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

Goal state before:
pp: 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
⊢ f ((aeval pb.gen) (minpoly K pb.gen)) = 0

Tactic:
minpoly.aeval,

Goal state after:
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
⊢ f 0 = 0
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: 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)
x y : L
this✝ : FiniteDimensional K ↥K⟮x, y⟯
this : Algebra.IsSeparable K ↥K⟮x, y⟯
α : L
hα : α ∈ K⟮x, y⟯
e : K⟮α⟯ = K⟮x, y⟯
⊢ x ∈ {x, y}

Tactic:
by simp

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: 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)
x y : L
this✝ : FiniteDimensional K ↥K⟮x, y⟯
this : Algebra.IsSeparable K ↥K⟮x, y⟯
α : L
hα : α ∈ K⟮x, y⟯
e : K⟮α⟯ = K⟮x, y⟯
hx : x ∈ K⟮α⟯
⊢ y ∈ {x, y}

Tactic:
by simp

Goal state after:
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
⊢ ∀ (a : A), a • 0 = 0

Tactic:
by simp

Goal state after:
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
⊢ ∀ (a : A) (x y : S ⊗[R] Ω[A⁄R]), a • (x + y) = a • x + a • y

Tactic:
by simp

Goal state after:
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
x : S ⊗[R] Ω[A⁄R]
⊢ 0 • x = 0

Tactic:
by induction x <;> simp [*]

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case mp
K : Type u_1
A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
H : ∀ (i : MaximalSpectrum A), FormallyEtale K (A ⧸ i.asIdeal)
this✝¹ : Module.Finite K A
this✝ : IsReduced A
this : IsArtinianRing A
v : MaximalSpectrum A → A := fun i ↦ (IsArtinianRing.equivPi A).symm (Pi.single i 1)
⊢ ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Algebra.IsSeparable K (Ai i)

Tactic:
exact ⟨_, inferInstance, _, _, _, (IsArtinianRing.equivPi A).restrictScalars K,
      fun I ↦ (iff_isSeparable _ _).mp inferInstance⟩

Goal state after:
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
⊢ (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) 1 = 0

Tactic:
rw [← (algebraMap A B).map_one]
    refine (h.out.lift_eq _ _).trans ?_
    dsimp
    rw [Derivation.map_one_eq_zero, TensorProduct.tmul_zero]

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul
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
x✝ y✝ : A
⊢ x✝ • x • 1 ⊗ₜ[R] (D R A) y✝ = x ⊗ₜ[R] (x✝ • (D R A) y✝)

Tactic:
TensorProduct.smul_tmul',

Goal state after:
case tmul
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
x✝ y✝ : A
⊢ x✝ • (x • 1) ⊗ₜ[R] (D R A) y✝ = x ⊗ₜ[R] (x✝ • (D R A) y✝)
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ I ^ 1 ≤ I

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
m n : ℕ
x✝ : m ≤ n
⊢ 1 m ≡ 1 n [SMOD I ^ m • ⊤]

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
h : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
y : AdicCompletion I N
⊢ Submodule.map (↑(LinearEquiv.refl R N)) (I ^ 0 • ⊤) = ⊤

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
h : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
y : AdicCompletion I N
n : ℕ
xn : { m // Submodule.Quotient.mk (f m) = (eval I N n) y }
⊢ (factor ⋯) ((eval I N (n + 1)) y) = (eval I N n) y

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
h : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
y : AdicCompletion I N
x : (n : ℕ) → { m // Submodule.Quotient.mk (f m) = (eval I N n) y } :=
  fun n ↦ Nat.recAux ⟨0, ⋯⟩ (fun n xn ↦ (fun z hz ↦ ⟨z, ⋯⟩) (Classical.choose ⋯) ⋯) n
n : ℕ
⊢ (factor ⋯) ((eval I N (n + 1)) y) = (eval I N n) y

Tactic:
by simp

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

Goal state before:
pp: R✝¹ : Type u_1
S✝ : Type u_2
inst✝⁹ : CommRing R✝¹
inst✝⁸ : CommRing S✝
I✝ : Ideal R✝¹
M : Type u_3
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R✝¹ M
R✝ : Type u_4
S : Type u_5
inst✝⁵ : NonAssocSemiring R✝
inst✝⁴ : 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
R : Type u_6
A : Type u_7
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Algebra R S
f : (n : ℕ) → A →ₐ[R] S ⧸ I ^ n
hf : ∀ {m n : ℕ} (hle : m ≤ n), (Ideal.Quotient.factorₐ R ⋯).comp (f n) = f m
x : A
⊢ I ^ 1 ≤ I

Tactic:
by simp

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

Goal state before:
pp: S : Type u_5
inst✝¹ : CommRing S
I : Ideal S
inst✝ : IsAdicComplete I S
n : ℕ
x : AdicCompletion I S
⊢ (Ideal.Quotient.mk (I ^ n • ⊤)) ((ofAlgEquiv I).symm x) = (eval I S n) x

Tactic:
← of_ofAlgEquiv_symm I x

Goal state after:
S : Type u_5
inst✝¹ : CommRing S
I : Ideal S
inst✝ : IsAdicComplete I S
n : ℕ
x : AdicCompletion I S
⊢ (Ideal.Quotient.mk (I ^ n • ⊤)) ((ofAlgEquiv I).symm x) = (eval I S n) ((of I S) ((ofAlgEquiv I).symm x))
Module: Mathlib.RingTheory.Smooth.AdicCompletion

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra R S
I : Ideal S
f : A →ₐ[R] S ⧸ I
inst✝ : FormallySmooth R A
⊢ Subsingleton (S ⧸ I ^ 0)

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra R S
I : Ideal S
f : A →ₐ[R] S ⧸ I
inst✝ : FormallySmooth R A
m : ℕ
T : Type u_3 := S ⧸ I ^ (m + 1 + 1)
J : Ideal T := Ideal.map (Ideal.Quotient.mkₐ R (I ^ (m + 1 + 1))) (I ^ (m + 1))
q : A →ₐ[R] T ⧸ J := (↑(DoubleQuot.quotQuotEquivQuotOfLEₐ R ⋯).symm).comp (liftAdicCompletionAux (m + 1))
⊢ m + 1 + 1 ≤ (m + 1) * (m + 1 + 1)

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : FormallySmooth R A
I : Ideal S
inst✝ : IsAdicComplete I S
f : A →ₐ[R] S ⧸ I
⊢ ∃ g, (Ideal.Quotient.mkₐ R I).comp g = f

Tactic:
obtain ⟨g, hg⟩ := exists_adicCompletionEvalOneₐ_comp_eq f

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : FormallySmooth R A
I : Ideal S
inst✝ : IsAdicComplete I S
f : A →ₐ[R] S ⧸ I
g : A →ₐ[R] AdicCompletion I S
hg : (AlgHom.restrictScalars R (AdicCompletion.evalOneₐ I)).comp g = f
⊢ ∃ g, (Ideal.Quotient.mkₐ R I).comp g = f
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
P✝ : PreSubmersivePresentation R S ι σ
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
⊢ ∀ (a : σ') (b : σ), Sum.inl (Q.map a) ≠ Sum.inr (P.map b)

Tactic:
by 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✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁷ : CommRing T
inst✝⁶ : Algebra R T
inst✝⁵ : Algebra S T
inst✝⁴ : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝³ : Finite ι
inst✝² : Finite ι'
inst✝¹ : Finite σ
inst✝ : Finite σ'
⊢ (Q.comp P).dimension = Q.dimension + P.dimension

Tactic:
simp only [Presentation.dimension]

Goal state after:
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁷ : CommRing T
inst✝⁶ : Algebra R T
inst✝⁵ : Algebra S T
inst✝⁴ : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝³ : Finite ι
inst✝² : Finite ι'
inst✝¹ : Finite σ
inst✝ : Finite σ'
⊢ Nat.card (ι' ⊕ ι) - Nat.card (σ' ⊕ σ) = Nat.card ι' - Nat.card σ' + (Nat.card ι - Nat.card σ)
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
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
ι : Type u_5
κ : Type u_6
σ : Type u_7
v : ι → M
a : κ → ι
b : σ → ι
hg : Function.Surjective ⇑g
hsa : ∀ (i : κ), s (v (a i)) = 0
hlib : LinearIndependent R (⇑s ∘ v ∘ b)
hab : Codisjoint (Set.range a) (Set.range b)
hsp : ⊤ ≤ span R (Set.range v)
⊢ ⊤ ≤ span R (Set.range (⇑g ∘ v ∘ a))

Tactic:
apply top_le_span_of_aux hs hfg (Sum.elim (v ∘ a) (v ∘ b)) hg hsa hlib

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
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
ι : Type u_5
κ : Type u_6
σ : Type u_7
v : ι → M
a : κ → ι
b : σ → ι
hg : Function.Surjective ⇑g
hsa : ∀ (i : κ), s (v (a i)) = 0
hlib : LinearIndependent R (⇑s ∘ v ∘ b)
hab : Codisjoint (Set.range a) (Set.range b)
hsp : ⊤ ≤ span R (Set.range v)
⊢ ⊤ ≤ span R (Set.range (Sum.elim (v ∘ a) (v ∘ b)))
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: 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
r : R
inst✝ : IsLocalization.Away r S
⊢ (fun x ↦ r ^ x) 1 = r

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Presentation R S ι σ
x : σ
⊢ ↑(MvPolynomial.coeffs (P.relation x)) ⊆ ↑(MvPolynomial.coeffs (P.relation x))

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Presentation R S ι σ
R₀ : Type u_5
inst✝⁶ : CommRing R₀
inst✝⁵ : Algebra R₀ R
inst✝⁴ : Algebra R₀ S
inst✝³ : IsScalarTower R₀ R S
inst✝² : P.HasCoeffs R₀
inst✝¹ : Finite ι
inst✝ : Finite σ
val✝ : Fintype σ
⊢ Ideal.span ↑(Finset.image (relationOfHasCoeffs R₀) Finset.univ) = Ideal.span (Set.range (relationOfHasCoeffs R₀))

Tactic:
by simp

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

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
⊢ (↑(MvPolynomial.coeffs (P.σ ↑⋯.unit⁻¹))).Finite

Tactic:
by simp

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

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
⊢ ∀ i ∈ Set.univ, (↑(P.jacobianRelations i).coeffs).Finite

Tactic:
by simp

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

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
⊢ ∀ i ∉ Set.univ, ↑(P.jacobianRelations i).coeffs = ∅

Tactic:
by simp

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

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ W'.negY ![W'.dblX P, W'.negDblY P, W'.dblZ P] = P y ^ 4

Tactic:
negY_eq,

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ -W'.negDblY P - W'.a₁ * W'.dblX P - W'.a₃ * W'.dblZ P = P y ^ 4
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: case a
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'.negY ![W'.dblX P, W'.negDblY P, W'.dblZ P] * P z ^ 2 + W'.negDblY P * P z ^ 2 -
      ((eval P) W'.polynomialX ^ 3 + -(eval P) W'.polynomialX ^ 3) =
    0

Tactic:
negY_eq,

Goal state after:
case a
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'.negDblY P - W'.a₁ * W'.dblX P - W'.a₃ * W'.dblZ P) * P z ^ 2 + W'.negDblY P * P z ^ 2 -
      ((eval P) W'.polynomialX ^ 3 + -(eval P) W'.polynomialX ^ 3) =
    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
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
⊢ (algebraMap P.Ring S) (P.σ g) = g

Tactic:
by simp

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
x : ((localizationAway T g).comp P).toExtension.Cotangent
⊢ (cotangentCompAwaySec g P x) (cMulXSubOneCotangent T g) = x

Tactic:
rw [← basisCotangentAway_apply _ (), cotangentCompAwaySec, Module.Basis.constr_basis]

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
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
x : MvPolynomial ι R ⧸ Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b))
⊢ (Ideal.Quotient.mk (Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b))))
      (if x = 0 then 0 else if x = -1 then -1 else Function.surjInv ⋯ x) =
    x

Tactic:
by
    split_ifs
    · next h => subst h; rfl
    · next h => subst h; rfl
    · simp [Function.surjInv_eq]

Goal state after:
Module: Mathlib.RingTheory.Smooth.NoetherianDescent

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁶ : CommRing R
A : Type u
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra A B
D : DescentAux A B
inst✝¹ : Finite D.vars
inst✝ : Finite D.rels
⊢ D.P.coeffs.Finite

pp: case refine_2
R : Type u_1
inst✝⁶ : CommRing R
A : Type u
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra A B
D : DescentAux A B
inst✝¹ : Finite D.vars
inst✝ : Finite D.rels
⊢ (⋃ i, ↑(D.h i).coeffs).Finite

pp: case refine_3
R : Type u_1
inst✝⁶ : CommRing R
A : Type u
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra A B
D : DescentAux A B
inst✝¹ : Finite D.vars
inst✝ : Finite D.rels
⊢ (⋃ i, ⋃ x ∈ (D.q i).coeffs, ↑(coeffs x)).Finite

pp: case refine_4
R : Type u_1
inst✝⁶ : CommRing R
A : Type u
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra A B
D : DescentAux A B
inst✝¹ : Finite D.vars
inst✝ : Finite D.rels
⊢ (⋃ i, ⋃ x ∈ (D.p i).coeffs, ↑x.coeffs).Finite

Tactic:
· exact Presentation.finite_coeffs

Goal state after:
case refine_2
R : Type u_1
inst✝⁶ : CommRing R
A : Type u
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra A B
D : DescentAux A B
inst✝¹ : Finite D.vars
inst✝ : Finite D.rels
⊢ (⋃ i, ↑(D.h i).coeffs).Finite

case refine_3
R : Type u_1
inst✝⁶ : CommRing R
A : Type u
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra A B
D : DescentAux A B
inst✝¹ : Finite D.vars
inst✝ : Finite D.rels
⊢ (⋃ i, ⋃ x ∈ (D.q i).coeffs, ↑(coeffs x)).Finite

case refine_4
R : Type u_1
inst✝⁶ : CommRing R
A : Type u
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra A B
D : DescentAux A B
inst✝¹ : Finite D.vars
inst✝ : Finite D.rels
⊢ (⋃ i, ⋃ x ∈ (D.p i).coeffs, ↑x.coeffs).Finite
Module: Mathlib.RingTheory.RingHom.StandardSmooth

Goal state before:
pp: n : ℕ
R S T : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.IsStandardSmoothOfRelativeDimension n R T
this : Algebra.IsStandardSmoothOfRelativeDimension n S (S ⊗[R] T)
⊢ IsStandardSmoothOfRelativeDimension n (algebraMap S (S ⊗[R] T))

Tactic:
by
      rw [RingHom.IsStandardSmoothOfRelativeDimension]
      convert! this; ext; simp_rw [Algebra.smul_def]; rfl

Goal state after:
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
⊢ P.ker ≤ (Submodule.subtype P.ker).range

Tactic:
by simp

Goal state after:
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
inst✝ : Smooth A B
⊢ ∃ A₀ B₀ x x_1 x_2 x_3 x_4,
    Function.Injective ⇑(algebraMap A₀ A) ∧ FiniteType R A₀ ∧ Smooth A₀ B₀ ∧ Nonempty (B ≃ₐ[A] A ⊗[A₀] B₀)

Tactic:
obtain ⟨A₀, B₀, _, _, hA₀, _, _⟩ := exists_subalgebra_fg R A B
  use A₀, B₀, inferInstance, inferInstance, inferInstance, inferInstance, inferInstance,
    Subtype.val_injective, ⟨A₀.fg_top.mpr hA₀⟩, inferInstance

Goal state after:
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
hF : Irreducible F
i : ι
H : AlgebraicIndependent k fun x ↦ a ↑x
hc : a '' {i}ᶜ = Set.range fun x ↦ a ↑x
⊢ ∀ (x : ι), x ≠ i ↔ x ∈ {j | j ≠ i}

Tactic:
by simp

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

Goal state before:
pp: X : Scheme
K : Type u
inst✝³ : Field K
f : X ⟶ Spec (CommRingCat.of K)
inst✝² : IsIntegral X
inst✝¹ : UniversallyClosed f
inst✝ : LocallyOfFiniteType f
x : ↥X
U : TopologicalSpace.Opens ↥X
hU : U ∈ X.affineOpens
hxU : x ∈ ↑U
this✝¹ : Field ↑Γ(Spec (CommRingCat.of K), ⊤) := ⋯.toField
this✝ : Field ↑Γ(X, ⊤) := ⋯.toField
this : Nonempty ↥↑U
⊢ U ≤ f ⁻¹ᵁ ⊤

Tactic:
by simp

Goal state after: