awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := relativeGluingData ⋯
F : (directedCover X).I₀ ⥤ Scheme := D.functor
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun (i : (↑X.affineOpens : Type u)) ↦ (↑i : X.Opens)) ⋯).toPreZeroHypercover.1
h :
  (cocone X).pt.Opens = X.Opens →
    HEq.{u + 2} (α := Category.{u, u} (cocone X).pt.Opens)
        (Preorder.smallCategory (TopologicalSpace.Opens ↥(cocone X).pt)) (β := Category.{u, u} X.Opens)
        (Preorder.smallCategory (TopologicalSpace.Opens ↥X)) →
      (↑(cocone X).pt.toPresheafedSpace : TopCat) = (↑X.toPresheafedSpace : TopCat) →
        (cocone X).pt.toPresheafedSpace = X.toPresheafedSpace →
          (cocone X).pt.toLocallyRingedSpace = X.toLocallyRingedSpace →
            (cocone X).pt = X → Eq (α := (colimit F).Hom (cocone X).pt) (colimit.desc F (cocone X)) D.toBase
⊢ (↑(colimit.desc F (cocone X) ⁻¹ᵁ (↑U : X.Opens)) : Set ↥(colimit F)) =
    (↑(Hom.opensRange (colimit.ι F U)) : Set ↥(colimit F))
have h₁ : (cocone X).pt.Opens = X.Opens → (↑U : X.Opens) = Hom.opensRange ((directedCover X).f U)
Tactic: convert! congr($(D.toBase_preimage_eq_opensRange_ι U).1)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := relativeGluingData ⋯
F : (directedCover X).I₀ ⥤ Scheme := D.functor
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun (i : (↑X.affineOpens : Type u)) ↦ (↑i : X.Opens)) ⋯).toPreZeroHypercover.1
e_1 : (cocone X).pt.Opens = X.Opens
e_2 :
  HEq.{u + 2} (α := Category.{u, u} (cocone X).pt.Opens)
    (Preorder.smallCategory (TopologicalSpace.Opens ↥(cocone X).pt)) (β := Category.{u, u} X.Opens)
    (Preorder.smallCategory (TopologicalSpace.Opens ↥X))
e_2_1 : (↑(cocone X).pt.toPresheafedSpace : TopCat) = (↑X.toPresheafedSpace : TopCat)
e_4 : (cocone X).pt.toPresheafedSpace = X.toPresheafedSpace
e_2_2 : (cocone X).pt.toLocallyRingedSpace = X.toLocallyRingedSpace
e_2_3 : (cocone X).pt = X
⊢ Eq (α := (colimit F).Hom (cocone X).pt) (colimit.desc F (cocone X)) D.toBase
have h : Eq (α :=
  (colimit F).Hom
    (Cocone.pt (J := (directedCover X).I₀) (F := F)
      { pt := X, ι := { app := fun (U : X.AffineZariskiSite) ↦ sorry.fromSpec, naturality := sorry } }))
  (colimit.desc F { pt := X, ι := { app := fun (U : X.AffineZariskiSite) ↦ sorry.fromSpec, naturality := sorry } }) D.toBase
Tactic: delta cocone
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := relativeGluingData ⋯
F : (directedCover X).I₀ ⥤ Scheme := D.functor
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun (i : (↑X.affineOpens : Type u)) ↦ (↑i : X.Opens)) ⋯).toPreZeroHypercover.1
e_1 : (cocone X).pt.Opens = X.Opens
e_2 :
  HEq.{u + 2} (α := Category.{u, u} (cocone X).pt.Opens)
    (Preorder.smallCategory (TopologicalSpace.Opens ↥(cocone X).pt)) (β := Category.{u, u} X.Opens)
    (Preorder.smallCategory (TopologicalSpace.Opens ↥X))
e_2_1 : (↑(cocone X).pt.toPresheafedSpace : TopCat) = (↑X.toPresheafedSpace : TopCat)
e_4 : (cocone X).pt.toPresheafedSpace = X.toPresheafedSpace
e_2_2 : (cocone X).pt.toLocallyRingedSpace = X.toLocallyRingedSpace
e_2_3 : (cocone X).pt = X
⊢ Eq (α :=
    (colimit F).Hom
      (Cocone.pt (J := (directedCover X).I₀) (F := F)
        { pt := X, ι := { app := fun (U : X.AffineZariskiSite) ↦ ⋯.fromSpec, naturality := ⋯ } }))
    (colimit.desc F { pt := X, ι := { app := fun (U : X.AffineZariskiSite) ↦ ⋯.fromSpec, naturality := ⋯ } }) D.toBase
have h : ∀ (U : X.AffineZariskiSite),
  sorry.fromSpec =
    D.natTrans.app U ≫ NatTrans.app (Cover.functorOfLocallyDirectedHomBase (P := IsOpenImmersion) (directedCover X)) U
Tactic: congr with U
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := relativeGluingData ⋯
F : (directedCover X).I₀ ⥤ Scheme := D.functor
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun (i : (↑X.affineOpens : Type u)) ↦ (↑i : X.Opens)) ⋯).toPreZeroHypercover.1
e : pullback (X := colimit F) (colimit.desc F (cocone X)) (↑U : X.Opens).ι ≅ (↑(↑U : X.Opens) : Scheme) :=
  IsOpenImmersion.isoOfRangeEq (pullback.fst (colimit.desc F (cocone X)) (↑U : X.Opens).ι)
    ((IsAffineOpen.isoSpec ⋯).hom ≫ colimit.ι F U) ⋯
⊢ IsIso (X := pullback (X := colimit F) (colimit.desc F (cocone X)) (↑U : X.Opens).ι)
    (pullback.snd (colimit.desc F (cocone X)) (↑U : X.Opens).ι)
have h : pullback.snd (colimit.desc F (cocone X)) (↑U : X.Opens).ι = e.hom
Tactic: convert! (inferInstance : IsIso e.hom)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski (goal)
Goal:
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := relativeGluingData ⋯
F : (directedCover X).I₀ ⥤ Scheme := D.functor
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun (i : (↑X.affineOpens : Type u)) ↦ (↑i : X.Opens)) ⋯).toPreZeroHypercover.1
e : pullback (X := colimit F) (colimit.desc F (cocone X)) (↑U : X.Opens).ι ≅ (↑(↑U : X.Opens) : Scheme) :=
  IsOpenImmersion.isoOfRangeEq (pullback.fst (colimit.desc F (cocone X)) (↑U : X.Opens).ι)
    ((IsAffineOpen.isoSpec ⋯).hom ≫ colimit.ι F U) ⋯
⊢ pullback.snd (colimit.desc F (cocone X)) (↑U : X.Opens).ι = e.hom
have h : e.inv ≫ pullback.snd (colimit.desc F (cocone X)) (↑U : X.Opens).ι ≫ (↑U : X.Opens).ι = (↑U : X.Opens).ι
Tactic: rw [← cancel_mono U.1.ι, ← Iso.inv_comp_eq]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R S : CommRingCat
f : R ⟶ S
x x_1 : (TopologicalSpace.Opens (↑(↑(sheafedSpaceObj R).toPresheafedSpace : TopCat) : Type u))ᵒᵖ
x_2 : x ⟶ x_1
⊢ (sheafedSpaceObj R).presheaf.map x_2 ≫
      CommRingCat.ofHom
        (comap (CommRingCat.Hom.hom f) (unop x_1) ((TopologicalSpace.Opens.map (topMap f)).obj (unop x_1)) ⋯) =
    CommRingCat.ofHom
        (comap (CommRingCat.Hom.hom f) (unop x) ((TopologicalSpace.Opens.map (topMap f)).obj (unop x)) ⋯) ≫
      (Functor.obj (TopCat.Presheaf.pushforward CommRingCat (topMap f)) (sheafedSpaceObj S).presheaf).map x_2
have hf : ∀ (x_3 : (↑((sheafedSpaceObj R).presheaf.obj x) : Type u)),
  Eq (α :=
    (↑((Functor.obj (TopCat.Presheaf.pushforward CommRingCat (topMap f)) (sheafedSpaceObj S).presheaf).obj x_1) :
      Type u))
    ((CommRingCat.Hom.hom
            ((sheafedSpaceObj R).presheaf.map x_2 ≫
              CommRingCat.ofHom
                (comap (CommRingCat.Hom.hom f) (unop x_1) ((TopologicalSpace.Opens.map (topMap f)).obj (unop x_1)) sorry)) :
          (a : (↑((sheafedSpaceObj R).presheaf.obj x) : Type u)) →
            (↑(((TopCat.Presheaf.pushforward CommRingCat (topMap f)).obj (sheafedSpaceObj S).presheaf).obj x_1) :
              Type u))
        x_3 :
      (↑((Functor.obj (TopCat.Presheaf.pushforward CommRingCat (topMap f)) (sheafedSpaceObj S).presheaf).obj x_1) :
        Type u))
    ((CommRingCat.Hom.hom
            (CommRingCat.ofHom
                (comap (CommRingCat.Hom.hom f) (unop x) ((TopologicalSpace.Opens.map (topMap f)).obj (unop x)) sorry) ≫
              (Functor.obj (TopCat.Presheaf.pushforward CommRingCat (topMap f)) (sheafedSpaceObj S).presheaf).map x_2) :
          (a : (↑((sheafedSpaceObj R).presheaf.obj x) : Type u)) →
            (↑(((TopCat.Presheaf.pushforward CommRingCat (topMap f)).obj (sheafedSpaceObj S).presheaf).obj x_1) :
              Type u))
        x_3 :
      (↑((Functor.obj (TopCat.Presheaf.pushforward CommRingCat (topMap f)) (sheafedSpaceObj S).presheaf).obj x_1) :
        Type u))
Tactic: ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R : CommRingCat
⊢ sheafedSpaceMap (𝟙 R) = 𝟙 (sheafedSpaceObj R)
have w : (sheafedSpaceMap (𝟙 R)).hom.base = (InducedCategory.Hom.hom (𝟙 (sheafedSpaceObj R))).base
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R : CommRingCat
w : (sheafedSpaceMap (𝟙 R)).hom.base = (InducedCategory.Hom.hom (𝟙 (sheafedSpaceObj R))).base
⊢ sheafedSpaceMap (𝟙 R) = 𝟙 (sheafedSpaceObj R)
have h : (sheafedSpaceMap (𝟙 R)).hom.c ≫ Functor.whiskerRight (eqToHom sorry) (sheafedSpaceObj R).presheaf =
  (InducedCategory.Hom.hom (𝟙 (sheafedSpaceObj R))).c
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R : CommRingCat
⊢ (sheafedSpaceMap (𝟙 R)).hom.c ≫ Functor.whiskerRight (eqToHom ⋯) (sheafedSpaceObj R).presheaf =
    (InducedCategory.Hom.hom (𝟙 (sheafedSpaceObj R))).c
have h : ∀ (U : TopologicalSpace.Opens (↑(↑(sheafedSpaceObj R).toPresheafedSpace : TopCat) : Type u))
  (x : (↑((sheafedSpaceObj R).presheaf.obj (op U)) : Type u)),
  Eq (α :=
    (↑(((TopologicalSpace.Opens.map (InducedCategory.Hom.hom (𝟙 (sheafedSpaceObj R))).base).op ⋙
              (sheafedSpaceObj R).presheaf).obj
          (op U)) :
      Type u))
    ((CommRingCat.Hom.hom
            (NatTrans.app
              ((sheafedSpaceMap (𝟙 R)).hom.c ≫ Functor.whiskerRight (eqToHom sorry) (sheafedSpaceObj R).presheaf) (op U)) :
          (a : (↑((sheafedSpaceObj R).presheaf.obj (op U)) : Type u)) →
            (↑(((TopologicalSpace.Opens.map (𝟙 (sheafedSpaceObj R)).hom.base).op ⋙ (sheafedSpaceObj R).presheaf).obj
                  (op U)) :
              Type u))
        x :
      (↑(((TopologicalSpace.Opens.map (InducedCategory.Hom.hom (𝟙 (sheafedSpaceObj R))).base).op ⋙
                (sheafedSpaceObj R).presheaf).obj
            (op U)) :
        Type u))
    ((CommRingCat.Hom.hom ((InducedCategory.Hom.hom (𝟙 (sheafedSpaceObj R))).c.app (op U)) :
          (a : (↑((sheafedSpaceObj R).presheaf.obj (op U)) : Type u)) →
            (↑(((TopologicalSpace.Opens.map (𝟙 (sheafedSpaceObj R)).hom.base).op ⋙ (sheafedSpaceObj R).presheaf).obj
                  (op U)) :
              Type u))
        x :
      (↑(((TopologicalSpace.Opens.map (InducedCategory.Hom.hom (𝟙 (sheafedSpaceObj R))).base).op ⋙
                (sheafedSpaceObj R).presheaf).obj
            (op U)) :
        Type u))
Tactic: ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
⊢ sheafedSpaceMap (f ≫ g) = sheafedSpaceMap g ≫ sheafedSpaceMap f
have w : (sheafedSpaceMap (f ≫ g)).hom.base = (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.base
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
w : (sheafedSpaceMap (f ≫ g)).hom.base = (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.base
⊢ sheafedSpaceMap (f ≫ g) = sheafedSpaceMap g ≫ sheafedSpaceMap f
have h : (sheafedSpaceMap (f ≫ g)).hom.c ≫ Functor.whiskerRight (eqToHom sorry) (sheafedSpaceObj T).presheaf =
  (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.c
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
⊢ (sheafedSpaceMap (f ≫ g)).hom.c ≫ Functor.whiskerRight (eqToHom ⋯) (sheafedSpaceObj T).presheaf =
    (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.c
have h : ∀ (U : TopologicalSpace.Opens (↑(↑(sheafedSpaceObj R).toPresheafedSpace : TopCat) : Type u))
  (x : (↑((sheafedSpaceObj R).presheaf.obj (op U)) : Type u)),
  Eq (α :=
    (↑(((TopologicalSpace.Opens.map (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.base).op ⋙
              (sheafedSpaceObj T).presheaf).obj
          (op U)) :
      Type u))
    ((CommRingCat.Hom.hom
            (NatTrans.app
              ((sheafedSpaceMap (f ≫ g)).hom.c ≫ Functor.whiskerRight (eqToHom sorry) (sheafedSpaceObj T).presheaf)
              (op U)) :
          (a : (↑((sheafedSpaceObj R).presheaf.obj (op U)) : Type u)) →
            (↑(((TopologicalSpace.Opens.map (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.base).op ⋙
                      (sheafedSpaceObj T).presheaf).obj
                  (op U)) :
              Type u))
        x :
      (↑(((TopologicalSpace.Opens.map (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.base).op ⋙
                (sheafedSpaceObj T).presheaf).obj
            (op U)) :
        Type u))
    ((CommRingCat.Hom.hom ((sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.c.app (op U)) :
          (a : (↑((sheafedSpaceObj R).presheaf.obj (op U)) : Type u)) →
            (↑(((TopologicalSpace.Opens.map (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.base).op ⋙
                      (sheafedSpaceObj T).presheaf).obj
                  (op U)) :
              Type u))
        x :
      (↑(((TopologicalSpace.Opens.map (sheafedSpaceMap g ≫ sheafedSpaceMap f).hom.base).op ⋙
                (sheafedSpaceObj T).presheaf).obj
            (op U)) :
        Type u))
Tactic: ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
X : RingedSpace
R : CommRingCat
α β : X ⟶ sheafedSpaceObj R
w : α.hom.base = β.hom.base
r : (↑R : Type u)
U : TopologicalSpace.Opens (PrimeSpectrum (↑R : Type u)) := PrimeSpectrum.basicOpen r
⊢ (TopologicalSpace.Opens.map α.hom.base).op.obj (op U) = (TopologicalSpace.Opens.map β.hom.base).op.obj (op U)
have rw : (TopologicalSpace.Opens.map β.hom.base).op.obj (op U) = (TopologicalSpace.Opens.map β.hom.base).op.obj (op U)
Tactic: rw [w]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
X : RingedSpace
R : CommRingCat
α β : X ⟶ sheafedSpaceObj R
w : α.hom.base = β.hom.base
h :
  ∀ (r : (↑R : Type u)),
    let U : TopologicalSpace.Opens (PrimeSpectrum (↑R : Type u)) := PrimeSpectrum.basicOpen r;
    Eq (α := CommRingCat.of (↑R : Type u) ⟶ X.presheaf.obj ((TopologicalSpace.Opens.map β.hom.base).op.obj (op U)))
      ((CommRingCat.ofHom
            (algebraMap (↑R : Type u)
              (Functor.obj
                (ObjectProperty.FullSubcategory.obj (P :=
                  Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
                  (structureSheafInType (↑R : Type u) (↑R : Type u)))
                (op U))) ≫
          α.hom.c.app (op U)) ≫
        X.presheaf.map (eqToHom ⋯))
      (CommRingCat.ofHom
          (algebraMap (↑R : Type u)
            (Functor.obj
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
                (structureSheafInType (↑R : Type u) (↑R : Type u)))
              (op U))) ≫
        β.hom.c.app (op U))
w₁ : α.hom.base = β.hom.base
⊢ α = β
have h₁ : α.hom.c ≫ Functor.whiskerRight (eqToHom sorry) X.presheaf = β.hom.c
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
X : RingedSpace
R : CommRingCat
α β : X ⟶ sheafedSpaceObj R
w : α.hom.base = β.hom.base
h :
  ∀ (r : (↑R : Type u)),
    let U : TopologicalSpace.Opens (PrimeSpectrum (↑R : Type u)) := PrimeSpectrum.basicOpen r;
    Eq (α := CommRingCat.of (↑R : Type u) ⟶ X.presheaf.obj ((TopologicalSpace.Opens.map β.hom.base).op.obj (op U)))
      ((CommRingCat.ofHom
            (algebraMap (↑R : Type u)
              (Functor.obj
                (ObjectProperty.FullSubcategory.obj (P :=
                  Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
                  (structureSheafInType (↑R : Type u) (↑R : Type u)))
                (op U))) ≫
          α.hom.c.app (op U)) ≫
        X.presheaf.map (eqToHom ⋯))
      (CommRingCat.ofHom
          (algebraMap (↑R : Type u)
            (Functor.obj
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
                (structureSheafInType (↑R : Type u) (↑R : Type u)))
              (op U))) ≫
        β.hom.c.app (op U))
⊢ α.hom.c ≫ Functor.whiskerRight (eqToHom ⋯) X.presheaf = β.hom.c
have h₁ : ∀ (i : (↑R : Type u)),
  Eq (α :=
    (sheafedSpaceObj R).presheaf.obj (op (PrimeSpectrum.basicOpen i)) ⟶
      (Functor.obj (TopCat.Sheaf.pushforward CommRingCat β.hom.base) (SheafedSpace.sheaf X)).obj (P :=
            Presheaf.IsSheaf
              (Opens.grothendieckTopology (↑(↑(sheafedSpaceObj R).toPresheafedSpace : TopCat) : Type u))).obj
        (op (PrimeSpectrum.basicOpen i)))
    ((α.hom.c ≫ Functor.whiskerRight (eqToHom sorry) X.presheaf).app (op (PrimeSpectrum.basicOpen i)))
    (β.hom.c.app (op (PrimeSpectrum.basicOpen i)))
Tactic: apply ((TopCat.Sheaf.pushforward _ β.hom.base).obj X.sheaf).hom_ext _
      PrimeSpectrum.isBasis_basic_opens
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
X : RingedSpace
R : CommRingCat
α β : X ⟶ sheafedSpaceObj R
w : α.hom.base = β.hom.base
h :
  ∀ (r : (↑R : Type u)),
    let U : TopologicalSpace.Opens (PrimeSpectrum (↑R : Type u)) := PrimeSpectrum.basicOpen r;
    Eq (α := CommRingCat.of (↑R : Type u) ⟶ X.presheaf.obj ((TopologicalSpace.Opens.map β.hom.base).op.obj (op U)))
      ((CommRingCat.ofHom
            (algebraMap (↑R : Type u)
              (Functor.obj
                (ObjectProperty.FullSubcategory.obj (P :=
                  Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
                  (structureSheafInType (↑R : Type u) (↑R : Type u)))
                (op U))) ≫
          α.hom.c.app (op U)) ≫
        X.presheaf.map (eqToHom ⋯))
      (CommRingCat.ofHom
          (algebraMap (↑R : Type u)
            (Functor.obj
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
                (structureSheafInType (↑R : Type u) (↑R : Type u)))
              (op U))) ≫
        β.hom.c.app (op U))
r : (↑R : Type u)
⊢ Eq (α :=
    (sheafedSpaceObj R).presheaf.obj (op (PrimeSpectrum.basicOpen r)) ⟶
      (Functor.obj (TopCat.Sheaf.pushforward CommRingCat β.hom.base) (SheafedSpace.sheaf X)).obj (P :=
            Presheaf.IsSheaf
              (Opens.grothendieckTopology (↑(↑(sheafedSpaceObj R).toPresheafedSpace : TopCat) : Type u))).obj
        (op (PrimeSpectrum.basicOpen r)))
    ((α.hom.c ≫ Functor.whiskerRight (eqToHom ⋯) X.presheaf).app (op (PrimeSpectrum.basicOpen r)))
    (β.hom.c.app (op (PrimeSpectrum.basicOpen r)))
have h₁ : Eq (α :=
  CommRingCat.of (↑R : Type u) ⟶
    (Functor.obj (TopCat.Sheaf.pushforward CommRingCat β.hom.base) (SheafedSpace.sheaf X)).obj (P :=
          Presheaf.IsSheaf
            (Opens.grothendieckTopology (↑(↑(sheafedSpaceObj R).toPresheafedSpace : TopCat) : Type u))).obj
      (op (PrimeSpectrum.basicOpen r)))
  (CommRingCat.ofHom
      (algebraMap (↑R : Type u)
        (↑(Functor.obj (C := (TopologicalSpace.Opens (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
                (structureSheaf (↑R : Type u)))
              (op (PrimeSpectrum.basicOpen r))) :
          Type u)) ≫
    (α.hom.c ≫ Functor.whiskerRight (eqToHom sorry) X.presheaf).app (op (PrimeSpectrum.basicOpen r)))
  (CommRingCat.ofHom
      (algebraMap (↑R : Type u)
        (↑(Functor.obj (C := (TopologicalSpace.Opens (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
                (structureSheaf (↑R : Type u)))
              (op (PrimeSpectrum.basicOpen r))) :
          Type u)) ≫
    β.hom.c.app (op (PrimeSpectrum.basicOpen r)))
Tactic: apply (StructureSheaf.to_basicOpen_epi R r).1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R S : CommRingCat
f : R ⟶ S
p : PrimeSpectrum (↑S : Type u)
⊢ toStalk (↑R : Type u) (PrimeSpectrum.comap (CommRingCat.Hom.hom f) p) ≫
      PresheafedSpace.Hom.stalkMap (Spec.sheafedSpaceMap f).hom p =
    f ≫ toStalk (↑S : Type u) p
have rw : Eq (α := CommRingCat.of (↑R : Type u) ⟶ (Spec.sheafedSpaceObj S).presheaf.stalk p)
  (CategoryStruct.comp (Y :=
    CommRingCat.of
      (Functor.obj (C := (TopologicalSpace.Opens (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
          (structureSheafInType (↑R : Type u) (↑R : Type u)))
        (op ⊤)))
    (CommRingCat.ofHom
      (algebraMap (↑R : Type u)
        (Functor.obj (C := (TopologicalSpace.Opens (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u))ᵒᵖ)
          (ObjectProperty.FullSubcategory.obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
            (structureSheafInType (↑R : Type u) (↑R : Type u)))
          (op ⊤))))
    (TopCat.Presheaf.germ (structurePresheafInCommRingCat (↑R : Type u)) ⊤
        (PrimeSpectrum.comap (CommRingCat.Hom.hom f) p) trivial ≫
      PresheafedSpace.Hom.stalkMap (Spec.sheafedSpaceMap f).hom p))
  (f ≫
    CategoryStruct.comp (Y :=
      CommRingCat.of
        (Functor.obj (C := (TopologicalSpace.Opens (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u))ᵒᵖ)
          (ObjectProperty.FullSubcategory.obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
            (structureSheafInType (↑S : Type u) (↑S : Type u)))
          (op ⊤)))
      (CommRingCat.ofHom
        (algebraMap (↑S : Type u)
          (Functor.obj (C := (TopologicalSpace.Opens (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
              (structureSheafInType (↑S : Type u) (↑S : Type u)))
            (op ⊤))))
      (TopCat.Presheaf.germ (structurePresheafInCommRingCat (↑S : Type u)) ⊤ p trivial))
Tactic: rw [← algebraMap_germ ⊤ p trivial, ← algebraMap_germ ⊤ (PrimeSpectrum.comap f.hom p) trivial,
    Category.assoc]