(** *** Search using Choose *)
Section Fix_f.
Variable u : term.
Hypothesis proc_u : proc u.
Hypothesis total_u : forall n, u (enc n) ≡ none \/ exists s, u (enc n) ≡ oenc (Some s).
Definition Re : term := Eval cbv in
.\ "s"; !C (.\ "n"; !u "n" (.\ "t"; !Eq "s" "t") !F).

Lemma H_rec n s : ((lambda (((u 0) (lambda ((Eq (tenc s)) 0))) F)) (enc n) ≡ u (enc n) (lambda (Eq (tenc s) 0)) F).