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9655094 | 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 | 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
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