| import OAI.NumberTheory.GaussianMoat.Smoothing |
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| namespace OAI |
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| namespace GaussianMoat |
| open scoped BigOperators Classical |
| open Filter |
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| noncomputable def commonBlocks (a : ℝ) (m W : ℕ) (J : ℕ → Finset ℕ) : List ℕ := |
| windowBlocks a m J W++accurateBlock m (1/20) |
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| noncomputable def middleBlocks (a : ℝ) (m w : ℕ) (J : ℕ → Finset ℕ) : List ℕ := |
| accurateBlock m ((100:ℝ)^w/4)++windowBlocks a m J w |
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| lemma windowScale_tendsto (w : ℕ) : |
| Tendsto (fun m : ℕ => (100:ℝ)^w*Real.exp m) atTop atTop := |
| (Real.tendsto_exp_atTop.comp tendsto_natCast_atTop_atTop).const_mul_atTop (by positivity) |
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| lemma dyadic_prime_bounds (j : ℕ) : |
| ∀ p∈(dyadicSieve j).primes, (2:ℝ)^j≤(p:ℝ) ∧ (p:ℝ)≤2*(2:ℝ)^j := by |
| intro p hp |
| have hh := (PrimeLogMass.mem_dyadicBatch j p).mp hp |
| constructor |
| · exact_mod_cast hh.1 |
| · have hh' : p≤2^(j+1) := hh.2.1.le |
| rw [pow_succ] at hh' |
| exact_mod_cast (by simpa only [mul_comm] using hh' : p≤2*2^j) |
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| lemma dyadic_dense_nonempty {δ : ℝ} (hδ : 0<δ) {j : ℕ} |
| (hd : δ*(2:ℝ)^j/Real.log ((2:ℝ)^(j+1))≤(PrimeLogMass.dyadicBatch j).card) : |
| (dyadicSieve j).primes.Nonempty := by |
| apply Finset.card_pos.mp |
| have hp : 0<δ*(2:ℝ)^j/Real.log ((2:ℝ)^(j+1)) := by rw [Real.log_pow]; positivity |
| exact_mod_cast hp.trans_le hd |
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| lemma dyadic_mean_bounds {X : ℝ} (hX : 2≤X) {j : ℕ} |
| (hne : (dyadicSieve j).primes.Nonempty) |
| (hlo : X≤j*Real.log 2) (hhi : j*Real.log 2≤21/20*X) : |
| 1≤(dyadicSieve j).meanLog ∧ (dyadicSieve j).meanLog≤2*X := by |
| have hh := (dyadicSieve j).meanLog_bounds hne (by positivity) (dyadic_prime_bounds j) |
| rw [Real.log_mul (by norm_num) (by positivity),Real.log_pow] at hh |
| have hl2 := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2) |
| constructor <;> linarith only [hh.1,hh.2,hlo,hhi,hX,hl2] |
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| lemma commonBlocks_length {a : ℝ} {m W : ℕ} (J : ℕ → Finset ℕ) (hm : 2≤Real.exp m) |
| (hJ : ∀ w<W, ∀ j∈J w, (j:ℝ)*Real.log 2≤21/20*((100:ℝ)^w*Real.exp m)) : |
| ((commonBlocks a m W J).length:ℝ)+2≤ |
| (W:ℝ)*(60*(100:ℝ)^W*Real.exp m+4*(m:ℝ)^2)+4*(m:ℝ)^2+2 := by |
| have hh := windowBlocks_length_bound_of_bins (a := a) J hm hJ |
| simp only [commonBlocks,List.length_append,accurateBlock,List.length_ofFn, |
| accurateIterations,Nat.cast_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] |
| linarith only [hh] |
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| lemma middleBlocks_length_le {a : ℝ} {m W w : ℕ} (J : ℕ → Finset ℕ) (hw : w<W) : |
| (middleBlocks a m w J).length≤(windowBlocks a m J W).length := by |
| obtain ⟨pre,he⟩ := windowBlocks_split a m J hw |
| simp only [middleBlocks,he,List.length_append] |
| omega |
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|
| theorem eventually_window_tail_bounds {D a : ℝ} (hD : 1≤D) (ha : 0≤a) (ha1 : a≤1) |
| (W w : ℕ) (hw : w<W) : |
| ∀ᶠ m : ℕ in atTop, ∀ J : ℕ → Finset ℕ, |
| (∀ i<W, ∀ j∈J i, (j:ℝ)*Real.log 2≤21/20*((100:ℝ)^i*Real.exp m)) → |
| ((commonBlocks a m W J).length:ℝ)+2≤Real.exp (Real.exp m) ∧ |
| (windowSmoothing W m:ℝ)≤Real.exp (Real.exp ((1/100)*Real.exp m)) ∧ |
| ((windowBlocks a m J w).length+2:ℝ)≤Real.exp (Real.exp (((100:ℝ)^w/50)*Real.exp m)) ∧ |
| Real.log 36+2*Real.log (D*((accurateBlock m (1/20)).sum+windowSmoothing W m))≤ |
| Real.exp (3/50*((100:ℝ)^w*Real.exp m)) ∧ |
| Real.log 36+2*Real.log (D*((middleBlocks a m w J++accurateBlock m (1/20)).sum+windowSmoothing W m))≤ |
| Real.exp (13/50*((100:ℝ)^w*Real.exp m)) := by |
| have hN := eventually_mul_le_double_exp (a := 1/100) (by norm_num) (20*(100:ℝ)^W+2) |
| have hlen := eventually_mul_le_double_exp (a := 1/100) (by norm_num) 1 |
| have ht := windowScale_tendsto w |
| have hbot := ht.eventually (eventually_suffix_cost (D := D) (a := 1/20) (b := 3/50) |
| (C := 20*(100:ℝ)^W+2) (by linarith only [hD]) (by norm_num) (by norm_num)) |
| have hrest := ht.eventually (eventually_suffix_cost (D := D) (a := 1/4) (b := 13/50) |
| (C := 20*(100:ℝ)^W+2) (by linarith only [hD]) (by norm_num) (by norm_num)) |
| have hm2 := (windowScale_tendsto 0).eventually (eventually_ge_atTop (2:ℝ)) |
| filter_upwards [eventually_window_length W,hN,hlen,hbot,hrest,hm2] with m hm hN hlen hbot hrest hm2 |
| intro J hJ |
| simp only [one_mul] at hlen |
| simp only [pow_zero,one_mul] at hm2 |
| have hl := (commonBlocks_length (a := a) J hm2 hJ).trans hm |
| have hpow := window_power_one w |
| have hX : Real.exp (m:ℝ)≤(100:ℝ)^w*Real.exp m := by nlinarith only [hpow,Real.exp_pos (m:ℝ)] |
| have hNsmall : (windowSmoothing W m:ℝ)≤Real.exp (Real.exp ((1/100)*Real.exp m)) := by |
| have hN' := (windowSmoothing_bounds W m).2.trans (Real.exp_le_exp.mpr hN) |
| linarith only [hN'] |
| have hlw : ((windowBlocks a m J w).length+2:ℝ)≤Real.exp (Real.exp (((100:ℝ)^w/50)*Real.exp m)) := by |
| have hh : (windowBlocks a m J w).length≤(commonBlocks a m W J).length := by |
| obtain ⟨pre,he⟩ := windowBlocks_split a m J hw |
| simp only [commonBlocks,he,List.length_append]; omega |
| have harg : Real.exp (m:ℝ)≤Real.exp (((100:ℝ)^w/50)*Real.exp m) := by |
| apply hlen.trans (Real.exp_le_exp.mpr _) |
| nlinarith only [hpow,Real.exp_pos (m:ℝ)] |
| exact (add_le_add (by exact_mod_cast hh : ((windowBlocks a m J w).length:ℝ)≤(commonBlocks a m W J).length) (le_refl (2:ℝ))) |>.trans (hl.trans (Real.exp_le_exp.mpr harg)) |
| have hNs : (windowSmoothing W m:ℝ)+1≤Real.exp ((20*(100:ℝ)^W+2)*((100:ℝ)^w*Real.exp m)) := |
| (windowSmoothing_bounds W m).2.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hX (by positivity))) |
| have hbLen : ((accurateBlock m (1/20)).length+2:ℝ)≤Real.exp ((100:ℝ)^w*Real.exp m) := by |
| have hh : (accurateBlock m (1/20)).length≤(commonBlocks a m W J).length := by simp [commonBlocks] |
| exact (add_le_add (by exact_mod_cast hh : ((accurateBlock m (1/20)).length:ℝ)≤(commonBlocks a m W J).length) (le_refl (2:ℝ))) |>.trans (hl.trans (Real.exp_le_exp.mpr hX)) |
| have hrLen : ((middleBlocks a m w J++accurateBlock m (1/20)).length+2:ℝ)≤Real.exp ((100:ℝ)^w*Real.exp m) := by |
| have hh := middleBlocks_length_le (a := a) (m := m) J hw |
| have hh' : (middleBlocks a m w J++accurateBlock m (1/20)).length≤(commonBlocks a m W J).length := by |
| simp only [commonBlocks,List.length_append]; omega |
| exact (add_le_add (by exact_mod_cast hh' : ((middleBlocks a m w J++accurateBlock m (1/20)).length:ℝ)≤(commonBlocks a m W J).length) (le_refl (2:ℝ))) |>.trans (hl.trans (Real.exp_le_exp.mpr hX)) |
| have hbOps : ∀ n∈accurateBlock m (1/20), (n:ℝ)≤Real.exp (Real.exp ((1/20)*((100:ℝ)^w*Real.exp m))) := by |
| apply accurateBlock_bound (accurateGrid_le_one m) |
| exact (mul_le_of_le_one_left (Real.exp_nonneg _) (accurateGrid_le_one m)).trans |
| (Real.exp_le_exp.mpr (by linarith only [hX])) |
| have hrOps : ∀ n∈middleBlocks a m w J++accurateBlock m (1/20), (n:ℝ)≤Real.exp (Real.exp ((1/4)*((100:ℝ)^w*Real.exp m))) := by |
| intro n hn |
| rcases List.mem_append.mp hn with hn|hn |
| · rcases List.mem_append.mp hn with hn|hn |
| · apply accurateBlock_bound (accurateGrid_le_one m) _ n hn |
| simpa only [div_mul_eq_mul_div,mul_assoc,one_mul] using |
| (mul_le_of_le_one_left (Real.exp_nonneg (((100:ℝ)^w/4)*Real.exp m)) (accurateGrid_le_one m)) |
| · exact (smaller_window_operations ha ha1 m w J (fun i hi => hJ i (hi.trans hw)) n hn).trans |
| (Real.exp_le_exp.mpr (Real.exp_le_exp.mpr (by nlinarith only [show 0≤(100:ℝ)^w*Real.exp m by positivity] : ((100:ℝ)^w/50)*Real.exp m≤(1/4)*((100:ℝ)^w*Real.exp m)))) |
| · exact (hbOps n hn).trans (Real.exp_le_exp.mpr (Real.exp_le_exp.mpr (by nlinarith only [show 0≤(100:ℝ)^w*Real.exp m by positivity] : (1/20)*((100:ℝ)^w*Real.exp m)≤(1/4)*((100:ℝ)^w*Real.exp m)))) |
| exact ⟨hl,hNsmall,hlw, |
| hbot _ _ (by have := windowSmoothing_pos W m; omega) hbLen hbOps hNs, |
| hrest _ _ (by have := windowSmoothing_pos W m; omega) hrLen hrOps hNs⟩ |
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| lemma floor_size_lower {s : SplitSieve} {g U X : ℝ} {l : ℕ} |
| (hX : 0<X) (hm : 0<s.meanLog) (hm' : s.meanLog≤2*X) |
| (hscale : Real.exp (X/8)≤bandScale g U l) (hlarge : 4*X≤Real.exp (X/8)) : |
| Real.exp (X/8)/(4*X)≤(bandSize s g U l:ℝ) := by |
| have hf := Nat.lt_floor_add_one (bandScale g U l/s.meanLog) |
| have hh := mul_lt_mul_of_pos_right hf hm |
| rw [div_mul_cancel₀ _ hm.ne',add_mul,one_mul] at hh |
| have hb := mul_le_mul_of_nonneg_left hm' (Nat.cast_nonneg (bandSize s g U l)) |
| change (bandScale g U l)<((bandSize s g U l:ℝ)*s.meanLog)+s.meanLog at hh |
| apply (div_le_iff₀ (by positivity)).mpr |
| nlinarith only [hf,hh,hb,hscale,hlarge,hm'] |
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| theorem eventually_window_bands {D a δ : ℝ} (hD : 1≤D) (ha : 0≤a) (ha1 : a≤1) |
| (hδ : 0<δ) (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) → |
| ∀ j∈J w, |
| Nonempty (EntropyBand (dyadicSieve j) D (middleBlocks a m w J) 0 |
| (bandSize (dyadicSieve j) (accurateGrid m) (Real.exp (((100:ℝ)^w/4)*Real.exp m)) (accurateIterations m)) |
| (91*accurateGrid m)) ∧ |
| Nonempty (EntropyBand (dyadicSieve j) D (accurateBlock m (1/20)) (windowSmoothing W m) |
| (bandSize (dyadicSieve j) (accurateGrid m) (Real.exp ((1/20)*Real.exp m)) (accurateIterations m)) |
| (91*accurateGrid m)) ∧ |
| Real.exp (((100:ℝ)^w*Real.exp m)/8)/(4*((100:ℝ)^w*Real.exp m))≤ |
| (bandSize (dyadicSieve j) (accurateGrid m) (Real.exp (((100:ℝ)^w/4)*Real.exp m)) (accurateIterations m):ℝ) := by |
| have hwp := window_power_one w |
| have hp : (0:ℝ)<100^w := by positivity |
| have hmid := eventually_accurate_band (D := D) (A := (100:ℝ)^w) (B := 2*(100:ℝ)^w) |
| (a := (100:ℝ)^w/4) (b := (100:ℝ)^w/50) hD hp (by positivity) (by linarith only [hp]) |
| (by linarith only [hp]) hδ |
| have hbot := eventually_accurate_band (D := D) (A := 1) (B := 2*(100:ℝ)^w) |
| (a := 1/20) (b := 1/100) hD (by norm_num) (by norm_num) (by norm_num) (by norm_num) hδ |
| have hscale := eventually_accurate_scale (a := (100:ℝ)^w/4) (by positivity) 0 |
| have hlarge := eventually_mul_le_double_exp (a := (100:ℝ)^w/8) (by positivity) (4*(100:ℝ)^w) |
| have hm2 := (windowScale_tendsto 0).eventually (eventually_ge_atTop (2:ℝ)) |
| filter_upwards [hmid,hbot,hscale,hlarge,eventually_window_tail_bounds hD ha ha1 W w hw,hm2] |
| with m hmid hbot hscale hlarge htail hm2 |
| intro J hJ j hj |
| simp only [pow_zero,one_mul] at hm2 |
| have hdata := hJ w hw j hj |
| have hne := dyadic_dense_nonempty hδ hdata.2.2 |
| have hX2 : 2≤(100:ℝ)^w*Real.exp m := by nlinarith only [hm2,hwp] |
| have hm := dyadic_mean_bounds hX2 hne hdata.1 hdata.2.1 |
| have hpj := dyadic_prime_bounds j |
| have hdense : δ*(2:ℝ)^j/Real.log (2*(2:ℝ)^j)≤(dyadicSieve j).primes.card := by |
| change δ*(2:ℝ)^j/Real.log (2*(2:ℝ)^j)≤(PrimeLogMass.dyadicBatch j).card |
| simpa only [pow_succ,mul_comm] using hdata.2.2 |
| have hT : Real.exp ((100:ℝ)^w*Real.exp m)≤(2:ℝ)^j := by |
| rw [←Real.exp_log (by positivity : (0:ℝ)<2^j),Real.log_pow] |
| exact Real.exp_le_exp.mpr hdata.1 |
| have hThi : Real.log ((2:ℝ)^j)≤(2*(100:ℝ)^w)*Real.exp m := by |
| rw [Real.log_pow] |
| nlinarith only [hdata.2.1,hX2] |
| obtain ⟨_,hN,hlen,_,_⟩ := htail J (fun i hi k hk => (hJ i hi k hk).2.1) |
| have hmid' := hmid (dyadicSieve j) ((2:ℝ)^j) [] (windowBlocks a m J w) 0 |
| hne hpj hdense hT hThi |
| (smaller_window_operations ha ha1 m w J (fun i hi k hk => (hJ i (hi.trans hw) k hk).2.1)) |
| (by simp only [Nat.cast_zero]; positivity) hlen |
| have hTwo : (2:ℝ)≤Real.exp (Real.exp ((1/100)*Real.exp m)) := by |
| have he : 1≤Real.exp ((1/100)*Real.exp (m:ℝ)) := Real.one_le_exp_iff.mpr (by positivity) |
| linarith only [Real.add_one_le_exp (Real.exp ((1/100)*Real.exp (m:ℝ))),he] |
| have hbot' := hbot (dyadicSieve j) ((2:ℝ)^j) [] [] (windowSmoothing W m) |
| hne hpj hdense |
| ((Real.exp_le_exp.mpr (by nlinarith only [hwp,Real.exp_pos (m:ℝ)] : 1*Real.exp m≤(100:ℝ)^w*Real.exp m)).trans hT) |
| hThi (by simp only [List.not_mem_nil,IsEmpty.forall_iff,implies_true]) hN |
| (by simpa only [List.length_nil,Nat.cast_zero,zero_add] using hTwo) |
| refine ⟨?_,?_,?_⟩ |
| · simpa only [List.nil_append,accurateBlock,middleBlocks] using hmid' |
| · simpa only [List.nil_append,List.append_nil,accurateBlock] using hbot' |
| · apply floor_size_lower (by linarith only [hX2]) (by linarith only [hm.1]) hm.2 |
| · simpa only [pow_zero,one_mul,div_div,show (4:ℝ)*2=8 by norm_num,div_mul_eq_mul_div] using hscale |
| · simpa only [mul_assoc,div_mul_eq_mul_div] using hlarge |
|
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| end GaussianMoat |
|
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| end OAI |
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