l3lab/ntp-mathlib huggingface dataset:
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        

Proof state:
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
⊢ Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) R (Sieve.pullback ψ (Sieve.pullback (Sigma.desc f) Z))

Next tactic:
suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by 

Proof state:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
this :
  Sieve.pullback ψ (Sieve.pullback F Z) ∈
    GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) R
⊢ Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) R (Sieve.pullback ψ (Sieve.pullback (Sigma.desc f) Z))

Next tactic:
assumption
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption
        

Proof state:
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
⊢ Sieve.pullback ψ (Sieve.pullback F Z) ∈
    GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) R

Next tactic:
apply GrothendieckTopology.pullback_stable'
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption
        apply GrothendieckTopology.pullback_stable'
        

Proof state:
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
⊢ Sieve.pullback F Z ∈
    GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) (∐ fun i => X i)

Next tactic:
suffices Coverage.saturate ((extensiveCoverage C) ⊔ (regularCoverage C)) Xs
          (Z.pullback F) by assumption
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption
        apply GrothendieckTopology.pullback_stable'
        suffices Coverage.saturate ((extensiveCoverage C) ⊔ (regularCoverage C)) Xs
          (Z.pullback F) by 

Proof state:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
this : Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) Xs (Sieve.pullback F Z)
⊢ Sieve.pullback F Z ∈
    GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) (∐ fun i => X i)

Next tactic:
assumption
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption
        apply GrothendieckTopology.pullback_stable'
        suffices Coverage.saturate ((extensiveCoverage C) ⊔ (regularCoverage C)) Xs
          (Z.pullback F) by assumption
        

Proof state:
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
⊢ Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) Xs (Sieve.pullback F Z)

Next tactic:
suffices : Sieve.generate (Presieve.ofArrows X φ) ≤ Z.pullback F
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption
        apply GrothendieckTopology.pullback_stable'
        suffices Coverage.saturate ((extensiveCoverage C) ⊔ (regularCoverage C)) Xs
          (Z.pullback F) by assumption
        suffices : Sieve.generate (Presieve.ofArrows X φ) ≤ Z.pullback F
        · 

Proof state:
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
this : Sieve.generate (Presieve.ofArrows X φ) ≤ Sieve.pullback F Z
⊢ Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) Xs (Sieve.pullback F Z)

Next tactic:
apply Coverage.saturate_of_superset _ this
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption
        apply GrothendieckTopology.pullback_stable'
        suffices Coverage.saturate ((extensiveCoverage C) ⊔ (regularCoverage C)) Xs
          (Z.pullback F) by assumption
        suffices : Sieve.generate (Presieve.ofArrows X φ) ≤ Z.pullback F
        · apply Coverage.saturate_of_superset _ this
          

Proof state:
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
this : Sieve.generate (Presieve.ofArrows X φ) ≤ Sieve.pullback F Z
⊢ Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) (∐ X) (Sieve.generate (Presieve.ofArrows X φ))

Next tactic:
apply Coverage.saturate.of
/-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
    ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
    (coherentTopology C) := by
  ext B S
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · induction h with
    | of Y T hT =>
      apply Coverage.saturate.of
      simp only [Coverage.sup_covering, Set.mem_union] at hT
      exact Or.elim hT
        (fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
        (fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
    | top => apply Coverage.saturate.top
    | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption]
  · induction h with
    | of Y T hT =>
      obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
      let φ := fun (i : I) ↦ Sigma.ι X i
      let F := Sigma.desc f
      let Z := Sieve.generate T
      let Xs := (∐ fun (i : I) => X i)
      let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
      apply Coverage.saturate.transitive Y Zf
      · apply Coverage.saturate.of
        simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
          Set.mem_setOf_eq]
        exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩
      · intro R g hZfg
        dsimp at hZfg
        rw [Presieve.ofArrows_pUnit] at hZfg
        obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg
        induction hW
        rw [← hW', Sieve.pullback_comp Z]
        suffices Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
          ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption
        apply GrothendieckTopology.pullback_stable'
        suffices Coverage.saturate ((extensiveCoverage C) ⊔ (regularCoverage C)) Xs
          (Z.pullback F) by assumption
        suffices : Sieve.generate (Presieve.ofArrows X φ) ≤ Z.pullback F
        · apply Coverage.saturate_of_superset _ this
          apply Coverage.saturate.of
          

Proof state:
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : C := ∐ fun i => X i
Zf : Sieve Y := Sieve.generate (Presieve.ofArrows (fun x => Xs) fun x => F)
R : C
g : R ⟶ Y
W : C
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => X i
hW' : ψ ≫ Sigma.desc f = g
this : Sieve.generate (Presieve.ofArrows X φ) ≤ Sieve.pullback F Z
⊢ Presieve.ofArrows X φ ∈ Coverage.covering (extensiveCoverage C ⊔ regularCoverage C) (∐ X)

Next tactic:
simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
            Set.mem_setOf_eq]