-- Mathlib/RingTheory/WittVector/Frobenius.lean
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
this :
  ↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p * ↑p ^ n =
    ↑p ^ j * ↑p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) * ↑p ^ (n - i - v p (j + 1))
⊢ ↑p ^ j * ↑p * (↑((p ^ (n - i)).choose (j + 1)) * (↑p ^ i * (↑p ^ n)⁻¹)) =
    ↑((p ^ (n - i)).choose (j + 1)) / ↑p ^ (n - i - v p (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p

Tactic:
have aux : ∀ k : ℕ, (p : ℚ)^ k ≠ 0 := by
  intro; apply pow_ne_zero; exact mod_cast hp.1.ne_zero

Result:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
this :
  ↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p * ↑p ^ n =
    ↑p ^ j * ↑p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) * ↑p ^ (n - i - v p (j + 1))
aux : ∀ (k : ℕ), ↑p ^ k ≠ 0
⊢ ↑p ^ j * ↑p * (↑((p ^ (n - i)).choose (j + 1)) * (↑p ^ i * (↑p ^ n)⁻¹)) =
    ↑((p ^ (n - i)).choose (j + 1)) / ↑p ^ (n - i - v p (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ m)
⊢ ↑p ^ n * ⅟↑p ^ n = 1

Tactic:
rw [← mul_pow, mul_invOf_self, one_pow]

Result:
no goals

Proof state:
case h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
this :
  ↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p * ↑p ^ n =
    ↑p ^ j * ↑p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) * ↑p ^ (n - i - v p (j + 1))
k✝ : ℕ
⊢ ↑p ≠ 0

Tactic:
exact mod_cast hp.1.ne_zero

Result:
no goals

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
⊢ C (↑p ^ i) * (X i ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p i)) ^ p ^ (n - i) *
        C (⅟↑p ^ n) -
      C (↑p ^ i) * X i ^ p ^ (n + 1 - i) * C (⅟↑p ^ n) =
    ∑ i_1 ∈ range (p ^ (n - i)),
      C ↑p *
        (MvPolynomial.map (Int.castRingHom ℚ))
          ((X i ^ p) ^ (p ^ (n - i) - (i_1 + 1)) * frobeniusPolyAux p i ^ (i_1 + 1) *
            C ↑((p ^ (n - i)).choose (i_1 + 1) / p ^ (n - i - v p (i_1 + 1)) * p ^ (i_1 - v p (i_1 + 1))))

Tactic:
rw [add_comm (X i ^ p), add_pow, sum_range_succ', pow_zero, tsub_zero, Nat.choose_zero_right,
  one_mul, Nat.cast_one, mul_one, mul_add, add_mul, Nat.succ_sub (le_of_lt hi),
  Nat.succ_eq_add_one (n - i), pow_succ', pow_mul, add_sub_cancel_right, mul_sum, sum_mul]

Result:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
⊢ ∑ i_1 ∈ range (p ^ (n - i)),
      C (↑p ^ i) *
          ((C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p i)) ^ (i_1 + 1) *
              (X i ^ p) ^ (p ^ (n - i) - (i_1 + 1)) *
            ↑((p ^ (n - i)).choose (i_1 + 1))) *
        C (⅟↑p ^ n) =
    ∑ i_1 ∈ range (p ^ (n - i)),
      C ↑p *
        (MvPolynomial.map (Int.castRingHom ℚ))
          ((X i ^ p) ^ (p ^ (n - i) - (i_1 + 1)) * frobeniusPolyAux p i ^ (i_1 + 1) *
            C ↑((p ^ (n - i)).choose (i_1 + 1) / p ^ (n - i - v p (i_1 + 1)) * p ^ (i_1 - v p (i_1 + 1))))

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
⊢ (∑ x ∈ range n, C (↑p ^ x) * (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ x) ^ p ^ (n - x)) *
        C (⅟↑p ^ n) -
      (∑ x ∈ range n, C (↑p ^ x) * X x ^ p ^ (n + 1 - x)) * C (⅟↑p ^ n) =
    C ↑p *
      (MvPolynomial.map (Int.castRingHom ℚ))
        (∑ i ∈ range n,
          ∑ j ∈ range (p ^ (n - i)),
            (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *
              C ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p (j + 1)) * p ^ (j - v p (j + 1))))

Tactic:
simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]

Result:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
⊢ ∑ x ∈ range n,
      (C (↑p ^ x) * (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ x) ^ p ^ (n - x) * C (⅟↑p ^ n) -
        C (↑p ^ x) * X x ^ p ^ (n + 1 - x) * C (⅟↑p ^ n)) =
    ∑ x ∈ range n,
      ∑ i ∈ range (p ^ (n - x)),
        C ↑p *
          (MvPolynomial.map (Int.castRingHom ℚ))
            ((X x ^ p) ^ (p ^ (n - x) - (i + 1)) * frobeniusPolyAux p x ^ (i + 1) *
              C ↑((p ^ (n - x)).choose (i + 1) / p ^ (n - x - v p (i + 1)) * p ^ (i - v p (i + 1))))

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i ∈ range n
⊢ C (↑p ^ i) * (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ i) ^ p ^ (n - i) * C (⅟↑p ^ n) -
      C (↑p ^ i) * X i ^ p ^ (n + 1 - i) * C (⅟↑p ^ n) =
    ∑ i_1 ∈ range (p ^ (n - i)),
      C ↑p *
        (MvPolynomial.map (Int.castRingHom ℚ))
          ((X i ^ p) ^ (p ^ (n - i) - (i_1 + 1)) * frobeniusPolyAux p i ^ (i_1 + 1) *
            C ↑((p ^ (n - i)).choose (i_1 + 1) / p ^ (n - i - v p (i_1 + 1)) * p ^ (i_1 - v p (i_1 + 1))))

Tactic:
rw [mem_range] at hi

Result:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
⊢ C (↑p ^ i) * (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ i) ^ p ^ (n - i) * C (⅟↑p ^ n) -
      C (↑p ^ i) * X i ^ p ^ (n + 1 - i) * C (⅟↑p ^ n) =
    ∑ i_1 ∈ range (p ^ (n - i)),
      C ↑p *
        (MvPolynomial.map (Int.castRingHom ℚ))
          ((X i ^ p) ^ (p ^ (n - i) - (i_1 + 1)) * frobeniusPolyAux p i ^ (i_1 + 1) *
            C ↑((p ^ (n - i)).choose (i_1 + 1) / p ^ (n - i - v p (i_1 + 1)) * p ^ (i_1 - v p (i_1 + 1))))

Proof state:
case h.h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
⊢ C ↑p ^ (j + 1) * (C ↑((p ^ (n - i)).choose (j + 1)) * (C ↑p ^ i * C ⅟↑p ^ n)) =
    C ((Int.castRingHom ℚ) ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p (j + 1)) * p ^ (j - v p (j + 1)))) * C ↑p

Tactic:
simp only [← RingHom.map_pow, ← C_mul]

Result:
case h.h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
⊢ C (↑p ^ (j + 1) * (↑((p ^ (n - i)).choose (j + 1)) * (↑p ^ i * ⅟↑p ^ n))) =
    C ((Int.castRingHom ℚ) ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p (j + 1)) * p ^ (j - v p (j + 1))) * ↑p)

Proof state:
case h.h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
⊢ ↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p * ↑p ^ n =
    ↑p ^ j * ↑p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) * ↑p ^ (n - i - v p (j + 1))

Tactic:
rw [mul_comm _ (p : ℚ), mul_assoc, mul_assoc, ← pow_add,
  map_frobeniusPoly.key₂ p hi.le hj, Nat.cast_mul, Nat.cast_pow]

Result:
case h.h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
⊢ ↑p * (↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ (i + j + (n - i - v p (j + 1)))) =
    ↑p ^ j * ↑p * (↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ i) * ↑p ^ (n - i - v p (j + 1))