awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial.{v, u} C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₁.tgt)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.tgt X₂.tgt)
      π(π(π₁ ≫ π₁, prod.map (π₁ ≫ f₁.f) π₁ ≫ g₁.F) ≫ f₁.F, π(π₁ ≫ π₂, prod.map (π₂ ≫ f₂.f) π₂ ≫ g₂.F) ≫ f₂.F) π₁)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.tgt X₂.tgt)
      π(CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.src Z₁.tgt) (prod.map π₁ π₁)
          (π(π₁, prod.map f₁.f (𝟙 Z₁.tgt) ≫ g₁.F) ≫ f₁.F),
        CategoryStruct.comp (Y := Limits.prod.{v, u} X₂.src Z₂.tgt) (prod.map π₂ π₂)
          (π(π₁, prod.map f₂.f (𝟙 Z₂.tgt) ≫ g₂.F) ≫ f₂.F))
      π₁)
have hF : Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₁.tgt)
  (π(π₁ ≫ π₁, prod.map (π₁ ≫ f₁.f) π₁ ≫ g₁.F) ≫ f₁.F)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.src Z₁.tgt) (prod.map π₁ π₁)
    (π(π₁, prod.map f₁.f (𝟙 Z₁.tgt) ≫ g₁.F) ≫ f₁.F))
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial.{v, u} C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₂.tgt)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.tgt X₂.tgt)
      π(π(π₁ ≫ π₁, prod.map (π₁ ≫ f₁.f) π₁ ≫ g₁.F) ≫ f₁.F, π(π₁ ≫ π₂, prod.map (π₂ ≫ f₂.f) π₂ ≫ g₂.F) ≫ f₂.F) π₂)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.tgt X₂.tgt)
      π(CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.src Z₁.tgt) (prod.map π₁ π₁)
          (π(π₁, prod.map f₁.f (𝟙 Z₁.tgt) ≫ g₁.F) ≫ f₁.F),
        CategoryStruct.comp (Y := Limits.prod.{v, u} X₂.src Z₂.tgt) (prod.map π₂ π₂)
          (π(π₁, prod.map f₂.f (𝟙 Z₂.tgt) ≫ g₂.F) ≫ f₂.F))
      π₂)
have hF : Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₂.tgt)
  (π(π₁ ≫ π₂, prod.map (π₂ ≫ f₂.f) π₂ ≫ g₂.F) ≫ f₂.F)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X₂.src Z₂.tgt) (prod.map π₂ π₂)
    (π(π₁, prod.map f₂.f (𝟙 Z₂.tgt) ≫ g₂.F) ≫ f₂.F))
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial.{v, u} C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₁.tgt)
    (π(π₁ ≫ π₁, prod.map (π₁ ≫ f₁.f) π₁ ≫ g₁.F) ≫ f₁.F)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.src Z₁.tgt) (prod.map π₁ π₁)
      (π(π₁, prod.map f₁.f (𝟙 Z₁.tgt) ≫ g₁.F) ≫ f₁.F))
have hF : Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₁.tgt)
  (π(π₁ ≫ π₁, prod.map (π₁ ≫ f₁.f) π₁ ≫ g₁.F) ≫ f₁.F)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.src Z₁.tgt) (prod.map π₁ π₁) π(π₁, prod.map f₁.f (𝟙 Z₁.tgt) ≫ g₁.F) ≫
    f₁.F)
Tactic: rw [← Category.assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial.{v, u} C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₁.tgt)
    (π(π₁ ≫ π₁, prod.map (π₁ ≫ f₁.f) π₁ ≫ g₁.F) ≫ f₁.F)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.src Z₁.tgt) (prod.map π₁ π₁)
        π(π₁, prod.map f₁.f (𝟙 Z₁.tgt) ≫ g₁.F) ≫
      f₁.F)
have hF : Eq (α :=
  Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶
    Limits.prod.{v, u} X₁.src Y₁.tgt)
  π(π₁ ≫ π₁, prod.map (π₁ ≫ f₁.f) π₁ ≫ g₁.F)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X₁.src Z₁.tgt) (prod.map π₁ π₁) π(π₁, prod.map f₁.f (𝟙 Z₁.tgt) ≫ g₁.F))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial.{v, u} C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₂.tgt)
    (π(π₁ ≫ π₂, prod.map (π₂ ≫ f₂.f) π₂ ≫ g₂.F) ≫ f₂.F)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X₂.src Z₂.tgt) (prod.map π₂ π₂)
      (π(π₁, prod.map f₂.f (𝟙 Z₂.tgt) ≫ g₂.F) ≫ f₂.F))
have hF : Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₂.tgt)
  (π(π₁ ≫ π₂, prod.map (π₂ ≫ f₂.f) π₂ ≫ g₂.F) ≫ f₂.F)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X₂.src Z₂.tgt) (prod.map π₂ π₂) π(π₁, prod.map f₂.f (𝟙 Z₂.tgt) ≫ g₂.F) ≫
    f₂.F)
Tactic: rw [← Category.assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial.{v, u} C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶ X₂.tgt)
    (π(π₁ ≫ π₂, prod.map (π₂ ≫ f₂.f) π₂ ≫ g₂.F) ≫ f₂.F)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X₂.src Z₂.tgt) (prod.map π₂ π₂)
        π(π₁, prod.map f₂.f (𝟙 Z₂.tgt) ≫ g₂.F) ≫
      f₂.F)
have hF : Eq (α :=
  Limits.prod.{v, u} (Limits.prod.{v, u} X₁.src X₂.src) (Limits.prod.{v, u} Z₁.tgt Z₂.tgt) ⟶
    Limits.prod.{v, u} X₂.src Y₂.tgt)
  π(π₁ ≫ π₂, prod.map (π₂ ≫ f₂.f) π₂ ≫ g₂.F)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X₂.src Z₂.tgt) (prod.map π₂ π₂) π(π₁, prod.map f₂.f (𝟙 Z₂.tgt) ≫ g₂.F))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq (α := tensorObj (𝒞 := instCategory) ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) X ⟶ Y)
    ((𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) ⊗ₘ f) ≫ Iso.hom (λ_ Y)) (Iso.hom (λ_ X) ≫ f)
have hf : Eq (α := (tensorObj (𝒞 := instCategory) ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) X).src ⟶ Y.src)
  (Hom.f ((𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) ⊗ₘ f) ≫ Iso.hom (λ_ Y))) (Hom.f (Iso.hom (λ_ X) ≫ f))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
hf :
  Eq (α := (tensorObj (𝒞 := instCategory) ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) X).src ⟶ Y.src)
    (Hom.f ((𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) ⊗ₘ f) ≫ Iso.hom (λ_ Y))) (Hom.f (Iso.hom (λ_ X) ≫ f))
⊢ Eq (α := tensorObj (𝒞 := instCategory) ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) X ⟶ Y)
    ((𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) ⊗ₘ f) ≫ Iso.hom (λ_ Y)) (Iso.hom (λ_ X) ≫ f)
have hF : Eq (α :=
  Limits.prod.{v, u} (tensorObj (𝒞 := instCategory) ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) X).src Y.tgt ⟶
    (tensorObj (𝒞 := instCategory) ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) X).tgt)
  (Hom.F ((𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) ⊗ₘ f) ≫ Iso.hom (λ_ Y))) (Hom.F (Iso.hom (λ_ X) ≫ f))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq (α :=
    Limits.prod.{v, u} (tensorObj (𝒞 := instCategory) ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) X).src Y.tgt ⟶
      (tensorObj (𝒞 := instCategory) ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) X).tgt)
    (Hom.F ((𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) ⊗ₘ f) ≫ Iso.hom (λ_ Y))) (Hom.F (Iso.hom (λ_ X) ≫ f))
have hF : Eq (α :=
  Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶
    Limits.prod.{v, u} (terminal.{v, u} C) X.tgt)
  π(terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt), π(π₁ ≫ π₂, π₂) ≫ f.F)
  π(terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt),
    prod.map π₂ (𝟙 Y.tgt) ≫ f.F)
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq (α :=
    Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶
      Limits.prod.{v, u} (terminal.{v, u} C) X.tgt)
    π(terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt), π(π₁ ≫ π₂, π₂) ≫ f.F)
    π(terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt),
      prod.map π₂ (𝟙 Y.tgt) ≫ f.F)
have hF : CategoryStruct.comp (Y := Limits.prod.{v, u} (terminal.{v, u} C) X.tgt)
    π(terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt), π(π₁ ≫ π₂, π₂) ≫ f.F)
    π₂ =
  CategoryStruct.comp (Y := Limits.prod.{v, u} (terminal.{v, u} C) X.tgt)
    π(terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt),
      prod.map π₂ (𝟙 Y.tgt) ≫ f.F)
    π₂
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ CategoryStruct.comp (Y := Limits.prod.{v, u} (terminal.{v, u} C) X.tgt)
      π(terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt), π(π₁ ≫ π₂, π₂) ≫ f.F)
      π₂ =
    CategoryStruct.comp (Y := Limits.prod.{v, u} (terminal.{v, u} C) X.tgt)
      π(terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt),
        prod.map π₂ (𝟙 Y.tgt) ≫ f.F)
      π₂
have hF : Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶ X.tgt) (π(π₁ ≫ π₂, π₂) ≫ f.F)
  (prod.map π₂ (𝟙 Y.tgt) ≫ f.F)
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶ X.tgt) (π(π₁ ≫ π₂, π₂) ≫ f.F)
    (prod.map π₂ (𝟙 Y.tgt) ≫ f.F)
have hF : Eq.{v + 1} (α :=
  Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶ Limits.prod.{v, u} X.src Y.tgt)
  π(π₁ ≫ π₂, π₂) (prod.map π₂ (𝟙 Y.tgt))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq.{v + 1} (α :=
    Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶ Limits.prod.{v, u} X.src Y.tgt)
    π(π₁ ≫ π₂, π₂) (prod.map π₂ (𝟙 Y.tgt))
have hF : Eq.{v + 1} (α := Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶ X.src)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) π(π₁ ≫ π₂, π₂) π₁)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) (prod.map π₂ (𝟙 Y.tgt)) π₁)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
hF :
  Eq.{v + 1} (α := Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶ X.src)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) π(π₁ ≫ π₂, π₂) π₁)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) (prod.map π₂ (𝟙 Y.tgt)) π₁)
⊢ Eq.{v + 1} (α :=
    Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶ Limits.prod.{v, u} X.src Y.tgt)
    π(π₁ ≫ π₂, π₂) (prod.map π₂ (𝟙 Y.tgt))
have hF₁ : Eq.{v + 1} (α := Limits.prod.{v, u} (Limits.prod.{v, u} (terminal.{v, u} C) X.src) Y.tgt ⟶ Y.tgt)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) π(π₁ ≫ π₂, π₂) π₂)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) (prod.map π₂ (𝟙 Y.tgt)) π₂)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq (α := tensorObj (𝒞 := instCategory) X ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) ⟶ Y)
    ((f ⊗ₘ 𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))) ≫ Iso.hom (ρ_ Y)) (Iso.hom (ρ_ X) ≫ f)
have hf : Eq (α := (tensorObj (𝒞 := instCategory) X ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))).src ⟶ Y.src)
  (Hom.f ((f ⊗ₘ 𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))) ≫ Iso.hom (ρ_ Y))) (Hom.f (Iso.hom (ρ_ X) ≫ f))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
hf :
  Eq (α := (tensorObj (𝒞 := instCategory) X ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))).src ⟶ Y.src)
    (Hom.f ((f ⊗ₘ 𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))) ≫ Iso.hom (ρ_ Y))) (Hom.f (Iso.hom (ρ_ X) ≫ f))
⊢ Eq (α := tensorObj (𝒞 := instCategory) X ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory)) ⟶ Y)
    ((f ⊗ₘ 𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))) ≫ Iso.hom (ρ_ Y)) (Iso.hom (ρ_ X) ≫ f)
have hF : Eq (α :=
  Limits.prod.{v, u} (tensorObj (𝒞 := instCategory) X ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))).src Y.tgt ⟶
    (tensorObj (𝒞 := instCategory) X ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))).tgt)
  (Hom.F ((f ⊗ₘ 𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))) ≫ Iso.hom (ρ_ Y))) (Hom.F (Iso.hom (ρ_ X) ≫ f))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq (α :=
    Limits.prod.{v, u} (tensorObj (𝒞 := instCategory) X ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))).src Y.tgt ⟶
      (tensorObj (𝒞 := instCategory) X ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))).tgt)
    (Hom.F ((f ⊗ₘ 𝟙 ((𝟙_ (Dial.{v, u} C)) (𝒞 := instCategory))) ≫ Iso.hom (ρ_ Y))) (Hom.F (Iso.hom (ρ_ X) ≫ f))
have hF : Eq (α :=
  Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶
    Limits.prod.{v, u} X.tgt (terminal.{v, u} C))
  π(π(π₁ ≫ π₁, π₂) ≫ f.F, terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt))
  π(prod.map π₁ (𝟙 Y.tgt) ≫ f.F,
    terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt))
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq (α :=
    Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶
      Limits.prod.{v, u} X.tgt (terminal.{v, u} C))
    π(π(π₁ ≫ π₁, π₂) ≫ f.F, terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt))
    π(prod.map π₁ (𝟙 Y.tgt) ≫ f.F,
      terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt))
have hF : CategoryStruct.comp (Y := Limits.prod.{v, u} X.tgt (terminal.{v, u} C))
    π(π(π₁ ≫ π₁, π₂) ≫ f.F, terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt))
    π₁ =
  CategoryStruct.comp (Y := Limits.prod.{v, u} X.tgt (terminal.{v, u} C))
    π(prod.map π₁ (𝟙 Y.tgt) ≫ f.F,
      terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt))
    π₁
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ CategoryStruct.comp (Y := Limits.prod.{v, u} X.tgt (terminal.{v, u} C))
      π(π(π₁ ≫ π₁, π₂) ≫ f.F, terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt))
      π₁ =
    CategoryStruct.comp (Y := Limits.prod.{v, u} X.tgt (terminal.{v, u} C))
      π(prod.map π₁ (𝟙 Y.tgt) ≫ f.F,
        terminal.from (Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt))
      π₁
have hF : Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶ X.tgt) (π(π₁ ≫ π₁, π₂) ≫ f.F)
  (prod.map π₁ (𝟙 Y.tgt) ≫ f.F)
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq (α := Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶ X.tgt) (π(π₁ ≫ π₁, π₂) ≫ f.F)
    (prod.map π₁ (𝟙 Y.tgt) ≫ f.F)
have hF : Eq.{v + 1} (α :=
  Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶ Limits.prod.{v, u} X.src Y.tgt)
  π(π₁ ≫ π₁, π₂) (prod.map π₁ (𝟙 Y.tgt))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
⊢ Eq.{v + 1} (α :=
    Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶ Limits.prod.{v, u} X.src Y.tgt)
    π(π₁ ≫ π₁, π₂) (prod.map π₁ (𝟙 Y.tgt))
have hF : Eq.{v + 1} (α := Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶ X.src)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) π(π₁ ≫ π₁, π₂) π₁)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) (prod.map π₁ (𝟙 Y.tgt)) π₁)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y : Dial.{v, u} C
f : X ⟶ Y
hF :
  Eq.{v + 1} (α := Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶ X.src)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) π(π₁ ≫ π₁, π₂) π₁)
    (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) (prod.map π₁ (𝟙 Y.tgt)) π₁)
⊢ Eq.{v + 1} (α :=
    Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶ Limits.prod.{v, u} X.src Y.tgt)
    π(π₁ ≫ π₁, π₂) (prod.map π₁ (𝟙 Y.tgt))
have hF₁ : Eq.{v + 1} (α := Limits.prod.{v, u} (Limits.prod.{v, u} X.src (terminal.{v, u} C)) Y.tgt ⟶ Y.tgt)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) π(π₁ ≫ π₁, π₂) π₂)
  (CategoryStruct.comp (Y := Limits.prod.{v, u} X.src Y.tgt) (prod.map π₁ (𝟙 Y.tgt)) π₂)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
W X Y Z : Dial.{v, u} C
⊢ (Iso.hom (α_ W X Y) ⊗ₘ 𝟙 Z) ≫ Iso.hom (α_ W (tensorObj (𝒞 := instCategory) X Y) Z) ≫ (𝟙 W ⊗ₘ Iso.hom (α_ X Y Z)) =
    Iso.hom (α_ (tensorObj (𝒞 := instCategory) W X) Y Z) ≫ Iso.hom (α_ W X (tensorObj (𝒞 := instCategory) Y Z))
have hf : Hom.f
    ((Iso.hom (α_ W X Y) ⊗ₘ 𝟙 Z) ≫ Iso.hom (α_ W (tensorObj (𝒞 := instCategory) X Y) Z) ≫ (𝟙 W ⊗ₘ Iso.hom (α_ X Y Z))) =
  (Iso.hom (α_ (tensorObj (𝒞 := instCategory) W X) Y Z) ≫ Iso.hom (α_ W X (tensorObj (𝒞 := instCategory) Y Z))).f
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Dialectica.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasFiniteProducts.{v, u} C
inst_2 : HasPullbacks.{v, u} C
W X Y Z : Dial.{v, u} C
hf :
  Hom.f
      ((Iso.hom (α_ W X Y) ⊗ₘ 𝟙 Z) ≫
        Iso.hom (α_ W (tensorObj (𝒞 := instCategory) X Y) Z) ≫ (𝟙 W ⊗ₘ Iso.hom (α_ X Y Z))) =
    (Iso.hom (α_ (tensorObj (𝒞 := instCategory) W X) Y Z) ≫ Iso.hom (α_ W X (tensorObj (𝒞 := instCategory) Y Z))).f
⊢ (Iso.hom (α_ W X Y) ⊗ₘ 𝟙 Z) ≫ Iso.hom (α_ W (tensorObj (𝒞 := instCategory) X Y) Z) ≫ (𝟙 W ⊗ₘ Iso.hom (α_ X Y Z)) =
    Iso.hom (α_ (tensorObj (𝒞 := instCategory) W X) Y Z) ≫ Iso.hom (α_ W X (tensorObj (𝒞 := instCategory) Y Z))
have hF : Hom.F
    ((Iso.hom (α_ W X Y) ⊗ₘ 𝟙 Z) ≫ Iso.hom (α_ W (tensorObj (𝒞 := instCategory) X Y) Z) ≫ (𝟙 W ⊗ₘ Iso.hom (α_ X Y Z))) =
  (Iso.hom (α_ (tensorObj (𝒞 := instCategory) W X) Y Z) ≫ Iso.hom (α_ W X (tensorObj (𝒞 := instCategory) Y Z))).F
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.DifferentialObject (goal)
Goal:
S : Type u_1
inst : AddMonoidWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
X Y : DifferentialObject.{v, u, u_1} S C
h : X = Y
⊢ Hom.f (eqToHom h) = eqToHom ⋯
have subst : Hom.f (eqToHom sorry) = eqToHom sorry
Tactic: subst h
Cleared hypotheses: Y, h
-- Lean 4 proof step from Mathlib.CategoryTheory.DifferentialObject (goal)
Goal:
S : Type u_1
inst : AddMonoidWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
X : DifferentialObject.{v, u, u_1} S C
⊢ Hom.f (eqToHom ⋯) = eqToHom ⋯
have rw : Hom.f (𝟙 X) = 𝟙 X.obj
Tactic: rw [eqToHom_refl, eqToHom_refl]
-- Lean 4 proof step from Mathlib.CategoryTheory.DifferentialObject (goal)
Goal:
S : Type u_1
inst : AddMonoidWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
X Y : DifferentialObject.{v, u, u_1} S C
f : X ≅ Y
⊢ f.hom.f ≫ f.inv.f = 𝟙 X.obj
have rw : (f.hom ≫ f.inv).f = 𝟙 X.obj
Tactic: rw [← comp_f, Iso.hom_inv_id, id_f]
-- Lean 4 proof step from Mathlib.CategoryTheory.DifferentialObject (goal)
Goal:
S : Type u_1
inst : AddMonoidWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
X Y : DifferentialObject.{v, u, u_1} S C
f : X ≅ Y
rw : (f.hom ≫ f.inv).f = 𝟙 X.obj
⊢ f.hom.f ≫ f.inv.f = 𝟙 X.obj
have rw₁ : Hom.f (𝟙 X) = 𝟙 X.obj
Tactic: rw [← comp_f, Iso.hom_inv_id, id_f]
-- Lean 4 proof step from Mathlib.CategoryTheory.DifferentialObject (goal)
Goal:
S : Type u_1
inst : AddMonoidWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
X Y : DifferentialObject.{v, u, u_1} S C
f : X ≅ Y
rw : (f.hom ≫ f.inv).f = 𝟙 X.obj
rw₁ : Hom.f (𝟙 X) = 𝟙 X.obj
⊢ f.hom.f ≫ f.inv.f = 𝟙 X.obj
have rw₂ : Eq.{v + 1} (α := X.obj ⟶ X.obj) (𝟙 X.obj) (𝟙 X.obj)
Tactic: rw [← comp_f, Iso.hom_inv_id, id_f]
-- Lean 4 proof step from Mathlib.CategoryTheory.DifferentialObject (goal)
Goal:
S : Type u_1
inst : AddMonoidWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
X Y : DifferentialObject.{v, u, u_1} S C
f : X ≅ Y
⊢ f.inv.f ≫ f.hom.f = 𝟙 Y.obj
have rw : (f.inv ≫ f.hom).f = 𝟙 Y.obj
Tactic: rw [← comp_f, Iso.inv_hom_id, id_f]
-- Lean 4 proof step from Mathlib.CategoryTheory.DifferentialObject (goal)
Goal:
S : Type u_1
inst : AddMonoidWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
X Y : DifferentialObject.{v, u, u_1} S C
f : X ≅ Y
rw : (f.inv ≫ f.hom).f = 𝟙 Y.obj
⊢ f.inv.f ≫ f.hom.f = 𝟙 Y.obj
have rw₁ : Hom.f (𝟙 Y) = 𝟙 Y.obj
Tactic: rw [← comp_f, Iso.inv_hom_id, id_f]
-- Lean 4 proof step from Mathlib.CategoryTheory.DifferentialObject (goal)
Goal:
S : Type u_1
inst : AddMonoidWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
X Y : DifferentialObject.{v, u, u_1} S C
f : X ≅ Y
rw : (f.inv ≫ f.hom).f = 𝟙 Y.obj
rw₁ : Hom.f (𝟙 Y) = 𝟙 Y.obj
⊢ f.inv.f ≫ f.hom.f = 𝟙 Y.obj
have rw₂ : Eq.{v + 1} (α := Y.obj ⟶ Y.obj) (𝟙 Y.obj) (𝟙 Y.obj)
Tactic: rw [← comp_f, Iso.inv_hom_id, id_f]