src/Pure/axclass.ML
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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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User interfaces for axiomatic type classes.
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*)
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signature AX_CLASS =
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sig
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  val quiet_mode: bool ref
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  val add_classrel_thms: thm list -> theory -> theory
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  val add_arity_thms: thm list -> theory -> theory
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  val add_axclass: bclass * xclass list -> (string * string) list
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    -> theory -> theory
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  val add_axclass_i: bclass * class list -> (string * term) list
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    -> theory -> theory
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  val add_inst_subclass: xclass * xclass -> string list -> thm list
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    -> tactic option -> theory -> theory
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  val add_inst_subclass_i: class * class -> string list -> thm list
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    -> tactic option -> theory -> theory
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  val add_inst_arity: xstring * xsort list * xclass list -> string list
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    -> thm list -> tactic option -> theory -> theory
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  val add_inst_arity_i: string * sort list * class list -> string list
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    -> thm list -> tactic option -> theory -> theory
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  val axclass_tac: theory -> thm list -> tactic
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  val prove_subclass: theory -> class * class -> thm list
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    -> tactic option -> thm
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  val prove_arity: theory -> string * sort list * class -> thm list
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    -> tactic option -> thm
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  val goal_subclass: theory -> xclass * xclass -> thm list
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  val goal_arity: theory -> xstring * xsort list * xclass -> thm list
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end;
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structure AxClass : AX_CLASS =
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struct
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(** utilities **)
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(* messages *)
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val quiet_mode = ref false;
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fun message s = if ! quiet_mode then () else writeln s;
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(* type vars *)
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fun map_typ_frees f (Type (t, tys)) = Type (t, map (map_typ_frees f) tys)
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  | map_typ_frees f (TFree a) = f a
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  | map_typ_frees _ a = a;
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val map_term_tfrees = map_term_types o map_typ_frees;
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fun aT S = TFree ("'a", S);
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fun dest_varT (TFree (x, S)) = ((x, ~1), S)
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  | dest_varT (TVar xi_S) = xi_S
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  | dest_varT T = raise TYPE ("dest_varT", [T], []);
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(* get axioms and theorems *)
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fun get_ax thy name =
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  Some (get_axiom thy name) handle THEORY _ => None;
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val get_axioms = mapfilter o get_ax;
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val is_def = Logic.is_equals o #prop o rep_thm;
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fun witnesses thy names thms =
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  flat (map (PureThy.get_thms thy) names) @ thms @ filter is_def (map snd (axioms_of thy));
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(** abstract syntax operations **)
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(* subclass relations as terms *)
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fun mk_classrel (c1, c2) = Logic.mk_inclass (aT [c1], c2);
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fun dest_classrel tm =
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  let
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    fun err () = raise TERM ("dest_classrel", [tm]);
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    val (ty, c2) = Logic.dest_inclass tm handle TERM _ => err ();
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    val c1 = (case dest_varT ty of (_, [c]) => c | _ => err ())
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      handle TYPE _ => err ();
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  in
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    (c1, c2)
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  end;
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(* arities as terms *)
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fun mk_arity (t, ss, c) =
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  let
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    val names = tl (variantlist (replicate (length ss + 1) "'", []));
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    val tfrees = ListPair.map TFree (names, ss);
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  in
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    Logic.mk_inclass (Type (t, tfrees), c)
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  end;
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fun dest_arity tm =
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  let
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    fun err () = raise TERM ("dest_arity", [tm]);
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    val (ty, c) = Logic.dest_inclass tm handle TERM _ => err ();
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    val (t, tvars) =
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      (case ty of
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        Type (t, tys) => (t, map dest_varT tys handle TYPE _ => err ())
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      | _ => err ());
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    val ss =
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      if null (gen_duplicates eq_fst tvars)
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      then map snd tvars else err ();
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  in
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    (t, ss, c)
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  end;
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(** add theorems as axioms **)
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fun prep_thm_axm thy thm =
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  let
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    fun err msg = raise THM ("prep_thm_axm: " ^ msg, 0, [thm]);
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    val {sign, hyps, prop, ...} = rep_thm thm;
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  in
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    if not (Sign.subsig (sign, sign_of thy)) then
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      err "theorem not of same theory"
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    else if not (null (extra_shyps thm)) orelse not (null hyps) then
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      err "theorem may not contain hypotheses"
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    else prop
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  end;
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(*theorems expressing class relations*)
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fun add_classrel_thms thms thy =
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  let
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    fun prep_thm thm =
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      let
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        val prop = prep_thm_axm thy thm;
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        val (c1, c2) = dest_classrel prop handle TERM _ =>
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          raise THM ("add_classrel_thms: theorem is not a class relation", 0, [thm]);
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      in (c1, c2) end;
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  in
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    Theory.add_classrel (map prep_thm thms) thy
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  end;
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(*theorems expressing arities*)
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fun add_arity_thms thms thy =
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  let
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    fun prep_thm thm =
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      let
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        val prop = prep_thm_axm thy thm;
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        val (t, ss, c) = dest_arity prop handle TERM _ =>
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          raise THM ("add_arity_thms: theorem is not an arity", 0, [thm]);
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      in (t, ss, [c]) end;
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  in
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    Theory.add_arities (map prep_thm thms) thy
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  end;
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(** add axiomatic type classes **)
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(* errors *)
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fun err_not_logic c =
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  error ("Axiomatic class " ^ quote c ^ " not subclass of " ^ quote logicC);
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fun err_bad_axsort ax c =
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  error ("Sort constraint in axiom " ^ quote ax ^ " not supersort of " ^ quote c);
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fun err_bad_tfrees ax =
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  error ("More than one type variable in axiom " ^ quote ax);
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(* ext_axclass *)
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fun ext_axclass int prep_axm (raw_class, raw_super_classes) raw_axioms old_thy =
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  let
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    val old_sign = sign_of old_thy;
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    val axioms = map (prep_axm old_sign) raw_axioms;
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    val class = Sign.full_name old_sign raw_class;
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    val thy =
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      (if int then Theory.add_classes else Theory.add_classes_i)
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        [(raw_class, raw_super_classes)] old_thy;
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    val sign = sign_of thy;
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    val super_classes =
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      if int then map (Sign.intern_class sign) raw_super_classes
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      else raw_super_classes;
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    (* prepare abstract axioms *)
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    fun abs_axm ax =
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      if null (term_tfrees ax) then
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        Logic.mk_implies (Logic.mk_inclass (aT logicS, class), ax)
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      else map_term_tfrees (K (aT [class])) ax;
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    val abs_axioms = map (apsnd abs_axm) axioms;
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    (* prepare introduction orule *)
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    val _ =
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      if Sign.subsort sign ([class], logicS) then ()
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      else err_not_logic class;
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    fun axm_sort (name, ax) =
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      (case term_tfrees ax of
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        [] => []
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      | [(_, S)] =>
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          if Sign.subsort sign ([class], S) then S
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          else err_bad_axsort name class
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      | _ => err_bad_tfrees name);
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    val axS = Sign.norm_sort sign (logicC :: flat (map axm_sort axioms))
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    val int_axm = Logic.close_form o map_term_tfrees (K (aT axS));
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    fun inclass c = Logic.mk_inclass (aT axS, c);
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    val intro_axm = Logic.list_implies
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      (map inclass super_classes @ map (int_axm o snd) axioms, inclass class);
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  in
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    thy
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    |> PureThy.add_axioms_i (map Attribute.none ((raw_class ^ "I", intro_axm) :: abs_axioms))
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  end;
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(* external interfaces *)
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val add_axclass = ext_axclass true read_axm;
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val add_axclass_i = ext_axclass false cert_axm;
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(** prove class relations and type arities **)
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(* class_axms *)
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fun class_axms thy =
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  let
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    val classes = Sign.classes (sign_of thy);
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    val intros = map (fn c => c ^ "I") classes;
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  in
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    map (class_triv thy) classes @
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    get_axioms thy intros
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  end;
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(* axclass_tac *)
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(*(1) repeatedly resolve goals of form "OFCLASS(ty, c_class)",
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      try class_trivs first, then "cI" axioms
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  (2) rewrite goals using user supplied definitions
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  (3) repeatedly resolve goals with user supplied non-definitions*)
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fun axclass_tac thy thms =
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  let
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    val defs = filter is_def thms;
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    val non_defs = filter_out is_def thms;
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  in
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    TRY (REPEAT_FIRST (resolve_tac (class_axms thy))) THEN
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    TRY (rewrite_goals_tac defs) THEN
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    TRY (REPEAT_FIRST (fn i => assume_tac i ORELSE resolve_tac non_defs i))
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  end;
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(* provers *)
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fun prove term_of str_of thy sig_prop thms usr_tac =
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  let
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    val sign = sign_of thy;
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    val goal = cterm_of sign (term_of sig_prop);
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    val tac = axclass_tac thy thms THEN (if_none usr_tac all_tac);
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  in
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    prove_goalw_cterm [] goal (K [tac])
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  end
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  handle ERROR => error ("The error(s) above occurred while trying to prove "
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    ^ quote (str_of (sign_of thy, sig_prop)));
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val prove_subclass =
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  prove mk_classrel (fn (sg, c1_c2) => Sign.str_of_classrel sg c1_c2);
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val prove_arity =
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  prove mk_arity (fn (sg, (t, Ss, c)) => Sign.str_of_arity sg (t, Ss, [c]));
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(** add proved subclass relations and arities **)
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fun intrn_classrel sg c1_c2 =
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  pairself (Sign.intern_class sg) c1_c2;
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fun ext_inst_subclass int raw_c1_c2 names thms usr_tac thy =
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  let
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    val c1_c2 =
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      if int then intrn_classrel (sign_of thy) raw_c1_c2
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      else raw_c1_c2;
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  in
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    message ("Proving class inclusion " ^
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      quote (Sign.str_of_classrel (sign_of thy) c1_c2) ^ " ...");
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    add_classrel_thms
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      [prove_subclass thy c1_c2 (witnesses thy names thms) usr_tac] thy
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  end;
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fun intrn_arity sg intrn (t, Ss, x) =
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  (Sign.intern_tycon sg t, map (Sign.intern_sort sg) Ss, intrn sg x);
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fun ext_inst_arity int (raw_t, raw_Ss, raw_cs) names thms usr_tac thy =
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  let
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    val sign = sign_of thy;
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    val (t, Ss, cs) =
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      if int then intrn_arity sign Sign.intern_sort (raw_t, raw_Ss, raw_cs)
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      else (raw_t, raw_Ss, raw_cs);
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    val wthms = witnesses thy names thms;
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    fun prove c =
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     (message ("Proving type arity " ^
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        quote (Sign.str_of_arity sign (t, Ss, [c])) ^ " ...");
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        prove_arity thy (t, Ss, c) wthms usr_tac);
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  in
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    add_arity_thms (map prove cs) thy
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  end;
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val add_inst_subclass = ext_inst_subclass true;
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val add_inst_subclass_i = ext_inst_subclass false;
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val add_inst_arity = ext_inst_arity true;
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val add_inst_arity_i = ext_inst_arity false;
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404
dd3d3d6467db axiomatic type class 'package' for Pure (alpha version);
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3949
c60ff6d0a6b8 fixed goal_XXX;
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(* make goals (for interactive use) *)
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c60ff6d0a6b8 fixed goal_XXX;
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fun mk_goal term_of thy sig_prop =
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  goalw_cterm [] (cterm_of (sign_of thy) (term_of sig_prop));
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c60ff6d0a6b8 fixed goal_XXX;
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fun goal_subclass thy =
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  mk_goal (mk_classrel o intrn_classrel (sign_of thy)) thy;
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c60ff6d0a6b8 fixed goal_XXX;
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fun goal_arity thy =
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  mk_goal (mk_arity o intrn_arity (sign_of thy) Sign.intern_class) thy;
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c60ff6d0a6b8 fixed goal_XXX;
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404
dd3d3d6467db axiomatic type class 'package' for Pure (alpha version);
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end;