File size: 3,394 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 79 80 81 82 83 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordRepresentationHistory
import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordRunLaw
import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorStoredSpace
namespace OAI
/-! # The complete word draw buffer fits below the branch workspace -/
noncomputable section
namespace LowSpaceSubsetSum.WordSamplingBounds
variable {n : ℕ}
theorem family_width (n : ℕ) : 2*RepresentationSetup.blockHalf n 0 ≤ n := by
unfold RepresentationSetup.blockHalf
simp only [Matrix.cons_val_zero]
omega
theorem family_codes (n : ℕ) :
2^(2*RepresentationSetup.blockHalf n 0) ≤ compressionDictionaryCapacity n := by
apply (Nat.pow_le_pow_right (by decide)
(show 2*RepresentationSetup.blockHalf n 0 ≤ 2*(2*RepresentationSetup.blockHalf n 0) by omega)).trans
exact RepresentationSetup.separator_table_rate_le_capacity
theorem history_normalize (n : ℕ) : WordRealization.historyReads n =
4*n^4+10*n^3+n+44*2^(2*RepresentationSetup.blockHalf n 0)*
(2*RepresentationSetup.blockHalf n 0)*n^3 := by
unfold WordRealization.historyReads
ring
theorem history_reads (I : Instance n) (hn : 10000 ≤ n) :
WordRealization.historyReads n ≤ inputScale I^5*(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 hn1 : 1 ≤ u := by omega
have h34 : n^3 ≤ u^4 :=
(Nat.pow_le_pow_left hnu 3).trans (Nat.pow_le_pow_right hn1 (by decide : 3 ≤ 4))
have hn4 : n ≤ u^4 := by
exact hnu.trans (by simpa only [pow_one] using
Nat.pow_le_pow_right hn1 (by decide : 1 ≤ 4))
have hnfour : n^4 ≤ u^4 := Nat.pow_le_pow_left hnu 4
have hf : 44*2^(2*RepresentationSetup.blockHalf n 0)*
(2*RepresentationSetup.blockHalf n 0)*n^3 ≤ 44*u^4*C := by
calc
_ ≤ 44*C*u*u^3 := by
gcongr
· exact family_codes n
· exact (family_width n).trans hnu
_ = _ := by ring
have hp : WordRealization.historyReads n ≤ 59*u^4*(C+1) := by
rw [history_normalize]
nlinarith only [h34,hn4,hnfour,hf]
apply hp.trans
calc
_ ≤ u*u^4*(C+1) := Nat.mul_le_mul_right _ (Nat.mul_le_mul_right _ (by omega))
_ = _ := by rw [← pow_succ']
theorem word_width (I : Instance n) : 3*n+1 ≤ wordWidth I := by
unfold wordWidth inputScale
omega
def representation (I : Instance n) (hn : 10000 ≤ n) :
WordRealization.Realization (wordWidth I) (WordRealization.historyReads n)
((canonicalRepresentationData I).zeroSetup.bucketHistoryCoupling hn) :=
WordRealization.bucketHistory _ hn _ (word_width I)
theorem representation_probability (I : Instance n) (hn : 10000 ≤ n)
(P : Option ((canonicalRepresentationData I).zeroSetup.BucketTrialHistory (by omega)) → Prop) :
(FiniteLaw.uniform : FiniteLaw (Fin (WordRealization.historyReads n) → Fin (2^wordWidth I))).probability
(fun words => P (WordProgram.run (WordProgram.finiteStream words)
(representation I hn).program 0).1) =
((canonicalRepresentationData I).zeroSetup.bucketHistoryCoupling hn).boundedLaw.probability P := by
rw [← WordProgram.run_probability _ _ (representation I hn).bounded]
exact (representation I hn).faithful P
end LowSpaceSubsetSum.WordSamplingBounds
end
end OAI
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