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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordRejection
import OAI.Combinatorics.LowSpaceSubsetSum.Search.StoppedSamplerLaw
namespace OAI
/-! # Exact fair-word refinements of the stopped sampler constructors -/
noncomputable section
namespace LowSpaceSubsetSum.WordRealization
open WordProgram
attribute [local instance] Classical.propDecidable
structure Realization (w N : ℕ) {α : Type} [Fintype α] (S : SamplerCoupling α) where
program : Program w (Option α)
faithful : ∀ P, WordProgram.probability program P = S.boundedLaw.probability P
bounded : WordProgram.Bounded program N
variable {w N M : ℕ} {α β : Type} [Fintype α] [Fintype β]
def pure (a : α) : Realization w 0 (SamplerCoupling.pure a) where
program := .pure (some a)
faithful P := by
change (if P (some a) then (1 : ℝ) else 0) =
((FiniteLaw.uniform (Ω := Unit)).map (fun _ => some a)).probability P
rw [FiniteLaw.map_probability]
by_cases h : P (some a)
· simp only [h, ↓reduceIte, FiniteLaw.probability_true]
· simp only [h, ↓reduceIte, FiniteLaw.probability, Finset.sum_const_zero]
bounded := Bounded.pure _ 0
def map {S : SamplerCoupling α} (X : Realization w N S) (f : α → β) :
Realization w N (S.map f) where
program := WordProgram.map (Option.map f) X.program
faithful P := by
rw [WordProgram.probability_map, X.faithful]
dsimp only [SamplerCoupling.boundedLaw, SamplerCoupling.map]
rw [FiniteLaw.map_probability, FiniteLaw.map_probability]
bounded := by
simpa only [WordProgram.map, Nat.add_zero] using
WordProgram.bind_bound X.program N X.bounded (fun a => .pure (a.map f)) 0
(fun _ => Bounded.pure _ 0)
def continuation (B : α → Type) [∀ a, Fintype (B a)]
(T : ∀ a, SamplerCoupling (B a)) (Y : ∀ a, Realization w M (T a)) :
Option α → Program w (Option (Sigma B))
| none => .pure none
| some a => WordProgram.map (Option.map (fun b => ⟨a,b⟩)) (Y a).program
omit [Fintype α] in
theorem continuation_probability (B : α → Type) [∀ a, Fintype (B a)]
(T : ∀ a, SamplerCoupling (B a)) (Y : ∀ a, Realization w M (T a))
(P : Option (Sigma B) → Prop) (a : Option α) :
WordProgram.probability (continuation B T Y a) P =
StoppedSamplerLaw.nextProbability B T P a := by
cases a with
| none => rfl
| some a =>
exact (WordProgram.probability_map _ (Y a).program P).trans ((Y a).faithful _)
def bind {S : SamplerCoupling α} (X : Realization w N S)
(B : α → Type) [∀ a, Fintype (B a)] (T : ∀ a, SamplerCoupling (B a))
(Y : ∀ a, Realization w M (T a)) : Realization w (N+M) (S.bind B T) where
program := WordProgram.bind X.program (continuation B T Y)
faithful P := by
change WordProgram.expectation (WordProgram.bind X.program (continuation B T Y))
(fun x => if P x then 1 else 0) = _
rw [WordProgram.expectation_bind]
change WordProgram.expectation X.program
(fun a => WordProgram.probability (continuation B T Y a) P) = _
simp only [continuation_probability]
rw [WordProgram.expectation_of_probability X.program S.boundedLaw X.faithful]
exact (StoppedSamplerLaw.bind_probability S B T P).symm
bounded := by
apply WordProgram.bind_bound X.program N X.bounded _ M
intro a
cases a with
| none => exact Bounded.pure none M
| some a =>
simpa only [continuation, WordProgram.map, Nat.add_zero] using
WordProgram.bind_bound (Y a).program M (Y a).bounded
(fun x => .pure (x.map (fun b => (⟨a,b⟩ : Sigma B)))) 0 (fun _ => Bounded.pure _ 0)
def rejection (w k : ℕ) (hk : k ≤ w) (N : ℕ)
(A : Fin (2^k) → Prop) [DecidablePred A] [Nonempty {a : Fin (2^k) // A a}] :
Realization w N (firstAcceptCoupling A N) where
program := WordRejection.sample w k N A
faithful := WordRejection.sample_probability w k hk N A
bounded := WordRejection.sample_bounded w k N A
end LowSpaceSubsetSum.WordRealization
end
end OAI
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