| import Mathlib | |
| import OAI.Probability.SKBarriers.Interpolation.AdaptiveODERegular | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open scoped BigOperators NNReal | |
| open Set Metric | |
| namespace SK.Analytic | |
| theorem exists_floored_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) | |
| (ε : ℝ) (hε : 0 < ε) (q₀ : Fin (k+1) → ℝ) : | |
| ∃ q : ℝ → (Fin (k+1) → ℝ), q 0 = q₀ ∧ | |
| ∀ t ∈ Icc (0:ℝ) 1, HasDerivWithinAt q | |
| (R (Real.sqrt (1-t),positiveGapScales k ε (q t))) (Icc (0:ℝ) 1) t := by | |
| let G := fun p : ℝ × (Fin (k+1) → ℝ) => R (p.1,positiveGapScales k ε p.2) | |
| have hG : LocallyLipschitz G := regular_positiveGapScales R hR ε hε | |
| obtain ⟨K,hK⟩ := LocallyLipschitzOn.exists_lipschitzOnWith_of_compact | |
| (isCompact_Icc.prod (isCompact_closedBall q₀ 1)) hG.locallyLipschitzOn | |
| have hs (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : Real.sqrt (1-t) ∈ Icc (0:ℝ) 1 := by | |
| refine ⟨Real.sqrt_nonneg _,?_⟩ | |
| apply Real.sqrt_le_one.mpr | |
| linarith [ht.1] | |
| let F := fun t q => G (Real.sqrt (1-t),q) | |
| have hf : IsPicardLindelof F (⟨0,by norm_num⟩ : Icc (0:ℝ) 1) q₀ 1 0 1 K := by | |
| constructor | |
| · intro t ht | |
| apply LipschitzOnWith.of_dist_le_mul | |
| intro x hx y hy | |
| simpa only [F,Prod.dist_eq,dist_self,max_eq_right dist_nonneg] using | |
| hK.dist_le_mul (Real.sqrt (1-t),x) ⟨hs t ht,hx⟩ | |
| (Real.sqrt (1-t),y) ⟨hs t ht,hy⟩ | |
| · intro x _ | |
| exact (hG.continuous.comp ((Real.continuous_sqrt.comp | |
| (continuous_const.sub continuous_id)).prodMk continuous_const)).continuousOn | |
| · intro t _ x _ | |
| apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ) ≤ 1)).mpr | |
| intro j | |
| rw [Real.norm_eq_abs,abs_of_nonneg (hRb _ j).1] | |
| exact (hRb _ j).2 | |
| · norm_num | |
| exact hf.exists_eq_forall_mem_Icc_hasDerivWithinAt₀ | |
| theorem monotoneOn_unit_of_deriv_nonneg {f g : ℝ → ℝ} | |
| (hf : ∀ t ∈ Icc (0:ℝ) 1, HasDerivWithinAt f (g t) (Icc (0:ℝ) 1) t) | |
| (hg : ∀ t ∈ Icc (0:ℝ) 1, 0 ≤ g t) : MonotoneOn f (Icc (0:ℝ) 1) := by | |
| apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 1) | |
| (fun t ht => (hf t ht).continuousWithinAt) | |
| · intro t ht | |
| exact (hf t (interior_subset ht)).mono interior_subset | |
| · intro t ht | |
| exact hg t (interior_subset ht) | |
| end SK.Analytic | |
| end | |
| end | |
| end | |
| end OAI | |