mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case w
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
hx' : (Ideal.Quotient.mk I) ((aeval x) (minpoly K pb.gen)) = 0
u : B
hu : (aeval x) (derivative (minpoly K pb.gen)) * u + 1 ∈ I
⊢ eval x (map (algebraMap K B) (minpoly K pb.gen)) +
      eval x (derivative (map (algebraMap K B) (minpoly K pb.gen))) * (u * (aeval x) (minpoly K pb.gen)) =
    0

pp: case w.hy
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
hx' : (Ideal.Quotient.mk I) ((aeval x) (minpoly K pb.gen)) = 0
u : B
hu : (aeval x) (derivative (minpoly K pb.gen)) * u + 1 ∈ I
⊢ (u * (aeval x) (minpoly K pb.gen)) ^ 2 = 0

Tactic:
derivative_map,

Goal state after:
case w
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
hx' : (Ideal.Quotient.mk I) ((aeval x) (minpoly K pb.gen)) = 0
u : B
hu : (aeval x) (derivative (minpoly K pb.gen)) * u + 1 ∈ I
⊢ eval x (map (algebraMap K B) (minpoly K pb.gen)) +
      eval x (map (algebraMap K B) (derivative (minpoly K pb.gen))) * (u * (aeval x) (minpoly K pb.gen)) =
    0

case w.hy
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
hx' : (Ideal.Quotient.mk I) ((aeval x) (minpoly K pb.gen)) = 0
u : B
hu : (aeval x) (derivative (minpoly K pb.gen)) * u + 1 ∈ I
⊢ (u * (aeval x) (minpoly K pb.gen)) ^ 2 = 0
Module: Mathlib.RingTheory.Smooth.Local

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : IsLocalRing S
inst✝¹¹ : Algebra R S
P : Type u_3
K : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : CommRing P
inst✝⁸ : Algebra R P
inst✝⁷ : Algebra P S
inst✝⁶ : IsScalarTower R P S
inst✝⁵ : Algebra S K
inst✝⁴ : Algebra P K
inst✝³ : IsScalarTower P S K
inst✝² : FormallySmooth R P
inst✝¹ : Module.Free P Ω[P⁄R]
inst✝ : Module.Finite P Ω[P⁄R]
h₁ : Function.Surjective ⇑(algebraMap P S)
h₂ : (RingHom.ker (algebraMap P S)).FG
h₃ : maximalIdeal S ≤ RingHom.ker (algebraMap S K)
f : ResidueField S →ₐ[S] K := Ideal.Quotient.liftₐ (maximalIdeal S) (ofId S K) h₃
this✝¹ : Algebra (ResidueField S) K := f.toAlgebra
this✝ : IsScalarTower S (ResidueField S) K
this : IsScalarTower P (ResidueField S) K
⊢ FormallySmooth R S ↔ Function.Injective ⇑(cotangentComplexBaseChange R S P K)

Tactic:
FormallySmooth.iff_injective_cotangentComplexBaseChange_residueField P h₁ h₂,

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : IsLocalRing S
inst✝¹¹ : Algebra R S
P : Type u_3
K : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : CommRing P
inst✝⁸ : Algebra R P
inst✝⁷ : Algebra P S
inst✝⁶ : IsScalarTower R P S
inst✝⁵ : Algebra S K
inst✝⁴ : Algebra P K
inst✝³ : IsScalarTower P S K
inst✝² : FormallySmooth R P
inst✝¹ : Module.Free P Ω[P⁄R]
inst✝ : Module.Finite P Ω[P⁄R]
h₁ : Function.Surjective ⇑(algebraMap P S)
h₂ : (RingHom.ker (algebraMap P S)).FG
h₃ : maximalIdeal S ≤ RingHom.ker (algebraMap S K)
f : ResidueField S →ₐ[S] K := Ideal.Quotient.liftₐ (maximalIdeal S) (ofId S K) h₃
this✝¹ : Algebra (ResidueField S) K := f.toAlgebra
this✝ : IsScalarTower S (ResidueField S) K
this : IsScalarTower P (ResidueField S) K
⊢ Function.Injective ⇑(cotangentComplexBaseChange R S P (ResidueField S)) ↔
    Function.Injective ⇑(cotangentComplexBaseChange R S P K)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
this : Algebra.IsPushout R A S B
⊢ B ⊗[A] Ω[A⁄R] ≃ₗ[B] Ω[B⁄S]

Tactic:
let e₁ : B ⊗[A] Ω[A⁄R] ≃ₗ[A] Ω[A⁄R] ⊗[R] S :=
    AlgebraTensorModule.congr (Algebra.IsPushout.equiv R A S B).symm.toLinearEquiv (.refl _ _)
      ≪≫ₗ _root_.TensorProduct.comm _ _ _ ≪≫ₗ AlgebraTensorModule.cancelBaseChange ..

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
this : Algebra.IsPushout R A S B
e₁ : B ⊗[A] Ω[A⁄R] ≃ₗ[A] Ω[A⁄R] ⊗[R] S :=
  AlgebraTensorModule.congr (↑(Algebra.IsPushout.equiv R A S B).symm) (LinearEquiv.refl A Ω[A⁄R]) ≪≫ₗ
      TensorProduct.comm A (A ⊗[R] S) Ω[A⁄R] ≪≫ₗ
    AlgebraTensorModule.cancelBaseChange R A A Ω[A⁄R] S
⊢ B ⊗[A] Ω[A⁄R] ≃ₗ[B] Ω[B⁄S]
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.tmul.tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
this : Algebra.IsPushout R A S B
e₁ : B ⊗[A] Ω[A⁄R] ≃ₗ[A] Ω[A⁄R] ⊗[R] S :=
  AlgebraTensorModule.congr (↑(Algebra.IsPushout.equiv R A S B).symm) (LinearEquiv.refl A Ω[A⁄R]) ≪≫ₗ
      TensorProduct.comm A (A ⊗[R] S) Ω[A⁄R] ≪≫ₗ
    AlgebraTensorModule.cancelBaseChange R A A Ω[A⁄R] S
e₂ : B ⊗[A] Ω[A⁄R] ≃ₗ[R] Ω[B⁄S] :=
  LinearEquiv.restrictScalars R e₁ ≪≫ₗ TensorProduct.comm R Ω[A⁄R] S ≪≫ₗ
    LinearEquiv.restrictScalars R (tensorKaehlerEquivBase R S A B)
a : A
b : S
y : Ω[A⁄R]
x : A
z : S
⊢ e₂ ((Algebra.IsPushout.equiv R A S B) (a ⊗ₜ[R] b) • (Algebra.IsPushout.equiv R A S B) (x ⊗ₜ[R] z) ⊗ₜ[A] y) =
    (Algebra.IsPushout.equiv R A S B) (a ⊗ₜ[R] b) • e₂ ((Algebra.IsPushout.equiv R A S B) (x ⊗ₜ[R] z) ⊗ₜ[A] y)

Tactic:
suffices b • z • a • x • KaehlerDifferential.map R S A B y =
      (algebraMap A B a * algebraMap S B b) • z • x • KaehlerDifferential.map R S A B y by
    simpa [e₂, e₁, smul_tmul', Algebra.IsPushout.equiv_tmul, ← mul_smul,
      Algebra.IsPushout.equiv_symm_algebraMap_left, Algebra.IsPushout.equiv_symm_algebraMap_right]
  simp only [← mul_smul, ← @algebraMap_smul S _ B, ← @algebraMap_smul A _ B]
  ring_nf

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

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
I : Ideal R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type w
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Surjective ⇑f
x : AdicCauchySequence I N
n : ℕ
y yₙ : M
hy : f y = ↑x (n + 1)
hyₙ : f yₙ = ↑x n
⊢ f (yₙ - y) ∈ I ^ n • Submodule.map f ⊤

Tactic:
Submodule.map_top,

Goal state after:
R : Type u
inst✝⁴ : CommRing R
I : Ideal R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type w
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Surjective ⇑f
x : AdicCauchySequence I N
n : ℕ
y yₙ : M
hy : f y = ↑x (n + 1)
hyₙ : f yₙ = ↑x n
⊢ f (yₙ - y) ∈ I ^ n • f.range
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) • Submodule.map g ⊤

Tactic:
Submodule.map_top,

Goal state after:
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) • g.range
Module: Mathlib.RingTheory.Smooth.StandardSmoothCotangent

Goal state before:
pp: R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
κ : Type u_5
f : κ → ι
hf : Function.Injective f
hcompl : IsCompl (Set.range f) (Set.range P.map)
⊢ Basis κ S Ω[S⁄R]

Tactic:
apply P.cotangentSpaceBasis.ofSplitExact (sectionCotangent_comp P)
    Extension.exact_cotangentComplex_toKaehler Extension.toKaehler_surjective hf (b := P.map)

Goal state after:
case hsa
R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
κ : Type u_5
f : κ → ι
hf : Function.Injective f
hcompl : IsCompl (Set.range f) (Set.range P.map)
⊢ ∀ (i : κ), P.sectionCotangent (P.cotangentSpaceBasis (f i)) = 0

case hlib
R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
κ : Type u_5
f : κ → ι
hf : Function.Injective f
hcompl : IsCompl (Set.range f) (Set.range P.map)
⊢ LinearIndependent S (⇑P.sectionCotangent ∘ ⇑P.cotangentSpaceBasis ∘ P.map)

case hab
R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
κ : Type u_5
f : κ → ι
hf : Function.Injective f
hcompl : IsCompl (Set.range f) (Set.range P.map)
⊢ Codisjoint (Set.range f) (Set.range P.map)
Module: Mathlib.RingTheory.Extension.Cotangent.Free

Goal state before:
pp: case hsp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ Submodule.map (↑(Finsupp.linearEquivFunOnFinite S S σ)) (Submodule.map (P.cotangentRestrict ⋯) ⊤) = ⊤

Tactic:
Submodule.map_top,

Goal state after:
case hsp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ Submodule.map (↑(Finsupp.linearEquivFunOnFinite S S σ)) (P.cotangentRestrict ⋯).range = ⊤
Module: Mathlib.RingTheory.Extension.Cotangent.Free

Goal state before:
pp: case hsp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ Submodule.map ↑(Finsupp.linearEquivFunOnFinite S S σ) ⊤ = ⊤

Tactic:
Submodule.map_top,

Goal state after:
case hsp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ (↑(Finsupp.linearEquivFunOnFinite S S σ)).range = ⊤
Module: Mathlib.RingTheory.Smooth.StandardSmoothOfFree

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Smooth R S
⊢ ∃ s, Ideal.span s = ⊤ ∧ ∀ x ∈ s, IsStandardSmooth R (Localization.Away x)

Tactic:
choose f hf₁ hf₂ using IsSmoothAt.exists_notMem_isStandardSmooth R (S := S)
  /- #adaptation_note Before https://github.com/leanprover/lean4/pull/13166
  (replacing grind's canonicalizer with a type-directed normalizer), the original proof was:
  `refine ⟨Set.range (fun p : PrimeSpectrum S ↦ f p.asIdeal), ?_, by grind⟩`
  It is not yet clear whether this is due to defeq abuse in Mathlib or a problem in the new
  canonicalizer; a minimization would help. -/
  refine ⟨Set.range (fun p : PrimeSpectrum S ↦ f p.asIdeal), ?_, by
    simp only [Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff]; intro; apply hf₂⟩
  simp [← PrimeSpectrum.iSup_basicOpen_eq_top_iff, TopologicalSpace.Opens.ext_iff, Set.ext_iff]
  grind

Goal state after:
Module: Mathlib.RingTheory.RingHom.StandardSmooth

Goal state before:
pp: n : ℕ
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S
this✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite
ι σ : Type
w✝¹ : _root_.Finite σ
w✝ : _root_.Finite ι
P : Algebra.SubmersivePresentation R S ι σ
e✝ : P.dimension = n
e₀ : σ ⊕ Fin n ≃ ι :=
  ((Equiv.ofInjective P.map ⋯).sumCongr (Finite.equivFinOfCardEq ⋯).symm).trans (Equiv.Set.sumCompl (Set.range P.map))
e : MvPolynomial σ (MvPolynomial (Fin n) R) ≃ₐ[R] P.Ring :=
  (MvPolynomial.sumAlgEquiv R σ (Fin n)).symm.trans (MvPolynomial.renameEquiv R e₀)
φ : MvPolynomial (Fin n) R →ₐ[R] P.Ring :=
  (↑e).comp (IsScalarTower.toAlgHom R (MvPolynomial (Fin n) R) (MvPolynomial σ (MvPolynomial (Fin n) R)))
algInst✝¹ : Algebra (MvPolynomial (Fin n) R) P.Ring := φ.toAlgebra
algInst✝ : Algebra (MvPolynomial (Fin n) R) S := ((algebraMap P.Ring S).comp φ.toRingHom).toAlgebra
scalarTowerInst✝ : IsScalarTower (MvPolynomial (Fin n) R) P.Ring S :=
  IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap (MvPolynomial (Fin n) R) S))
this✝ : IsScalarTower R (MvPolynomial (Fin n) R) P.Ring
this : IsScalarTower R (MvPolynomial (Fin n) R) S
⊢ ∃ g, g.Etale

Tactic:
refine ⟨IsScalarTower.toAlgHom _ _ _, ?_⟩

Goal state after:
n : ℕ
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S
this✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite
ι σ : Type
w✝¹ : _root_.Finite σ
w✝ : _root_.Finite ι
P : Algebra.SubmersivePresentation R S ι σ
e✝ : P.dimension = n
e₀ : σ ⊕ Fin n ≃ ι :=
  ((Equiv.ofInjective P.map ⋯).sumCongr (Finite.equivFinOfCardEq ⋯).symm).trans (Equiv.Set.sumCompl (Set.range P.map))
e : MvPolynomial σ (MvPolynomial (Fin n) R) ≃ₐ[R] P.Ring :=
  (MvPolynomial.sumAlgEquiv R σ (Fin n)).symm.trans (MvPolynomial.renameEquiv R e₀)
φ : MvPolynomial (Fin n) R →ₐ[R] P.Ring :=
  (↑e).comp (IsScalarTower.toAlgHom R (MvPolynomial (Fin n) R) (MvPolynomial σ (MvPolynomial (Fin n) R)))
algInst✝¹ : Algebra (MvPolynomial (Fin n) R) P.Ring := φ.toAlgebra
algInst✝ : Algebra (MvPolynomial (Fin n) R) S := ((algebraMap P.Ring S).comp φ.toRingHom).toAlgebra
scalarTowerInst✝ : IsScalarTower (MvPolynomial (Fin n) R) P.Ring S :=
  IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap (MvPolynomial (Fin n) R) S))
this✝ : IsScalarTower R (MvPolynomial (Fin n) R) P.Ring
this : IsScalarTower R (MvPolynomial (Fin n) R) S
⊢ (IsScalarTower.toAlgHom R (MvPolynomial (Fin n) R) S).Etale
Module: Mathlib.RingTheory.Extension.Cotangent.Basis

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
inst✝ : Nontrivial S
r : Unit ⊕ σ
⊢ D.basis r = Extension.Cotangent.mk ⟨D.pres.relation r, ⋯⟩

Tactic:
obtain (r | r) := r
  · rw [basis_inl, cotangentEquivProd_symm_apply]
    exact cotangentCompLocalizationAwayEquiv_symm_inr _ _ _
  · rw [basis_inr, cotangentEquivProd_symm_apply, cotangentCompLocalizationAwayEquiv_symm_inl,
      basisLeft, Module.Basis.map_apply, tensorCotangentEquiv_symm_apply,
      LinearMap.liftBaseChange_tmul, one_smul, Extension.Cotangent.map_mk]
    simp only [Extension.Hom.toAlgHom_apply, Hom.toExtensionHom_toRingHom, AlgHom.toRingHom_eq_coe]
    congr! 2 with x
    simp [pres, Presentation.comp_relation_inr, kerGen, presLeft, Generators.toComp_toAlgHom]
    rfl

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
⊢ ∃ P' b, P'.val ∘ Sum.inr = P.val ∧ ∀ (r : Unit ⊕ σ), b r = Extension.Cotangent.mk ⟨P'.relation r, ⋯⟩

Tactic:
suffices hJ : P.ker ≤ J ⊔ P.ker • P.ker by
    obtain ⟨g, hgmem, hg⟩ := Submodule.exists_sub_one_mem_and_smul_le_of_fg_of_le_sup hJfg le_rfl hJ
    let D : Aux P b₀ := { f := f, hf := hf, g := g, hgmem := hgmem, hg := hg }
    exact ⟨D.pres, D.basis, D.pres_val_comp_inr, D.basis_apply⟩

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
⊢ P.ker ≤ J ⊔ P.ker • P.ker
Module: Mathlib.RingTheory.Smooth.NoetherianDescent

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
A : Type u
B : Type u_2
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra A B
n : ℕ
inst✝ : IsStandardSmoothOfRelativeDimension n A B
⊢ ∃ A₀ B₀ x x_1, A₀.FG ∧ IsStandardSmoothOfRelativeDimension n (↥A₀) B₀ ∧ Nonempty (B ≃ₐ[A] A ⊗[↥A₀] B₀)

Tactic:
obtain ⟨ι, σ, _, _, P, hP⟩ := IsStandardSmoothOfRelativeDimension.out (n := n) (R := A) (S := B)

Goal state after:
R : Type u_1
inst✝⁵ : CommRing R
A : Type u
B : Type u_2
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra A B
n : ℕ
inst✝ : IsStandardSmoothOfRelativeDimension n A B
ι σ : Type
w✝¹ : Finite σ
w✝ : Finite ι
P : SubmersivePresentation A B ι σ
hP : P.dimension = n
⊢ ∃ A₀ B₀ x x_1, A₀.FG ∧ IsStandardSmoothOfRelativeDimension n (↥A₀) B₀ ∧ Nonempty (B ≃ₐ[A] A ⊗[↥A₀] B₀)
Module: Mathlib.RingTheory.Extension.Cotangent.Basis

Goal state before:
pp: 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
x✝ : σ
⊢ Extension.Cotangent.mk (v x✝) = b₀ x✝

Tactic:
exact hf _

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
⊢ Smooth f ↔
    ∀ (x : ↥X),
      ∃ U,
        ∃ (_ : IsAffineOpen U),
          ∃ V,
            ∃ (_ : IsAffineOpen V) (_ : x ∈ V) (e : V ≤ f ⁻¹ᵁ U),
              (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V e)).IsStandardSmooth

Tactic:
have : HasRingHomProperty @Smooth.{u} (Locally IsStandardSmooth) := by
    convert! (inferInstance : HasRingHomProperty (@Smooth.{u}) RingHom.Smooth)
    ext f
    rw [RingHom.smooth_iff_locally_isStandardSmooth]
  rw [HasRingHomProperty.iff_exists_appLE_locally (P := @Smooth)]
  · congr!
    simp [Subtype.exists]
    grind [Scheme.affineOpens]
  · exact isStandardSmooth_stableUnderCompositionWithLocalizationAway.left
  · exact isStandardSmooth_respectsIso

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
u v : R
⊢ W'.negAddY (u • P) (v • Q) = (u * v) ^ 2 * W'.negAddY P Q

Tactic:
by
  simp only [negAddY, smul_fin3_ext]
  ring1

Goal state after: