| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.ClockEndpoint |
| import OAI.Probability.SKBarriers.Coverage.FiniteCoverageMixing |
| import OAI.Probability.SKBarriers.Dynamics.GreedyEventual |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Topology |
| open Classical MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem mixedMass_le_pairStrip {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) |
| {a₀ a₁ q a δ L : ℝ} (ha₀ : 0<a₀) (haq : a₁<q) (ha : 0<a) |
| (hgap : 2/(n:ℝ)<a₁-a₀) (hL : 0≤L) (htail : (n:ℝ)*Real.exp (-L)≤(1/8:ℝ)) |
| (hcov : (gibbsFiniteLaw β J).prob (fun x => (gibbsFiniteLaw β J).prob (fun y => q≤|overlap x y|)<a)≤δ) : |
| continuousMixedMass β J (Real.exp L)≤δ+(16*((⌈Real.exp (2*L)⌉₊:ℝ)+1)/a)*replicaGibbsMass β J (pairStripSet n a₀ a₁) ∧ |
| discreteMixedMass β J ⌊Real.exp L⌋₊≤δ+(16*((⌈Real.exp (2*L)⌉₊:ℝ)+1)/a)*replicaGibbsMass β J (pairStripSet n a₀ a₁) := by |
| have hnum : (⌊Real.exp L⌋₊:ℝ)+1≤(⌈Real.exp (2*L)⌉₊:ℝ)+1 := by |
| have H : ⌊Real.exp L⌋₊≤⌈Real.exp (2*L)⌉₊ := |
| (Nat.floor_mono (Real.exp_le_exp.mpr (by linarith only [hL]))).trans (Nat.floor_le_ceil _) |
| exact_mod_cast Nat.add_le_add_right H 1 |
| have HC (v : Config n) : |
| (gibbsFiniteLaw β J).prob (fun x => a₁<overlap v x ∧ continuousDistance β J x (Real.exp L)≤(1/4:ℝ))≤ |
| (8*((⌈Real.exp (2*L)⌉₊:ℝ)+1))*(gibbsFiniteLaw β J).prob (fun x => a₀≤overlap v x ∧ overlap v x≤a₁) := by |
| convert mass_continuous_mixed_high_le_strip hn β J v ha₀ hgap L htail using 1 <;> simp only [gibbs_prob_eq_sum,Nat.cast_add,Nat.cast_one] |
| · apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> rfl |
| · congr 1 |
| apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> rfl |
| have HD (v : Config n) : |
| (gibbsFiniteLaw β J).prob (fun x => a₁<overlap v x ∧ discreteDistance β J x ⌊Real.exp L⌋₊≤(1/4:ℝ))≤ |
| (8*((⌈Real.exp (2*L)⌉₊:ℝ)+1))*(gibbsFiniteLaw β J).prob (fun x => a₀≤overlap v x ∧ overlap v x≤a₁) := by |
| have H := mass_mixed_high_le_strip hn β J v ha₀ hgap ⌊Real.exp L⌋₊ |
| have HH : (gibbsFiniteLaw β J).prob (fun x => a₁<overlap v x ∧ discreteDistance β J x ⌊Real.exp L⌋₊≤(1/4:ℝ))≤ |
| (4*((⌊Real.exp L⌋₊:ℝ)+1))*(gibbsFiniteLaw β J).prob (fun x => a₀≤overlap v x ∧ overlap v x≤a₁) := by |
| convert H using 1 <;> simp only [gibbs_prob_eq_sum,Nat.cast_add,Nat.cast_one] |
| · apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> rfl |
| · congr 1 |
| apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> rfl |
| refine HH.trans (mul_le_mul_of_nonneg_right ?_ ((gibbsFiniteLaw β J).prob_nonneg _)) |
| nlinarith only [hnum,(Nat.cast_nonneg (⌈Real.exp (2*L)⌉₊) : (0:ℝ)≤_)] |
| have general (E : Config n → Prop) |
| (HE : ∀v,(gibbsFiniteLaw β J).prob (fun x => a₁<overlap v x ∧ E x)≤ |
| (8*((⌈Real.exp (2*L)⌉₊:ℝ)+1))*(gibbsFiniteLaw β J).prob (fun x => a₀≤overlap v x ∧ overlap v x≤a₁)) : |
| (gibbsFiniteLaw β J).prob E≤δ+(16*((⌈Real.exp (2*L)⌉₊:ℝ)+1)/a)*replicaGibbsMass β J (pairStripSet n a₀ a₁) := by |
| have H := covered_mixed_mass_le_signed_average β J E haq ha hcov |
| have HB := (gibbsFiniteLaw β J).expect_mono HE |
| rw [FiniteLaw.expect_const_mul,pairStrip_expect] at HB |
| refine H.trans (add_le_add_right ((mul_le_mul_of_nonneg_left HB (by positivity : 0≤2/a)).trans_eq ?_) δ) |
| ring |
| constructor |
| · simpa only [gibbs_prob_eq_sum,continuousMixedMass] using general _ HC |
| · simpa only [gibbs_prob_eq_sum,discreteMixedMass] using general _ HD |
|
|
| theorem pairStrip_numerical_error {L c v : ℝ} (hL : 0≤L) (hv : v≤Real.exp (-(c+10)*L)) : |
| Real.exp (-3*L)+(16*((⌈Real.exp (2*L)⌉₊:ℝ)+1)/Real.exp (-c*L))*v≤ |
| Real.exp (-3*L)+48*Real.exp (-8*L) := by |
| apply add_le_add_right |
| have HM := horizon_size_bound hL |
| have HN : 0≤16*((⌈Real.exp (2*L)⌉₊:ℝ)+1)/Real.exp (-c*L) := by positivity |
| calc |
| _ ≤ (16*((⌈Real.exp (2*L)⌉₊:ℝ)+1)/Real.exp (-c*L))*Real.exp (-(c+10)*L) := mul_le_mul_of_nonneg_left hv HN |
| _ ≤ (16*(3*Real.exp (2*L))/Real.exp (-c*L))*Real.exp (-(c+10)*L) := |
| mul_le_mul_of_nonneg_right (div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left HM (by norm_num)) (Real.exp_pos _).le) (Real.exp_pos _).le |
| _ = _ := by |
| rw [← mul_assoc 16 3,show (16:ℝ)*3=48 by norm_num,mul_div_assoc,← Real.exp_sub,mul_assoc,← Real.exp_add] |
| congr 2 |
| ring |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
