src/HOL/Tools/datatype_rep_proofs.ML
author wenzelm
Tue Jan 12 13:54:51 1999 +0100 (1999-01-12)
changeset 6092 d9db67970c73
parent 5696 c2c2214f8037
child 6171 cd237a10cbf8
permissions -rw-r--r--
eliminated tthm type and Attribute structure;
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(*  Title:      HOL/Tools/datatype_rep_proofs.ML
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    ID:         $Id$
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    Author:     Stefan Berghofer
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    Copyright   1998  TU Muenchen
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Definitional introduction of datatypes
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Proof of characteristic theorems:
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 - injectivity of constructors
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 - distinctness of constructors (internal version)
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 - induction theorem
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*)
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signature DATATYPE_REP_PROOFS =
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sig
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  val representation_proofs : bool -> DatatypeAux.datatype_info Symtab.table ->
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    string list -> (int * (string * DatatypeAux.dtyp list *
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      (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
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        (string * mixfix) list -> (string * mixfix) list list -> theory ->
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          theory * thm list list * thm list list * thm
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end;
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structure DatatypeRepProofs : DATATYPE_REP_PROOFS =
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struct
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open DatatypeAux;
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val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
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(* figure out internal names *)
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val image_name = Sign.intern_const (sign_of Set.thy) "op ``";
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val UNIV_name = Sign.intern_const (sign_of Set.thy) "UNIV";
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val inj_name = Sign.intern_const (sign_of Fun.thy) "inj";
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val inj_on_name = Sign.intern_const (sign_of Fun.thy) "inj_on";
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val inv_name = Sign.intern_const (sign_of Fun.thy) "inv";
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fun exh_thm_of (dt_info : datatype_info Symtab.table) tname =
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  #exhaustion (the (Symtab.lookup (dt_info, tname)));
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(******************************************************************************)
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fun representation_proofs flat_names (dt_info : datatype_info Symtab.table)
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      new_type_names descr sorts types_syntax constr_syntax thy =
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  let
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    val Univ_thy = the (get_thy "Univ" thy);
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    val node_name = Sign.intern_tycon (sign_of Univ_thy) "node";
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    val [In0_name, In1_name, Scons_name, Leaf_name, Numb_name] =
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      map (Sign.intern_const (sign_of Univ_thy))
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        ["In0", "In1", "Scons", "Leaf", "Numb"];
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    val [In0_inject, In1_inject, Scons_inject, Leaf_inject, In0_eq, In1_eq,
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      In0_not_In1, In1_not_In0] = map (get_thm Univ_thy)
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        ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject", "In0_eq",
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         "In1_eq", "In0_not_In1", "In1_not_In0"];
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    val descr' = flat descr;
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    val big_name = space_implode "_" new_type_names;
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    val thy1 = add_path flat_names big_name thy;
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    val big_rec_name = big_name ^ "_rep_set";
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    val rep_set_names = map (Sign.full_name (sign_of thy1))
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      (if length descr' = 1 then [big_rec_name] else
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        (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
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          (1 upto (length descr'))));
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    val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
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    val leafTs' = get_nonrec_types descr' sorts;
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    val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names (leafTs', []);
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    val leafTs = leafTs' @ (map (fn n => TFree (n, the (assoc (sorts, n)))) unneeded_vars);
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    val recTs = get_rec_types descr' sorts;
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    val newTs = take (length (hd descr), recTs);
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    val oldTs = drop (length (hd descr), recTs);
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    val sumT = if null leafTs then HOLogic.unitT
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      else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;
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    val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT]));
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    val UnivT = HOLogic.mk_setT Univ_elT;
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    val In0 = Const (In0_name, Univ_elT --> Univ_elT);
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    val In1 = Const (In1_name, Univ_elT --> Univ_elT);
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    val Leaf = Const (Leaf_name, sumT --> Univ_elT);
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    (* make injections needed for embedding types in leaves *)
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    fun mk_inj T' x =
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      let
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        fun mk_inj' T n i =
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          if n = 1 then x else
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          let val n2 = n div 2;
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              val Type (_, [T1, T2]) = T
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          in
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            if i <= n2 then
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              Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
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            else
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              Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
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          end
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      in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
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      end;
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    (* make injections for constructors *)
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    fun mk_univ_inj ts = access_bal (ap In0, ap In1, if ts = [] then
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        Const ("arbitrary", Univ_elT)
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      else
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        foldr1 (HOLogic.mk_binop Scons_name) ts);
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    (************** generate introduction rules for representing set **********)
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    val _ = message "Constructing representing sets...";
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    (* make introduction rule for a single constructor *)
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    fun make_intr s n (i, (_, cargs)) =
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      let
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        fun mk_prem (DtRec k, (j, prems, ts)) =
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              let val free_t = mk_Free "x" Univ_elT j
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              in (j + 1, (HOLogic.mk_mem (free_t,
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                Const (nth_elem (k, rep_set_names), UnivT)))::prems, free_t::ts)
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              end
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          | mk_prem (dt, (j, prems, ts)) =
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              let val T = typ_of_dtyp descr' sorts dt
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              in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
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              end;
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        val (_, prems, ts) = foldr mk_prem (cargs, (1, [], []));
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        val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
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          (mk_univ_inj ts n i, Const (s, UnivT)))
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      in Logic.list_implies (map HOLogic.mk_Trueprop prems, concl)
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      end;
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    val consts = map (fn s => Const (s, UnivT)) rep_set_names;
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    val intr_ts = flat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
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      map (make_intr rep_set_name (length constrs))
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        ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names));
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    val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
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      setmp InductivePackage.quiet_mode (!quiet_mode)
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        (InductivePackage.add_inductive_i false true big_rec_name false true false
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           consts intr_ts [] []) thy1;
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    (********************************* typedef ********************************)
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    val thy3 = add_path flat_names big_name (foldl (fn (thy, ((((name, mx), tvs), c), name')) =>
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      setmp TypedefPackage.quiet_mode true
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        (TypedefPackage.add_typedef_i_no_def name' (name, tvs, mx) c [] []
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          (Some (QUIET_BREADTH_FIRST (has_fewer_prems 1) (resolve_tac rep_intrs 1)))) thy)
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            (parent_path flat_names thy2, types_syntax ~~ tyvars ~~ (take (length newTs, consts)) ~~
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              new_type_names));
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    (*********************** definition of constructors ***********************)
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    val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
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    val rep_names = map (curry op ^ "Rep_") new_type_names;
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    val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
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      (1 upto (length (flat (tl descr))));
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    val all_rep_names = map (Sign.intern_const (sign_of thy3)) rep_names @
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      map (Sign.full_name (sign_of thy3)) rep_names';
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    (* isomorphism declarations *)
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    val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
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      (oldTs ~~ rep_names');
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    (* constructor definitions *)
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    fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
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      let
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        fun constr_arg (dt, (j, l_args, r_args)) =
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          let val T = typ_of_dtyp descr' sorts dt;
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              val free_t = mk_Free "x" T j
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          in (case dt of
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              DtRec m => (j + 1, free_t::l_args, (Const (nth_elem (m, all_rep_names),
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                T --> Univ_elT) $ free_t)::r_args)
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            | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
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          end;
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        val (_, l_args, r_args) = foldr constr_arg (cargs, (1, [], []));
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        val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
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        val abs_name = Sign.intern_const (sign_of thy) ("Abs_" ^ tname);
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        val rep_name = Sign.intern_const (sign_of thy) ("Rep_" ^ tname);
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        val lhs = list_comb (Const (cname, constrT), l_args);
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        val rhs = mk_univ_inj r_args n i;
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        val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
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        val def_name = (Sign.base_name cname) ^ "_def";
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        val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
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          (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
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        val thy' = thy |>
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          Theory.add_consts_i [(cname', constrT, mx)] |>
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          Theory.add_defs_i [(def_name, def)];
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      in (thy', defs @ [get_axiom thy' def_name], eqns @ [eqn], i + 1)
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      end;
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    (* constructor definitions for datatype *)
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    fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
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        ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
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      let
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        val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
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        val sg = sign_of thy;
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        val rep_const = cterm_of sg
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          (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
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        val cong' = cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong;
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        val dist = cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma;
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        val (thy', defs', eqns', _) = foldl ((make_constr_def tname T) (length constrs))
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          ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
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      in
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        (parent_path flat_names thy', defs', eqns @ [eqns'],
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          rep_congs @ [cong'], dist_lemmas @ [dist])
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      end;
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    val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = foldl dt_constr_defs
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      ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
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        hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
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    (*********** isomorphisms for new types (introduced by typedef) ***********)
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    val _ = message "Proving isomorphism properties...";
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    (* get axioms from theory *)
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    val newT_iso_axms = map (fn s =>
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      (get_axiom thy4 ("Abs_" ^ s ^ "_inverse"),
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       get_axiom thy4 ("Rep_" ^ s ^ "_inverse"),
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       get_axiom thy4 ("Rep_" ^ s))) new_type_names;
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    (*------------------------------------------------*)
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    (* prove additional theorems:                     *)
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    (*  inj_on dt_Abs_i rep_set_i  and  inj dt_Rep_i  *)
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    (*------------------------------------------------*)
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    fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) =
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      let
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        val sg = sign_of thy4;
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        val RepT = T --> Univ_elT;
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        val Rep_name = Sign.intern_const sg ("Rep_" ^ s);
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        val AbsT = Univ_elT --> T;
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        val Abs_name = Sign.intern_const sg ("Abs_" ^ s);
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        val inj_on_Abs_thm = prove_goalw_cterm [] (cterm_of sg
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          (HOLogic.mk_Trueprop (Const (inj_on_name, [AbsT, UnivT] ---> HOLogic.boolT) $
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            Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT))))
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              (fn _ => [rtac inj_on_inverseI 1, etac thm1 1]);
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        val inj_Rep_thm = prove_goalw_cterm [] (cterm_of sg
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          (HOLogic.mk_Trueprop (Const (inj_name, RepT --> HOLogic.boolT) $
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            Const (Rep_name, RepT))))
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              (fn _ => [rtac inj_inverseI 1, rtac thm2 1])
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      in (inj_on_Abs_thm, inj_Rep_thm) end;
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    val newT_iso_inj_thms = map prove_newT_iso_inj_thm
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      (new_type_names ~~ newT_iso_axms ~~ newTs ~~
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        take (length newTs, rep_set_names));
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    (********* isomorphisms between existing types and "unfolded" types *******)
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    (*---------------------------------------------------------------------*)
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    (* isomorphisms are defined using primrec-combinators:                 *)
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    (* generate appropriate functions for instantiating primrec-combinator *)
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    (*                                                                     *)
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    (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 ((Leaf h) $ y))        *)
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    (*                                                                     *)
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    (* also generate characteristic equations for isomorphisms             *)
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    (*                                                                     *)
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    (*   e.g.  dt_Rep_i (cons h t) = In1 ((dt_Rep_j h) $ (dt_Rep_i t))     *)
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    (*---------------------------------------------------------------------*)
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    fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
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      let
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        val argTs = map (typ_of_dtyp descr' sorts) cargs;
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        val T = nth_elem (k, recTs);
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        val rep_name = nth_elem (k, all_rep_names);
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        val rep_const = Const (rep_name, T --> Univ_elT);
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        val constr = Const (cname, argTs ---> T);
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        fun process_arg ks' ((i2, i2', ts), dt) =
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          let val T' = typ_of_dtyp descr' sorts dt
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          in (case dt of
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              DtRec j => if j mem ks' then
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                  (i2 + 1, i2' + 1, ts @ [mk_Free "y" Univ_elT i2'])
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                else
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                  (i2 + 1, i2', ts @ [Const (nth_elem (j, all_rep_names),
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                    T' --> Univ_elT) $ mk_Free "x" T' i2])
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            | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)]))
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          end;
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        val (i2, i2', ts) = foldl (process_arg ks) ((1, 1, []), cargs);
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        val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
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        val ys = map (mk_Free "y" Univ_elT) (1 upto (i2' - 1));
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   292
        val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
berghofe@5177
   293
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   294
        val (_, _, ts') = foldl (process_arg []) ((1, 1, []), cargs);
berghofe@5177
   295
        val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
berghofe@5177
   296
          (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
berghofe@5177
   297
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   298
      in (fs @ [f], eqns @ [eqn], i + 1) end;
berghofe@5177
   299
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   300
    (* define isomorphisms for all mutually recursive datatypes in list ds *)
berghofe@5177
   301
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   302
    fun make_iso_defs (ds, (thy, char_thms)) =
berghofe@5177
   303
      let
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   304
        val ks = map fst ds;
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   305
        val (_, (tname, _, _)) = hd ds;
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   306
        val {rec_rewrites, rec_names, ...} = the (Symtab.lookup (dt_info, tname));
berghofe@5177
   307
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   308
        fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
berghofe@5177
   309
          let
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   310
            val (fs', eqns', _) = foldl (make_iso_def k ks (length constrs))
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   311
              ((fs, eqns, 1), constrs);
berghofe@5177
   312
            val iso = (nth_elem (k, recTs), nth_elem (k, all_rep_names))
berghofe@5177
   313
          in (fs', eqns', isos @ [iso]) end;
berghofe@5177
   314
        
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   315
        val (fs, eqns, isos) = foldl process_dt (([], [], []), ds);
berghofe@5177
   316
        val fTs = map fastype_of fs;
berghofe@5177
   317
        val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
berghofe@5177
   318
          equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
berghofe@5177
   319
            list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
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   320
        val thy' = Theory.add_defs_i defs thy;
berghofe@5177
   321
        val def_thms = map (get_axiom thy') (map fst defs);
berghofe@5177
   322
berghofe@5177
   323
        (* prove characteristic equations *)
berghofe@5177
   324
oheimb@5553
   325
        val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
berghofe@5177
   326
        val char_thms' = map (fn eqn => prove_goalw_cterm rewrites
berghofe@5177
   327
          (cterm_of (sign_of thy') eqn) (fn _ => [rtac refl 1])) eqns;
berghofe@5177
   328
berghofe@5177
   329
      in (thy', char_thms' @ char_thms) end;
berghofe@5177
   330
berghofe@5661
   331
    val (thy5, iso_char_thms) = foldr make_iso_defs
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   332
      (tl descr, (add_path flat_names big_name thy4, []));
berghofe@5177
   333
berghofe@5177
   334
    (* prove isomorphism properties *)
berghofe@5177
   335
berghofe@5177
   336
    (* prove  x : dt_rep_set_i --> x : range dt_Rep_i *)
berghofe@5177
   337
berghofe@5177
   338
    fun mk_iso_t (((set_name, iso_name), i), T) =
berghofe@5177
   339
      let val isoT = T --> Univ_elT
berghofe@5177
   340
      in HOLogic.imp $ 
berghofe@5177
   341
        HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
berghofe@5177
   342
          (if i < length newTs then Const ("True", HOLogic.boolT)
berghofe@5177
   343
           else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
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   344
             Const (image_name, [isoT, HOLogic.mk_setT T] ---> UnivT) $
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   345
               Const (iso_name, isoT) $ Const (UNIV_name, HOLogic.mk_setT T)))
berghofe@5177
   346
      end;
berghofe@5177
   347
berghofe@5177
   348
    val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
berghofe@5177
   349
      (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
berghofe@5177
   350
berghofe@5177
   351
    val newT_Abs_inverse_thms = map (fn (iso, _, _) => iso RS subst) newT_iso_axms;
berghofe@5177
   352
berghofe@5177
   353
    (* all the theorems are proved by one single simultaneous induction *)
berghofe@5177
   354
berghofe@5177
   355
    val iso_thms = if length descr = 1 then [] else
berghofe@5177
   356
      drop (length newTs, split_conj_thm
berghofe@5177
   357
        (prove_goalw_cterm [] (cterm_of (sign_of thy5) iso_t) (fn _ =>
berghofe@5177
   358
           [indtac rep_induct 1,
berghofe@5177
   359
            REPEAT (rtac TrueI 1),
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   360
            REPEAT (EVERY
berghofe@5177
   361
              [REPEAT (etac rangeE 1),
berghofe@5177
   362
               REPEAT (eresolve_tac newT_Abs_inverse_thms 1),
berghofe@5177
   363
               TRY (hyp_subst_tac 1),
berghofe@5177
   364
               rtac (sym RS range_eqI) 1,
berghofe@5177
   365
               resolve_tac iso_char_thms 1])])));
berghofe@5177
   366
berghofe@5177
   367
    val Abs_inverse_thms = newT_Abs_inverse_thms @ (map (fn r =>
berghofe@5177
   368
      r RS mp RS f_inv_f RS subst) iso_thms);
berghofe@5177
   369
berghofe@5177
   370
    (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
berghofe@5177
   371
berghofe@5177
   372
    fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
berghofe@5177
   373
      let
berghofe@5177
   374
        val (_, (tname, _, _)) = hd ds;
berghofe@5177
   375
        val {induction, ...} = the (Symtab.lookup (dt_info, tname));
berghofe@5177
   376
berghofe@5177
   377
        fun mk_ind_concl (i, _) =
berghofe@5177
   378
          let
berghofe@5177
   379
            val T = nth_elem (i, recTs);
berghofe@5177
   380
            val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT);
berghofe@5177
   381
            val rep_set_name = nth_elem (i, rep_set_names)
berghofe@5177
   382
          in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
berghofe@5177
   383
                HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
berghofe@5177
   384
                  HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
berghofe@5177
   385
              HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
berghofe@5177
   386
          end;
berghofe@5177
   387
berghofe@5177
   388
        val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
berghofe@5177
   389
oheimb@5553
   390
        val rewrites = map mk_meta_eq iso_char_thms;
berghofe@5177
   391
        val inj_thms' = map (fn r => r RS injD) inj_thms;
berghofe@5177
   392
berghofe@5177
   393
        val inj_thm = prove_goalw_cterm [] (cterm_of (sign_of thy5)
berghofe@5177
   394
          (HOLogic.mk_Trueprop (mk_conj ind_concl1))) (fn _ =>
berghofe@5177
   395
            [indtac induction 1,
berghofe@5177
   396
             REPEAT (EVERY
berghofe@5177
   397
               [rtac allI 1, rtac impI 1,
berghofe@5177
   398
                exh_tac (exh_thm_of dt_info) 1,
berghofe@5177
   399
                REPEAT (EVERY
berghofe@5177
   400
                  [hyp_subst_tac 1,
berghofe@5177
   401
                   rewrite_goals_tac rewrites,
berghofe@5177
   402
                   REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
berghofe@5177
   403
                   (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
berghofe@5177
   404
                   ORELSE (EVERY
berghofe@5177
   405
                     [REPEAT (etac Scons_inject 1),
berghofe@5177
   406
                      REPEAT (dresolve_tac
berghofe@5177
   407
                        (inj_thms' @ [Leaf_inject, Inl_inject, Inr_inject]) 1),
berghofe@5177
   408
                      REPEAT (EVERY [etac allE 1, dtac mp 1, atac 1]),
berghofe@5177
   409
                      TRY (hyp_subst_tac 1),
berghofe@5177
   410
                      rtac refl 1])])])]);
berghofe@5177
   411
berghofe@5177
   412
        val inj_thms'' = map (fn r =>
berghofe@5177
   413
          r RS (allI RS (inj_def RS meta_eq_to_obj_eq RS iffD2)))
berghofe@5177
   414
            (split_conj_thm inj_thm);
berghofe@5177
   415
berghofe@5177
   416
        val elem_thm = prove_goalw_cterm [] (cterm_of (sign_of thy5)
berghofe@5177
   417
          (HOLogic.mk_Trueprop (mk_conj ind_concl2))) (fn _ =>
berghofe@5177
   418
            [indtac induction 1,
berghofe@5177
   419
             rewrite_goals_tac rewrites,
berghofe@5177
   420
             REPEAT (EVERY
berghofe@5177
   421
               [resolve_tac rep_intrs 1,
berghofe@5177
   422
                REPEAT ((atac 1) ORELSE (resolve_tac elem_thms 1))])]);
berghofe@5177
   423
berghofe@5177
   424
      in (inj_thms @ inj_thms'', elem_thms @ (split_conj_thm elem_thm))
berghofe@5177
   425
      end;
berghofe@5177
   426
berghofe@5177
   427
    val (iso_inj_thms, iso_elem_thms) = foldr prove_iso_thms
berghofe@5177
   428
      (tl descr, (map snd newT_iso_inj_thms, map #3 newT_iso_axms));
berghofe@5177
   429
berghofe@5177
   430
    (******************* freeness theorems for constructors *******************)
berghofe@5177
   431
berghofe@5661
   432
    val _ = message "Proving freeness of constructors...";
berghofe@5177
   433
berghofe@5177
   434
    (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
berghofe@5177
   435
    
berghofe@5177
   436
    fun prove_constr_rep_thm eqn =
berghofe@5177
   437
      let
berghofe@5177
   438
        val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
oheimb@5553
   439
        val rewrites = constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
berghofe@5177
   440
      in prove_goalw_cterm [] (cterm_of (sign_of thy5) eqn) (fn _ =>
berghofe@5177
   441
        [resolve_tac inj_thms 1,
berghofe@5177
   442
         rewrite_goals_tac rewrites,
berghofe@5177
   443
         rtac refl 1,
berghofe@5177
   444
         resolve_tac rep_intrs 2,
berghofe@5177
   445
         REPEAT (resolve_tac iso_elem_thms 1)])
berghofe@5177
   446
      end;
berghofe@5177
   447
berghofe@5177
   448
    (*--------------------------------------------------------------*)
berghofe@5177
   449
    (* constr_rep_thms and rep_congs are used to prove distinctness *)
berghofe@5177
   450
    (* of constructors internally.                                  *)
berghofe@5177
   451
    (* the external version uses dt_case which is not defined yet   *)
berghofe@5177
   452
    (*--------------------------------------------------------------*)
berghofe@5177
   453
berghofe@5177
   454
    val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
berghofe@5177
   455
berghofe@5177
   456
    val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
berghofe@5177
   457
      dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
berghofe@5177
   458
        (constr_rep_thms ~~ dist_lemmas);
berghofe@5177
   459
berghofe@5177
   460
    (* prove injectivity of constructors *)
berghofe@5177
   461
berghofe@5177
   462
    fun prove_constr_inj_thm rep_thms t =
berghofe@5177
   463
      let val inj_thms = Scons_inject::(map make_elim
berghofe@5177
   464
        ((map (fn r => r RS injD) iso_inj_thms) @
berghofe@5177
   465
          [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject]))
berghofe@5177
   466
      in prove_goalw_cterm [] (cterm_of (sign_of thy5) t) (fn _ =>
berghofe@5177
   467
        [rtac iffI 1,
berghofe@5177
   468
         REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
berghofe@5177
   469
         dresolve_tac rep_congs 1, dtac box_equals 1,
berghofe@5177
   470
         REPEAT (resolve_tac rep_thms 1),
berghofe@5177
   471
         REPEAT (eresolve_tac inj_thms 1),
berghofe@5177
   472
         hyp_subst_tac 1,
berghofe@5177
   473
         REPEAT (resolve_tac [conjI, refl] 1)])
berghofe@5177
   474
      end;
berghofe@5177
   475
berghofe@5177
   476
    val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
berghofe@5177
   477
      ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
berghofe@5177
   478
berghofe@5661
   479
    val thy6 = store_thmss "inject" new_type_names
berghofe@5661
   480
      constr_inject (parent_path flat_names thy5);
berghofe@5177
   481
berghofe@5177
   482
    (*************************** induction theorem ****************************)
berghofe@5177
   483
berghofe@5661
   484
    val _ = message "Proving induction rule for datatypes...";
berghofe@5177
   485
berghofe@5177
   486
    val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
berghofe@5177
   487
      (map (fn r => r RS inv_f_f RS subst) (drop (length newTs, iso_inj_thms)));
berghofe@5177
   488
    val Rep_inverse_thms' = map (fn r => r RS inv_f_f)
berghofe@5177
   489
      (drop (length newTs, iso_inj_thms));
berghofe@5177
   490
berghofe@5177
   491
    fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
berghofe@5177
   492
      let
berghofe@5177
   493
        val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT) $
berghofe@5177
   494
          mk_Free "x" T i;
berghofe@5177
   495
berghofe@5177
   496
        val Abs_t = if i < length newTs then
berghofe@5177
   497
            Const (Sign.intern_const (sign_of thy6)
berghofe@5177
   498
              ("Abs_" ^ (nth_elem (i, new_type_names))), Univ_elT --> T)
berghofe@5177
   499
          else Const (inv_name, [T --> Univ_elT, Univ_elT] ---> T) $
berghofe@5177
   500
            Const (nth_elem (i, all_rep_names), T --> Univ_elT)
berghofe@5177
   501
berghofe@5177
   502
      in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
berghofe@5177
   503
            Const (nth_elem (i, rep_set_names), UnivT)) $
berghofe@5177
   504
              (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
berghofe@5177
   505
          concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
berghofe@5177
   506
      end;
berghofe@5177
   507
berghofe@5177
   508
    val (indrule_lemma_prems, indrule_lemma_concls) =
berghofe@5177
   509
      foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
berghofe@5177
   510
berghofe@5177
   511
    val cert = cterm_of (sign_of thy6);
berghofe@5177
   512
berghofe@5177
   513
    val indrule_lemma = prove_goalw_cterm [] (cert
berghofe@5177
   514
      (Logic.mk_implies
berghofe@5177
   515
        (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
berghofe@5177
   516
         HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls)))) (fn prems =>
berghofe@5177
   517
           [cut_facts_tac prems 1, REPEAT (etac conjE 1),
berghofe@5177
   518
            REPEAT (EVERY
berghofe@5177
   519
              [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
berghofe@5177
   520
               etac mp 1, resolve_tac iso_elem_thms 1])]);
berghofe@5177
   521
berghofe@5177
   522
    val Ps = map head_of (dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
berghofe@5177
   523
    val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
berghofe@5177
   524
      map (Free o apfst fst o dest_Var) Ps;
berghofe@5177
   525
    val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
berghofe@5177
   526
berghofe@5177
   527
    val dt_induct = prove_goalw_cterm [] (cert
berghofe@5177
   528
      (DatatypeProp.make_ind descr sorts)) (fn prems =>
berghofe@5177
   529
        [rtac indrule_lemma' 1, indtac rep_induct 1,
berghofe@5177
   530
         EVERY (map (fn (prem, r) => (EVERY
berghofe@5177
   531
           [REPEAT (eresolve_tac Abs_inverse_thms 1),
berghofe@5177
   532
            simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
berghofe@5177
   533
            DEPTH_SOLVE_1 (ares_tac [prem] 1)]))
oheimb@5553
   534
              (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
berghofe@5177
   535
berghofe@5661
   536
    val thy7 = thy6 |>
berghofe@5661
   537
      Theory.add_path big_name |>
wenzelm@6092
   538
      PureThy.add_thms [(("induct", dt_induct), [])] |>
berghofe@5661
   539
      Theory.parent_path;
berghofe@5177
   540
berghofe@5177
   541
  in (thy7, constr_inject, dist_rewrites, dt_induct)
berghofe@5177
   542
  end;
berghofe@5177
   543
berghofe@5177
   544
end;