mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
X✝ Y✝ : C
x✝¹ : Retract X✝ Y✝
x✝ : ⊤ Y✝
⊢ ⊤ X✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
X : C
⊢ P.retractClosure X ↔ ∃ Y, ∃ (_ : P Y), Nonempty (Retract X Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P Q : ObjectProperty C
h : P ≤ Q
⊢ P.retractClosure ≤ Q.retractClosure

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
inst✝ : P.IsStableUnderRetracts
⊢ P.retractClosure = P

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P Q : ObjectProperty C
inst✝ : Q.IsStableUnderRetracts
h : P ≤ Q
⊢ Q.retractClosure ≤ Q

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
⊢ P.isoClosure.retractClosure = P.retractClosure

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
⊢ ∀ {X Y : C} (x : Retract X Y), P.retractClosure Y → P.retractClosure X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝ : LocallySmall.{w, v, u} C
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁ : Q ≤ P
h₂ : P ≤ Q.isoClosure
α : Type (max u v) := (X : Subtype Q) × { p // p ≫ p = p }
X Y : C
h : Retract Y X
hX : Q X
⊢ (h.r ≫ h.i) ≫ h.r ≫ h.i = h.r ≫ h.i

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
⊢ ∀ (f g : Arrow C) (x : f ≅ g), W.colimitsOfShape J f.hom → W.colimitsOfShape J g.hom

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
J : Type u_2
inst✝¹ : Preorder J
inst✝ : OrderBot J
F : J ⥤ C
c : Cocone F
hc : IsColimit c
h : ∀ (j : J), W (F.map (homOfLE ⋯))
x✝² x✝¹ : J
x✝ : x✝² ⟶ x✝¹
⊢ ((Functor.const J).obj (F.obj ⊥)).map x✝ ≫ F.map (homOfLE ⋯) = F.map (homOfLE ⋯) ≫ F.map x✝

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
J : Type u_2
inst✝¹ : Preorder J
inst✝ : OrderBot J
F : J ⥤ C
c : Cocone F
hc : IsColimit c
h : ∀ (j : J), W (F.map (homOfLE ⋯))
⊢ ∀ (j : J),
    (constCocone J (F.obj ⊥)).ι.app j ≫ c.ι.app ⊥ =
      { app := fun k ↦ F.map (homOfLE ⋯), naturality := ⋯ }.app j ≫ c.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝ : LocallySmall.{w, v, u} C
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁ : Q ≤ P
h₂ : P ≤ Q.isoClosure
α : Type (max u v) := (X : Subtype Q) × { p // p ≫ p = p }
g : {X Y : C} → Retract Y X → Q X → α := fun {X Y} h hX ↦ ⟨⟨X, hX⟩, ⟨h.r ≫ h.i, ⋯⟩⟩
R : α → Prop := fun a ↦ ∃ X Y h, ∃ (hX : Q X), g h hX = a
X Y : Subtype R → C
h : (a : Subtype R) → Retract (Y a) (X a)
hX : ∀ (a : Subtype R), ∃ (hX : Q (X a)), g (h a) hX = ↑a
y x : C
hx : Q x
r : Retract y x
⊢ (r.r ≫ r.i) ≫ r.r ≫ r.i = r.r ≫ r.i

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
h : W.IsStableUnderColimitsOfShape J
X Y : C
X₁✝ X₂✝ : J ⥤ C
c₁✝ : Cocone X₁✝
c₂✝ : Cocone X₂✝
h₁ : IsColimit c₁✝
h₂ : IsColimit c₂✝
f : X₁✝ ⟶ X₂✝
hf : W.functorCategory J f
⊢ ∀ (j : J), c₁✝.ι.app j ≫ h₁.desc { pt := c₂✝.pt, ι := f ≫ c₂✝.ι } = f.app j ≫ c₂✝.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝ : LocallySmall.{w, v, u} C
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁ : Q ≤ P
h₂ : P ≤ Q.isoClosure
α : Type (max u v) := (X : Subtype Q) × { p // p ≫ p = p }
g : {X Y : C} → Retract Y X → Q X → α := fun {X Y} h hX ↦ ⟨⟨X, hX⟩, ⟨h.r ≫ h.i, ⋯⟩⟩
R : α → Prop := fun a ↦ ∃ X Y h, ∃ (hX : Q X), g h hX = a
X Y : Subtype R → C
h : (a : Subtype R) → Retract (Y a) (X a)
hX : ∀ (a : Subtype R), ∃ (hX : Q (X a)), g (h a) hX = ↑a
y x : C
hx : Q x
r : Retract y x
⊢ ∃ a, ∃ (h₁ : Q (X a)), g (h a) h₁ = g r hx

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: J : Type v
inst✝² : SmallCategory J
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i ↦ f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsColimit c₁
t₂ : IsColimit c₂
s : Cocone (Discrete.functor f)
m : (extendCofan c₁ c₂).pt ⟶ s.pt
w : ∀ (j : Discrete (Fin (n + 1))), (extendCofan c₁ c₂).ι.app j ≫ m = s.ι.app j
⊢ m =
    ↑(BinaryCofan.IsColimit.desc' t₂ (s.ι.app { as := 0 })
        (t₁.desc { pt := s.pt, ι := Discrete.natTrans fun i ↦ s.ι.app { as := i.as.succ } }))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasBinaryCoproducts C
inst✝ : HasInitial C
n : ℕ
K : Discrete (Fin n) ⥤ C
⊢ HasColimit K

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesColimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesColimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteCoproducts C
f : Fin 0 → C
⊢ PreservesColimit (Discrete.functor f) F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝ : LocallySmall.{w, v, u} C
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁✝ : Q ≤ P
h₂✝ : P ≤ Q.isoClosure
α : Type (max u v) := (X : Subtype Q) × { p // p ≫ p = p }
g : {X Y : C} → Retract Y X → Q X → α := fun {X Y} h hX ↦ ⟨⟨X, hX⟩, ⟨h.r ≫ h.i, ⋯⟩⟩
R : α → Prop := fun a ↦ ∃ X Y h, ∃ (hX : Q X), g h hX = a
X Y : Subtype R → C
h : (a : Subtype R) → Retract (Y a) (X a)
hX : ∀ (a : Subtype R), ∃ (hX : Q (X a)), g (h a) hX = ↑a
y : C
a : Subtype R
h₁ hx : Q (X a)
r : Retract y (X a)
h₂ : g (h a) h₁ = g r hx
⊢ (h a).r ≫ (h a).i = r.r ≫ r.i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝ : LocallySmall.{w, v, u} C
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁✝ : Q ≤ P
h₂✝ : P ≤ Q.isoClosure
α : Type (max u v) := (X : Subtype Q) × { p // p ≫ p = p }
g : {X Y : C} → Retract Y X → Q X → α := fun {X Y} h hX ↦ ⟨⟨X, hX⟩, ⟨h.r ≫ h.i, ⋯⟩⟩
R : α → Prop := fun a ↦ ∃ X Y h, ∃ (hX : Q X), g h hX = a
X Y : Subtype R → C
h : (a : Subtype R) → Retract (Y a) (X a)
hX : ∀ (a : Subtype R), ∃ (hX : Q (X a)), g (h a) hX = ↑a
y : C
a : Subtype R
h₁ hx : Q (X a)
r : Retract y (X a)
h₂ : g (h a) h₁ = g r hx
hri : (h a).r ≫ (h a).i = r.r ≫ r.i
⊢ (r.i ≫ (h a).r) ≫ (h a).i ≫ r.r = 𝟙 y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝ : LocallySmall.{w, v, u} C
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁✝ : Q ≤ P
h₂✝ : P ≤ Q.isoClosure
α : Type (max u v) := (X : Subtype Q) × { p // p ≫ p = p }
g : {X Y : C} → Retract Y X → Q X → α := fun {X Y} h hX ↦ ⟨⟨X, hX⟩, ⟨h.r ≫ h.i, ⋯⟩⟩
R : α → Prop := fun a ↦ ∃ X Y h, ∃ (hX : Q X), g h hX = a
X Y : Subtype R → C
h : (a : Subtype R) → Retract (Y a) (X a)
hX : ∀ (a : Subtype R), ∃ (hX : Q (X a)), g (h a) hX = ↑a
y : C
a : Subtype R
h₁ hx : Q (X a)
r : Retract y (X a)
h₂ : g (h a) h₁ = g r hx
hri : (h a).r ≫ (h a).i = r.r ≫ r.i
⊢ ((h a).i ≫ r.r) ≫ r.i ≫ (h a).r = 𝟙 (Y ⟨↑a, ⋯⟩)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Retract

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝ : LocallySmall.{w, v, u} C
⊢ ∃ Q, ∃ (_ : ObjectProperty.Small.{w, v, u} Q), P.retractClosure ≤ Q.isoClosure

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
⊢ W.IsStableUnderColimitsOfShape J ↔ W.colimitsOfShape J ≤ W

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝¹ : Category.{v_1, u_1} J
inst✝ : W.IsStableUnderColimitsOfShape J
⊢ W.colimitsOfShape J ≤ W

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
F₁ F₂ : J ⥤ C
c₁ : Cocone F₁
c₂ : Cocone F₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : F₁ ⟶ F₂
x✝ : ∀ (j : J), IsIso (f.app j)
φ : c₁.pt ⟶ c₂.pt
hφ : ∀ (j : J), c₁.ι.app j ≫ φ = f.app j ≫ c₂.ι.app j
this : IsIso f
j : J
⊢ c₁.ι.app j ≫ φ ≫ h₂.desc { pt := c₁.pt, ι := inv f ≫ c₁.ι } = c₁.ι.app j ≫ 𝟙 c₁.pt

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
F₁ F₂ : J ⥤ C
c₁ : Cocone F₁
c₂ : Cocone F₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : F₁ ⟶ F₂
x✝ : ∀ (j : J), IsIso (f.app j)
φ : c₁.pt ⟶ c₂.pt
hφ : ∀ (j : J), c₁.ι.app j ≫ φ = f.app j ≫ c₂.ι.app j
this : IsIso f
⊢ ∀ (j : J), c₂.ι.app j ≫ h₂.desc { pt := c₁.pt, ι := inv f ≫ c₁.ι } ≫ φ = c₂.ι.app j ≫ 𝟙 c₂.pt

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
F₁ F₂ : J ⥤ C
c₁ : Cocone F₁
c₂ : Cocone F₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : F₁ ⟶ F₂
x✝ : ∀ (j : J), IsIso (f.app j)
φ : c₁.pt ⟶ c₂.pt
hφ : ∀ (j : J), c₁.ι.app j ≫ φ = f.app j ≫ c₂.ι.app j
⊢ MorphismProperty.isomorphisms C φ

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
X Y : C
f : X ⟶ Y
⊢ coproducts.{w, v, u} W f ↔ ∃ J, W.colimitsOfShape (Discrete J) f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
X Y : C
f : X ⟶ Y
J : Type w'
hf : W.colimitsOfShape (Discrete J) f
inst✝ : Small.{w, w'} J
⊢ coproducts.{w, v, u} W f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : W f
h : IsInitial (⊥_ Discrete PUnit.{w + 1})
c₁ : Cocone (Discrete.functor fun x ↦ X₁) := coconeOfDiagramInitial h (Discrete.functor fun x ↦ X₁)
c₂ : Cocone (Discrete.functor fun x ↦ X₂) := coconeOfDiagramInitial h (Discrete.functor fun x ↦ X₂)
hc₁ : IsColimit c₁
hc₂ : IsColimit c₂
x✝ : Discrete PUnit.{w + 1}
⊢ (Discrete.functor fun x ↦ X₁).obj x✝ ⟶ (Discrete.functor fun x ↦ X₂).obj x✝

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : W f
h : IsInitial (⊥_ Discrete PUnit.{w + 1})
c₁ : Cocone (Discrete.functor fun x ↦ X₁) := coconeOfDiagramInitial h (Discrete.functor fun x ↦ X₁)
c₂ : Cocone (Discrete.functor fun x ↦ X₂) := coconeOfDiagramInitial h (Discrete.functor fun x ↦ X₂)
hc₁ : IsColimit c₁
hc₂ : IsColimit c₂
x : Discrete PUnit.{w + 1}
⊢ x = ⊥_ Discrete PUnit.{w + 1}

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : W f
h : IsInitial (⊥_ Discrete PUnit.{w + 1})
c₁ : Cocone (Discrete.functor fun x ↦ X₁) := coconeOfDiagramInitial h (Discrete.functor fun x ↦ X₁)
c₂ : Cocone (Discrete.functor fun x ↦ X₂) := coconeOfDiagramInitial h (Discrete.functor fun x ↦ X₂)
hc₁ : IsColimit c₁
hc₂ : IsColimit c₂
x : Discrete PUnit.{w + 1}
⊢ c₁.ι.app x ≫ hc₁.desc { pt := c₂.pt, ι := (Discrete.natTrans fun x ↦ f) ≫ c₂.ι } = c₁.ι.app x ≫ f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
⊢ W ≤ W.colimitsOfShape (Discrete PUnit.{w + 1})

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
⊢ Monotone coproducts.{w, v, u}

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
X✝ Y✝ Z : C
inst✝¹ : HasPullbacks C
X Y S X' Y' S' : C
f : X ⟶ S
g : Y ⟶ S
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
⊢ lift (fst (fst (fst f' g') i₁) (fst (snd f' g') i₂) ≫ snd (fst f' g') i₁)
        (snd (fst (fst f' g') i₁) (fst (snd f' g') i₂) ≫ snd (snd f' g') i₂) ⋯ ≫
      lift (lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (fst f g) ⋯) (lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (snd f g) ⋯) ⋯ =
    𝟙 (pullback (fst (fst f' g') i₁) (fst (snd f' g') i₂))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
X✝ Y✝ Z : C
inst✝¹ : HasPullbacks C
X Y S X' Y' S' : C
f : X ⟶ S
g : Y ⟶ S
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
⊢ lift (lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (fst f g) ⋯) (lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (snd f g) ⋯) ⋯ ≫
      lift (fst (fst (fst f' g') i₁) (fst (snd f' g') i₂) ≫ snd (fst f' g') i₁)
        (snd (fst (fst f' g') i₁) (fst (snd f' g') i₂) ≫ snd (snd f' g') i₂) ⋯ =
    𝟙 (pullback f g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasPullbacks C
X Y S X' Y' S' : C
f : X ⟶ S
g : Y ⟶ S
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
⊢ map f g f' g' i₁ i₂ i₃ e₁ e₂ =
    (pullbackFstFstIso f g f' g' i₁ i₂ i₃ e₁ e₂).inv ≫
      snd (fst (fst f' g') i₁) (fst (snd f' g') i₂) ≫ fst (snd f' g') i₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasPullbacks C
X Y S X' Y' S' : C
f : X ⟶ S
g : Y ⟶ S
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
⊢ lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (fst f g) ⋯ ≫ fst (fst f' g') i₁ =
    lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (snd f g) ⋯ ≫ fst (snd f' g') i₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasPullbacks C
X Y S X' Y' S' : C
f : X ⟶ S
g : Y ⟶ S
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
⊢ (pullbackFstFstIso f g f' g' i₁ i₂ i₃ e₁ e₂).symm.hom ≫ fst (fst (fst f' g') i₁) (fst (snd f' g') i₂) =
    lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (fst f g) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasPullbacks C
X Y S X' Y' S' : C
f : X ⟶ S
g : Y ⟶ S
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
⊢ (pullbackFstFstIso f g f' g' i₁ i₂ i₃ e₁ e₂).symm.hom ≫ snd (fst (fst f' g') i₁) (fst (snd f' g') i₂) =
    lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (snd f g) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X Y Z S : C
f : X ⟶ S
g : Y ⟶ S
h : Z ⟶ S
⊢ map (fst f g) (fst f h) g h (snd f g) (snd f h) f ⋯ ⋯ ≫ fst g h = fst (fst f g) (fst f h) ≫ snd f g

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : W.RespectsIso
hW :
  ∀ (X₁ X₂ : J → C) [inst : HasProduct X₁] [inst_1 : HasProduct X₂] (f : (j : J) → X₁ j ⟶ X₂ j),
    (∀ (j : J), W (f j)) → W (Limits.Pi.map f)
X₁ X₂ : Discrete J ⥤ C
c₁ : Cone X₁
c₂ : Cone X₂
hc₁ : IsLimit c₁
hc₂ : IsLimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory (Discrete J) f
α : c₁.pt ⟶ c₂.pt
hα : ∀ (j : Discrete J), α ≫ c₂.π.app j = c₁.π.app j ≫ f.app j
⊢ W α

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X Y Z S : C
f : X ⟶ S
g : Y ⟶ S
h : Z ⟶ S
c : PullbackCone (fst g h) (snd f g)
m : c.pt ⟶ pullback (fst f g) (fst f h)
hfst : m ≫ map (fst f g) (fst f h) g h (snd f g) (snd f h) f ⋯ ⋯ = c.fst
hsnd : m ≫ fst (fst f g) (fst f h) = c.snd
⊢ m ≫ fst (fst f g) (fst f h) = lift c.snd (lift (c.snd ≫ fst f g) (c.fst ≫ snd g h) ⋯) ⋯ ≫ fst (fst f g) (fst f h)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesColimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesColimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteCoproducts C
n : ℕ
⊢ ∀ (f : Fin (n + 1) → C), PreservesColimit (Discrete.functor f) F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesColimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesColimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteCoproducts C
n : ℕ
K : Discrete (Fin n) ⥤ C
⊢ PreservesColimit K F

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : W.RespectsIso
hW :
  ∀ (X₁ X₂ : J → C) [inst : HasCoproduct X₁] [inst_1 : HasCoproduct X₂] (f : (j : J) → X₁ j ⟶ X₂ j),
    (∀ (j : J), W (f j)) → W (Limits.Sigma.map f)
X₁ X₂ : Discrete J ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
hc₁ : IsColimit c₁
hc₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory (Discrete J) f
α : c₁.pt ⟶ c₂.pt
hα : ∀ (j : Discrete J), c₁.ι.app j ≫ α = f.app j ≫ c₂.ι.app j
⊢ W α

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.FullSubcategory

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
P : ObjectProperty C
J : Type w
inst✝³ : Category.{w', w} J
inst✝² : HasColimitsOfShape J P.FullSubcategory
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : PreservesColimitsOfShape J P.ι
⊢ P.colimitsOfShape J ≤ P

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.FullSubcategory

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
P : ObjectProperty C
J : Type w
inst✝³ : Category.{w', w} J
inst✝² : HasLimitsOfShape J P.FullSubcategory
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : PreservesLimitsOfShape J P.ι
⊢ P.limitsOfShape J ≤ P

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.FullSubcategory

Goal state before:
pp: J : Type w
inst✝⁶ : Category.{w', w} J
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝³ : HasLimitsOfShape J C
inst✝² : PreservesLimitsOfShape J F
inst✝¹ : F.Full
inst✝ : F.Faithful
⊢ F.essImage.strictLimitsOfShape J ≤ F.essImage

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.FullSubcategory

Goal state before:
pp: J : Type w
inst✝⁶ : Category.{w', w} J
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝³ : HasColimitsOfShape J C
inst✝² : PreservesColimitsOfShape J F
inst✝¹ : F.Full
inst✝ : F.Faithful
⊢ F.essImage.strictColimitsOfShape J ≤ F.essImage

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
inst✝ : IsStableUnderCoproducts.{w, v, u} W
⊢ coproducts.{w, v, u} W ≤ W

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : IsStableUnderCoproducts.{w, v, u} Q
⊢ coproducts.{w, v, u} P ≤ Q ↔ P ≤ Q

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝ : Category.{v', u'} J
Q : ObjectProperty C
h : P ≤ Q
⊢ P.strictColimitsOfShape J ≤ Q.strictColimitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u'
inst✝¹ : Category.{v', u'} J
inst✝ : Nonempty J
⊢ ⊥.strictColimitsOfShape J = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
J' : Type u''
inst✝ : Category.{v'', u''} J'
X : C
p : P.ColimitOfShape J X
j : J
⊢ P.ι.obj { obj := p.diag.obj j, property := ⋯ } ⟶ X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
J' : Type u''
inst✝ : Category.{v'', u''} J'
X : C
p : P.ColimitOfShape J X
X✝ Y✝ : J
f : X✝ ⟶ Y✝
⊢ (CostructuredArrow.mk (p.ι.app X✝)).left.obj ⟶ (CostructuredArrow.mk (p.ι.app Y✝)).left.obj

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝ : Category.{v', u'} J
⊢ P.strictColimitsOfShape J ≤ P.colimitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u'
inst✝¹ : Category.{v', u'} J
inst✝ : Nonempty J
⊢ ⊥.colimitsOfShape J = ⊥

Tactic:
by

Goal state after: