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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 82 83 84 85 86 87 88 89 90 91 92 93 94 | 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
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