awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.IsStableUnderComposition
ι : Type w
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
hf : Presieve.ofArrows X f ∈ J.coverings S
σ : ι → Type w'
Y : (i : ι) → σ i → C
g : (i : ι) → (j : σ i) → Y i j ⟶ X i
hg : ∀ (i : ι), Membership.mem (γ := Set (Presieve.{v, u} (X i))) (J.coverings (X i)) (Presieve.ofArrows (Y i) (g i))
ι' : Type (max u v) := (↑(Presieve.ofArrows X f).uncurry : Type (max u v))
σ' : (i : ι') → Type (max u v) :=
  fun (i : ι') ↦
    (↑(Presieve.uncurry (Presieve.ofArrows (Y (Presieve.ofArrows.idx ⋯)) (g (Presieve.ofArrows.idx ⋯)))) :
      Type (max u v))
α : Type (max u v) :=
  (↑(Presieve.uncurry
        (Presieve.ofArrows
          (fun (p : (i : ι) × σ i) ↦ Y (p.fst (β := fun (i : ι) ↦ σ i)) (p.snd (β := fun (i : ι) ↦ σ i)))
          fun (x : (i : ι) × σ i) ↦
          g (x.fst (β := fun (i : ι) ↦ σ i)) (x.snd (β := fun (i : ι) ↦ σ i)) ≫ f (x.fst (β := fun (i : ι) ↦ σ i)))) :
    Type (max u v))
τ' : (a : α) → Type (max u v) :=
  fun (a : α) ↦
    (↑(Presieve.uncurry
          (Presieve.ofArrows (Y ((Presieve.ofArrows.idx ⋯).fst (β := fun (i : ι) ↦ σ i)))
            (g ((Presieve.ofArrows.idx ⋯).fst (β := fun (i : ι) ↦ σ i))))) :
      Type (max u v))
fib : (i : ι' ⊕ α) → Type (max u v) := fun (i : ι' ⊕ α) ↦ Sum.elim (fun (i : ι') ↦ σ' i) (fun (i : α) ↦ Unit ⊕ τ' i) i
incl : (p : ι' ⊕ α) → ι :=
  fun (p : ι' ⊕ α) ↦
    Sum.elim (fun (i : ι') ↦ Presieve.ofArrows.idx ⋯)
      (fun (i : α) ↦ (Presieve.ofArrows.idx ⋯).fst (β := fun (i : ι) ↦ σ i)) p
fibincl : (i : ι' ⊕ α) → (j : fib i) → σ (incl i) :=
  fun (i : ι' ⊕ α) (j : fib i) ↦
    match (motive := (i : ι' ⊕ α) → (j : fib i) → σ (incl i)) i, j with
    | Sum.inl i, j => Presieve.ofArrows.idx ⋯
    | Sum.inr i, j =>
      Sum.elim (fun (x : Unit) ↦ (Presieve.ofArrows.idx ⋯).snd (β := fun (i : ι) ↦ σ i))
        (fun (i_1 : τ' i) ↦ Presieve.ofArrows.idx ⋯) j
i : ι' ⊕ α
⊢ Membership.mem (γ := Set (Presieve.{v, u} (X (incl i)))) (J.coverings (X (incl i)))
    (Presieve.ofArrows (fun (j : fib i) ↦ Y (incl i) (fibincl i j))
      ((fun (i : ι' ⊕ α) (j : fib i) ↦ g (incl i) (fibincl i j)) i))
have h : Eq (α := Presieve.{v, u} (X (incl i)))
  (Presieve.ofArrows (fun (j : fib i) ↦ Y (incl i) (fibincl i j))
    ((fun (i : ι' ⊕ α) (j : fib i) ↦ g (incl i) (fibincl i j)) i))
  (Presieve.ofArrows (Y (incl i)) (g (incl i)))
Tactic: convert! hg (incl i)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.IsStableUnderComposition
ι : Type w
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
hf : Presieve.ofArrows X f ∈ J.coverings S
σ : ι → Type w'
Y : (i : ι) → σ i → C
g : (i : ι) → (j : σ i) → Y i j ⟶ X i
hg : ∀ (i : ι), Membership.mem (γ := Set (Presieve.{v, u} (X i))) (J.coverings (X i)) (Presieve.ofArrows (Y i) (g i))
ι' : Type (max u v) := (↑(Presieve.ofArrows X f).uncurry : Type (max u v))
σ' : (i : ι') → Type (max u v) :=
  fun (i : ι') ↦
    (↑(Presieve.uncurry (Presieve.ofArrows (Y (Presieve.ofArrows.idx ⋯)) (g (Presieve.ofArrows.idx ⋯)))) :
      Type (max u v))
α : Type (max u v) :=
  (↑(Presieve.uncurry
        (Presieve.ofArrows
          (fun (p : (i : ι) × σ i) ↦ Y (p.fst (β := fun (i : ι) ↦ σ i)) (p.snd (β := fun (i : ι) ↦ σ i)))
          fun (x : (i : ι) × σ i) ↦
          g (x.fst (β := fun (i : ι) ↦ σ i)) (x.snd (β := fun (i : ι) ↦ σ i)) ≫ f (x.fst (β := fun (i : ι) ↦ σ i)))) :
    Type (max u v))
τ' : (a : α) → Type (max u v) :=
  fun (a : α) ↦
    (↑(Presieve.uncurry
          (Presieve.ofArrows (Y ((Presieve.ofArrows.idx ⋯).fst (β := fun (i : ι) ↦ σ i)))
            (g ((Presieve.ofArrows.idx ⋯).fst (β := fun (i : ι) ↦ σ i))))) :
      Type (max u v))
fib : (i : ι' ⊕ α) → Type (max u v) := fun (i : ι' ⊕ α) ↦ Sum.elim (fun (i : ι') ↦ σ' i) (fun (i : α) ↦ Unit ⊕ τ' i) i
incl : (p : ι' ⊕ α) → ι :=
  fun (p : ι' ⊕ α) ↦
    Sum.elim (fun (i : ι') ↦ Presieve.ofArrows.idx ⋯)
      (fun (i : α) ↦ (Presieve.ofArrows.idx ⋯).fst (β := fun (i : ι) ↦ σ i)) p
fibincl : (i : ι' ⊕ α) → (j : fib i) → σ (incl i) :=
  fun (i : ι' ⊕ α) (j : fib i) ↦
    match (motive := (i : ι' ⊕ α) → (j : fib i) → σ (incl i)) i, j with
    | Sum.inl i, j => Presieve.ofArrows.idx ⋯
    | Sum.inr i, j =>
      Sum.elim (fun (x : Unit) ↦ (Presieve.ofArrows.idx ⋯).snd (β := fun (i : ι) ↦ σ i))
        (fun (i_1 : τ' i) ↦ Presieve.ofArrows.idx ⋯) j
i : ι' ⊕ α
⊢ Eq (α := Presieve.{v, u} (X (incl i)))
    (Presieve.ofArrows (fun (j : fib i) ↦ Y (incl i) (fibincl i j))
      ((fun (i : ι' ⊕ α) (j : fib i) ↦ g (incl i) (fibincl i j)) i))
    (Presieve.ofArrows (Y (incl i)) (g (incl i)))
have h : ∀ (T : C) (u : T ⟶ X (incl i)) (hu : Presieve.ofArrows (Y (incl i)) (g (incl i)) u),
  Presieve.ofArrows (fun (j : fib i) ↦ Y (incl i) (fibincl i j))
    ((fun (i : ι' ⊕ α) (j : fib i) ↦ g (incl i) (fibincl i j)) i) u
Tactic: refine le_antisymm (fun T u ⟨p⟩ ↦ .mk _) fun T u hu ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.IsStableUnderBaseChange
X Y : C
f : X ⟶ Y
R : Presieve.{v, u} Y
hR : R ∈ J.coverings Y
inst_2 : R.HasPullbacks f
⊢ Presieve.pullbackArrows f R ∈ J.coverings X
have obtain : ∀ (ι : Type (max u v)) (Z : ι → C) (g : (i : ι) → Z i ⟶ Y) (hR : Presieve.ofArrows Z g ∈ J.coverings Y)
  [inst_3 : (Presieve.ofArrows Z g).HasPullbacks f], Presieve.pullbackArrows f (Presieve.ofArrows Z g) ∈ J.coverings X
Tactic: obtain ⟨ι, Z, g, rfl⟩ := R.exists_eq_ofArrows
Cleared hypotheses: R, hR, inst_2
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.IsStableUnderBaseChange
X Y : C
f : X ⟶ Y
ι : Type (max u v)
Z : ι → C
g : (i : ι) → Z i ⟶ Y
hR : Presieve.ofArrows Z g ∈ J.coverings Y
inst_2 : (Presieve.ofArrows Z g).HasPullbacks f
⊢ Presieve.pullbackArrows f (Presieve.ofArrows Z g) ∈ J.coverings X
have this : ∀ (i : ι), Limits.HasPullback (g i) f
Tactic: have (i : ι) : Limits.HasPullback (g i) f := Presieve.hasPullback f (Presieve.ofArrows.mk i)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.IsStableUnderBaseChange
X Y : C
f : X ⟶ Y
ι : Type (max u v)
Z : ι → C
g : (i : ι) → Z i ⟶ Y
hR : Presieve.ofArrows Z g ∈ J.coverings Y
inst_2 : (Presieve.ofArrows Z g).HasPullbacks f
this : ∀ (i : ι), Limits.HasPullback (g i) f
⊢ Presieve.pullbackArrows f (Presieve.ofArrows Z g) ∈ J.coverings X
have rw : (Presieve.ofArrows (fun (i : ι) ↦ Limits.pullback (g i) f) fun (x : ι) ↦ Limits.pullback.snd (g x) f) ∈ J.coverings X
Tactic: rw [← Presieve.ofArrows_pullback]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
K : Precoverage.{v, u} C
⊢ comap (𝟭 C) K = K
have coverings : ∀ (x : C) (x_1 : Presieve.{v, u} x), x_1 ∈ (comap (𝟭 C) K).coverings x ↔ x_1 ∈ K.coverings x
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v, v_1, u, u_1} C D
G : Functor.{v_1, v_2, u_1, u_2} D E
J : Precoverage.{v_2, u_2} E
⊢ comap (F ⋙ G) J = comap F (comap G J)
have coverings : ∀ (X : C) (R : Presieve.{v, u} X), R ∈ (comap (F ⋙ G) J).coverings X ↔ R ∈ (comap F (comap G J)).coverings X
Tactic: ext X R
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v, v_1, u, u_1} C D
G : Functor.{v_1, v_2, u_1, u_2} D E
J : Precoverage.{v_2, u_2} E
X : C
R : Presieve.{v, u} X
⊢ R ∈ (comap (F ⋙ G) J).coverings X ↔ R ∈ (comap F (comap G J)).coverings X
have coverings : ∀ (ι : Type (max u v)) (Y : ι → C) (f : (i : ι) → Y i ⟶ X),
  Presieve.ofArrows Y f ∈ (comap (F ⋙ G) J).coverings X ↔ Presieve.ofArrows Y f ∈ (comap F (comap G J)).coverings X
Tactic: obtain ⟨ι, Y, f, rfl⟩ := R.exists_eq_ofArrows
Cleared hypotheses: R
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
J K : Precoverage.{v_1, u_1} D
inst_2 : J.IsStableUnderComposition
ι : Type (max u v)
S : C
Y : ι → C
f : (i : ι) → Y i ⟶ S
hf : Presieve.ofArrows Y f ∈ (comap F J).coverings S
σ : ι → Type (max u v)
Z : (i : ι) → σ i → C
g : (i : ι) → (j : σ i) → Z i j ⟶ Y i
hg :
  ∀ (i : ι),
    Membership.mem (γ := Set (Presieve.{v, u} (Y i))) ((comap F J).coverings (Y i)) (Presieve.ofArrows (Z i) (g i))
⊢ (Presieve.ofArrows (fun (p : (i : ι) × σ i) ↦ Z (p.fst (β := fun (i : ι) ↦ σ i)) (p.snd (β := fun (i : ι) ↦ σ i)))
      fun (x : (i : ι) × σ i) ↦
      g (x.fst (β := fun (i : ι) ↦ σ i)) (x.snd (β := fun (i : ι) ↦ σ i)) ≫ f (x.fst (β := fun (i : ι) ↦ σ i))) ∈
    (comap F J).coverings S
have simp : ∀
  (hf :
    Membership.mem (γ := Set (Presieve.{v_1, u_1} (F.obj S))) (J.coverings (F.obj S))
      (Presieve.ofArrows (fun (i : ι) ↦ F.obj (Y i)) fun (i : ι) ↦ F.map (f i)))
  (hg :
    ∀ (i : ι),
      Membership.mem (γ := Set (Presieve.{v_1, u_1} (F.obj (Y i)))) (J.coverings (F.obj (Y i)))
        (Presieve.ofArrows (fun (i_1 : σ i) ↦ F.obj (Z i i_1)) fun (i_1 : σ i) ↦ F.map (g i i_1))),
  Membership.mem (γ := Set (Presieve.{v_1, u_1} (F.obj S))) (J.coverings (F.obj S))
    (Presieve.ofArrows
      (fun (i : (i : ι) × σ i) ↦ F.obj (Z (i.fst (β := fun (i : ι) ↦ σ i)) (i.snd (β := fun (i : ι) ↦ σ i))))
      fun (i : (i : ι) × σ i) ↦
      F.map (g (i.fst (β := fun (i : ι) ↦ σ i)) (i.snd (β := fun (i : ι) ↦ σ i))) ≫
        F.map (f (i.fst (β := fun (i : ι) ↦ σ i))))
Tactic: simp only [mem_comap_iff, Presieve.map_ofArrows, Functor.map_comp] at hf hg ⊢
Cleared hypotheses: hf, hg
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
J K : Precoverage.{v_1, u_1} D
inst_2 : PreservesLimitsOfShape WalkingCospan F
inst_3 : J.IsStableUnderBaseChange
ι : Type (max u v)
S : C
Y : ι → C
f : (i : ι) → Y i ⟶ S
hf : Presieve.ofArrows Y f ∈ (comap F J).coverings S
Z : C
g : Z ⟶ S
P : ι → C
p₁ : (i : ι) → P i ⟶ Z
p₂ : (i : ι) → P i ⟶ Y i
h : ∀ (i : ι), IsPullback (p₁ i) (p₂ i) g (f i)
⊢ Presieve.ofArrows P p₁ ∈ (comap F J).coverings Z
have simp : ∀
  (hf :
    Membership.mem (γ := Set (Presieve.{v_1, u_1} (F.obj S))) (J.coverings (F.obj S))
      (Presieve.ofArrows (fun (i : ι) ↦ F.obj (Y i)) fun (i : ι) ↦ F.map (f i))),
  Membership.mem (γ := Set (Presieve.{v_1, u_1} (F.obj Z))) (J.coverings (F.obj Z))
    (Presieve.ofArrows (fun (i : ι) ↦ F.obj (P i)) fun (i : ι) ↦ F.map (p₁ i))
Tactic: simp only [mem_comap_iff, Presieve.map_ofArrows] at hf ⊢
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
J K : Precoverage.{v_1, u_1} D
inst_2 : CreatesLimitsOfShape WalkingCospan F
inst_3 : J.HasPullbacks
X Y : C
R : Presieve.{v, u} Y
f : X ⟶ Y
hR : R ∈ (comap F J).coverings Y
⊢ R.HasPullbacks f
have refine : ∀ {Z : C} {g : Z ⟶ Y} (hg : R g), HasPullback g f
Tactic: refine ⟨fun {Z g} hg ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
J K : Precoverage.{v_1, u_1} D
inst_2 : CreatesLimitsOfShape WalkingCospan F
inst_3 : J.HasPullbacks
X Y : C
R : Presieve.{v, u} Y
f : X ⟶ Y
hR : R ∈ (comap F J).coverings Y
Z : C
g : Z ⟶ Y
hg : R g
⊢ HasPullback g f
have this : (Presieve.map F R).HasPullbacks (F.map f)
Tactic: have : (Presieve.map F R).HasPullbacks (F.map f) := J.hasPullbacks_of_mem (F.map f) hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
J K : Precoverage.{v_1, u_1} D
inst_2 : CreatesLimitsOfShape WalkingCospan F
inst_3 : J.HasPullbacks
X Y : C
R : Presieve.{v, u} Y
f : X ⟶ Y
hR : R ∈ (comap F J).coverings Y
Z : C
g : Z ⟶ Y
hg : R g
this : (Presieve.map F R).HasPullbacks (F.map f)
⊢ HasPullback g f
have this₁ : HasPullback (F.map g) (F.map f)
Tactic: have : HasPullback (F.map g) (F.map f) := (R.map F).hasPullback _ (R.map_map hg)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (hypothesis)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} C D
X : C
R : Presieve.{v_1, u_1} X
inst_2 : F.PreservesPairwisePullbacks R
inst_3 : R.HasPairwisePullbacks
Y Z : D
f : Y ⟶ F.obj X
h : map F R f
g : Z ⟶ F.obj X
h_1 : map F R g
Y_1 : C
u : Y_1 ⟶ X
hf : R u
Y_2 : C
u_1 : Y_2 ⟶ X
hg : R u_1
⊢ HasPullback (F.map u) (F.map u_1)
have this : HasPullback u u_1
Tactic: have := Presieve.HasPairwisePullbacks.has_pullbacks hf hg
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Precoverage (hypothesis)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} C D
X : C
R : Presieve.{v_1, u_1} X
inst_2 : F.PreservesPairwisePullbacks R
inst_3 : R.HasPairwisePullbacks
Y Z : D
f : Y ⟶ F.obj X
h : map F R f
g : Z ⟶ F.obj X
h_1 : map F R g
Y_1 : C
u : Y_1 ⟶ X
hf : R u
Y_2 : C
u_1 : Y_2 ⟶ X
hg : R u_1
this : HasPullback u u_1
⊢ HasPullback (F.map u) (F.map u_1)
have this₁ : PreservesLimit (cospan u u_1) F
Tactic: have := F.preservesLimit_cospan_of_mem_presieve _ hf hg
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
⊢ J.toGrothendieck =
    sInf
      {K : GrothendieckTopology.{u_1, u_2} C |
        ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
have refine_1 : J.toGrothendieck ≤
  sInf
    {K : GrothendieckTopology.{u_1, u_2} C |
      ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
refine_1 :
  J.toGrothendieck ≤
    sInf
      {K : GrothendieckTopology.{u_1, u_2} C |
        ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
⊢ J.toGrothendieck =
    sInf
      {K : GrothendieckTopology.{u_1, u_2} C |
        ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
have refine_2 : sInf
    {K : GrothendieckTopology.{u_1, u_2} C |
      ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X} ≤
  J.toGrothendieck
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
⊢ J.toGrothendieck ≤
    sInf
      {K : GrothendieckTopology.{u_1, u_2} C |
        ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
have refine_1 : ∀ (b : GrothendieckTopology.{u_1, u_2} C),
  Membership.mem (γ := Set (GrothendieckTopology.{u_1, u_2} C))
      {K : GrothendieckTopology.{u_1, u_2} C |
        ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
      b →
    J.toGrothendieck ≤ b
Tactic: rw [le_sInf_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
K : GrothendieckTopology.{u_1, u_2} C
hK :
  Membership.mem (γ := Set (GrothendieckTopology.{u_1, u_2} C))
    {K : GrothendieckTopology.{u_1, u_2} C |
      ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
    K
⊢ J.toGrothendieck ≤ K
have refine_1 : ∀ (i : C), (J.toGrothendieck : (a : C) → Set (Sieve a)) i ≤ (K : (a : C) → Set (Sieve a)) i
Tactic: _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
K : GrothendieckTopology.{u_1, u_2} C
hK :
  Membership.mem (γ := Set (GrothendieckTopology.{u_1, u_2} C))
    {K : GrothendieckTopology.{u_1, u_2} C |
      ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
    K
i : C
⊢ (J.toGrothendieck : (a : C) → Set (Sieve a)) i ≤ (K : (a : C) → Set (Sieve a)) i
have refine_1 : ∀ ⦃a : Sieve.{u_1, u_2} i⦄, a ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) i → a ∈ (K : (a : C) → Set (Sieve a)) i
Tactic: _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
K : GrothendieckTopology.{u_1, u_2} C
hK :
  Membership.mem (γ := Set (GrothendieckTopology.{u_1, u_2} C))
    {K : GrothendieckTopology.{u_1, u_2} C |
      ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
    K
i : C
a : Sieve.{u_1, u_2} i
hS : a ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) i
⊢ a ∈ (K : (a : C) → Set (Sieve a)) i
have refine_1 : ∀ (X : C) (S : Presieve.{u_1, u_2} X) (hS : S ∈ J.coverings X), Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X
Tactic: induction hS with
    | of _ _ hS => exact hK _ hS
    | top => exact K.top_mem _
    | pullback _ _ _ _ _ ih => exact K.pullback_stable _ ih
    | transitive _ _ _ _ _ ih1 ih2 => exact K.transitive ih1 _ ih2
Cleared hypotheses: hS
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
K : GrothendieckTopology.{u_1, u_2} C
hK :
  Membership.mem (γ := Set (GrothendieckTopology.{u_1, u_2} C))
    {K : GrothendieckTopology.{u_1, u_2} C |
      ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
    K
i : C
a : Sieve.{u_1, u_2} i
hS : a ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) i
refine_1 :
  ∀ (X : C) (S : Presieve.{u_1, u_2} X) (hS : S ∈ J.coverings X), Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X
⊢ a ∈ (K : (a : C) → Set (Sieve a)) i
have refine_1₁ : ∀ (X : C), ⊤ ∈ (K : (a : C) → Set (Sieve a)) X
Tactic: induction hS with
    | of _ _ hS => exact hK _ hS
    | top => exact K.top_mem _
    | pullback _ _ _ _ _ ih => exact K.pullback_stable _ ih
    | transitive _ _ _ _ _ ih1 ih2 => exact K.transitive ih1 _ ih2
Cleared hypotheses: hS
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
K : GrothendieckTopology.{u_1, u_2} C
hK :
  Membership.mem (γ := Set (GrothendieckTopology.{u_1, u_2} C))
    {K : GrothendieckTopology.{u_1, u_2} C |
      ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
    K
i : C
a : Sieve.{u_1, u_2} i
hS : a ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) i
refine_1 :
  ∀ (X : C) (S : Presieve.{u_1, u_2} X) (hS : S ∈ J.coverings X), Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X
refine_1₁ : ∀ (X : C), ⊤ ∈ (K : (a : C) → Set (Sieve a)) X
⊢ a ∈ (K : (a : C) → Set (Sieve a)) i
have refine_1₂ : ∀ (X : C) (S : Sieve.{u_1, u_2} X),
  J.Saturate X S →
    ∀ (Y : C) (f : Y ⟶ X) (ih : S ∈ (K : (a : C) → Set (Sieve a)) X),
      Sieve.pullback f S ∈ (K : (a : C) → Set (Sieve a)) Y
Tactic: induction hS with
    | of _ _ hS => exact hK _ hS
    | top => exact K.top_mem _
    | pullback _ _ _ _ _ ih => exact K.pullback_stable _ ih
    | transitive _ _ _ _ _ ih1 ih2 => exact K.transitive ih1 _ ih2
Cleared hypotheses: hS
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_2
inst : Category.{u_1, u_2} C
J : Precoverage.{u_1, u_2} C
K : GrothendieckTopology.{u_1, u_2} C
hK :
  Membership.mem (γ := Set (GrothendieckTopology.{u_1, u_2} C))
    {K : GrothendieckTopology.{u_1, u_2} C |
      ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X}
    K
i : C
a : Sieve.{u_1, u_2} i
hS : a ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) i
refine_1 :
  ∀ (X : C) (S : Presieve.{u_1, u_2} X) (hS : S ∈ J.coverings X), Sieve.generate S ∈ (K : (a : C) → Set (Sieve a)) X
refine_1₁ : ∀ (X : C), ⊤ ∈ (K : (a : C) → Set (Sieve a)) X
refine_1₂ :
  ∀ (X : C) (S : Sieve.{u_1, u_2} X),
    J.Saturate X S →
      ∀ (Y : C) (f : Y ⟶ X) (ih : S ∈ (K : (a : C) → Set (Sieve a)) X),
        Sieve.pullback f S ∈ (K : (a : C) → Set (Sieve a)) Y
⊢ a ∈ (K : (a : C) → Set (Sieve a)) i
have refine_1₃ : ∀ (X : C) (S R : Sieve.{u_1, u_2} X),
  J.Saturate X S →
    (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → J.Saturate Y (Sieve.pullback f R)) →
      ∀ (ih1 : S ∈ (K : (a : C) → Set (Sieve a)) X)
        (ih2 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ (K : (a : C) → Set (Sieve a)) Y),
        R ∈ (K : (a : C) → Set (Sieve a)) X
Tactic: induction hS with
    | of _ _ hS => exact hK _ hS
    | top => exact K.top_mem _
    | pullback _ _ _ _ _ ih => exact K.pullback_stable _ ih
    | transitive _ _ _ _ _ ih1 ih2 => exact K.transitive ih1 _ ih2
Cleared hypotheses: hS
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
⊢ Presieve.IsSheaf J.toGrothendieck P ↔
    ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
have mp : Presieve.IsSheaf J.toGrothendieck P →
  ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
mp :
  Presieve.IsSheaf J.toGrothendieck P →
    ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
⊢ Presieve.IsSheaf J.toGrothendieck P ↔
    ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
have mpr : (∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows) →
  Presieve.IsSheaf J.toGrothendieck P
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
H : Presieve.IsSheaf J.toGrothendieck P
x x_1 : C
x_2 : x_1 ⟶ x
x_3 : Presieve.{u_2, u_3} x
hR : x_3 ∈ J.coverings x
⊢ Presieve.IsSheafFor P (Sieve.pullback x_2 (Sieve.generate x_3)).arrows
have mp : Sieve.generate (Sieve.pullback x_2 (Sieve.generate x_3)).arrows ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) x_1
Tactic: apply H.isSheafFor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck (goal)
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
P : Functor.{u_2, u_1, u_3, u_1 + 1} Cᵒᵖ (Type u_1)
H : Presieve.IsSheaf J.toGrothendieck P
x x_1 : C
x_2 : x_1 ⟶ x
x_3 : Presieve.{u_2, u_3} x
hR : x_3 ∈ J.coverings x
⊢ Sieve.generate (Sieve.pullback x_2 (Sieve.generate x_3)).arrows ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) x_1
have mp : Sieve.pullback x_2 (Sieve.generate x_3) ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) x_1
Tactic: rw [Sieve.generate_sieve]