mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Ring.Subsemiring.MulOpposite

Goal state before:
pp: ι : Sort u_1
R : Type u_2
inst✝ : NonAssocSemiring R
S : Subsemiring R
a✝ b✝ : Rᵐᵒᵖ
hx : a✝ ∈ S.op.carrier
hy : b✝ ∈ S.op.carrier
⊢ a✝ + b✝ ∈ S.op.carrier

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Subsemiring.MulOpposite

Goal state before:
pp: ι : Sort u_1
R : Type u_2
inst✝ : NonAssocSemiring R
S : Subsemiring Rᵐᵒᵖ
a✝ b✝ : R
hx : a✝ ∈ S.unop.carrier
hy : b✝ ∈ S.unop.carrier
⊢ a✝ + b✝ ∈ S.unop.carrier

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : Semiring R
f : R[X]
p : Ideal R[X] := Ideal.span {g | ∃ i, g = C (f.coeff i)}
n : ℕ
_hn : n ∈ f.support
⊢ C (f.coeff n) ∈ p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Subsemiring.MulOpposite

Goal state before:
pp: R : Type u_2
inst✝ : NonAssocSemiring R
s : Set R
⊢ (closure s).op = closure (MulOpposite.unop ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Subsemiring.MulOpposite

Goal state before:
pp: R : Type u_2
inst✝ : NonAssocSemiring R
s : Set Rᵐᵒᵖ
⊢ (closure s).unop = closure (MulOpposite.op ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : Semiring R
f : R[X]
⊢ f ∈ Ideal.span {g | ∃ i, g = C (f.coeff i)}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
r : R
hr : Prime r
⊢ Prime (C r)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
σ : Type v
r : R
inst✝¹ : CommRing R
inst✝ : Finite σ
⊢ Prime (C r) ↔ Prime r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
σ : Type v
inst✝ : CommRing R
r : R
hr : Prime r
h : C r = 0
⊢ r = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
σ : Type v
inst✝ : CommRing R
r : R
hr : Prime r
h : IsUnit (C r)
⊢ IsUnit r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
σ : Type v
inst✝ : CommRing R
r : R
hr : Prime r
s : Finset σ
a' b' : MvPolynomial (↥s) R
hd : (algebraMap R (MvPolynomial σ R)) r ∣ (rename Subtype.val) a' * (rename Subtype.val) b'
⊢ (algebraMap R (MvPolynomial (↥s) R)) r ∣ a' * b'

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
σ : Type v
inst✝ : CommRing R
r : R
hr : Prime r
a b : MvPolynomial σ R
hd : C r ∣ a * b
⊢ C r ∣ a ∨ C r ∣ b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
σ : Type v
inst✝ : CommRing R
s : Set σ
p : MvPolynomial (↑s) R
eqv : MvPolynomial (↑sᶜ) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=
  (sumAlgEquiv R ↑sᶜ ↑s).symm.trans (renameEquiv R ((Equiv.sumComm ↑sᶜ ↑s).trans (Equiv.Set.sumCompl s)))
⊢ rename Subtype.val = (↑eqv).comp (Algebra.algHom R (MvPolynomial (↑s) R) (MvPolynomial (↑sᶜ) (MvPolynomial (↑s) R)))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Semiring A
inst✝³ : Algebra R S
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
hA : FiniteType R A
s : Finset A
hS : adjoin R ↑s = ⊤
b : A
⊢ adjoin R ↑s ≤ Subalgebra.restrictScalars R (adjoin S ↑s)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Semiring A
inst✝³ : Algebra R S
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
hA : FiniteType R A
⊢ FiniteType S A

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
A : Type uA
B : Type uB
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Semiring B
inst✝² : Algebra R A
inst✝¹ : Algebra R B
inst✝ : FiniteType R A
f : A →ₐ[R] B
hf : Surjective ⇑f
⊢ ⊤.FG

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
σ : Type v
inst✝ : CommRing R
s : Set σ
p : MvPolynomial (↑s) R
⊢ Prime ((rename Subtype.val) p) ↔ Prime p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : CommSemiring S
inst✝⁹ : Semiring A
inst✝⁸ : Semiring B
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : FiniteType R S
⊢ FiniteType R S[X]

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R ↑s = I.degreeLE ↑N
k : ℕ
h : N < k
x✝ : R
hx : x✝ ∈ I.leadingCoeffNth k
hxm : x✝ ∉ M
⊢ ¬M < I.leadingCoeffNth k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Semiring A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : Algebra R B
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
ι : Type u_1
inst✝¹ : Finite ι
inst✝ : FiniteType R S
⊢ FiniteType R (MvPolynomial ι S)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Semiring A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : Algebra R B
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
ι : Type u_1
inst✝¹ : Finite ι
inst✝ : FiniteType R S
⊢ FiniteType R (FreeAlgebra S ι)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: case mp
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Finset A
hs : adjoin R ↑s = ⊤
⊢ Surjective ⇑((FreeAlgebra.lift R) Subtype.val)

Tactic:
← Set.range_eq_univ,

Goal state after:
case mp
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Finset A
hs : adjoin R ↑s = ⊤
⊢ Set.range ⇑((FreeAlgebra.lift R) Subtype.val) = Set.univ
Module: Mathlib.Algebra.Ring.Subring.MulOpposite

Goal state before:
pp: ι : Sort u_1
R : Type u_2
inst✝ : NonAssocRing R
S : Subring R
⊢ ∀ {x : Rᵐᵒᵖ}, x ∈ S.op.carrier → -x ∈ S.op.carrier

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ FiniteType R A ↔ ∃ s f, Surjective ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Subring.MulOpposite

Goal state before:
pp: ι : Sort u_1
R : Type u_2
inst✝ : NonAssocRing R
S : Subring Rᵐᵒᵖ
⊢ ∀ {x : R}, x ∈ S.unop.carrier → -x ∈ S.unop.carrier

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ FiniteType R S ↔ ∃ s f, Surjective ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Subring.MulOpposite

Goal state before:
pp: R : Type u_2
inst✝ : NonAssocRing R
s : Set R
⊢ (closure s).op = closure (MulOpposite.unop ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Subring.MulOpposite

Goal state before:
pp: R : Type u_2
inst✝ : NonAssocRing R
s : Set Rᵐᵒᵖ
⊢ (closure s).unop = closure (MulOpposite.op ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Algebra.Subalgebra.MulOpposite

Goal state before:
pp: R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ (Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s)

Tactic:
apply toSubsemiring_injective
  simp_rw [Algebra.adjoin, op_toSubsemiring, Subsemiring.op_closure, Set.preimage_union]
  congr with x
  simp_rw [Set.mem_preimage, Set.mem_range, MulOpposite.algebraMap_apply]
  congr!
  rw [← MulOpposite.op_injective.eq_iff (b := x.unop), MulOpposite.op_unop]

Goal state after:
Module: Mathlib.Algebra.Algebra.Subalgebra.MulOpposite

Goal state before:
pp: R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ (Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Algebra.Subalgebra.MulOpposite

Goal state before:
pp: R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set Aᵐᵒᵖ
⊢ (Algebra.adjoin R s).unop = Algebra.adjoin R (MulOpposite.op ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ FiniteType R A ↔ ∃ ι x f, Surjective ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ FiniteType R S ↔ ∃ ι x f, Surjective ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
ι : Type uS
hfintype : Fintype ι
f : MvPolynomial ι R →ₐ[R] S
hsur : Surjective ⇑f
equiv : MvPolynomial ι R ≃ₐ[R] MvPolynomial (Fin (Fintype.card ι)) R
⊢ Surjective ⇑(f.comp ↑equiv.symm)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ FiniteType R S ↔ ∃ n f, Surjective ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
hA : FiniteType R A
hB : FiniteType R B
⊢ ⊤.FG

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

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
h : FiniteType R S
inst✝ : IsNoetherianRing R
s : Finset S
hs : adjoin R ↑s = ⊤
⊢ Surjective ⇑(MvPolynomial.aeval Subtype.val).toRingHom

Tactic:
← Set.range_eq_univ,

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
h : FiniteType R S
inst✝ : IsNoetherianRing R
s : Finset S
hs : adjoin R ↑s = ⊤
⊢ Set.range ⇑(MvPolynomial.aeval Subtype.val).toRingHom = Set.univ
Module: Mathlib.RingTheory.FiniteType

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
h : FiniteType R S
inst✝ : IsNoetherianRing R
⊢ IsNoetherianRing S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
inst✝² : CommSemiring R
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
t : Set A
h : t.Finite
⊢ adjoin R ↑h.toFinset = adjoin R t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type uR
inst✝² : CommSemiring R
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
t : Set A
h : t.Finite
⊢ FiniteType R ↥(adjoin R t)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : Algebra A B
⊢ (algebraMap A B).FiniteType ↔ Algebra.FiniteType A B

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: A : Type u_1
inst✝ : CommRing A
⊢ (RingHom.id A).FiniteType

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : CommRing C
f : A →+* B
g : B →+* C
hf : f.FiniteType
hg : Surjective ⇑g
⊢ (g.comp f).FiniteType

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
hf : Surjective ⇑f
⊢ f.FiniteType

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R ↑s = I.degreeLE ↑N
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE ↑N → x ∈ Ideal.span ↑s
this : Submodule.span R[X] ↑s = Ideal.span ↑s
k : ℕ
ih : ∀ m < k, ∀ ⦃p : R[X]⦄, p ∈ I → p.natDegree = m → p ∈ Ideal.span ↑s
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
⊢ p ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : CommRing C
g : B →+* C
f : A →+* B
hg : g.FiniteType
hf : f.FiniteType
⊢ (g.comp f).FiniteType

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : CommRing C
f : A →+* B
g : B →+* C
h : (g.comp f).FiniteType
⊢ g.FiniteType

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : CommRing C
f : A →+* B
g : B →+* C
h : (g.comp f).FiniteType
⊢ g.FiniteType

Tactic:
by
  algebraize [f, g, g.comp f]
  exact Algebra.FiniteType.of_restrictScalars_finiteType A B C

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
inst✝ : AddMonoid M
S : Set R[M]
hS : adjoin R S = ⊤
⊢ adjoin R (⋃ f ∈ S, of' R M '' ↑f.coeff.support) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
inst✝ : AddMonoid M
S : Set R[M]
hS : adjoin R S = ⊤
this : of' R M '' ⋃ f ∈ S, ↑f.coeff.support = ⋃ f ∈ S, of' R M '' ↑f.coeff.support
⊢ adjoin R (of' R M '' ⋃ f ∈ S, ↑f.coeff.support) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
inst✝ : AddMonoid M
S : Set R[M]
hS : adjoin R S = ⊤
⊢ adjoin R (of' R M '' ⋃ f ∈ S, ↑f.coeff.support) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommRing R
inst✝ : AddMonoid M
S : Finset R[M]
hS : adjoin R ↑S = ⊤
this : DecidableEq M := Classical.decEq M
⊢ ↑(S.biUnion fun f ↦ f.coeff.support) = ⋃ f ∈ S, ↑f.coeff.support

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommRing R
inst✝ : AddMonoid M
h : FiniteType R R[M]
⊢ ∃ G, adjoin R (of' R M '' ↑G) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R ↑s = I.degreeLE ↑N
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE ↑N → x ∈ Ideal.span ↑s
this : Submodule.span R[X] ↑s = Ideal.span ↑s
k : ℕ
ih : ∀ m < k, ∀ ⦃p : R[X]⦄, p ∈ I → p.natDegree = m → p ∈ Ideal.span ↑s
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
⊢ 0 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R ↑s = I.degreeLE ↑N
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE ↑N → x ∈ Ideal.span ↑s
this✝¹ : Submodule.span R[X] ↑s = Ideal.span ↑s
k : ℕ
ih : ∀ m < k, ∀ ⦃p : R[X]⦄, p ∈ I → p.natDegree = m → p ∈ Ideal.span ↑s
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this✝ : 0 ≠ 1
this : Nontrivial R
⊢ p.leadingCoeff ∈ I.leadingCoeffNth N

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
m : M
this : m ∈ closure S
m₁ m₂ : M
hx✝ : m₁ ∈ closure S
hy✝ : m₂ ∈ closure S
P₁ : MvPolynomial (↑S) R
hP₁ : (MvPolynomial.aeval fun s ↦ of' R M ↑s) P₁ = (of R M) (Multiplicative.ofAdd m₁)
P₂ : MvPolynomial (↑S) R
hP₂ : (MvPolynomial.aeval fun s ↦ of' R M ↑s) P₂ = (of R M) (Multiplicative.ofAdd m₂)
⊢ (MvPolynomial.aeval fun s ↦ of' R M ↑s) (P₁ * P₂) = (of R M) (Multiplicative.ofAdd (m₁ + m₂))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
⊢ Surjective ⇑(MvPolynomial.aeval fun s ↦ of' R M ↑s)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : AddMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
m : M
this : m ∈ closure S
m₁ m₂ : M
hx✝ : m₁ ∈ closure S
hy✝ : m₂ ∈ closure S
P₁ : FreeAlgebra R ↑S
hP₁ : ((FreeAlgebra.lift R) fun s ↦ of' R M ↑s) P₁ = (of R M) (Multiplicative.ofAdd m₁)
P₂ : FreeAlgebra R ↑S
hP₂ : ((FreeAlgebra.lift R) fun s ↦ of' R M ↑s) P₂ = (of R M) (Multiplicative.ofAdd m₂)
⊢ ((FreeAlgebra.lift R) fun s ↦ of' R M ↑s) (P₁ * P₂) = (of R M) (Multiplicative.ofAdd (m₁ + m₂))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : AddMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
⊢ Surjective ⇑((FreeAlgebra.lift R) fun s ↦ of' R M ↑s)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : AddMonoid M
inst✝ : CommRing R
h : AddMonoid.FG M
⊢ FiniteType R R[M]

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : AddMonoid M
inst✝¹ : CommRing R
inst✝ : Nontrivial R
h : FiniteType R R[M]
S : Finset M
hS : adjoin R (of' R M '' ↑S) = ⊤
m : M
⊢ of' R M m ∈ Subalgebra.toSubmodule (adjoin R (of' R M '' ↑S))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R ↑s = I.degreeLE ↑N
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE ↑N → x ∈ Ideal.span ↑s
this✝¹ : Submodule.span R[X] ↑s = Ideal.span ↑s
k : ℕ
ih : ∀ m < k, ∀ ⦃p : R[X]⦄, p ∈ I → p.natDegree = m → p ∈ Ideal.span ↑s
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this✝ : 0 ≠ 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ ↑N
hlqp : q.leadingCoeff = p.leadingCoeff
⊢ q ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : AddMonoid M
inst✝¹ : CommRing R
inst✝ : Nontrivial R
⊢ FiniteType R R[M] ↔ AddMonoid.FG M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
G : Type u_3
inst✝² : AddGroup G
inst✝¹ : CommRing R
inst✝ : Nontrivial R
⊢ FiniteType R R[G] ↔ AddGroup.FG G

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Monoid M
S : Set R[M]
hS : adjoin R S = ⊤
⊢ adjoin R (⋃ f ∈ S, ⇑(of R M) '' ↑f.coeff.support) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Monoid M
S : Set R[M]
hS : adjoin R S = ⊤
this : ⇑(of R M) '' ⋃ f ∈ S, ↑f.coeff.support = ⋃ f ∈ S, ⇑(of R M) '' ↑f.coeff.support
⊢ adjoin R (⇑(of R M) '' ⋃ f ∈ S, ↑f.coeff.support) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Monoid M
S : Set R[M]
hS : adjoin R S = ⊤
⊢ adjoin R (⇑(of R M) '' ⋃ f ∈ S, ↑f.coeff.support) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommRing R
inst✝ : Monoid M
S : Finset R[M]
hS : adjoin R ↑S = ⊤
this : DecidableEq M := Classical.decEq M
⊢ ↑(S.biUnion fun f ↦ f.coeff.support) = ⋃ f ∈ S, ↑f.coeff.support

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommRing R
inst✝ : Monoid M
h : FiniteType R R[M]
⊢ ∃ G, adjoin R (⇑(of R M) '' ↑G) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R ↑s = I.degreeLE ↑N
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE ↑N → x ∈ Ideal.span ↑s
this✝¹ : Submodule.span R[X] ↑s = Ideal.span ↑s
k : ℕ
ih : ∀ m < k, ∀ ⦃p : R[X]⦄, p ∈ I → p.natDegree = m → p ∈ Ideal.span ↑s
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this✝ : 0 ≠ 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ ↑N
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
⊢ p.degree = (q * X ^ (k - q.natDegree)).degree

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
m : M
this : m ∈ closure S
m₁ m₂ : M
hx✝ : m₁ ∈ closure S
hy✝ : m₂ ∈ closure S
P₁ : MvPolynomial (↑S) R
hP₁ : (MvPolynomial.aeval fun s ↦ (of R M) ↑s) P₁ = (of R M) m₁
P₂ : MvPolynomial (↑S) R
hP₂ : (MvPolynomial.aeval fun s ↦ (of R M) ↑s) P₂ = (of R M) m₂
⊢ (MvPolynomial.aeval fun s ↦ (of R M) ↑s) (P₁ * P₂) = (of R M) (m₁ * m₂)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FiniteType

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
⊢ Surjective ⇑(MvPolynomial.aeval fun s ↦ (of R M) ↑s)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R ↑s = I.degreeLE ↑N
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE ↑N → x ∈ Ideal.span ↑s
this✝¹ : Submodule.span R[X] ↑s = Ideal.span ↑s
k : ℕ
ih : ∀ m < k, ∀ ⦃p : R[X]⦄, p ∈ I → p.natDegree = m → p ∈ Ideal.span ↑s
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this✝ : 0 ≠ 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ ↑N
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
h1 : p.degree = (q * X ^ (k - q.natDegree)).degree
⊢ p.leadingCoeff = (q * X ^ (k - q.natDegree)).leadingCoeff

Tactic:
by

Goal state after: