src/HOL/Tools/Datatype/datatype_prop.ML

tuned;

(*  Title:      HOL/Tools/Datatype/datatype_prop.ML
    Author:     Stefan Berghofer, TU Muenchen

Datatype package: characteristic properties of datatypes.
*)

signature DATATYPE_PROP =
sig
  include DATATYPE_COMMON
  val indexify_names: string list -> string list
  val make_tnames: typ list -> string list
  val make_injs : descr list -> (string * sort) list -> term list list
  val make_distincts : descr list ->
    (string * sort) list -> (int * term list) list (*no symmetric inequalities*)
  val make_ind : descr list -> (string * sort) list -> term
  val make_casedists : descr list -> (string * sort) list -> term list
  val make_primrec_Ts : descr list -> (string * sort) list ->
    string list -> typ list * typ list
  val make_primrecs : string list -> descr list ->
    (string * sort) list -> theory -> term list
  val make_cases : string list -> descr list ->
    (string * sort) list -> theory -> term list list
  val make_splits : string list -> descr list ->
    (string * sort) list -> theory -> (term * term) list
  val make_case_combs : string list -> descr list ->
    (string * sort) list -> theory -> string -> term list
  val make_weak_case_congs : string list -> descr list ->
    (string * sort) list -> theory -> term list
  val make_case_congs : string list -> descr list ->
    (string * sort) list -> theory -> term list
  val make_nchotomys : descr list ->
    (string * sort) list -> term list
end;

structure Datatype_Prop : DATATYPE_PROP =
struct

fun indexify_names names =
  let
    fun index (x :: xs) tab =
      (case AList.lookup (op =) tab x of
        NONE => if member (op =) xs x then (x ^ "1") :: index xs ((x, 2) :: tab) else x :: index xs tab
      | SOME i => (x ^ 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, _)) = unprefix "'" name
      | type_name (Type (name, _)) =
          let val name' = Long_Name.base_name name
          in if Lexicon.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 (Datatype_Aux.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 @{const_name HOL.conj})
             (map HOLogic.mk_eq (frees ~~ frees')))))
        end;
  in
    map2 (fn d => fn T => fold_rev (make_inj T) (#3 (snd d)) [])
      (hd descr) (take (length (hd descr)) (Datatype_Aux.get_rec_types descr' sorts))
  end;


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

fun make_distincts descr sorts =
  let
    val descr' = flat descr;
    val recTs = Datatype_Aux.get_rec_types descr' sorts;
    val newTs = take (length (hd descr)) recTs;

    fun prep_constr (cname, cargs) = (cname, map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs);

    fun make_distincts' _ [] = []
      | make_distincts' T ((cname, cargs)::constrs) =
          let
            val frees = map Free ((make_tnames cargs) ~~ cargs);
            val t = list_comb (Const (cname, cargs ---> T), frees);

            fun make_distincts'' (cname', cargs') =
              let
                val frees' = map Free ((map ((op ^) o (rpair "'"))
                  (make_tnames cargs')) ~~ cargs');
                val t' = list_comb (Const (cname', cargs' ---> T), frees')
              in
                HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t, t'))
              end

          in map make_distincts'' constrs @ make_distincts' T constrs end;

  in
    map2 (fn ((_, (_, _, constrs))) => fn T =>
      (length constrs, make_distincts' T (map prep_constr constrs))) (hd descr) newTs
  end;


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

fun make_ind descr sorts =
  let
    val descr' = flat descr;
    val recTs = Datatype_Aux.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 (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 (Datatype_Aux.body_index dt) U $
                  Datatype_Aux.app_bnds (Free (s, T)) (length Us)))
          end;

        val recs = filter Datatype_Aux.is_rec_type cargs;
        val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
        val recTs' = map (Datatype_Aux.typ_of_dtyp descr' sorts) recs;
        val tnames = Name.variant_list pnames (make_tnames Ts);
        val rec_tnames = map fst (filter (Datatype_Aux.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 = maps (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 @{const_name HOL.conj})
      (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' = flat descr;

    fun make_casedist_prem T (cname, cargs) =
      let
        val Ts = map (Datatype_Aux.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
    map2 make_casedist (hd descr)
      (take (length (hd descr)) (Datatype_Aux.get_rec_types descr' sorts))
  end;

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

fun make_primrec_Ts descr sorts used =
  let
    val descr' = flat descr;

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

    val reccomb_fn_Ts = maps (fn (i, (_, _, constrs)) =>
      map (fn (_, cargs) =>
        let
          val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
          val recs = filter (Datatype_Aux.is_rec_type o fst) (cargs ~~ Ts);

          fun mk_argT (dt, T) =
            binder_types T ---> nth rec_result_Ts (Datatype_Aux.body_index dt);

          val argTs = Ts @ map mk_argT recs
        in argTs ---> 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' = flat descr;
    val recTs = Datatype_Aux.get_rec_types descr' sorts;
    val used = List.foldr Misc_Legacy.add_typ_tfree_names [] recTs;

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

    val rec_fns = map (uncurry (Datatype_Aux.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 (cname, cargs) (ts, f::fs) =
      let
        val recs = filter Datatype_Aux.is_rec_type cargs;
        val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
        val recTs' = map (Datatype_Aux.typ_of_dtyp descr' sorts) recs;
        val tnames = make_tnames Ts;
        val rec_tnames = map fst (filter (Datatype_Aux.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,
              nth reccombs (Datatype_Aux.body_index dt) $ Datatype_Aux.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
    fold (fn ((dt, T), comb_t) => fold (make_primrec T comb_t) (#3 (snd dt)))
      (descr' ~~ recTs ~~ reccombs) ([], rec_fns)
    |> fst
  end;

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

fun make_case_combs new_type_names descr sorts thy fname =
  let
    val descr' = flat descr;
    val recTs = Datatype_Aux.get_rec_types descr' sorts;
    val used = List.foldr Misc_Legacy.add_typ_tfree_names [] recTs;
    val newTs = take (length (hd descr)) recTs;
    val T' = TFree (singleton (Name.variant_list used) "'t", HOLogic.typeS);

    val case_fn_Ts = map (fn (i, (_, _, constrs)) =>
      map (fn (_, cargs) =>
        let val Ts = map (Datatype_Aux.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 (Datatype_Aux.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' = flat descr;
    val recTs = Datatype_Aux.get_rec_types descr' sorts;
    val newTs = take (length (hd descr)) recTs;

    fun make_case T comb_t ((cname, cargs), f) =
      let
        val Ts = map (Datatype_Aux.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;


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

fun make_splits new_type_names descr sorts thy =
  let
    val descr' = flat descr;
    val recTs = Datatype_Aux.get_rec_types descr' sorts;
    val used' = List.foldr Misc_Legacy.add_typ_tfree_names [] recTs;
    val newTs = take (length (hd descr)) recTs;
    val T' = TFree (singleton (Name.variant_list 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 (Datatype_Aux.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 (fold_rev (fn Free (s, T) => fn t => HOLogic.mk_all (s, T, t)) frees
                (HOLogic.imp $ eqn $ P') :: t1s,
              fold_rev (fn Free (s, T) => fn t => HOLogic.mk_exists (s, T, t)) frees
                (HOLogic.conj $ eqn $ (HOLogic.Not $ P')) :: t2s)
          end;

        val (t1s, t2s) = fold_rev process_constr (constrs ~~ fs) ([], []);
        val lhs = P $ (comb_t $ Free ("x", T))
      in
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, Datatype_Aux.mk_conj t1s)),
         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, HOLogic.Not $ Datatype_Aux.mk_disj t2s)))
      end

  in map make_split ((hd descr) ~~ newTs ~~
    (make_case_combs new_type_names descr sorts thy "f"))
  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 = binder_types (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' = flat descr;
    val recTs = Datatype_Aux.get_rec_types descr' sorts;
    val newTs = take (length (hd descr)) recTs;

    fun mk_eqn T (cname, cargs) =
      let
        val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
        val tnames = Name.variant_list ["v"] (make_tnames Ts);
        val frees = tnames ~~ Ts
      in
        fold_rev (fn (s, T') => fn t => HOLogic.mk_exists (s, T', t)) frees
          (HOLogic.mk_eq (Free ("v", T),
            list_comb (Const (cname, Ts ---> T), map Free frees)))
      end

  in map (fn ((_, (_, _, constrs)), T) =>
    HOLogic.mk_Trueprop (HOLogic.mk_all ("v", T, Datatype_Aux.mk_disj (map (mk_eqn T) constrs))))
      (hd descr ~~ newTs)
  end;


open Datatype_Aux;

end;