| import Mathlib |
| import OAI.Probability.SKBarriers.Coverage.CoverageScale |
| import OAI.Probability.SKBarriers.Dynamics.GreedyNumerics |
| import OAI.Probability.SKBarriers.Dynamics.BankCardinality |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Topology |
| open Classical MeasureTheory Filter Set |
| namespace SK.Analytic |
|
|
| def scheduledBankSize (c : ℕ → ℝ) (B n : ℕ) {p : ℕ} (j : Fin p) : ℕ := |
| ⌈Real.exp ((c (j.val%B)+1)*levelLogScale n)⌉₊ |
|
|
| theorem scheduledBankSize_pos (c : ℕ → ℝ) (B n : ℕ) {p : ℕ} (j : Fin p) : |
| 0<scheduledBankSize c B n j := Nat.ceil_pos.mpr (Real.exp_pos _) |
|
|
| theorem eventually_exp_dominates_linear (A : ℝ) : |
| ∀ᶠ n : ℕ in atTop,A*levelLogScale n≤Real.exp (levelLogScale n) := by |
| have H := ((tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 1 zero_lt_one).comp stretchedLogScale_tendsto).const_mul A |
| have H0 : Tendsto (fun n => A*levelLogScale n*Real.exp (-levelLogScale n)) atTop (𝓝 0) := by |
| simpa only [levelLogScale,Real.rpow_one,neg_one_mul,mul_assoc,mul_zero,Function.comp_apply] using H |
| filter_upwards [H0.eventually_le_const zero_lt_one] with n hn |
| have HH := mul_le_mul_of_nonneg_right hn (Real.exp_pos (levelLogScale n)).le |
| simpa only [mul_assoc,← Real.exp_add,neg_add_cancel,Real.exp_zero,mul_one,one_mul] using HH |
|
|
| theorem greedy_level_count_bound {B : ℕ} (hB : 0<B) {t : ℝ} (ht : 0<t) (n : ℕ) : |
| (B*greedyBlockCount B t n:ℕ)≤(t/8)*levelLogScale n := by |
| have hb : (0:ℝ)<B := by exact_mod_cast hB |
| have H := mul_le_mul_of_nonneg_left (Nat.floor_le (show 0≤(t/(8*B))*levelLogScale n by unfold levelLogScale; positivity)) hb.le |
| change (B:ℝ)*(greedyBlockCount B t n:ℝ)≤(B:ℝ)*((t/(8*B))*levelLogScale n) at H |
| rw [Nat.cast_mul] |
| convert H using 1 |
| field_simp |
|
|
| theorem bank_miss_exp_bound (c L : ℝ) (_index : ℕ) : |
| Real.exp (-(Real.exp (-c*L))*(⌈Real.exp ((c+1)*L)⌉₊:ℝ))≤Real.exp (-Real.exp L) := by |
| have HK : Real.exp ((c+1)*L)≤(⌈Real.exp ((c+1)*L)⌉₊:ℝ) := Nat.le_ceil _ |
| apply Real.exp_le_exp.mpr |
| have H := mul_le_mul_of_nonneg_left HK (Real.exp_pos (-c*L)).le |
| have he : Real.exp (-c*L)*Real.exp ((c+1)*L)=Real.exp L := by |
| rw [← Real.exp_add]; congr 1; ring |
| rw [he] at H |
| linarith only [H] |
|
|
| theorem scheduled_bank_coverage_sum {p B n : ℕ} (hB : 0<B) (c : ℕ → ℝ) (hc : ∀j,0≤c j) |
| (hL : 1≤levelLogScale n) |
| (hdom : (bankPrefix c B+5)*levelLogScale n≤Real.exp (levelLogScale n)) : |
| (∑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))))≤ |
| 2*p*Real.exp (-5*levelLogScale n) := by |
| have hK (j : Fin p) : (scheduledBankSize c B n j:ℝ)≤Real.exp ((c (j.val%B)+2)*levelLogScale n) := ceil_exp_bound (hc _) hL |
| calc |
| _ ≤ ∑_j : Fin p,2*Real.exp (-5*levelLogScale n) := by |
| apply Finset.sum_le_sum |
| intro j _ |
| have HA := bankTuple_card_bound hB c (scheduledBankSize c B n) (levelLogScale n) hK j |
| have hAj : bankPrefix c (j.val%B)≤bankPrefix c B := bankPrefix_mono c hc (Nat.mod_lt _ hB).le |
| have Hmiss : Real.exp (-(Real.exp (-c (j.val%B)*levelLogScale n))*(scheduledBankSize c B n j))≤ |
| Real.exp (-(bankPrefix c (j.val%B)+5)*levelLogScale n) := by |
| refine (bank_miss_exp_bound (c (j.val%B)) (levelLogScale n) 0).trans ?_ |
| apply Real.exp_le_exp.mpr |
| have HH := mul_le_mul_of_nonneg_right (add_le_add hAj (le_refl (5:ℝ))) (by linarith only [hL]) |
| linarith only [HH,hdom] |
| have H := mul_le_mul HA (add_le_add (le_refl (Real.exp (-(bankPrefix c (j.val%B)+5)*levelLogScale n))) Hmiss) (by positivity) (Real.exp_pos _).le |
| refine H.trans_eq ?_ |
| rw [← two_mul,mul_comm (Real.exp _) (2*_),mul_assoc,← Real.exp_add] |
| congr 2 |
| ring |
| _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring |
|
|
| theorem scheduled_bank_locking_cost {p B n : ℕ} (hB : 0<B) (c : ℕ → ℝ) (hc : ∀j,0≤c j) |
| (hL : 5≤levelLogScale n) (hp : (p:ℝ)≤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≤Real.exp (-4*levelLogScale n) := by |
| let A := bankPrefix c B |
| let L := levelLogScale n |
| let N : ℝ := Fintype.card (BankIndex (scheduledBankSize c B n (p:=p))) |
| have hA : 0≤A := bankPrefix_nonneg c hc B |
| have hL0 : 0≤L := by dsimp only [L]; linarith only [hL] |
| have hN0 : 0≤N := Nat.cast_nonneg _ |
| have hN : N≤Real.exp ((A+1)*L) := by |
| refine (bankIndex_card_bound hB c hc (scheduledBankSize c B n) hL0 |
| (fun j => ceil_exp_bound (hc _) (by linarith only [hL]))).trans ?_ |
| calc |
| _ ≤ Real.exp L*Real.exp (A*L) := mul_le_mul_of_nonneg_right hp (Real.exp_pos _).le |
| _ = _ := by rw [← Real.exp_add]; congr 1; ring |
| have he1 : 1≤Real.exp ((A+1)*L) := Real.one_le_exp_iff.mpr (by positivity) |
| have HN3 : N^3≤Real.exp (3*(A+1)*L) := by |
| have H := pow_le_pow_left₀ hN0 hN 3 |
| simpa only [← Real.exp_nat_mul,Nat.cast_ofNat,mul_assoc] using H |
| have HN2 : N^2≤Real.exp (3*(A+1)*L) := by |
| have H := pow_le_pow_left₀ hN0 hN 2 |
| have HH : (Real.exp ((A+1)*L))^2≤(Real.exp ((A+1)*L))^3 := pow_le_pow_right₀ he1 (by decide) |
| exact (H.trans HH).trans_eq (by rw [← Real.exp_nat_mul]; congr 1; ring) |
| have HC : (8*N^3+48*N^2)*v≤56*Real.exp (-17*L) := by |
| calc |
| _ ≤ (56*Real.exp (3*(A+1)*L))*Real.exp (-(3*A+20)*L) := |
| mul_le_mul (by linarith only [HN3,HN2]) hv hv0 (by positivity) |
| _ = _ := by rw [mul_assoc,← Real.exp_add]; congr 2; ring |
| have HE56 : (56:ℝ)≤Real.exp (13*L) := by |
| have H := Real.add_one_le_exp (13*L) |
| dsimp only [L] at H |
| linarith only [H,hL] |
| refine HC.trans ?_ |
| calc |
| _ ≤ Real.exp (13*L)*Real.exp (-17*L) := mul_le_mul_of_nonneg_right HE56 (Real.exp_pos _).le |
| _ = _ := by rw [← Real.exp_add]; congr 1; ring |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
