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f205175 | 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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.BoundaryCount
namespace OAI
section
section CurrentModule5
noncomputable section
open Set
open scoped Classical BigOperators
namespace CriticalPercolation.Sharpness
open FiniteProduct ClusterDeletion
variable {E V : Type*} [Fintype E]
lemma continuous_probability (F : Set E → Prop) :
Continuous (fun p : ℝ => probability p F) := by
unfold probability expect
apply continuous_finsetSum
intro ω _
apply Continuous.mul _ continuous_const
exact continuous_iff_continuousAt.mpr (fun p => (hasDerivAt_density p ω).continuousAt)
lemma probability_monotoneOn (F : Set E → Prop)
(hF : ∀ s t, s ⊆ t → F s → F t) :
MonotoneOn (fun p : ℝ => probability p F) (Icc 0 1) := by
apply monotoneOn_of_deriv_nonneg (convex_Icc 0 1)
(continuous_probability F).continuousOn
· intro p hp
rw [interior_Icc] at hp
exact (hasDerivAt_probability hp F hF).differentiableAt.differentiableWithinAt
· intro p hp
rw [interior_Icc] at hp
rw [(hasDerivAt_probability hp F hF).deriv]
exact Finset.sum_nonneg fun e _ => probability_nonneg ⟨hp.1.le,hp.2.le⟩ _
variable (G : BondGraph V E)
omit [Fintype E] in
lemma boundary_event_mono (o : V) (R : Set V) (e : E) {ω ω' : Set E}
(h : ω ⊆ ω') :
boundaryReach G R (delete (G.openGraph ω) R) o e →
boundaryReach G R (delete (G.openGraph ω') R) o e := by
rintro ⟨x,y,he,hx,hy,hox⟩
have hdel : delete (G.openGraph ω) R ≤ delete (G.openGraph ω') R := by
intro a b hh
exact ⟨G.openGraph_mono h hh.1,hh.2⟩
exact ⟨x,y,he,hx,hy,hox.mono hdel⟩
lemma boundaryFlux_eq_sum (p : ℝ) (o : V) (R : Set V) :
boundaryFlux G p o R = p * ∑ e, probability p
(fun ω => boundaryReach G R (delete (G.openGraph ω) R) o e) := by
simp only [boundaryFlux,boundaryCount,expect_sum,probability]
lemma boundaryFlux_nonneg {p : ℝ} (hp : p ∈ Icc 0 1) (o : V) (R : Set V) :
0 ≤ boundaryFlux G p o R := by
rw [boundaryFlux_eq_sum]
exact mul_nonneg hp.1 (Finset.sum_nonneg fun e _ => probability_nonneg hp _)
lemma boundaryFlux_monotoneOn (o : V) (R : Set V) :
MonotoneOn (fun p : ℝ => boundaryFlux G p o R) (Icc 0 1) := by
intro p hp q hq hpq
simp only [boundaryFlux_eq_sum]
apply mul_le_mul hpq _ (Finset.sum_nonneg fun e _ => probability_nonneg hp _) hq.1
apply Finset.sum_le_sum
intro e _
exact probability_monotoneOn _ (fun _ _ h => boundary_event_mono G o R e h) hp hq hpq
theorem uniform_escape_lower_bound {p q : ℝ} (hp : p ∈ Ioo 0 1)
(hpq : p < q) (hq : q < 1) (o : V) (T : Set V)
(hflux : ∀ R : Set V, T ⊆ R → o ∉ R → 1 ≤ boundaryFlux G p o R) :
(q-p)/(1+q-p) ≤ probability q (reachesTarget G o T) := by
let F := reachesTarget G o T
have hF : ∀ s t, s ⊆ t → F s → F t := fun _ _ h => reachesTarget_mono G o T h
have hq0 : 0 < q := hp.1.trans hpq
obtain ⟨c,hc,hderiv⟩ := exists_deriv_eq_slope
(fun r => probability r F) hpq (continuous_probability F).continuousOn
(fun r hr => (hasDerivAt_probability ⟨hp.1.trans hr.1,hr.2.trans hq⟩ F hF
).differentiableAt.differentiableWithinAt)
have hc01 : c ∈ Ioo 0 1 := ⟨hp.1.trans hc.1,hc.2.trans hq⟩
have hd := differential_inequality G hc01 o T (a := 1) (fun R hTR ho =>
(hflux R hTR ho).trans (boundaryFlux_monotoneOn G o R
⟨hp.1.le,hp.2.le⟩ ⟨hc01.1.le,hc01.2.le⟩ hc.1.le))
have hmono := probability_monotoneOn F hF ⟨hc01.1.le,hc01.2.le⟩
⟨hq0.le,hq.le⟩ hc.2.le
have hdnonneg : 0 ≤ deriv (fun r => probability r F) c := by
rw [(hasDerivAt_probability hc01 F hF).deriv]
exact Finset.sum_nonneg fun e _ => probability_nonneg ⟨hc01.1.le,hc01.2.le⟩ _
have hfactor : c*(1-c) ≤ 1 := by nlinarith [sq_nonneg c]
have hds : c*(1-c)*deriv (fun r => probability r F) c ≤
deriv (fun r => probability r F) c := by
simpa only [one_mul] using mul_le_mul_of_nonneg_right hfactor hdnonneg
have hlow : 1-probability q F ≤ deriv (fun r => probability r F) c := by
change 1*(1-probability c F) ≤ _ at hd
linarith
have hdiff := mul_le_mul_of_nonneg_right hlow (sub_pos.mpr hpq).le
have heq := (eq_div_iff (sub_pos.mpr hpq).ne').mp hderiv
have hnonneg := probability_nonneg ⟨hp.1.le,hp.2.le⟩ F
apply (div_le_iff₀ (by linarith : 0 < 1+q-p)).mpr
change q-p ≤ probability q F * (1+q-p)
nlinarith
end CriticalPercolation.Sharpness
end
end CurrentModule5
end
end OAI
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