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9655094 | 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 | import OAI.Analysis.CircleDomains.Modulus.PhysicalPieceLength
namespace OAI
/-! Finite physical length of the actual collar concatenation. The bound
is proved before applying allowed-path stability. -/
open Set Function MeasureTheory
open scoped ENNReal Topology
noncomputable section
namespace CircleDomainRigidity
variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] {G : Set Y} {U : Set X}
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
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 _)
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⟩
end CircleDomainRigidity
end
end OAI
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