(* Identifiers and addresses are just drawn from the naturals.*)
Definition ident := nat.
Definition addr  := nat.
(* Values are either integers or pointers (addresses) *)
Inductive val : Set :=
| int_val : nat -> val
| ptr_val : addr -> val.
Definition locals := ShareMap.map ident val.
Instance Join_locals: Join locals := ShareMap.Join_map _ _ (Join_equiv _).
Instance locals_sa : Sep_alg locals := ShareMap.sa_map _ _ _.
(* We characterize expressions as functions from local variable environments whcih are monotone with respect to increasing the environment. *)
Definition expr :=
{ f: locals -> option val |
forall l l' v, join_sub l l' -> f l = Some v -> f l' = Some v }.
Definition mem := ShareMap.map addr val.
Instance Join_mem: Join mem := ShareMap.Join_map _ _ (Join_equiv _).
Instance mem_sa : Sep_alg mem := ShareMap.sa_map _ _ _.
(* The syntax of commands.  These are pretty basic imperative language constructs. *)
Inductive cmd :=
| skip
| seq : cmd -> cmd -> cmd
| call : ident -> ident -> list expr -> cmd
| ret : expr -> cmd
| load : ident -> expr -> cmd
| store : expr -> expr -> cmd
| assign : ident -> expr -> cmd
| cif : expr -> cmd -> cmd -> cmd
| while : expr -> cmd -> cmd.
(* A function declaration consists of a list of formal arguments (local identifiers which will receive the actual arguments), a list of local variables together with their initial values, and a command implementing the function body.  Finally we include a syntactic validity fact -- the formals and declared locals are disjoint. *)
Inductive fun_decl :=
{ fnd_formals : list ident
; fnd_locals  : list (ident * val)
; fnd_cmd     : cmd
; fnd_valid   : NoDup (fnd_formals ++ map (@fst _ _) fnd_locals)
}.
(* A program is a mapping from function identifiers to function declarations. *)
Definition program_unit := nat -> option fun_decl.
(** Operational semantics **)
(* Controls capture the control stack of imperative languages.  kseq c k       means "execute c, then run k" kcall i rho k  means "when a return is executed, unwind to this point, restore local varibles rho, and update i with the return value" kext i f vs k  means "I tried to call funcion f with arguments vs, but f isn't defined" knil           means "safely halted" *)
Inductive ctl :=
| kseq : cmd -> ctl -> ctl
| kcall : ident -> locals -> ctl -> ctl
| kext  : ident -> ident -> list val -> ctl -> ctl
| knil : ctl.
(* We define a small step semantics which manipulate states of this form.*)
Definition state := (ctl * locals * mem)%type.
(* In local environment rho, the list of expressions es have values vs *)
Fixpoint evaluate_exprs (rho:locals) (es:list expr) (vs:list val) : Prop :=
match es, vs with
| e::es', v::vs' => proj1_sig e rho = Some v /\ evaluate_exprs rho es' vs'
| nil, nil => True
| _, _ => False
end.
(* the funtion make_locals builds the initial local variable environment at function entry *)
Section function_entry_locals.
Variable vals:list val.
Variable fd:fun_decl.
Let actuals := combine (fnd_formals fd) vals.
Definition make_locals := ShareMap.build_map _ _ _ (actuals ++ fnd_locals fd).
End function_entry_locals.
(* unwind_return drops all the leading kseq constructions to finde the correct function return position *)
Fixpoint unwind_return (k:ctl) : option (ident * locals * ctl):=
match k with
| kseq _ k' => unwind_return k'
| kcall i rho k => Some (i,rho,k)
| kext _ _ _ _ => None
| knil => None
end.
(* Main small step semantics definition. We send state (k,rho,m) to state (k',rho',m') when we execute command c (in program p). *)
Inductive step (pu:program_unit) :
forall (k:ctl) (rho:locals) (m:mem)
(c:cmd)
(k':ctl) (rho':locals) (m':mem), Prop :=
| step_skip : forall k rho m,
step pu  k rho m
skip
k rho m
| step_seq : forall k rho m c1 c2,
step pu k rho m
(seq c1 c2)
(kseq c1 (kseq c2 k)) rho m
| step_if_false : forall k rho m e c1 c2,
proj1_sig e rho = Some (int_val 0) ->
step pu k rho m
(cif e c1 c2)
(kseq c2 k) rho m
| step_if_true : forall k rho m x e c1 c2,
x <> 0 ->
proj1_sig e rho = Some (int_val x) ->
step pu k rho m
(cif e c1 c2)
(kseq c1 k) rho m
| step_while_false : forall k rho m e c,
proj1_sig e rho = Some (int_val 0) ->
step pu k rho m
(while e c)
k rho m
| step_while_true : forall k rho m e c x,
x <> 0 ->
proj1_sig e rho = Some (int_val x) ->
step pu k rho m
(while e c)
(kseq c (kseq (while e c) k)) rho m
| step_ret : forall k rho m e v i rho' k' rho'',
proj1_sig e rho = Some v ->
unwind_return k = (Some (i,rho',k')) ->
ShareMap.map_upd _ _ _ i v rho' = Some rho'' ->
step pu k rho m
(ret e)
k' rho''  m
| step_call_internal : forall k rho m x f exps vals fd,
evaluate_exprs rho exps vals ->
pu f = Some fd ->
step pu k rho m
(call x f exps)
(kseq (fnd_cmd fd) (kcall x rho k))
(make_locals vals fd) m
| step_call_external : forall k rho m x f exps vals,
pu f = None ->
evaluate_exprs rho exps vals ->
step pu k rho m
(call x f exps)
(kext x f vals k) rho m
| step_assign : forall k rho rho' m i e v,
proj1_sig e rho = Some v ->
ShareMap.map_upd _ _ _ i v rho = Some rho' ->
step pu k rho m
(assign i e)
k rho' m
| step_load : forall k rho rho' m i e a v,
proj1_sig e rho = Some (ptr_val a) ->
ShareMap.map_val _ _ a m = Some v ->
ShareMap.map_upd _ _ _ i v rho = Some rho' ->
step pu k rho m
(load i e)
k rho' m
| step_store : forall k rho m e1 a e2 v m',
proj1_sig e1 rho = Some (ptr_val a) ->
proj1_sig e2 rho = Some v ->
ShareMap.map_upd _ _ _ a v m = Some m' ->
step pu k rho m
(store e1 e2)
k rho m'.
(* Pull off the outermost command and execute it *)
Inductive step' (pu:program_unit) :
forall (st st':state), Prop :=
| step'_step : forall c k rho m k' rho' m',
step  pu k rho m c k' rho' m' ->
step' pu (kseq c k,rho,m) (k',rho',m').

(* Evaluating a list of expressions is functional. *)
Lemma evaluate_exprs_fun : forall rho exps vals1 vals2,
evaluate_exprs rho exps vals1 ->
evaluate_exprs rho exps vals2 ->
vals1 = vals2.