l3lab/ntp-mathlib huggingface dataset:
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
⊢ B ≤ Sx

Next tactic:
rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        

Proof state:
case intro.intro.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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
Y : C
f : Y ⟶ X
Z : C
e1 : Y ⟶ Z
e2 : Z ⟶ X
he2 : (Cover.sieve S).arrows e2
he1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1
hee : e1 ≫ e2 = f
⊢ ((fun a => ↑a) Sx).arrows f

Next tactic:
rw [← hee]
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        

Proof state:
case intro.intro.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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
Y : C
f : Y ⟶ X
Z : C
e1 : Y ⟶ Z
e2 : Z ⟶ X
he2 : (Cover.sieve S).arrows e2
he1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1
hee : e1 ≫ e2 = f
⊢ ((fun a => ↑a) Sx).arrows (e1 ≫ e2)

Next tactic:
apply leOfHom (h1 ⟨_, _, he2⟩)
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h1 ⟨_, _, he2⟩)
        

Proof state:
case intro.intro.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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
Y : C
f : Y ⟶ X
Z : C
e1 : Y ⟶ Z
e2 : Z ⟶ X
he2 : (Cover.sieve S).arrows e2
he1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1
hee : e1 ≫ e2 = f
⊢ ((fun a => ↑a) (W { Y := Z, f := e2, hf := he2 })).arrows e1

Next tactic:
exact he1
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h1 ⟨_, _, he2⟩)
        exact he1)
  

Proof state:
case h
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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
ex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)
⊢ ∃ h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
let ey : B ⟶ Sy :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h2 ⟨_, _, he2⟩)
        exact he1)
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h1 ⟨_, _, he2⟩)
        exact he1)
  let ey : B ⟶ Sy :=
    homOfLE
      (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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
ex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)
⊢ B ≤ Sy

Next tactic:
rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h1 ⟨_, _, he2⟩)
        exact he1)
  let ey : B ⟶ Sy :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        

Proof state:
case intro.intro.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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
ex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)
Y : C
f : Y ⟶ X
Z : C
e1 : Y ⟶ Z
e2 : Z ⟶ X
he2 : (Cover.sieve S).arrows e2
he1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1
hee : e1 ≫ e2 = f
⊢ ((fun a => ↑a) Sy).arrows f

Next tactic:
rw [← hee]
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h1 ⟨_, _, he2⟩)
        exact he1)
  let ey : B ⟶ Sy :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        

Proof state:
case intro.intro.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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
ex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)
Y : C
f : Y ⟶ X
Z : C
e1 : Y ⟶ Z
e2 : Z ⟶ X
he2 : (Cover.sieve S).arrows e2
he1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1
hee : e1 ≫ e2 = f
⊢ ((fun a => ↑a) Sy).arrows (e1 ≫ e2)

Next tactic:
apply leOfHom (h2 ⟨_, _, he2⟩)
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h1 ⟨_, _, he2⟩)
        exact he1)
  let ey : B ⟶ Sy :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h2 ⟨_, _, he2⟩)
        

Proof state:
case intro.intro.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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
ex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)
Y : C
f : Y ⟶ X
Z : C
e1 : Y ⟶ Z
e2 : Z ⟶ X
he2 : (Cover.sieve S).arrows e2
he1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1
hee : e1 ≫ e2 = f
⊢ ((fun a => ↑a) (W { Y := Z, f := e2, hf := he2 })).arrows e1

Next tactic:
exact he1
/-- `P⁺` is always separated. -/
theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : (J.plusObj P).obj (op X))
    (h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
  -- First, we choose representatives for x and y.
  obtain ⟨Sx, x, rfl⟩ := exists_rep x
  obtain ⟨Sy, y, rfl⟩ := exists_rep y
  simp only [res_mk_eq_mk_pullback] at h
  -- Next, using our assumption,
  -- choose covers over which the pullbacks of these representatives become equal.
  choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
  -- To prove equality, it suffices to prove that there exists a cover over which
  -- the representatives become equal.
  rw [eq_mk_iff_exists]
  -- Construct the cover over which the representatives become equal by combining the various
  -- covers chosen above.
  let B : J.Cover X := S.bind W
  use B
  -- Prove that this cover refines the two covers over which our representatives are defined
  -- and use these proofs.
  let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h1 ⟨_, _, he2⟩)
        exact he1)
  let ey : B ⟶ Sy :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h2 ⟨_, _, he2⟩)
        exact he1)
  

Proof state:
case h
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)
X : C
P : Cᵒᵖ ⥤ D
S Sx : Cover J X
x : Meq P Sx
Sy : Cover J X
y : Meq P Sy
h : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)
W : (I : Cover.Arrow S) → Cover J I.Y
h1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx
h2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy
hh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)
B : Cover J X := Cover.bind S W
ex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)
ey : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)
⊢ ∃ h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
use ex, ey