mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case e'_2.e_a
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
⊢ Set.univ ⊆ zeroLocus (Set.range 0)

Tactic:
]

Goal state after:
case e'_2.e_a
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
⊢ Set.univ ⊆ zeroLocus (Set.range 0)
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)
⊢ (∀ (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)

Tactic:
]

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: 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
⊢ ⋂ i, zeroLocus {g.val i} = zeroLocus {g.val i}

Tactic:
]

Goal state after:
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
⊢ ⋂ i, zeroLocus {g.val i} = zeroLocus {g.val i}
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: 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 (Set.range g.val) = zeroLocus {g.val i}

Tactic:
rw [Set.range_eq_iUnion, zeroLocus_iUnion]

Goal state after:
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
⊢ ⋂ i, zeroLocus {g.val i} = zeroLocus {g.val i}
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 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
    ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map
      (pullback.fst (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f j) S)))

Tactic:
]

Goal state after:
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 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
    ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map
      (pullback.fst (Hom.asOverProp f S) (Hom.asOverProp (𝒰.f j) S)))
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 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
    (pullback.fst
      ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map (Hom.asOverProp f S))
      ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map
        (Hom.asOverProp (𝒰.f j) S)))

Tactic:
]

Goal state after:
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 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
    (pullback.fst
      ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map (Hom.asOverProp f S))
      ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map
        (Hom.asOverProp (𝒰.f j) S)))
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case neg
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
j : Fin n
hij : ¬i = j
⊢ zeroLocus {g.val i} ⊆ Set.univ

Tactic:
]

Goal state after:
case neg
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
j : Fin n
hij : ¬i = j
⊢ zeroLocus {g.val i} ⊆ Set.univ
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)
⊢ (∀ (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)

Tactic:
]

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)
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✝¹ : Module.Free R M
this✝ : Module.Finite R M
x : PrimeSpectrum R
this : zeroLocus (Set.range g.val) = zeroLocus {g.val i}
⊢ (¬∀ i_1 < Module.finrank R M,
        (LinearMap.charpoly ((Algebra.lmul R M) ((algebraMap R[X] M) f))).coeff i_1 ∈ x.asIdeal) ↔
    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:
]

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}
⊢ (¬∀ i_1 < Module.finrank R M,
        (LinearMap.charpoly ((Algebra.lmul R M) ((algebraMap R[X] M) f))).coeff i_1 ∈ x.asIdeal) ↔
    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
    ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map
      (pullback.snd (Hom.asOverProp (𝒰.f j) S) (Hom.asOverProp f S)))

Tactic:
]

Goal state after:
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
    ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map
      (pullback.snd (Hom.asOverProp (𝒰.f j) S) (Hom.asOverProp f S)))
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
    (pullback.snd
      ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map
        (Hom.asOverProp (𝒰.f j) S))
      ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map (Hom.asOverProp f S)))

Tactic:
]

Goal state after:
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
    (pullback.snd
      ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map
        (Hom.asOverProp (𝒰.f j) S))
      ((MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S).map (Hom.asOverProp f S)))
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: S : Type u_1
U : Set S
inst✝ : TopologicalSpace S
⊢ (∃ V, IsCompact V.carrier ∧ id '' V.carrier = U) ↔ IsOpen U ∧ IsCompact U

Tactic:
]

Goal state after:
S : Type u_1
U : Set S
inst✝ : TopologicalSpace S
⊢ (∃ V, IsCompact V.carrier ∧ id '' V.carrier = U) ↔ IsOpen U ∧ IsCompact U
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)
U : Set S
⊢ IsCompactOpenCovered f U ↔ ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U

Tactic:
]

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)
U : Set S
⊢ IsCompactOpenCovered f U ↔ ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case hP₃.inr.refine_1
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
⊢ comap Polynomial.C '' (zeroLocus ↑(Ideal.span (Set.range c.val)) \ zeroLocus {f}) =
    comap Polynomial.C '' (zeroLocus ↑(Ideal.span (Set.range c.val)) \ zeroLocus {f})

Tactic:
]

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)
U : Set S
x✝ : ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U
V : Opens ((i : ι) × X i)
hc✝ : IsCompact V.carrier
heq : (fun p ↦ f p.fst p.snd) '' V.carrier = U
s : Set ι
t : (i : ι) → Set (X i)
hs : s.Finite
hc : ∀ (i : ι), IsCompact (t i)
heq' : s.sigma t = V.carrier
i : ι
hi : i ∈ s
⊢ IsOpen (Sigma.mk i ⁻¹' s.sigma t)

Tactic:
]

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)
U : Set S
x✝ : ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U
V : Opens ((i : ι) × X i)
hc✝ : IsCompact V.carrier
heq : (fun p ↦ f p.fst p.snd) '' V.carrier = U
s : Set ι
t : (i : ι) → Set (X i)
hs : s.Finite
hc : ∀ (i : ι), IsCompact (t i)
heq' : s.sigma t = V.carrier
i : ι
hi : i ∈ s
⊢ IsOpen (Sigma.mk i ⁻¹' s.sigma t)
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✝¹ : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst✝ : Finite κ
s : κ → Set S
hs : ⋃ i, s i = U
H : ∀ (i : κ), IsCompactOpenCovered f (s i)
⊢ ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U

Tactic:
]

Goal state after:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst✝ : Finite κ
s : κ → Set S
hs : ⋃ i, s i = U
H : ∀ (i : κ), IsCompactOpenCovered f (s i)
⊢ ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U
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✝¹ : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst✝ : Finite κ
s : κ → Set S
hs : ⋃ i, s i = U
H : ∀ (i : κ), IsCompactOpenCovered f (s i)
i : κ
⊢ (∃ V, IsCompact V.carrier ∧ (fun x ↦ f x.fst x.snd) '' V.carrier = s i) ↔
    ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = s i

Tactic:
]

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

Goal state before:
pp: case hxy
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
k : Fin n
a✝ : k ∈ Finset.univ
⊢ (if k = j then c.val j %ₘ c.val i else c.val k).degree ≤ (c.val k).degree

Tactic:
]

Goal state after:
case hxy
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
k : Fin n
a✝ : k ∈ Finset.univ
⊢ (if k = j then c.val j %ₘ c.val i else c.val k).degree ≤ (c.val k).degree