(*  Title:      Pure/axclass.ML

     ID:         $Id$

     Author:     Markus Wenzel, TU Muenchen

     License:    GPL (GNU GENERAL PUBLIC LICENSE)

 Axiomatic type class package.

 *)

 signature AX_CLASS =

 sig

   val quiet_mode: bool ref

   val print_axclasses: theory -> unit

   val add_classrel_thms: thm list -> theory -> theory

   val add_arity_thms: thm list -> theory -> theory

   val add_axclass: bclass * xclass list -> ((bstring * string) * Args.src list) list

     -> theory -> theory * {intro: thm, axioms: thm list}

   val add_axclass_i: bclass * class list -> ((bstring * term) * theory attribute list) list

     -> theory -> theory * {intro: thm, axioms: thm list}

   val add_inst_subclass_x: xclass * xclass -> string list -> thm list

     -> tactic option -> theory -> theory

   val add_inst_subclass: xclass * xclass -> tactic -> theory -> theory

   val add_inst_subclass_i: class * class -> tactic -> theory -> theory

   val add_inst_arity_x: xstring * string list * string -> string list

     -> thm list -> tactic option -> theory -> theory

   val add_inst_arity: xstring * string list * string -> tactic -> theory -> theory

   val add_inst_arity_i: string * sort list * sort -> tactic -> theory -> theory

   val default_intro_classes_tac: thm list -> int -> tactic

   val axclass_tac: thm list -> tactic

   val instance_subclass_proof: (xclass * xclass) * Comment.text -> bool -> theory -> ProofHistory.T

   val instance_subclass_proof_i: (class * class) * Comment.text -> bool -> theory -> ProofHistory.T

   val instance_arity_proof: (xstring * string list * xclass) * Comment.text

     -> bool -> theory -> ProofHistory.T

   val instance_arity_proof_i: (string * sort list * class) * Comment.text

     -> bool -> theory -> ProofHistory.T

   val setup: (theory -> theory) list

 end;

 structure AxClass : AX_CLASS =

 struct

 (** utilities **)

 (* messages *)

 val quiet_mode = ref false;

 fun message s = if ! quiet_mode then () else writeln s;

 (* type vars *)

 fun map_typ_frees f (Type (t, tys)) = Type (t, map (map_typ_frees f) tys)

   | map_typ_frees f (TFree a) = f a

   | map_typ_frees _ a = a;

 val map_term_tfrees = map_term_types o map_typ_frees;

 fun aT S = TFree ("'a", S);

 fun dest_varT (TFree (x, S)) = ((x, ~1), S)

   | dest_varT (TVar xi_S) = xi_S

   | dest_varT T = raise TYPE ("dest_varT", [T], []);

 (* get axioms and theorems *)

 val is_def = Logic.is_equals o #prop o rep_thm;

 fun witnesses thy names thms =

   PureThy.get_thmss thy names @ thms @ filter is_def (map snd (axioms_of thy));

 (** abstract syntax operations **)

 (* names *)

 fun intro_name c = c ^ "I";

 val introN = "intro";

 val axiomsN = "axioms";

 (* subclass relations as terms *)

 fun mk_classrel (c1, c2) = Logic.mk_inclass (aT [c1], c2);

 fun dest_classrel tm =

   let

     fun err () = raise TERM ("dest_classrel", [tm]);

     val (ty, c2) = Logic.dest_inclass tm handle TERM _ => err ();

     val c1 = (case dest_varT ty of (_, [c]) => c | _ => err ())

       handle TYPE _ => err ();

   in (c1, c2) end;

 (* arities as terms *)

 fun mk_arity (t, ss, c) =

   let

     val tfrees = ListPair.map TFree (Term.invent_type_names [] (length ss), ss);

   in Logic.mk_inclass (Type (t, tfrees), c) end;

 fun dest_arity tm =

   let

     fun err () = raise TERM ("dest_arity", [tm]);

     val (ty, c) = Logic.dest_inclass tm handle TERM _ => err ();

     val (t, tvars) =

       (case ty of

         Type (t, tys) => (t, map dest_varT tys handle TYPE _ => err ())

       | _ => err ());

     val ss =

       if null (gen_duplicates eq_fst tvars)

       then map snd tvars else err ();

   in (t, ss, c) end;

 (** add theorems as axioms **)

 fun prep_thm_axm thy thm =

   let

     fun err msg = raise THM ("prep_thm_axm: " ^ msg, 0, [thm]);

     val {sign, hyps, prop, ...} = rep_thm thm;

   in

     if not (Sign.subsig (sign, Theory.sign_of thy)) then

       err "theorem not of same theory"

     else if not (null (extra_shyps thm)) orelse not (null hyps) then

       err "theorem may not contain hypotheses"

     else prop

   end;

 (*theorems expressing class relations*)

 fun add_classrel_thms thms thy =

   let

     fun prep_thm thm =

       let

         val prop = prep_thm_axm thy thm;

         val (c1, c2) = dest_classrel prop handle TERM _ =>

           raise THM ("add_classrel_thms: theorem is not a class relation", 0, [thm]);

       in (c1, c2) end;

   in Theory.add_classrel_i (map prep_thm thms) thy end;

 (*theorems expressing arities*)

 fun add_arity_thms thms thy =

   let

     fun prep_thm thm =

       let

         val prop = prep_thm_axm thy thm;

         val (t, ss, c) = dest_arity prop handle TERM _ =>

           raise THM ("add_arity_thms: theorem is not an arity", 0, [thm]);

       in (t, ss, [c]) end;

   in Theory.add_arities_i (map prep_thm thms) thy end;

 (** axclass info **)

 (* data kind 'Pure/axclasses' *)

 type axclass_info =

   {super_classes: class list,

     intro: thm,

     axioms: thm list};

 structure AxclassesArgs =

 struct

   val name = "Pure/axclasses";

   type T = axclass_info Symtab.table;

   val empty = Symtab.empty;

   val copy = I;

   val prep_ext = I;

   fun merge (tab1, tab2) = Symtab.merge (K true) (tab1, tab2);

   fun print sg tab =

     let

       val ext_class = Sign.cond_extern sg Sign.classK;

       val ext_thm = PureThy.cond_extern_thm_sg sg;

       fun pretty_class c cs = Pretty.block

         (Pretty.str (ext_class c) :: Pretty.str " <" :: Pretty.brk 1 ::

           Pretty.breaks (map (Pretty.str o ext_class) cs));

       fun pretty_thms name thms = Pretty.big_list (name ^ ":")

         (map (Display.pretty_thm_sg sg) thms);

       fun pretty_axclass (name, {super_classes, intro, axioms}) = Pretty.block (Pretty.fbreaks

         [pretty_class name super_classes, pretty_thms introN [intro], pretty_thms axiomsN axioms]);

     in Pretty.writeln (Pretty.chunks (map pretty_axclass (Symtab.dest tab))) end;

 end;

 structure AxclassesData = TheoryDataFun(AxclassesArgs);

 val print_axclasses = AxclassesData.print;

 (* get and put data *)

 fun lookup_axclass_info_sg sg c = Symtab.lookup (AxclassesData.get_sg sg, c);

 fun get_axclass_info thy c =

   (case lookup_axclass_info_sg (Theory.sign_of thy) c of

     None => error ("Unknown axclass " ^ quote c)

   | Some info => info);

 fun put_axclass_info c info thy =

   thy |> AxclassesData.put (Symtab.update ((c, info), AxclassesData.get thy));

 (** add axiomatic type classes **)

 (* errors *)

 fun err_not_logic c =

   error ("Axiomatic class " ^ quote c ^ " not subclass of " ^ quote logicC);

 fun err_bad_axsort ax c =

   error ("Sort constraint in axiom " ^ quote ax ^ " not supersort of " ^ quote c);

 fun err_bad_tfrees ax =

   error ("More than one type variable in axiom " ^ quote ax);

 (* ext_axclass *)

 fun ext_axclass prep_class prep_axm prep_att (bclass, raw_super_classes) raw_axioms_atts thy =

   let

     val sign = Theory.sign_of thy;

     val class = Sign.full_name sign bclass;

     val super_classes = map (prep_class sign) raw_super_classes;

     val axms = map (prep_axm sign o fst) raw_axioms_atts;

     val atts = map (map (prep_att thy) o snd) raw_axioms_atts;

     (*declare class*)

     val class_thy =

       thy |> Theory.add_classes_i [(bclass, super_classes)];

     val class_sign = Theory.sign_of class_thy;

     (*prepare abstract axioms*)

     fun abs_axm ax =

       if null (term_tfrees ax) then

         Logic.mk_implies (Logic.mk_inclass (aT logicS, class), ax)

       else map_term_tfrees (K (aT [class])) ax;

     val abs_axms = map (abs_axm o #2) axms;

     (*prepare introduction rule*)

     val _ = if Sign.subsort class_sign ([class], logicS) then () else err_not_logic class;

     fun axm_sort (name, ax) =

       (case term_tfrees ax of

         [] => []

       | [(_, S)] => if Sign.subsort class_sign ([class], S) then S else err_bad_axsort name class

       | _ => err_bad_tfrees name);

     val axS = Sign.norm_sort class_sign (logicC :: flat (map axm_sort axms));

     val int_axm = Logic.close_form o map_term_tfrees (K (aT axS));

     fun inclass c = Logic.mk_inclass (aT axS, c);

     val intro_axm = Logic.list_implies

       (map inclass super_classes @ map (int_axm o #2) axms, inclass class);

     (*declare axioms and rule*)

     val (axms_thy, ([intro], [axioms])) =

       class_thy

       |> Theory.add_path bclass

       |> PureThy.add_axioms_i [Thm.no_attributes (introN, intro_axm)]

       |>>> PureThy.add_axiomss_i [Thm.no_attributes (axiomsN, abs_axms)];

     val info = {super_classes = super_classes, intro = intro, axioms = axioms};

     (*store info*)

     val final_thy =

       axms_thy

       |> (#1 o PureThy.add_thms ((map #1 axms ~~ axioms) ~~ atts))

       |> Theory.parent_path

       |> (#1 o PureThy.add_thms [Thm.no_attributes (intro_name bclass, intro)])

       |> put_axclass_info class info;

   in (final_thy, {intro = intro, axioms = axioms}) end;

 (* external interfaces *)

 val add_axclass = ext_axclass Sign.intern_class Theory.read_axm Attrib.global_attribute;

 val add_axclass_i = ext_axclass (K I) Theory.cert_axm (K I);

 (** prove class relations and type arities **)

 (* class_axms *)

 fun class_axms sign =

   let val classes = Sign.classes sign in

     map (Thm.class_triv sign) classes @

     mapfilter (apsome #intro o lookup_axclass_info_sg sign) classes

   end;

 (* intro_classes *)

 fun intro_classes_tac facts i st =

   ((Method.insert_tac facts THEN'

     REPEAT_ALL_NEW (resolve_tac (class_axms (Thm.sign_of_thm st)))) i

     THEN Tactic.distinct_subgoals_tac) st;    (*affects all subgoals!?*)

 val intro_classes_method =

   ("intro_classes", Method.no_args (Method.METHOD (HEADGOAL o intro_classes_tac)),

     "back-chain introduction rules of axiomatic type classes");

 (* default method *)

 fun default_intro_classes_tac [] i = intro_classes_tac [] i

   | default_intro_classes_tac _ _ = Tactical.no_tac;    (*no error message!*)

 fun default_tac rules ctxt facts =

   HEADGOAL (Method.some_rule_tac rules ctxt facts ORELSE' default_intro_classes_tac facts);

 val default_method =

   ("default", Method.thms_ctxt_args (Method.METHOD oo default_tac), "apply some intro/elim rule");

 (* axclass_tac *)

 (*(1) repeatedly resolve goals of form "OFCLASS(ty, c_class)",

       try class_trivs first, then "cI" axioms

   (2) rewrite goals using user supplied definitions

   (3) repeatedly resolve goals with user supplied non-definitions*)

 fun axclass_tac thms =

   let

     val defs = filter is_def thms;

     val non_defs = filter_out is_def thms;

   in

     HEADGOAL (intro_classes_tac []) THEN

     TRY (rewrite_goals_tac defs) THEN

     TRY (REPEAT_FIRST (fn i => assume_tac i ORELSE resolve_tac non_defs i))

   end;

 (* provers *)

 fun prove mk_prop str_of sign prop thms usr_tac =

   (Tactic.prove sign [] [] (mk_prop prop) (K (axclass_tac thms THEN (if_none usr_tac all_tac)))

     handle ERROR => error ("The error(s) above occurred while trying to prove " ^

      quote (str_of sign prop))) |> Drule.standard;

 val prove_subclass =

   prove mk_classrel (fn sg => fn c1_c2 => Sign.str_of_classrel sg c1_c2);

 val prove_arity =

   prove mk_arity (fn sg => fn (t, Ss, c) => Sign.str_of_arity sg (t, Ss, [c]));

 (** add proved subclass relations and arities **)

 (* prepare classes and arities *)

 fun read_class sg c = Sign.certify_class sg (Sign.intern_class sg c);

 fun cert_classrel sg cc = Library.pairself (Sign.certify_class sg) cc;

 fun read_classrel sg cc = Library.pairself (read_class sg) cc;

 fun check_tycon sg t =

   let val {tycons, abbrs, ...} = Type.rep_tsig (Sign.tsig_of sg) in

     if is_some (Symtab.lookup (abbrs, t)) then

       error ("Illegal type abbreviation: " ^ quote t)

     else if is_none (Symtab.lookup (tycons, t)) then

       error ("Undeclared type constructor: " ^ quote t)

     else t

   end;

 fun prep_arity prep_tycon prep_sort prep_x sg (t, Ss, x) =

   (check_tycon sg (prep_tycon sg t), map (prep_sort sg) Ss, prep_x sg x);

 val read_arity = prep_arity Sign.intern_tycon Sign.read_sort Sign.read_sort;

 val cert_arity = prep_arity (K I) Sign.certify_sort Sign.certify_sort;

 val read_simple_arity = prep_arity Sign.intern_tycon Sign.read_sort Sign.intern_class;

 fun cert_simple_arity arg = prep_arity (K I) Sign.certify_sort (K I) arg;

 (* old-style instance declarations *)

 fun ext_inst_subclass prep_classrel raw_c1_c2 names thms usr_tac thy =

   let

     val sign = Theory.sign_of thy;

     val c1_c2 = prep_classrel sign raw_c1_c2;

   in

     message ("Proving class inclusion " ^ quote (Sign.str_of_classrel sign c1_c2) ^ " ...");

     thy |> add_classrel_thms [prove_subclass sign c1_c2 (witnesses thy names thms) usr_tac]

   end;

 fun ext_inst_arity prep_arity (raw_t, raw_Ss, raw_cs) names thms usr_tac thy =

   let

     val sign = Theory.sign_of thy;

     val (t, Ss, cs) = prep_arity sign (raw_t, raw_Ss, raw_cs);

     val wthms = witnesses thy names thms;

     fun prove c =

      (message ("Proving type arity " ^

         quote (Sign.str_of_arity sign (t, Ss, [c])) ^ " ...");

         prove_arity sign (t, Ss, c) wthms usr_tac);

   in add_arity_thms (map prove cs) thy end;

 fun sane_inst_subclass prep sub tac = ext_inst_subclass prep sub [] [] (Some tac);

 fun sane_inst_arity prep arity tac = ext_inst_arity prep arity [] [] (Some tac);

 val add_inst_subclass_x = ext_inst_subclass read_classrel;

 val add_inst_subclass = sane_inst_subclass read_classrel;

 val add_inst_subclass_i = sane_inst_subclass cert_classrel;

 val add_inst_arity_x = ext_inst_arity read_arity;

 val add_inst_arity = sane_inst_arity read_arity;

 val add_inst_arity_i = sane_inst_arity cert_arity;

 (** instance proof interfaces **)

 fun inst_attribute add_thms (thy, thm) = (add_thms [thm] thy, thm);

 fun inst_proof mk_prop add_thms (sig_prop, comment) int thy =

   thy

   |> IsarThy.theorem_i Drule.internalK (None, []) ((("", [inst_attribute add_thms]),

     (mk_prop (Theory.sign_of thy) sig_prop, ([], []))), comment) int;

 val instance_subclass_proof = inst_proof (mk_classrel oo read_classrel) add_classrel_thms;

 val instance_subclass_proof_i = inst_proof (mk_classrel oo cert_classrel) add_classrel_thms;

 val instance_arity_proof = inst_proof (mk_arity oo read_simple_arity) add_arity_thms;

 val instance_arity_proof_i = inst_proof (mk_arity oo cert_simple_arity) add_arity_thms;

 (** package setup **)

 (* setup theory *)

 val setup =

  [AxclassesData.init,

   Method.add_methods [intro_classes_method, default_method]];

 (* outer syntax *)

 local structure P = OuterParse and K = OuterSyntax.Keyword in

 val axclassP =

   OuterSyntax.command "axclass" "define axiomatic type class" K.thy_decl

     (((P.name -- Scan.optional ((P.$$$ "\\<subseteq>" || P.$$$ "<") |--

         P.!!! (P.list1 P.xname)) []) --| P.marg_comment)

       -- Scan.repeat (P.spec_name --| P.marg_comment)

       >> (fn (cls, axs) => Toplevel.theory (#1 o add_axclass cls axs)));

 val instanceP =

   OuterSyntax.command "instance" "prove type arity or subclass relation" K.thy_goal

     ((P.xname -- ((P.$$$ "\\<subseteq>" || P.$$$ "<") |-- P.xname)

         -- P.marg_comment >> instance_subclass_proof ||

       (P.xname -- (P.$$$ "::" |-- P.simple_arity) >> P.triple2) -- P.marg_comment

         >> instance_arity_proof)

       >> (Toplevel.print oo Toplevel.theory_to_proof));

 val _ = OuterSyntax.add_parsers [axclassP, instanceP];

 end;

 end;