(*  Title:      HOL/Tools/typedef_package.ML

    ID:         $Id$

    Author:     Markus Wenzel, TU Muenchen

    License:    GPL (GNU GENERAL PUBLIC LICENSE)

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

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";

(** type declarations **)

fun add_typedecls decls thy =

  let

    val full = Sign.full_name (Theory.sign_of thy);

    fun arity_of (raw_name, args, mx) =

      (full (Syntax.type_name raw_name mx), replicate (length args) HOLogic.typeS, HOLogic.typeS);

  in

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

      thy

      |> PureThy.add_typedecls decls

      |> Theory.add_arities_i (map arity_of decls)

    else thy |> PureThy.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 witn_names @ witn_thms;

    val tac =

      witn1_tac THEN

      TRY (rewrite_goals_tac (filter is_def thms)) THEN

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

      if_none witn2_tac (TRY (ALLGOALS (CLASET' blast_tac)));

  in

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

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

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

(* prepare_typedef *)

fun read_term sg used s =

  #1 (Thm.read_def_cterm (sg, K None, K None) used true (s, HOLogic.typeT));

fun cert_term sg _ t = Thm.cterm_of sg 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 sign = Theory.sign_of thy;

    val full = Sign.full_name sign;

    (*rhs*)

    val full_name = full name;

    val cset = prep_term sign 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 sign 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 Term.read_int);

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

    (*lhs*)

    val lhs_tfrees = map (fn v => (v, if_none (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) = if_none 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

      |> 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

    (if_none 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;

(** 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.!!!

      (((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, if_none 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;