ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.BraidedCategory.hexagon_forward
class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] where
    braiding : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X
  braiding_naturality_right :
    ∀ (X : C) {Y Z : C} (f : Y ⟶ Z),
      X ◁ f ≫ (braiding X Z).hom = (braiding X Y).hom ≫ f ▷ X := by
    aesop_cat
  braiding_naturality_left :
    ∀ {X Y : C} (f : X ⟶ Y) (Z : C),
      f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by
    aesop_cat
    hexagon_forward :
    ∀ X Y Z : C,
      (α_ X Y Z).hom ≫ (braiding X (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom =
        ((braiding X Y).hom ▷ Z) ≫ (α_ Y X Z).hom ≫ (Y ◁ (braiding X Z).hom) := by
    aesop_cat
    hexagon_reverse :
    ∀ X Y Z : C,
      (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =
        (X ◁ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ▷ Y) := by
    aesop_cat
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [self : CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (α_ X Y Z).hom ≫ (β_ X (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom = (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (β_ X Z).hom
-- CategoryTheory.BraidedCategory.braiding_naturality_left
class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] where
    braiding : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X
  braiding_naturality_right :
    ∀ (X : C) {Y Z : C} (f : Y ⟶ Z),
      X ◁ f ≫ (braiding X Z).hom = (braiding X Y).hom ≫ f ▷ X := by
    aesop_cat
  braiding_naturality_left :
    ∀ {X Y : C} (f : X ⟶ Y) (Z : C),
      f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by
    aesop_cat
    hexagon_forward :
    ∀ X Y Z : C,
      (α_ X Y Z).hom ≫ (braiding X (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom =
        ((braiding X Y).hom ▷ Z) ≫ (α_ Y X Z).hom ≫ (Y ◁ (braiding X Z).hom) := by
    aesop_cat
    hexagon_reverse :
    ∀ X Y Z : C,
      (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =
        (X ◁ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ▷ Y) := by
    aesop_cat
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [self : CategoryTheory.BraidedCategory C] {X Y : C} (f : X ⟶ Y) (Z : C)
Goal: f ▷ Z ≫ (β_ Y Z).hom = (β_ X Z).hom ≫ Z ◁ f
-- CategoryTheory.BraidedCategory.braiding_naturality_right
class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] where
    braiding : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X
  braiding_naturality_right :
    ∀ (X : C) {Y Z : C} (f : Y ⟶ Z),
      X ◁ f ≫ (braiding X Z).hom = (braiding X Y).hom ≫ f ▷ X := by
    aesop_cat
  braiding_naturality_left :
    ∀ {X Y : C} (f : X ⟶ Y) (Z : C),
      f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by
    aesop_cat
    hexagon_forward :
    ∀ X Y Z : C,
      (α_ X Y Z).hom ≫ (braiding X (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom =
        ((braiding X Y).hom ▷ Z) ≫ (α_ Y X Z).hom ≫ (Y ◁ (braiding X Z).hom) := by
    aesop_cat
    hexagon_reverse :
    ∀ X Y Z : C,
      (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =
        (X ◁ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ▷ Y) := by
    aesop_cat
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [self : CategoryTheory.BraidedCategory C] (X : C) {Y Z : C} (f : Y ⟶ Z)
Goal: X ◁ f ≫ (β_ X Z).hom = (β_ X Y).hom ≫ f ▷ X
-- CategoryTheory.BraidedCategory.braiding_tensor_left
@[simp, reassoc]
theorem braiding_tensor_left (X Y Z : C) :
    (β_ (X ⊗ Y) Z).hom  =
      (α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫
        (β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (β_ (X ⊗ Y) Z).hom = (α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom
-- CategoryTheory.BraidedCategory.braiding_tensor_right
@[simp, reassoc]
theorem braiding_tensor_right (X Y Z : C) :
    (β_ X (Y ⊗ Z)).hom  =
      (α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
        Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (β_ X (Y ⊗ Z)).hom = (α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv
-- CategoryTheory.BraidedCategory.braiding_inv_tensor_left
@[simp, reassoc]
theorem braiding_inv_tensor_left (X Y Z : C) :
    (β_ (X ⊗ Y) Z).inv  =
      (α_ Z X Y).inv ≫ (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫
        X ◁ (β_ Y Z).inv ≫ (α_ X Y Z).inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (β_ (X ⊗ Y) Z).inv = (α_ Z X Y).inv ≫ (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫ X ◁ (β_ Y Z).inv ≫ (α_ X Y Z).inv
-- CategoryTheory.BraidedCategory.braiding_inv_tensor_right
@[simp, reassoc]
theorem braiding_inv_tensor_right (X Y Z : C) :
    (β_ X (Y ⊗ Z)).inv  =
      (α_ Y Z X).hom ≫ Y ◁ (β_ X Z).inv ≫ (α_ Y X Z).inv ≫
        (β_ X Y).inv ▷ Z ≫ (α_ X Y Z).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (β_ X (Y ⊗ Z)).inv = (α_ Y Z X).hom ≫ Y ◁ (β_ X Z).inv ≫ (α_ Y X Z).inv ≫ (β_ X Y).inv ▷ Z ≫ (α_ X Y Z).hom
-- CategoryTheory.BraidedCategory.braiding_naturality
@[reassoc (attr := simp)]
theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
    (f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y')
Goal: (f ⊗ g) ≫ (β_ Y Y').hom = (β_ X X').hom ≫ (g ⊗ f)
-- CategoryTheory.BraidedCategory.braiding_inv_naturality_right
@[reassoc (attr := simp)]
theorem braiding_inv_naturality_right (X : C) {Y Z : C} (f : Y ⟶ Z) :
    X ◁ f ≫ (β_ Z X).inv = (β_ Y X).inv ≫ f ▷ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C) {Y Z : C} (f : Y ⟶ Z)
Goal: X ◁ f ≫ (β_ Z X).inv = (β_ Y X).inv ≫ f ▷ X
-- CategoryTheory.BraidedCategory.braiding_inv_naturality_left
@[reassoc (attr := simp)]
theorem braiding_inv_naturality_left {X Y : C} (f : X ⟶ Y) (Z : C) :
    f ▷ Z ≫ (β_ Z Y).inv = (β_ Z X).inv ≫ Z ◁ f 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {X Y : C} (f : X ⟶ Y) (Z : C)
Goal: f ▷ Z ≫ (β_ Z Y).inv = (β_ Z X).inv ≫ Z ◁ f
-- CategoryTheory.BraidedCategory.braiding_inv_naturality
@[reassoc (attr := simp)]
theorem braiding_inv_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
    (f ⊗ g) ≫ (β_ Y' Y).inv = (β_ X' X).inv ≫ (g ⊗ f) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y')
Goal: (f ⊗ g) ≫ (β_ Y' Y).inv = (β_ X' X).inv ≫ (g ⊗ f)
-- CategoryTheory.BraidedCategory.yang_baxter
@[reassoc]
theorem yang_baxter (X Y Z : C) :
    (α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
    Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom =
      X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫
      (α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (α_ X Y Z).inv ≫
    (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom =
  X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom
-- CategoryTheory.BraidedCategory.yang_baxter'
theorem yang_baxter' (X Y Z : C) :
    (β_ X Y).hom ▷ Z ⊗≫ Y ◁ (β_ X Z).hom ⊗≫ (β_ Y Z).hom ▷ X =
      𝟙 _ ⊗≫ (X ◁ (β_ Y Z).hom ⊗≫ (β_ X Z).hom ▷ Y ⊗≫ Z ◁ (β_ X Y).hom) ⊗≫ 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (β_ X Y).hom ▷ Z ⊗≫ Y ◁ (β_ X Z).hom ⊗≫ (β_ Y Z).hom ▷ X =
  𝟙 ((X ⊗ Y) ⊗ Z) ⊗≫ (X ◁ (β_ Y Z).hom ⊗≫ (β_ X Z).hom ▷ Y ⊗≫ Z ◁ (β_ X Y).hom) ⊗≫ 𝟙 ((Z ⊗ Y) ⊗ X)
-- CategoryTheory.BraidedCategory.yang_baxter_iso
theorem yang_baxter_iso (X Y Z : C) :
    (α_ X Y Z).symm ≪≫ whiskerRightIso (β_ X Y) Z ≪≫ α_ Y X Z ≪≫
    whiskerLeftIso Y (β_ X Z) ≪≫ (α_ Y Z X).symm ≪≫
    whiskerRightIso (β_ Y Z) X ≪≫ (α_ Z Y X) =
      whiskerLeftIso X (β_ Y Z) ≪≫ (α_ X Z Y).symm ≪≫
      whiskerRightIso (β_ X Z) Y ≪≫ α_ Z X Y ≪≫
      whiskerLeftIso Z (β_ X Y) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (α_ X Y Z).symm ≪≫
    CategoryTheory.MonoidalCategory.whiskerRightIso (β_ X Y) Z ≪≫
      α_ Y X Z ≪≫
        CategoryTheory.MonoidalCategory.whiskerLeftIso Y (β_ X Z) ≪≫
          (α_ Y Z X).symm ≪≫ CategoryTheory.MonoidalCategory.whiskerRightIso (β_ Y Z) X ≪≫ α_ Z Y X =
  CategoryTheory.MonoidalCategory.whiskerLeftIso X (β_ Y Z) ≪≫
    (α_ X Z Y).symm ≪≫
      CategoryTheory.MonoidalCategory.whiskerRightIso (β_ X Z) Y ≪≫
        α_ Z X Y ≪≫ CategoryTheory.MonoidalCategory.whiskerLeftIso Z (β_ X Y)
-- CategoryTheory.BraidedCategory.hexagon_forward_iso
theorem hexagon_forward_iso (X Y Z : C) :
    α_ X Y Z ≪≫ β_ X (Y ⊗ Z) ≪≫ α_ Y Z X =
      whiskerRightIso (β_ X Y) Z ≪≫ α_ Y X Z ≪≫ whiskerLeftIso Y (β_ X Z) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: α_ X Y Z ≪≫ β_ X (Y ⊗ Z) ≪≫ α_ Y Z X =
  CategoryTheory.MonoidalCategory.whiskerRightIso (β_ X Y) Z ≪≫
    α_ Y X Z ≪≫ CategoryTheory.MonoidalCategory.whiskerLeftIso Y (β_ X Z)
-- CategoryTheory.BraidedCategory.hexagon_reverse_iso
theorem hexagon_reverse_iso (X Y Z : C) :
    (α_ X Y Z).symm ≪≫ β_ (X ⊗ Y) Z ≪≫ (α_ Z X Y).symm =
      whiskerLeftIso X (β_ Y Z) ≪≫ (α_ X Z Y).symm ≪≫ whiskerRightIso (β_ X Z) Y 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (α_ X Y Z).symm ≪≫ β_ (X ⊗ Y) Z ≪≫ (α_ Z X Y).symm =
  CategoryTheory.MonoidalCategory.whiskerLeftIso X (β_ Y Z) ≪≫
    (α_ X Z Y).symm ≪≫ CategoryTheory.MonoidalCategory.whiskerRightIso (β_ X Z) Y
-- CategoryTheory.BraidedCategory.hexagon_forward_inv
@[reassoc]
theorem hexagon_forward_inv (X Y Z : C) :
    (α_ Y Z X).inv ≫ (β_ X (Y ⊗ Z)).inv ≫ (α_ X Y Z).inv =
      Y ◁ (β_ X Z).inv ≫ (α_ Y X Z).inv ≫ (β_ X Y).inv ▷ Z 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (α_ Y Z X).inv ≫ (β_ X (Y ⊗ Z)).inv ≫ (α_ X Y Z).inv = Y ◁ (β_ X Z).inv ≫ (α_ Y X Z).inv ≫ (β_ X Y).inv ▷ Z
-- CategoryTheory.BraidedCategory.hexagon_reverse_inv
@[reassoc]
theorem hexagon_reverse_inv (X Y Z : C) :
    (α_ Z X Y).hom ≫ (β_ (X ⊗ Y) Z).inv ≫ (α_ X Y Z).hom =
      (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫ X ◁ (β_ Y Z).inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y Z : C)
Goal: (α_ Z X Y).hom ≫ (β_ (X ⊗ Y) Z).inv ≫ (α_ X Y Z).hom = (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫ X ◁ (β_ Y Z).inv
-- CategoryTheory.braiding_leftUnitor_aux₁
theorem braiding_leftUnitor_aux₁ (X : C) :
    (α_ (𝟙_ C) (𝟙_ C) X).hom ≫
        (𝟙_ C ◁ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ▷ _) =
      ((λ_ _).hom ▷ X) ≫ (β_ X (𝟙_ C)).inv 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (α_ (𝟙_ C) (𝟙_ C) X).hom ≫ 𝟙_ C ◁ (β_ X (𝟙_ C)).inv ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ (λ_ X).hom ▷ 𝟙_ C =
  (λ_ (𝟙_ C)).hom ▷ X ≫ (β_ X (𝟙_ C)).inv
-- CategoryTheory.braiding_leftUnitor_aux₂
theorem braiding_leftUnitor_aux₂ (X : C) :
    ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ ((λ_ X).hom ▷ 𝟙_ C) = (ρ_ X).hom ▷ 𝟙_ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (β_ X (𝟙_ C)).hom ▷ 𝟙_ C ≫ (λ_ X).hom ▷ 𝟙_ C = (ρ_ X).hom ▷ 𝟙_ C
-- CategoryTheory.braiding_leftUnitor
@[reassoc]
theorem braiding_leftUnitor (X : C) : (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom = (ρ_ X).hom 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom = (ρ_ X).hom
-- CategoryTheory.braiding_rightUnitor_aux₁
theorem braiding_rightUnitor_aux₁ (X : C) :
    (α_ X (𝟙_ C) (𝟙_ C)).inv ≫
        ((β_ (𝟙_ C) X).inv ▷ 𝟙_ C) ≫ (α_ _ X _).hom ≫ (_ ◁ (ρ_ X).hom) =
      (X ◁ (ρ_ _).hom) ≫ (β_ (𝟙_ C) X).inv 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (α_ X (𝟙_ C) (𝟙_ C)).inv ≫ (β_ (𝟙_ C) X).inv ▷ 𝟙_ C ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ 𝟙_ C ◁ (ρ_ X).hom =
  X ◁ (ρ_ (𝟙_ C)).hom ≫ (β_ (𝟙_ C) X).inv
-- CategoryTheory.braiding_rightUnitor_aux₂
theorem braiding_rightUnitor_aux₂ (X : C) :
    (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) = 𝟙_ C ◁ (λ_ X).hom 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: 𝟙_ C ◁ (β_ (𝟙_ C) X).hom ≫ 𝟙_ C ◁ (ρ_ X).hom = 𝟙_ C ◁ (λ_ X).hom
-- CategoryTheory.braiding_rightUnitor
@[reassoc]
theorem braiding_rightUnitor (X : C) : (β_ (𝟙_ C) X).hom ≫ (ρ_ X).hom = (λ_ X).hom 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (β_ (𝟙_ C) X).hom ≫ (ρ_ X).hom = (λ_ X).hom
-- CategoryTheory.braiding_tensorUnit_left
@[reassoc, simp]
theorem braiding_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).hom = (λ_ X).hom ≫ (ρ_ X).inv 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (β_ (𝟙_ C) X).hom = (λ_ X).hom ≫ (ρ_ X).inv
-- CategoryTheory.braiding_inv_tensorUnit_left
@[reassoc, simp]
theorem braiding_inv_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).inv = (ρ_ X).hom ≫ (λ_ X).inv 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (β_ (𝟙_ C) X).inv = (ρ_ X).hom ≫ (λ_ X).inv
-- CategoryTheory.leftUnitor_inv_braiding
@[reassoc]
theorem leftUnitor_inv_braiding (X : C) : (λ_ X).inv ≫ (β_ (𝟙_ C) X).hom = (ρ_ X).inv 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (λ_ X).inv ≫ (β_ (𝟙_ C) X).hom = (ρ_ X).inv
-- CategoryTheory.rightUnitor_inv_braiding
@[reassoc]
theorem rightUnitor_inv_braiding (X : C) : (ρ_ X).inv ≫ (β_ X (𝟙_ C)).hom = (λ_ X).inv 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (ρ_ X).inv ≫ (β_ X (𝟙_ C)).hom = (λ_ X).inv
-- CategoryTheory.braiding_tensorUnit_right
@[reassoc, simp]
theorem braiding_tensorUnit_right (X : C) : (β_ X (𝟙_ C)).hom = (ρ_ X).hom ≫ (λ_ X).inv 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (β_ X (𝟙_ C)).hom = (ρ_ X).hom ≫ (λ_ X).inv
-- CategoryTheory.braiding_inv_tensorUnit_right
@[reassoc, simp]
theorem braiding_inv_tensorUnit_right (X : C) : (β_ X (𝟙_ C)).inv = (λ_ X).hom ≫ (ρ_ X).inv 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (β_ X (𝟙_ C)).inv = (λ_ X).hom ≫ (ρ_ X).inv
-- CategoryTheory.SymmetricCategory.symmetry
class SymmetricCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] extends
    BraidedCategory.{v} C where
  -- braiding symmetric:
  symmetry : ∀ X Y : C, (β_ X Y).hom ≫ (β_ Y X).hom = 𝟙 (X ⊗ Y) := by aesop_cat
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [self : CategoryTheory.SymmetricCategory C] (X Y : C)
Goal: (β_ X Y).hom ≫ (β_ Y X).hom = 𝟙 (X ⊗ Y)
-- CategoryTheory.SymmetricCategory.braiding_swap_eq_inv_braiding
lemma SymmetricCategory.braiding_swap_eq_inv_braiding {C : Type u₁}
    [Category.{v₁} C] [MonoidalCategory C] [SymmetricCategory C] (X Y : C) :
    (β_ Y X).hom = (β_ X Y).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.SymmetricCategory C] (X Y : C)
Goal: (β_ Y X).hom = (β_ X Y).inv
-- CategoryTheory.LaxBraidedFunctor.braided
structure LaxBraidedFunctor extends LaxMonoidalFunctor C D where
  braided : ∀ X Y : C, μ X Y ≫ map (β_ X Y).hom = (β_ (obj X) (obj Y)).hom ≫ μ Y X := by aesop_cat
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (self : CategoryTheory.LaxBraidedFunctor C D) (X Y : C)
Goal: self.μ X Y ≫ self.map (β_ X Y).hom = (β_ (self.obj X) (self.obj Y)).hom ≫ self.μ Y X
-- CategoryTheory.LaxBraidedFunctor.id_obj
@[simps!]
def id : LaxBraidedFunctor C C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C)
Goal: (CategoryTheory.LaxBraidedFunctor.id C).obj X = X
-- CategoryTheory.LaxBraidedFunctor.id_μ
@[simps!]
def id : LaxBraidedFunctor C C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X Y : C)
Goal: (CategoryTheory.LaxBraidedFunctor.id C).μ X Y = 𝟙 (X ⊗ Y)
-- CategoryTheory.LaxBraidedFunctor.id_map
@[simps!]
def id : LaxBraidedFunctor C C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C]
Goal: ∀ {X Y : C} (f : X ⟶ Y), (CategoryTheory.LaxBraidedFunctor.id C).map f = f
-- CategoryTheory.LaxBraidedFunctor.id_ε
@[simps!]
def id : LaxBraidedFunctor C C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C]
Goal: (CategoryTheory.LaxBraidedFunctor.id C).ε = 𝟙 (𝟙_ C)
-- CategoryTheory.LaxBraidedFunctor.comp_ε
@[simps!]
def comp (F : LaxBraidedFunctor C D) (G : LaxBraidedFunctor D E) : LaxBraidedFunctor C E 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] [CategoryTheory.BraidedCategory E] (F : CategoryTheory.LaxBraidedFunctor C D) (G : CategoryTheory.LaxBraidedFunctor D E)
Goal: (F.comp G).ε = G.ε ≫ G.map F.ε
-- CategoryTheory.LaxBraidedFunctor.comp_map
@[simps!]
def comp (F : LaxBraidedFunctor C D) (G : LaxBraidedFunctor D E) : LaxBraidedFunctor C E 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] [CategoryTheory.BraidedCategory E] (F : CategoryTheory.LaxBraidedFunctor C D) (G : CategoryTheory.LaxBraidedFunctor D E)
Goal: ∀ {X Y : C} (f : X ⟶ Y), (F.comp G).map f = G.map (F.map f)
-- CategoryTheory.LaxBraidedFunctor.comp_obj
@[simps!]
def comp (F : LaxBraidedFunctor C D) (G : LaxBraidedFunctor D E) : LaxBraidedFunctor C E 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] [CategoryTheory.BraidedCategory E] (F : CategoryTheory.LaxBraidedFunctor C D) (G : CategoryTheory.LaxBraidedFunctor D E) (X : C)
Goal: (F.comp G).obj X = G.obj (F.obj X)
-- CategoryTheory.LaxBraidedFunctor.comp_μ
@[simps!]
def comp (F : LaxBraidedFunctor C D) (G : LaxBraidedFunctor D E) : LaxBraidedFunctor C E 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] [CategoryTheory.BraidedCategory E] (F : CategoryTheory.LaxBraidedFunctor C D) (G : CategoryTheory.LaxBraidedFunctor D E) (X Y : C)
Goal: (F.comp G).μ X Y = G.μ (F.obj X) (F.obj Y) ≫ G.map (F.μ X Y)
-- CategoryTheory.LaxBraidedFunctor.ext'
@[ext]
lemma ext' {F G : LaxBraidedFunctor C D} {α β : F ⟶ G} (w : ∀ X : C, α.app X = β.app X) : α = β 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {F G : CategoryTheory.LaxBraidedFunctor C D} {α β : F ⟶ G} (w : ∀ (X : C), α.app X = β.app X)
Goal: α = β
-- CategoryTheory.LaxBraidedFunctor.comp_toNatTrans
@[simp]
theorem comp_toNatTrans {F G H : LaxBraidedFunctor C D} {α : F ⟶ G} {β : G ⟶ H} :
    (α ≫ β).toNatTrans = @CategoryStruct.comp (C ⥤ D) _ _ _ _ α.toNatTrans β.toNatTrans 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {F G H : CategoryTheory.LaxBraidedFunctor C D} {α : F ⟶ G} {β : G ⟶ H}
Goal: (α ≫ β).toNatTrans = α.toNatTrans ≫ β.toNatTrans
-- CategoryTheory.LaxBraidedFunctor.mkIso_hom
@[simps]
def mkIso {F G : LaxBraidedFunctor C D} (i : F.toLaxMonoidalFunctor ≅ G.toLaxMonoidalFunctor) :
    F ≅ G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {F G : CategoryTheory.LaxBraidedFunctor C D} (i : F.toLaxMonoidalFunctor ≅ G.toLaxMonoidalFunctor)
Goal: (CategoryTheory.LaxBraidedFunctor.mkIso i).hom = i.hom
-- CategoryTheory.LaxBraidedFunctor.mkIso_inv
@[simps]
def mkIso {F G : LaxBraidedFunctor C D} (i : F.toLaxMonoidalFunctor ≅ G.toLaxMonoidalFunctor) :
    F ≅ G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {F G : CategoryTheory.LaxBraidedFunctor C D} (i : F.toLaxMonoidalFunctor ≅ G.toLaxMonoidalFunctor)
Goal: (CategoryTheory.LaxBraidedFunctor.mkIso i).inv = i.inv
-- CategoryTheory.BraidedFunctor.braided
structure BraidedFunctor extends MonoidalFunctor C D where
  -- Note this is stated differently than for `LaxBraidedFunctor`.
  -- We move the `μ X Y` to the right hand side,
  -- so that this makes a good `@[simp]` lemma.
  braided : ∀ X Y : C, map (β_ X Y).hom = inv (μ X Y) ≫ (β_ (obj X) (obj Y)).hom ≫ μ Y X := by
    aesop_cat
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (self : CategoryTheory.BraidedFunctor C D) (X Y : C)
Goal: self.map (β_ X Y).hom = CategoryTheory.inv (self.μ X Y) ≫ (β_ (self.obj X) (self.obj Y)).hom ≫ self.μ Y X