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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
H : W.isLocal.isLocal.TransfiniteCompositionOfShape κ.ord.ToType (reflection W Z κ) :=
  transfiniteCompositionOfShapeReflection W Z κ
X Y : C
f : X ⟶ Y
hf : W f
left✝ : IsCardinalPresentable X κ
right✝ : IsCardinalPresentable Y κ
j₁ : κ.ord.ToType
g₁ : Y ⟶ H.F.obj j₁
j₂ : κ.ord.ToType
g₂ : Y ⟶ H.F.obj j₂
h :
  (fun g ↦ f ≫ g) (g₁ ≫ { pt := reflectionObj W Z κ, ι := H.incl }.ι.app j₁) =
    (fun g ↦ f ≫ g) (g₂ ≫ { pt := reflectionObj W Z κ, ι := H.incl }.ι.app j₂)
⊢ (g₁ ≫ H.F.map (homOfLE ⋯)) ≫ H.incl.app (max j₁ j₂) = g₁ ≫ { pt := reflectionObj W Z κ, ι := H.incl }.ι.app j₁ ∧
    (g₂ ≫ H.F.map (homOfLE ⋯)) ≫ H.incl.app (max j₁ j₂) = g₂ ≫ { pt := reflectionObj W Z κ, ι := H.incl }.ι.app j₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.SharplyLT.Basic

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
inst✝¹ : Fact κ₁.IsRegular
inst✝ : Fact κ₂.IsRegular
h₀ : κ₁ < κ₂
X : Type w
Y : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)
hY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂
m : (B : Set X) → (hB : HasCardinalLT (↑B) κ₂) → (C : SetCardinalLT κ₁ ↑B) → C ∈ Y B hB → X
A : Set X
hA : HasCardinalLT (↑A) κ₂
j : κ₁.ord.ToType
⊢ HasCardinalLT (↑(transfiniteIterate (φ Y m) j A)) κ₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.SharplyLT.Basic

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
inst✝ : Fact κ₁.IsRegular
X : Type w
Y : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)
m : (B : Set X) → (hB : HasCardinalLT (↑B) κ₂) → (C : SetCardinalLT κ₁ ↑B) → C ∈ Y B hB → X
A : Set X
this✝ : Nonempty κ₁.ord.ToType
this : OrderBot κ₁.ord.ToType := WellFoundedLT.toOrderBot κ₁.ord.ToType
⊢ A ⊆ transfiniteIterate (φ Y m) ⊥ A

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
H : W.isLocal.isLocal.TransfiniteCompositionOfShape κ.ord.ToType (reflection W Z κ) :=
  transfiniteCompositionOfShapeReflection W Z κ
X Y : C
f : X ⟶ Y
hf : W f
left✝ : IsCardinalPresentable X κ
right✝ : IsCardinalPresentable Y κ
g₁ g₂ : Y ⟶ reflectionObj W Z κ
h : (fun g ↦ f ≫ g) g₁ = (fun g ↦ f ≫ g) g₂
⊢ ∃ j g₁' g₂', g₁' ≫ H.incl.app j = g₁ ∧ g₂' ≫ H.incl.app j = g₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.SharplyLT.Basic

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
inst✝ : Fact κ₁.IsRegular
X : Type w
Y : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)
m : (B : Set X) → (hB : HasCardinalLT (↑B) κ₂) → (C : SetCardinalLT κ₁ ↑B) → C ∈ Y B hB → X
A : Set X
⊢ A ⊆ ⋃ j, transfiniteIterate (φ Y m) j A

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝² : SmallCategory J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)
ι : Type w
D : ι → DiagramWithUniqueTerminal J κ
hι : HasCardinalLT ι κ
m₀ : ι → J
t₀ : (i : ι) → (D i).top ⟶ m₀ i
hm₀ : ∀ (i : ι), IsEmpty (m₀ i ⟶ (D i).top)
m₁ : J := max m₀ hι
t₁ : (i : ι) → m₀ i ⟶ m₁ := fun i ↦ toMax m₀ hι i
u : (i : ι) → (D i).top ⟶ m₁ := fun i ↦ t₀ i ≫ t₁ i
S : Type (max 0 w) := { x // (D x.1).P x.2.2 ∧ (D x.2.1).P x.2.2 }
shape : MultispanShape :=
  { L := { x // (D x.1).P x.2.2 ∧ (D x.2.1).P x.2.2 }, R := PUnit.{w + 1}, fst := fun x ↦ PUnit.unit,
    snd := fun x ↦ PUnit.unit }
index : MultispanIndex shape J :=
  { left := fun x ↦ (↑x).2.2, right := fun x ↦ m₁, fst := fun x ↦ (D (↑x).1).isTerminal.lift ⋯ ≫ u (↑x).1,
    snd := fun x ↦ (D (↑x).2.1).isTerminal.lift ⋯ ≫ u (↑x).2.1 }
hκ : Cardinal.aleph0 ≤ κ
hL : HasCardinalLT shape.L κ
hR : HasCardinalLT shape.R κ
⊢ HasCardinalLT (Arrow (WalkingMultispan shape)) κ

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.SharplyLT.Basic

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
inst✝² : Fact κ₁.IsRegular
inst✝¹ : Fact κ₂.IsRegular
h₀ : κ₁ < κ₂
X : Type w
inst✝ : PartialOrder X
Y : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)
hY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂
hY' : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), IsCofinal (Y B hB)
m : (B : Set X) → (hB : HasCardinalLT (↑B) κ₂) → (C : SetCardinalLT κ₁ ↑B) → C ∈ Y B hB → X
hm : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂) (C : SetCardinalLT κ₁ ↑B) (hC : C ∈ Y B hB), ∀ b ∈ ↑C, ↑b ≤ m B hB C hC
A : Set X
hA : HasCardinalLT (↑A) κ₂
this :
  ∀ ⦃K : Type w⦄ (j : κ₁.ord.ToType) (f : K → ↑(transfiniteIterate (φ Y m) j A)),
    HasCardinalLT K κ₁ → ∃ x, ∀ (k : K), ↑(f k) ≤ ↑x
K : Type w
f : K → ↑(⋃ j, transfiniteIterate (φ Y m) j A)
hK : HasCardinalLT K κ₁
k : K
⊢ ∃ j, ↑(f k) ∈ transfiniteIterate (φ Y m) j A

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
H : W.isLocal.isLocal.TransfiniteCompositionOfShape κ.ord.ToType (reflection W Z κ) :=
  transfiniteCompositionOfShapeReflection W Z κ
X Y : C
f : X ⟶ Y
hf : W f
left✝ : IsCardinalPresentable X κ
right✝ : IsCardinalPresentable Y κ
j : κ.ord.ToType
g₁ g₂ : Y ⟶ H.F.obj j
h : f ≫ g₁ ≫ H.incl.app j = f ≫ g₂ ≫ H.incl.app j
⊢ (f ≫ g₁) ≫ { pt := reflectionObj W Z κ, ι := H.incl }.ι.app j =
    (f ≫ g₂) ≫ { pt := reflectionObj W Z κ, ι := H.incl }.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.SharplyLT.Basic

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
inst✝² : Fact κ₁.IsRegular
inst✝¹ : Fact κ₂.IsRegular
h₀ : κ₁ < κ₂
X : Type w
inst✝ : PartialOrder X
Y : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)
hY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂
hY' : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), IsCofinal (Y B hB)
m : (B : Set X) → (hB : HasCardinalLT (↑B) κ₂) → (C : SetCardinalLT κ₁ ↑B) → C ∈ Y B hB → X
hm : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂) (C : SetCardinalLT κ₁ ↑B) (hC : C ∈ Y B hB), ∀ b ∈ ↑C, ↑b ≤ m B hB C hC
A : Set X
hA : HasCardinalLT (↑A) κ₂
this :
  ∀ ⦃K : Type w⦄ (j : κ₁.ord.ToType) (f : K → ↑(transfiniteIterate (φ Y m) j A)),
    HasCardinalLT K κ₁ → ∃ x, ∀ (k : K), ↑(f k) ≤ ↑x
⊢ IsCardinalFiltered (↑(⋃ j, transfiniteIterate (φ Y m) j A)) κ₁

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
H : W.isLocal.isLocal.TransfiniteCompositionOfShape κ.ord.ToType (reflection W Z κ) :=
  transfiniteCompositionOfShapeReflection W Z κ
X Y : C
f : X ⟶ Y
hf : W f
left✝ : IsCardinalPresentable X κ
right✝ : IsCardinalPresentable Y κ
j : κ.ord.ToType
g₁ g₂ : Y ⟶ H.F.obj j
h : f ≫ g₁ ≫ H.incl.app j = f ≫ g₂ ≫ H.incl.app j
k : κ.ord.ToType
u : j ⟶ k
hk : (f ≫ g₁) ≫ H.F.map u = (f ≫ g₂) ≫ H.F.map u
⊢ f ≫ g₁ ≫ H.F.map u = f ≫ g₂ ≫ H.F.map u

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝² : SmallCategory J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)
ι : Type w
D : ι → DiagramWithUniqueTerminal J κ
hι : HasCardinalLT ι κ
m₀ : ι → J
t₀ : (i : ι) → (D i).top ⟶ m₀ i
hm₀ : ∀ (i : ι), IsEmpty (m₀ i ⟶ (D i).top)
m₁ : J := max m₀ hι
t₁ : (i : ι) → m₀ i ⟶ m₁ := fun i ↦ toMax m₀ hι i
u : (i : ι) → (D i).top ⟶ m₁ := fun i ↦ t₀ i ≫ t₁ i
S : Type (max 0 w) := { x // (D x.1).P x.2.2 ∧ (D x.2.1).P x.2.2 }
shape : MultispanShape :=
  { L := { x // (D x.1).P x.2.2 ∧ (D x.2.1).P x.2.2 }, R := PUnit.{w + 1}, fst := fun x ↦ PUnit.unit,
    snd := fun x ↦ PUnit.unit }
index : MultispanIndex shape J :=
  { left := fun x ↦ (↑x).2.2, right := fun x ↦ m₁, fst := fun x ↦ (D (↑x).1).isTerminal.lift ⋯ ≫ u (↑x).1,
    snd := fun x ↦ (D (↑x).2.1).isTerminal.lift ⋯ ≫ u (↑x).2.1 }
hκ : Cardinal.aleph0 ≤ κ
hL : HasCardinalLT shape.L κ
hR : HasCardinalLT shape.R κ
hshape : HasCardinalLT (Arrow (WalkingMultispan shape)) κ
c : Multicofork index := cocone index.multispan hshape
i₁ i₂ : ι
j : J
h₁ : (D i₁).P j
h₂ : (D i₂).P j
⊢ (D i₁).isTerminal.lift h₁ ≫ (fun i ↦ u i ≫ c.π PUnit.unit) i₁ =
    (D i₂).isTerminal.lift h₂ ≫ (fun i ↦ u i ≫ c.π PUnit.unit) i₂

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝² : SmallCategory J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)
ι : Type w
D : ι → DiagramWithUniqueTerminal J κ
hι : HasCardinalLT ι κ
⊢ ∃ m u,
    (∀ (i : ι), IsEmpty (m ⟶ (D i).top)) ∧
      ∀ (i₁ i₂ : ι) (j : J) (hj₁ : (D i₁).P j) (hj₂ : (D i₂).P j),
        (D i₁).isTerminal.lift hj₁ ≫ u i₁ = (D i₂).isTerminal.lift hj₂ ≫ u i₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.SharplyLT.Basic

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
inst✝² : Fact κ₁.IsRegular
inst✝¹ : Fact κ₂.IsRegular
h₀ : κ₁ < κ₂
X : Type w
inst✝ : PartialOrder X
Y : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)
hY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂
hY' : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), IsCofinal (Y B hB)
m : (B : Set X) → (hB : HasCardinalLT (↑B) κ₂) → (C : SetCardinalLT κ₁ ↑B) → C ∈ Y B hB → X
hm : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂) (C : SetCardinalLT κ₁ ↑B) (hC : C ∈ Y B hB), ∀ b ∈ ↑C, ↑b ≤ m B hB C hC
A : Set X
hA : HasCardinalLT (↑A) κ₂
⊢ IsCardinalFiltered (↑(⋃ j, transfiniteIterate (φ Y m) j A)) κ₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.SharplyLT.Basic

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
inst✝³ : Fact κ₁.IsRegular
inst✝² : Fact κ₂.IsRegular
h₀ : κ₁ < κ₂
h : ∀ (X : Type w), HasCardinalLT X κ₂ → ∃ Y, HasCardinalLT (↑Y) κ₂ ∧ IsCofinal Y
X : Type w
inst✝¹ : PartialOrder X
inst✝ : IsCardinalFiltered X κ₁
A : Set X
hA : HasCardinalLT (↑A) κ₂
Y : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)
hY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂
hY' : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), IsCofinal (Y B hB)
m : (B : Set X) → (hB : HasCardinalLT (↑B) κ₂) → (C : SetCardinalLT κ₁ ↑B) → C ∈ Y B hB → X
hm : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂) (C : SetCardinalLT κ₁ ↑B) (hC : C ∈ Y B hB), ∀ b ∈ ↑C, ↑b ≤ m B hB C hC
⊢ HasCardinalLT κ₁.ord.ToType κ₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.SharplyLT.Basic

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
inst✝³ : Fact κ₁.IsRegular
inst✝² : Fact κ₂.IsRegular
h₀ : κ₁ < κ₂
h : ∀ (X : Type w), HasCardinalLT X κ₂ → ∃ Y, HasCardinalLT (↑Y) κ₂ ∧ IsCofinal Y
X : Type w
inst✝¹ : PartialOrder X
inst✝ : IsCardinalFiltered X κ₁
A : Set X
hA : HasCardinalLT (↑A) κ₂
⊢ ∃ B, A ⊆ B ∧ IsCardinalFiltered (↑B) κ₁ ∧ HasCardinalLT (↑B) κ₂

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
hX : HasCardinalLT X κ
inst✝ : Fact κ.IsRegular
J : Type u
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
F : J ⥤ Type u
c : Cocone F
hc : IsColimit c
this : IsFiltered J
j : J
f g : X ⟶ F.obj j
h : f ≫ c.ι.app j = g ≫ c.ι.app j
k : ToType X → J
a : (x : ToType X) → j ⟶ k x
hk : ∀ (x : ToType X), (hom (F.map (a x))) (f.hom' x) = (hom (F.map (a x))) (g.hom' x)
⊢ ∀ (x : ToType X), (hom (F.map (a x))) ((hom f) x) = (hom (F.map (a x))) ((hom g) x)

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
hX : HasCardinalLT X κ
inst✝ : Fact κ.IsRegular
J : Type u
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
F : J ⥤ Type u
c : Cocone F
hc : IsColimit c
this : IsFiltered J
j : J
f g : X ⟶ F.obj j
h : f ≫ c.ι.app j = g ≫ c.ι.app j
k : ToType X → J
a : (x : ToType X) → j ⟶ k x
hk : ∀ (x : ToType X), (hom (F.map (a x))) ((hom f) x) = (hom (F.map (a x))) ((hom g) x)
φ : X → (j ⟶ IsCardinalFiltered.max k hX) := fun x ↦ a x ≫ IsCardinalFiltered.toMax k hX x
x : ToType X
⊢ a x ≫ (fun x ↦ IsCardinalFiltered.toMax k hX x ≫ IsCardinalFiltered.coeqHom φ hX) x = IsCardinalFiltered.toCoeq φ hX

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
H : W.isLocal.isLocal.TransfiniteCompositionOfShape κ.ord.ToType (reflection W Z κ) :=
  transfiniteCompositionOfShapeReflection W Z κ
X Y : C
f : X ⟶ Y
hf : W f
left✝ : IsCardinalPresentable X κ
right✝ : IsCardinalPresentable Y κ
j : κ.ord.ToType
g : X ⟶ H.F.obj j
g' : Y ⟶ (iteration W Z κ).obj (Order.succ j)
hg' : f ≫ g' = g ≫ (iteration W Z κ).map (homOfLE ⋯)
⊢ (fun g ↦ f ≫ g) (g' ≫ H.incl.app (Order.succ j)) = g ≫ { pt := reflectionObj W Z κ, ι := H.incl }.ι.app j

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
hX : HasCardinalLT X κ
inst✝ : Fact κ.IsRegular
J : Type u
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
F : J ⥤ Type u
c : Cocone F
hc : IsColimit c
this : IsFiltered J
j : J
f g : X ⟶ F.obj j
h : f ≫ c.ι.app j = g ≫ c.ι.app j
k : ToType X → J
a : (x : ToType X) → j ⟶ k x
hk : ∀ (x : ToType X), (hom (F.map (a x))) ((hom f) x) = (hom (F.map (a x))) ((hom g) x)
⊢ ∃ l c b, ∀ (x : ToType X), a x ≫ b x = c

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝² : SmallCategory J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)
ι : Type w
D : ι → DiagramWithUniqueTerminal J κ
hι : HasCardinalLT ι κ
m : J
u : (i : ι) → (D i).top ⟶ m
⊢ ∀ {i j : J} {f : i ⟶ j},
    ((D₁ D hι m).W ⊔ MorphismProperty.ofHoms fun x ↦ (D x.fst).isTerminal.lift ⋯ ≫ u x.fst) f → (D₁ D hι m).P i

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
⊢ W.isLocal (reflectionObj W Z κ)

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
hX : HasCardinalLT X κ
inst✝ : Fact κ.IsRegular
J : Type u
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
F : J ⥤ Type u
c✝ : Cocone F
hc : IsColimit c✝
this : IsFiltered J
j : J
f g : X ⟶ F.obj j
h : f ≫ c✝.ι.app j = g ≫ c✝.ι.app j
k : ToType X → J
a : (x : ToType X) → j ⟶ k x
hk : ∀ (x : ToType X), (hom (F.map (a x))) ((hom f) x) = (hom (F.map (a x))) ((hom g) x)
l : J
b : j ⟶ l
c : (x : ToType X) → k x ⟶ l
hl : ∀ (x : ToType X), a x ≫ c x = b
⊢ f ≫ F.map b = g ≫ F.map b

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
hX : HasCardinalLT X κ
inst✝ : Fact κ.IsRegular
J : Type u
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
F : J ⥤ Type u
c : Cocone F
hc : IsColimit c
⊢ IsColimit ((coyoneda.obj (op X)).mapCocone c)

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
ι : Type u
A : ι → HasCardinalLT.Set X κ
hι : Cardinal.mk ι < κ
⊢ HasCardinalLT ι κ

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
hκ : Cardinal.aleph0 ≤ κ
⊢ IsFilteredOrEmpty (HasCardinalLT.Set X κ)

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
⊢ Monotone Subtype.val

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
hκ : Cardinal.aleph0 ≤ κ
this : IsFiltered (HasCardinalLT.Set X κ)
x : (cocone X κ).pt
⊢ x ∈ ⋯.functor.obj ⟨{x}, ⋯⟩

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
⊢ W.isLocal.ι.IsRightAdjoint

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝² : SmallCategory J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)
ι : Type w
D : ι → DiagramWithUniqueTerminal J κ
hι : HasCardinalLT ι κ
m : J
u : (i : ι) → (D i).top ⟶ m
⊢ ∀ {i j : J} {f : i ⟶ j},
    ((D₁ D hι m).W ⊔ MorphismProperty.ofHoms fun x ↦ (D x.fst).isTerminal.lift ⋯ ≫ u x.fst) f → (D₁ D hι m).P j

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
W : MorphismProperty C
κ : Cardinal.{w}
inst✝³ : Fact κ.IsRegular
inst✝² : IsSmall.{w, v, u} W
inst✝¹ : LocallySmall.{w, v, u} C
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
inst✝ : HasColimitsOfSize.{w, w, v, u} C
⊢ Nonempty κ.ord.ToType

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
hκ : Cardinal.aleph0 ≤ κ
⊢ IsColimit (cocone X κ)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
W : MorphismProperty C
κ : Cardinal.{w}
inst✝³ : Fact κ.IsRegular
inst✝² : IsSmall.{w, v, u} W
inst✝¹ : LocallySmall.{w, v, u} C
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
inst✝ : HasColimitsOfSize.{w, w, v, u} C
⊢ W.isLocal.ι.IsRightAdjoint

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
W : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : IsCardinalLocallyPresentable C κ
inst✝ : IsSmall.{w, v, u} W
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
⊢ IsCardinalLocallyPresentable W.isLocal.FullSubcategory κ

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
x✝ : IsCardinalPresentable X κ
this :
  ∀ (J : Type u) [inst : SmallCategory J] [IsCardinalFiltered J κ], PreservesColimitsOfShape J (coyoneda.obj (op X))
A : Set X
hA : HasCardinalLT (↑A) κ
f : (HasCardinalLT.Set.functor X κ ⋙ coyoneda.obj (op X)).obj ⟨A, hA⟩
hf : (hom (((coyoneda.obj (op X)).mapCocone (HasCardinalLT.Set.cocone X κ)).ι.app ⟨A, hA⟩)) f = 𝟙 X
⊢ A = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Profunctor.Basic

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
P : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D
X : C
Y Y' Y'' : D
g : Y ⟶ Y'
g' : Y' ⟶ Y''
⊢ P.map (𝟙 X) (g ≫ g') = P.map (𝟙 X) g' ≫ P.map (𝟙 X) g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Profunctor.Basic

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
P : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D
X X' X'' : C
Y : D
f : X ⟶ X'
f' : X' ⟶ X''
⊢ P.map (f ≫ f') (𝟙 Y) = P.map f (𝟙 Y) ≫ P.map f' (𝟙 Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Profunctor.Basic

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
P : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D
X X' : C
Y Y' : D
f : X ⟶ X'
g : Y ⟶ Y'
⊢ P.map (𝟙 X) g ≫ P.map f (𝟙 Y) = P.map f g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Profunctor.Basic

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
P : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D
X X' : C
Y Y' : D
f : X ⟶ X'
g : Y ⟶ Y'
⊢ P.map f (𝟙 Y') ≫ P.map (𝟙 X') g = P.map f g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Precoherent C
⊢ ∀ ⦃X Y : C⦄ (f : Y ⟶ X),
    ∀ S ∈ {S | ∃ α, ∃ (_ : Finite α), ∃ X_1 π, S = Presieve.ofArrows X_1 π ∧ EffectiveEpiFamily X_1 π},
      ∃ T ∈ {S | ∃ α, ∃ (_ : Finite α), ∃ X π, S = Presieve.ofArrows X π ∧ EffectiveEpiFamily X π},
        T.FactorsThruAlong S f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x ↦ Z) fun x ↦ π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ Presieve.singleton h = Presieve.ofArrows (fun x ↦ W) fun x ↦ h

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝¹ : SmallCategory J
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
ι : Type w
D : ι → DiagramWithUniqueTerminal J κ
hι : HasCardinalLT ι κ
m : J
u : (i : ι) → (D i).top ⟶ m
hD : ∀ {i : ι}, ¬(D i).P m
f : m ⟶ m
hf : (D₂ D hι u).W f
⊢ f = 𝟙 m

Tactic:
by

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

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
⊢ IsCardinalPresentable X κ ↔ HasCardinalLT X κ

Tactic:
by

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

Goal state before:
pp: X✝ X : Type u
⊢ IsPresentable.{u, u, u + 1} X

Tactic:
by

Goal state after: