type t =
  | Relation of string * Term.t list
  | Equal of Term.t * Term.t
  | Bottom
  | Neg of t
  | Conj of t * t
  | Disj of t * t
  | Impl of t * t
  | Iff of t * t
  | Forall of t
  | Exists of t

let rec match_term t phi psi =
  match (phi, psi) with
  | Relation (r1, ts1), Relation (r2, ts2) ->
      if r1 = r2 && List.length ts1 = List.length ts2 then
        List.map2 (Term.match_term t) ts1 ts2
        |> List.fold_left Term.merge_result None
      else MatchErr
  | Equal (t1, s1), Equal (t2, s2) ->
      Term.(merge_result (match_term t t1 t2) (match_term t s1 s2))
  | Bottom, Bottom -> None
  | Neg phi, Neg psi -> match_term t phi psi
  | Conj (phi1, psi1), Conj (phi2, psi2) ->
      Term.merge_result (match_term t phi1 phi2) (match_term t psi1 psi2)
  | Disj (phi1, psi1), Disj (phi2, psi2) ->
      Term.merge_result (match_term t phi1 phi2) (match_term t psi1 psi2)
  | Impl (phi1, psi1), Impl (phi2, psi2) ->
      Term.merge_result (match_term t phi1 phi2) (match_term t psi1 psi2)
  | Iff (phi1, psi1), Iff (phi2, psi2) ->
      Term.merge_result (match_term t phi1 phi2) (match_term t psi1 psi2)
  | Forall phi, Forall psi -> match_term (Term.inc_var t) phi psi
  | Exists phi, Exists psi -> match_term (Term.inc_var t) phi psi
  | _ -> MatchErr

let rec occurs t = function
  | Relation (_, ts) -> List.exists (Term.occurs t) ts
  | Equal (t1, t2) -> Term.occurs t t1 || Term.occurs t t2
  | Bottom -> false
  | Neg phi -> occurs t phi
  | Conj (phi, psi) -> occurs t phi || occurs t psi
  | Disj (phi, psi) -> occurs t phi || occurs t psi
  | Impl (phi, psi) -> occurs t phi || occurs t psi
  | Iff (phi, psi) -> occurs t phi || occurs t psi
  | Forall phi -> occurs (Term.inc_var t) phi
  | Exists phi -> occurs (Term.inc_var t) phi