| import OAI.Analysis.CircleDomains.Modulus.PhysicalPieceLength |
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| namespace OAI |
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| /-! Finite physical length of the actual collar concatenation. The bound |
| is proved before applying allowed-path stability. -/ |
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| open Set Function MeasureTheory |
| open scoped ENNReal Topology |
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| noncomputable section |
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| namespace CircleDomainRigidity |
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| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] {G : Set Y} {U : Set X} |
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| theorem physicalDyadicLength_collarPieces_le (e : G ≃ₜ U) (hU : IsOpen U) (y₀ : Y) |
| (γ : ℕ → FinitePhysicalPath e y₀) (η : C(PathTime, X)) |
| (hmatch : ∀ n s, η (affinePathTime (collarTime (n + 1)) (collarTime n) s) = (γ n).val s) |
| (k : ℕ) (i : Fin (2 ^ k)) : |
| physicalDyadicLength e y₀ k i η ≤ |
| ∑' n, physicalTimeLength e y₀ (γ n) |
| (Icc (rescalePathTime (collarTime (n + 1)) (collarTime n) (dyadicLeft k i)) |
| (rescalePathTime (collarTime (n + 1)) (collarTime n) (dyadicRight k i))) := by |
| by_cases hη : η ∈ pathsInsideOn U (dyadicLeft k i) (dyadicRight k i) |
| · rw [physicalDyadicLength, physicalDyadicPath_of_mem e y₀ k i η hη, |
| collar_variation_sum] |
| apply ENNReal.tsum_le_tsum |
| intro n |
| exact physicalSubpath_piece_length e hU y₀ (γ n) η |
| (collarTime_strictAnti (Nat.lt_succ_self n)) (hmatch n) (dyadicLeft_le_right k i) hη |
| · simp only [physicalDyadicLength, physicalDyadicPath, totalPhysicalSubpath, dite_eq_right hη] |
| have hz : eVariationOn (ContinuousMap.const PathTime y₀) univ = 0 := |
| (eVariationOn.eq_zero_iff _).mpr fun _ _ _ _ => edist_self _ |
| rw [hz] |
| exact bot_le |
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| theorem physicalDomainLength_collarPieces_le (e : G ≃ₜ U) (hU : IsOpen U) (y₀ : Y) |
| (γ : ℕ → FinitePhysicalPath e y₀) (η : C(PathTime, X)) |
| (hmatch : ∀ n s, η (affinePathTime (collarTime (n + 1)) (collarTime n) s) = (γ n).val s) : |
| physicalDomainLength e y₀ η ≤ ∑' n, physicalDomainLength e y₀ (γ n).val := by |
| apply iSup_le |
| intro k |
| calc |
| physicalStageLength e y₀ k η ≤ |
| ∑ i : Fin (2 ^ k), ∑' n, physicalTimeLength e y₀ (γ n) |
| (Icc (rescalePathTime (collarTime (n + 1)) (collarTime n) (dyadicLeft k i)) |
| (rescalePathTime (collarTime (n + 1)) (collarTime n) (dyadicRight k i))) := |
| Finset.sum_le_sum fun i _ => physicalDyadicLength_collarPieces_le e hU y₀ γ η hmatch k i |
| _ = ∑' n, ∑ i : Fin (2 ^ k), physicalTimeLength e y₀ (γ n) |
| (Icc (rescalePathTime (collarTime (n + 1)) (collarTime n) (dyadicLeft k i)) |
| (rescalePathTime (collarTime (n + 1)) (collarTime n) (dyadicRight k i))) := by |
| simpa only [tsum_fintype] using (ENNReal.tsum_comm (f := fun i : Fin (2 ^ k) => |
| fun n => physicalTimeLength e y₀ (γ n) |
| (Icc (rescalePathTime (collarTime (n + 1)) (collarTime n) (dyadicLeft k i)) |
| (rescalePathTime (collarTime (n + 1)) (collarTime n) (dyadicRight k i))))) |
| _ ≤ _ := by |
| apply ENNReal.tsum_le_tsum |
| intro n |
| rw [sum_monotone_dyadic_mass _ _ |
| (rescalePathTime_monotone (collarTime_strictAnti (Nat.lt_succ_self n))), |
| ← physicalTimeLength_univ e y₀ (γ n)] |
| exact measure_mono (subset_univ _) |
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| def finitePhysicalCollarConcatenation (e : G ≃ₜ U) (hU : IsOpen U) (y₀ : Y) |
| (p : X) (γ : ℕ → FinitePhysicalPath e y₀) |
| (hjoin : ∀ n, (γ (n + 1)).val ⊤ = (γ n).val ⊥) |
| (htail : ∀ V ∈ 𝓝 p, ∃ N : ℕ, ∀ n ≥ N, ∀ s, (γ n).val s ∈ V) |
| (hfinite : (∑' n, physicalDomainLength e y₀ (γ n).val) < ∞) : |
| FinitePhysicalPath e y₀ := |
| ⟨collarConcatenation p (fun n => (γ n).val) hjoin htail, |
| (physicalDomainLength_collarPieces_le e hU y₀ γ _ |
| (collarConcatenation_affine p (fun n => (γ n).val) hjoin htail)).trans_lt hfinite⟩ |
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| end CircleDomainRigidity |
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| end |
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| end OAI |
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