openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /FirstExitFragmentLimits.lean
| import OAI.Analysis.CircleDomains.Modulus.FirstExitFragmentBudget | |
| namespace OAI | |
| /-! Joint compact hit-position limits and their genuine limiting cost | |
| budget for the actual first-exit fragment array. -/ | |
| open Set Function Filter MeasureTheory | |
| open scoped ENNReal Topology | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| variable {X Y : Type*} [MetricSpace X] [CompactSpace X] [MetricSpace Y] | |
| [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] | |
| [MeasurableSpace C(PathTime, X)] | |
| theorem exists_firstExit_fragment_limits | |
| (d : PathCodeData X Y) (hU : IsOpen d.quotientDomain) | |
| (Γ : ℕ → TaggedPhysicalPath d) (V : ℕ → Set X) | |
| (hcut : ∀n j, j ≤ n → firstExitTime (Γ n).1.val (V (j+1)) < | |
| firstExitTime (Γ n).1.val (V j)) | |
| (hV : ∀j, IsOpen (V j)) (hanti : Antitone V) | |
| (hstart : ∀n, (Γ n).1.val ⊥ ∈ V (n+1)) (hend : ∀n, (Γ n).1.val ⊤ ∉ V 0) | |
| (g : Y → ℝ≥0∞) (hg : Measurable g) (m : X → ℝ≥0∞) | |
| (hm : ∀n j, j ≤ n+1 → m ((Γ n).1.val (firstExitTime (Γ n).1.val (V j))) = 0) | |
| (A : ℝ≥0∞) (hA : Tendsto (fun n => physicalTollCost d g m (Γ n).1) atTop (𝓝 A)) : | |
| ∃ (s : ℕ → ℕ) (x : ℕ → X) (c : ℕ → ℝ≥0∞), StrictMono s ∧ | |
| (∀j, x j ∈ frontier (V j)) ∧ | |
| (∀j, Tendsto (fun n => (firstExitFragmentArray d hU Γ V hcut (s n) j).1.val ⊥) | |
| atTop (𝓝 (x (j+1)))) ∧ | |
| (∀j, Tendsto (fun n => (firstExitFragmentArray d hU Γ V hcut (s n) j).1.val ⊤) | |
| atTop (𝓝 (x j))) ∧ | |
| (∀j, Tendsto (fun n => physicalTollCost d g m | |
| (firstExitFragmentArray d hU Γ V hcut (s n) j).1) atTop (𝓝 (c j))) ∧ | |
| (∑'j, c j) ≤ A := by | |
| let hit : ℕ → ℕ → X := fun n j => (Γ n).1.val (firstExitTime (Γ n).1.val (V j)) | |
| let cost : ℕ → ℕ → ℝ≥0∞ := fun n j => | |
| physicalTollCost d g m (firstExitFragmentArray d hU Γ V hcut n j).1 | |
| obtain ⟨s, _, x, c, hs, _, hx, hc⟩ := exists_joint_fragment_limits hit hit cost | |
| have hindex (j : ℕ) : ∀ᶠn in atTop, j ≤ s n := | |
| hs.tendsto_atTop.eventually_ge_atTop j | |
| have hxfront (j : ℕ) : x j ∈ frontier (V j) := by | |
| apply isClosed_frontier.mem_of_tendsto (hx j) | |
| filter_upwards [hindex j] with n hn | |
| exact firstExit_frontier (Γ (s n)).1.val (hV j) | |
| (hanti (hn.trans (Nat.le_succ _)) (hstart (s n))) | |
| (fun h => hend (s n) (hanti (Nat.zero_le j) h)) | |
| have hstartlim (j : ℕ) : Tendsto | |
| (fun n => (firstExitFragmentArray d hU Γ V hcut (s n) j).1.val ⊥) | |
| atTop (𝓝 (x (j+1))) := by | |
| apply (hx (j+1)).congr' | |
| filter_upwards [hindex j] with n hn | |
| exact (firstExitFragmentArray_endpoints d hU Γ V hcut hn).1.symm | |
| have hstoplim (j : ℕ) : Tendsto | |
| (fun n => (firstExitFragmentArray d hU Γ V hcut (s n) j).1.val ⊤) | |
| atTop (𝓝 (x j)) := by | |
| apply (hx j).congr' | |
| filter_upwards [hindex j] with n hn | |
| exact (firstExitFragmentArray_endpoints d hU Γ V hcut hn).2.symm | |
| refine ⟨s, x, c, hs, hxfront, hstartlim, hstoplim, hc, ?_⟩ | |
| apply fragment_limit_budget_of_eventually (fun n j => cost (s n) j) c | |
| (fun n => physicalTollCost d g m (Γ (s n)).1) A hc (hA.comp hs.tendsto_atTop) | |
| intro F | |
| filter_upwards [hindex (F.sup id)] with n hn | |
| apply firstExitFragmentArray_finite_budget d hU Γ V hcut hV hanti hend g hg m hm F | |
| intro j hj | |
| exact (Finset.le_sup (f := id) hj).trans hn | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |