l3lab/ntp-mathlib huggingface dataset:
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y
⊢ ∃ k f g, (F.map f) x = (F.map g) y

Next tactic:
erw [T.hom.w, T.hom.w] at h
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : G.ι.app i x = G.ι.app j y
⊢ ∃ k f g, (F.map f) x = (F.map g) y

Next tactic:
replace h := Quot.exact _ h
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x } { fst := j, snd := y }
⊢ ∃ k f g, (F.map f) x = (F.map g) y

Next tactic:
suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by 

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x } { fst := j, snd := y }
this :
  ∀ (a b : (j : J) × (forget C).obj (F.obj j)),
    EqvGen (Types.Quot.Rel (F ⋙ forget C)) a b → ∃ k f g, (F.map f) a.snd = (F.map g) b.snd
⊢ ∃ k f g, (F.map f) x = (F.map g) y

Next tactic:
exact this ⟨i, x⟩ ⟨j, y⟩ h
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x } { fst := j, snd := y }
⊢ ∀ (a b : (j : J) × (forget C).obj (F.obj j)),
    EqvGen (Types.Quot.Rel (F ⋙ forget C)) a b → ∃ k f g, (F.map f) a.snd = (F.map g) b.snd

Next tactic:
intro a b h
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  intro a b h
  

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h✝ : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x } { fst := j, snd := y }
a b : (j : J) × (forget C).obj (F.obj j)
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) a b
⊢ ∃ k f g, (F.map f) a.snd = (F.map g) b.snd

Next tactic:
induction h
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  intro a b h
  induction h
  

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x✝ : (forget C).obj (F.obj i)
y✝ : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x✝ } { fst := j, snd := y✝ }
a b x y : (j : J) × (forget C).obj (F.obj j)
hh : Types.Quot.Rel (F ⋙ forget C) x y
⊢ ∃ k f g, (F.map f) x.snd = (F.map g) y.snd

Next tactic:
case rel x y hh =>
    obtain ⟨e, he⟩ := hh
    use y.1, e, 𝟙 _
    simpa using he.symm
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  intro a b h
  induction h
  case rel x y hh =>
    

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x✝ : (forget C).obj (F.obj i)
y✝ : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x✝ } { fst := j, snd := y✝ }
a b x y : (j : J) × (forget C).obj (F.obj j)
hh : Types.Quot.Rel (F ⋙ forget C) x y
⊢ ∃ k f g, (F.map f) x.snd = (F.map g) y.snd

Next tactic:
obtain ⟨e, he⟩ := hh
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  intro a b h
  induction h
  case rel x y hh =>
    obtain ⟨e, he⟩ := hh
    

Proof state:
case intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x✝ : (forget C).obj (F.obj i)
y✝ : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x✝ } { fst := j, snd := y✝ }
a b x y : (j : J) × (forget C).obj (F.obj j)
e : x.fst ⟶ y.fst
he : y.snd = (F ⋙ forget C).map e x.snd
⊢ ∃ k f g, (F.map f) x.snd = (F.map g) y.snd

Next tactic:
use y.1, e, 𝟙 _
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  intro a b h
  induction h
  case rel x y hh =>
    obtain ⟨e, he⟩ := hh
    use y.1, e, 𝟙 _
    

Proof state:
case h
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x✝ : (forget C).obj (F.obj i)
y✝ : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x✝ } { fst := j, snd := y✝ }
a b x y : (j : J) × (forget C).obj (F.obj j)
e : x.fst ⟶ y.fst
he : y.snd = (F ⋙ forget C).map e x.snd
⊢ (F.map e) x.snd = (F.map (𝟙 y.fst)) y.snd

Next tactic:
simpa using he.symm
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  intro a b h
  induction h
  case rel x y hh =>
    obtain ⟨e, he⟩ := hh
    use y.1, e, 𝟙 _
    simpa using he.symm
  

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x✝ : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x✝ } { fst := j, snd := y }
a b x : (j : J) × (forget C).obj (F.obj j)
⊢ ∃ k f g, (F.map f) x.snd = (F.map g) x.snd

Next tactic:
case refl x =>
    exact ⟨x.1, 𝟙 _, 𝟙 _, rfl⟩
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  intro a b h
  induction h
  case rel x y hh =>
    obtain ⟨e, he⟩ := hh
    use y.1, e, 𝟙 _
    simpa using he.symm
  case refl x =>
    

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x✝ : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x✝ } { fst := j, snd := y }
a b x : (j : J) × (forget C).obj (F.obj j)
⊢ ∃ k f g, (F.map f) x.snd = (F.map g) x.snd

Next tactic:
exact ⟨x.1, 𝟙 _, 𝟙 _, rfl⟩
theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
    (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
    ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by
  let E := (forget C).mapCocone D
  let hE : IsColimit E := isColimitOfPreserves _ hD
  let G := Types.colimitCocone.{v, v} (F ⋙ forget C)
  let hG := Types.colimitCoconeIsColimit.{v, v} (F ⋙ forget C)
  let T : E ≅ G := hE.uniqueUpToIso hG
  let TX : E.pt ≅ G.pt := (Cocones.forget _).mapIso T
  apply_fun TX.hom at h
  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h
  erw [T.hom.w, T.hom.w] at h
  replace h := Quot.exact _ h
  suffices
    ∀ (a b : Σj, F.obj j) (_ : EqvGen (Limits.Types.Quot.Rel.{v, v} (F ⋙ forget C)) a b),
      ∃ (k : _) (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2
    by exact this ⟨i, x⟩ ⟨j, y⟩ h
  intro a b h
  induction h
  case rel x y hh =>
    obtain ⟨e, he⟩ := hh
    use y.1, e, 𝟙 _
    simpa using he.symm
  case refl x =>
    exact ⟨x.1, 𝟙 _, 𝟙 _, rfl⟩
  

Proof state:
case symm
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x } { fst := j, snd := y }
a b x✝ y✝ : (j : J) × (forget C).obj (F.obj j)
a✝ : EqvGen (Types.Quot.Rel (F ⋙ forget C)) x✝ y✝
a_ih✝ : ∃ k f g, (F.map f) x✝.snd = (F.map g) y✝.snd
⊢ ∃ k f g, (F.map f) y✝.snd = (F.map g) x✝.snd
case trans
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ConcreteCategory C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : PreservesColimit F (forget C)
inst✝ : IsFiltered J
D : Cocone F
i j : J
hD : IsColimit D
x : (forget C).obj (F.obj i)
y : (forget C).obj (F.obj j)
E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
hE : IsColimit E := isColimitOfPreserves (forget C) hD
G : Cocone (F ⋙ forget C) := Types.colimitCocone (F ⋙ forget C)
hG : IsColimit (Types.colimitCocone (F ⋙ forget C)) := Types.colimitCoconeIsColimit (F ⋙ forget C)
T : E ≅ G := IsColimit.uniqueUpToIso hE hG
TX : E.pt ≅ G.pt := (Cocones.forget (F ⋙ forget C)).mapIso T
h : EqvGen (Types.Quot.Rel (F ⋙ forget C)) { fst := i, snd := x } { fst := j, snd := y }
a b x✝ y✝ z✝ : (j : J) × (forget C).obj (F.obj j)
a✝¹ : EqvGen (Types.Quot.Rel (F ⋙ forget C)) x✝ y✝
a✝ : EqvGen (Types.Quot.Rel (F ⋙ forget C)) y✝ z✝
a_ih✝¹ : ∃ k f g, (F.map f) x✝.snd = (F.map g) y✝.snd
a_ih✝ : ∃ k f g, (F.map f) y✝.snd = (F.map g) z✝.snd
⊢ ∃ k f g, (F.map f) x✝.snd = (F.map g) z✝.snd

Next tactic:
case symm x y _ hh =>
    obtain ⟨k, f, g, hh⟩ := hh
    exact ⟨k, g, f, hh.symm⟩