Section ProcessSemantics.
Variable this : nid.
Fixpoint step (W : world) A (s1 : state) (p1 : proc this W A)
sc (s2 : state) (p2 : proc this W A) : Prop :=
match sc, p1 with
(* Action - make a step *)
| ActStep, Act a => exists v pf, @a_step _ _ _ a s1 pf s2 v /\ p2 = Ret v
(* Sequencing - apply a continuation *)
| SeqRet, Seq _ (Ret v) k => s2 = s1 /\ p2 = k v
| SeqStep sc', Seq _ p' k1 =>
exists p'', step s1 p' sc' s2 p'' /\ p2 = Seq p'' k1
(* Injection of a non-reduced term *)
| InjectRet, Inject V K pf (Ret v) =>
exists s1', [/\ s2 = s1, p2 = Ret v & extends pf s1 s1']
| InjectStep sc', Inject V K pf t1' =>
exists s1' s2' s t2',
[/\ p2 = Inject pf t2', s1 = s1' \+ s, s2 = s2' \+ s,
s1' \In Coh V & step s1' t1' sc' s2' t2']
(* Imposing an inductive invariant on a non-reduced term *)
| WithInvRet, WithInv p inv ii pf (Ret v) =>
exists s1', [/\ s2 = s1, p2 = Ret v & s1 = s1']
| WithInvStep sc', WithInv p inv ii pf t1' =>
exists t2', p2 = WithInv p inv ii pf t2' /\
step s1 t1' sc' s2 t2'
| _, _ => False
end.
Fixpoint good (W : world) A (p : proc this W A) sc  : Prop :=
match sc, p with
| ActStep, Act _ => True
| SeqRet, Seq _ (Ret _) _ => True
| SeqStep sc', Seq _ p' _ => good p' sc'
| InjectStep sc', Inject _ _ _ p' => good p' sc'
| InjectRet, Inject _ _ _ (Ret _) => True
| WithInvStep sc', WithInv _ _ _ _ p' => good p' sc'
| WithInvRet, WithInv _ _ _ _ (Ret _) => True
| _, _ => False
end.
(*  [Safety in small-step semantics]  The safety (in order to make the following step) with respect to the schedule is defined inductively on the shape of the program and the schedule. Omitting the schedule is not a good idea, at it's required in order to "sequentialize" the execution of the program structure. Once it's dropped, this structure is lost.  *)
Fixpoint safe (W : world) A (p : proc this W A) sc (s : state)  : Prop :=
match sc, p with
| ActStep, Act a => a_safe a s
| SeqRet, Seq _ (Ret _) _ => True
| SeqStep sc', Seq _ p' _ => safe p' sc' s
| InjectStep sc', Inject V K pf p' =>
exists s', extends pf s s' /\ safe p' sc' s'
| InjectRet, Inject V K pf (Ret _) => exists s', extends pf s s'
| WithInvStep sc', WithInv _ _ _ _ p' => safe p' sc' s
| WithInvRet, WithInv _ _ _ _ (Ret _) => True
| _, _ => True
end.
Definition pstep (W : world) A s1 (p1 : proc this W A) sc s2 p2 :=
[/\ s1 \In Coh W, safe p1 sc s1 & step s1 p1 sc s2 p2].
(* Some sanity lemmas wrt. stepping *)
(* Some view lemmas for processes and corresponding schedules *)

Lemma stepInject V W K A (em : injects V W K)
s1 (t : proc this V A) sc s2 (q : proc this W A) :
pstep s1 (Inject em t) sc s2 q <->
(exists s1' v, [/\ sc = InjectRet, t = Ret v, q = Ret v, s2 = s1 &
extends em s1 s1']) \/
exists sc' t' s1' s2' s,
[/\ sc = InjectStep sc', q = Inject em t',
s1 = s1' \+ s, s2 = s2' \+ s, s1 \In Coh W &
pstep s1' t sc' s2' t'].