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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 81 | 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
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