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

Goal state before:
pp: case refine_2
X : Scheme
R A : CommRingCat
x : ↥X
inst✝ : X.IsGermInjectiveAt x
U : X.Opens
hxU : x ∈ U
φ : A ⟶ X.presheaf.stalk x
φRA : R ⟶ A
φRX : R ⟶ Γ(X, U)
hφRA : (CommRingCat.Hom.hom φRA).FiniteType
e✝ : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU
V : X.Opens
hxV : x ∈ V
iVU : V ≤ U
hV : (CommRingCat.Hom.hom φ).range ≤ (CommRingCat.Hom.hom (X.presheaf.germ V x hxV)).range
V' : X.Opens
hxV' : x ∈ V'
hV' : IsAffineOpen V'
iV'V : V' ≤ V
H : Function.Injective ⇑(ConcreteCategory.hom (X.presheaf.germ V' x hxV'))
f : X.presheaf.obj (Opposite.op V') ⟶ X.presheaf.stalk x := X.presheaf.germ V' x hxV'
hf' : (CommRingCat.Hom.hom (X.presheaf.germ V x hxV)).range ≤ (CommRingCat.Hom.hom f).range
e : ↑(X.presheaf.obj (Opposite.op V')) ≃+* ↥(ConcreteCategory.hom (X.presheaf.germ V' x hxV')).range :=
  RingEquiv.ofLeftInverse ⋯
a : ↑R
⊢ (ConcreteCategory.hom (X.presheaf.germ U x ⋯)) (φRX.hom' a) = (ConcreteCategory.hom (φRX ≫ X.presheaf.germ U x hxU)) a

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.FunctionField

Goal state before:
pp: R : CommRingCat
inst✝ : IsDomain ↑R
⊢ PrimeSpectrum.zeroLocus ↑⊥.asIdeal = PrimeSpectrum.zeroLocus {0}

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.FunctionField

Goal state before:
pp: X : Scheme
inst✝ : IsIntegral X
f : ↑X.functionField
hf : f ≠ 0
U : Opens ↥X
hU : genericPoint ↥X ∈ U
g : ToType (X.presheaf.obj (op U))
hg : (ConcreteCategory.hom (X.presheaf.germ U (genericPoint ↥X) hU)) g = f
A : Opens ↥X
hA : A ∈ X.affineOpens
hxA : genericPoint ↥X ∈ ↑A
hAU : ↑A ⊆ ↑U
this : Nonempty ↥↑A
gA : ↑Γ(X, A) := (ConcreteCategory.hom (X.presheaf.map (homOfLE hAU).op)) g
⊢ (ConcreteCategory.hom (X.presheaf.germ A (genericPoint ↥X) hxA))
      ((ConcreteCategory.hom (X.presheaf.map (homOfLE hAU).op)) g) =
    (ConcreteCategory.hom (X.presheaf.germ A (genericPoint ↥X) hxA))
      ((ConcreteCategory.hom (X.presheaf.map (homOfLE hAU).op)) g)

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: X Y : Scheme
f : X.PartialMap Y
U : X.Opens
hU : Dense ↑U
hU' : U ≤ f.domain
x : ↥X
hx : x ∈ U
⊢ ⟨x, ⋯⟩ = (ConcreteCategory.hom ((TopologicalSpace.Opens.toTopCat ↑X.toPresheafedSpace).map (homOfLE hU'))) ⟨x, hx⟩

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
H : ∀ (j : 𝒰.I₀), ∃ i' f, IsEmpty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
⊢ False

Tactic:
choose i' f hf using H

Goal state after:
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
i' : 𝒰.I₀ → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
⊢ False
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: X Y S : Scheme
inst✝⁵ : X.Over S
inst✝⁴ : Y.Over S
inst✝³ : IsReduced X
inst✝² : IsSeparated (Y ↘ S)
f : X.PartialMap Y
g : X ⟶ Y
inst✝¹ : IsOver S f
inst✝ : Hom.IsOver g S
⊢ X.homOfLE ⋯ ≫ f.hom = (f.domain ⊓ ⊤).ι ≫ g ↔ (X.isoOfEq ⋯).hom ≫ f.hom = (f.domain ⊓ ⊤).ι ≫ g

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: case mp
X Y S : Scheme
inst✝¹ : X.Over S
inst✝ : Y.Over S
f : X ⤏ Y
h : IsOver S f
⊢ f.compHom (Y ↘ S) = Hom.toRationalMap (X ↘ S)

Tactic:
obtain ⟨g, hg, e⟩ := f.exists_partialMap_over S

Goal state after:
case mp
X Y S : Scheme
inst✝¹ : X.Over S
inst✝ : Y.Over S
f : X ⤏ Y
h : IsOver S f
g : X.PartialMap Y
hg : PartialMap.IsOver S g
e : g.toRationalMap = f
⊢ f.compHom (Y ↘ S) = Hom.toRationalMap (X ↘ S)
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: case e
X Y Z S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝ : IrreducibleSpace ↥X
f✝ : X ⤏ Y
f g : X.PartialMap Y
W : X.Opens
hW : Dense ↑W
hWl : W ≤ f.domain
hWr : W ≤ g.domain
e : (f.restrict W hW hWl).hom = (g.restrict W hW hWr).hom
⊢ (f.restrict W hW hWl).domain = (g.restrict W hW hWr).domain

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: X Y Z S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝³ : X.Over S
inst✝² : Y.Over S
inst✝¹ : IsIntegral X
inst✝ : LocallyOfFiniteType (Y ↘ S)
⊢ (fun f ↦ f ≫ Y ↘ S = Spec X.functionField ↘ S) = fun f ↦ f ≫ Y ↘ S = X.fromSpecStalk (genericPoint ↥X) ≫ X ↘ S

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: X Y Z S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
f : X ⤏ Y
⊢ { I₀ := ↑{x | ∃ g, ∃ (_ : g.toRationalMap = f), g.domain = x}, X := fun U ↦ ↑↑U,
        f := fun U ↦ X.homOfLE ⋯ }.presieve₀ ∈
    (precoverage IsOpenImmersion).coverings ↑f.domain

Tactic:
presieve₀_mem_precoverage_iff

Goal state after:
X Y Z S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
f : X ⤏ Y
⊢ (∀ (x : ↥↑f.domain),
      ∃ i,
        x ∈
          Set.range
            ⇑({ I₀ := ↑{x | ∃ g, ∃ (_ : g.toRationalMap = f), g.domain = x}, X := fun U ↦ ↑↑U,
                    f := fun U ↦ X.homOfLE ⋯ }.f
                i)) ∧
    ∀
      (i :
        { I₀ := ↑{x | ∃ g, ∃ (_ : g.toRationalMap = f), g.domain = x}, X := fun U ↦ ↑↑U, f := fun U ↦ X.homOfLE ⋯ }.I₀),
      IsOpenImmersion
        ({ I₀ := ↑{x | ∃ g, ∃ (_ : g.toRationalMap = f), g.domain = x}, X := fun U ↦ ↑↑U, f := fun U ↦ X.homOfLE ⋯ }.f
          i)
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case inr
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝¹ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this✝ : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this : ∀ (i' : Over i), IsAffine (F.obj i')
⊢ Nonempty ↥c.pt

Tactic:
have (i' : _) : Nonempty (F.obj i') := H i'.hom

Goal state after:
case inr
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝² : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this✝¹ : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this✝ : ∀ (i' : Over i), IsAffine (F.obj i')
this : ∀ (i' : Over i), Nonempty ↥(F.obj i')
⊢ Nonempty ↥c.pt
Module: Mathlib.AlgebraicGeometry.Cover.Directed

Goal state before:
pp: P : MorphismProperty Scheme
X : Scheme
𝒰 : Cover (precoverage P) X
inst✝³ : Category.{v_1, ?u.75} 𝒰.I₀
inst✝² : 𝒰.LocallyDirected
inst✝¹ : P.IsStableUnderBaseChange
inst✝ : P.HasOfPostcompProperty P
i j : 𝒰.I₀
⊢ { I₀ := (k : 𝒰.I₀) × (k ⟶ i) × (k ⟶ j), X := fun k ↦ 𝒰.X k.fst,
        f := fun k ↦ pullback.lift (𝒰.trans k.snd.1) (𝒰.trans k.snd.2) ⋯ }.presieve₀ ∈
    (precoverage P).coverings (pullback (𝒰.f i) (𝒰.f j))

Tactic:
presieve₀_mem_precoverage_iff

Goal state after:
P : MorphismProperty Scheme
X : Scheme
𝒰 : Cover (precoverage P) X
inst✝³ : Category.{v_1, ?u.75} 𝒰.I₀
inst✝² : 𝒰.LocallyDirected
inst✝¹ : P.IsStableUnderBaseChange
inst✝ : P.HasOfPostcompProperty P
i j : 𝒰.I₀
⊢ (∀ (x : ↥(pullback (𝒰.f i) (𝒰.f j))),
      ∃ i_1,
        x ∈
          Set.range
            ⇑({ I₀ := (k : 𝒰.I₀) × (k ⟶ i) × (k ⟶ j), X := fun k ↦ 𝒰.X k.fst,
                    f := fun k ↦ pullback.lift (𝒰.trans k.snd.1) (𝒰.trans k.snd.2) ⋯ }.f
                i_1)) ∧
    ∀
      (i_1 :
        { I₀ := (k : 𝒰.I₀) × (k ⟶ i) × (k ⟶ j), X := fun k ↦ 𝒰.X k.fst,
            f := fun k ↦ pullback.lift (𝒰.trans k.snd.1) (𝒰.trans k.snd.2) ⋯ }.I₀),
      P
        ({ I₀ := (k : 𝒰.I₀) × (k ⟶ i) × (k ⟶ j), X := fun k ↦ 𝒰.X k.fst,
              f := fun k ↦ pullback.lift (𝒰.trans k.snd.1) (𝒰.trans k.snd.2) ⋯ }.f
          i_1)
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: case H
X Y : Scheme
inst✝ : IrreducibleSpace ↥X
f : X.PartialMap Y
⊢ X.homOfLE ⋯ ≫ f.hom = (X.isoOfEq ⋯).hom ≫ f.hom

Tactic:
rfl

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

Goal state before:
pp: case a
X Y : Scheme
f : X ⟶ Y
hf : UniversallyInjective f
x : ↥(pullback.diagonalObj f)
⊢ (𝟙 X) ((pullback.fst f f) x) = (pullback.fst f f) x

Tactic:
rfl

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
⊢ diagonal (@Surjective) f ↔ Surjective (pullback.diagonal f)

Tactic:
rfl

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
tfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)
⊢ (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →
    ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f

Tactic:
intro ⟨h_inj, hf⟩ K _ g₁ g₂ hg

Goal state after:
X Y : Scheme
f : X ⟶ Y
tfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)
h_inj : Injective ⇑f
hf : ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable
K : Type u
inst✝ : Field K
g₁ g₂ : Spec (CommRingCat.of K) ⟶ X
hg : (fun g ↦ g ≫ f) g₁ = (fun g ↦ g ≫ f) g₂
⊢ g₁ = g₂
Module: Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
tfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)
tfae_3_to_2 :
  (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →
    ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f
tfae_2_to_4 : (∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f) → Surjective (pullback.diagonal f)
hf : Function.Surjective ⇑(pullback.diagonal f)
this : UniversallyInjective f
x : ↥X
algInst✝ : Algebra ↑(Y.residueField (f x)) ↑(X.residueField x) :=
  (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).toAlgebra
σ₁ σ₂ : Field.Emb ↑(Y.residueField (f x)) ↑(X.residueField x)
g₁ : Spec (CommRingCat.of (AlgebraicClosure ↑(X.residueField x))) ⟶ X :=
  (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X).symm ⟨x, CommRingCat.ofHom σ₁.toRingHom⟩
g₂ : Spec (CommRingCat.of (AlgebraicClosure ↑(X.residueField x))) ⟶ X :=
  (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X).symm ⟨x, CommRingCat.ofHom σ₂.toRingHom⟩
q : Spec (CommRingCat.of (AlgebraicClosure ↑(X.residueField x))) ⟶ pullback f f := pullback.lift g₁ g₂ ⋯
q_fst : q ≫ pullback.fst f f = g₁
⊢ (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X) g₁ =
    (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X) g₂

Tactic:
have q_snd : q ≫ pullback.snd f f = g₂ := pullback.lift_snd _ _ _

Goal state after:
X Y : Scheme
f : X ⟶ Y
tfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)
tfae_3_to_2 :
  (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →
    ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f
tfae_2_to_4 : (∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f) → Surjective (pullback.diagonal f)
hf : Function.Surjective ⇑(pullback.diagonal f)
this : UniversallyInjective f
x : ↥X
algInst✝ : Algebra ↑(Y.residueField (f x)) ↑(X.residueField x) :=
  (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).toAlgebra
σ₁ σ₂ : Field.Emb ↑(Y.residueField (f x)) ↑(X.residueField x)
g₁ : Spec (CommRingCat.of (AlgebraicClosure ↑(X.residueField x))) ⟶ X :=
  (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X).symm ⟨x, CommRingCat.ofHom σ₁.toRingHom⟩
g₂ : Spec (CommRingCat.of (AlgebraicClosure ↑(X.residueField x))) ⟶ X :=
  (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X).symm ⟨x, CommRingCat.ofHom σ₂.toRingHom⟩
q : Spec (CommRingCat.of (AlgebraicClosure ↑(X.residueField x))) ⟶ pullback f f := pullback.lift g₁ g₂ ⋯
q_fst : q ≫ pullback.fst f f = g₁
q_snd : q ≫ pullback.snd f f = g₂
⊢ (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X) g₁ =
    (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X) g₂
Module: Mathlib.AlgebraicGeometry.Cover.Directed

Goal state before:
pp: P : MorphismProperty Scheme
X✝ : Scheme
𝒰✝ : X✝.OpenCover
inst✝³ : Category.{v_1, ?u.14} 𝒰✝.I₀
inst✝² : Cover.LocallyDirected 𝒰✝
S : Scheme
X : Over S
𝒰 : X.left.OpenCover
inst✝¹ : Category.{v_2, ?u.87} 𝒰.I₀
inst✝ : Cover.LocallyDirected 𝒰
Y : Over S
g : (i : 𝒰.I₀) → 𝒰.X i ⟶ Y.left
h : ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), Cover.trans 𝒰 hij ≫ g j = g i
w : ∀ (i : 𝒰.I₀), g i ≫ Y.hom = 𝒰.f i ≫ X.hom
⊢ 𝒰.glueMorphismsOfLocallyDirected g ⋯ ≫ Y.hom = X.hom

Tactic:
apply 𝒰.hom_ext

Goal state after:
P : MorphismProperty Scheme
X✝ : Scheme
𝒰✝ : X✝.OpenCover
inst✝³ : Category.{v_1, ?u.14} 𝒰✝.I₀
inst✝² : Cover.LocallyDirected 𝒰✝
S : Scheme
X : Over S
𝒰 : X.left.OpenCover
inst✝¹ : Category.{v_2, ?u.91} 𝒰.I₀
inst✝ : Cover.LocallyDirected 𝒰
Y : Over S
g : (i : 𝒰.I₀) → 𝒰.X i ⟶ Y.left
h : ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), Cover.trans 𝒰 hij ≫ g j = g i
w : ∀ (i : 𝒰.I₀), g i ≫ Y.hom = 𝒰.f i ≫ X.hom
⊢ ∀ (x : 𝒰.I₀), 𝒰.f x ≫ 𝒰.glueMorphismsOfLocallyDirected g ⋯ ≫ Y.hom = 𝒰.f x ≫ X.hom
Module: Mathlib.AlgebraicGeometry.Cover.Directed

Goal state before:
pp: P : MorphismProperty Scheme
X : Scheme
⊢ { I₀ := ↑X.affineOpens, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.presieve₀ ∈ (precoverage IsOpenImmersion).coverings X

Tactic:
presieve₀_mem_precoverage_iff

Goal state after:
P : MorphismProperty Scheme
X : Scheme
⊢ (∀ (x : ↥X), ∃ i, x ∈ Set.range ⇑({ I₀ := ↑X.affineOpens, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.f i)) ∧
    ∀ (i : { I₀ := ↑X.affineOpens, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.I₀),
      IsOpenImmersion ({ I₀ := ↑X.affineOpens, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.f i)
Module: Mathlib.AlgebraicGeometry.Cover.Directed

Goal state before:
pp: P : MorphismProperty Scheme
X : Scheme
i✝ j✝ : X.directedAffineCover.I₀
⊢ i✝ ≤ j✝ ↔ i✝ ≤ j✝

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.RelativeGluing

Goal state before:
pp: case h.refine_2
J : Type u_1
inst✝² : Category.{u_2, u_1} J
F G : J ⥤ Scheme
s : F ⟶ G
inst✝¹ : Quiver.IsThin J
hs : NatTrans.Equifibered s
H : ∀ {i j : J} (hij : i ⟶ j), Function.Injective ⇑(F.map hij)
inst✝ : (G ⋙ forget).IsLocallyDirected
i j k : J
fi : i ⟶ k
fj : j ⟶ k
xi : (F ⋙ forget).obj i
xj : (F ⋙ forget).obj j
heq : (ConcreteCategory.hom (↾⇑(F.map fi))) xi = (ConcreteCategory.hom (↾⇑(F.map fj))) xj
l : J
fli : l ⟶ i
flj : l ⟶ j
x : (G ⋙ forget).obj l
hi : (ConcreteCategory.hom ((G ⋙ forget).map fli)) x = (s.app i) xi
hj : (ConcreteCategory.hom ((G ⋙ forget).map flj)) x = (s.app j) xj
e : F.obj l ≅ pullback (s.app i) (G.map fli) := ⋯.isoPullback
z : ↥(pullback (s.app i) (G.map fli))
h1 : (pullback.fst (s.app i) (G.map fli)) z = xi
h2 : (pullback.snd (s.app i) (G.map fli)) z = x
⊢ (ConcreteCategory.hom ((F ⋙ forget).map flj)) (e.inv z) = xj

Tactic:
· apply H fj
      simp only [Functor.comp_obj, forget_obj, Functor.comp_map, forget_map,
        ConcreteCategory.hom_ofHom, TypeCat.Fun.coe_mk, ← Scheme.Hom.comp_apply,
        Category.assoc, ← Functor.map_comp, show flj ≫ fj = fli ≫ fi by subsingleton]
      dsimp at heq
      simp [e, Functor.map_comp, ← heq, h1]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Cover.Over

Goal state before:
pp: P : MorphismProperty Scheme
S : Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝³ : W.Over S
inst✝² : X.Over S
inst✝¹ : Cover.Over S 𝒰
inst✝ : Hom.IsOver f S
⊢ { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOver f S) (Hom.asOver (𝒰.f x) S)).left,
        f := fun x ↦ Over.Hom.left (pullback.fst (Hom.asOver f S) (Hom.asOver (𝒰.f x) S)) }.presieve₀ ∈
    (precoverage P).coverings W

Tactic:
presieve₀_mem_precoverage_iff

Goal state after:
P : MorphismProperty Scheme
S : Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝³ : W.Over S
inst✝² : X.Over S
inst✝¹ : Cover.Over S 𝒰
inst✝ : Hom.IsOver f S
⊢ (∀ (x : ↥W),
      ∃ i,
        x ∈
          Set.range
            ⇑({ I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOver f S) (Hom.asOver (𝒰.f x) S)).left,
                    f := fun x ↦ Over.Hom.left (pullback.fst (Hom.asOver f S) (Hom.asOver (𝒰.f x) S)) }.f
                i)) ∧
    ∀
      (i :
        { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOver f S) (Hom.asOver (𝒰.f x) S)).left,
            f := fun x ↦ Over.Hom.left (pullback.fst (Hom.asOver f S) (Hom.asOver (𝒰.f x) S)) }.I₀),
      P
        ({ I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOver f S) (Hom.asOver (𝒰.f x) S)).left,
              f := fun x ↦ Over.Hom.left (pullback.fst (Hom.asOver f S) (Hom.asOver (𝒰.f x) S)) }.f
          i)
Module: Mathlib.AlgebraicGeometry.Cover.Over

Goal state before:
pp: P : MorphismProperty Scheme
S : Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝³ : W.Over S
inst✝² : X.Over S
inst✝¹ : Cover.Over S 𝒰
inst✝ : Hom.IsOver f S
⊢ { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)).left,
        f := fun x ↦ Over.Hom.left (pullback.snd (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)) }.presieve₀ ∈
    (precoverage P).coverings W

Tactic:
presieve₀_mem_precoverage_iff

Goal state after:
P : MorphismProperty Scheme
S : Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝³ : W.Over S
inst✝² : X.Over S
inst✝¹ : Cover.Over S 𝒰
inst✝ : Hom.IsOver f S
⊢ (∀ (x : ↥W),
      ∃ i,
        x ∈
          Set.range
            ⇑({ I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)).left,
                    f := fun x ↦ Over.Hom.left (pullback.snd (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)) }.f
                i)) ∧
    ∀
      (i :
        { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)).left,
            f := fun x ↦ Over.Hom.left (pullback.snd (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)) }.I₀),
      P
        ({ I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)).left,
              f := fun x ↦ Over.Hom.left (pullback.snd (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)) }.f
          i)
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R₀ : Type u_1
R : Type u_2
inst✝² : CommRing R₀
inst✝¹ : CommRing R
inst✝ : Algebra R₀ R
n : ℕ
S : InductionObj R n
⊢ Statement R₀ R n S

Tactic:
revert R₀

Goal state after:
R : Type u_2
inst✝ : CommRing R
n : ℕ
S : InductionObj R n
⊢ ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R], Statement R₀ R n S
Module: Mathlib.AlgebraicGeometry.Cover.Over

Goal state before:
pp: P : MorphismProperty Scheme
S : Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝⁶ : W.Over S
inst✝⁵ : X.Over S
inst✝⁴ : Cover.Over S 𝒰
inst✝³ : Hom.IsOver f S
Q : MorphismProperty Scheme
inst✝² : Q.HasOfPostcompProperty Q
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
⊢ { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f x) S)).left,
        f := fun x ↦ (pullback.fst (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f x) S)).left }.presieve₀ ∈
    (precoverage P).coverings W

Tactic:
presieve₀_mem_precoverage_iff

Goal state after:
P : MorphismProperty Scheme
S : Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝⁶ : W.Over S
inst✝⁵ : X.Over S
inst✝⁴ : Cover.Over S 𝒰
inst✝³ : Hom.IsOver f S
Q : MorphismProperty Scheme
inst✝² : Q.HasOfPostcompProperty Q
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
⊢ (∀ (x : ↥W),
      ∃ i,
        x ∈
          Set.range
            ⇑({ I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f x) S)).left,
                    f := fun x ↦ (pullback.fst (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f x) S)).left }.f
                i)) ∧
    ∀
      (i :
        { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f x) S)).left,
            f := fun x ↦ (pullback.fst (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f x) S)).left }.I₀),
      P
        ({ I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f x) S)).left,
              f := fun x ↦ (pullback.fst (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f x) S)).left }.f
          i)
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case pos
R✝ : Type u_2
inst✝² : CommRing R✝
n : ℕ
R : Type u_2
inst✝¹ : CommRing R
g : InductionObj R n
i : Fin n
hi : (g.val i).Monic
hi_min : ∀ (j : Fin n), j ≠ i → g.val j = 0
R₀✝ : Type u_1
R₀ : CommRing R₀✝
inst✝ : Algebra R₀✝ R
f : R[X]
M : Type u_2 := R[X] ⧸ Ideal.span {g.val i}
this✝ : Module.Free R M
this : Module.Finite R M
x : PrimeSpectrum R
⊢ zeroLocus {g.val i} ⊆ zeroLocus {g.val i}

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.Cover.Over

Goal state before:
pp: P : MorphismProperty Scheme
S : Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝⁶ : W.Over S
inst✝⁵ : X.Over S
inst✝⁴ : Cover.Over S 𝒰
inst✝³ : Hom.IsOver f S
Q : MorphismProperty Scheme
inst✝² : Q.HasOfPostcompProperty Q
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
⊢ { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left,
        f := fun x ↦ (pullback.snd (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left }.presieve₀ ∈
    (precoverage P).coverings W

Tactic:
presieve₀_mem_precoverage_iff

Goal state after:
P : MorphismProperty Scheme
S : Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝⁶ : W.Over S
inst✝⁵ : X.Over S
inst✝⁴ : Cover.Over S 𝒰
inst✝³ : Hom.IsOver f S
Q : MorphismProperty Scheme
inst✝² : Q.HasOfPostcompProperty Q
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
⊢ (∀ (x : ↥W),
      ∃ i,
        x ∈
          Set.range
            ⇑({ I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left,
                    f := fun x ↦ (pullback.snd (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left }.f
                i)) ∧
    ∀
      (i :
        { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left,
            f := fun x ↦ (pullback.snd (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left }.I₀),
      P
        ({ I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left,
              f := fun x ↦ (pullback.snd (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left }.f
          i)