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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 43 44 45 46 47 48 49 50 51 52 53 54 | import OAI.Geometry.RCD.Transport.CompressedPlanUnique
namespace OAI
/-! # Comparison of two arbitrary finite compression constants -/
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped ENNReal NNReal Topology
noncomputable section
namespace RegularPointCoordinates.NoncollapsedRCD
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[ProperSpace X] [SeparableSpace X] {K : ℝ} {n : ℕ}
[IsLocallyFiniteMeasure (hausdorff n X)] (hX : NoncollapsedRCD K n X)
include hX
attribute [local instance] pathBorel pathBorelSpace
theorem compressed_plan_unique_finite (hn : 2 ≤ n) (p : X) {R : ℝ} (hR : 0 ≤ R)
{ν₀ ν₁ : Measure X}
(π ρ : Measure C(Time,X)) [IsProbabilityMeasure π] [IsProbabilityMeasure ρ]
(hgπ : ∀ᵐ γ ∂π, IsGeodesic γ) (hgρ : ∀ᵐ γ ∂ρ, IsGeodesic γ)
(hpπ : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0,γ 1))))
(hpρ : IsOptimalCoupling ν₀ ν₁ (ρ.map (fun γ => (γ 0,γ 1))))
{C D : ℝ≥0∞} (hC : C ≠ ∞) (hD : D ≠ ∞)
(hcπ : ∀ t : Time, π.map (fun γ => γ t) ≤ C • hausdorff n X)
(hcρ : ∀ t : Time, ρ.map (fun γ => γ t) ≤ D • hausdorff n X)
(hs₀ : ∀ᵐ x ∂ν₀, dist x p ≤ R) (hs₁ : ∀ᵐ x ∂ν₁, dist x p ≤ R) : π=ρ := by
let B : ℝ≥0 := ⟨C.toReal+D.toReal+1,by positivity⟩
have hB : 0 < B := by change 0 < C.toReal+D.toReal+1; positivity
have hCB : C ≤ ENNReal.ofReal B := by
rw [←ENNReal.ofReal_toReal hC]
apply ENNReal.ofReal_le_ofReal
change C.toReal ≤ C.toReal+D.toReal+1
linarith [ENNReal.toReal_nonneg (a:=D)]
have hDB : D ≤ ENNReal.ofReal B := by
rw [←ENNReal.ofReal_toReal hD]
apply ENNReal.ofReal_le_ofReal
change D.toReal ≤ C.toReal+D.toReal+1
linarith [ENNReal.toReal_nonneg (a:=C)]
apply hX.compressed_plan_unique hn p hR π ρ hgπ hgρ hpπ hpρ hB _ _ hs₀ hs₁
· intro t
apply (hcπ t).trans
intro S
exact mul_le_mul' hCB le_rfl
· intro t
apply (hcρ t).trans
intro S
exact mul_le_mul' hDB le_rfl
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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