File size: 3,332 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedRealization
import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordRunLaw
namespace OAI
/-! # Running the prepared action on actual word-generated seeds -/
noncomputable section
namespace LowSpaceSubsetSum.WordSeedRealization
variable {w N : ℕ} {α β : Type} [Fintype α] {S : SamplerCoupling α}
def run (X : Realization w N S) (words : Fin N → Fin (2^w)) :=
WordProgram.run (WordProgram.finiteStream words) X.program 0
theorem run_reads (X : Realization w N S) (words : Fin N → Fin (2^w)) :
(run X words).2 ≤ N := by
simpa only [run,Nat.zero_add] using
(WordProgram.run_bound X.program N X.bounded (WordProgram.finiteStream words) 0).2
theorem run_probability (X : Realization w N S) (P : Option α → Prop) :
(FiniteLaw.uniform : FiniteLaw (Fin N → Fin (2^w))).probability
(fun words => P (observe S (run X words).1)) = S.boundedLaw.probability P := by
unfold run
exact (WordProgram.run_probability X.program N X.bounded
(fun s => P (observe S s))).symm.trans
((WordProgram.probability_map (observe S) X.program P).symm.trans (X.faithful P))
def execute (X : Realization w N S) (action : S.Seed → β) (failure : β)
(words : Fin N → Fin (2^w)) : β :=
match (run X words).1 with
| none => failure
| some s => action s.val
/-- The action's value (which may itself contain its work counter) and
the separate number of fair words read. This does not charge continuations
as free machine operations. -/
def executeCounted (X : Realization w N S) (action : S.Seed → β) (failure : β)
(words : Fin N → Fin (2^w)) : β × ℕ :=
let out := run X words
(match out.1 with | none => failure | some s => action s.val,out.2)
theorem executeCounted_value (X : Realization w N S) (action : S.Seed → β) (failure : β)
(words : Fin N → Fin (2^w)) :
(executeCounted X action failure words).1 = execute X action failure words := rfl
theorem executeCounted_reads (X : Realization w N S) (action : S.Seed → β) (failure : β)
(words : Fin N → Fin (2^w)) : (executeCounted X action failure words).2 ≤ N :=
run_reads X words
theorem execute_success (X : Realization w N S) (action : S.Seed → β) (failure : β)
(words : Fin N → Fin (2^w)) (s : SuccessfulSeed S)
(h : (run X words).1 = some s) : execute X action failure words = action s.val := by
simp only [execute,h]
theorem probability_success (X : Realization w N S) (test : S.Seed → Bool)
(P : α → Prop)
(h : ∀ s a, S.bounded s = some a → P a → test s = true) :
S.boundedLaw.probability (fun a => ∃ x, a = some x ∧ P x) ≤
(FiniteLaw.uniform : FiniteLaw (Fin N → Fin (2^w))).probability
(fun words => execute X test false words = true) := by
rw [←run_probability X (fun a => ∃ x, a = some x ∧ P x)]
apply FiniteLaw.probability_mono
intro words hw
cases hs : (run X words).1 with
| none =>
simp only [hs,observe,Option.bind_none,reduceCtorEq,false_and,exists_false] at hw
| some s =>
obtain ⟨a,ha,hpa⟩ := hw
have hb : S.bounded s.val = some a := by
simpa only [hs,observe,Option.bind_some] using ha
exact (execute_success X test false words s hs).trans (h s.val a hb hpa)
end LowSpaceSubsetSum.WordSeedRealization
end
end OAI
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