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5711324 | 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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import OAI.Analysis.CircleDomains.Selection.ApproximateFiniteConcatenation
namespace OAI
/-! In a chain of limiting junctions an equal consecutive pair requires
no extra path or pause. Only distinct pairs are connected, and every
actually used junction is paid once. -/
open Set Function MeasureTheory
open scoped ENNReal
noncomputable section
namespace CircleDomainRigidity
variable {X Y : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]
[MeasurableSpace C(PathTime, X)]
theorem exists_allowed_collapsing_junction_prefix
(d : PathCodeData X Y) (hU : IsOpen d.quotientDomain)
(ν : Measure X) [NullSingletonClass ν]
(N : ∀n : ℕ, (Fin (n+1) → ℕ) → Set (Fin (n+1) → UnitAddCircle))
(g : Y → ℝ≥0∞) (hg : Measurable g) (m : X → ℝ≥0∞)
(hpoint : ∀i x, volume {t | (t, x) ∈ d.levels i} ≤ d.levelWeights i x)
(F : Set X) (n : ℕ) (p : Fin (n+1) → X)
(A : Fin (n+1) → ℝ≥0∞) (J : Fin n → ℝ≥0∞)
(hpaid : ∀i : Fin n, m (p i.castSucc) ≤ J i)
(hfirst : ∀ε : ℝ, 0 < ε →
∃ (ρ : FinitePhysicalPath d.domainEquiv d.basePoint) (θ : ℕ → TagSlot),
AllowedTaggedPath d ν N (ρ.val, θ) ∧ ρ.val ⊥ ∈ F ∧ ρ.val ⊤ = p 0 ∧
interiorPhysicalTollCost d g m ρ ≤ A 0 + ENNReal.ofReal ε)
(hstep : ∀i : Fin n, p i.castSucc ≠ p i.succ → ∀ε : ℝ, 0 < ε →
∃ (ρ : FinitePhysicalPath d.domainEquiv d.basePoint) (θ : ℕ → TagSlot),
AllowedTaggedPath d ν N (ρ.val, θ) ∧
ρ.val ⊥ = p i.castSucc ∧ ρ.val ⊤ = p i.succ ∧
interiorPhysicalTollCost d g m ρ ≤ A i.succ + ENNReal.ofReal ε)
{η : ℝ} (hη : 0 < η) :
∃ (ρ : FinitePhysicalPath d.domainEquiv d.basePoint) (θ : ℕ → TagSlot),
AllowedTaggedPath d ν N (ρ.val, θ) ∧ ρ.val ⊥ ∈ F ∧ ρ.val ⊤ = p (Fin.last n) ∧
interiorPhysicalTollCost d g m ρ ≤ (∑i, A i) + (∑i, J i) + ENNReal.ofReal η := by
classical
induction n generalizing η with
| zero =>
obtain ⟨ρ, θ, hρ, hρ0, hρ1, hc⟩ := hfirst η hη
exact ⟨ρ, θ, hρ, hρ0, hρ1, by simpa using hc⟩
| succ n ih =>
have hh : 0 < η/2 := by positivity
let p' : Fin (n+1) → X := fun i => p i.castSucc
let A' : Fin (n+1) → ℝ≥0∞ := fun i => A i.castSucc
let J' : Fin n → ℝ≥0∞ := fun i => J i.castSucc
have hp' (i : Fin n) : m (p' i.castSucc) ≤ J' i := hpaid i.castSucc
have hf' : ∀ε : ℝ, 0 < ε →
∃ (ρ : FinitePhysicalPath d.domainEquiv d.basePoint) (θ : ℕ → TagSlot),
AllowedTaggedPath d ν N (ρ.val, θ) ∧ ρ.val ⊥ ∈ F ∧ ρ.val ⊤ = p' 0 ∧
interiorPhysicalTollCost d g m ρ ≤ A' 0 + ENNReal.ofReal ε := hfirst
have hs' (i : Fin n) (hne : p' i.castSucc ≠ p' i.succ) (ε : ℝ) (hε : 0 < ε) :
∃ (ρ : FinitePhysicalPath d.domainEquiv d.basePoint) (θ : ℕ → TagSlot),
AllowedTaggedPath d ν N (ρ.val, θ) ∧
ρ.val ⊥ = p' i.castSucc ∧ ρ.val ⊤ = p' i.succ ∧
interiorPhysicalTollCost d g m ρ ≤ A' i.succ + ENNReal.ofReal ε :=
hstep i.castSucc hne ε hε
obtain ⟨α, θα, hα, hα0, hα1, hαc⟩ := ih p' A' J' hp' hf' hs' hh
have hsA := Fin.sum_univ_castSucc A
have hsJ := Fin.sum_univ_castSucc J
by_cases heq : p (Fin.last n).castSucc = p (Fin.last (n+1))
· refine ⟨α, θα, hα, hα0, hα1.trans heq, ?_⟩
have he : ENNReal.ofReal (η/2) ≤ ENNReal.ofReal η :=
ENNReal.ofReal_le_ofReal (by linarith)
calc
interiorPhysicalTollCost d g m α ≤ (∑i, A' i) + (∑i, J' i) + ENNReal.ofReal η :=
hαc.trans (add_le_add le_rfl he)
_ ≤ ((∑i, A' i) + A (Fin.last (n+1))) +
((∑i, J' i) + J (Fin.last n)) + ENNReal.ofReal η :=
add_le_add (add_le_add (le_add_right le_rfl) (le_add_right le_rfl)) le_rfl
_ = (∑i, A i) + (∑i, J i) + ENNReal.ofReal η := by
rw [hsA, hsJ]
· obtain ⟨β, θβ, hβ, hβ0, hβ1, hβc⟩ := hstep (Fin.last n) heq (η/2) hh
obtain ⟨ρ, θ, hρ, hρ0, hρ1, hc⟩ := exists_allowed_joined_junction_cost
d hU ν N (α, θα) (β, θβ) hα hβ (hα1.trans hβ0.symm) g hg m hpoint
refine ⟨ρ, θ, hρ, hρ0.symm ▸ hα0, hρ1.trans hβ1, ?_⟩
have herr : ENNReal.ofReal (η/2) + ENNReal.ofReal (η/2) = ENNReal.ofReal η := by
rw [← ENNReal.ofReal_add hh.le hh.le]
congr 1
ring
calc
interiorPhysicalTollCost d g m ρ =
interiorPhysicalTollCost d g m α + interiorPhysicalTollCost d g m β + m (p' (Fin.last n)) := by
rw [hc, hα1]
_ ≤ ((∑i, A' i) + (∑i, J' i) + ENNReal.ofReal (η/2)) +
(A (Fin.last (n+1)) + ENNReal.ofReal (η/2)) + J (Fin.last n) :=
add_le_add (add_le_add hαc hβc) (hpaid (Fin.last n))
_ = (∑i, A i) + (∑i, J i) + ENNReal.ofReal η := by
rw [hsA, hsJ, ← herr]
dsimp [A', J']
ac_rfl
end CircleDomainRigidity
end
end OAI
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