(*  Title:      HOL/Tools/typedef_package.ML

     ID:         $Id$

     Author:     Markus Wenzel and Stefan Berghofer, TU Muenchen

 Gordon/HOL-style type definitions.

 *)

 signature TYPEDEF_PACKAGE =

 sig

   val quiet_mode: bool ref

   val add_typedecls: (bstring * string list * mixfix) list -> theory -> theory

   val add_typedef_x: string -> bstring * string list * mixfix ->

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

   val add_typedef: bool -> string option -> bstring * string list * mixfix ->

     string -> (bstring * bstring) option -> tactic -> theory -> theory *

     {type_definition: thm, set_def: thm option, Rep: thm, Rep_inverse: thm,

       Abs_inverse: thm, Rep_inject: thm, Abs_inject: thm, Rep_cases: thm, Abs_cases: thm,

       Rep_induct: thm, Abs_induct: thm}

   val add_typedef_i: bool -> string option -> bstring * string list * mixfix ->

     term -> (bstring * bstring) option -> tactic -> theory -> theory *

     {type_definition: thm, set_def: thm option, Rep: thm, Rep_inverse: thm,

       Abs_inverse: thm, Rep_inject: thm, Abs_inject: thm, Rep_cases: thm, Abs_cases: thm,

       Rep_induct: thm, Abs_induct: thm}

   val typedef_proof: (bool * string) * (bstring * string list * mixfix) * string

     * (string * string) option -> bool -> theory -> ProofHistory.T

   val typedef_proof_i: (bool * string) * (bstring * string list * mixfix) * term

     * (string * string) option -> bool -> theory -> ProofHistory.T

   val setup: (theory -> theory) list

 end;

 structure TypedefPackage: TYPEDEF_PACKAGE =

 struct

 (** theory context references **)

 val type_definitionN = "Typedef.type_definition";

 val Rep = thm "type_definition.Rep";

 val Rep_inverse = thm "type_definition.Rep_inverse";

 val Abs_inverse = thm "type_definition.Abs_inverse";

 val Rep_inject = thm "type_definition.Rep_inject";

 val Abs_inject = thm "type_definition.Abs_inject";

 val Rep_cases = thm "type_definition.Rep_cases";

 val Abs_cases = thm "type_definition.Abs_cases";

 val Rep_induct = thm "type_definition.Rep_induct";

 val Abs_induct = thm "type_definition.Abs_induct";

 (** theory data **)

 structure TypedefData = TheoryDataFun

 (struct

   val name = "HOL/typedef";

   type T = (typ * typ * string * string) Symtab.table;

   val empty = Symtab.empty;

   val copy = I;

   val extend = I;

   fun merge _ (tabs: T * T) = Symtab.merge (op =) tabs;

   fun print _ _ = ();

 end);

 fun put_typedef newT oldT Abs_name Rep_name thy =

   TypedefData.put (Symtab.update_new

     ((fst (dest_Type newT), (newT, oldT, Abs_name, Rep_name)),

      TypedefData.get thy)) thy;

 (** type declarations **)

 fun add_typedecls decls thy =

   let

     fun arity_of (raw_name, args, mx) =

       (Sign.full_name thy (Syntax.type_name raw_name mx),

         replicate (length args) HOLogic.typeS, HOLogic.typeS);

   in

     if can (Theory.assert_super HOL.thy) thy then

       thy |> Theory.add_typedecls decls

       |> Theory.add_arities_i (map arity_of decls)

     else thy |> Theory.add_typedecls decls

   end;

 (** type definitions **)

 (* messages *)

 val quiet_mode = ref false;

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

 (* prove_nonempty -- tactical version *)        (*exception ERROR*)

 fun prove_nonempty thy cset goal (witn1_tac, witn_names, witn_thms, witn2_tac) =

   let

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

     val thms = PureThy.get_thmss thy (map Name witn_names) @ witn_thms;

     val tac =

       witn1_tac THEN

       TRY (rewrite_goals_tac (List.filter is_def thms)) THEN

       TRY (REPEAT_FIRST (resolve_tac (filter_out is_def thms))) THEN

       getOpt (witn2_tac, TRY (ALLGOALS (CLASET' blast_tac)));

   in

     message ("Proving non-emptiness of set " ^ quote (string_of_cterm cset) ^ " ...");

     Tactic.prove thy [] [] goal (K tac)

   end

   handle ERROR => error ("Failed to prove non-emptiness of " ^ quote (string_of_cterm cset));

 (* prepare_typedef *)

 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_typedef name =

   error ("The error(s) above occurred in typedef " ^ quote name);

 fun prepare_typedef prep_term def name (t, vs, mx) raw_set opt_morphs thy =

   let

     val _ = Theory.requires thy "Typedef" "typedefs";

     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_Trueprop (HOLogic.mk_exists ("x", oldT, HOLogic.mk_mem (Free ("x", oldT), A)));

     val goal = mk_nonempty set;

     val vname = take_suffix Symbol.is_digit (Symbol.explode name)

       |> apfst implode |> apsnd (#1 o Library.read_int);

     val goal_pat = mk_nonempty (Var (vname, setT));

     (*lhs*)

     val lhs_tfrees = map (fn v => (v, getOpt (assoc (rhs_tfrees, v), HOLogic.typeS))) vs;

     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 setC = Const (full_name, setT);

     val RepC = Const (full Rep_name, newT --> oldT);

     val AbsC = Const (full Abs_name, oldT --> newT);

     val x_new = Free ("x", newT);

     val y_old = Free ("y", oldT);

     val set' = if def then setC else set;

     val typedef_name = "type_definition_" ^ name;

     val typedefC =

       Const (type_definitionN, (newT --> oldT) --> (oldT --> newT) --> setT --> HOLogic.boolT);

     val typedef_prop =

       Logic.mk_implies (goal, HOLogic.mk_Trueprop (typedefC $ RepC $ AbsC $ set'));

     fun typedef_result (theory, nonempty) =

       theory

       |> put_typedef newT oldT (full Abs_name) (full Rep_name)

       |> add_typedecls [(t, vs, mx)]

       |> Theory.add_consts_i

        ((if def then [(name, setT, NoSyn)] else []) @

         [(Rep_name, newT --> oldT, NoSyn),

          (Abs_name, oldT --> newT, NoSyn)])

       |> (if def then (apsnd (SOME o hd) oo (PureThy.add_defs_i false o map Thm.no_attributes))

            [Logic.mk_defpair (setC, set)]

           else rpair NONE)

       |>>> PureThy.add_axioms_i [((typedef_name, typedef_prop),

           [apsnd (fn cond_axm => Drule.standard (nonempty RS cond_axm))])]

       |>> Theory.add_finals_i false [RepC, AbsC]

       |> (fn (theory', (set_def, [type_definition])) =>

         let

           fun make th = Drule.standard (th OF [type_definition]);

           val (theory'', [Rep, Rep_inverse, Abs_inverse, Rep_inject, Abs_inject,

               Rep_cases, Abs_cases, Rep_induct, Abs_induct]) =

             theory'

             |> Theory.add_path name

             |> PureThy.add_thms

               ([((Rep_name, make Rep), []),

                 ((Rep_name ^ "_inverse", make Rep_inverse), []),

                 ((Abs_name ^ "_inverse", make Abs_inverse), []),

                 ((Rep_name ^ "_inject", make Rep_inject), []),

                 ((Abs_name ^ "_inject", make Abs_inject), []),

                 ((Rep_name ^ "_cases", make Rep_cases),

                   [RuleCases.case_names [Rep_name], InductAttrib.cases_set_global full_name]),

                 ((Abs_name ^ "_cases", make Abs_cases),

                   [RuleCases.case_names [Abs_name], InductAttrib.cases_type_global full_tname]),

                 ((Rep_name ^ "_induct", make Rep_induct),

                   [RuleCases.case_names [Rep_name], InductAttrib.induct_set_global full_name]),

                 ((Abs_name ^ "_induct", make Abs_induct),

                   [RuleCases.case_names [Abs_name], InductAttrib.induct_type_global full_tname])])

             |>> Theory.parent_path;

           val result = {type_definition = type_definition, set_def = set_def,

             Rep = Rep, Rep_inverse = Rep_inverse, Abs_inverse = Abs_inverse,

             Rep_inject = Rep_inject, Abs_inject = Abs_inject, Rep_cases = Rep_cases,

             Abs_cases = Abs_cases, Rep_induct = Rep_induct, Abs_induct = Abs_induct};

         in ((theory'', type_definition), result) end);

     (* errors *)

     fun show_names pairs = commas_quote (map fst pairs);

     val illegal_vars =

       if null (term_vars set) andalso null (term_tvars set) then []

       else ["Illegal schematic variable(s) on rhs"];

     val dup_lhs_tfrees =

       (case duplicates lhs_tfrees of [] => []

       | dups => ["Duplicate type variables on lhs: " ^ show_names dups]);

     val extra_rhs_tfrees =

       (case gen_rems (op =) (rhs_tfrees, lhs_tfrees) of [] => []

       | extras => ["Extra type variables on rhs: " ^ show_names extras]);

     val illegal_frees =

       (case term_frees set of [] => []

       | xs => ["Illegal variables on rhs: " ^ show_names (map dest_Free xs)]);

     val errs = illegal_vars @ dup_lhs_tfrees @ extra_rhs_tfrees @ illegal_frees;

     val _ = if null errs then () else error (cat_lines errs);

     (*test theory errors now!*)

     val test_thy = Theory.copy thy;

     val _ = (test_thy,

       setmp quick_and_dirty true (SkipProof.make_thm test_thy) goal) |> typedef_result;

   in (cset, goal, goal_pat, typedef_result) end

   handle ERROR => err_in_typedef name;

 (* add_typedef interfaces *)

 fun gen_typedef prep_term def name typ set opt_morphs tac1 names thms tac2 thy =

   let

     val (cset, goal, _, typedef_result) =

       prepare_typedef prep_term def name typ set opt_morphs thy;

     val non_empty = prove_nonempty thy cset goal (tac1, names, thms, tac2);

     val ((thy', _), result) = (thy, non_empty) |> typedef_result;

   in (thy', result) end;

 fun sane_typedef prep_term def opt_name typ set opt_morphs tac =

   gen_typedef prep_term def

     (getOpt (opt_name, #1 typ)) typ set opt_morphs all_tac [] [] (SOME tac);

 fun add_typedef_x name typ set names thms tac =

   #1 o gen_typedef read_term true name typ set NONE (Tactic.rtac exI 1) names thms tac;

 val add_typedef = sane_typedef read_term;

 val add_typedef_i = sane_typedef cert_term;

 (* typedef_proof interface *)

 fun gen_typedef_proof prep_term ((def, name), typ, set, opt_morphs) int thy =

   let

     val (_, goal, goal_pat, att_result) =

       prepare_typedef prep_term def name typ set opt_morphs thy;

     val att = #1 o att_result;

   in thy |> IsarThy.theorem_i Drule.internalK (("", [att]), (goal, ([goal_pat], []))) int end;

 val typedef_proof = gen_typedef_proof read_term;

 val typedef_proof_i = gen_typedef_proof cert_term;

 (** trivial code generator **)

 fun typedef_codegen thy gr dep brack t =

   let

     fun mk_fun s T ts =

       let

         val (gr', _) = Codegen.invoke_tycodegen thy dep false (gr, T);

         val (gr'', ps) =

           foldl_map (Codegen.invoke_codegen thy dep true) (gr', ts);

         val id = Codegen.mk_const_id thy s

       in SOME (gr'', Codegen.mk_app brack (Pretty.str id) ps) end;

     fun get_name (Type (tname, _)) = tname

       | get_name _ = "";

     fun lookup f T = getOpt (Option.map f (Symtab.lookup

       (TypedefData.get thy, get_name T)), "")

   in

     (case strip_comb t of

        (Const (s, Type ("fun", [T, U])), ts) =>

          if lookup #4 T = s andalso

            is_none (Codegen.get_assoc_type thy (get_name T))

          then mk_fun s T ts

          else if lookup #3 U = s andalso

            is_none (Codegen.get_assoc_type thy (get_name U))

          then mk_fun s U ts

          else NONE

      | _ => NONE)

   end;

 fun mk_tyexpr [] s = Pretty.str s

   | mk_tyexpr [p] s = Pretty.block [p, Pretty.str (" " ^ s)]

   | mk_tyexpr ps s = Pretty.list "(" (") " ^ s) ps;

 fun typedef_tycodegen thy gr dep brack (Type (s, Ts)) =

       (case Symtab.lookup (TypedefData.get thy, s) of

          NONE => NONE

        | SOME (newT as Type (tname, Us), oldT, Abs_name, Rep_name) =>

            if isSome (Codegen.get_assoc_type thy tname) then NONE else

            let

              val Abs_id = Codegen.mk_const_id thy Abs_name;

              val Rep_id = Codegen.mk_const_id thy Rep_name;

              val ty_id = Codegen.mk_type_id thy s;

              val (gr', qs) = foldl_map

                (Codegen.invoke_tycodegen thy dep (length Ts = 1)) (gr, Ts);

              val gr'' = Graph.add_edge (Abs_id, dep) gr' handle Graph.UNDEF _ =>

                let

                  val (gr'', p :: ps) = foldl_map

                    (Codegen.invoke_tycodegen thy Abs_id false)

                    (Graph.add_edge (Abs_id, dep)

                       (Graph.new_node (Abs_id, (NONE, "")) gr'), oldT :: Us);

                  val s =

                    Pretty.string_of (Pretty.block [Pretty.str "datatype ",

                      mk_tyexpr ps ty_id,

                      Pretty.str " =", Pretty.brk 1, Pretty.str (Abs_id ^ " of"),

                      Pretty.brk 1, p, Pretty.str ";"]) ^ "\n\n" ^

                    Pretty.string_of (Pretty.block [Pretty.str ("fun " ^ Rep_id),

                      Pretty.brk 1, Pretty.str ("(" ^ Abs_id), Pretty.brk 1,

                      Pretty.str "x) = x;"]) ^ "\n\n" ^

                    (if "term_of" mem !Codegen.mode then

                       Pretty.string_of (Pretty.block [Pretty.str "fun ",

                         Codegen.mk_term_of thy false newT, Pretty.brk 1,

                         Pretty.str ("(" ^ Abs_id), Pretty.brk 1,

                         Pretty.str "x) =", Pretty.brk 1,

                         Pretty.block [Pretty.str ("Const (\"" ^ Abs_name ^ "\","),

                           Pretty.brk 1, Codegen.mk_type false (oldT --> newT),

                           Pretty.str ")"], Pretty.str " $", Pretty.brk 1,

                         Codegen.mk_term_of thy false oldT, Pretty.brk 1,

                         Pretty.str "x;"]) ^ "\n\n"

                     else "") ^

                    (if "test" mem !Codegen.mode then

                       Pretty.string_of (Pretty.block [Pretty.str "fun ",

                         Codegen.mk_gen thy false [] "" newT, Pretty.brk 1,

                         Pretty.str "i =", Pretty.brk 1,

                         Pretty.block [Pretty.str (Abs_id ^ " ("),

                           Codegen.mk_gen thy false [] "" oldT, Pretty.brk 1,

                           Pretty.str "i);"]]) ^ "\n\n"

                     else "")

                in Graph.map_node Abs_id (K (NONE, s)) gr'' end

            in

              SOME (gr'', mk_tyexpr qs ty_id)

            end)

   | typedef_tycodegen thy gr dep brack _ = NONE;

 val setup =

   [TypedefData.init,

    Codegen.add_codegen "typedef" typedef_codegen,

    Codegen.add_tycodegen "typedef" typedef_tycodegen];

 (** outer syntax **)

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

 val typedeclP =

   OuterSyntax.command "typedecl" "type declaration (HOL)" K.thy_decl

     (P.type_args -- P.name -- P.opt_infix >> (fn ((vs, t), mx) =>

       Toplevel.theory (add_typedecls [(t, vs, mx)])));

 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_typedef_proof ((((((def, opt_name), (vs, t)), mx), A), morphs)) =

   typedef_proof ((def, getOpt (opt_name, Syntax.type_name t mx)), (t, vs, mx), A, morphs);

 val typedefP =

   OuterSyntax.command "typedef" "HOL type definition (requires non-emptiness proof)" K.thy_goal

     (typedef_proof_decl >> (Toplevel.print oo (Toplevel.theory_to_proof o mk_typedef_proof)));

 val _ = OuterSyntax.add_keywords ["morphisms"];

 val _ = OuterSyntax.add_parsers [typedeclP, typedefP];

 end;

 end;