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e4d9d2b | 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 | import OAI.Geometry.RCD.Transport.ContinuousLengthDistortion
import OAI.Geometry.RCD.Sobolev.WeakEndpointWeights
namespace OAI
/-! # Reduced distortion as a bounded continuous path test -/
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped ENNReal NNReal Topology BoundedContinuousFunction
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X]
def pathDistortion (K N t : ℝ) (hK : K≤0) (hN : 1<N) (ht : 0≤t) (ht1 : t≤1) :
C(Time,X) →ᵇ ℝ :=
BoundedContinuousFunction.mkOfBound
⟨fun γ => (sigma K N t (dist (γ 0) (γ 1))).toReal,by
have hh : Continuous (fun γ : C(Time,X) => sigma K N t (dist (γ 0) (γ 1))) :=
(continuous_sigma_nonpos_length hK hN ht ht1).comp
((continuous_eval_const 0).nndist (continuous_eval_const 1))
exact continuous_iff_continuousAt.mpr (fun γ =>
(ENNReal.continuousAt_toReal (sigma_nonpos_ne_top (t:=t)
(d:=dist (γ 0) (γ 1)) hK hN dist_nonneg)).comp
(f:=fun η : C(Time,X) => sigma K N t (dist (η 0) (η 1))) hh.continuousAt)⟩
1 (by
intro γ η
have hb (ζ : C(Time,X)) : 0≤(sigma K N t (dist (ζ 0) (ζ 1))).toReal ∧
(sigma K N t (dist (ζ 0) (ζ 1))).toReal≤1 := by
refine ⟨ENNReal.toReal_nonneg,?_⟩
exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top
(sigma_nonpos_le_time hK hN ht ht1 dist_nonneg)).trans
(by simpa only [ENNReal.toReal_ofReal ht] using ht1)
change |(sigma K N t (dist (γ 0) (γ 1))).toReal-
(sigma K N t (dist (η 0) (η 1))).toReal|≤1
exact abs_le.mpr ⟨by linarith [(hb γ).1,(hb η).2],by linarith [(hb γ).2,(hb η).1]⟩)
end RegularPointCoordinates
end
end OAI
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