Theorem: WittVector.mul_polyOfInterest_aux3
File: Mathlib/RingTheory/WittVector/MulCoeff.lean

Proof steps:
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
⊢ wittPolyProd p (n + 1) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +
      remainder p n
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ wittPolyProd p (n + 1) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +
      remainder p n
tactic: have mvpz : (p : 𝕄) ^ (n + 1) = MvPolynomial.C ((p : ℤ) ^ (n + 1)) := by norm_cast
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ wittPolyProd p (n + 1) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +
      remainder p n
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
      ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      remainder p n
tactic: rw [wittPolyProd, wittPolynomial, map_sum, map_sum]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
      ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      remainder p n
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1)) *
      ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      remainder p n
tactic: conv_lhs =>
    arg 1
    rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,
      rename_C, rename_X, ← mvpz]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1)) *
      ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      remainder p n
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1)) *
      (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      remainder p n
tactic: conv_lhs =>
    arg 2
    rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,
      rename_C, rename_X, ← mvpz]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1)) *
      (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      remainder p n
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1)) *
      (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
              ↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      remainder p n
tactic: conv_rhs =>
    enter [1, 1, 2, 2]
    rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,
      rename_C, rename_X, ← mvpz]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1)) *
      (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
              ↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          ∑ x ∈ range (n + 1 + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      remainder p n
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1)) *
      (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
              ↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
            ↑p ^ (n + 1) * X (0, n + 1)) +
      remainder p n
tactic: conv_rhs =>
    enter [1, 2, 2]
    rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,
      rename_C, rename_X, ← mvpz]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1)) *
      (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (1, n + 1)) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) *
            (∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
              ↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) *
          (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
            ↑p ^ (n + 1) * X (0, n + 1)) +
      remainder p n
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      ((∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          (↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          (↑p ^ (n + 1) * X (0, n + 1) *
              ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
            ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) +
        (↑p ^ (n + 1) * X (1, n + 1) *
            ∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
          ↑p ^ (n + 1) * X (1, n + 1) * (↑p ^ (n + 1) * X (0, n + 1))) +
      remainder p n
tactic: simp only [add_mul, mul_add]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      ((∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          (↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          (↑p ^ (n + 1) * X (0, n + 1) *
              ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
            ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) +
        (↑p ^ (n + 1) * X (1, n + 1) *
            ∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
          ↑p ^ (n + 1) * X (1, n + 1) * (↑p ^ (n + 1) * X (0, n + 1))) +
      remainder p n
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      ((∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          (↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) =
    remainder p n +
      (-(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          (↑p ^ (n + 1) * X (0, n + 1) *
              ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
            ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) +
        (↑p ^ (n + 1) * X (1, n + 1) *
            ∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
          ↑p ^ (n + 1) * X (1, n + 1) * (↑p ^ (n + 1) * X (0, n + 1))))
tactic: rw [add_comm _ (remainder p n)]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ↑p ^ (n + 1) * X (0, n + 1) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      ((∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          (↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) =
    remainder p n +
      (-(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          (↑p ^ (n + 1) * X (0, n + 1) *
              ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
            ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) +
        (↑p ^ (n + 1) * X (1, n + 1) *
            ∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
          ↑p ^ (n + 1) * X (1, n + 1) * (↑p ^ (n + 1) * X (0, n + 1))))
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
        ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      (↑p ^ (n + 1) * X (0, n + 1) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ((∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
            (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1)))) =
    remainder p n +
      (-(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
        (↑p ^ (n + 1) * X (0, n + 1) *
            ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
          (↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1)) +
            (↑p ^ (n + 1) * X (1, n + 1) *
                ∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
              ↑p ^ (n + 1) * X (1, n + 1) * (↑p ^ (n + 1) * X (0, n + 1))))))
tactic: simp only [add_assoc]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
        ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      (↑p ^ (n + 1) * X (0, n + 1) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        ((∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
            (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1)))) =
    remainder p n +
      (-(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
        (↑p ^ (n + 1) * X (0, n + 1) *
            ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
          (↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1)) +
            (↑p ^ (n + 1) * X (1, n + 1) *
                ∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
              ↑p ^ (n + 1) * X (1, n + 1) * (↑p ^ (n + 1) * X (0, n + 1))))))
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ ↑p ^ (n + 1) * X (0, n + 1) *
        ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      ((∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          (↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
      (↑p ^ (n + 1) * X (0, n + 1) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        (↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1)) +
          (↑p ^ (n + 1) * X (1, n + 1) *
              ∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
            ↑p ^ (n + 1) * X (1, n + 1) * (↑p ^ (n + 1) * X (0, n + 1)))))
tactic: apply congrArg (Add.add _)
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ ↑p ^ (n + 1) * X (0, n + 1) *
        ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
      ((∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
          (↑p ^ (n + 1) * X (1, n + 1)) +
        ↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1))) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
      (↑p ^ (n + 1) * X (0, n + 1) *
          ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
        (↑p ^ (n + 1) * X (0, n + 1) * (↑p ^ (n + 1) * X (1, n + 1)) +
          (↑p ^ (n + 1) * X (1, n + 1) *
              ∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) +
            ↑p ^ (n + 1) * X (1, n + 1) * (↑p ^ (n + 1) * X (0, n + 1)))))
state_after: no goals
tactic: ring