mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution

Goal state before:
pp: C✝ : Type u₁
inst✝¹⁶ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹⁵ : Category.{v₂, u₂} V✝
inst✝¹⁴ : MonoidalCategory C✝
inst✝¹³ : MonoidalCategory V✝
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
V : Type u₂
inst✝¹¹ : Category.{v₂, u₂} V
inst✝¹⁰ : MonoidalCategory C
inst✝⁹ : MonoidalCategory V
D : Type u₃
inst✝⁸ : Category.{v₃, u₃} D
inst✝⁷ : MonoidalCategoryStruct D
inst✝⁶ : LawfulDayConvolutionMonoidalCategoryStruct C V D
inst✝⁵ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝⁴ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝³ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
inst✝² :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow ((𝟭 C).prod (Functor.fromPUnit (𝟙_ C))) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C × C), Limits.PreservesColimitsOfShape (CostructuredArrow ((tensor C).prod (𝟭 C)) d) (tensorRight v)
x y : D
fst✝ snd✝ : C
⊢ ((corepresentableBy ((ι C V D).obj x) ((ι C V D).obj y)).homEquiv
          (map (𝟙 ((ι C V D).obj x)) (𝟙 ((ι C V D).obj y)))).app
      (fst✝, snd✝) =
    ((corepresentableBy ((ι C V D).obj x) ((ι C V D).obj y)).homEquiv (𝟙 ((ι C V D).obj (x ⊗ y)))).app (fst✝, snd✝)

Tactic:
simp only [externalProductBifunctor_obj_obj, Functor.comp_obj, tensor_obj,
        corepresentableBy_homEquiv_apply_app, unit_app_map_app, NatTrans.id_app,
        tensorHom_id, id_whiskerRight, Category.id_comp]

Goal state after:
C✝ : Type u₁
inst✝¹⁶ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹⁵ : Category.{v₂, u₂} V✝
inst✝¹⁴ : MonoidalCategory C✝
inst✝¹³ : MonoidalCategory V✝
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
V : Type u₂
inst✝¹¹ : Category.{v₂, u₂} V
inst✝¹⁰ : MonoidalCategory C
inst✝⁹ : MonoidalCategory V
D : Type u₃
inst✝⁸ : Category.{v₃, u₃} D
inst✝⁷ : MonoidalCategoryStruct D
inst✝⁶ : LawfulDayConvolutionMonoidalCategoryStruct C V D
inst✝⁵ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝⁴ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝³ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
inst✝² :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow ((𝟭 C).prod (Functor.fromPUnit (𝟙_ C))) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C × C), Limits.PreservesColimitsOfShape (CostructuredArrow ((tensor C).prod (𝟭 C)) d) (tensorRight v)
x y : D
fst✝ snd✝ : C
⊢ (unit ((ι C V D).obj x) ((ι C V D).obj y)).app (fst✝, snd✝) =
    (unit ((ι C V D).obj x) ((ι C V D).obj y)).app (fst✝, snd✝) ≫ 𝟙 (((ι C V D).obj (x ⊗ y)).obj (fst✝ ⊗ snd✝))
Module: Mathlib.CategoryTheory.MorphismProperty.LocalClosure

Goal state before:
pp: case comp
C : Type u
inst✝³ : Category.{v, u} C
K : Precoverage C
P Q : MorphismProperty C
X✝¹ Y✝¹ : C
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : K.IsStableUnderBaseChange
inst✝ : HasPullbacks C
Y X Z : C
f : X ⟶ Z
X✝ Y✝ : C
f' : X✝ ⟶ Y✝
hf' : sourceLocalClosure K P f'
R : Presieve X✝
hR : R ∈ K.coverings X✝
U✝ : C
g' : U✝ ⟶ X✝
hg' : R g'
ih : ∀ (W : C) (g : Y ⟶ Y✝) (fst : W ⟶ X✝) (snd : W ⟶ Y), IsPullback fst snd f' g → sourceLocalClosure K P snd
W : C
g : Y ⟶ Y✝
fst : W ⟶ U✝
snd : W ⟶ Y
h✝ : IsPullback fst snd (g' ≫ f') g
u : W ⟶ pullback g f' := pullback.lift snd (fst ≫ g') ⋯
h : IsPullback fst (u ≫ pullback.fst g f') (g' ≫ f') g
this : snd = u ≫ pullback.fst g f'
e : W ≅ pullback g' (pullback.snd g f') := ⋯.isoPullback
⊢ sourceLocalClosure K P (u ≫ pullback.fst g f')

Tactic:
← (sourceLocalClosure K P).cancel_left_of_respectsIso e.inv,

Goal state after:
case comp
C : Type u
inst✝³ : Category.{v, u} C
K : Precoverage C
P Q : MorphismProperty C
X✝¹ Y✝¹ : C
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : K.IsStableUnderBaseChange
inst✝ : HasPullbacks C
Y X Z : C
f : X ⟶ Z
X✝ Y✝ : C
f' : X✝ ⟶ Y✝
hf' : sourceLocalClosure K P f'
R : Presieve X✝
hR : R ∈ K.coverings X✝
U✝ : C
g' : U✝ ⟶ X✝
hg' : R g'
ih : ∀ (W : C) (g : Y ⟶ Y✝) (fst : W ⟶ X✝) (snd : W ⟶ Y), IsPullback fst snd f' g → sourceLocalClosure K P snd
W : C
g : Y ⟶ Y✝
fst : W ⟶ U✝
snd : W ⟶ Y
h✝ : IsPullback fst snd (g' ≫ f') g
u : W ⟶ pullback g f' := pullback.lift snd (fst ≫ g') ⋯
h : IsPullback fst (u ≫ pullback.fst g f') (g' ≫ f') g
this : snd = u ≫ pullback.fst g f'
e : W ≅ pullback g' (pullback.snd g f') := ⋯.isoPullback
⊢ sourceLocalClosure K P (e.inv ≫ u ≫ pullback.fst g f')
Module: Mathlib.CategoryTheory.MorphismProperty.LocalClosure

Goal state before:
pp: case comp
C : Type u
inst✝³ : Category.{v, u} C
K : Precoverage C
P Q : MorphismProperty C
X✝¹ Y✝¹ : C
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : K.IsStableUnderBaseChange
inst✝ : HasPullbacks C
Y X Z : C
f : X ⟶ Z
X✝ Y✝ : C
f' : X✝ ⟶ Y✝
hf' : sourceLocalClosure K P f'
R : Presieve X✝
hR : R ∈ K.coverings X✝
U✝ : C
g' : U✝ ⟶ X✝
hg' : R g'
ih : ∀ (W : C) (g : Y ⟶ Y✝) (fst : W ⟶ X✝) (snd : W ⟶ Y), IsPullback fst snd f' g → sourceLocalClosure K P snd
W : C
g : Y ⟶ Y✝
fst : W ⟶ U✝
snd : W ⟶ Y
h : IsPullback fst snd (g' ≫ f') g
⊢ sourceLocalClosure K P snd

Tactic:
let u : W ⟶ pullback g f' := pullback.lift snd (fst ≫ g') (by simp [h.w.symm])
      have : snd = u ≫ pullback.fst g f' := by simp [u]
      rw [this] at h ⊢
      let e : W ≅ pullback g' (pullback.snd g f') :=
        IsPullback.isoPullback (.of_bot h (by simp [u]) (.flip <| .of_hasPullback _ _))
      rw [← (sourceLocalClosure K P).cancel_left_of_respectsIso e.inv, ← Category.assoc]
      refine .comp _ (ih _ _ _ _ (.flip (.of_hasPullback _ _))) _
        (K.pullbackArrows_mem (pullback.snd _ _) hR) _ ?_
      simpa [e, u] using .mk _ _ hg'

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Ind

Goal state before:
pp: case refine_2.refine_5
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.underObj.ind (CategoryTheory.Under.mk f)
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
pres : ColimitPresentation J (CategoryTheory.Under.mk f)
hpres : ∀ (i : J), P.underObj (pres.diag.obj i)
j : J
⊢ { app := fun j ↦ (pres.diag.obj j).hom, naturality := ⋯ }.app j ≫
      (Functor.whiskerRight pres.ι (CategoryTheory.Under.forget X)).app j =
    f

Tactic:
· simp

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

Goal state before:
pp: case a.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 : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| 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 ≫
    coequalizer.desc (AssociatorBimod.invAux M N P') ⋯

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 : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| (α_ M.X N.X P.X).hom

pp: case a.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 : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| M.X ◁
    (N.X ◁ f.hom ≫
      colimit.ι (parallelPair (N.actRight ▷ P'.X) ((α_ N.X Y.X P'.X).hom ≫ N.X ◁ P'.actLeft)) WalkingParallelPair.one)

Tactic:
rw [coequalizer.π_desc]

Goal state after:
case a.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 : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| AssociatorBimod.invAux M N P'

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 : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| (α_ M.X N.X P.X).hom

case a.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 : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| M.X ◁
    (N.X ◁ f.hom ≫
      colimit.ι (parallelPair (N.actRight ▷ P'.X) ((α_ N.X Y.X P'.X).hom ≫ N.X ◁ P'.actLeft)) WalkingParallelPair.one)
Module: Mathlib.CategoryTheory.MorphismProperty.LocalClosure

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
K : Precoverage C
P : MorphismProperty C
inst✝⁵ : P.RespectsIso
inst✝⁴ : P.RespectsLeft K.morphismProperty
inst✝³ : K.HasIsos
inst✝² : K.IsStableUnderBaseChange
inst✝¹ : K.IsStableUnderComposition
inst✝ : K.HasPullbacks
X Y : C
f : X ⟶ Y
R : Presieve X
hR : R ∈ K.coverings X
h : ∀ (U : C) (g : U ⟶ X), R g → P (g ≫ f)
⊢ ∀ (U : C) (g : U ⟶ X), R g → sourceLocalClosure K P (g ≫ f)

Tactic:
by grind

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.LocalEpi

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : ObjectProperty C
⊢ P.localEpi.IsStableUnderCobaseChange

Tactic:
refine .mk' fun X Y S f g _ hg T hT u v huv ↦ ?_
  refine pushout.hom_ext (hg hT ?_) huv
  simp [pushout.condition_assoc, huv]

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.NoetherianObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : C
h : ∀ (f : ℕ → Subobject X), ¬StrictMono f
⊢ isNoetherianObject.Is X

Tactic:
rw [ObjectProperty.is_iff, isNoetherianObject, WellFoundedGT,
    isWellFounded_iff, RelEmbedding.wellFounded_iff_isEmpty]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
X : C
h : ∀ (f : ℕ → Subobject X), ¬StrictMono f
⊢ IsEmpty ((fun x1 x2 ↦ x1 > x2) ↪r fun x1 x2 ↦ x2 < x1)
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a.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)
X Y : Mon C
M N : Bimod X Y
f : M ⟶ N
| coequalizer.π (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ μ) ≫ coequalizer.desc M.actRight ⋯

pp: case a.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)
X Y : Mon C
M N : Bimod X Y
f : M ⟶ N
| f.hom

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)
X Y : Mon C
M N : Bimod X Y
f : M ⟶ N
| RightUnitorBimod.inv N

Tactic:
rw [coequalizer.π_desc]

Goal state after:
case a.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)
X Y : Mon C
M N : Bimod X Y
f : M ⟶ N
| M.actRight

case a.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)
X Y : Mon C
M N : Bimod X Y
f : M ⟶ N
| f.hom

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)
X Y : Mon C
M N : Bimod X Y
f : M ⟶ N
| RightUnitorBimod.inv N
Module: Mathlib.CategoryTheory.ObjectProperty.ClosureShift

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
P : ObjectProperty C
A : Type u_2
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : P.IsStableUnderShift A
a : A
X Y : C
hY : P Y
i : X ⟶ Y
r : Y ⟶ X
w : i ≫ r = 𝟙 X
⊢ (shiftFunctor C a).map i ≫ (shiftFunctor C a).map r = 𝟙 ((shiftFunctor C a).obj X)

Tactic:
by grind

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

Goal state before:
pp: case a.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 (TensorBimod.actRight M' N ▷ P.X)
        ((α_ (coequalizer (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft)) Y.X P.X).hom ≫
          coequalizer (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) ◁ P.actLeft))
      WalkingParallelPair.one ≫
    coequalizer.desc (AssociatorBimod.homAux M' N P) ⋯

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
| (α_ M.X N.X P.X).inv

pp: case a.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 ≫
      colimit.ι (parallelPair (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft)) WalkingParallelPair.one) ▷
    P.X

Tactic:
rw [coequalizer.π_desc]

Goal state after:
case a.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
| AssociatorBimod.homAux M' N P

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
| (α_ M.X N.X P.X).inv

case a.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 ≫
      colimit.ι (parallelPair (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft)) WalkingParallelPair.one) ▷
    P.X
Module: Mathlib.CategoryTheory.Preadditive.EndoFunctor

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : F.Additive
A₁ A₂ : Coalgebra F
n : ℕ
α : A₁ ⟶ A₂
⊢ n • A₁.str ≫ F.map α.f = (n • α.f) ≫ A₂.str

Tactic:
Endofunctor.Coalgebra.Hom.h,

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : F.Additive
A₁ A₂ : Coalgebra F
n : ℕ
α : A₁ ⟶ A₂
⊢ n • α.f ≫ A₂.str = (n • α.f) ≫ A₂.str
Module: Mathlib.CategoryTheory.Preadditive.EndoFunctor

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : F.Additive
A₁ A₂ : Coalgebra F
r : ℤ
α : A₁ ⟶ A₂
⊢ r • A₁.str ≫ F.map α.f = (r • α.f) ≫ A₂.str

Tactic:
Endofunctor.Coalgebra.Hom.h,

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : F.Additive
A₁ A₂ : Coalgebra F
r : ℤ
α : A₁ ⟶ A₂
⊢ r • α.f ≫ A₂.str = (r • α.f) ≫ A₂.str
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 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 : 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 intros; ext i; exact Functor.LaxBraided.braided _ _

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

Goal state before:
pp: case pos
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 : β
k : α
w : j ∈ g ⁻¹' {f k}
h : f k = g j
⊢ (z (f k) ⟨j, ⋯⟩ ⟨k, ⋯⟩ ≫ eqToHom ⋯) ≫ eqToHom ⋯ = z (f k) ⟨j, w⟩ ⟨k, ⋯⟩

pp: case neg
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 : β
k : α
w : j ∈ g ⁻¹' {f k}
h : ¬f k = g j
⊢ 0 ≫ eqToHom ⋯ = z (f k) ⟨j, w⟩ ⟨k, ⋯⟩

Tactic:
· simp

Goal state after:
case neg
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 : β
k : α
w : j ∈ g ⁻¹' {f k}
h : ¬f k = g j
⊢ 0 ≫ eqToHom ⋯ = z (f k) ⟨j, w⟩ ⟨k, ⋯⟩
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: case w
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)
w : (⨁ fun b ↦ s (g b)) ⟶ ⨁ fun c ↦ s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∀ (x : β) (hx : x ∈ g ⁻¹' {h c}),
    eqToHom ⋯ ≫
        biproduct.ι (fun a ↦ s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b ↦ s (g b)) x ≫ biproduct.ι (fun b ↦ s (g b)) x) ≫ w) ≫
              biproduct.π (fun b ↦ s (h b)) c =
      eqToHom ⋯ ≫
        biproduct.ι (fun a ↦ s (f a)) a ≫
          z ≫
            biproduct.π (fun b ↦ s (g b)) ↑⟨x, hx⟩ ≫
              biproduct.ι (fun a ↦ s (g a)) ↑⟨x, hx⟩ ≫ w ≫ biproduct.π (fun b ↦ s (h b)) c

pp: case w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)
w : (⨁ fun b ↦ s (g b)) ⟶ ⨁ fun c ↦ s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∀ x ∉ g ⁻¹' {h c},
    eqToHom ⋯ ≫
        biproduct.ι (fun a ↦ s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b ↦ s (g b)) x ≫ biproduct.ι (fun b ↦ s (g b)) x) ≫ w) ≫
              biproduct.π (fun b ↦ s (h b)) c =
      0

Tactic:
· simp

Goal state after:
case w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)
w : (⨁ fun b ↦ s (g b)) ⟶ ⨁ fun c ↦ s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∀ x ∉ g ⁻¹' {h c},
    eqToHom ⋯ ≫
        biproduct.ι (fun a ↦ s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b ↦ s (g b)) x ≫ biproduct.ι (fun b ↦ s (g b)) x) ≫ w) ≫
              biproduct.π (fun b ↦ s (h b)) c =
      0
Module: Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
inst✝² : ∀ (i : ι), InvariantBasisNumber (End (s i))
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
i : (⨁ fun a ↦ s (f a)) ≅ ⨁ fun b ↦ s (g b)
c : ι
⊢ ↑(f ⁻¹' {c}) ≃ ↑(g ⁻¹' {c})

Tactic:
apply Nonempty.some

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
inst✝² : ∀ (i : ι), InvariantBasisNumber (End (s i))
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
i : (⨁ fun a ↦ s (f a)) ≅ ⨁ fun b ↦ s (g b)
c : ι
⊢ Nonempty (↑(f ⁻¹' {c}) ≃ ↑(g ⁻¹' {c}))
Module: Mathlib.CategoryTheory.Preadditive.OfBiproducts

Goal state before:
pp: case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.fst = (f ≫ biprod.inr) ≫ biprod.fst

pp: case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.snd = (f ≫ biprod.inr) ≫ biprod.snd

Tactic:
· simp

Goal state after:
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.snd = (f ≫ biprod.inr) ≫ biprod.snd