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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordRealization
import OAI.Combinatorics.LowSpaceSubsetSum.Counting.FiniteProductSamplerCoupling
namespace OAI
/-! # Exact bounded word programs for range and permutation calls -/
noncomputable section
namespace LowSpaceSubsetSum.WordRealization
def enlarge {w N M : ℕ} {α : Type} [Fintype α] {S : SamplerCoupling α}
(X : Realization w N S) (h : N ≤ M) : Realization w M S where
program := X.program
faithful := X.faithful
bounded := by
have hb := WordProgram.bound_add X.program N X.bounded (M-N)
simpa only [Nat.sub_add_cancel h] using hb
def prod {w N M : ℕ} {α β : Type} [Fintype α] [Fintype β]
{S : SamplerCoupling α} {T : SamplerCoupling β}
(X : Realization w N S) (Y : Realization w M T) :
Realization w (N+M) (S.prod T) :=
map (bind X (fun _ => β) (fun _ => T) (fun _ => Y)) (Equiv.sigmaEquivProd α β)
def range (w n D : ℕ) (hD : 1 < D) (hw : Nat.clog 2 D ≤ w) :
Realization w (n^3) (rangeCoupling hD n) := by
let : Nonempty (Fin D) := ⟨⟨0, by omega⟩⟩
let : Nonempty {x : Fin (2 ^ Nat.clog 2 D) // rangeEligible D x} :=
⟨(rangeEligibleEquiv D).symm ⟨0, by omega⟩⟩
exact map (rejection w (Nat.clog 2 D) hw (n^3) (rangeEligible D)) (rangeEligibleEquiv D)
def fin (w n D : ℕ) [NeZero D] (hw : Nat.clog 2 D ≤ w) :
Realization w (n^3) (finCoupling n D) := by
unfold finCoupling
split
next h => exact range w n D h hw
next h => exact enlarge (pure (⟨0, Nat.pos_of_ne_zero (NeZero.ne D)⟩ : Fin D)) (Nat.zero_le _)
def pi (w N : ℕ) : (k : ℕ) → (A : Fin k → Type) → [∀ i, Fintype (A i)] →
(S : ∀ i, SamplerCoupling (A i)) → (∀ i, Realization w N (S i)) →
Realization w (k*N) (SamplerCoupling.pi k A S)
| 0, A, _, _, _ => by
simpa only [Nat.zero_mul, SamplerCoupling.pi] using
(pure (w := w) (fun i : Fin 0 => (Fin.elim0 i : A i)))
| k+1, A, _, S, X => by
have h := map (prod (X 0)
(pi w N k (fun i => A i.succ) (fun i => S i.succ) (fun i => X i.succ)))
(Fin.consEquiv A)
simpa only [SamplerCoupling.pi, Nat.add_mul, Nat.one_mul, Nat.add_comm] using h
def permutation (w n : ℕ) : (k : ℕ) → Nat.clog 2 k ≤ w →
Realization w (k*n^3) (permutationCoupling n k)
| 0, _ => by
simpa only [Nat.zero_mul, permutationCoupling] using (pure (w := w) (Equiv.refl (Fin 0)))
| k+1, hw => by
have hk : Nat.clog 2 k ≤ w :=
(Nat.clog_mono_right 2 (by omega : k ≤ k+1)).trans hw
have h := map (prod (fin w n (k+1) hw) (permutation w n k hk))
Equiv.Perm.decomposeFin.symm
simpa only [permutationCoupling, Nat.add_mul, Nat.one_mul, Nat.add_comm] using h
def finiteChoice (w n : ℕ) (A : Type) [Fintype A] [Nonempty A]
(hw : Nat.clog 2 (Fintype.card A) ≤ w) :
Realization w (n^3) (finiteChoiceCoupling n A) :=
map (fin w n (Fintype.card A) hw) (Fintype.equivFin A).symm
def finitePermutation (w n : ℕ) (A : Type) [Fintype A] [DecidableEq A]
(hw : Nat.clog 2 (Fintype.card A) ≤ w) :
Realization w (Fintype.card A*n^3) (finitePermutationCoupling n A) :=
map (permutation w n (Fintype.card A) hw) (Equiv.permCongr (Fintype.equivFin A).symm)
end LowSpaceSubsetSum.WordRealization
end
end OAI
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