File size: 3,445 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 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedMeasured
import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedExecution
namespace OAI
/-! # Sequential composition of measured seed generation and a stored action
The generated seed stays resident while the prepared action runs. Generation
work and action work are both charged, including the exhaustion branch.
-/
noncomputable section
namespace LowSpaceSubsetSum.WordSampledAction
open WordSeedRealization WordSeedMeasured
variable {w N : ℕ} {α : Type} [Fintype α] {S : SamplerCoupling α}
structure Result where
answer : Bool
work : ℕ
cells : ℕ
cursor : ℕ
def run {X : Realization w N S} (E : Execution X)
(action : S.Seed → (Bool × ℕ) × ℕ)
(stream : ℕ → Fin (2^w)) (cursor : ℕ) : Result :=
let generated := E.run stream cursor
match generated.accepted with
| none => ⟨false,generated.work+1,max generated.cells 8,generated.cursor⟩
| some accepted =>
let out := action accepted.seed.val
⟨out.1.1,generated.work+out.1.2+1,
max generated.cells (generated.retained+out.2+8),generated.cursor⟩
theorem value {X : Realization w N S} (E : Execution X)
(action : S.Seed → (Bool × ℕ) × ℕ)
(stream : ℕ → Fin (2^w)) (cursor : ℕ) :
(run E action stream cursor).answer =
match (WordProgram.run stream X.program cursor).1 with
| none => false | some s => (action s.val).1.1 := by
have h := congrArg Prod.fst (E.value stream cursor)
rw [←h]
cases hg : (E.run stream cursor).accepted <;>
simp only [run,hg,Option.map_none,Option.map_some]
theorem finite_value {X : Realization w N S} (E : Execution X)
(action : S.Seed → (Bool × ℕ) × ℕ) (words : Fin N → Fin (2^w)) :
(run E action (WordProgram.finiteStream words) 0).answer =
WordSeedRealization.execute X (fun s => (action s).1.1) false words :=
value E action _ 0
theorem cursor_eq {X : Realization w N S} (E : Execution X)
(action : S.Seed → (Bool × ℕ) × ℕ)
(stream : ℕ → Fin (2^w)) (cursor : ℕ) :
(run E action stream cursor).cursor = (WordProgram.run stream X.program cursor).2 := by
have h := congrArg Prod.snd (E.value stream cursor)
rw [←h]
cases hg : (E.run stream cursor).accepted <;> simp only [run,hg]
theorem cursor_le {X : Realization w N S} (E : Execution X)
(action : S.Seed → (Bool × ℕ) × ℕ)
(stream : ℕ → Fin (2^w)) (cursor : ℕ) :
(run E action stream cursor).cursor ≤ cursor+N := by
rw [cursor_eq]
exact (WordProgram.run_bound X.program N X.bounded stream cursor).2
theorem bounded {X : Realization w N S} (E : Execution X)
(action : S.Seed → (Bool × ℕ) × ℕ) (W C R V D : ℕ)
(hE : E.Bounded W C R) (hV : ∀ s, (action s).1.2 ≤ V)
(hD : ∀ s, (action s).2 ≤ D)
(stream : ℕ → Fin (2^w)) (cursor : ℕ) :
(run E action stream cursor).work ≤ W+V+1 ∧
(run E action stream cursor).cells ≤ max C (R+D+8) := by
have he := hE stream cursor
cases ha : (E.run stream cursor).accepted with
| none =>
simp only [run,ha]
exact ⟨by omega,max_le (he.2.1.trans (le_max_left _ _))
((show 8 ≤ R+D+8 by omega).trans (le_max_right _ _))⟩
| some a =>
simp only [run,ha]
have hv := hV a.seed.val
have hd := hD a.seed.val
exact ⟨by omega,max_le_max he.2.1 (by omega)⟩
end LowSpaceSubsetSum.WordSampledAction
end
end OAI
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