mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W'.negY Q * P z
⊢ W'.dblX P = 0

Tactic:
apply eq_zero_of_ne_zero_of_mul_right_eq_zero hPz

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W'.negY Q * P z
⊢ W'.dblX P * P z = 0
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝¹ : NonAssocSemiring R
inst✝ : CommRing S
I : Ideal S
f : (n : ℕ) → R →+* S ⧸ I ^ n
hf : ∀ {m n : ℕ} (hle : m ≤ n), (Ideal.Quotient.factorPow I hle).comp (f n) = f m
n : ℕ
⊢ (↑(evalₐ I n)).comp (liftRingHom I f ⋯) = f n

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Exactness

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
I : Ideal R
M : Type u
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : Type u
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
g : N →ₗ[R] P
hf : Function.Injective ⇑f
hfg : Function.Exact ⇑f ⇑g
hg : Function.Surjective ⇑g
k : ℕ
hkn : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCauchySequence I N
hker : ∀ (n : ℕ), g (↑x n) ∈ I ^ n • ⊤
x✝ n : ℕ
⊢ g (↑x (k + n)) ∈ I ^ (k + n) • ⊤

Tactic:
exact hker (k + n)

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R✝ : Type u_1
S✝ : Type u_2
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S✝
I✝ : Ideal R✝
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_4
S : Type u_5
inst✝² : NonAssocSemiring R
inst✝¹ : CommRing S
I : Ideal S
f : (n : ℕ) → R →+* S ⧸ I ^ n
hf : ∀ {m n : ℕ} (hle : m ≤ n), (Ideal.Quotient.factorPow I hle).comp (f n) = f m
inst✝ : IsAdicComplete I S
x✝¹ x✝ : S
⊢ (↑(ofLinearEquiv I S)).toFun (x✝¹ * x✝) = (↑(ofLinearEquiv I S)).toFun x✝¹ * (↑(ofLinearEquiv I S)).toFun x✝

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Finite σ
T : Type u_1
ι' : Type u_2
σ' : Type u_3
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : Finite σ'
Q : SubmersivePresentation S T ι' σ'
P : SubmersivePresentation R S ι σ
⊢ IsUnit ((algebraMap S T) P.jacobian * Q.jacobian)

Tactic:
IsUnit.mul_iff

Goal state after:
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Finite σ
T : Type u_1
ι' : Type u_2
σ' : Type u_3
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : Finite σ'
Q : SubmersivePresentation S T ι' σ'
P : SubmersivePresentation R S ι σ
⊢ IsUnit ((algebraMap S T) P.jacobian) ∧ IsUnit Q.jacobian
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Finite σ
T : Type u_1
ι' : Type u_2
σ' : Type u_3
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : Finite σ'
Q : SubmersivePresentation S T ι' σ'
P : SubmersivePresentation R S ι σ
⊢ IsUnit (Q.comp P.toPreSubmersivePresentation).jacobian

Tactic:
rw [comp_jacobian_eq_jacobian_smul_jacobian, Algebra.smul_def, IsUnit.mul_iff]
    exact ⟨RingHom.isUnit_map _ <| P.jacobian_isUnit, Q.jacobian_isUnit⟩

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
Q : Generators S T Unit := localizationAway T g
algInst✝ : Algebra P.Ring ((localizationAway T g).comp P).Ring := ((localizationAway T g).toComp P).toAlgHom.toAlgebra
f : P.Ring ⧸ P.ker ^ 2 := (Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)
π : ((localizationAway T g).comp P).Ring →ₐ[R] Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)) :=
  compLocalizationAwayAlgHom T g P
x : ↥P.toExtension.ker
hx :
  (lift
        (↑S
          (↑(LinearMap.ringLmapEquivSelf T T (P.toExtension.Cotangent →ₗ[S] (Q.comp P).toExtension.Cotangent)).symm
            (Extension.Cotangent.map (Q.toComp P).toExtensionHom))))
      (((TensorProduct.mk S T P.toExtension.Cotangent) 1) (Extension.Cotangent.mk x)) =
    0
⊢ ((TensorProduct.mk S T P.toExtension.Cotangent) 1) (Extension.Cotangent.mk x) = 0

Tactic:
suffices h : algebraMap P.Ring (Localization.Away f) x.val = 0 by
    rw [IsScalarTower.algebraMap_apply _ (P.Ring ⧸ P.ker ^ 2) _,
      IsLocalization.map_eq_zero_iff (Submonoid.powers f) (Localization.Away f)] at h
    obtain ⟨⟨m, ⟨n, rfl⟩⟩, hm⟩ := h
    rw [IsLocalizedModule.eq_zero_iff (Submonoid.powers g)]
    use ⟨g ^ n, n, rfl⟩
    dsimp [f] at hm
    rw [← map_pow, ← map_mul, Ideal.Quotient.eq_zero_iff_mem] at hm
    simp only [Submonoid.smul_def]
    rw [show g = algebraMap P.Ring S (P.σ g) by simp, ← map_pow, algebraMap_smul, ← map_smul,
      Extension.Cotangent.mk_eq_zero_iff]
    simpa using! hm

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
Q : Generators S T Unit := localizationAway T g
algInst✝ : Algebra P.Ring ((localizationAway T g).comp P).Ring := ((localizationAway T g).toComp P).toAlgHom.toAlgebra
f : P.Ring ⧸ P.ker ^ 2 := (Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)
π : ((localizationAway T g).comp P).Ring →ₐ[R] Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)) :=
  compLocalizationAwayAlgHom T g P
x : ↥P.toExtension.ker
hx :
  (lift
        (↑S
          (↑(LinearMap.ringLmapEquivSelf T T (P.toExtension.Cotangent →ₗ[S] (Q.comp P).toExtension.Cotangent)).symm
            (Extension.Cotangent.map (Q.toComp P).toExtensionHom))))
      (((TensorProduct.mk S T P.toExtension.Cotangent) 1) (Extension.Cotangent.mk x)) =
    0
⊢ (algebraMap P.Ring (Localization.Away f)) ↑x = 0
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
x : ((localizationAway T g).comp P).toExtension.Cotangent
hx : (Extension.Cotangent.map ((localizationAway T g).ofComp P).toExtensionHom) x = cMulXSubOneCotangent T g
⊢ (cotangentCompLocalizationAwayEquiv g P hx).symm (0, cMulXSubOneCotangent T g) = x

Tactic:
simpa [cotangentCompLocalizationAwayEquiv, Function.Exact.splitSurjectiveEquiv] using
    cotangentCompAwaySec_apply g P x

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.Basis

Goal state before:
pp: case inr
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : FinitePresentation R S
α : Type u_4
P : Generators R S α
inst✝ : Finite α
σ : Type u_5
b₀ : Module.Basis σ S P.toExtension.Cotangent
h✝ : Nontrivial S
f : P.toExtension.Cotangent → ↥P.toExtension.ker
hf : ∀ (b : P.toExtension.Cotangent), Extension.Cotangent.mk (f b) = b
v : σ → ↥P.ker := fun i ↦ f (b₀ i)
J : Ideal P.Ring := Ideal.span (Set.range (Subtype.val ∘ v))
hJfg : P.ker.FG
hJ : J ≤ P.ker
⊢ ⊤ ≤ Submodule.span P.Ring (Set.range v) ⊔ Extension.Cotangent.mk.ker

Tactic:
simp only [← LinearMap.map_le_map_iff, Submodule.map_span, ← Set.range_comp,
    Function.comp_def, ← Submodule.restrictScalars_span P.Ring S P.algebraMap_surjective]

Goal state after:
case inr
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : FinitePresentation R S
α : Type u_4
P : Generators R S α
inst✝ : Finite α
σ : Type u_5
b₀ : Module.Basis σ S P.toExtension.Cotangent
h✝ : Nontrivial S
f : P.toExtension.Cotangent → ↥P.toExtension.ker
hf : ∀ (b : P.toExtension.Cotangent), Extension.Cotangent.mk (f b) = b
v : σ → ↥P.ker := fun i ↦ f (b₀ i)
J : Ideal P.Ring := Ideal.span (Set.range (Subtype.val ∘ v))
hJfg : P.ker.FG
hJ : J ≤ P.ker
⊢ Submodule.map Extension.Cotangent.mk ⊤ ≤
    Submodule.restrictScalars P.Ring (Submodule.span S (Set.range fun x ↦ Extension.Cotangent.mk (v x)))
Module: Mathlib.AlgebraicGeometry.Morphisms.Smooth

Goal state before:
pp: n m : ℕ
X Y : Scheme
f : X ⟶ Y
Z : Scheme
g : Y ⟶ Z
hf : SmoothOfRelativeDimension n f
hg : SmoothOfRelativeDimension m g
x : ↥X
U₂ : Z.Opens
hU₂ : IsAffineOpen U₂
V₂ : Y.Opens
hV₂ : IsAffineOpen V₂
hfx₂ : f x ∈ V₂
e₂ : V₂ ≤ g ⁻¹ᵁ U₂
hf₂ : IsStandardSmoothOfRelativeDimension m (CommRingCat.Hom.hom (Scheme.Hom.appLE g U₂ V₂ e₂))
U₁' : Y.Opens
hU₁' : IsAffineOpen U₁'
V₁' : X.Opens
hV₁' : IsAffineOpen V₁'
hx₁' : x ∈ V₁'
e₁' : V₁' ≤ f ⁻¹ᵁ U₁'
hf₁' : IsStandardSmoothOfRelativeDimension n (CommRingCat.Hom.hom (Scheme.Hom.appLE f U₁' V₁' e₁'))
r : ↑Γ(Y, ↑⟨V₂, hV₂⟩)
s : ↑Γ(X, ↑⟨V₁', hV₁'⟩)
hx₁ : x ∈ X.basicOpen s
e₁ : X.basicOpen s ≤ f ⁻¹ᵁ Y.basicOpen r
hf₁ :
  IsStandardSmoothOfRelativeDimension n (CommRingCat.Hom.hom (Scheme.Hom.appLE f (Y.basicOpen r) (X.basicOpen s) e₁))
e : X.basicOpen s ≤ (f ≫ g) ⁻¹ᵁ U₂
heq :
  Scheme.Hom.appLE (f ≫ g) U₂ (X.basicOpen s) e =
    Scheme.Hom.appLE g U₂ V₂ e₂ ≫
      CommRingCat.ofHom (algebraMap ↑Γ(Y, V₂) ↑Γ(Y, Y.basicOpen r)) ≫
        Scheme.Hom.appLE f (Y.basicOpen r) (X.basicOpen s) e₁
⊢ ∃ U,
    ∃ (_ : IsAffineOpen U),
      ∃ V,
        ∃ (_ : IsAffineOpen V) (_ : x ∈ V) (e : V ≤ (f ≫ g) ⁻¹ᵁ U),
          IsStandardSmoothOfRelativeDimension (n + m) (CommRingCat.Hom.hom (Scheme.Hom.appLE (f ≫ g) U V e))

Tactic:
refine ⟨U₂, hU₂, X.basicOpen s, hV₁'.basicOpen s, hx₁, e, heq ▸ ?_⟩

Goal state after:
n m : ℕ
X Y : Scheme
f : X ⟶ Y
Z : Scheme
g : Y ⟶ Z
hf : SmoothOfRelativeDimension n f
hg : SmoothOfRelativeDimension m g
x : ↥X
U₂ : Z.Opens
hU₂ : IsAffineOpen U₂
V₂ : Y.Opens
hV₂ : IsAffineOpen V₂
hfx₂ : f x ∈ V₂
e₂ : V₂ ≤ g ⁻¹ᵁ U₂
hf₂ : IsStandardSmoothOfRelativeDimension m (CommRingCat.Hom.hom (Scheme.Hom.appLE g U₂ V₂ e₂))
U₁' : Y.Opens
hU₁' : IsAffineOpen U₁'
V₁' : X.Opens
hV₁' : IsAffineOpen V₁'
hx₁' : x ∈ V₁'
e₁' : V₁' ≤ f ⁻¹ᵁ U₁'
hf₁' : IsStandardSmoothOfRelativeDimension n (CommRingCat.Hom.hom (Scheme.Hom.appLE f U₁' V₁' e₁'))
r : ↑Γ(Y, ↑⟨V₂, hV₂⟩)
s : ↑Γ(X, ↑⟨V₁', hV₁'⟩)
hx₁ : x ∈ X.basicOpen s
e₁ : X.basicOpen s ≤ f ⁻¹ᵁ Y.basicOpen r
hf₁ :
  IsStandardSmoothOfRelativeDimension n (CommRingCat.Hom.hom (Scheme.Hom.appLE f (Y.basicOpen r) (X.basicOpen s) e₁))
e : X.basicOpen s ≤ (f ≫ g) ⁻¹ᵁ U₂
heq :
  Scheme.Hom.appLE (f ≫ g) U₂ (X.basicOpen s) e =
    Scheme.Hom.appLE g U₂ V₂ e₂ ≫
      CommRingCat.ofHom (algebraMap ↑Γ(Y, V₂) ↑Γ(Y, Y.basicOpen r)) ≫
        Scheme.Hom.appLE f (Y.basicOpen r) (X.basicOpen s) e₁
⊢ IsStandardSmoothOfRelativeDimension (n + m)
    (CommRingCat.Hom.hom
      (Scheme.Hom.appLE g U₂ V₂ e₂ ≫
        CommRingCat.ofHom (algebraMap ↑Γ(Y, V₂) ↑Γ(Y, Y.basicOpen r)) ≫
          Scheme.Hom.appLE f (Y.basicOpen r) (X.basicOpen s) e₁))
Module: Mathlib.AlgebraicGeometry.Morphisms.Smooth

Goal state before:
pp: case inr
X : Scheme
K : Type u
inst✝³ : Field K
inst✝² : PerfectField K
inst✝¹ : IsReduced X
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFinitePresentation f
this✝ :
  ∀ {X : Scheme} [IsReduced X] (f : X ⟶ Spec (CommRingCat.of K)) [inst : LocallyOfFinitePresentation f],
    CompactSpace ↥X → Dense ↑(smoothLocus f)
H : ¬CompactSpace ↥X
x : ↥X
U : X.Opens
hU : U ∈ X.affineOpens
hxU : x ∈ ↑U
this : U.ι ⟨x, hxU⟩ ∈ closure ↑(smoothLocus f)
⊢ x ∈ closure ↑(smoothLocus f)

Tactic:
U.ι_apply

Goal state after:
case inr
X : Scheme
K : Type u
inst✝³ : Field K
inst✝² : PerfectField K
inst✝¹ : IsReduced X
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFinitePresentation f
this✝ :
  ∀ {X : Scheme} [IsReduced X] (f : X ⟶ Spec (CommRingCat.of K)) [inst : LocallyOfFinitePresentation f],
    CompactSpace ↥X → Dense ↑(smoothLocus f)
H : ¬CompactSpace ↥X
x : ↥X
U : X.Opens
hU : U ∈ X.affineOpens
hxU : x ∈ ↑U
this : ↑⟨x, hxU⟩ ∈ closure ↑(smoothLocus f)
⊢ x ∈ closure ↑(smoothLocus f)
Module: Mathlib.AlgebraicGeometry.Morphisms.LocalIso

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
⊢ IsZariskiLocalAtSource @IsLocalIso

Tactic:
by
  rw [eq_sourceLocalClosure_isOpenImmersion]
  infer_instance

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
⊢ IsMultiplicative @IsLocalIso

Tactic:
by
  rw [eq_sourceLocalClosure_isOpenImmersion]
  infer_instance

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
⊢ IsStableUnderBaseChange @IsLocalIso

Tactic:
by
  rw [eq_sourceLocalClosure_isOpenImmersion]
  infer_instance

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

Goal state before:
pp: case inr.refine_2
P P' : MorphismProperty Scheme
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
inst✝ : IsZariskiLocalAtTarget P
H :
  ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),
    P' (Spec.map φ) → P (pullback.fst (Spec.map φ) g) → P g
R : CommRingCat
X Y✝ : Scheme
f : X ⟶ Spec R
g : Y✝ ⟶ Spec R
hf : (P' ⊓ @QuasiCompact) f
h : P (pullback.fst f g)
this : ∀ {X : Scheme} (f : X ⟶ Spec R), (P' ⊓ @QuasiCompact) f → P (pullback.fst f g) → (∃ T, X = Spec T) → P g
hX : ¬∃ T, X = Spec T
x✝ : CompactSpace ↥X
Y : Scheme
p : Y ⟶ X
hsurj : Surjective p
hP' : IsLocalIso p
hY : IsAffine Y
⊢ P (pullback.fst ((Y.isoSpec.inv ≫ p) ≫ f) g)

Tactic:
· rw [← pullbackRightPullbackFstIso_inv_fst f g _, P.cancel_left_of_respectsIso]
      exact P.pullback_fst _ _ h

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

Goal state before:
pp: P : MorphismProperty Scheme
inst✝¹ : P.DescendsAlong (@Surjective ⊓ @Flat ⊓ @QuasiCompact)
inst✝ : IsZariskiLocalAtTarget P
R : CommRingCat
X Y : Scheme
f : X ⟶ Spec R
g : Y ⟶ Spec R
h₃ : LocallyOfFinitePresentation f
h₁ : Surjective f
h₂ : Flat f
H : P (pullback.fst f g)
V : X.Opens
hV : IsCompact V.carrier
e : ⇑f '' ↑V = Set.univ
⊢ P g

Tactic:
refine MorphismProperty.of_isPullback_of_descendsAlong (Q := @Surjective ⊓ @Flat ⊓ @QuasiCompact)
    (.paste_vert (.of_hasPullback V.ι _) (.of_hasPullback f g)) ⟨⟨?_, inferInstance⟩,
      (quasiCompact_iff_compactSpace _).mpr (isCompact_iff_compactSpace.mp hV)⟩ ?_

Goal state after:
case refine_1
P : MorphismProperty Scheme
inst✝¹ : P.DescendsAlong (@Surjective ⊓ @Flat ⊓ @QuasiCompact)
inst✝ : IsZariskiLocalAtTarget P
R : CommRingCat
X Y : Scheme
f : X ⟶ Spec R
g : Y ⟶ Spec R
h₃ : LocallyOfFinitePresentation f
h₁ : Surjective f
h₂ : Flat f
H : P (pullback.fst f g)
V : X.Opens
hV : IsCompact V.carrier
e : ⇑f '' ↑V = Set.univ
⊢ Surjective (V.ι ≫ f)

case refine_2
P : MorphismProperty Scheme
inst✝¹ : P.DescendsAlong (@Surjective ⊓ @Flat ⊓ @QuasiCompact)
inst✝ : IsZariskiLocalAtTarget P
R : CommRingCat
X Y : Scheme
f : X ⟶ Spec R
g : Y ⟶ Spec R
h₃ : LocallyOfFinitePresentation f
h₁ : Surjective f
h₂ : Flat f
H : P (pullback.fst f g)
V : X.Opens
hV : IsCompact V.carrier
e : ⇑f '' ↑V = Set.univ
⊢ P (pullback.fst V.ι (pullback.fst f g))
Module: Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
H : IsIntegral R s
t : R
ht : t ∈ R⁰
ht' : t • s ∈ (algebraMap R S).range
i : ℕ
hi : i < (minpoly R s).natDegree
⊢ t ^ ((minpoly R s).natDegree - i) • t ^ i • s ^ i ∈ (algebraMap R S).range

Tactic:
← smul_pow

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
H : IsIntegral R s
t : R
ht : t ∈ R⁰
ht' : t • s ∈ (algebraMap R S).range
i : ℕ
hi : i < (minpoly R s).natDegree
⊢ t ^ ((minpoly R s).natDegree - i) • (t • s) ^ i ∈ (algebraMap R S).range
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
h_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)
f : ↥R[x] ⧸ Ideal.map (algebraMap R ↥R[x]) I →+* S ⧸ Ideal.map (algebraMap R S) I :=
  Ideal.Quotient.lift (Ideal.map (algebraMap R ↥R[x]) I)
    ((Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)).comp (algebraMap (↥R[x]) S)) ⋯
y : S ⧸ Ideal.map (algebraMap R S) I
z : S
hz : (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y
hx :
  (algebraMap R S) 1 ∈
    Ideal.map (algebraMap R S) (comap (algebraMap R S) (conductor R x)) ⊔ Ideal.map (algebraMap R S) I
⊢ 1 ∈ conductor R x ⊔ Ideal.map (algebraMap R S) I

Tactic:
map_one

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
h_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)
f : ↥R[x] ⧸ Ideal.map (algebraMap R ↥R[x]) I →+* S ⧸ Ideal.map (algebraMap R S) I :=
  Ideal.Quotient.lift (Ideal.map (algebraMap R ↥R[x]) I)
    ((Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)).comp (algebraMap (↥R[x]) S)) ⋯
y : S ⧸ Ideal.map (algebraMap R S) I
z : S
hz : (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y
hx : 1 ∈ Ideal.map (algebraMap R S) (comap (algebraMap R S) (conductor R x)) ⊔ Ideal.map (algebraMap R S) I
⊢ 1 ∈ conductor R x ⊔ Ideal.map (algebraMap R S) I
Module: Mathlib.FieldTheory.RatFunc.AsPolynomial

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
f : L
h : Transcendental K f
u : K⟮X⟯
| ↑((algEquivOfTranscendental f h) u)

Tactic:
rw [← num_div_denom u]

Goal state after:
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
f : L
h : Transcendental K f
u : K⟮X⟯
| ↑((algEquivOfTranscendental f h) ((algebraMap K[X] K⟮X⟯) u.num / (algebraMap K[X] K⟮X⟯) u.denom))
Module: Mathlib.FieldTheory.RatFunc.AsPolynomial

Goal state before:
pp: K✝ : Type u
K : Type u_1
inst✝ : Field K
⊢ (Ideal.span {X}).IsPrime

Tactic:
by rw [Ideal.span_singleton_prime]; exacts [Polynomial.prime_X, Polynomial.X_ne_zero]

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Weakly

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
p✝ : Ideal R
q : Ideal S
inst✝² : q.IsPrime
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : WeaklyQuasiFiniteAt R p
f : T ≃ₐ[R] S
⊢ p = Ideal.comap (↑f.symm) (Ideal.comap f.toRingEquiv.toRingHom p)

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Weakly

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal S
inst✝⁴ : p.IsPrime
inst✝³ : QuasiFiniteAt R (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p))) p)
A : Type u_4
inst✝² : CommRing A
inst✝¹ : Algebra R A
q : Ideal (A ⊗[R] S)
inst✝ : q.IsPrime
hq : p = Ideal.comap TensorProduct.includeRight.toRingHom q
φ : A ⊗[R] S →ₐ[A] (A ⧸ Ideal.under A q) ⊗[R] (S ⧸ Ideal.map (algebraMap R S) (Ideal.under R p)) :=
  TensorProduct.map (Ideal.Quotient.mkₐ A (Ideal.under A q))
    (Ideal.Quotient.mkₐ R (Ideal.map (algebraMap R S) (Ideal.under R p)))
hφ₁ : Function.Surjective ⇑φ
hφ₂ : RingHom.ker φ.toRingHom = Ideal.map (algebraMap A (A ⊗[R] S)) (Ideal.under A q)
hφ₃ :
  φ.comp TensorProduct.includeRight.toRingHom =
    TensorProduct.includeRight.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p)))
this✝¹ : (Ideal.map φ.toRingHom q).IsPrime
this✝ : QuasiFiniteAt (A ⧸ Ideal.under A q) (Ideal.map φ.toRingHom q)
this : QuasiFiniteAt A (Ideal.map φ.toRingHom q)
e : (A ⊗[R] S ⧸ Ideal.map (algebraMap A (A ⊗[R] S)) (Ideal.under A q)) ≃ₐ[A]
  (A ⧸ Ideal.under A q) ⊗[R] (S ⧸ Ideal.map (algebraMap R S) (Ideal.under R p)) :=
  ((Ideal.map (algebraMap A (A ⊗[R] S)) (Ideal.under A q)).quotientEquivAlg (RingHom.ker φ) AlgEquiv.refl ⋯).trans
    (Ideal.quotientKerAlgEquivOfSurjective hφ₁)
⊢ Ideal.map φ.toRingHom q =
    Ideal.map e.toRingEquiv (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap A (A ⊗[R] S)) (Ideal.under A q))) q)

Tactic:
rw [← Ideal.map_coe e.toRingEquiv, Ideal.map_map]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal S
inst✝⁴ : p.IsPrime
inst✝³ : QuasiFiniteAt R (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p))) p)
A : Type u_4
inst✝² : CommRing A
inst✝¹ : Algebra R A
q : Ideal (A ⊗[R] S)
inst✝ : q.IsPrime
hq : p = Ideal.comap TensorProduct.includeRight.toRingHom q
φ : A ⊗[R] S →ₐ[A] (A ⧸ Ideal.under A q) ⊗[R] (S ⧸ Ideal.map (algebraMap R S) (Ideal.under R p)) :=
  TensorProduct.map (Ideal.Quotient.mkₐ A (Ideal.under A q))
    (Ideal.Quotient.mkₐ R (Ideal.map (algebraMap R S) (Ideal.under R p)))
hφ₁ : Function.Surjective ⇑φ
hφ₂ : RingHom.ker φ.toRingHom = Ideal.map (algebraMap A (A ⊗[R] S)) (Ideal.under A q)
hφ₃ :
  φ.comp TensorProduct.includeRight.toRingHom =
    TensorProduct.includeRight.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p)))
this✝¹ : (Ideal.map φ.toRingHom q).IsPrime
this✝ : QuasiFiniteAt (A ⧸ Ideal.under A q) (Ideal.map φ.toRingHom q)
this : QuasiFiniteAt A (Ideal.map φ.toRingHom q)
e : (A ⊗[R] S ⧸ Ideal.map (algebraMap A (A ⊗[R] S)) (Ideal.under A q)) ≃ₐ[A]
  (A ⧸ Ideal.under A q) ⊗[R] (S ⧸ Ideal.map (algebraMap R S) (Ideal.under R p)) :=
  ((Ideal.map (algebraMap A (A ⊗[R] S)) (Ideal.under A q)).quotientEquivAlg (RingHom.ker φ) AlgEquiv.refl ⋯).trans
    (Ideal.quotientKerAlgEquivOfSurjective hφ₁)
⊢ Ideal.map φ.toRingHom q =
    Ideal.map ((↑e.toRingEquiv).comp (Ideal.Quotient.mk (Ideal.map (algebraMap A (A ⊗[R] S)) (Ideal.under A q)))) q
Module: Mathlib.RingTheory.QuasiFinite.Weakly

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
q : Ideal S
inst✝⁴ : q.IsPrime
inst✝³ : p.IsPrime
inst✝² : q.LiesOver p
Q : Ideal (p.Fiber S)
inst✝¹ : Q.IsPrime
hQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q
inst✝ : QuasiFiniteAt p.ResidueField Q
Sq : Type u_2 := Localization.AtPrime q
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime q) := Localization.AtPrime.algebraOfLiesOver p q
φ₁ : p.ResidueField →ₐ[R] Sq ⧸ Ideal.map (algebraMap R Sq) p :=
  Ideal.quotientMapₐ (Ideal.map (algebraMap R Sq) p) (IsScalarTower.toAlgHom R (Localization.AtPrime p) Sq) ⋯
φ₂ : p.Fiber S →ₐ[R] Sq ⧸ Ideal.map (algebraMap R Sq) p :=
  TensorProduct.lift φ₁ (IsScalarTower.toAlgHom R S (Sq ⧸ Ideal.map (algebraMap R Sq) p)) ⋯
y : p.Fiber S
hy : y ∈ Q.primeCompl
r : R
hrp : r ∉ p
s : S
H : r • y = 1 ⊗ₜ[R] s
this : IsUnit ((algebraMap R (Sq ⧸ Ideal.map (algebraMap R Sq) p)) r * φ₂ y)
⊢ IsUnit (φ₂ ↑⟨y, hy⟩)

Tactic:
IsUnit.mul_iff

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
q : Ideal S
inst✝⁴ : q.IsPrime
inst✝³ : p.IsPrime
inst✝² : q.LiesOver p
Q : Ideal (p.Fiber S)
inst✝¹ : Q.IsPrime
hQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q
inst✝ : QuasiFiniteAt p.ResidueField Q
Sq : Type u_2 := Localization.AtPrime q
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime q) := Localization.AtPrime.algebraOfLiesOver p q
φ₁ : p.ResidueField →ₐ[R] Sq ⧸ Ideal.map (algebraMap R Sq) p :=
  Ideal.quotientMapₐ (Ideal.map (algebraMap R Sq) p) (IsScalarTower.toAlgHom R (Localization.AtPrime p) Sq) ⋯
φ₂ : p.Fiber S →ₐ[R] Sq ⧸ Ideal.map (algebraMap R Sq) p :=
  TensorProduct.lift φ₁ (IsScalarTower.toAlgHom R S (Sq ⧸ Ideal.map (algebraMap R Sq) p)) ⋯
y : p.Fiber S
hy : y ∈ Q.primeCompl
r : R
hrp : r ∉ p
s : S
H : r • y = 1 ⊗ₜ[R] s
this : IsUnit ((algebraMap R (Sq ⧸ Ideal.map (algebraMap R Sq) p)) r) ∧ IsUnit (φ₂ y)
⊢ IsUnit (φ₂ ↑⟨y, hy⟩)