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)
a b c d : D
⊢ ((α_ a b c).hom ⊗ₘ 𝟙 d) ≫ (α_ a (b ⊗ c) d).hom ≫ (𝟙 a ⊗ₘ (α_ b c d).hom) = (α_ (a ⊗ b) c d).hom ≫ (α_ a b (c ⊗ d)).hom

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)
a b : D
⊢ (α_ a (𝟙_ D) b).hom ≫ (𝟙 a ⊗ₘ (λ_ b).hom) = (ρ_ a).hom ⊗ₘ 𝟙 b

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁹ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝⁸ : Category.{v₂, u₂} V✝
inst✝⁷ : MonoidalCategory C✝
inst✝⁶ : MonoidalCategory V✝
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
D : Type u₃
inst✝¹ : Category.{v₃, u₃} D
inst✝ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
d d' : D
⊢ ∀ ⦃X Y : C × C⦄ (f : X ⟶ Y),
    ((ι C V D).obj d ⊠ (ι C V D).obj d').map f ≫ convolutionUnitApp V d d' Y.1 Y.2 =
      convolutionUnitApp V d d' X.1 X.2 ≫ (tensor C ⋙ (ι C V D).obj (tensorObj C V d d')).map f

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁹ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝⁸ : Category.{v₂, u₂} V✝
inst✝⁷ : MonoidalCategory C✝
inst✝⁶ : MonoidalCategory V✝
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
D : Type u₃
inst✝¹ : Category.{v₃, u₃} D
inst✝ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
d d' : D
⊢ StructuredArrow.hom
        (Functor.LeftExtension.mk (DayConvolution.convolution ((ι C V D).obj d) ((ι C V D).obj d'))
          (DayConvolution.unit ((ι C V D).obj d) ((ι C V D).obj d'))) ≫
      ((Functor.whiskeringLeft (C × C) C V).obj (tensor C)).map (tensorObjIsoConvolution C V d d').symm.hom =
    StructuredArrow.hom
      (Functor.LeftExtension.mk ((ι C V D).obj (tensorObj C V d d'))
        { app := fun x ↦ convolutionUnitApp V d d' x.1 x.2, naturality := ⋯ })

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
H✝ : P ≤ isFinitelyPresentable C
X Y : C
f : X ⟶ Y
inst✝ : IsFinitelyAccessibleCategory (Under X)
H : ∀ {Z : C} (p : X ⟶ Z) (g : Z ⟶ Y), isFinitelyPresentable C p → p ≫ g = f → ∃ W u v, u ≫ v = g ∧ P (p ≫ u)
Z : Under X
g : Z ⟶ CategoryTheory.Under.mk f
hZ : IsFinitelyPresentable Z
W : C
u : Z.right ⟶ W
v : W ⟶ Y
huv : u ≫ v = Under.Hom.right g
hW : P (Z.hom ≫ u)
⊢ CategoryTheory.Under.homMk u ⋯ ≫ CategoryTheory.Under.homMk v ⋯ = g

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁸ : Category.{v₂, u₂} V
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory V
D : Type u₃
inst✝⁵ : Category.{v₃, u₃} D
inst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
inst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
d₁ d₁' d₂ d₂' : D
f : d₁ ⟶ d₂
f' : d₁' ⟶ d₂'
⊢ (ι C V D).map (f ⊗ₘ f') = DayConvolution.map ((ι C V D).map f) ((ι C V D).map 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 : MorphismProperty C
H : P ≤ isFinitelyPresentable C
X Y : C
f : X ⟶ Y
inst✝ : IsFinitelyAccessibleCategory (Under X)
⊢ P.ind f ↔ ∀ {Z : C} (p : X ⟶ Z) (g : Z ⟶ Y), isFinitelyPresentable C p → p ≫ g = f → ∃ W u v, u ≫ v = g ∧ P (p ≫ u)

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝¹³ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹² : Category.{v₂, u₂} V✝
inst✝¹¹ : MonoidalCategory C✝
inst✝¹⁰ : MonoidalCategory V✝
C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁸ : Category.{v₂, u₂} V
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory V
D : Type u₃
inst✝⁵ : Category.{v₃, u₃} D
inst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
inst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
this : MonoidalCategoryStruct D := mkMonoidalCategoryStruct C V D
⊢ ∀ {d₁ d₂ d₁' d₂' : D} (f₁ : d₁ ⟶ d₁') (f₂ : d₂ ⟶ d₂') (x y : C),
    (DayConvolution.unit ((ι C V D).obj d₁) ((ι C V D).obj d₂)).app (x, y) ≫ ((ι C V D).map (f₁ ⊗ₘ f₂)).app (x ⊗ y) =
      (((ι C V D).map f₁).app x ⊗ₘ ((ι C V D).map f₂).app y) ≫
        (DayConvolution.unit ((ι C V D).obj d₁') ((ι C V D).obj d₂')).app (x, y)

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝¹³ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹² : Category.{v₂, u₂} V✝
inst✝¹¹ : MonoidalCategory C✝
inst✝¹⁰ : MonoidalCategory V✝
C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁸ : Category.{v₂, u₂} V
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory V
D : Type u₃
inst✝⁵ : Category.{v₃, u₃} D
inst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
inst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
this : MonoidalCategoryStruct D := mkMonoidalCategoryStruct C V D
⊢ ∀ (d₁ : D) {d₂ d₂' : D} (f₂ : d₂ ⟶ d₂') (x y : C),
    (DayConvolution.unit ((ι C V D).obj d₁) ((ι C V D).obj d₂)).app (x, y) ≫ ((ι C V D).map (d₁ ◁ f₂)).app (x ⊗ y) =
      ((ι C V D).obj d₁).obj x ◁ ((ι C V D).map f₂).app y ≫
        (DayConvolution.unit ((ι C V D).obj d₁) ((ι C V D).obj d₂')).app (x, y)

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝¹³ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹² : Category.{v₂, u₂} V✝
inst✝¹¹ : MonoidalCategory C✝
inst✝¹⁰ : MonoidalCategory V✝
C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁸ : Category.{v₂, u₂} V
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory V
D : Type u₃
inst✝⁵ : Category.{v₃, u₃} D
inst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
inst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
this : MonoidalCategoryStruct D := mkMonoidalCategoryStruct C V D
⊢ ∀ {d₁ d₁' : D} (f₁ : d₁ ⟶ d₁') (d₂ : D) (x y : C),
    (DayConvolution.unit ((ι C V D).obj d₁) ((ι C V D).obj d₂)).app (x, y) ≫ ((ι C V D).map (f₁ ▷ d₂)).app (x ⊗ y) =
      ((ι C V D).map f₁).app x ▷ ((ι C V D).obj d₂).obj y ≫
        (DayConvolution.unit ((ι C V D).obj d₁') ((ι C V D).obj d₂)).app (x, y)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
P : MorphismProperty C
inst✝⁴ : ∀ (X : C), IsFinitelyAccessibleCategory (Under X)
inst✝³ : HasPushouts C
inst✝² : P.IsStableUnderComposition
inst✝¹ : P.IsStableUnderCobaseChange
inst✝ : P.PreIndSpreads
H : P ≤ isFinitelyPresentable C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
T : C
p : X ⟶ T
u : T ⟶ Z
hp : isFinitelyPresentable C p
hpu : p ≫ u = f ≫ g
J₁ : Type w
w✝³ : SmallCategory J₁
w✝² : IsFiltered J₁
D₁ : J₁ ⥤ C
s₁ : (Functor.const J₁).obj X ⟶ D₁
t₁ : D₁ ⟶ (Functor.const J₁).obj Y
ht₁ : IsColimit { pt := Y, ι := t₁ }
h₁ : ∀ (j : J₁), P (s₁.app j) ∧ s₁.app j ≫ t₁.app j = f
J₂ : Type w
w✝¹ : SmallCategory J₂
w✝ : IsFiltered J₂
D₂ : J₂ ⥤ C
s₂ : (Functor.const J₂).obj Y ⟶ D₂
t₂ : D₂ ⟶ (Functor.const J₂).obj Z
ht₂ : IsColimit { pt := Z, ι := t₂ }
h₂ : ∀ (j : J₂), P (s₂.app j) ∧ s₂.app j ≫ t₂.app j = g
this : IsFinitelyPresentable (CategoryTheory.Under.mk p)
⊢ ∀ (j : J₂), ((Functor.const J₂).map f ≫ s₂).app j ≫ { pt := Z, ι := t₂ }.ι.app j = p ≫ u

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝¹³ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹² : Category.{v₂, u₂} V✝
inst✝¹¹ : MonoidalCategory C✝
inst✝¹⁰ : MonoidalCategory V✝
C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁸ : Category.{v₂, u₂} V
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory V
D : Type u₃
inst✝⁵ : Category.{v₃, u₃} D
inst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
inst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
this : MonoidalCategoryStruct D := mkMonoidalCategoryStruct C V D
d₁ d₂ d₃ : D
x₁ x₂ x₃ : C
⊢ (DayConvolution.unit ((ι C V D).obj d₁) ((ι C V D).obj d₂)).app (x₁, x₂) ▷ ((ι C V D).obj d₃).obj x₃ ≫
      (DayConvolution.unit ((ι C V D).obj (d₁ ⊗ d₂)) ((ι C V D).obj d₃)).app (x₁ ⊗ x₂, x₃) ≫
        ((ι C V D).map (α_ d₁ d₂ d₃).hom).app ((x₁ ⊗ x₂) ⊗ x₃) =
    (α_ (((ι C V D).obj d₁).obj x₁) (((ι C V D).obj d₂, (ι C V D).obj d₃).1.obj (x₂, x₃).1)
          (((ι C V D).obj d₂, (ι C V D).obj d₃).2.obj (x₂, x₃).2)).hom ≫
      ((ι C V D).obj d₁).obj x₁ ◁ (DayConvolution.unit ((ι C V D).obj d₂) ((ι C V D).obj d₃)).app (x₂, x₃) ≫
        (DayConvolution.unit ((ι C V D).obj d₁) ((ι C V D).obj (d₂ ⊗ d₃))).app (x₁, x₂ ⊗ x₃) ≫
          ((ι C V D).obj (d₁ ⊗ d₂ ⊗ d₃)).map (α_ (x₁, x₂ ⊗ x₃).1 x₂ x₃).inv

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝¹³ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹² : Category.{v₂, u₂} V✝
inst✝¹¹ : MonoidalCategory C✝
inst✝¹⁰ : MonoidalCategory V✝
C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁸ : Category.{v₂, u₂} V
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory V
D : Type u₃
inst✝⁵ : Category.{v₃, u₃} D
inst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
inst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
this : MonoidalCategoryStruct D := mkMonoidalCategoryStruct C V D
x✝¹ : D
x✝ : C
⊢ DayConvolutionUnit.can ▷ ((ι C V D).obj x✝¹).obj x✝ ≫
      (DayConvolution.unit ((ι C V D).obj (𝟙_ D)) ((ι C V D).obj x✝¹)).app (𝟙_ C, x✝) ≫
        ((ι C V D).map (λ_ x✝¹).hom).app (𝟙_ C ⊗ x✝) =
    (λ_ (((ι C V D).obj x✝¹).obj x✝)).hom ≫ ((ι C V D).obj x✝¹).map (λ_ x✝).inv

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✝
E : K.ZeroHypercover X✝
h : ∀ (U : C) (g : U ⟶ X✝), Presieve.ofArrows E.X E.f g → sourceLocalClosure K P (g ≫ f)
ih :
  ∀ (U : C) (g : U ⟶ X✝),
    Presieve.ofArrows E.X E.f g → ∃ R ∈ K.coverings U, ∀ (U_1 : C) (g_1 : U_1 ⟶ U), R g_1 → P (g_1 ≫ g ≫ f)
S : (i : E.I₀) → Presieve (E.X i)
h' : ∀ (i : E.I₀) (U : C) (g : U ⟶ E.X i), S i g → P (g ≫ E.f i ≫ f)
F : (i : E.I₀) → K.ZeroHypercover (E.X i)
hF : ∀ (i : E.I₀), S i = Presieve.ofArrows (F i).X (F i).f
U : C
g : U ⟶ X✝
x✝ : (E.bind F).presieve₀ g
j : (E.bind F).I₀
⊢ S j.fst ((F j.fst).f j.snd)

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝¹³ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹² : Category.{v₂, u₂} V✝
inst✝¹¹ : MonoidalCategory C✝
inst✝¹⁰ : MonoidalCategory V✝
C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁸ : Category.{v₂, u₂} V
inst✝⁷ : MonoidalCategory C
inst✝⁶ : MonoidalCategory V
D : Type u₃
inst✝⁵ : Category.{v₃, u₃} D
inst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D
inst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
this : MonoidalCategoryStruct D := mkMonoidalCategoryStruct C V D
x✝¹ : D
x✝ : C
⊢ ((ι C V D).obj x✝¹).obj x✝ ◁ DayConvolutionUnit.can ≫
      (DayConvolution.unit ((ι C V D).obj x✝¹) ((ι C V D).obj (𝟙_ D))).app (x✝, 𝟙_ C) ≫
        ((ι C V D).map (ρ_ x✝¹).hom).app (x✝ ⊗ 𝟙_ C) =
    (ρ_ (((ι C V D).obj x✝¹).obj x✝)).hom ≫ ((ι C V D).obj x✝¹).map (ρ_ x✝).inv

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
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

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
X Z : C
x✝ : P Z
⊢ Function.Injective fun g ↦ 𝟙 X ≫ g

Tactic:
by

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
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : P.localEpi f
hg : P.localEpi g
T : C
hT : P T
x✝¹ x✝ : Z✝ ⟶ T
huv : (fun g_1 ↦ (f ≫ g) ≫ g_1) x✝¹ = (fun g_1 ↦ (f ≫ g) ≫ g_1) x✝
⊢ (fun g ↦ f ≫ g) ((fun g_1 ↦ g ≫ g_1) x✝¹) = (fun g ↦ f ≫ g) ((fun g_1 ↦ g ≫ g_1) x✝)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
X Y : C
f : X ⟶ Y
inst✝ : Epi f
⊢ P.localEpi f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
⊢ ⊤.localEpi = MorphismProperty.epimorphisms C

Tactic:
by

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.isoClosure.localEpi = P.localEpi

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
⊢ sourceLocalClosure K P f ↔ ∃ R ∈ K.coverings X, ∀ (U : C) (g : U ⟶ X), R g → P (g ≫ f)

Tactic:
by

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
W : MorphismProperty C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
x✝ : W f
hfg : P.localEpi (f ≫ g)
T : C
hT : P T
u v : Z ⟶ T
huv : (fun g_1 ↦ g ≫ g_1) u = (fun g_1 ↦ g ≫ g_1) v
⊢ (fun g_1 ↦ (f ≫ g) ≫ g_1) u = (fun g_1 ↦ (f ≫ g) ≫ g_1) v

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
P : MorphismProperty C
inst✝⁴ : ∀ (X : C), IsFinitelyAccessibleCategory (Under X)
inst✝³ : HasPushouts C
inst✝² : P.IsStableUnderComposition
inst✝¹ : P.IsStableUnderCobaseChange
inst✝ : P.PreIndSpreads
H : P ≤ isFinitelyPresentable C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
T : C
p : X ⟶ T
u : T ⟶ Z
hp : isFinitelyPresentable C p
hpu : p ≫ u = f ≫ g
J₁ : Type w
w✝³ : SmallCategory J₁
w✝² : IsFiltered J₁
D₁ : J₁ ⥤ C
s₁ : (Functor.const J₁).obj X ⟶ D₁
t₁ : D₁ ⟶ (Functor.const J₁).obj Y
ht₁ : IsColimit { pt := Y, ι := t₁ }
h₁ : ∀ (j : J₁), P (s₁.app j) ∧ s₁.app j ≫ t₁.app j = f
J₂ : Type w
w✝¹ : SmallCategory J₂
w✝ : IsFiltered J₂
D₂ : J₂ ⥤ C
s₂ : (Functor.const J₂).obj Y ⟶ D₂
t₂ : D₂ ⟶ (Functor.const J₂).obj Z
ht₂ : IsColimit { pt := Z, ι := t₂ }
h₂ : ∀ (j : J₂), P (s₂.app j) ∧ s₂.app j ≫ t₂.app j = g
this : IsFinitelyPresentable (CategoryTheory.Under.mk p)
j₂ : J₂
q : T ⟶ D₂.obj j₂
hcomp : p ≫ q = ((Functor.const J₂).map f ≫ s₂).app j₂
hu : q ≫ { pt := Z, ι := t₂ }.ι.app j₂ = u
j₁ : J₁
W : C
f' : D₁.obj j₁ ⟶ W
g' : W ⟶ D₂.obj j₂
h : IsPushout ({ pt := Y, ι := t₁ }.ι.app j₁) f' (s₂.app j₂) g'
hf' : P f'
D' : Under j₁ ⥤ C := (Under.post D₁ ⋙ Under.pushout f') ⋙ CategoryTheory.Under.forget W
c' : Cocone D' :=
  ((Under.pushout f' ⋙ CategoryTheory.Under.forget W).mapCocone ({ pt := Y, ι := t₁ }.underPost j₁)).extend
    h.isoPushout.inv
hc' : IsColimit c' :=
  IsColimit.extendIso h.isoPushout.inv
    (isColimitOfPreserves (Under.pushout f' ⋙ CategoryTheory.Under.forget W) (ht₁.underPost j₁))
k l : Under j₁
a : k ⟶ l
⊢ ((Functor.const (Under j₁)).obj X).map a ≫ s₁.app l.right ≫ pushout.inl ((Under.post D₁).obj l).hom f' =
    (s₁.app k.right ≫ pushout.inl ((Under.post D₁).obj k).hom f') ≫ D'.map a

Tactic:
by

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
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
hg : P.isLocal g
hfg : P.localEpi (f ≫ g)
T : C
hT : P T
u v : Z ⟶ T
huv : (fun g ↦ f ≫ g) ((fun g_1 ↦ g ≫ g_1) u) = (fun g ↦ f ≫ g) ((fun g_1 ↦ g ≫ g_1) v)
⊢ (fun g_1 ↦ (f ≫ g) ≫ g_1) ?m.71 = (fun g_1 ↦ (f ≫ g) ≫ g_1) ?m.72

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : C
⊢ IsArtinianObject X ↔ ∀ (f : ℕ →o (Subobject X)ᵒᵈ), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : C
⊢ IsArtinianObject X ↔ ∀ (f : ℕ → Subobject X), ¬StrictAnti f

Tactic:
by

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
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
hg : P.isLocal g
hfg : P.localEpi (f ≫ g)
T : C
hT : P T
u v : Y ⟶ T
huv : (fun g ↦ f ≫ g) u = (fun g ↦ f ≫ g) v
⊢ u = v

Tactic:
by

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:
by

Goal state after: