openAI-math-dataset / lean /OAI /Probability /SKBarriers /Interpolation /AdaptiveEndpointAlgebra.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace SK.Analytic | |
| theorem weighted_square_shift_bound {I : Type} [Fintype I] | |
| (w Q a : I → ℝ) (hw : ∀ i, 0 ≤ w i) (hs : ∑ i, w i = 1) | |
| {δ : ℝ} (hδ : 0 ≤ δ) (ha : ∀ i, a i ∈ Set.Icc 0 δ) | |
| (hQ : ∀ i, Q i ≤ 1+δ) : | |
| (∑ i, w i*(Q i)^2)-(∑ i, w i*(Q i-a i)^2) ≤ 2*δ*(1+δ) := by | |
| calc | |
| _ = ∑ i, w i*((Q i)^2-(Q i-a i)^2) := by | |
| simp only [mul_sub,Finset.sum_sub_distrib] | |
| _ ≤ ∑ i, w i*(2*δ*(1+δ)) := by | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| apply mul_le_mul_of_nonneg_left _ (hw i) | |
| have H1 := mul_le_mul_of_nonneg_right (hQ i) (ha i).1 | |
| have H2 := mul_le_mul_of_nonneg_left (ha i).2 (by linarith : 0 ≤ 1+δ) | |
| nlinarith [sq_nonneg (a i)] | |
| _ = _ := by rw [← Finset.sum_mul,hs,one_mul] | |
| theorem adaptive_endpoint_lower_algebra {I : Type} [Fintype I] | |
| (i₀ : I) (w Q a : I → ℝ) (hw : ∀ i, 0 ≤ w i) (hs : ∑ i, w i = 1) | |
| {δ c ε f₀ f₁ p : ℝ} (hδ : 0 ≤ δ) (hc : 0 ≤ c) | |
| (ha : ∀ i, a i ∈ Set.Icc 0 δ) (hQ : ∀ i, Q i ≤ 1+δ) | |
| (hpath : f₁-f₀ ≤ c*(-1+2*(Q i₀-a i₀)-∑ i, w i*(Q i-a i)^2+ε)) | |
| (hfield : f₀ ≤ p+2*c*δ) : | |
| f₁+c*(1-2*Q i₀+∑ i, w i*(Q i)^2)-c*(ε+2*δ*(1+δ)+2*δ) ≤ p := by | |
| have H := mul_le_mul_of_nonneg_left (weighted_square_shift_bound w Q a hw hs hδ ha hQ) hc | |
| have H0 := mul_nonneg hc (ha i₀).1 | |
| nlinarith | |
| end SK.Analytic | |
| end | |
| end | |
| end | |
| end OAI | |