l3lab/ntp-mathlib huggingface dataset:
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPoly p n) = frobeniusPolyRat p n

Next tactic:
rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ X n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
    (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ n)

Next tactic:
refine Nat.strong_induction_on n ?_
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; 

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ ∀ (n : ℕ),
    (∀ 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)) →
      X n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
        (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ n)

Next tactic:
clear n
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ ∀ (n : ℕ),
    (∀ 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)) →
      X n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
        (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ n)

Next tactic:
intro n IH
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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)
⊢ X n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
    (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ n)

Next tactic:
rw [xInTermsOfW_eq]
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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)
⊢ X n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
    (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1))
      ((X n - ∑ i in range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) * C (⅟↑p ^ n))

Next tactic:
simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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)
⊢ X n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
    ((bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (X n) -
        ∑ x in range n, C (↑p ^ x) * (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ x) ^ p ^ (n - x)) *
      C (⅟↑p ^ n)

Next tactic:
have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by 

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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

Next tactic:
rw [← mul_pow, mul_invOf_self, one_pow]
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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 n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
    ((bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (X n) -
        ∑ x in range n, C (↑p ^ x) * (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ x) ^ p ^ (n - x)) *
      C (⅟↑p ^ n)

Next tactic:
rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
    sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
    add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ,
    mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
    ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
    add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
    add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
  rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
    sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
    add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ,
    mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
    ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
    add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
    add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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 in 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 in range n, C (↑p ^ x) * X x ^ p ^ (n + 1 - x)) * C (⅟↑p ^ n) =
    C ↑p *
      (MvPolynomial.map (Int.castRingHom ℚ))
        (∑ i in range n,
          ∑ j in range (p ^ (n - i)),
            (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *
              C
                ↑(Nat.choose (p ^ (n - i)) (j + 1) /
                      p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *
                    p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })))

Next tactic:
simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
  rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
    sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
    add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ,
    mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
    ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
    add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
    add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
  simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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 in 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 in range n,
      ∑ x_1 in range (p ^ (n - x)),
        C ↑p *
          (MvPolynomial.map (Int.castRingHom ℚ))
            ((X x ^ p) ^ (p ^ (n - x) - (x_1 + 1)) * frobeniusPolyAux p x ^ (x_1 + 1) *
              C
                ↑(Nat.choose (p ^ (n - x)) (x_1 + 1) /
                      p ^ (n - x - v p { val := x_1 + 1, property := (_ : 0 < Nat.succ x_1) }) *
                    p ^ (x_1 - v p { val := x_1 + 1, property := (_ : 0 < Nat.succ x_1) })))

Next tactic:
apply sum_congr rfl
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
  rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
    sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
    add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ,
    mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
    ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
    add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
    add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
  simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]
  apply sum_congr rfl
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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_1 in range (p ^ (n - x)),
        C ↑p *
          (MvPolynomial.map (Int.castRingHom ℚ))
            ((X x ^ p) ^ (p ^ (n - x) - (x_1 + 1)) * frobeniusPolyAux p x ^ (x_1 + 1) *
              C
                ↑(Nat.choose (p ^ (n - x)) (x_1 + 1) /
                      p ^ (n - x - v p { val := x_1 + 1, property := (_ : 0 < Nat.succ x_1) }) *
                    p ^ (x_1 - v p { val := x_1 + 1, property := (_ : 0 < Nat.succ x_1) })))

Next tactic:
intro i hi
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
  rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
    sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
    add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ,
    mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
    ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
    add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
    add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
  simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]
  apply sum_congr rfl
  intro i hi
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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) =
    ∑ x in range (p ^ (n - i)),
      C ↑p *
        (MvPolynomial.map (Int.castRingHom ℚ))
          ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *
            C
              ↑(Nat.choose (p ^ (n - i)) (x + 1) /
                    p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *
                  p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))

Next tactic:
rw [mem_range] at hi
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
  rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
    sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
    add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ,
    mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
    ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
    add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
    add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
  simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]
  apply sum_congr rfl
  intro i hi
  rw [mem_range] at hi
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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) =
    ∑ x in range (p ^ (n - i)),
      C ↑p *
        (MvPolynomial.map (Int.castRingHom ℚ))
          ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *
            C
              ↑(Nat.choose (p ^ (n - i)) (x + 1) /
                    p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *
                  p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))

Next tactic:
rw [← IH i hi]
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
  rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
    sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
    add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ,
    mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
    ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
    add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
    add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
  simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]
  apply sum_congr rfl
  intro i hi
  rw [mem_range] at hi
  rw [← IH i hi]
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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) * (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) =
    ∑ x in range (p ^ (n - i)),
      C ↑p *
        (MvPolynomial.map (Int.castRingHom ℚ))
          ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *
            C
              ↑(Nat.choose (p ^ (n - i)) (x + 1) /
                    p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *
                  p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))

Next tactic:
clear IH
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
  rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
    Int.cast_ofNat, frobeniusPolyRat]
  refine Nat.strong_induction_on n ?_; clear n
  intro n IH
  rw [xInTermsOfW_eq]
  simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
  have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
  rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
    sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
    add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ,
    mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
    ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
    add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
    add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
  simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]
  apply sum_congr rfl
  intro i hi
  rw [mem_range] at hi
  rw [← IH i hi]
  clear IH
  

Proof state:
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
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) =
    ∑ x in range (p ^ (n - i)),
      C ↑p *
        (MvPolynomial.map (Int.castRingHom ℚ))
          ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *
            C
              ↑(Nat.choose (p ^ (n - i)) (x + 1) /
                    p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *
                  p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))

Next 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, mul_sum, sum_mul]