(* Boolean test selecting formulas which describe a constructible set, *)
(* i.e. formulas without quantifiers.                                  *)
(* The quantifier elimination check. *)
(* Boolean holds predicate for quantifier free formulas *)
Definition qf_eval e := fix loop (f : formula R) : bool :=
match f with
| Bool b => b
| t1 == t2 => (eval e t1 == eval e t2)%bool
| Unit t1 => eval e t1 \in unit
| f1 /\ f2 => loop f1 && loop f2
| f1 \/ f2 => loop f1 || loop f2
| f1 ==> f2 => (loop f1 ==> loop f2)%bool
| ~ f1 => ~~ loop f1
|_ => false
end%T.

(* qf_eval is equivalent to holds *)
Lemma qf_evalP e f : qf_form f -> reflect (holds e f) (qf_eval e f).
Implicit Type bc : seq (term R) * seq (term R).