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

Goal state before:
pp: J : Type w
inst✝¹ : SmallCategory J
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
j : J
⊢ __Diagram✝.IsTerminal j

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
j : J
⊢ ∀ (j_1 : J) (hj : (Diagram.single j).IsTerminal j_1), j_1 = j

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 u_1
D : ι → Diagram J κ
hι : HasCardinalLT ι κ
i✝ j✝ : J
f✝ : i✝ ⟶ j✝
hf : (⨆ i, (D i).W) f✝
⊢ (⨆ i, (D i).P) i✝

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 u_1
D : ι → Diagram J κ
hι : HasCardinalLT ι κ
i✝ j✝ : J
f✝ : i✝ ⟶ j✝
hf : (⨆ i, (D i).W) f✝
⊢ (⨆ i, (D i).P) j✝

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
h : κ ≤ κ'
x✝ : IsCardinalPresentable J κ'
X : Subtype (J.PropSetWithTop κ')
f : J ⟶ (functorOfPredicateSet (J.PropSetWithTop κ')).obj X
hf : f ≫ (J.coconeWithTop κ').ι.app X = ObjectProperty.homMk (PartOrdEmb.ofHom WithTop.coeOrderHom)
⊢ ⇑(OrderEmbedding.subtype fun x ↦ x ∈ ↑X) ∘ ⇑(ConcreteCategory.hom f) = ⇑WithTop.coeOrderHom

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
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : EssentiallySmall.{w, v', u'} J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
Z : C
x✝ : W.isLocal.colimitsOfShape J Z
X Y : C
f : X ⟶ Y
hf : W f
p : W.isLocal.ColimitOfShape J Z
left✝ : IsCardinalPresentable X κ
right✝ : IsCardinalPresentable Y κ
j : J
g : Y ⟶ p.diag.obj j
⊢ (fun g ↦ f ≫ g) (g ≫ p.ι.app j) = (fun g ↦ f ≫ g) g ≫ { pt := Z, ι := p.ι }.ι.app 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
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : EssentiallySmall.{w, v', u'} J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
Z : C
x✝ : W.isLocal.colimitsOfShape J Z
X Y : C
f : X ⟶ Y
hf : W f
p : W.isLocal.ColimitOfShape J Z
⊢ Function.Bijective fun g ↦ f ≫ g

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
D₁ D₂ : Diagram J κ
⊢ ∀ {i j : J} {f : i ⟶ j}, (D₁.W ⊔ D₂.W) f → (D₁.P ⊔ D₂.P) i

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : HasCardinalFilteredColimits C κ
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
⊢ PreservesColimitsOfShape J W.isLocal.ι

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝¹ : MorphismProperty.IsSmall.{w, v, u} W
inst✝ : LocallySmall.{w, v, u} C
⊢ Small.{w, max u v} (D₁ W Z)

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
h : κ ≤ κ'
⊢ IsCardinalPresentable J κ' ↔ HasCardinalLT (↑J.obj) κ'

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝¹ : HasCoproduct obj₁
inst✝ : HasCoproduct obj₂
d : D₁ W Z
⊢ Sigma.ι obj₁ d ≫ t W Z = (↑d.fst).hom ≫ Sigma.ι obj₂ d

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
j : ↑J.obj
⊢ ∀ (a : ↑{j}), a ≤ ⟨j, ⋯⟩

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝¹ : HasCoproduct obj₁
inst✝ : HasCoproduct obj₂
X Y : C
f : X ⟶ Y
hf : W f
g : X ⟶ Z
⊢ ιLeft f hf g ≫ t W Z = f ≫ ιRight f hf g

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
j : ↑J.obj
⊢ HasTerminal ↑{j}

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
D₁ D₂ : Diagram J κ
⊢ ∀ {i j : J} {f : i ⟶ j}, (D₁.W ⊔ D₂.W) f → (D₁.P ⊔ D₂.P) j

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁴ : HasCoproduct D₁.obj₁
inst✝³ : HasCoproduct D₁.obj₂
inst✝² : HasPushouts C
inst✝¹ : MorphismProperty.IsSmall.{w, v, u} W
inst✝ : LocallySmall.{w, v, u} C
⊢ Small.{w, max u v} (D₂ W Z)

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:
choose m₀ t₀ hm₀ using fun i ↦ hJ (D i).top

Goal state after:
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)
⊢ ∃ 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
Module: Mathlib.CategoryTheory.Presentable.OrthogonalReflection

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁴ : HasCoproduct D₁.obj₁
inst✝³ : HasCoproduct D₁.obj₂
inst✝² : HasPushouts C
inst✝¹ : MorphismProperty.IsSmall.{w, v, u} W
inst✝ : LocallySmall.{w, v, u} C
⊢ Small.{w, max u v} (D₂.multispanShape W Z).L

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
K : Type u
α : K → Subtype J.PropSet
hK : HasCardinalLT K κ
t : (k : K) → ↑↑(α k) := fun k ↦ ⊤_ ↑↑(α k)
m : ↑J.obj := IsCardinalFiltered.max (fun k ↦ ↑(t k)) hK
S : Set ↑J.obj := (⋃ k, ↑(α k)) ∪ {m}
⊢ m ∈ S

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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
X Y : C
f : X ⟶ Y
hf : W f
g₁ g₂ : Y ⟶ Z
hg : f ≫ g₁ = f ≫ g₂
⊢ f ≫ g₁ ≫ toStep W Z = f ≫ g₂ ≫ toStep W Z

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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
X Y : C
f : X ⟶ Y
hf : W f
g₁ g₂ : Y ⟶ Z
hg : f ≫ g₁ = f ≫ g₂
⊢ g₁ ≫ toSucc W Z = g₂ ≫ toSucc W Z

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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
X Y : C
f : X ⟶ Y
hf : W f
g : X ⟶ Z
⊢ f ≫ D₁.ιRight f hf g ≫ pushout.inl (D₁.t W Z) (D₁.l W Z) ≫ fromStep W Z = g ≫ toSucc W Z

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
K : Type u
α : K → Subtype J.PropSet
hK : HasCardinalLT K κ
t : (k : K) → ↑↑(α k) := fun k ↦ ⊤_ ↑↑(α k)
m : ↑J.obj := IsCardinalFiltered.max (fun k ↦ ↑(t k)) hK
S : Set ↑J.obj := (⋃ k, ↑(α k)) ∪ {m}
s : ↑J.obj
hs✝ : s ∈ S
k : K
hs : s ∈ ↑(α k)
⊢ s ⟶ ↑(t k)

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
K : Type u
α : K → Subtype J.PropSet
hK : HasCardinalLT K κ
t : (k : K) → ↑↑(α k) := fun k ↦ ⊤_ ↑↑(α k)
m : ↑J.obj := IsCardinalFiltered.max (fun k ↦ ↑(t k)) hK
S : Set ↑J.obj := (⋃ k, ↑(α k)) ∪ {m}
⊢ ∀ (a : ↑S), a ≤ ⟨m, ⋯⟩

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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
T : C
hT : W.isLocal T
g : Z ⟶ T
f : (d : D₁ W Z) → (↑d.fst).right ⟶ T
hf : ∀ (d : D₁ W Z), (fun g ↦ (↑d.fst).hom ≫ g) (f d) = d.snd ≫ g
d : (D₂.multispanShape W Z).L
⊢ (fun g ↦ (↑d.fst).hom ≫ g)
      ((D₂.multispanIndex W Z).fst d ≫
        match (D₂.multispanShape W Z).fst d with
        | PUnit.unit => pushout.desc (Sigma.desc f) g ⋯) =
    (fun g ↦ (↑d.fst).hom ≫ g)
      ((D₂.multispanIndex W Z).snd d ≫
        match (D₂.multispanShape W Z).snd d with
        | PUnit.unit => pushout.desc (Sigma.desc f) g ⋯)

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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
T : C
hT : W.isLocal T
g : Z ⟶ T
f : (d : D₁ W Z) → (↑d.fst).right ⟶ T
hf : ∀ (d : D₁ W Z), (fun g ↦ (↑d.fst).hom ≫ g) (f d) = d.snd ≫ g
⊢ (fun g ↦ toSucc W Z ≫ g)
      (Multicoequalizer.desc (D₂.multispanIndex W Z) T
        (fun x ↦
          match x with
          | PUnit.unit => pushout.desc (Sigma.desc f) g ⋯)
        ⋯) =
    g

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
K : Type u
α : K → Subtype J.PropSet
hK : Cardinal.mk K < κ
⊢ ∃ j, ∀ (k : K), α k ≤ 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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
⊢ W.isLocal.isLocal (toSucc W Z)

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
J✝ J : CardinalDirectedPoset κ
a : ↑J.obj
⊢ a ∈ {a}

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
⊢ hasCardinalLTWithTerminal κ ≤ isCardinalPresentable (CardinalDirectedPoset κ) κ

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
X K : Type u
f : K → SetCardinalLT κ X
hK : Cardinal.mk K < κ
⊢ HasCardinalLT K 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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
x✝ : IsIso (toSucc W Z)
X Y : C
f : X ⟶ Y
hf : W f
g₁ g₂ : Y ⟶ Z
h : (fun g ↦ f ≫ g) g₁ = (fun g ↦ f ≫ g) g₂
⊢ f ≫ g₁ ≫ toStep W Z = f ≫ g₂ ≫ toStep W Z

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
h : κ₁ ≤ κ₂
inst✝ : IsCardinalAccessibleCategory (CardinalDirectedPoset κ₁) κ₂
X : Type w
hX : HasCardinalLT X κ₂
J : Type w
w✝¹ : SmallCategory J
w✝ : IsCardinalFiltered J κ₂
p : (isCardinalPresentable (CardinalDirectedPoset κ₁) κ₂).ColimitOfShape J (setCardinalLT κ₁ X)
this : IsCardinalFiltered J κ₁
j : J
⊢ HasCardinalLT (↑(p.diag.obj j).obj) κ₂

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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
x✝ : IsIso (toSucc W Z)
X Y : C
f : X ⟶ Y
hf : W f
g : X ⟶ Z
hZ : toStep W Z ≫ fromStep W Z ≫ inv (toSucc W Z) = 𝟙 Z
⊢ (fun g ↦ f ≫ g) (D₁.ιRight f hf g ≫ pushout.inl (D₁.t W Z) (D₁.l W Z) ≫ fromStep W Z ≫ inv (toSucc W Z)) = g

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
h : κ₁ ≤ κ₂
inst✝ : IsCardinalAccessibleCategory (CardinalDirectedPoset κ₁) κ₂
X : Type w
hX : HasCardinalLT X κ₂
J : Type w
w✝¹ : SmallCategory J
w✝ : IsCardinalFiltered J κ₂
p : (isCardinalPresentable (CardinalDirectedPoset κ₁) κ₂).ColimitOfShape J (setCardinalLT κ₁ X)
this : IsCardinalFiltered J κ₁
hp : ∀ (j : J), HasCardinalLT (↑(p.diag.obj j).obj) κ₂
j : X → J
y : (x : X) → ToType (p.diag.obj (j x))
hy : ∀ (x : X), (ConcreteCategory.hom (p.ι.app (j x))) (y x) = SetCardinalLT.singleton κ₁ x
j' : J := IsCardinalFiltered.max j hX
y' : X → ↑(p.diag.obj j').obj := fun x ↦ (ConcreteCategory.hom (p.diag.map (IsCardinalFiltered.toMax j hX x))) (y x)
x : X
⊢ (ConcreteCategory.hom (p.ι.app j')) (y' x) = SetCardinalLT.singleton κ₁ x

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
h : κ₁ ≤ κ₂
inst✝ : IsCardinalAccessibleCategory (CardinalDirectedPoset κ₁) κ₂
X : Type w
hX : HasCardinalLT X κ₂
⊢ ∃ Y, HasCardinalLT (↑Y) κ₂ ∧ IsCofinal Y

Tactic:
by

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

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
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), 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
B : Set X
hB : HasCardinalLT (↑B) κ₂
T : Type w
f : T → ↑B
hT : HasCardinalLT T κ₁
C₀ : SetCardinalLT κ₁ ↑B := ⟨Set.range f, ⋯⟩
C : SetCardinalLT κ₁ ↑B
hC : C ∈ Y B hB
hC' : C₀ ≤ C
⊢ m B hB C hC ∈ {m B hB ↑⟨C, hC⟩ ⋯}

Tactic:
by

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

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
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), 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
B : Set X
hB : HasCardinalLT (↑B) κ₂
T : Type w
f : T → ↑B
hT : HasCardinalLT T κ₁
C₀ : SetCardinalLT κ₁ ↑B := ⟨Set.range f, ⋯⟩
C : SetCardinalLT κ₁ ↑B
hC : C ∈ Y B hB
hC' : C₀ ≤ C
t : T
⊢ f t ∈ ↑C₀

Tactic:
by

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

Goal state before:
pp: κ₁ κ₂ : Cardinal.{w}
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), 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
B : Set X
hB : HasCardinalLT (↑B) κ₂
T : Type w
f : T → ↑B
hT : HasCardinalLT T κ₁
⊢ ∃ a, ∀ (t : T), ↑(f t) ≤ ↑a

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
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
⊢ IsIso (toSucc W Z) ↔ W.isLocal Z

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
this : HasCardinalLT (ι × (i : ι) × Subtype (D i).P) κ
i₁ i₂ : ι
j : J
left✝¹ : (D (i₁, i₂, j).1).P (i₁, i₂, j).2.2
right✝¹ : (D (i₁, i₂, j).2.1).P (i₁, i₂, j).2.2
j' : J
left✝ : (D (i₁, i₂, j').1).P (i₁, i₂, j').2.2
right✝ : (D (i₁, i₂, j').2.1).P (i₁, i₂, j').2.2
h : i₁ = i₁ ∧ i₂ = i₂ ∧ ⟨j, right✝¹⟩ ≍ ⟨j', right✝⟩
⊢ j = j'

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁵ : HasPushouts C
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝³ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝² : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : HasIterationOfShape κ.ord.ToType C
⊢ (succStruct W Z).prop ≤ W.isLocal.isLocal

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
j : κ.ord.ToType
⊢ ¬IsMax j

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
X Y : C
f : X ⟶ Y
hf : W f
j : κ.ord.ToType
g₁ g₂ : Y ⟶ (iteration W Z κ).obj j
hg : f ≫ g₁ = f ≫ g₂
⊢ g₁ ≫ (iteration W Z κ).map (homOfLE ⋯) = g₂ ≫ (iteration W Z κ).map (homOfLE ⋯)

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
⊢ HasCardinalLT shape.L κ

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)
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
B : Set X
hB : HasCardinalLT (↑B) κ₂
b : X
hb : b ∈ B
C : SetCardinalLT κ₁ ↑B
hC : C ∈ Y B hB
hC' : ⟨{⟨b, hb⟩}, ⋯⟩ ≤ C
⊢ ⟨b, hb⟩ ∈ ↑⟨{⟨b, hb⟩}, ⋯⟩

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
X Y : C
f : X ⟶ Y
hf : W f
j : κ.ord.ToType
g : X ⟶ (iteration W Z κ).obj j
g' : Y ⟶ succ W ((iteration W Z κ).obj j)
hg' : f ≫ g' = g ≫ toSucc W ((iteration W Z κ).obj j)
⊢ f ≫ g' ≫ (iterationObjSuccIso W Z κ j).inv =
    g ≫ toSucc W ((iteration W Z κ).obj j) ≫ (iterationObjSuccIso W Z κ j).inv

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)
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
B : Set X
⊢ B ⊆ φ Y m B

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
X Y : C
f : X ⟶ Y
hf : W f
j : κ.ord.ToType
g : X ⟶ (iteration W Z κ).obj j
⊢ ∃ g', f ≫ g' = g ≫ (iteration W Z κ).map (homOfLE ⋯)

Tactic:
by

Goal state after: