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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | 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
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