l3lab/ntp-mathlib huggingface dataset:
theorem inj_of_sep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) : Function.Injective ((J.toPlus P).app (op X)) := by
  intro x y h
  simp only [toPlus_eq_mk] at h
  rw [eq_mk_iff_exists] at h
  obtain ⟨W, h1, h2, hh⟩ := h
  

Proof state:
case intro.intro.intro
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
x y : (forget D).obj (P.obj (op X))
W : Cover J X
h1 h2 : W ⟶ ⊤
hh : Meq.refine (Meq.mk ⊤ x) h1 = Meq.refine (Meq.mk ⊤ y) h2
⊢ x = y

Next tactic:
apply hsep X W
theorem inj_of_sep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) : Function.Injective ((J.toPlus P).app (op X)) := by
  intro x y h
  simp only [toPlus_eq_mk] at h
  rw [eq_mk_iff_exists] at h
  obtain ⟨W, h1, h2, hh⟩ := h
  apply hsep X W
  

Proof state:
case intro.intro.intro.a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
x y : (forget D).obj (P.obj (op X))
W : Cover J X
h1 h2 : W ⟶ ⊤
hh : Meq.refine (Meq.mk ⊤ x) h1 = Meq.refine (Meq.mk ⊤ y) h2
⊢ ∀ (I : Cover.Arrow W), (P.map I.f.op) x = (P.map I.f.op) y

Next tactic:
intro I
theorem inj_of_sep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) : Function.Injective ((J.toPlus P).app (op X)) := by
  intro x y h
  simp only [toPlus_eq_mk] at h
  rw [eq_mk_iff_exists] at h
  obtain ⟨W, h1, h2, hh⟩ := h
  apply hsep X W
  intro I
  

Proof state:
case intro.intro.intro.a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
x y : (forget D).obj (P.obj (op X))
W : Cover J X
h1 h2 : W ⟶ ⊤
hh : Meq.refine (Meq.mk ⊤ x) h1 = Meq.refine (Meq.mk ⊤ y) h2
I : Cover.Arrow W
⊢ (P.map I.f.op) x = (P.map I.f.op) y

Next tactic:
apply_fun fun e => e I at hh
theorem inj_of_sep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) : Function.Injective ((J.toPlus P).app (op X)) := by
  intro x y h
  simp only [toPlus_eq_mk] at h
  rw [eq_mk_iff_exists] at h
  obtain ⟨W, h1, h2, hh⟩ := h
  apply hsep X W
  intro I
  apply_fun fun e => e I at hh
  

Proof state:
case intro.intro.intro.a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
x y : (forget D).obj (P.obj (op X))
W : Cover J X
h1 h2 : W ⟶ ⊤
I : Cover.Arrow W
hh : ↑(Meq.refine (Meq.mk ⊤ x) h1) I = ↑(Meq.refine (Meq.mk ⊤ y) h2) I
⊢ (P.map I.f.op) x = (P.map I.f.op) y

Next tactic:
exact hh
/-- An auxiliary definition to be used in the proof of `exists_of_sep` below.
  Given a compatible family of local sections for `P⁺`, and representatives of said sections,
  construct a compatible family of local sections of `P` over the combination of the covers
  associated to the representatives.
  The separatedness condition is used to prove compatibility among these local sections of `P`. -/
def meqOfSep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) (T : ∀ I : S.Arrow, J.Cover I.Y)
    (t : ∀ I : S.Arrow, Meq P (T I)) (ht : ∀ I : S.Arrow, s I = mk (t I)) : Meq P (S.bind T) where
  val I := t I.fromMiddle I.toMiddle
  property := by
    

Proof state:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
T : (I : Cover.Arrow S) → Cover J I.Y
t : (I : Cover.Arrow S) → Meq P (T I)
ht : ∀ (I : Cover.Arrow S), ↑s I = mk (t I)
⊢ ∀ (I : Cover.Relation (Cover.bind S T)),
    (P.map I.g₁.op) ((fun I => ↑(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.fst I)) =
      (P.map I.g₂.op) ((fun I => ↑(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.snd I))

Next tactic:
intro II
/-- An auxiliary definition to be used in the proof of `exists_of_sep` below.
  Given a compatible family of local sections for `P⁺`, and representatives of said sections,
  construct a compatible family of local sections of `P` over the combination of the covers
  associated to the representatives.
  The separatedness condition is used to prove compatibility among these local sections of `P`. -/
def meqOfSep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) (T : ∀ I : S.Arrow, J.Cover I.Y)
    (t : ∀ I : S.Arrow, Meq P (T I)) (ht : ∀ I : S.Arrow, s I = mk (t I)) : Meq P (S.bind T) where
  val I := t I.fromMiddle I.toMiddle
  property := by
    intro II
    

Proof state:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
T : (I : Cover.Arrow S) → Cover J I.Y
t : (I : Cover.Arrow S) → Meq P (T I)
ht : ∀ (I : Cover.Arrow S), ↑s I = mk (t I)
II : Cover.Relation (Cover.bind S T)
⊢ (P.map II.g₁.op) ((fun I => ↑(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.fst II)) =
    (P.map II.g₂.op) ((fun I => ↑(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.snd II))

Next tactic:
apply inj_of_sep P hsep
/-- An auxiliary definition to be used in the proof of `exists_of_sep` below.
  Given a compatible family of local sections for `P⁺`, and representatives of said sections,
  construct a compatible family of local sections of `P` over the combination of the covers
  associated to the representatives.
  The separatedness condition is used to prove compatibility among these local sections of `P`. -/
def meqOfSep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) (T : ∀ I : S.Arrow, J.Cover I.Y)
    (t : ∀ I : S.Arrow, Meq P (T I)) (ht : ∀ I : S.Arrow, s I = mk (t I)) : Meq P (S.bind T) where
  val I := t I.fromMiddle I.toMiddle
  property := by
    intro II
    apply inj_of_sep P hsep
    

Proof state:
case a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
T : (I : Cover.Arrow S) → Cover J I.Y
t : (I : Cover.Arrow S) → Meq P (T I)
ht : ∀ (I : Cover.Arrow S), ↑s I = mk (t I)
II : Cover.Relation (Cover.bind S T)
⊢ ((toPlus J P).app (op II.Z))
      ((P.map II.g₁.op) ((fun I => ↑(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.fst II))) =
    ((toPlus J P).app (op II.Z))
      ((P.map II.g₂.op) ((fun I => ↑(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.snd II)))

Next tactic:
rw [← comp_apply, ← comp_apply, (J.toPlus P).naturality, (J.toPlus P).naturality, comp_apply,
      comp_apply]
/-- An auxiliary definition to be used in the proof of `exists_of_sep` below.
  Given a compatible family of local sections for `P⁺`, and representatives of said sections,
  construct a compatible family of local sections of `P` over the combination of the covers
  associated to the representatives.
  The separatedness condition is used to prove compatibility among these local sections of `P`. -/
def meqOfSep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) (T : ∀ I : S.Arrow, J.Cover I.Y)
    (t : ∀ I : S.Arrow, Meq P (T I)) (ht : ∀ I : S.Arrow, s I = mk (t I)) : Meq P (S.bind T) where
  val I := t I.fromMiddle I.toMiddle
  property := by
    intro II
    apply inj_of_sep P hsep
    rw [← comp_apply, ← comp_apply, (J.toPlus P).naturality, (J.toPlus P).naturality, comp_apply,
      comp_apply]
    

Proof state:
case a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
T : (I : Cover.Arrow S) → Cover J I.Y
t : (I : Cover.Arrow S) → Meq P (T I)
ht : ∀ (I : Cover.Arrow S), ↑s I = mk (t I)
II : Cover.Relation (Cover.bind S T)
⊢ ((plusObj J P).map II.g₁.op)
      (((toPlus J P).app (op II.Y₁))
        ((fun I => ↑(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.fst II))) =
    ((plusObj J P).map II.g₂.op)
      (((toPlus J P).app (op II.Y₂))
        ((fun I => ↑(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.snd II)))

Next tactic:
erw [toPlus_apply (T II.fst.fromMiddle) (t II.fst.fromMiddle) II.fst.toMiddle,
      toPlus_apply (T II.snd.fromMiddle) (t II.snd.fromMiddle) II.snd.toMiddle, ← ht, ← ht, ←
      comp_apply, ← comp_apply, ← (J.plusObj P).map_comp, ← (J.plusObj P).map_comp]
/-- An auxiliary definition to be used in the proof of `exists_of_sep` below.
  Given a compatible family of local sections for `P⁺`, and representatives of said sections,
  construct a compatible family of local sections of `P` over the combination of the covers
  associated to the representatives.
  The separatedness condition is used to prove compatibility among these local sections of `P`. -/
def meqOfSep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) (T : ∀ I : S.Arrow, J.Cover I.Y)
    (t : ∀ I : S.Arrow, Meq P (T I)) (ht : ∀ I : S.Arrow, s I = mk (t I)) : Meq P (S.bind T) where
  val I := t I.fromMiddle I.toMiddle
  property := by
    intro II
    apply inj_of_sep P hsep
    rw [← comp_apply, ← comp_apply, (J.toPlus P).naturality, (J.toPlus P).naturality, comp_apply,
      comp_apply]
    erw [toPlus_apply (T II.fst.fromMiddle) (t II.fst.fromMiddle) II.fst.toMiddle,
      toPlus_apply (T II.snd.fromMiddle) (t II.snd.fromMiddle) II.snd.toMiddle, ← ht, ← ht, ←
      comp_apply, ← comp_apply, ← (J.plusObj P).map_comp, ← (J.plusObj P).map_comp]
    

Proof state:
case a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
T : (I : Cover.Arrow S) → Cover J I.Y
t : (I : Cover.Arrow S) → Meq P (T I)
ht : ∀ (I : Cover.Arrow S), ↑s I = mk (t I)
II : Cover.Relation (Cover.bind S T)
⊢ ((plusObj J P).map ((Cover.Arrow.toMiddle (Cover.Relation.fst II)).f.op ≫ II.g₁.op))
      (↑s (Cover.Arrow.fromMiddle (Cover.Relation.fst II))) =
    ((plusObj J P).map ((Cover.Arrow.toMiddle (Cover.Relation.snd II)).f.op ≫ II.g₂.op))
      (↑s (Cover.Arrow.fromMiddle (Cover.Relation.snd II)))

Next tactic:
rw [← op_comp, ← op_comp]
/-- An auxiliary definition to be used in the proof of `exists_of_sep` below.
  Given a compatible family of local sections for `P⁺`, and representatives of said sections,
  construct a compatible family of local sections of `P` over the combination of the covers
  associated to the representatives.
  The separatedness condition is used to prove compatibility among these local sections of `P`. -/
def meqOfSep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) (T : ∀ I : S.Arrow, J.Cover I.Y)
    (t : ∀ I : S.Arrow, Meq P (T I)) (ht : ∀ I : S.Arrow, s I = mk (t I)) : Meq P (S.bind T) where
  val I := t I.fromMiddle I.toMiddle
  property := by
    intro II
    apply inj_of_sep P hsep
    rw [← comp_apply, ← comp_apply, (J.toPlus P).naturality, (J.toPlus P).naturality, comp_apply,
      comp_apply]
    erw [toPlus_apply (T II.fst.fromMiddle) (t II.fst.fromMiddle) II.fst.toMiddle,
      toPlus_apply (T II.snd.fromMiddle) (t II.snd.fromMiddle) II.snd.toMiddle, ← ht, ← ht, ←
      comp_apply, ← comp_apply, ← (J.plusObj P).map_comp, ← (J.plusObj P).map_comp]
    rw [← op_comp, ← op_comp]
    

Proof state:
case a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
T : (I : Cover.Arrow S) → Cover J I.Y
t : (I : Cover.Arrow S) → Meq P (T I)
ht : ∀ (I : Cover.Arrow S), ↑s I = mk (t I)
II : Cover.Relation (Cover.bind S T)
⊢ ((plusObj J P).map (II.g₁ ≫ (Cover.Arrow.toMiddle (Cover.Relation.fst II)).f).op)
      (↑s (Cover.Arrow.fromMiddle (Cover.Relation.fst II))) =
    ((plusObj J P).map (II.g₂ ≫ (Cover.Arrow.toMiddle (Cover.Relation.snd II)).f).op)
      (↑s (Cover.Arrow.fromMiddle (Cover.Relation.snd II)))

Next tactic:
let IR : S.Relation :=
      ⟨_, _, _, II.g₁ ≫ II.fst.toMiddleHom, II.g₂ ≫ II.snd.toMiddleHom, II.fst.fromMiddleHom,
        II.snd.fromMiddleHom, II.fst.from_middle_condition, II.snd.from_middle_condition, by
          simpa only [Category.assoc, II.fst.middle_spec, II.snd.middle_spec] using II.w⟩
/-- An auxiliary definition to be used in the proof of `exists_of_sep` below.
  Given a compatible family of local sections for `P⁺`, and representatives of said sections,
  construct a compatible family of local sections of `P` over the combination of the covers
  associated to the representatives.
  The separatedness condition is used to prove compatibility among these local sections of `P`. -/
def meqOfSep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) (T : ∀ I : S.Arrow, J.Cover I.Y)
    (t : ∀ I : S.Arrow, Meq P (T I)) (ht : ∀ I : S.Arrow, s I = mk (t I)) : Meq P (S.bind T) where
  val I := t I.fromMiddle I.toMiddle
  property := by
    intro II
    apply inj_of_sep P hsep
    rw [← comp_apply, ← comp_apply, (J.toPlus P).naturality, (J.toPlus P).naturality, comp_apply,
      comp_apply]
    erw [toPlus_apply (T II.fst.fromMiddle) (t II.fst.fromMiddle) II.fst.toMiddle,
      toPlus_apply (T II.snd.fromMiddle) (t II.snd.fromMiddle) II.snd.toMiddle, ← ht, ← ht, ←
      comp_apply, ← comp_apply, ← (J.plusObj P).map_comp, ← (J.plusObj P).map_comp]
    rw [← op_comp, ← op_comp]
    let IR : S.Relation :=
      ⟨_, _, _, II.g₁ ≫ II.fst.toMiddleHom, II.g₂ ≫ II.snd.toMiddleHom, II.fst.fromMiddleHom,
        II.snd.fromMiddleHom, II.fst.from_middle_condition, II.snd.from_middle_condition, by
          

Proof state:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
T : (I : Cover.Arrow S) → Cover J I.Y
t : (I : Cover.Arrow S) → Meq P (T I)
ht : ∀ (I : Cover.Arrow S), ↑s I = mk (t I)
II : Cover.Relation (Cover.bind S T)
⊢ (II.g₁ ≫ Cover.Arrow.toMiddleHom (Cover.Relation.fst II)) ≫ Cover.Arrow.fromMiddleHom (Cover.Relation.fst II) =
    (II.g₂ ≫ Cover.Arrow.toMiddleHom (Cover.Relation.snd II)) ≫ Cover.Arrow.fromMiddleHom (Cover.Relation.snd II)

Next tactic:
simpa only [Category.assoc, II.fst.middle_spec, II.snd.middle_spec] using II.w
theorem exists_of_sep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) :
    ∃ t : (J.plusObj P).obj (op X), Meq.mk S t = s := by
  

Proof state:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
⊢ ∃ t, Meq.mk S t = s

Next tactic:
have inj : ∀ X : C, Function.Injective ((J.toPlus P).app (op X)) := inj_of_sep _ hsep
theorem exists_of_sep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) :
    ∃ t : (J.plusObj P).obj (op X), Meq.mk S t = s := by
  have inj : ∀ X : C, Function.Injective ((J.toPlus P).app (op X)) := inj_of_sep _ hsep
  -- Choose representatives for the given local sections.
  

Proof state:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
inj : ∀ (X : C), Function.Injective ⇑((toPlus J P).app (op X))
⊢ ∃ t, Meq.mk S t = s

Next tactic:
choose T t ht using fun I => exists_rep (s I)
theorem exists_of_sep (P : Cᵒᵖ ⥤ D)
    (hsep :
      ∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
        (∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
    (X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) :
    ∃ t : (J.plusObj P).obj (op X), Meq.mk S t = s := by
  have inj : ∀ X : C, Function.Injective ((J.toPlus P).app (op X)) := inj_of_sep _ hsep
  -- Choose representatives for the given local sections.
  choose T t ht using fun I => exists_rep (s I)
  -- Construct a large cover over which we will define a representative that will
  -- provide the gluing of the given local sections.
  

Proof state:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),
    (∀ (I : Cover.Arrow S), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
S : Cover J X
s : Meq (plusObj J P) S
inj : ∀ (X : C), Function.Injective ⇑((toPlus J P).app (op X))
T : (I : Cover.Arrow S) → Cover J I.Y
t : (I : Cover.Arrow S) → Meq P (T I)
ht : ∀ (I : Cover.Arrow S), ↑s I = mk (t I)
⊢ ∃ t, Meq.mk S t = s

Next tactic:
let B : J.Cover X := S.bind T