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

Goal state before:
pp: case succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : SimplicialObject C
f✝ f : X ⟶ Y
n✝ : ℕ
i : Fin (n✝ + 1)
x✝ : i ∈ Finset.univ
⊢ (kernelSubobject (X.δ i.succ)).factorThru (Finset.univ.inf fun k ↦ kernelSubobject (X.δ k.succ)).arrow ⋯ ≫
      (kernelSubobject (X.δ i.succ)).arrow ≫ X.δ i.succ ≫ f.app (op ⦋n✝⦌) =
    0

Tactic:
kernelSubobject_arrow_comp_assoc,

Goal state after:
case succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : SimplicialObject C
f✝ f : X ⟶ Y
n✝ : ℕ
i : Fin (n✝ + 1)
x✝ : i ∈ Finset.univ
⊢ (kernelSubobject (X.δ i.succ)).factorThru (Finset.univ.inf fun k ↦ kernelSubobject (X.δ k.succ)).arrow ⋯ ≫
      0 ≫ f.app (op ⦋n✝⦌) =
    0
Module: Mathlib.CategoryTheory.Idempotents.Basic

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Abelian D
⊢ IsIdempotentComplete D

Tactic:
rw [isIdempotentComplete_iff_idempotents_have_kernels]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Abelian D
⊢ ∀ (X : D) (p : X ⟶ X), p ≫ p = p → HasKernel p
Module: Mathlib.CategoryTheory.Idempotents.Karoubi

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
P Q : Karoubi C
f : P.Hom Q
⊢ f.f ≫ Q.p = f.f

Tactic:
rw [← f.comm, assoc, assoc, Q.idem]

Goal state after:
Module: Mathlib.CategoryTheory.Idempotents.Karoubi

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : Karoubi C
p : P ⟶ P
hp : p ≫ p = p
⊢ ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p

Tactic:
simp only [hom_ext_iff, comp_f] at hp

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : Karoubi C
p : P ⟶ P
hp : p.f ≫ p.f = p.f
⊢ ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p
Module: Mathlib.CategoryTheory.Idempotents.Basic

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
ε : C ≌ D
⊢ IsIdempotentComplete D → IsIdempotentComplete C

Tactic:
exact Equivalence.isIdempotentComplete ε.symm

Goal state after:
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ ∑ x ∈ Finset.univ ×ˢ Finset.univ, ((-1) ^ ↑x.2 • X.δ x.2) ≫ ((-1) ^ ↑x.1 • X.δ x.1) = 0

Tactic:
let P := Fin (n + 2) × Fin (n + 3)

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
⊢ ∑ x ∈ Finset.univ ×ˢ Finset.univ, ((-1) ^ ↑x.2 • X.δ x.2) ≫ ((-1) ^ ↑x.1 • X.δ x.1) = 0
Module: Mathlib.CategoryTheory.Idempotents.FunctorCategories

Goal state before:
pp: case h.left
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
P Q : Karoubi (J ⥤ C)
f : P ⟶ Q
X : J
inst✝ : IsIdempotentComplete C
F : J ⥤ C
p : F ⟶ F
hp : p ≫ p = p
hC : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
this : ∀ (j : J), HasEqualizer (𝟙 (F.obj j)) (p.app j)
Y : J ⥤ C := ⋯
i : Y ⟶ F := ⋯
e : F ⟶ Y := ⋯
⊢ i ≫ e = 𝟙 Y

pp: case h.right
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
P Q : Karoubi (J ⥤ C)
f : P ⟶ Q
X : J
inst✝ : IsIdempotentComplete C
F : J ⥤ C
p : F ⟶ F
hp : p ≫ p = p
hC : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
this : ∀ (j : J), HasEqualizer (𝟙 (F.obj j)) (p.app j)
Y : J ⥤ C := ⋯
i : Y ⟶ F := ⋯
e : F ⟶ Y := ⋯
⊢ e ≫ i = p

Tactic:
· ext j
    dsimp
    rw [assoc, equalizer.lift_ι, ← equalizer.condition, id_comp, comp_id]

Goal state after:
case h.right
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
P Q : Karoubi (J ⥤ C)
f : P ⟶ Q
X : J
inst✝ : IsIdempotentComplete C
F : J ⥤ C
p : F ⟶ F
hp : p ≫ p = p
hC : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
this : ∀ (j : J), HasEqualizer (𝟙 (F.obj j)) (p.app j)
Y : J ⥤ C := ⋯
i : Y ⟶ F := ⋯
e : F ⟶ Y := ⋯
⊢ e ≫ i = p
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: n : ℕ
i : Fin (n + 3)
j : Fin (n + 2)
H : j.succ < i
⊢ (j.castLT ⋯).castSucc < i.pred ⋯

Tactic:
rw [Fin.castSucc_castLT, ← Fin.succ_lt_succ_iff, Fin.succ_pred]
    exact H

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case h
n : ℕ
Δ : SimplexCategory
θ : Δ ⟶ ⦋n + 1⦌
hθ : ¬Function.Surjective ⇑(Hom.toOrderHom θ)
i : Fin (⦋n + 1⦌.len + 1)
hi : ¬∃ a, (Hom.toOrderHom θ) a = i
⊢ ∃ θ', θ = θ' ≫ δ i

Tactic:
exact eq_comp_δ_of_not_surjective' θ i (not_exists.mp hi)

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

Goal state before:
pp: S T : SSet
f : S ⟶ T
n : ℕ
i : Fin (n + 2)
x : S _⦋n + 1⦌
⊢ (ConcreteCategory.hom (f.app (op ⦋n⦌))) ((ConcreteCategory.hom (SimplicialObject.δ S i)) x) =
    (ConcreteCategory.hom (SimplicialObject.δ T i)) ((ConcreteCategory.hom (f.app (op ⦋n + 1⦌))) x)

Tactic:
change (S.δ i ≫ f.app (op ⦋n⦌)) x = (f.app (op ⦋n + 1⦌) ≫ T.δ i) x
  exact congr_hom (SimplicialObject.δ_naturality f i) x

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
⊢ ∀ {X Y : C} (f : X ⟶ Y),
    (Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).map f ≫
        (HomologicalComplex.Hom.isoOfComponents
            (fun x ↦ Iso.refl (((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj Y).X x)) ⋯).hom =
      (HomologicalComplex.Hom.isoOfComponents
            (fun x ↦ Iso.refl (((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj X).X x)) ⋯).hom ≫
        ChainComplex.alternatingConst.map f

Tactic:
intros; ext; simp

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

Goal state before:
pp: n : ℕ
i j : Fin (n + 2)
h : i < j
x✝ : ↥{i, j}ᶜ
m : Fin (n + 1)
hl : j.succAbove m ∈ {i, j}ᶜ
k : Fin n
hk : (i.castPred ⋯).succAbove k = m
⊢ (fun k ↦ ⟨j.succAbove ((i.castPred ⋯).succAbove k), ⋯⟩) k = ⟨j.succAbove m, hl⟩

Tactic:
simp [hk]

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
k : Fin (n + 1)
hk : j.rev = k
⊢ h ed j.castSucc ≫ X.left.δ i.succ = ((𝟭 (SimplicialObject C)).obj X.left).δ i ≫ h ed j

Tactic:
obtain ⟨l, hl⟩ : ∃ l, i.val = j + 1 + l := by
      rw [Fin.castSucc_lt_iff_succ_le, Fin.le_def] at hij
      obtain ⟨l, hl⟩ := Nat.le.dest hij
      exact ⟨l, by grind⟩

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑i = ↑j + 1 + l
⊢ h ed j.castSucc ≫ X.left.δ i.succ = ((𝟭 (SimplicialObject C)).obj X.left).δ i ≫ h ed j
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
x : Δ[n] _⦋n⦌
⊢ x ∈ Δ[n].nonDegenerate n ↔ x = objEquiv.symm (𝟙 ⦋n⦌)

Tactic:
refine ⟨fun h ↦ ?_, ?_⟩

Goal state after:
case refine_1
n : ℕ
x : Δ[n] _⦋n⦌
h : x ∈ Δ[n].nonDegenerate n
⊢ x = objEquiv.symm (𝟙 ⦋n⦌)

case refine_2
n : ℕ
x : Δ[n] _⦋n⦌
⊢ x = objEquiv.symm (𝟙 ⦋n⦌) → x ∈ Δ[n].nonDegenerate n
Module: Mathlib.CategoryTheory.GradedObject.Trifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} C₄
inst✝¹ : Category.{v_5, u_5} C₁₂
inst✝ : Category.{v_6, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
α : F ⟶ F'
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
X₁ Y₁ : GradedObject I₁ C₁
φ : X₁ ⟶ Y₁
⊢ (mapTrifunctor F I₁ I₂ I₃).map φ ≫
      { app := fun X₂ ↦ { app := fun x x_1 ↦ ((α.app (Y₁ x_1.1)).app (X₂ x_1.2.1)).app (x x_1.2.2), naturality := ⋯ },
        naturality := ⋯ } =
    { app := fun X₂ ↦ { app := fun x x_1 ↦ ((α.app (X₁ x_1.1)).app (X₂ x_1.2.1)).app (x x_1.2.2), naturality := ⋯ },
        naturality := ⋯ } ≫
      (mapTrifunctor F' I₁ I₂ I₃).map φ

Tactic:
by
    ext X₂ X₃ ⟨i₁, i₂, i₃⟩
    dsimp
    simp only [← NatTrans.comp_app, NatTrans.naturality]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L : HomologicalComplex₂ C c₁ c₂
f : K ⟶ L
⊢ ∀ (i j : I₂),
    c₂.Rel i j →
      { f := fun j ↦ (f.f j).f i, comm' := ⋯ } ≫ L.flip.d i j =
        K.flip.d i j ≫ { f := fun j_1 ↦ (f.f j_1).f j, comm' := ⋯ }

Tactic:
intros; ext; simp

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

Goal state before:
pp: 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
⊢ K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0

Tactic:
by
  by_cases h₁ : c₁₂.Rel i₁₂ i₁₂'
  · by_cases h₂ : c₁₂.Rel i₁₂' i₁₂''
    · ext ⟨i₁, i₂⟩ h
      simp only [totalAux.ιMapObj_D₁_assoc, comp_zero]
      by_cases h₃ : c₁.Rel i₁ (c₁.next i₁)
      · rw [totalAux.d₁_eq K c₁₂ h₃ i₂ i₁₂']; swap
        · rw [← ComplexShape.next_π₁ c₂ c₁₂ h₃ i₂, ← c₁₂.next_eq' h₁, h]
        simp only [Linear.units_smul_comp, assoc, totalAux.ιMapObj_D₁]
        by_cases h₄ : c₁.Rel (c₁.next i₁) (c₁.next (c₁.next i₁))
        · rw [totalAux.d₁_eq K c₁₂ h₄ i₂ i₁₂'', Linear.comp_units_smul,
            d_f_comp_d_f_assoc, zero_comp, smul_zero, smul_zero]
          rw [← ComplexShape.next_π₁ c₂ c₁₂ h₄, ← ComplexShape.next_π₁ c₂ c₁₂ h₃,
            h, c₁₂.next_eq' h₁, c₁₂.next_eq' h₂]
        · rw [K.d₁_eq_zero _ _ _ _ h₄, comp_zero, smul_zero]
      · rw [K.d₁_eq_zero c₁₂ _ _ _ h₃, zero_comp]
    · rw [K.D₁_shape c₁₂ _ _ h₂, comp_zero]
  · rw [K.D₁_shape c₁₂ _ _ h₁, zero_comp]

Goal state after:
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
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫ (mapBifunctorMap f₁ f₂ F c).f j =
    K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫
      ((K₁.mapBifunctor K₂ F c).d j (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j +
          hom₁ h₁ f₂ F c j (c.prev j) ≫ (L₁.mapBifunctor L₂ F c).d (c.prev j) j +
        (mapBifunctorMap f₁' f₂ F c).f j)

Tactic:
simp only [ι_mapBifunctorMap, h₁.comm i₁, dNext_eq_dFrom_fromNext, dFrom, fromNext,
      AddMonoidHom.mk'_apply, prevD_eq_toPrev_dTo, toPrev, dTo, Functor.map_add,
      Functor.map_comp, NatTrans.app_add, NatTrans.comp_app,
      Preadditive.add_comp, assoc, HomologicalComplex₂.total_d,
      Functor.mapBifunctorHomologicalComplex_obj_obj_toGradedObject, Preadditive.comp_add,
      HomologicalComplex₂.ι_D₁_assoc,
      HomologicalComplex₂.ι_D₂_assoc, add_left_inj]

Goal state after:
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
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ (F.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂) ≫
        (F.map (h₁.hom (c₁.next i₁) i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂ j h +
      (F.map (h₁.hom i₁ (c₁.prev i₁))).app (K₂.X i₂) ≫
        (F.map (L₁.d (c₁.prev i₁) i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂ j h =
    (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₁ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j +
        (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j +
      (K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫
          hom₁ h₁ f₂ F c j (c.prev j) ≫ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).D₁ c (c.prev j) j +
        K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫
          hom₁ h₁ f₂ F c j (c.prev j) ≫ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).D₂ c (c.prev j) j)
Module: Mathlib.Algebra.Homology.CochainComplexOpposite

Goal state before:
pp: case e_a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy f g
n : ℤ
⊢ ((dNext (-n)) h.hom).op = (prevD n) fun p q ↦ (h.hom (-q) (-p)).op

Tactic:
· rw [dNext_eq _ (i' := - (n - 1)) (by dsimp; lia)]
      symm
      exact prevD_eq _ (j' := n - 1) (by simp)

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: case pos.e_a
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
⊢ (K.extendXIso e ⋯).hom ≫ (dNext i) h.hom ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)

Tactic:
by_cases hi : c.Rel i (c.next i)

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : c.Rel i (c.next i)
⊢ (K.extendXIso e ⋯).hom ≫ (dNext i) h.hom ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)

case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : ¬c.Rel i (c.next i)
⊢ (K.extendXIso e ⋯).hom ≫ (dNext i) h.hom ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: case e_a
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
this :
  (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) h.hom + (prevD (e.f i)) h.hom + (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
⊢ (dNext (e.f i)) h.hom =
    (K.extendXIso e ⋯).hom ≫
      ((dNext i) fun i j ↦ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom) ≫
        (L.extendXIso e ⋯).inv

Tactic:
by_cases hi : c.Rel i (c.next i)

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
this :
  (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) h.hom + (prevD (e.f i)) h.hom + (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
hi : c.Rel i (c.next i)
⊢ (dNext (e.f i)) h.hom =
    (K.extendXIso e ⋯).hom ≫
      ((dNext i) fun i j ↦ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom) ≫
        (L.extendXIso e ⋯).inv

case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
this :
  (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) h.hom + (prevD (e.f i)) h.hom + (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
hi : ¬c.Rel i (c.next i)
⊢ (dNext (e.f i)) h.hom =
    (K.extendXIso e ⋯).hom ≫
      ((dNext i) fun i j ↦ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom) ≫
        (L.extendXIso e ⋯).inv
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (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✝⁸ : 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✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : ¬c₁.Rel i₁ (c₁.next i₁)
⊢ (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ i₃ (c₂.π c₃ c₂₃ (i₂, i₃)) ⋯) ≫ 0 =
    d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'

Tactic:
d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₁,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (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✝⁸ : 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✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : ¬c₁.Rel i₁ (c₁.next i₁)
⊢ (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ i₃ (c₂.π c₃ c₂₃ (i₂, i₃)) ⋯) ≫ 0 = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
L : CochainComplex C ℤ
d : ℤ
inst✝¹ : L.IsStrictlyGE d
inst✝ : ∀ (n : ℤ), Injective (L.X n)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomologicalComplex.Acyclic K
X : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) (↑n + d - 1)}
x₀ : ↑(X 0) := ⟨0, ⋯⟩
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1)) := fun n α ↦ ⟨↑α + Cochain.single ⋯.choose (-1), ⋯⟩
hφ : ∀ (k : ℕ) (x : ↑(X k)), (↑(φ k x)).EqUpTo (↑x) (d + ↑k)
⊢ Nonempty (Homotopy f 0)

Tactic:
refine ⟨(Cochain.equivHomotopy f 0).symm ⟨limitSequence φ hφ x₀, ?_⟩⟩

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
L : CochainComplex C ℤ
d : ℤ
inst✝¹ : L.IsStrictlyGE d
inst✝ : ∀ (n : ℤ), Injective (L.X n)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomologicalComplex.Acyclic K
X : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) (↑n + d - 1)}
x₀ : ↑(X 0) := ⟨0, ⋯⟩
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1)) := fun n α ↦ ⟨↑α + Cochain.single ⋯.choose (-1), ⋯⟩
hφ : ∀ (k : ℕ) (x : ↑(X k)), (↑(φ k x)).EqUpTo (↑x) (d + ↑k)
⊢ Cochain.ofHom f = δ (-1) 0 (limitSequence φ hφ x₀) + Cochain.ofHom 0
Module: Mathlib.Algebra.Homology.HomotopyFiber

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
α : Type u_2
c : ComplexShape α
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : DecidableRel c.Rel
inst✝ : HasHomotopyFiber φ
i j : α
hij : c.symm.Rel i j
⊢ HasBinaryBiproduct (((opFunctor C c).obj (op G)).X j) (((opFunctor C c).obj (op F)).X i)

Tactic:
by
    have := HasHomotopyFiber.hasBinaryBiproduct φ j i hij
    dsimp
    infer_instance

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
⊢ X.IsLE n ↔ ∀ (i : ℤ), n < i → IsZero ((homologyFunctor C i).obj X)

Tactic:
constructor
  · rintro ⟨K, e, _⟩ i hi
    apply ((K.exactAt_of_isLE n i hi).isZero_homology).of_iso
    exact (homologyFunctor C i).mapIso e ≪≫ (homologyFunctorFactors C i).app K
  · intro hX
    have : (Q.objPreimage X).IsLE n := by
      rw [CochainComplex.isLE_iff]
      intro i hi
      rw [HomologicalComplex.exactAt_iff_isZero_homology]
      apply (hX i hi).of_iso
      exact (homologyFunctorFactors C i).symm.app _ ≪≫
        (homologyFunctor C i).mapIso (Q.objObjPreimageIso X)
    exact ⟨(Q.objPreimage X).truncLE n, (Q.objObjPreimageIso X).symm ≪≫
      (asIso (Q.map ((Q.objPreimage X).ιTruncLE n))).symm, inferInstance⟩

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.rlp ≤ T.pushouts.rlp

Tactic:
rw [← le_llp_iff_le_rlp]
    exact T.pushouts_le_llp_rlp

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.rlp ≤ T.retracts.rlp

Tactic:
rw [← le_llp_iff_le_rlp]
    exact T.retracts_le_llp_rlp

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : CategoryWithCofibrations C
inst✝³ : CategoryWithFibrations C
X Y : C
f : X ⟶ Y
inst✝² : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
inst✝¹ : (weakEquivalences C).IsStableUnderRetracts
inst✝ : IsIso f
⊢ WeakEquivalence f

Tactic:
have h := MorphismProperty.factorizationData (trivialCofibrations C) (fibrations C) f
  rw [weakEquivalence_iff]
  exact MorphismProperty.of_retract (RetractArrow.ofLeftLiftingProperty h.fac) h.hi.2

Goal state after: