File size: 4,175 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.RegisterControl
namespace OAI
namespace MatchingFPRAS.RegisterStructured
open RegisterProc TreeData TreeMachine RegisterControl
variable {R : Type} [DecidableEq R]
inductive Select where | test | left | right | halt deriving DecidableEq, Inhabited
instance : Fintype Select := ⟨{.test,.left,.right,.halt},by intro s; cases s <;> simp⟩
def selectCode (i : R) (P Q : Proc (K R)) : Select → RegisterControl.Instr R Select
| .test => .branch i .left .right
| .left => .call P .halt
| .right => .call Q .halt
| .halt => .halt
def select (i : R) (P Q : Proc (K R)) : Proc (K R) := compile (selectCode i P Q)
lemma select_nil (i : R) (P Q : Proc (K R)) (E F : Env R) (n : ℕ)
(hi : E i=.nil) (h : Exec P (encodeEnv E) (encodeEnv F) n) :
Exec (select i P Q) (encodeEnv E) (encodeEnv F) (n+3) := by
have hh : RegisterControl.Runs (selectCode i P Q) (some Select.halt) F F 1 :=
Runs.step _ _ _ _ _ 0 1 (Effect.halt _) (Runs.done _)
have hl : RegisterControl.Runs (selectCode i P Q) (some Select.left) E F ((n+1)+1) :=
Runs.step _ _ _ _ _ _ _ (Effect.call _ _ _ _ n h) hh
have ht : RegisterControl.Runs (selectCode i P Q) (some Select.test) E F (1+((n+1)+1)) :=
Runs.step _ _ _ _ _ _ _ (Effect.nilBranch E i _ _ hi) hl
exact (compile_exec _ _ _ _ ht).mono (by omega)
lemma select_cons (i : R) (P Q : Proc (K R)) (E F : Env R) (n : ℕ) (x y : Data)
(hi : E i=.cons x y) (h : Exec Q (encodeEnv E) (encodeEnv F) n) :
Exec (select i P Q) (encodeEnv E) (encodeEnv F) (n+3) := by
have hh : RegisterControl.Runs (selectCode i P Q) (some Select.halt) F F 1 :=
Runs.step _ _ _ _ _ 0 1 (Effect.halt _) (Runs.done _)
have hr : RegisterControl.Runs (selectCode i P Q) (some Select.right) E F ((n+1)+1) :=
Runs.step _ _ _ _ _ _ _ (Effect.call _ _ _ _ n h) hh
have ht : RegisterControl.Runs (selectCode i P Q) (some Select.test) E F (1+((n+1)+1)) :=
Runs.step _ _ _ _ _ _ _ (Effect.consBranch E i _ _ x y hi) hr
exact (compile_exec _ _ _ _ ht).mono (by omega)
inductive Loop where | test | body | halt deriving DecidableEq, Inhabited
instance : Fintype Loop := ⟨{.test,.body,.halt},by intro l; cases l <;> simp⟩
def loopCode (i : R) (P : Proc (K R)) : Loop → RegisterControl.Instr R Loop
| .test => .branch i .halt .body
| .body => .call P .test
| .halt => .halt
def loop (i : R) (P : Proc (K R)) : Proc (K R) := compile (loopCode i P)
lemma loop_exec (i : R) (P : Proc (K R)) (E : ℕ → Env R) (n B : ℕ)
(hc : ∀ k<n, E k i ≠ .nil)
(hb : ∀ k<n, Exec P (encodeEnv (E k)) (encodeEnv (E (k+1))) B)
(he : E n i=.nil) :
Exec (loop i P) (encodeEnv (E 0)) (encodeEnv (E n)) (n*(B+2)+2) := by
have aux : ∀ n (E : ℕ → Env R), (∀ k<n,E k i≠.nil) →
(∀ k<n,Exec P (encodeEnv (E k)) (encodeEnv (E (k+1))) B) → E n i=.nil →
RegisterControl.Runs (loopCode i P) (some Loop.test) (E 0) (E n) (n*(B+2)+2) := by
intro n
induction n with
| zero =>
intro E _nonempty _body he
have hh : RegisterControl.Runs (loopCode i P) (some Loop.halt) (E 0) (E 0) 1 :=
Runs.step _ _ _ _ _ 0 1 (Effect.halt _) (Runs.done _)
simpa only [Nat.zero_mul,Nat.zero_add] using RegisterControl.Runs.step (M:=loopCode i P) Loop.test (E 0) (some Loop.halt) (E 0) (E 0) 1 1 (Effect.nilBranch _ i Loop.halt Loop.body he) hh
| succ n ih =>
intro E hc hb he
have ht := ih (fun k=>E (k+1)) (fun k hk=>hc (k+1) (by omega))
(fun k hk=>hb (k+1) (by omega)) he
have hr : RegisterControl.Runs (loopCode i P) (some Loop.body) (E 0) (E (n+1))
((B+1)+(n*(B+2)+2)) :=
Runs.step _ _ _ _ _ _ _ (Effect.call _ _ _ _ B (hb 0 (by omega))) ht
cases hx : E 0 i with
| nil => exact False.elim (hc 0 (by omega) hx)
| cons x y =>
have hh : RegisterControl.Runs (loopCode i P) (some Loop.test) (E 0) (E (n+1))
(1+((B+1)+(n*(B+2)+2))) :=
Runs.step _ _ _ _ _ _ _ (Effect.consBranch _ i _ _ x y hx) hr
convert hh using 1; ring
exact compile_exec _ _ _ _ (aux n E hc hb he)
end MatchingFPRAS.RegisterStructured
end OAI
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