src/Pure/sorts.ML
author wenzelm
Thu Feb 23 15:49:40 2012 +0100 (2012-02-23)
changeset 46614 165886a4fe64
parent 45595 fe57d786fd5b
child 47005 421760a1efe7
permissions -rw-r--r--
clarified Graph.restrict (formerly Graph.subgraph) based on public graph operations;
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(*  Title:      Pure/sorts.ML
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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 make: sort list -> sort Ord_List.T
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  val subset: sort Ord_List.T * sort Ord_List.T -> bool
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  val union: sort Ord_List.T -> sort Ord_List.T -> sort Ord_List.T
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  val subtract: sort Ord_List.T -> sort Ord_List.T -> sort Ord_List.T
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  val remove_sort: sort -> sort Ord_List.T -> sort Ord_List.T
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  val insert_sort: sort -> sort Ord_List.T -> sort Ord_List.T
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  val insert_typ: typ -> sort Ord_List.T -> sort Ord_List.T
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  val insert_typs: typ list -> sort Ord_List.T -> sort Ord_List.T
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  val insert_term: term -> sort Ord_List.T -> sort Ord_List.T
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  val insert_terms: term list -> sort Ord_List.T -> sort Ord_List.T
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  type algebra
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  val classes_of: algebra -> serial Graph.T
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  val arities_of: algebra -> (class * sort list) list Symtab.table
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  val all_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 minimal_sorts: algebra -> sort list -> sort Ord_List.T
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  val add_class: Proof.context -> class * class list -> algebra -> algebra
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  val add_classrel: Proof.context -> class * class -> algebra -> algebra
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  val add_arities: Proof.context -> string * (class * sort list) list -> algebra -> algebra
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  val empty_algebra: algebra
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  val merge_algebra: Proof.context -> algebra * algebra -> algebra
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  val subalgebra: Proof.context -> (class -> bool) -> (class * string -> sort list option)
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    -> algebra -> (sort -> sort) * algebra
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  type class_error
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  val class_error: Proof.context -> class_error -> string
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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 meet_sort: algebra -> typ * sort
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    -> sort Vartab.table -> sort Vartab.table   (*exception CLASS_ERROR*)
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  val meet_sort_typ: algebra -> typ * sort -> typ -> typ   (*exception CLASS_ERROR*)
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  val of_sort: algebra -> typ * sort -> bool
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  val of_sort_derivation: algebra ->
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    {class_relation: typ -> 'a * class -> class -> 'a,
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     type_constructor: string * typ list -> ('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 classrel_derivation: algebra ->
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    ('a * class -> class -> 'a) -> 'a * class -> class -> 'a  (*exception CLASS_ERROR*)
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  val witness_sorts: algebra -> string list -> (typ * 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 make = Ord_List.make Term_Ord.sort_ord;
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val subset = Ord_List.subset Term_Ord.sort_ord;
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val union = Ord_List.union Term_Ord.sort_ord;
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val subtract = Ord_List.subtract Term_Ord.sort_ord;
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val remove_sort = Ord_List.remove Term_Ord.sort_ord;
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val insert_sort = Ord_List.insert Term_Ord.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, Ss) of ars represents the arity t::(Ss)c.  "Coregularity" of
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    the arities structure requires that for any two declarations
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    t::(Ss1)c1 and t::(Ss2)c2 such that 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 * sort list) list Symtab.table};
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fun classes_of (Algebra {classes, ...}) = classes;
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fun arities_of (Algebra {arities, ...}) = arities;
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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 super_classes = Graph.immediate_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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fun minimal_sorts algebra raw_sorts =
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  let
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    fun le S1 S2 = sort_le algebra (S1, S2);
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    val sorts = make (map (minimize_sort algebra) raw_sorts);
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  in sorts |> filter_out (fn S => exists (fn S' => le S' S andalso not (le S S')) sorts) end;
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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 ctxt css =
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  error (cat_lines (map (fn cs =>
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    "Cycle in class relation: " ^ Syntax.string_of_classrel ctxt cs) css));
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fun add_class ctxt (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 ctxt 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 ctxt (SOME (c1, c2)) = " for classes " ^ Syntax.string_of_classrel ctxt [c1, c2];
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fun err_conflict ctxt t cc (c, Ss) (c', Ss') =
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  error ("Conflict of type arities" ^ for_classes ctxt cc ^ ":\n  " ^
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    Syntax.string_of_arity ctxt (t, Ss, [c]) ^ " and\n  " ^
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    Syntax.string_of_arity ctxt (t, Ss', [c']));
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fun coregular ctxt algebra t (c, 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 ctxt t (SOME (c1, c2)) (c, Ss) (c', Ss')
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    | NONE => (c, Ss) :: ars)
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  end;
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fun complete algebra (c, Ss) = map (rpair Ss) (c :: super_classes algebra c);
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fun insert ctxt algebra t (c, Ss) ars =
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  (case AList.lookup (op =) ars c of
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    NONE => coregular ctxt algebra t (c, 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)
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      then coregular ctxt algebra t (c, Ss) (remove (op =) (c, Ss') ars)
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      else err_conflict ctxt t NONE (c, Ss) (c, Ss'));
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in
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fun insert_ars ctxt algebra t = fold_rev (insert ctxt algebra t);
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fun insert_complete_ars ctxt 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 (insert_ars ctxt algebra t) (map (complete algebra) ars);
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  in Symtab.update (t, ars') arities end;
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fun add_arities ctxt arg algebra =
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  algebra |> map_arities (insert_complete_ars ctxt algebra arg);
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fun add_arities_table ctxt algebra =
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  Symtab.fold (fn (t, ars) => insert_complete_ars ctxt algebra (t, ars));
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end;
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(* classrel *)
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fun rebuild_arities ctxt algebra = algebra |> map_arities (fn arities =>
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  Symtab.empty
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  |> add_arities_table ctxt algebra arities);
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fun add_classrel ctxt rel = rebuild_arities ctxt 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 ctxt 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 ctxt
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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 ctxt css;
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    val algebra0 = make_algebra (classes', Symtab.empty);
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    val arities' =
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      (case (pointer_eq (classes1, classes2), pointer_eq (arities1, arities2)) of
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        (true, true) => arities1
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      | (true, false) =>  (*no completion*)
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          (arities1, arities2) |> Symtab.join (fn t => fn (ars1, ars2) =>
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            if pointer_eq (ars1, ars2) then raise Symtab.SAME
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            else insert_ars ctxt algebra0 t ars2 ars1)
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      | (false, true) =>  (*unary completion*)
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          Symtab.empty
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          |> add_arities_table ctxt algebra0 arities1
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      | (false, false) => (*binary completion*)
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          Symtab.empty
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          |> add_arities_table ctxt algebra0 arities1
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          |> add_arities_table ctxt algebra0 arities2);
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  in make_algebra (classes', arities') end;
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(* algebra projections *)  (* FIXME potentially violates abstract type integrity *)
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fun subalgebra ctxt 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 t (c, Ss) =
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      if P c then
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        (case sargs (c, t) of
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          SOME sorts =>
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            SOME (c, Ss |> map2 (curry (inter_sort algebra)) sorts |> map restrict_sort)
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        | NONE => NONE)
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      else NONE;
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    val classes' = classes |> Graph.restrict P;
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    val arities' = arities |> Symtab.map (map_filter o restrict_arity);
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  in (restrict_sort, rebuild_arities ctxt (make_algebra (classes', arities'))) end;
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(** sorts of types **)
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(* errors -- performance tuning via delayed message composition *)
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datatype class_error =
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  No_Classrel of class * class |
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  No_Arity of string * class |
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  No_Subsort of sort * sort;
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fun class_error ctxt (No_Classrel (c1, c2)) =
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      "No class relation " ^ Syntax.string_of_classrel ctxt [c1, c2]
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  | class_error ctxt (No_Arity (a, c)) =
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      "No type arity " ^ Syntax.string_of_arity ctxt (a, [], [c])
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  | class_error ctxt (No_Subsort (S1, S2)) =
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      "Cannot derive subsort relation " ^
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        Syntax.string_of_sort ctxt S1 ^ " < " ^ Syntax.string_of_sort ctxt S2;
wenzelm@19529
   333
wenzelm@19578
   334
exception CLASS_ERROR of class_error;
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wenzelm@19578
   336
wenzelm@19578
   337
(* mg_domain *)
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wenzelm@19645
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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 (No_Arity (a, c))
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   345
      | 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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   352
wenzelm@19529
   353
wenzelm@26639
   354
(* meet_sort *)
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wenzelm@26639
   356
fun meet_sort algebra =
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  let
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    fun inters S S' = inter_sort algebra (S, S');
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   359
    fun meet _ [] = I
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   360
      | meet (TFree (_, S)) S' =
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   361
          if sort_le algebra (S, S') then I
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   362
          else raise CLASS_ERROR (No_Subsort (S, S'))
wenzelm@26639
   363
      | meet (TVar (v, S)) S' =
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   364
          if sort_le algebra (S, S') then I
wenzelm@26639
   365
          else Vartab.map_default (v, S) (inters S')
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   366
      | meet (Type (a, Ts)) S = fold2 meet Ts (mg_domain algebra a S);
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   367
  in uncurry meet end;
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   368
haftmann@28665
   369
fun meet_sort_typ algebra (T, S) =
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  let val tab = meet_sort algebra (T, S) Vartab.empty;
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   371
  in Term.map_type_tvar (fn (v, _) => TVar (v, (the o Vartab.lookup tab) v)) end;
haftmann@28665
   372
wenzelm@26639
   373
wenzelm@19529
   374
(* of_sort *)
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   375
wenzelm@19645
   376
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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   380
      | ofS (TVar (_, S), S') = sort_le algebra (S, S')
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   381
      | ofS (Type (a, Ts), S) =
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          let val Ss = mg_domain algebra a S in
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   383
            ListPair.all ofS (Ts, Ss)
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   384
          end handle CLASS_ERROR _ => false;
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  in ofS end;
wenzelm@19529
   386
wenzelm@19529
   387
haftmann@27498
   388
(* animating derivations *)
haftmann@27498
   389
wenzelm@32791
   390
fun of_sort_derivation algebra {class_relation, type_constructor, type_variable} =
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   391
  let
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   392
    val arities = arities_of algebra;
wenzelm@19578
   393
wenzelm@36104
   394
    fun weaken T D1 S2 =
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   395
      let val S1 = map snd D1 in
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   396
        if S1 = S2 then map fst D1
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   397
        else
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   398
          S2 |> map (fn c2 =>
wenzelm@36104
   399
            (case D1 |> find_first (fn (_, c1) => class_le algebra (c1, c2)) of
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   400
              SOME d1 => class_relation T d1 c2
wenzelm@36105
   401
            | NONE => raise CLASS_ERROR (No_Subsort (S1, S2))))
wenzelm@36104
   402
      end;
wenzelm@19529
   403
wenzelm@36103
   404
    fun derive (_, []) = []
wenzelm@44338
   405
      | derive (Type (a, Us), S) =
wenzelm@19529
   406
          let
wenzelm@19645
   407
            val Ss = mg_domain algebra a S;
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   408
            val dom = map2 (fn U => fn S => derive (U, S) ~~ S) Us Ss;
wenzelm@19529
   409
          in
wenzelm@19529
   410
            S |> map (fn c =>
wenzelm@19529
   411
              let
wenzelm@37248
   412
                val Ss' = the (AList.lookup (op =) (Symtab.lookup_list arities a) c);
wenzelm@36102
   413
                val dom' = map (fn ((U, d), S') => weaken U d S' ~~ S') ((Us ~~ dom) ~~ Ss');
wenzelm@37248
   414
              in type_constructor (a, Us) dom' c end)
wenzelm@19529
   415
          end
wenzelm@36103
   416
      | derive (T, S) = weaken T (type_variable T) S;
wenzelm@36103
   417
  in derive end;
wenzelm@19529
   418
wenzelm@35961
   419
fun classrel_derivation algebra class_relation =
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   420
  let
wenzelm@35961
   421
    fun path (x, c1 :: c2 :: cs) = path (class_relation (x, c1) c2, c2 :: cs)
wenzelm@35961
   422
      | path (x, _) = x;
wenzelm@35961
   423
  in
wenzelm@35961
   424
    fn (x, c1) => fn c2 =>
wenzelm@35961
   425
      (case Graph.irreducible_paths (classes_of algebra) (c1, c2) of
wenzelm@36105
   426
        [] => raise CLASS_ERROR (No_Classrel (c1, c2))
wenzelm@35961
   427
      | cs :: _ => path (x, cs))
wenzelm@35961
   428
  end;
wenzelm@35961
   429
wenzelm@19529
   430
wenzelm@19529
   431
(* witness_sorts *)
wenzelm@19529
   432
wenzelm@19645
   433
fun witness_sorts algebra types hyps sorts =
wenzelm@19529
   434
  let
wenzelm@19645
   435
    fun le S1 S2 = sort_le algebra (S1, S2);
wenzelm@31946
   436
    fun get S2 (T, S1) = if le S1 S2 then SOME (T, S2) else NONE;
wenzelm@19645
   437
    fun mg_dom t S = SOME (mg_domain algebra t S) handle CLASS_ERROR _ => NONE;
wenzelm@19529
   438
wenzelm@19578
   439
    fun witn_sort _ [] solved_failed = (SOME (propT, []), solved_failed)
wenzelm@19578
   440
      | witn_sort path S (solved, failed) =
wenzelm@19578
   441
          if exists (le S) failed then (NONE, (solved, failed))
wenzelm@19529
   442
          else
wenzelm@31946
   443
            (case get_first (get S) solved of
wenzelm@19578
   444
              SOME w => (SOME w, (solved, failed))
wenzelm@19529
   445
            | NONE =>
wenzelm@31946
   446
                (case get_first (get S) hyps of
wenzelm@19578
   447
                  SOME w => (SOME w, (w :: solved, failed))
wenzelm@19584
   448
                | NONE => witn_types path types S (solved, failed)))
wenzelm@19529
   449
wenzelm@19578
   450
    and witn_sorts path x = fold_map (witn_sort path) x
wenzelm@19529
   451
wenzelm@19578
   452
    and witn_types _ [] S (solved, failed) = (NONE, (solved, S :: failed))
wenzelm@19578
   453
      | witn_types path (t :: ts) S solved_failed =
wenzelm@19529
   454
          (case mg_dom t S of
wenzelm@19529
   455
            SOME SS =>
wenzelm@19529
   456
              (*do not descend into stronger args (achieving termination)*)
wenzelm@19529
   457
              if exists (fn D => le D S orelse exists (le D) path) SS then
wenzelm@19578
   458
                witn_types path ts S solved_failed
wenzelm@19529
   459
              else
wenzelm@19578
   460
                let val (ws, (solved', failed')) = witn_sorts (S :: path) SS solved_failed in
wenzelm@19529
   461
                  if forall is_some ws then
wenzelm@19529
   462
                    let val w = (Type (t, map (#1 o the) ws), S)
wenzelm@19578
   463
                    in (SOME w, (w :: solved', failed')) end
wenzelm@19578
   464
                  else witn_types path ts S (solved', failed')
wenzelm@19529
   465
                end
wenzelm@19578
   466
          | NONE => witn_types path ts S solved_failed);
wenzelm@19529
   467
wenzelm@19584
   468
  in map_filter I (#1 (witn_sorts [] sorts ([], []))) end;
wenzelm@19529
   469
wenzelm@19514
   470
end;