File size: 3,299 Bytes
e5a97c6 | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSamplingBounds
import OAI.Combinatorics.LowSpaceSubsetSum.Search.PreparedAsymptotics
namespace OAI
/-! # Finite random-word addresses fit the fixed machine word -/
noncomputable section
namespace LowSpaceSubsetSum.WordAddressBounds
def roundReads (n : ℕ) : ℕ := n*n^3+n^3+WordRealization.historyReads n
def totalReads {n : ℕ} (I : Instance n) : ℕ := inputScale I^120*roundReads n
theorem round_reads {n : ℕ} (I : Instance n) (hn : 10000 ≤ n) :
roundReads n ≤ inputScale I^6*(compressionDictionaryCapacity n+1) := by
let u := inputScale I
let C := compressionDictionaryCapacity n
have hu : 10000 ≤ u := by unfold u inputScale; omega
have hnu : n ≤ u := by unfold u inputScale; omega
have hp : n*n^3+n^3 ≤ 2*u^4 := by
have h1 : n*n^3 ≤ u^4 := by
simpa only [←pow_succ'] using Nat.mul_le_mul hnu (Nat.pow_le_pow_left hnu 3)
have h2 : n^3 ≤ u^4 := (Nat.pow_le_pow_left hnu 3).trans
(Nat.pow_le_pow_right (by omega) (by decide : 3 ≤ 4))
omega
have hh := WordSamplingBounds.history_reads I hn
change WordRealization.historyReads n ≤ u^5*(C+1) at hh
have hs : 2*u^4 ≤ u^5*(C+1) := by
have h1 : 2*u^4 ≤ u*u^4 := Nat.mul_le_mul_right _ (by omega)
have h2 : u^5 ≤ u^5*(C+1) := Nat.le_mul_of_pos_right _ (by omega)
rw [←pow_succ'] at h1
exact h1.trans h2
have hround : roundReads n ≤ 2*(u^5*(C+1)) := by
unfold roundReads
omega
apply hround.trans
calc
_ ≤ u*(u^5*(C+1)) := Nat.mul_le_mul_right _ (by omega)
_ = _ := by rw [←Nat.mul_assoc,←pow_succ']
theorem total_reads {n : ℕ} (I : Instance n) (hn : 10000 ≤ n) :
totalReads I ≤ inputScale I^126*(compressionDictionaryCapacity n+1) := by
have h := Nat.mul_le_mul_left (inputScale I^120) (round_reads I hn)
simpa only [totalReads,←Nat.mul_assoc,←pow_add] using h
theorem total_limit {n : ℕ} (I : Instance n) (hn : 10000 ≤ n)
(hscale : Real.logb 2 (inputScale I : ℝ) ≤ (n : ℝ)/1000000000) :
totalReads I < 2^n := by
have hu : (0 : ℝ) < inputScale I := by
exact_mod_cast (show 0 < inputScale I by unfold inputScale; omega)
have hC : (1 : ℝ) ≤ compressionDictionaryCapacity n := by
exact_mod_cast (show 1 ≤ compressionDictionaryCapacity n from Nat.one_le_two_pow)
have hc := PreparedAsymptotics.capacity_upper n
have hsum : (compressionDictionaryCapacity n : ℝ)+1 ≤
(2 : ℝ)^((199/1000 : ℝ)*n+1) := by
rw [Real.rpow_add (by norm_num : (0 : ℝ) < 2),Real.rpow_one]
linarith only [hC,hc]
have hs := mul_le_mul_of_nonneg_left hsum
(show 0 ≤ (inputScale I : ℝ)^126 by positivity)
have ht := guarded_polynomial_exponential hn hu hscale 126
(a := (199/1000 : ℝ)) (by norm_num)
have hb : ((totalReads I : ℕ) : ℝ) ≤
(inputScale I : ℝ)^126*((compressionDictionaryCapacity n : ℝ)+1) := by
exact_mod_cast total_reads I hn
have hfin := hb.trans_lt (hs.trans_lt ht)
exact_mod_cast hfin
theorem total_fits {n : ℕ} (I : Instance n) (hn : 10000 ≤ n)
(hscale : Real.logb 2 (inputScale I : ℝ) ≤ (n : ℝ)/1000000000) :
totalReads I < 2^wordWidth I := cutoff_fits_word I (total_limit I hn hscale)
end LowSpaceSubsetSum.WordAddressBounds
end
end OAI
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