mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.Homotopy

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
inst✝ : Preadditive V
u v w : Arrow V
f✝ g✝ : u ⟶ v
f g : v ⟶ w
h : RightHomotopy f g
e : u ⟶ v
⊢ Hom.right (e ≫ f) - Hom.right (e ≫ g) = (Hom.right e ≫ h.hom) ≫ w.hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
inst✝ : Preadditive V
⊢ ∀ ⦃X Y : Arrow V⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y),
    rightHomotopic V f₁ f₂ → rightHomotopic V g₁ g₂ → rightHomotopic V (f₁ + g₁) (f₂ + g₂)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
inst✝ : Preadditive V
u v : Arrow V
f g : u ⟶ v
h : (quotient V).map f = (quotient V).map g
⊢ (quotient V).map f = (quotient V).map g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
inst✝ : Preadditive V
u v : Arrow V
f g : u ⟶ v
x✝ : Nonempty (RightHomotopy f g)
h : RightHomotopy f g
⊢ (quotient V).map f = (quotient V).map g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
inst✝ : Preadditive V
u v : Arrow V
f : u ⟶ v
h : (quotient V).map f = 0
⊢ (quotient V).map f = (quotient V).map 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
inst✝ : Preadditive V
u v : Arrow V
f : u ⟶ v
x✝ : Nonempty (RightHomotopy f 0)
h : RightHomotopy f 0
⊢ (quotient V).map f = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u v : Arrow V
f : u ⟶ v
inst✝ : IsIso (Hom.right f)
Z✝ : RightFreyd V
g₁ g₂ : v ⟶ Z✝.as
eq : (quotient V).map f ≫ (quotient V).map g₁ = (quotient V).map f ≫ (quotient V).map g₂
h : RightHomotopy (f ≫ g₁) (f ≫ g₂) := homotopyOfEq (f ≫ g₁) (f ≫ g₂) eq
⊢ Hom.right g₁ - Hom.right g₂ = (inv (Hom.right f) ≫ h.hom) ≫ Z✝.as.hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u v : Arrow V
f : u ⟶ v
inst✝ : IsIso (Hom.right f)
Z✝ : RightFreyd V
g₁ g₂ : (quotient V).obj v ⟶ Z✝
eq : (quotient V).map f ≫ g₁ = (quotient V).map f ≫ g₂
⊢ g₁ = g₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u v : Arrow V
f g : u ⟶ v
inst✝ : HasZeroObject V
x✝³ x✝² : V
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ (rightFunctor V).map (x✝¹ + x✝) = (rightFunctor V).map x✝¹ + (rightFunctor V).map x✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u v : Arrow V
f g : u ⟶ v
inst✝ : HasZeroObject V
⊢ (functor V).Additive

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u✝ v : Arrow V
f g : u✝ ⟶ v
inst✝ : HasZeroObject V
X✝ Y✝ : V
u : (rightFunctor V).obj X✝ ⟶ (rightFunctor V).obj Y✝
⊢ (rightFunctor V).map (Hom.right u) = u

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u v : Arrow V
f g : u ⟶ v
inst✝ : HasZeroObject V
X✝ Y✝ : V
a : (functor V).obj X✝ ⟶ (functor V).obj Y✝
⊢ ∃ a_1, (functor V).map a_1 = a

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
inst✝ : (i : I) → MonoidalCategory (C i)
A B X Y : (i : I) → C i
f : A ⟶ B
g : X ⟶ Y
⊢ f ⊗ₘ g = f ▷ X ≫ B ◁ g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u v : Arrow V
f✝ g✝ : u ⟶ v
inst✝ : HasZeroObject V
x✝¹ x✝ : V
f g : x✝¹ ⟶ x✝
eq : (functor V).map f = (functor V).map g
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u✝ v✝ : Arrow V
f✝ g : u✝ ⟶ v✝
inst✝ : HasBinaryBiproducts V
u v : Arrow V
f : u ⟶ v
⊢ Hom.right (f ≫ π f) - Hom.right 0 = biprod.inr ≫ (cokernel f).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
inst✝¹ : (i : I) → MonoidalCategory (C i)
inst✝ : (i : I) → BraidedCategory (C i)
X Y Z : (i : I) → C i
⊢ (α_ X Y Z).hom ≫ (isoMk fun i ↦ β_ (X i) ((Y ⊗ Z) i)).hom ≫ (α_ Y Z X).hom =
    (isoMk fun i ↦ β_ (X i) (Y i)).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (isoMk fun i ↦ β_ (X i) (Z i)).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u✝ v✝ : Arrow V
f✝ g : u✝ ⟶ v✝
inst✝ : HasBinaryBiproducts V
u v : Arrow V
f : u ⟶ v
⊢ IsIso (Hom.right (π f))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u✝ v✝ : Arrow V
f✝ g✝ : u✝ ⟶ v✝
inst✝ : HasBinaryBiproducts V
u v : Arrow V
f : u ⟶ v
w : Arrow V
g : v ⟶ w
h : RightHomotopy (f ≫ g) 0
⊢ biprod.inl ≫ biprod.desc (Hom.left g) h.hom ≫ w.hom = biprod.inl ≫ (cokernel f).hom ≫ Hom.right g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
inst✝¹ : (i : I) → MonoidalCategory (C i)
inst✝ : (i : I) → BraidedCategory (C i)
X Y Z : (i : I) → C i
⊢ (α_ X Y Z).inv ≫ (isoMk fun i ↦ β_ ((X ⊗ Y) i) (Z i)).hom ≫ (α_ Z X Y).inv =
    X ◁ (isoMk fun i ↦ β_ (Y i) (Z i)).hom ≫ (α_ X Z Y).inv ≫ (isoMk fun i ↦ β_ (X i) (Z i)).hom ▷ Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
inst✝¹ : (i : I) → MonoidalCategory (C i)
inst✝ : (i : I) → SymmetricCategory (C i)
X Y : (i : I) → C i
⊢ (β_ X Y).hom ≫ (β_ Y X).hom = 𝟙 (X ⊗ Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u✝ v✝ : Arrow V
f✝ g✝ : u✝ ⟶ v✝
inst✝ : HasBinaryBiproducts V
u v : Arrow V
f : u ⟶ v
w : Arrow V
g : v ⟶ w
h : RightHomotopy (f ≫ g) 0
⊢ biprod.inr ≫ biprod.desc (Hom.left g) h.hom ≫ w.hom = biprod.inr ≫ (cokernel f).hom ≫ Hom.right g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁴ : (i : I) → Category.{v₁, u₁} (C i)
inst✝³ : (i : I) → MonoidalCategory (C i)
D : Type u_1
inst✝² : Category.{v_1, u_1} D
inst✝¹ : MonoidalCategory D
F : (i : I) → D ⥤ C i
inst✝ : (i : I) → (F i).OplaxMonoidal
X : D
⊢ (λ_ ((Functor.pi' F).obj X)).inv =
    (Functor.pi' F).map (λ_ X).inv ≫
      (fun i ↦ Functor.OplaxMonoidal.δ (F i) (𝟙_ D) X) ≫ (fun i ↦ Functor.OplaxMonoidal.η (F i)) ▷ (Functor.pi' F).obj X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁴ : (i : I) → Category.{v₁, u₁} (C i)
inst✝³ : (i : I) → MonoidalCategory (C i)
D : Type u_1
inst✝² : Category.{v_1, u_1} D
inst✝¹ : MonoidalCategory D
F : (i : I) → D ⥤ C i
inst✝ : (i : I) → (F i).OplaxMonoidal
X : D
⊢ (ρ_ ((Functor.pi' F).obj X)).inv =
    (Functor.pi' F).map (ρ_ X).inv ≫
      (fun i ↦ Functor.OplaxMonoidal.δ (F i) X (𝟙_ D)) ≫ (Functor.pi' F).obj X ◁ fun i ↦ Functor.OplaxMonoidal.η (F i)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
inst✝ : HasBinaryBiproducts V
u v : Arrow V
f : u ⟶ v
w : Arrow V
g : v ⟶ w
h : RightHomotopy (f ≫ g) 0
⊢ π f ≫ desc f g h = g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u✝ v✝ : Arrow V
f✝ g✝¹ : u✝ ⟶ v✝
inst✝ : HasBinaryBiproducts V
u v : Arrow V
f : u ⟶ v
w : Arrow V
g✝ : v ⟶ w
h : RightHomotopy (f ≫ g✝) 0
A✝ : RightFreyd V
g : v ⟶ A✝.as
hg : (quotient V).map f ≫ (quotient V).map g = 0
⊢ (quotient V).map (π f) ≫ (quotient V).map (desc f g (homotopyOfEq (f ≫ g) 0 hg)) = (quotient V).map g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁶ : (i : I) → Category.{v₁, u₁} (C i)
inst✝⁵ : (i : I) → MonoidalCategory (C i)
inst✝⁴ : (i : I) → BraidedCategory (C i)
D : Type u_1
inst✝³ : Category.{v_1, u_1} D
inst✝² : MonoidalCategory D
inst✝¹ : BraidedCategory D
F : (i : I) → D ⥤ C i
inst✝ : (i : I) → (F i).LaxBraided
⊢ ∀ (X Y : D),
    Functor.LaxMonoidal.μ (Functor.pi' F) X Y ≫ (Functor.pi' F).map (β_ X Y).hom =
      (β_ ((Functor.pi' F).obj X) ((Functor.pi' F).obj Y)).hom ≫ Functor.LaxMonoidal.μ (Functor.pi' F) Y X

Tactic:
intros; ext i; exact Functor.LaxBraided.braided _ _

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁶ : (i : I) → Category.{v₁, u₁} (C i)
inst✝⁵ : (i : I) → MonoidalCategory (C i)
inst✝⁴ : (i : I) → BraidedCategory (C i)
D : Type u_1
inst✝³ : Category.{v_1, u_1} D
inst✝² : MonoidalCategory D
inst✝¹ : BraidedCategory D
F : (i : I) → D ⥤ C i
inst✝ : (i : I) → (F i).LaxBraided
⊢ ∀ (X Y : D),
    Functor.LaxMonoidal.μ (Functor.pi' F) X Y ≫ (Functor.pi' F).map (β_ X Y).hom =
      (β_ ((Functor.pi' F).obj X) ((Functor.pi' F).obj Y)).hom ≫ Functor.LaxMonoidal.μ (Functor.pi' F) Y X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u✝ v✝ : Arrow V
f✝ g✝¹ : u✝ ⟶ v✝
inst✝ : HasBinaryBiproducts V
u v : Arrow V
f : u ⟶ v
w : Arrow V
g✝ : v ⟶ w
h : RightHomotopy (f ≫ g✝) 0
A✝ : RightFreyd V
g : (quotient V).obj v ⟶ A✝
hg : (quotient V).map f ≫ g = 0
⊢ Nonempty { l // (quotient V).map (π f) ≫ l = g }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁴ : (i : I) → Category.{v₁, u₁} (C i)
inst✝³ : (i : I) → MonoidalCategory (C i)
D : I → Type u₂
inst✝² : (i : I) → Category.{v₂, u₂} (D i)
inst✝¹ : (i : I) → MonoidalCategory (D i)
F : (i : I) → D i ⥤ C i
inst✝ : (i : I) → (F i).OplaxMonoidal
X : (i : I) → D i
⊢ (λ_ ((Functor.pi F).obj X)).inv =
    (Functor.pi F).map (λ_ X).inv ≫
      (fun i ↦ Functor.OplaxMonoidal.δ (F i) ((𝟙_ ((i : I) → D i)) i) (X i)) ≫
        (fun i ↦ Functor.OplaxMonoidal.η (F i)) ▷ (Functor.pi F).obj X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Comma

Goal state before:
pp: A : Type u₁
inst✝⁷ : Category.{v₁, u₁} A
inst✝⁶ : Preadditive A
B : Type u₂
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Preadditive B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
inst✝² : Preadditive T
L : A ⥤ T
inst✝¹ : L.Additive
R : B ⥤ T
inst✝ : R.Additive
u v : Comma L R
α β : u ⟶ v
⊢ L.map (α.left + β.left) ≫ v.hom = u.hom ≫ R.map (α.right + β.right)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u✝ v✝ : Arrow V
f✝¹ g : u✝ ⟶ v✝
inst✝ : HasBinaryBiproducts V
u v : Arrow V
f✝ : u ⟶ v
X✝ Y✝ : RightFreyd V
f : X✝ ⟶ Y✝
⊢ Nonempty (ColimitCocone (parallelPair f 0))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁴ : (i : I) → Category.{v₁, u₁} (C i)
inst✝³ : (i : I) → MonoidalCategory (C i)
D : I → Type u₂
inst✝² : (i : I) → Category.{v₂, u₂} (D i)
inst✝¹ : (i : I) → MonoidalCategory (D i)
F : (i : I) → D i ⥤ C i
inst✝ : (i : I) → (F i).OplaxMonoidal
X : (i : I) → D i
⊢ (ρ_ ((Functor.pi F).obj X)).inv =
    (Functor.pi F).map (ρ_ X).inv ≫
      (fun i ↦ Functor.OplaxMonoidal.δ (F i) (X i) ((𝟙_ ((i : I) → D i)) i)) ≫
        (Functor.pi F).obj X ◁ fun i ↦ Functor.OplaxMonoidal.η (F i)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Comma

Goal state before:
pp: A : Type u₁
inst✝⁷ : Category.{v₁, u₁} A
inst✝⁶ : Preadditive A
B : Type u₂
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Preadditive B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
inst✝² : Preadditive T
L : A ⥤ T
inst✝¹ : L.Additive
R : B ⥤ T
inst✝ : R.Additive
u v : Comma L R
α β : u ⟶ v
⊢ L.map (α.left - β.left) ≫ v.hom = u.hom ≫ R.map (α.right - β.right)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
    f.hom ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)

pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| colimit.ι
    (parallelPair (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))
      ((α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))).hom ≫
        M'.X ◁ TensorBimod.actLeft N P))
    WalkingParallelPair.one

Tactic:
whisker_exchange

Goal state after:
case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| f.hom ▷ (N.X ⊗ P.X) ≫ M'.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)

case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| colimit.ι
    (parallelPair (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))
      ((α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))).hom ≫
        M'.X ◁ TensorBimod.actLeft N P))
    WalkingParallelPair.one
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
z : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)
i : ι
j : ↑(g ⁻¹' {i})
k : ↑(f ⁻¹' {i})
⊢ s i = s (f ↑k)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
z : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)
i : ι
j : ↑(g ⁻¹' {i})
k : ↑(f ⁻¹' {i})
⊢ s (g ↑j) = s i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
z : (i : ι) → Matrix (↑(g ⁻¹' {i})) (↑(f ⁻¹' {i})) (End (s i))
j : α
k : β
h : f j = g k
⊢ k ∈ g ⁻¹' {f j}

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
z : (i : ι) → Matrix (↑(g ⁻¹' {i})) (↑(f ⁻¹' {i})) (End (s i))
j : α
k : β
h : f j = g k
⊢ j ∈ f ⁻¹' {f j}

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
z : (i : ι) → Matrix (↑(g ⁻¹' {i})) (↑(f ⁻¹' {i})) (End (s i))
j : α
k : β
h : f j = g k
⊢ s (f j) = s (g k)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
z : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)
⊢ (fun z ↦ biproduct.matrix fun j k ↦ if h : f j = g k then z (f j) ⟨k, ⋯⟩ ⟨j, ⋯⟩ ≫ eqToHom ⋯ else 0)
      ((fun z i j k ↦ eqToHom ⋯ ≫ biproduct.components z ↑k ↑j ≫ eqToHom ⋯) z) =
    z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁶ : (i : I) → Category.{v₁, u₁} (C i)
inst✝⁵ : (i : I) → MonoidalCategory (C i)
inst✝⁴ : (i : I) → BraidedCategory (C i)
D : I → Type u₂
inst✝³ : (i : I) → Category.{v₂, u₂} (D i)
inst✝² : (i : I) → MonoidalCategory (D i)
inst✝¹ : (i : I) → BraidedCategory (D i)
F : (i : I) → D i ⥤ C i
inst✝ : (i : I) → (F i).LaxBraided
⊢ ∀ (X Y : (i : I) → D i),
    Functor.LaxMonoidal.μ (Functor.pi F) X Y ≫ (Functor.pi F).map (β_ X Y).hom =
      (β_ ((Functor.pi F).obj X) ((Functor.pi F).obj Y)).hom ≫ Functor.LaxMonoidal.μ (Functor.pi F) Y X

Tactic:
intros; ext i; exact Functor.LaxBraided.braided _ _

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Comma

Goal state before:
pp: A : Type u₁
inst✝⁷ : Category.{v₁, u₁} A
inst✝⁶ : Preadditive A
B : Type u₂
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Preadditive B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
inst✝² : Preadditive T
L : A ⥤ T
inst✝¹ : L.Additive
R : B ⥤ T
inst✝ : R.Additive
u v : Comma L R
x✝² x✝¹ x✝ : u ⟶ v
⊢ x✝² + x✝¹ + x✝ = x✝² + (x✝¹ + x✝)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁶ : (i : I) → Category.{v₁, u₁} (C i)
inst✝⁵ : (i : I) → MonoidalCategory (C i)
inst✝⁴ : (i : I) → BraidedCategory (C i)
D : I → Type u₂
inst✝³ : (i : I) → Category.{v₂, u₂} (D i)
inst✝² : (i : I) → MonoidalCategory (D i)
inst✝¹ : (i : I) → BraidedCategory (D i)
F : (i : I) → D i ⥤ C i
inst✝ : (i : I) → (F i).LaxBraided
⊢ ∀ (X Y : (i : I) → D i),
    Functor.LaxMonoidal.μ (Functor.pi F) X Y ≫ (Functor.pi F).map (β_ X Y).hom =
      (β_ ((Functor.pi F).obj X) ((Functor.pi F).obj Y)).hom ≫ Functor.LaxMonoidal.μ (Functor.pi F) Y X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Comma

Goal state before:
pp: A : Type u₁
inst✝⁷ : Category.{v₁, u₁} A
inst✝⁶ : Preadditive A
B : Type u₂
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Preadditive B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
inst✝² : Preadditive T
L : A ⥤ T
inst✝¹ : L.Additive
R : B ⥤ T
inst✝ : R.Additive
u v : Comma L R
x✝ : u ⟶ v
⊢ 0 + x✝ = x✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Comma

Goal state before:
pp: A : Type u₁
inst✝⁷ : Category.{v₁, u₁} A
inst✝⁶ : Preadditive A
B : Type u₂
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Preadditive B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
inst✝² : Preadditive T
L : A ⥤ T
inst✝¹ : L.Additive
R : B ⥤ T
inst✝ : R.Additive
u v : Comma L R
x✝ : u ⟶ v
⊢ x✝ + 0 = x✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Comma

Goal state before:
pp: A : Type u₁
inst✝⁷ : Category.{v₁, u₁} A
inst✝⁶ : Preadditive A
B : Type u₂
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Preadditive B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
inst✝² : Preadditive T
L : A ⥤ T
inst✝¹ : L.Additive
R : B ⥤ T
inst✝ : R.Additive
u v : Comma L R
n : ℕ
α : u ⟶ v
⊢ L.map (n • α.left) ≫ v.hom = u.hom ≫ R.map (n • α.right)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
z : (i : ι) → Matrix (↑(g ⁻¹' {i})) (↑(f ⁻¹' {i})) (End (s i))
⊢ (fun z i j k ↦ eqToHom ⋯ ≫ biproduct.components z ↑k ↑j ≫ eqToHom ⋯)
      ((fun z ↦ biproduct.matrix fun j k ↦ if h : f j = g k then z (f j) ⟨k, ⋯⟩ ⟨j, ⋯⟩ ≫ eqToHom ⋯ else 0) z) =
    z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝⁶ : (i : I) → Category.{v₁, u₁} (C i)
inst✝⁵ : (i : I) → MonoidalCategory (C i)
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : MonoidalCategory D
F G : D ⥤ ((i : I) → C i)
inst✝² : F.LaxMonoidal
inst✝¹ : G.LaxMonoidal
τ : (i : I) → F ⋙ eval C i ⟶ G ⋙ eval C i
inst✝ : ∀ (i : I), NatTrans.IsMonoidal (τ i)
⊢ Functor.LaxMonoidal.ε F ≫ (NatTrans.pi' τ).app (𝟙_ D) = Functor.LaxMonoidal.ε G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Comma

Goal state before:
pp: A : Type u₁
inst✝⁷ : Category.{v₁, u₁} A
inst✝⁶ : Preadditive A
B : Type u₂
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Preadditive B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
inst✝² : Preadditive T
L : A ⥤ T
inst✝¹ : L.Additive
R : B ⥤ T
inst✝ : R.Additive
u v : Comma L R
⊢ ∀ (x : u ⟶ v), 0 • x = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝³ : Preadditive C
inst✝² : HasFiniteBiproducts C
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
w z : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)
⊢ o.matrixDecomposition.toFun (w + z) = o.matrixDecomposition.toFun w + o.matrixDecomposition.toFun z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
⊢ whiskerRight f (N.tensorBimod P) =
    (M.associatorBimod N P).inv ≫ whiskerRight (whiskerRight f N) P ≫ (M'.associatorBimod N P).hom

Tactic:
by

Goal state after: