awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
x : ↥X
hx : x ∈ U.toOpens
⊢ (∃ (V : X.AffineZariskiSite),
      (∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen (f * g) = V.toOpens) ∧
        ∃ (x_1 : V ⟶ U), x ∈ V.toOpens) ↔
    ∃ i ∈ s, x ∈ X.basicOpen i
have mp : (∃ (V : X.AffineZariskiSite),
    (∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen (f * g) = V.toOpens) ∧ ∃ (x_1 : V ⟶ U), x ∈ V.toOpens) →
  ∃ i ∈ s, x ∈ X.basicOpen i
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
x : ↥X
hx : x ∈ U.toOpens
mp :
  (∃ (V : X.AffineZariskiSite),
      (∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen (f * g) = V.toOpens) ∧
        ∃ (x_1 : V ⟶ U), x ∈ V.toOpens) →
    ∃ i ∈ s, x ∈ X.basicOpen i
⊢ (∃ (V : X.AffineZariskiSite),
      (∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen (f * g) = V.toOpens) ∧
        ∃ (x_1 : V ⟶ U), x ∈ V.toOpens) ↔
    ∃ i ∈ s, x ∈ X.basicOpen i
have mpr : (∃ i ∈ s, x ∈ X.basicOpen i) →
  ∃ (V : X.AffineZariskiSite),
    (∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen (f * g) = V.toOpens) ∧ ∃ (x_1 : V ⟶ U), x ∈ V.toOpens
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
s : Set (↑Γ(X, U.toOpens) : Type u)
x : ↥X
hx : x ∈ U.toOpens
⊢ (∃ (V : X.AffineZariskiSite),
      (∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen (f * g) = V.toOpens) ∧
        ∃ (x_1 : V ⟶ U), x ∈ V.toOpens) →
    ∃ i ∈ s, x ∈ X.basicOpen i
have mp : (∃ (V : X.AffineZariskiSite),
    (∃ f ∈ s, ∃ (g : (↑Γ(X, U.toOpens) : Type u)), X.basicOpen f ⊓ X.basicOpen g = V.toOpens) ∧
      ∃ (x_1 : V ⟶ U), x ∈ V.toOpens) →
  ∃ i ∈ s, x ∈ X.basicOpen i
Tactic: simp only [basicOpen_mul]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
S : Sieve U
⊢ S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    Eq (α := Ideal (↑Γ(X, U.toOpens) : Type u)) (Ideal.span (sectionsOfPresieve S.arrows)) ⊤
have rw : S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
  Sieve.generate (U.presieveOfSections (sectionsOfPresieve S.arrows)) ∈
    (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U
Tactic: rw [← generate_presieveOfSections_mem_grothendieckTopology, presieveOfSections_sectionsOfPresieve,
    Sieve.generate_sieve]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
S : Sieve U
rw :
  S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    Sieve.generate (U.presieveOfSections (sectionsOfPresieve S.arrows)) ∈
      (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U
⊢ S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    Eq (α := Ideal (↑Γ(X, U.toOpens) : Type u)) (Ideal.span (sectionsOfPresieve S.arrows)) ⊤
have rw₁ : S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
  Sieve.generate S.arrows ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U
Tactic: rw [← generate_presieveOfSections_mem_grothendieckTopology, presieveOfSections_sectionsOfPresieve,
    Sieve.generate_sieve]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U : X.AffineZariskiSite
S : Sieve U
rw :
  S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    Sieve.generate (U.presieveOfSections (sectionsOfPresieve S.arrows)) ∈
      (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U
rw₁ :
  S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    Sieve.generate S.arrows ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U
⊢ S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
    Eq (α := Ideal (↑Γ(X, U.toOpens) : Type u)) (Ideal.span (sectionsOfPresieve S.arrows)) ⊤
have rw₂ : S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U ↔
  S ∈ (grothendieckTopology X : (a : X.AffineZariskiSite) → Set (Sieve a)) U
Tactic: rw [← generate_presieveOfSections_mem_grothendieckTopology, presieveOfSections_sectionsOfPresieve,
    Sieve.generate_sieve]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
⊢ PreZeroHypercover.presieve₀
      { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
        f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι } ∈
    (precoverage IsOpenImmersion).coverings X
have rw : (∀ (x : ↥X),
    ∃ (i :
      PreZeroHypercover.I₀ (C := Scheme) (S := X)
        { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
          f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι }),
      Membership.mem (γ := Set ↥X)
        (Set.range (ι :=
          ↥(PreZeroHypercover.X
              { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
                f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
              i))
          ⇑(PreZeroHypercover.f
              { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
                f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
              i))
        x) ∧
  ∀
    (i :
      PreZeroHypercover.I₀ (C := Scheme) (S := X)
        { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
          f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι }),
    IsOpenImmersion
      (PreZeroHypercover.f
        { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
          f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
        i)
Tactic: rw [presieve₀_mem_precoverage_iff]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
⊢ (∀ (x : ↥X),
      ∃ (i :
        PreZeroHypercover.I₀ (C := Scheme) (S := X)
          { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
            f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι }),
        Membership.mem (γ := Set ↥X)
          (Set.range (ι :=
            ↥(PreZeroHypercover.X
                { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
                  f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
                i))
            ⇑(PreZeroHypercover.f
                { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
                  f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
                i))
          x) ∧
    ∀
      (i :
        PreZeroHypercover.I₀ (C := Scheme) (S := X)
          { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
            f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι }),
      IsOpenImmersion
        (PreZeroHypercover.f
          { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
            f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
          i)
have refine : ∀ (x : ↥X),
  ∃ (i :
    PreZeroHypercover.I₀ (C := Scheme) (S := X)
      { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
        f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι }),
    Membership.mem (γ := Set ↥X)
      (Set.range (ι :=
        ↥(PreZeroHypercover.X
            { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
              f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
            i))
        ⇑(PreZeroHypercover.f
            { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
              f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
            i))
      x
Tactic: refine ⟨fun x ↦ ?_, inferInstance⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
x : ↥X
⊢ ∃ (i :
    PreZeroHypercover.I₀ (C := Scheme) (S := X)
      { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
        f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι }),
    Membership.mem (γ := Set ↥X)
      (Set.range (ι :=
        ↥(PreZeroHypercover.X
            { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
              f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
            i))
        ⇑(PreZeroHypercover.f
            { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
              f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
            i))
      x
have obtain : ∀ (U : (↑X.affineOpens : Type u)) (hxU : x ∈ (↑U : X.Opens)),
  ∃ (i :
    PreZeroHypercover.I₀ (C := Scheme) (S := X)
      { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
        f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι }),
    Membership.mem (γ := Set ↥X)
      (Set.range (ι :=
        ↥(PreZeroHypercover.X
            { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
              f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
            i))
        ⇑(PreZeroHypercover.f
            { I₀ := X.AffineZariskiSite, X := fun (U : X.AffineZariskiSite) ↦ (↑(↑U : X.Opens) : Scheme),
              f := fun (U : X.AffineZariskiSite) ↦ (↑U : X.Opens).ι : PreZeroHypercover X }
            i))
      x
Tactic: obtain ⟨U, hxU⟩ := TopologicalSpace.Opens.mem_iSup.mp
      ((iSup_affineOpens_eq_top X).ge (Set.mem_univ x))
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (hypothesis)
Goal:
X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U : X.Opens).ι) x
⊢ ∃ (k : (directedCover X).I₀) (hki : k ⟶ U) (hkj : k ⟶ V) (y : ↥((directedCover X).X k)),
    (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x
have haU : a ∈ (↑U : X.Opens)
Tactic: have haU : a ∈ U.1 := (pullback.fst U.1.ι V.1.ι x).2
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U : X.Opens).ι) x
haU : a ∈ (↑U : X.Opens)
⊢ Membership.mem (α := ↥X) (↑V : X.Opens)
    ((pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U : X.Opens).ι) x : ↥X)
have rw : Membership.mem (α := ↥X) (↑V : X.Opens)
  ((pullback.snd ((directedCover X).f U) ((directedCover X).f V) ≫ (directedCover X).f V) x : ↥X)
Tactic: rw [pullback.condition]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (hypothesis)
Goal:
X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U : X.Opens).ι) x
haU : a ∈ (↑U : X.Opens)
⊢ ∃ (k : (directedCover X).I₀) (hki : k ⟶ U) (hkj : k ⟶ V) (y : ↥((directedCover X).X k)),
    (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x
have haV : a ∈ (↑V : X.Opens)
Tactic: have haV : a ∈ V.1 := by unfold a; rw [pullback.condition]; exact (pullback.snd U.1.ι V.1.ι x).2
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U : X.Opens).ι) x
haU : a ∈ (↑U : X.Opens)
haV : a ∈ (↑V : X.Opens)
⊢ ∃ (k : (directedCover X).I₀) (hki : k ⟶ U) (hkj : k ⟶ V) (y : ↥((directedCover X).X k)),
    (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x
have obtain : ∀ (f : (↑Γ(X, (↑U : X.Opens)) : Type u)) (g : (↑Γ(X, (↑V : X.Opens)) : Type u)) (e : X.basicOpen f = X.basicOpen g)
  (hxf : a ∈ X.basicOpen f),
  ∃ (k : (directedCover X).I₀) (hki : k ⟶ U) (hkj : k ⟶ V) (y : ↥((directedCover X).X k)),
    (pullback.lift (X.homOfLE sorry) (X.homOfLE sorry) sorry) y = x
Tactic: obtain ⟨f, g, e, hxf⟩ := exists_basicOpen_le_affine_inter U.2 V.2 _ ⟨haU, haV⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U : X.Opens).ι) x
haU : a ∈ (↑U : X.Opens)
haV : a ∈ (↑V : X.Opens)
f : (↑Γ(X, (↑U : X.Opens)) : Type u)
g : (↑Γ(X, (↑V : X.Opens)) : Type u)
e : X.basicOpen f = X.basicOpen g
hxf : a ∈ X.basicOpen f
⊢ ∃ (k : (directedCover X).I₀) (hki : k ⟶ U) (hkj : k ⟶ V) (y : ↥((directedCover X).X k)),
    (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x
have refine : (pullback.lift (X.homOfLE sorry) (X.homOfLE sorry) sorry)
    (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ (↑(basicOpen U f) : X.Opens)) a hxf) =
  x
Tactic: refine ⟨U.basicOpen f, homOfLE (U.basicOpen_le f), eqToHom (Subtype.ext (by exact e)) ≫
      homOfLE (V.basicOpen_le g), ⟨a, hxf⟩, ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U : X.Opens).ι) x
haU : a ∈ (↑U : X.Opens)
haV : a ∈ (↑V : X.Opens)
f : (↑Γ(X, (↑U : X.Opens)) : Type u)
g : (↑Γ(X, (↑V : X.Opens)) : Type u)
e : X.basicOpen f = X.basicOpen g
hxf : a ∈ X.basicOpen f
⊢ (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯)
      (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ (↑(basicOpen U f) : X.Opens)) a hxf) =
    x
have a₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: apply (pullback.fst _ _ ≫ U.1.ι).isOpenEmbedding.injective
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U : X.Opens).ι) x
haU : a ∈ (↑U : X.Opens)
haV : a ∈ (↑V : X.Opens)
f : (↑Γ(X, (↑U : X.Opens)) : Type u)
g : (↑Γ(X, (↑V : X.Opens)) : Type u)
e : X.basicOpen f = X.basicOpen g
hxf : a ∈ X.basicOpen f
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have a₁ : a =
  (↑((TopCat.Hom.hom
            (PresheafedSpace.Hom.base
              (LocallyRingedSpace.Hom.toHom (Hom.toLRSHom' (pullback.fst (↑U : X.Opens).ι (↑V : X.Opens).ι)))) :
          (a : ↥(pullback (↑U : X.Opens).ι (↑V : X.Opens).ι)) → ↥(↑(↑U : X.Opens) : Scheme))
        x) :
    ↥X)
Tactic: simp only [pullback.lift_fst_assoc, homOfLE_ι, Opens.ι_apply]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : X.AffineZariskiSite
f : U ⟶ V
⊢ (toOpensFunctor X ⋙ Functor.rightOp X.presheaf ⋙ Scheme.Spec).map f ≫ ⋯.fromSpec =
    ⋯.fromSpec ≫ CategoryTheory.Functor.map (Functor.obj (Functor.const X.AffineZariskiSite) X) f
have dsimp : Spec.map (X.presheaf.map ((toOpensFunctor X).map f).op) ≫ sorry.fromSpec = sorry.fromSpec ≫ 𝟙 X
Tactic: dsimp
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
U V : X.AffineZariskiSite
f : U ⟶ V
⊢ Spec.map (X.presheaf.map ((toOpensFunctor X).map f).op) ≫ ⋯.fromSpec = ⋯.fromSpec ≫ 𝟙 X
have rw : sorry.fromSpec = sorry.fromSpec ≫ 𝟙 X
Tactic: rw [V.2.map_fromSpec U.2]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
⊢ NatTrans.Coequifibered α ↔
    ∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
      IsLocalization.Away (R := (↑(F.obj (Opposite.op U)) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := ((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op U))
                (α.app (Opposite.op U)) :
              (a : (↑(((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op U)) : Type u)) →
                (↑(F.obj (Opposite.op U)) : Type u))
            f :
          (↑(F.obj (Opposite.op U)) : Type u))
        (↑(F.obj (Opposite.op (U.basicOpen f))) : Type u)
have trans : NatTrans.Coequifibered α ↔
  ∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
    IsPushout (X.presheaf.map (homOfLE sorry).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
      (F.map (homOfLE sorry).op)
Tactic: trans ∀ (U : X.AffineZariskiSite) (f : Γ(X, U.1)),
    IsPushout (X.presheaf.map (homOfLE (X.basicOpen_le f)).op)
      (α.app _) (α.app (.op (U.basicOpen f))) (F.map (homOfLE (U.basicOpen_le f)).op)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
trans :
  NatTrans.Coequifibered α ↔
    ∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
      IsPushout (X.presheaf.map (homOfLE ⋯).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
        (F.map (homOfLE ⋯).op)
⊢ NatTrans.Coequifibered α ↔
    ∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
      IsLocalization.Away (R := (↑(F.obj (Opposite.op U)) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := ((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op U))
                (α.app (Opposite.op U)) :
              (a : (↑(((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op U)) : Type u)) →
                (↑(F.obj (Opposite.op U)) : Type u))
            f :
          (↑(F.obj (Opposite.op U)) : Type u))
        (↑(F.obj (Opposite.op (U.basicOpen f))) : Type u)
have trans₁ : (∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
    IsPushout (X.presheaf.map (homOfLE sorry).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
      (F.map (homOfLE sorry).op)) ↔
  ∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
    IsLocalization.Away (R := (↑(F.obj (Opposite.op U)) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (X := ((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op U))
              (α.app (Opposite.op U)) :
            (a : (↑(((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op U)) : Type u)) →
              (↑(F.obj (Opposite.op U)) : Type u))
          f :
        (↑(F.obj (Opposite.op U)) : Type u))
      (↑(F.obj (Opposite.op (U.basicOpen f))) : Type u)
Tactic: trans ∀ (U : X.AffineZariskiSite) (f : Γ(X, U.1)),
    IsPushout (X.presheaf.map (homOfLE (X.basicOpen_le f)).op)
      (α.app _) (α.app (.op (U.basicOpen f))) (F.map (homOfLE (U.basicOpen_le f)).op)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
⊢ NatTrans.Coequifibered α ↔
    ∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
      IsPushout (X.presheaf.map (homOfLE ⋯).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
        (F.map (homOfLE ⋯).op)
have refine : ∀
  (H :
    ∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
      IsPushout (X.presheaf.map (homOfLE sorry).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
        (F.map (homOfLE sorry).op))
  (x x : X.AffineZariskiSiteᵒᵖ) (V U : X.AffineZariskiSite) (x : Opposite.op V ⟶ Opposite.op U)
  (f : (↑Γ(X, (Opposite.unop (Opposite.op V)).toOpens) : Type u))
  (hf : X.basicOpen f = (Opposite.unop (Opposite.op U)).toOpens),
  IsPushout (X := ((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op U))
    (((toOpensFunctor X).op ⋙ X.presheaf).map (Opposite.op { down := { down := sorry } })) (α.app (Opposite.op V))
    (α.app (Opposite.op U)) (F.map (Opposite.op { down := { down := sorry } }))
Tactic: refine ⟨fun H U f ↦ H (homOfLE (U.basicOpen_le f)).op, fun H ⟨V⟩ ⟨U⟩ ⟨f, hf⟩ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
H :
  ∀ (U : X.AffineZariskiSite) (f : (↑Γ(X, (↑U : X.Opens)) : Type u)),
    IsPushout (X.presheaf.map (homOfLE ⋯).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
      (F.map (homOfLE ⋯).op)
x x_1 : X.AffineZariskiSiteᵒᵖ
V U : X.AffineZariskiSite
x_2 : Opposite.op V ⟶ Opposite.op U
f : (↑Γ(X, (Opposite.unop (Opposite.op V)).toOpens) : Type u)
hf : X.basicOpen f = (Opposite.unop (Opposite.op U)).toOpens
⊢ IsPushout (X := ((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op U))
    (((toOpensFunctor X).op ⋙ X.presheaf).map (Opposite.op { down := { down := ⋯ } })) (α.app (Opposite.op V))
    (α.app (Opposite.op U)) (F.map (Opposite.op { down := { down := ⋯ } }))
have obtain : ∀ (x : Opposite.op V ⟶ Opposite.op (V.basicOpen f))
  (hf : X.basicOpen f = (Opposite.unop (Opposite.op (V.basicOpen f))).toOpens),
  IsPushout (X := ((toOpensFunctor X).op ⋙ X.presheaf).obj (Opposite.op (V.basicOpen f)))
    (((toOpensFunctor X).op ⋙ X.presheaf).map (Opposite.op { down := { down := sorry } })) (α.app (Opposite.op V))
    (α.app (Opposite.op (V.basicOpen f))) (F.map (Opposite.op { down := { down := sorry } }))
Tactic: obtain rfl : V.basicOpen f = U := Subtype.ext hf
Cleared hypotheses: U, x_2, hf