awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
c : idx × ℕ →₀ ℕ
i : ℕ
hi : i < n
simp :
  HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
        ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) ((Int.castRingHom ℚ : (a : ℤ) → ℚ) (↑p : ℤ)) :
          MvPolynomial (idx × ℕ) ℚ)
        i *
      HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
        (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
        (p ^ (n - i)) =
    HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ) ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (↑p : ℚ)) i *
      HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
        (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
        (p ^ (n - i))
⊢ (map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
      (HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
        ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
        (wittStructureInt p Φ i ^ p ^ (n - i))) =
    HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
      ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
      (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
        (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
        (p ^ (n - i)))

Tactic:
simp only [IH i hi, map_mul, map_pow, map_C]

Premises used: map_pow, map_mul, MvPolynomial.map_C
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
c : idx × ℕ →₀ ℕ
i : ℕ
hi : i < n
⊢ HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
        ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) ((Int.castRingHom ℚ : (a : ℤ) → ℚ) (↑p : ℤ)) :
          MvPolynomial (idx × ℕ) ℚ)
        i *
      HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
        (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
        (p ^ (n - i)) =
    HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ) ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (↑p : ℚ)) i *
      HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
        (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
        (p ^ (n - i))

Tactic:
rfl
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
c : idx × ℕ →₀ ℕ
⊢ Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
    (∑ i ∈ Finset.range n,
        HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
          ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
          (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
            (p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)

Tactic:
have sum_induction_steps :
      map (Int.castRingHom ℚ)
        (∑ i ∈ range n, C ((p : ℤ) ^ i) * wittStructureInt p Φ i ^ p ^ (n - i)) =
      ∑ i ∈ range n,
        C ((p : ℚ) ^ i) * wittStructureRat p (map (Int.castRingHom ℚ) Φ) i ^ p ^ (n - i) := by
    rw [map_sum]
    apply Finset.sum_congr rfl
    intro i hi
    rw [Finset.mem_range] at hi
    simp only [IH i hi, map_mul, map_pow, map_C]
    rfl
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
c : idx × ℕ →₀ ℕ
sum_induction_steps :
  Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
    (∑ i ∈ Finset.range n,
        HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
          ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
          (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
            (p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
norm_cast norm_cast₁ norm_cast₂ norm_cast₃ : p ^ n = p ^ n
⊢ HPow.hPow (α := ℚ) (↑p : ℚ) n = (↑(↑(p ^ n) : ℤ) : ℚ)

Tactic:
norm_cast
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
c : idx × ℕ →₀ ℕ
sum_induction_steps :
  Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
    (∑ i ∈ Finset.range n,
        HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
          ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
          (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
            (p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
⊢ Ne (α := ℤ) (↑(p ^ n) : ℤ) 0

Tactic:
mod_cast pow_ne_zero n hp.1.ne_zero

Premises used: congrArg, cast, pow_ne_zero, Ne, Nat.cast, HPow.hPow, Nat.Prime.ne_zero, Nat.cast_inj._simp_1, Nat.cast_zero, Eq.symm, Not, Fact.out, Eq.trans
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
c : idx × ℕ →₀ ℕ
sum_induction_steps :
  Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
    (∑ i ∈ Finset.range n,
        HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
          ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
          (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
            (p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
⊢ Ne (α := ℤ) (↑(p ^ n) : ℤ) 0

Tactic:
exact mod_cast pow_ne_zero n hp.1.ne_zero

Premises used: pow_ne_zero, Nat.Prime.ne_zero, Fact.out
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
c : idx × ℕ →₀ ℕ
sum_induction_steps :
  Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
    (∑ i ∈ Finset.range n,
        HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
          ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
          (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
            (p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
h :
  ∀ (c : idx × ℕ →₀ ℕ),
    (↑(p ^ n) : ℤ) ∣
      coeff c
        (HSub.hSub (α := MvPolynomial (idx × ℕ) ℤ) (β := MvPolynomial (idx × ℕ) ℤ)
          ((bind₁ fun (i : idx) ↦
                  (rename fun (i_1 : ℕ) ↦ (i, i_1) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
                (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
              Φ :
            MvPolynomial (idx × ℕ) ℤ)
          (∑ i ∈ Finset.range n,
              HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
                ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
                (wittStructureInt p Φ i ^ p ^ (n - i)) :
            MvPolynomial (idx × ℕ) ℤ))
⊢ (↑(p ^ n) : ℤ) ∣
    coeff c
      (HSub.hSub (α := MvPolynomial (idx × ℕ) ℤ) (β := MvPolynomial (idx × ℕ) ℤ)
        ((bind₁ fun (i : idx) ↦
                (rename fun (i_1 : ℕ) ↦ (i, i_1) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
              (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
            Φ :
          MvPolynomial (idx × ℕ) ℤ)
        (∑ i ∈ Finset.range n,
            HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
              ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
              (wittStructureInt p Φ i ^ p ^ (n - i)) :
          MvPolynomial (idx × ℕ) ℤ))

Tactic:
revert c
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
sum_induction_steps :
  Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
    (∑ i ∈ Finset.range n,
        HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
          ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
          (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
            (p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
h :
  (C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (↑(p ^ n) : ℤ) ∣
    HSub.hSub (α := MvPolynomial (idx × ℕ) ℤ) (β := MvPolynomial (idx × ℕ) ℤ)
      ((bind₁ fun (i : idx) ↦
              (rename fun (i_1 : ℕ) ↦ (i, i_1) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
            (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
          Φ :
        MvPolynomial (idx × ℕ) ℤ)
      (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i)) :
        MvPolynomial (idx × ℕ) ℤ)
⊢ ∀ (c : idx × ℕ →₀ ℕ),
    (↑(p ^ n) : ℤ) ∣
      coeff c
        (HSub.hSub (α := MvPolynomial (idx × ℕ) ℤ) (β := MvPolynomial (idx × ℕ) ℤ)
          ((bind₁ fun (i : idx) ↦
                  (rename fun (i_1 : ℕ) ↦ (i, i_1) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
                (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
              Φ :
            MvPolynomial (idx × ℕ) ℤ)
          (∑ i ∈ Finset.range n,
              HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
                ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
                (wittStructureInt p Φ i ^ p ^ (n - i)) :
            MvPolynomial (idx × ℕ) ℤ))

Tactic:
rw [← C_dvd_iff_dvd_coeff]

Premises used: MvPolynomial.C_dvd_iff_dvd_coeff
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
sum_induction_steps :
  Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
    (∑ i ∈ Finset.range n,
        HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
          ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
          (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
            (p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
⊢ (C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (↑(p ^ n) : ℤ) ∣
    HSub.hSub (α := MvPolynomial (idx × ℕ) ℤ) (β := MvPolynomial (idx × ℕ) ℤ)
      ((bind₁ fun (i : idx) ↦
              (rename fun (i_1 : ℕ) ↦ (i, i_1) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
            (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
          Φ :
        MvPolynomial (idx × ℕ) ℤ)
      (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i)) :
        MvPolynomial (idx × ℕ) ℤ)

Tactic:
C_p_pow_dvd_bind₁_rename_wittPolynomial_sub_sum Φ n IH

Premises used: C_p_pow_dvd_bind₁_rename_wittPolynomial_sub_sum
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n,
    Eq (α := MvPolynomial (idx × ℕ) ℚ)
      ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ) (wittStructureInt p Φ m) :
        MvPolynomial (idx × ℕ) ℚ)
      (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) m :
        MvPolynomial (idx × ℕ) ℚ)
sum_induction_steps :
  Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
    (∑ i ∈ Finset.range n,
        HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
          ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
          (HPow.hPow (α := MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ) i)
            (p ^ (n - i))) :
      MvPolynomial (idx × ℕ) ℚ)
⊢ (C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (↑(p ^ n) : ℤ) ∣
    HSub.hSub (α := MvPolynomial (idx × ℕ) ℤ) (β := MvPolynomial (idx × ℕ) ℤ)
      ((bind₁ fun (i : idx) ↦
              (rename fun (i_1 : ℕ) ↦ (i, i_1) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
            (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
          Φ :
        MvPolynomial (idx × ℕ) ℤ)
      (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℤ)
            ((C : (a : ℤ) → MvPolynomial (idx × ℕ) ℤ) (HPow.hPow (α := ℤ) (↑p : ℤ) i) : MvPolynomial (idx × ℕ) ℤ)
            (wittStructureInt p Φ i ^ p ^ (n - i)) :
        MvPolynomial (idx × ℕ) ℤ)

Tactic:
exact C_p_pow_dvd_bind₁_rename_wittPolynomial_sub_sum Φ n IH

Premises used: C_p_pow_dvd_bind₁_rename_wittPolynomial_sub_sum
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
a :
  Eq (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        ((bind₁ (wittStructureInt p Φ) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
          MvPolynomial (idx × ℕ) ℤ) :
      MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        ((bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
              (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
            Φ :
          MvPolynomial (idx × ℕ) ℤ) :
      MvPolynomial (idx × ℕ) ℚ)
⊢ Eq (α := MvPolynomial (idx × ℕ) ℤ)
    ((bind₁ (wittStructureInt p Φ) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
      MvPolynomial (idx × ℕ) ℤ)
    ((bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
          (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
        Φ :
      MvPolynomial (idx × ℕ) ℤ)

Tactic:
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective

Premises used: Rat, MvPolynomial.map_injective, Int.castRingHom, Int.cast_injective
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
⊢ Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((bind₁ (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ)) :
          (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ)
        (W_ ℚ n) :
      MvPolynomial (idx × ℕ) ℚ)
    ((bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ) (W_ ℚ n) :
          (a : MvPolynomial idx ℚ) → MvPolynomial (idx × ℕ) ℚ)
        ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ : MvPolynomial idx ℚ) :
      MvPolynomial (idx × ℕ) ℚ)

Tactic:
have := wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
this :
  Eq.{u_2 + 1} (α := MvPolynomial (idx × ℕ) ℚ)
    ((bind₁ (wittStructureRat p ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ)) :
          (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ)
        (W_ ℚ n) :
      MvPolynomial (idx × ℕ) ℚ)
    ((bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ) (W_ ℚ n) :
          (a : MvPolynomial idx ℚ) → MvPolynomial (idx × ℕ) ℚ)
        ((map (Int.castRingHom ℚ) : (a : MvPolynomial idx ℤ) → MvPolynomial idx ℚ) Φ : MvPolynomial idx ℚ) :
      MvPolynomial (idx × ℕ) ℚ)
⊢ Eq (α := MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        ((bind₁ (wittStructureInt p Φ) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
          MvPolynomial (idx × ℕ) ℤ) :
      MvPolynomial (idx × ℕ) ℚ)
    ((map (Int.castRingHom ℚ) : (a : MvPolynomial (idx × ℕ) ℤ) → MvPolynomial (idx × ℕ) ℚ)
        ((bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
              (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
            Φ :
          MvPolynomial (idx × ℕ) ℤ) :
      MvPolynomial (idx × ℕ) ℚ)

Tactic:
simpa only [map_bind₁, ← eval₂Hom_map_hom, eval₂Hom_C_left, map_rename, map_wittPolynomial,
    AlgHom.coe_toRingHom, map_wittStructureInt]

Premises used: MvPolynomial.map_bind₁, AlgHom.coe_toRingHom, MvPolynomial.map_rename, MvPolynomial.eval₂Hom_map_hom, map_wittPolynomial, MvPolynomial.eval₂Hom_C_left, map_wittStructureInt
-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
φ : ℕ → MvPolynomial (idx × ℕ) ℤ
h :
  ∀ (n : ℕ),
    Eq (α := MvPolynomial (idx × ℕ) ℤ)
      ((bind₁ φ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) : MvPolynomial (idx × ℕ) ℤ)
      ((bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℤ) → MvPolynomial (idx × ℕ) ℤ) (W_ ℤ n) :
            (a : MvPolynomial idx ℤ) → MvPolynomial (idx × ℕ) ℤ)
          Φ :
        MvPolynomial (idx × ℕ) ℤ)
h₁ : ∀ (k : ℕ), φ k = wittStructureInt p Φ k
⊢ φ = wittStructureInt p Φ

Tactic:
funext k