| import OAI.Combinatorics.EditDistance.LowerParameters |
|
|
| namespace OAI |
|
|
| /-! |
| # Inverting recursive depth for all sufficiently large lengths |
| -/ |
|
|
| namespace EditDepth |
| open Filter |
|
|
| noncomputable def radius (t : ℝ) := Real.sqrt (t / Real.log t) |
| noncomputable def depth (t : ℝ) : ℕ := ⌊radius t / 100⌋₊ |
|
|
| lemma radius_nonneg (t : ℝ) : 0 ≤ radius t := Real.sqrt_nonneg _ |
|
|
| lemma eventually_radius_large (K : ℕ) : |
| ∀ᶠ t : ℝ in atTop, (100 : ℝ)*(K+2) ≤ radius t := by |
| let R : ℝ := 100*((K : ℝ)+2) |
| have hR : 0 < R := by dsimp [R]; positivity |
| have he : 0 < 1/R^2 := by positivity |
| have hlog := Real.isLittleO_log_id_atTop.def he |
| have hl := Real.tendsto_log_atTop.eventually (eventually_ge_atTop (1 : ℝ)) |
| filter_upwards [hlog,hl,eventually_ge_atTop (1 : ℝ)] with t ht hlt htp |
| have tp : 0 < t := by linarith |
| have lp : 0 < Real.log t := by linarith |
| have ht' : Real.log t ≤ t/R^2 := by |
| simpa only [Function.id_def,Real.norm_eq_abs,abs_of_pos tp,abs_of_pos lp,div_eq_mul_inv,one_mul,mul_comm] using ht |
| have hs : R^2 ≤ t/Real.log t := by |
| apply (le_div_iff₀ lp).mpr |
| have hb := (le_div_iff₀ (sq_pos_of_pos hR)).mp ht' |
| nlinarith |
| exact Real.le_sqrt_of_sq_le hs |
|
|
| lemma depth_lower {t : ℝ} (ht : 200 ≤ radius t) : radius t/200 ≤ (depth t : ℝ) := by |
| have hf := Nat.lt_floor_add_one (radius t/100) |
| change radius t/100 < (depth t : ℝ)+1 at hf |
| linarith |
|
|
| lemma depth_upper (t : ℝ) : (depth t : ℝ) ≤ radius t/100 := |
| Nat.floor_le (div_nonneg (radius_nonneg t) (by norm_num)) |
|
|
| lemma depth_bounds {t : ℝ} (ht : 1 ≤ t) (hlt : 1 ≤ Real.log t) |
| (hr : 200 ≤ radius t) (hll : Real.log (Real.log t) ≤ Real.log t/6) |
| (hlarge : 12*Real.log (200 : ℝ) ≤ Real.log t) : |
| 1000*(depth t : ℝ)^2*Real.log (depth t : ℝ) ≤ t ∧ |
| Real.sqrt (t*Real.log t)/1200 ≤ (1/2 : ℝ)*(depth t : ℝ)*Real.log (depth t : ℝ) := by |
| have tp : 0 < t := by linarith |
| have lp : 0 < Real.log t := by linarith |
| have rp : 0 < radius t := by linarith |
| have kp : 0 < (depth t : ℝ) := lt_of_lt_of_le (by positivity) (depth_lower hr) |
| have hr2 : radius t^2 = t/Real.log t := Real.sq_sqrt (div_nonneg tp.le lp.le) |
| have hr2' : radius t^2*Real.log t = t := (eq_div_iff lp.ne').mp hr2 |
| have hrle : radius t ≤ t := by |
| have hdiv : t/Real.log t ≤ t := (div_le_iff₀ lp).mpr (by nlinarith) |
| nlinarith [radius_nonneg t] |
| have hkle : (depth t : ℝ) ≤ t := by |
| have hh := depth_upper t |
| nlinarith [radius_nonneg t] |
| have logupper : Real.log (depth t : ℝ) ≤ Real.log t := Real.log_le_log kp hkle |
| have loglower : Real.log t/3 ≤ Real.log (depth t : ℝ) := by |
| have hlog := Real.log_le_log (by positivity : 0 < radius t/200) (depth_lower hr) |
| have logR : Real.log (radius t) = (Real.log t - Real.log (Real.log t))/2 := by |
| dsimp [radius] |
| rw [Real.log_sqrt (div_nonneg tp.le lp.le),Real.log_div tp.ne' lp.ne'] |
| rw [Real.log_div rp.ne' (by norm_num),logR] at hlog |
| linarith |
| have hsq : (depth t : ℝ)^2 ≤ radius t^2/10000 := by |
| have hh := depth_upper t |
| nlinarith [sq_nonneg (radius t/100-(depth t : ℝ))] |
| have hsqrt : Real.sqrt (t*Real.log t) = radius t*Real.log t := by |
| apply (Real.sqrt_eq_iff_mul_self_eq (by positivity) (by positivity)).mpr |
| nlinarith [congrArg (fun x : ℝ => x*Real.log t) hr2'] |
| constructor |
| · calc |
| _ ≤ 1000*(depth t : ℝ)^2*Real.log t := mul_le_mul_of_nonneg_left logupper (by positivity) |
| _ ≤ t := by |
| have hh := mul_le_mul_of_nonneg_right hsq lp.le |
| nlinarith |
| · rw [hsqrt] |
| have hh : radius t/200*(Real.log t/3) ≤ (depth t : ℝ)*Real.log (depth t : ℝ) := by |
| gcongr |
| · exact depth_lower hr |
| linarith |
|
|
| theorem eventually_choose_depth (K : ℕ) : ∀ᶠ t : ℝ in atTop, |
| K ≤ depth t ∧ 1000*(depth t : ℝ)^2*Real.log (depth t : ℝ) ≤ t ∧ |
| Real.sqrt (t*Real.log t)/1200 ≤ (1/2 : ℝ)*(depth t : ℝ)*Real.log (depth t : ℝ) := by |
| have hsmall := Real.isLittleO_log_id_atTop.def (by norm_num : (0 : ℝ)<1/6) |
| have hsmall' := Real.tendsto_log_atTop.eventually hsmall |
| have hl := Real.tendsto_log_atTop.eventually (eventually_ge_atTop (1 : ℝ)) |
| have hh := Real.tendsto_log_atTop.eventually (eventually_ge_atTop (12*Real.log (200 : ℝ))) |
| filter_upwards [eventually_radius_large K,hsmall',hl,hh,eventually_ge_atTop (1 : ℝ)] with t hr hs hl hh ht |
| have hK : (K : ℝ) ≤ radius t/100 := by nlinarith |
| have hr' : 200 ≤ radius t := by have := Nat.cast_nonneg (α := ℝ) K; nlinarith |
| have hll : Real.log (Real.log t) ≤ Real.log t/6 := by |
| have hlog := Real.log_nonneg hl |
| have hpos : 0 ≤ Real.log t := by linarith |
| simpa only [Function.id_def,Real.norm_eq_abs,abs_of_nonneg hlog,abs_of_nonneg hpos, |
| div_eq_mul_inv,one_mul,mul_comm] using hs |
| exact ⟨Nat.le_floor hK,depth_bounds ht hl hr' hll hh⟩ |
|
|
| end EditDepth |
|
|
| end OAI |
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|
