| import OAI.Analysis.CircleDomains.Modulus.RetaggedCost |
| import OAI.Analysis.CircleDomains.Selection.PreterminalTolls |
|
|
| namespace OAI |
|
|
| /-! A positive perturbation of an admissible coding metric gives a genuine |
| physical-plus-preterminal cost lower bound on every allowed path. -/ |
|
|
| open Set MeasureTheory |
| open scoped ENNReal |
|
|
| noncomputable section |
|
|
| namespace CircleDomainRigidity |
|
|
| variable {X Y D : Type*} [MetricSpace X] [MetricSpace Y] [TopologicalSpace D] |
| [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] |
| [MeasurableSpace D] [BorelSpace D] [MeasurableSpace C(PathTime, X)] |
|
|
| def physicalAtomicCost (d : PathCodeData X Y) (p : D → X) (Γ : BasePathCode d) |
| (g : Y → ℝ≥0∞) (m : D → ℝ≥0∞) : ℝ≥0∞ := |
| (∫⁻ y, g y ∂baseCodeOccupation d Γ) + |
| ∑' s, m s * preterminalVisitCount Γ.val.1 (p s) |
|
|
| omit [TopologicalSpace D] [MeasurableSpace D] [BorelSpace D] in |
| theorem nonTupleCost_le_physicalAtomicCost (d : PathCodeData X Y) |
| (p : D → X) (c : D → ℝ≥0∞) (Γ : BasePathCode d) |
| (h : PathCostSpace Y D → ℝ≥0∞) (g : Y → ℝ≥0∞) (m : D → ℝ≥0∞) |
| (hg : ∀ y, h (costPhysical y) ≤ g y) |
| (hm : ∀ s, c s * h (costAtomic s) ≤ m s) : |
| nonTupleCost d p c Γ h ≤ physicalAtomicCost d p Γ g m := by |
| apply add_le_add (lintegral_mono hg) |
| apply ENNReal.tsum_le_tsum |
| intro s |
| calc |
| (c s * preterminalVisitCount Γ.val.1 (p s)) * h (costAtomic s) = |
| (c s * h (costAtomic s)) * preterminalVisitCount Γ.val.1 (p s) := by ac_rfl |
| _ ≤ m s * preterminalVisitCount Γ.val.1 (p s) := mul_le_mul' (hm s) le_rfl |
|
|
| theorem finite_code_of_physicalAtomicCost (d : PathCodeData X Y) |
| (p : D → X) (c : D → ℝ≥0∞) (Γ : BasePathCode d) |
| (g : Y → ℝ≥0∞) (m : D → ℝ≥0∞) {δ : ℝ≥0∞} |
| (hδ : 0 < δ) (hm : ∀ s, δ * c s ≤ m s) |
| (hcost : physicalAtomicCost d p Γ g m < ∞) : |
| pathCostMeasure d p c Γ univ < ∞ := by |
| rw [pathCostMeasure_finite_iff] |
| apply finite_preterminal_sum_of_positive_tolls Γ.val.1 p hδ hm |
| exact lt_of_le_of_lt le_add_self hcost |
|
|
| theorem physicalAtomicCost_ge_one (d : PathCodeData X Y) |
| (p : D → X) (c : D → ℝ≥0∞) (ν : Measure X) (Γ : BasePathCode d) |
| (h : PathCostSpace Y D → ℝ≥0∞) (hh : Measurable h) |
| (hΓ : EndpointCode d ν (tupleInfiniteExclusions (h ∘ costTuple)) Γ.val) |
| (hadm : CostAdmissible d p c ν h) |
| (g : Y → ℝ≥0∞) (m : D → ℝ≥0∞) {δ : ℝ≥0∞} (hδ : 0 < δ) |
| (hg : ∀ y, h (costPhysical y) ≤ g y) |
| (hm : ∀ s, c s * h (costAtomic s) ≤ m s) |
| (hpos : ∀ s, δ * c s ≤ m s) : |
| 1 ≤ physicalAtomicCost d p Γ g m := by |
| by_cases hfinite : physicalAtomicCost d p Γ g m < ∞ |
| · exact (nonTupleCost_ge_one_of_admissible d p c ν Γ h hh hΓ |
| (finite_code_of_physicalAtomicCost d p c Γ g m hδ hpos hfinite) hadm).trans |
| (nonTupleCost_le_physicalAtomicCost d p c Γ h g m hg hm) |
| · rw [top_unique (not_lt.mp hfinite)] |
| exact le_top |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
