Definition initialized_temp (id : positive) (t : PTree.t (type * bool)) :=
match (t ! id) with
| Some (ty, _) =>
PTree.set id (ty, true) t
| None => t
end.
Fixpoint update_temp (t : PTree.t (type * bool)) (s : statement) :=
match s with
| Sskip | Scontinue | Sbreak => t
| Sassign e1 e2 => t (*already there?*)
| Sset id e2 => (initialized_temp id t)
| Ssequence s1 s2 => let t' := update_temp t s1 in
update_temp t' s2
| Sifthenelse b s1 s2 => join_te (update_temp t s1) (update_temp t s2)
| Sloop _ _ => t
| Sswitch e ls => update_temp_labeled t ls
| Scall (Some id) _ _ => (initialized_temp id t)
| _ => t  (* et cetera *)
end
with update_temp_labeled (t : PTree.t (type * bool)) (ls : labeled_statements) :=
match ls with
| LSnil => t
| LScons _ s ls' =>
join_te (update_temp t s) (update_temp_labeled t ls')
end.

(* Specifications: *)
Lemma initialized_temp_eq : forall t v r gt gs i,
initialized i (mk_tycontext t v r gt gs) = mk_tycontext (initialized_temp i t) v r gt gs.
Lemma update_temp_eq : forall t v r gt gs s,
update_tycon (mk_tycontext t v r gt gs) s = (mk_tycontext (update_temp t s) v r gt gs)
with
update_temp_labeled_eq : forall t v r gt gs s,
join_tycon_labeled s (mk_tycontext t v r gt gs) = (mk_tycontext (update_temp_labeled t s) v r gt gs).