| import OAI.Analysis.CircleDomains.Selection.FiniteChainPotential | |
| import OAI.Analysis.CircleDomains.Uniformization.BandLimits | |
| namespace OAI | |
| /-! Real-valued oscillation and topological bands of the truncated chain | |
| distance. The closure argument retains all cluster values. -/ | |
| open Set Filter | |
| open scoped Topology ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| variable {X : Type*} | |
| def realChainPotential (R : X → X → ℝ≥0∞ → Prop) (E : Set X) (x : X) : ℝ := | |
| (chainPotential R E x).toReal | |
| theorem chainPotential_ne_top (R : X → X → ℝ≥0∞ → Prop) (E : Set X) (x : X) : | |
| chainPotential R E x ≠ ∞ := | |
| (lt_of_le_of_lt (chainPotential_le_one R E x) (by simp)).ne | |
| theorem realChainPotential_mem_Icc (R : X → X → ℝ≥0∞ → Prop) (E : Set X) (x : X) : | |
| realChainPotential R E x ∈ Icc 0 1 := by | |
| refine ⟨ENNReal.toReal_nonneg, ?_⟩ | |
| change (chainPotential R E x).toReal ≤ 1 | |
| simpa only [ENNReal.toReal_one] using | |
| ENNReal.toReal_mono (by simp : (1 : ℝ≥0∞) ≠ ∞) (chainPotential_le_one R E x) | |
| theorem realChainPotential_difference (R : X → X → ℝ≥0∞ → Prop) | |
| (hR : ∀ x y a, R x y a → R y x a) {E : Set X} {x y : X} {a : ℝ≥0∞} | |
| (h : R x y a) : ENNReal.ofReal |realChainPotential R E y - realChainPotential R E x| ≤ 4 * a := by | |
| by_cases ha : a = ∞ | |
| · simp [ha] | |
| have hcost : (4 : ℝ≥0∞) * a ≠ ∞ := ENNReal.mul_ne_top (by simp) ha | |
| obtain ⟨hyx, hxy⟩ := chainPotential_two_sided R hR (E := E) h | |
| have hyxr := ENNReal.toReal_mono | |
| (ENNReal.add_ne_top.mpr ⟨chainPotential_ne_top R E x, hcost⟩) hyx | |
| have hxyr := ENNReal.toReal_mono | |
| (ENNReal.add_ne_top.mpr ⟨chainPotential_ne_top R E y, hcost⟩) hxy | |
| rw [ENNReal.toReal_add (chainPotential_ne_top R E x) hcost] at hyxr | |
| rw [ENNReal.toReal_add (chainPotential_ne_top R E y) hcost] at hxyr | |
| apply (ENNReal.ofReal_le_iff_le_toReal hcost).mpr | |
| apply abs_le.mpr | |
| change -(4 * a).toReal ≤ (chainPotential R E y).toReal - (chainPotential R E x).toReal ∧ | |
| (chainPotential R E y).toReal - (chainPotential R E x).toReal ≤ (4 * a).toReal | |
| constructor <;> linarith | |
| variable [TopologicalSpace X] [FirstCountableTopology X] | |
| theorem clusterGraph_oscillation_of_neighborhood {G V : Set X} {f : X → ℝ} | |
| {s : X} {a b : ℝ} {c : ℝ≥0∞} (hV : V ∈ 𝓝 s) | |
| (hbound : ∀ x ∈ G ∩ V, ∀ y ∈ G ∩ V, ENNReal.ofReal |f x - f y| ≤ c) | |
| (ha : (s, a) ∈ clusterGraph G f) (hb : (s, b) ∈ clusterGraph G f) : | |
| ENNReal.ofReal |a - b| ≤ c := by | |
| obtain ⟨v, hfv, hv⟩ := (mem_clusterGraph_iff_mapClusterPt.mp ha).exists_seq_tendsto | |
| obtain ⟨w, hfw, hw⟩ := (mem_clusterGraph_iff_mapClusterPt.mp hb).exists_seq_tendsto | |
| have hvg := (tendsto_nhdsWithin_iff.mp hv).2 | |
| have hwg := (tendsto_nhdsWithin_iff.mp hw).2 | |
| have hvv := (tendsto_nhdsWithin_iff.mp hv).1.eventually hV | |
| have hwv := (tendsto_nhdsWithin_iff.mp hw).1.eventually hV | |
| apply le_of_tendsto (ENNReal.tendsto_ofReal ((hfv.sub hfw).abs)) | |
| filter_upwards [hvg, hwg, hvv, hwv] with n hvg hwg hvv hwv | |
| exact hbound (v n) ⟨hvg, hvv⟩ (w n) ⟨hwg, hwv⟩ | |
| theorem clusterGraph_oscillation_of_jumps (R : X → X → ℝ≥0∞ → Prop) | |
| (hR : ∀ x y a, R x y a → R y x a) (E G : Set X) | |
| {s : X} {V : ℕ → Set X} (hV : ∀ j, V j ∈ 𝓝 s) (c : ℕ → ℝ≥0∞) | |
| (hj : ∀ j, ∀ x ∈ G ∩ V j, ∀ y ∈ G ∩ V j, R x y (c j)) | |
| {a b : ℝ} (ha : (s, a) ∈ clusterGraph G (realChainPotential R E)) | |
| (hb : (s, b) ∈ clusterGraph G (realChainPotential R E)) : | |
| ENNReal.ofReal |a - b| ≤ 4 * limsup c atTop := by | |
| have hc (j : ℕ) : ENNReal.ofReal |a - b| ≤ 4 * c j := | |
| clusterGraph_oscillation_of_neighborhood (hV j) | |
| (fun x hx y hy => realChainPotential_difference R hR (hj j y hy x hx)) ha hb | |
| have hl : ENNReal.ofReal |a - b| ≤ limsup (fun j => (4 : ℝ≥0∞) * c j) atTop := by | |
| apply le_limsup_of_le (h := ?_) | |
| intro t ht | |
| obtain ⟨j, hj⟩ := ht.exists | |
| exact (hc j).trans hj | |
| have hm := ENNReal.limsup_mul_le' (f := atTop) | |
| (u := fun _ : ℕ => (4 : ℝ≥0∞)) (v := c) | |
| (Or.inl (by simp)) (Or.inl (by simp)) | |
| simp only [limsup_const] at hm | |
| exact hl.trans hm | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |