awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Localization
Goal:
R : Type u_1
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsNoetherianRing R
inst_4 : Module.Finite R M
p : Ideal R
hp : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (annihilator R M)) p
prime : p.IsPrime
Rₚ : Type u_1 := Localization.AtPrime p
this : Nontrivial (LocalizedModule p.primeCompl M)
q : Ideal Rₚ
hq : Membership.mem (γ := Set (Ideal Rₚ)) (associatedPrimes Rₚ (LocalizedModule p.primeCompl M)) q
q_prime : q.IsPrime
⊢ Membership.mem (γ := Set (Ideal R)) (associatedPrimes R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)

Tactic:
simp only [← preimage_comap_associatedPrimes_eq_associatedPrimes_of_isLocalizedModule p.primeCompl
    Rₚ (LocalizedModule.mkLinearMap p.primeCompl M), Set.mem_preimage] at hq

Premises used: Module.associatedPrimes.preimage_comap_associatedPrimes_eq_associatedPrimes_of_isLocalizedModule, LocalizedModule.mkLinearMap, Ideal.primeCompl, Set.mem_preimage
-- Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Localization
Goal:
R : Type u_1
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsNoetherianRing R
inst_4 : Module.Finite R M
p : Ideal R
hp : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (annihilator R M)) p
prime : p.IsPrime
Rₚ : Type u_1 := Localization.AtPrime p
this : Nontrivial (LocalizedModule p.primeCompl M)
q : Ideal Rₚ
q_prime : q.IsPrime
hq : Membership.mem (γ := Set (Ideal R)) (associatedPrimes R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
⊢ LE.le (α := Ideal R) (annihilator R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)

Tactic:
have ann_le : Module.annihilator R M ≤ Ideal.comap (algebraMap R Rₚ) q :=
    le_of_eq_of_le Submodule.annihilator_top.symm (IsAssociatedPrime.annihilator_le hq)
-- Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Localization
Goal:
R : Type u_1
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsNoetherianRing R
inst_4 : Module.Finite R M
p : Ideal R
hp : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (annihilator R M)) p
prime : p.IsPrime
Rₚ : Type u_1 := Localization.AtPrime p
this : Nontrivial (LocalizedModule p.primeCompl M)
q : Ideal Rₚ
q_prime : q.IsPrime
hq : Membership.mem (γ := Set (Ideal R)) (associatedPrimes R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
ann_le : LE.le (α := Ideal R) (annihilator R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
⊢ Disjoint (↑p.primeCompl : Set R) (↑(Ideal.under R q) : Set R)

Tactic:
have := (IsLocalization.disjoint_under_iff p.primeCompl Rₚ q).mpr q_prime.ne_top
-- Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Localization
Goal:
R : Type u_1
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsNoetherianRing R
inst_4 : Module.Finite R M
p : Ideal R
hp : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (annihilator R M)) p
prime : p.IsPrime
Rₚ : Type u_1 := Localization.AtPrime p
this_1 : Nontrivial (LocalizedModule p.primeCompl M)
q : Ideal Rₚ
q_prime : q.IsPrime
hq : Membership.mem (γ := Set (Ideal R)) (associatedPrimes R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
ann_le : LE.le (α := Ideal R) (annihilator R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
this : Disjoint (↑p.primeCompl : Set R) (↑(Ideal.under R q) : Set R)
⊢ Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q ≤ p

Tactic:
simpa only [Ideal.primeCompl, Submonoid.coe_set_mk, Subsemigroup.coe_set_mk,
      Set.disjoint_compl_left_iff_subset] using this

Premises used: Set.disjoint_compl_left_iff_subset, Subsemigroup.coe_set_mk, Ideal.primeCompl, Submonoid.coe_set_mk
-- Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Localization
Goal:
R : Type u_1
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsNoetherianRing R
inst_4 : Module.Finite R M
p : Ideal R
hp : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (annihilator R M)) p
prime : p.IsPrime
Rₚ : Type u_1 := Localization.AtPrime p
this : Nontrivial (LocalizedModule p.primeCompl M)
q : Ideal Rₚ
q_prime : q.IsPrime
hq : Membership.mem (γ := Set (Ideal R)) (associatedPrimes R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
ann_le : LE.le (α := Ideal R) (annihilator R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
⊢ Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q ≤ p

Tactic:
have le : Ideal.comap (algebraMap R Rₚ) q ≤ p := by
    have := (IsLocalization.disjoint_under_iff p.primeCompl Rₚ q).mpr q_prime.ne_top
    simpa only [Ideal.primeCompl, Submonoid.coe_set_mk, Subsemigroup.coe_set_mk,
      Set.disjoint_compl_left_iff_subset] using this
-- Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Localization
Goal:
R : Type u_1
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsNoetherianRing R
inst_4 : Module.Finite R M
p : Ideal R
hp : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (annihilator R M)) p
prime : p.IsPrime
Rₚ : Type u_1 := Localization.AtPrime p
this : Nontrivial (LocalizedModule p.primeCompl M)
q : Ideal Rₚ
q_prime : q.IsPrime
hq : Membership.mem (γ := Set (Ideal R)) (associatedPrimes R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
ann_le : LE.le (α := Ideal R) (annihilator R M) (Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q)
le : Ideal.comap (F := R →+* Rₚ) (algebraMap R Rₚ) q ≤ p
⊢ Membership.mem (γ := Set (Ideal R)) (associatedPrimes R M) p

Tactic:
simpa [Minimal.eq_of_le hp.out ⟨IsAssociatedPrime.isPrime hq, ann_le⟩ le] using hq

Premises used: Membership.mem.out, IsAssociatedPrime.isPrime, Minimal.eq_of_le, And.intro
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
I : (i : ι) → Ideal (R i)
r : (i : ι) → R i
⊢ r ∈ pi (R := R) I ↔ ∀ (i : ι), r i ∈ I i

Tactic:
Iff.rfl

Premises used: Set.Mem, Iff.rfl, setOf
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
r : (i : ι) → R i
h :
  ∀ (x : (i : ι) → R i),
    Membership.mem (γ := Ideal ((i : ι) → R i)) (pi (R := R) fun (x : ι) ↦ span {r x}) x ↔
      Membership.mem (γ := Ideal ((i : ι) → R i)) (span {r}) x
⊢ Eq (α := Ideal ((i : ι) → R i)) (pi (R := R) fun (x : ι) ↦ span {r x}) (span {r})

Tactic:
ext
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
r x : (i : ι) → R i
h :
  (∀ (i : ι), Membership.mem (γ := Ideal (R i)) (span {r i}) (x i)) ↔
    Membership.mem (γ := Ideal ((i : ι) → R i)) (span {r}) x
h₁ : (∀ (i : ι), ∃ (a : R i), a * r i = x i) ↔ ∃ (a : (i : ι) → R i), a * r = x
h₂ : (∀ (i : ι), ∃ (a : R i), a * r i = x i) ↔ ∃ (a : (i : ι) → R i), ∀ (x_1 : ι), (a * r) x_1 = x x_1
h₃ :
  (∃ (f : (x : ι) → R x), ∀ (x_1 : ι), f x_1 * r x_1 = x x_1) ↔ ∃ (a : (i : ι) → R i), ∀ (x_1 : ι), (a * r) x_1 = x x_1
⊢ Membership.mem (γ := Ideal ((i : ι) → R i)) (pi (R := R) fun (x : ι) ↦ span {r x}) x ↔
    Membership.mem (γ := Ideal ((i : ι) → R i)) (span {r}) x

Tactic:
simp_rw [mem_pi, mem_span_singleton', funext_iff, Classical.skolem, Pi.mul_def]

Premises used: Classical.skolem, Ideal.mem_span_singleton', funext_iff, Pi.mul_def, Ideal.mem_pi
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
α : Type u_2
β : Type u_3
F : Type u_4
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
I J : (i : ι) → Ideal (R i)
inst_1 : ∀ (i : ι), (I i).IsTwoSided
⊢ (pi (R := R) I).IsTwoSided

Tactic:
⟨fun _b hb i ↦ mul_mem_right _ _ (hb i)⟩

Premises used: Ideal.mul_mem_right, Ideal.IsTwoSided.mk
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
I : (i : ι) → Ideal (R i)
inst_1 : DecidableEq ι
i : ι
r : R i
hr : r ∈ I i
j : ι
inl : Pi.single (M := R) i r i ∈ I i
inr : ∀ (ne : i ≠ j), Pi.single (M := R) i r j ∈ I j
⊢ Pi.single (M := R) i r j ∈ I j

Tactic:
obtain rfl | ne := eq_or_ne i j

Premises used: eq_or_ne, rfl
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
I : (i : ι) → Ideal (R i)
inst_1 : DecidableEq ι
i : ι
r : R i
hr : r ∈ I i
⊢ Pi.single (M := R) i r i ∈ I i

Tactic:
simpa
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
I : (i : ι) → Ideal (R i)
inst_1 : DecidableEq ι
i : ι
r : R i
hr : r ∈ I i
j : ι
ne : i ≠ j
⊢ Pi.single (M := R) i r j ∈ I j

Tactic:
simp [ne]
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
I J : (i : ι) → Ideal (R i)
le : pi (R := R) I ≤ pi (R := R) J
i : ι
r : R i
hr : r ∈ I i
⊢ r ∈ J i

Tactic:
simpa using le (single_mem_pi hr) i

Premises used: Ideal.single_mem_pi
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
ι : Type u_1
R : ι → Type u_5
inst : (i : ι) → Semiring (R i)
I J : (i : ι) → Ideal (R i)
le : pi (R := R) I ≤ pi (R := R) J
i : ι
r : R i
hr : r ∈ I i
⊢ r ∈ J i

Tactic:
classical simpa using le (single_mem_pi hr) i
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
simp_rw : ∑ x ∈ Finset.range (k + 1), (↑(k.choose x) : α) * (a ^ x * b ^ (k - x)) ∈ I
⊢ (a + b) ^ k ∈ I

Tactic:
simp_rw [hab.add_pow, ← Nat.cast_comm]

Premises used: Nat.cast_comm, Commute.add_pow
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
apply : ∀ c ∈ Finset.range (k + 1), (↑(k.choose c) : α) * (a ^ c * b ^ (k - c)) ∈ I
⊢ ∑ x ∈ Finset.range (k + 1), (↑(k.choose x) : α) * (a ^ x * b ^ (k - x)) ∈ I

Tactic:
apply I.sum_mem

Premises used: Submodule.sum_mem
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
_ : c ∈ Finset.range (k + 1) → (↑(k.choose c) : α) * (a ^ c * b ^ (k - c)) ∈ I
⊢ c ∈ Finset.range (k + 1) → (↑(k.choose c) : α) * (a ^ c * b ^ (k - c)) ∈ I

Tactic:
_
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
h : c ∈ Finset.range (k + 1)
a₁ : a ^ c * b ^ (k - c) ∈ I
⊢ (↑(k.choose c) : α) * (a ^ c * b ^ (k - c)) ∈ I

Tactic:
apply mul_mem_left

Premises used: Ideal.mul_mem_left
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
h : c ∈ Finset.range (k + 1)
pos : ∀ (h : m ≤ c), a ^ c * b ^ (k - c) ∈ I
neg : ∀ (h : ¬m ≤ c), a ^ c * b ^ (k - c) ∈ I
⊢ a ^ c * b ^ (k - c) ∈ I

Tactic:
by_cases h : m ≤ c
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
h_1 : c ∈ Finset.range (k + 1)
h : m ≤ c
pos : b ^ (k - c) * a ^ c ∈ I
⊢ a ^ c * b ^ (k - c) ∈ I

Tactic:
rw [hab.pow_pow]

Premises used: Commute.pow_pow
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
h_1 : c ∈ Finset.range (k + 1)
h : m ≤ c
⊢ b ^ (k - c) * a ^ c ∈ I

Tactic:
I.mul_mem_left _ (I.pow_mem_of_pow_mem ha h)

Premises used: Set.Mem, Ideal.pow_mem_of_pow_mem, Ideal.mul_mem_left
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
h_1 : c ∈ Finset.range (k + 1)
h : m ≤ c
⊢ b ^ (k - c) * a ^ c ∈ I

Tactic:
exact I.mul_mem_left _ (I.pow_mem_of_pow_mem ha h)

Premises used: Ideal.pow_mem_of_pow_mem, Ideal.mul_mem_left
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
h_1 : c ∈ Finset.range (k + 1)
h : ¬m ≤ c
⊢ a ^ c * b ^ (k - c) ∈ I

Tactic:
I.mul_mem_left _ (I.pow_mem_of_pow_mem hb ?_)

Premises used: Ideal.pow_mem_of_pow_mem, Ideal.mul_mem_left
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
h_1 : c ∈ Finset.range (k + 1)
h : ¬m ≤ c
neg : n ≤ k - c
⊢ a ^ c * b ^ (k - c) ∈ I

Tactic:
refine I.mul_mem_left _ (I.pow_mem_of_pow_mem hb ?_)

Premises used: Ideal.pow_mem_of_pow_mem, Ideal.mul_mem_left
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
hab : Commute a b
c : ℕ
h_1 : c ∈ Finset.range (k + 1)
h : ¬m ≤ c
⊢ n ≤ k - c

Tactic:
lia
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_6
inst : Semiring α
I : Ideal α
a b : α
m n : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hab : Commute a b
rw rw₁ : m + n - 1 ≤ m + n - 1
⊢ m + n ≤ m + n - 1 + 1

Tactic:
rw [← Nat.sub_le_iff_le_add]

Premises used: Nat.sub_le_iff_le_add
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
a b : α
inst : CommSemiring α
I : Ideal α
m n k : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
hk : m + n ≤ k + 1
⊢ (a + b) ^ k ∈ I

Tactic:
I.add_pow_mem_of_pow_mem_of_le_of_commute ha hb hk (Commute.all ..)

Premises used: Ideal.add_pow_mem_of_pow_mem_of_le_of_commute, Commute.all
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
a b : α
inst : CommSemiring α
I : Ideal α
m n : ℕ
ha : a ^ m ∈ I
hb : b ^ n ∈ I
⊢ (a + b) ^ (m + n - 1) ∈ I

Tactic:
I.add_pow_add_pred_mem_of_pow_mem_of_commute ha hb (Commute.all ..)

Premises used: Ideal.add_pow_add_pred_mem_of_pow_mem_of_commute, Commute.all
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
⊢ Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) 0)) : Set α))
    (HPow.hPow (α := α) (Multiset.sum 0) (Multiset.card.{u_2} (α := α) 0 * n + 1))

Tactic:
simp
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
⊢ Membership.mem (γ := Ideal α)
    (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) (a ::ₘ s))) : Set α))
    ((a ::ₘ s).sum ^ ((a ::ₘ s).card * n + 1))

Tactic:
?_
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
s : Multiset α
n : ℕ
cons :
  ∀ (a : α) (s : Multiset α)
    (hs :
      Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
        (s.sum ^ (s.card * n + 1))),
    Membership.mem (γ := Ideal α)
      (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) (a ::ₘ s))) : Set α))
      ((a ::ₘ s).sum ^ ((a ::ₘ s).card * n + 1))
⊢ Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))

Tactic:
induction s using Multiset.induction_on with
  | empty => simp
  | cons a s hs => ?_
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
cons :
  Membership.mem (γ := Ideal α)
    (span (insert (a ^ (n + 1)) (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)))
    (∑ m ∈ Finset.range ((s.card + 1) * n + 1 + 1),
      a ^ m * s.sum ^ ((s.card + 1) * n + 1 - m) * (↑(((s.card + 1) * n + 1).choose m) : α))
⊢ Membership.mem (γ := Ideal α)
    (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) (a ::ₘ s))) : Set α))
    ((a ::ₘ s).sum ^ ((a ::ₘ s).card * n + 1))

Tactic:
simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]

Premises used: Multiset.toFinset_cons, Multiset.map_cons, Multiset.card_cons, add_pow, Multiset.sum_cons, Finset.coe_insert
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
⊢ Membership.mem (γ := Ideal α)
    (span (insert (a ^ (n + 1)) (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)))
    (∑ m ∈ Finset.range ((s.card + 1) * n + 1 + 1),
      a ^ m * s.sum ^ ((s.card + 1) * n + 1 - m) * (↑(((s.card + 1) * n + 1).choose m) : α))

Tactic:
Submodule.sum_mem _ ?_

Premises used: Set.Mem, Submodule.sum_mem
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
cons :
  ∀ c ∈ Finset.range ((s.card + 1) * n + 1 + 1),
    Membership.mem (γ := Submodule α α)
      (span (insert (a ^ (n + 1)) (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)))
      (a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α))
⊢ Membership.mem (γ := Ideal α)
    (span (insert (a ^ (n + 1)) (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)))
    (∑ m ∈ Finset.range ((s.card + 1) * n + 1 + 1),
      a ^ m * s.sum ^ ((s.card + 1) * n + 1 - m) * (↑(((s.card + 1) * n + 1).choose m) : α))

Tactic:
refine Submodule.sum_mem _ ?_

Premises used: Submodule.sum_mem
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
c : ℕ
_hc : c ∈ Finset.range ((s.card + 1) * n + 1 + 1)
cons :
  ∃ (a_1 : α) (z : α),
    Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)) z ∧
      a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) = a_1 * a ^ (n + 1) + z
⊢ Membership.mem (γ := Submodule α α)
    (span (insert (a ^ (n + 1)) (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)))
    (a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α))

Tactic:
rw [mem_span_insert]

Premises used: Ideal.mem_span_insert
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
c : ℕ
_hc : c ∈ Finset.range ((s.card + 1) * n + 1 + 1)
pos :
  ∀ (h : n + 1 ≤ c),
    ∃ (a_1 : α) (z : α),
      Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
          z ∧
        a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) = a_1 * a ^ (n + 1) + z
neg :
  ∀ (h : c < n + 1),
    ∃ (a_1 : α) (z : α),
      Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
          z ∧
        a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) = a_1 * a ^ (n + 1) + z
⊢ ∃ (a_1 : α) (z : α),
    Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)) z ∧
      a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) = a_1 * a ^ (n + 1) + z

Tactic:
by_cases! h : n + 1 ≤ c
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
c : ℕ
_hc : c ∈ Finset.range ((s.card + 1) * n + 1 + 1)
h : n + 1 ≤ c
⊢ ∃ (a_1 : α) (z : α),
    Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)) z ∧
      a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) = a_1 * a ^ (n + 1) + z

Tactic:
⟨a ^ (c - (n + 1)) * s.sum ^ ((Multiset.card s + 1) * n + 1 - c) *
      ((Multiset.card s + 1) * n + 1).choose c, 0, Submodule.zero_mem _, ?_⟩

Premises used: Multiset.sum, Nat.choose, HMul.hMul, HSub.hSub, Nat.cast, HPow.hPow, HAdd.hAdd, Submodule.zero_mem, And.intro, Exists.intro, Multiset.card, OfNat.ofNat
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
c : ℕ
_hc : c ∈ Finset.range ((s.card + 1) * n + 1 + 1)
h : n + 1 ≤ c
pos :
  a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) =
    a ^ (c - (n + 1)) * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) * a ^ (n + 1) + 0
⊢ ∃ (a_1 : α) (z : α),
    Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α)) z ∧
      a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) = a_1 * a ^ (n + 1) + z

Tactic:
refine ⟨a ^ (c - (n + 1)) * s.sum ^ ((Multiset.card s + 1) * n + 1 - c) *
      ((Multiset.card s + 1) * n + 1).choose c, 0, Submodule.zero_mem _, ?_⟩
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
c : ℕ
_hc : c ∈ Finset.range ((s.card + 1) * n + 1 + 1)
h : n + 1 ≤ c
pos :
  a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) =
    a ^ (n + 1) * (a ^ (c - (n + 1)) * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α)) +
      0
⊢ a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) =
    a ^ (c - (n + 1)) * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) * a ^ (n + 1) + 0

Tactic:
rw [mul_comm _ (a ^ (n + 1))]

Premises used: HPow.hPow, HAdd.hAdd, mul_comm, OfNat.ofNat
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
inst_1 : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (s.sum ^ (s.card * n + 1))
c : ℕ
_hc : c ∈ Finset.range ((s.card + 1) * n + 1 + 1)
h : n + 1 ≤ c
pos :
  a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) =
    a ^ (n + 1) * a ^ (c - (n + 1)) * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) + 0
⊢ a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) =
    a ^ (n + 1) * (a ^ (c - (n + 1)) * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α)) +
      0

Tactic:
simp_rw [← mul_assoc]

Premises used: mul_assoc