File size: 3,278 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.WordSeedPermutationExecution
import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedPrimeInitialization
namespace OAI
/-! # Actual word-driven compression sampler
The permutation generator retains its range candidate buffers. The prime
width is computed by one multiplication and one division before its
modulus and bounded rejection routine. Exhaustion remains `none`.
-/
noncomputable section
namespace LowSpaceSubsetSum.WordSeedMeasured
private def compressionCore {n : ℕ} (D : CompressionData n) (hn : 168000000 ≤ n)
(w : ℕ) (hp : Nat.clog 2 n ≤ w) (hq : compressionPrimeExponent n ≤ w) :
Execution (WordSeedRealization.compression D hn w hp hq) :=
let ell := 249*n/1000
bind (permutation w n n hp) (fun _ => CompressionSetup.PrimeSpace n)
(fun _ => primeCoupling (by unfold compressionPrimeExponent; omega) n)
(fun _ => WordSeedRealization.prime w n ell (by omega) hq)
(fun _ => primeInitialized w n ell (by omega) hq)
def compression {n : ℕ} (D : CompressionData n) (hn : 168000000 ≤ n)
(w : ℕ) (hp : Nat.clog 2 n ≤ w) (hq : compressionPrimeExponent n ≤ w) :
Execution (WordSeedRealization.compression D hn w hp hq) where
run stream cursor :=
let out := (compressionCore D hn w hp hq).run stream cursor
⟨out.accepted,out.cursor,out.work+6,max 8 (out.cells+8),out.retained+1⟩
value stream cursor := (compressionCore D hn w hp hq).value stream cursor
def compressionWork (n : ℕ) : ℕ :=
n*(13*n^3+5*n+64)+
(n^3*(12*2^((compressionPrimeExponent n+1)/2)+
12*(compressionPrimeExponent n+1)+11)+3*compressionPrimeExponent n+17)+9
theorem compression_bounded {n : ℕ} (D : CompressionData n) (hn : 168000000 ≤ n)
(w : ℕ) (hp : Nat.clog 2 n ≤ w) (hq : compressionPrimeExponent n ≤ w) :
(compression D hn w hp hq).Bounded (compressionWork n)
((n+3)*(2*n^3+256)) ((n+2)*(n^3+64)) := by
have hh := bind_bounded (permutation w n n hp)
(fun _ => CompressionSetup.PrimeSpace n)
(fun _ => primeCoupling (by unfold compressionPrimeExponent; omega) n)
(fun _ => WordSeedRealization.prime w n (compressionPrimeExponent n)
(by unfold compressionPrimeExponent; omega) hq)
(fun _ => primeInitialized w n (compressionPrimeExponent n)
(by unfold compressionPrimeExponent; omega) hq)
_ _ _ _ _ _ (permutation_bounded w n n hp)
(fun _ => primeInitialized_bounded w n (compressionPrimeExponent n)
(by unfold compressionPrimeExponent; omega) hq)
intro stream cursor
have h := hh stream cursor
dsimp only [compression,compressionCore]
change _+6 ≤ compressionWork n ∧ max 8 (_+8) ≤ (n+3)*(2*n^3+256) ∧
_+1 ≤ (n+2)*(n^3+64)
refine ⟨?_,?_,?_⟩
· have hwork := Nat.add_le_add_right h.1 6
apply hwork.trans
unfold compressionWork
omega
· apply max_le
· nlinarith
· have hc := Nat.add_le_add_right h.2.1 8
apply hc.trans
have hA : (n+1)*(2*n^3+256)+8 ≤ (n+3)*(2*n^3+256) := by nlinarith
have hB : ((n+1)*(n^3+64)+(2*n^3+124)+8)+8 ≤
(n+3)*(2*n^3+256) := by nlinarith
omega
· have hr := Nat.add_le_add_right h.2.2 1
exact hr.trans (by nlinarith)
end LowSpaceSubsetSum.WordSeedMeasured
end
end OAI
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