src/Pure/sorts.ML
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(*  Title:      Pure/sorts.ML
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    ID:         $Id$
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    Author:     Markus Wenzel and Stefan Berghofer, TU Muenchen
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The order-sorted algebra of type classes.
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Classes denote (possibly empty) collections of types that are
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partially ordered by class inclusion. They are represented
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symbolically by strings.
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Sorts are intersections of finitely many classes. They are represented
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by lists of classes.  Normal forms of sorts are sorted lists of
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minimal classes (wrt. current class inclusion).
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*)
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signature SORTS =
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sig
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  val union: sort list -> sort list -> sort list
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  val subtract: sort list -> sort list -> sort list
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  val remove_sort: sort -> sort list -> sort list
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  val insert_sort: sort -> sort list -> sort list
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  val insert_typ: typ -> sort list -> sort list
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  val insert_typs: typ list -> sort list -> sort list
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  val insert_term: term -> sort list -> sort list
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  val insert_terms: term list -> sort list -> sort list
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  type algebra
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  val rep_algebra: algebra ->
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   {classes: serial Graph.T,
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    arities: (class * (class * sort list)) list Symtab.table}
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  val all_classes: algebra -> class list
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  val minimal_classes: algebra -> class list
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  val super_classes: algebra -> class -> class list
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  val class_less: algebra -> class * class -> bool
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  val class_le: algebra -> class * class -> bool
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  val sort_eq: algebra -> sort * sort -> bool
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  val sort_le: algebra -> sort * sort -> bool
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  val sorts_le: algebra -> sort list * sort list -> bool
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  val inter_sort: algebra -> sort * sort -> sort
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  val minimize_sort: algebra -> sort -> sort
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  val complete_sort: algebra -> sort -> sort
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  val certify_class: algebra -> class -> class    (*exception TYPE*)
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  val certify_sort: algebra -> sort -> sort       (*exception TYPE*)
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  val add_class: Pretty.pp -> class * class list -> algebra -> algebra
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  val add_classrel: Pretty.pp -> class * class -> algebra -> algebra
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  val add_arities: Pretty.pp -> string * (class * sort list) list -> algebra -> algebra
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  val empty_algebra: algebra
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  val merge_algebra: Pretty.pp -> algebra * algebra -> algebra
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  val subalgebra: Pretty.pp -> (class -> bool) -> (class * string -> sort list)
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    -> algebra -> (sort -> sort) * algebra
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  type class_error
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  val msg_class_error: Pretty.pp -> class_error -> string
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  val class_error: Pretty.pp -> class_error -> 'a
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  exception CLASS_ERROR of class_error
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  val mg_domain: algebra -> string -> sort -> sort list   (*exception CLASS_ERROR*)
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  val of_sort: algebra -> typ * sort -> bool
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  val of_sort_derivation: Pretty.pp -> algebra ->
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    {class_relation: 'a * class -> class -> 'a,
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     type_constructor: string -> ('a * class) list list -> class -> 'a,
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     type_variable: typ -> ('a * class) list} ->
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    typ * sort -> 'a list   (*exception CLASS_ERROR*)
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  val witness_sorts: algebra -> string list -> sort list -> sort list -> (typ * sort) list
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end;
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structure Sorts: SORTS =
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struct
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(** ordered lists of sorts **)
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val op union = OrdList.union Term.sort_ord;
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val subtract = OrdList.subtract Term.sort_ord;
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val remove_sort = OrdList.remove Term.sort_ord;
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val insert_sort = OrdList.insert Term.sort_ord;
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fun insert_typ (TFree (_, S)) Ss = insert_sort S Ss
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  | insert_typ (TVar (_, S)) Ss = insert_sort S Ss
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  | insert_typ (Type (_, Ts)) Ss = insert_typs Ts Ss
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and insert_typs [] Ss = Ss
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  | insert_typs (T :: Ts) Ss = insert_typs Ts (insert_typ T Ss);
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fun insert_term (Const (_, T)) Ss = insert_typ T Ss
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  | insert_term (Free (_, T)) Ss = insert_typ T Ss
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  | insert_term (Var (_, T)) Ss = insert_typ T Ss
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  | insert_term (Bound _) Ss = Ss
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  | insert_term (Abs (_, T, t)) Ss = insert_term t (insert_typ T Ss)
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  | insert_term (t $ u) Ss = insert_term t (insert_term u Ss);
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fun insert_terms [] Ss = Ss
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  | insert_terms (t :: ts) Ss = insert_terms ts (insert_term t Ss);
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(** order-sorted algebra **)
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(*
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  classes: graph representing class declarations together with proper
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    subclass relation, which needs to be transitive and acyclic.
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  arities: table of association lists of all type arities; (t, ars)
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    means that type constructor t has the arities ars; an element
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    (c, (c0, Ss)) of ars represents the arity t::(Ss)c being derived
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    via c0 <= c.  "Coregularity" of the arities structure requires
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    that for any two declarations t::(Ss1)c1 and t::(Ss2)c2 such that
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    c1 <= c2 holds Ss1 <= Ss2.
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*)
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datatype algebra = Algebra of
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 {classes: serial Graph.T,
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  arities: (class * (class * sort list)) list Symtab.table};
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fun rep_algebra (Algebra args) = args;
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val classes_of = #classes o rep_algebra;
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val arities_of = #arities o rep_algebra;
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fun make_algebra (classes, arities) =
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  Algebra {classes = classes, arities = arities};
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fun map_classes f (Algebra {classes, arities}) = make_algebra (f classes, arities);
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fun map_arities f (Algebra {classes, arities}) = make_algebra (classes, f arities);
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(* classes *)
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fun all_classes (Algebra {classes, ...}) = Graph.all_preds classes (Graph.maximals classes);
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val minimal_classes = Graph.minimals o classes_of;
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val super_classes = Graph.imm_succs o classes_of;
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(* class relations *)
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val class_less = Graph.is_edge o classes_of;
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fun class_le algebra (c1, c2) = c1 = c2 orelse class_less algebra (c1, c2);
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(* sort relations *)
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fun sort_le algebra (S1, S2) =
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  S1 = S2 orelse forall (fn c2 => exists (fn c1 => class_le algebra (c1, c2)) S1) S2;
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fun sorts_le algebra (Ss1, Ss2) =
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  ListPair.all (sort_le algebra) (Ss1, Ss2);
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fun sort_eq algebra (S1, S2) =
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  sort_le algebra (S1, S2) andalso sort_le algebra (S2, S1);
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(* intersection *)
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fun inter_class algebra c S =
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  let
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    fun intr [] = [c]
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      | intr (S' as c' :: c's) =
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          if class_le algebra (c', c) then S'
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          else if class_le algebra (c, c') then intr c's
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          else c' :: intr c's
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  in intr S end;
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fun inter_sort algebra (S1, S2) =
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  sort_strings (fold (inter_class algebra) S1 S2);
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(* normal forms *)
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fun minimize_sort _ [] = []
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  | minimize_sort _ (S as [_]) = S
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  | minimize_sort algebra S =
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      filter (fn c => not (exists (fn c' => class_less algebra (c', c)) S)) S
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      |> sort_distinct string_ord;
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fun complete_sort algebra =
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  Graph.all_succs (classes_of algebra) o minimize_sort algebra;
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(* certify *)
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fun certify_class algebra c =
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  if can (Graph.get_node (classes_of algebra)) c then c
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  else raise TYPE ("Undeclared class: " ^ quote c, [], []);
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fun certify_sort classes = minimize_sort classes o map (certify_class classes);
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(** build algebras **)
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(* classes *)
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fun err_dup_class c = error ("Duplicate declaration of class: " ^ quote c);
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fun err_cyclic_classes pp css =
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  error (cat_lines (map (fn cs =>
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    "Cycle in class relation: " ^ Pretty.string_of_classrel pp cs) css));
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fun add_class pp (c, cs) = map_classes (fn classes =>
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  let
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    val classes' = classes |> Graph.new_node (c, serial ())
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      handle Graph.DUP dup => err_dup_class dup;
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    val classes'' = classes' |> fold Graph.add_edge_trans_acyclic (map (pair c) cs)
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      handle Graph.CYCLES css => err_cyclic_classes pp css;
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  in classes'' end);
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(* arities *)
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local
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fun for_classes _ NONE = ""
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  | for_classes pp (SOME (c1, c2)) =
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      " for classes " ^ Pretty.string_of_classrel pp [c1, c2];
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fun err_conflict pp t cc (c, Ss) (c', Ss') =
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  error ("Conflict of type arities" ^ for_classes pp cc ^ ":\n  " ^
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    Pretty.string_of_arity pp (t, Ss, [c]) ^ " and\n  " ^
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    Pretty.string_of_arity pp (t, Ss', [c']));
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fun coregular pp algebra t (c, (c0, Ss)) ars =
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  let
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    fun conflict (c', (_, Ss')) =
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      if class_le algebra (c, c') andalso not (sorts_le algebra (Ss, Ss')) then
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        SOME ((c, c'), (c', Ss'))
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      else if class_le algebra (c', c) andalso not (sorts_le algebra (Ss', Ss)) then
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        SOME ((c', c), (c', Ss'))
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      else NONE;
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  in
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    (case get_first conflict ars of
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      SOME ((c1, c2), (c', Ss')) => err_conflict pp t (SOME (c1, c2)) (c, Ss) (c', Ss')
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    | NONE => (c, (c0, Ss)) :: ars)
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  end;
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fun complete algebra (c0, Ss) = map (rpair (c0, Ss)) (c0 :: super_classes algebra c0);
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fun insert pp algebra t (c, (c0, Ss)) ars =
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  (case AList.lookup (op =) ars c of
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    NONE => coregular pp algebra t (c, (c0, Ss)) ars
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  | SOME (_, Ss') =>
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      if sorts_le algebra (Ss, Ss') then ars
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      else if sorts_le algebra (Ss', Ss) then
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        coregular pp algebra t (c, (c0, Ss))
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          (filter_out (fn (c'', (_, Ss'')) => c = c'' andalso Ss'' = Ss') ars)
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      else err_conflict pp t NONE (c, Ss) (c, Ss'));
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fun insert_ars pp algebra (t, ars) arities =
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  let val ars' =
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    Symtab.lookup_list arities t
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    |> fold_rev (fold_rev (insert pp algebra t)) (map (complete algebra) ars)
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  in Symtab.update (t, ars') arities end;
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in
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fun add_arities pp arg algebra = algebra |> map_arities (insert_ars pp algebra arg);
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fun add_arities_table pp algebra =
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  Symtab.fold (fn (t, ars) => insert_ars pp algebra (t, map snd ars));
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end;
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(* classrel *)
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fun rebuild_arities pp algebra = algebra |> map_arities (fn arities =>
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  Symtab.empty
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  |> add_arities_table pp algebra arities);
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fun add_classrel pp rel = rebuild_arities pp o map_classes (fn classes =>
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  classes |> Graph.add_edge_trans_acyclic rel
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    handle Graph.CYCLES css => err_cyclic_classes pp css);
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(* empty and merge *)
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val empty_algebra = make_algebra (Graph.empty, Symtab.empty);
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fun merge_algebra pp
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   (Algebra {classes = classes1, arities = arities1},
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    Algebra {classes = classes2, arities = arities2}) =
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  let
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    val classes' = Graph.merge_trans_acyclic (op =) (classes1, classes2)
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      handle Graph.DUP c => err_dup_class c
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          | Graph.CYCLES css => err_cyclic_classes pp css;
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    val algebra0 = make_algebra (classes', Symtab.empty);
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    val arities' = Symtab.empty
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      |> add_arities_table pp algebra0 arities1
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      |> add_arities_table pp algebra0 arities2;
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  in make_algebra (classes', arities') end;
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(* subalgebra *)
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fun subalgebra pp P sargs (algebra as Algebra {classes, arities}) =
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  let
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    val restrict_sort = minimize_sort algebra o filter P o Graph.all_succs classes;
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    fun restrict_arity tyco (c, (_, Ss)) =
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      if P c then
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        SOME (c, (c, Ss |> map2 (curry (inter_sort algebra)) (sargs (c, tyco))
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          |> map restrict_sort))
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      else NONE;
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    val classes' = classes |> Graph.subgraph P;
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    val arities' = arities |> Symtab.map' (map_filter o restrict_arity);
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  in (restrict_sort, rebuild_arities pp (make_algebra (classes', arities'))) end;
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(** sorts of types **)
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(* errors *)
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datatype class_error = NoClassrel of class * class | NoArity of string * class;
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fun msg_class_error pp (NoClassrel (c1, c2)) =
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      "No class relation " ^ Pretty.string_of_classrel pp [c1, c2]
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  | msg_class_error pp (NoArity (a, c)) =
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      "No type arity " ^ Pretty.string_of_arity pp (a, [], [c]);
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fun class_error pp = error o msg_class_error pp;
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exception CLASS_ERROR of class_error;
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(* mg_domain *)
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fun mg_domain algebra a S =
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  let
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    val arities = arities_of algebra;
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    fun dom c =
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      (case AList.lookup (op =) (Symtab.lookup_list arities a) c of
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        NONE => raise CLASS_ERROR (NoArity (a, c))
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      | SOME (_, Ss) => Ss);
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    fun dom_inter c Ss = ListPair.map (inter_sort algebra) (dom c, Ss);
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  in
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    (case S of
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      [] => raise Fail "Unknown domain of empty intersection"
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    | c :: cs => fold dom_inter cs (dom c))
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  end;
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(* of_sort *)
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fun of_sort algebra =
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  let
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    fun ofS (_, []) = true
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      | ofS (TFree (_, S), S') = sort_le algebra (S, S')
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      | ofS (TVar (_, S), S') = sort_le algebra (S, S')
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      | ofS (Type (a, Ts), S) =
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          let val Ss = mg_domain algebra a S in
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            ListPair.all ofS (Ts, Ss)
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          end handle CLASS_ERROR _ => false;
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  in ofS end;
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(* of_sort_derivation *)
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fun of_sort_derivation pp algebra {class_relation, type_constructor, type_variable} =
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  let
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    val {classes, arities} = rep_algebra algebra;
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    fun weaken_path (x, c1 :: c2 :: cs) =
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          weaken_path (class_relation (x, c1) c2, c2 :: cs)
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      | weaken_path (x, _) = x;
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    fun weaken (x, c1) c2 =
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      (case Graph.irreducible_paths classes (c1, c2) of
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        [] => raise CLASS_ERROR (NoClassrel (c1, c2))
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      | cs :: _ => weaken_path (x, cs));
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    fun weakens S1 S2 = S2 |> map (fn c2 =>
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      (case S1 |> find_first (fn (_, c1) => class_le algebra (c1, c2)) of
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        SOME d1 => weaken d1 c2
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      | NONE => error ("Cannot derive subsort relation " ^
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          Pretty.string_of_sort pp (map #2 S1) ^ " < " ^ Pretty.string_of_sort pp S2)));
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    fun derive _ [] = []
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      | derive (Type (a, Ts)) S =
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          let
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            val Ss = mg_domain algebra a S;
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            val dom = map2 (fn T => fn S => derive T S ~~ S) Ts Ss;
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          in
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            S |> map (fn c =>
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              let
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                val (c0, Ss') = the (AList.lookup (op =) (Symtab.lookup_list arities a) c);
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                val dom' = map2 (fn d => fn S' => weakens d S' ~~ S') dom Ss';
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              in weaken (type_constructor a dom' c0, c0) c end)
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          end
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      | derive T S = weakens (type_variable T) S;
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  in uncurry derive end;
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(* witness_sorts *)
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fun witness_sorts algebra types hyps sorts =
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  let
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    fun le S1 S2 = sort_le algebra (S1, S2);
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    fun get_solved S2 (T, S1) = if le S1 S2 then SOME (T, S2) else NONE;
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    fun get_hyp S2 S1 = if le S1 S2 then SOME (TFree ("'hyp", S1), S2) else NONE;
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    fun mg_dom t S = SOME (mg_domain algebra t S) handle CLASS_ERROR _ => NONE;
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    fun witn_sort _ [] solved_failed = (SOME (propT, []), solved_failed)
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      | witn_sort path S (solved, failed) =
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          if exists (le S) failed then (NONE, (solved, failed))
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          else
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            (case get_first (get_solved S) solved of
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              SOME w => (SOME w, (solved, failed))
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            | NONE =>
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                (case get_first (get_hyp S) hyps of
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                  SOME w => (SOME w, (w :: solved, failed))
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                | NONE => witn_types path types S (solved, failed)))
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    and witn_sorts path x = fold_map (witn_sort path) x
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    and witn_types _ [] S (solved, failed) = (NONE, (solved, S :: failed))
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      | witn_types path (t :: ts) S solved_failed =
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          (case mg_dom t S of
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            SOME SS =>
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              (*do not descend into stronger args (achieving termination)*)
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              if exists (fn D => le D S orelse exists (le D) path) SS then
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                witn_types path ts S solved_failed
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              else
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                let val (ws, (solved', failed')) = witn_sorts (S :: path) SS solved_failed in
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                  if forall is_some ws then
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                    let val w = (Type (t, map (#1 o the) ws), S)
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                    in (SOME w, (w :: solved', failed')) end
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                  else witn_types path ts S (solved', failed')
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                end
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          | NONE => witn_types path ts S solved_failed);
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  in map_filter I (#1 (witn_sorts [] sorts ([], []))) end;
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end;