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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : F.IsCardinalAccessible κ
κ' : Cardinal.{w}
inst✝ : Fact κ'.IsRegular
h : κ ≤ κ'
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ'
⊢ PreservesColimitsOfShape J F

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
F G : C ⥤ D
e : F ≅ G
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : F.IsCardinalAccessible κ
J : Type w
x✝ : SmallCategory J
hκ : IsCardinalFiltered J κ
⊢ PreservesColimitsOfShape J G

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Topology.Basic

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
inst✝ : TopologicalSpace R
X Y : TopCat
f : X ⟶ Y
⊢ (𝟭 TopCat).map f ≫ TopCat.ofHom { toFun := fun x ↦ Finsupp.single x 1, continuous_toFun := ⋯ } =
    TopCat.ofHom { toFun := fun x ↦ Finsupp.single x 1, continuous_toFun := ⋯ } ≫
      (free R ⋙ forget₂ (TopModuleCat R) TopCat).map f

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
F✝ G✝ : C ⥤ D
e : F✝ ≅ G✝
κ : Cardinal.{w}
inst✝³ : Fact κ.IsRegular
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : F.IsCardinalAccessible κ
inst✝ : G.IsCardinalAccessible κ
⊢ (F ⋙ G).IsCardinalAccessible κ

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F✝ G : C ⥤ D
e : F✝ ≅ G
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
A : C
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
F : J ⥤ C
c : Cocone F
hc : IsColimit c
h : IsFiltered J
j : J
⊢ IsIso ((((const C).obj A).mapCocone c).ι.app j)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F✝ G : C ⥤ D
e : F✝ ≅ G
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
A : C
J : Type w
x✝³ : SmallCategory J
x✝² : IsCardinalFiltered J κ
F : J ⥤ C
c : Cocone F
hc : IsColimit c
h : IsFiltered J
this : ∀ (j : J), IsIso ((((const C).obj A).mapCocone c).ι.app j)
x✝¹ : J
x✝ : ⋯.some ⟶ x✝¹
⊢ IsIso ((F ⋙ (const C).obj A).map x✝)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F✝ G : C ⥤ D
e : F✝ ≅ G
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
A : C
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
F : J ⥤ C
c : Cocone F
hc : IsColimit c
⊢ IsColimit (((const C).obj A).mapCocone c)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Topology.Basic

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
inst✝ : TopologicalSpace R
X : TopModuleCat R
⊢ Continuous ((Finsupp.lift (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat) R ↑X.toModuleCat) id).toFun

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : IsAccessible.{w, v₁, v₂, u₁, u₂} F
inst✝ : IsAccessible.{w, v₂, v₃, u₂, u₃} G
κF : Cardinal.{w}
w✝¹ : Fact κF.IsRegular
h✝¹ : F.IsCardinalAccessible κF
κG : Cardinal.{w}
w✝ : Fact κG.IsRegular
h✝ : G.IsCardinalAccessible κG
this : Fact (max κF κG).IsRegular
⊢ κF ≤ max κF κG

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : IsAccessible.{w, v₁, v₂, u₁, u₂} F
inst✝ : IsAccessible.{w, v₂, v₃, u₂, u₃} G
κF : Cardinal.{w}
w✝¹ : Fact κF.IsRegular
h✝¹ : F.IsCardinalAccessible κF
κG : Cardinal.{w}
w✝ : Fact κG.IsRegular
h✝ : G.IsCardinalAccessible κG
this✝ : Fact (max κF κG).IsRegular
this : F.IsCardinalAccessible (max κF κG)
⊢ κG ≤ max κF κG

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : IsAccessible.{w, v₁, v₂, u₁, u₂} F
inst✝ : IsAccessible.{w, v₂, v₃, u₂, u₃} G
⊢ IsAccessible.{w, v₁, v₃, u₁, u₃} (F ⋙ G)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
A : C
⊢ IsAccessible.{w, v₁, v₁, u₁, u₁} ((const C).obj A)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
X✝ Y : C
e : X✝ ≅ Y
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
X : (isCardinalPresentable C κ).FullSubcategory
⊢ IsCardinalPresentable ((isCardinalPresentable C κ).ι.obj X) κ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Topology.Basic

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
inst✝ : TopologicalSpace R
X Y : TopModuleCat R
f : X ⟶ Y
⊢ (forget₂ (TopModuleCat R) TopCat ⋙ free R).map f ≫
      ConcreteCategory.ofHom
        { toLinearMap := (Finsupp.lift (↑((𝟭 (TopModuleCat R)).obj Y).toModuleCat) R ↑Y.toModuleCat) id, cont := ⋯ } =
    ConcreteCategory.ofHom
        { toLinearMap := (Finsupp.lift (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat) R ↑X.toModuleCat) id, cont := ⋯ } ≫
      (𝟭 (TopModuleCat R)).map f

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
κ' : Cardinal.{w}
inst✝ : Fact κ'.IsRegular
h : κ ≤ κ'
⊢ isCardinalPresentable C κ ≤ isCardinalPresentable C κ'

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
X Y : C
e✝ : X ≅ Y
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
X✝ Y✝ : C
e : X✝ ≅ Y✝
hX : isCardinalPresentable C κ X✝
⊢ isCardinalPresentable C κ Y✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Topology.Basic

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
inst✝ : TopologicalSpace R
X : TopCat
⊢ (free R).map
        ({ app := fun X ↦ TopCat.ofHom { toFun := fun x ↦ Finsupp.single x 1, continuous_toFun := ⋯ },
              naturality := ⋯ }.app
          X) ≫
      {
            app := fun X ↦
              ConcreteCategory.ofHom
                { toLinearMap := (Finsupp.lift (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat) R ↑X.toModuleCat) id,
                  cont := ⋯ },
            naturality := ⋯ }.app
        ((free R).obj X) =
    𝟙 ((free R).obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
inst✝ : IsCardinalPresentable X κ
e : C ≌ C'
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
Y : J ⥤ C'
this : PreservesColimitsOfShape J (coyoneda.obj (op X))
⊢ ∀ {X_1 Y : C'} (f : X_1 ⟶ Y),
    (coyoneda.obj (op (e.functor.obj X)) ⋙ uliftFunctor.{v₁, v₃}).map f ≫
        (Equiv.ulift.trans ((e.toAdjunction.homEquiv X Y).trans Equiv.ulift.symm)).toIso.hom =
      (Equiv.ulift.trans ((e.toAdjunction.homEquiv X X_1).trans Equiv.ulift.symm)).toIso.hom ≫
        (e.inverse ⋙ coyoneda.obj (op X) ⋙ uliftFunctor.{v₃, v₁}).map f

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
inst✝ : IsCardinalPresentable X κ
e : C ≌ C'
⊢ IsCardinalPresentable (e.functor.obj X) κ

Tactic:
refine ⟨fun J _ _ ↦ ⟨fun {Y} ↦ ?_⟩⟩
  have := preservesColimitsOfShape_of_isCardinalPresentable X κ J
  suffices PreservesColimit Y (coyoneda.obj (op (e.functor.obj X)) ⋙ uliftFunctor.{v₁}) from
    ⟨fun {c} hc ↦ ⟨isColimitOfReflects uliftFunctor.{v₁}
        (isColimitOfPreserves (coyoneda.obj (op (e.functor.obj X)) ⋙ uliftFunctor.{v₁}) hc)⟩⟩
  have iso : coyoneda.obj (op (e.functor.obj X)) ⋙ uliftFunctor.{v₁} ≅
    e.inverse ⋙ coyoneda.obj (op X) ⋙ uliftFunctor.{v₃} :=
    NatIso.ofComponents (fun Z ↦
      (Equiv.ulift.trans ((e.toAdjunction.homEquiv X Z).trans Equiv.ulift.symm)).toIso) (by
        intro _ _ f
        ext ⟨g⟩
        simp [Adjunction.homEquiv_unit])
  exact preservesColimit_of_natIso Y iso.symm

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

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
inst✝ : IsCardinalPresentable X κ
e : C ≌ C'
⊢ IsCardinalPresentable (e.functor.obj X) κ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Topology.Basic

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
inst✝ : TopologicalSpace R
X : TopModuleCat R
⊢ { app := fun X ↦ TopCat.ofHom { toFun := fun x ↦ Finsupp.single x 1, continuous_toFun := ⋯ }, naturality := ⋯ }.app
        ((forget₂ (TopModuleCat R) TopCat).obj X) ≫
      (forget₂ (TopModuleCat R) TopCat).map
        ({
              app := fun X ↦
                ConcreteCategory.ofHom
                  { toLinearMap := (Finsupp.lift (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat) R ↑X.toModuleCat) id,
                    cont := ⋯ },
              naturality := ⋯ }.app
          X) =
    𝟙 ((forget₂ (TopModuleCat R) TopCat).obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
F : C ⥤ C'
inst✝ : F.IsEquivalence
⊢ IsCardinalPresentable (F.obj X) κ ↔ IsCardinalPresentable X κ

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : Type u_1
inst✝¹ : Category.{v_1, u_1} J
D : J ⥤ C
c : Cone D
hc : IsLimit c
X : C
inst✝ : PreservesColimit D.op (yoneda.obj X)
f : c.pt ⟶ X
⊢ ∃ j p, c.π.app j ≫ p = f

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : Type u_1
inst✝² : Category.{v_1, u_1} J
D : J ⥤ C
inst✝¹ : IsCofiltered J
c : Cone D
hc : IsLimit c
X : C
inst✝ : PreservesColimit D.op (yoneda.obj X)
i j : J
f : D.obj i ⟶ X
g : D.obj j ⟶ X
h : c.π.app i ≫ f = c.π.app j ≫ g
⊢ ∃ k u v, D.map u ≫ f = D.map v ≫ g

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Ring.LinearAlgebra

Goal state before:
pp: A B C D : CommRingCat
hA : IsField ↑A
f : A ⟶ B
g : A ⟶ C
inl : B ⟶ D
inr : C ⟶ D
inst✝¹ : Nontrivial ↑B
inst✝ : Nontrivial ↑C
h : IsPushout f g inl inr
⊢ Nontrivial ↑D

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : Type u_1
inst✝² : Category.{v_1, u_1} J
D : J ⥤ C
inst✝¹ : IsCofiltered J
c : Cone D
hc : IsLimit c
X : C
inst✝ : PreservesColimit D.op (yoneda.obj X)
i : J
f g : D.obj i ⟶ X
h : c.π.app i ≫ f = c.π.app i ≫ g
⊢ ∃ j a, D.map a ≫ f = D.map a ≫ g

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝⁴ : Fact κ.IsRegular
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
inst✝² : IsCardinalPresentable X κ
inst✝¹ : EssentiallySmall.{w, v_1, u_1} J
inst✝ : IsCardinalFiltered J κ
F : J ⥤ C
c : Cocone F
hc : IsColimit c
f : X ⟶ c.pt
⊢ ∃ j f', f' ≫ c.ι.app j = f

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝⁴ : Fact κ.IsRegular
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
inst✝² : IsCardinalPresentable X κ
inst✝¹ : EssentiallySmall.{w, v_1, u_1} J
inst✝ : IsCardinalFiltered J κ
F : J ⥤ C
c : Cocone F
hc : IsColimit c
i₁ i₂ : J
f₁ : X ⟶ F.obj i₁
f₂ : X ⟶ F.obj i₂
hf : f₁ ≫ c.ι.app i₁ = f₂ ≫ c.ι.app i₂
⊢ ∃ j u v, f₁ ≫ F.map u = f₂ ≫ F.map v

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝⁴ : Fact κ.IsRegular
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
inst✝² : IsCardinalPresentable X κ
inst✝¹ : EssentiallySmall.{w, v_1, u_1} J
inst✝ : IsCardinalFiltered J κ
F : J ⥤ C
c : Cocone F
hc : IsColimit c
i : J
f₁ f₂ : X ⟶ F.obj i
hf : f₁ ≫ c.ι.app i = f₂ ≫ c.ι.app i
⊢ ∃ j u, f₁ ≫ F.map u = f₂ ≫ F.map u

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
⊢ IsCardinalPresentable X κ ↔ (uliftCoyoneda.{t, v₁, u₁}.obj (op X)).IsCardinalAccessible κ

Tactic:
by

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

Goal state before:
pp: T : Type u
κ : Cardinal.{w}
hT : HasCardinalLT T κ
hκ : Cardinal.aleph0 ≤ κ
⊢ HasCardinalLT (Arrow (WalkingParallelFamily T)) κ

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝² : Category.{v, u} J
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
inst✝¹ : IsCardinalFiltered J κ
A : Type v'
inst✝ : Category.{u', v'} A
F : A ⥤ J
hA : HasCardinalLT (Arrow A) κ
this✝¹ : Small.{w, max u' v'} (Arrow A)
this✝ : Small.{w, v'} A
this : LocallySmall.{w, u', v'} A
e : A ≌ ShrinkHoms.{w} (Shrink.{w, v'} A) := (Shrink.equivalence A).trans (ShrinkHoms.equivalence (Shrink.{w, v'} A))
⊢ HasCardinalLT (Arrow (ShrinkHoms.{w} (Shrink.{w, v'} A))) κ

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝² : Category.{v, u} J
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
inst✝¹ : IsCardinalFiltered J κ
A : Type v'
inst✝ : Category.{u', v'} A
F : A ⥤ J
hA : HasCardinalLT (Arrow A) κ
⊢ Cocone F

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v, u} J
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
K : Type u'
S : K → J
hS : HasCardinalLT K κ
⊢ HasCardinalLT (Arrow (Discrete K)) κ

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v, u} J
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
K : Type u'
S : K → J
hS : HasCardinalLT K κ
k : K
⊢ HasCardinalLT (Arrow (Discrete K)) κ

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v, u} J
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
ι : Type v'
j : J
k : ι → J
f : (i : ι) → j ⟶ k i
hι : HasCardinalLT ι κ
φ : ι → (j ⟶ max k hι) := fun i ↦ f i ≫ toMax k hι i
⊢ ∀ (i : ι), f i ≫ (fun i ↦ toMax k hι i ≫ coeqHom φ hι) i = toCoeq φ hι

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v, u} J
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
ι : Type v'
j : J
k : ι → J
f : (i : ι) → j ⟶ k i
hι : HasCardinalLT ι κ
⊢ ∃ m a b, ∀ (i : ι), f i ≫ a i = b

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝³ : Category.{v, u} J
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
inst✝² : IsCardinalFiltered J κ
A : Type w
inst✝¹ : SmallCategory A
inst✝ : FinCategory A
F : A ⥤ J
⊢ HasCardinalLT (Arrow A) κ

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v, u} J
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
⊢ IsFiltered J

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝ : Category.{v, u} J
⊢ IsCardinalFiltered J Cardinal.aleph0 ↔ IsFiltered J

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝¹ : Preorder J
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
h : ∀ ⦃K : Type w⦄ (s : K → J), Cardinal.mk K < κ → ∃ j, ∀ (k : K), s k ≤ j
A : Type w
x✝ : SmallCategory A
F : A ⥤ J
hA : HasCardinalLT (Arrow A) κ
⊢ Cardinal.mk A < κ

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝¹ : Preorder J
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
h : ∀ ⦃K : Type w⦄ (s : K → J), Cardinal.mk K < κ → ∃ j, ∀ (k : K), s k ≤ j
A : Type w
x✝ : SmallCategory A
F : A ⥤ J
hA : HasCardinalLT (Arrow A) κ
⊢ Nonempty (Cocone F)

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{w}
hκ : Fact κ.IsRegular
ι : Type w
f : ι → κ.ord.ToType
hs : Cardinal.mk ι < κ
x✝ : { x // x ∈ Set.range f }
⊢ ∃ a, (fun i ↦ ⟨f i, ⋯⟩) a = x✝

Tactic:
by

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

Goal state before:
pp: κ : Cardinal.{w}
hκ : Fact κ.IsRegular
ι : Type w
f : ι → κ.ord.ToType
hs : Cardinal.mk ι < κ
⊢ ∃ j, ∀ (k : ι), f k ≤ j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Ring.Small

Goal state before:
pp: P Q : ObjectProperty CommRingCat
inst✝ : ObjectProperty.EssentiallySmall.{u, u, u + 1} Q
hPQ : ∀ (S : CommRingCat), P S → ∃ R, Q R ∧ ∃ f, (Hom.hom f).FiniteType
⊢ ObjectProperty.EssentiallySmall.{u, u, u + 1} P

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝² : Category.{v, u} J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
j₀ : J
A : Type w
x✝ : SmallCategory A
F : A ⥤ Under j₀
hA : HasCardinalLT (Arrow A) κ
this : IsFiltered J
c : Cocone (F ⋙ Under.forget j₀) := cocone (F ⋙ Under.forget j₀) hA
x : A → (j₀ ⟶ IsFiltered.max j₀ c.pt) := fun a ↦ (F.obj a).hom ≫ c.ι.app a ≫ IsFiltered.rightToMax j₀ c.pt
hκ' : HasCardinalLT A κ
a : A
⊢ (F.obj a).hom ≫ c.ι.app a ≫ IsFiltered.rightToMax j₀ c.pt ≫ coeqHom x hκ' =
    (((Functor.const A).obj (Under.mk (toCoeq x hκ'))).obj a).hom

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝² : Category.{v, u} J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
j₀ : J
A : Type w
x✝ : SmallCategory A
F : A ⥤ Under j₀
hA : HasCardinalLT (Arrow A) κ
this : IsFiltered J
c : Cocone (F ⋙ Under.forget j₀) := cocone (F ⋙ Under.forget j₀) hA
x : A → (j₀ ⟶ IsFiltered.max j₀ c.pt) := fun a ↦ (F.obj a).hom ≫ c.ι.app a ≫ IsFiltered.rightToMax j₀ c.pt
hκ' : HasCardinalLT A κ
a b : A
f : a ⟶ b
⊢ F.map f ≫ Under.homMk (c.ι.app b ≫ IsFiltered.rightToMax j₀ c.pt ≫ coeqHom x hκ') ⋯ =
    Under.homMk (c.ι.app a ≫ IsFiltered.rightToMax j₀ c.pt ≫ coeqHom x hκ') ⋯ ≫
      ((Functor.const A).obj (Under.mk (toCoeq x hκ'))).map f

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝² : Category.{v, u} J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
j₀ : J
A : Type w
x✝ : SmallCategory A
F : A ⥤ Under j₀
hA : HasCardinalLT (Arrow A) κ
⊢ Cocone F

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Ring.Small

Goal state before:
pp: P Q : ObjectProperty CommRingCat
inst✝ : ObjectProperty.EssentiallySmall.{u, u, u + 1} Q
hPQ : ∀ (S : CommRingCat), P S → ∃ s, Ideal.span s = ⊤ ∧ ∀ f ∈ s, Q (of (Localization.Away f))
Q' : ObjectProperty CommRingCat
w✝ : ObjectProperty.Small.{u, u, u + 1} Q'
hQ'Q : Q' ≤ Q
hQQ' : Q ≤ Q'.isoClosure
f : (n : ℕ) × (R : Fin n → Subtype Q') × Subring ((i : Fin n) → ↑↑(R i)) → CommRingCat := fun S ↦ of ↥S.snd.snd
S : CommRingCat
hS : P S
s : Set ↑S
hs : Ideal.span s = ⊤
H : ∀ f ∈ s, Q (of (Localization.Away f))
hs' : s.Finite
S' : ↑s → CommRingCat
hS' : ∀ (f : ↑s), Q' (S' f)
e : ∀ (f : ↑s), Nonempty (of (Localization.Away ↑f) ≅ S' f)
φ : ↑S →+* (i : Fin (Nat.card ↑s)) → ↑(S' ((Finite.equivFin ↑s).symm i)) :=
  ((RingEquiv.piCongrRight fun i ↦ ⋯.some.commRingCatIsoToRingEquiv).trans
          (RingEquiv.piCongrLeft (fun x ↦ ↑(S' x)) (Finite.equivFin ↑s).symm).symm).toRingHom.comp
    (algebraMap (↑S) ((i : ↑s) → ↑(of (Localization.Away ↑i))))
⊢ Function.Injective ⇑φ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Ring.Small

Goal state before:
pp: P Q : ObjectProperty CommRingCat
inst✝ : ObjectProperty.EssentiallySmall.{u, u, u + 1} Q
hPQ : ∀ (S : CommRingCat), P S → ∃ s, Ideal.span s = ⊤ ∧ ∀ f ∈ s, Q (of (Localization.Away f))
⊢ ObjectProperty.EssentiallySmall.{u, u, u + 1} P

Tactic:
by

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

Goal state before:
pp: J₁ : Type u
J₂ : Type u'
inst✝⁵ : Category.{v, u} J₁
inst✝⁴ : Category.{v', u'} J₂
κ : Cardinal.{w}
inst✝³ : Fact κ.IsRegular
inst✝² : IsCardinalFiltered J₁ κ
inst✝¹ : IsCardinalFiltered J₂ κ
A✝ : Type w
inst✝ : SmallCategory A✝
F : A✝ ⥤ J₁ × J₂
hC : HasCardinalLT (Arrow A✝) κ
c₁ : Cocone (F ⋙ Prod.fst J₁ J₂) := cocone (F ⋙ Prod.fst J₁ J₂) hC
c₂ : Cocone (F ⋙ Prod.snd J₁ J₂) := cocone (F ⋙ Prod.snd J₁ J₂) hC
i j : A✝
f : i ⟶ j
⊢ F.map f ≫ (c₁.ι.app j, c₂.ι.app j) = (c₁.ι.app i, c₂.ι.app i) ≫ ((Functor.const A✝).obj (c₁.pt, c₂.pt)).map f

Tactic:
by

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

Goal state before:
pp: J₁ : Type u
J₂ : Type u'
inst✝⁵ : Category.{v, u} J₁
inst✝⁴ : Category.{v', u'} J₂
κ : Cardinal.{w}
inst✝³ : Fact κ.IsRegular
inst✝² : IsCardinalFiltered J₁ κ
inst✝¹ : IsCardinalFiltered J₂ κ
A✝ : Type w
inst✝ : SmallCategory A✝
F : A✝ ⥤ J₁ × J₂
hC : HasCardinalLT (Arrow A✝) κ
⊢ Cocone F

Tactic:
by

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

Goal state before:
pp: ι : Type u'
J : ι → Type u
inst✝³ : (i : ι) → Category.{v, u} (J i)
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : ∀ (i : ι), IsCardinalFiltered (J i) κ
A✝ : Type w
inst✝ : SmallCategory A✝
F : A✝ ⥤ ((i : ι) → J i)
hC : HasCardinalLT (Arrow A✝) κ
c : (i : ι) → Cocone (F ⋙ Pi.eval J i) := fun i ↦ cocone (F ⋙ Pi.eval J i) hC
X Y : A✝
f : X ⟶ Y
⊢ (F.map f ≫ fun i ↦ (c i).ι.app Y) = (fun i ↦ (c i).ι.app X) ≫ ((Functor.const A✝).obj fun i ↦ (c i).pt).map f

Tactic:
by

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

Goal state before:
pp: ι : Type u'
J : ι → Type u
inst✝³ : (i : ι) → Category.{v, u} (J i)
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : ∀ (i : ι), IsCardinalFiltered (J i) κ
A✝ : Type w
inst✝ : SmallCategory A✝
F : A✝ ⥤ ((i : ι) → J i)
hC : HasCardinalLT (Arrow A✝) κ
⊢ Cocone F

Tactic:
by

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

Goal state before:
pp: J : Type u
inst✝ : Category.{v, u} J
κ : Cardinal.{w}
h₁ : ∀ ⦃ι : Type w⦄ (j : ι → J), HasCardinalLT ι κ → ∃ k, ∀ (i : ι), Nonempty (j i ⟶ k)
h₂ : ∀ ⦃ι : Type w⦄ ⦃j k : J⦄ (f : ι → (j ⟶ k)), HasCardinalLT ι κ → ∃ l a b, ∀ (i : ι), f i ≫ a = b
ι : Type w
j : J
k : ι → J
f : (i : ι) → j ⟶ k i
hι : HasCardinalLT ι κ
l : J
hl : ∀ (i : ι), Nonempty (k i ⟶ l)
a : (i : ι) → k i ⟶ l := fun i ↦ ⋯.some
m : J
b : l ⟶ m
c : j ⟶ m
hm : ∀ (i : ι), (f i ≫ a i) ≫ b = c
⊢ ∀ (i : ι), f i ≫ (fun i ↦ a i ≫ b) i = c

Tactic:
by

Goal state after: