include "utilities/extralib.ma".
include "common/IOMonad.ma".
include "common/Integers.ma".
include "common/Events.ma".
include "common/Mem.ma".

record execstep (outty:Type[0]) (inty:outty → Type[0]) : Type[1] ≝
{ genv  : Type[0]
; state : Type[0]
; is_final : state → option int
; mem_of_state : state → mem
; step : genv → state → IO outty inty (trace×state)
}.

let rec repeat (n:nat) (outty:Type[0]) (inty:outty → Type[0]) (exec:execstep outty inty)
               (g:genv ?? exec) (s:state ?? exec)
 : IO outty inty (trace × (state ?? exec)) ≝
match n with
[ O ⇒ Value ??? 〈E0, s〉
| S n' ⇒ ! 〈t1,s1〉 ← step ?? exec g s;
         repeat n' ?? exec g s1
].

let rec trace_map (A,B:Type[0]) (f:A → res (trace × B))
                  (l:list A) on l : res (trace × (list B)) ≝
match l with
[ nil ⇒ OK ? 〈E0, [ ]〉
| cons h t ⇒
    do 〈tr,h'〉 ← f h;
    do 〈tr',t'〉 ← trace_map … f t;
    OK ? 〈tr ⧺ tr',h'::t'〉
].

(* A (possibly non-terminating) execution.   *)
coinductive execution (state:Type[0]) (output:Type[0]) (input:output → Type[0]) : Type[0] ≝
| e_stop : trace → int → mem → execution state output input
| e_step : trace → state → execution state output input → execution state output input
| e_wrong : errmsg → execution state output input
| e_interact : ∀o:output. (input o → execution state output input) → execution state output input.

(* This definition is slightly awkward because of the need to provide resumptions.
   It records the last trace/state passed in, then recursively processes the next
   state. *)

let corec exec_inf_aux (output:Type[0]) (input:output → Type[0])
                       (exec:execstep output input) (ge:genv ?? exec)
                       (s:IO output input (trace×(state ?? exec)))
                       : execution ??? ≝
match s with
[ Wrong m ⇒ e_wrong ??? m
| Value v ⇒ match v with [ pair t s' ⇒ 
    match is_final ?? exec s' with
    [ Some r ⇒ e_stop ??? t r (mem_of_state ?? exec s')
    | None ⇒ e_step ??? t s' (exec_inf_aux ??? ge (step ?? exec ge s')) ] ]
| Interact out k' ⇒ e_interact ??? out (λv. exec_inf_aux ??? ge (k' v))
].

lemma execution_cases: ∀o,i,s.∀e:execution s o i.
 e = match e with [ e_stop tr r m ⇒ e_stop ??? tr r m
 | e_step tr s e ⇒ e_step ??? tr s e
 | e_wrong m ⇒ e_wrong ??? m | e_interact o k ⇒ e_interact ??? o k ].
#o #i #s #e cases e; //; qed.

axiom exec_inf_aux_unfold: ∀o,i,exec,ge,s. exec_inf_aux o i exec ge s =
match s with
[ Wrong m ⇒ e_wrong ??? m
| Value v ⇒ match v with [ pair t s' ⇒ 
    match is_final ?? exec s' with
    [ Some r ⇒ e_stop ??? t r (mem_of_state ?? exec s')
    | None ⇒ e_step ??? t s' (exec_inf_aux ??? ge (step ?? exec ge s')) ] ]
| Interact out k' ⇒ e_interact ??? out (λv. exec_inf_aux ??? ge (k' v))
].
(*
#exec #ge #s >(execution_cases ? (exec_inf_aux …)) cases s
[ #o #k
| #x cases x #tr #s' (* XXX Can't unfold exec_inf_aux here *)
| ] 
whd in ⊢ (??%%); //;
qed.
*)

record fullexec (outty:Type[0]) (inty:outty → Type[0]) : Type[1] ≝
{ es1 :> execstep outty inty
; program : Type[0]
; make_initial_state : program → res (genv ?? es1 × (state ?? es1))
}.

definition exec_inf : ∀o,i.∀fx:fullexec o i. ∀p:program ?? fx. execution (state ?? fx) o i ≝
λo,i,fx,p.
  match make_initial_state ?? fx p with 
  [ OK gs ⇒ exec_inf_aux ?? fx (\fst gs) (Value … 〈E0,\snd gs〉)
  | Error m ⇒ e_wrong ??? m
  ].


record execstep' (outty:Type[0]) (inty:outty → Type[0]) : Type[1] ≝
{ es0 :> execstep outty inty
; initial : state ?? es0 → Prop
}.


alias symbol "and" (instance 2) = "logical and".
record related_semantics : Type[1] ≝
{ output : Type[0]
; input : output → Type[0]
; sem1 : execstep' output input
; sem2 : execstep' output input
; ge1 : genv ?? sem1
; ge2 : genv ?? sem2
; match_states : state ?? sem1 → state ?? sem2 → Prop
; match_initial_states : ∀st1. (initial ?? sem1) st1 → ∃st2. (initial ?? sem2) st2 ∧ match_states st1 st2
; match_final_states : ∀st1,st2,r. match_states st1 st2 → (is_final ?? sem1) st1 = Some ? r → (is_final ?? sem2) st2 = Some ? r
}.



record simulation : Type[1] ≝ 
{ sems :> related_semantics
; sim : ∀st1,st2,t,st1'.
        P_io' ?? (trace × (state ?? (sem1 sems))) (λr. r = 〈t,st1'〉) (step ?? (sem1 sems) (ge1 sems) st1) →
        match_states sems st1 st2 → 
        ∃st2':(state ?? (sem2 sems)).(∃n:nat.P_io' ?? (trace × (state ?? (sem2 sems))) (λr. r = 〈t,st2'〉) (repeat n ?? (sem2 sems) (ge2 sems) st2)) ∧
          match_states sems st1' st2'
}.