| import OAI.Computability.FourierTransform.UniformBounds |
|
|
| namespace OAI |
|
|
| /- asymptotic analysis, one n for both length functions -/ |
| namespace PowerSaving |
| open RAM Bench Selection Slant Filter Asymptotics Real |
| noncomputable section |
|
|
| namespace Skies |
|
|
| abbrev Tm (n : ℕ) : ℝ := log (n:ℝ) |
| abbrev Qm (n : ℕ) : ℝ := log (Tm n) |
|
|
| lemma Ttop : Tendsto Tm atTop atTop := Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop |
| lemma Qtop : Tendsto Qm atTop atTop := Real.tendsto_log_atTop.comp Ttop |
| lemma log2_min : 2/3 ≤ log (2:ℝ) := by |
| have hh := le_log_one_add_of_nonneg (show (0:ℝ)≤1 by norm_num) |
| norm_num at * |
| exact hh |
|
|
| |
| lemma stack_lim (n N : ℕ) (hn : 0<n) (h : n≤N) (hk : N ≤ 2*n) (ht : 1≤Tm n) : |
| (stackN N:ℝ) ≤ 10 * Tm n := by |
| let k := Nat.log 2 (2*N) |
| have H : 2^k ≤ 2*N := Nat.pow_log_le_self _ (by omega) |
| have H0 : 2^k ≤ 4*n := by omega |
| have he : 0<(n:ℝ) := by exact_mod_cast hn |
| have HH := log_le_log (by positivity : 0<(2:ℝ)^k) |
| (show (2:ℝ)^k ≤ 4*n by exact_mod_cast H0) |
| rw [log_pow,log_mul (by norm_num) he.ne'] at HH |
| have H4 := log_le_sub_one_of_pos (show (0:ℝ)<4 by norm_num) |
| change (((k+1:ℕ):ℝ)) ≤ _ |
| push_cast |
| unfold Tm at * |
| nlinarith [log2_min,show 0≤(k:ℝ) by positivity] |
|
|
| lemma order_pow_lim (n N : ℕ) (hn : 0<n) (h : n≤N) (hk : N ≤ 2*n) (ht : 1 ≤ Tm n) : |
| (2:ℝ)^orderE N ≤ 10^6*Tm n^2 := by |
| have hh : (2:ℝ)^orderE N ≤ 8*harvestN N := by exact_mod_cast orderE_small N |
| refine hh.trans ?_ |
| unfold harvestN silo; push_cast |
| have hg := stack_lim n N hn h hk ht |
| calc |
| _ ≤ 8*(16*(14*Tm n))^2 := by gcongr; linarith |
| _ ≤ _ := by nlinarith [sq_nonneg (Tm n)] |
|
|
| lemma order_lim (n N : ℕ) (hn : 0<n) (h : n≤N) (hk : N ≤ 2*n) (ht : 1 ≤ Tm n) (hu : 1 ≤ Qm n) : |
| (orderE N:ℝ) ≤ 3*10^6 * Qm n := by |
| have H := log_le_log (by positivity : 0<(2:ℝ)^orderE N) (order_pow_lim n N hn h hk ht) |
| have he : 0<Tm n := by linarith |
| rw [log_pow,log_mul (by norm_num) (by positivity),log_pow,log_pow] at H |
| have H4 := log_le_sub_one_of_pos (show 0 < (10:ℝ) by norm_num) |
| unfold Qm at * |
| have h0 : 0 ≤ (orderE N:ℝ) := by positivity |
| push_cast at H |
| nlinarith [log2_min] |
|
|
| lemma score_lim (n N : ℕ) (hn : 0<n) (h : n≤N) (hk : N ≤ 2*n) (ht : 1 ≤ Tm n) (hu : 1 ≤ Qm n) : |
| (score (orderE N):ℝ)+1 ≤ 10^32*Qm n^2 := by |
| have H := order_lim n N hn h hk ht hu |
| have hk : (score (orderE N):ℝ) ≤ 20000000*((orderE N:ℝ)+1)^2+7 := by |
| exact_mod_cast score_le (orderE N) |
| calc |
| _ ≤ (20000000*((orderE N:ℝ)+1)^2+7)+1 := by gcongr |
| _ ≤ (20000000*(4*10^6*Qm n)^2+7)+1 := by gcongr; linarith |
| _ ≤ _ := by nlinarith |
|
|
| lemma hills_lim (n N : ℕ) (hn : 0<n) (h : n≤N) (hk : N ≤ 2*n) (ht : 1 ≤ Tm n) : |
| (hills (barge N).l:ℝ) ≤ 100*Tm n^alpha := by |
| have he : 0<Tm n := by linarith |
| have H := stack_lim n N hn h hk ht |
| have G : ((barge N).l:ℝ) ≤ stackN N := by |
| exact_mod_cast (barge_spec N (lt_of_lt_of_le hn h)).count |
| have hg := alpha_pos |
| refine (hills_real _).trans ?_ |
| calc |
| _ ≤ 2*(11*Tm n)^alpha := by gcongr; linarith |
| _ = 2*(11^alpha*Tm n^alpha) := by rw [mul_rpow (by norm_num) he.le] |
| _ ≤ 2*(11^(1:ℝ)*Tm n^alpha) := by gcongr <;> first | exact alpha_lt.le | norm_num |
| _ ≤ _ := by |
| rw [rpow_one] |
| have hTT : 0 < Tm n^alpha := by positivity |
| linarith |
|
|
| def flight (n : ℕ) := (n:ℝ)*(Tm n)^alpha*Qm n^2 |
|
|
| lemma leg_lim (n N : ℕ) (hn : 0<n) (h : n≤N) (hk : N ≤ 2*n) (ht : 1 ≤ Tm n) (hu : 1 ≤ Qm n) : |
| (leg N:ℝ) ≤ 10^40*flight n := by |
| have hg : (barge N).vol ≤ 8*n := by |
| have hh := (barge_spec N (by omega)).upper |
| omega |
| have hh : ((barge N).vol:ℝ) ≤ 8*n := by exact_mod_cast hg |
| have hT : 0 < Tm n := by linarith |
| unfold leg; push_cast |
| calc |
| _ ≤ (10^32 * Qm n^2)*((8*n)*(100*Tm n^alpha)) := |
| mul_le_mul (score_lim n N hn h hk ht hu) |
| (mul_le_mul hh (hills_lim n N hn h hk ht) (by positivity) (by positivity)) |
| (by positivity) (by positivity) |
| _ = 8*100*10^32*flight n := by unfold flight; ring |
| _ ≤ _ := by |
| have hh : 0 ≤ flight n := by unfold flight; positivity |
| gcongr; norm_num |
|
|
| lemma flight_big (n : ℕ) (ht : 1 ≤ Tm n) (hu : 1 ≤ Qm n) : |
| (n:ℝ) ≤ flight n := by |
| have H := Real.one_le_rpow ht alpha_pos.le |
| unfold flight |
| calc |
| (n:ℝ) = n*1*1 := by ring |
| _ ≤ _ := by gcongr; nlinarith |
|
|
| |
| lemma dwarf {v : ℕ→ℝ} (h : Tendsto v atTop atTop) (d : ℕ) {p c : ℝ} (hp : 0<p) (hc : 0<c) : |
| ∀ᶠ n in atTop, log (v n)^d ≤ c*v n^p := by |
| have H := (isLittleO_log_rpow_rpow_atTop (d:ℝ) hp).comp_tendsto h |
| have hg := H.def hc |
| filter_upwards [hg,h.eventually_ge_atTop 1] with n hn he |
| have he' : 0<v n := by linarith |
| change ‖log (v n)^(d:ℝ)‖ ≤ c*‖v n^p‖ at hn |
| rw [rpow_natCast,Real.norm_of_nonneg (pow_nonneg (Real.log_nonneg he) _), |
| Real.norm_of_nonneg (by positivity)] at hn |
| exact hn |
|
|
| variable {N : ℕ→ℕ} |
| |
| lemma docks_ev (h : ∀ n:ℕ,n ≤ N n ∧ N n ≤ 2*n) : |
| ∀ᶠ n in atTop, Ports (N n) := by |
| have H := dwarf (v:=fun n => (n:ℝ)) tendsto_natCast_atTop_atTop 2 |
| (c:=1/10^6) (p:=1) (by norm_num) (by norm_num) |
| filter_upwards [H,eventually_gt_atTop 0,Ttop.eventually_ge_atTop 1] with n hh hn ht |
| rw [rpow_one] at hh |
| have hh' : (Tm n)^2 ≤ 1/10^6*n := hh |
| have hh1 := order_pow_lim n (N n) hn (h n).1 (h n).2 ht |
| have hh2 : (2:ℝ)^(orderE (N n)) ≤ n := by linarith |
| exact (show 2^orderE (N n) ≤ n by exact_mod_cast hh2).trans (h n).1 |
|
|
| lemma polylog_ev (h : ∀ n:ℕ,n ≤ N n ∧ N n ≤ 2*n) (d : ℕ) : |
| ∀ᶠ n in atTop, ((stackN (N n):ℝ)+2)^d ≤ n := by |
| have h0 : 0< (12:ℝ)^d := by positivity |
| have H := dwarf (v:=fun n => (n:ℝ)) tendsto_natCast_atTop_atTop d |
| (c:=((12:ℝ)^d)⁻¹) (p:=1) (by norm_num) (by positivity) |
| filter_upwards [H,eventually_gt_atTop 0,Ttop.eventually_ge_atTop 1] with n hh hn ht |
| rw [rpow_one] at hh |
| have hh' : (Tm n)^d ≤ ((12:ℝ)^d)⁻¹*n := hh |
| have hh1 := stack_lim n (N n) hn (h n).1 (h n).2 ht |
| calc |
| _ ≤ (12*Tm n)^d := by gcongr; linarith |
| _ ≤ _ := by |
| rw [mul_pow] |
| calc |
| _ ≤ _ := mul_le_mul_of_nonneg_left hh' h0.le |
| _ = _ := by rw [← mul_assoc,mul_inv_cancel₀ h0.ne',one_mul] |
|
|
| lemma wind_ev (h : ∀ n:ℕ,n ≤ N n ∧ N n ≤ 2*n) (d : ℕ) : |
| ∀ᶠ n in atTop,(fairwind d (N n):ℝ)+1 ≤ 10^41*flight n := by |
| filter_upwards [docks_ev h,polylog_ev h d,eventually_gt_atTop 0,Ttop.eventually_ge_atTop 1, |
| Qtop.eventually_ge_atTop 1] with n hh hk hn ht hu |
| unfold fairwind Bench.surf |
| rw [ite_eq_left hh,Nat.add_zero] |
| push_cast |
| have hp := leg_lim n (N n) hn (h n).1 (h n).2 ht hu |
| have he := flight_big n ht hu |
| have he' : 1≤(n:ℝ) := by exact_mod_cast hn |
| have hx : (N n:ℝ) ≤ 2*n := by exact_mod_cast (h n).2 |
| linarith |
|
|
| lemma lunch_big (a b : Ration) (h : ∀ n:ℕ,n ≤ N n ∧ N n ≤ 2*n) : |
| (fun n : ℕ => (lunch a b N n:ℝ)) =O[atTop] flight := by |
| let W := lunch a b N |
| refine isBigO_iff.mpr ⟨(a.c:ℝ)*10^41+1+(b.c:ℝ)*10^41,?_⟩ |
| filter_upwards [wind_ev h a.d,wind_ev h b.d,eventually_gt_atTop 0,Ttop.eventually_ge_atTop 1, |
| Qtop.eventually_ge_atTop 1] with n hh hk hn ht hu |
| have he := flight_big n ht hu |
| have h1 : 1≤(n:ℝ) := by exact_mod_cast hn |
| rw [Real.norm_of_nonneg (by positivity),Real.norm_of_nonneg (by linarith)] |
| unfold lunch calorie |
| push_cast; simp only [add_mul,one_mul,mul_assoc] |
| gcongr |
| exact h1.trans he |
|
|
| end Skies |
| end |
| end PowerSaving |
|
|
| end OAI |
|
|
