l3lab/ntp-mathlib huggingface dataset:
/--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
    sInf {J | K ≤ ofGrothendieck _ J } := by
  apply le_antisymm
  · apply le_sInf; intro J hJ
    intro X S hS
    induction hS with
    | of X S hS => apply hJ; assumption
    | top => apply J.top_mem
    | transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2
  · 

Proof state:
case a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
⊢ sInf {J | K ≤ ofGrothendieck C J} ≤ toGrothendieck C K

Next tactic:
apply sInf_le
/--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
    sInf {J | K ≤ ofGrothendieck _ J } := by
  apply le_antisymm
  · apply le_sInf; intro J hJ
    intro X S hS
    induction hS with
    | of X S hS => apply hJ; assumption
    | top => apply J.top_mem
    | transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2
  · apply sInf_le
    

Proof state:
case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
⊢ toGrothendieck C K ∈ {J | K ≤ ofGrothendieck C J}

Next tactic:
intro X S hS
/--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
    sInf {J | K ≤ ofGrothendieck _ J } := by
  apply le_antisymm
  · apply le_sInf; intro J hJ
    intro X S hS
    induction hS with
    | of X S hS => apply hJ; assumption
    | top => apply J.top_mem
    | transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2
  · apply sInf_le
    intro X S hS
    

Proof state:
case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ S ∈ covering (ofGrothendieck C (toGrothendieck C K)) X

Next tactic:
apply saturate.of _ _ hS
instance : SemilatticeSup (Coverage C) where
  sup x y :=
  { covering := fun B ↦ x.covering B ∪ y.covering B
    pullback := by
      

Proof state:
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
⊢ ∀ ⦃X Y : C⦄ (f : Y ⟶ X),
    ∀ S ∈ (fun B => covering x B ∪ covering y B) X,
      ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f

Next tactic:
rintro X Y f S (hx | hy)
instance : SemilatticeSup (Coverage C) where
  sup x y :=
  { covering := fun B ↦ x.covering B ∪ y.covering B
    pullback := by
      rintro X Y f S (hx | hy)
      · 

Proof state:
case inl
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
X Y : C
f : Y ⟶ X
S : Presieve X
hx : S ∈ covering x X
⊢ ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f

Next tactic:
obtain ⟨T, hT⟩ := x.pullback f S hx
instance : SemilatticeSup (Coverage C) where
  sup x y :=
  { covering := fun B ↦ x.covering B ∪ y.covering B
    pullback := by
      rintro X Y f S (hx | hy)
      · obtain ⟨T, hT⟩ := x.pullback f S hx
        

Proof state:
case inl.intro
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
X Y : C
f : Y ⟶ X
S : Presieve X
hx : S ∈ covering x X
T : Presieve Y
hT : T ∈ covering x Y ∧ Presieve.FactorsThruAlong T S f
⊢ ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f

Next tactic:
exact ⟨T, Or.inl hT.1, hT.2⟩
instance : SemilatticeSup (Coverage C) where
  sup x y :=
  { covering := fun B ↦ x.covering B ∪ y.covering B
    pullback := by
      rintro X Y f S (hx | hy)
      · obtain ⟨T, hT⟩ := x.pullback f S hx
        exact ⟨T, Or.inl hT.1, hT.2⟩
      · 

Proof state:
case inr
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
X Y : C
f : Y ⟶ X
S : Presieve X
hy : S ∈ covering y X
⊢ ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f

Next tactic:
obtain ⟨T, hT⟩ := y.pullback f S hy
instance : SemilatticeSup (Coverage C) where
  sup x y :=
  { covering := fun B ↦ x.covering B ∪ y.covering B
    pullback := by
      rintro X Y f S (hx | hy)
      · obtain ⟨T, hT⟩ := x.pullback f S hx
        exact ⟨T, Or.inl hT.1, hT.2⟩
      · obtain ⟨T, hT⟩ := y.pullback f S hy
        

Proof state:
case inr.intro
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
X Y : C
f : Y ⟶ X
S : Presieve X
hy : S ∈ covering y X
T : Presieve Y
hT : T ∈ covering y Y ∧ Presieve.FactorsThruAlong T S f
⊢ ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f

Next tactic:
exact ⟨T, Or.inr hT.1, hT.2⟩
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  

Proof state:
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
⊢ IsSheaf (toGrothendieck C K) P ↔ ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R

Next tactic:
constructor
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · 

Proof state:
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
⊢ IsSheaf (toGrothendieck C K) P → ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R

Next tactic:
intro H X R hR
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    

Proof state:
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : IsSheaf (toGrothendieck C K) P
X : C
R : Presieve X
hR : R ∈ covering K X
⊢ IsSheafFor P R

Next tactic:
rw [Presieve.isSheafFor_iff_generate]
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    

Proof state:
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : IsSheaf (toGrothendieck C K) P
X : C
R : Presieve X
hR : R ∈ covering K X
⊢ IsSheafFor P (Sieve.generate R).arrows

Next tactic:
apply H _ <| saturate.of _ _ hR
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · 

Proof state:
case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
⊢ (∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R) → IsSheaf (toGrothendieck C K) P

Next tactic:
intro H X S hS
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    

Proof state:
case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
⊢ IsSheafFor P S.arrows

Next tactic:
suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      

Proof state:
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
this : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f S).arrows
⊢ IsSheafFor P S.arrows

Next tactic:
simpa using this (f := 𝟙 _)
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    

Proof state:
case mpr.of
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f (Sieve.generate S)).arrows

Next tactic:
| of X S hS =>
      intro Y f
      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
      apply Presieve.isSheafFor_of_factorsThru (S := T)
      · intro Z g hg
        obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
        exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
      · apply H; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      

Proof state:
case mpr.of
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f (Sieve.generate S)).arrows

Next tactic:
intro Y f
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      intro Y f
      

Proof state:
case mpr.of
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
⊢ IsSheafFor P (Sieve.pullback f (Sieve.generate S)).arrows

Next tactic:
obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      intro Y f
      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
      

Proof state:
case mpr.of.intro.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ IsSheafFor P (Sieve.pullback f (Sieve.generate S)).arrows

Next tactic:
apply Presieve.isSheafFor_of_factorsThru (S := T)
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      intro Y f
      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
      apply Presieve.isSheafFor_of_factorsThru (S := T)
      · 

Proof state:
case mpr.of.intro.intro.H
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ FactorsThru T (Sieve.pullback f (Sieve.generate S)).arrows

Next tactic:
intro Z g hg
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      intro Y f
      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
      apply Presieve.isSheafFor_of_factorsThru (S := T)
      · intro Z g hg
        

Proof state:
case mpr.of.intro.intro.H
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
Z : C
g : Z ⟶ Y
hg : T g
⊢ ∃ W i e, (Sieve.pullback f (Sieve.generate S)).arrows e ∧ i ≫ e = g

Next tactic:
obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      intro Y f
      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
      apply Presieve.isSheafFor_of_factorsThru (S := T)
      · intro Z g hg
        obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
        

Proof state:
case mpr.of.intro.intro.H.intro.intro.intro.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
Z : C
g : Z ⟶ Y
hg : T g
W : C
i : Z ⟶ W
e : W ⟶ X
h1 : S e
h2 : i ≫ e = g ≫ f
⊢ ∃ W i e, (Sieve.pullback f (Sieve.generate S)).arrows e ∧ i ≫ e = g

Next tactic:
exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      intro Y f
      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
      apply Presieve.isSheafFor_of_factorsThru (S := T)
      · intro Z g hg
        obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
        exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by 

Proof state:
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
Z : C
g : Z ⟶ Y
hg : T g
W : C
i : Z ⟶ W
e : W ⟶ X
h1 : S e
h2 : i ≫ e = g ≫ f
⊢ 𝟙 Z ≫ g = g

Next tactic:
simp
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      intro Y f
      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
      apply Presieve.isSheafFor_of_factorsThru (S := T)
      · intro Z g hg
        obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
        exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
      · 

Proof state:
case mpr.of.intro.intro.hS
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ IsSheafFor P T

Next tactic:
apply H
/--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
    Presieve.IsSheaf (toGrothendieck _ K) P ↔
    (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
  constructor
  · intro H X R hR
    rw [Presieve.isSheafFor_iff_generate]
    apply H _ <| saturate.of _ _ hR
  · intro H X S hS
    -- This is the key point of the proof:
    -- We must generalize the induction in the correct way.
    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
      simpa using this (f := 𝟙 _)
    induction hS with
    | of X S hS =>
      intro Y f
      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
      apply Presieve.isSheafFor_of_factorsThru (S := T)
      · intro Z g hg
        obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
        exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
      · apply H; 

Proof state:
case mpr.of.intro.intro.hS.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ T ∈ covering K Y

Next tactic:
assumption