openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /HilbertGrowthComparison.lean
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeHilbertDegree | |
| namespace OAI | |
| namespace PiExponentJets.W24 | |
| open Polynomial | |
| theorem natDegree_le_of_eventual_polynomial_le (p q : Polynomial ℚ) | |
| (hp : 0 < p.leadingCoeff) (N : ℕ) | |
| (hle : ∀ n : ℕ, N ≤ n → p.eval (n : ℚ) ≤ q.eval (n : ℚ)) : | |
| p.natDegree ≤ q.natDegree := by | |
| by_contra hn | |
| have hdeg : q.degree < p.degree := degree_lt_degree (Nat.lt_of_not_ge hn) | |
| have hnonneg := W27.polynomial_leadingCoeff_nonneg_of_eventually_nonneg | |
| (q - p) N (fun n hn => by simpa only [eval_sub] using sub_nonneg.mpr (hle n hn)) | |
| rw [leadingCoeff_sub_of_degree_lt' hdeg] at hnonneg | |
| exact (not_le_of_gt hp) (neg_nonneg.mp hnonneg) | |
| theorem natDegree_affine_comp_le (p : Polynomial ℚ) (a b : ℚ) : | |
| (p.comp (C a * X + C b)).natDegree ≤ p.natDegree := by | |
| have ha : (C a * X + C b : Polynomial ℚ).natDegree ≤ 1 := by | |
| rw [natDegree_add_C] | |
| simpa only [natDegree_X] using natDegree_C_mul_le a (X : Polynomial ℚ) | |
| exact (natDegree_comp_le).trans ((Nat.mul_le_mul_left _ ha).trans_eq (Nat.mul_one _)) | |
| theorem natDegree_eq_of_polynomial_growth_bounds | |
| (q lower upper : Polynomial ℚ) (d : ℕ) | |
| (hql : 0 < q.leadingCoeff) (hll : 0 < lower.leadingCoeff) | |
| (hld : lower.natDegree = d) (hud : upper.natDegree ≤ d) | |
| (a b c e : ℚ) (N : ℕ) | |
| (hlo : ∀ n : ℕ, N ≤ n → lower.eval (n : ℚ) ≤ q.eval (a * n + b)) | |
| (hup : ∀ n : ℕ, N ≤ n → q.eval (n : ℚ) ≤ upper.eval (c * n + e)) : | |
| q.natDegree = d := by | |
| have hlow : lower.natDegree ≤ (q.comp (C a * X + C b)).natDegree := | |
| natDegree_le_of_eventual_polynomial_le lower _ hll N (by | |
| simpa only [eval_comp, eval_add, eval_mul, eval_C, eval_X] using hlo) | |
| have hupp : q.natDegree ≤ (upper.comp (C c * X + C e)).natDegree := | |
| natDegree_le_of_eventual_polynomial_le q _ hql N (by | |
| simpa only [eval_comp, eval_add, eval_mul, eval_C, eval_X] using hup) | |
| exact le_antisymm | |
| (hupp.trans ((natDegree_affine_comp_le upper c e).trans hud)) | |
| (hld ▸ hlow.trans (natDegree_affine_comp_le q a b)) | |
| end PiExponentJets.W24 | |
| end OAI | |