openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Word /WordRepresentationDraws.lean
| 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 | |