awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : ¬c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : c₂.Rel i₂ (c₂.next i₂)
h₄ : c₁₂.π c₃ c₄ (c₁.π c₂ c₁₂ (i₁, c₂.next i₂), i₃) = j'
pos : c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = c₁₂.π c₃ c₄ (c₁.π c₂ c₁₂ (i₁, c₂.next i₂), i₃)
⊢ c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
have pos₁ : c₁₂.π c₃ c₄ (c₁.π c₂ c₁₂ (i₁, c₂.next i₂), i₃) = c₁₂.π c₃ c₄ (c₁.π c₂ c₁₂ (i₁, c₂.next i₂), i₃)
Tactic: rw [← h₄, ComplexShape.next_π₂ c₁ c₁₂ i₁ h₃]
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : ¬c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ Eq (α :=
    (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
      X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
    (0 :
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
    (d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' :
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
have neg : Eq.{v_4 + 1} (α :=
  (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶ X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
  0 0
Tactic: rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃]
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : ¬c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
⊢ NatTrans.app (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)) (K₃.X i₃) ≫
      mapBifunctor.d₁ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ (c₁.π c₂ c₁₂ (i₁, i₂)) i₃ j' =
    HAdd.hAdd (α :=
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (β :=
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' :
        (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' :
        (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
have neg : ¬c₂.Rel i₂ (c₂.next i₂)
Tactic: rw [mapBifunctor.d₁_eq_zero _ _ _ _ _ _ _ h₁, comp_zero,
      d₁_eq_zero, d₂_eq_zero, zero_add]
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : ¬c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
neg : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ NatTrans.app (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)) (K₃.X i₃) ≫
      mapBifunctor.d₁ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ (c₁.π c₂ c₁₂ (i₁, i₂)) i₃ j' =
    HAdd.hAdd (α :=
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (β :=
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' :
        (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' :
        (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
have neg₁ : ¬c₁.Rel i₁ (c₁.next i₁)
Tactic: rw [mapBifunctor.d₁_eq_zero _ _ _ _ _ _ _ h₁, comp_zero,
      d₁_eq_zero, d₂_eq_zero, zero_add]
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : ¬c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₂.Rel i₂ (c₂.next i₂)
⊢ False
have neg : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
Tactic: apply h₁
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (hypothesis)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : ¬c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₂.Rel i₂ (c₂.next i₂)
⊢ c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
have this : c₁₂.Rel (c₁.π c₂ c₁₂ (i₁, i₂)) (c₁.π c₂ c₁₂ (i₁, c₂.next i₂))
Tactic: have := ComplexShape.rel_π₂ c₁ c₁₂ i₁ h₂
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : ¬c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₁.Rel i₁ (c₁.next i₁)
⊢ False
have neg : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
Tactic: apply h₁
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (hypothesis)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : ¬c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₁.Rel i₁ (c₁.next i₁)
⊢ c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
have this : c₁₂.Rel (c₁.π c₂ c₁₂ (i₁, i₂)) (c₁.π c₂ c₁₂ (c₁.next i₁, i₂))
Tactic: have := ComplexShape.rel_π₁ c₂ c₁₂ h₂ i₂
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_5, u_3} C₁₂
inst_5 : Category.{v_6, u_4} C₂₃
inst_6 : HasZeroMorphisms.{v_1, u_1} C₁
inst_7 : HasZeroMorphisms.{v_2, u_2} C₂
inst_8 : HasZeroMorphisms.{v_3, u_5} C₃
inst_9 : Preadditive.{v_5, u_3} C₁₂
inst_10 : Preadditive.{v_6, u_4} C₂₃
inst_11 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
F : C₁ ⥤ CategoryTheory.Functor.{v_6, v_4, u_4, u_6} C₂₃ C₄
G₂₃ : C₂ ⥤ CategoryTheory.Functor.{v_3, v_6, u_5, u_4} C₃ C₂₃
inst_12 : F₁₂.PreservesZeroMorphisms
inst_13 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_14 : G.Additive
inst_15 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst_16 : G₂₃.PreservesZeroMorphisms
inst_17 : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst_18 : F.PreservesZeroMorphisms
inst_19 : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
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_20 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst_21 : TotalComplexShape c₁ c₂ c₁₂
inst_22 : TotalComplexShape c₁₂ c₃ c₄
inst_23 : TotalComplexShape c₂ c₃ c₂₃
inst_24 : TotalComplexShape c₁ c₂₃ c₄
inst_25 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_26 : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst_27 : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst_28 : DecidableEq ι₂₃
inst_29 : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
i₂₃ : ι₂₃
j : ι₄
h₂₃ : c₂.π c₃ c₂₃ (i₂, i₃) = i₂₃
h : c₁.π c₂₃ c₄ (i₁, i₂₃) = j
⊢ c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
have rw : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = c₁₂.π c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃)
Tactic: rw [← h, ← h₂₃, ← ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄]
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_6, u_4} C₂₃
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_6, u_4} C₂₃
inst_9 : Preadditive.{v_4, u_6} C₄
F : C₁ ⥤ CategoryTheory.Functor.{v_6, v_4, u_4, u_6} C₂₃ C₄
G₂₃ : C₂ ⥤ CategoryTheory.Functor.{v_3, v_6, u_5, u_4} C₃ C₂₃
inst_10 : G₂₃.PreservesZeroMorphisms
inst_11 : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst_12 : F.PreservesZeroMorphisms
inst_13 : ∀ (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_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : TotalComplexShape c₂ c₃ c₂₃
inst_18 : TotalComplexShape c₁ c₂₃ c₄
inst_19 : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst_20 : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst_21 : DecidableEq ι₂₃
inst_22 : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
i₂₃ : ι₂₃
j : ι₄
h₂₃ : c₂.π c₃ c₂₃ (i₂, i₃) = i₂₃
h : c₁.π c₂₃ c₄ (i₁, i₂₃) = j
⊢ Eq (α :=
    (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶ X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
    (ι F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j ⋯ :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
    ((F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ i₃ i₂₃ h₂₃) ≫
        K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ i₂₃ j h :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
have subst : ∀ (h : c₁.π c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j),
  Eq (α :=
    (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶ X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
    (ι F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j sorry :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
    ((F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ i₃ (c₂.π c₃ c₂₃ (i₂, i₃)) sorry) ≫
        K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂.π c₃ c₂₃ (i₂, i₃)) j h :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
Tactic: subst h₂₃
Cleared hypotheses: i₂₃, h₂₃, h
-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorAssociator (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_6, u_4} C₂₃
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_6, u_4} C₂₃
inst_9 : Preadditive.{v_4, u_6} C₄
F : C₁ ⥤ CategoryTheory.Functor.{v_6, v_4, u_4, u_6} C₂₃ C₄
G₂₃ : C₂ ⥤ CategoryTheory.Functor.{v_3, v_6, u_5, u_4} C₃ C₂₃
inst_10 : G₂₃.PreservesZeroMorphisms
inst_11 : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst_12 : F.PreservesZeroMorphisms
inst_13 : ∀ (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_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : TotalComplexShape c₂ c₃ c₂₃
inst_18 : TotalComplexShape c₁ c₂₃ c₄
inst_19 : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst_20 : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst_21 : DecidableEq ι₂₃
inst_22 : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j : ι₄
h : ¬c₁.Rel i₁ (c₁.next i₁)
⊢ Eq (α :=
    (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶ X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
    (d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
    (0 :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
have dsimp : HSMul.hSMul (β :=
    (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶ X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j)
    (c₁.ε₁ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)))
    (CategoryStruct.comp (Y := (F.obj (K₁.X (c₁.next i₁))).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)))
        (NatTrans.app (F.map (K₁.d i₁ (c₁.next i₁))) ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)))
        (ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ (c₁.next i₁) i₂ i₃ j) :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j) =
  0
Tactic: dsimp [d₁]