| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.ThresholdObstruction |
|
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| namespace OAI |
|
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| section |
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|
| noncomputable section |
| open scoped BigOperators Topology |
| open Classical MeasureTheory Filter Set |
| namespace SK.Analytic |
|
|
| def greedySlack (t ρ : ℝ) : ℝ := (3*t/4)*ρ^19+2*ρ^100+ρ^20+3*ρ^10000 |
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| theorem greedyScale_of_slack {n B H s : ℕ} {t ρ b b' q : ℝ} |
| (hn : 0<n) (hB : 0<B) (hH : H<B) (hs : 0<s) (hpack : ⌈2/b^2⌉₊≤H) |
| (ht : 0<t) (hb : 0<b) (hpack' : 3*t<b^2/2) (hcut : 3*t<b) |
| (hρ : 0<ρ) (hρ1 : ρ<1) |
| (hη : (n:ℝ)^(-(1:ℝ)/100)=ρ^100) (hInv : 1/(n:ℝ)=ρ^10000) |
| (hp : (B*s:ℕ)*ρ≤t/8) |
| (hF : greedySlack t ρ<ρ^8) (hFt : greedySlack t ρ<t) |
| (hFb : 2*t+greedySlack t ρ<b') (hFq : 2*t+2*greedySlack t ρ<q) |
| (hwidth : ρ^8<t/2) : GreedyScaleBounds n B H (B*s) t ρ (2/(n:ℝ)) b b' q := by |
| have hρ8 : ρ^8≤ρ := pow_le_of_le_one hρ.le hρ1.le (by decide) |
| have h20 : ρ^10000≤ρ^20 := pow_le_pow_of_le_one hρ.le hρ1.le (by decide) |
| have h100 : ρ^100<ρ^20 := pow_lt_pow_right_of_lt_one₀ hρ hρ1 (by decide) |
| have he : 6*(B*s:ℕ)*ρ^20≤(3*t/4)*ρ^19 := by |
| have h := mul_le_mul_of_nonneg_right hp (show 0≤6*ρ^19 by positivity) |
| nlinarith only [h] |
| have hρ20 : 0≤ρ^20 := by positivity |
| have hi : 0≤ρ^10000 := by positivity |
| have hη0 : 0≤ρ^100 := by positivity |
| have hd : 2/(n:ℝ)=2*ρ^10000 := by rw [← hInv]; ring |
| have hp0 : 0≤6*(B*s:ℕ)*ρ^20 := by positivity |
| have hF0 : 0≤greedySlack t ρ := by unfold greedySlack; positivity |
| have heF : 6*(B*s:ℕ)*ρ^20+2*ρ^100+ρ^20+3*ρ^10000≤greedySlack t ρ := by |
| unfold greedySlack; linarith only [he] |
| constructor |
| · exact Nat.mul_pos hB hs |
| · exact hB |
| · exact hH |
| · exact hpack |
| · exact ht |
| · exact hρ |
| · positivity |
| · exact hb |
| · exact hpack' |
| · exact hcut |
| · have := heF.trans_lt hFt; linarith only [this,hρ20,hi,hη0] |
| · rw [hη]; linarith only [heF,hFb,hρ20,hi] |
| · rw [hη]; nlinarith only [heF,hF,hρ8,hρ20,hi,hρ] |
| · exact hwidth |
| · rw [hη,hd]; linarith only [heF,hF,hi] |
| · rw [hη,hd]; linarith only [heF,hF,hρ8,hρ] |
| · rw [hd]; linarith only [h20] |
| · rw [hη]; linarith only [h100] |
| · rw [hη]; linarith only [heF,hFq,hi,hη0,hp0] |
| · norm_num only [Nat.cast_mul] at hp ⊢; linarith only [hp,ht] |
|
|
| def levelLogScale (n : ℕ) : ℝ := (n:ℝ)^kappa |
| def levelRho (n : ℕ) : ℝ := (n:ℝ)^(-kappa) |
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| theorem levelRho_pos {n : ℕ} (hn : 0<n) : 0<levelRho n := Real.rpow_pos_of_pos (by exact_mod_cast hn) _ |
| theorem levelRho_tendsto : Tendsto levelRho atTop (𝓝 0) := |
| (tendsto_rpow_neg_atTop (by norm_num [kappa] : 0<kappa)).comp (tendsto_natCast_atTop_atTop (R:=ℝ)) |
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| theorem levelRho_pow {n : ℕ} (hn : 0<n) (a : ℕ) : (levelRho n)^a=(n:ℝ)^(-((a:ℝ)*kappa)) := by |
| unfold levelRho |
| rw [← Real.rpow_natCast,← Real.rpow_mul (by positivity : (0:ℝ)≤n)] |
| congr 1 |
| ring |
|
|
| theorem levelRho_inv {n : ℕ} (hn : 0<n) : 1/(n:ℝ)=(levelRho n)^10000 := by |
| rw [levelRho_pow hn] |
| norm_num [kappa,Real.rpow_neg_one] |
|
|
| theorem levelRho_eta {n : ℕ} (hn : 0<n) : (n:ℝ)^(-(1:ℝ)/100)=(levelRho n)^100 := by |
| rw [levelRho_pow hn]; norm_num [kappa] |
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| theorem levelLogScale_mul_rho {n : ℕ} (hn : 0<n) : levelLogScale n*levelRho n=1 := by |
| unfold levelLogScale levelRho |
| rw [← Real.rpow_add (by exact_mod_cast hn : (0:ℝ)<n)] |
| simp |
|
|
| def greedyBlockCount (B : ℕ) (t : ℝ) (n : ℕ) : ℕ := ⌊(t/(8*B))*levelLogScale n⌋₊ |
|
|
| theorem greedyBlockCount_upper {n B : ℕ} (hn : 0<n) (hB : 0<B) {t : ℝ} (ht : 0<t) : |
| (B*greedyBlockCount B t n:ℕ)*levelRho n≤t/8 := by |
| have hB' : (0:ℝ)<B := by exact_mod_cast hB |
| have h := Nat.floor_le (show 0≤(t/(8*B))*levelLogScale n by unfold levelLogScale; positivity) |
| have h' := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left h hB'.le) (levelRho_pos hn).le |
| have he : (B:ℝ)*((t/(8*B))*levelLogScale n)*levelRho n=t/8 := by |
| rw [mul_assoc (B:ℝ),mul_assoc,levelLogScale_mul_rho hn,mul_one] |
| field_simp |
| simpa only [greedyBlockCount,Nat.cast_mul,he] using h' |
|
|
| theorem greedySlack_tendsto (t : ℝ) : Tendsto (fun n => greedySlack t (levelRho n)) atTop (𝓝 0) := by |
| have H := (((levelRho_tendsto.pow 19).const_mul (3*t/4)).add ((levelRho_tendsto.pow 100).const_mul 2)).add |
| (levelRho_tendsto.pow 20) |>.add ((levelRho_tendsto.pow 10000).const_mul 3) |
| simpa only [greedySlack,zero_pow (by decide : 19≠0),zero_pow (by decide : 100≠0),zero_pow (by decide : 20≠0),zero_pow (by decide : 10000≠0),mul_zero,add_zero] using H |
|
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| theorem greedySlack_div_tendsto (t : ℝ) : Tendsto (fun n => greedySlack t (levelRho n)/(levelRho n)^8) atTop (𝓝 0) := by |
| have H : Tendsto (fun n => (3*t/4)*(levelRho n)^11+2*(levelRho n)^92+(levelRho n)^12+3*(levelRho n)^9992) atTop (𝓝 0) := by |
| convert (((levelRho_tendsto.pow 11).const_mul (3*t/4)).add ((levelRho_tendsto.pow 92).const_mul 2)).add |
| (levelRho_tendsto.pow 12) |>.add ((levelRho_tendsto.pow 9992).const_mul 3) using 1 |
| norm_num |
| apply H.congr' |
| filter_upwards [eventually_gt_atTop (0:ℕ)] with n hn |
| have hρ := (levelRho_pos hn).ne' |
| unfold greedySlack |
| field_simp |
|
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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