mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case h₁.h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ ((((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
              pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
                pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
            pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
              (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
          pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
        I.glueDataObjι i) ≫
      (↑i).ι =
    (((𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
            pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
              (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
          pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
        I.glueDataObjι i) ≫
      (↑i).ι

Tactic:
]

Goal state after:
case h₁.h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ ((((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
              pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
                pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
            pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
              (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
          pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
        I.glueDataObjι i) ≫
      (↑i).ι =
    (((𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
            pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
              (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
          pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
        I.glueDataObjι i) ≫
      (↑i).ι
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: X Y : Scheme
S R : CommRingCat
φ : S ⟶ R
inst✝ : QuasiCompact (Spec.map φ)
⊢ closure (Set.range (PrimeSpectrum.comap (CommRingCat.Hom.hom (φ ≫ (ΓSpecIso R).inv)))) ⊆
    closure (Set.range ⇑(Spec.map φ))

Tactic:
]

Goal state after:
X Y : Scheme
S R : CommRingCat
φ : S ⟶ R
inst✝ : QuasiCompact (Spec.map φ)
⊢ closure (Set.range (PrimeSpectrum.comap (CommRingCat.Hom.hom (φ ≫ (ΓSpecIso R).inv)))) ⊆
    closure (Set.range ⇑(Spec.map φ))
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case h₁.h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) ≫ (↑i ⊓ ↑j).ι =
    pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) ≫ (↑i ⊓ ↑j).ι

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: case a
X Y : Scheme
f : X ⟶ Y
inst✝ : QuasiCompact f
⊢ Set.range ⇑f ⊆ ↑(ker f).support

Tactic:
]

Goal state after:
case a
X Y : Scheme
f : X ⟶ Y
inst✝ : QuasiCompact f
⊢ Set.range ⇑f ⊆ ↑(ker f).support
Module: Mathlib.AlgebraicGeometry.PullbackCarrier

Goal state before:
pp: case h
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
X' Y' S' : Scheme
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
t : ↥(pullback f g)
⊢ (pullback.fst f g ≫ i₁) t = (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂ ≫ pullback.fst f' g') t

Tactic:
]

Goal state after:
case h
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
X' Y' S' : Scheme
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
t : ↥(pullback f g)
⊢ (pullback.fst f g ≫ i₁) t = (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂ ≫ pullback.fst f' g') t
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case h₁
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ (((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
            pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
              pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
          pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
            (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
        pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      (↑i ⊓ ↑k).ι =
    ((𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
          pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
            (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
        pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      (↑i ⊓ ↑k).ι

Tactic:
]

Goal state after:
case h₁
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ (((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
            pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
              pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
          pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
            (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
        pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      (↑i ⊓ ↑k).ι =
    ((𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
          pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
            (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
        pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      (↑i ⊓ ↑k).ι
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case h₁
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) ≫ (↑i ⊓ ↑k).ι =
    pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) ≫ (↑i ⊓ ↑k).ι

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.PullbackCarrier

Goal state before:
pp: case h
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
X' Y' S' : Scheme
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
t : ↥(pullback f g)
⊢ (pullback.snd f g ≫ i₂) t = (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂ ≫ pullback.snd f' g') t

Tactic:
]

Goal state after:
case h
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
X' Y' S' : Scheme
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
t : ↥(pullback f g)
⊢ (pullback.snd f g ≫ i₂) t = (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂ ≫ pullback.snd f' g') t
Module: Mathlib.AlgebraicGeometry.Morphisms.Affine

Goal state before:
pp: case hs
X : Scheme
U : X.Opens
s : Set ↑Γ(X, U)
hs : Ideal.span s = ⊤
hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Morphisms.AffineAnd

Goal state before:
pp: case right
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝¹ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
x✝ : affineAnd (fun {R S} [CommRing R] [CommRing S] ↦ Q) f
hX : IsAffine X
hf : Q (CommRingCat.Hom.hom (Scheme.Hom.appTop f))
⊢ Q (CommRingCat.Hom.hom (Scheme.Hom.app f ((Y.basicOpen r).ι ''ᵁ ⊤)))

Tactic:
]

Goal state after:
case right
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝¹ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
x✝ : affineAnd (fun {R S} [CommRing R] [CommRing S] ↦ Q) f
hX : IsAffine X
hf : Q (CommRingCat.Hom.hom (Scheme.Hom.appTop f))
⊢ Q (CommRingCat.Hom.hom (Scheme.Hom.app f ((Y.basicOpen r).ι ''ᵁ ⊤)))
Module: Mathlib.AlgebraicGeometry.Morphisms.AffineAnd

Goal state before:
pp: case right
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝¹ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
x✝ : affineAnd (fun {R S} [CommRing R] [CommRing S] ↦ Q) f
hX : IsAffine X
hf : Q (CommRingCat.Hom.hom (Scheme.Hom.appTop f))
⊢ Q (CommRingCat.Hom.hom (Scheme.Hom.app f (Y.basicOpen r)))

Tactic:
]

Goal state after:
case right
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝¹ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
x✝ : affineAnd (fun {R S} [CommRing R] [CommRing S] ↦ Q) f
hX : IsAffine X
hf : Q (CommRingCat.Hom.hom (Scheme.Hom.appTop f))
⊢ Q (CommRingCat.Hom.hom (Scheme.Hom.app f (Y.basicOpen r)))
Module: Mathlib.AlgebraicGeometry.Morphisms.Affine

Goal state before:
pp: case hs₂
X : Scheme
U : X.Opens
s : Set ↑Γ(X, U)
hs : Ideal.span s = ⊤
hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)
j : ↑Γ(X, U)
hj : j ∈ s
⊢ IsAffineOpen
    (X.basicOpen ((ConcreteCategory.hom (Scheme.Hom.appIso U.ι ⊤).inv) ((ConcreteCategory.hom U.topIso.inv) j)))

Tactic:
]

Goal state after:
case hs₂
X : Scheme
U : X.Opens
s : Set ↑Γ(X, U)
hs : Ideal.span s = ⊤
hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)
j : ↑Γ(X, U)
hj : j ∈ s
⊢ IsAffineOpen
    (X.basicOpen ((ConcreteCategory.hom (Scheme.Hom.appIso U.ι ⊤).inv) ((ConcreteCategory.hom U.topIso.inv) j)))
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
h : U ≤ V
this : IsIso (X.homOfLE ⋯)
⊢ X.homOfLE ⋯ ≫ X.homOfLE ⋯ = 𝟙 ↑(↑U ⊓ ↑V)

Tactic:
]

Goal state after:
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
h : U ≤ V
this : IsIso (X.homOfLE ⋯)
⊢ X.homOfLE ⋯ ≫ X.homOfLE ⋯ = 𝟙 ↑(↑U ⊓ ↑V)
Module: Mathlib.AlgebraicGeometry.Morphisms.AffineAnd

Goal state before:
pp: case right
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝¹ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
x✝ : affineAnd (fun {R S} [CommRing R] [CommRing S] ↦ Q) f
hX : IsAffine X
hf : Q (CommRingCat.Hom.hom (Scheme.Hom.appTop f))
⊢ Q
    (CommRingCat.Hom.hom
      (CommRingCat.ofHom
        (IsLocalization.Away.map (↑Γ(Y, Y.basicOpen r))
          (↑Γ(X, X.basicOpen ((ConcreteCategory.hom (Scheme.Hom.appTop f)) r)))
          (CommRingCat.Hom.hom (Scheme.Hom.appTop f)) r)))

Tactic:
]

Goal state after:
case right
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝¹ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
x✝ : affineAnd (fun {R S} [CommRing R] [CommRing S] ↦ Q) f
hX : IsAffine X
hf : Q (CommRingCat.Hom.hom (Scheme.Hom.appTop f))
⊢ Q
    (CommRingCat.Hom.hom
      (CommRingCat.ofHom
        (IsLocalization.Away.map (↑Γ(Y, Y.basicOpen r))
          (↑Γ(X, X.basicOpen ((ConcreteCategory.hom (Scheme.Hom.appTop f)) r)))
          (CommRingCat.Hom.hom (Scheme.Hom.appTop f)) r)))
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
h : U ≤ V
this✝ : IsIso (X.homOfLE ⋯)
H : inv (X.homOfLE ⋯) = X.homOfLE ⋯
this :
  inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫
      I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯) ≫ I.glueData.ι V =
    I.glueData.ι U
⊢ I.glueDataObjMap h ≫ I.glueData.ι V = I.glueData.ι U

Tactic:
]

Goal state after:
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
h : U ≤ V
this✝ : IsIso (X.homOfLE ⋯)
H : inv (X.homOfLE ⋯) = X.homOfLE ⋯
this :
  inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫
      I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯) ≫ I.glueData.ι V =
    I.glueData.ι U
⊢ I.glueDataObjMap h ≫ I.glueData.ι V = I.glueData.ι U
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
h : U ≤ V
this✝ : IsIso (X.homOfLE ⋯)
H : inv (X.homOfLE ⋯) = X.homOfLE ⋯
this :
  inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫
      I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯) ≫ I.glueData.ι V =
    I.glueData.ι U
⊢ I.glueDataObjMap h ≫ I.glueData.ι V =
    𝟙 (I.glueData.U U) ≫
      inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫
        I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯) ≫ I.glueData.ι V

Tactic:
]

Goal state after:
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
h : U ≤ V
this✝ : IsIso (X.homOfLE ⋯)
H : inv (X.homOfLE ⋯) = X.homOfLE ⋯
this :
  inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫
      I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯) ≫ I.glueData.ι V =
    I.glueData.ι U
⊢ I.glueDataObjMap h ≫ I.glueData.ι V =
    𝟙 (I.glueData.U U) ≫
      inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫
        I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯) ≫ I.glueData.ι V
Module: Mathlib.AlgebraicGeometry.Morphisms.Affine

Goal state before:
pp: case to_basicOpen
X✝ Y✝ Z : Scheme
f✝ : X✝ ⟶ Y✝
g : Y✝ ⟶ Z
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
H this : IsAffine X
⊢ IsAffineOpen (X.basicOpen ((ConcreteCategory.hom (Scheme.Hom.app f ⊤)) r))

Tactic:
]

Goal state after:
case to_basicOpen
X✝ Y✝ Z : Scheme
f✝ : X✝ ⟶ Y✝
g : Y✝ ⟶ Z
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
H this : IsAffine X
⊢ IsAffineOpen (X.basicOpen ((ConcreteCategory.hom (Scheme.Hom.app f ⊤)) r))
Module: Mathlib.AlgebraicGeometry.Morphisms.AffineAnd

Goal state before:
pp: case hs
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝ : IsAffine Y
f : X ⟶ Y
s : Finset ↑Γ(Y, ⊤)
hf : ∀ (r : ↥s), IsAffine ↑(f ⁻¹ᵁ Y.basicOpen ↑r) ∧ Q (CommRingCat.Hom.hom (Scheme.Hom.appTop (f ∣_ Y.basicOpen ↑r)))
hs : Ideal.span (⇑(CommRingCat.Hom.hom (Scheme.Hom.appTop f)) '' ↑s) = ⊤
⊢ Ideal.span (⇑(ConcreteCategory.hom (Scheme.Hom.appTop f)) '' ↑s) = ⊤

Tactic:
]

Goal state after:
case hs
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝ : IsAffine Y
f : X ⟶ Y
s : Finset ↑Γ(Y, ⊤)
hf : ∀ (r : ↥s), IsAffine ↑(f ⁻¹ᵁ Y.basicOpen ↑r) ∧ Q (CommRingCat.Hom.hom (Scheme.Hom.appTop (f ∣_ Y.basicOpen ↑r)))
hs : Ideal.span (⇑(CommRingCat.Hom.hom (Scheme.Hom.appTop f)) '' ↑s) = ⊤
⊢ Ideal.span (⇑(ConcreteCategory.hom (Scheme.Hom.appTop f)) '' ↑s) = ⊤
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case e_a
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
h : U ≤ V
this✝ : IsIso (X.homOfLE ⋯)
H : inv (X.homOfLE ⋯) = X.homOfLE ⋯
this :
  inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫
      I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯) ≫ I.glueData.ι V =
    I.glueData.ι U
⊢ I.glueDataObjMap h ≫ I.glueDataObjι V =
    ((inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫ I.glueDataT U V) ≫
        pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯)) ≫
      I.glueDataObjι V

Tactic:
]

Goal state after:
case e_a
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
h : U ≤ V
this✝ : IsIso (X.homOfLE ⋯)
H : inv (X.homOfLE ⋯) = X.homOfLE ⋯
this :
  inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫
      I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯) ≫ I.glueData.ι V =
    I.glueData.ι U
⊢ I.glueDataObjMap h ≫ I.glueDataObjι V =
    ((inv (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫ I.glueDataT U V) ≫
        pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯)) ≫
      I.glueDataObjι V
Module: Mathlib.AlgebraicGeometry.Morphisms.Affine

Goal state before:
pp: case of_basicOpenCover
X✝ Y✝ Z : Scheme
f✝ : X✝ ⟶ Y✝
g : Y✝ ⟶ Z
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
S : Finset ↑Γ(Y, ⊤)
hS' : ∀ (r : ↥S), IsAffine ↑(f ⁻¹ᵁ Y.basicOpen ↑r)
hS : Ideal.span (⇑(CommRingCat.Hom.hom (Scheme.Hom.appTop f)) '' ↑S) = ⊤
⊢ IsAffine X

Tactic:
]

Goal state after:
case of_basicOpenCover
X✝ Y✝ Z : Scheme
f✝ : X✝ ⟶ Y✝
g : Y✝ ⟶ Z
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
S : Finset ↑Γ(Y, ⊤)
hS' : ∀ (r : ↥S), IsAffine ↑(f ⁻¹ᵁ Y.basicOpen ↑r)
hS : Ideal.span (⇑(CommRingCat.Hom.hom (Scheme.Hom.appTop f)) '' ↑S) = ⊤
⊢ IsAffine X
Module: Mathlib.AlgebraicGeometry.PullbackCarrier

Goal state before:
pp: case h
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
X' Y' S' : Scheme
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
z : ↥(pullback f' g')
x : ↥X
hx : i₁ x = (pullback.fst f' g') z
y : ↥Y
hy : i₂ y = (pullback.snd f' g') z
T₁ : Triplet (pullback.fst f' g') i₁ := Triplet.mk' z x ⋯
w₁ : ↥(pullback (pullback.fst f' g') i₁)
hw₁ : (pullback.fst (pullback.fst f' g') i₁) w₁ = T₁.x ∧ (pullback.snd (pullback.fst f' g') i₁) w₁ = T₁.y
T₂ : Triplet (pullback.snd f' g') i₂ := Triplet.mk' z y ⋯
w₂ : ↥(pullback (pullback.snd f' g') i₂)
hw₂ : (pullback.fst (pullback.snd f' g') i₂) w₂ = T₂.x ∧ (pullback.snd (pullback.snd f' g') i₂) w₂ = T₂.y
T : Triplet (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂) := Triplet.mk' w₁ w₂ ⋯
t : ↥(pullback (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂))
left✝ : (pullback.fst (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂)) t = T.x
ht₂ : (pullback.snd (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂)) t = T.y
⊢ (pullback.snd (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂) ≫
        pullback.fst (pullback.snd f' g') i₂)
      t =
    z

Tactic:
]

Goal state after:
case h
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
X' Y' S' : Scheme
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
z : ↥(pullback f' g')
x : ↥X
hx : i₁ x = (pullback.fst f' g') z
y : ↥Y
hy : i₂ y = (pullback.snd f' g') z
T₁ : Triplet (pullback.fst f' g') i₁ := Triplet.mk' z x ⋯
w₁ : ↥(pullback (pullback.fst f' g') i₁)
hw₁ : (pullback.fst (pullback.fst f' g') i₁) w₁ = T₁.x ∧ (pullback.snd (pullback.fst f' g') i₁) w₁ = T₁.y
T₂ : Triplet (pullback.snd f' g') i₂ := Triplet.mk' z y ⋯
w₂ : ↥(pullback (pullback.snd f' g') i₂)
hw₂ : (pullback.fst (pullback.snd f' g') i₂) w₂ = T₂.x ∧ (pullback.snd (pullback.snd f' g') i₂) w₂ = T₂.y
T : Triplet (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂) := Triplet.mk' w₁ w₂ ⋯
t : ↥(pullback (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂))
left✝ : (pullback.fst (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂)) t = T.x
ht₂ : (pullback.snd (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂)) t = T.y
⊢ (pullback.snd (pullback.fst (pullback.fst f' g') i₁) (pullback.fst (pullback.snd f' g') i₂) ≫
        pullback.fst (pullback.snd f' g') i₂)
      t =
    z
Module: Mathlib.AlgebraicGeometry.Morphisms.Affine

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
U : ↥Y → Y.Opens
hxU : ∀ (x : ↥Y), x ∈ U x
hU : ∀ (x : ↥Y), IsAffineOpen (U x)
hfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)
⊢ ∀ (i : ↥Y), IsAffine ↑(f ⁻¹ᵁ ↑⟨U i, ⋯⟩)

pp: X Y : Scheme
f : X ⟶ Y
U : ↥Y → Y.Opens
hxU : ∀ (x : ↥Y), x ∈ U x
hU : ∀ (x : ↥Y), IsAffineOpen (U x)
hfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)
⊢ ⨆ i, ↑⟨U i, ⋯⟩ = ⊤

Tactic:
]

Goal state after:
X Y : Scheme
f : X ⟶ Y
U : ↥Y → Y.Opens
hxU : ∀ (x : ↥Y), x ∈ U x
hU : ∀ (x : ↥Y), IsAffineOpen (U x)
hfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)
⊢ ∀ (i : ↥Y), IsAffine ↑(f ⁻¹ᵁ ↑⟨U i, ⋯⟩)

X Y : Scheme
f : X ⟶ Y
U : ↥Y → Y.Opens
hxU : ∀ (x : ↥Y), x ∈ U x
hU : ∀ (x : ↥Y), IsAffineOpen (U x)
hfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)
⊢ ⨆ i, ↑⟨U i, ⋯⟩ = ⊤
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: X : Scheme
I : X.IdealSheafData
ia : I.glueData.J
a : ↥(I.glueDataObj ia)
ib : I.glueData.J
b : ↥(I.glueDataObj ib)
e :
  I.gluedTo ((ConcreteCategory.hom (forget.map (I.glueData.ι ia))) a) =
    I.gluedTo ((ConcreteCategory.hom (forget.map (I.glueData.ι ib))) b)
this : (I.glueDataObjι ia ≫ (↑ia).ι) a = (I.glueDataObjι ib ≫ (↑ib).ι) b
⊢ ↑((I.glueDataObjι ia) a) = ↑((I.glueDataObjι ib) b)

Tactic:
]

Goal state after:
X : Scheme
I : X.IdealSheafData
ia : I.glueData.J
a : ↥(I.glueDataObj ia)
ib : I.glueData.J
b : ↥(I.glueDataObj ib)
e :
  I.gluedTo ((ConcreteCategory.hom (forget.map (I.glueData.ι ia))) a) =
    I.gluedTo ((ConcreteCategory.hom (forget.map (I.glueData.ι ib))) b)
this : (I.glueDataObjι ia ≫ (↑ia).ι) a = (I.glueDataObjι ib ≫ (↑ib).ι) b
⊢ ↑((I.glueDataObjι ia) a) = ↑((I.glueDataObjι ib) b)
Module: Mathlib.AlgebraicGeometry.PullbackCarrier

Goal state before:
pp: case h.fst
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
x : (fun X ↦ X) (Types.PullbackObj (forget.map f) (forget.map g))
z : ↥(pullback f g)
h1 : (pullback.fst f g) z = (↑x).1
h2 : (pullback.snd f g) z = (↑x).2
⊢ (ConcreteCategory.hom (forget.map (pullback.fst f g))) z = (↑x).1

Tactic:
]

Goal state after:
case h.fst
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
x : (fun X ↦ X) (Types.PullbackObj (forget.map f) (forget.map g))
z : ↥(pullback f g)
h1 : (pullback.fst f g) z = (↑x).1
h2 : (pullback.snd f g) z = (↑x).2
⊢ (ConcreteCategory.hom (forget.map (pullback.fst f g))) z = (↑x).1