awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
R : Type u_2
inst : CommRing R
inst_1 : Fintype σ
k : ℕ
a : ℕ × ℕ
ha : a ∈ @antidiagonal _ instAddMonoid _ k
x : σ
h : Membership.mem (γ := Finset σ) univ x
s : Finset σ
hs : Membership.mem (γ := Finset (Finset σ)) (powersetCard a.1 univ) s
rw :
  (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) (k - a.1) =
    (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) a.2
⊢ (HPow.hPow (α := MvPolynomial σ R) (-1) #s * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) (k - #s) =
    (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) a.2
have rw₁ : (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) *
    HPow.hPow (α := MvPolynomial σ R) (X x) (a.1 + a.2 - a.1) =
  (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) a.2
Tactic: rw [mem_powersetCard_univ.mp hs, ← mem_antidiagonal.mp ha, add_sub_self_left]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
R : Type u_2
inst : CommRing R
inst_1 : Fintype σ
k : ℕ
a : ℕ × ℕ
ha : a ∈ @antidiagonal _ instAddMonoid _ k
x : σ
h : Membership.mem (γ := Finset σ) univ x
s : Finset σ
hs : Membership.mem (γ := Finset (Finset σ)) (powersetCard a.1 univ) s
rw :
  (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) (k - a.1) =
    (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) a.2
rw₁ :
  (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) *
      HPow.hPow (α := MvPolynomial σ R) (X x) (a.1 + a.2 - a.1) =
    (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) a.2
⊢ (HPow.hPow (α := MvPolynomial σ R) (-1) #s * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) (k - #s) =
    (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) a.2
have rw₂ : (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) a.2 =
  (HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * ∏ a ∈ s, X a) * HPow.hPow (α := MvPolynomial σ R) (X x) a.2
Tactic: rw [mem_powersetCard_univ.mp hs, ← mem_antidiagonal.mp ha, add_sub_self_left]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
R : Type u_2
inst : CommRing R
inst_1 : Fintype σ
inst_2 : DecidableEq σ
k : ℕ
⊢ Eq (α := MvPolynomial σ R)
    (∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2)
    (∑ t ∈ pairs σ k with #t.1 < k, weight σ R k t)
have rw : Eq (α := MvPolynomial σ R)
  (∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k, ∑ A ∈ powersetCard a.1 univ, ∑ j : σ, weight σ R k (A, j))
  (∑ t ∈ pairs σ k with #t.1 < k, weight σ R k t)
Tactic: rw [← sum_congr rfl (fun a ha ↦ esymm_mul_psum_summand_to_weight σ R k a (mem_filter.mp ha).left),
    sum_filter_pairs_eq_sum_filter_antidiagonal_powersetCard_sum σ R k]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
R : Type u_2
inst : CommRing R
inst_1 : Fintype σ
inst_2 : DecidableEq σ
k : ℕ
rw :
  Eq (α := MvPolynomial σ R)
    (∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k, ∑ A ∈ powersetCard a.1 univ, ∑ j : σ, weight σ R k (A, j))
    (∑ t ∈ pairs σ k with #t.1 < k, weight σ R k t)
⊢ Eq (α := MvPolynomial σ R)
    (∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2)
    (∑ t ∈ pairs σ k with #t.1 < k, weight σ R k t)
have rw₁ : Eq (α := MvPolynomial σ R)
  (∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k, ∑ A ∈ powersetCard a.1 univ, ∑ j : σ, weight σ R k (A, j))
  (∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k, ∑ A ∈ powersetCard a.1 univ, ∑ j : σ, weight σ R k (A, j))
Tactic: rw [← sum_congr rfl (fun a ha ↦ esymm_mul_psum_summand_to_weight σ R k a (mem_filter.mp ha).left),
    sum_filter_pairs_eq_sum_filter_antidiagonal_powersetCard_sum σ R k]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
⊢ HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
have rw : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
    -1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
  0
Tactic: rw [NewtonIdentities.esymm_to_weight σ R k, NewtonIdentities.esymm_mul_psum_to_weight σ R k,
    eq_comm, ← sub_eq_zero, sub_eq_add_neg, neg_mul_eq_neg_mul,
    neg_eq_neg_one_mul ((-1 : MvPolynomial σ R) ^ k)]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      -1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
have nth_rw : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
    HPow.hPow (α := MvPolynomial σ R) (β := ℕ) (-1) 1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
  0
Tactic: nth_rw 2 [← pow_one (-1 : MvPolynomial σ R)]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (β := ℕ) (-1) 1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
have rw : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
    HPow.hPow (α := MvPolynomial σ R) (-1) (1 + k) *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
  0
Tactic: rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
rw :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (1 + k) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (β := ℕ) (-1) 1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
have rw₁ : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
  0
Tactic: rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
rw :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (1 + k) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₁ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (β := ℕ) (-1) 1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
have rw₂ : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
    HAdd.hAdd (α := MvPolynomial σ R) (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t)
      (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t) =
  0
Tactic: rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
rw :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (1 + k) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₁ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₂ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t)
        (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t) =
    0
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (β := ℕ) (-1) 1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
have rw₃ : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
    ∑ x ∈ {t ∈ NewtonIdentities.pairs σ k | #t.1 < k}.disjUnion ({t ∈ NewtonIdentities.pairs σ k | #t.1 = k}) sorry,
      NewtonIdentities.weight σ R k x =
  0
Tactic: rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
rw :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (1 + k) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₁ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₂ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t)
        (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t) =
    0
rw₃ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ x ∈ {t ∈ NewtonIdentities.pairs σ k | #t.1 < k}.disjUnion ({t ∈ NewtonIdentities.pairs σ k | #t.1 = k}) ⋯,
        NewtonIdentities.weight σ R k x =
    0
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (β := ℕ) (-1) 1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
have rw₄ : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * ∑ x ∈ NewtonIdentities.pairs σ k, NewtonIdentities.weight σ R k x = 0
Tactic: rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
rw :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (1 + k) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₁ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₂ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t)
        (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t) =
    0
rw₃ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ x ∈ {t ∈ NewtonIdentities.pairs σ k | #t.1 < k}.disjUnion ({t ∈ NewtonIdentities.pairs σ k | #t.1 = k}) ⋯,
        NewtonIdentities.weight σ R k x =
    0
rw₄ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * ∑ x ∈ NewtonIdentities.pairs σ k, NewtonIdentities.weight σ R k x = 0
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (β := ℕ) (-1) 1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
have rw₅ : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * 0 = 0
Tactic: rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
rw :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (1 + k) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₁ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
rw₂ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t)
        (∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t) =
    0
rw₃ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ x ∈ {t ∈ NewtonIdentities.pairs σ k | #t.1 < k}.disjUnion ({t ∈ NewtonIdentities.pairs σ k | #t.1 = k}) ⋯,
        NewtonIdentities.weight σ R k x =
    0
rw₄ :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * ∑ x ∈ NewtonIdentities.pairs σ k, NewtonIdentities.weight σ R k x = 0
rw₅ : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * 0 = 0
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      HPow.hPow (α := MvPolynomial σ R) (β := ℕ) (-1) 1 * HPow.hPow (α := MvPolynomial σ R) (-1) k *
        ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
have rw₆ : Eq (α := MvPolynomial σ R) 0 0
Tactic: rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
⊢ HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
have rw : HPow.hPow (α := MvPolynomial σ R) (-1) k *
    ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
    ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
Tactic: classical
  rw [NewtonIdentities.esymm_to_weight σ R k, NewtonIdentities.esymm_mul_psum_to_weight σ R k,
    eq_comm, ← sub_eq_zero, sub_eq_add_neg, neg_mul_eq_neg_mul,
    neg_eq_neg_one_mul ((-1 : MvPolynomial σ R) ^ k)]
  nth_rw 2 [← pow_one (-1 : MvPolynomial σ R)]
  rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
rw :
  HPow.hPow (α := MvPolynomial σ R) (-1) k *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
⊢ HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
have rw₁ : HPow.hPow (α := MvPolynomial σ R) (-1) k *
    ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
    ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t
Tactic: classical
  rw [NewtonIdentities.esymm_to_weight σ R k, NewtonIdentities.esymm_mul_psum_to_weight σ R k,
    eq_comm, ← sub_eq_zero, sub_eq_add_neg, neg_mul_eq_neg_mul,
    neg_eq_neg_one_mul ((-1 : MvPolynomial σ R) ^ k)]
  nth_rw 2 [← pow_one (-1 : MvPolynomial σ R)]
  rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]
-- Lean 4 proof step from Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities (goal)
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
rw :
  HPow.hPow (α := MvPolynomial σ R) (-1) k *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
rw₁ :
  HPow.hPow (α := MvPolynomial σ R) (-1) k *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t
⊢ HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
have rw₂ : HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
    ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t =
  HPow.hPow (α := MvPolynomial σ R) (-1) k *
    ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t
Tactic: classical
  rw [NewtonIdentities.esymm_to_weight σ R k, NewtonIdentities.esymm_mul_psum_to_weight σ R k,
    eq_comm, ← sub_eq_zero, sub_eq_add_neg, neg_mul_eq_neg_mul,
    neg_eq_neg_one_mul ((-1 : MvPolynomial σ R) ^ k)]
  nth_rw 2 [← pow_one (-1 : MvPolynomial σ R)]
  rw [← pow_add, add_comm 1 k, ← left_distrib,
    ← sum_disjUnion (NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq σ k),
    NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k, NewtonIdentities.weight_sum σ R k,
    neg_one_pow_mul_eq_zero_iff.mpr rfl]