(*  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 thynames_of_arity: theory -> class * 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 *  (*classrel theorems*)

  ((class * sort list) * (thm * string)) list Symtab.table;  (*arity theorems with theory name*)

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 = Theory_Data_PP

(

  type T = axclasses * (instances * inst_params);

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

  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 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 thynames_of_arity thy (c, a) =

  Symtab.lookup_list (#2 (get_instances thy)) a

  |> map_filter (fn ((c', _), (_, name)) => if c = c' then SOME name else NONE)

  |> rev;

fun insert_arity_completions thy (t, ((c, Ss), (th, thy_name))) 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 completions = map (fn c1 => (Sorts.weaken algebra

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

        |> Thm.close_derivation, c1)) super_class_completions;

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

      completions arities;

  in (null completions, arities') end;

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

  let

    val arity' = (t, ((c, Ss), (th, Context.theory_name thy)));

  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 (finished, arities') = arities

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

  in

    if forall I finished 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, _)) =

  if is_some (class_of_param thy c)

  then 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

  else 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 (ProofContext.read_class (ProofContext.init 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.qualified_path true (Binding.name 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_global

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

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

      #> snd

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

    ||> Sign.restore_naming thy

  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 raw_th thy =

  let

    val th = Thm.strip_shyps (Thm.transfer thy raw_th);

    val prop = Thm.plain_prop_of th;

    fun err () = raise THM ("add_classrel: malformed class relation", 0, [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 raw_th thy =

  let

    val th = Thm.strip_shyps (Thm.transfer thy raw_th);

    val prop = Thm.plain_prop_of th;

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

  in

    thy

    |> fold (snd oo declare_overloaded) missing_params

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

    |> put_arity ((t, Ss, c), Thm.close_derivation (Drule.unconstrainTs 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 = ProofContext.cert_arity ctxt 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)

        |> Balanced_Tree.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_CRITICAL 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 = Logic.mk_of_sort (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 [((Thm.def_binding 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.qualified_path true bconst

      |> PureThy.note_thmss ""

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

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

         ((Binding.name axiomsN, []),

           [(map (fn th => Drule.export_without_context (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.qualified_path false 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 o ProofContext.init) 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 (ProofContext.read_class o ProofContext.init) read_classrel;

val axiomatize_classrel = ax_classrel cert_classrel;

val axiomatize_classrel_cmd = ax_classrel read_classrel;

val axiomatize_arity = ax_arity ProofContext.cert_arity;

val axiomatize_arity_cmd = ax_arity ProofContext.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 (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

          (Syntax.init_pretty_global thy)

          (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;