-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
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 : ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
X : C
S : Sieve.{u_2, u_3} X
hS : S ∈ (J.toGrothendieck : (a : C) → Set (Sieve a)) X
mpr :
  ∀ (X : C) (S : Presieve.{u_2, u_3} X) (hS : S ∈ J.coverings X) ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate S)).arrows
mpr₁ : ∀ (X : C) ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f ⊤).arrows
mpr₂ :
  ∀ (X : C) (S : Sieve.{u_2, u_3} X) (hS : J.Saturate X S) (Y : C) (f : Y ⟶ X)
    (ih : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
    Presieve.IsSheafFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
mpr₃ :
  ∀ (X : C) (R S : Sieve.{u_2, u_3} X) (hS : J.Saturate X R)
    (h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S))
    (H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f R).arrows)
    (H2 :
      ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
        R.arrows f →
          ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSheafFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows)
    ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows

Tactic:
induction hS with
    | of X S hS =>
      exact fun _ _ => H S hS
    | top =>
      simp [Presieve.isSheafFor_top P]
    | pullback X S hS _ f ih =>
      intro Y f
      rw [← S.pullback_comp]
      exact ih (f ≫ _)
    | transitive X R S hS h 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 => ?_⟩
      · intro x t₁ t₂ h₁ h₂
        refine (H1 f).ext (fun Z g hg => ?_)
        refine (H2 hg (𝟙 _)).ext (fun ZZ gg hgg => ?_)
        simp only [Sieve.pullback_id, Sieve.pullback_apply] at hgg
        simp only [← comp_apply]
        rw [← P.map_comp, ← op_comp, h₁, h₂]
        simpa only [Sieve.pullback_apply, Category.assoc] using hgg
      let y : ∀ ⦃Z : C⦄ (g : Z ⟶ Y),
        ((S.pullback (g ≫ f)).pullback (𝟙 _)).arrows.FamilyOfElements P :=
        fun Z g ZZ gg hgg => x (gg ≫ g) (by simpa using hgg)
      have hy : ∀ ⦃Z : C⦄ (g : Z ⟶ Y), (y g).Compatible := by
        intro Z g Y₁ Y₂ ZZ g₁ g₂ f₁ f₂ h₁ h₂ h
        rw [hx]
        rw [reassoc_of% h]
      choose z hz using fun ⦃Z : C⦄ ⦃g : Z ⟶ Y⦄ (hg : R.pullback f g) =>
        H2' hg (𝟙 _) (y g) (hy g)
      let q : (R.pullback f).arrows.FamilyOfElements P := fun Z g hg => z hg
      have hq : q.Compatible := by
        intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
        apply (H2 h₁ g₁).ext
        intro ZZ gg hgg
        simp only [← comp_apply]
        rw [← P.map_comp, ← P.map_comp, ← op_comp, ← op_comp, hz, hz]
        · dsimp [y]; congr 1; simp only [Category.assoc, h]
        · simpa [reassoc_of% h] using hgg
        · simpa using hgg
      obtain ⟨t, ht⟩ := H1' f q hq
      refine ⟨t, fun Z g hg => ?_⟩
      refine (H1 (g ≫ f)).ext (fun ZZ gg hgg => ?_)
      rw [← comp_apply _ (P.map gg.op), ← P.map_comp, ← op_comp, ht]
      on_goal 2 => simpa using hgg
      refine (H2 hgg (𝟙 _)).ext (fun ZZZ ggg hggg => ?_)
      rw [← comp_apply _ (P.map ggg.op), ← P.map_comp, ← op_comp, hz]
      on_goal 2 => simpa using hggg
      refine (H2 hgg ggg).ext (fun ZZZZ gggg _ => ?_)
      rw [← comp_apply _ (P.map gggg.op), ← P.map_comp, ← op_comp]
      apply hx
      simp

Premises used: CategoryTheory.Sieve.pullback_apply, CategoryTheory.Category.assoc, CategoryTheory.op_comp, CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor, CategoryTheory.Sieve.pullback_comp, CategoryTheory.comp_apply, CategoryTheory.Functor.map_comp, CategoryTheory.Sieve.pullback_id, CategoryTheory.Presieve.isSheafFor_top