| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.OrderedMixerCells |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.BackgroundPreparationCells |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.StoredLeafCells |
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationLeafArrays |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| The ordered mixer arrays and completed background rows stay shared through |
| all eight finite-set profile constructions. |
| -/ |
|
|
| noncomputable section |
| namespace LowSpaceSubsetSum.LeafArraysCells |
|
|
| variable {n : ℕ} |
|
|
| def run (I : Instance n) (data : RepresentationPrefixArrays.Data n) : |
| (Option (RepresentationLeafArrays.Leaves n) × ℕ) × ℕ := |
| let ordered := OrderedMixerCells.run data.layout |
| match ordered.1.1 with |
| | none => ((none, ordered.1.2+1), ordered.2+16*n+128) |
| | some domains => |
| let background := BackgroundPreparationCells.run I data.layout.permutation data.guesses |
| match background.1.1 with |
| | none => ((none, ordered.1.2+background.1.2+2), ordered.2+background.2+16*n+128) |
| | some rows => |
| let out := StoredLeafCells.run domains data.partitions rows data.guesses |
| ((some out.1.1, ordered.1.2+background.1.2+out.1.2+3), |
| ordered.2+background.2+out.2+16*n+128) |
|
|
| theorem value (I : Instance n) (data : RepresentationPrefixArrays.Data n) : |
| (run I data).1 = RepresentationLeafArrays.run I data := by |
| have ho₁ := congrArg Prod.fst (OrderedMixerCells.value data.layout) |
| have ho₂ := congrArg Prod.snd (OrderedMixerCells.value data.layout) |
| have hb₁ := congrArg Prod.fst (BackgroundPreparationCells.value I data.layout.permutation data.guesses) |
| have hb₂ := congrArg Prod.snd (BackgroundPreparationCells.value I data.layout.permutation data.guesses) |
| simp only [run, RepresentationLeafArrays.run, ho₁, ho₂, hb₁, hb₂] |
| split |
| · simp_all only |
| · split |
| · simp_all only |
| · simp_all only [StoredLeafCells.value] |
|
|
| theorem cells (I : Instance n) (data : RepresentationPrefixArrays.Data n) |
| (hl : ∀ j : Fin 3, data.layout.domains[j.val].length ≤ n) : |
| (run I data).2 ≤ 16384*(n+1)^2 := by |
| have ho := OrderedMixerCells.cells data.layout hl |
| have hb := BackgroundPreparationCells.cells I data.layout.permutation data.guesses |
| unfold run |
| dsimp only |
| split |
| · nlinarith only [ho, Nat.zero_le (n^2)] |
| next domains hd => |
| split |
| · nlinarith only [ho, hb, Nat.zero_le (n^2)] |
| next rows hr => |
| have hs : (RepresentationBackgroundPreparation.run I data.layout.permutation data.guesses).1 = |
| some rows := by |
| rw [← BackgroundPreparationCells.value I data.layout.permutation data.guesses] |
| exact hr |
| have hw := StoredLeafCells.cells domains data.partitions rows data.guesses |
| (RepresentationBackgroundPreparation.lengths I data.layout.permutation data.guesses rows hs) |
| change _+_+_+16*n+128 ≤ _ |
| nlinarith only [ho, hb, hw, Nat.zero_le (n^2)] |
|
|
| end LowSpaceSubsetSum.LeafArraysCells |
|
|
| end |
|
|
| end OAI |
|
|
