l3lab/ntp-mathlib huggingface dataset:
theorem span_singleton_eq_span_singleton {α : Type u} [CommRing α] [IsDomain α] {x y : α} :
    span ({x} : Set α) = span ({y} : Set α) ↔ Associated x y := by
  rw [← dvd_dvd_iff_associated, le_antisymm_iff, and_comm]
  apply and_congr <;> 

Proof state:
case h₂
α✝ : Type u
β : Type v
a b : α✝
inst✝² : CommSemiring α✝
I : Ideal α✝
α : Type u
inst✝¹ : CommRing α
inst✝ : IsDomain α
x y : α
⊢ span {x} ≤ span {y} ↔ y ∣ x

Next tactic:
rw [span_singleton_le_span_singleton]
theorem span_singleton_mul_right_unit {a : α} (h2 : IsUnit a) (x : α) :
    span ({x * a} : Set α) = span {x} := by 

Proof state:
α : Type u
β : Type v
a✝ b : α
inst✝ : CommSemiring α
I : Ideal α
a : α
h2 : IsUnit a
x : α
⊢ span {x * a} = span {x}

Next tactic:
rw [mul_comm, span_singleton_mul_left_unit h2]
theorem span_singleton_eq_top {x} : span ({x} : Set α) = ⊤ ↔ IsUnit x := by
  

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
x : α
⊢ span {x} = ⊤ ↔ IsUnit x

Next tactic:
rw [isUnit_iff_dvd_one, ← span_singleton_le_span_singleton, span_singleton_one, eq_top_iff]
theorem span_singleton_prime {p : α} (hp : p ≠ 0) : IsPrime (span ({p} : Set α)) ↔ Prime p := by
  

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
p : α
hp : p ≠ 0
⊢ IsPrime (span {p}) ↔ Prime p

Next tactic:
simp [isPrime_iff, Prime, span_singleton_eq_top, hp, mem_span_singleton]
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
⊢ y ∈ I

Next tactic:
let J : Ideal α := Submodule.span α (insert x ↑I)
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
⊢ y ∈ I

Next tactic:
have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
      

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
IJ : I ≤ J
⊢ y ∈ I

Next tactic:
have xJ : x ∈ J := Ideal.subset_span (Set.mem_insert x I)
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
      have xJ : x ∈ J := Ideal.subset_span (Set.mem_insert x I)
      

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
IJ : I ≤ J
xJ : x ∈ J
⊢ y ∈ I

Next tactic:
cases' isMaximal_iff.1 H with _ oJ
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
      have xJ : x ∈ J := Ideal.subset_span (Set.mem_insert x I)
      cases' isMaximal_iff.1 H with _ oJ
      

Proof state:
case intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
IJ : I ≤ J
xJ : x ∈ J
left✝ : 1 ∉ I
oJ : ∀ (J : Ideal α) (x : α), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J
⊢ y ∈ I

Next tactic:
specialize oJ J x IJ hx xJ
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
      have xJ : x ∈ J := Ideal.subset_span (Set.mem_insert x I)
      cases' isMaximal_iff.1 H with _ oJ
      specialize oJ J x IJ hx xJ
      

Proof state:
case intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
IJ : I ≤ J
xJ : x ∈ J
left✝ : 1 ∉ I
oJ : 1 ∈ J
⊢ y ∈ I

Next tactic:
rcases Submodule.mem_span_insert.mp oJ with ⟨a, b, h, oe⟩
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
      have xJ : x ∈ J := Ideal.subset_span (Set.mem_insert x I)
      cases' isMaximal_iff.1 H with _ oJ
      specialize oJ J x IJ hx xJ
      rcases Submodule.mem_span_insert.mp oJ with ⟨a, b, h, oe⟩
      

Proof state:
case intro.intro.intro.intro
α : Type u
β : Type v
a✝ b✝ : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
IJ : I ≤ J
xJ : x ∈ J
left✝ : 1 ∉ I
oJ : 1 ∈ J
a b : α
h : b ∈ Submodule.span α ↑I
oe : 1 = a • x + b
⊢ y ∈ I

Next tactic:
obtain F : y * 1 = y * (a • x + b) := congr_arg (fun g : α => y * g) oe
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
      have xJ : x ∈ J := Ideal.subset_span (Set.mem_insert x I)
      cases' isMaximal_iff.1 H with _ oJ
      specialize oJ J x IJ hx xJ
      rcases Submodule.mem_span_insert.mp oJ with ⟨a, b, h, oe⟩
      obtain F : y * 1 = y * (a • x + b) := congr_arg (fun g : α => y * g) oe
      

Proof state:
case intro.intro.intro.intro
α : Type u
β : Type v
a✝ b✝ : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
IJ : I ≤ J
xJ : x ∈ J
left✝ : 1 ∉ I
oJ : 1 ∈ J
a b : α
h : b ∈ Submodule.span α ↑I
oe : 1 = a • x + b
F : y * 1 = y * (a • x + b)
⊢ y ∈ I

Next tactic:
rw [← mul_one y, F, mul_add, mul_comm, smul_eq_mul, mul_assoc]
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
      have xJ : x ∈ J := Ideal.subset_span (Set.mem_insert x I)
      cases' isMaximal_iff.1 H with _ oJ
      specialize oJ J x IJ hx xJ
      rcases Submodule.mem_span_insert.mp oJ with ⟨a, b, h, oe⟩
      obtain F : y * 1 = y * (a • x + b) := congr_arg (fun g : α => y * g) oe
      rw [← mul_one y, F, mul_add, mul_comm, smul_eq_mul, mul_assoc]
      

Proof state:
case intro.intro.intro.intro
α : Type u
β : Type v
a✝ b✝ : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
IJ : I ≤ J
xJ : x ∈ J
left✝ : 1 ∉ I
oJ : 1 ∈ J
a b : α
h : b ∈ Submodule.span α ↑I
oe : 1 = a • x + b
F : y * 1 = y * (a • x + b)
⊢ a * (x * y) + y * b ∈ I

Next tactic:
refine' Submodule.add_mem I (I.mul_mem_left a hxy) (Submodule.smul_mem I y _)
theorem IsMaximal.isPrime {I : Ideal α} (H : I.IsMaximal) : I.IsPrime :=
  ⟨H.1.1, @fun x y hxy =>
    or_iff_not_imp_left.2 fun hx => by
      let J : Ideal α := Submodule.span α (insert x ↑I)
      have IJ : I ≤ J := Set.Subset.trans (subset_insert _ _) subset_span
      have xJ : x ∈ J := Ideal.subset_span (Set.mem_insert x I)
      cases' isMaximal_iff.1 H with _ oJ
      specialize oJ J x IJ hx xJ
      rcases Submodule.mem_span_insert.mp oJ with ⟨a, b, h, oe⟩
      obtain F : y * 1 = y * (a • x + b) := congr_arg (fun g : α => y * g) oe
      rw [← mul_one y, F, mul_add, mul_comm, smul_eq_mul, mul_assoc]
      refine' Submodule.add_mem I (I.mul_mem_left a hxy) (Submodule.smul_mem I y _)
      

Proof state:
case intro.intro.intro.intro
α : Type u
β : Type v
a✝ b✝ : α
inst✝ : CommSemiring α
I✝ I : Ideal α
H : IsMaximal I
x y : α
hxy : x * y ∈ I
hx : x ∉ I
J : Ideal α := Submodule.span α (insert x ↑I)
IJ : I ≤ J
xJ : x ∈ J
left✝ : 1 ∉ I
oJ : 1 ∈ J
a b : α
h : b ∈ Submodule.span α ↑I
oe : 1 = a • x + b
F : y * 1 = y * (a • x + b)
⊢ b ∈ I

Next tactic:
rwa [Submodule.span_eq] at h
theorem span_singleton_lt_span_singleton [IsDomain α] {x y : α} :
    span ({x} : Set α) < span ({y} : Set α) ↔ DvdNotUnit y x := by
  

Proof state:
α : Type u
β : Type v
a b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : IsDomain α
x y : α
⊢ span {x} < span {y} ↔ DvdNotUnit y x

Next tactic:
rw [lt_iff_le_not_le, span_singleton_le_span_singleton, span_singleton_le_span_singleton,
    dvd_and_not_dvd_iff]
theorem factors_decreasing [IsDomain α] (b₁ b₂ : α) (h₁ : b₁ ≠ 0) (h₂ : ¬IsUnit b₂) :
    span ({b₁ * b₂} : Set α) < span {b₁} :=
  lt_of_le_not_le
    (Ideal.span_le.2 <| singleton_subset_iff.2 <| Ideal.mem_span_singleton.2 ⟨b₂, rfl⟩) fun h =>
    h₂ <| isUnit_of_dvd_one <|
        (mul_dvd_mul_iff_left h₁).1 <| by 

Proof state:
α : Type u
β : Type v
a b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : IsDomain α
b₁ b₂ : α
h₁ : b₁ ≠ 0
h₂ : ¬IsUnit b₂
h : span {b₁} ≤ span {b₁ * b₂}
⊢ b₁ * b₂ ∣ b₁ * 1

Next tactic:
rwa [mul_one, ← Ideal.span_singleton_le_span_singleton]
theorem pow_mem_of_mem (ha : a ∈ I) (n : ℕ) (hn : 0 < n) : a ^ n ∈ I :=
  Nat.casesOn n (Not.elim (by 

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
ha : a ∈ I
n : ℕ
hn : 0 < n
⊢ ¬0 < Nat.zero

Next tactic:
decide
theorem IsPrime.mul_mem_iff_mem_or_mem {I : Ideal α} (hI : I.IsPrime) :
    ∀ {x y : α}, x * y ∈ I ↔ x ∈ I ∨ y ∈ I := @fun x y =>
  ⟨hI.mem_or_mem, by
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
hI : IsPrime I
x y : α
⊢ x ∈ I ∨ y ∈ I → x * y ∈ I

Next tactic:
rintro (h | h)
theorem IsPrime.mul_mem_iff_mem_or_mem {I : Ideal α} (hI : I.IsPrime) :
    ∀ {x y : α}, x * y ∈ I ↔ x ∈ I ∨ y ∈ I := @fun x y =>
  ⟨hI.mem_or_mem, by
    rintro (h | h)
    

Proof state:
case inl
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
hI : IsPrime I
x y : α
h : x ∈ I
⊢ x * y ∈ I
case inr
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
hI : IsPrime I
x y : α
h : y ∈ I
⊢ x * y ∈ I

Next tactic:
exacts [I.mul_mem_right y h, I.mul_mem_left x h]
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  

Proof state:
α : Type u
β : Type v
a b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
s : Multiset α
n : ℕ
⊢ Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))

Next tactic:
induction' s using Multiset.induction_on with a s hs
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · 

Proof state:
case empty
α : Type u
β : Type v
a b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
⊢ Multiset.sum 0 ^ (Multiset.card 0 * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) 0))

Next tactic:
simp
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  

Proof state:
case cons
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
⊢ Multiset.sum (a ::ₘ s) ^ (Multiset.card (a ::ₘ s) * n + 1) ∈
    span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) (a ::ₘ s)))

Next tactic:
simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  

Proof state:
case cons
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
⊢ ∑ m in Finset.range ((Multiset.card s + 1) * n + 1 + 1),
      a ^ m * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - m) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) m) ∈
    span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))

Next tactic:
refine' Submodule.sum_mem _ _
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  refine' Submodule.sum_mem _ _
  

Proof state:
case cons
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
⊢ ∀ c ∈ Finset.range ((Multiset.card s + 1) * n + 1 + 1),
    a ^ c * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) ∈
      span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))

Next tactic:
intro c _hc
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  refine' Submodule.sum_mem _ _
  intro c _hc
  

Proof state:
case cons
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
c : ℕ
_hc : c ∈ Finset.range ((Multiset.card s + 1) * n + 1 + 1)
⊢ a ^ c * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) ∈
    span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))

Next tactic:
rw [mem_span_insert]
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  refine' Submodule.sum_mem _ _
  intro c _hc
  rw [mem_span_insert]
  

Proof state:
case cons
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
c : ℕ
_hc : c ∈ Finset.range ((Multiset.card s + 1) * n + 1 + 1)
⊢ ∃ a_1,
    ∃ z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)),
      a ^ c * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) =
        a_1 * a ^ (n + 1) + z

Next tactic:
by_cases h : n + 1 ≤ c
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  refine' Submodule.sum_mem _ _
  intro c _hc
  rw [mem_span_insert]
  by_cases h : n + 1 ≤ c
  · 

Proof state:
case pos
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
c : ℕ
_hc : c ∈ Finset.range ((Multiset.card s + 1) * n + 1 + 1)
h : n + 1 ≤ c
⊢ ∃ a_1,
    ∃ z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)),
      a ^ c * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) =
        a_1 * a ^ (n + 1) + z

Next 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 _, _⟩
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  refine' Submodule.sum_mem _ _
  intro c _hc
  rw [mem_span_insert]
  by_cases h : n + 1 ≤ c
  · 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 _, _⟩
    

Proof state:
case pos
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
c : ℕ
_hc : c ∈ Finset.range ((Multiset.card s + 1) * n + 1 + 1)
h : n + 1 ≤ c
⊢ a ^ c * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) =
    a ^ (c - (n + 1)) * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) *
          ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) *
        a ^ (n + 1) +
      0

Next tactic:
rw [mul_comm _ (a ^ (n + 1))]
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  refine' Submodule.sum_mem _ _
  intro c _hc
  rw [mem_span_insert]
  by_cases h : n + 1 ≤ c
  · 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 _, _⟩
    rw [mul_comm _ (a ^ (n + 1))]
    

Proof state:
case pos
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
c : ℕ
_hc : c ∈ Finset.range ((Multiset.card s + 1) * n + 1 + 1)
h : n + 1 ≤ c
⊢ a ^ c * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) =
    a ^ (n + 1) *
        (a ^ (c - (n + 1)) * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) *
          ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c)) +
      0

Next tactic:
simp_rw [← mul_assoc]
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  refine' Submodule.sum_mem _ _
  intro c _hc
  rw [mem_span_insert]
  by_cases h : n + 1 ≤ c
  · 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 _, _⟩
    rw [mul_comm _ (a ^ (n + 1))]
    simp_rw [← mul_assoc]
    

Proof state:
case pos
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
c : ℕ
_hc : c ∈ Finset.range ((Multiset.card s + 1) * n + 1 + 1)
h : n + 1 ≤ c
⊢ a ^ c * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) =
    a ^ (n + 1) * a ^ (c - (n + 1)) * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) *
        ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) +
      0

Next tactic:
rw [← pow_add, add_zero, add_tsub_cancel_of_le h]
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
    s.sum ^ (Multiset.card s * n + 1) ∈
    span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
  induction' s using Multiset.induction_on with a s hs
  · simp
  simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
    Multiset.card_cons, add_pow]
  refine' Submodule.sum_mem _ _
  intro c _hc
  rw [mem_span_insert]
  by_cases h : n + 1 ≤ c
  · 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 _, _⟩
    rw [mul_comm _ (a ^ (n + 1))]
    simp_rw [← mul_assoc]
    rw [← pow_add, add_zero, add_tsub_cancel_of_le h]
  · 

Proof state:
case neg
α : Type u
β : Type v
a✝ b : α
inst✝¹ : CommSemiring α
I : Ideal α
inst✝ : DecidableEq α
n : ℕ
a : α
s : Multiset α
hs : Multiset.sum s ^ (Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))
c : ℕ
_hc : c ∈ Finset.range ((Multiset.card s + 1) * n + 1 + 1)
h : ¬n + 1 ≤ c
⊢ ∃ a_1,
    ∃ z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)),
      a ^ c * Multiset.sum s ^ ((Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((Multiset.card s + 1) * n + 1) c) =
        a_1 * a ^ (n + 1) + z

Next tactic:
use 0