| import Mathlib.Algebra.Polynomial.Degree.Lemmas |
| import Mathlib.NumberTheory.BernoulliPolynomials |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W24 |
|
|
| open Polynomial Finset |
|
|
| noncomputable def powerPrefixPolynomial (d : ℕ) : Polynomial ℚ := |
| C ((d + 1 : ℚ)⁻¹) * (Polynomial.bernoulli (d + 1) - C (_root_.bernoulli (d + 1))) |
|
|
| theorem powerPrefixPolynomial_eval (d n : ℕ) : |
| (powerPrefixPolynomial d).eval (n : ℚ) = ∑ j ∈ range n, (j : ℚ) ^ d := by |
| rw [powerPrefixPolynomial, eval_mul, eval_C, eval_sub, eval_C] |
| rw [← Polynomial.sum_range_pow_eq_bernoulli_sub n d] |
| have hd : (d + 1 : ℚ) ≠ 0 := by positivity |
| exact inv_mul_cancel_left₀ hd _ |
|
|
| theorem powerPrefixPolynomial_degree_le (d : ℕ) : |
| (powerPrefixPolynomial d).natDegree ≤ d + 1 := by |
| apply natDegree_le_iff_coeff_eq_zero.mpr |
| intro j hj |
| have hj0 : j ≠ 0 := by omega |
| simp only [powerPrefixPolynomial, coeff_C_mul, coeff_sub, |
| Polynomial.coeff_bernoulli, ite_eq_right (by omega : ¬j ≤ d + 1), |
| coeff_C, ite_eq_right hj0, sub_zero, mul_zero] |
|
|
| theorem powerPrefixPolynomial_top_coefficient (d : ℕ) : |
| (powerPrefixPolynomial d).coeff (d + 1) = (d + 1 : ℚ)⁻¹ := by |
| simp [powerPrefixPolynomial, coeff_C_mul, Polynomial.coeff_bernoulli] |
|
|
| noncomputable def polynomialPrefix (p : Polynomial ℚ) : Polynomial ℚ := |
| ∑ i ∈ range (p.natDegree + 1), C (p.coeff i) * powerPrefixPolynomial i |
|
|
| theorem polynomialPrefix_eval (p : Polynomial ℚ) (n : ℕ) : |
| (polynomialPrefix p).eval (n : ℚ) = ∑ j ∈ range n, p.eval (j : ℚ) := by |
| simp only [polynomialPrefix, eval_finsetSum, eval_mul, eval_C, |
| powerPrefixPolynomial_eval, Finset.mul_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| exact (Polynomial.eval_eq_sum_range (p := p) (j : ℚ)).symm |
|
|
| theorem polynomialPrefix_degree_le (p : Polynomial ℚ) : |
| (polynomialPrefix p).natDegree ≤ p.natDegree + 1 := by |
| apply Polynomial.natDegree_sum_le_of_forall_le |
| intro i hi |
| exact (natDegree_C_mul_le _ _).trans |
| ((powerPrefixPolynomial_degree_le i).trans (by |
| have hii := Finset.mem_range.mp hi |
| omega)) |
|
|
| theorem polynomialPrefix_top_coefficient (p : Polynomial ℚ) : |
| (polynomialPrefix p).coeff (p.natDegree + 1) = |
| p.leadingCoeff * (p.natDegree + 1 : ℚ)⁻¹ := by |
| rw [polynomialPrefix, finsetSum_coeff] |
| rw [Finset.sum_eq_single p.natDegree] |
| · rw [coeff_C_mul, powerPrefixPolynomial_top_coefficient, coeff_natDegree] |
| · intro i hi hne |
| have hz : (powerPrefixPolynomial i).coeff (p.natDegree + 1) = 0 := |
| coeff_eq_zero_of_natDegree_lt ((powerPrefixPolynomial_degree_le i).trans_lt (by |
| have hii := Finset.mem_range.mp hi |
| omega)) |
| rw [coeff_C_mul, hz, mul_zero] |
| · intro h |
| exact (h (Finset.mem_range.mpr (Nat.lt_succ_self _))).elim |
|
|
| theorem polynomialPrefix_natDegree (p : Polynomial ℚ) (hp : p ≠ 0) : |
| (polynomialPrefix p).natDegree = p.natDegree + 1 := by |
| apply natDegree_eq_of_le_of_coeff_ne_zero (polynomialPrefix_degree_le p) |
| rw [polynomialPrefix_top_coefficient] |
| exact mul_ne_zero (leadingCoeff_ne_zero.mpr hp) (inv_ne_zero (by positivity)) |
|
|
| theorem exists_eventual_cumulative_polynomial |
| (values : ℕ → ℕ) (p : Polynomial ℚ) (hp0 : p ≠ 0) (N : ℕ) |
| (hp : ∀ n : ℕ, N ≤ n → p.eval (n : ℚ) = (values n : ℚ)) : |
| ∃ q : Polynomial ℚ, q.natDegree = p.natDegree + 1 ∧ |
| ∀ n : ℕ, N ≤ n → q.eval (n : ℚ) = |
| ((∑ j ∈ range (n + 1), values j : ℕ) : ℚ) := by |
| let c : ℚ := (∑ j ∈ range N, (values j : ℚ)) - (polynomialPrefix p).eval (N : ℚ) |
| let q : Polynomial ℚ := (polynomialPrefix p).comp (X + C 1) + C c |
| have hsum (m : ℕ) (hm : N ≤ m) : |
| (polynomialPrefix p).eval (m : ℚ) + c = ∑ j ∈ range m, (values j : ℚ) := by |
| induction m, hm using Nat.le_induction with |
| | base => simp [c] |
| | succ m hm ih => |
| rw [polynomialPrefix_eval, Finset.sum_range_succ, |
| ← polynomialPrefix_eval, hp m hm, Finset.sum_range_succ] |
| linarith |
| refine ⟨q, ?_, ?_⟩ |
| · simp only [q, natDegree_add_C, natDegree_comp, natDegree_X, mul_one, |
| polynomialPrefix_natDegree p hp0] |
| · intro n hn |
| simpa only [q, eval_add, eval_comp, eval_X, eval_C, Nat.cast_add, Nat.cast_one, |
| Nat.cast_sum] using hsum (n + 1) (by omega) |
|
|
| end PiExponentJets.W24 |
|
|
| end OAI |
|
|
