mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Localization.BousfieldTransfiniteComposition

Goal state before:
pp: case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
P : ObjectProperty C
J : Type w
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
inst✝¹ : OrderBot J
inst✝ : WellFoundedLT J
X Y : C
f : X ⟶ Y
x✝ : P.isLocal.transfiniteCompositionsOfShape J f
Z : C
hZ : P Z
hf : P.isLocal.TransfiniteCompositionOfShape J f
g₁ g₂ : Y ⟶ Z
h : f ≫ g₁ = f ≫ g₂
j : J
⊢ hf.incl.app j ≫ g₁ = hf.incl.app j ≫ g₂

Tactic:
induction j using SuccOrder.limitRecOn with

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Monoidal.Functor

Goal state before:
pp: case refine_2
C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Category.{v_3, u_3} E
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory D
inst✝⁵ : MonoidalCategory E
L : C ⥤ D
W : MorphismProperty C
inst✝⁴ : L.IsLocalization W
inst✝³ : L.Monoidal
F : D ⥤ E
G : C ⥤ E
inst✝² : G.Monoidal
inst✝¹ : W.ContainsIdentities
inst✝ : Lifting L W G F
e : L ⋙ F ≅ G := Lifting.iso L W G F
⊢ LaxMonoidal.ofBifunctor.leftMapₗ F =
    LaxMonoidal.ofBifunctor.topMapₗ (εIso G ≪≫ e.symm.app (𝟙_ C) ≪≫ F.mapIso (εIso L).symm).hom ≫
      (curriedTensorPreIsoPost L W F G).hom.app (𝟙_ D) ≫ LaxMonoidal.ofBifunctor.bottomMapₗ F

pp: case refine_3
C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Category.{v_3, u_3} E
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory D
inst✝⁵ : MonoidalCategory E
L : C ⥤ D
W : MorphismProperty C
inst✝⁴ : L.IsLocalization W
inst✝³ : L.Monoidal
F : D ⥤ E
G : C ⥤ E
inst✝² : G.Monoidal
inst✝¹ : W.ContainsIdentities
inst✝ : Lifting L W G F
e : L ⋙ F ≅ G := Lifting.iso L W G F
⊢ LaxMonoidal.ofBifunctor.leftMapᵣ F =
    LaxMonoidal.ofBifunctor.topMapᵣ (εIso G ≪≫ e.symm.app (𝟙_ C) ≪≫ F.mapIso (εIso L).symm).hom ≫
      ((flipFunctor D D E).map (curriedTensorPreIsoPost L W F G).hom).app (𝟙_ D) ≫ LaxMonoidal.ofBifunctor.bottomMapᵣ F

Tactic:
· refine natTrans_ext L W (fun X₂ ↦ ?_)
    have := NatTrans.congr_app ((curriedTensorPreIsoPost L W F G).hom.naturality (εIso L).inv)
      (L.obj X₂)
    dsimp [e] at this ⊢
    monoidal_simps
    simp [reassoc_of% this, curriedTensorPreIsoPost_hom_app_app, ← comp_whiskerRight_assoc,
      -comp_whiskerRight, tensorHom_def, ← whisker_exchange_assoc, ← map_comp]

Goal state after:
case refine_3
C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Category.{v_3, u_3} E
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory D
inst✝⁵ : MonoidalCategory E
L : C ⥤ D
W : MorphismProperty C
inst✝⁴ : L.IsLocalization W
inst✝³ : L.Monoidal
F : D ⥤ E
G : C ⥤ E
inst✝² : G.Monoidal
inst✝¹ : W.ContainsIdentities
inst✝ : Lifting L W G F
e : L ⋙ F ≅ G := Lifting.iso L W G F
⊢ LaxMonoidal.ofBifunctor.leftMapᵣ F =
    LaxMonoidal.ofBifunctor.topMapᵣ (εIso G ≪≫ e.symm.app (𝟙_ C) ≪≫ F.mapIso (εIso L).symm).hom ≫
      ((flipFunctor D D E).map (curriedTensorPreIsoPost L W F G).hom).app (𝟙_ D) ≫ LaxMonoidal.ofBifunctor.bottomMapᵣ F
Module: Mathlib.CategoryTheory.Monad.Monadicity

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₁, u₂} D
G : D ⥤ C
F : C ⥤ D
adj : F ⊣ G
inst✝² : HasCoequalizerOfIsSplitPair G
inst✝¹ : PreservesColimitOfIsSplitPair G
inst✝ : ReflectsColimitOfIsSplitPair G
X : adj.toMonad.Algebra
⊢ IsIso ((comparisonAdjunction adj).unit.app X).f

Tactic:
comparisonAdjunction_unit_f

Goal state after:
C : Type u₁
D : Type u₂
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₁, u₂} D
G : D ⥤ C
F : C ⥤ D
adj : F ⊣ G
inst✝² : HasCoequalizerOfIsSplitPair G
inst✝¹ : PreservesColimitOfIsSplitPair G
inst✝ : ReflectsColimitOfIsSplitPair G
X : adj.toMonad.Algebra
⊢ IsIso ((beckCoequalizer X).desc (unitCofork X))
Module: Mathlib.CategoryTheory.Monad.Monadicity

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₁, u₂} D
G : D ⥤ C
F : C ⥤ D
adj : F ⊣ G
inst✝² : HasReflexiveCoequalizers D
inst✝¹ : G.ReflectsIsomorphisms
inst✝ : PreservesColimitOfIsReflexivePair G
X : adj.toMonad.Algebra
⊢ IsIso ((comparisonAdjunction adj).unit.app X).f

Tactic:
comparisonAdjunction_unit_f

Goal state after:
C : Type u₁
D : Type u₂
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₁, u₂} D
G : D ⥤ C
F : C ⥤ D
adj : F ⊣ G
inst✝² : HasReflexiveCoequalizers D
inst✝¹ : G.ReflectsIsomorphisms
inst✝ : PreservesColimitOfIsReflexivePair G
X : adj.toMonad.Algebra
⊢ IsIso ((beckCoequalizer X).desc (unitCofork X))
Module: Mathlib.CategoryTheory.Monoidal.Mod

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
M : C
inst✝ : MonObj M
X : C
h₁ h₂ : ModObj M X
H : γ = γ
⊢ h₁ = h₂

Tactic:
by
  cases h₁
  cases h₂
  subst H
  rfl

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
R S T : Mon C
P : Bimod R S
Q : Bimod S T
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)
⊢ actLeft P Q ▷ T.X ≫ actRight P Q = (α_ R.X (X P Q) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q

Tactic:
refine (cancel_epi ((tensorLeft _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 ?_

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
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)
⊢ (tensorLeft R.X ⋙ tensorRight T.X).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
      actLeft P Q ▷ T.X ≫ actRight P Q =
    (tensorLeft R.X ⋙ tensorRight T.X).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
      (α_ R.X (X P Q) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q
Module: Mathlib.CategoryTheory.Monoidal.Closed.Ideal

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : CartesianMonoidalCategory C
inst✝³ : Reflective i
inst✝² : MonoidalClosed C
inst✝¹ : CartesianMonoidalCategory D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) (reflector i)
ir : reflector i ⊣ i := reflectorAdjunction i
L : C ⥤ D := reflector i
η : 𝟭 C ⟶ L ⋙ i := ir.unit
ε : i ⋙ L ⟶ 𝟭 D := ir.counit
B : D
A : C
⊢ i.essImage (A ⟹ i.obj B)

Tactic:
let q : i.obj (L.obj (A ⟹ i.obj B)) ⟶ A ⟹ i.obj B := by
    apply MonoidalClosed.curry (ir.homEquiv _ _ _)
    apply _ ≫ (ir.homEquiv _ _).symm ((ihom.ev A).app (i.obj B))
    exact prodComparison L A _ ≫ (_ ◁ (ε.app _)) ≫ inv (prodComparison _ _ _)

Goal state after:
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : CartesianMonoidalCategory C
inst✝³ : Reflective i
inst✝² : MonoidalClosed C
inst✝¹ : CartesianMonoidalCategory D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) (reflector i)
ir : reflector i ⊣ i := reflectorAdjunction i
L : C ⥤ D := reflector i
η : 𝟭 C ⟶ L ⋙ i := ir.unit
ε : i ⋙ L ⟶ 𝟭 D := ir.counit
B : D
A : C
q : i.obj (L.obj (A ⟹ i.obj B)) ⟶ A ⟹ i.obj B :=
  curry
    ((ir.homEquiv (A ⊗ i.obj (L.obj (A ⟹ i.obj B))) B)
      ((prodComparison L A (i.obj (L.obj (A ⟹ i.obj B))) ≫
          L.obj A ◁ ε.app (L.obj (A ⟹ i.obj B)) ≫ inv (prodComparison (reflector i) A (A ⟹ i.obj B))) ≫
        (ir.homEquiv ((ihom A ⋙ tensorLeft A).obj (i.obj B)) B).symm ((ihom.ev A).app (i.obj B))))
⊢ i.essImage (A ⟹ i.obj B)
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided

Goal state before:
pp: C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H F
inst✝⁵ : DayConvolution G (H ⊛ F)
inst✝⁴ : DayConvolution (G ⊛ H) F
inst✝³ : DayConvolution G F
inst✝² : DayConvolution (G ⊛ F) H
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution G (F ⊛ H)
z x y : C
⊢ (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)) ≫
          (unit G (H ⊛ F)).app (y, z ⊗ x) ≫ (G ⊛ H ⊛ F).map ((α_ y z x).inv ≫ (β_ (y ⊗ z) x).hom ≫ (α_ x y z).inv) =
    (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ (unit F H).app (x, z) ≫
          G.obj y ◁ (braiding F H).hom.app (x ⊗ z) ≫
            (unit G (H ⊛ F)).app (y, x ⊗ z) ≫ (G ⊛ H ⊛ F).map ((α_ y x z).inv ≫ (β_ y x).hom ▷ z)

Tactic:
have := unit_app_braiding_hom_app F H x z =≫ (H ⊛ F).map (β_ z x).inv

Goal state after:
C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H F
inst✝⁵ : DayConvolution G (H ⊛ F)
inst✝⁴ : DayConvolution (G ⊛ H) F
inst✝³ : DayConvolution G F
inst✝² : DayConvolution (G ⊛ F) H
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution G (F ⊛ H)
z x y : C
this :
  ((unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z)) ≫ (H ⊛ F).map (β_ z x).inv =
    ((β_ ((H, F).2.obj (z, x).2) ((H, F).1.obj (z, x).1)).hom ≫
        (unit H F).app (z, x) ≫ (H ⊛ F).map (β_ (z, x).1 (z, x).2).hom) ≫
      (H ⊛ F).map (β_ z x).inv
⊢ (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)) ≫
          (unit G (H ⊛ F)).app (y, z ⊗ x) ≫ (G ⊛ H ⊛ F).map ((α_ y z x).inv ≫ (β_ (y ⊗ z) x).hom ≫ (α_ x y z).inv) =
    (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ (unit F H).app (x, z) ≫
          G.obj y ◁ (braiding F H).hom.app (x ⊗ z) ≫
            (unit G (H ⊛ F)).app (y, x ⊗ z) ≫ (G ⊛ H ⊛ F).map ((α_ y x z).inv ≫ (β_ y x).hom ▷ z)
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided

Goal state before:
pp: C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H F
inst✝⁵ : DayConvolution G (H ⊛ F)
inst✝⁴ : DayConvolution (G ⊛ H) F
inst✝³ : DayConvolution G F
inst✝² : DayConvolution (G ⊛ F) H
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution G (F ⊛ H)
z x y : C
this :
  (unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z) ≫ (H ⊛ F).map (β_ z x).inv =
    (β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)
⊢ (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)) ≫
          (unit G (H ⊛ F)).app (y, z ⊗ x) ≫ (G ⊛ H ⊛ F).map ((α_ y z x).inv ≫ (β_ (y ⊗ z) x).hom ≫ (α_ x y z).inv) =
    (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ (unit F H).app (x, z) ≫
          G.obj y ◁ (braiding F H).hom.app (x ⊗ z) ≫
            (unit G (H ⊛ F)).app (y, x ⊗ z) ≫ (G ⊛ H ⊛ F).map ((α_ y x z).inv ≫ (β_ y x).hom ▷ z)

Tactic:
← this,

Goal state after:
C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H F
inst✝⁵ : DayConvolution G (H ⊛ F)
inst✝⁴ : DayConvolution (G ⊛ H) F
inst✝³ : DayConvolution G F
inst✝² : DayConvolution (G ⊛ F) H
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution G (F ⊛ H)
z x y : C
this :
  (unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z) ≫ (H ⊛ F).map (β_ z x).inv =
    (β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)
⊢ (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ ((unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z) ≫ (H ⊛ F).map (β_ z x).inv) ≫
          (unit G (H ⊛ F)).app (y, z ⊗ x) ≫ (G ⊛ H ⊛ F).map ((α_ y z x).inv ≫ (β_ (y ⊗ z) x).hom ≫ (α_ x y z).inv) =
    (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ (unit F H).app (x, z) ≫
          G.obj y ◁ (braiding F H).hom.app (x ⊗ z) ≫
            (unit G (H ⊛ F)).app (y, x ⊗ z) ≫ (G ⊛ H ⊛ F).map ((α_ y x z).inv ≫ (β_ y x).hom ▷ z)
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁴ : Category.{v₂, u₂} V
inst✝³ : MonoidalCategory C
inst✝² : MonoidalCategory V
inst✝¹ : MonoidalClosed V
F G H : C ⥤ V
inst✝ : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
x y : C × C
f : x ⟶ y
this :
  F.obj x.1 ◁ (ℌ.π x.2 x.1 ≫ (ihom (F.obj x.1)).map (G.map (f.1 ▷ x.2))) =
    F.obj x.1 ◁ (ℌ.π x.2 y.1 ≫ (MonoidalClosed.pre (F.map f.1)).app (G.obj (y.1 ⊗ x.2)))
⊢ (F.map f.1 ▷ H.obj x.2 ≫ F.obj y.1 ◁ H.map f.2) ≫ MonoidalClosed.uncurry (ℌ.π y.2 y.1) =
    MonoidalClosed.uncurry (ℌ.π x.2 x.1) ≫ G.map (f.1 ⊗ₘ f.2)

Tactic:
simp only [whiskerLeft_comp] at this

Goal state after:
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁴ : Category.{v₂, u₂} V
inst✝³ : MonoidalCategory C
inst✝² : MonoidalCategory V
inst✝¹ : MonoidalClosed V
F G H : C ⥤ V
inst✝ : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
x y : C × C
f : x ⟶ y
this :
  F.obj x.1 ◁ ℌ.π x.2 x.1 ≫ F.obj x.1 ◁ (ihom (F.obj x.1)).map (G.map (f.1 ▷ x.2)) =
    F.obj x.1 ◁ ℌ.π x.2 y.1 ≫ F.obj x.1 ◁ (MonoidalClosed.pre (F.map f.1)).app (G.obj (y.1 ⊗ x.2))
⊢ (F.map f.1 ▷ H.obj x.2 ≫ F.obj y.1 ◁ H.map f.2) ≫ MonoidalClosed.uncurry (ℌ.π y.2 y.1) =
    MonoidalClosed.uncurry (ℌ.π x.2 x.1) ≫ G.map (f.1 ⊗ₘ f.2)
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided

Goal state before:
pp: case e_a
C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H (F ⊛ G)
inst✝⁵ : DayConvolution H F
inst✝⁴ : DayConvolution (H ⊛ F) G
inst✝³ : DayConvolution H G
inst✝² : DayConvolution F (H ⊛ G)
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution (F ⊛ H) G
x y z : C
⊢ ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)) ▷ G.obj y ≫
      (unit (H ⊛ F) G).app (z ⊗ x, y) ≫ ((H ⊛ F) ⊛ G).map ((α_ z x y).hom ≫ (β_ z (x ⊗ y)).hom ≫ (α_ x y z).hom) =
    (unit F H).app (x, z) ▷ G.obj y ≫
      (braiding F H).hom.app (x ⊗ z) ▷ G.obj y ≫
        (unit (H ⊛ F) G).app (x ⊗ z, y) ≫ ((H ⊛ F) ⊛ G).map ((α_ x z y).hom ≫ x ◁ (β_ z y).hom)

Tactic:
have := unit_app_braiding_hom_app F H x z =≫ (H ⊛ F).map (β_ z x).inv

Goal state after:
case e_a
C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H (F ⊛ G)
inst✝⁵ : DayConvolution H F
inst✝⁴ : DayConvolution (H ⊛ F) G
inst✝³ : DayConvolution H G
inst✝² : DayConvolution F (H ⊛ G)
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution (F ⊛ H) G
x y z : C
this :
  ((unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z)) ≫ (H ⊛ F).map (β_ z x).inv =
    ((β_ ((H, F).2.obj (z, x).2) ((H, F).1.obj (z, x).1)).hom ≫
        (unit H F).app (z, x) ≫ (H ⊛ F).map (β_ (z, x).1 (z, x).2).hom) ≫
      (H ⊛ F).map (β_ z x).inv
⊢ ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)) ▷ G.obj y ≫
      (unit (H ⊛ F) G).app (z ⊗ x, y) ≫ ((H ⊛ F) ⊛ G).map ((α_ z x y).hom ≫ (β_ z (x ⊗ y)).hom ≫ (α_ x y z).hom) =
    (unit F H).app (x, z) ▷ G.obj y ≫
      (braiding F H).hom.app (x ⊗ z) ▷ G.obj y ≫
        (unit (H ⊛ F) G).app (x ⊗ z, y) ≫ ((H ⊛ F) ⊛ G).map ((α_ x z y).hom ≫ x ◁ (β_ z y).hom)
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided

Goal state before:
pp: case e_a
C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H (F ⊛ G)
inst✝⁵ : DayConvolution H F
inst✝⁴ : DayConvolution (H ⊛ F) G
inst✝³ : DayConvolution H G
inst✝² : DayConvolution F (H ⊛ G)
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution (F ⊛ H) G
x y z : C
this :
  (unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z) ≫ (H ⊛ F).map (β_ z x).inv =
    (β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)
⊢ ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)) ▷ G.obj y ≫
      (unit (H ⊛ F) G).app (z ⊗ x, y) ≫ ((H ⊛ F) ⊛ G).map ((α_ z x y).hom ≫ (β_ z (x ⊗ y)).hom ≫ (α_ x y z).hom) =
    (unit F H).app (x, z) ▷ G.obj y ≫
      (braiding F H).hom.app (x ⊗ z) ▷ G.obj y ≫
        (unit (H ⊛ F) G).app (x ⊗ z, y) ≫ ((H ⊛ F) ⊛ G).map ((α_ x z y).hom ≫ x ◁ (β_ z y).hom)

Tactic:
← this,

Goal state after:
case e_a
C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H (F ⊛ G)
inst✝⁵ : DayConvolution H F
inst✝⁴ : DayConvolution (H ⊛ F) G
inst✝³ : DayConvolution H G
inst✝² : DayConvolution F (H ⊛ G)
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution (F ⊛ H) G
x y z : C
this :
  (unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z) ≫ (H ⊛ F).map (β_ z x).inv =
    (β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)
⊢ ((unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z) ≫ (H ⊛ F).map (β_ z x).inv) ▷ G.obj y ≫
      (unit (H ⊛ F) G).app (z ⊗ x, y) ≫ ((H ⊛ F) ⊛ G).map ((α_ z x y).hom ≫ (β_ z (x ⊗ y)).hom ≫ (α_ x y z).hom) =
    (unit F H).app (x, z) ▷ G.obj y ≫
      (braiding F H).hom.app (x ⊗ z) ▷ G.obj y ≫
        (unit (H ⊛ F) G).app (x ⊗ z, y) ≫ ((H ⊛ F) ⊛ G).map ((α_ x z y).hom ≫ x ◁ (β_ z y).hom)
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Braided

Goal state before:

Tactic:
_ = 𝟙 Y ⊗≫ η_ X Y ▷ Y ⊗≫ ((β_ Y X).inv ▷ Y ⊗≫ Y ◁ (β_ X Y).hom) ≫ Y ◁ ε_ X Y ⊗≫ 𝟙 Y := by
      monoidal
    _ = 𝟙 Y ⊗≫ η_ X Y ▷ Y ⊗≫ (𝟙 ((X ⊗ Y) ⊗ Y) ⊗≫ X ◁ (β_ Y Y).hom ⊗≫ (β_ X Y).hom ▷ Y
        ⊗≫ Y ◁ (β_ Y X).inv ⊗≫ (β_ Y Y).inv ▷ X ⊗≫ 𝟙 (Y ⊗ Y ⊗ X)) ⊗≫ Y ◁ ε_ X Y ⊗≫ 𝟙 Y := by
      congr 3
      on_goal 2 => simp [monoidalComp]
      simp only [monoidalComp, MonoidalCoherence.assoc_iso, MonoidalCoherence.whiskerRight_iso,
        MonoidalCoherence.refl_iso, whiskerRightIso_refl, Iso.trans_refl,
        MonoidalCoherence.assoc'_iso, Iso.refl_trans, Iso.symm_hom, comp_id, id_comp]
      iterate 2 rw [← IsIso.eq_inv_comp]
      repeat rw [← assoc]
      iterate 4 rw [← IsIso.comp_inv_eq]
      simpa using (yang_baxter Y X Y).symm
    _ = 𝟙 Y ⊗≫ (η_ X Y ▷ Y ≫ (β_ (X ⊗ Y) Y).hom) ⊗≫ ((β_ Y (Y ⊗ X)).inv ≫ Y ◁ ε_ X Y) ⊗≫ 𝟙 Y := by
      simp [monoidalComp, braiding_tensor_left_hom, braiding_tensor_right_inv]
    _ = _ := by
      rw [braiding_naturality_left, ← braiding_inv_naturality_left]
      simp [monoidalComp]

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

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁵ : Category.{v₂, u₂} V
inst✝⁴ : MonoidalCategory C
inst✝³ : MonoidalCategory V
U : C ⥤ V
inst✝² : DayConvolutionUnit U
F : C ⥤ V
inst✝¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ : DayConvolution U F
x : C
⊢ (leftUnitor U F).inv.app x =
    (λ_ (F.obj x)).inv ≫ can ▷ F.obj x ≫ (DayConvolution.unit U F).app (𝟙_ C, x) ≫ (U ⊛ F).map (λ_ x).hom

Tactic:
dsimp [leftUnitor, Coyoneda.fullyFaithful, corepresentableByLeft,
    leftUnitorCorepresentingIso, Functor.CorepresentableBy.ofIso,
    Functor.corepresentableByEquiv]

Goal state after:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁵ : Category.{v₂, u₂} V
inst✝⁴ : MonoidalCategory C
inst✝³ : MonoidalCategory V
U : C ⥤ V
inst✝² : DayConvolutionUnit U
F : C ⥤ V
inst✝¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ : DayConvolution U F
x : C
⊢ (λ_ (F.obj x)).inv ≫
      ((ConcreteCategory.hom
                (𝟙
                  (prod.leftInverseUnitor C ⋙ fromPUnit (𝟙_ V) ⊠ F ⟶
                    prod.leftInverseUnitor C ⋙ (fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ U ⊛ F)))
              ((prod.leftUnitorEquivalence C).congrLeft.fullyFaithfulFunctor.homEquiv
                (((U ⊠ F).homEquivOfIsLeftKanExtension (extensionUnitLeft U (φ U) F) (tensor C ⋙ U ⊛ F))
                  (((U ⊛ F).homEquivOfIsLeftKanExtension (DayConvolution.unit U F) (U ⊛ F)) (𝟙 (U ⊛ F)))))).app
          x ≫
        (U ⊛ F).map (λ_ x).hom =
    (λ_ (F.obj x)).inv ≫ can ▷ F.obj x ≫ (DayConvolution.unit U F).app (𝟙_ C, x) ≫ (U ⊛ F).map (λ_ x).hom