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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
x : PrimeSpectrum R
⊢ (evalRingHom 0).comp C = RingHom.id R

Tactic:
ext; simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.Artinian

Goal state before:
pp: X : Scheme
ι : Type u_1
U : ι → X.Opens
hU : TopologicalSpace.IsOpenCover U
hU' : ∀ (i : ι), IsAffineOpen (U i)
⊢ IsLocallyArtinian X ↔ ∀ (i : ι), IsArtinianRing ↑Γ(X, U i)

Tactic:
by
  refine ⟨fun _ _ ↦ IsLocallyArtinian.isArtinianRing_presheaf_obj ⟨_, hU' _⟩, fun H ↦ ?_⟩
  rw [isLocallyArtinian_iff_openCover (X.openCoverOfIsOpenCover U hU)]
  have : ∀ i, IsLocallyArtinian (Spec Γ(X, U i)) := by simpa
  exact fun i ↦ .of_isImmersion (hU' _).isoSpec.hom

Goal state after:
Module: Mathlib.AlgebraicGeometry.Artinian

Goal state before:
pp: X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
inst✝¹ : DiscreteTopology ↥X
inst✝ : IsReduced X
this : ∀ {X : Scheme} [DiscreteTopology ↥X] [IsReduced X], Subsingleton ↥X → IsLocallyArtinian X
hX : ¬Subsingleton ↥X
⊢ ⨆ x, { carrier := {x}, is_open' := ⋯ } = ⊤

Tactic:
ext; simp

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
σ : Type u_1
x : PrimeSpectrum R
⊢ (eval₂Hom (RingHom.id R) 0).comp C = RingHom.id R

Tactic:
ext; simp

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

Goal state before:
pp: case inr
X Y : Scheme
f : X ⟶ Y
H₁ : UniversallyClosed f
H₂ : IsAffineHom f
this : ∀ {X Y : Scheme} {f : X ⟶ Y}, UniversallyClosed f → IsAffineHom f → (∃ R, Y = Spec R) → IsIntegralHom f
hY : ¬∃ R, Y = Spec R
⊢ IsIntegralHom f

Tactic:
rw [IsZariskiLocalAtTarget.iff_of_openCover (P := @IsIntegralHom) Y.affineCover]
    rw [IsZariskiLocalAtTarget.iff_of_openCover (P := @UniversallyClosed) Y.affineCover] at H₁
    rw [IsZariskiLocalAtTarget.iff_of_openCover (P := @IsAffineHom) Y.affineCover] at H₂
    exact fun _ ↦ this inferInstance inferInstance ⟨_, rfl⟩

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineSpace

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

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Y : Type u_2
inst✝¹ : DecidableEq X
inst✝ : Zero Y
x₁ x₂ : X
y : Y
⊢ (single x₁ y) x₂ = if x₂ = x₁ then y else 0

Tactic:
by
  simp_rw [DFunLike.coe, single, Pi.single_apply]

Goal state after:
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: case left
X : Type u_1
inst✝¹ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝ : AddMonoid Y
f g : X → Y
hf : f ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}
hg : g ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}
⊢ Function.support (f + g) ⊆ U

pp: case right
X : Type u_1
inst✝¹ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝ : AddMonoid Y
f g : X → Y
hf : f ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}
hg : g ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}
⊢ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support (f + g)).Finite

Tactic:
· intro x hx
      contrapose hx
      simp [notMem_support.1 fun a ↦ hx (hf.1 a), notMem_support.1 fun a ↦ hx (hg.1 a)]

Goal state after:
case right
X : Type u_1
inst✝¹ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝ : AddMonoid Y
f g : X → Y
hf : f ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}
hg : g ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}
⊢ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support (f + g)).Finite
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})
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
⊢ comap Polynomial.C ∘ comap (mapRingHom q₁.toRingHom) = comap ↑q₁ ∘ comap Polynomial.C

Tactic:
ext; simp

Goal state after:
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})
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
⊢ comap Polynomial.C ∘ comap (mapRingHom q₂.toRingHom) = comap ↑q₂ ∘ comap Polynomial.C

Tactic:
ext; simp

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

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

Tactic:
intro e

Goal state after:
case mpr
X Y S : Scheme
inst✝¹ : X.Over S
inst✝ : Y.Over S
f : X ⤏ Y
e : f.compHom (Y ↘ S) = Hom.toRationalMap (X ↘ S)
⊢ IsOver S f
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: X Y Z : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X ⤏ Y
inst✝ : f.IsDominant
h : Y ⟶ Z
⊢ f.comp (Hom.toRationalMap h) = f.compHom h

Tactic:
simp [comp_def, PartialMap.comp_toPartialMap]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case e'_2.refine_3
I : Type u
inst✝¹ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f✝ : X ⟶ S
c : Cone D
hc : IsLimit c
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
j k : Over i
f : j ⟶ k
V : ((opensDiagram D i U).obj k).Opens
hV : IsAffineOpen V
e_1✝ : (opensDiagram D i U).obj j = ↑(D.map j.hom ⁻¹ᵁ U)
x : ↥((opensDiagram D i U).obj j)
⊢ x ∈ ↑((D.map j.hom ⁻¹ᵁ U).ι ⁻¹ᵁ D.map (Over.Hom.left f) ⁻¹ᵁ (D.map k.hom ⁻¹ᵁ U).ι ''ᵁ V) →
    ((opensDiagram D i U).map f) x ∈ V

Tactic:
· rintro ⟨⟨_, hU⟩, hV, rfl⟩
      convert! hV
      exact Subtype.ext (by simp)

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
s : Set (TopologicalSpace.Opens ↥c.pt)
hs : s ⊆ {x | ∃ i V, ∃ (_ : IsAffineOpen V), c.π.app i ⁻¹ᵁ V = x}
hsf : s.Finite
hU : IsCompact ↑(sSup s)
⊢ ∃ i V, IsCompact ↑V ∧ c.π.app i ⁻¹ᵁ V = sSup s

Tactic:
have : Finite s := hsf

Goal state after:
I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
s : Set (TopologicalSpace.Opens ↥c.pt)
hs : s ⊆ {x | ∃ i V, ∃ (_ : IsAffineOpen V), c.π.app i ⁻¹ᵁ V = x}
hsf : s.Finite
hU : IsCompact ↑(sSup s)
this : Finite ↑s
⊢ ∃ i V, IsCompact ↑V ∧ c.π.app i ⁻¹ᵁ V = sSup s
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: P : MorphismProperty Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝² : Category.{v_2, u_2} 𝒰.I₀
inst✝¹ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone i).pt ≅ ((MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)).mapCocone (d.cocone j)).pt :=
  (d.isColimit i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)) (d.isColimit j))
    (D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm)
a : J
⊢ (MorphismProperty.Over.map ⊤ ⋯).map
        ((D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
                (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
                    (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm).hom.app
          a) ≫
      (MorphismProperty.Over.mapPullbackAdj P ⊤ (Cover.trans 𝒰 hij) ⋯ trivial).counit.app
          ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).obj a) ≫
        (d.cocone j).ι.app a =
    (MorphismProperty.Over.map ⊤ ⋯).map ((d.cocone i).ι.app a) ≫ d.transitionMap hij

Tactic:
cocone_ι_transitionMap

Goal state after:
P : MorphismProperty Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝² : Category.{v_2, u_2} 𝒰.I₀
inst✝¹ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone i).pt ≅ ((MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)).mapCocone (d.cocone j)).pt :=
  (d.isColimit i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)) (d.isColimit j))
    (D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm)
a : J
⊢ (MorphismProperty.Over.map ⊤ ⋯).map
        ((D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
                (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
                    (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm).hom.app
          a) ≫
      (MorphismProperty.Over.mapPullbackAdj P ⊤ (Cover.trans 𝒰 hij) ⋯ trivial).counit.app
          ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).obj a) ≫
        (d.cocone j).ι.app a =
    (d.trans hij).app a ≫ (d.cocone j).ι.app a
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✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : Finite ι
inst✝ : TopologicalSpace S
hf : ∀ (i : ι), IsSpectralMap (f i)
U : Set S
hs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)
hU : IsOpen U
hc : IsCompact U
⊢ IsCompactOpenCovered f U

Tactic:
refine ⟨.univ, Set.finite_univ, fun i _ ↦ ⟨f i ⁻¹' U, hU.preimage (hf i).1⟩,
    fun i _ ↦ hc.preimage_of_isOpen (hf i) hU, subset_antisymm (by simp) fun x hx ↦ ?_⟩
  obtain ⟨i, y, rfl⟩ := hs x hx
  simpa using ⟨i, y, hx, rfl⟩

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

Goal state before:
pp: case e'_4
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) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
l : Fin n
m : ℕ
hlj : l = j
⊢ c.val j ≠ 0

Tactic:
intro e

Goal state after:
case e'_4
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) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
l : Fin n
m : ℕ
hlj : l = j
e : c.val j = 0
⊢ False
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: 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) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
l : Fin n
m : ℕ
hlj : ¬l = j
⊢ ¬c.val i = 0

Tactic:
intro e

Goal state after:
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) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
l : Fin n
m : ℕ
hlj : ¬l = j
e : c.val i = 0
⊢ False
Module: Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange

Goal state before:
pp: case r
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
C✝ C' C : VariableChange R
⊢ -C.r * ↑C.u⁻¹ ^ 2 * ↑C.u ^ 2 + C.r = 0

Tactic:
linear_combination -C.r * pow_mul_pow_eq_one 2 C.u.inv_mul

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact ↑U
s : ↑Γ(D.obj i, U)
hs : (ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U)) s = 0
this : CompactSpace ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
⊢ (ConcreteCategory.hom (Scheme.Hom.appTop (Scheme.Hom.resLE (c.π.app i) (D.map (𝟙 i) ⁻¹ᵁ U) (c.π.app i ⁻¹ᵁ U) ⋯)))
      ((ConcreteCategory.hom ((D.obj i).presheaf.map (homOfLE ⋯).op)) s) =
    (CommRingCat.Hom.hom (c.pt.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U)) s)

Tactic:
rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply]

Goal state after:
I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact ↑U
s : ↑Γ(D.obj i, U)
hs : (ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U)) s = 0
this : CompactSpace ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
⊢ (ConcreteCategory.hom
        ((D.obj i).presheaf.map (homOfLE ⋯).op ≫
          Scheme.Hom.appTop (Scheme.Hom.resLE (c.π.app i) (D.map (𝟙 i) ⁻¹ᵁ U) (c.π.app i ⁻¹ᵁ U) ⋯)))
      s =
    (ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U ≫ c.pt.presheaf.map (homOfLE ⋯).op)) s
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case e'_2
I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact ↑U
s : ↑Γ(D.obj i, U)
hs : (ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U)) s = 0
this : CompactSpace ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  (ConcreteCategory.hom
        (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)))
      ((ConcreteCategory.hom ((D.obj i).presheaf.map (homOfLE ⋯).op)) s) =
    0
hf' : Over.Hom.left f = j.hom
⊢ (ConcreteCategory.hom (Scheme.Hom.app (D.map (Over.Hom.left f)) U)) s =
    (ConcreteCategory.hom ((D.obj j.left).presheaf.map (homOfLE ⋯).op))
      ((ConcreteCategory.hom
          (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)))
        ((ConcreteCategory.hom ((D.obj i).presheaf.map (homOfLE ⋯).op)) s))

Tactic:
rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply]

Goal state after:
case e'_2
I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact ↑U
s : ↑Γ(D.obj i, U)
hs : (ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U)) s = 0
this : CompactSpace ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  (ConcreteCategory.hom
        (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)))
      ((ConcreteCategory.hom ((D.obj i).presheaf.map (homOfLE ⋯).op)) s) =
    0
hf' : Over.Hom.left f = j.hom
⊢ (ConcreteCategory.hom (Scheme.Hom.app (D.map (Over.Hom.left f)) U)) s =
    (ConcreteCategory.hom
        ((D.obj i).presheaf.map (homOfLE ⋯).op ≫
          Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯) ≫
            (D.obj j.left).presheaf.map (homOfLE ⋯).op))
      s
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
x y : R
⊢ W'.Nonsingular x (W'.negY x y) ↔ W'.Nonsingular x y

Tactic:
rw [nonsingular_iff, equation_neg, ← negY, negY_negY, ← @ne_comm _ y, nonsingular_iff]
  exact and_congr_right' <| (iff_congr not_and_or.symm not_and_or.symm).mpr <|
    not_congr <| and_congr_left fun h => by rw [← h]

Goal state after: