Theorem: WittVector.peval_polyOfInterest'
File: Mathlib/RingTheory/WittVector/MulCoeff.lean

tactic: rw [peval_polyOfInterest]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
⊢ peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i] =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
⊢ (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -
        y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coeff i ^ p ^ (n + 1 - i) -
      x.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * y.coeff i ^ p ^ (n + 1 - i) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)

tactic: have : (p : k) = 0 := CharP.cast_eq_zero k p
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
⊢ (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -
        y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coeff i ^ p ^ (n + 1 - i) -
      x.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * y.coeff i ^ p ^ (n + 1 - i) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
this : ↑p = 0
⊢ (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -
        y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coeff i ^ p ^ (n + 1 - i) -
      x.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * y.coeff i ^ p ^ (n + 1 - i) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)

tactic: simp only [this, Nat.cast_pow, ne_eq, add_eq_zero, and_false, zero_pow, zero_mul, add_zero,
  not_false_eq_true]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
this : ↑p = 0
⊢ (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -
        y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coeff i ^ p ^ (n + 1 - i) -
      x.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * y.coeff i ^ p ^ (n + 1 - i) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
this : ↑p = 0
⊢ (x * y).coeff (n + 1) - y.coeff (n + 1) * ∑ x_1 ∈ range (n + 1 + 1), 0 ^ x_1 * x.coeff x_1 ^ p ^ (n + 1 - x_1) -
      x.coeff (n + 1) * ∑ x ∈ range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)

tactic: have sum_zero_pow_mul_pow_p (y : 𝕎 k) : ∑ x ∈ range (n + 1 + 1),
    (0 : k) ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0 ^ p ^ (n + 1) := by
  rw [Finset.sum_eq_single_of_mem 0] <;> simp (config := { contextual := true })
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
this : ↑p = 0
⊢ (x * y).coeff (n + 1) - y.coeff (n + 1) * ∑ x_1 ∈ range (n + 1 + 1), 0 ^ x_1 * x.coeff x_1 ^ p ^ (n + 1 - x_1) -
      x.coeff (n + 1) * ∑ x ∈ range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
this : ↑p = 0
sum_zero_pow_mul_pow_p :
  ∀ (y : 𝕎 k), ∑ x ∈ range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0 ^ p ^ (n + 1)
⊢ (x * y).coeff (n + 1) - y.coeff (n + 1) * ∑ x_1 ∈ range (n + 1 + 1), 0 ^ x_1 * x.coeff x_1 ^ p ^ (n + 1 - x_1) -
      x.coeff (n + 1) * ∑ x ∈ range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)

tactic: congr <;> apply sum_zero_pow_mul_pow_p
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
this : ↑p = 0
sum_zero_pow_mul_pow_p :
  ∀ (y : 𝕎 k), ∑ x ∈ range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0 ^ p ^ (n + 1)
⊢ (x * y).coeff (n + 1) - y.coeff (n + 1) * ∑ x_1 ∈ range (n + 1 + 1), 0 ^ x_1 * x.coeff x_1 ^ p ^ (n + 1 - x_1) -
      x.coeff (n + 1) * ∑ x ∈ range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
state_after: no goals

tactic: rw [Finset.sum_eq_single_of_mem 0] <;> simp (config := { contextual := true })
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y✝ : 𝕎 k
this : ↑p = 0
y : 𝕎 k
⊢ ∑ x ∈ range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0 ^ p ^ (n + 1)
state_after: no goals