| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedPreparedBind |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.LayoutPreparationCells |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| The mixer counts, order and prime widths are obtained by the counted stored |
| preparation. They are not evaluated implicitly while selecting a sampler. |
| -/ |
|
|
| noncomputable section |
| namespace LowSpaceSubsetSum.WordLayoutPreparation |
|
|
| variable {n : ℕ} |
|
|
| abbrev Layout (I : Instance n) := |
| WordSeedMeasured.Accepted (canonicalRepresentationData I).zeroSetup.layoutCoupling |
|
|
| abbrev Prepared (I : Instance n) (a : Layout I) := |
| {d : RepresentationLayoutPreparation.Data n // |
| (LayoutPreparationCells.run I a.seed.val).1.1 = some d} |
|
|
| theorem source (I : Instance n) (a : Layout I) : |
| (canonicalRepresentationData I).zeroSetup.layoutCoupling.bounded a.seed.val = some a.value := |
| (WordSeedRealization.successful_agrees a.seed).trans (congrArg some a.correct.symm) |
|
|
| theorem exists_data (I : Instance n) (a : Layout I) : Nonempty (Prepared I a) := by |
| obtain ⟨d,hd,_⟩ := RepresentationLayoutPreparation.complete |
| (canonicalRepresentationData I) a.seed.val a.value.1 a.value.2 (source I a) |
| exact ⟨⟨d,(congrArg Prod.fst (LayoutPreparationCells.value I a.seed.val)).trans hd⟩⟩ |
|
|
| def compute (I : Instance n) (a : Layout I) : (Prepared I a × ℕ) × ℕ := |
| let out := LayoutPreparationCells.run I a.seed.val |
| match h : out.1.1 with |
| | none => False.elim (by |
| obtain ⟨d⟩ := exists_data I a |
| cases h.symm.trans d.property) |
| | some d => ((⟨d,h⟩,out.1.2+1),out.2+8) |
|
|
| theorem fields (I : Instance n) (a : Layout I) (d : Prepared I a) : |
| (∀ i : Fin n, d.val.permutation[i.val] = a.value.1 i) ∧ |
| (∀ i : Fin 4, d.val.parameters.widths[i.val] = |
| (canonicalRepresentationData I).zeroSetup.samplerPrimeWidths a.value i) := by |
| have hd := (congrArg Prod.fst (LayoutPreparationCells.value I a.seed.val)).symm.trans d.property |
| obtain ⟨e,he,hv,_,hw⟩ := RepresentationLayoutPreparation.value |
| (canonicalRepresentationData I) a.seed.val d.val hd |
| have heq : (e,d.val.signs.semantic) = a.value := Option.some.inj (he.symm.trans (source I a)) |
| constructor |
| · intro i |
| have hi := congrArg (fun z => z.1 i) heq |
| exact (hv i).trans hi |
| · intro i |
| exact (hw i).trans (congrArg |
| (fun z => (canonicalRepresentationData I).zeroSetup.samplerPrimeWidths z i) heq) |
|
|
| theorem work (I : Instance n) (hn : 10000 ≤ n) (a : Layout I) : |
| (compute I a).1.2 ≤ n*n^3+10*(n+1)^2+4*n+7+ |
| inputScale I^6*2^mixerSize n := by |
| have hw := RepresentationLayoutPreparation.work (canonicalRepresentationData I) hn a.seed.val |
| change (RepresentationLayoutPreparation.run (canonicalRepresentationData I) a.seed.val).2 ≤ |
| n*n^3+10*(n+1)^2+4*n+6+inputScale I^6*2^mixerSize n at hw |
| have hv := congrArg Prod.snd (LayoutPreparationCells.value I a.seed.val) |
| rw [←hv] at hw |
| obtain ⟨d⟩ := exists_data I a |
| unfold compute |
| dsimp only |
| split |
| next h => cases h.symm.trans d.property |
| · dsimp only |
| omega |
|
|
| theorem cells (I : Instance n) (hn : 10000 ≤ n) (a : Layout I) : |
| (compute I a).2 ≤ inputScale I^10*(compressionDictionaryCapacity n+1)+8 := by |
| have hc := LayoutPreparationCells.cells I hn a.seed.val |
| obtain ⟨d⟩ := exists_data I a |
| unfold compute |
| dsimp only |
| split |
| next h => cases h.symm.trans d.property |
| · dsimp only |
| omega |
|
|
| end LowSpaceSubsetSum.WordLayoutPreparation |
|
|
| end |
|
|
| end OAI |
|
|
