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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.CoverageScale
import OAI.Probability.SKBarriers.Dynamics.GreedyNumerics
namespace OAI
section
noncomputable section
open scoped BigOperators Topology
open Classical MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem high_probability_inter (C D : (n : ℕ) → Set (Disorder n))
(hD : ∀n,MeasurableSet (D n))
(hC1 : Tendsto (fun n => (disorderLaw n).real (C n)) atTop (𝓝 1))
(hD1 : Tendsto (fun n => (disorderLaw n).real (D n)) atTop (𝓝 1)) :
Tendsto (fun n => (disorderLaw n).real (C n ∩ D n)) atTop (𝓝 1) := by
apply tendsto_of_tendsto_of_tendsto_of_le_of_le
(g:=fun n => (disorderLaw n).real (C n)+(disorderLaw n).real (D n)-1)
(h:=fun _ => (1:ℝ))
· convert (hC1.add hD1).sub_const 1 using 1; norm_num
· exact tendsto_const_nhds
· intro n
have H := measureReal_union_add_inter (μ:=disorderLaw n) (s:=C n) (hD n)
have H1 : (disorderLaw n).real (C n ∪ D n)≤1 := measureReal_le_one
linarith only [H,H1]
· intro n; exact measureReal_le_one
def finiteGoodEvent (C : ℕ → (n : ℕ) → Set (Disorder n)) : ℕ → (n : ℕ) → Set (Disorder n)
| 0,_ => univ
| j+1,n => finiteGoodEvent C j n ∩ C j n
theorem finiteGoodEvent_mem (C : ℕ → (n : ℕ) → Set (Disorder n)) (B n : ℕ) (J : Disorder n) :
J∈finiteGoodEvent C B n ↔ ∀j<B,J∈C j n := by
induction B with
| zero => simp [finiteGoodEvent]
| succ B ih =>
simp only [finiteGoodEvent,mem_inter_iff,ih]
constructor
· rintro ⟨h,hB⟩ j hj
rcases Nat.lt_succ_iff_lt_or_eq.mp hj with hj|rfl
· exact h j hj
· exact hB
· intro h; exact ⟨fun j hj => h j (by omega),h B (by omega)⟩
theorem finiteGoodEvent_measurable (C : ℕ → (n : ℕ) → Set (Disorder n))
(hC : ∀j n,MeasurableSet (C j n)) (B n : ℕ) : MeasurableSet (finiteGoodEvent C B n) := by
induction B with
| zero => exact MeasurableSet.univ
| succ B ih => exact ih.inter (hC B n)
theorem finiteGoodEvent_tendsto (C : ℕ → (n : ℕ) → Set (Disorder n))
(hC : ∀j n,MeasurableSet (C j n))
(hC1 : ∀j,Tendsto (fun n => (disorderLaw n).real (C j n)) atTop (𝓝 1)) (B : ℕ) :
Tendsto (fun n => (disorderLaw n).real (finiteGoodEvent C B n)) atTop (𝓝 1) := by
induction B with
| zero => simpa only [finiteGoodEvent,probReal_univ] using (tendsto_const_nhds (x:=(1:ℝ)))
| succ B ih => exact high_probability_inter _ _ (hC B) ih (hC1 B)
theorem real_markov_bound {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
{f : Ω → ℝ} (hf : Integrable f μ) (hm : Measurable f) (h0 : ∀x,0≤f x)
{a : ℝ} (ha : 0<a) : μ.real {x | a<f x}≤(∫x,f x ∂μ)/a := by
have hA : MeasurableSet {x | a<f x} := measurableSet_lt measurable_const hm
apply (le_div_iff₀ ha).mpr
have H : (∫x,({x | a<f x}).indicator (fun _ => a) x ∂μ)≤∫x,f x ∂μ := by
apply integral_mono ((integrable_const a).indicator hA) hf
intro x
by_cases hx : a<f x
· simpa only [Set.indicator_of_mem (show x∈{x | a<f x} from hx)] using hx.le
· simpa only [Set.indicator_of_notMem (show x∉{x | a<f x} from hx)] using h0 x
simpa only [integral_indicator_const _ hA,smul_eq_mul,mul_comm] using H
theorem stretched_exp_decay {c : ℝ} (hc : 0<c) :
Tendsto (fun n => Real.exp (-c*levelLogScale n)) atTop (𝓝 0) := by
exact Real.tendsto_exp_atBot.comp (stretchedLogScale_tendsto.const_mul_atTop_of_neg (neg_neg_of_pos hc))
theorem faster_stretched_decay {u : ℝ} (hu : kappa<u) (A K : ℝ) :
Tendsto (fun n : ℕ => K*Real.exp (A*levelLogScale n-(n:ℝ)^u)) atTop (𝓝 0) := by
have hu0 : 0<u := (by norm_num [kappa] : (0:ℝ)<kappa).trans hu
have H : Tendsto (fun n : ℕ => Real.exp (-(1/2:ℝ)*(n:ℝ)^u)) atTop (𝓝 0) :=
Real.tendsto_exp_atBot.comp (((tendsto_rpow_atTop hu0).comp (tendsto_natCast_atTop_atTop (R:=ℝ))).const_mul_atTop_of_neg (by norm_num))
have H0 : Tendsto (fun n : ℕ => Real.exp (A*levelLogScale n-(n:ℝ)^u)) atTop (𝓝 0) := by
apply squeeze_zero' (Eventually.of_forall (fun n => (Real.exp_pos _).le)) _ H
filter_upwards [eventually_const_rpow_le hu A (by norm_num : (0:ℝ)<1/2)] with n hn
apply Real.exp_le_exp.mpr
change A*(n:ℝ)^kappa-(n:ℝ)^u≤-(1/2:ℝ)*(n:ℝ)^u
linarith only [hn]
simpa only [mul_zero] using H0.const_mul K
theorem high_probability_small_replica {d : ℕ} (β : ℝ) (S : (n : ℕ) → Finset (ReplicaConfig n d))
{u : ℝ} (hu : kappa<u) (K A : ℝ)
(h : ∀ᶠ n : ℕ in atTop,(∫J,replicaGibbsMass β J (S n) ∂disorderLaw n)≤K*Real.exp (-(n:ℝ)^u)) :
Tendsto (fun n => (disorderLaw n).real {J | replicaGibbsMass β J (S n)≤Real.exp (-A*levelLogScale n)}) atTop (𝓝 1) := by
have HC : Tendsto (fun n => (disorderLaw n).real {J | Real.exp (-A*levelLogScale n)<replicaGibbsMass β J (S n)}) atTop (𝓝 0) := by
apply squeeze_zero' (Eventually.of_forall (fun n => measureReal_nonneg)) _ (faster_stretched_decay hu A K)
filter_upwards [h] with n hn
refine (real_markov_bound (disorderLaw n) (replicaGibbsMass_integrable β (S n))
(continuous_replicaGibbsMass β (S n)).measurable (fun J => replicaGibbsMass_nonneg β J (S n)) (Real.exp_pos _)).trans ?_
calc
_ ≤ (K*Real.exp (-(n:ℝ)^u))/Real.exp (-A*levelLogScale n) := div_le_div_of_nonneg_right hn (Real.exp_pos _).le
_ = K*Real.exp (A*levelLogScale n-(n:ℝ)^u) := by rw [mul_div_assoc,← Real.exp_sub]; congr 2; ring
have H := (tendsto_const_nhds (x:=(1:ℝ))).sub HC
convert H using 1
· funext n
have hM : MeasurableSet {J | Real.exp (-A*levelLogScale n)<replicaGibbsMass β J (S n)} :=
measurableSet_lt measurable_const (continuous_replicaGibbsMass β (S n)).measurable
rw [← probReal_compl_eq_one_sub hM]
congr 1
ext J
simp only [mem_ofPred_eq,mem_compl_iff,not_lt]
· norm_num
end SK.Analytic
end
end
end OAI
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