| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.ClockEndpoint |
| import OAI.Probability.SKBarriers.Dynamics.GreedyEventual |
| import OAI.Probability.SKBarriers.Dynamics.BankNumerics |
| import OAI.Probability.SKBarriers.Dynamics.BankDynamics |
| |
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Topology |
| open Classical MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def bankEndpointError (n : ℕ) : ℝ := |
| 4*(9*Real.exp (-2*levelLogScale n)+(n:ℝ)*Real.exp (-levelLogScale n)) |
|
|
| theorem scheduled_bank_total {n p B : ℕ} (hB : 0<B) (c : ℕ → ℝ) (hc : ∀j,0≤c j) |
| (hL : 5≤levelLogScale n) (hp : (p:ℝ)≤Real.exp (levelLogScale n)) |
| (hdom : (bankPrefix c B+5)*levelLogScale n≤Real.exp (levelLogScale n)) |
| {v : ℝ} (hv0 : 0≤v) (hv : v≤Real.exp (-(3*bankPrefix c B+20)*levelLogScale n)) : |
| (8*(Fintype.card (BankIndex (scheduledBankSize c B n (p:=p))):ℝ)^3+ |
| 48*(Fintype.card (BankIndex (scheduledBankSize c B n (p:=p))):ℝ)^2)*v+ |
| (∑j : Fin p,(Fintype.card (BankTuple (scheduledBankSize c B n) (blockPrior p B) j):ℝ)* |
| (Real.exp (-(bankPrefix c (j.val%B)+5)*levelLogScale n)+ |
| Real.exp (-(Real.exp (-c (j.val%B)*levelLogScale n))*(scheduledBankSize c B n j))))≤ |
| 3*Real.exp (-4*levelLogScale n) := by |
| have HL := scheduled_bank_locking_cost hB c hc hL hp hv0 hv |
| have HC := scheduled_bank_coverage_sum (p:=p) hB c hc (by linarith only [hL]) hdom |
| have HP : 2*(p:ℝ)*Real.exp (-5*levelLogScale n)≤2*Real.exp (-4*levelLogScale n) := by |
| calc |
| _ ≤ 2*Real.exp (levelLogScale n)*Real.exp (-5*levelLogScale n) := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hp (by norm_num)) (Real.exp_pos _).le |
| _ = _ := by rw [mul_assoc,← Real.exp_add] |
| linarith only [HL,HC,HP] |
|
|
| theorem identity_quenched_bound {n B H : ℕ} (hn : 0<n) (hB : 0<B) |
| (β : ℝ) (J : Disorder n) (μ : ProbabilityMeasure ℝ) (t b b' q : ℝ) |
| (c : ℕ → ℝ) (hc : ∀j,0≤c j) |
| (hcov : ∀j<B,∀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)) |
| (hv : replicaGibbsMass β J (lockingUnion n μ t (levelRho n) q b b')≤Real.exp (-(3*bankPrefix c B+20)*levelLogScale n)) |
| (hL : 5≤levelLogScale n) |
| (hp : (B*greedyBlockCount B t n:ℕ)≤Real.exp (levelLogScale n)) |
| (hdom : (bankPrefix c B+5)*levelLogScale n≤Real.exp (levelLogScale n)) |
| (hsmall : 3*Real.exp (-4*levelLogScale n)<1) |
| (G : GreedyScaleBounds n B H (B*greedyBlockCount B t n) t (levelRho n) (2/(n:ℝ)) b b' q) |
| (hcount : ∀m≤⌈Real.exp (2*levelLogScale n)⌉₊, |
| (B*greedyBlockCount B t n:ℕ)*((4*(levelRho n)^8/(levelRho n)+1/(n:ℝ))/(levelRho n)^4)+ |
| (m+1)*(2*(6*(B*greedyBlockCount B t n:ℕ)*(levelRho n)^20)/(levelRho n)+1/(n:ℝ))^(greedyBlockCount B t n)<1) : |
| continuousMixedMass β J (timeScale n)≤bankEndpointError n ∧ |
| discreteMixedMass β J ⌊timeScale n⌋₊≤bankEndpointError n := by |
| let p := B*greedyBlockCount B t n |
| let K : Fin p → ℕ := scheduledBankSize c B n |
| let a : Fin p → ℝ := fun j => Real.exp (-c (j.val%B)*levelLogScale n) |
| let δ : Fin p → ℝ := fun j => Real.exp (-(bankPrefix c (j.val%B)+5)*levelLogScale n) |
| let S := lockingUnion n μ t (levelRho n) q b b' |
| have hK : ∀j,0<K j := scheduledBankSize_pos c B n |
| obtain ⟨z,hz,hzmass⟩ := exists_fixed_good_bank β J K (blockPrior p B) (blockPrior_not_self p B) t q a δ |
| (fun j => (Real.exp_pos _).le) S (fun j w => hcov (j.val%B) (Nat.mod_lt _ hB) (fun x => ∀i,|overlap (w i) x|≤3*t)) |
| (3*Real.exp (-4*levelLogScale n)) |
| (scheduled_bank_total hB c hc hL hp hdom (replicaGibbsMass_nonneg β J S) hv) hsmall |
| let v := initialBankReference (natBankPool K z) (natBankDefault K hK z) B t q |
| have HC : ∀m≤⌈Real.exp (2*levelLogScale n)⌉₊, |
| (∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) m x)≤9*Real.exp (-2*levelLogScale n) := by |
| intro m hm |
| have HM := bank_stationary_crossing hn β J μ B H (greedyBlockCount B t n) m t (levelRho n) b b' q G (by simpa only [Nat.cast_mul] using hcount m hm) K hK z hz |
| have HK : (m:ℝ)+1≤3*Real.exp (2*levelLogScale n) := by |
| have hi : (m:ℝ)≤(⌈Real.exp (2*levelLogScale n)⌉₊:ℝ) := by exact_mod_cast hm |
| exact (add_le_add hi (le_refl (1:ℝ))).trans (horizon_size_bound (by linarith only [hL])) |
| change _≤((m:ℝ)+1)*(gibbsFiniteLaw β J).prob (fun x => bankCoverageBad K (blockPrior p B) t q z x ∨ bankVertexBad (fun i : BankIndex K => i.1.val) S z x) at HM |
| refine HM.trans ?_ |
| calc |
| _ ≤ (3*Real.exp (2*levelLogScale n))*(3*Real.exp (-4*levelLogScale n)) := |
| mul_le_mul HK hzmass ((gibbsFiniteLaw β J).prob_nonneg _) (by positivity) |
| _ = _ := by rw [mul_mul_mul_comm,← Real.exp_add] |
| have HH := mixedMass_bounds_of_crossing hn β J (levelLogScale n) (9*Real.exp (-2*levelLogScale n)) (by linarith only [hL]) (by positivity) v HC |
| refine ⟨HH.1,HH.2.trans ?_⟩ |
| dsimp only [bankEndpointError] |
| have hp0 : 0≤(n:ℝ)*Real.exp (-levelLogScale n) := by positivity |
| linarith only [hp0] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
