awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
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
rw₆ rw₇ : Eq (α := MvPolynomial σ R) 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

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]

Premises used: Iff.mpr, _private.0.MvPolynomial.NewtonIdentities.disjUnion_filter_pairs_eq_pairs, pow_add, Finset.sum_disjUnion, _private.0.MvPolynomial.NewtonIdentities.weight_sum, add_comm, OfNat.ofNat, neg_one_pow_mul_eq_zero_iff, _private.0.MvPolynomial.NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq, rfl, left_distrib
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ := Fintype.card σ
this :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    0
⊢ Eq (α := MvPolynomial σ R)
    (∑ a ∈ @antidiagonal _ instAddMonoid _ (Fintype.card σ),
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2)
    0

Tactic:
simpa using this
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ := Fintype.card σ
suffices :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    0
⊢ Eq (α := MvPolynomial σ R)
    (∑ a ∈ @antidiagonal _ instAddMonoid _ (Fintype.card σ),
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2)
    0

Tactic:
suffices (-1 : MvPolynomial σ R) ^ (k + 1) *
      ∑ a ∈ antidiagonal k, (-1) ^ a.fst * esymm σ R a.fst * psum σ R a.snd = 0 by
    simpa using this
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ := Fintype.card σ
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    0

Tactic:
simp [k, ← sum_filter_add_sum_filter_not (antidiagonal k) (fun a ↦ a.fst < k),
    ← mul_esymm_eq_sum, mul_add, ← mul_assoc, ← pow_add, mul_comm ↑k (esymm σ R k)]

Premises used: mul_assoc, pow_add, MvPolynomial.esymm, Prod.fst, MvPolynomial.mul_esymm_eq_sum, Finset.sum_filter_add_sum_filter_not, LT.lt, mul_comm, mul_add, Finset.HasAntidiagonal.antidiagonal
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
simp :
  psum σ R k =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * (↑k : MvPolynomial σ R) * esymm σ R k -
      ∑ x ∈ @antidiagonal _ instAddMonoid _ k with x.1 < k ∧ 0 < x.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2
⊢ psum σ R k =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * (↑k : MvPolynomial σ R) * esymm σ R k -
      ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 ∈ Set.Ioo 0 k,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2

Tactic:
simp only [Set.Ioo, Set.mem_setOf_eq, and_comm]

Premises used: Set.mem_setOf_eq, and_comm, Set.Ioo
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
⊢ 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

Tactic:
have hesymm := mul_esymm_eq_sum σ R k
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  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
⊢ HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)

Tactic:
rw [← (sum_filter_add_sum_filter_not {a ∈ antidiagonal k | a.fst < k}
    (fun a ↦ 0 < a.fst) (fun a ↦ (-1) ^ a.fst * esymm σ R a.fst * psum σ R a.snd))] at hesymm

Premises used: HMul.hMul, MvPolynomial.esymm, Prod.fst, HPow.hPow, MvPolynomial.psum, Finset.sum_filter_add_sum_filter_not, LT.lt, OfNat.ofNat, Finset.filter, Prod.snd, Finset.HasAntidiagonal.antidiagonal, Neg.neg
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
⊢ Eq (α := MvPolynomial σ R)
    ((fun (x : MvPolynomial σ R) ↦
        x -
          HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
            ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
              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)))
    ((fun (x : MvPolynomial σ R) ↦
        x -
          HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
            ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
              HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2)
      (HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        HAdd.hAdd (α := MvPolynomial σ R)
          (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
            HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
          (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
            HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)))

Tactic:
have sub_both_sides := congrArg (· - (-1 : MvPolynomial σ R) ^ (k + 1) *
    ∑ a ∈ {a ∈ antidiagonal k | a.fst < k} with 0 < a.fst,
    (-1) ^ a.fst * esymm σ R a.fst * psum σ R a.snd) hesymm
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides :
  Eq (α := MvPolynomial σ R)
    ((fun (x : MvPolynomial σ R) ↦
        x -
          HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
            ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
              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)))
    ((fun (x : MvPolynomial σ R) ↦
        x -
          HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
            ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
              HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2)
      (HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        HAdd.hAdd (α := MvPolynomial σ R)
          (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
            HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
          (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
            HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.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 ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2

Tactic:
simp only [left_distrib, add_sub_cancel_left] at sub_both_sides

Premises used: add_sub_cancel_left, left_distrib
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) -
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
⊢ Eq (α := MvPolynomial σ R)
    ((fun (x : MvPolynomial σ R) ↦ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * x)
      (HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) -
        HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
          ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
            HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2))
    ((fun (x : MvPolynomial σ R) ↦ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * x)
      (HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2))

Tactic:
have sub_both_sides := congrArg ((-1 : MvPolynomial σ R) ^ (k + 1) * ·) sub_both_sides
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides_1 :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) -
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  Eq (α := MvPolynomial σ R)
    ((fun (x : MvPolynomial σ R) ↦ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * x)
      (HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) -
        HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
          ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
            HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2))
    ((fun (x : MvPolynomial σ R) ↦ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * x)
      (HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2))
⊢ HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * (↑k : MvPolynomial σ R) * esymm σ R k -
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k ∧ ¬0 < a.1,
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2

Tactic:
simp only [mul_sub_left_distrib, ← mul_assoc, ← pow_add, Even.neg_one_pow ⟨k + 1, rfl⟩, one_mul,
    filter_filter (fun a : ℕ × ℕ ↦ a.fst < k) (fun a ↦ ¬0 < a.fst)]
    at sub_both_sides

Premises used: Finset.filter_filter, mul_assoc, pow_add, mul_sub_left_distrib, Even.neg_one_pow, Prod.fst, HAdd.hAdd, LT.lt, Exists.intro, OfNat.ofNat, Not, one_mul, rfl
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides_1 :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) -
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * (↑k : MvPolynomial σ R) * esymm σ R k -
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k ∧ ¬0 < a.1,
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
h₁ :
  ∀ (a : ℕ × ℕ),
    a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k ∧ ¬0 < a.1} ↔ Membership.mem (γ := Finset (ℕ × ℕ)) {(0, k)} a
⊢ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k ∧ ¬0 < a.1} = {(0, k)}

Tactic:
ext a
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides_1 :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) -
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * (↑k : MvPolynomial σ R) * esymm σ R k -
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k ∧ ¬0 < a.1,
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
h₁ : a.1 + a.2 = k ∧ a.1 < k ∧ ¬0 < a.1 ↔ a = (0, k)
⊢ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k ∧ ¬0 < a.1} ↔ Membership.mem (γ := Finset (ℕ × ℕ)) {(0, k)} a

Tactic:
rw [mem_filter, mem_antidiagonal, mem_singleton]

Premises used: Finset.mem_singleton, Finset.HasAntidiagonal.mem_antidiagonal, Finset.mem_filter
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides_1 :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) -
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * (↑k : MvPolynomial σ R) * esymm σ R k -
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k ∧ ¬0 < a.1,
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
⊢ (0, k).1 + (0, k).2 = k ∧ (0, k).1 < k ∧ ¬LT.lt (α := ℕ) 0 (0, k).1

Tactic:
lia
-- Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
Goal:
σ : Type u_1
inst : Fintype σ
R : Type u_2
inst_1 : CommRing R
k : ℕ
h : 0 < k
hesymm :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      HAdd.hAdd (α := MvPolynomial σ R)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
        (∑ x ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < x.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides_1 :
  HMul.hMul (α := MvPolynomial σ R) (↑k : MvPolynomial σ R) (esymm σ R k) -
      HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
        ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
          HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) *
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with ¬0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  HPow.hPow (α := MvPolynomial σ R) (-1) (k + 1) * (↑k : MvPolynomial σ R) * esymm σ R k -
      ∑ a ∈ {a ∈ @antidiagonal _ instAddMonoid _ k | a.1 < k} with 0 < a.1,
        HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ @antidiagonal _ instAddMonoid _ k with a.1 < k ∧ ¬0 < a.1,
      HPow.hPow (α := MvPolynomial σ R) (-1) a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
h₁ : a.1 + a.2 = k ∧ a.1 < k ∧ ¬0 < a.1 → a = (0, k)
⊢ a.1 + a.2 = k ∧ a.1 < k ∧ ¬0 < a.1 ↔ a = (0, k)

Tactic:
refine ⟨?_, by rintro rfl; lia⟩