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135052a | 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 Mathlib
import OAI.Probability.SKBarriers.Coverage.FiniteCoverage
import OAI.Probability.SKBarriers.Dynamics.GreedyNumerics
import OAI.Probability.SKBarriers.Dynamics.BankCardinality
namespace OAI
section
noncomputable section
open scoped BigOperators Topology
open Classical MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem gibbs_prob_eq_finiteMass {n : ℕ} (β : ℝ) (J : Disorder n) (E : Config n → Prop) :
(gibbsFiniteLaw β J).prob E=finiteMass (gibbs β J) (Finset.univ.filter E) := by
unfold FiniteLaw.prob FiniteLaw.expect gibbsFiniteLaw finiteMass
simp only [mul_ite,mul_one,mul_zero,Finset.sum_filter]
theorem gibbs_coverage_predicate {n : ℕ} (β : ℝ) (J : Disorder n) (q a δ : ℝ)
(h : ∀ S : Finset (Config n),finiteMass (gibbs β J) (S.filter (fun x =>
finiteMass (gibbs β J) (S.filter (fun y => q≤|overlap x y|))<a))<δ)
(E : Config n → Prop) :
(gibbsFiniteLaw β J).prob (fun x => E x ∧ (gibbsFiniteLaw β J).prob (fun y => E y ∧ q≤|overlap x y|)<a)≤δ := by
have H := h (Finset.univ.filter E)
simp_rw [Finset.filter_filter] at H
refine le_trans ?_ H.le
apply le_of_eq
rw [gibbs_prob_eq_finiteMass]
congr 1
ext x
simp only [Finset.mem_filter,Finset.mem_univ,true_and,gibbs_prob_eq_finiteMass]
apply and_congr_right
intro _
apply Iff.of_eq
apply congrArg (fun r : ℝ => r<a)
apply congrArg (finiteMass (gibbs β J))
ext y
simp only [Finset.mem_filter,Finset.mem_univ,true_and]
def coveragePrefix (f : ℝ → ℝ) : ℕ → ℝ
| 0 => 0
| j+1 => coveragePrefix f j+f (coveragePrefix f j+5)+2
theorem coveragePrefix_nonneg (f : ℝ → ℝ) (hf : ∀x,0<f x) (j : ℕ) : 0≤coveragePrefix f j := by
induction j with
| zero => exact le_rfl
| succ j ih => simp only [coveragePrefix]; linarith only [ih,hf (coveragePrefix f j+5)]
theorem bankPrefix_coverage (f : ℝ → ℝ) (j : ℕ) :
bankPrefix (fun j => f (coveragePrefix f j+5)) j=coveragePrefix f j := by
induction j with
| zero => rfl
| succ j ih => simp only [bankPrefix,coveragePrefix,ih]
theorem exists_coverage_schedule (β q : ℝ)
(h : ∀D : ℝ,0<D → ∃c : ℝ,0<c ∧ ∃C : (n : ℕ) → Set (Disorder n),
(∀n,MeasurableSet (C n)) ∧ Tendsto (fun n => (disorderLaw n).real (C n)) atTop (𝓝 1) ∧
∀n J,J∈C n → ∀S : Finset (Config n),finiteMass (gibbs β J) (S.filter (fun x =>
finiteMass (gibbs β J) (S.filter (fun y => q≤|overlap x y|))<Real.exp (-c*levelLogScale n)))<Real.exp (-D*levelLogScale n)) :
∃c : ℕ → ℝ, (∀j,0<c j) ∧ ∃C : ℕ → (n : ℕ) → Set (Disorder n),
(∀j n,MeasurableSet (C j n)) ∧
(∀j,Tendsto (fun n => (disorderLaw n).real (C j n)) atTop (𝓝 1)) ∧
∀j n J,J∈C j n → ∀E : Config n → Prop,
(gibbsFiniteLaw β J).prob (fun x => E x ∧ (gibbsFiniteLaw β J).prob (fun y => E y ∧ q≤|overlap x y|)<Real.exp (-c j*levelLogScale n))≤
Real.exp (-(bankPrefix c j+5)*levelLogScale n) := by
choose f hf C hC hC1 hcov using (fun D : ℝ => h (max D 1) (lt_of_lt_of_le zero_lt_one (le_max_right _ _)))
let c : ℕ → ℝ := fun j => f (coveragePrefix f j+5)
refine ⟨c,fun j => hf _,fun j => C (coveragePrefix f j+5),fun j n => hC _ n,fun j => hC1 _,?_⟩
intro j n J hJ E
have HD : max (coveragePrefix f j+5) 1=coveragePrefix f j+5 := max_eq_left (by linarith only [coveragePrefix_nonneg f hf j])
have H := gibbs_coverage_predicate β J q _ _ (hcov (coveragePrefix f j+5) n J hJ) E
simpa only [HD,c,bankPrefix_coverage] using H
end SK.Analytic
end
end
end OAI
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