Module: Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact

Goal state before:
pp: case refine_1
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
this :
  ∀ (R : CommRingCat) (𝒰 : Cover (propQCPrecoverage P) (Spec R)),
    (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀ → Presieve.IsSheafFor F 𝒰.presieve₀
h𝒰 : ¬((∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀)
𝒱 : AffineCover P (Spec R)
f✝ : 𝒱.cover ⟶ 𝒰.forgetQc
hfin : Finite 𝒱.I₀
ho : ∀ (j : 𝒱.cover.I₀), IsOpenImmersion (f✝.h₀ j)
V : Scheme
f : V ⟶ Spec R
i : V.affineOneHypercover.I₀
⊢ Presieve.IsSheafFor F
    (Sieve.pullback (V.affineOneHypercover.f i)
        (Sieve.generate
          (Presieve.ofArrows (Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).X
            (Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).f))).arrows

pp: case refine_2
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
this :
  ∀ (R : CommRingCat) (𝒰 : Cover (propQCPrecoverage P) (Spec R)),
    (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀ → Presieve.IsSheafFor F 𝒰.presieve₀
h𝒰 : ¬((∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀)
𝒱 : AffineCover P (Spec R)
f✝ : 𝒱.cover ⟶ 𝒰.forgetQc
hfin : Finite 𝒱.I₀
ho : ∀ (j : 𝒱.cover.I₀), IsOpenImmersion (f✝.h₀ j)
V : Scheme
f : V ⟶ Spec R
i j : V.affineOneHypercover.I₀
k : V.affineOneHypercover.I₁ i j
⊢ Presieve.IsSeparatedFor F
    (Sieve.pullback (V.affineOneHypercover.p₁ k ≫ V.affineOneHypercover.f i)
        (Sieve.generate
          (Presieve.ofArrows (Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).X
            (Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).f))).arrows

Tactic:
· rw [← Sieve.pullbackArrows_comm, ← Presieve.ofArrows_pullback,
            ← Presieve.isSheafFor_iff_generate]
          let 𝒰' := (𝒱.cover.pullback₂ f).pullback₂ (V.affineCover.f i)
          exact this _ (.ofQuasiCompactCover 𝒰' (qc := by dsimp [𝒰']; infer_instance))
            ⟨fun j ↦ .of_isIso (pullbackLeftPullbackSndIso (𝒱.f j) f (V.affineCover.f i)).hom, hfin⟩

Goal state after:
case refine_2
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
this :
  ∀ (R : CommRingCat) (𝒰 : Cover (propQCPrecoverage P) (Spec R)),
    (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀ → Presieve.IsSheafFor F 𝒰.presieve₀
h𝒰 : ¬((∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀)
𝒱 : AffineCover P (Spec R)
f✝ : 𝒱.cover ⟶ 𝒰.forgetQc
hfin : Finite 𝒱.I₀
ho : ∀ (j : 𝒱.cover.I₀), IsOpenImmersion (f✝.h₀ j)
V : Scheme
f : V ⟶ Spec R
i j : V.affineOneHypercover.I₀
k : V.affineOneHypercover.I₁ i j
⊢ Presieve.IsSeparatedFor F
    (Sieve.pullback (V.affineOneHypercover.p₁ k ≫ V.affineOneHypercover.f i)
        (Sieve.generate
          (Presieve.ofArrows (Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).X
            (Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).f))).arrows