l3lab/ntp-mathlib huggingface dataset:
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      cases I; rfl
  · rintro ⟨S, h1, h2, e⟩
    apply Concrete.colimit_rep_eq_of_exists
    use op S, h1.op, h2.op
    apply Concrete.multiequalizer_ext
    intro i
    apply_fun fun ee => ee i at e
    convert e
    all_goals
      dsimp [diagram]
      rw [← comp_apply, Multiequalizer.lift_ι]
      erw [Meq.equiv_symm_eq_apply]
      cases i; 

Proof state:
case h.e'_2.h.mk
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✝ T : Cover J X
x : Meq P S✝
y : Meq P T
S : Cover J X
h1 : S ⟶ S✝
h2 : S ⟶ T
Y✝ : C
f✝ : Y✝ ⟶ X
hf✝ : (Cover.sieve (op S).unop).arrows f✝
e : ↑(Meq.refine x h1) { Y := Y✝, f := f✝, hf := hf✝ } = ↑(Meq.refine y h2) { Y := Y✝, f := f✝, hf := hf✝ }
e_1✝ :
  (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) { Y := Y✝, f := f✝, hf := hf✝ }) =
    (forget D).obj (P.obj (op { Y := Y✝, f := f✝, hf := hf✝ }.Y))
⊢ ↑x (Cover.Arrow.map { Y := Y✝, f := f✝, hf := hf✝ } h1) =
    ↑x
      { Y := { Y := Y✝, f := f✝, hf := hf✝ }.Y, f := { Y := Y✝, f := f✝, hf := hf✝ }.f,
        hf := (_ : ((fun a => ↑a) S✝).arrows { Y := Y✝, f := f✝, hf := hf✝ }.f) }

Next tactic:
rfl
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      cases I; rfl
  · rintro ⟨S, h1, h2, e⟩
    apply Concrete.colimit_rep_eq_of_exists
    use op S, h1.op, h2.op
    apply Concrete.multiequalizer_ext
    intro i
    apply_fun fun ee => ee i at e
    convert e
    all_goals
      

Proof state:
case h.e'_3.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✝ T : Cover J X
x : Meq P S✝
y : Meq P T
S : Cover J X
h1 : S ⟶ S✝
h2 : S ⟶ T
i : (Cover.index (op S).unop P).L
e : ↑(Meq.refine x h1) i = ↑(Meq.refine y h2) i
e_1✝ : (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i) = (forget D).obj (P.obj (op i.Y))
⊢ (Multiequalizer.ι (Cover.index (op S).unop P) i) (((diagram J P X).map h2.op) ((Meq.equiv P T).symm y)) =
    ↑(Meq.refine y h2) i

Next tactic:
dsimp [diagram]
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      cases I; rfl
  · rintro ⟨S, h1, h2, e⟩
    apply Concrete.colimit_rep_eq_of_exists
    use op S, h1.op, h2.op
    apply Concrete.multiequalizer_ext
    intro i
    apply_fun fun ee => ee i at e
    convert e
    all_goals
      dsimp [diagram]
      

Proof state:
case h.e'_3.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✝ T : Cover J X
x : Meq P S✝
y : Meq P T
S : Cover J X
h1 : S ⟶ S✝
h2 : S ⟶ T
i : (Cover.index (op S).unop P).L
e : ↑(Meq.refine x h1) i = ↑(Meq.refine y h2) i
e_1✝ : (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i) = (forget D).obj (P.obj (op i.Y))
⊢ (Multiequalizer.ι (Cover.index S P) i)
      ((Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index T P))
          (fun I => Multiequalizer.ι (Cover.index T P) (Cover.Arrow.map I h2))
          (_ :
            ∀ (I : (Cover.index (op S).unop P).R),
              Multiequalizer.ι (Cover.index (op T).unop P)
                    (MulticospanIndex.fstTo (Cover.index (op T).unop P) (Cover.Relation.map I h2.op.unop)) ≫
                  MulticospanIndex.fst (Cover.index (op T).unop P) (Cover.Relation.map I h2.op.unop) =
                Multiequalizer.ι (Cover.index (op T).unop P)
                    (MulticospanIndex.sndTo (Cover.index (op T).unop P) (Cover.Relation.map I h2.op.unop)) ≫
                  MulticospanIndex.snd (Cover.index (op T).unop P) (Cover.Relation.map I h2.op.unop)))
        ((Meq.equiv P T).symm y)) =
    ↑y { Y := i.Y, f := i.f, hf := (_ : ((fun a => ↑a) T).arrows i.f) }

Next tactic:
rw [← comp_apply, Multiequalizer.lift_ι]
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      cases I; rfl
  · rintro ⟨S, h1, h2, e⟩
    apply Concrete.colimit_rep_eq_of_exists
    use op S, h1.op, h2.op
    apply Concrete.multiequalizer_ext
    intro i
    apply_fun fun ee => ee i at e
    convert e
    all_goals
      dsimp [diagram]
      rw [← comp_apply, Multiequalizer.lift_ι]
      

Proof state:
case h.e'_3.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✝ T : Cover J X
x : Meq P S✝
y : Meq P T
S : Cover J X
h1 : S ⟶ S✝
h2 : S ⟶ T
i : (Cover.index (op S).unop P).L
e : ↑(Meq.refine x h1) i = ↑(Meq.refine y h2) i
e_1✝ : (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i) = (forget D).obj (P.obj (op i.Y))
⊢ (Multiequalizer.ι (Cover.index T P) (Cover.Arrow.map i h2)) ((Meq.equiv P T).symm y) =
    ↑y { Y := i.Y, f := i.f, hf := (_ : ((fun a => ↑a) T).arrows i.f) }

Next tactic:
erw [Meq.equiv_symm_eq_apply]
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      cases I; rfl
  · rintro ⟨S, h1, h2, e⟩
    apply Concrete.colimit_rep_eq_of_exists
    use op S, h1.op, h2.op
    apply Concrete.multiequalizer_ext
    intro i
    apply_fun fun ee => ee i at e
    convert e
    all_goals
      dsimp [diagram]
      rw [← comp_apply, Multiequalizer.lift_ι]
      erw [Meq.equiv_symm_eq_apply]
      

Proof state:
case h.e'_3.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✝ T : Cover J X
x : Meq P S✝
y : Meq P T
S : Cover J X
h1 : S ⟶ S✝
h2 : S ⟶ T
i : (Cover.index (op S).unop P).L
e : ↑(Meq.refine x h1) i = ↑(Meq.refine y h2) i
e_1✝ : (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i) = (forget D).obj (P.obj (op i.Y))
⊢ ↑y (Cover.Arrow.map i h2) = ↑y { Y := i.Y, f := i.f, hf := (_ : ((fun a => ↑a) T).arrows i.f) }

Next tactic:
cases i
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      cases I; rfl
  · rintro ⟨S, h1, h2, e⟩
    apply Concrete.colimit_rep_eq_of_exists
    use op S, h1.op, h2.op
    apply Concrete.multiequalizer_ext
    intro i
    apply_fun fun ee => ee i at e
    convert e
    all_goals
      dsimp [diagram]
      rw [← comp_apply, Multiequalizer.lift_ι]
      erw [Meq.equiv_symm_eq_apply]
      cases i; 

Proof state:
case h.e'_3.h.mk
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✝ T : Cover J X
x : Meq P S✝
y : Meq P T
S : Cover J X
h1 : S ⟶ S✝
h2 : S ⟶ T
Y✝ : C
f✝ : Y✝ ⟶ X
hf✝ : (Cover.sieve (op S).unop).arrows f✝
e : ↑(Meq.refine x h1) { Y := Y✝, f := f✝, hf := hf✝ } = ↑(Meq.refine y h2) { Y := Y✝, f := f✝, hf := hf✝ }
e_1✝ :
  (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) { Y := Y✝, f := f✝, hf := hf✝ }) =
    (forget D).obj (P.obj (op { Y := Y✝, f := f✝, hf := hf✝ }.Y))
⊢ ↑y (Cover.Arrow.map { Y := Y✝, f := f✝, hf := hf✝ } h2) =
    ↑y
      { Y := { Y := Y✝, f := f✝, hf := hf✝ }.Y, f := { Y := Y✝, f := f✝, hf := hf✝ }.f,
        hf := (_ : ((fun a => ↑a) T).arrows { Y := Y✝, f := f✝, hf := hf✝ }.f) }

Next tactic:
rfl
/-- `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.
  

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 : Cover J X
x y : (forget D).obj ((plusObj J P).obj (op X))
h : ∀ (I : Cover.Arrow S), ((plusObj J P).map I.f.op) x = ((plusObj J P).map I.f.op) y
⊢ x = y

Next tactic:
obtain ⟨Sx, x, rfl⟩ := exists_rep x
/-- `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
  

Proof state:
case 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 : Cover J X
y : (forget D).obj ((plusObj J P).obj (op X))
Sx : Cover J X
x : Meq P Sx
h : ∀ (I : Cover.Arrow S), ((plusObj J P).map I.f.op) (mk x) = ((plusObj J P).map I.f.op) y
⊢ mk x = y

Next tactic:
obtain ⟨Sy, y, rfl⟩ := exists_rep y
/-- `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
  

Proof state:
case 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), ((plusObj J P).map I.f.op) (mk x) = ((plusObj J P).map I.f.op) (mk y)
⊢ mk x = mk y

Next tactic:
simp only [res_mk_eq_mk_pullback] at h
/-- `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.
  

Proof state:
case 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)
⊢ mk x = mk y

Next tactic:
choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)
/-- `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.
  

Proof state:
case 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)
⊢ mk x = mk y

Next tactic:
rw [eq_mk_iff_exists]
/-- `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.
  

Proof state:
case 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)
⊢ ∃ W h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
let B : J.Cover X := S.bind W
/-- `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
  

Proof state:
case 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
⊢ ∃ W h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
use B
/-- `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.
  

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
⊢ ∃ h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
let ex : B ⟶ Sx :=
    homOfLE
      (by
        rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩
        rw [← hee]
        apply leOfHom (h1 ⟨_, _, he2⟩)
        exact he1)