(*  Title:      HOL/Tools/typedef.ML

    Author:     Markus Wenzel and Stefan Berghofer, TU Muenchen

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

represented by a non-empty set.

*)

signature TYPEDEF =

sig

  type info =

   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string, axiom_name: string} *

   {inhabited: thm, 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 transform_info: morphism -> info -> info

  val get_info: Proof.context -> string -> info list

  val get_info_global: theory -> string -> info list

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

  val setup: theory -> theory

  val add_typedef: bool -> binding option -> binding * (string * sort) list * mixfix ->

    term -> (binding * binding) option -> tactic -> local_theory -> (string * info) * local_theory

  val add_typedef_global: bool -> binding option -> binding * (string * sort) list * mixfix ->

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

  val typedef: (bool * binding) * (binding * (string * sort) list * mixfix) * term *

    (binding * binding) option -> local_theory -> Proof.state

  val typedef_cmd: (bool * binding) * (binding * (string * string option) list * mixfix) * string *

    (binding * binding) option -> local_theory -> Proof.state

end;

structure Typedef: TYPEDEF =

struct

(** type definitions **)

(* theory data *)

type info =

  (*global part*)

  {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string, axiom_name: string} *

  (*local part*)

  {inhabited: thm, 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};

fun transform_info phi (info: info) =

  let

    val thm = Morphism.thm phi;

    val (global_info, {inhabited, type_definition,

      set_def, Rep, Rep_inverse, Abs_inverse, Rep_inject, Abs_inject,

      Rep_cases, Abs_cases, Rep_induct, Abs_induct}) = info;

  in

    (global_info,

     {inhabited = thm inhabited, type_definition = thm type_definition,

      set_def = Option.map thm set_def, Rep = thm Rep, Rep_inverse = thm Rep_inverse,

      Abs_inverse = thm Abs_inverse, Rep_inject = thm Rep_inject, Abs_inject = thm Abs_inject,

      Rep_cases = thm Rep_cases, Abs_cases = thm Abs_cases, Rep_induct = thm Rep_induct,

      Abs_induct = thm Abs_induct})

  end;

structure Data = Generic_Data

(

  type T = info list Symtab.table;

  val empty = Symtab.empty;

  val extend = I;

  fun merge data = Symtab.merge_list (K true) data;

);

val get_info = Symtab.lookup_list o Data.get o Context.Proof;

val get_info_global = Symtab.lookup_list o Data.get o Context.Theory;

fun put_info name info = Data.map (Symtab.cons_list (name, info));

(* global interpretation *)

structure Typedef_Interpretation = Interpretation(type T = string val eq = op =);

val interpretation = Typedef_Interpretation.interpretation;

val setup = Typedef_Interpretation.init;

(* primitive typedef axiomatization -- for fresh typedecl *)

fun mk_inhabited A =

  let val T = HOLogic.dest_setT (Term.fastype_of A)

  in HOLogic.mk_Trueprop (HOLogic.exists_const T $ Abs ("x", T, HOLogic.mk_mem (Bound 0, A))) end;

fun mk_typedef newT oldT RepC AbsC A =

  let

    val typedefC =

      Const (@{const_name type_definition},

        (newT --> oldT) --> (oldT --> newT) --> HOLogic.mk_setT oldT --> HOLogic.boolT);

  in Logic.mk_implies (mk_inhabited A, HOLogic.mk_Trueprop (typedefC $ RepC $ AbsC $ A)) end;

fun primitive_typedef typedef_name newT oldT Rep_name Abs_name A lthy =

  let

    (* errors *)

    fun show_names pairs = commas_quote (map fst pairs);

    val lhs_tfrees = Term.add_tfreesT newT [];

    val rhs_tfrees = Term.add_tfreesT oldT [];

    val _ =

      (case fold (remove (op =)) lhs_tfrees rhs_tfrees of [] => ()

      | extras => error ("Extra type variables in representing set: " ^ show_names extras));

    val _ =

      (case Term.add_frees A [] of [] => []

      | xs => error ("Illegal variables in representing set: " ^ show_names xs));

    (* axiomatization *)

    val ((RepC, AbsC), consts_lthy) = lthy

      |> Local_Theory.background_theory_result

        (Sign.declare_const lthy ((Rep_name, newT --> oldT), NoSyn) ##>>

          Sign.declare_const lthy ((Abs_name, oldT --> newT), NoSyn));

    val typedef_deps = Term.add_consts A [];

    val ((axiom_name, axiom), axiom_lthy) = consts_lthy

      |> Local_Theory.background_theory_result

        (Thm.add_axiom consts_lthy (typedef_name, mk_typedef newT oldT RepC AbsC A) ##>

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

          Theory.add_deps consts_lthy "" (dest_Const AbsC) typedef_deps);

  in ((RepC, AbsC, axiom_name, axiom), axiom_lthy) end;

(* prepare_typedef *)

fun prepare_typedef prep_term def_set name (tname, raw_args, mx) raw_set opt_morphs lthy =

  let

    val full_name = Local_Theory.full_name lthy name;

    val bname = Binding.name_of name;

    (* rhs *)

    val tmp_ctxt = lthy |> fold (Variable.declare_typ o TFree) raw_args;

    val set = prep_term tmp_ctxt raw_set;

    val tmp_ctxt' = tmp_ctxt |> Variable.declare_term set;

    val setT = Term.fastype_of set;

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

      error ("Not a set type: " ^ quote (Syntax.string_of_typ lthy setT));

    val goal = mk_inhabited set;

    val goal_pat = mk_inhabited (Var (the_default (bname, 0) (Lexicon.read_variable bname), setT));

    (* lhs *)

    val args = map (Proof_Context.check_tfree tmp_ctxt') raw_args;

    val (newT, typedecl_lthy) = lthy

      |> Typedecl.typedecl (tname, args, mx)

      ||> Variable.declare_term set;

    val Type (full_tname, type_args) = newT;

    val lhs_tfrees = map Term.dest_TFree type_args;

    (* set definition *)

    val ((set', set_def), set_lthy) =

      if def_set then

        typedecl_lthy

        |> Local_Theory.define ((name, NoSyn), ((Thm.def_binding name, []), set))

        |>> (fn (set', (_, set_def)) => (set', SOME set_def))

      else ((set, NONE), typedecl_lthy);

    (* axiomatization *)

    val (Rep_name, Abs_name) =

      (case opt_morphs of

        NONE => (Binding.prefix_name "Rep_" name, Binding.prefix_name "Abs_" name)

      | SOME morphs => morphs);

    val typedef_name = Binding.prefix_name "type_definition_" name;

    val ((RepC, AbsC, axiom_name, typedef), typedef_lthy) =

      let

        val thy = Proof_Context.theory_of set_lthy;

        val cert = Thm.cterm_of thy;

        val ((defs, _), A) =

          Local_Defs.export_cterm set_lthy (Proof_Context.init_global thy) (cert set')

          ||> Thm.term_of;

        val ((RepC, AbsC, axiom_name, axiom), axiom_lthy) = set_lthy

          |> primitive_typedef typedef_name newT oldT Rep_name Abs_name A;

        val cert = Thm.cterm_of (Proof_Context.theory_of axiom_lthy);

        val typedef =

          Local_Defs.contract axiom_lthy defs (cert (mk_typedef newT oldT RepC AbsC set')) axiom;

      in ((RepC, AbsC, axiom_name, typedef), axiom_lthy) end;

    val alias_lthy = typedef_lthy

      |> Local_Theory.const_alias Rep_name (#1 (Term.dest_Const RepC))

      |> Local_Theory.const_alias Abs_name (#1 (Term.dest_Const AbsC));

    (* result *)

    fun note_qualify ((b, atts), th) =

      Local_Theory.note ((Binding.qualify false bname b, map (Attrib.internal o K) atts), [th])

      #>> (fn (_, [th']) => th');

    fun typedef_result inhabited lthy1 =

      let

        val cert = Thm.cterm_of (Proof_Context.theory_of lthy1);

        val inhabited' =

          Local_Defs.contract lthy1 (the_list set_def) (cert (mk_inhabited set')) inhabited;

        val typedef' = inhabited' RS typedef;

        fun make th = Goal.norm_result (typedef' RS th);

        val (((((((((((_, [type_definition]), Rep), Rep_inverse), Abs_inverse), Rep_inject),

            Abs_inject), Rep_cases), Abs_cases), Rep_induct), Abs_induct), lthy2) = lthy1

          |> Local_Theory.note ((typedef_name, []), [typedef'])

          ||>> note_qualify ((Rep_name, []), make @{thm type_definition.Rep})

          ||>> note_qualify ((Binding.suffix_name "_inverse" Rep_name, []),

              make @{thm type_definition.Rep_inverse})

          ||>> note_qualify ((Binding.suffix_name "_inverse" Abs_name, []),

              make @{thm type_definition.Abs_inverse})

          ||>> note_qualify ((Binding.suffix_name "_inject" Rep_name, []),

              make @{thm type_definition.Rep_inject})

          ||>> note_qualify ((Binding.suffix_name "_inject" Abs_name, []),

              make @{thm type_definition.Abs_inject})

          ||>> note_qualify ((Binding.suffix_name "_cases" Rep_name,

                [Rule_Cases.case_names [Binding.name_of Rep_name], Induct.cases_pred full_name]),

              make @{thm type_definition.Rep_cases})

          ||>> note_qualify ((Binding.suffix_name "_cases" Abs_name,

                [Rule_Cases.case_names [Binding.name_of Abs_name], Induct.cases_type full_tname]),

              make @{thm type_definition.Abs_cases})

          ||>> note_qualify ((Binding.suffix_name "_induct" Rep_name,

                [Rule_Cases.case_names [Binding.name_of Rep_name], Induct.induct_pred full_name]),

              make @{thm type_definition.Rep_induct})

          ||>> note_qualify ((Binding.suffix_name "_induct" Abs_name,

                [Rule_Cases.case_names [Binding.name_of Abs_name], Induct.induct_type full_tname]),

              make @{thm type_definition.Abs_induct});

        val info =

          ({rep_type = oldT, abs_type = newT, Rep_name = #1 (Term.dest_Const RepC),

            Abs_name = #1 (Term.dest_Const AbsC), axiom_name = axiom_name},

           {inhabited = inhabited, 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

        lthy2

        |> Local_Theory.declaration {syntax = false, pervasive = true}

          (fn phi => put_info full_tname (transform_info phi info))

        |> Local_Theory.background_theory (Typedef_Interpretation.data full_tname)

        |> pair (full_tname, info)

      end;

  in ((goal, goal_pat, typedef_result), alias_lthy) end

  handle ERROR msg =>

    cat_error msg ("The error(s) above occurred in typedef " ^ Binding.print name);

(* add_typedef: tactic interface *)

fun add_typedef def opt_name typ set opt_morphs tac lthy =

  let

    val name = the_default (#1 typ) opt_name;

    val ((goal, _, typedef_result), lthy') =

      prepare_typedef Syntax.check_term def name typ set opt_morphs lthy;

    val inhabited =

      Goal.prove lthy' [] [] goal (K tac)

      |> Goal.norm_result |> Thm.close_derivation;

  in typedef_result inhabited lthy' end;

fun add_typedef_global def opt_name typ set opt_morphs tac =

  Named_Target.theory_init

  #> add_typedef def opt_name typ set opt_morphs tac

  #> Local_Theory.exit_result_global (apsnd o transform_info);

(* typedef: proof interface *)

local

fun gen_typedef prep_term prep_constraint ((def, name), (b, raw_args, mx), set, opt_morphs) lthy =

  let

    val args = map (apsnd (prep_constraint lthy)) raw_args;

    val ((goal, goal_pat, typedef_result), lthy') =

      prepare_typedef prep_term def name (b, args, mx) set opt_morphs lthy;

    fun after_qed [[th]] = snd o typedef_result th;

  in Proof.theorem NONE after_qed [[(goal, [goal_pat])]] lthy' end;

in

val typedef = gen_typedef Syntax.check_term (K I);

val typedef_cmd = gen_typedef Syntax.read_term Typedecl.read_constraint;

end;

(** outer syntax **)

val _ =

  Outer_Syntax.local_theory_to_proof "typedef" "HOL type definition (requires non-emptiness proof)"

    Keyword.thy_goal

    (Scan.optional (Parse.$$$ "(" |--

        ((Parse.$$$ "open" >> K false) -- Scan.option Parse.binding ||

          Parse.binding >> (fn s => (true, SOME s))) --| Parse.$$$ ")") (true, NONE) --

      (Parse.type_args_constrained -- Parse.binding) --

        Parse.opt_mixfix -- (Parse.$$$ "=" |-- Parse.term) --

        Scan.option (Parse.$$$ "morphisms" |-- Parse.!!! (Parse.binding -- Parse.binding))

    >> (fn ((((((def, opt_name), (args, t)), mx), A), morphs)) =>

        typedef_cmd ((def, the_default t opt_name), (t, args, mx), A, morphs)));

end;