Theorem: MvPolynomial.psum_eq_mul_esymm_sub_sum
File: Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean

tactic: simp only [Set.Ioo, Set.mem_setOf_eq, and_comm]
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => a.1 ∈ Set.Ioo 0 k) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2

tactic: have hesymm := mul_esymm_eq_sum σ R k
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => a.1 < k) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2

tactic: rw [← (sum_filter_add_sum_filter_not ((antidiagonal k).filter (fun a ↦ 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
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => a.1 < k) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2

tactic: simp only [left_distrib, add_sub_cancel_left] at sub_both_sides
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides :
  (fun x =>
        x -
          (-1) ^ (k + 1) *
            ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
              (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2)
      (↑k * esymm σ R k) =
    (fun x =>
        x -
          (-1) ^ (k + 1) *
            ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
              (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2)
      ((-1) ^ (k + 1) *
        (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
            (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
          ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
            (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2))
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2

tactic: simp only [mul_sub_left_distrib, ← mul_assoc, ← pow_add, Even.neg_one_pow ⟨k + 1, rfl⟩, one_mul,
  not_le, lt_one_iff, filter_filter (fun a : ℕ × ℕ ↦ a.fst < k) (fun a ↦ ¬0 < a.fst)]
  at sub_both_sides
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (fun x => (-1) ^ (k + 1) * x)
      (↑k * esymm σ R k -
        (-1) ^ (k + 1) *
          ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
            (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2) =
    (fun x => (-1) ^ (k + 1) * x)
      ((-1) ^ (k + 1) *
        ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2)
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2

tactic: have : filter (fun a ↦ a.fst < k ∧ ¬0 < a.fst) (antidiagonal k) = {(0, k)} := by
  ext a
  rw [mem_filter, mem_antidiagonal, mem_singleton]
  refine ⟨?_, by rintro rfl; omega⟩
  rintro ⟨ha, ⟨_, ha0⟩⟩
  rw [← ha, Nat.eq_zero_of_not_pos ha0, zero_add, ← Nat.eq_zero_of_not_pos ha0]
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2

tactic: rw [this, sum_singleton] at sub_both_sides
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (0, k).1 * esymm σ R (0, k).1 * psum σ R (0, k).2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2

tactic: simp only [_root_.pow_zero, esymm_zero, mul_one, one_mul, filter_filter] at sub_both_sides
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (0, k).1 * esymm σ R (0, k).1 * psum σ R (0, k).2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 =
    psum σ R k
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2

tactic: exact sub_both_sides.symm
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 =
    psum σ R k
⊢ psum σ R k =
    (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
state_after: no goals

tactic: ext a
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
⊢ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
state_after: case a
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
⊢ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) ↔ a ∈ {(0, k)}

tactic: rw [mem_filter, mem_antidiagonal, mem_singleton]
state_before: case a
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
⊢ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) ↔ a ∈ {(0, k)}
state_after: case a
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
⊢ a.1 + a.2 = k ∧ a.1 < k ∧ ¬0 < a.1 ↔ a = (0, k)

tactic: refine ⟨?_, by rintro rfl; omega⟩
state_before: case a
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
⊢ a.1 + a.2 = k ∧ a.1 < k ∧ ¬0 < a.1 ↔ a = (0, k)
state_after: case a
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
⊢ a.1 + a.2 = k ∧ a.1 < k ∧ ¬0 < a.1 → a = (0, k)

tactic: rintro ⟨ha, ⟨_, ha0⟩⟩
state_before: case a
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
⊢ a.1 + a.2 = k ∧ a.1 < k ∧ ¬0 < a.1 → a = (0, k)
state_after: case a.intro.intro
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
ha : a.1 + a.2 = k
left✝ : a.1 < k
ha0 : ¬0 < a.1
⊢ a = (0, k)

tactic: rw [← ha, Nat.eq_zero_of_not_pos ha0, zero_add, ← Nat.eq_zero_of_not_pos ha0]
state_before: case a.intro.intro
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
ha : a.1 + a.2 = k
left✝ : a.1 < k
ha0 : ¬0 < a.1
⊢ a = (0, k)
state_after: no goals

tactic: rintro rfl
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
a : ℕ × ℕ
⊢ a = (0, k) → a.1 + a.2 = k ∧ a.1 < k ∧ ¬0 < a.1
state_after: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
⊢ (0, k).1 + (0, k).2 = k ∧ (0, k).1 < k ∧ ¬0 < (0, k).1

tactic: omega
state_before: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm :
  ↑k * esymm σ R k =
    (-1) ^ (k + 1) *
      (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +
        ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ :
  ↑k * esymm σ R k -
      (-1) ^ (k + 1) *
        ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
          (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    (-1) ^ (k + 1) *
      ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides :
  (-1) ^ (k + 1) * ↑k * esymm σ R k -
      ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)),
        (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 =
    ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
⊢ (0, k).1 + (0, k).2 = k ∧ (0, k).1 < k ∧ ¬0 < (0, k).1
state_after: no goals