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

Goal state before:
pp: case refine_5
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsSeparable 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
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ ∀ (r : K),
    (↑↑{ toFun := fun x ↦ (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯,
                map_add' := ⋯ }).toFun
        ((algebraMap K L) r) =
      (algebraMap K B) r

Tactic:
intro r
    change g _ (algebraMap K _ r) = _
    rw [AlgHom.commutes]

Goal state after:
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
x✝ : ↥(RingHom.ker (algebraMap P S))
⊢ (⇑(AlgebraTensorModule.cancelBaseChange P (ResidueField S) K K Ω[P⁄R]) ∘
        ⇑(LinearMap.baseChange K (cotangentComplexBaseChange R S P (ResidueField S))) ∘
          ⇑(AlgebraTensorModule.cancelBaseChange P (ResidueField S) K K ↥(RingHom.ker (algebraMap P S))).symm)
      (1 ⊗ₜ[P] x✝) =
    (cotangentComplexBaseChange R S P K) (1 ⊗ₜ[P] x✝)

Tactic:
Function.comp_apply,

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
x✝ : ↥(RingHom.ker (algebraMap P S))
⊢ (AlgebraTensorModule.cancelBaseChange P (ResidueField S) K K Ω[P⁄R])
      ((LinearMap.baseChange K (cotangentComplexBaseChange R S P (ResidueField S)))
        ((AlgebraTensorModule.cancelBaseChange P (ResidueField S) K K ↥(RingHom.ker (algebraMap P S))).symm
          (1 ⊗ₜ[P] x✝))) =
    (cotangentComplexBaseChange R S P K) (1 ⊗ₜ[P] x✝)
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : IsSepClosed K
x : K
p : PrimeSpectrum K
this : Field (K ⧸ p.asIdeal) := Ideal.Quotient.field p.asIdeal
⊢ (AlgEquiv.ofBijective (ofId K (K ⧸ p.asIdeal)) ⋯).symm ((Ideal.Quotient.mk p.asIdeal) x) = x

Tactic:
apply (AlgEquiv.ofBijective (ofId K (K ⧸ p.asIdeal))
    (IsSepClosed.algebraMap_bijective _ _)).injective

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : IsSepClosed K
x : K
p : PrimeSpectrum K
this : Field (K ⧸ p.asIdeal) := ⋯
⊢ (AlgEquiv.ofBijective (ofId K (K ⧸ p.asIdeal)) ⋯)
      ((AlgEquiv.ofBijective (ofId K (K ⧸ p.asIdeal)) ⋯).symm ((Ideal.Quotient.mk p.asIdeal) x)) =
    (AlgEquiv.ofBijective (ofId K (K ⧸ p.asIdeal)) ⋯) x
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✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
⊢ ∀ (y : Ω[A⁄R]),
    (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
        (x ⊗ₜ[R] y) =
      (↑R LinearMap.id) (x ⊗ₜ[R] y)

Tactic:
y

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
y : Ω[A⁄R]
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] y) =
    (↑R LinearMap.id) (x ⊗ₜ[R] y)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing A
inst✝ : Algebra R A
s : S
a : A
⊢ (D S (S ⊗[R] A)) (s ⊗ₜ[R] a) = s ⊗ₜ[R] 1 • (D S (S ⊗[R] A)) ((algebraMap A (S ⊗[R] A)) a)

Tactic:
rw [show s ⊗ₜ 1 = algebraMap S (S ⊗ A) s by simp, Algebra.TensorProduct.right_algebraMap_apply,
    algebraMap_smul, ← Derivation.map_smul, smul_tmul', smul_eq_mul, mul_one]

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : Type u_4
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
x : AdicCompletion I M
⊢ (map I g) ((map I f) x) = (map I (g ∘ₗ f)) x

Tactic:
by
  change (map I g ∘ₗ map I f) x = map I (g ∘ₗ f) x
  rw [map_comp]

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

Goal state before:
pp: R : Type u
inst✝⁶ : CommRing R
I : Ideal R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
k : ℕ
hk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCompletion I M
a : AdicCauchySequence I M
hx : (map I f) ((mk I M) a) = 0
n : ℕ
x✝ : n ≥ 42
⊢ f (↑a (n + k)) ∈ I ^ n • (I ^ k • ⊤ ⊓ f.range)

Tactic:
show n = n + k - k by lia

Goal state after:
R : Type u
inst✝⁶ : CommRing R
I : Ideal R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
k : ℕ
hk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCompletion I M
a : AdicCauchySequence I M
hx : (map I f) ((mk I M) a) = 0
n : ℕ
x✝ : n ≥ 42
⊢ f (↑a (n + k)) ∈ I ^ (n + k - k) • (I ^ k • ⊤ ⊓ f.range)
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
A : Type u_4
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Algebra R S
f : S →ₐ[R] A
hf : Function.Surjective ⇑f
⊢ Function.Surjective (⇑(Ideal.quotientKerAlgEquivOfSurjective hf) ∘ ⇑(evalOneₐ (RingHom.ker f)))

Tactic:
exact (AlgEquiv.surjective _).comp (evalOneₐ_surjective _)

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
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Fintype σ
inst✝ : Fintype σ'
⊢ ((aeval (Q.comp P).val).mapMatrix (Q.comp P).jacobiMatrix.toBlocks₁₁).det =
    ((algebraMap Q.Ring T).mapMatrix Q.jacobiMatrix).det

Tactic:
congr

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
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Fintype σ
inst✝ : Fintype σ'
⊢ (aeval (Q.comp P).val).mapMatrix (Q.comp P).jacobiMatrix.toBlocks₁₁ = (algebraMap Q.Ring T).mapMatrix Q.jacobiMatrix
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
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Fintype σ
inst✝ : Fintype σ'
⊢ ((aeval (Q.comp P).val).mapMatrix (Q.comp P).jacobiMatrix.toBlocks₂₂).det =
    ((algebraMap S T).mapMatrix ((algebraMap P.Ring S).mapMatrix P.jacobiMatrix)).det

Tactic:
congr

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
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Fintype σ
inst✝ : Fintype σ'
⊢ (aeval (Q.comp P).val).mapMatrix (Q.comp P).jacobiMatrix.toBlocks₂₂ =
    (algebraMap S T).mapMatrix ((algebraMap P.Ring S).mapMatrix P.jacobiMatrix)
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
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Finite σ
inst✝ : Finite σ'
val✝¹ : Fintype σ'
val✝ : Fintype σ
⊢ (aeval (Q.comp P).val)
      (Matrix.fromBlocks (Q.comp P).jacobiMatrix.toBlocks₁₁ 0 (Q.comp P).jacobiMatrix.toBlocks₂₁
          (Q.comp P).jacobiMatrix.toBlocks₂₂).det =
    (aeval (Q.comp P).val) ((Q.comp P).jacobiMatrix.toBlocks₁₁.det * (Q.comp P).jacobiMatrix.toBlocks₂₂.det)

Tactic:
congr

Goal state after:
case e_6
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
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Finite σ
inst✝ : Finite σ'
val✝¹ : Fintype σ'
val✝ : Fintype σ
⊢ (Matrix.fromBlocks (Q.comp P).jacobiMatrix.toBlocks₁₁ 0 (Q.comp P).jacobiMatrix.toBlocks₂₁
        (Q.comp P).jacobiMatrix.toBlocks₂₂).det =
    (Q.comp P).jacobiMatrix.toBlocks₁₁.det * (Q.comp P).jacobiMatrix.toBlocks₂₂.det
Module: Mathlib.LinearAlgebra.Basis.Exact

Goal state before:
pp: R : Type u_1
M : Type u_2
K : Type u_3
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup K
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R K
inst✝ : Module R P
f : K →ₗ[R] M
g : M →ₗ[R] P
hfg : Function.Exact ⇑f ⇑g
p q : Submodule R M
hpq : IsCompl p q
hmap : map g q = ⊤
⊢ Set.SurjOn (⇑(p.projectionOnto q hpq)) (Set.range ⇑f) Set.univ

Tactic:
← LinearMap.coe_range,

Goal state after:
R : Type u_1
M : Type u_2
K : Type u_3
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup K
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R K
inst✝ : Module R P
f : K →ₗ[R] M
g : M →ₗ[R] P
hfg : Function.Exact ⇑f ⇑g
p q : Submodule R M
hpq : IsCompl p q
hmap : map g q = ⊤
⊢ Set.SurjOn (⇑(p.projectionOnto q hpq)) (↑f.range) Set.univ
Module: Mathlib.RingTheory.Smooth.StandardSmooth

Goal state before:
pp: n : ℕ
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
inst✝ : Finite ι
P : SubmersivePresentation R S ι σ
hP : P.dimension = n
⊢ (P.reindex (Fintype.equivFin ι).symm (Fintype.equivFin σ).symm).dimension = n

Tactic:
simp [hP]

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

Goal state before:
pp: case a
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
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway T g).comp P).Ring
hx : x ∈ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker
⊢ ∀ s ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1},
    x * s ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
y

Goal state after:
case a
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
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway T g).comp P).Ring
hx : x ∈ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker
y : ((localizationAway T g).comp P).Ring
⊢ y ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} →
    x * y ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a
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
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway T g).comp P).Ring
hx : x ∈ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker
y : ((localizationAway T g).comp P).Ring
hy : y ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
⊢ x * y ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
simp [hsple hy]

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

Goal state before:
pp: case a.a
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
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway T g).comp P).Ring
hx : x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
⊢ ∀ s ∈ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker,
    x * s ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
y

Goal state after:
case a.a
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
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway T g).comp P).Ring
hx : x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
y : ((localizationAway T g).comp P).Ring
⊢ y ∈ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker →
    x * y ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a
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
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway T g).comp P).Ring
hx : x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
⊢ ∀ s ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1},
    x * s ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
y

Goal state after:
case a
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
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway T g).comp P).Ring
hx : x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
y : ((localizationAway T g).comp P).Ring
⊢ y ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} →
    x * y ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Smooth.NoetherianDescent

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
A : Type u
B : Type u_2
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra A B
D : DescentAux A B
p : D.vars → MvPolynomial D.vars ↥(subalgebra R D)
hp : ∀ (i : D.vars), (MvPolynomial.map (algebraMap (↥(subalgebra R D)) A)) (p i) = D.h i
⊢ ∃ σ₀,
    (Ideal.Quotient.mkₐ (↥(subalgebra R D))
              (Ideal.span (Set.range (Presentation.relationOfHasCoeffs ↥(subalgebra R D))))).kerSquareLift.comp
        σ₀ =
      AlgHom.id (↥(subalgebra R D)) (Presentation.ModelOfHasCoeffs ↥(subalgebra R D))

Tactic:
refine ⟨?_, ?_⟩

Goal state after:
case refine_1
R : Type u_1
inst✝⁴ : CommRing R
A : Type u
B : Type u_2
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra A B
D : DescentAux A B
p : D.vars → MvPolynomial D.vars ↥(subalgebra R D)
hp : ∀ (i : D.vars), (MvPolynomial.map (algebraMap (↥(subalgebra R D)) A)) (p i) = D.h i
⊢ Presentation.ModelOfHasCoeffs ↥(subalgebra R D) →ₐ[↥(subalgebra R D)]
    MvPolynomial D.vars ↥(subalgebra R D) ⧸
      RingHom.ker
          (Ideal.Quotient.mkₐ (↥(subalgebra R D))
            (Ideal.span (Set.range (Presentation.relationOfHasCoeffs ↥(subalgebra R D))))) ^
        2

case refine_2
R : Type u_1
inst✝⁴ : CommRing R
A : Type u
B : Type u_2
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra A B
D : DescentAux A B
p : D.vars → MvPolynomial D.vars ↥(subalgebra R D)
hp : ∀ (i : D.vars), (MvPolynomial.map (algebraMap (↥(subalgebra R D)) A)) (p i) = D.h i
⊢ (Ideal.Quotient.mkₐ (↥(subalgebra R D))
            (Ideal.span (Set.range (Presentation.relationOfHasCoeffs ↥(subalgebra R D))))).kerSquareLift.comp
      ?refine_1 =
    AlgHom.id (↥(subalgebra R D)) (Presentation.ModelOfHasCoeffs ↥(subalgebra R D))
Module: Mathlib.FieldTheory.SeparablyGenerated

Goal state before:
pp: k : Type u_1
K : Type u_2
ι : Type u_3
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
a : ι → K
F : MvPolynomial ι k
hFa : (aeval a) F = 0
i : ι
⊢ ((AlgHom.restrictScalars k (Polynomial.aeval (a i))).comp
        ((Polynomial.mapAlgHom (aeval fun j ↦ ⟨a ↑j, ⋯⟩)).comp
          ((↑(optionEquivLeft k { b // b ≠ i })).comp ↑(renameEquiv k (Equiv.optionSubtypeNe i).symm))))
      F =
    (aeval a) F

Tactic:
congr

Goal state after:
k : Type u_1
K : Type u_2
ι : Type u_3
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
a : ι → K
F : MvPolynomial ι k
hFa : (aeval a) F = 0
i : ι
⊢ (AlgHom.restrictScalars k (Polynomial.aeval (a i))).comp
      ((Polynomial.mapAlgHom (aeval fun j ↦ ⟨a ↑j, ⋯⟩)).comp
        ((↑(optionEquivLeft k { b // b ≠ i })).comp ↑(renameEquiv k (Equiv.optionSubtypeNe i).symm))) =
    aeval a
Module: Mathlib.FieldTheory.SeparablyGenerated

Goal state before:
pp: case refine_2
k : Type u_1
K : Type u_2
ι : Type u_3
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
a : ι → K
F : MvPolynomial ι k
HF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree
σ : ι →₀ ℕ
hσ : σ ∈ F.support
i : ι
hσi : σ i ≠ 0
H : (F.toPolynomialAdjoinImageCompl a i).coeff (σ i) = 0
F₀ : Polynomial (MvPolynomial { b // b ≠ i } k) :=
  (optionEquivLeft k { b // b ≠ i }) ((renameEquiv k (Equiv.optionSubtypeNe i).symm) F)
⊢ (aeval a) ((rename Subtype.val) (F₀.coeff (σ i))) = 0

Tactic:
apply_fun Subalgebra.val _ at H

Goal state after:
case refine_2
k : Type u_1
K : Type u_2
ι : Type u_3
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
a : ι → K
F : MvPolynomial ι k
HF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree
σ : ι →₀ ℕ
hσ : σ ∈ F.support
i : ι
hσi : σ i ≠ 0
F₀ : Polynomial (MvPolynomial { b // b ≠ i } k) :=
  (optionEquivLeft k { b // b ≠ i }) ((renameEquiv k (Equiv.optionSubtypeNe i).symm) F)
H :
  (Algebra.adjoin k (a '' {i}ᶜ)).val ((F.toPolynomialAdjoinImageCompl a i).coeff (σ i)) =
    (Algebra.adjoin k (a '' {i}ᶜ)).val 0
⊢ (aeval a) ((rename Subtype.val) (F₀.coeff (σ i))) = 0
Module: Mathlib.FieldTheory.SeparablyGenerated

Goal state before:
pp: case refine_2
k : Type u_1
K : Type u_2
ι : Type u_3
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
a : ι → K
F : MvPolynomial ι k
HF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree
σ : ι →₀ ℕ
hσ : σ ∈ F.support
i : ι
hσi : σ i ≠ 0
F₀ : Polynomial (MvPolynomial { b // b ≠ i } k) :=
  (optionEquivLeft k { b // b ≠ i }) ((renameEquiv k (Equiv.optionSubtypeNe i).symm) F)
H :
  (Algebra.adjoin k (a '' {i}ᶜ)).val
      ((aeval fun j ↦ ⟨a ↑j, ⋯⟩)
        (((optionEquivLeft k { b // b ≠ i }) ((renameEquiv k (Equiv.optionSubtypeNe i).symm) F)).coeff (σ i))) =
    0
⊢ ((aeval a).comp (rename Subtype.val)) (F₀.coeff (σ i)) =
    ((Algebra.adjoin k (a '' {i}ᶜ)).val.comp (aeval fun j ↦ ⟨a ↑j, ⋯⟩))
      (((optionEquivLeft k { b // b ≠ i }) ((renameEquiv k (Equiv.optionSubtypeNe i).symm) F)).coeff (σ i))

Tactic:
congr

Goal state after:
case refine_2
k : Type u_1
K : Type u_2
ι : Type u_3
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
a : ι → K
F : MvPolynomial ι k
HF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree
σ : ι →₀ ℕ
hσ : σ ∈ F.support
i : ι
hσi : σ i ≠ 0
F₀ : Polynomial (MvPolynomial { b // b ≠ i } k) :=
  (optionEquivLeft k { b // b ≠ i }) ((renameEquiv k (Equiv.optionSubtypeNe i).symm) F)
H :
  (Algebra.adjoin k (a '' {i}ᶜ)).val
      ((aeval fun j ↦ ⟨a ↑j, ⋯⟩)
        (((optionEquivLeft k { b // b ≠ i }) ((renameEquiv k (Equiv.optionSubtypeNe i).symm) F)).coeff (σ i))) =
    0
⊢ (aeval a).comp (rename Subtype.val) = (Algebra.adjoin k (a '' {i}ᶜ)).val.comp (aeval fun j ↦ ⟨a ↑j, ⋯⟩)