| import OAI.Geometry.IsometricImmersion.Comparison.ActualComparisonEnergy | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped ContDiff Topology Interval | |
| namespace SmoothLocal.Hyperbolic | |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE | |
| theorem movingSourceNorm_le_of_pointwise_bound | |
| {f : Coord → ℝ} {xl xr a b speed t F : ℝ} | |
| (hspeed : 0 ≤ speed) (ht : t ∈ Icc a b) | |
| (horder : inwardLeft xl a speed t ≤ inwardRight xr a speed t) | |
| (hF : 0 ≤ F) | |
| (hbound : ∀ p ∈ shrinkingSlab xl xr a b speed, |f p| ≤ F) : | |
| movingSourceNorm xl xr a speed f t ≤ F*Real.sqrt (xr-xl) := by | |
| let l := inwardLeft xl a speed t | |
| let r := inwardRight xr a speed t | |
| have hlen : r-l ≤ xr-xl := by | |
| have hi : 0 ≤ speed*(t-a) := mul_nonneg hspeed (sub_nonneg.mpr ht.1) | |
| dsimp [l,r,inwardLeft,inwardRight] | |
| linarith | |
| have hi := intervalIntegral.norm_integral_le_of_norm_le_const | |
| (a := l) (b := r) (C := F^2) (f := fun x => f (coordinatePoint x t)^2) (by | |
| intro x hx | |
| have hx0 : x ∈ Ioc l r := by | |
| rw [uIoc_of_le (show l ≤ r from horder)] at hx | |
| exact hx | |
| have hb := hbound _ (point_mem_shrinkingSlab ht ⟨hx0.1.le,hx0.2⟩) | |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] | |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hF).mpr hb) | |
| have hsquare : (∫ x in l..r, f (coordinatePoint x t)^2) ≤ F^2*(xr-xl) := by | |
| apply (le_abs_self _).trans | |
| apply hi.trans | |
| rw [abs_of_nonneg (sub_nonneg.mpr horder)] | |
| exact mul_le_mul_of_nonneg_left hlen (sq_nonneg F) | |
| have hsqrt := Real.sqrt_le_sqrt hsquare | |
| simpa only [movingSourceNorm,movingIntervalIntegral,l,r, | |
| Real.sqrt_mul (sq_nonneg F),Real.sqrt_sq hF] using hsqrt | |
| theorem integrated_movingSourceNorm_le_of_pointwise_bound | |
| {f : Coord → ℝ} {xl xr a b speed F : ℝ} | |
| (hab : a ≤ b) (hspeed : 0 ≤ speed) (hF : 0 ≤ F) | |
| (horder : ∀ t ∈ Icc a b, inwardLeft xl a speed t ≤ inwardRight xr a speed t) | |
| (hbound : ∀ p ∈ shrinkingSlab xl xr a b speed, |f p| ≤ F) : | |
| (∫ t in a..b, movingSourceNorm xl xr a speed f t) ≤ | |
| F*Real.sqrt (xr-xl)*(b-a) := by | |
| have hi := intervalIntegral.norm_integral_le_of_norm_le_const | |
| (a := a) (b := b) (C := F*Real.sqrt (xr-xl)) | |
| (f := movingSourceNorm xl xr a speed f) (by | |
| intro t ht | |
| have ht0 : t ∈ Ioc a b := by simpa only [uIoc_of_le hab] using ht | |
| have ht1 : t ∈ Icc a b := ⟨ht0.1.le,ht0.2⟩ | |
| rw [Real.norm_eq_abs,abs_of_nonneg (show 0 ≤ movingSourceNorm xl xr a speed f t from Real.sqrt_nonneg _)] | |
| exact movingSourceNorm_le_of_pointwise_bound hspeed ht1 (horder t ht1) hF hbound) | |
| exact (le_abs_self _).trans (by simpa only [Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hab)] using hi) | |
| end SmoothLocal.Hyperbolic | |
| end | |
| end OAI | |