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695bd70 | 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 | import Mathlib
import OAI.Geometry.SmoothYau.Smoothness.UniformJetBoundsFiberComp
namespace OAI
noncomputable section
open Set Filter
open scoped Topology ContDiff
namespace YauCounterexamples
lemma compact_normal_forward_bound (g : SmoothMetric NormalWaveSpace NormalWaveSpace)
{K : Set NormalWaveSpace} (hK : IsCompact K) {R : ℝ} (hR : 0 ≤ R) :
∃ C : ℝ, 1 ≤ C ∧ ∀ q ∈ metricFrameSet g K, ∀ x : Fin 3 → ℝ, ‖x‖ ≤ R →
‖normalJetMap q.1 q.2 ((metricChristoffel g q.1).bilinearComp q.2 q.2)
(normalWaveEquiv x)-q.1‖ ≤ C*‖x‖ := by
let F := fun w : NormalWaveParameter × (Fin 3 → ℝ) =>
normalJetMap w.1.1 w.1.2 ((metricChristoffel g w.1.1).bilinearComp w.1.2 w.1.2) (normalWaveEquiv w.2)
have hF : ContDiff ℝ ∞ F := (contDiff_normalJetFamily g).comp
(contDiff_fst.prodMk (normalWaveEquiv.contDiff.comp contDiff_snd))
have hL : IsCompact (metricFrameSet g K ×ˢ Metric.closedBall (0 : Fin 3 → ℝ) R) :=
(metricFrameSet_isCompact g hK).prod (isCompact_closedBall 0 R)
obtain ⟨C,hC,hb⟩ := compact_fiber_jet_bound F hF hL 1
refine ⟨C,hC,?_⟩
intro q hq x hx
have hFq : ContDiff ℝ ∞ (fun x => F (q,x)) := hF.comp (contDiff_const.prodMk contDiff_id)
have hd (v) (hv : v ∈ Metric.closedBall (0 : Fin 3 → ℝ) R) :
‖fderiv ℝ (fun x => F (q,x)) v‖ ≤ C := by
rw [←norm_iteratedFDeriv_one]
exact hb (q,v) ⟨hq,hv⟩ 1 le_rfl
have hh := (convex_closedBall (0 : Fin 3 → ℝ) R).norm_image_sub_le_of_norm_fderiv_le
(fun v _ => hFq.differentiable (by simp) v) hd (Metric.mem_closedBall_self hR)
(show x ∈ Metric.closedBall (0 : Fin 3 → ℝ) R by simpa only [Metric.mem_closedBall,dist_zero_right] using hx)
simpa only [F,map_zero,normalJetMap_zero,sub_zero] using hh
lemma gaussian_coordinate_to_physical {c C n : ℝ} (hc : 0 ≤ c) (hC : 0 < C) (hn : 0 ≤ n)
{a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hba : b ≤ C*a) :
Real.exp (-c*n*a^2) ≤ Real.exp (-(c/C^2)*n*b^2) := by
apply Real.exp_le_exp.mpr
have hsq : b^2 ≤ C^2*a^2 := by
calc
b^2 ≤ (C*a)^2 := by
simpa only [pow_two] using mul_le_mul hba hba hb (mul_nonneg hC.le ha)
_ = C^2*a^2 := mul_pow C a 2
have hm := mul_le_mul_of_nonneg_left hsq (div_nonneg (mul_nonneg hc hn) (sq_nonneg C))
have hnC : C^2 ≠ 0 := pow_ne_zero _ hC.ne'
have he : (c*n/C^2)*(C^2*a^2) = c*n*a^2 := by field_simp
rw [he] at hm
calc
-c*n*a^2 = -(c*n*a^2) := by ring
_ ≤ -((c*n/C^2)*b^2) := neg_le_neg hm
_ = -(c/C^2)*n*b^2 := by ring
end YauCounterexamples
end
end OAI
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