src/HOL/Tools/datatype_prop.ML
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(*  Title:      HOL/Tools/datatype_prop.ML
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    Author:     Stefan Berghofer, TU Muenchen
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Characteristic properties of datatypes.
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*)
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signature DATATYPE_PROP =
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sig
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  val indexify_names: string list -> string list
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  val make_tnames: typ list -> string list
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  val make_injs : DatatypeAux.descr list -> (string * sort) list -> term list list
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  val make_distincts : DatatypeAux.descr list ->
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    (string * sort) list -> (int * term list) list (*no symmetric inequalities*)
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  val make_ind : DatatypeAux.descr list -> (string * sort) list -> term
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  val make_casedists : DatatypeAux.descr list -> (string * sort) list -> term list
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  val make_primrec_Ts : DatatypeAux.descr list -> (string * sort) list ->
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    string list -> typ list * typ list
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  val make_primrecs : string list -> DatatypeAux.descr list ->
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    (string * sort) list -> theory -> term list
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  val make_cases : string list -> DatatypeAux.descr list ->
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    (string * sort) list -> theory -> term list list
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  val make_splits : string list -> DatatypeAux.descr list ->
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    (string * sort) list -> theory -> (term * term) list
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  val make_weak_case_congs : string list -> DatatypeAux.descr list ->
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    (string * sort) list -> theory -> term list
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  val make_case_congs : string list -> DatatypeAux.descr list ->
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    (string * sort) list -> theory -> term list
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  val make_nchotomys : DatatypeAux.descr list ->
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    (string * sort) list -> term list
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end;
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structure DatatypeProp : DATATYPE_PROP =
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struct
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open DatatypeAux;
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fun indexify_names names =
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  let
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    fun index (x :: xs) tab =
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      (case AList.lookup (op =) tab x of
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        NONE => if member (op =) xs x then (x ^ "1") :: index xs ((x, 2) :: tab) else x :: index xs tab
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      | SOME i => (x ^ string_of_int i) :: index xs ((x, i + 1) :: tab))
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    | index [] _ = [];
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  in index names [] end;
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fun make_tnames Ts =
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  let
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    fun type_name (TFree (name, _)) = implode (tl (explode name))
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      | type_name (Type (name, _)) = 
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          let val name' = Sign.base_name name
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          in if Syntax.is_identifier name' then name' else "x" end;
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  in indexify_names (map type_name Ts) end;
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(************************* injectivity of constructors ************************)
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fun make_injs descr sorts =
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  let
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    val descr' = flat descr;
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    fun make_inj T (cname, cargs) =
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      if null cargs then I else
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        let
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          val Ts = map (typ_of_dtyp descr' sorts) cargs;
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          val constr_t = Const (cname, Ts ---> T);
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          val tnames = make_tnames Ts;
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          val frees = map Free (tnames ~~ Ts);
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          val frees' = map Free ((map ((op ^) o (rpair "'")) tnames) ~~ Ts);
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        in cons (HOLogic.mk_Trueprop (HOLogic.mk_eq
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          (HOLogic.mk_eq (list_comb (constr_t, frees), list_comb (constr_t, frees')),
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           foldr1 (HOLogic.mk_binop "op &")
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             (map HOLogic.mk_eq (frees ~~ frees')))))
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        end;
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  in
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    map2 (fn d => fn T => fold_rev (make_inj T) (#3 (snd d)) [])
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      (hd descr) (Library.take (length (hd descr), get_rec_types descr' sorts))
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  end;
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(************************* distinctness of constructors ***********************)
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fun make_distincts descr sorts =
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  let
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    val descr' = flat descr;
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    val recTs = get_rec_types descr' sorts;
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    val newTs = Library.take (length (hd descr), recTs);
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    fun prep_constr (cname, cargs) = (cname, map (typ_of_dtyp descr' sorts) cargs);
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    fun make_distincts' _ [] = []
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      | make_distincts' T ((cname, cargs)::constrs) =
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          let
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            val frees = map Free ((make_tnames cargs) ~~ cargs);
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            val t = list_comb (Const (cname, cargs ---> T), frees);
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            fun make_distincts'' (cname', cargs') =
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              let
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                val frees' = map Free ((map ((op ^) o (rpair "'"))
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                  (make_tnames cargs')) ~~ cargs');
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                val t' = list_comb (Const (cname', cargs' ---> T), frees')
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              in
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                HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t, t'))
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              end
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          in map make_distincts'' constrs @ make_distincts' T constrs end;
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  in
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    map2 (fn ((_, (_, _, constrs))) => fn T =>
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      (length constrs, make_distincts' T (map prep_constr constrs))) (hd descr) newTs
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  end;
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(********************************* induction **********************************)
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fun make_ind descr sorts =
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  let
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    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val pnames = if length descr' = 1 then ["P"]
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      else map (fn i => "P" ^ string_of_int i) (1 upto length descr');
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    fun make_pred i T =
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      let val T' = T --> HOLogic.boolT
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      in Free (List.nth (pnames, i), T') end;
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    fun make_ind_prem k T (cname, cargs) =
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      let
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        fun mk_prem ((dt, s), T) =
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          let val (Us, U) = strip_type T
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          in list_all (map (pair "x") Us, HOLogic.mk_Trueprop
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            (make_pred (body_index dt) U $ app_bnds (Free (s, T)) (length Us)))
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          end;
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        val recs = List.filter is_rec_type cargs;
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        val Ts = map (typ_of_dtyp descr' sorts) cargs;
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        val recTs' = map (typ_of_dtyp descr' sorts) recs;
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        val tnames = Name.variant_list pnames (make_tnames Ts);
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        val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
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        val frees = tnames ~~ Ts;
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        val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
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      in list_all_free (frees, Logic.list_implies (prems,
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        HOLogic.mk_Trueprop (make_pred k T $ 
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          list_comb (Const (cname, Ts ---> T), map Free frees))))
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      end;
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    val prems = List.concat (map (fn ((i, (_, _, constrs)), T) =>
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      map (make_ind_prem i T) constrs) (descr' ~~ recTs));
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    val tnames = make_tnames recTs;
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    val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
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      (map (fn (((i, _), T), tname) => make_pred i T $ Free (tname, T))
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        (descr' ~~ recTs ~~ tnames)))
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  in Logic.list_implies (prems, concl) end;
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(******************************* case distinction *****************************)
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fun make_casedists descr sorts =
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  let
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    val descr' = List.concat descr;
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    fun make_casedist_prem T (cname, cargs) =
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      let
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        val Ts = map (typ_of_dtyp descr' sorts) cargs;
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        val frees = Name.variant_list ["P", "y"] (make_tnames Ts) ~~ Ts;
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        val free_ts = map Free frees
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      in list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop
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        (HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))),
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          HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT))))
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      end;
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    fun make_casedist ((_, (_, _, constrs)), T) =
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      let val prems = map (make_casedist_prem T) constrs
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      in Logic.list_implies (prems, HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT)))
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      end
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  in map make_casedist
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    ((hd descr) ~~ Library.take (length (hd descr), get_rec_types descr' sorts))
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  end;
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(*************** characteristic equations for primrec combinator **************)
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fun make_primrec_Ts descr sorts used =
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  let
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    val descr' = List.concat descr;
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    val rec_result_Ts = map TFree (Name.variant_list used (replicate (length descr') "'t") ~~
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      replicate (length descr') HOLogic.typeS);
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    val reccomb_fn_Ts = List.concat (map (fn (i, (_, _, constrs)) =>
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      map (fn (_, cargs) =>
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        let
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          val Ts = map (typ_of_dtyp descr' sorts) cargs;
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          val recs = List.filter (is_rec_type o fst) (cargs ~~ Ts);
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          fun mk_argT (dt, T) =
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            binder_types T ---> List.nth (rec_result_Ts, body_index dt);
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          val argTs = Ts @ map mk_argT recs
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        in argTs ---> List.nth (rec_result_Ts, i)
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        end) constrs) descr');
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  in (rec_result_Ts, reccomb_fn_Ts) end;
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fun make_primrecs new_type_names descr sorts thy =
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  let
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    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val used = foldr OldTerm.add_typ_tfree_names [] recTs;
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    val (rec_result_Ts, reccomb_fn_Ts) = make_primrec_Ts descr sorts used;
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    val rec_fns = map (uncurry (mk_Free "f"))
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      (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));
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    val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
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    val reccomb_names = map (Sign.intern_const thy)
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      (if length descr' = 1 then [big_reccomb_name] else
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        (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
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          (1 upto (length descr'))));
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    val reccombs = map (fn ((name, T), T') => list_comb
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      (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))
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        (reccomb_names ~~ recTs ~~ rec_result_Ts);
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    fun make_primrec T comb_t ((ts, f::fs), (cname, cargs)) =
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      let
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        val recs = List.filter is_rec_type cargs;
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        val Ts = map (typ_of_dtyp descr' sorts) cargs;
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        val recTs' = map (typ_of_dtyp descr' sorts) recs;
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        val tnames = make_tnames Ts;
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        val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
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        val frees = map Free (tnames ~~ Ts);
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        val frees' = map Free (rec_tnames ~~ recTs');
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        fun mk_reccomb ((dt, T), t) =
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          let val (Us, U) = strip_type T
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          in list_abs (map (pair "x") Us,
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            List.nth (reccombs, body_index dt) $ app_bnds t (length Us))
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          end;
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        val reccombs' = map mk_reccomb (recs ~~ recTs' ~~ frees')
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      in (ts @ [HOLogic.mk_Trueprop (HOLogic.mk_eq
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        (comb_t $ list_comb (Const (cname, Ts ---> T), frees),
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         list_comb (f, frees @ reccombs')))], fs)
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      end
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  in fst (Library.foldl (fn (x, ((dt, T), comb_t)) =>
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    Library.foldl (make_primrec T comb_t) (x, #3 (snd dt)))
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      (([], rec_fns), descr' ~~ recTs ~~ reccombs))
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  end;
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(****************** make terms of form  t_case f1 ... fn  *********************)
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fun make_case_combs new_type_names descr sorts thy fname =
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  let
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    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val used = foldr OldTerm.add_typ_tfree_names [] recTs;
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    val newTs = Library.take (length (hd descr), recTs);
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    val T' = TFree (Name.variant used "'t", HOLogic.typeS);
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    val case_fn_Ts = map (fn (i, (_, _, constrs)) =>
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      map (fn (_, cargs) =>
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        let val Ts = map (typ_of_dtyp descr' sorts) cargs
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        in Ts ---> T' end) constrs) (hd descr);
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    val case_names = map (fn s =>
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      Sign.intern_const thy (s ^ "_case")) new_type_names
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  in
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    map (fn ((name, Ts), T) => list_comb
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      (Const (name, Ts @ [T] ---> T'),
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        map (uncurry (mk_Free fname)) (Ts ~~ (1 upto length Ts))))
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          (case_names ~~ case_fn_Ts ~~ newTs)
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  end;
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(**************** characteristic equations for case combinator ****************)
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fun make_cases new_type_names descr sorts thy =
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  let
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    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val newTs = Library.take (length (hd descr), recTs);
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    fun make_case T comb_t ((cname, cargs), f) =
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      let
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        val Ts = map (typ_of_dtyp descr' sorts) cargs;
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        val frees = map Free ((make_tnames Ts) ~~ Ts)
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      in HOLogic.mk_Trueprop (HOLogic.mk_eq
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        (comb_t $ list_comb (Const (cname, Ts ---> T), frees),
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         list_comb (f, frees)))
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      end
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  in map (fn (((_, (_, _, constrs)), T), comb_t) =>
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    map (make_case T comb_t) (constrs ~~ (snd (strip_comb comb_t))))
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      ((hd descr) ~~ newTs ~~ (make_case_combs new_type_names descr sorts thy "f"))
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  end;
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(*************************** the "split" - equations **************************)
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fun make_splits new_type_names descr sorts thy =
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  let
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    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val used' = foldr OldTerm.add_typ_tfree_names [] recTs;
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    val newTs = Library.take (length (hd descr), recTs);
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    val T' = TFree (Name.variant used' "'t", HOLogic.typeS);
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    val P = Free ("P", T' --> HOLogic.boolT);
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    fun make_split (((_, (_, _, constrs)), T), comb_t) =
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      let
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        val (_, fs) = strip_comb comb_t;
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        val used = ["P", "x"] @ (map (fst o dest_Free) fs);
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        fun process_constr (((cname, cargs), f), (t1s, t2s)) =
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          let
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            val Ts = map (typ_of_dtyp descr' sorts) cargs;
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            val frees = map Free (Name.variant_list used (make_tnames Ts) ~~ Ts);
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            val eqn = HOLogic.mk_eq (Free ("x", T),
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              list_comb (Const (cname, Ts ---> T), frees));
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            val P' = P $ list_comb (f, frees)
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          in ((foldr (fn (Free (s, T), t) => HOLogic.mk_all (s, T, t))
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                (HOLogic.imp $ eqn $ P') frees)::t1s,
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              (foldr (fn (Free (s, T), t) => HOLogic.mk_exists (s, T, t))
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                (HOLogic.conj $ eqn $ (HOLogic.Not $ P')) frees)::t2s)
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          end;
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        val (t1s, t2s) = foldr process_constr ([], []) (constrs ~~ fs);
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        val lhs = P $ (comb_t $ Free ("x", T))
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      in
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        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, mk_conj t1s)),
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         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, HOLogic.Not $ mk_disj t2s)))
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      end
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  in map make_split ((hd descr) ~~ newTs ~~
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    (make_case_combs new_type_names descr sorts thy "f"))
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  end;
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(************************* additional rules for TFL ***************************)
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fun make_weak_case_congs new_type_names descr sorts thy =
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  let
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    val case_combs = make_case_combs new_type_names descr sorts thy "f";
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    fun mk_case_cong comb =
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      let 
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        val Type ("fun", [T, _]) = fastype_of comb;
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        val M = Free ("M", T);
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        val M' = Free ("M'", T);
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      in
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        Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')),
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          HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb $ M')))
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      end
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  in
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    map mk_case_cong case_combs
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  end;
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(*---------------------------------------------------------------------------
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 * Structure of case congruence theorem looks like this:
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 *
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 *    (M = M') 
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 *    ==> (!!x1,...,xk. (M' = C1 x1..xk) ==> (f1 x1..xk = g1 x1..xk)) 
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 *    ==> ... 
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 *    ==> (!!x1,...,xj. (M' = Cn x1..xj) ==> (fn x1..xj = gn x1..xj)) 
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 *    ==>
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 *      (ty_case f1..fn M = ty_case g1..gn M')
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 *---------------------------------------------------------------------------*)
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fun make_case_congs new_type_names descr sorts thy =
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  let
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    val case_combs = make_case_combs new_type_names descr sorts thy "f";
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    val case_combs' = make_case_combs new_type_names descr sorts thy "g";
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    fun mk_case_cong ((comb, comb'), (_, (_, _, constrs))) =
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      let
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        val Type ("fun", [T, _]) = fastype_of comb;
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        val (_, fs) = strip_comb comb;
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        val (_, gs) = strip_comb comb';
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        val used = ["M", "M'"] @ map (fst o dest_Free) (fs @ gs);
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        val M = Free ("M", T);
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        val M' = Free ("M'", T);
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   383
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   384
        fun mk_clause ((f, g), (cname, _)) =
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   385
          let
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   386
            val (Ts, _) = strip_type (fastype_of f);
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8f3e1ddb50e6 replaced Term.variant(list) by Name.variant(_list);
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            val tnames = Name.variant_list used (make_tnames Ts);
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            val frees = map Free (tnames ~~ Ts)
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          in
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   390
            list_all_free (tnames ~~ Ts, Logic.mk_implies
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              (HOLogic.mk_Trueprop
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                (HOLogic.mk_eq (M', list_comb (Const (cname, Ts ---> T), frees))),
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               HOLogic.mk_Trueprop
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   394
                (HOLogic.mk_eq (list_comb (f, frees), list_comb (g, frees)))))
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          end
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   396
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   397
      in
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   398
        Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')) ::
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   399
          map mk_clause (fs ~~ gs ~~ constrs),
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   400
            HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb' $ M')))
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   401
      end
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   402
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   403
  in
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   404
    map mk_case_cong (case_combs ~~ case_combs' ~~ hd descr)
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   405
  end;
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   406
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   407
(*---------------------------------------------------------------------------
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   408
 * Structure of exhaustion theorem looks like this:
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   409
 *
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   410
 *    !v. (? y1..yi. v = C1 y1..yi) | ... | (? y1..yj. v = Cn y1..yj)
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   411
 *---------------------------------------------------------------------------*)
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   412
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   413
fun make_nchotomys descr sorts =
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   414
  let
15570
8d8c70b41bab Move towards standard functions.
skalberg
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   415
    val descr' = List.concat descr;
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   416
    val recTs = get_rec_types descr' sorts;
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8d8c70b41bab Move towards standard functions.
skalberg
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diff changeset
   417
    val newTs = Library.take (length (hd descr), recTs);
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   418
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   419
    fun mk_eqn T (cname, cargs) =
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   420
      let
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   421
        val Ts = map (typ_of_dtyp descr' sorts) cargs;
20071
8f3e1ddb50e6 replaced Term.variant(list) by Name.variant(_list);
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   422
        val tnames = Name.variant_list ["v"] (make_tnames Ts);
5177
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   423
        val frees = tnames ~~ Ts
0d3a168e4d44 New datatype definition package
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   424
      in
15574
b1d1b5bfc464 Removed practically all references to Library.foldr.
skalberg
parents: 15570
diff changeset
   425
        foldr (fn ((s, T'), t) => HOLogic.mk_exists (s, T', t))
b1d1b5bfc464 Removed practically all references to Library.foldr.
skalberg
parents: 15570
diff changeset
   426
          (HOLogic.mk_eq (Free ("v", T),
b1d1b5bfc464 Removed practically all references to Library.foldr.
skalberg
parents: 15570
diff changeset
   427
            list_comb (Const (cname, Ts ---> T), map Free frees))) frees
5177
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   428
      end
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   429
0d3a168e4d44 New datatype definition package
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   430
  in map (fn ((_, (_, _, constrs)), T) =>
0d3a168e4d44 New datatype definition package
berghofe
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diff changeset
   431
    HOLogic.mk_Trueprop (HOLogic.mk_all ("v", T, mk_disj (map (mk_eqn T) constrs))))
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berghofe
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   432
      (hd descr ~~ newTs)
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   433
  end;
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   434
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   435
end;