| import OAI.Analysis.CircleDomains.Topology.ResidualClosedIntervals |
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| namespace OAI |
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| /-! Incidence bounds for the countable image exposed by discarded constant |
| tags. The estimate charges parameter times, and hence retains multiplicity. -/ |
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| open Set Function MeasureTheory |
| open scoped ENNReal Classical |
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| noncomputable section |
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| namespace CircleDomainRigidity |
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| variable {T X Z : Type*} |
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| def timeIncidence (R : Set (Z × X)) (γ : T → X) (A : Set T) : Set Z := |
| {z | ∃ t ∈ A, (z, γ t) ∈ R} |
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| def timeToll (γ : T → X) (A : Set T) (m : X → ℝ≥0∞) : ℝ≥0∞ := |
| ∑' t : A, m (γ t) |
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| theorem timeToll_eq_indicator (γ : T → X) (A : Set T) (m : X → ℝ≥0∞) : |
| timeToll γ A m = ∑' t, if t ∈ A then m (γ t) else 0 := by |
| exact tsum_subtype A (fun t => m (γ t)) |
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| theorem timeIncidence_iUnion {ι : Type*} (R : Set (Z × X)) (γ : T → X) |
| (A : ι → Set T) : timeIncidence R γ (⋃ i, A i) = ⋃ i, timeIncidence R γ (A i) := by |
| ext z |
| simp only [timeIncidence, mem_ofPred_eq, mem_iUnion] |
| constructor |
| · rintro ⟨t, ⟨i, hi⟩, ht⟩ |
| exact ⟨i, t, hi, ht⟩ |
| · rintro ⟨i, t, hi, ht⟩ |
| exact ⟨t, ⟨i, hi⟩, ht⟩ |
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| theorem timeToll_mono (γ : T → X) (m : X → ℝ≥0∞) {A C : Set T} |
| (h : A ⊆ C) : timeToll γ A m ≤ timeToll γ C m := |
| ENNReal.tsum_mono_subtype (fun t => m (γ t)) h |
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| theorem timeToll_iUnion {ι : Type*} (γ : T → X) (m : X → ℝ≥0∞) |
| (A : ι → Set T) (hA : Pairwise (Disjoint on A)) : |
| timeToll γ (⋃ i, A i) m = ∑' i, timeToll γ (A i) m := by |
| exact ENNReal.tsum_biUnion (f := fun t => m (γ t)) (fun i _ j _ hij => hA hij) |
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| variable [MeasurableSpace Z] |
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| theorem countable_image_incidence_bound (R : Set (Z × X)) (γ : T → X) |
| (A : Set T) (ν : Measure Z) (m : X → ℝ≥0∞) |
| (hpoint : ∀ x, ν {z | (z, x) ∈ R} ≤ m x) (hA : (γ '' A).Countable) : |
| ν (timeIncidence R γ A) ≤ timeToll γ A m := by |
| let : Countable (γ '' A) := hA.to_subtype |
| have he : timeIncidence R γ A = ⋃ x : γ '' A, {z | (z, x.val) ∈ R} := by |
| ext z |
| constructor |
| · rintro ⟨t, ht, hr⟩ |
| exact mem_iUnion.mpr ⟨⟨γ t, ⟨t, ht, rfl⟩⟩, hr⟩ |
| · intro hz |
| obtain ⟨x, hx⟩ := mem_iUnion.mp hz |
| obtain ⟨t, ht, he⟩ := x.property |
| exact ⟨t, ht, he ▸ hx⟩ |
| let f : A → γ '' A := fun t => ⟨γ t, ⟨t, t.property, rfl⟩⟩ |
| have hf : Surjective f := by |
| intro x |
| obtain ⟨t, ht, he⟩ := x.property |
| exact ⟨⟨t, ht⟩, Subtype.ext he⟩ |
| calc |
| ν (timeIncidence R γ A) ≤ ∑' x : γ '' A, ν {z | (z, x.val) ∈ R} := by |
| rw [he] |
| exact measure_iUnion_le _ |
| _ ≤ ∑' x : γ '' A, m x := ENNReal.tsum_le_tsum fun x => hpoint x |
| _ ≤ timeToll γ A m := ENNReal.tsum_le_tsum_comp_of_surjective hf (fun x => m x) |
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| theorem countable_image_deleted_incidence_bound (R : Set (Z × X)) (P : Set X) |
| (γ : T → X) (A : Set T) (ν : Measure Z) (m : X → ℝ≥0∞) |
| (hpoint : ∀ x, ν {z | (z, x) ∈ R} ≤ m x) (hA : (γ '' A).Countable) : |
| ν (timeIncidence {q | q ∈ R ∧ q.2 ∉ P} γ A) ≤ timeToll γ A (deletedWeight P m) := by |
| apply countable_image_incidence_bound _ γ A ν _ _ hA |
| intro x |
| by_cases hx : x ∈ P |
| · simp [deletedWeight, hx] |
| · simpa [deletedWeight, hx] using hpoint x |
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| theorem timeIncidence_bound_of_disjoint_cover {ι : Type*} [Countable ι] |
| (R : Set (Z × X)) (γ : T → X) (A : Set T) (C : ι → Set T) |
| (ν : Measure Z) (m : X → ℝ≥0∞) (hcover : A ⊆ ⋃ i, C i) |
| (hsub : ∀ i, C i ⊆ A) (hdis : Pairwise (Disjoint on C)) |
| (hbound : ∀ i, ν (timeIncidence R γ (C i)) ≤ timeToll γ (C i) m) : |
| ν (timeIncidence R γ A) ≤ timeToll γ A m := by |
| have he : A = ⋃ i, C i := hcover.antisymm (iUnion_subset hsub) |
| rw [he, timeIncidence_iUnion, timeToll_iUnion γ m C hdis] |
| exact (measure_iUnion_le _).trans (ENNReal.tsum_le_tsum hbound) |
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| end CircleDomainRigidity |
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| end |
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| end OAI |
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