊢ ∃ P, P.IsVeryGood ∧ Nonempty (P.LeftHomotopy f g)

Tactic:
obtain ⟨P, _, ⟨h⟩⟩ := h.exists_good_cylinder
  let fac := MorphismProperty.factorizationData (trivialCofibrations C) (fibrations C) P.π
  let P' : Cylinder X :=
    { I := fac.Z
      i₀ := P.i₀ ≫ fac.i
      i₁ := P.i₁ ≫ fac.i
      π := fac.p
      weakEquivalence_π := weakEquivalence_of_precomp_of_fac fac.fac }
  have : Cofibration P'.i := by
    rw [show P'.i = P.i ≫ fac.i by cat_disch]
    infer_instance
  have sq : CommSq h.h fac.i (terminal.from _) (terminal.from _) := { }
  exact ⟨P', { }, ⟨{ h := sq.lift }⟩ ⟩

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a b : ℤ
h : a + 1 = b
⊢ t.triangleLEGE a b h ≅ t.triangleLTGE b

Tactic:
by
  refine Triangle.functorIsoMk _ _ (t.truncLEIsoTruncLT a b h) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_
  · cat_disch
  · cat_disch
  · ext
    dsimp [truncGEδLE]
    simp only [Category.assoc, Category.id_comp, ← Functor.map_comp,
      Iso.inv_hom_id_app, Functor.map_id, Category.comp_id]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
hg₂ :
  ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₂ = 0), Nonempty { X₁ := I, X₂ := Y, X₃ := Z, f := f, g := g₂, zero := w }.Splitting
I₁ : C
w✝ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
⊢ ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₁ ≫ g₂ = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Z, f := f, g := g₁ ≫ g₂, zero := w }.Splitting

Tactic:
obtain ⟨I₂, _, f₂, w₂, ⟨σ₂⟩⟩ := hg₂

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₁ ≫ g₂ = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Z, f := f, g := g₁ ≫ g₂, zero := w }.Splitting
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n) isStrictlyGE_iff._auto_1
⊢ IsZero ((mappingCone (𝟙 (I K)) ⊞ L).X i)

Tactic:
refine IsZero.of_iso ?_ ((HomologicalComplex.eval C (ComplexShape.up ℤ) i).mapBiprod _ _)

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n) isStrictlyGE_iff._auto_1
⊢ IsZero
    ((HomologicalComplex.eval C (ComplexShape.up ℤ) i).obj (mappingCone (𝟙 (I K))) ⊞
      (HomologicalComplex.eval C (ComplexShape.up ℤ) i).obj L)
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
f₀ f₁ : X ⟶ Y
h✝ : LeftHomotopyRel (f₀ ≫ g) (f₁ ≫ g)
P : Cylinder X
left✝ : P.IsGood
h : P.LeftHomotopy (f₀ ≫ g) (f₁ ≫ g)
sq : CommSq (coprod.desc f₀ f₁) P.i g h.h
⊢ P.i₁ ≫ sq.lift = f₁

Tactic:
have := coprod.inr ≫= sq.fac_left
          rwa [P.inr_i_assoc, coprod.inr_desc] at this

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case refine_2
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
n : ℤ
hf : ∀ i < n, QuasiIsoAt f i
inst✝¹ : Mono f
inst✝ : Mono (homologyMap f n)
i : ℤ
hi : i < n
⊢ { X₁ := { X₁ := K, X₂ := L, X₃ := cokernel f, f := f, g := cokernel.π f, zero := ⋯ }.X₂.homology i,
        X₂ := { X₁ := K, X₂ := L, X₃ := cokernel f, f := f, g := cokernel.π f, zero := ⋯ }.X₃.homology i,
        X₃ := { X₁ := K, X₂ := L, X₃ := cokernel f, f := f, g := cokernel.π f, zero := ⋯ }.X₁.homology (i + 1),
        f := homologyMap { X₁ := K, X₂ := L, X₃ := cokernel f, f := f, g := cokernel.π f, zero := ⋯ }.g i,
        g := ⋯.δ i (i + 1) ⋯, zero := ⋯ }.g =
    0

Tactic:
rw [← ((shortExact f).homology_exact₁ i (i + 1) (by simp)).mono_g_iff]

Goal state after:
case refine_2
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
n : ℤ
hf : ∀ i < n, QuasiIsoAt f i
inst✝¹ : Mono f
inst✝ : Mono (homologyMap f n)
i : ℤ
hi : i < n
⊢ Mono
    { X₁ := { X₁ := K, X₂ := L, X₃ := cokernel f, f := f, g := cokernel.π f, zero := ⋯ }.X₃.homology i,
        X₂ := { X₁ := K, X₂ := L, X₃ := cokernel f, f := f, g := cokernel.π f, zero := ⋯ }.X₁.homology (i + 1),
        X₃ := { X₁ := K, X₂ := L, X₃ := cokernel f, f := f, g := cokernel.π f, zero := ⋯ }.X₂.homology (i + 1),
        f := ⋯.δ i (i + 1) ⋯,
        g := homologyMap { X₁ := K, X₂ := L, X₃ := cokernel f, f := f, g := cokernel.π f, zero := ⋯ }.f (i + 1),
        zero := ⋯ }.g
Module: Mathlib.CategoryTheory.GuitartExact.KanExtension

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
D : Type u₅
inst✝⁵ : Category.{v₁, u₁} C₁
inst✝⁴ : Category.{v₂, u₂} C₂
inst✝³ : Category.{v₃, u₃} C₃
inst✝² : Category.{v₄, u₄} C₄
inst✝¹ : Category.{v₅, u₅} D
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
F : C₂ ⥤ D
X₃ : C₃
inst✝ : (w.costructuredArrowRightwards X₃).Final
⊢ L.HasPointwiseLeftKanExtensionAt (T ⋙ F) X₃ ↔ R.HasPointwiseLeftKanExtensionAt F (B.obj X₃)

Tactic:
dsimp [Functor.HasPointwiseLeftKanExtensionAt]
  rw [← Functor.Final.hasColimit_comp_iff (w.costructuredArrowRightwards X₃)]
  rfl

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
H : Type u₃
inst✝⁷ : Category.{v₁, u₁} C₁
inst✝⁶ : Category.{v₂, u₂} C₂
inst✝⁵ : Category.{v₃, u₃} H
D₁ : Type u₄
D₂ : Type u₅
inst✝⁴ : Category.{v₄, u₄} D₁
inst✝³ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : C₂ ⥤ H
F₁ : D₁ ⥤ H
α₁ : Φ.functor ⋙ F ⟶ L₁ ⋙ F₁
F₂ : D₂ ⥤ H
α₂ : F ⟶ L₂ ⋙ F₂
inst✝ : F₁.IsRightDerivedFunctor α₁ W₁
X : C₁
⊢ α₁.app X ≫ (Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂).app (L₁.obj X) =
    α₂.app (Φ.functor.obj X) ≫ F₂.map ((CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom.app X)

Tactic:
by
  simpa using congr_app (Φ.rightDerivedFunctorComparison_fac L₁ L₂ F F₁ α₁ F₂ α₂) X

Goal state after: