-- Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean
theorem frobenius_frobeniusRotation {a₁ a₂ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
    frobenius (frobeniusRotation p ha₁ ha₂) * a₁ = frobeniusRotation p ha₁ ha₂ * a₂

Proof state:
case h.succ
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
⊢ (frobenius (frobeniusRotation p ha₁ ha₂) * a₁).coeff (n + 1) = (frobeniusRotation p ha₁ ha₂ * a₂).coeff (n + 1)

Tactic:
simp only [nthRemainder_spec, WittVector.coeff_frobenius_charP, frobeniusRotationCoeff,
  frobeniusRotation]

Result:
case h.succ
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
⊢ (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff (n + 1) ^ p * a₁.coeff 0 ^ p ^ (n + 1) +
        a₁.coeff (n + 1) * ((mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff 0 ^ p) ^ p ^ (n + 1) +
      nthRemainder p n (truncateFun (n + 1) (frobenius (mk p (frobeniusRotationCoeff p ha₁ ha₂))))
        (truncateFun (n + 1) a₁) =
    (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff (n + 1) * a₂.coeff 0 ^ p ^ (n + 1) +
        a₂.coeff (n + 1) * (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff 0 ^ p ^ (n + 1) +
      nthRemainder p n (truncateFun (n + 1) (mk p (frobeniusRotationCoeff p ha₁ ha₂))) (truncateFun (n + 1) a₂)

Proof state:
case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
⊢ (frobenius (frobeniusRotation p ha₁ ha₂) * a₁).coeff n = (frobeniusRotation p ha₁ ha₂ * a₂).coeff n

Tactic:
rcases n with - | n

Result:
case h.zero
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
⊢ (frobenius (frobeniusRotation p ha₁ ha₂) * a₁).coeff 0 = (frobeniusRotation p ha₁ ha₂ * a₂).coeff 0

case h.succ
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
⊢ (frobenius (frobeniusRotation p ha₁ ha₂) * a₁).coeff (n + 1) = (frobeniusRotation p ha₁ ha₂ * a₂).coeff (n + 1)

Proof state:
case h.e'_2.h.e'_6.h.e'_7
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
this :
  succNthVal p n a₁ a₂ (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) ha₁ ha₂ ^ p * a₁.coeff 0 ^ p ^ (n + 1) +
        a₁.coeff (n + 1) * (solution p a₁ a₂ ^ p) ^ p ^ (n + 1) +
      nthRemainder p n (fun v => frobeniusRotationCoeff p ha₁ ha₂ ↑v ^ p) (truncateFun (n + 1) a₁) =
    succNthVal p n a₁ a₂ (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) ha₁ ha₂ * a₂.coeff 0 ^ p ^ (n + 1) +
        a₂.coeff (n + 1) * solution p a₁ a₂ ^ p ^ (n + 1) +
      nthRemainder p n (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) (truncateFun (n + 1) a₂)
⊢ truncateFun (n + 1) (frobenius (mk p (frobeniusRotationCoeff p ha₁ ha₂))) = fun v =>
    frobeniusRotationCoeff p ha₁ ha₂ ↑v ^ p

Tactic:
apply TruncatedWittVector.ext

Result:
case h.e'_2.h.e'_6.h.e'_7.h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
this :
  succNthVal p n a₁ a₂ (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) ha₁ ha₂ ^ p * a₁.coeff 0 ^ p ^ (n + 1) +
        a₁.coeff (n + 1) * (solution p a₁ a₂ ^ p) ^ p ^ (n + 1) +
      nthRemainder p n (fun v => frobeniusRotationCoeff p ha₁ ha₂ ↑v ^ p) (truncateFun (n + 1) a₁) =
    succNthVal p n a₁ a₂ (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) ha₁ ha₂ * a₂.coeff 0 ^ p ^ (n + 1) +
        a₂.coeff (n + 1) * solution p a₁ a₂ ^ p ^ (n + 1) +
      nthRemainder p n (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) (truncateFun (n + 1) a₂)
⊢ ∀ (i : Fin (n + 1)),
    TruncatedWittVector.coeff i (truncateFun (n + 1) (frobenius (mk p (frobeniusRotationCoeff p ha₁ ha₂)))) =
      TruncatedWittVector.coeff i fun v => frobeniusRotationCoeff p ha₁ ha₂ ↑v ^ p

Proof state:
case h.succ
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
⊢ (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff (n + 1) ^ p * a₁.coeff 0 ^ p ^ (n + 1) +
        a₁.coeff (n + 1) * ((mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff 0 ^ p) ^ p ^ (n + 1) +
      nthRemainder p n (truncateFun (n + 1) (frobenius (mk p (frobeniusRotationCoeff p ha₁ ha₂))))
        (truncateFun (n + 1) a₁) =
    (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff (n + 1) * a₂.coeff 0 ^ p ^ (n + 1) +
        a₂.coeff (n + 1) * (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff 0 ^ p ^ (n + 1) +
      nthRemainder p n (truncateFun (n + 1) (mk p (frobeniusRotationCoeff p ha₁ ha₂))) (truncateFun (n + 1) a₂)

Tactic:
have :=
  succNthVal_spec' p n a₁ a₂ (fun i : Fin (n + 1) => frobeniusRotationCoeff p ha₁ ha₂ i.val)
    ha₁ ha₂

Result:
case h.succ
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
this :
  succNthVal p n a₁ a₂ (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) ha₁ ha₂ ^ p * a₁.coeff 0 ^ p ^ (n + 1) +
        a₁.coeff (n + 1) * (frobeniusRotationCoeff p ha₁ ha₂ ↑0 ^ p) ^ p ^ (n + 1) +
      nthRemainder p n (fun v => frobeniusRotationCoeff p ha₁ ha₂ ↑v ^ p) (truncateFun (n + 1) a₁) =
    succNthVal p n a₁ a₂ (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) ha₁ ha₂ * a₂.coeff 0 ^ p ^ (n + 1) +
        a₂.coeff (n + 1) * frobeniusRotationCoeff p ha₁ ha₂ ↑0 ^ p ^ (n + 1) +
      nthRemainder p n (fun i => frobeniusRotationCoeff p ha₁ ha₂ ↑i) (truncateFun (n + 1) a₂)
⊢ (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff (n + 1) ^ p * a₁.coeff 0 ^ p ^ (n + 1) +
        a₁.coeff (n + 1) * ((mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff 0 ^ p) ^ p ^ (n + 1) +
      nthRemainder p n (truncateFun (n + 1) (frobenius (mk p (frobeniusRotationCoeff p ha₁ ha₂))))
        (truncateFun (n + 1) a₁) =
    (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff (n + 1) * a₂.coeff 0 ^ p ^ (n + 1) +
        a₂.coeff (n + 1) * (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff 0 ^ p ^ (n + 1) +
      nthRemainder p n (truncateFun (n + 1) (mk p (frobeniusRotationCoeff p ha₁ ha₂))) (truncateFun (n + 1) a₂)

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
⊢ frobenius (frobeniusRotation p ha₁ ha₂) * a₁ = frobeniusRotation p ha₁ ha₂ * a₂

Tactic:
ext n

Result:
case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
n : ℕ
⊢ (frobenius (frobeniusRotation p ha₁ ha₂) * a₁).coeff n = (frobeniusRotation p ha₁ ha₂ * a₂).coeff n

Proof state:
case h.zero
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : Field k
inst✝¹ : CharP k p
inst✝ : IsAlgClosed k
a₁ a₂ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
⊢ (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff 0 ^ p * a₁.coeff 0 =
    (mk p (frobeniusRotationCoeff p ha₁ ha₂)).coeff 0 * a₂.coeff 0

Tactic:
apply solution_spec' _ ha₁

Result:
no goals