Fixpoint simplify_domain (d : domain) : domain :=
match d with
| And ld =>
let l := foldr (fun d acc =>
let d' := simplify_domain d in
match d' with
| Always => acc
| And l => cat l acc
| Never => Never :: nil
| _ => d' :: acc
end) nil ld in
match l with
| nil => Always
| d :: nil => d
| _ => And l
end
| Forall e1 e2 d =>
let d' := simplify_domain d in
match d' with
| Always => Always
| Never => Never
| _ => Forall e1 e2 d'
end
| Forone e d =>
let d' := simplify_domain d in
match d' with
| Always => Always
| Never => Never
| _ => Forone e d'
end
| Locally n d =>
let d' := simplify_domain d in
match d' with
| Always => Always
| Never => Never
| _ => Locally n d'
end
| Locally2 m n d =>
let d' := simplify_domain d in
match d' with
| Always => Always
| Never => Never
| _ => Locally2 m n d'
end
| ForallWide n e1 e2 d =>
let d' := simplify_domain d in
match d' with
| Always => Always
| Never => Never
| _ => ForallWide n e1 e2 d'
end
| _ => d
end.

(* Specifications: *)
Lemma simplify_domain_correct :
forall d l,
interp_domain l (simplify_domain d) -> interp_domain l d.