| import Mathlib | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open scoped BigOperators | |
| def polynomialBudget (P : Polynomial ℝ) : ℝ := ∑n ∈ P.support,|P.coeff n| | |
| lemma polynomialBudget_nonneg (P : Polynomial ℝ) : 0≤polynomialBudget P := | |
| Finset.sum_nonneg (fun _ _ => abs_nonneg _) | |
| lemma eval_abs_le_polynomialBudget (P : Polynomial ℝ) {ρ : ℝ} (h0 : 0≤ρ) (h1 : ρ≤1) : | |
| |P.eval ρ|≤polynomialBudget P := by | |
| rw [Polynomial.eval_eq_sum] | |
| unfold Polynomial.sum polynomialBudget | |
| apply (Finset.abs_sum_le_sum_abs _ _).trans | |
| apply Finset.sum_le_sum | |
| intro n _ | |
| rw [abs_mul,abs_pow,abs_of_nonneg h0] | |
| exact mul_le_of_le_one_right (abs_nonneg _) (pow_le_one₀ h0 h1) | |
| lemma scaledCoefficient_bound (P : Polynomial ℝ) (n : ℕ) {r L ρ : ℝ} | |
| (hl0 : 0≤r*L) (hl1 : r*L≤1) (h0 : 0≤ρ) (h1 : ρ≤1) : | |
| |(r*L)^n*P.eval ρ|≤polynomialBudget P := by | |
| rw [abs_mul,abs_pow,abs_of_nonneg hl0] | |
| exact (mul_le_mul_of_nonneg_right (pow_le_one₀ hl0 hl1) (abs_nonneg _)).trans | |
| (by simpa using eval_abs_le_polynomialBudget P h0 h1) | |
| end LogConcaveSampling | |
| end | |
| end | |
| end | |
| end OAI | |