| 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 | |