| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordRepresentationHistory |
| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordRunLaw |
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorStoredSpace |
|
|
| namespace OAI |
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|
| /-! |
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|
| 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 |
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|
| end |
|
|
| end OAI |
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|
