awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.Equivalence (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_1 : GaloisCategory.{u₁, u₂} C
inst_2 : FiberFunctor F
F' : Functor.{u₂, u₁, u₁, u₁ + 1} C FintypeCat := F ⋙ FintypeCat.uSwitch
this_1 : FiberFunctor F' := ⋯
this : (functorToContAction F').EssSurj
f : Aut F ≃ₜ* Aut F' := autEquivAutWhiskerRight F FintypeCat.uSwitchEquivalence.fullyFaithfulFunctor
equiv : ContAction FintypeCat (FV := fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj) (CV :=
    fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (Aut F') ≌
  ContAction FintypeCat (FV := fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj) (CV :=
    fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (Aut F) :=
  Equivalence.trans
    (Equivalence.mapContAction (FV := fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
      (CV := fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (FW :=
      fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj) (CW :=
      fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (Aut F') FintypeCat.uSwitchEquivalence ⋯ ⋯)
    (ContAction.resEquiv FintypeCat (FV := fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
      (CV := fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) f)
⊢ (functorToContAction F).EssSurj
have have : ∀ (this : functorToContAction F ≅ functorToContAction F' ⋙ equiv.functor), (functorToContAction F).EssSurj
Tactic: have : functorToContAction F ≅ functorToContAction F' ⋙ equiv.functor :=
    NatIso.ofComponents
      (fun X ↦ ObjectProperty.isoMk _ (Action.mkIso (FintypeCat.uSwitchEquivalence.unitIso.app _)
      (fun g ↦ FintypeCat.uSwitchEquivalence.unitIso.hom.naturality (g.hom.app X))))
      (fun f ↦ by
        ext : 2
        exact FintypeCat.uSwitchEquivalence.unitIso.hom.naturality (F.map f))
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y X
inst_6 : HasTensor.{u_1, v_1, u_2} X Z
inst_7 : HasTensor.{u_1, v_1, u_2} Z X
f : Y ⟶ Z
⊢ Eq (α := tensorObj.{u_1, v_1, u_2} X Y ⟶ tensorObj.{u_1, v_1, u_2} Z X)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X Z) (whiskerLeft X f) (Iso.hom (braiding X Z)))
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Y X) (Iso.hom (braiding X Y)) (whiskerRight f X))
have dsimp : Eq (α := tensorObj.{u_1, v_1, u_2} X Y ⟶ tensorObj.{u_1, v_1, u_2} Z X)
  (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X Z) (whiskerLeft X f) fun (k : I) ↦
    tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (X i) (Z j)) ≫ ιTensorObj Z X j i k sorry)
  (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Y X)
    (fun (k : I) ↦ tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (X i) (Y j)) ≫ ιTensorObj Y X j i k sorry)
    (whiskerRight f X))
Tactic: dsimp [braiding]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} Y Z
inst_5 : HasTensor.{u_1, v_1, u_2} Z Y
inst_6 : HasTensor.{u_1, v_1, u_2} X Z
inst_7 : HasTensor.{u_1, v_1, u_2} Z X
f : X ⟶ Y
⊢ Eq (α := tensorObj.{u_1, v_1, u_2} X Z ⟶ tensorObj.{u_1, v_1, u_2} Z Y)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Y Z) (whiskerRight f Z) (Iso.hom (braiding Y Z)))
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Z X) (Iso.hom (braiding X Z)) (whiskerLeft Z f))
have dsimp : Eq (α := tensorObj.{u_1, v_1, u_2} X Z ⟶ tensorObj.{u_1, v_1, u_2} Z Y)
  (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Y Z) (whiskerRight f Z) fun (k : I) ↦
    tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (Y i) (Z j)) ≫ ιTensorObj Z Y j i k sorry)
  (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Z X)
    (fun (k : I) ↦ tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (X i) (Z j)) ≫ ιTensorObj Z X j i k sorry)
    (whiskerLeft Z f))
Tactic: dsimp [braiding]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y X
inst_6 : HasTensor.{u_1, v_1, u_2} Y Z
inst_7 : HasTensor.{u_1, v_1, u_2} Z X
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y Z) X
inst_12 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} Z X)
inst_13 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y X) Z
inst_14 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} X Z)
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y Z X
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y Z X
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y X Z
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
⊢ i₂ + i₃ + i₁ = k
have rw : i₁ + (i₂ + i₃) = k
Tactic: rw [add_comm _ i₁, ← add_assoc, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y X
inst_6 : HasTensor.{u_1, v_1, u_2} Y Z
inst_7 : HasTensor.{u_1, v_1, u_2} Z X
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y Z) X
inst_12 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} Z X)
inst_13 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y X) Z
inst_14 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} X Z)
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y Z X
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y Z X
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y X Z
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
rw : i₁ + (i₂ + i₃) = k
rw₁ : i₁ + i₂ + i₃ = k
⊢ i₂ + i₃ + i₁ = k
have rw₂ : k = k
Tactic: rw [add_comm _ i₁, ← add_assoc, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y X
inst_6 : HasTensor.{u_1, v_1, u_2} Y Z
inst_7 : HasTensor.{u_1, v_1, u_2} Z X
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y Z) X
inst_12 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} Z X)
inst_13 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y X) Z
inst_14 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} X Z)
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y Z X
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y Z X
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y X Z
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z k)
    (ιTensorObj₃' X Y Z i₁ i₂ i₃ k h)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z) k)
      (Iso.hom (associator X Y Z) k)
      (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y Z) X k)
        (tensorObjDesc fun (i j : I) (hij : i + j = k) ↦
          Iso.hom (β_ (X i) (tensorObj.{u_1, v_1, u_2} Y Z j)) ≫ ιTensorObj (tensorObj.{u_1, v_1, u_2} Y Z) X j i k ⋯)
        (Iso.hom (associator Y Z X) k)))
have rw : MonoidalCategoryStruct.whiskerRight (Iso.hom (β_ (X i₁) (Y i₂))) (Z i₃) ≫
  Iso.hom (MonoidalCategoryStruct.associator (Y i₂) (X i₁) (Z i₃)) ≫
    MonoidalCategoryStruct.whiskerLeft (Y i₂) (Iso.hom (β_ (X i₁) (Z i₃))) ≫ ιTensorObj₃ Y Z X i₂ i₃ i₁ k sorry
Tactic: rw [ιTensorObj₃'_associator_hom_assoc, ιTensorObj₃_eq X Y Z i₁ i₂ i₃ k h _ rfl,
    assoc, ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.id_tensorHom,
    BraidedCategory.braiding_naturality_assoc,
    BraidedCategory.braiding_tensor_right_hom, assoc, assoc, assoc, assoc, Iso.hom_inv_id_assoc,
    MonoidalCategory.tensorHom_id,
    ← ιTensorObj₃'_eq_assoc Y Z X i₂ i₃ i₁ k (by rw [add_comm _ i₁, ← add_assoc, h]) _ rfl,
    ιTensorObj₃'_associator_hom, Iso.inv_hom_id_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y X
inst_6 : HasTensor.{u_1, v_1, u_2} Y Z
inst_7 : HasTensor.{u_1, v_1, u_2} Z X
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y Z) X
inst_12 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} Z X)
inst_13 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y X) Z
inst_14 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} X Z)
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y Z X
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y Z X
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y X Z
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
rw : i₁ + i₂ + i₃ = k
⊢ i₂ + i₁ + i₃ = k
have rw₁ : k = k
Tactic: rw [add_comm i₂ i₁, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y Z
inst_6 : HasTensor.{u_1, v_1, u_2} Z X
inst_7 : HasTensor.{u_1, v_1, u_2} Z Y
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y)
inst_12 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y
inst_13 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y)
inst_14 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Z X Y
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Z X Y
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Z Y
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
⊢ i₃ + i₁ + i₂ = k
have rw : i₃ + (i₁ + i₂) = k
Tactic: rw [add_assoc, add_comm i₃, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y Z
inst_6 : HasTensor.{u_1, v_1, u_2} Z X
inst_7 : HasTensor.{u_1, v_1, u_2} Z Y
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y)
inst_12 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y
inst_13 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y)
inst_14 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Z X Y
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Z X Y
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Z Y
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
rw : i₃ + (i₁ + i₂) = k
rw₁ : i₁ + i₂ + i₃ = k
⊢ i₃ + i₁ + i₂ = k
have rw₂ : k = k
Tactic: rw [add_assoc, add_comm i₃, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y Z
inst_6 : HasTensor.{u_1, v_1, u_2} Z X
inst_7 : HasTensor.{u_1, v_1, u_2} Z Y
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y)
inst_12 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y
inst_13 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y)
inst_14 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Z X Y
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Z X Y
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Z Y
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z) k)
    (ιTensorObj₃ X Y Z i₁ i₂ i₃ k h)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z k)
      (Iso.inv (associator X Y Z) k)
      (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y) k)
        (tensorObjDesc fun (i j : I) (hij : i + j = k) ↦
          Iso.hom (β_ (tensorObj.{u_1, v_1, u_2} X Y i) (Z j)) ≫ ιTensorObj Z (tensorObj.{u_1, v_1, u_2} X Y) j i k ⋯)
        (Iso.inv (associator Z X Y) k)))
have rw : MonoidalCategoryStruct.whiskerLeft (X i₁) (Iso.hom (β_ (Y i₂) (Z i₃))) ≫
  Iso.inv (MonoidalCategoryStruct.associator (X i₁) (Z i₃) (Y i₂)) ≫
    MonoidalCategoryStruct.whiskerRight (Iso.hom (β_ (X i₁) (Z i₃))) (Y i₂) ≫ ιTensorObj₃' Z X Y i₃ i₁ i₂ k sorry
Tactic: rw [ιTensorObj₃_associator_inv_assoc, ιTensorObj₃'_eq X Y Z i₁ i₂ i₃ k h _ rfl, assoc,
    ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.tensorHom_id,
    BraidedCategory.braiding_naturality_assoc,
    BraidedCategory.braiding_tensor_left_hom, assoc, assoc, assoc, assoc, Iso.inv_hom_id_assoc,
    MonoidalCategory.id_tensorHom,
    ← ιTensorObj₃_eq_assoc Z X Y i₃ i₁ i₂ k (by rw [add_assoc, add_comm i₃, h]) _ rfl,
    ιTensorObj₃_associator_inv, Iso.hom_inv_id_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y Z
inst_6 : HasTensor.{u_1, v_1, u_2} Z X
inst_7 : HasTensor.{u_1, v_1, u_2} Z Y
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y)
inst_12 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y
inst_13 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y)
inst_14 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Z X Y
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Z X Y
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Z Y
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
⊢ i₁ + i₃ + i₂ = k
have rw : i₁ + (i₃ + i₂) = k
Tactic: rw [add_assoc, add_comm i₃, ← add_assoc, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y Z
inst_6 : HasTensor.{u_1, v_1, u_2} Z X
inst_7 : HasTensor.{u_1, v_1, u_2} Z Y
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y)
inst_12 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y
inst_13 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y)
inst_14 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Z X Y
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Z X Y
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Z Y
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
rw : i₁ + (i₃ + i₂) = k
⊢ i₁ + i₃ + i₂ = k
have rw₁ : i₁ + (i₂ + i₃) = k
Tactic: rw [add_assoc, add_comm i₃, ← add_assoc, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y Z
inst_6 : HasTensor.{u_1, v_1, u_2} Z X
inst_7 : HasTensor.{u_1, v_1, u_2} Z Y
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y)
inst_12 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y
inst_13 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y)
inst_14 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Z X Y
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Z X Y
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Z Y
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
rw : i₁ + (i₃ + i₂) = k
rw₁ : i₁ + (i₂ + i₃) = k
rw₂ : i₁ + i₂ + i₃ = k
⊢ i₁ + i₃ + i₂ = k
have rw₃ : k = k
Tactic: rw [add_assoc, add_comm i₃, ← add_assoc, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Braiding (goal)
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y Z
inst_6 : HasTensor.{u_1, v_1, u_2} Z X
inst_7 : HasTensor.{u_1, v_1, u_2} Z Y
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y)
inst_12 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y
inst_13 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y)
inst_14 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Z X Y
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Z X Y
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Z Y
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z) k)
    (ιTensorObj₃ X Y Z i₁ i₂ i₃ k h)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y) k)
      (whiskerLeft X
        (fun (k : I) ↦
          tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (Y i) (Z j)) ≫ ιTensorObj Z Y j i k ⋯)
        k)
      (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y k)
        (Iso.inv (associator X Z Y) k)
        (whiskerRight
          (fun (k : I) ↦
            tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (X i) (Z j)) ≫ ιTensorObj Z X j i k ⋯)
          Y k)))
have rw : MonoidalCategoryStruct.whiskerLeft (X i₁) (Iso.hom (β_ (Y i₂) (Z i₃))) ≫
  Iso.inv (MonoidalCategoryStruct.associator (X i₁) (Z i₃) (Y i₂)) ≫
    MonoidalCategoryStruct.whiskerRight (Iso.hom (β_ (X i₁) (Z i₃))) (Y i₂) ≫ ιTensorObj₃' Z X Y i₃ i₁ i₂ k sorry
Tactic: rw [ιTensorObj₃_eq X Y Z i₁ i₂ i₃ k h _ rfl, assoc, ι_tensorHom_assoc,
    ← MonoidalCategory.id_tensorHom,
    MonoidalCategory.tensorHom_comp_tensorHom_assoc, id_comp, ι_tensorObjDesc,
    categoryOfGradedObjects_id, MonoidalCategory.id_tensor_comp, assoc,
    MonoidalCategory.id_tensorHom, MonoidalCategory.id_tensorHom,
    ← ιTensorObj₃_eq_assoc X Z Y i₁ i₃ i₂ k
      (by rw [add_assoc, add_comm i₃, ← add_assoc, h]) (i₂ + i₃) (add_comm _ _),
    ιTensorObj₃_associator_inv_assoc,
    ιTensorObj₃'_eq X Z Y i₁ i₃ i₂ k (by rw [add_assoc, add_comm i₃, ← add_assoc, h]) _ rfl,
    assoc, ι_tensorHom, categoryOfGradedObjects_id, ← MonoidalCategory.tensorHom_id,
    ← MonoidalCategory.comp_tensor_id_assoc,
    ι_tensorObjDesc, MonoidalCategory.comp_tensor_id, assoc,
    MonoidalCategory.tensorHom_id, MonoidalCategory.tensorHom_id,
    ← ιTensorObj₃'_eq Z X Y i₃ i₁ i₂ k (by rw [add_assoc, add_comm i₃, h])
      (i₁ + i₃) (add_comm _ _)]