(*  Title:      HOL/Tools/datatype_prop.ML

    ID:         $Id$

    Author:     Stefan Berghofer, TU Muenchen

Characteristic properties of datatypes.

*)

signature DATATYPE_PROP =

sig

  val dtK : int ref

  val indexify_names: string list -> string list

  val make_tnames: typ list -> string list

  val make_injs : DatatypeAux.descr list -> (string * sort) list -> term list list

  val make_ind : DatatypeAux.descr list -> (string * sort) list -> term

  val make_casedists : DatatypeAux.descr list -> (string * sort) list -> term list

  val make_primrec_Ts : DatatypeAux.descr list -> (string * sort) list ->

    string list -> typ list * typ list

  val make_primrecs : string list -> DatatypeAux.descr list ->

    (string * sort) list -> theory -> term list

  val make_cases : string list -> DatatypeAux.descr list ->

    (string * sort) list -> theory -> term list list

  val make_distincts : string list -> DatatypeAux.descr list ->

    (string * sort) list -> theory -> term list list

  val make_splits : string list -> DatatypeAux.descr list ->

    (string * sort) list -> theory -> (term * term) list

  val make_size : DatatypeAux.descr list -> (string * sort) list ->

    theory -> term list

  val make_weak_case_congs : string list -> DatatypeAux.descr list ->

    (string * sort) list -> theory -> term list

  val make_case_congs : string list -> DatatypeAux.descr list ->

    (string * sort) list -> theory -> term list

  val make_nchotomys : DatatypeAux.descr list ->

    (string * sort) list -> term list

end;

structure DatatypeProp : DATATYPE_PROP =

struct

open DatatypeAux;

(*the kind of distinctiveness axioms depends on number of constructors*)

val dtK = ref 7;

fun indexify_names names =

  let

    fun index (x :: xs) tab =

      (case AList.lookup (op =) tab x of

        NONE => if x mem xs then (x ^ "1") :: index xs ((x, 2) :: tab) else x :: index xs tab

      | SOME i => (x ^ Library.string_of_int i) :: index xs ((x, i + 1) :: tab))

    | index [] _ = [];

  in index names [] end;

fun make_tnames Ts =

  let

    fun type_name (TFree (name, _)) = implode (tl (explode name))

      | type_name (Type (name, _)) = 

          let val name' = Sign.base_name name

          in if Syntax.is_identifier name' then name' else "x" end;

  in indexify_names (map type_name Ts) end;

(************************* injectivity of constructors ************************)

fun make_injs descr sorts =

  let

    val descr' = flat descr;

    fun make_inj T (cname, cargs) =

      if null cargs then I else

        let

          val Ts = map (typ_of_dtyp descr' sorts) cargs;

          val constr_t = Const (cname, Ts ---> T);

          val tnames = make_tnames Ts;

          val frees = map Free (tnames ~~ Ts);

          val frees' = map Free ((map ((op ^) o (rpair "'")) tnames) ~~ Ts);

        in cons (HOLogic.mk_Trueprop (HOLogic.mk_eq

          (HOLogic.mk_eq (list_comb (constr_t, frees), list_comb (constr_t, frees')),

           foldr1 (HOLogic.mk_binop "op &")

             (map HOLogic.mk_eq (frees ~~ frees')))))

        end;

  in

    map2 (fn d => fn T => fold_rev (make_inj T) (#3 (snd d)) [])

      (hd descr) (Library.take (length (hd descr), get_rec_types descr' sorts))

  end;

(********************************* induction **********************************)

fun make_ind descr sorts =

  let

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val pnames = if length descr' = 1 then ["P"]

      else map (fn i => "P" ^ string_of_int i) (1 upto length descr');

    fun make_pred i T =

      let val T' = T --> HOLogic.boolT

      in Free (List.nth (pnames, i), T') end;

    fun make_ind_prem k T (cname, cargs) =

      let

        fun mk_prem ((dt, s), T) =

          let val (Us, U) = strip_type T

          in list_all (map (pair "x") Us, HOLogic.mk_Trueprop

            (make_pred (body_index dt) U $ app_bnds (Free (s, T)) (length Us)))

          end;

        val recs = List.filter is_rec_type cargs;

        val Ts = map (typ_of_dtyp descr' sorts) cargs;

        val recTs' = map (typ_of_dtyp descr' sorts) recs;

        val tnames = Name.variant_list pnames (make_tnames Ts);

        val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));

        val frees = tnames ~~ Ts;

        val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');

      in list_all_free (frees, Logic.list_implies (prems,

        HOLogic.mk_Trueprop (make_pred k T $ 

          list_comb (Const (cname, Ts ---> T), map Free frees))))

      end;

    val prems = List.concat (map (fn ((i, (_, _, constrs)), T) =>

      map (make_ind_prem i T) constrs) (descr' ~~ recTs));

    val tnames = make_tnames recTs;

    val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")

      (map (fn (((i, _), T), tname) => make_pred i T $ Free (tname, T))

        (descr' ~~ recTs ~~ tnames)))

  in Logic.list_implies (prems, concl) end;

(******************************* case distinction *****************************)

fun make_casedists descr sorts =

  let

    val descr' = List.concat descr;

    fun make_casedist_prem T (cname, cargs) =

      let

        val Ts = map (typ_of_dtyp descr' sorts) cargs;

        val frees = Name.variant_list ["P", "y"] (make_tnames Ts) ~~ Ts;

        val free_ts = map Free frees

      in list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop

        (HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))),

          HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT))))

      end;

    fun make_casedist ((_, (_, _, constrs)), T) =

      let val prems = map (make_casedist_prem T) constrs

      in Logic.list_implies (prems, HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT)))

      end

  in map make_casedist

    ((hd descr) ~~ Library.take (length (hd descr), get_rec_types descr' sorts))

  end;

(*************** characteristic equations for primrec combinator **************)

fun make_primrec_Ts descr sorts used =

  let

    val descr' = List.concat descr;

    val rec_result_Ts = map TFree (Name.variant_list used (replicate (length descr') "'t") ~~

      replicate (length descr') HOLogic.typeS);

    val reccomb_fn_Ts = List.concat (map (fn (i, (_, _, constrs)) =>

      map (fn (_, cargs) =>

        let

          val Ts = map (typ_of_dtyp descr' sorts) cargs;

          val recs = List.filter (is_rec_type o fst) (cargs ~~ Ts);

          fun mk_argT (dt, T) =

            binder_types T ---> List.nth (rec_result_Ts, body_index dt);

          val argTs = Ts @ map mk_argT recs

        in argTs ---> List.nth (rec_result_Ts, i)

        end) constrs) descr');

  in (rec_result_Ts, reccomb_fn_Ts) end;

fun make_primrecs new_type_names descr sorts thy =

  let

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val used = foldr add_typ_tfree_names [] recTs;

    val (rec_result_Ts, reccomb_fn_Ts) = make_primrec_Ts descr sorts used;

    val rec_fns = map (uncurry (mk_Free "f"))

      (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));

    val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";

    val reccomb_names = map (Sign.intern_const thy)

      (if length descr' = 1 then [big_reccomb_name] else

        (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)

          (1 upto (length descr'))));

    val reccombs = map (fn ((name, T), T') => list_comb

      (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))

        (reccomb_names ~~ recTs ~~ rec_result_Ts);

    fun make_primrec T comb_t ((ts, f::fs), (cname, cargs)) =

      let

        val recs = List.filter is_rec_type cargs;

        val Ts = map (typ_of_dtyp descr' sorts) cargs;

        val recTs' = map (typ_of_dtyp descr' sorts) recs;

        val tnames = make_tnames Ts;

        val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));

        val frees = map Free (tnames ~~ Ts);

        val frees' = map Free (rec_tnames ~~ recTs');

        fun mk_reccomb ((dt, T), t) =

          let val (Us, U) = strip_type T

          in list_abs (map (pair "x") Us,

            List.nth (reccombs, body_index dt) $ app_bnds t (length Us))

          end;

        val reccombs' = map mk_reccomb (recs ~~ recTs' ~~ frees')

      in (ts @ [HOLogic.mk_Trueprop (HOLogic.mk_eq

        (comb_t $ list_comb (Const (cname, Ts ---> T), frees),

         list_comb (f, frees @ reccombs')))], fs)

      end

  in fst (Library.foldl (fn (x, ((dt, T), comb_t)) =>

    Library.foldl (make_primrec T comb_t) (x, #3 (snd dt)))

      (([], rec_fns), descr' ~~ recTs ~~ reccombs))

  end;

(****************** make terms of form  t_case f1 ... fn  *********************)

fun make_case_combs new_type_names descr sorts thy fname =

  let

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val used = foldr add_typ_tfree_names [] recTs;

    val newTs = Library.take (length (hd descr), recTs);

    val T' = TFree (Name.variant used "'t", HOLogic.typeS);

    val case_fn_Ts = map (fn (i, (_, _, constrs)) =>

      map (fn (_, cargs) =>

        let val Ts = map (typ_of_dtyp descr' sorts) cargs

        in Ts ---> T' end) constrs) (hd descr);

    val case_names = map (fn s =>

      Sign.intern_const thy (s ^ "_case")) new_type_names

  in

    map (fn ((name, Ts), T) => list_comb

      (Const (name, Ts @ [T] ---> T'),

        map (uncurry (mk_Free fname)) (Ts ~~ (1 upto length Ts))))

          (case_names ~~ case_fn_Ts ~~ newTs)

  end;

(**************** characteristic equations for case combinator ****************)

fun make_cases new_type_names descr sorts thy =

  let

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val newTs = Library.take (length (hd descr), recTs);

    fun make_case T comb_t ((cname, cargs), f) =

      let

        val Ts = map (typ_of_dtyp descr' sorts) cargs;

        val frees = map Free ((make_tnames Ts) ~~ Ts)

      in HOLogic.mk_Trueprop (HOLogic.mk_eq

        (comb_t $ list_comb (Const (cname, Ts ---> T), frees),

         list_comb (f, frees)))

      end

  in map (fn (((_, (_, _, constrs)), T), comb_t) =>

    map (make_case T comb_t) (constrs ~~ (snd (strip_comb comb_t))))

      ((hd descr) ~~ newTs ~~ (make_case_combs new_type_names descr sorts thy "f"))

  end;

(************************* distinctness of constructors ***********************)

fun make_distincts new_type_names descr sorts thy =

  let

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val newTs = Library.take (length (hd descr), recTs);

    (**** number of constructors < dtK : C_i ... ~= C_j ... ****)

    fun make_distincts_1 _ [] = []

      | make_distincts_1 T ((cname, cargs)::constrs) =

          let

            val Ts = map (typ_of_dtyp descr' sorts) cargs;

            val frees = map Free ((make_tnames Ts) ~~ Ts);

            val t = list_comb (Const (cname, Ts ---> T), frees);

            fun make_distincts' [] = []

              | make_distincts' ((cname', cargs')::constrs') =

                  let

                    val Ts' = map (typ_of_dtyp descr' sorts) cargs';

                    val frees' = map Free ((map ((op ^) o (rpair "'"))

                      (make_tnames Ts')) ~~ Ts');

                    val t' = list_comb (Const (cname', Ts' ---> T), frees')

                  in

                    (HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t, t')))::

                    (HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t', t)))::

                      (make_distincts' constrs')

                  end

          in (make_distincts' constrs) @ (make_distincts_1 T constrs)

          end;

  in map (fn (((_, (_, _, constrs)), T), tname) =>

      if length constrs < !dtK then make_distincts_1 T constrs else [])

        ((hd descr) ~~ newTs ~~ new_type_names)

  end;

(*************************** the "split" - equations **************************)

fun make_splits new_type_names descr sorts thy =

  let

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val used' = foldr add_typ_tfree_names [] recTs;

    val newTs = Library.take (length (hd descr), recTs);

    val T' = TFree (Name.variant used' "'t", HOLogic.typeS);

    val P = Free ("P", T' --> HOLogic.boolT);

    fun make_split (((_, (_, _, constrs)), T), comb_t) =

      let

        val (_, fs) = strip_comb comb_t;

        val used = ["P", "x"] @ (map (fst o dest_Free) fs);

        fun process_constr (((cname, cargs), f), (t1s, t2s)) =

          let

            val Ts = map (typ_of_dtyp descr' sorts) cargs;

            val frees = map Free (Name.variant_list used (make_tnames Ts) ~~ Ts);

            val eqn = HOLogic.mk_eq (Free ("x", T),

              list_comb (Const (cname, Ts ---> T), frees));

            val P' = P $ list_comb (f, frees)

          in ((foldr (fn (Free (s, T), t) => HOLogic.mk_all (s, T, t))

                (HOLogic.imp $ eqn $ P') frees)::t1s,

              (foldr (fn (Free (s, T), t) => HOLogic.mk_exists (s, T, t))

                (HOLogic.conj $ eqn $ (HOLogic.Not $ P')) frees)::t2s)

          end;

        val (t1s, t2s) = foldr process_constr ([], []) (constrs ~~ fs);

        val lhs = P $ (comb_t $ Free ("x", T))

      in

        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, mk_conj t1s)),

         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, HOLogic.Not $ mk_disj t2s)))

      end

  in map make_split ((hd descr) ~~ newTs ~~

    (make_case_combs new_type_names descr sorts thy "f"))

  end;

(******************************* size functions *******************************)

fun make_size descr sorts thy =

  let

    val descr' = flat descr;

    val recTs = get_rec_types descr' sorts;

    val Const (size_name, _) = HOLogic.size_const dummyT;

    val size_names = replicate (length (hd descr)) size_name @

      map (Sign.intern_const thy) (indexify_names

        (map (fn T => name_of_typ T ^ "_size") (Library.drop (length (hd descr), recTs))));

    val size_consts = map (fn (s, T) =>

      Const (s, T --> HOLogic.natT)) (size_names ~~ recTs);

    fun plus (t1, t2) = Const ("HOL.plus_class.plus", HOLogic.natT --> HOLogic.natT --> HOLogic.natT) $ t1 $ t2;

    fun make_size_eqn size_const T (cname, cargs) =

      let

        val recs = filter is_rec_type cargs;

        val Ts = map (typ_of_dtyp descr' sorts) cargs;

        val recTs = map (typ_of_dtyp descr' sorts) recs;

        val tnames = make_tnames Ts;

        val rec_tnames = map fst (filter (is_rec_type o snd) (tnames ~~ cargs));

        val ts = map (fn ((r, s), T) => nth size_consts (dest_DtRec r) $

          Free (s, T)) (recs ~~ rec_tnames ~~ recTs);

        val t = if ts = [] then HOLogic.zero else

          foldl1 plus (ts @ [HOLogic.Suc_zero])

      in

        HOLogic.mk_Trueprop (HOLogic.mk_eq (size_const $

          list_comb (Const (cname, Ts ---> T), map Free (tnames ~~ Ts)), t))

      end

  in

    List.concat (map (fn (((_, (_, _, constrs)), size_const), T) =>

      map (make_size_eqn size_const T) constrs) (descr' ~~ size_consts ~~ recTs))

  end;

(************************* additional rules for TFL ***************************)

fun make_weak_case_congs new_type_names descr sorts thy =

  let

    val case_combs = make_case_combs new_type_names descr sorts thy "f";

    fun mk_case_cong comb =

      let 

        val Type ("fun", [T, _]) = fastype_of comb;

        val M = Free ("M", T);

        val M' = Free ("M'", T);

      in

        Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')),

          HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb $ M')))

      end

  in

    map mk_case_cong case_combs

  end;

(*---------------------------------------------------------------------------

 * Structure of case congruence theorem looks like this:

 *

 *    (M = M') 

 *    ==> (!!x1,...,xk. (M' = C1 x1..xk) ==> (f1 x1..xk = g1 x1..xk)) 

 *    ==> ... 

 *    ==> (!!x1,...,xj. (M' = Cn x1..xj) ==> (fn x1..xj = gn x1..xj)) 

 *    ==>

 *      (ty_case f1..fn M = ty_case g1..gn M')

 *---------------------------------------------------------------------------*)

fun make_case_congs new_type_names descr sorts thy =

  let

    val case_combs = make_case_combs new_type_names descr sorts thy "f";

    val case_combs' = make_case_combs new_type_names descr sorts thy "g";

    fun mk_case_cong ((comb, comb'), (_, (_, _, constrs))) =

      let

        val Type ("fun", [T, _]) = fastype_of comb;

        val (_, fs) = strip_comb comb;

        val (_, gs) = strip_comb comb';

        val used = ["M", "M'"] @ map (fst o dest_Free) (fs @ gs);

        val M = Free ("M", T);

        val M' = Free ("M'", T);

        fun mk_clause ((f, g), (cname, _)) =

          let

            val (Ts, _) = strip_type (fastype_of f);

            val tnames = Name.variant_list used (make_tnames Ts);

            val frees = map Free (tnames ~~ Ts)

          in

            list_all_free (tnames ~~ Ts, Logic.mk_implies

              (HOLogic.mk_Trueprop

                (HOLogic.mk_eq (M', list_comb (Const (cname, Ts ---> T), frees))),

               HOLogic.mk_Trueprop

                (HOLogic.mk_eq (list_comb (f, frees), list_comb (g, frees)))))

          end

      in

        Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')) ::

          map mk_clause (fs ~~ gs ~~ constrs),

            HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb' $ M')))

      end

  in

    map mk_case_cong (case_combs ~~ case_combs' ~~ hd descr)

  end;

(*---------------------------------------------------------------------------

 * Structure of exhaustion theorem looks like this:

 *

 *    !v. (? y1..yi. v = C1 y1..yi) | ... | (? y1..yj. v = Cn y1..yj)

 *---------------------------------------------------------------------------*)

fun make_nchotomys descr sorts =

  let

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val newTs = Library.take (length (hd descr), recTs);

    fun mk_eqn T (cname, cargs) =

      let

        val Ts = map (typ_of_dtyp descr' sorts) cargs;

        val tnames = Name.variant_list ["v"] (make_tnames Ts);

        val frees = tnames ~~ Ts

      in

        foldr (fn ((s, T'), t) => HOLogic.mk_exists (s, T', t))

          (HOLogic.mk_eq (Free ("v", T),

            list_comb (Const (cname, Ts ---> T), map Free frees))) frees

      end

  in map (fn ((_, (_, _, constrs)), T) =>

    HOLogic.mk_Trueprop (HOLogic.mk_all ("v", T, mk_disj (map (mk_eqn T) constrs))))

      (hd descr ~~ newTs)

  end;

end;