l3lab/ntp-mathlib huggingface dataset:
/--
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; assumption
      · 

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
⊢ ∀ ⦃Y_1 : C⦄ ⦃f_1 : Y_1 ⟶ Y⦄,
    (Sieve.pullback f (Sieve.generate S)).arrows f_1 → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R T f_1

Next tactic:
intro Z g _
/--
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; assumption
      · intro Z g _
        

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
a✝ : (Sieve.pullback f (Sieve.generate S)).arrows g
⊢ ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R T g

Next tactic:
obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
/--
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; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        

Proof state:
case mpr.of.intro.intro.h.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
a✝ : (Sieve.pullback f (Sieve.generate S)).arrows g
R : Presieve Z
hR1 : R ∈ covering K Z
hR2 : FactorsThruAlong R T g
⊢ ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R T g

Next tactic:
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 =>
      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⟩
    

Proof state:
case mpr.top
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
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X✝), IsSheafFor P (Sieve.pullback f ⊤).arrows

Next tactic:
| top => intros; simpa using Presieve.isSheafFor_top_sieve _
/--
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; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
    | top => 

Proof state:
case mpr.top
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
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X✝), IsSheafFor P (Sieve.pullback f ⊤).arrows

Next tactic:
intros
/--
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; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
    | top => intros; 

Proof state:
case mpr.top
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✝ Y✝ : C
f✝ : Y✝ ⟶ X✝
⊢ IsSheafFor P (Sieve.pullback f✝ ⊤).arrows

Next tactic:
simpa using Presieve.isSheafFor_top_sieve _
/--
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; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
    | top => intros; simpa using Presieve.isSheafFor_top_sieve _
    | transitive X R S _ _ H1 H2 =>
      

Proof state:
case mpr.transitive
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
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f R).arrows
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), IsSheafFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f 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
      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⟩
    | top => intros; simpa using Presieve.isSheafFor_top_sieve _
    | transitive X R S _ _ H1 H2 =>
      intro Y f
      

Proof state:
case mpr.transitive
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
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f R).arrows
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), IsSheafFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
Y : C
f : Y ⟶ X
⊢ IsSheafFor P (Sieve.pullback f S).arrows

Next tactic:
simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
/--
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; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
    | top => intros; simpa using Presieve.isSheafFor_top_sieve _
    | transitive X R S _ _ H1 H2 =>
      intro Y f
      simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
      

Proof state:
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X : C},
    ∀ R ∈ covering K X,
      IsSeparatedFor P R ∧
        ∀ (x : FamilyOfElements P R), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
H1 :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    IsSeparatedFor P (Sieve.pullback f R).arrows ∧
      ∀ (x : FamilyOfElements P (Sieve.pullback f R).arrows),
        FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
        IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows ∧
          ∀ (x : FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
            FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
⊢ IsSeparatedFor P (Sieve.pullback f S).arrows ∧
    ∀ (x : FamilyOfElements P (Sieve.pullback f S).arrows),
      FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t

Next tactic:
choose H1 H1' using H1
/--
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; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
    | top => intros; simpa using Presieve.isSheafFor_top_sieve _
    | transitive X R S _ _ H1 H2 =>
      intro Y f
      simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
      choose H1 H1' using H1
      

Proof state:
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X : C},
    ∀ R ∈ covering K X,
      IsSeparatedFor P R ∧
        ∀ (x : FamilyOfElements P R), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
        IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows ∧
          ∀ (x : FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
            FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : FamilyOfElements P (Sieve.pullback f R).arrows),
    FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
⊢ IsSeparatedFor P (Sieve.pullback f S).arrows ∧
    ∀ (x : FamilyOfElements P (Sieve.pullback f S).arrows),
      FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t

Next tactic:
choose H2 H2' using H2
/--
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; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
    | top => intros; simpa using Presieve.isSheafFor_top_sieve _
    | transitive X R S _ _ H1 H2 =>
      intro Y f
      simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
      choose H1 H1' using H1
      choose H2 H2' using H2
      

Proof state:
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X : C},
    ∀ R ∈ covering K X,
      IsSeparatedFor P R ∧
        ∀ (x : FamilyOfElements P R), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : FamilyOfElements P (Sieve.pullback f R).arrows),
    FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
⊢ IsSeparatedFor P (Sieve.pullback f S).arrows ∧
    ∀ (x : FamilyOfElements P (Sieve.pullback f S).arrows),
      FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t

Next tactic:
refine ⟨?_, fun x hx => ?_⟩
/--
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; assumption
      · intro Z g _
        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
        refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
    | top => intros; simpa using Presieve.isSheafFor_top_sieve _
    | transitive X R S _ _ H1 H2 =>
      intro Y f
      simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
      choose H1 H1' using H1
      choose H2 H2' using H2
      refine ⟨?_, fun x hx => ?_⟩
      · 

Proof state:
case mpr.transitive.refine_1
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X : C},
    ∀ R ∈ covering K X,
      IsSeparatedFor P R ∧
        ∀ (x : FamilyOfElements P R), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : FamilyOfElements P (Sieve.pullback f R).arrows),
    FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
⊢ IsSeparatedFor P (Sieve.pullback f S).arrows

Next tactic:
intro x t₁ t₂ h₁ h₂