l3lab/ntp-mathlib huggingface dataset:
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · 

Proof state:
case inl
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
⊢ (-1) ^ card t.1 * ((∏ a in t.1, X a) * X t.2 ^ (k - card t.1)) +
      (-1) ^ card (MvPolynomial.NewtonIdentities.pairMap σ t).1 *
        ((∏ a in (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *
          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - card (MvPolynomial.NewtonIdentities.pairMap σ t).1)) =
    0

Next tactic:
rw [pairMap_of_snd_mem_fst σ h1]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    

Proof state:
case inl
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
⊢ (-1) ^ card t.1 * ((∏ a in t.1, X a) * X t.2 ^ (k - card t.1)) +
      (-1) ^ card (erase t.1 t.2, t.2).1 *
        ((∏ a in (erase t.1 t.2, t.2).1, X a) * X (erase t.1 t.2, t.2).2 ^ (k - card (erase t.1 t.2, t.2).1)) =
    0

Next tactic:
simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    

Proof state:
case inl
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
⊢ (-1) ^ card t.1 * ((∏ a in erase t.1 t.2, X a) * (X t.2 * X t.2 ^ (k - card t.1))) +
      (-1) ^ (card t.1 - 1) * ((∏ a in erase t.1 t.2, X a) * X t.2 ^ (k - (card t.1 - 1))) =
    0

Next tactic:
nth_rewrite 1 [← pow_one (X t.snd)]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    

Proof state:
case inl
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
⊢ (-1) ^ card t.1 * ((∏ a in erase t.1 t.2, X a) * (X t.2 ^ 1 * X t.2 ^ (k - card t.1))) +
      (-1) ^ (card t.1 - 1) * ((∏ a in erase t.1 t.2, X a) * X t.2 ^ (k - (card t.1 - 1))) =
    0

Next tactic:
simp only [← pow_add, add_comm]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    

Proof state:
case inl
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
⊢ (-1) ^ card t.1 * ((∏ a in erase t.1 t.2, X a) * X t.2 ^ (k - card t.1 + 1)) +
      (-1) ^ (card t.1 - 1) * ((∏ a in erase t.1 t.2, X a) * X t.2 ^ (k - (card t.1 - 1))) =
    0

Next tactic:
have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
    

Proof state:
case inl
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
h3 : 1 ≤ card t.1
⊢ (-1) ^ card t.1 * ((∏ a in erase t.1 t.2, X a) * X t.2 ^ (k - card t.1 + 1)) +
      (-1) ^ (card t.1 - 1) * ((∏ a in erase t.1 t.2, X a) * X t.2 ^ (k - (card t.1 - 1))) =
    0

Next tactic:
rw [← tsub_tsub_assoc h.left h3, ← neg_neg ((-1 : MvPolynomial σ R) ^ (card t.fst - 1)),
      h2 (card t.fst - 1), Nat.sub_add_cancel h3]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
    rw [← tsub_tsub_assoc h.left h3, ← neg_neg ((-1 : MvPolynomial σ R) ^ (card t.fst - 1)),
      h2 (card t.fst - 1), Nat.sub_add_cancel h3]
    

Proof state:
case inl
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
h3 : 1 ≤ card t.1
⊢ (-1) ^ card t.1 * ((∏ a in erase t.1 t.2, X a) * X t.2 ^ (k - (card t.1 - 1))) +
      -(-1) ^ card t.1 * ((∏ a in erase t.1 t.2, X a) * X t.2 ^ (k - (card t.1 - 1))) =
    0

Next tactic:
simp
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
    rw [← tsub_tsub_assoc h.left h3, ← neg_neg ((-1 : MvPolynomial σ R) ^ (card t.fst - 1)),
      h2 (card t.fst - 1), Nat.sub_add_cancel h3]
    simp
  · 

Proof state:
case inr
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (-1) ^ card t.1 * ((∏ a in t.1, X a) * X t.2 ^ (k - card t.1)) +
      (-1) ^ card (MvPolynomial.NewtonIdentities.pairMap σ t).1 *
        ((∏ a in (MvPolynomial.NewtonIdentities.pairMap σ t).1, X a) *
          X (MvPolynomial.NewtonIdentities.pairMap σ t).2 ^ (k - card (MvPolynomial.NewtonIdentities.pairMap σ t).1)) =
    0

Next tactic:
rw [pairMap_of_snd_nmem_fst σ h1]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
    rw [← tsub_tsub_assoc h.left h3, ← neg_neg ((-1 : MvPolynomial σ R) ^ (card t.fst - 1)),
      h2 (card t.fst - 1), Nat.sub_add_cancel h3]
    simp
  · rw [pairMap_of_snd_nmem_fst σ h1]
    

Proof state:
case inr
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (-1) ^ card t.1 * ((∏ a in t.1, X a) * X t.2 ^ (k - card t.1)) +
      (-1) ^ card (cons t.2 t.1 h1, t.2).1 *
        ((∏ a in (cons t.2 t.1 h1, t.2).1, X a) * X (cons t.2 t.1 h1, t.2).2 ^ (k - card (cons t.2 t.1 h1, t.2).1)) =
    0

Next tactic:
simp only [mul_comm, mul_assoc (∏ a in t.fst, X a), card_cons, prod_cons]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
    rw [← tsub_tsub_assoc h.left h3, ← neg_neg ((-1 : MvPolynomial σ R) ^ (card t.fst - 1)),
      h2 (card t.fst - 1), Nat.sub_add_cancel h3]
    simp
  · rw [pairMap_of_snd_nmem_fst σ h1]
    simp only [mul_comm, mul_assoc (∏ a in t.fst, X a), card_cons, prod_cons]
    

Proof state:
case inr
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (∏ a in t.1, X a) * (X t.2 ^ (k - card t.1) * (-1) ^ card t.1) +
      (∏ a in t.1, X a) * (X t.2 * X t.2 ^ (k - (card t.1 + 1)) * (-1) ^ (card t.1 + 1)) =
    0

Next tactic:
nth_rewrite 2 [← pow_one (X t.snd)]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
    rw [← tsub_tsub_assoc h.left h3, ← neg_neg ((-1 : MvPolynomial σ R) ^ (card t.fst - 1)),
      h2 (card t.fst - 1), Nat.sub_add_cancel h3]
    simp
  · rw [pairMap_of_snd_nmem_fst σ h1]
    simp only [mul_comm, mul_assoc (∏ a in t.fst, X a), card_cons, prod_cons]
    nth_rewrite 2 [← pow_one (X t.snd)]
    

Proof state:
case inr
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (∏ a in t.1, X a) * (X t.2 ^ (k - card t.1) * (-1) ^ card t.1) +
      (∏ a in t.1, X a) * (X t.2 ^ 1 * X t.2 ^ (k - (card t.1 + 1)) * (-1) ^ (card t.1 + 1)) =
    0

Next tactic:
simp only [← pow_add, ← Nat.add_sub_assoc (Nat.lt_of_le_of_ne h.left (mt h.right h1)), add_comm]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
    rw [← tsub_tsub_assoc h.left h3, ← neg_neg ((-1 : MvPolynomial σ R) ^ (card t.fst - 1)),
      h2 (card t.fst - 1), Nat.sub_add_cancel h3]
    simp
  · rw [pairMap_of_snd_nmem_fst σ h1]
    simp only [mul_comm, mul_assoc (∏ a in t.fst, X a), card_cons, prod_cons]
    nth_rewrite 2 [← pow_one (X t.snd)]
    simp only [← pow_add, ← Nat.add_sub_assoc (Nat.lt_of_le_of_ne h.left (mt h.right h1)), add_comm]
    

Proof state:
case inr
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (∏ a in t.1, X a) * (X t.2 ^ (k - card t.1) * (-1) ^ card t.1) +
      (∏ a in t.1, X a) * (X t.2 ^ (k + 1 - succ (card t.1)) * (-1) ^ (card t.1 + 1)) =
    0

Next tactic:
rw [← neg_neg ((-1 : MvPolynomial σ R) ^ card t.fst), h2]
private theorem weight_add_weight_pairMap {k : ℕ} (t : Finset σ × σ) (h : t ∈ pairs σ k) :
    weight σ R k t + weight σ R k (pairMap σ t) = 0 := by
  rw [weight, weight]
  rw [mem_pairs] at h
  have h2 (n : ℕ) : -(-1 : MvPolynomial σ R) ^ n = (-1) ^ (n + 1) := by
    rw [← neg_one_mul ((-1 : MvPolynomial σ R) ^ n), pow_add, pow_one, mul_comm]
  rcases (em (t.snd ∈ t.fst)) with h1 | h1
  · rw [pairMap_of_snd_mem_fst σ h1]
    simp only [← prod_erase_mul t.fst (fun j ↦ (X j : MvPolynomial σ R)) h1,
      mul_assoc (∏ a in erase t.fst t.snd, X a), card_erase_of_mem h1]
    nth_rewrite 1 [← pow_one (X t.snd)]
    simp only [← pow_add, add_comm]
    have h3 : 1 ≤ card t.fst := lt_iff_add_one_le.mp (card_pos.mpr ⟨t.snd, h1⟩)
    rw [← tsub_tsub_assoc h.left h3, ← neg_neg ((-1 : MvPolynomial σ R) ^ (card t.fst - 1)),
      h2 (card t.fst - 1), Nat.sub_add_cancel h3]
    simp
  · rw [pairMap_of_snd_nmem_fst σ h1]
    simp only [mul_comm, mul_assoc (∏ a in t.fst, X a), card_cons, prod_cons]
    nth_rewrite 2 [← pow_one (X t.snd)]
    simp only [← pow_add, ← Nat.add_sub_assoc (Nat.lt_of_le_of_ne h.left (mt h.right h1)), add_comm]
    rw [← neg_neg ((-1 : MvPolynomial σ R) ^ card t.fst), h2]
    

Proof state:
case inr
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
t : Finset σ × σ
h : card t.1 ≤ k ∧ (card t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (∏ a in t.1, X a) * (X t.2 ^ (k - card t.1) * -(-1) ^ (card t.1 + 1)) +
      (∏ a in t.1, X a) * (X t.2 ^ (k + 1 - succ (card t.1)) * (-1) ^ (card t.1 + 1)) =
    0

Next tactic:
simp
private theorem sum_filter_pairs_eq_sum_powersetCard_sum (k : ℕ)
    (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = k) (pairs σ k), f t) =
    ∑ A in powersetCard k univ, (∑ j in A, f (A, j)) := by
  

Proof state:
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
f : Finset σ × σ → MvPolynomial σ R
⊢ ∑ t in filter (fun t => card t.1 = k) (MvPolynomial.NewtonIdentities.pairs σ k), f t =
    ∑ A in powersetCard k univ, ∑ j in A, f (A, j)

Next tactic:
apply sum_finset_product
private theorem sum_filter_pairs_eq_sum_powersetCard_sum (k : ℕ)
    (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = k) (pairs σ k), f t) =
    ∑ A in powersetCard k univ, (∑ j in A, f (A, j)) := by
  apply sum_finset_product
  

Proof state:
case h
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
f : Finset σ × σ → MvPolynomial σ R
⊢ ∀ (p : Finset σ × σ),
    p ∈ filter (fun t => card t.1 = k) (MvPolynomial.NewtonIdentities.pairs σ k) ↔ p.1 ∈ powersetCard k univ ∧ p.2 ∈ p.1

Next tactic:
aesop
private theorem sum_filter_pairs_eq_sum_powersetCard_mem_filter_antidiagonal_sum (k : ℕ) (a : ℕ × ℕ)
    (ha : a ∈ (antidiagonal k).filter (fun a ↦ a.fst < k)) (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = a.fst) (pairs σ k), f t) =
    ∑ A in powersetCard a.fst univ, (∑ j, f (A, j)) := by
  

Proof state:
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ filter (fun a => a.1 < k) (antidiagonal k)
f : Finset σ × σ → MvPolynomial σ R
⊢ ∑ t in filter (fun t => card t.1 = a.1) (MvPolynomial.NewtonIdentities.pairs σ k), f t =
    ∑ A in powersetCard a.1 univ, ∑ j : σ, f (A, j)

Next tactic:
apply sum_finset_product
private theorem sum_filter_pairs_eq_sum_powersetCard_mem_filter_antidiagonal_sum (k : ℕ) (a : ℕ × ℕ)
    (ha : a ∈ (antidiagonal k).filter (fun a ↦ a.fst < k)) (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = a.fst) (pairs σ k), f t) =
    ∑ A in powersetCard a.fst univ, (∑ j, f (A, j)) := by
  apply sum_finset_product
  

Proof state:
case h
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ filter (fun a => a.1 < k) (antidiagonal k)
f : Finset σ × σ → MvPolynomial σ R
⊢ ∀ (p : Finset σ × σ),
    p ∈ filter (fun t => card t.1 = a.1) (MvPolynomial.NewtonIdentities.pairs σ k) ↔
      p.1 ∈ powersetCard a.1 univ ∧ p.2 ∈ univ

Next tactic:
simp only [mem_filter, mem_powersetCard_univ, mem_univ, and_true, and_iff_right_iff_imp]
private theorem sum_filter_pairs_eq_sum_powersetCard_mem_filter_antidiagonal_sum (k : ℕ) (a : ℕ × ℕ)
    (ha : a ∈ (antidiagonal k).filter (fun a ↦ a.fst < k)) (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = a.fst) (pairs σ k), f t) =
    ∑ A in powersetCard a.fst univ, (∑ j, f (A, j)) := by
  apply sum_finset_product
  simp only [mem_filter, mem_powersetCard_univ, mem_univ, and_true, and_iff_right_iff_imp]
  

Proof state:
case h
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ filter (fun a => a.1 < k) (antidiagonal k)
f : Finset σ × σ → MvPolynomial σ R
⊢ ∀ (p : Finset σ × σ), card p.1 = a.1 → p ∈ MvPolynomial.NewtonIdentities.pairs σ k

Next tactic:
rintro p hp
private theorem sum_filter_pairs_eq_sum_powersetCard_mem_filter_antidiagonal_sum (k : ℕ) (a : ℕ × ℕ)
    (ha : a ∈ (antidiagonal k).filter (fun a ↦ a.fst < k)) (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = a.fst) (pairs σ k), f t) =
    ∑ A in powersetCard a.fst univ, (∑ j, f (A, j)) := by
  apply sum_finset_product
  simp only [mem_filter, mem_powersetCard_univ, mem_univ, and_true, and_iff_right_iff_imp]
  rintro p hp
  

Proof state:
case h
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ filter (fun a => a.1 < k) (antidiagonal k)
f : Finset σ × σ → MvPolynomial σ R
p : Finset σ × σ
hp : card p.1 = a.1
⊢ p ∈ MvPolynomial.NewtonIdentities.pairs σ k

Next tactic:
have : card p.fst ≤ k := by apply le_of_lt; aesop
private theorem sum_filter_pairs_eq_sum_powersetCard_mem_filter_antidiagonal_sum (k : ℕ) (a : ℕ × ℕ)
    (ha : a ∈ (antidiagonal k).filter (fun a ↦ a.fst < k)) (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = a.fst) (pairs σ k), f t) =
    ∑ A in powersetCard a.fst univ, (∑ j, f (A, j)) := by
  apply sum_finset_product
  simp only [mem_filter, mem_powersetCard_univ, mem_univ, and_true, and_iff_right_iff_imp]
  rintro p hp
  have : card p.fst ≤ k := by 

Proof state:
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ filter (fun a => a.1 < k) (antidiagonal k)
f : Finset σ × σ → MvPolynomial σ R
p : Finset σ × σ
hp : card p.1 = a.1
⊢ card p.1 ≤ k

Next tactic:
apply le_of_lt
private theorem sum_filter_pairs_eq_sum_powersetCard_mem_filter_antidiagonal_sum (k : ℕ) (a : ℕ × ℕ)
    (ha : a ∈ (antidiagonal k).filter (fun a ↦ a.fst < k)) (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = a.fst) (pairs σ k), f t) =
    ∑ A in powersetCard a.fst univ, (∑ j, f (A, j)) := by
  apply sum_finset_product
  simp only [mem_filter, mem_powersetCard_univ, mem_univ, and_true, and_iff_right_iff_imp]
  rintro p hp
  have : card p.fst ≤ k := by apply le_of_lt; 

Proof state:
case a
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ filter (fun a => a.1 < k) (antidiagonal k)
f : Finset σ × σ → MvPolynomial σ R
p : Finset σ × σ
hp : card p.1 = a.1
⊢ card p.1 < k

Next tactic:
aesop
private theorem sum_filter_pairs_eq_sum_powersetCard_mem_filter_antidiagonal_sum (k : ℕ) (a : ℕ × ℕ)
    (ha : a ∈ (antidiagonal k).filter (fun a ↦ a.fst < k)) (f : Finset σ × σ → MvPolynomial σ R) :
    (∑ t in filter (fun t ↦ card t.fst = a.fst) (pairs σ k), f t) =
    ∑ A in powersetCard a.fst univ, (∑ j, f (A, j)) := by
  apply sum_finset_product
  simp only [mem_filter, mem_powersetCard_univ, mem_univ, and_true, and_iff_right_iff_imp]
  rintro p hp
  have : card p.fst ≤ k := by apply le_of_lt; aesop
  

Proof state:
case h
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ filter (fun a => a.1 < k) (antidiagonal k)
f : Finset σ × σ → MvPolynomial σ R
p : Finset σ × σ
hp : card p.1 = a.1
this : card p.1 ≤ k
⊢ p ∈ MvPolynomial.NewtonIdentities.pairs σ k

Next tactic:
aesop