-- Module: Mathlib.RingTheory.WittVector.Frobenius
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    HPow.hPow (α := MvPolynomial ℕ ℚ) (X m) p +
        HMul.hMul (α := MvPolynomial ℕ ℚ) (β := MvPolynomial ℕ ℚ)
          ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (↑p : ℚ) : MvPolynomial ℕ ℚ)
          ((MvPolynomial.map (Int.castRingHom ℚ) : (a : MvPolynomial ℕ ℤ) → MvPolynomial ℕ ℚ) (frobeniusPolyAux p m) :
            MvPolynomial ℕ ℚ) =
      (bind₁ (wittPolynomial p ℚ ∘ fun (n : ℕ) ↦ n + 1) : (a : MvPolynomial ℕ ℚ) → MvPolynomial ℕ ℚ) (xInTermsOfW p ℚ m)
h1 :
  HPow.hPow (α := ℚ) (↑p : ℚ) n *
      HPow.hPow (α := ℚ) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (↑p : ℚ) _) n =
    1
i : ℕ
hi : i < n
rw :
  HMul.hMul (β := MvPolynomial ℕ ℚ)
        (HMul.hMul (α := MvPolynomial ℕ ℚ)
          ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial ℕ ℚ)
          ((HPow.hPow (α := MvPolynomial ℕ ℚ) (X i) p +
              HMul.hMul (α := MvPolynomial ℕ ℚ) (β := MvPolynomial ℕ ℚ)
                ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (↑p : ℚ) : MvPolynomial ℕ ℚ)
                ((MvPolynomial.map (Int.castRingHom ℚ) : (a : MvPolynomial ℕ ℤ) → MvPolynomial ℕ ℚ)
                    (frobeniusPolyAux p i) :
                  MvPolynomial ℕ ℚ)) ^
            p ^ (n - i)))
        ((C : (a : ℚ) → MvPolynomial ℕ ℚ)
            (HPow.hPow (α := ℚ) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (↑p : ℚ) _) n) :
          MvPolynomial ℕ ℚ) -
      HMul.hMul (β := MvPolynomial ℕ ℚ)
        (HMul.hMul (α := MvPolynomial ℕ ℚ)
          ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial ℕ ℚ)
          (HPow.hPow (α := MvPolynomial ℕ ℚ) (X i) (p ^ (n + 1 - i))))
        ((C : (a : ℚ) → MvPolynomial ℕ ℚ)
            (HPow.hPow (α := ℚ) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (↑p : ℚ) _) n) :
          MvPolynomial ℕ ℚ) =
    ∑ i_1 ∈ range (p ^ (n - i)),
      HMul.hMul (α := MvPolynomial ℕ ℚ) (β := MvPolynomial ℕ ℚ)
        ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (↑p : ℚ) : MvPolynomial ℕ ℚ)
        ((MvPolynomial.map (Int.castRingHom ℚ) : (a : MvPolynomial ℕ ℤ) → MvPolynomial ℕ ℚ)
            (HPow.hPow (α := MvPolynomial ℕ ℤ) (X i) p ^ (p ^ (n - i) - (i_1 + 1)) * frobeniusPolyAux p i ^ (i_1 + 1) *
              (C : (a : ℤ) → MvPolynomial ℕ ℤ)
                (↑((p ^ (n - i)).choose (i_1 + 1) / p ^ (n - i - v p (i_1 + 1)) * p ^ (i_1 - v p (i_1 + 1))) : ℤ)) :
          MvPolynomial ℕ ℚ)
⊢ HMul.hMul (β := MvPolynomial ℕ ℚ)
        (HMul.hMul (α := MvPolynomial ℕ ℚ)
          ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial ℕ ℚ)
          (HPow.hPow (α := MvPolynomial ℕ ℚ)
            ((bind₁ (wittPolynomial p ℚ ∘ fun (n : ℕ) ↦ n + 1) : (a : MvPolynomial ℕ ℚ) → MvPolynomial ℕ ℚ)
                (xInTermsOfW p ℚ i) :
              MvPolynomial ℕ ℚ)
            (p ^ (n - i))))
        ((C : (a : ℚ) → MvPolynomial ℕ ℚ)
            (HPow.hPow (α := ℚ) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (↑p : ℚ) _) n) :
          MvPolynomial ℕ ℚ) -
      HMul.hMul (β := MvPolynomial ℕ ℚ)
        (HMul.hMul (α := MvPolynomial ℕ ℚ)
          ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial ℕ ℚ)
          (HPow.hPow (α := MvPolynomial ℕ ℚ) (X i) (p ^ (n + 1 - i))))
        ((C : (a : ℚ) → MvPolynomial ℕ ℚ)
            (HPow.hPow (α := ℚ) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (↑p : ℚ) _) n) :
          MvPolynomial ℕ ℚ) =
    ∑ i_1 ∈ range (p ^ (n - i)),
      HMul.hMul (α := MvPolynomial ℕ ℚ) (β := MvPolynomial ℕ ℚ)
        ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (↑p : ℚ) : MvPolynomial ℕ ℚ)
        ((MvPolynomial.map (Int.castRingHom ℚ) : (a : MvPolynomial ℕ ℤ) → MvPolynomial ℕ ℚ)
            (HPow.hPow (α := MvPolynomial ℕ ℤ) (X i) p ^ (p ^ (n - i) - (i_1 + 1)) * frobeniusPolyAux p i ^ (i_1 + 1) *
              (C : (a : ℤ) → MvPolynomial ℕ ℤ)
                (↑((p ^ (n - i)).choose (i_1 + 1) / p ^ (n - i - v p (i_1 + 1)) * p ^ (i_1 - v p (i_1 + 1))) : ℤ)) :
          MvPolynomial ℕ ℚ)

Tactic:
rw [← IH i hi]