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

Goal state before:
pp: R M : Type u
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
S : Type u
inst✝² : CommRing S
N : Type u
inst✝¹ : AddCommGroup N
inst✝ : Module S N
σ : R →+* S
f : M →ₛₗ[σ] N
U : Opens ↑(PrimeSpectrum.Top R)
V : Opens ↑(PrimeSpectrum.Top S)
hUV : V.carrier ⊆ PrimeSpectrum.comap σ ⁻¹' U.carrier
s : (x : ↥U) → Localizations M ↑x
hs✝ : (isLocallyFraction R M).pred s
this✝¹ : Module R N := Module.compHom N σ
this✝ : Algebra R S := σ.toAlgebra
this : IsScalarTower R S N
p : ↑(PrimeSpectrum.Top S)
hpV : p ∈ V
W : Opens ↑(PrimeSpectrum.Top R)
m : ↑⟨PrimeSpectrum.comap σ p, ⋯⟩ ∈ W
iWU : W ⟶ U
a : M
b : R
h_frac : ∀ (x : ↥W), ∃ (hs : b ∉ (↑x).asIdeal), (fun x ↦ s (iWU x)) x = LocalizedModule.mk a ⟨b, hs⟩
q : ↑(PrimeSpectrum.Top S)
hqW : q ∈ ↑{ carrier := PrimeSpectrum.comap σ ⁻¹' W.carrier, is_open' := ⋯ }
hqV : q ∈ ↑V
hs : b ∉ (↑⟨PrimeSpectrum.comap σ q, hqW⟩).asIdeal
H : s ⟨PrimeSpectrum.comap σ q, ⋯⟩ = LocalizedModule.mk a ⟨b, hs⟩
⊢ (Localizations.comapFun f q) (s ⟨PrimeSpectrum.comap σ q, ⋯⟩) = LocalizedModule.mk (f a) ⟨σ b, hs⟩

Tactic:
H

Goal state after:
R M : Type u
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
S : Type u
inst✝² : CommRing S
N : Type u
inst✝¹ : AddCommGroup N
inst✝ : Module S N
σ : R →+* S
f : M →ₛₗ[σ] N
U : Opens ↑(PrimeSpectrum.Top R)
V : Opens ↑(PrimeSpectrum.Top S)
hUV : V.carrier ⊆ PrimeSpectrum.comap σ ⁻¹' U.carrier
s : (x : ↥U) → Localizations M ↑x
hs✝ : (isLocallyFraction R M).pred s
this✝¹ : Module R N := Module.compHom N σ
this✝ : Algebra R S := σ.toAlgebra
this : IsScalarTower R S N
p : ↑(PrimeSpectrum.Top S)
hpV : p ∈ V
W : Opens ↑(PrimeSpectrum.Top R)
m : ↑⟨PrimeSpectrum.comap σ p, ⋯⟩ ∈ W
iWU : W ⟶ U
a : M
b : R
h_frac : ∀ (x : ↥W), ∃ (hs : b ∉ (↑x).asIdeal), (fun x ↦ s (iWU x)) x = LocalizedModule.mk a ⟨b, hs⟩
q : ↑(PrimeSpectrum.Top S)
hqW : q ∈ ↑{ carrier := PrimeSpectrum.comap σ ⁻¹' W.carrier, is_open' := ⋯ }
hqV : q ∈ ↑V
hs : b ∉ (↑⟨PrimeSpectrum.comap σ q, hqW⟩).asIdeal
H : s ⟨PrimeSpectrum.comap σ q, ⋯⟩ = LocalizedModule.mk a ⟨b, hs⟩
⊢ (Localizations.comapFun f q) (LocalizedModule.mk a ⟨b, hs⟩) = LocalizedModule.mk (f a) ⟨σ b, hs⟩
Module: Mathlib.CategoryTheory.GlueData

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v, u₁} C
C' : Type u₂
inst✝⁴ : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
inst✝³ : HasMulticoequalizer D.diagram
inst✝² : PreservesColimit D.diagram.multispan F
inst✝¹ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F
i j : D.J
inst✝ : ReflectsLimit (cospan (D.ι i) (D.ι j)) F
hc : IsLimit ((D.mapGlueData F).vPullbackCone i j)
⊢ IsLimit (D.vPullbackCone i j)

Tactic:
apply isLimitOfReflects F

Goal state after:
C : Type u₁
inst✝⁵ : Category.{v, u₁} C
C' : Type u₂
inst✝⁴ : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
inst✝³ : HasMulticoequalizer D.diagram
inst✝² : PreservesColimit D.diagram.multispan F
inst✝¹ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F
i j : D.J
inst✝ : ReflectsLimit (cospan (D.ι i) (D.ι j)) F
hc : IsLimit ((D.mapGlueData F).vPullbackCone i j)
⊢ IsLimit (F.mapCone (D.vPullbackCone i j))
Module: Mathlib.Topology.Gluing

Goal state before:
pp: D : GlueData
h : MkCore
i j k : h.J
x : ↑(h.U i)
hx : x ∈ h.V i j
hx' : x ∈ h.V i k
⊢ ⟨(⟨↑((ConcreteCategory.hom (h.t k i))
              ⟨↑((ConcreteCategory.hom (h.t j k)) ⟨↑((ConcreteCategory.hom (h.t i j)) ⟨x, hx⟩), ⋯⟩), ⋯⟩),
          ⋯⟩,
        (ConcreteCategory.hom (h.t k i))
          ⟨↑((ConcreteCategory.hom (h.t j k)) ⟨↑((ConcreteCategory.hom (h.t i j)) ⟨x, hx⟩), ⋯⟩), ⋯⟩),
      ⋯⟩ =
    ⟨(⟨x, hx⟩, ⟨x, hx'⟩), ⋯⟩

Tactic:
rw [Subtype.mk_eq_mk, Prod.mk_inj, Subtype.mk_eq_mk, Subtype.ext_iff, and_self_iff]

Goal state after:
D : GlueData
h : MkCore
i j k : h.J
x : ↑(h.U i)
hx : x ∈ h.V i j
hx' : x ∈ h.V i k
⊢ ↑((ConcreteCategory.hom (h.t k i))
        ⟨↑((ConcreteCategory.hom (h.t j k)) ⟨↑((ConcreteCategory.hom (h.t i j)) ⟨x, hx⟩), ⋯⟩), ⋯⟩) =
    x
Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing

Goal state before:
pp: case w.refine_2
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(pullback (D.f i j) (D.f i k))
this :
  ⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =
    ⇑(TopCat.Hom.hom (D.t k i).base) ∘
      ⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base) ∘ ⇑(TopCat.Hom.hom (inv (D.t' k i j)).base)
⊢ Function.RightInverse ⇑(ConcreteCategory.hom (D.t' k i j).base) ⇑(TopCat.Hom.hom (inv (D.t' k i j)).base)

Tactic:
intro x
      rw [← ConcreteCategory.comp_apply, ← comp_base, IsIso.hom_inv_id, id_base,
        ConcreteCategory.id_apply]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Gluing

Goal state before:
pp: X : Scheme
𝒰 : X.OpenCover
F : Scheme ⥤ PresheafedSpace CommRingCat :=
  forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace ⋙ SheafedSpace.forgetToPresheafedSpace
⊢ IsIso (F.map (fromGlued 𝒰))

Tactic:
change IsIso 𝒰.fromGlued.toPshHom
    apply PresheafedSpace.IsOpenImmersion.to_iso

Goal state after:
Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)
⊢ ⇑(ConcreteCategory.hom (D.ι j).base) ⁻¹' ⇑(ConcreteCategory.hom (D.ι i).base) '' ↑U =
    ⇑(ConcreteCategory.hom (D.f j i).base) ''
      ⇑(ConcreteCategory.hom (D.t j i).base) ⁻¹' ⇑(ConcreteCategory.hom (D.f i j).base) ⁻¹' ↑U

Tactic:
rw [← show _ = (𝖣.ι i).base from 𝖣.ι_gluedIso_inv (PresheafedSpace.forget _) i, ←
    show _ = (𝖣.ι j).base from 𝖣.ι_gluedIso_inv (PresheafedSpace.forget _) j]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)
⊢ ⇑(ConcreteCategory.hom ((D.mapGlueData (forget C)).ι j ≫ (D.gluedIso (forget C)).inv)) ⁻¹'
      ⇑(ConcreteCategory.hom ((D.mapGlueData (forget C)).ι i ≫ (D.gluedIso (forget C)).inv)) '' ↑U =
    ⇑(ConcreteCategory.hom (D.f j i).base) ''
      ⇑(ConcreteCategory.hom (D.t j i).base) ⁻¹' ⇑(ConcreteCategory.hom (D.f i j).base) ⁻¹' ↑U
Module: Mathlib.AlgebraicGeometry.Gluing

Goal state before:
pp: J : Type w
inst✝² : Category.{v, w} J
F : J ⥤ Scheme
inst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst✝ : (F ⋙ forget).IsLocallyDirected
i j k : J
x : ↥(pullback (V F i j).ι (V F i k).ι)
k₁ : (k : J) × (k ⟶ i) × (k ⟶ j)
k₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)
l : J
hli : l ⟶ k₁.fst
hlk : l ⟶ k₂.fst
z : (F ⋙ forget).obj l
hy₁ : (F.map k₁.snd.1) ((ConcreteCategory.hom ((F ⋙ forget).map hli)) z) = ↑((pullback.fst (V F i j).ι (V F i k).ι) x)
hy₂ : (F.map k₂.snd.1) ((ConcreteCategory.hom ((F ⋙ forget).map hlk)) z) = ↑((pullback.snd (V F i j).ι (V F i k).ι) x)
⊢ ∃ l fi fj fk α z,
    IsOpenImmersion α ∧
      α ≫ pullback.fst (V F i j).ι (V F i k).ι = (Hom.isoOpensRange (F.map fi)).hom ≫ (F.obj i).homOfLE ⋯ ∧
        α ≫ pullback.snd (V F i j).ι (V F i k).ι = (Hom.isoOpensRange (F.map fi)).hom ≫ (F.obj i).homOfLE ⋯ ∧ α z = x

Tactic:
let α : F.obj l ⟶ pullback (V F i j).ι (V F i k).ι :=
    pullback.lift
      ((F.map (hli ≫ k₁.2.1)).isoOpensRange.hom ≫ Scheme.homOfLE _
        (le_iSup_of_le ⟨l, hli ≫ k₁.2.1, hli ≫ k₁.2.2⟩ le_rfl))
      ((F.map (hli ≫ k₁.2.1)).isoOpensRange.hom ≫ Scheme.homOfLE _
        (le_iSup_of_le ⟨l, hli ≫ k₁.2.1, hlk ≫ k₂.2.2⟩ le_rfl))
      (by simp)

Goal state after:
J : Type w
inst✝² : Category.{v, w} J
F : J ⥤ Scheme
inst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst✝ : (F ⋙ forget).IsLocallyDirected
i j k : J
x : ↥(pullback (V F i j).ι (V F i k).ι)
k₁ : (k : J) × (k ⟶ i) × (k ⟶ j)
k₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)
l : J
hli : l ⟶ k₁.fst
hlk : l ⟶ k₂.fst
z : (F ⋙ forget).obj l
hy₁ : (F.map k₁.snd.1) ((ConcreteCategory.hom ((F ⋙ forget).map hli)) z) = ↑((pullback.fst (V F i j).ι (V F i k).ι) x)
hy₂ : (F.map k₂.snd.1) ((ConcreteCategory.hom ((F ⋙ forget).map hlk)) z) = ↑((pullback.snd (V F i j).ι (V F i k).ι) x)
α : F.obj l ⟶ pullback (V F i j).ι (V F i k).ι :=
  pullback.lift ((Hom.isoOpensRange (F.map (hli ≫ k₁.snd.1))).hom ≫ (F.obj i).homOfLE ⋯)
    ((Hom.isoOpensRange (F.map (hli ≫ k₁.snd.1))).hom ≫ (F.obj i).homOfLE ⋯) ⋯
⊢ ∃ l fi fj fk α z,
    IsOpenImmersion α ∧
      α ≫ pullback.fst (V F i j).ι (V F i k).ι = (Hom.isoOpensRange (F.map fi)).hom ≫ (F.obj i).homOfLE ⋯ ∧
        α ≫ pullback.snd (V F i j).ι (V F i k).ι = (Hom.isoOpensRange (F.map fi)).hom ≫ (F.obj i).homOfLE ⋯ ∧ α z = x
Module: Mathlib.AlgebraicGeometry.Gluing

Goal state before:
pp: J : Type w
inst✝³ : Category.{v, w} J
F : J ⥤ Scheme
inst✝² : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst✝¹ : (F ⋙ forget).IsLocallyDirected
inst✝ : Quiver.IsThin J
i j : J
k₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)
U : (F.obj i).Opens
h₁ : Hom.opensRange (F.map k₁.snd.1) ≤ U
h₂ : Hom.opensRange (F.map k₂.snd.1) ≤ U
x : ↥(pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
l : J
hli : l ⟶ k₁.fst
hlj : l ⟶ k₂.fst
y : (F ⋙ forget).obj l
hy₁ :
  (ConcreteCategory.hom ((F ⋙ forget).map hli)) y =
    (pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫ (Hom.isoOpensRange (F.map k₁.snd.1)).inv) x
hy₂ :
  (ConcreteCategory.hom ((F ⋙ forget).map hlj)) y =
    (pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫ (Hom.isoOpensRange (F.map k₂.snd.1)).inv) x
α : F.obj l ⟶ pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) :=
  pullback.lift (F.map hli ≫ (Hom.isoOpensRange (F.map k₁.snd.1)).hom)
    (F.map hlj ≫ (Hom.isoOpensRange (F.map k₂.snd.1)).hom) ⋯
this✝ : IsOpenImmersion α
this : α y = x
⊢ (Hom.opensRange α).ι ≫
      pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
        (Hom.isoOpensRange (F.map k₁.snd.1)).inv ≫ F.map k₁.snd.2 =
    (Hom.opensRange α).ι ≫
      pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
        (Hom.isoOpensRange (F.map k₂.snd.1)).inv ≫ F.map k₂.snd.2

Tactic:
rw [← cancel_epi α.isoOpensRange.hom]

Goal state after:
J : Type w
inst✝³ : Category.{v, w} J
F : J ⥤ Scheme
inst✝² : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst✝¹ : (F ⋙ forget).IsLocallyDirected
inst✝ : Quiver.IsThin J
i j : J
k₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)
U : (F.obj i).Opens
h₁ : Hom.opensRange (F.map k₁.snd.1) ≤ U
h₂ : Hom.opensRange (F.map k₂.snd.1) ≤ U
x : ↥(pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
l : J
hli : l ⟶ k₁.fst
hlj : l ⟶ k₂.fst
y : (F ⋙ forget).obj l
hy₁ :
  (ConcreteCategory.hom ((F ⋙ forget).map hli)) y =
    (pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫ (Hom.isoOpensRange (F.map k₁.snd.1)).inv) x
hy₂ :
  (ConcreteCategory.hom ((F ⋙ forget).map hlj)) y =
    (pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫ (Hom.isoOpensRange (F.map k₂.snd.1)).inv) x
α : F.obj l ⟶ pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) :=
  pullback.lift (F.map hli ≫ (Hom.isoOpensRange (F.map k₁.snd.1)).hom)
    (F.map hlj ≫ (Hom.isoOpensRange (F.map k₂.snd.1)).hom) ⋯
this✝ : IsOpenImmersion α
this : α y = x
⊢ (Hom.isoOpensRange α).hom ≫
      (Hom.opensRange α).ι ≫
        pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
          (Hom.isoOpensRange (F.map k₁.snd.1)).inv ≫ F.map k₁.snd.2 =
    (Hom.isoOpensRange α).hom ≫
      (Hom.opensRange α).ι ≫
        pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
          (Hom.isoOpensRange (F.map k₂.snd.1)).inv ≫ F.map k₂.snd.2
Module: Mathlib.AlgebraicGeometry.Gluing

Goal state before:
pp: J : Type w
inst✝⁴ : Category.{v, w} J
F : J ⥤ Scheme
inst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst✝² : (F ⋙ forget).IsLocallyDirected
inst✝¹ : Quiver.IsThin J
inst✝ : Small.{u, w} J
i j k : Shrink.{u, w} J
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
l : J
fi : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
fj : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
fk : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
α : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
z : ↥(F.obj l)
hα : IsOpenImmersion α
hα₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hα₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
e : α z = x
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
rw [← cancel_epi α.isoOpensRange.hom]

Goal state after:
J : Type w
inst✝⁴ : Category.{v, w} J
F : J ⥤ Scheme
inst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst✝² : (F ⋙ forget).IsLocallyDirected
inst✝¹ : Quiver.IsThin J
inst✝ : Small.{u, w} J
i j k : Shrink.{u, w} J
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
l : J
fi : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
fj : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
fk : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
α : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
z : ↥(F.obj l)
hα : IsOpenImmersion α
hα₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hα₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
e : α z = x
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s✝ s : PullbackCone f g
m : s.pt ⟶ (PullbackCone.mk (p1 𝒰 f g) (p2 𝒰 f g) ⋯).pt
h₁ : m ≫ (PullbackCone.mk (p1 𝒰 f g) (p2 𝒰 f g) ⋯).fst = s.fst
h₂ : m ≫ (PullbackCone.mk (p1 𝒰 f g) (p2 𝒰 f g) ⋯).snd = s.snd
⊢ m = gluedLift 𝒰 f g s

Tactic:
PullbackCone.mk_π_app

Goal state after:
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s✝ s : PullbackCone f g
m : s.pt ⟶ (PullbackCone.mk (p1 𝒰 f g) (p2 𝒰 f g) ⋯).pt
h₁ : m ≫ p1 𝒰 f g = s.fst
h₂ : m ≫ p2 𝒰 f g = s.snd
⊢ m = gluedLift 𝒰 f g s
Module: Mathlib.AlgebraicGeometry.Morphisms.Basic

Goal state before:
pp: P : MorphismProperty Scheme
inst✝² : IsZariskiLocalAtSource P
X Y : Scheme
f : X ⟶ Y
inst✝¹ : IsZariskiLocalAtTarget P
inst✝ : P.RespectsRight IsOpenImmersion
⊢ P f ↔ ∀ (x : ↥X), ∃ U V, ∃ (_ : x ∈ V.carrier) (e : V ≤ f ⁻¹ᵁ U), P (Scheme.Hom.resLE f U V e)

Tactic:
refine ⟨fun hf x ↦ ⟨⊤, ⊤, trivial, by simp, resLE _ hf⟩, fun hf ↦ ?_⟩
  choose U V hxU e hf using hf
  rw [IsZariskiLocalAtSource.iff_of_iSup_eq_top (fun x : X ↦ V x) (P := P)]
  · intro x
    rw [← Scheme.Hom.resLE_comp_ι _ (e x)]
    exact MorphismProperty.RespectsRight.postcomp (Q := @IsOpenImmersion) _ inferInstance _ (hf x)
  · rw [eq_top_iff]
    rintro x -
    simp only [Opens.mem_iSup]
    use x, hxU x

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

Goal state before:
pp: case restrict
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
this : Q.IsLocal := isLocal_affineProperty P
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
⊢ targetAffineLocally Q f → targetAffineLocally Q (f ∣_ U)

Tactic:
H

Goal state after:
case restrict
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
this : Q.IsLocal := isLocal_affineProperty P
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
H : targetAffineLocally Q f
⊢ targetAffineLocally Q (f ∣_ U)
Module: Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap

Goal state before:
pp: X Y Z : Scheme
f✝ : X ⟶ Y
g : Y ⟶ Z
this : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ Function.Surjective).RespectsIso
α β : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
ι : Type u_1
U : ι → Opens β
⊢ IsOpenCover U →
    Continuous[inst✝¹, inst✝] f →
      (∀ (i : ι), Function.Surjective ((U i).carrier.restrictPreimage f)) → Function.Surjective f

Tactic:
H

Goal state after:
X Y Z : Scheme
f✝ : X ⟶ Y
g : Y ⟶ Z
this : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ Function.Surjective).RespectsIso
α β : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
ι : Type u_1
U : ι → Opens β
H : IsOpenCover U
⊢ Continuous[inst✝¹, inst✝] f →
    (∀ (i : ι), Function.Surjective ((U i).carrier.restrictPreimage f)) → Function.Surjective f
Module: Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap

Goal state before:
pp: X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
e : X ⟶ Y
inst✝ : Surjective e
hge : e ≫ g = f
⊢ Set.range ⇑f = Set.range ⇑g

Tactic:
by
  rw [← hge]
  simp [Set.range_comp]

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

Goal state before:
pp: case of_sSup_eq_top
P : MorphismProperty Scheme
hP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f
X Y : Scheme
f : X ⟶ Y
ι : Type u
U : ι → Y.Opens
hU : iSup U = ⊤
⊢ (∀ (i : ι), P.universally (f ∣_ U i)) → P.universally f

Tactic:
H

Goal state after:
case of_sSup_eq_top
P : MorphismProperty Scheme
hP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f
X Y : Scheme
f : X ⟶ Y
ι : Type u
U : ι → Y.Opens
hU : iSup U = ⊤
H : ∀ (i : ι), P.universally (f ∣_ U i)
⊢ P.universally f
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] ↦ P
hP' : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
X Y S : Scheme
inst✝² : IsAffine X
inst✝¹ : IsAffine Y
inst✝ : IsAffine S
f : X ⟶ S
g : Y ⟶ S
H : P (CommRingCat.Hom.hom (Scheme.Hom.appTop g))
⊢ P (CommRingCat.Hom.hom (Scheme.Hom.appTop (pullback.fst (AffineScheme.ofHom f) (AffineScheme.ofHom g)).hom))

Tactic:
have := congr_arg Quiver.Hom.unop
      (PreservesPullback.iso_hom_fst AffineScheme.Γ.rightOp (AffineScheme.ofHom f)
        (AffineScheme.ofHom g))

Goal state after:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] ↦ P
hP' : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
X Y S : Scheme
inst✝² : IsAffine X
inst✝¹ : IsAffine Y
inst✝ : IsAffine S
f : X ⟶ S
g : Y ⟶ S
H : P (CommRingCat.Hom.hom (Scheme.Hom.appTop g))
this :
  ((PreservesPullback.iso AffineScheme.Γ.rightOp (AffineScheme.ofHom f) (AffineScheme.ofHom g)).hom ≫
        pullback.fst (AffineScheme.Γ.rightOp.map (AffineScheme.ofHom f))
          (AffineScheme.Γ.rightOp.map (AffineScheme.ofHom g))).unop =
    (AffineScheme.Γ.rightOp.map (pullback.fst (AffineScheme.ofHom f) (AffineScheme.ofHom g))).unop
⊢ P (CommRingCat.Hom.hom (Scheme.Hom.appTop (pullback.fst (AffineScheme.ofHom f) (AffineScheme.ofHom g)).hom))
Module: Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks

Goal state before:
pp: case h₂
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
⊢ (algebraMap R T).SurjectiveOnStalks → (algebraMap S (TensorProduct R S T)).SurjectiveOnStalks

Tactic:
H

Goal state after:
case h₂
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
H : (algebraMap R T).SurjectiveOnStalks
⊢ (algebraMap S (TensorProduct R S T)).SurjectiveOnStalks
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case to_basicOpen
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
⊢ sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
    sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ Y.basicOpen r)

Tactic:
H

Goal state after:
case to_basicOpen
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑Γ(Y, ⊤)
H : sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f
⊢ sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ Y.basicOpen r)
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact

Goal state before:
pp: X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
⊢ ∀ (U : Set ↥Z), IsOpen U → IsCompact U → IsCompact (⇑(f ≫ g) ⁻¹' U)

Tactic:
intro U hU hU'

Goal state after:
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
U : Set ↥Z
hU : IsOpen U
hU' : IsCompact U
⊢ IsCompact (⇑(f ≫ g) ⁻¹' U)
Module: Mathlib.RingTheory.Ideal.Height

Goal state before:
pp: case mp
R : Type u_1
inst✝³ : CommRing R
inst✝² : FiniteRingKrullDim R
inst✝¹ : IsLocalRing R
I : Ideal R
inst✝ : I.IsPrime
⊢ ↑I.height = ringKrullDim R → I = IsLocalRing.maximalIdeal R

pp: case mpr
R : Type u_1
inst✝³ : CommRing R
inst✝² : FiniteRingKrullDim R
inst✝¹ : IsLocalRing R
I : Ideal R
inst✝ : I.IsPrime
⊢ I = IsLocalRing.maximalIdeal R → ↑I.height = ringKrullDim R

Tactic:
· intro h
    exact IsLocalRing.eq_maximalIdeal (Ideal.isMaximal_of_height_eq_ringKrullDim h)

Goal state after:
case mpr
R : Type u_1
inst✝³ : CommRing R
inst✝² : FiniteRingKrullDim R
inst✝¹ : IsLocalRing R
I : Ideal R
inst✝ : I.IsPrime
⊢ I = IsLocalRing.maximalIdeal R → ↑I.height = ringKrullDim R
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case localizationAwayPreserves
P : MorphismProperty Scheme
inst✝⁹ : IsZariskiLocalAtTarget P
inst✝⁸ : IsZariskiLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
⊢ (fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))) f →
    (fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))) (IsLocalization.Away.map R' S' f r)

Tactic:
H

Goal state after:
case localizationAwayPreserves
P : MorphismProperty Scheme
inst✝⁹ : IsZariskiLocalAtTarget P
inst✝⁸ : IsZariskiLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
H : P (Spec.map (CommRingCat.ofHom f))
⊢ P (Spec.map (CommRingCat.ofHom (IsLocalization.Away.map R' S' f r)))