File: Mathlib/CategoryTheory/Monoidal/Types/Basic.lean

full_name: CategoryTheory.typesChosenFiniteProducts
code: instance typesChosenFiniteProducts : ChosenFiniteProducts (Type u) 
goal: CategoryTheory.ChosenFiniteProducts (Type u)

full_name: CategoryTheory.tensor_apply
code: @[simp]
theorem tensor_apply {W X Y Z : Type u} (f : W ⟶ X) (g : Y ⟶ Z) (p : W ⊗ Y) :
    (f ⊗ g) p = (f p.1, g p.2) 
goal: (f ⊗ g) p = (f p.1, g p.2)

full_name: CategoryTheory.whiskerLeft_apply
code: @[simp]
theorem whiskerLeft_apply (X : Type u) {Y Z : Type u} (f : Y ⟶ Z) (p : X ⊗ Y) :
    (X ◁ f) p = (p.1, f p.2) 
goal: (X ◁ f) p = (p.1, f p.2)

full_name: CategoryTheory.whiskerRight_apply
code: @[simp]
theorem whiskerRight_apply {Y Z : Type u} (f : Y ⟶ Z) (X : Type u) (p : Y ⊗ X) :
    (f ▷ X) p = (f p.1, p.2) 
goal: (f ▷ X) p = (f p.1, p.2)

full_name: CategoryTheory.leftUnitor_hom_apply
code: @[simp]
theorem leftUnitor_hom_apply {X : Type u} {x : X} {p : PUnit} :
    ((λ_ X).hom : 𝟙_ (Type u) ⊗ X → X) (p, x) = x 
goal: (λ_ X).hom (p, x) = x

full_name: CategoryTheory.leftUnitor_inv_apply
code: @[simp]
theorem leftUnitor_inv_apply {X : Type u} {x : X} :
    ((λ_ X).inv : X ⟶ 𝟙_ (Type u) ⊗ X) x = (PUnit.unit, x) 
goal: (λ_ X).inv x = (PUnit.unit, x)

full_name: CategoryTheory.rightUnitor_hom_apply
code: @[simp]
theorem rightUnitor_hom_apply {X : Type u} {x : X} {p : PUnit} :
    ((ρ_ X).hom : X ⊗ 𝟙_ (Type u) → X) (x, p) = x 
goal: (ρ_ X).hom (x, p) = x

full_name: CategoryTheory.rightUnitor_inv_apply
code: @[simp]
theorem rightUnitor_inv_apply {X : Type u} {x : X} :
    ((ρ_ X).inv : X ⟶ X ⊗ 𝟙_ (Type u)) x = (x, PUnit.unit) 
goal: (ρ_ X).inv x = (x, PUnit.unit)

full_name: CategoryTheory.associator_hom_apply
code: @[simp]
theorem associator_hom_apply {X Y Z : Type u} {x : X} {y : Y} {z : Z} :
    ((α_ X Y Z).hom : (X ⊗ Y) ⊗ Z → X ⊗ Y ⊗ Z) ((x, y), z) = (x, (y, z)) 
goal: (α_ X Y Z).hom ((x, y), z) = (x, y, z)

full_name: CategoryTheory.associator_inv_apply
code: @[simp]
theorem associator_inv_apply {X Y Z : Type u} {x : X} {y : Y} {z : Z} :
    ((α_ X Y Z).inv : X ⊗ Y ⊗ Z → (X ⊗ Y) ⊗ Z) (x, (y, z)) = ((x, y), z) 
goal: (α_ X Y Z).inv (x, y, z) = ((x, y), z)

full_name: CategoryTheory.associator_hom_apply_1
code: @[simp] theorem associator_hom_apply_1 {X Y Z : Type u} {x} :
    (((α_ X Y Z).hom : (X ⊗ Y) ⊗ Z → X ⊗ Y ⊗ Z) x).1 = x.1.1 
goal: ((α_ X Y Z).hom x).1 = x.1.1

full_name: CategoryTheory.associator_hom_apply_2_1
code: @[simp] theorem associator_hom_apply_2_1 {X Y Z : Type u} {x} :
    (((α_ X Y Z).hom : (X ⊗ Y) ⊗ Z → X ⊗ Y ⊗ Z) x).2.1 = x.1.2 
goal: ((α_ X Y Z).hom x).2.1 = x.1.2

full_name: CategoryTheory.associator_hom_apply_2_2
code: @[simp] theorem associator_hom_apply_2_2 {X Y Z : Type u} {x} :
    (((α_ X Y Z).hom : (X ⊗ Y) ⊗ Z → X ⊗ Y ⊗ Z) x).2.2 = x.2 
goal: ((α_ X Y Z).hom x).2.2 = x.2

full_name: CategoryTheory.associator_inv_apply_1_1
code: @[simp] theorem associator_inv_apply_1_1 {X Y Z : Type u} {x} :
    (((α_ X Y Z).inv : X ⊗ Y ⊗ Z → (X ⊗ Y) ⊗ Z) x).1.1 = x.1 
goal: ((α_ X Y Z).inv x).1.1 = x.1

full_name: CategoryTheory.associator_inv_apply_1_2
code: @[simp] theorem associator_inv_apply_1_2 {X Y Z : Type u} {x} :
    (((α_ X Y Z).inv : X ⊗ Y ⊗ Z → (X ⊗ Y) ⊗ Z) x).1.2 = x.2.1 
goal: ((α_ X Y Z).inv x).1.2 = x.2.1

full_name: CategoryTheory.associator_inv_apply_2
code: @[simp] theorem associator_inv_apply_2 {X Y Z : Type u} {x} :
    (((α_ X Y Z).inv : X ⊗ Y ⊗ Z → (X ⊗ Y) ⊗ Z) x).2 = x.2.2 
goal: ((α_ X Y Z).inv x).2 = x.2.2

full_name: CategoryTheory.MonoidalFunctor.mapPi
code: /-- If `F` is a monoidal functor out of `Type`, it takes the (n+1)st cartesian power
of a type to the image of that type, tensored with the image of the nth cartesian power. -/
noncomputable def MonoidalFunctor.mapPi {C : Type*} [Category C] [MonoidalCategory C]
    (F : MonoidalFunctor (Type _) C) (n : ℕ) (β : Type*) :
    F.obj (Fin (n + 1) → β) ≅ F.obj β ⊗ F.obj (Fin n → β) 
goal: F.obj (Fin (n + 1) → β) ≅ F.obj β ⊗ F.obj (Fin n → β)