File size: 3,041 Bytes
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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | 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
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