Section hoas. Import HOAS_Notations.
Definition Eval :term := Eval cbn in
(λ u, !!(ext eva)
(!!mu (λ n, !!(ext doesHaltIn) u n)) u !!I !!I).
End hoas.

Lemma Eval_eval (s t : term) : lambda t -> Eval (ext s) == t -> exists t', ext t' = t /\ eval s t'.