(*  Title:      Pure/axclass.ML

    ID:         $Id$

    Author:     Markus Wenzel, TU Muenchen

Type classes as parameter records and predicates, with explicit

definitions and proofs.

*)

signature AX_CLASS =

sig

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

    ((bstring * attribute list) * term list) list -> theory -> class * theory

  val define_class_params: bstring * class list -> ((bstring * typ) * mixfix) list ->

    (theory -> (string * typ) list -> ((bstring * attribute list) * term list) list) ->

    string list -> theory ->

    (class * ((string * typ) list * thm list)) * 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_definition: 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 params_of_class: theory -> class -> string * (string * typ) list

  val print_axclasses: theory -> unit

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

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

  val axiomatize_class: bstring * xstring list -> theory -> theory

  val axiomatize_class_i: bstring * class list -> theory -> theory

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

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

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

  val axiomatize_arity_i: arity -> theory -> theory

  type cache

  val cache: cache

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

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));

(* setup data *)

structure AxClassData = TheoryDataFun

(

  type T = axclasses * instances;

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

  val copy = I;

  val extend = I;

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

    (merge_axclasses pp (axclasses1, axclasses2), (merge_instances (instances1, instances2)));

);

(* maintain axclasses *)

val get_axclasses = #1 o AxClassData.get;

fun map_axclasses f = AxClassData.map (apfst f);

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

fun get_definition 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));

fun params_of_class thy class = (Name.aT, #params (get_definition thy class));

(* maintain instances *)

val get_instances = #2 o AxClassData.get;

fun map_instances f = AxClassData.map (apsnd f);

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 put_arity ((t, Ss, c), th) = map_instances (fn (classrel, arities) =>

  (classrel, arities |> Symtab.insert_list (eq_fst op =) (t, ((c, Ss), th))));

(* print data *)

fun print_axclasses thy =

  let

    val axclasses = #1 (get_axclasses thy);

    val ctxt = ProofContext.init thy;

    fun pretty_axclass (class, AxClass {def, intro, axioms, params}) =

      Pretty.block (Pretty.fbreaks

       [Pretty.block

          [Pretty.str "class ", Syntax.pretty_sort ctxt [class], Pretty.str ":"],

        Pretty.strs ("parameters:" :: params_of thy class),

        ProofContext.pretty_fact ctxt ("def", [def]),

        ProofContext.pretty_fact ctxt (introN, [intro]),

        ProofContext.pretty_fact ctxt (axiomsN, axioms)]);

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

(** instances **)

(* class relations *)

fun cert_classrel thy raw_rel =

  let

    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 " ^ Sign.string_of_sort thy [c1] ^ " lacks parameter(s) " ^

          commas_quote xs ^ " of " ^ Sign.string_of_sort thy [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;

(* 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), Drule.unconstrainTs th)

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

  in

    thy

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

    |> put_arity ((t, Ss, c), 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)) (fn _ => 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 [((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 []) (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) params raw_specs thy =

  let

    val ctxt = ProofContext.init thy;

    val pp = ProofContext.pp ctxt;

    (* prepare specification *)

    val bconst = 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 prep_axiom t =

      (case Term.add_tfrees t [] of

        [(a, S)] =>

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

          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)

      | [] => t

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

      |> 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_inclass (Term.aT [])) super @ flat axiomss;

    val class_eq =

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

    val ([def], def_thy) =

      thy

      |> Sign.primitive_class (bclass, super)

      |> PureThy.add_defs_i false [((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

      |> PureThy.note_thmss_qualified "" bconst

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

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

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

    (* params *)

    val params_typs = map (fn param =>

      let

        val ty = Sign.the_const_type thy param;

        val _ = case Term.typ_tvars ty

         of [_] => ()

          | _ => error ("Exactly one type variable required in parameter " ^ quote param);

        val ty' = Term.map_type_tvar (fn _ => TFree (Name.aT, [class])) ty;

      in (param, ty') end) params;

    (* result *)

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

    val result_thy =

      facts_thy

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

      |> Sign.add_path bconst

      |> PureThy.note_thmss_i "" (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;

(** axclasses with implicit parameter handling **)

fun define_class_params (name, raw_superclasses) raw_consts raw_dep_axioms other_consts thy =

  let

    val superclasses = map (Sign.certify_class thy) raw_superclasses;

    val consts = (map o apfst o apsnd) (Sign.certify_typ thy) raw_consts;

    val prefix = Logic.const_of_class name;

    fun mk_const_name c = NameSpace.map_base (NameSpace.append prefix)

      (Sign.full_name thy c);

    fun add_const ((c, ty), syn) =

      Sign.add_consts_authentic [] [(c, ty, syn)]

      #> pair (mk_const_name c, ty);

    fun mk_axioms cs thy =

      raw_dep_axioms thy cs

      |> (map o apsnd o map) (Sign.cert_prop thy)

      |> rpair thy;

    fun add_constraint class (c, ty) =

      Sign.add_const_constraint (c, SOME

        (Term.map_type_tfree (fn (v, _) => TFree (v, [class])) ty));

  in

    thy

    |> Sign.add_path prefix

    |> fold_map add_const consts

    ||> Sign.restore_naming thy

    |-> (fn cs => mk_axioms cs

    #-> (fn axioms_prop => define_class (name, superclasses)

           (map fst cs @ other_consts) axioms_prop

    #-> (fn class => `(fn thy => get_definition thy class)

    #-> (fn {axioms, ...} => fold (add_constraint class) cs

    #> pair (class, (cs, axioms))))))

  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_i (map (rpair []) (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.read_class read_classrel;

val axiomatize_class_i = ax_class Sign.certify_class cert_classrel;

val axiomatize_classrel = ax_classrel read_classrel;

val axiomatize_classrel_i = ax_classrel cert_classrel;

val axiomatize_arity = ax_arity Sign.read_arity;

val axiomatize_arity_i = ax_arity Sign.cert_arity;

end;

(** explicit derivations -- cached **)

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

val cache = Types Typtab.empty;

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 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, Drule.sort_triv thy (T, S) ~~ S));

        val ths = (typ, sort) |> Sorts.of_sort_derivation (Sign.pp 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: " ^

        Sign.string_of_typ thy typ ^ " :: " ^ Sign.string_of_sort 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_inclass (typ, c))))

        |> Thm.adjust_maxidx_thm ~1);

  in (ths, cache') end;

end;

end;