awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 + (c - (n + 1))) * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) + 0
pos₁ :
  a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) =
    a ^ (n + 1 + (c - (n + 1))) * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α)
pos₂ pos₃ :
  a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α) =
    a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α)
⊢ 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:
rw [← pow_add, add_zero, add_tsub_cancel_of_le h]

Premises used: pow_add, add_zero, add_tsub_cancel_of_le
-- 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 : c < n + 1
h₁ :
  ∃ (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) = 0 * 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:
use 0
-- 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 : c < n + 1
h₁ :
  ∃ (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) = z
⊢ ∃ (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) = 0 * a ^ (n + 1) + z

Tactic:
simp_rw [zero_mul, zero_add]

Premises used: MulZeroClass.zero_mul, zero_add
-- 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 : c < n + 1
⊢ ∃ (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) = z

Tactic:
⟨_, ?_, rfl⟩

Premises used: And.intro, Exists.intro, rfl
-- 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 : c < n + 1
h₁ :
  Membership.mem (γ := Ideal α) (span (↑(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))
⊢ ∃ (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) = z

Tactic:
refine ⟨_, ?_, rfl⟩
-- 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 : c < n + 1
⊢ c ≤ n

Tactic:
replace h : c ≤ n := Nat.lt_succ_iff.mp h
-- 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 : c ≤ n
rw : s.card * n + 1 * n + 1 - c = s.card * n + 1 + (n - c)
rw₁ : s.card * n + n + 1 - c = s.card * n + 1 + (n - c)
rw₂ : s.card * n + (n + 1) - c = s.card * n + 1 + (n - c)
rw₃ : s.card * n + (1 + n) - c = s.card * n + 1 + (n - c)
rw₄ : s.card * n + 1 + n - c = s.card * n + 1 + (n - c)
rw₅ rw₆ : s.card * n + 1 + (n - c) = s.card * n + 1 + (n - c)
⊢ (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)

Tactic:
rw [add_mul, one_mul, add_assoc, add_comm n 1, ← add_assoc, add_tsub_assoc_of_le h]

Premises used: add_mul, add_tsub_assoc_of_le, add_assoc, add_comm, OfNat.ofNat, one_mul
-- 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 : c ≤ n
⊢ (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)

Tactic:
have : (Multiset.card s + 1) * n + 1 - c = Multiset.card s * n + 1 + (n - c) := by
      rw [add_mul, one_mul, add_assoc, add_comm n 1, ← add_assoc, add_tsub_assoc_of_le h]
-- 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 : c ≤ n
this : (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
h₁ :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (a ^ c * (s.sum ^ (s.card * n + 1) * s.sum ^ (n - c)) * (↑(((s.card + 1) * n + 1).choose c) : α))
⊢ Membership.mem (γ := Ideal α) (span (↑(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 [this, pow_add]

Premises used: pow_add
-- 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 : c ≤ n
this : (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
h₁ :
  Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (a ^ c * s.sum ^ (n - c) * (↑(((s.card + 1) * n + 1).choose c) : α) * s.sum ^ (s.card * n + 1))
⊢ Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (a ^ c * (s.sum ^ (s.card * n + 1) * s.sum ^ (n - c)) * (↑(((s.card + 1) * n + 1).choose c) : α))

Tactic:
simp_rw [mul_assoc, mul_comm (s.sum ^ (Multiset.card s * n + 1)), ← mul_assoc]

Premises used: Multiset.sum, HMul.hMul, mul_assoc, HPow.hPow, HAdd.hAdd, mul_comm, 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 : c ≤ n
this : (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
⊢ Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (a ^ c * s.sum ^ (n - c) * (↑(((s.card + 1) * n + 1).choose c) : α) * s.sum ^ (s.card * n + 1))

Tactic:
mul_mem_left _ _ hs

Premises used: Set.Mem, Ideal.mul_mem_left
-- 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 : c ≤ n
this : (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
⊢ Membership.mem (γ := Ideal α) (span (↑(Multiset.toFinset (Multiset.map (fun (x : α) ↦ x ^ (n + 1)) s)) : Set α))
    (a ^ c * s.sum ^ (n - c) * (↑(((s.card + 1) * n + 1).choose c) : α) * s.sum ^ (s.card * n + 1))

Tactic:
exact mul_mem_left _ _ hs

Premises used: Ideal.mul_mem_left
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
ι : Type u_5
s : Finset ι
f : ι → α
n : ℕ
⊢ Membership.mem (γ := Ideal α) (span ((fun (i : ι) ↦ f i ^ (n + 1)) '' (↑s : Set ι)))
    (HPow.hPow (α := α) (∑ i ∈ s, f i) (s.card * n + 1))

Tactic:
simpa only [Multiset.card_map, Multiset.map_map, comp_apply, Multiset.toFinset_map,
    Finset.coe_image, Finset.val_toFinset] using pow_multiset_sum_mem_span_pow (s.1.map f) n

Premises used: Ideal.pow_multiset_sum_mem_span_pow, Finset.val_toFinset, Multiset.map, Finset.coe_image, Finset.val, Multiset.map_map, Function.comp_apply, Multiset.toFinset_map, Multiset.card_map
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
ι : Type u_5
s : Finset ι
f : ι → α
n : ℕ
⊢ Membership.mem (γ := Ideal α) (span ((fun (i : ι) ↦ f i ^ (n + 1)) '' (↑s : Set ι)))
    (HPow.hPow (α := α) (∑ i ∈ s, f i) (s.card * n + 1))

Tactic:
classical
  simpa only [Multiset.card_map, Multiset.map_map, comp_apply, Multiset.toFinset_map,
    Finset.coe_image, Finset.val_toFinset] using pow_multiset_sum_mem_span_pow (s.1.map f) n

Premises used: Finset.val_toFinset, Finset.coe_image, Multiset.map_map, Function.comp_apply, Multiset.toFinset_map, Multiset.card_map
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
n : ℕ
rw : Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ n) '' s)) 1
⊢ Eq (α := Ideal α) (span ((fun (x : α) ↦ x ^ n) '' s)) ⊤

Tactic:
rw [eq_top_iff_one]

Premises used: Ideal.eq_top_iff_one
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
n : ℕ
zero : Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' s)) 1
succ : ∀ (n : ℕ), Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ n) '' s)) 1

Tactic:
rcases n with - | n
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
zero :
  ∀ (hs : Eq.{u_2 + 1} (α := Ideal α) (span ∅) ⊤),
    Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' ∅)) 1
zero₁ : ∀ (x : α) (hx : x ∈ s), Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' s)) 1
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' s)) 1

Tactic:
obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s

Premises used: Set.eq_empty_or_nonempty, rfl
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
hs : Eq.{u_2 + 1} (α := Ideal α) (span ∅) ⊤
zero : Membership.mem (γ := Ideal α) ⊤ 1
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' ∅)) 1

Tactic:
rw [Set.image_empty, hs]

Premises used: Set.image_empty
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
hs : Eq.{u_2 + 1} (α := Ideal α) (span ∅) ⊤
⊢ Membership.mem (γ := Ideal α) ⊤ 1

Tactic:
trivial
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
x : α
hx : x ∈ s
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' s)) 1

Tactic:
subset_span ⟨_, hx, pow_zero _⟩

Premises used: Ideal.subset_span, pow_zero, And.intro, Exists.intro
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
x : α
hx : x ∈ s
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' s)) 1

Tactic:
exact subset_span ⟨_, hx, pow_zero _⟩
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
n : ℕ
succ :
  ∀
    (hs :
      ∃ (l : (↑s : Type u_2) →₀ α),
        Eq (α := α)
          ((Finsupp.linearCombination α (Subtype.val (p := fun (x : α) ↦ x ∈ s)) : (a : (↑s : Type u_2) →₀ α) → α) l :
            α)
          (1 : α))
    (n : ℕ), Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1

Tactic:
rw [eq_top_iff_one, span, Finsupp.mem_span_iff_linearCombination] at hs

Premises used: Finsupp.mem_span_iff_linearCombination, Ideal.eq_top_iff_one, Ideal.span
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs :
  ∃ (l : (↑s : Type u_2) →₀ α),
    Eq (α := α)
      ((Finsupp.linearCombination α (Subtype.val (p := fun (x : α) ↦ x ∈ s)) : (a : (↑s : Type u_2) →₀ α) → α) l : α)
      (1 : α)
n : ℕ
succ :
  ∀ (f : (↑s : Type u_2) →₀ α)
    (hf :
      Eq (α := α)
        ((Finsupp.linearCombination α (Subtype.val (p := fun (x : α) ↦ x ∈ s)) : (a : (↑s : Type u_2) →₀ α) → α) f : α)
        (1 : α)),
    Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1

Tactic:
rcases hs with ⟨f, hf⟩
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf :
  Eq (α := α)
    ((Finsupp.linearCombination α (Subtype.val (p := fun (x : α) ↦ x ∈ s)) : (a : (↑s : Type u_2) →₀ α) → α) f : α)
    (1 : α)
⊢ Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1

Tactic:
simp only [Finsupp.linearCombination, Finsupp.coe_lsum, Finsupp.sum, LinearMap.coe_smulRight,
    LinearMap.id_coe, id_eq, smul_eq_mul] at hf

Premises used: LinearMap.id_coe, Finsupp.sum, Finsupp.coe_lsum, LinearMap.coe_smulRight, smul_eq_mul, Finsupp.linearCombination, id_eq
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
⊢ Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    (HPow.hPow (α := α) (∑ i ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α))
      (f.support.card * n + 1))

Tactic:
have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    (HPow.hPow (α := α) (∑ i ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α))
      (f.support.card * n + 1))
⊢ Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1

Tactic:
rw [hf, one_pow] at this

Premises used: one_pow
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1

Tactic:
span_le.mpr ?_ this

Premises used: Ideal.span_le, Iff.mpr, Set.Mem
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
succ :
  (fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
      (↑f.support : Set (↑s : Type u_2)) ⊆
    (↑(span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) : Set α)
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1

Tactic:
refine span_le.mpr ?_ this

Premises used: Ideal.span_le, Iff.mpr
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
a : α
hx :
  Membership.mem (γ := Set α)
    ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
      (↑f.support : Set (↑s : Type u_2)))
    a
⊢ ∃ x ∈ (↑f.support : Set (↑s : Type u_2)),
    HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α) ^ (n + 1) = a

Tactic:
simp_rw [Set.mem_image] at hx

Premises used: Set.mem_image
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
a : α
hx :
  ∃ x ∈ (↑f.support : Set (↑s : Type u_2)),
    HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α) ^ (n + 1) = a
succ :
  ∀ x ∈ (↑f.support : Set (↑s : Type u_2)),
    HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α) ^ (n + 1) ∈
      (↑(span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) : Set α)
⊢ a ∈ (↑(span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) : Set α)

Tactic:
rcases hx with ⟨x, _, rfl⟩

Premises used: rfl
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
x : (↑s : Type u_2)
left : x ∈ (↑f.support : Set (↑s : Type u_2))
rw : (↑x : α) ^ (n + 1) ∈ (↑(span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) : Set α)
⊢ LE.le (α := Ideal α) (span {(↑x : α) ^ (n + 1)}) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s))

Tactic:
rw [span_le, Set.singleton_subset_iff]

Premises used: Ideal.span_le, Set.singleton_subset_iff
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
x : (↑s : Type u_2)
left : x ∈ (↑f.support : Set (↑s : Type u_2))
⊢ (↑x : α) ^ (n + 1) ∈ (↑(span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) : Set α)

Tactic:
subset_span ⟨x, x.prop, rfl⟩

Premises used: Ideal.subset_span, Subtype.prop, And.intro, Exists.intro, Subtype.val, rfl
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
x : (↑s : Type u_2)
left : x ∈ (↑f.support : Set (↑s : Type u_2))
⊢ (↑x : α) ^ (n + 1) ∈ (↑(span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) : Set α)

Tactic:
exact subset_span ⟨x, x.prop, rfl⟩
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
x : (↑s : Type u_2)
left : x ∈ (↑f.support : Set (↑s : Type u_2))
⊢ LE.le (α := Ideal α) (span {(↑x : α) ^ (n + 1)}) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s))

Tactic:
have : span ({(x : α) ^ (n + 1)} : Set α) ≤ span ((fun x : α => x ^ (n + 1)) '' s) := by
    rw [span_le, Set.singleton_subset_iff]
    exact subset_span ⟨x, x.prop, rfl⟩
-- Module: Mathlib.RingTheory.Ideal.Basic
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
n : ℕ
f : (↑s : Type u_2) →₀ α
hf : Eq (α := α) (∑ x ∈ f.support, HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α)) 1
this_1 :
  Membership.mem (γ := Ideal α)
    (span
      ((fun (i : (↑s : Type u_2)) ↦ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) i : α) (↑i : α) ^ (n + 1)) ''
        (↑f.support : Set (↑s : Type u_2))))
    1
x : (↑s : Type u_2)
left : x ∈ (↑f.support : Set (↑s : Type u_2))
this : LE.le (α := Ideal α) (span {(↑x : α) ^ (n + 1)}) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s))
⊢ HMul.hMul (α := α) ((f : (a : (↑s : Type u_2)) → α) x : α) (↑x : α) ^ (n + 1) ∈
    (↑(span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) : Set α)

Tactic:
this ?_

Premises used: Set.Mem