| import OAI.Analysis.RCDHeat.CompressionIntegral | |
| namespace OAI | |
| /-! # A common bounded region for a plan with bounded endpoints -/ | |
| open Set Filter MeasureTheory Metric | |
| open scoped ENNReal Topology | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] | |
| attribute [local instance] pathBorel pathBorelSpace | |
| omit [MeasurableSpace X] [BorelSpace X] in | |
| theorem geodesic_plan_ball_support (π : Measure C(Time,X)) | |
| (hgeo : ∀ᵐ γ ∂π, IsGeodesic γ) (p : X) {R : ℝ} | |
| (h0 : ∀ᵐ γ ∂π, dist p (γ 0) ≤ R) (h1 : ∀ᵐ γ ∂π, dist p (γ 1) ≤ R) : | |
| ∀ᵐ γ ∂π, ∀ s : Time, γ s ∈ closedBall p (3*R) := by | |
| filter_upwards [hgeo,h0,h1] with γ hγ h0 h1 | |
| intro s | |
| have hlength : dist (γ 0) (γ 1) ≤ 2*R := by | |
| have hh := dist_triangle (γ 0) p (γ 1) | |
| rw [dist_comm (γ 0) p] at hh | |
| linarith | |
| have hdist : dist (γ s) (γ 0) = (s:ℝ)*dist (γ 0) (γ 1) := by | |
| simpa [abs_of_nonneg s.property.1] using hγ s 0 | |
| rw [mem_closedBall,dist_comm] | |
| have hh := dist_triangle p (γ 0) (γ s) | |
| rw [dist_comm (γ 0) (γ s),hdist] at hh | |
| have hmul := mul_le_mul_of_nonneg_right s.property.2 (dist_nonneg (x := γ 0) (y := γ 1)) | |
| linarith | |
| theorem geodesic_plan_eval_ball (π : Measure C(Time,X)) | |
| (hgeo : ∀ᵐ γ ∂π, IsGeodesic γ) (p : X) {R : ℝ} | |
| (h0 : ∀ᵐ γ ∂π, dist p (γ 0) ≤ R) (h1 : ∀ᵐ γ ∂π, dist p (γ 1) ≤ R) | |
| (s : Time) : ∀ᵐ x ∂π.map (fun γ => γ s), x ∈ closedBall p (3*R) := by | |
| apply (ae_map_iff (measurable_path_eval s).aemeasurable measurableSet_closedBall).mpr | |
| exact (geodesic_plan_ball_support π hgeo p h0 h1).mono fun γ hγ => hγ s | |
| end RegularPointCoordinates | |
| end OAI | |