| import Mathlib |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators |
| namespace SK.Analytic |
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| theorem integral_square_ge_square {r : ℝ → ℝ} |
| (hr : ContinuousOn r (Set.Icc (0:ℝ) 1)) : |
| (∫ u in (0:ℝ)..1, r u)^2 ≤ ∫ u in (0:ℝ)..1, (r u)^2 := by |
| let d := ∫ u in (0:ℝ)..1, r u |
| have hi := hr.intervalIntegrable_of_Icc (μ := volume) (by norm_num : (0:ℝ) ≤ 1) |
| have hi2 := (hr.pow 2).intervalIntegrable_of_Icc (μ := volume) (by norm_num : (0:ℝ) ≤ 1) |
| have H := intervalIntegral.integral_nonneg_of_forall (μ := volume) (by norm_num : (0:ℝ) ≤ 1) |
| (fun u => sq_nonneg (r u-d)) |
| have he : (fun u => (r u-d)^2) = (fun u => (r u)^2-2*d*r u+d^2) := by |
| funext u; ring |
| rw [he,intervalIntegral.integral_add (f := fun u => (r u)^2-2*d*r u) (g := fun _ => d^2) |
| (hi2.sub (hi.const_mul (2*d))) intervalIntegrable_const, |
| intervalIntegral.integral_sub (f := fun u => (r u)^2) (g := fun u => 2*d*r u) hi2 (hi.const_mul (2*d)),intervalIntegral.integral_const_mul, |
| intervalIntegral.integral_const] at H |
| change 0 ≤ (∫ u in (0:ℝ)..1, (r u)^2)-2*d*d+(1-0)*d^2 at H |
| dsimp only [d] at H |
| nlinarith |
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| theorem adaptive_integrated_bound {k : ℕ} (f : ℝ → ℝ) (q : ℝ → Fin (k+1) → ℝ) |
| (r : ℝ → Fin (k+1) → ℝ) (e : ℝ → ℝ) (w : Fin (k+1) → ℝ) |
| (hw : ∀ j, 0 ≤ w j) {c ε : ℝ} (hc : 0 ≤ c) |
| (hf : ContinuousOn f (Set.Icc (0:ℝ) 1)) |
| (hr : ∀ j, ContinuousOn (fun u => r u j) (Set.Icc (0:ℝ) 1)) |
| (hq : ∀ u ∈ Set.Icc (0:ℝ) 1, HasDerivWithinAt q (r u) (Set.Icc (0:ℝ) 1) u) |
| (hD : ∀ u ∈ Set.Ioo (0:ℝ) 1, HasDerivAt f |
| (c*(-1+2*r u (Fin.last k)-∑ j, w j*(r u j)^2+e u)) u) |
| (he : ∀ u ∈ Set.Ioo (0:ℝ) 1, e u ≤ ε) : |
| f 1-f 0 ≤ c*(-1+2*(q 1 (Fin.last k)-q 0 (Fin.last k))- |
| ∑ j, w j*(q 1 j-q 0 j)^2+ε) := by |
| have hqc := HasDerivWithinAt.continuousOn hq |
| have hqi (j : Fin (k+1)) := (continuous_apply j).comp_continuousOn hqc |
| have hri (j : Fin (k+1)) := (hr j).intervalIntegrable_of_Icc (μ := volume) (by norm_num : (0:ℝ) ≤ 1) |
| have hr2 (j : Fin (k+1)) := ((hr j).pow 2).intervalIntegrable_of_Icc (μ := volume) (by norm_num : (0:ℝ) ≤ 1) |
| have hint (j : Fin (k+1)) : (∫ u in (0:ℝ)..1, r u j) = q 1 j-q 0 j := by |
| apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le (by norm_num) (hqi j) _ (hri j) |
| intro u hu |
| exact hasDerivAt_pi.mp ((hq u ⟨hu.1.le,hu.2.le⟩).hasDerivAt (Icc_mem_nhds hu.1 hu.2)) j |
| let g := fun u => c*(-1+2*r u (Fin.last k)-∑ j, w j*(r u j)^2+ε) |
| have hg : ContinuousOn g (Set.Icc (0:ℝ) 1) := by |
| exact continuousOn_const.mul (((continuousOn_const.add (continuousOn_const.mul (hr _))).sub |
| (continuousOn_finsetSum _ (fun j _ => continuousOn_const.mul ((hr j).pow 2)))).add continuousOn_const) |
| have H : f 1-f 0 ≤ ∫ u in (0:ℝ)..1, g u := by |
| apply intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le (by norm_num) hf |
| (fun u hu => (hD u hu).hasDerivWithinAt) hg.integrableOn_Icc |
| intro u hu |
| exact mul_le_mul_of_nonneg_left (add_le_add (le_refl _) (he u hu)) hc |
| have hs : (∫ u in (0:ℝ)..1, ∑ j, w j*(r u j)^2) = |
| ∑ j, w j*(∫ u in (0:ℝ)..1, (r u j)^2) := by |
| rw [intervalIntegral.integral_finsetSum] |
| · simp only [intervalIntegral.integral_const_mul] |
| · intro j _; exact (hr2 j).const_mul (w j) |
| have hsum : IntervalIntegrable (fun u => ∑ j, w j*(r u j)^2) volume 0 1 := |
| (continuousOn_finsetSum _ (fun j _ => continuousOn_const.mul ((hr j).pow 2))).intervalIntegrable_of_Icc (by norm_num) |
| have hgint : (∫ u in (0:ℝ)..1, g u) = |
| c*(-1+2*(q 1 (Fin.last k)-q 0 (Fin.last k))- |
| ∑ j, w j*(∫ u in (0:ℝ)..1, (r u j)^2)+ε) := by |
| dsimp only [g] |
| rw [intervalIntegral.integral_const_mul, |
| intervalIntegral.integral_add ((intervalIntegrable_const.add ((hri _).const_mul 2)).sub hsum) intervalIntegrable_const, |
| intervalIntegral.integral_sub (intervalIntegrable_const.add ((hri _).const_mul 2)) hsum, |
| intervalIntegral.integral_add intervalIntegrable_const ((hri _).const_mul 2), |
| intervalIntegral.integral_const_mul,hs,hint] |
| simp |
| rw [hgint] at H |
| apply H.trans |
| apply mul_le_mul_of_nonneg_left _ hc |
| have hx : (∑ j, w j*(q 1 j-q 0 j)^2) ≤ |
| ∑ j, w j*(∫ u in (0:ℝ)..1, (r u j)^2) := by |
| apply Finset.sum_le_sum |
| intro j _ |
| have HH := integral_square_ge_square (hr j) |
| rw [hint] at HH |
| exact mul_le_mul_of_nonneg_left HH (hw j) |
| linarith |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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