| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedPrimitives |
| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordRepresentationDraws |
|
|
| namespace OAI |
|
|
| /-! # Successful seeds for signs, partitions and separator families -/ |
|
|
| noncomputable section |
| namespace LowSpaceSubsetSum.WordSeedRealization |
|
|
| def bitsSeed (n : ℕ) (f : Fin n → Bool) : SuccessfulSeed (bitArrayCoupling n) := |
| ⟨f,f,rfl⟩ |
|
|
| def bits (w n : ℕ) (hw : 1 ≤ w) : Realization w n (bitArrayCoupling n) where |
| program := WordProgram.map |
| (fun f => some (bitsSeed n (fun i => finTwoEquiv (ExactWordDraw.low w 1 (f i))))) |
| (WordTuple.sample n) |
| faithful P := by |
| have he : WordProgram.map (observe (bitArrayCoupling n)) |
| (WordProgram.map |
| (fun f => some (bitsSeed n (fun i => finTwoEquiv (ExactWordDraw.low w 1 (f i))))) |
| (WordTuple.sample n)) = (WordRealization.bits w n hw).program := by |
| rw [map_map] |
| rfl |
| rw [he] |
| exact (WordRealization.bits w n hw).faithful P |
| bounded := by |
| simpa only [WordProgram.map,Nat.add_zero] using |
| WordProgram.bind_bound (WordTuple.sample (w := w) n) n (WordTuple.bounded n) |
| (fun f => .pure (some (bitsSeed n |
| (fun i => finTwoEquiv (ExactWordDraw.low w 1 (f i)))))) 0 |
| (fun _ => WordProgram.Bounded.pure _ 0) |
|
|
| def signs (w n : ℕ) (hw : 1 ≤ w) : Realization w n (signCoupling n) := |
| map (bits w n hw) (bitsSubsetEquiv n) |
|
|
| def subset (w n g : ℕ) (K : Finset (Fin g)) (hw : Nat.clog 2 g ≤ w) : |
| Realization w (g*n^3) (subsetCoupling n g K) := |
| map (permutation w n g hw) (permutationBlock K) |
|
|
| def fixedSize (w n g e : ℕ) (he : e ≤ g) (hw : Nat.clog 2 g ≤ w) : |
| Realization w (g*n^3) (fixedSizeCoupling n g e he) := |
| map (subset w n g (initialSubset g e he) hw) (initialSubsetEquiv g e he) |
|
|
| def families {g : ℕ} (S : SeparatorExpectation.RoundedParameters g) |
| (w n : ℕ) (hw : Nat.clog 2 g ≤ w) : |
| Realization w (4*(11*2^g*(g*n^3))) (S.samplerCoupling n) := |
| pi w (11*2^g*(g*n^3)) 4 _ _ (fun i => |
| enlarge (pi w (g*n^3) (S.counts i) _ _ |
| (fun _ => fixedSize w n g (S.e i) (S.size_le i) hw)) |
| (Nat.mul_le_mul_right _ (S.counts_le i))) |
|
|
| def layout {n : ℕ} (R : RepresentationSetup n) (w : ℕ) |
| (hp : Nat.clog 2 n ≤ w) (hb : 1 ≤ w) : |
| Realization w (n*n^3+n) R.layoutCoupling := |
| prod (permutation w n n hp) (signs w n hb) |
|
|
| def guesses {n : ℕ} (R : RepresentationSetup n) (w : ℕ) |
| (hw : Nat.clog 2 (CanonicalGuesses.width n) ≤ w) : |
| Realization w (3*n^3) R.guessesCoupling := |
| pi w (n^3) 3 _ _ (fun _ => map (fin w n (CanonicalGuesses.width n) hw) |
| (CanonicalGuesses.encoding n)) |
|
|
| def partition {n : ℕ} (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) |
| (i : Fin 3) (w : ℕ) |
| (hw : Nat.clog 2 (RepresentationSetup.blockCount i * |
| (2*RepresentationSetup.blockHalf n i)) ≤ w) : |
| Realization w ((RepresentationSetup.blockCount i * |
| (2*RepresentationSetup.blockHalf n i))*n^3) (CanonicalPartition.coupling R z i) := |
| map (map (permutation w n _ hw) (Equiv.permCongr (CanonicalPartition.ranks R z i).symm)) |
| (coloringOfPermutation (CanonicalPartition.base R z i)) |
|
|
| end LowSpaceSubsetSum.WordSeedRealization |
|
|
| end |
|
|
| end OAI |
|
|
