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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 eq_set: sort list * sort list -> bool
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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 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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{classrel: 'a * class -> class -> 'a,
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constructor: string -> ('a * class) list list -> class -> 'a,
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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 eq_set = OrdList.eq_set Term.sort_ord;
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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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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 form *)
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fun norm_sort _ [] = []
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| norm_sort _ (S as [_]) = S
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| norm_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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(* 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 = norm_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_classes cs =
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error ("Duplicate declaration of class(es): " ^ commas_quote cs);
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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_classes [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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wenzelm@19645
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279 |
handle Graph.DUPS cs => err_dup_classes cs
|
wenzelm@19645
|
280 |
| Graph.CYCLES css => err_cyclic_classes pp css;
|
wenzelm@19645
|
281 |
val algebra0 = make_algebra (classes', Symtab.empty);
|
wenzelm@19645
|
282 |
val arities' = Symtab.empty
|
wenzelm@19645
|
283 |
|> add_arities_table pp algebra0 arities1
|
wenzelm@19645
|
284 |
|> add_arities_table pp algebra0 arities2;
|
wenzelm@19645
|
285 |
in make_algebra (classes', arities') end;
|
wenzelm@19645
|
286 |
|
wenzelm@21933
|
287 |
|
wenzelm@21933
|
288 |
(* subalgebra *)
|
wenzelm@21933
|
289 |
|
haftmann@22181
|
290 |
fun subalgebra pp P sargs (algebra as Algebra {classes, arities}) =
|
haftmann@19952
|
291 |
let
|
wenzelm@21933
|
292 |
val restrict_sort = norm_sort algebra o filter P o Graph.all_succs classes;
|
haftmann@22181
|
293 |
fun restrict_arity tyco (c, (_, Ss)) =
|
haftmann@22181
|
294 |
if P c then
|
haftmann@22181
|
295 |
SOME (c, (c, Ss |> map2 (curry (inter_sort algebra)) (sargs (c, tyco))
|
haftmann@22181
|
296 |
|> map restrict_sort))
|
haftmann@22181
|
297 |
else NONE;
|
wenzelm@21933
|
298 |
val classes' = classes |> Graph.subgraph P;
|
haftmann@22181
|
299 |
val arities' = arities |> Symtab.map' (map_filter o restrict_arity);
|
wenzelm@21933
|
300 |
in (restrict_sort, rebuild_arities pp (make_algebra (classes', arities'))) end;
|
haftmann@20465
|
301 |
|
wenzelm@19645
|
302 |
|
wenzelm@2956
|
303 |
|
wenzelm@2956
|
304 |
(** sorts of types **)
|
wenzelm@2956
|
305 |
|
wenzelm@19578
|
306 |
(* errors *)
|
wenzelm@19578
|
307 |
|
wenzelm@19578
|
308 |
datatype class_error = NoClassrel of class * class | NoArity of string * class;
|
wenzelm@19578
|
309 |
|
haftmann@22196
|
310 |
fun msg_class_error pp (NoClassrel (c1, c2)) =
|
haftmann@22196
|
311 |
"No class relation " ^ Pretty.string_of_classrel pp [c1, c2]
|
haftmann@22196
|
312 |
| msg_class_error pp (NoArity (a, c)) =
|
haftmann@22196
|
313 |
"No type arity " ^ Pretty.string_of_arity pp (a, [], [c]);
|
haftmann@22196
|
314 |
|
haftmann@22196
|
315 |
fun class_error pp = error o msg_class_error pp;
|
wenzelm@19578
|
316 |
|
wenzelm@19578
|
317 |
exception CLASS_ERROR of class_error;
|
wenzelm@19578
|
318 |
|
wenzelm@19578
|
319 |
|
wenzelm@7643
|
320 |
(* mg_domain *)
|
wenzelm@7643
|
321 |
|
wenzelm@19645
|
322 |
fun mg_domain algebra a S =
|
wenzelm@16881
|
323 |
let
|
wenzelm@19645
|
324 |
val arities = arities_of algebra;
|
wenzelm@16881
|
325 |
fun dom c =
|
wenzelm@18931
|
326 |
(case AList.lookup (op =) (Symtab.lookup_list arities a) c of
|
wenzelm@19578
|
327 |
NONE => raise CLASS_ERROR (NoArity (a, c))
|
wenzelm@19524
|
328 |
| SOME (_, Ss) => Ss);
|
wenzelm@19645
|
329 |
fun dom_inter c Ss = ListPair.map (inter_sort algebra) (dom c, Ss);
|
wenzelm@16881
|
330 |
in
|
wenzelm@16881
|
331 |
(case S of
|
wenzelm@19529
|
332 |
[] => raise Fail "Unknown domain of empty intersection"
|
wenzelm@16881
|
333 |
| c :: cs => fold dom_inter cs (dom c))
|
wenzelm@16881
|
334 |
end;
|
wenzelm@2956
|
335 |
|
wenzelm@2956
|
336 |
|
wenzelm@2990
|
337 |
(* of_sort *)
|
wenzelm@2990
|
338 |
|
wenzelm@19645
|
339 |
fun of_sort algebra =
|
wenzelm@2990
|
340 |
let
|
wenzelm@2990
|
341 |
fun ofS (_, []) = true
|
wenzelm@19645
|
342 |
| ofS (TFree (_, S), S') = sort_le algebra (S, S')
|
wenzelm@19645
|
343 |
| ofS (TVar (_, S), S') = sort_le algebra (S, S')
|
wenzelm@2990
|
344 |
| ofS (Type (a, Ts), S) =
|
wenzelm@19645
|
345 |
let val Ss = mg_domain algebra a S in
|
wenzelm@2990
|
346 |
ListPair.all ofS (Ts, Ss)
|
wenzelm@19578
|
347 |
end handle CLASS_ERROR _ => false;
|
wenzelm@2990
|
348 |
in ofS end;
|
wenzelm@2990
|
349 |
|
wenzelm@2990
|
350 |
|
wenzelm@19529
|
351 |
(* of_sort_derivation *)
|
wenzelm@2956
|
352 |
|
wenzelm@19645
|
353 |
fun of_sort_derivation pp algebra {classrel, constructor, variable} =
|
wenzelm@19529
|
354 |
let
|
wenzelm@19645
|
355 |
val {classes, arities} = rep_algebra algebra;
|
haftmann@19952
|
356 |
fun weaken_path (x, c1 :: c2 :: cs) =
|
haftmann@19952
|
357 |
weaken_path (classrel (x, c1) c2, c2 :: cs)
|
wenzelm@19578
|
358 |
| weaken_path (x, _) = x;
|
wenzelm@19578
|
359 |
fun weaken (x, c1) c2 =
|
wenzelm@19578
|
360 |
(case Graph.irreducible_paths classes (c1, c2) of
|
wenzelm@19578
|
361 |
[] => raise CLASS_ERROR (NoClassrel (c1, c2))
|
wenzelm@19578
|
362 |
| cs :: _ => weaken_path (x, cs));
|
wenzelm@19578
|
363 |
|
wenzelm@19529
|
364 |
fun weakens S1 S2 = S2 |> map (fn c2 =>
|
wenzelm@19645
|
365 |
(case S1 |> find_first (fn (_, c1) => class_le algebra (c1, c2)) of
|
wenzelm@19529
|
366 |
SOME d1 => weaken d1 c2
|
wenzelm@19529
|
367 |
| NONE => error ("Cannot derive subsort relation " ^
|
wenzelm@19529
|
368 |
Pretty.string_of_sort pp (map #2 S1) ^ " < " ^ Pretty.string_of_sort pp S2)));
|
wenzelm@19529
|
369 |
|
wenzelm@19529
|
370 |
fun derive _ [] = []
|
wenzelm@19529
|
371 |
| derive (Type (a, Ts)) S =
|
wenzelm@19529
|
372 |
let
|
wenzelm@19645
|
373 |
val Ss = mg_domain algebra a S;
|
wenzelm@19529
|
374 |
val dom = map2 (fn T => fn S => derive T S ~~ S) Ts Ss;
|
wenzelm@19529
|
375 |
in
|
wenzelm@19529
|
376 |
S |> map (fn c =>
|
wenzelm@19529
|
377 |
let
|
wenzelm@19529
|
378 |
val (c0, Ss') = the (AList.lookup (op =) (Symtab.lookup_list arities a) c);
|
wenzelm@19529
|
379 |
val dom' = map2 (fn d => fn S' => weakens d S' ~~ S') dom Ss';
|
wenzelm@19529
|
380 |
in weaken (constructor a dom' c0, c0) c end)
|
wenzelm@19529
|
381 |
end
|
wenzelm@19529
|
382 |
| derive T S = weakens (variable T) S;
|
wenzelm@19529
|
383 |
in uncurry derive end;
|
wenzelm@19529
|
384 |
|
wenzelm@19529
|
385 |
|
wenzelm@19529
|
386 |
(* witness_sorts *)
|
wenzelm@2956
|
387 |
|
wenzelm@19645
|
388 |
fun witness_sorts algebra types hyps sorts =
|
wenzelm@7643
|
389 |
let
|
wenzelm@19645
|
390 |
fun le S1 S2 = sort_le algebra (S1, S2);
|
skalberg@15531
|
391 |
fun get_solved S2 (T, S1) = if le S1 S2 then SOME (T, S2) else NONE;
|
skalberg@15531
|
392 |
fun get_hyp S2 S1 = if le S1 S2 then SOME (TFree ("'hyp", S1), S2) else NONE;
|
wenzelm@19645
|
393 |
fun mg_dom t S = SOME (mg_domain algebra t S) handle CLASS_ERROR _ => NONE;
|
wenzelm@7643
|
394 |
|
wenzelm@19578
|
395 |
fun witn_sort _ [] solved_failed = (SOME (propT, []), solved_failed)
|
wenzelm@19578
|
396 |
| witn_sort path S (solved, failed) =
|
wenzelm@19578
|
397 |
if exists (le S) failed then (NONE, (solved, failed))
|
wenzelm@7643
|
398 |
else
|
wenzelm@7643
|
399 |
(case get_first (get_solved S) solved of
|
wenzelm@19578
|
400 |
SOME w => (SOME w, (solved, failed))
|
skalberg@15531
|
401 |
| NONE =>
|
wenzelm@7643
|
402 |
(case get_first (get_hyp S) hyps of
|
wenzelm@19578
|
403 |
SOME w => (SOME w, (w :: solved, failed))
|
wenzelm@19584
|
404 |
| NONE => witn_types path types S (solved, failed)))
|
wenzelm@7643
|
405 |
|
wenzelm@19578
|
406 |
and witn_sorts path x = fold_map (witn_sort path) x
|
wenzelm@7643
|
407 |
|
wenzelm@19578
|
408 |
and witn_types _ [] S (solved, failed) = (NONE, (solved, S :: failed))
|
wenzelm@19578
|
409 |
| witn_types path (t :: ts) S solved_failed =
|
wenzelm@7643
|
410 |
(case mg_dom t S of
|
skalberg@15531
|
411 |
SOME SS =>
|
wenzelm@7643
|
412 |
(*do not descend into stronger args (achieving termination)*)
|
wenzelm@7643
|
413 |
if exists (fn D => le D S orelse exists (le D) path) SS then
|
wenzelm@19578
|
414 |
witn_types path ts S solved_failed
|
wenzelm@7643
|
415 |
else
|
wenzelm@19578
|
416 |
let val (ws, (solved', failed')) = witn_sorts (S :: path) SS solved_failed in
|
wenzelm@17756
|
417 |
if forall is_some ws then
|
wenzelm@18931
|
418 |
let val w = (Type (t, map (#1 o the) ws), S)
|
wenzelm@19578
|
419 |
in (SOME w, (w :: solved', failed')) end
|
wenzelm@19578
|
420 |
else witn_types path ts S (solved', failed')
|
wenzelm@7643
|
421 |
end
|
wenzelm@19578
|
422 |
| NONE => witn_types path ts S solved_failed);
|
wenzelm@7643
|
423 |
|
wenzelm@19584
|
424 |
in map_filter I (#1 (witn_sorts [] sorts ([], []))) end;
|
wenzelm@2956
|
425 |
|
wenzelm@2956
|
426 |
end;
|