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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | 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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