openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /HilbertDifferenceDegree.lean
| import Mathlib.Algebra.Polynomial.Degree.SmallDegree | |
| import Mathlib.Algebra.Polynomial.Taylor | |
| import Mathlib.Tactic | |
| namespace OAI | |
| namespace PiExponentJets.W27 | |
| open Polynomial | |
| theorem difference_taylor_coefficient (p : Polynomial ℚ) (s : ℕ) | |
| (hp : p.natDegree = s+1) (a : ℚ) : | |
| (p - Polynomial.taylor a p).coeff s = -a * (s+1 : ℚ) * p.leadingCoeff := by | |
| have hd : (Polynomial.hasseDeriv s p).natDegree ≤ 1 := by | |
| simpa [hp] using Polynomial.natDegree_hasseDeriv_le p s | |
| have hlin := Polynomial.eq_X_add_C_of_natDegree_le_one hd | |
| have h0 : (Polynomial.hasseDeriv s p).coeff 0 = p.coeff s := by | |
| simp [Polynomial.hasseDeriv_coeff] | |
| have h1 : (Polynomial.hasseDeriv s p).coeff 1 = (s+1 : ℚ) * p.leadingCoeff := by | |
| rw [Polynomial.hasseDeriv_coeff] | |
| simp only [Nat.add_comm 1 s, Nat.choose_succ_self_right, Nat.cast_add, Nat.cast_one] | |
| rw [← hp, Polynomial.coeff_natDegree] | |
| rw [Polynomial.coeff_sub, Polynomial.taylor_coeff, hlin] | |
| simp only [Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] | |
| rw [h0, h1] | |
| ring | |
| theorem difference_taylor_natDegree_le (p : Polynomial ℚ) (s : ℕ) | |
| (hp : p.natDegree = s+1) (a : ℚ) : | |
| (p - Polynomial.taylor a p).natDegree ≤ s := by | |
| apply Polynomial.natDegree_le_iff_coeff_eq_zero.mpr | |
| intro n hn | |
| rw [Polynomial.coeff_sub] | |
| by_cases he : n = s+1 | |
| · subst n | |
| rw [← hp, Polynomial.coeff_natDegree, Polynomial.coeff_taylor_natDegree, sub_self] | |
| · have hlt : p.natDegree < n := by omega | |
| rw [Polynomial.coeff_eq_zero_of_natDegree_lt hlt, | |
| Polynomial.coeff_eq_zero_of_natDegree_lt (by simpa using hlt), sub_self] | |
| theorem backward_difference_degree_and_leading | |
| (p : Polynomial ℚ) (s : ℕ) (hp : p.natDegree = s+1) | |
| (d : ℚ) (hd : d ≠ 0) : | |
| (p - Polynomial.taylor (-d) p).natDegree = s ∧ | |
| (p - Polynomial.taylor (-d) p).leadingCoeff = d * (s+1 : ℚ) * p.leadingCoeff := by | |
| have hp0 : p ≠ 0 := by | |
| intro hz | |
| simp [hz] at hp | |
| have hs : (s+1 : ℚ) ≠ 0 := by positivity | |
| have hc : (p - Polynomial.taylor (-d) p).coeff s = | |
| d * (s+1 : ℚ) * p.leadingCoeff := by | |
| simpa using difference_taylor_coefficient p s hp (-d) | |
| have hcn : (p - Polynomial.taylor (-d) p).coeff s ≠ 0 := by | |
| rw [hc] | |
| exact mul_ne_zero (mul_ne_zero hd hs) (Polynomial.leadingCoeff_ne_zero.mpr hp0) | |
| have hdeg := Polynomial.natDegree_eq_of_le_of_coeff_ne_zero | |
| (difference_taylor_natDegree_le p s hp (-d)) hcn | |
| refine ⟨hdeg, ?_⟩ | |
| simpa only [Polynomial.leadingCoeff, hdeg] using hc | |
| theorem backward_difference_multiplicity | |
| (p : Polynomial ℚ) (s : ℕ) (hp : p.natDegree = s+1) | |
| (d : ℚ) (hd : d ≠ 0) : | |
| (p - Polynomial.taylor (-d) p).leadingCoeff * | |
| ((p - Polynomial.taylor (-d) p).natDegree.factorial : ℚ) = | |
| d * (p.leadingCoeff * (p.natDegree.factorial : ℚ)) := by | |
| obtain ⟨hdeg, hl⟩ := backward_difference_degree_and_leading p s hp d hd | |
| rw [hl, hdeg, hp, Nat.factorial_succ] | |
| push_cast | |
| ring | |
| theorem backward_difference_eq_comp (p : Polynomial ℚ) (d : ℚ) : | |
| p - Polynomial.taylor (-d) p = p - p.comp (Polynomial.X - Polynomial.C d) := by | |
| simp only [Polynomial.taylor_apply, Polynomial.C_neg, sub_eq_add_neg] | |
| end PiExponentJets.W27 | |
| end OAI | |