-- Module: Mathlib.RingTheory.WittVector.Frobenius
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
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
apply :
  ∀ x ∈ range (p ^ (n - i)),
    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)
          (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) ^
                (x + 1) *
              HPow.hPow (α := MvPolynomial ℕ ℚ) (X i) p ^ (p ^ (n - i) - (x + 1)) *
            (↑((p ^ (n - i)).choose (x + 1)) : MvPolynomial ℕ ℚ)))
        ((C : (a : ℚ) → MvPolynomial ℕ ℚ)
            (HPow.hPow (α := ℚ) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne (↑p : ℚ) _) n) :
          MvPolynomial ℕ ℚ) =
      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) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *
              (C : (a : ℤ) → MvPolynomial ℕ ℤ)
                (↑((p ^ (n - i)).choose (x + 1) / p ^ (n - i - v p (x + 1)) * p ^ (x - v p (x + 1))) : ℤ)) :
          MvPolynomial ℕ ℚ)
⊢ Eq (α := MvPolynomial ℕ ℚ)
    (∑ i_1 ∈ range (p ^ (n - i)),
      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)
          (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) ^
                (i_1 + 1) *
              HPow.hPow (α := MvPolynomial ℕ ℚ) (X i) p ^ (p ^ (n - i) - (i_1 + 1)) *
            (↑((p ^ (n - i)).choose (i_1 + 1)) : MvPolynomial ℕ ℚ)))
        ((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:
apply sum_congr rfl

Premises used: Finset.sum_congr, rfl