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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : Flat f
⊢ GeneralizingMap ⇑f

Tactic:
have := HasRingHomProperty.of_isZariskiLocalAtSource_of_isZariskiLocalAtTarget.{u}
    (topologically GeneralizingMap)
  change topologically GeneralizingMap f
  rw [HasRingHomProperty.iff_appLE (P := topologically GeneralizingMap)]
  intro U V e
  algebraize [(f.appLE U V e).hom]
  apply Algebra.HasGoingDown.iff_generalizingMap_primeSpectrumComap.mp
  convert! Algebra.HasGoingDown.of_flat
  exact HasRingHomProperty.appLE @Flat f ‹_› U V e

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

Goal state before:
pp: case convert_2
X Y : Scheme
f : X ⟶ Y
S K : CommRingCat
hK : IsField ↑K
inst✝¹ : IsIntegral (Spec K)
inst✝ : Subsingleton ↥(Spec K)
φ : K ⟶ S
R : CommRingCat
ψ : K ⟶ R
x✝ : HasPullback (Spec.map ψ) (Spec.map φ)
algInst✝¹ : Algebra ↑K ↑S := (CommRingCat.Hom.hom φ).toAlgebra
algInst✝ : Algebra ↑K ↑R := (CommRingCat.Hom.hom ψ).toAlgebra
this : Field ↑K := hK.toField
⊢ topologically (@IsOpenMap) (Spec.map (CommRingCat.ofHom (algebraMap (↑R) (TensorProduct ↑K ↑R ↑S))))

Tactic:
exact PrimeSpectrum.isOpenMap_comap_algebraMap_tensorProduct_of_field

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n : Type u
S X : Scheme
f : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
Adjunction.homEquiv_symm_apply

Goal state after:
n : Type u
S X : Scheme
f : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n : Type u
S : Scheme
inst✝ : IsAffine S
⊢ (𝔸(n; S).toSpecΓ ≫
        Spec.map (CommRingCat.ofHom (eval₂Hom (CommRingCat.Hom.hom (Scheme.Hom.appTop (𝔸(n; S) ↘ S))) (coord S)))) ≫
      homOfVector (Spec.map (CommRingCat.ofHom C) ≫ S.isoSpec.inv)
        (⇑(ConcreteCategory.hom (Scheme.ΓSpecIso (CommRingCat.of (MvPolynomial n ↑Γ(S, ⊤)))).inv) ∘ X) =
    𝟙 𝔸(n; S)

Tactic:
ext1
        · simp only [Category.assoc, homOfVector_over, Category.id_comp]
          rw [← Spec.map_comp_assoc, ← CommRingCat.ofHom_comp, eval₂Hom_comp_C,
            CommRingCat.ofHom_hom, ← Scheme.toSpecΓ_naturality_assoc]
          simp [Scheme.isoSpec]
        · simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Connected

Goal state before:
pp: I : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} I
inst✝¹ : IsConnected I
inst✝ : Category.{v_2, u_2} C
F G : I ⥤ C
α : F ⟶ G
cF : Cocone F
cG : Cocone G
f : cF ⟶ (Cocone.precompose α).obj cG
hf : ∀ (i : I), IsPushout (cF.ι.app i) (α.app i) f.hom (cG.ι.app i)
hcF : IsColimit cF
s : Cocone G
j : I
⊢ cG.ι.app j ≫ ⋯.desc (hcF.desc ((Cocone.precompose α).obj s)) (s.ι.app (Classical.arbitrary I)) ⋯ = s.ι.app j

Tactic:
let f (i : _) : cG.pt ⟶ s.pt :=
      (hf i).desc (hcF.desc ((Cocone.precompose α).obj s)) (s.ι.app i) (by simp)
    have (i j : _) : f i = f j := by
      refine constant_of_preserves_morphisms f (fun j₁ j₂ g ↦ ?_) i j
      refine (hf j₁).hom_ext (by simp [f]) ?_
      rw [IsPushout.inr_desc, ← cG.w g, Category.assoc, IsPushout.inr_desc, Cocone.w]
    change _ ≫ f _ = _
    rw [this _ j]
    simp [f]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered

Goal state before:
pp: case mpr
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'
i : J
x : F.obj i
i' : J
x' : F.obj i'
eq✝ : c.ι i x = c.ι i' x'
k : J
f : i ⟶ k
f' : i' ⟶ k
eq : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
⊢ F.ιColimitType k ((ConcreteCategory.hom (F.map f)) x) = F.ιColimitType i' x'

Tactic:
eq,

Goal state after:
case mpr
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'
i : J
x : F.obj i
i' : J
x' : F.obj i'
eq✝ : c.ι i x = c.ι i' x'
k : J
f : i ⟶ k
f' : i' ⟶ k
eq : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
⊢ F.ιColimitType k ((ConcreteCategory.hom (F.map f')) x') = F.ιColimitType i' x'
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: case mp.inl
n : Type u
S : Scheme
h : IsIntegralHom (𝔸(n; S) ↘ S)
h✝ : IsEmpty ↥S
⊢ IsEmpty ↥S ∨ IsEmpty n

pp: case mp.inr
n : Type u
S : Scheme
h : IsIntegralHom (𝔸(n; S) ↘ S)
h✝ : Nonempty ↥S
⊢ IsEmpty ↥S ∨ IsEmpty n

Tactic:
· exact .inl ‹_›

Goal state after:
case mp.inr
n : Type u
S : Scheme
h : IsIntegralHom (𝔸(n; S) ↘ S)
h✝ : Nonempty ↥S
⊢ IsEmpty ↥S ∨ IsEmpty n
Module: Mathlib.AlgebraicGeometry.AlgebraicCycle.Basic

Goal state before:
pp: X : Scheme
R : Type u_1
inst✝¹ : Semiring R
N : Type u_2
inst✝ : DecidableEq N
wx : ↥X → N
c : AlgebraicCycle X R
⊢ map (𝟙 X) wx wx c = c

Tactic:
apply Function.locallyFinsupp.map_id

Goal state after:
X : Scheme
R : Type u_1
inst✝¹ : Semiring R
N : Type u_2
inst✝ : DecidableEq N
wx : ↥X → N
c : AlgebraicCycle X R
⊢ ∀ (z : ↥X), ↑(mapCoeff (𝟙 X) wx wx z) = 1
Module: Mathlib.AlgebraicGeometry.SpreadingOut

Goal state before:
pp: X : Scheme
R A : CommRingCat
U : X.Opens
x : ↥X
hxU : x ∈ U
φ : A ⟶ X.presheaf.stalk x
φRA : R ⟶ A
φRX : R ⟶ Γ(X, U)
e : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU
this✝² : Algebra ↑R ↑A := (CommRingCat.Hom.hom φRA).toAlgebra
s : Finset ↑A
hs : Algebra.adjoin ↑R ↑s = ⊤
W : ↑A → TopologicalSpace.Opens ↥X
hxW : ∀ (t : ↑A), x ∈ W t
f : (t : ↑A) → ToType (X.presheaf.obj (Opposite.op (W t)))
hf : ∀ (t : ↑A), (ConcreteCategory.hom (X.presheaf.germ (W t) x ⋯)) (f t) = (ConcreteCategory.hom φ) t
H : x ∈ s.inf W ⊓ U
this✝¹ : Algebra ↑R ↑Γ(X, U) := (CommRingCat.Hom.hom φRX).toAlgebra
this✝ : Algebra ↑R ↑(X.presheaf.stalk x) := (CommRingCat.Hom.hom (φRX ≫ X.presheaf.germ U x hxU)).toAlgebra
this : Algebra ↑R ↑(X.presheaf.obj (Opposite.op (s.inf W ⊓ U))) :=
  (CommRingCat.Hom.hom (φRX ≫ X.presheaf.map (homOfLE ⋯).op)).toAlgebra
φ' : ↑A →ₐ[↑R] ↑(X.presheaf.stalk x) :=
  let __src := CommRingCat.Hom.hom φ;
  { toRingHom := __src, commutes' := ⋯ }
ψ : ↑Γ(X, s.inf W ⊓ U) →ₐ[↑R] ↑(X.presheaf.stalk x) :=
  let __src := CommRingCat.Hom.hom (X.presheaf.germ (s.inf W ⊓ U) x H);
  { toRingHom := __src, commutes' := ⋯ }
i : ↑A
hi : i ∈ ↑s
⊢ ψ.toRingHom ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) (f i)) = (ConcreteCategory.hom φ) i

Tactic:
exact (X.presheaf.germ_res_apply _ _ _ _).trans (hf _)

Goal state after:
Module: Mathlib.AlgebraicGeometry.FunctionField

Goal state before:
pp: R : CommRingCat
inst✝ : IsDomain ↑R
⊢ IsGenericPoint ⊥ Set.univ

Tactic:
isGenericPoint_def

Goal state after:
R : CommRingCat
inst✝ : IsDomain ↑R
⊢ closure {⊥} = Set.univ
Module: Mathlib.AlgebraicGeometry.Morphisms.Flat

Goal state before:
pp: case h₁
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
UX : X.Opens
hUST : UT ≤ f ⁻¹ᵁ US
hUSX : UX ≤ iX ⁻¹ᵁ US
UY : Y.Opens
hUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT
ι : Type u
inst✝ : Finite ι
VX : ι → X.Opens
hVU : iSup VX = UX
hV : ∀ (i : ι), IsIso (pushoutSection H hUST ⋯ ⋯)
hV' : ∀ (i j : ι), Mono (pushoutSection H hUST ⋯ ⋯)
hT : (CommRingCat.Hom.hom (Scheme.Hom.appLE f US UT hUST)).Flat
D : CategoryTheory.Pairwise ι ⥤ X.Opens := ⋯
h : iSup D.obj = UX
c₀ : Cocone D := ⋯
F : (CategoryTheory.Pairwise ι)ᵒᵖ ⥤ CommRingCat := ⋯
G : X.Opens ⥤ Y.Opens := ⋯
αF : F ⟶ D.op ⋙ G.op ⋙ Y.presheaf := ⋯
c : Cone F := ⋯
this : PreservesFiniteLimits (Under.pushout (Scheme.Hom.appLE f US UT hUST))
val✝ : Fintype ι
hc : IsLimit c := ⋯
c'₀ : Cocone (D ⋙ G) := ⋯
c' : Cone (D.op ⋙ G.op ⋙ Y.presheaf) := ⋯
hc' : IsLimit c' := ⋯
HαF₂ : ∀ (i j : ι), Mono (αF.app (op (Pairwise.pair i j)))
f₁ : c.pt ⟶ c'.pt := ⋯
f₂ : c'.pt ⟶ c.pt := ⋯
e : c.pt ≅ c'.pt := ⋯
e_3✝ : pushout (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST) = c.pt
e_4✝ : Γ(Y, UY) = c'.pt
i : (CategoryTheory.Pairwise ι)ᵒᵖ
⊢ pushout.inr (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST) ≫
      pushoutSection H hUST hUSX hUY ≫ c'.π.app i =
    pushout.inr (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST) ≫ ((Cone.postcompose αF).obj c).π.app i

Tactic:
· simp [αF, c, c']

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

Goal state before:
pp: X Y : Scheme
inst✝¹ : IsReduced X
inst✝ : Y.IsSeparated
f : X ⤏ Y
⊢ f.toPartialMap.toRationalMap = f

Tactic:
by
  obtain ⟨f, rfl⟩ := PartialMap.toRationalMap_surjective f
  trans (f.toRationalMap.toPartialMap.restrict _
    f.dense_domain f.le_domain_toRationalMap).toRationalMap
  · simp
  · congr 1
    exact PartialMap.ext _ f rfl (by simpa using f.toPartialMap_toRationalMap_restrict)

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case refine_2.refine_1
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
hT₂ : comap Polynomial.C '' (zeroLocus (Set.range e₂.val) \ zeroLocus {(mapRingHom ↑q₂) f}) = T₂.toSet
hT₂deg : ∀ x ∈ T₂, x.n ≤ e₂.degBound
x✝ : Invertible (q₁ c) := { invOf := IsLocalization.Away.invSelf c, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
he₁span :
  coeffSubmodule R₀ e₁ ^ e₁.powBound =
    ⅟(q₁ c ^ e₁.powBound) • Submodule.map q₁.toLinearMap (span R₀ ({c} ∪ ⋃ i, coeff(e.val i)) ^ e₁.powBound)
g₁ : (x : BasicConstructibleSetData (Localization.Away c)) → Fin x.n → R
hg₁ : ∀ x ∈ T₁, ∀ (i : Fin x.n), g₁ x i ∈ ↑(span R₀ ({c} ∪ ⋃ i, coeff(e.val i)) ^ e₁.powBound)
hq₁g₁ : ∀ x ∈ T₁, ∀ (i : Fin x.n), q₁.toLinearMap (g₁ x i) = q₁ c ^ e₁.powBound * x.g i
n₁ : Localization.Away c → ℕ
f₁ : Localization.Away c → R
hf₁ : ∀ (x : Localization.Away c), x * q₁ c ^ n₁ x = q₁ (f₁ x)
g₂ : (x : BasicConstructibleSetData (R ⧸ Ideal.span {c})) → Fin x.n → R
hg₂ : ∀ x ∈ T₂, ∀ (i : Fin x.n), g₂ x i ∈ ↑(coeffSubmodule R₀ e ^ e₂.powBound)
hq₂g₂ : ∀ x ∈ T₂, ∀ (i : Fin x.n), q₂.toLinearMap (g₂ x i) = x.g i
f₂ : R ⧸ Ideal.span {c} → R
hf₂ : ∀ (x : R ⧸ Ideal.span {c}), q₂ (f₂ x) = x
S₁ : Finset (BasicConstructibleSetData R) := Finset.image (fun x ↦ { f := c * f₁ x.f, n := x.n, g := g₁ x }) T₁
S₂ : Finset (BasicConstructibleSetData R) :=
  Finset.image (fun x ↦ { f := f₂ x.f, n := x.n + 1, g := Fin.cons c (g₂ x) }) T₂
x : BasicConstructibleSetData (Localization.Away c)
hx : x ∈ T₁
k : Fin x.n
⊢ ↑(span R₀ ({c} ∪ ⋃ i, coeff(e.val i)) ^ e₁.powBound) ⊆ ↑(coeffSubmodule R₀ e ^ e.powBound)

Tactic:
norm_cast
      calc
        span R₀ ({c} ∪ ⋃ i, coeff(e i)) ^ e₁.powBound
        _ ≤ span R₀ (⋃ i, coeff(e i)) ^ e₁.powBound := by
          gcongr; simpa [Set.insert_subset_iff] using ⟨_, _, hi.symm⟩
        _ ≤ e.coeffSubmodule R₀ ^ e.powBound := by
          unfold coeffSubmodule powBound
          gcongr
          · exact one_le_coeffSubmodule
          · exact Set.subset_union_right
          · lia

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case hP₂.refine_1
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
⊢ x ∈ comap Polynomial.C '' (zeroLocus (Set.range g.val) \ zeroLocus {f}) ↔
    x ∈
      ConstructibleSetData.toSet
        (Finset.image
          (fun j ↦
            { f := (LinearMap.charpoly ((Algebra.lmul R M) ((Ideal.Quotient.mk (Ideal.span {g.val i})) f))).coeff j,
              n := 0, g := 0 })
          (Finset.range (Module.finrank R M)))

Tactic:
have : zeroLocus (Set.range g.val) = zeroLocus {g.1 i} := by
        rw [Set.range_eq_iUnion, zeroLocus_iUnion]
        refine (Set.iInter_subset _ _).antisymm (Set.subset_iInter fun j ↦ ?_)
        by_cases hij : i = j
        · subst hij; rfl
        · rw [hi_min j (.symm hij), zeroLocus_singleton_zero]; exact Set.subset_univ _

Goal state after:
case hP₂.refine_1
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✝¹ : Module.Free R M
this✝ : Module.Finite R M
x : PrimeSpectrum R
this : zeroLocus (Set.range g.val) = zeroLocus {g.val i}
⊢ x ∈ comap Polynomial.C '' (zeroLocus (Set.range g.val) \ zeroLocus {f}) ↔
    x ∈
      ConstructibleSetData.toSet
        (Finset.image
          (fun j ↦
            { f := (LinearMap.charpoly ((Algebra.lmul R M) ((Ideal.Quotient.mk (Ideal.span {g.val i})) f))).coeff j,
              n := 0, g := 0 })
          (Finset.range (Module.finrank R M)))
Module: Mathlib.AlgebraicGeometry.Cover.Over

Goal state before:
pp: case refine_2
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)
j :
  { 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
      j)

Tactic:
· simp only [← CategoryTheory.Over.forget_map]
      rw [MorphismProperty.Comma.toCommaMorphism_eq_hom,
        ← MorphismProperty.Comma.forget_map, ← Functor.comp_map]
      rw [← PreservesPullback.iso_hom_snd, P.cancel_left_of_respectsIso]
      exact P.pullback_snd _ _ (𝒰.map_prop j)

Goal state after:
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝ : (i : ι) → TopologicalSpace (X i)
B : (i : ι) → Set (Opens (X i))
hB : ∀ (i : ι), IsBasis (B i)
hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier
U : Set S
s : Set ι
hs : s.Finite
V : (i : ι) → i ∈ s → Opens (X i)
hc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier
hunion : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
Us : (i : ↑s) → Set (Opens (X ↑i))
UsB : ∀ (i : ↑s), Us i ⊆ B ↑i
hUsf : ∀ (i : ↑s), (Us i).Finite
hUs : ∀ (i : ↑s), V ↑i ⋯ = sSup (Us i)
σ : Type (max u_2 u_3) := (i : ↑s) × ↑(Us i)
this✝ : Finite ↑s
this : ∀ (i : ↑s), Finite ↑(Us i)
x : S
x✝ : ∃ i, ∃ (i_1 : i ∈ s), x ∈ f i '' ↑(V i i_1)
i : ι
hi : i ∈ s
h : X i
hmem : ∃ i_1, h ∈ ⋃ (_ : i_1 ∈ Us ⟨i, hi⟩), ↑i_1
heq : f i h = x
⊢ ∃ i, x ∈ (fun a ↦ f (↑i.fst) a) '' ↑↑i.snd

Tactic:
obtain ⟨a, ha⟩ := hmem

Goal state after:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝ : (i : ι) → TopologicalSpace (X i)
B : (i : ι) → Set (Opens (X i))
hB : ∀ (i : ι), IsBasis (B i)
hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier
U : Set S
s : Set ι
hs : s.Finite
V : (i : ι) → i ∈ s → Opens (X i)
hc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier
hunion : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
Us : (i : ↑s) → Set (Opens (X ↑i))
UsB : ∀ (i : ↑s), Us i ⊆ B ↑i
hUsf : ∀ (i : ↑s), (Us i).Finite
hUs : ∀ (i : ↑s), V ↑i ⋯ = sSup (Us i)
σ : Type (max u_2 u_3) := (i : ↑s) × ↑(Us i)
this✝ : Finite ↑s
this : ∀ (i : ↑s), Finite ↑(Us i)
x : S
x✝ : ∃ i, ∃ (i_1 : i ∈ s), x ∈ f i '' ↑(V i i_1)
i : ι
hi : i ∈ s
h : X i
heq : f i h = x
a : Opens (X i)
ha : h ∈ ⋃ (_ : a ∈ Us ⟨i, hi⟩), ↑a
⊢ ∃ i, x ∈ (fun a ↦ f (↑i.fst) a) '' ↑↑i.snd
Module: Mathlib.AlgebraicGeometry.Cover.QuasiCompact

Goal state before:
pp: S : Scheme
𝒰✝¹ 𝒰✝ : PreZeroHypercover S
K : Precoverage Scheme
X : Scheme
𝒰 : PreZeroHypercover X
inst✝¹ : QuasiCompactCover 𝒰
f : (x : 𝒰.I₀) → PreZeroHypercover (𝒰.X x)
inst✝ : ∀ (x : 𝒰.I₀), QuasiCompactCover (f x)
U : X.Opens
hU : IsAffineOpen U
⊢ IsCompactOpenCovered (fun x ↦ ⇑((𝒰.bind f).f x)) ↑U

Tactic:
by
    obtain ⟨s, hs, V, hcV, hU⟩ := hU.isCompactOpenCovered 𝒰
    have (i) (hi) : IsCompactOpenCovered ((f i).f ·) (V i hi) :=
      isCompactOpenCovered_of_isCompact (f i) (hcV i hi)
    choose t ht W hcW hV using this
    have : Finite s := hs
    have (i) (hi) : Finite (t i hi) := ht i hi
    refine .of_finite (κ := Σ (i : s), t i.1 i.2) (fun p ↦ ⟨p.1, p.2⟩) (fun p ↦ W _ p.1.2 _ p.2.2)
      (fun p ↦ hcW ..) ?_
    simpa [← hV, Set.iUnion_sigma, Set.iUnion_subtype, Set.image_iUnion, Set.image_image] using! hU

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝ : F.ReflectsFiniteEffectiveEpiFamilies
X✝ Y✝ : C
x✝ : X✝ ⟶ Y✝
h : EffectiveEpiFamily (fun x ↦ F.obj X✝) fun x ↦ F.map x✝
⊢ EffectiveEpi x✝

Tactic:
have := F.finite_effectiveEpiFamily_of_map _ _ h

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝ : F.ReflectsFiniteEffectiveEpiFamilies
X✝ Y✝ : C
x✝ : X✝ ⟶ Y✝
h : EffectiveEpiFamily (fun x ↦ F.obj X✝) fun x ↦ F.map x✝
this : EffectiveEpiFamily (fun a ↦ X✝) fun a ↦ x✝
⊢ EffectiveEpi x✝
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝⁴ : IsAffine X
inst✝³ : IsAffine Y
π : X ⟶ Y
inst✝² : Surjective π
inst✝¹ : Flat π
f : X ⟶ S
hf : pullback.fst π π ≫ f = pullback.snd π π ≫ f
inst✝ : IsAffine S
⊢ EffectiveEpi (AffineScheme.equivCommRingCat.functor.map (AffineScheme.ofHom π))

Tactic:
apply CommRingCat.Opposite.effectiveEpi_of_faithfullyFlat

Goal state after:
X Y S : Scheme
inst✝⁴ : IsAffine X
inst✝³ : IsAffine Y
π : X ⟶ Y
inst✝² : Surjective π
inst✝¹ : Flat π
f : X ⟶ S
hf : pullback.fst π π ≫ f = pullback.snd π π ≫ f
inst✝ : IsAffine S
⊢ (CommRingCat.Hom.hom (AffineScheme.equivCommRingCat.functor.map (AffineScheme.ofHom π)).unop).FaithfullyFlat
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case h₀
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.499} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.499} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : d.glued.left.OpenCover := d.relativeGluingData.cover
s : Cocone D
m : d.gluedCocone.pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone.ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
a : J
⊢ (Comma.Hom.hom ((d.cocone i).ι.app a)).left ≫ colimit.ι d.relativeGluingData.functor i ≫ m.left =
    pullback.fst (D.obj a).hom (𝒰.f i) ≫ (s.ι.app a).left

Tactic:
← congr($(hm a).left)

Goal state after:
case h₀
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.499} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.499} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : d.glued.left.OpenCover := d.relativeGluingData.cover
s : Cocone D
m : d.gluedCocone.pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone.ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
a : J
⊢ (Comma.Hom.hom ((d.cocone i).ι.app a)).left ≫ colimit.ι d.relativeGluingData.functor i ≫ m.left =
    pullback.fst (D.obj a).hom (𝒰.f i) ≫ (d.gluedCocone.ι.app a ≫ m).left
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝³ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝² : IsCofiltered I
inst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
inst✝ : CompactSpace ↥(D.obj i)
s : ↑Γ(D.obj i, ⊤)
hs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0
x : ↥(D.obj i)
U : TopologicalSpace.Opens ↥(D.obj i)
hU : IsAffineOpen U
hxU : x ∈ ↑U
this : ∀ (j : Over i), IsAffine ((opensDiagram D i U).obj j)
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hj :
  (ConcreteCategory.hom (Scheme.Hom.appTop ((opensDiagram D i U).map f)))
      ((ConcreteCategory.hom (Scheme.Hom.appTop ((opensDiagramι D i U).app (Over.mk (𝟙 i))))) s) =
    0
⊢ Over.Hom.left f = j.hom

Tactic:
simpa using Over.w f

Goal state after: