# File pcpodef_package.ML

    (*  Title:      HOLCF/pcpodef_package.ML
        ID:         $Id: pcpodef_package.ML,v 1.5 2005/09/13 20:21:46 wenzelm Exp $
        Author:     Brian Huffman

    Gordon/HOL-style type definitions for HOLCF.
    *)

    signature PCPODEF_PACKAGE =
    sig
      val quiet_mode: bool ref
      val pcpodef_proof: (bool * string) * (bstring * string list * mixfix) * string
        * (string * string) option -> theory -> Proof.state
      val pcpodef_proof_i: (bool * string) * (bstring * string list * mixfix) * term
        * (string * string) option -> theory -> Proof.state
      val cpodef_proof: (bool * string) * (bstring * string list * mixfix) * string
        * (string * string) option -> theory -> Proof.state
      val cpodef_proof_i: (bool * string) * (bstring * string list * mixfix) * term
        * (string * string) option -> theory -> Proof.state
    end;

    structure PcpodefPackage: PCPODEF_PACKAGE =
    struct

    (** theory context references **)

    val typedef_po = thm "typedef_po";
    val typedef_cpo = thm "typedef_cpo";
    val typedef_pcpo = thm "typedef_pcpo";
    val typedef_lub = thm "typedef_lub";
    val typedef_thelub = thm "typedef_thelub";
    val cont_Rep = thm "typedef_cont_Rep";
    val cont_Abs = thm "typedef_cont_Abs";
    val Rep_strict = thm "typedef_Rep_strict";
    val Abs_strict = thm "typedef_Abs_strict";
    val Rep_defined = thm "typedef_Rep_defined";
    val Abs_defined = thm "typedef_Abs_defined";

    (** type definitions **)

    (* messages *)

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

    (* prepare_cpodef *)

    fun read_term thy used s =
      #1 (Thm.read_def_cterm (thy, K NONE, K NONE) used true (s, HOLogic.typeT));

    fun cert_term thy _ t = Thm.cterm_of thy t handle TERM (msg, _) => error msg;

    fun err_in_cpodef name =
      error ("The error(s) above occurred in cpodef " ^ quote name);

    fun adm_const T = Const ("Adm.adm", (T --> HOLogic.boolT) --> HOLogic.boolT);
    fun mk_adm (x, T, P) = adm_const T $ absfree (x, T, P);

    fun gen_prepare_pcpodef prep_term pcpo def name (t, vs, mx) raw_set opt_morphs thy =
      let
        val full = Sign.full_name thy;

        (*rhs*)
        val full_name = full name;
        val cset = prep_term thy vs raw_set;
        val {T = setT, t = set, ...} = Thm.rep_cterm cset;
        val rhs_tfrees = term_tfrees set;
        val oldT = HOLogic.dest_setT setT handle TYPE _ =>
          error ("Not a set type: " ^ quote (Sign.string_of_typ thy setT));
        fun mk_nonempty A =
          HOLogic.mk_exists ("x", oldT, HOLogic.mk_mem (Free ("x", oldT), A));
        fun mk_admissible A =
          mk_adm ("x", oldT, HOLogic.mk_mem (Free ("x", oldT), A));
        fun mk_UU_mem A = HOLogic.mk_mem (Const ("Pcpo.UU", oldT), A);
        val goal = if pcpo
          then HOLogic.mk_Trueprop (HOLogic.mk_conj (mk_UU_mem set, mk_admissible set))
          else HOLogic.mk_Trueprop (HOLogic.mk_conj (mk_nonempty set, mk_admissible set));

        (*lhs*)
        val lhs_tfrees = map (fn v => (v, AList.lookup (op =) rhs_tfrees v |> the_default HOLogic.typeS)) vs;
        val lhs_sorts = map snd lhs_tfrees;
        val tname = Syntax.type_name t mx;
        val full_tname = full tname;
        val newT = Type (full_tname, map TFree lhs_tfrees);

        val (Rep_name, Abs_name) = getOpt (opt_morphs, ("Rep_" ^ name, "Abs_" ^ name));
        val RepC = Const (full Rep_name, newT --> oldT);
        fun lessC T = Const ("Porder.op <<", T --> T --> HOLogic.boolT);
        val less_def = ("less_" ^ name ^ "_def", Logic.mk_equals (lessC newT,
          Abs ("x", newT, Abs ("y", newT, lessC oldT $ (RepC $ Bound 1) $ (RepC $ Bound 0)))));

        fun option_fold_rule NONE = I
          | option_fold_rule (SOME def) = fold_rule [def];

        fun make_po tac theory =
          theory
          |> TypedefPackage.add_typedef_i def (SOME name) (t, vs, mx) set opt_morphs tac
          |>> AxClass.add_inst_arity_i (full_tname, lhs_sorts, ["Porder.sq_ord"])
              (AxClass.intro_classes_tac [])
          |>>> PureThy.add_defs_i true [Thm.no_attributes less_def]
          |> (fn (theory', ({type_definition, set_def, ...}, [less_definition])) =>
            theory'
            |> AxClass.add_inst_arity_i (full_tname, lhs_sorts, ["Porder.po"])
                (Tactic.rtac (typedef_po OF [type_definition, less_definition]) 1)
            |> rpair (type_definition, less_definition, set_def));

        fun make_cpo admissible (theory, defs as (type_def, less_def, set_def)) =
          let
            val admissible' = option_fold_rule set_def admissible;
            val cpo_thms = [type_def, less_def, admissible'];
            val (theory', _) =
              theory
              |> AxClass.add_inst_arity_i (full_tname, lhs_sorts, ["Pcpo.cpo"])
                (Tactic.rtac (typedef_cpo OF cpo_thms) 1)
              |> Theory.add_path name
              |> PureThy.add_thms
                ([(("adm_" ^ name, admissible'), []),
                  (("cont_" ^ Rep_name, cont_Rep OF cpo_thms), []),
                  (("cont_" ^ Abs_name, cont_Abs OF cpo_thms), []),
                  (("lub_"    ^ name, typedef_lub    OF cpo_thms), []),
                  (("thelub_" ^ name, typedef_thelub OF cpo_thms), [])])
              |>> Theory.parent_path;
          in (theory', defs) end;

        fun make_pcpo UUmem (theory, defs as (type_def, less_def, set_def)) =
          let
            val UUmem' = option_fold_rule set_def UUmem;
            val pcpo_thms = [type_def, less_def, UUmem'];
            val (theory', _) =
              theory
              |> AxClass.add_inst_arity_i (full_tname, lhs_sorts, ["Pcpo.pcpo"])
                (Tactic.rtac (typedef_pcpo OF pcpo_thms) 1)
              |> Theory.add_path name
              |> PureThy.add_thms
                ([((Rep_name ^ "_strict", Rep_strict OF pcpo_thms), []),
                  ((Abs_name ^ "_strict", Abs_strict OF pcpo_thms), []),
                  ((Rep_name ^ "_defined", Rep_defined OF pcpo_thms), []),
                  ((Abs_name ^ "_defined", Abs_defined OF pcpo_thms), [])
                  ])
              |>> Theory.parent_path;
          in (theory', defs) end;

        fun pcpodef_result (theory, UUmem_admissible) =
          let
            val UUmem = UUmem_admissible RS conjunct1;
            val admissible = UUmem_admissible RS conjunct2;
          in
            theory
            |> make_po (Tactic.rtac exI 1 THEN Tactic.rtac UUmem 1)
            |> make_cpo admissible
            |> make_pcpo UUmem
            |> (fn (theory', (type_def, _, _)) => (theory', type_def))
          end;

        fun cpodef_result (theory, nonempty_admissible) =
          let
            val nonempty = nonempty_admissible RS conjunct1;
            val admissible = nonempty_admissible RS conjunct2;
          in
            theory
            |> make_po (Tactic.rtac nonempty 1)
            |> make_cpo admissible
            |> (fn (theory', (type_def, _, _)) => (theory', type_def))
          end;

      in (goal, if pcpo then pcpodef_result else cpodef_result) end
      handle ERROR => err_in_cpodef name;

    (* cpodef_proof interface *)

    fun gen_pcpodef_proof prep_term pcpo ((def, name), typ, set, opt_morphs) thy =
      let
        val (goal, att) =
          gen_prepare_pcpodef prep_term pcpo def name typ set opt_morphs thy;
      in IsarThy.theorem_i Drule.internalK ("", [att]) (goal, ([], [])) thy end;

    val pcpodef_proof = gen_pcpodef_proof read_term true;
    val pcpodef_proof_i = gen_pcpodef_proof cert_term true;

    val cpodef_proof = gen_pcpodef_proof read_term false;
    val cpodef_proof_i = gen_pcpodef_proof cert_term false;

    (** outer syntax **)

    local structure P = OuterParse and K = OuterKeyword in

    (* copied from HOL/Tools/typedef_package.ML *)
    val typedef_proof_decl =
      Scan.optional (P.$$$ "(" |--
          ((P.$$$ "open" >> K false) -- Scan.option P.name || P.name >> (fn s => (true, SOME s)))
            --| P.$$$ ")") (true, NONE) --
        (P.type_args -- P.name) -- P.opt_infix -- (P.$$$ "=" |-- P.term) --
        Scan.option (P.$$$ "morphisms" |-- P.!!! (P.name -- P.name));

    fun mk_pcpodef_proof pcpo ((((((def, opt_name), (vs, t)), mx), A), morphs)) =
      (if pcpo then pcpodef_proof else cpodef_proof)
        ((def, getOpt (opt_name, Syntax.type_name t mx)), (t, vs, mx), A, morphs);

    val pcpodefP =
      OuterSyntax.command "pcpodef" "HOLCF type definition (requires admissibility proof)" K.thy_goal
        (typedef_proof_decl >>
          (Toplevel.print oo (Toplevel.theory_to_proof o mk_pcpodef_proof true)));

    val cpodefP =
      OuterSyntax.command "cpodef" "HOLCF type definition (requires admissibility proof)" K.thy_goal
        (typedef_proof_decl >>
          (Toplevel.print oo (Toplevel.theory_to_proof o mk_pcpodef_proof false)));

    val _ = OuterSyntax.add_parsers [pcpodefP, cpodefP];

    end;

    end;