(*  Title:      Pure/axclass.ML

     Author:     Markus Wenzel, TU Muenchen

 Type classes defined as predicates, associated with a record of

 parameters.

 *)

 signature AX_CLASS =

 sig

   val define_class: binding * class list -> string list ->

     (Thm.binding * term list) list -> theory -> class * theory

   val add_classrel: thm -> theory -> theory

   val add_arity: thm -> theory -> theory

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

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

   val get_info: theory -> class ->

     {def: thm, intro: thm, axioms: thm list, params: (string * typ) list}

   val class_intros: theory -> thm list

   val class_of_param: theory -> string -> class option

   val cert_classrel: theory -> class * class -> class * class

   val read_classrel: theory -> xstring * xstring -> class * class

   val axiomatize_class: binding * class list -> theory -> theory

   val axiomatize_class_cmd: binding * xstring list -> theory -> theory

   val axiomatize_classrel: (class * class) list -> theory -> theory

   val axiomatize_classrel_cmd: (xstring * xstring) list -> theory -> theory

   val axiomatize_arity: arity -> theory -> theory

   val axiomatize_arity_cmd: xstring * string list * string -> theory -> theory

   val instance_name: string * class -> string

   val declare_overloaded: string * typ -> theory -> term * theory

   val define_overloaded: binding -> string * term -> theory -> thm * theory

   val unoverload: theory -> thm -> thm

   val overload: theory -> thm -> thm

   val unoverload_conv: theory -> conv

   val overload_conv: theory -> conv

   val unoverload_const: theory -> string * typ -> string

   val lookup_inst_param: Consts.T -> ((string * string) * 'a) list -> string * typ -> 'a option

   val param_of_inst: theory -> string * string -> string

   val inst_of_param: theory -> string -> (string * string) option

   val arity_property: theory -> class * string -> string -> string list

   type cache

   val of_sort: theory -> typ * sort -> cache -> thm list * cache  (*exception Sorts.CLASS_ERROR*)

   val cache: cache

   val introN: string

   val axiomsN: string

 end;

 structure AxClass: AX_CLASS =

 struct

 (** theory data **)

 (* class parameters (canonical order) *)

 type param = string * class;

 fun add_param pp ((x, c): param) params =

   (case AList.lookup (op =) params x of

     NONE => (x, c) :: params

   | SOME c' => error ("Duplicate class parameter " ^ quote x ^

       " for " ^ Pretty.string_of_sort pp [c] ^

       (if c = c' then "" else " and " ^ Pretty.string_of_sort pp [c'])));

 fun merge_params _ ([], qs) = qs

   | merge_params pp (ps, qs) =

       fold_rev (fn q => if member (op =) ps q then I else add_param pp q) qs ps;

 (* axclasses *)

 val introN = "intro";

 val superN = "super";

 val axiomsN = "axioms";

 datatype axclass = AxClass of

  {def: thm,

   intro: thm,

   axioms: thm list,

   params: (string * typ) list};

 type axclasses = axclass Symtab.table * param list;

 fun make_axclass ((def, intro, axioms), params) = AxClass

   {def = def, intro = intro, axioms = axioms, params = params};

 fun merge_axclasses pp ((tab1, params1), (tab2, params2)) : axclasses =

   (Symtab.merge (K true) (tab1, tab2), merge_params pp (params1, params2));

 (* instances *)

 val classrel_prefix = "classrel_";

 val arity_prefix = "arity_";

 type instances =

   ((class * class) * thm) list *

   ((class * sort list) * thm) list Symtab.table;

 fun merge_instances ((classrel1, arities1): instances, (classrel2, arities2)) =

  (merge (eq_fst op =) (classrel1, classrel2),

   Symtab.join (K (merge (eq_fst op =))) (arities1, arities2));

 (* instance parameters *)

 type inst_params =

   (string * thm) Symtab.table Symtab.table

     (*constant name ~> type constructor ~> (constant name, equation)*)

   * (string * string) Symtab.table; (*constant name ~> (constant name, type constructor)*)

 fun merge_inst_params ((const_param1, param_const1), (const_param2, param_const2)) =

   (Symtab.join  (K (Symtab.merge (K true))) (const_param1, const_param2),

     Symtab.merge (K true) (param_const1, param_const2));

 (* setup data *)

 structure AxClassData = TheoryDataFun

 (

   type T = axclasses * (instances * inst_params);

   val empty = ((Symtab.empty, []), (([], Symtab.empty), (Symtab.empty, Symtab.empty)));

   val copy = I;

   val extend = I;

   fun merge pp ((axclasses1, (instances1, inst_params1)), (axclasses2, (instances2, inst_params2))) =

     (merge_axclasses pp (axclasses1, axclasses2),

       (merge_instances (instances1, instances2), merge_inst_params (inst_params1, inst_params2)));

 );

 (* maintain axclasses *)

 val get_axclasses = #1 o AxClassData.get;

 val map_axclasses = AxClassData.map o apfst;

 val lookup_def = Symtab.lookup o #1 o get_axclasses;

 fun get_info thy c =

   (case lookup_def thy c of

     SOME (AxClass info) => info

   | NONE => error ("No such axclass: " ^ quote c));

 fun class_intros thy =

   let

     fun add_intro c =

       (case lookup_def thy c of SOME (AxClass {intro, ...}) => cons intro | _ => I);

     val classes = Sign.all_classes thy;

   in map (Thm.class_triv thy) classes @ fold add_intro classes [] end;

 fun get_params thy pred =

   let val params = #2 (get_axclasses thy);

   in fold (fn (x, c) => if pred c then cons x else I) params [] end;

 fun params_of thy c = get_params thy (fn c' => c' = c);

 fun all_params_of thy S = get_params thy (fn c => Sign.subsort thy (S, [c]));

 fun class_of_param thy = AList.lookup (op =) (#2 (get_axclasses thy));

 (* maintain instances *)

 fun instance_name (a, c) = Long_Name.base_name c ^ "_" ^ Long_Name.base_name a;

 val get_instances = #1 o #2 o AxClassData.get;

 val map_instances = AxClassData.map o apsnd o apfst;

 fun the_classrel thy (c1, c2) =

   (case AList.lookup (op =) (#1 (get_instances thy)) (c1, c2) of

     SOME th => Thm.transfer thy th

   | NONE => error ("Unproven class relation " ^

       Syntax.string_of_classrel (ProofContext.init thy) [c1, c2]));

 fun put_classrel arg = map_instances (fn (classrel, arities) =>

   (insert (eq_fst op =) arg classrel, arities));

 fun the_arity thy a (c, Ss) =

   (case AList.lookup (op =) (Symtab.lookup_list (#2 (get_instances thy)) a) (c, Ss) of

     SOME th => Thm.transfer thy th

   | NONE => error ("Unproven type arity " ^

       Syntax.string_of_arity (ProofContext.init thy) (a, Ss, [c])));

 fun arity_property thy (c, a) x =

   Symtab.lookup_list (snd (get_instances thy)) a

   |> map_filter (fn ((c', _), th) => if c = c'

       then AList.lookup (op =) (Thm.get_tags th) x else NONE)

   |> rev;

 fun insert_arity_completions thy (t, ((c, Ss), th)) arities =

   let

     val algebra = Sign.classes_of thy;

     val super_class_completions = Sign.super_classes thy c

       |> filter_out (fn c1 => exists (fn ((c2, Ss2), _) => c1 = c2

           andalso Sorts.sorts_le algebra (Ss2, Ss)) (Symtab.lookup_list arities t))

     val tags = Thm.get_tags th;

     val completions = map (fn c1 => (Sorts.weaken algebra

       (fn (th, c2) => fn c3 => th RS the_classrel thy (c2, c3)) (th, c) c1

         |> Thm.map_tags (K tags) |> Thm.close_derivation, c1)) super_class_completions;

     val arities' = fold (fn (th1, c1) => Symtab.cons_list (t, ((c1, Ss), th1)))

       completions arities;

   in (completions, arities') end;

 fun put_arity ((t, Ss, c), th) thy =

   let

     val th' = (Thm.map_tags o AList.default (op =))

       (Markup.theory_nameN, Context.theory_name thy) th;

     val arity' = (t, ((c, Ss), th'));

   in

     thy

     |> map_instances (fn (classrel, arities) => (classrel,

       arities

       |> Symtab.insert_list (eq_fst op =) arity'

       |> insert_arity_completions thy arity'

       |> snd))

   end;

 fun complete_arities thy =

   let

     val arities = snd (get_instances thy);

     val (completions, arities') = arities

       |> fold_map (insert_arity_completions thy) (Symtab.dest_list arities)

       |>> flat;

   in if null completions

     then NONE

     else SOME (thy |> map_instances (fn (classrel, _) => (classrel, arities')))

   end;

 val _ = Context.>> (Context.map_theory (Theory.at_begin complete_arities));

 (* maintain instance parameters *)

 val get_inst_params = #2 o #2 o AxClassData.get;

 val map_inst_params = AxClassData.map o apsnd o apsnd;

 fun get_inst_param thy (c, tyco) =

   case Symtab.lookup ((the_default Symtab.empty o Symtab.lookup (fst (get_inst_params thy))) c) tyco

    of SOME c' => c'

     | NONE => error ("No instance parameter for constant " ^ quote c

         ^ " on type constructor " ^ quote tyco);

 fun add_inst_param (c, tyco) inst = (map_inst_params o apfst

       o Symtab.map_default (c, Symtab.empty)) (Symtab.update_new (tyco, inst))

   #> (map_inst_params o apsnd) (Symtab.update_new (fst inst, (c, tyco)));

 val inst_of_param = Symtab.lookup o snd o get_inst_params;

 val param_of_inst = fst oo get_inst_param;

 fun inst_thms thy = (Symtab.fold (Symtab.fold (cons o snd o snd) o snd) o fst)

   (get_inst_params thy) [];

 fun get_inst_tyco consts = try (fst o dest_Type o the_single o Consts.typargs consts);

 fun unoverload thy = MetaSimplifier.simplify true (inst_thms thy);

 fun overload thy = MetaSimplifier.simplify true (map Thm.symmetric (inst_thms thy));

 fun unoverload_conv thy = MetaSimplifier.rewrite true (inst_thms thy);

 fun overload_conv thy = MetaSimplifier.rewrite true (map Thm.symmetric (inst_thms thy));

 fun lookup_inst_param consts params (c, T) = case get_inst_tyco consts (c, T)

  of SOME tyco => AList.lookup (op =) params (c, tyco)

   | NONE => NONE;

 fun unoverload_const thy (c_ty as (c, _)) =

   case class_of_param thy c

    of SOME class => (case get_inst_tyco (Sign.consts_of thy) c_ty

        of SOME tyco => try (param_of_inst thy) (c, tyco) |> the_default c

         | NONE => c)

     | NONE => c;

 (** instances **)

 (* class relations *)

 fun cert_classrel thy raw_rel =

   let

     val string_of_sort = Syntax.string_of_sort_global thy;

     val (c1, c2) = pairself (Sign.certify_class thy) raw_rel;

     val _ = Sign.primitive_classrel (c1, c2) (Theory.copy thy);

     val _ =

       (case subtract (op =) (all_params_of thy [c1]) (all_params_of thy [c2]) of

         [] => ()

       | xs => raise TYPE ("Class " ^ string_of_sort [c1] ^ " lacks parameter(s) " ^

           commas_quote xs ^ " of " ^ string_of_sort [c2], [], []));

   in (c1, c2) end;

 fun read_classrel thy raw_rel =

   cert_classrel thy (pairself (Sign.read_class thy) raw_rel)

     handle TYPE (msg, _, _) => error msg;

 (* declaration and definition of instances of overloaded constants *)

 fun inst_tyco_of thy (c, T) = case get_inst_tyco (Sign.consts_of thy) (c, T)

  of SOME tyco => tyco

   | NONE => error ("Illegal type for instantiation of class parameter: "

       ^ quote (c ^ " :: " ^ Syntax.string_of_typ_global thy T));

 fun declare_overloaded (c, T) thy =

   let

     val class = case class_of_param thy c

      of SOME class => class

       | NONE => error ("Not a class parameter: " ^ quote c);

     val tyco = inst_tyco_of thy (c, T);

     val name_inst = instance_name (tyco, class) ^ "_inst";

     val c' = Long_Name.base_name c ^ "_" ^ Long_Name.base_name tyco;

     val T' = Type.strip_sorts T;

   in

     thy

     |> Sign.mandatory_path name_inst

     |> Sign.declare_const [] ((Binding.name c', T'), NoSyn)

     |-> (fn const' as Const (c'', _) =>

       Thm.add_def false true

         (Binding.name (Thm.def_name c'), Logic.mk_equals (Const (c, T'), const'))

       #>> Thm.varifyT

       #-> (fn thm => add_inst_param (c, tyco) (c'', thm)

       #> PureThy.add_thm ((Binding.name c', thm), [Thm.kind_internal])

       #> snd

       #> Sign.restore_naming thy

       #> pair (Const (c, T))))

   end;

 fun define_overloaded b (c, t) thy =

   let

     val T = Term.fastype_of t;

     val tyco = inst_tyco_of thy (c, T);

     val (c', eq) = get_inst_param thy (c, tyco);

     val prop = Logic.mk_equals (Const (c', T), t);

     val b' = Thm.def_binding_optional

       (Binding.name (Long_Name.base_name c ^ "_" ^ Long_Name.base_name tyco)) b;

   in

     thy

     |> Thm.add_def false false (b', prop)

     |>> (fn thm =>  Drule.transitive_thm OF [eq, thm])

   end;

 (* primitive rules *)

 fun add_classrel th thy =

   let

     fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);

     val prop = Thm.plain_prop_of (Thm.transfer thy th);

     val rel = Logic.dest_classrel prop handle TERM _ => err ();

     val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();

   in

     thy

     |> Sign.primitive_classrel (c1, c2)

     |> put_classrel ((c1, c2), Thm.close_derivation (Drule.unconstrainTs th))

     |> perhaps complete_arities

   end;

 fun add_arity th thy =

   let

     fun err () = raise THM ("add_arity: malformed type arity", 0, [th]);

     val prop = Thm.plain_prop_of (Thm.transfer thy th);

     val (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();

     val T = Type (t, map TFree (Name.names Name.context Name.aT Ss));

     val missing_params = Sign.complete_sort thy [c]

       |> maps (these o Option.map #params o try (get_info thy))

       |> filter_out (fn (const, _) => can (get_inst_param thy) (const, t))

       |> (map o apsnd o map_atyps) (K T);

     val _ = map (Sign.certify_sort thy) Ss = Ss orelse err ();

     val th' = Drule.unconstrainTs th;

   in

     thy

     |> fold (snd oo declare_overloaded) missing_params

     |> Sign.primitive_arity (t, Ss, [c])

     |> put_arity ((t, Ss, c), th')

   end;

 (* tactical proofs *)

 fun prove_classrel raw_rel tac thy =

   let

     val ctxt = ProofContext.init thy;

     val (c1, c2) = cert_classrel thy raw_rel;

     val th = Goal.prove ctxt [] [] (Logic.mk_classrel (c1, c2)) (K tac) handle ERROR msg =>

       cat_error msg ("The error(s) above occurred while trying to prove class relation " ^

         quote (Syntax.string_of_classrel ctxt [c1, c2]));

   in

     thy

     |> PureThy.add_thms [((Binding.name

         (prefix classrel_prefix (Logic.name_classrel (c1, c2))), th), [])]

     |-> (fn [th'] => add_classrel th')

   end;

 fun prove_arity raw_arity tac thy =

   let

     val ctxt = ProofContext.init thy;

     val arity = Sign.cert_arity thy raw_arity;

     val names = map (prefix arity_prefix) (Logic.name_arities arity);

     val props = Logic.mk_arities arity;

     val ths = Goal.prove_multi ctxt [] [] props

       (fn _ => Goal.precise_conjunction_tac (length props) 1 THEN tac) handle ERROR msg =>

         cat_error msg ("The error(s) above occurred while trying to prove type arity " ^

           quote (Syntax.string_of_arity ctxt arity));

   in

     thy

     |> PureThy.add_thms (map (rpair []) (map Binding.name names ~~ ths))

     |-> fold add_arity

   end;

 (** class definitions **)

 fun split_defined n eq =

   let

     val intro =

       (eq RS Drule.equal_elim_rule2)

       |> Conjunction.curry_balanced n

       |> n = 0 ? Thm.eq_assumption 1;

     val dests =

       if n = 0 then []

       else

         (eq RS Drule.equal_elim_rule1)

         |> BalancedTree.dest (fn th =>

           (th RS Conjunction.conjunctionD1, th RS Conjunction.conjunctionD2)) n;

   in (intro, dests) end;

 fun define_class (bclass, raw_super) raw_params raw_specs thy =

   let

     val ctxt = ProofContext.init thy;

     val pp = Syntax.pp ctxt;

     (* class *)

     val bconst = Binding.map_name Logic.const_of_class bclass;

     val class = Sign.full_name thy bclass;

     val super = Sign.minimize_sort thy (Sign.certify_sort thy raw_super);

     fun check_constraint (a, S) =

       if Sign.subsort thy (super, S) then ()

       else error ("Sort constraint of type variable " ^

         setmp show_sorts true (Pretty.string_of_typ pp) (TFree (a, S)) ^

         " needs to be weaker than " ^ Pretty.string_of_sort pp super);

     (* params *)

     val params = raw_params |> map (fn p =>

       let

         val T = Sign.the_const_type thy p;

         val _ =

           (case Term.add_tvarsT T [] of

             [((a, _), S)] => check_constraint (a, S)

           | _ => error ("Exactly one type variable expected in class parameter " ^ quote p));

         val T' = Term.map_type_tvar (fn _ => TFree (Name.aT, [class])) T;

       in (p, T') end);

     (* axioms *)

     fun prep_axiom t =

       (case Term.add_tfrees t [] of

         [(a, S)] => check_constraint (a, S)

       | [] => ()

       | _ => error ("Multiple type variables in class axiom:\n" ^ Pretty.string_of_term pp t);

       t

       |> Term.map_types (Term.map_atyps (fn TFree _ => Term.aT [] | U => U))

       |> Logic.close_form);

     val axiomss = map (map (prep_axiom o Sign.cert_prop thy) o snd) raw_specs;

     val name_atts = map fst raw_specs;

     (* definition *)

     val conjs = map (curry Logic.mk_of_class (Term.aT [])) super @ flat axiomss;

     val class_eq =

       Logic.mk_equals (Logic.mk_of_class (Term.aT [], class), Logic.mk_conjunction_balanced conjs);

     val ([def], def_thy) =

       thy

       |> Sign.primitive_class (bclass, super)

       |> PureThy.add_defs false [((Binding.map_name Thm.def_name bconst, class_eq), [])];

     val (raw_intro, (raw_classrel, raw_axioms)) =

       split_defined (length conjs) def ||> chop (length super);

     (* facts *)

     val class_triv = Thm.class_triv def_thy class;

     val ([(_, [intro]), (_, classrel), (_, axioms)], facts_thy) =

       def_thy

       |> Sign.mandatory_path (Binding.name_of bconst)

       |> PureThy.note_thmss ""

         [((Binding.name introN, []), [([Drule.standard raw_intro], [])]),

          ((Binding.name superN, []), [(map Drule.standard raw_classrel, [])]),

          ((Binding.name axiomsN, []),

            [(map (fn th => Drule.standard (class_triv RS th)) raw_axioms, [])])]

       ||> Sign.restore_naming def_thy;

     (* result *)

     val axclass = make_axclass ((def, intro, axioms), params);

     val result_thy =

       facts_thy

       |> fold put_classrel (map (pair class) super ~~ classrel)

       |> Sign.add_path (Binding.name_of bconst)

       |> PureThy.note_thmss "" (name_atts ~~ map Thm.simple_fact (unflat axiomss axioms)) |> snd

       |> Sign.restore_naming facts_thy

       |> map_axclasses (fn (axclasses, parameters) =>

         (Symtab.update (class, axclass) axclasses,

           fold (fn (x, _) => add_param pp (x, class)) params parameters));

   in (class, result_thy) end;

 (** axiomatizations **)

 local

 fun axiomatize prep mk name add raw_args thy =

   let

     val args = prep thy raw_args;

     val specs = mk args;

     val names = name args;

   in

     thy

     |> PureThy.add_axioms (map (rpair []) (map Binding.name names ~~ specs))

     |-> fold add

   end;

 fun ax_classrel prep =

   axiomatize (map o prep) (map Logic.mk_classrel)

     (map (prefix classrel_prefix o Logic.name_classrel)) add_classrel;

 fun ax_arity prep =

   axiomatize prep Logic.mk_arities

     (map (prefix arity_prefix) o Logic.name_arities) add_arity;

 fun class_const c =

   (Logic.const_of_class c, Term.itselfT (Term.aT []) --> propT);

 fun ax_class prep_class prep_classrel (bclass, raw_super) thy =

   let

     val class = Sign.full_name thy bclass;

     val super = map (prep_class thy) raw_super |> Sign.minimize_sort thy;

   in

     thy

     |> Sign.primitive_class (bclass, super)

     |> ax_classrel prep_classrel (map (fn c => (class, c)) super)

     |> Theory.add_deps "" (class_const class) (map class_const super)

   end;

 in

 val axiomatize_class = ax_class Sign.certify_class cert_classrel;

 val axiomatize_class_cmd = ax_class Sign.read_class read_classrel;

 val axiomatize_classrel = ax_classrel cert_classrel;

 val axiomatize_classrel_cmd = ax_classrel read_classrel;

 val axiomatize_arity = ax_arity Sign.cert_arity;

 val axiomatize_arity_cmd = ax_arity Sign.read_arity;

 end;

 (** explicit derivations -- cached **)

 datatype cache = Types of (class * thm) list Typtab.table;

 local

 fun lookup_type (Types cache) = AList.lookup (op =) o Typtab.lookup_list cache;

 fun insert_type T der (Types cache) = Types (Typtab.insert_list (eq_fst op =) (T, der) cache);

 fun derive_type _ (_, []) = []

   | derive_type thy (typ, sort) =

       let

         val certT = Thm.ctyp_of thy;

         val vars = Term.fold_atyps

             (fn T as TFree (_, S) => insert (eq_fst op =) (T, S)

               | T as TVar (_, S) => insert (eq_fst op =) (T, S)

               | _ => I) typ [];

         val hyps = vars

           |> map (fn (T, S) => (T, Thm.of_sort (certT T, S) ~~ S));

         val ths = (typ, sort)

           |> Sorts.of_sort_derivation (Syntax.pp_global thy) (Sign.classes_of thy)

            {class_relation =

               fn (th, c1) => fn c2 => th RS the_classrel thy (c1, c2),

             type_constructor =

               fn a => fn dom => fn c =>

                 let val Ss = map (map snd) dom and ths = maps (map fst) dom

                 in ths MRS the_arity thy a (c, Ss) end,

             type_variable =

               the_default [] o AList.lookup (op =) hyps};

       in ths end;

 in

 fun of_sort thy (typ, sort) cache =

   let

     val sort' = filter (is_none o lookup_type cache typ) sort;

     val ths' = derive_type thy (typ, sort')

       handle ERROR msg => cat_error msg ("The error(s) above occurred for sort derivation: " ^

         Syntax.string_of_typ_global thy typ ^ " :: " ^ Syntax.string_of_sort_global thy sort');

     val cache' = cache |> fold (insert_type typ) (sort' ~~ ths');

     val ths =

       sort |> map (fn c =>

         Goal.finish (the (lookup_type cache' typ c) RS

           Goal.init (Thm.cterm_of thy (Logic.mk_of_class (typ, c))))

         |> Thm.adjust_maxidx_thm ~1);

   in (ths, cache') end;

 end;

 val cache = Types Typtab.empty;

 end;