| 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 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 |
|
|
