| import OAI.NumberTheory.GaussianMoat.CommonBlocks |
|
|
| namespace OAI |
|
|
| namespace GaussianMoat |
| open scoped BigOperators Classical |
| open Filter |
|
|
| lemma eventually_length_smoothing : |
| ∀ᶠ X : ℝ in atTop, ∀ p L N : ℝ, |
| 0<p → p≤2*Real.exp (21/20*X) → 0≤L → L≤Real.exp (21/50*X) → |
| Real.exp (20*X)≤N → L≤N ∧ L/(N+1)≤((1/4)/p)/2 := by |
| have hh := eventually_mul_exp_le_exp (a := 147/100) (b := 20) (by norm_num) 16 |
| filter_upwards [hh,eventually_ge_atTop (1:ℝ)] with X hh hX |
| intro p L N hp hp' hL hL' hN |
| have hNpos : 0<N+1 := by linarith only [hN,Real.exp_pos (20*X)] |
| have hm := mul_le_mul hp' hL' hL (by positivity : 0≤2*Real.exp (21/20*X)) |
| rw [mul_assoc,←Real.exp_add,show 21/20*X+21/50*X=147/100*X by ring] at hm |
| refine ⟨hL'.trans ((Real.exp_le_exp.mpr (by linarith only [hX])).trans hN),?_⟩ |
| apply (div_le_iff₀ hNpos).mpr |
| have he : ((1/4:ℝ)/p)/2=(1/8)/p := by ring |
| rw [he] |
| apply (mul_le_mul_iff_left₀ hp).mp |
| have heq : (1/8/p)*(N+1)*p=(N+1)/8 := by field_simp |
| rw [heq] |
| nlinarith only [hm,hh,hN] |
|
|
| theorem eventually_window_batch {D a δ : ℝ} {K : ℕ} |
| (hD : 1≤D) (ha : 0<a) (ha1 : a≤1) (haδ : 2*a≤δ) |
| (hgap : a≤(2:ℝ)^K*topCoefficient a) (W w : ℕ) (hw : w<W) : |
| ∀ᶠ m : ℕ in atTop, ∀ J : ℕ → Finset ℕ, |
| (∀ i<W, ∀ j∈J i, ((100:ℝ)^i*Real.exp m)≤j*Real.log 2 ∧ |
| j*Real.log 2≤21/20*((100:ℝ)^i*Real.exp m) ∧ |
| δ*(2:ℝ)^j/Real.log ((2:ℝ)^(j+1))≤(PrimeLogMass.dyadicBatch j).card) → |
| (∀ i∈J w, ∀ j∈J w, i<j → i+K≤j) → |
| ∀ j∈J w, ∀ E : ℝ, E≤(topCoefficient a/4)*(2:ℝ)^j/10000 → |
| Nonempty (BatchCertificate D (commonBlocks a m W J) (windowSmoothing W m) |
| (dyadicSieve j) (batchWordLength j) E |
| ((topCoefficient a/4)/(10000000*Real.log ((2:ℝ)^j)))) := by |
| have hδ : 0<δ := by linarith only [ha,haδ] |
| have hc : 0<topCoefficient a/4 := by exact div_pos (topCoefficient_pos ha) (by norm_num) |
| have ht := windowScale_tendsto w |
| have hε : 0 < min (1/1024:ℝ) (Real.exp (-100)/(512*640000)) := by positivity |
| have herr := eventually_accurate_error (2*(100:ℝ)^w) hε |
| filter_upwards [eventually_window_bands hD ha.le ha1 hδ W w hw, |
| eventually_window_tail_bounds hD ha.le ha1 W w hw, |
| ht.eventually (eventually_window_top hD ha haδ hgap), |
| ht.eventually (eventually_integer_batch_bounds hD), |
| ht.eventually eventually_coverage_hazard,ht.eventually eventually_coverage_data_cost, |
| ht.eventually (eventually_package_displacement hc), |
| ht.eventually eventually_length_smoothing,herr,ht.eventually (eventually_ge_atTop (1600:ℝ))] |
| with m hbands htail htop hint hhazard hccost hpcost hsmooth herr hX |
| intro J hJ hsep j hj E hE |
| let X : ℝ := (100:ℝ)^w*Real.exp m |
| let s := dyadicSieve j |
| have hdata := hJ w hw j hj |
| have hne : s.primes.Nonempty := dyadic_dense_nonempty hδ hdata.2.2 |
| have hpj := dyadic_prime_bounds j |
| have hm := dyadic_mean_bounds (by linarith only [hX]) hne hdata.1 hdata.2.1 |
| have hTpos : (0:ℝ)<2^j := by positivity |
| have hT : Real.exp X≤(2:ℝ)^j := by |
| rw [←Real.exp_log hTpos,Real.log_pow] |
| exact Real.exp_le_exp.mpr hdata.1 |
| have hprime : ∀ i : s.SignedIndex, Real.exp X≤(s.signedPrime i:ℝ) ∧ |
| (s.signedPrime i:ℝ)≤2*Real.exp (21/20*X) := by |
| intro i |
| have hp := hpj _ (s.enumeration i.2).property |
| refine ⟨hT.trans hp.1,hp.2.trans ?_⟩ |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| rw [←Real.exp_log hTpos,Real.log_pow] |
| exact Real.exp_le_exp.mpr hdata.2.1 |
| obtain ⟨pre,hpre⟩ := windowBlocks_split a m J hw |
| obtain ⟨hband,hbsize⟩ := htop (J w) pre hsep (hJ w hw) j hj |
| obtain ⟨qtop⟩ := hband |
| obtain ⟨hmid,hbot,hbmid⟩ := hbands J hJ j hj |
| obtain ⟨qmid⟩ := hmid |
| obtain ⟨qbot⟩ := hbot |
| obtain ⟨_,_,_,hbotcost,hrestcost⟩ := htail J (fun i hi k hk => (hJ i hi k hk).2.1) |
| have herrors : (91*accurateGrid m)*s.meanLog≤ min (1/1024:ℝ) (Real.exp (-100)/(512*640000)) := by |
| have hh := mul_le_mul_of_nonneg_left hm.2 (by have := accurateGrid_pos m; positivity : 0≤91*accurateGrid m) |
| exact hh.trans (by simpa only [mul_assoc] using herr) |
| have hcoverage := coverage_from_scalar_guards s D X (windowSmoothing W m) |
| (middleBlocks a m w J) (accurateBlock m (1/20)) _ _ _ qmid qbot |
| (by linarith only [hm.1]) (by have := windowSmoothing_pos W m; omega) |
| (by linarith only [hX]) hprime |
| (herrors.trans (min_le_right _ _)) (herrors.trans (min_le_left _ _)) |
| (hccost _ _ hbotcost hbmid) (fun i => hhazard _ ((Real.exp_pos X).trans_le (hprime i).1) (hprime i).2) |
| obtain ⟨hn,hshort,hnL,hnL',hnmax,hLmax⟩ := hint j hdata.1 hdata.2.1 |
| have hN : Real.exp (20*X)≤(windowSmoothing W m:ℝ) := by |
| apply le_trans _ (windowSmoothing_bounds W m).1 |
| apply Real.exp_le_exp.mpr |
| have hh := mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num : (1:ℝ)≤100) hw.le) (Real.exp_nonneg (m:ℝ)) |
| dsimp only [X] |
| nlinarith only [hh] |
| have hNLength : batchWordLength j≤windowSmoothing W m := by |
| have h := hsmooth ((2:ℝ)^j) (batchWordLength j) (windowSmoothing W m) hTpos |
| (by |
| have hh : (2:ℝ)^j≤Real.exp (21/20*X) := by |
| rw [←Real.exp_log hTpos,Real.log_pow] |
| exact Real.exp_le_exp.mpr hdata.2.1 |
| linarith only [hh,Real.exp_nonneg (21/20*X)]) |
| (Nat.cast_nonneg _) hLmax hN |
| exact_mod_cast h.1 |
| refine ⟨{ |
| pre := pre++topBlocks a (J w) |
| rest := middleBlocks a m w J++accurateBlock m (1/20) |
| schedule := by simp only [commonBlocks,middleBlocks,hpre,List.append_assoc] |
| T := (2:ℝ)^j |
| nonempty := hne |
| prime_scale := hpj |
| large := by linarith only [hT,Real.add_one_le_exp X,hX] |
| mean := hm.1 |
| b := bandSize s topParameter (a*(2:ℝ)^j) topIterations |
| n := packageRepetitions j |
| τ := 1/4 |
| A := X/400 |
| η := 1/100 |
| δ := 1/10000 |
| u := fun i => (s.signedPrime i:ℝ)*Real.exp (-X/200) |
| band := qtop |
| coverage := hcoverage |
| cap := qtop.size_cap (by linarith only [hm.1]) |
| repetitions := hn |
| length := batchWordLength_pos j |
| floor := by norm_num |
| gain := by linarith only [hX] |
| exceptions := fun i => by positivity |
| short := fun i => hshort.trans_le (hpj _ (s.enumeration i.2).property).1 |
| smooth_length := hNLength |
| smooth_error := fun i => (hsmooth _ _ _ ((Real.exp_pos X).trans_le (hprime i).1) |
| (hprime i).2 (Nat.cast_nonneg _) hLmax hN).2 |
| list := ?_ |
| rate := ?_ |
| }⟩ |
| · intro i |
| have hh := package_list_arithmetic hX hT ((Real.exp_pos X).trans_le (hprime i).1) |
| (hpj _ (s.enumeration i.2).property).2 hnL |
| dsimp only [s] at hh |
| convert hh using 1; congr 2; ring_nf |
| · apply package_rate_arithmetic (by linarith only [hX]) hTpos hc |
| (by simpa only [Real.log_pow] using hdata.1) |
| (by simpa only [Real.log_pow] using hdata.2.1) hm.2 |
| (by convert hbsize using 1; ring) hnL' |
| (hpcost _ _ _ (Nat.cast_nonneg _) hnmax hrestcost hT) hE |
|
|
| end GaussianMoat |
|
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| end OAI |
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