| import Mathlib |
| import OAI.Probability.SKBarriers.Interpolation.AdaptiveODEExistence |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators NNReal |
| open Set Metric |
| namespace SK.Analytic |
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| theorem exists_positive_adaptive_path {k : ℕ} |
| (R : (ℝ × (Fin (k+1) → ℝ)) → (Fin (k+1) → ℝ)) (hR : ContDiff ℝ 1 R) |
| (hRb : ∀ p j, R p j ∈ Icc (0:ℝ) 1) (hRm : ∀ p, Monotone (R p)) |
| (η : ℝ) (hη : 0 < η) : |
| ∃ q : ℝ → (Fin (k+1) → ℝ), |
| q 0 = (fun j => ((j.val:ℝ)+1)*η) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, HasDerivWithinAt q |
| (R (Real.sqrt (1-t),fun j => Real.sqrt (cumulativeGapMap k (q t) j))) (Icc (0:ℝ) 1) t) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ j, η ≤ cumulativeGapMap k (q t) j) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ j, q t j ≤ ((j.val:ℝ)+1)*η+t) := by |
| let q₀ : Fin (k+1) → ℝ := fun j => ((j.val:ℝ)+1)*η |
| let ε := η/2 |
| obtain ⟨q,hq0,hq⟩ := exists_floored_adaptive_path R hR hRb ε (by dsimp [ε]; positivity) q₀ |
| let v := fun t => R (Real.sqrt (1-t),positiveGapScales k ε (q t)) |
| have hGap0 (j : Fin (k+1)) : cumulativeGapMap k q₀ j = η := by |
| refine Fin.cases ?_ (fun i => ?_) j |
| · simp [q₀] |
| · simp only [cumulativeGapMap_succ,q₀,Fin.val_succ,Fin.val_castSucc,Nat.cast_add,Nat.cast_one] |
| ring |
| have hGapD (j : Fin (k+1)) (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : |
| HasDerivWithinAt (fun u => cumulativeGapMap k (q u) j) |
| (cumulativeGapMap k (v t) j) (Icc (0:ℝ) 1) t := by |
| exact ((ContinuousLinearMap.proj j).comp (cumulativeGapMap k)).hasFDerivAt.comp_hasDerivWithinAt t (hq t ht) |
| have hGapV (t : ℝ) (j : Fin (k+1)) : 0 ≤ cumulativeGapMap k (v t) j := by |
| refine Fin.cases ?_ (fun i => ?_) j |
| · exact (hRb _ 0).1 |
| · change 0 ≤ v t i.succ-v t i.castSucc |
| exact sub_nonneg.mpr (hRm _ (Fin.castSucc_le_succ i)) |
| have hGap (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) (j : Fin (k+1)) : |
| η ≤ cumulativeGapMap k (q t) j := by |
| have H := monotoneOn_unit_of_deriv_nonneg (hGapD j) (fun t _ => hGapV t j) |
| (show (0:ℝ) ∈ Icc 0 1 by norm_num) ht ht.1 |
| change cumulativeGapMap k (q 0) j ≤ cumulativeGapMap k (q t) j at H |
| rwa [hq0,hGap0] at H |
| have hscales (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : |
| positiveGapScales k ε (q t) = fun j => Real.sqrt (cumulativeGapMap k (q t) j) := by |
| funext j |
| rw [positiveGapScales,max_eq_right] |
| exact (by dsimp [ε]; linarith : ε ≤ η).trans (hGap t ht j) |
| refine ⟨q,hq0,?_,hGap,?_⟩ |
| · intro t ht |
| simpa only [hscales t ht] using hq t ht |
| · intro t ht j |
| have hD (u : ℝ) (hu : u ∈ Icc (0:ℝ) 1) : |
| HasDerivWithinAt (fun s => s-q s j) (1-v u j) (Icc (0:ℝ) 1) u := by |
| exact (hasDerivWithinAt_id u _).sub |
| ((ContinuousLinearMap.proj (R := ℝ) (φ := fun _ : Fin (k+1) => ℝ) j).hasFDerivAt.comp_hasDerivWithinAt u (hq u hu)) |
| have H := monotoneOn_unit_of_deriv_nonneg hD (fun u _ => sub_nonneg.mpr (hRb _ j).2) |
| (show (0:ℝ) ∈ Icc 0 1 by norm_num) ht ht.1 |
| change 0-q 0 j ≤ t-q t j at H |
| rw [hq0] at H |
| change 0-((j.val:ℝ)+1)*η ≤ t-q t j at H |
| 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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