src/Pure/axclass.ML
author wenzelm
Sun, 30 Apr 2006 22:50:05 +0200
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renamed add_axclass(_i) to define_axclass(_i); renamed get_info to get_definition; added axiomatize_class/classrel/arity (supercede Sign.add_classes/classrel/arities); tuned;
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(*  Title:      Pure/axclass.ML
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    ID:         $Id$
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    Author:     Markus Wenzel, TU Muenchen
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Type classes as parameter records and predicates, with explicit
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definitions and proofs.
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*)
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signature AX_CLASS =
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sig
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  val get_definition: theory -> class -> {def: thm, intro: thm, axioms: thm list}
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  val class_intros: theory -> thm list
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  val params_of: theory -> class -> string list
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  val all_params_of: theory -> sort -> string list
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  val print_axclasses: theory -> unit
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  val cert_classrel: theory -> class * class -> class * class
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  val read_classrel: theory -> xstring * xstring -> class * class
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  val add_classrel: thm -> theory -> theory
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  val add_arity: thm -> theory -> theory
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  val prove_classrel: class * class -> tactic -> theory -> theory
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  val prove_arity: string * sort list * sort -> tactic -> theory -> theory
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  val define_class: bstring * xstring list -> string list ->
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    ((bstring * Attrib.src list) * string list) list -> theory -> class * theory
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  val define_class_i: bstring * class list -> string list ->
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    ((bstring * attribute list) * term list) list -> theory -> class * theory
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  val axiomatize_class: bstring * xstring list -> theory -> theory
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  val axiomatize_class_i: bstring * class list -> theory -> theory
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  val axiomatize_classrel: (xstring * xstring) list -> theory -> theory
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  val axiomatize_classrel_i: (class * class) list -> theory -> theory
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  val axiomatize_arity: xstring * string list * string -> theory -> theory
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  val axiomatize_arity_i: arity -> theory -> theory
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  val of_sort: theory -> typ * sort -> thm list
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end;
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structure AxClass: AX_CLASS =
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struct
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(** theory data **)
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(* class parameters (canonical order) *)
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type param = string * class;
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fun add_param pp ((x, c): param) params =
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  (case AList.lookup (op =) params x of
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    NONE => (x, c) :: params
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  | SOME c' => error ("Duplicate class parameter " ^ quote x ^
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      " for " ^ Pretty.string_of_sort pp [c] ^
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      (if c = c' then "" else " and " ^ Pretty.string_of_sort pp [c'])));
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fun merge_params _ ([], qs) = qs
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  | merge_params pp (ps, qs) =
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      fold_rev (fn q => if member (op =) ps q then I else add_param pp q) qs ps;
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(* axclasses *)
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val introN = "intro";
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val superN = "super";
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val axiomsN = "axioms";
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datatype axclass = AxClass of
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 {def: thm,
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  intro: thm,
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  axioms: thm list};
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type axclasses = axclass Symtab.table * param list;
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fun make_axclass (def, intro, axioms) =
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  AxClass {def = def, intro = intro, axioms = axioms};
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fun merge_axclasses pp ((tab1, params1), (tab2, params2)) : axclasses =
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  (Symtab.merge (K true) (tab1, tab2), merge_params pp (params1, params2));
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(* instances *)
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val classrelN = "classrel";
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val arityN = "arity";
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datatype instances = Instances of
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 {classes: unit Graph.T,                 (*raw relation -- no closure!*)
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  classrel: ((class * class) * thm) list,
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  arities: ((class * sort list) * thm) list Symtab.table,
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  types: (class * thm) list Typtab.table};
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fun make_instances (classes, classrel, arities, types) =
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  Instances {classes = classes, classrel = classrel, arities = arities, types = types};
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fun map_instances f (Instances {classes, classrel, arities, types}) =
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  make_instances (f (classes, classrel, arities, types));
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fun merge_instances
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   (Instances {classes = classes1, classrel = classrel1, arities = arities1, types = types1},
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    Instances {classes = classes2, classrel = classrel2, arities = arities2, types = types2}) =
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  make_instances
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   (Graph.merge (K true) (classes1, classes2),
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    merge (eq_fst op =) (classrel1, classrel2),
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    Symtab.join (K (merge (eq_fst op =))) (arities1, arities2),
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    Typtab.join (K (merge (eq_fst op =))) (types1, types2));
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(* setup data *)
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structure AxClassData = TheoryDataFun
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(struct
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  val name = "Pure/axclass";
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  type T = axclasses * instances ref;
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  val empty : T =
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    ((Symtab.empty, []), ref (make_instances (Graph.empty, [], Symtab.empty, Typtab.empty)));
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  fun copy (axclasses, ref instances) : T = (axclasses, ref instances);
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  val extend = copy;
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  fun merge pp ((axclasses1, ref instances1), (axclasses2, ref instances2)) =
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    (merge_axclasses pp (axclasses1, axclasses2), ref (merge_instances (instances1, instances2)));
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  fun print _ _ = ();
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end);
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val _ = Context.add_setup AxClassData.init;
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(* retrieve axclasses *)
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val lookup_def = Symtab.lookup o #1 o #1 o AxClassData.get;
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fun get_definition thy c =
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  (case lookup_def thy c of
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    SOME (AxClass info) => info
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  | NONE => error ("Undefined type class " ^ quote c));
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fun class_intros thy =
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  let
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    fun add_intro c =
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      (case lookup_def thy c of SOME (AxClass {intro, ...}) => cons intro | _ => I);
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    val classes = Sign.classes thy;
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  in map (Thm.class_triv thy) classes @ fold add_intro classes [] end;
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(* retrieve parameters *)
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fun get_params thy pred =
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  let val params = #2 (#1 (AxClassData.get thy))
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  in fold (fn (x, c) => if pred c then cons x else I) params [] end;
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fun params_of thy c = get_params thy (fn c' => c' = c);
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fun all_params_of thy S = get_params thy (fn c => Sign.subsort thy (S, [c]));
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(* maintain instances *)
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val get_instances = AxClassData.get #> (fn (_, ref (Instances insts)) => insts);
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fun store_instance f thy (x, th) =
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  let
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    val th' = Drule.standard' th;
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    val _ = change (#2 (AxClassData.get thy)) (map_instances (f (x, th')));
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  in th' end;
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val store_classrel = store_instance (fn ((c1, c2), th) => fn (classes, classrel, arities, types) =>
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  (classes
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    |> Graph.default_node (c1, ())
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    |> Graph.default_node (c2, ())
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    |> Graph.add_edge (c1, c2),
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    insert (eq_fst op =) ((c1, c2), th) classrel, arities, types));
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val store_arity = store_instance (fn ((t, Ss, c), th) => fn (classes, classrel, arities, types) =>
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  (classes, classrel, arities |> Symtab.insert_list (eq_fst op =) (t, ((c, Ss), th)), types));
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val store_type = store_instance (fn ((T, c), th) => fn (classes, classrel, arities, types) =>
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  (classes, classrel, arities, types |> Typtab.insert_list (eq_fst op =) (T, (c, th))));
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(* print data *)
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fun print_axclasses thy =
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  let
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    val axclasses = #1 (#1 (AxClassData.get thy));
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    val ctxt = ProofContext.init thy;
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    fun pretty_axclass (class, AxClass {def, intro, axioms}) =
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      Pretty.block (Pretty.fbreaks
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       [Pretty.block
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          [Pretty.str "class ", ProofContext.pretty_sort ctxt [class], Pretty.str ":"],
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        Pretty.strs ("parameters:" :: params_of thy class),
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        ProofContext.pretty_fact ctxt ("def", [def]),
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        ProofContext.pretty_fact ctxt (introN, [intro]),
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        ProofContext.pretty_fact ctxt (axiomsN, axioms)]);
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  in Pretty.writeln (Pretty.chunks (map pretty_axclass (Symtab.dest axclasses))) end;
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(** instances **)
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(* class relations *)
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fun cert_classrel thy raw_rel =
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  let
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    val (c1, c2) = pairself (Sign.certify_class thy) raw_rel;
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    val _ = Type.add_classrel (Sign.pp thy) (c1, c2) (Sign.tsig_of thy);
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    val _ =
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      (case subtract (op =) (all_params_of thy [c1]) (all_params_of thy [c2]) of
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        [] => ()
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      | xs => raise TYPE ("Class " ^ Sign.string_of_sort thy [c1] ^ " lacks parameter(s) " ^
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          commas_quote xs ^ " of " ^ Sign.string_of_sort thy [c2], [], []));
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  in (c1, c2) end;
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fun read_classrel thy raw_rel =
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  cert_classrel thy (pairself (Sign.read_class thy) raw_rel)
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    handle TYPE (msg, _, _) => error msg;
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(* primitive rules *)
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fun add_classrel th thy =
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  let
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    fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);
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    val prop = Drule.plain_prop_of (Thm.transfer thy th);
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    val rel = Logic.dest_classrel prop handle TERM _ => err ();
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    val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();
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    val thy' = thy |> Sign.primitive_classrel (c1, c2);
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    val _ = store_classrel thy' ((c1, c2), Drule.unconstrainTs th);
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  in thy' end;
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fun add_arity th thy =
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  let
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    val prop = Drule.plain_prop_of (Thm.transfer thy th);
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    val (t, Ss, c) = Logic.dest_arity prop handle TERM _ =>
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      raise THM ("add_arity: malformed type arity", 0, [th]);
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    val thy' = thy |> Sign.primitive_arity (t, Ss, [c]);
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    val _ = store_arity thy' ((t, Ss, c), Drule.unconstrainTs th);
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  in thy' end;
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(* tactical proofs *)
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fun prove_classrel raw_rel tac thy =
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  let
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    val (c1, c2) = cert_classrel thy raw_rel;
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    val th = Goal.prove thy [] [] (Logic.mk_classrel (c1, c2)) (fn _ => tac) handle ERROR msg =>
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      cat_error msg ("The error(s) above occurred while trying to prove class relation " ^
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        quote (Sign.string_of_classrel thy [c1, c2]));
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  in
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    thy
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    |> PureThy.add_thms [((classrelN ^ "_" ^ serial_string (), th), [])]
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    |-> (fn [th'] => add_classrel th')
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  end;
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fun prove_arity raw_arity tac thy =
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  let
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    val arity = Sign.cert_arity thy raw_arity;
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    val props = Logic.mk_arities arity;
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    val ths = Goal.prove_multi thy [] [] props
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      (fn _ => Tactic.precise_conjunction_tac (length props) 1 THEN tac) handle ERROR msg =>
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        cat_error msg ("The error(s) above occurred while trying to prove type arity " ^
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          quote (Sign.string_of_arity thy arity));
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  in
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    thy
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    |> PureThy.add_thms (ths |> map (fn th => ((arityN ^ "_" ^ serial_string (), th), [])))
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    |-> fold add_arity
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  end;
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(** class definitions **)
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local
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fun def_class prep_class prep_att prep_propp
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    (bclass, raw_super) params raw_specs thy =
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  let
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    val ctxt = ProofContext.init thy;
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    val pp = ProofContext.pp ctxt;
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    (* prepare specification *)
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    val bconst = Logic.const_of_class bclass;
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    val class = Sign.full_name thy bclass;
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    val super = map (prep_class thy) raw_super |> Sign.certify_sort thy;
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    fun prep_axiom t =
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      (case Term.add_tfrees t [] of
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        [(a, S)] =>
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          if Sign.subsort thy (super, S) then t
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          else error ("Sort constraint of type variable " ^
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            setmp show_sorts true (Pretty.string_of_typ pp) (TFree (a, S)) ^
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            " needs to be weaker than " ^ Pretty.string_of_sort pp super)
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      | [] => t
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      | _ => error ("Multiple type variables in class axiom:\n" ^ Pretty.string_of_term pp t))
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      |> map_term_types (Term.map_atyps (fn TFree _ => Term.aT [] | U => U))
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      |> Logic.close_form;
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    val axiomss = prep_propp (ctxt, map (map (rpair ([], [])) o snd) raw_specs)
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      |> snd |> map (map (prep_axiom o fst));
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    val name_atts = Attrib.map_specs (prep_att thy) raw_specs |> map fst;
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    (* definition *)
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    val conjs = map (curry Logic.mk_inclass (Term.aT [])) super @ flat axiomss;
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    val class_eq =
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      Logic.mk_equals (Logic.mk_inclass (Term.aT [], class), Logic.mk_conjunction_list conjs);
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    val ([def], def_thy) =
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      thy
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      |> Sign.primitive_class (bclass, super)
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      |> PureThy.add_defs_i false [((Thm.def_name bconst, class_eq), [])];
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    val (raw_intro, (raw_classrel, raw_axioms)) =
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      (Conjunction.split_defined (length conjs) def) ||> chop (length super);
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    (* facts *)
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    val class_triv = Thm.class_triv def_thy class;
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    val ([(_, [intro]), (_, classrel), (_, axioms)], facts_thy) =
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      def_thy
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      |> PureThy.note_thmss_qualified "" bconst
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        [((introN, []), [([Drule.standard raw_intro], [])]),
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         ((superN, []), [(map Drule.standard raw_classrel, [])]),
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         ((axiomsN, []), [(map (fn th => Drule.standard (class_triv RS th)) raw_axioms, [])])];
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    val _ = map (store_classrel facts_thy) (map (pair class) super ~~ classrel);
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    (* result *)
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    val result_thy =
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      facts_thy
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      |> Sign.add_path bconst
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      |> PureThy.note_thmss_i "" (name_atts ~~ map Thm.simple_fact (unflat axiomss axioms)) |> snd
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      |> Sign.restore_naming facts_thy
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      |> AxClassData.map (apfst (fn (axclasses, parameters) =>
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        (Symtab.update (class, make_axclass (def, intro, axioms)) axclasses,
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          fold (fn x => add_param pp (x, class)) params parameters)));
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  in (class, result_thy) end;
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in
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val define_class = def_class Sign.read_class Attrib.attribute ProofContext.read_propp;
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val define_class_i = def_class Sign.certify_class (K I) ProofContext.cert_propp;
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end;
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(** axiomatizations **)
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local
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fun axiomatize kind add prep arg thy =
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  let val specs = arg |> prep thy |> map (fn prop => ((kind ^ "_" ^ serial_string (), prop), []))
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  in thy |> PureThy.add_axioms_i specs |-> fold add end;
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fun ax_classrel prep =
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  axiomatize classrelN add_classrel (fn thy => map (prep thy #> Logic.mk_classrel));
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fun ax_arity prep =
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  axiomatize arityN add_arity (fn thy => prep thy #> Logic.mk_arities);
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fun ax_class prep_class prep_classrel (bclass, raw_super) thy =
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  let
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    val class = Sign.full_name thy bclass;
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    val super = map (prep_class thy) raw_super |> Sign.certify_sort thy;
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  in
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    thy
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    |> Sign.primitive_class (bclass, super)
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    |> ax_classrel prep_classrel (map (fn c => (class, c)) super)
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  end;
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in
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val axiomatize_class = ax_class Sign.read_class read_classrel;
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val axiomatize_class_i = ax_class Sign.certify_class cert_classrel;
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val axiomatize_classrel = ax_classrel read_classrel;
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val axiomatize_classrel_i = ax_classrel cert_classrel;
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val axiomatize_arity = ax_arity Sign.read_arity;
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val axiomatize_arity_i = ax_arity Sign.cert_arity;
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end;
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(** explicit derivations -- cached **)
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local
19503
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fun derive_classrel thy (c1, c2) =
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  let
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    val {classes, classrel, ...} = get_instances thy;
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    val lookup = AList.lookup (op =) classrel;
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    fun derive [c, c'] = the (lookup (c, c'))
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      | derive (c :: c' :: cs) = derive [c, c'] RS derive (c' :: cs);
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  in
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    (case lookup (c1, c2) of
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      SOME rule => rule
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    | NONE =>
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        (case Graph.find_paths classes (c1, c2) of
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          [] => error ("Cannot derive class relation " ^ Sign.string_of_classrel thy [c1, c2])
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        | path :: _ => store_classrel thy ((c1, c2), derive path)))
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  end;
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fun weaken_subclass thy (c1, th) c2 =
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  if c1 = c2 then th
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  else th RS derive_classrel thy (c1, c2);
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   403
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   404
fun weaken_subsort thy S1 S2 = S2 |> map (fn c2 =>
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   405
  (case S1 |> find_first (fn (c1, _) => Sign.subsort thy ([c1], [c2])) of
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   406
    SOME c1 => weaken_subclass thy c1 c2
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  | NONE => error ("Cannot derive subsort relation " ^
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   408
      Sign.string_of_sort thy (map #1 S1) ^ " < " ^ Sign.string_of_sort thy S2)));
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fun apply_arity thy t dom c =
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   411
  let
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   412
    val {arities, ...} = get_instances thy;
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   413
    val subsort = Sign.subsort thy;
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    val Ss = map (map #1) dom;
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   415
  in
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   416
    (case Symtab.lookup_list arities t |> find_first (fn ((c', Ss'), _) =>
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        subsort ([c'], [c]) andalso ListPair.all subsort (Ss, Ss')) of
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      SOME ((c', Ss'), rule) =>
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        weaken_subclass thy (c', rule OF flat (map2 (weaken_subsort thy) dom Ss')) c
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    | NONE => error ("Cannot derive type arity " ^ Sign.string_of_arity thy (t, Ss, [c])))
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  end;
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   422
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fun derive_type thy hyps =
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  let
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   425
    fun derive (Type (a, Ts)) S =
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          let val Ss = Sign.arity_sorts thy a S
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   427
          in map (apply_arity thy a (map2 (fn T => fn S => S ~~ derive T S) Ts Ss)) S end
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      | derive (TFree (a, [])) S =
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   429
          weaken_subsort thy (the_default [] (AList.lookup (op =) hyps a)) S
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   430
      | derive T _ = error ("Illegal occurrence of type variable " ^
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          setmp show_sorts true (Sign.string_of_typ thy) T);
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   432
  in derive end;
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19511
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in
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19503
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   436
fun of_sort thy (typ, sort) =
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   437
  let
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   438
    fun lookup () = AList.lookup (op =) (Typtab.lookup_list (#types (get_instances thy)) typ);
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   439
    val sort' = filter (is_none o lookup ()) sort;
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   440
    val _ = conditional (not (null sort')) (fn () =>
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   441
      let
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   442
        val vars = Term.fold_atyps (insert (op =)) typ [];
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   443
        val renaming =
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   444
          map2 (fn T => fn a => (T, (a, case T of TFree (_, S) => S | TVar (_, S) => S)))
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   445
            vars (Term.invent_names [] "'a" (length vars));
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   446
        val typ' = typ |> Term.map_atyps
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   447
          (fn T => TFree (#1 (the (AList.lookup (op =) renaming T)), []));
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   448
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   449
        val hyps = renaming |> map (fn (_, (a, S)) => (a, S ~~ (S |> map (fn c =>
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   450
          Thm.assume (Thm.cterm_of thy (Logic.mk_inclass (TFree (a, []), c)))))));
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   451
        val inst = renaming |> map (fn (T, (a, S)) =>
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   452
          pairself (Thm.ctyp_of thy) (TVar ((a, 0), S), T));
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   453
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   454
        val ths =
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   455
          derive_type thy hyps typ' sort'
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   456
          |> map (Thm.instantiate (inst, []));
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   457
        val _ = map (store_type thy) (map (pair typ) sort' ~~ ths);
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   458
      in () end);
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   459
  in map (the o lookup ()) sort end;
10921826b160 instances data: mutable cache;
wenzelm
parents: 19482
diff changeset
   460
19511
b4bd790f9373 renamed add_axclass(_i) to define_axclass(_i);
wenzelm
parents: 19503
diff changeset
   461
end;
19418
03b01c9314fc reworded add_axclass(_i): canonical specification format,
wenzelm
parents: 19405
diff changeset
   462
15876
a67343c6ab2a sane interfaces for tactical instance proofs (do not expand defs of theory, proper handling of sort instances);
wenzelm
parents: 15853
diff changeset
   463
end;