mathlib-initiative/mathlib-tactics huggingface dataset:
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
hJ : HasCardinalLT (↑J.obj) κ'
h✝ : κ ≤ κ'
A : Type u
x✝¹ : SmallCategory A
x✝ : IsCardinalFiltered A κ'
F : A ⥤ CardinalDirectedPoset κ
c : Cocone F
this✝ : IsFiltered A
this : IsCardinalFiltered A κ
hc : IsColimit ((forget (CardinalDirectedPoset κ)).mapCocone c)
f : J ⟶ c.pt
j : ↑J.obj → A
g : (x : ↑J.obj) → (fun X ↦ X) ((F ⋙ forget (CardinalDirectedPoset κ)).obj (j x))
hg :
  ∀ (x : ↑J.obj),
    (ConcreteCategory.hom (((forget (CardinalDirectedPoset κ)).mapCocone c).ι.app (j x))) (g x) =
      (ConcreteCategory.hom f) x
m : A := IsCardinalFiltered.max j hJ
φ : ↑J.obj → ↑(F.obj m).obj := fun x ↦ (ConcreteCategory.hom (F.map (IsCardinalFiltered.toMax j hJ x))) (g x)
hφ : ∀ (x : ↑J.obj), (ConcreteCategory.hom f) x = (ConcreteCategory.hom (c.ι.app m)) (φ x)
x y : ↑(Opposite.unop (Opposite.op J)).1
h : φ x = φ y
⊢ (ConcreteCategory.hom f) x = (ConcreteCategory.hom f) y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
⊢ ((biproduct.desc fun i x k ↦ if h : i = k then eqToHom ⋯ else 0) ≫
      biproduct.lift fun i j x ↦ if h : j = i then eqToHom ⋯ else 0) =
    𝟙 (⨁ fun i ↦ (embedding C).obj (M.X i))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
j : J
x y : ↑(F.1 j)
⊢ { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } x ≤
      { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } y ↔
    x ≤ y

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ F.map f ≫ ofHom { toFun := ⇑(ConcreteCategory.hom (c.ι.app x✝)), inj' := ⋯, map_rel_iff' := ⋯ } =
    ofHom { toFun := ⇑(ConcreteCategory.hom (c.ι.app x✝¹)), inj' := ⋯, map_rel_iff' := ⋯ } ≫
      ((Functor.const J).obj { carrier := CoconePt hc, str := instPartialOrderCoconePt hc }).map f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
s : Cocone F
x y : CoconePt hc
h :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) x =
    (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) y
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Preadditive C
D : Type u₁
inst✝² : Category.{v₁, u₁} D
inst✝¹ : Preadditive D
F : Mat_ C ⥤ D
inst✝ : F.Additive
M : Mat_ C
i : M.ι
⊢ (additiveObjIsoBiproduct F M).hom ≫ biproduct.π (fun i ↦ F.obj ((embedding C).obj (M.X i))) i =
    F.map (M.isoBiproductEmbedding.hom ≫ biproduct.π (fun i ↦ (embedding C).obj (M.X i)) i)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Preadditive C
D : Type u₁
inst✝² : Category.{v₁, u₁} D
inst✝¹ : Preadditive D
F : Mat_ C ⥤ D
inst✝ : F.Additive
M : Mat_ C
i : M.ι
⊢ biproduct.ι (fun i ↦ F.obj ((embedding C).obj (M.X i))) i ≫ (additiveObjIsoBiproduct F M).inv =
    F.map (biproduct.ι (fun i ↦ (embedding C).obj (M.X i)) i ≫ M.isoBiproductEmbedding.inv)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : F.Additive
M N : Mat_ C
f : M ⟶ N
⊢ F.map f ≫ (additiveObjIsoBiproduct F N).hom =
    (additiveObjIsoBiproduct F M).hom ≫ biproduct.matrix fun i j ↦ F.map ((embedding C).map (f i j))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : F.Additive
M N : Mat_ C
f : M ⟶ N
⊢ (additiveObjIsoBiproduct F M).inv ≫ F.map f =
    (biproduct.matrix fun i j ↦ F.map ((embedding C).map (f i j))) ≫ (additiveObjIsoBiproduct F N).inv

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
s : Cocone F
j : J
x' y' : (F ⋙ forget PartOrdEmb).obj j
hx :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) x') =
    (ConcreteCategory.hom (s.ι.app j)) x'
hy :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) y') =
    (ConcreteCategory.hom (s.ι.app j)) y'
x✝ : (ConcreteCategory.hom (c.ι.app j)) x' ≤ (ConcreteCategory.hom (c.ι.app j)) y'
k : J
x y : ↑(F.obj k)
hx' : (ConcreteCategory.hom (c.ι.app k)) x = (ConcreteCategory.hom (c.ι.app j)) x'
hy' : (ConcreteCategory.hom (c.ι.app k)) y = (ConcreteCategory.hom (c.ι.app j)) y'
l : J
f : k ⟶ l
g : j ⟶ l
hl : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map g)) x'
l' : J
f' : k ⟶ l'
g' : j ⟶ l'
hl' : (ConcreteCategory.hom (F.map f')) y = (ConcreteCategory.hom (F.map g')) y'
m : J
a : l ⟶ m
h : (ConcreteCategory.hom (F.map (f ≫ a))) x ≤ (ConcreteCategory.hom (F.map (f ≫ a))) y
b : l' ⟶ m
h₁ : f ≫ a = f' ≫ b
h₂ : g ≫ a = g' ≫ b
⊢ (ConcreteCategory.hom (F.map (g ≫ a))) x' = (ConcreteCategory.hom (F.map (f ≫ a))) x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : C ⥤ D
inst✝ : F.Additive
X : Mat_ C
⊢ (biproduct.matrix fun i j ↦ F.map (𝟙 X i j)) = 𝟙 (⨁ fun i ↦ F.obj (X.X i))

Tactic:
ext i j
    by_cases h : j = i
    · subst h; simp
    · simp [h]

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : C ⥤ D
inst✝ : F.Additive
X : Mat_ C
⊢ (biproduct.matrix fun i j ↦ F.map (𝟙 X i j)) = 𝟙 (⨁ fun i ↦ F.obj (X.X i))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
s : Cocone F
j : J
x' y' : (F ⋙ forget PartOrdEmb).obj j
hx :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) x') =
    (ConcreteCategory.hom (s.ι.app j)) x'
hy :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) y') =
    (ConcreteCategory.hom (s.ι.app j)) y'
x✝ : (ConcreteCategory.hom (c.ι.app j)) x' ≤ (ConcreteCategory.hom (c.ι.app j)) y'
k : J
x y : ↑(F.obj k)
hx' : (ConcreteCategory.hom (c.ι.app k)) x = (ConcreteCategory.hom (c.ι.app j)) x'
hy' : (ConcreteCategory.hom (c.ι.app k)) y = (ConcreteCategory.hom (c.ι.app j)) y'
l : J
f : k ⟶ l
g : j ⟶ l
hl : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map g)) x'
l' : J
f' : k ⟶ l'
g' : j ⟶ l'
hl' : (ConcreteCategory.hom (F.map f')) y = (ConcreteCategory.hom (F.map g')) y'
m : J
a : l ⟶ m
h : (ConcreteCategory.hom (F.map (f ≫ a))) x ≤ (ConcreteCategory.hom (F.map (f ≫ a))) y
b : l' ⟶ m
h₁ : f ≫ a = f' ≫ b
h₂ : g ≫ a = g' ≫ b
⊢ (ConcreteCategory.hom (F.map (f ≫ a))) y = (ConcreteCategory.hom (F.map (g ≫ a))) y'

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
s : Cocone F
x y : CoconePt hc
⊢ { toFun := ⇑(ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))), inj' := ⋯ } x ≤
      { toFun := ⇑(ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))), inj' := ⋯ } y ↔
    x ≤ y

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
hJ : HasCardinalLT (↑J.obj) κ'
h✝ : κ ≤ κ'
A : Type u
x✝¹ : SmallCategory A
x✝ : IsCardinalFiltered A κ'
F : A ⥤ CardinalDirectedPoset κ
c : Cocone F
this✝ : IsFiltered A
this : IsCardinalFiltered A κ
hc : IsColimit ((forget (CardinalDirectedPoset κ)).mapCocone c)
j : A
f g : J ⟶ F.obj j
h : f ≫ c.ι.app j = g ≫ c.ι.app j
k : ↑J.obj → A
a : (x : ↑J.obj) → j ⟶ k x
hk :
  ∀ (x : ↑J.obj),
    (ConcreteCategory.hom (F.map (a x))) (f.hom.hom' x) = (ConcreteCategory.hom (F.map (a x))) (g.hom.hom' x)
φ : ↑J.obj → (j ⟶ IsCardinalFiltered.max k hJ) := fun x ↦ a x ≫ IsCardinalFiltered.toMax k hJ x
x : ↑J.obj
⊢ a x ≫ (fun x ↦ IsCardinalFiltered.toMax k hJ x ≫ IsCardinalFiltered.coeqHom φ hJ) x = IsCardinalFiltered.toCoeq φ hJ

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
s : Cocone F
j : J
⊢ (cocone hc).ι.app j ≫ ofHom (CoconePt.desc hc s) = s.ι.app 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
hJ : HasCardinalLT (↑J.obj) κ'
h✝ : κ ≤ κ'
A : Type u
x✝¹ : SmallCategory A
x✝ : IsCardinalFiltered A κ'
F : A ⥤ CardinalDirectedPoset κ
c : Cocone F
this✝ : IsFiltered A
this : IsCardinalFiltered A κ
hc : IsColimit ((forget (CardinalDirectedPoset κ)).mapCocone c)
j : A
f g : J ⟶ F.obj j
h : f ≫ c.ι.app j = g ≫ c.ι.app j
k : ↑J.obj → A
a : (x : ↑J.obj) → j ⟶ k x
hk :
  ∀ (x : ↑J.obj),
    (ConcreteCategory.hom (F.map (a x))) (f.hom.hom' x) = (ConcreteCategory.hom (F.map (a x))) (g.hom.hom' x)
⊢ ∃ l c b, ∀ (x : ↑J.obj), a x ≫ b x = c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
s : Cocone F
m : (cocone hc).pt ⟶ s.pt
hm : ∀ (j : J), (cocone hc).ι.app j ≫ m = s.ι.app j
⊢ m = ofHom (CoconePt.desc hc s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: α : PartOrdEmb
P : Set ↑α → Prop
X✝ Y✝ : Subtype P
f : X✝ ⟶ Y✝
x✝² x✝¹ : ↑↑X✝
x✝ : (fun x ↦ ⟨↑x, ⋯⟩) x✝² = (fun x ↦ ⟨↑x, ⋯⟩) x✝¹
⊢ x✝² = x✝¹

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: α : PartOrdEmb
P : Set ↑α → Prop
X✝ Y✝ : Subtype P
f : X✝ ⟶ Y✝
⊢ ∀ {a b : ↑↑X✝}, { toFun := fun x ↦ ⟨↑x, ⋯⟩, inj' := ⋯ } a ≤ { toFun := fun x ↦ ⟨↑x, ⋯⟩, inj' := ⋯ } b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: α : PartOrdEmb
P : Set ↑α → Prop
inst✝¹ : IsDirectedOrder (Subtype P)
inst✝ : Nonempty (Subtype P)
hP : ∀ (a : ↑α), ∃ J, P J ∧ a ∈ J
⊢ IsColimit ((forget PartOrdEmb).mapCocone (coconeOfPredicateSet P))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : F.Additive
L : Mat_ C ⥤ D
inst✝ : L.Additive
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ L.map f ≫
      (additiveObjIsoBiproduct L Y✝ ≪≫
          (biproduct.mapIso fun i ↦ α.app (Y✝.X i)) ≪≫
            (biproduct.mapIso fun i ↦ (embeddingLiftIso F).symm.app (Y✝.X i)) ≪≫
              (additiveObjIsoBiproduct (lift F) Y✝).symm).hom =
    (additiveObjIsoBiproduct L X✝ ≪≫
          (biproduct.mapIso fun i ↦ α.app (X✝.X i)) ≪≫
            (biproduct.mapIso fun i ↦ (embeddingLiftIso F).symm.app (X✝.X i)) ≪≫
              (additiveObjIsoBiproduct (lift F) X✝).symm).hom ≫
      (lift F).map f

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
h₁ h₂ : D.IsTerminal e
⊢ @lift J inst✝ κ D e h₁ = @lift J inst✝ κ D e h₂

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
h₁ h₂ : D.IsTerminal e
⊢ h₁ = h₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type u
inst✝ : Semiring R
⊢ ∀ {W X Y Z : Mat R} (f : Matrix W.obj X.obj R) (g : Matrix X.obj Y.obj R) (h : Matrix Y.obj Z.obj R),
    ((have this :=
          (have this := f;
            this) *
            have this := g;
            this;
        this) *
        have this := h;
        this) =
      (have this := f;
        this) *
        have this :=
          (have this := g;
            this) *
            have this := h;
            this;
        this

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
D : Diagram J κ
e : J
prop_id : D.W (𝟙 e)
h₁ : ∀ ⦃j : J⦄, D.P j → ∃ lift, D.W lift
h₂ : ∀ ⦃j : J⦄, D.P j → ∀ (l₁ l₂ : j ⟶ e), D.W l₁ → D.W l₂ → l₁ = l₂
h₃ : ∀ ⦃i j : J⦄ (f : i ⟶ j), D.W f → ∃ li lj, D.W li ∧ D.W lj ∧ f ≫ lj = li
i✝ j✝ : J
x✝ : i✝ ⟶ j✝
hf : D.W x✝
⊢ x✝ ≫ ⋯.choose = ⋯.choose

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
D₁ : Diagram J κ
top : J
h₁ : D₁.IsTerminal top
h₁' : ∀ (j : J) (hj : D₁.IsTerminal j), j = top
D₂ : Diagram J κ
top' : J
h₂ : D₂.IsTerminal top'
h₂' : ∀ (j : J) (hj : D₂.IsTerminal j), j = top'
hW :
  { toDiagram := D₁, top := top, isTerminal := h₁, uniq_terminal := h₁' }.W =
    { toDiagram := D₂, top := top', isTerminal := h₂, uniq_terminal := h₂' }.W
hP :
  { toDiagram := D₁, top := top, isTerminal := h₁, uniq_terminal := h₁' }.P =
    { toDiagram := D₂, top := top', isTerminal := h₂, uniq_terminal := h₂' }.P
⊢ D₁ = D₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: R : Type u
inst✝ : Semiring R
M : Mat R
i : M.obj
⊢ 𝟙 M i i = 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: R : Type u
inst✝ : Semiring R
M : Mat R
i j : M.obj
h : i ≠ j
⊢ 𝟙 M i j = 0

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
D₁ : Diagram J κ
top : J
h₁ : D₁.IsTerminal top
h₁' : ∀ (j : J) (hj : D₁.IsTerminal j), j = top
h₂ : D₁.IsTerminal top
h₂' : ∀ (j : J) (hj : D₁.IsTerminal j), j = top
hW :
  { toDiagram := D₁, top := top, isTerminal := h₁, uniq_terminal := h₁' }.W =
    { toDiagram := D₁, top := top, isTerminal := h₂, uniq_terminal := h₂' }.W
hP :
  { toDiagram := D₁, top := top, isTerminal := h₁, uniq_terminal := h₁' }.P =
    { toDiagram := D₁, top := top, isTerminal := h₂, uniq_terminal := h₂' }.P
⊢ h₁ = h₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat_ (SingleObj Rᵐᵒᵖ)
⊢ (fun i j ↦ MulOpposite.unop (𝟙 X i j)) = 𝟙 (FintypeCat.of X.ι)

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
hJ : HasCardinalLT (↑J.obj) κ'
h : κ ≤ κ'
A : Type u
x✝¹ : SmallCategory A
x✝ : IsCardinalFiltered A κ'
F : A ⥤ CardinalDirectedPoset κ
c : Cocone F
hc : IsColimit c
⊢ IsColimit ((coyoneda.obj (Opposite.op J)).mapCocone c)

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
D₁ D₂ : DiagramWithUniqueTerminal J κ
hW : D₁.W = D₂.W
hP : D₁.P = D₂.P
⊢ D₁ = D₂

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
x✝ : DiagramWithUniqueTerminal J κ
⊢ x✝.W ≤ x✝.W

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X✝ Y✝ Z✝ : Mat_ (SingleObj Rᵐᵒᵖ)
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ (fun i j ↦ MulOpposite.unop ((f ≫ g) i j)) = (fun i j ↦ MulOpposite.unop (f i j)) ≫ fun i j ↦ MulOpposite.unop (g i j)

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
x✝ : DiagramWithUniqueTerminal J κ
⊢ x✝.P ≤ x✝.P

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
S : Subtype (J.PropSetWithTop κ')
x✝ : ↑↑S
⊢ x✝ ≤ ⟨⊤, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X✝ Y✝ : Mat_ (SingleObj Rᵐᵒᵖ)
a₁✝ a₂✝ : X✝ ⟶ Y✝
w : (equivalenceSingleObjInverse R).map a₁✝ = (equivalenceSingleObjInverse R).map a₂✝
⊢ a₁✝ = a₂✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat R
⊢ (equivalenceSingleObjInverse R).obj { ι := X.obj, fintype := FintypeCat.fintype, X := fun x ↦ PUnit.unit } = X

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
K : Type u
α : K → Subtype (J.PropSetWithTop κ')
hK : HasCardinalLT K κ'
hκ' : Cardinal.aleph0 ≤ κ'
⊢ ⊤ ∈ (⋃ k, ↑(α k)) ∪ {⊤}

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
x✝¹ x✝ : DiagramWithUniqueTerminal J κ
h₁ : x✝¹.W ≤ x✝.W ∧ x✝¹.P ≤ x✝.P
h₂ : x✝.W ≤ x✝¹.W ∧ x✝.P ≤ x✝¹.P
⊢ x✝¹ = x✝

Tactic:
by

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

Goal state before:
pp: J : Type w
inst✝ : SmallCategory J
κ : Cardinal.{w}
D : DiagramWithUniqueTerminal J κ
⊢ functorMap ⋯ = 𝟙 D.top

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
⊢ ∀ {i j_1 : J} {f : i ⟶ j_1}, MorphismProperty.ofHoms (fun x ↦ 𝟙 j) f → ObjectProperty.ofObj (fun x ↦ j) i

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
K : Type u
α : K → Subtype (J.PropSetWithTop κ')
hK : Cardinal.mk K < κ'
⊢ ∃ j, ∀ (k : K), α k ≤ 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
⊢ ∀ {i j_1 : J} {f : i ⟶ j_1}, MorphismProperty.ofHoms (fun x ↦ 𝟙 j) f → ObjectProperty.ofObj (fun x ↦ j) j_1

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
j : ↑J.obj
⊢ ⊤ ∈ {⊤}

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : LocallySmall.{w, v, u} C
P : ObjectProperty C
inst✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
hP : P.IsCardinalFilteredGenerator κ
⊢ ObjectProperty.EssentiallySmall.{w + 1, v, u} ⊤

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
a : ↑J.obj
⊢ some a ∈ {↑a, ⊤}

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
⊢ Function.Surjective fun x ↦ ⟨Arrow.mk (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
j : J
x✝ : Unit
⊢ ObjectProperty.ofObj (fun x ↦ j) 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
x✝ : Subtype (ObjectProperty.ofObj fun x ↦ j)
k : J
hk : ObjectProperty.ofObj (fun x ↦ j) k
⊢ (fun x ↦ ⟨j, ⋯⟩) PUnit.unit = ⟨k, hk⟩

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
x✝ : Unit
⊢ (Diagram.single j).P 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
⊢ Function.Surjective fun x ↦ ⟨j, ⋯⟩

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : HasCardinalFilteredGenerator C κ
⊢ ∃ J x, Nonempty (C ≌ 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
⊢ none ∈ {↑(Classical.arbitrary ↑J.obj), ⊤}

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
a : ↑J.withTop.obj
⊢ ∃ S, J.PropSetWithTop κ' S ∧ a ∈ 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
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
j₁ : J
g₁ : Y ⟶ p.diag.obj j₁
j₂ : J
g₂ : Y ⟶ p.diag.obj j₂
h : f ≫ g₁ ≫ p.ι.app j₁ = f ≫ g₂ ≫ p.ι.app j₂
⊢ (f ≫ g₁) ≫ { pt := Z, ι := p.ι }.ι.app j₁ = (f ≫ g₂) ≫ { pt := Z, ι := p.ι }.ι.app j₂

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 X Y : J
⊢ 𝟙 j ≫ 𝟙 j = 𝟙 j

Tactic:
by

Goal state after: