awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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
⊢ ∃ (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
have 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
Tactic: by_cases! h : n + 1 ≤ c
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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
have 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
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 _, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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 ^ 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
have 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
Tactic: rw [mul_comm _ (a ^ (n + 1))]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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 ^ 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
have 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
Tactic: simp_rw [← mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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 ^ 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
have 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
Tactic: rw [← pow_add, add_zero, add_tsub_cancel_of_le h]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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
⊢ 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
have 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)
Tactic: rw [← pow_add, add_zero, add_tsub_cancel_of_le h]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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)
⊢ 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
have pos₂ : 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)
Tactic: rw [← pow_add, add_zero, add_tsub_cancel_of_le h]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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
⊢ ∃ (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
have 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
Tactic: use 0
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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) = 0 * a ^ (n + 1) + z
have 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
Tactic: simp_rw [zero_mul, zero_add]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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
have 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))
Tactic: refine ⟨_, ?_, rfl⟩
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (hypothesis)
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
⊢ 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))
have h₁ : c ≤ n
Tactic: replace h : c ≤ n := Nat.lt_succ_iff.mp h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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)
have rw : s.card * n + 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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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)
⊢ (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
have rw₁ : s.card * n + 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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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)
⊢ (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
have rw₂ : s.card * n + (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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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)
⊢ (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
have rw₃ : s.card * n + (1 + n) - 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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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)
⊢ (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
have rw₄ : s.card * n + 1 + n - 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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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)
⊢ (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c)
have rw₅ : s.card * n + 1 + (n - 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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (hypothesis)
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
⊢ 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))
have this : (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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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 ^ ((s.card + 1) * n + 1 - c) * (↑(((s.card + 1) * n + 1).choose c) : α))
have 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))
Tactic: rw [this, pow_add]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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 ^ (s.card * n + 1) * s.sum ^ (n - c)) * (↑(((s.card + 1) * n + 1).choose c) : α))
have 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))
Tactic: simp_rw [mul_assoc, mul_comm (s.sum ^ (Multiset.card s * n + 1)), ← mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
n : ℕ
⊢ Eq (α := Ideal α) (span ((fun (x : α) ↦ x ^ n) '' s)) ⊤
have rw : Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ n) '' s)) 1
Tactic: rw [eq_top_iff_one]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
n : ℕ
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ n) '' s)) 1
have zero : Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' s)) 1
Tactic: rcases n with - | n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ n) '' s)) 1
have succ : ∀ (n : ℕ), Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1
Tactic: rcases n with - | n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' s)) 1
have 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
Tactic: obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s
Cleared hypotheses: s, hs
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ HPow.hPow (β := ℕ) x 0) '' s)) 1
have zero₁ : ∀ (x : α) (hx : x ∈ s), 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
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
Goal:
α : Type u_2
inst : CommSemiring α
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
have zero : Membership.mem (γ := Ideal α) ⊤ 1
Tactic: rw [Set.image_empty, hs]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
Goal:
α : Type u_2
inst : CommSemiring α
s : Set α
hs : Eq (α := Ideal α) (span s) ⊤
n : ℕ
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1
have 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
Tactic: rw [eq_top_iff_one, span, Finsupp.mem_span_iff_linearCombination] at hs
Cleared hypotheses: hs, n
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (goal)
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
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1
have 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
Tactic: rcases hs with ⟨f, hf⟩
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (hypothesis)
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)
⊢ Membership.mem (γ := Ideal α) (span ((fun (x : α) ↦ x ^ (n + 1)) '' s)) 1
have hf₁ : 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
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (hypothesis)
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 (x : α) ↦ x ^ (n + 1)) '' s)) 1
have 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))
Tactic: have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Basic (hypothesis)
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 (x : α) ↦ x ^ (n + 1)) '' s)) 1
have this₁ : 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
Cleared hypotheses: this