mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered

Goal state before:
pp: case mp
J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : IsFiltered J
F : J ⥤ Type w₀
c : F.CoconeTypes
h :
  ∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
    c.ι j x = c.ι j' x' → ∃ k f f', (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
j : J
x x' : F.obj j
eq✝ : c.ι j x = c.ι j x'
k : J
f f' : j ⟶ k
eq : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
⊢ (ConcreteCategory.hom (F.map (f ≫ coeqHom f f'))) x = (ConcreteCategory.hom (F.map (f ≫ coeqHom f f'))) x'

Tactic:
map_comp,

Goal state after:
case mp
J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : IsFiltered J
F : J ⥤ Type w₀
c : F.CoconeTypes
h :
  ∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
    c.ι j x = c.ι j' x' → ∃ k f f', (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
j : J
x x' : F.obj j
eq✝ : c.ι j x = c.ι j x'
k : J
f f' : j ⟶ k
eq : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
⊢ (ConcreteCategory.hom (F.map f ≫ F.map (coeqHom f f'))) x = (ConcreteCategory.hom (F.map f ≫ F.map (coeqHom f f'))) x'
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : Algebra R₀ R
c : R
i : Fin n
e : InductionObj R n
hi : c = (e.val i).leadingCoeff
hc : c ≠ 0
q₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)
q₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ e.val }
degBound_e₁_le : e₁.degBound ≤ e.degBound
degBound_e₂_lt : e₂.degBound < e.degBound
H₁ : Statement R₀ (Localization.Away c) n e₁
H₂ : Statement R₀ (R ⧸ Ideal.span {c}) n e₂
f : R[X]
T₁ : ConstructibleSetData (Localization.Away c)
T₂ : ConstructibleSetData (R ⧸ Ideal.span {c})
hT₂ :
  comap Polynomial.C '' (zeroLocus (Set.range e₂.val) \ zeroLocus {(mapRingHom ↑q₂) f}) = T₂.toSet ∧
    (∀ x ∈ T₂, x.n ≤ e₂.degBound) ∧ ∀ x ∈ T₂, ∀ (i : Fin x.n), x.g i ∈ coeffSubmodule R₀ e₂ ^ e₂.powBound
hT₁ : comap Polynomial.C '' (zeroLocus (Set.range e₁.val) \ zeroLocus {(mapRingHom ↑q₁) f}) = T₁.toSet
hT₁deg : ∀ x ∈ T₁, x.n ≤ e₁.degBound
hT₁span : ∀ x ∈ T₁, ∀ (i : Fin x.n), x.g i ∈ coeffSubmodule R₀ e₁ ^ e₁.powBound
⊢ ∃ T,
    comap Polynomial.C '' (zeroLocus (Set.range e.val) \ zeroLocus {f}) = T.toSet ∧
      ∀ C ∈ T, C.n ≤ e.degBound ∧ ∀ (i : Fin C.n), C.g i ∈ coeffSubmodule R₀ e ^ e.powBound

Tactic:
obtain ⟨hT₂, hT₂deg, hT₂span⟩ := hT₂

Goal state after:
R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : Algebra R₀ R
c : R
i : Fin n
e : InductionObj R n
hi : c = (e.val i).leadingCoeff
hc : c ≠ 0
q₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)
q₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ e.val }
degBound_e₁_le : e₁.degBound ≤ e.degBound
degBound_e₂_lt : e₂.degBound < e.degBound
H₁ : Statement R₀ (Localization.Away c) n e₁
H₂ : Statement R₀ (R ⧸ Ideal.span {c}) n e₂
f : R[X]
T₁ : ConstructibleSetData (Localization.Away c)
T₂ : ConstructibleSetData (R ⧸ Ideal.span {c})
hT₁ : comap Polynomial.C '' (zeroLocus (Set.range e₁.val) \ zeroLocus {(mapRingHom ↑q₁) f}) = T₁.toSet
hT₁deg : ∀ x ∈ T₁, x.n ≤ e₁.degBound
hT₁span : ∀ x ∈ T₁, ∀ (i : Fin x.n), x.g i ∈ coeffSubmodule R₀ e₁ ^ e₁.powBound
hT₂ : comap Polynomial.C '' (zeroLocus (Set.range e₂.val) \ zeroLocus {(mapRingHom ↑q₂) f}) = T₂.toSet
hT₂deg : ∀ x ∈ T₂, x.n ≤ e₂.degBound
hT₂span : ∀ x ∈ T₂, ∀ (i : Fin x.n), x.g i ∈ coeffSubmodule R₀ e₂ ^ e₂.powBound
⊢ ∃ T,
    comap Polynomial.C '' (zeroLocus (Set.range e.val) \ zeroLocus {f}) = T.toSet ∧
      ∀ C ∈ T, C.n ≤ e.degBound ∧ ∀ (i : Fin C.n), C.g i ∈ coeffSubmodule R₀ e ^ e.powBound
Module: Mathlib.AlgebraicGeometry.Morphisms.Flat

Goal state before:
pp: case refine_1.single
X Y S T : Scheme
f : T ⟶ S
g : Y ⟶ X
iX : X ⟶ S
iY : Y ⟶ T
H : IsPullback g iY iX f
US : S.Opens
UT : T.Opens
hUST : UT ≤ f ⁻¹ᵁ US
UY : Y.Opens
ι : Type u
inst✝ : Finite ι
VX : ι → X.Opens
hT : (CommRingCat.Hom.hom (Scheme.Hom.appLE f US UT hUST)).Flat
D : CategoryTheory.Pairwise ι ⥤ X.Opens := Pairwise.diagram VX
hUSX : iSup VX ≤ iX ⁻¹ᵁ US
hUY : UY = g ⁻¹ᵁ iSup VX ⊓ iY ⁻¹ᵁ UT
hV : ∀ (i : ι), IsIso (pushoutSection H hUST ⋯ ⋯)
hV' : ∀ (i j : ι), Mono (pushoutSection H hUST ⋯ ⋯)
i : ι
⊢ D.obj (Pairwise.single i) ≤ iSup VX

pp: case refine_1.pair
X Y S T : Scheme
f : T ⟶ S
g : Y ⟶ X
iX : X ⟶ S
iY : Y ⟶ T
H : IsPullback g iY iX f
US : S.Opens
UT : T.Opens
hUST : UT ≤ f ⁻¹ᵁ US
UY : Y.Opens
ι : Type u
inst✝ : Finite ι
VX : ι → X.Opens
hT : (CommRingCat.Hom.hom (Scheme.Hom.appLE f US UT hUST)).Flat
D : CategoryTheory.Pairwise ι ⥤ X.Opens := Pairwise.diagram VX
hUSX : iSup VX ≤ iX ⁻¹ᵁ US
hUY : UY = g ⁻¹ᵁ iSup VX ⊓ iY ⁻¹ᵁ UT
hV : ∀ (i : ι), IsIso (pushoutSection H hUST ⋯ ⋯)
hV' : ∀ (i j : ι), Mono (pushoutSection H hUST ⋯ ⋯)
i j : ι
⊢ D.obj (Pairwise.pair i j) ≤ iSup VX

Tactic:
exacts [le_iSup VX _, inf_le_left.trans (le_iSup VX _)]

Goal state after:
Module: Mathlib.AlgebraicGeometry.SpreadingOut

Goal state before:
pp: case refine_1
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝¹ : LocallyOfFiniteType sY
x : ↥X
inst✝ : X.IsGermInjectiveAt x
y : ↥Y
e : sX x = sY y
φ : Y.presheaf.stalk y ⟶ X.presheaf.stalk x
h : Scheme.Hom.stalkMap sY y ≫ φ = S.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap sX x
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : sX x ∈ ↑U
hyU : sY y ∈ U
V : Y.Opens
hV : V ∈ Y.affineOpens
hyV : y ∈ ↑V
iVU : ↑V ⊆ (sY ⁻¹ᵁ U).carrier
this :
  Scheme.Hom.appLE sY U V iVU ≫ Y.presheaf.germ V y hyV ≫ φ = Scheme.Hom.app sX U ≫ X.presheaf.germ (sX ⁻¹ᵁ U) x hxU
W : X.Opens
hxW : x ∈ W
φ' : Γ(Y, V) ⟶ Γ(X, W)
i : W ≤ sX ⁻¹ᵁ U
hW : IsAffineOpen W
h₁ : Y.presheaf.germ V y hyV ≫ φ = φ' ≫ X.presheaf.germ W x hxW
h₂ : Scheme.Hom.app sX U ≫ X.presheaf.map i.hom.op = Scheme.Hom.appLE sY U V iVU ≫ φ'
⊢ Spec.map φ ≫ Y.fromSpecStalk y = Spec.map (φ' ≫ X.presheaf.germ W x hxW) ≫ IsAffineOpen.fromSpec hV

Tactic:
← h₁,

Goal state after:
case refine_1
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝¹ : LocallyOfFiniteType sY
x : ↥X
inst✝ : X.IsGermInjectiveAt x
y : ↥Y
e : sX x = sY y
φ : Y.presheaf.stalk y ⟶ X.presheaf.stalk x
h : Scheme.Hom.stalkMap sY y ≫ φ = S.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap sX x
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : sX x ∈ ↑U
hyU : sY y ∈ U
V : Y.Opens
hV : V ∈ Y.affineOpens
hyV : y ∈ ↑V
iVU : ↑V ⊆ (sY ⁻¹ᵁ U).carrier
this :
  Scheme.Hom.appLE sY U V iVU ≫ Y.presheaf.germ V y hyV ≫ φ = Scheme.Hom.app sX U ≫ X.presheaf.germ (sX ⁻¹ᵁ U) x hxU
W : X.Opens
hxW : x ∈ W
φ' : Γ(Y, V) ⟶ Γ(X, W)
i : W ≤ sX ⁻¹ᵁ U
hW : IsAffineOpen W
h₁ : Y.presheaf.germ V y hyV ≫ φ = φ' ≫ X.presheaf.germ W x hxW
h₂ : Scheme.Hom.app sX U ≫ X.presheaf.map i.hom.op = Scheme.Hom.appLE sY U V iVU ≫ φ'
⊢ Spec.map φ ≫ Y.fromSpecStalk y = Spec.map (Y.presheaf.germ V y hyV ≫ φ) ≫ IsAffineOpen.fromSpec hV
Module: Mathlib.AlgebraicGeometry.SpreadingOut

Goal state before:
pp: case refine_1
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝¹ : LocallyOfFiniteType sY
x : ↥X
inst✝ : X.IsGermInjectiveAt x
φ : Spec (X.presheaf.stalk x) ⟶ Y
h : φ ≫ sY = X.fromSpecStalk x ≫ sX
⊢ sX x = sY (φ (IsLocalRing.closedPoint ↑(X.presheaf.stalk x)))

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
case refine_1
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝¹ : LocallyOfFiniteType sY
x : ↥X
inst✝ : X.IsGermInjectiveAt x
φ : Spec (X.presheaf.stalk x) ⟶ Y
h : φ ≫ sY = X.fromSpecStalk x ≫ sX
⊢ sX x = (φ ≫ sY) (IsLocalRing.closedPoint ↑(X.presheaf.stalk x))
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: case e
X₁ X₂ X₃ Y : Scheme
inst✝⁴ : PreirreducibleSpace ↥X₁
inst✝³ : IrreducibleSpace ↥X₂
inst✝² : Nonempty ↥X₃
f : X₁.PartialMap X₂
inst✝¹ : IsDominant f.hom
g : X₂.PartialMap X₃
inst✝ : IsDominant g.hom
h : X₃.PartialMap Y
x✝ : ↥X₁
⊢ x✝ ∈ ↑((f.comp g).comp h).domain ↔ x✝ ∈ ↑(f.comp (g.comp h)).domain

pp: case H
X₁ X₂ X₃ Y : Scheme
inst✝⁴ : PreirreducibleSpace ↥X₁
inst✝³ : IrreducibleSpace ↥X₂
inst✝² : Nonempty ↥X₃
f : X₁.PartialMap X₂
inst✝¹ : IsDominant f.hom
g : X₂.PartialMap X₃
inst✝ : IsDominant g.hom
h : X₃.PartialMap Y
⊢ ((f.comp g).comp h).hom = (X₁.isoOfEq ⋯).hom ≫ (f.comp (g.comp h)).hom

Tactic:
· simp_rw [comp_domain, comp_hom, ← Category.assoc, Hom.comp_preimage, Hom.inv_preimage,
      ← Hom.comp_image, Hom.isoImage_hom_ι, Hom.comp_image, image_morphismRestrict_preimage]

Goal state after:
case H
X₁ X₂ X₃ Y : Scheme
inst✝⁴ : PreirreducibleSpace ↥X₁
inst✝³ : IrreducibleSpace ↥X₂
inst✝² : Nonempty ↥X₃
f : X₁.PartialMap X₂
inst✝¹ : IsDominant f.hom
g : X₂.PartialMap X₃
inst✝ : IsDominant g.hom
h : X₃.PartialMap Y
⊢ ((f.comp g).comp h).hom = (X₁.isoOfEq ⋯).hom ≫ (f.comp (g.comp h)).hom
Module: Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective

Goal state before:
pp: ⊢ @UniversallyInjective =
    (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] x ↦ Injective x).universally

Tactic:
ext X Y f

Goal state after:
X Y : Scheme
f : X ⟶ Y
⊢ UniversallyInjective f ↔
    (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] x ↦ Injective x).universally f
Module: Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective

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

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
case a
X Y : Scheme
f : X ⟶ Y
hf : UniversallyInjective f
x : ↥(pullback.diagonalObj f)
⊢ (pullback.diagonal f ≫ pullback.fst f f) ((pullback.fst f f) x) = (pullback.fst f f) x
Module: Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective

Goal state before:
pp: case a
X Y : Scheme
f : X ⟶ Y
hf : diagonal (@Surjective) f
x₁ x₂ : ↥X
e : f x₁ = f x₂
t : ↥X
ht₁ : (pullback.fst f f) ((pullback.diagonal f) t) = x₁
ht₂ : (pullback.snd f f) ((pullback.diagonal f) t) = x₂
⊢ (pullback.fst f f) ((pullback.diagonal f) t) = (pullback.snd f f) ((pullback.diagonal f) t)

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
case a
X Y : Scheme
f : X ⟶ Y
hf : diagonal (@Surjective) f
x₁ x₂ : ↥X
e : f x₁ = f x₂
t : ↥X
ht₁ : (pullback.fst f f) ((pullback.diagonal f) t) = x₁
ht₂ : (pullback.snd f f) ((pullback.diagonal f) t) = x₂
⊢ (pullback.diagonal f ≫ pullback.fst f f) t = (pullback.snd f f) ((pullback.diagonal f) t)
Module: Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective

Goal state before:
pp: case a
X Y : Scheme
f : X ⟶ Y
hf : diagonal (@Surjective) f
x₁ x₂ : ↥X
e : f x₁ = f x₂
t : ↥X
ht₁ : (pullback.fst f f) ((pullback.diagonal f) t) = x₁
ht₂ : (pullback.snd f f) ((pullback.diagonal f) t) = x₂
⊢ (pullback.diagonal f ≫ pullback.fst f f) t = (pullback.snd f f) ((pullback.diagonal f) t)

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
case a
X Y : Scheme
f : X ⟶ Y
hf : diagonal (@Surjective) f
x₁ x₂ : ↥X
e : f x₁ = f x₂
t : ↥X
ht₁ : (pullback.fst f f) ((pullback.diagonal f) t) = x₁
ht₂ : (pullback.snd f f) ((pullback.diagonal f) t) = x₂
⊢ (pullback.diagonal f ≫ pullback.fst f f) t = (pullback.diagonal f ≫ pullback.snd f f) t
Module: Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective

Goal state before:
pp: ⊢ @UniversallyInjective = diagonal @Surjective

Tactic:
by
  apply le_antisymm
  · intro X Y f hf
    refine ⟨fun x ↦ ⟨pullback.fst f f x, hf.1 _ _ _ (IsPullback.of_hasPullback f f) ?_⟩⟩
    rw [← Scheme.Hom.comp_apply, pullback.diagonal_fst]
    rfl
  · rw [← universally_eq_iff.mpr (inferInstance : IsStableUnderBaseChange (diagonal @Surjective)),
      universallyInjective_eq]
    apply universally_mono
    intro X Y f hf x₁ x₂ e
    obtain ⟨t, ht₁, ht₂⟩ := Scheme.Pullback.exists_preimage_pullback _ _ e
    obtain ⟨t, rfl⟩ := hf.1 t
    rw [← ht₁, ← ht₂, ← Scheme.Hom.comp_apply, ← Scheme.Hom.comp_apply, pullback.diagonal_fst,
      pullback.diagonal_snd]

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)
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
h : ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f
z : ↥(pullback.diagonalObj f)
φ : Spec ((pullback f f).residueField z) ⟶ pullback f f := (pullback f f).fromSpecResidueField z
hφ₁ : φ ≫ pullback.fst f f = φ ≫ pullback.snd f f
hφ₂ : φ = (φ ≫ pullback.fst f f) ≫ pullback.diagonal f
⊢ (pullback.diagonal f) ((φ ≫ pullback.fst f f) (IsLocalRing.closedPoint ↑((pullback f f).residueField z))) = z

Tactic:
← Scheme.Hom.comp_apply,

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
h : ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f
z : ↥(pullback.diagonalObj f)
φ : Spec ((pullback f f).residueField z) ⟶ pullback f f := (pullback f f).fromSpecResidueField z
hφ₁ : φ ≫ pullback.fst f f = φ ≫ pullback.snd f f
hφ₂ : φ = (φ ≫ pullback.fst f f) ≫ pullback.diagonal f
⊢ ((φ ≫ pullback.fst f f) ≫ pullback.diagonal f) (IsLocalRing.closedPoint ↑((pullback f f).residueField z)) = z
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
q_snd : q ≫ pullback.snd f f = g₂
u : ↥X
hu : (pullback.diagonal f) u = q (IsLocalRing.closedPoint ↑(CommRingCat.of (AlgebraicClosure ↑(X.residueField x))))
this :
  (pullback.fst f f) ((pullback.diagonal f) u) =
    (pullback.fst f f) (q (IsLocalRing.closedPoint ↑(CommRingCat.of (AlgebraicClosure ↑(X.residueField x)))))
⊢ u = x

Tactic:
← Scheme.Hom.comp_apply,

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
u : ↥X
hu : (pullback.diagonal f) u = q (IsLocalRing.closedPoint ↑(CommRingCat.of (AlgebraicClosure ↑(X.residueField x))))
this :
  (pullback.diagonal f ≫ pullback.fst f f) u =
    (pullback.fst f f) (q (IsLocalRing.closedPoint ↑(CommRingCat.of (AlgebraicClosure ↑(X.residueField x)))))
⊢ u = x
Module: Mathlib.AlgebraicGeometry.Cover.Directed

Goal state before:
pp: case refine_3
P : MorphismProperty Scheme
X : Scheme
𝒰 : X.OpenCover
inst✝ : Preorder 𝒰.I₀
hle : ∀ {i j : 𝒰.I₀}, i ≤ j ↔ Hom.opensRange (𝒰.f i) ≤ Hom.opensRange (𝒰.f j)
H : TopologicalSpace.Opens.IsBasis (Set.range fun i ↦ Hom.opensRange (𝒰.f i))
i j : 𝒰.I₀
x : ↥(pullback (𝒰.f i) (𝒰.f j))
this : (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i) x ∈ Hom.opensRange (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i)
k : 𝒰.I₀
h : (fun i ↦ Hom.opensRange (𝒰.f i)) k ≤ Hom.opensRange (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i)
y : ↥(𝒰.X k)
hy : (𝒰.f k) y = (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i) x
⊢ (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i)
      ((pullback.lift (IsOpenImmersion.lift (𝒰.f i) (𝒰.f k) ⋯) (IsOpenImmersion.lift (𝒰.f j) (𝒰.f k) ⋯) ⋯) y) =
    (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i) x

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
case refine_3
P : MorphismProperty Scheme
X : Scheme
𝒰 : X.OpenCover
inst✝ : Preorder 𝒰.I₀
hle : ∀ {i j : 𝒰.I₀}, i ≤ j ↔ Hom.opensRange (𝒰.f i) ≤ Hom.opensRange (𝒰.f j)
H : TopologicalSpace.Opens.IsBasis (Set.range fun i ↦ Hom.opensRange (𝒰.f i))
i j : 𝒰.I₀
x : ↥(pullback (𝒰.f i) (𝒰.f j))
this : (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i) x ∈ Hom.opensRange (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i)
k : 𝒰.I₀
h : (fun i ↦ Hom.opensRange (𝒰.f i)) k ≤ Hom.opensRange (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i)
y : ↥(𝒰.X k)
hy : (𝒰.f k) y = (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i) x
⊢ (pullback.lift (IsOpenImmersion.lift (𝒰.f i) (𝒰.f k) ⋯) (IsOpenImmersion.lift (𝒰.f j) (𝒰.f k) ⋯) ⋯ ≫
        pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i)
      y =
    (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i) x
Module: Mathlib.AlgebraicGeometry.RelativeGluing

Goal state before:
pp: 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 : (F.map fi) xi = (F.map fj) xj
⊢ (G.map fi) ((s.app i) xi) = (G.map fj) ((s.app j) xj)

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
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 : (F.map fi) xi = (F.map fj) xj
⊢ (s.app i ≫ G.map fi) xi = (G.map fj) ((s.app j) xj)
Module: Mathlib.AlgebraicGeometry.RelativeGluing

Goal state before:
pp: 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 : (F.map fi) xi = (F.map fj) xj
⊢ (s.app k) ((F.map fj) xj) = (G.map fj) ((s.app j) xj)

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
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 : (F.map fi) xi = (F.map fj) xj
⊢ (F.map fj ≫ s.app k) xj = (G.map fj) ((s.app j) xj)
Module: Mathlib.AlgebraicGeometry.RelativeGluing

Goal state before:
pp: S : Scheme
𝒰 : S.OpenCover
inst✝³ : Category.{?u.7, ?u.73} 𝒰.I₀
inst✝² : LocallyDirected 𝒰
d : RelativeGluingData 𝒰
inst✝¹ : Small.{u, ?u.73} 𝒰.I₀
inst✝ : Quiver.IsThin 𝒰.I₀
i j : d.cover.I₀
x : ↥(pullback (d.cover.f i) (d.cover.f j))
xi : ↥(d.cover.X i) := (pullback.fst (d.cover.f i) (d.cover.f j)) x
xj : ↥(d.cover.X j) := (pullback.snd (d.cover.f i) (d.cover.f j)) x
⊢ (𝒰.f i) ((d.natTrans.app i) ((pullback.fst (colimit.ι d.functor i) (colimit.ι d.functor j)) x)) =
    (𝒰.f j) ((d.natTrans.app j) ((pullback.snd (colimit.ι d.functor i) (colimit.ι d.functor j)) x))

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
S : Scheme
𝒰 : S.OpenCover
inst✝³ : Category.{?u.7, ?u.73} 𝒰.I₀
inst✝² : LocallyDirected 𝒰
d : RelativeGluingData 𝒰
inst✝¹ : Small.{u, ?u.73} 𝒰.I₀
inst✝ : Quiver.IsThin 𝒰.I₀
i j : d.cover.I₀
x : ↥(pullback (d.cover.f i) (d.cover.f j))
xi : ↥(d.cover.X i) := (pullback.fst (d.cover.f i) (d.cover.f j)) x
xj : ↥(d.cover.X j) := (pullback.snd (d.cover.f i) (d.cover.f j)) x
⊢ (d.natTrans.app i ≫ 𝒰.f i) ((pullback.fst (colimit.ι d.functor i) (colimit.ι d.functor j)) x) =
    (𝒰.f j) ((d.natTrans.app j) ((pullback.snd (colimit.ι d.functor i) (colimit.ι d.functor j)) x))
Module: Mathlib.AlgebraicGeometry.RelativeGluing

Goal state before:
pp: S : Scheme
𝒰 : S.OpenCover
inst✝³ : Category.{?u.7, ?u.73} 𝒰.I₀
inst✝² : LocallyDirected 𝒰
d : RelativeGluingData 𝒰
inst✝¹ : Small.{u, ?u.73} 𝒰.I₀
inst✝ : Quiver.IsThin 𝒰.I₀
i j : d.cover.I₀
x : ↥(pullback (d.cover.f i) (d.cover.f j))
xi : ↥(d.cover.X i) := (pullback.fst (d.cover.f i) (d.cover.f j)) x
xj : ↥(d.cover.X j) := (pullback.snd (d.cover.f i) (d.cover.f j)) x
⊢ (d.natTrans.app i ≫ 𝒰.f i) ((pullback.fst (colimit.ι d.functor i) (colimit.ι d.functor j)) x) =
    (𝒰.f j) ((d.natTrans.app j) ((pullback.snd (colimit.ι d.functor i) (colimit.ι d.functor j)) x))

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
S : Scheme
𝒰 : S.OpenCover
inst✝³ : Category.{?u.7, ?u.73} 𝒰.I₀
inst✝² : LocallyDirected 𝒰
d : RelativeGluingData 𝒰
inst✝¹ : Small.{u, ?u.73} 𝒰.I₀
inst✝ : Quiver.IsThin 𝒰.I₀
i j : d.cover.I₀
x : ↥(pullback (d.cover.f i) (d.cover.f j))
xi : ↥(d.cover.X i) := (pullback.fst (d.cover.f i) (d.cover.f j)) x
xj : ↥(d.cover.X j) := (pullback.snd (d.cover.f i) (d.cover.f j)) x
⊢ (pullback.fst (colimit.ι d.functor i) (colimit.ι d.functor j) ≫ d.natTrans.app i ≫ 𝒰.f i) x =
    (𝒰.f j) ((d.natTrans.app j) ((pullback.snd (colimit.ι d.functor i) (colimit.ι d.functor j)) x))
Module: Mathlib.AlgebraicGeometry.RelativeGluing

Goal state before:
pp: S : Scheme
𝒰 : S.OpenCover
inst✝³ : Category.{u_2, u_1} 𝒰.I₀
inst✝² : LocallyDirected 𝒰
d : RelativeGluingData 𝒰
inst✝¹ : Small.{u, u_1} 𝒰.I₀
inst✝ : Quiver.IsThin 𝒰.I₀
i j : 𝒰.I₀
xj : ↥(d.functor.obj j)
x✝ : (colimit.ι d.functor j) xj ∈ ⇑d.toBase ⁻¹' Set.range ⇑(𝒰.f i)
k : 𝒰.I₀
fi : k ⟶ i
fj : k ⟶ j
uk : ↥(𝒰.X k)
h✝ : (trans 𝒰 fj) uk = (d.natTrans.app j) xj
h : (𝒰.f i) ((trans 𝒰 fi) uk) = d.toBase ((colimit.ι d.functor j) xj)
⊢ (𝒰.f j) ((d.natTrans.app j) xj) = (𝒰.f j) ((trans 𝒰 fj) uk)

Tactic:
← Scheme.Hom.comp_apply,

Goal state after:
S : Scheme
𝒰 : S.OpenCover
inst✝³ : Category.{u_2, u_1} 𝒰.I₀
inst✝² : LocallyDirected 𝒰
d : RelativeGluingData 𝒰
inst✝¹ : Small.{u, u_1} 𝒰.I₀
inst✝ : Quiver.IsThin 𝒰.I₀
i j : 𝒰.I₀
xj : ↥(d.functor.obj j)
x✝ : (colimit.ι d.functor j) xj ∈ ⇑d.toBase ⁻¹' Set.range ⇑(𝒰.f i)
k : 𝒰.I₀
fi : k ⟶ i
fj : k ⟶ j
uk : ↥(𝒰.X k)
h✝ : (trans 𝒰 fj) uk = (d.natTrans.app j) xj
h : (𝒰.f i) ((trans 𝒰 fi) uk) = d.toBase ((colimit.ι d.functor j) xj)
⊢ (d.natTrans.app j ≫ 𝒰.f j) xj = (𝒰.f j) ((trans 𝒰 fj) uk)