File size: 3,537 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordPrimeRealization
import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorSamplerCoupling
namespace OAI
/-! # Word programs for signs, fixed-size samples, and separator families -/
noncomputable section
namespace LowSpaceSubsetSum.WordRealization
def bits (w n : ℕ) (hw : 1 ≤ w) : Realization w n (bitArrayCoupling n) where
program := WordProgram.map
(fun f => some (fun i => finTwoEquiv (ExactWordDraw.low w 1 (f i)))) (WordTuple.sample n)
faithful P := by
rw [WordProgram.probability_map, WordTuple.probability]
rw [ExactWordDraw.uniform_tuple w 1 hw n
(fun f => P (some (fun i => finTwoEquiv (f i))))]
let e : (Fin n → Fin 2) ≃ (Fin n → Bool) :=
Equiv.piCongrRight (fun _ => finTwoEquiv)
have he := FiniteLaw.map_equiv_uniform e
change (FiniteLaw.uniform : FiniteLaw (Fin n → Fin 2)).probability
(fun f => P (some (e f))) = _
have hp := congrArg (fun μ : FiniteLaw (Fin n → Bool) =>
μ.probability (fun f => P (some f))) he
rw [FiniteLaw.map_probability] at hp
apply hp.trans
change (FiniteLaw.uniform : FiniteLaw (Fin n → Bool)).probability (fun f => P (some f)) =
((FiniteLaw.uniform : FiniteLaw (Fin n → Bool)).map some).probability P
rw [FiniteLaw.map_probability]
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 (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.WordRealization
end
end OAI
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