openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Modulus /CountableConcatenationLength.lean
| import OAI.Analysis.CircleDomains.Topology.CountablePathConcatenation | |
| import OAI.Analysis.CircleDomains.Modulus.PathVariation | |
| namespace OAI | |
| /-! Exact arclength addition over the decreasing collar intervals. This is | |
| an extended-real identity and therefore does not assume rectifiability. -/ | |
| open Set Function Filter | |
| open scoped Topology ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| variable {X : Type*} [MetricSpace X] | |
| theorem collar_variation_partial (γ : PathTime → X) (N : ℕ) : | |
| (∑ n ∈ Finset.range N, eVariationOn γ (collarCell n)) = | |
| eVariationOn γ (Icc (collarTime N) ⊤) := by | |
| induction N with | |
| | zero => simp [collarTime_zero] | |
| | succ N ih => | |
| rw [Finset.sum_range_succ, ih, add_comm] | |
| have he := eVariationOn.Icc_add_Icc γ (s := univ) | |
| (collarTime_strictAnti (Nat.lt_succ_self N)).le (le_top : collarTime N ≤ ⊤) | |
| (mem_univ (collarTime N)) | |
| simpa only [univ_inter, collarCell] using he | |
| theorem collar_variation_sum (γ : C(PathTime, X)) : | |
| eVariationOn γ univ = ∑' n, eVariationOn γ (collarCell n) := by | |
| apply le_antisymm | |
| · have he := eVariationOn.eVariationOn_Ioc_eq_Icc_of_continuousWithinAt | |
| (γ.continuous.continuousAt (x := (⊥ : PathTime))).continuousWithinAt | |
| (b := (⊤ : PathTime)) | |
| rw [Icc_bot_top] at he | |
| rw [← he, eVariationOn] | |
| apply iSup_le | |
| rintro ⟨k, u, hu, hmem⟩ | |
| obtain ⟨N, hN⟩ := (collarTime_tendsto.eventually (gt_mem_nhds (hmem 0).1)).exists | |
| have hall : ∀ i, u i ∈ Icc (collarTime N) ⊤ := by | |
| intro i | |
| exact ⟨hN.le.trans (hu (Nat.zero_le i)), le_top⟩ | |
| exact (eVariationOn.sum_le hu hall).trans | |
| ((collar_variation_partial γ N).symm.le.trans (ENNReal.sum_le_tsum _)) | |
| · rw [ENNReal.tsum_eq_iSup_nat] | |
| apply iSup_le | |
| intro N | |
| rw [collar_variation_partial] | |
| exact eVariationOn.mono γ (subset_univ _) | |
| theorem collarConcatenation_variation (p : X) (γ : ℕ → C(PathTime, X)) hjoin htail : | |
| eVariationOn (collarConcatenation p γ hjoin htail) univ = | |
| ∑' n, eVariationOn (γ n) univ := by | |
| rw [collar_variation_sum] | |
| apply tsum_congr | |
| intro n | |
| let f := affinePathTime (collarTime (n + 1)) (collarTime n) | |
| have hf := affinePathTime_strictMono (collarTime_strictAnti (Nat.lt_succ_self n)) | |
| have he := eVariationOn.comp_eq_of_monotoneOn | |
| (collarConcatenation p γ hjoin htail) f (hf.monotone.monotoneOn univ) | |
| rw [image_univ, range_affinePathTime (collarTime_strictAnti (Nat.lt_succ_self n))] at he | |
| change eVariationOn _ (Icc (collarTime (n + 1)) (collarTime n)) = _ | |
| rw [← he] | |
| apply eVariationOn.congr | |
| intro t _ | |
| exact collarConcatenation_affine p γ hjoin htail n t | |
| theorem collarConcatenation_rectifiable (p : X) (γ : ℕ → C(PathTime, X)) hjoin htail | |
| (hfinite : (∑' n, eVariationOn (γ n) univ) < ∞) : | |
| BoundedVariationOn (collarConcatenation p γ hjoin htail) univ := by | |
| change eVariationOn _ _ ≠ ∞ | |
| rw [collarConcatenation_variation] | |
| exact hfinite.ne | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |