mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J✝ : Type u_1
I✝ : J✝ → Type u_2
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)
D✝ : J✝ ⥤ C
P✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)
J : Type w
I : J → Type w
inst✝³ : SmallCategory J
inst✝² : (j : J) → SmallCategory (I j)
D : J ⥤ C
P✝ : (j : J) → ColimitPresentation (I j) (D.obj j)
X : C
P : ColimitPresentation J X
Q : (j : J) → ColimitPresentation (I j) (P.diag.obj j)
inst✝¹ : ∀ (j : J), IsFiltered (I j)
inst✝ : ∀ (j : J) (i : I j), IsFinitelyPresentable ((Q j).diag.obj i)
k l : Total Q
u : k ⟶ l
⊢ { obj := fun k ↦ (Q k.fst).diag.obj k.snd, map := fun {k l} f ↦ f.hom, map_id := ⋯, map_comp := ⋯ }.map u ≫
      (Q l.fst).ι.app l.snd ≫ P.ι.app l.fst =
    ((Q k.fst).ι.app k.snd ≫ P.ι.app k.fst) ≫ ((Functor.const (Total Q)).obj X).map u

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
P : ObjectProperty C
ι : Type u_1
inst✝³ : Small.{w, u_1} ι
inst✝² : P.IsClosedUnderLimitsOfShape (Discrete ι)
inst✝¹ : HasProductsOfShape ι C
inst✝ : IsIPCOfShape.{w, u_1, v, u} ι C
⊢ P.ind.IsClosedUnderLimitsOfShape (Discrete ι)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J✝ : Type u_1
I✝ : J✝ → Type u_2
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)
D✝ : J✝ ⥤ C
P✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)
J : Type w
I : J → Type w
inst✝³ : SmallCategory J
inst✝² : (j : J) → SmallCategory (I j)
D : J ⥤ C
P✝ : (j : J) → ColimitPresentation (I j) (D.obj j)
X : C
P : ColimitPresentation J X
Q : (j : J) → ColimitPresentation (I j) (P.diag.obj j)
inst✝¹ : ∀ (j : J), IsFiltered (I j)
inst✝ : ∀ (j : J) (i : I j), IsFinitelyPresentable ((Q j).diag.obj i)
c : Cocone { obj := fun k ↦ (Q k.fst).diag.obj k.snd, map := fun {k l} f ↦ f.hom, map_id := ⋯, map_comp := ⋯ }
j : J
i i' : I j
u : i ⟶ i'
⊢ (Q j).diag.map u ≫ c.ι.app ⟨j, i'⟩ = c.ι.app ⟨j, i⟩ ≫ ((Functor.const (I j)).obj c.pt).map u

Tactic:
by

Goal state after:
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
⊢ 𝟙 x ⊗ₘ 𝟙 y = 𝟙 (x ⊗ y)

Tactic:
by

Goal state after:
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 y₁ y₂ : D
f : y₁ ⟶ y₂
⊢ 𝟙 x ⊗ₘ f = x ◁ f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J✝ : Type u_1
I✝ : J✝ → Type u_2
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)
D✝ : J✝ ⥤ C
P✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)
J : Type w
I : J → Type w
inst✝³ : SmallCategory J
inst✝² : (j : J) → SmallCategory (I j)
D : J ⥤ C
P✝ : (j : J) → ColimitPresentation (I j) (D.obj j)
X : C
P : ColimitPresentation J X
Q : (j : J) → ColimitPresentation (I j) (P.diag.obj j)
inst✝¹ : ∀ (j : J), IsFiltered (I j)
inst✝ : ∀ (j : J) (i : I j), IsFinitelyPresentable ((Q j).diag.obj i)
c : Cocone { obj := fun k ↦ (Q k.fst).diag.obj k.snd, map := fun {k l} f ↦ f.hom, map_id := ⋯, map_comp := ⋯ }
j j' : J
u : j ⟶ j'
⊢ P.diag.map u ≫ (Q j').isColimit.desc { pt := c.pt, ι := { app := fun i ↦ c.ι.app ⟨j', i⟩, naturality := ⋯ } } =
    (Q j).isColimit.desc { pt := c.pt, ι := { app := fun i ↦ c.ι.app ⟨j, i⟩, naturality := ⋯ } } ≫
      ((Functor.const J).obj c.pt).map u

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J✝ : Type u_1
I✝ : J✝ → Type u_2
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)
D✝ : J✝ ⥤ C
P✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)
J : Type w
I : J → Type w
inst✝³ : SmallCategory J
inst✝² : (j : J) → SmallCategory (I j)
D : J ⥤ C
P✝ : (j : J) → ColimitPresentation (I j) (D.obj j)
X : C
P : ColimitPresentation J X
Q : (j : J) → ColimitPresentation (I j) (P.diag.obj j)
inst✝¹ : ∀ (j : J), IsFiltered (I j)
inst✝ : ∀ (j : J) (i : I j), IsFinitelyPresentable ((Q j).diag.obj i)
c : Cocone { obj := fun k ↦ (Q k.fst).diag.obj k.snd, map := fun {k l} f ↦ f.hom, map_id := ⋯, map_comp := ⋯ }
x✝ : Total Q
j : J
i : I j
⊢ { pt := X, ι := { app := fun k ↦ (Q k.fst).ι.app k.snd ≫ P.ι.app k.fst, naturality := ⋯ } }.ι.app ⟨j, i⟩ ≫
      P.isColimit.desc
        { pt := c.pt,
          ι :=
            {
              app := fun j ↦
                (Q j).isColimit.desc { pt := c.pt, ι := { app := fun i ↦ c.ι.app ⟨j, i⟩, naturality := ⋯ } },
              naturality := ⋯ } } =
    c.ι.app ⟨j, i⟩

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
⊢ P ≤ P.ind

Tactic:
by

Goal state after:
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 y₁ : D
y₂ : x ⟶ y₁
f : D
⊢ y₂ ⊗ₘ 𝟙 f = y₂ ▷ f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.ind f
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
j : J
⊢ ((Under.lift D t).obj j).hom ≫ s.app j = (((Functor.const J).obj (CategoryTheory.Under.mk f)).obj j).hom

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.ind f
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
this : Nonempty J
⊢ ∀ (j : J), t.app j ≫ { pt := Y, ι := s }.ι.app j = f

Tactic:
by

Goal state after:
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)
X₁✝ Y₁✝ Z₁✝ X₂✝ Y₂✝ Z₂✝ : D
x✝³ : X₁✝ ⟶ Y₁✝
x✝² : X₂✝ ⟶ Y₂✝
x✝¹ : Y₁✝ ⟶ Z₁✝
x✝ : Y₂✝ ⟶ Z₂✝
⊢ (x✝³ ⊗ₘ x✝²) ≫ (x✝¹ ⊗ₘ x✝) = x✝³ ≫ x✝¹ ⊗ₘ x✝² ≫ x✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J✝ : Type u_1
I✝ : J✝ → Type u_2
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)
D✝ : J✝ ⥤ C
P✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)
J : Type w
I : J → Type w
inst✝³ : SmallCategory J
inst✝² : (j : J) → SmallCategory (I j)
D : J ⥤ C
P✝ : (j : J) → ColimitPresentation (I j) (D.obj j)
X : C
P : ColimitPresentation J X
Q : (j : J) → ColimitPresentation (I j) (P.diag.obj j)
inst✝¹ : ∀ (j : J), IsFiltered (I j)
inst✝ : ∀ (j : J) (i : I j), IsFinitelyPresentable ((Q j).diag.obj i)
c : Cocone { obj := fun k ↦ (Q k.fst).diag.obj k.snd, map := fun {k l} f ↦ f.hom, map_id := ⋯, map_comp := ⋯ }
m : { pt := X, ι := { app := fun k ↦ (Q k.fst).ι.app k.snd ≫ P.ι.app k.fst, naturality := ⋯ } }.pt ⟶ c.pt
hm :
  ∀ (j : Total Q),
    { pt := X, ι := { app := fun k ↦ (Q k.fst).ι.app k.snd ≫ P.ι.app k.fst, naturality := ⋯ } }.ι.app j ≫ m = c.ι.app j
⊢ m =
    P.isColimit.desc
      { pt := c.pt,
        ι :=
          { app := fun j ↦ (Q j).isColimit.desc { pt := c.pt, ι := { app := fun i ↦ c.ι.app ⟨j, i⟩, naturality := ⋯ } },
            naturality := ⋯ } }

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
K : Precoverage C
P Q : MorphismProperty C
h : P ≤ Q
inst✝ : Q.IsLocalAtSource K
⊢ sourceLocalClosure K P ≤ Q

Tactic:
by

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

Goal state before:
pp: 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
⊢ snd ≫ g = (fst ≫ g') ≫ f'

Tactic:
by

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

Goal state before:
pp: 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') ⋯
⊢ snd = u ≫ pullback.fst g f'

Tactic:
by

Goal state after:
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)
X₁✝ X₂✝ X₃✝ Y₁✝ Y₂✝ Y₃✝ : D
f₁ : X₁✝ ⟶ Y₁✝
f₂ : X₂✝ ⟶ Y₂✝
f₃ : X₃✝ ⟶ Y₃✝
⊢ ((f₁ ⊗ₘ f₂) ⊗ₘ f₃) ≫ (α_ Y₁✝ Y₂✝ Y₃✝).hom = (α_ X₁✝ X₂✝ X₃✝).hom ≫ (f₁ ⊗ₘ f₂ ⊗ₘ f₃)

Tactic:
by

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

Goal state before:
pp: 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'
⊢ fst ≫ g' = u ≫ pullback.snd g f'

Tactic:
by

Goal state after:
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
f : X✝ ⟶ Y✝
⊢ (𝟙 (𝟙_ D) ⊗ₘ f) ≫ (λ_ Y✝).hom = (λ_ X✝).hom ≫ f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
K : Precoverage C
P Q : MorphismProperty C
X✝ Y✝ : C
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : K.IsStableUnderBaseChange
inst✝ : HasPullbacks C
Y X W Z : C
g : Y ⟶ Z
f : X ⟶ Z
fst : W ⟶ X
snd : W ⟶ Y
h : IsPullback fst snd f g
hf : sourceLocalClosure K P f
⊢ sourceLocalClosure K P snd

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
⊢ P.ind f ↔ P.underObj.ind (CategoryTheory.Under.mk f)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X : C
⊢ P.ind.underObj = P.underObj.ind

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : P.RespectsLeft Q
X Y Z : C
i : X ⟶ Y
hi : Q i
f : Y ⟶ Z
x✝ : P.ind f
J : Type u_1
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj Y ⟶ D
s : D ⟶ (Functor.const J).obj Z
hs : IsColimit { pt := Z, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
j : J
⊢ ((Functor.const J).map i ≫ t).app j ≫ s.app j = i ≫ f

Tactic:
by

Goal state after:
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
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : P f
U : C
g : U ⟶ X✝
x✝ : Presieve.singleton (𝟙 X✝) g
⊢ P (𝟙 X✝ ≫ f)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : P.RespectsLeft Q
X Y Z : C
i : X ⟶ Y
hi : Q i
f : Y ⟶ Z
x✝ : P.ind f
J : Type u_1
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj Y ⟶ D
s : D ⟶ (Functor.const J).obj Z
hs : IsColimit { pt := Z, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ P.ind (i ≫ f)

Tactic:
refine ⟨J, ‹_›, ‹_›, D, (Functor.const J).map i ≫ t, s, hs, fun j ↦ ⟨?_, by simp [hst]⟩⟩

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : P.RespectsLeft Q
X Y Z : C
i : X ⟶ Y
hi : Q i
f : Y ⟶ Z
x✝ : P.ind f
J : Type u_1
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj Y ⟶ D
s : D ⟶ (Functor.const J).obj Z
hs : IsColimit { pt := Z, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
j : J
⊢ P (((Functor.const J).map i ≫ t).app j)
Module: Mathlib.CategoryTheory.MorphismProperty.Ind

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : P.RespectsLeft Q
X Y Z : C
i : X ⟶ Y
hi : Q i
f : Y ⟶ Z
x✝ : P.ind f
J : Type u_1
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj Y ⟶ D
s : D ⟶ (Functor.const J).obj Z
hs : IsColimit { pt := Z, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ P.ind (i ≫ f)

Tactic:
by

Goal state after:
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
f : X✝ ⟶ Y✝
⊢ (f ⊗ₘ 𝟙 (𝟙_ D)) ≫ (ρ_ Y✝).hom = (ρ_ X✝).hom ≫ f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : P.RespectsIso
X Y Z : C
i : Y ⟶ Z
hi : IsIso i
f : X ⟶ Y
x✝ : P.ind f
J : Type u_1
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ P.ind (f ≫ i)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : MorphismProperty C
X Y : C
g : X ⟶ Y
inst✝¹ : HasPushouts C
inst✝ : P.IsStableUnderCobaseChange
f : Under X
hf : P.underObj.ind f
⊢ P.underObj.ind ((Under.pushout g).obj f)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P Q : MorphismProperty C
inst✝¹ : P.IsStableUnderCobaseChange
inst✝ : HasPushouts C
⊢ P.ind.IsStableUnderCobaseChange

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P : MorphismProperty C
hp : P ≤ isFinitelyPresentable C
inst✝ : LocallySmall.{w, v, u} C
⊢ P.ind.ind = P.ind

Tactic:
by

Goal state after: