| import OAI.Geometry.RCD.Transport.CompressedPlanUnique |
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|
| namespace OAI |
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| /-! # Comparison of two arbitrary finite compression constants -/ |
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|
| 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 |
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| 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 |
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|
| end RegularPointCoordinates.NoncollapsedRCD |
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| end |
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| end OAI |
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