(*  Title:      HOL/Tools/typedef_package.ML

    ID:         $Id$

    Author:     Markus Wenzel and Stefan Berghofer, TU Muenchen

Gordon/HOL-style type definitions: create a new syntactic type

represented by a non-empty subset.

*)

signature TYPEDEF_PACKAGE =

sig

  type info =

   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,

    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 get_info: theory -> string -> info option

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

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

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

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

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

    * (string * string) option -> theory -> Proof.state

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

    * (string * string) option -> theory -> Proof.state

  val interpretation: (string -> theory -> theory) -> theory -> theory

  val setup: theory -> theory

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 definitions **)

(* theory data *)

type info =

 {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,

  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};

structure TypedefData = TheoryDataFun

(

  type T = info Symtab.table;

  val empty = Symtab.empty;

  val copy = I;

  val extend = I;

  fun merge _ tabs : T = Symtab.merge (K true) tabs;

);

val get_info = Symtab.lookup o TypedefData.get;

fun put_info name info = TypedefData.map (Symtab.update (name, info));

(* prepare_typedef *)

fun err_in_typedef msg name =

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

fun declare_type_name a = Variable.declare_constraints (Logic.mk_type (TFree (a, dummyS)));

structure TypedefInterpretation = InterpretationFun(type T = string val eq = op =);

val interpretation = TypedefInterpretation.interpretation;

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

  let

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

    val ctxt = ProofContext.init thy;

    val full = Sign.full_name thy;

    (*rhs*)

    val full_name = full name;

    val set = prep_term (ctxt |> fold declare_type_name vs) raw_set;

    val setT = Term.fastype_of set;

    val rhs_tfrees = Term.add_tfrees set [];

    val rhs_tfreesT = Term.add_tfreesT setT [];

    val oldT = HOLogic.dest_setT setT handle TYPE _ =>

      error ("Not a set type: " ^ quote (Syntax.string_of_typ ctxt 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 goal_pat = mk_nonempty (Var (the_default (name, 0) (Syntax.read_variable name), setT));

    (*lhs*)

    val defS = Sign.defaultS thy;

    val lhs_tfrees = map (fn v => (v, the_default defS (AList.lookup (op =) rhs_tfrees v))) vs;

    val args_setT = lhs_tfrees

      |> filter (member (op =) rhs_tfrees andf (not o member (op =) rhs_tfreesT))

      |> map TFree;

    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) = the_default ("Rep_" ^ name, "Abs_" ^ name) opt_morphs;

    val setT' = map Term.itselfT args_setT ---> setT;

    val setC = Term.list_comb (Const (full_name, setT'), map Logic.mk_type args_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'));

    val typedef_deps = Term.fold_aterms (fn Const c => insert (op =) c | _ => I) set' [];

    fun add_def eq thy =

      if def then

        thy

        |> PureThy.add_defs_i false [Thm.no_attributes eq]

        |-> (fn [th] => pair (SOME th))

      else (NONE, thy);

    fun typedef_result nonempty =

      ObjectLogic.typedecl (t, vs, mx)

      #> snd

      #> Sign.add_consts_i

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

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

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

      #> add_def (PrimitiveDefs.mk_defpair (setC, set))

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

          [apsnd (fn cond_axm => nonempty RS cond_axm)])]

      ##> Theory.add_deps "" (dest_Const RepC) typedef_deps

      ##> Theory.add_deps "" (dest_Const AbsC) typedef_deps

      #-> (fn (set_def, [type_definition]) => fn thy1 =>

        let

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

          val abs_inject = make Abs_inject;

          val abs_inverse = make Abs_inverse;

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

              Rep_cases, Abs_cases, Rep_induct, Abs_induct], thy2) =

            thy1

            |> Sign.add_path name

            |> PureThy.add_thms

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

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

                ((Abs_name ^ "_inverse", abs_inverse), []),

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

                ((Abs_name ^ "_inject", abs_inject), []),

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

                  [RuleCases.case_names [Rep_name], Induct.cases_pred full_name]),

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

                  [RuleCases.case_names [Abs_name], Induct.cases_type full_tname]),

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

                  [RuleCases.case_names [Rep_name], Induct.induct_pred full_name]),

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

                  [RuleCases.case_names [Abs_name], Induct.induct_type full_tname])])

            ||> Sign.parent_path;

          val info = {rep_type = oldT, abs_type = newT,

            Rep_name = full Rep_name, Abs_name = full Abs_name,

              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

          thy2

          |> put_info full_tname info

          |> TypedefInterpretation.data full_tname

          |> pair (full_tname, info)

        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 (op =) lhs_tfrees of [] => []

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

    val extra_rhs_tfrees =

      (case fold (remove (op =)) lhs_tfrees rhs_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

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

  in (set, goal, goal_pat, typedef_result) end

  handle ERROR msg => err_in_typedef msg name;

(* add_typedef interfaces *)

local

fun gen_typedef prep_term def opt_name typ set opt_morphs tac thy =

  let

    val name = the_default (#1 typ) opt_name;

    val (set, goal, _, typedef_result) =

      prepare_typedef prep_term def name typ set opt_morphs thy;

    val non_empty = Goal.prove_global thy [] [] goal (K tac)

      handle ERROR msg => cat_error msg

        ("Failed to prove non-emptiness of " ^ quote (Syntax.string_of_term_global thy set));

  in typedef_result non_empty thy end;

in

val add_typedef = gen_typedef Syntax.read_term;

val add_typedef_i = gen_typedef Syntax.check_term;

end;

(* Isar typedef interface *)

local

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

  let

    val (_, goal, goal_pat, typedef_result) =

      prepare_typedef prep_term def name typ set opt_morphs thy;

    fun after_qed [[th]] = ProofContext.theory (snd o typedef_result th);

  in Proof.theorem_i NONE after_qed [[(goal, [goal_pat])]] (ProofContext.init thy) end;

in

val typedef = gen_typedef Syntax.read_term;

val typedef_i = gen_typedef Syntax.check_term;

end;

(** outer syntax **)

local structure P = OuterParse and K = OuterKeyword in

val _ = OuterKeyword.keyword "morphisms";

val typedef_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 ((((((def, opt_name), (vs, t)), mx), A), morphs)) =

  typedef ((def, the_default (Syntax.type_name t mx) opt_name), (t, vs, mx), A, morphs);

val _ =

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

    (typedef_decl >> (Toplevel.print oo (Toplevel.theory_to_proof o mk_typedef)));

val setup = TypedefInterpretation.init;

end;

end;