| import OAI.Analysis.CircleDomains.Probability.AffineTimeMeasure | |
| import OAI.Analysis.CircleDomains.Selection.AffineTagRestriction | |
| import OAI.Analysis.CircleDomains.Modulus.PathLengthLocality | |
| namespace OAI | |
| /-! Exact arclength transformation of an affine restriction of a | |
| rectifiable metric path. -/ | |
| open Set Function MeasureTheory | |
| open scoped ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| variable {X : Type*} [MetricSpace X] | |
| def affineRectifiablePath (γ : RectifiablePath X) {a b : PathTime} (hab : a < b) : | |
| RectifiablePath X := | |
| ⟨affineRestrictedPath a b γ.val, | |
| lt_top_iff_ne_top.mp ((eVariationOn.comp_le_of_monotoneOn γ.val (affinePathTime a b) | |
| ((affinePathTime_strictMono hab).monotone.monotoneOn _) (fun _ _ => mem_univ _)).trans_lt | |
| (lt_top_iff_ne_top.mpr γ.property))⟩ | |
| theorem pathLengthMeasure_affine_Icc (γ : RectifiablePath X) {a b : PathTime} | |
| (hab : a < b) {c d : PathTime} (hcd : c ≤ d) : | |
| pathLengthMeasure (affineRectifiablePath γ hab) (Icc c d) = | |
| pathLengthMeasure γ (Icc (affinePathTime a b c) (affinePathTime a b d)) := by | |
| rw [pathLengthMeasure_Icc _ hcd, | |
| pathLengthMeasure_Icc _ ((affinePathTime_strictMono hab).monotone hcd)] | |
| change eVariationOn (γ.val ∘ affinePathTime a b) (Icc c d) = _ | |
| rw [eVariationOn.comp_eq_of_monotoneOn γ.val (affinePathTime a b) | |
| ((affinePathTime_strictMono hab).monotone.monotoneOn _), | |
| (affinePathTime a b).continuous.image_Icc_of_strictMono (affinePathTime_strictMono hab)] | |
| theorem pathLengthMeasure_affine (γ : RectifiablePath X) {a b : PathTime} (hab : a < b) : | |
| pathLengthMeasure (affineRectifiablePath γ hab) = | |
| ((pathLengthMeasure γ).restrict (Icc a b)).map (rescalePathTime a b) := by | |
| apply Measure.ext_of_Icc | |
| intro c d hcd | |
| rw [pathLengthMeasure_affine_Icc γ hab hcd, map_rescale_restrict_Icc _ hab] | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |