src/HOL/Tools/datatype_rep_proofs.ML
author wenzelm
Thu Aug 03 17:30:36 2006 +0200 (2006-08-03)
changeset 20328 5b240a4216b0
parent 20071 8f3e1ddb50e6
child 20820 58693343905f
permissions -rw-r--r--
RuleInsts.bires_inst_tac;
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(*  Title:      HOL/Tools/datatype_rep_proofs.ML
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    ID:         $Id$
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    Author:     Stefan Berghofer, 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
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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 -> DatatypeAux.descr list -> (string * sort) list ->
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      (string * mixfix) list -> (string * mixfix) list list -> attribute
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        -> theory -> (thm list list * thm list list * thm list list *
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          DatatypeAux.simproc_dist list * thm) * theory
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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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(** theory context references **)
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val f_myinv_f = thm "f_myinv_f";
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val myinv_f_f = thm "myinv_f_f";
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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 case_names_induct thy =
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  let
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    val Datatype_thy = ThyInfo.the_theory "Datatype" thy;
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    val node_name = "Datatype_Universe.node";
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    val In0_name = "Datatype_Universe.In0";
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    val In1_name = "Datatype_Universe.In1";
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    val Scons_name = "Datatype_Universe.Scons";
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    val Leaf_name = "Datatype_Universe.Leaf";
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    val Numb_name = "Datatype_Universe.Numb";
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    val Lim_name = "Datatype_Universe.Lim";
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    val Suml_name = "Datatype.Suml";
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    val Sumr_name = "Datatype.Sumr";
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    val [In0_inject, In1_inject, Scons_inject, Leaf_inject,
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         In0_eq, In1_eq, In0_not_In1, In1_not_In0,
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         Lim_inject, Suml_inject, Sumr_inject] = map (get_thm Datatype_thy o Name)
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        ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject",
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         "In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0",
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         "Lim_inject", "Suml_inject", "Sumr_inject"];
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    val descr' = List.concat 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 (Theory.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 branchTs = get_branching_types descr' sorts;
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    val branchT = if null branchTs then HOLogic.unitT
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      else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;
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    val arities = get_arities descr' \ 0;
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    val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names [] (leafTs' @ branchTs);
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    val leafTs = leafTs' @ (map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars);
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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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    val oldTs = Library.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, branchT]));
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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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    val Lim = Const (Lim_name, (branchT --> Univ_elT) --> 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 ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
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            else
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              Const ("Sum_Type.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 (fn t => In0 $ t, fn t => In1 $ t, 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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    (* function spaces *)
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    fun mk_fun_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
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            val n2 = n div 2;
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            val Type (_, [T1, T2]) = T;
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            fun mkT U = (U --> Univ_elT) --> T --> Univ_elT
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          in
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            if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i
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            else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
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          end
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      in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
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      end;
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    val mk_lim = foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t)));
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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 (dt, (j, prems, ts)) = (case strip_dtyp dt of
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            (dts, DtRec k) =>
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              let
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                val Ts = map (typ_of_dtyp descr' sorts) dts;
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                val free_t =
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                  app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
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              in (j + 1, list_all (map (pair "x") Ts,
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                  HOLogic.mk_Trueprop (HOLogic.mk_mem (free_t,
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                    Const (List.nth (rep_set_names, k), UnivT)))) :: prems,
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                mk_lim free_t Ts :: ts)
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              end
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          | _ =>
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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 (1, [], []) cargs;
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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 (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 = List.concat (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 (map (fn x => (("", x), [])) intr_ts) []) thy1;
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    (********************************* typedef ********************************)
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    val thy3 = add_path flat_names big_name (Library.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 false (SOME name') (name, tvs, mx) c NONE
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          (rtac exI 1 THEN
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            QUIET_BREADTH_FIRST (has_fewer_prems 1)
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            (resolve_tac rep_intrs 1))) thy |> snd)
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              (parent_path flat_names thy2, types_syntax ~~ tyvars ~~
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                (Library.take (length newTs, consts)) ~~ 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 (List.concat (tl descr))));
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    val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @
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      map (Sign.full_name (Theory.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 (strip_dtyp dt, strip_type T) of
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              ((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim
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                (Const (List.nth (all_rep_names, m), U --> Univ_elT) $
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                   app_bnds free_t (length Us)) Us :: 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 (1, [], []) cargs;
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        val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
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        val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
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        val rep_name = Sign.intern_const (Theory.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 ([def_thm], thy') =
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          thy
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          |> Theory.add_consts_i [(cname', constrT, mx)]
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          |> (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)];
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      in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) 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 = Theory.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' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);
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        val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);
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        val (thy', defs', eqns', _) = Library.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) = Library.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_thm thy4 (Name ("Abs_" ^ s ^ "_inverse")),
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       get_thm thy4 (Name ("Rep_" ^ s ^ "_inverse")),
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       get_thm thy4 (Name ("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 = Theory.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_Abs_thm = 
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            Goal.prove_global sg [] []
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              (HOLogic.mk_Trueprop 
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                (Const ("Fun.inj_on", [AbsT, UnivT] ---> HOLogic.boolT) $
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                 Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT)))
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              (fn _ => EVERY [rtac inj_on_inverseI 1, etac thm1 1]);
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        val setT = HOLogic.mk_setT T
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        val inj_Rep_thm =
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            Goal.prove_global sg [] []
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              (HOLogic.mk_Trueprop
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                (Const ("Fun.inj_on", [RepT, setT] ---> HOLogic.boolT) $
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                 Const (Rep_name, RepT) $ Const ("UNIV", setT)))
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              (fn _ => EVERY [rtac inj_inverseI 1, rtac thm2 1]);
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      in (inj_Abs_thm, inj_Rep_thm) end;
berghofe@5177
   297
berghofe@5177
   298
    val newT_iso_inj_thms = map prove_newT_iso_inj_thm
berghofe@5177
   299
      (new_type_names ~~ newT_iso_axms ~~ newTs ~~
skalberg@15570
   300
        Library.take (length newTs, rep_set_names));
berghofe@5177
   301
berghofe@5177
   302
    (********* isomorphisms between existing types and "unfolded" types *******)
berghofe@5177
   303
berghofe@5177
   304
    (*---------------------------------------------------------------------*)
berghofe@5177
   305
    (* isomorphisms are defined using primrec-combinators:                 *)
berghofe@5177
   306
    (* generate appropriate functions for instantiating primrec-combinator *)
berghofe@5177
   307
    (*                                                                     *)
berghofe@13641
   308
    (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)
berghofe@5177
   309
    (*                                                                     *)
berghofe@5177
   310
    (* also generate characteristic equations for isomorphisms             *)
berghofe@5177
   311
    (*                                                                     *)
berghofe@13641
   312
    (*   e.g.  dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)
berghofe@5177
   313
    (*---------------------------------------------------------------------*)
berghofe@5177
   314
berghofe@5177
   315
    fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
berghofe@5177
   316
      let
berghofe@5177
   317
        val argTs = map (typ_of_dtyp descr' sorts) cargs;
skalberg@15570
   318
        val T = List.nth (recTs, k);
skalberg@15570
   319
        val rep_name = List.nth (all_rep_names, k);
berghofe@5177
   320
        val rep_const = Const (rep_name, T --> Univ_elT);
berghofe@5177
   321
        val constr = Const (cname, argTs ---> T);
berghofe@5177
   322
berghofe@7015
   323
        fun process_arg ks' ((i2, i2', ts, Ts), dt) =
berghofe@13641
   324
          let
berghofe@13641
   325
            val T' = typ_of_dtyp descr' sorts dt;
berghofe@13641
   326
            val (Us, U) = strip_type T'
berghofe@13641
   327
          in (case strip_dtyp dt of
berghofe@13641
   328
              (_, DtRec j) => if j mem ks' then
skalberg@15574
   329
                  (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds
skalberg@15574
   330
                     (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
berghofe@13641
   331
                   Ts @ [Us ---> Univ_elT])
berghofe@5177
   332
                else
skalberg@15574
   333
                  (i2 + 1, i2', ts @ [mk_lim
skalberg@15574
   334
                     (Const (List.nth (all_rep_names, j), U --> Univ_elT) $
skalberg@15574
   335
                        app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)
berghofe@7015
   336
            | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
berghofe@5177
   337
          end;
berghofe@5177
   338
skalberg@15570
   339
        val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);
berghofe@5177
   340
        val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
berghofe@7015
   341
        val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
berghofe@5177
   342
        val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
berghofe@5177
   343
skalberg@15570
   344
        val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);
berghofe@5177
   345
        val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
berghofe@5177
   346
          (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
berghofe@5177
   347
berghofe@5177
   348
      in (fs @ [f], eqns @ [eqn], i + 1) end;
berghofe@5177
   349
berghofe@5177
   350
    (* define isomorphisms for all mutually recursive datatypes in list ds *)
berghofe@5177
   351
berghofe@5177
   352
    fun make_iso_defs (ds, (thy, char_thms)) =
berghofe@5177
   353
      let
berghofe@5177
   354
        val ks = map fst ds;
berghofe@5177
   355
        val (_, (tname, _, _)) = hd ds;
wenzelm@17412
   356
        val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);
berghofe@5177
   357
berghofe@5177
   358
        fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
berghofe@5177
   359
          let
skalberg@15570
   360
            val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))
berghofe@5177
   361
              ((fs, eqns, 1), constrs);
skalberg@15570
   362
            val iso = (List.nth (recTs, k), List.nth (all_rep_names, k))
berghofe@5177
   363
          in (fs', eqns', isos @ [iso]) end;
berghofe@5177
   364
        
skalberg@15570
   365
        val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);
berghofe@5177
   366
        val fTs = map fastype_of fs;
berghofe@5177
   367
        val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
berghofe@5177
   368
          equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
berghofe@5177
   369
            list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
haftmann@18358
   370
        val (def_thms, thy') = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;
berghofe@5177
   371
berghofe@5177
   372
        (* prove characteristic equations *)
berghofe@5177
   373
oheimb@5553
   374
        val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
wenzelm@20046
   375
        val char_thms' = map (fn eqn => Goal.prove_global thy' [] [] eqn
wenzelm@20046
   376
          (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
berghofe@5177
   377
berghofe@5177
   378
      in (thy', char_thms' @ char_thms) end;
berghofe@5177
   379
skalberg@15574
   380
    val (thy5, iso_char_thms) = foldr make_iso_defs
skalberg@15574
   381
      (add_path flat_names big_name thy4, []) (tl descr);
berghofe@5177
   382
berghofe@5177
   383
    (* prove isomorphism properties *)
berghofe@5177
   384
berghofe@7015
   385
    fun mk_funs_inv thm =
berghofe@7015
   386
      let
berghofe@13641
   387
        val {sign, prop, ...} = rep_thm thm;
wenzelm@16287
   388
        val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
wenzelm@16287
   389
          (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
berghofe@13641
   390
        val used = add_term_tfree_names (a, []);
berghofe@13641
   391
berghofe@13641
   392
        fun mk_thm i =
berghofe@13641
   393
          let
berghofe@13641
   394
            val Ts = map (TFree o rpair HOLogic.typeS)
wenzelm@20071
   395
              (Name.variant_list used (replicate i "'t"));
berghofe@13641
   396
            val f = Free ("f", Ts ---> U)
wenzelm@20046
   397
          in Goal.prove_global sign [] [] (Logic.mk_implies
berghofe@13641
   398
            (HOLogic.mk_Trueprop (HOLogic.list_all
berghofe@13641
   399
               (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
berghofe@13641
   400
             HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
wenzelm@17985
   401
               r $ (a $ app_bnds f i)), f))))
wenzelm@20046
   402
            (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1])
berghofe@13641
   403
          end
berghofe@13641
   404
      in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
berghofe@7015
   405
berghofe@5177
   406
    (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
berghofe@5177
   407
berghofe@5177
   408
    fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
berghofe@5177
   409
      let
berghofe@5177
   410
        val (_, (tname, _, _)) = hd ds;
wenzelm@17412
   411
        val {induction, ...} = the (Symtab.lookup dt_info tname);
berghofe@5177
   412
berghofe@5177
   413
        fun mk_ind_concl (i, _) =
berghofe@5177
   414
          let
skalberg@15570
   415
            val T = List.nth (recTs, i);
skalberg@15570
   416
            val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT);
skalberg@15570
   417
            val rep_set_name = List.nth (rep_set_names, i)
berghofe@5177
   418
          in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
berghofe@5177
   419
                HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
berghofe@5177
   420
                  HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
berghofe@5177
   421
              HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
berghofe@5177
   422
          end;
berghofe@5177
   423
berghofe@5177
   424
        val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
berghofe@5177
   425
oheimb@5553
   426
        val rewrites = map mk_meta_eq iso_char_thms;
berghofe@13641
   427
        val inj_thms' = map (fn r => r RS injD)
berghofe@13641
   428
          (map snd newT_iso_inj_thms @ inj_thms);
berghofe@5177
   429
wenzelm@20046
   430
        val inj_thm = Goal.prove_global thy5 [] []
wenzelm@17985
   431
          (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY
berghofe@11951
   432
            [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
berghofe@5177
   433
             REPEAT (EVERY
berghofe@5177
   434
               [rtac allI 1, rtac impI 1,
berghofe@5177
   435
                exh_tac (exh_thm_of dt_info) 1,
berghofe@5177
   436
                REPEAT (EVERY
berghofe@5177
   437
                  [hyp_subst_tac 1,
berghofe@5177
   438
                   rewrite_goals_tac rewrites,
berghofe@5177
   439
                   REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
berghofe@5177
   440
                   (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
berghofe@5177
   441
                   ORELSE (EVERY
berghofe@13641
   442
                     [REPEAT (eresolve_tac (Scons_inject ::
berghofe@13641
   443
                        map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
berghofe@13641
   444
                      REPEAT (cong_tac 1), rtac refl 1,
berghofe@13641
   445
                      REPEAT (atac 1 ORELSE (EVERY
berghofe@13641
   446
                        [REPEAT (rtac ext 1),
berghofe@13641
   447
                         REPEAT (eresolve_tac (mp :: allE ::
berghofe@13641
   448
                           map make_elim (Suml_inject :: Sumr_inject ::
berghofe@13641
   449
                             Lim_inject :: fun_cong :: inj_thms')) 1),
wenzelm@20046
   450
                         atac 1]))])])])]);
berghofe@5177
   451
paulson@6171
   452
        val inj_thms'' = map (fn r => r RS datatype_injI)
paulson@6171
   453
                             (split_conj_thm inj_thm);
berghofe@5177
   454
paulson@6171
   455
        val elem_thm = 
wenzelm@20046
   456
            Goal.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))
wenzelm@20046
   457
              (fn _ =>
wenzelm@20046
   458
               EVERY [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
wenzelm@20046
   459
                rewrite_goals_tac rewrites,
wenzelm@20046
   460
                REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
wenzelm@20046
   461
                  ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
berghofe@5177
   462
berghofe@11471
   463
      in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
berghofe@11471
   464
      end;
berghofe@11471
   465
skalberg@15574
   466
    val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms
skalberg@15574
   467
      ([], map #3 newT_iso_axms) (tl descr);
berghofe@11471
   468
    val iso_inj_thms = map snd newT_iso_inj_thms @ iso_inj_thms_unfolded;
berghofe@11471
   469
berghofe@11471
   470
    (* prove  x : dt_rep_set_i --> x : range dt_Rep_i *)
berghofe@11471
   471
berghofe@11471
   472
    fun mk_iso_t (((set_name, iso_name), i), T) =
berghofe@11471
   473
      let val isoT = T --> Univ_elT
berghofe@11471
   474
      in HOLogic.imp $ 
berghofe@11471
   475
        HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
berghofe@11471
   476
          (if i < length newTs then Const ("True", HOLogic.boolT)
berghofe@11471
   477
           else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
berghofe@11471
   478
             Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
berghofe@11471
   479
               Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
berghofe@5177
   480
      end;
berghofe@5177
   481
berghofe@11471
   482
    val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
berghofe@11471
   483
      (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
berghofe@11471
   484
berghofe@11471
   485
    (* all the theorems are proved by one single simultaneous induction *)
berghofe@11471
   486
berghofe@13641
   487
    val range_eqs = map (fn r => mk_meta_eq (r RS range_ex1_eq))
berghofe@13641
   488
      iso_inj_thms_unfolded;
berghofe@13641
   489
berghofe@11471
   490
    val iso_thms = if length descr = 1 then [] else
skalberg@15570
   491
      Library.drop (length newTs, split_conj_thm
wenzelm@20046
   492
        (Goal.prove_global thy5 [] [] iso_t (fn _ => EVERY
berghofe@13641
   493
           [(indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
berghofe@11471
   494
            REPEAT (rtac TrueI 1),
berghofe@13641
   495
            rewrite_goals_tac (mk_meta_eq choice_eq ::
berghofe@13641
   496
              symmetric (mk_meta_eq expand_fun_eq) :: range_eqs),
berghofe@13641
   497
            rewrite_goals_tac (map symmetric range_eqs),
berghofe@11471
   498
            REPEAT (EVERY
berghofe@13641
   499
              [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
skalberg@15570
   500
                 List.concat (map (mk_funs_inv o #1) newT_iso_axms)) 1),
berghofe@11471
   501
               TRY (hyp_subst_tac 1),
berghofe@11471
   502
               rtac (sym RS range_eqI) 1,
wenzelm@20046
   503
               resolve_tac iso_char_thms 1])])));
wenzelm@11435
   504
wenzelm@11435
   505
    val Abs_inverse_thms' =
wenzelm@11435
   506
      map #1 newT_iso_axms @
haftmann@18330
   507
      map2 (fn r_inj => fn r => f_myinv_f OF [r_inj, r RS mp])
haftmann@18330
   508
        iso_inj_thms_unfolded iso_thms;
wenzelm@11435
   509
skalberg@15570
   510
    val Abs_inverse_thms = List.concat (map mk_funs_inv Abs_inverse_thms');
berghofe@5177
   511
berghofe@5177
   512
    (******************* freeness theorems for constructors *******************)
berghofe@5177
   513
wenzelm@6427
   514
    val _ = message "Proving freeness of constructors ...";
berghofe@5177
   515
berghofe@5177
   516
    (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
berghofe@5177
   517
    
berghofe@5177
   518
    fun prove_constr_rep_thm eqn =
berghofe@5177
   519
      let
berghofe@5177
   520
        val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
berghofe@7015
   521
        val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
wenzelm@20046
   522
      in Goal.prove_global thy5 [] [] eqn (fn _ => EVERY
berghofe@5177
   523
        [resolve_tac inj_thms 1,
berghofe@5177
   524
         rewrite_goals_tac rewrites,
berghofe@5177
   525
         rtac refl 1,
berghofe@5177
   526
         resolve_tac rep_intrs 2,
wenzelm@20046
   527
         REPEAT (resolve_tac iso_elem_thms 1)])
berghofe@5177
   528
      end;
berghofe@5177
   529
berghofe@5177
   530
    (*--------------------------------------------------------------*)
berghofe@5177
   531
    (* constr_rep_thms and rep_congs are used to prove distinctness *)
berghofe@7015
   532
    (* of constructors.                                             *)
berghofe@5177
   533
    (*--------------------------------------------------------------*)
berghofe@5177
   534
berghofe@5177
   535
    val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
berghofe@5177
   536
berghofe@5177
   537
    val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
berghofe@5177
   538
      dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
berghofe@5177
   539
        (constr_rep_thms ~~ dist_lemmas);
berghofe@5177
   540
berghofe@7015
   541
    fun prove_distinct_thms (_, []) = []
berghofe@7015
   542
      | prove_distinct_thms (dist_rewrites', t::_::ts) =
berghofe@7015
   543
          let
wenzelm@20046
   544
            val dist_thm = Goal.prove_global thy5 [] [] t (fn _ =>
wenzelm@20046
   545
              EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
berghofe@7015
   546
          in dist_thm::(standard (dist_thm RS not_sym))::
berghofe@7015
   547
            (prove_distinct_thms (dist_rewrites', ts))
berghofe@7015
   548
          end;
berghofe@7015
   549
berghofe@7015
   550
    val distinct_thms = map prove_distinct_thms (dist_rewrites ~~
berghofe@7015
   551
      DatatypeProp.make_distincts new_type_names descr sorts thy5);
berghofe@7015
   552
berghofe@7015
   553
    val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
berghofe@7015
   554
      if length constrs < !DatatypeProp.dtK then FewConstrs dists
berghofe@7015
   555
      else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
berghofe@7015
   556
        constr_rep_thms ~~ rep_congs ~~ distinct_thms);
berghofe@7015
   557
berghofe@5177
   558
    (* prove injectivity of constructors *)
berghofe@5177
   559
berghofe@5177
   560
    fun prove_constr_inj_thm rep_thms t =
berghofe@13641
   561
      let val inj_thms = Scons_inject :: (map make_elim
berghofe@5177
   562
        ((map (fn r => r RS injD) iso_inj_thms) @
berghofe@13641
   563
          [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
berghofe@13641
   564
           Lim_inject, Suml_inject, Sumr_inject]))
wenzelm@20046
   565
      in Goal.prove_global thy5 [] [] t (fn _ => EVERY
berghofe@5177
   566
        [rtac iffI 1,
berghofe@5177
   567
         REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
berghofe@5177
   568
         dresolve_tac rep_congs 1, dtac box_equals 1,
berghofe@13641
   569
         REPEAT (resolve_tac rep_thms 1),
berghofe@5177
   570
         REPEAT (eresolve_tac inj_thms 1),
berghofe@13641
   571
         REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
berghofe@13641
   572
           REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
wenzelm@20046
   573
           atac 1]))])
berghofe@5177
   574
      end;
berghofe@5177
   575
berghofe@5177
   576
    val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
berghofe@5177
   577
      ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
berghofe@5177
   578
haftmann@18314
   579
    val ((constr_inject', distinct_thms'), thy6) =
haftmann@18314
   580
      thy5
haftmann@18314
   581
      |> parent_path flat_names
haftmann@18314
   582
      |> store_thmss "inject" new_type_names constr_inject
haftmann@18314
   583
      ||>> store_thmss "distinct" new_type_names distinct_thms;
berghofe@5177
   584
berghofe@5177
   585
    (*************************** induction theorem ****************************)
berghofe@5177
   586
wenzelm@6427
   587
    val _ = message "Proving induction rule for datatypes ...";
berghofe@5177
   588
berghofe@5177
   589
    val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
berghofe@11471
   590
      (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
berghofe@11471
   591
    val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
berghofe@5177
   592
berghofe@5177
   593
    fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
berghofe@5177
   594
      let
skalberg@15570
   595
        val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT) $
berghofe@5177
   596
          mk_Free "x" T i;
berghofe@5177
   597
berghofe@5177
   598
        val Abs_t = if i < length newTs then
wenzelm@6394
   599
            Const (Sign.intern_const (Theory.sign_of thy6)
skalberg@15570
   600
              ("Abs_" ^ (List.nth (new_type_names, i))), Univ_elT --> T)
wenzelm@11435
   601
          else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
skalberg@15570
   602
            Const (List.nth (all_rep_names, i), T --> Univ_elT)
berghofe@5177
   603
berghofe@5177
   604
      in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
skalberg@15570
   605
            Const (List.nth (rep_set_names, i), UnivT)) $
berghofe@5177
   606
              (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
berghofe@5177
   607
          concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
berghofe@5177
   608
      end;
berghofe@5177
   609
berghofe@5177
   610
    val (indrule_lemma_prems, indrule_lemma_concls) =
skalberg@15570
   611
      Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
berghofe@5177
   612
wenzelm@6394
   613
    val cert = cterm_of (Theory.sign_of thy6);
berghofe@5177
   614
wenzelm@20046
   615
    val indrule_lemma = Goal.prove_global thy6 [] []
berghofe@5177
   616
      (Logic.mk_implies
berghofe@5177
   617
        (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
wenzelm@17985
   618
         HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
wenzelm@17985
   619
           [REPEAT (etac conjE 1),
berghofe@5177
   620
            REPEAT (EVERY
berghofe@5177
   621
              [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
wenzelm@20046
   622
               etac mp 1, resolve_tac iso_elem_thms 1])]);
berghofe@5177
   623
wenzelm@8305
   624
    val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
berghofe@5177
   625
    val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
berghofe@5177
   626
      map (Free o apfst fst o dest_Var) Ps;
berghofe@5177
   627
    val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
berghofe@5177
   628
wenzelm@17985
   629
    val dt_induct_prop = DatatypeProp.make_ind descr sorts;
wenzelm@20046
   630
    val dt_induct = Goal.prove_global thy6 []
wenzelm@17985
   631
      (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
wenzelm@17985
   632
      (fn prems => EVERY
berghofe@13641
   633
        [rtac indrule_lemma' 1,
berghofe@13641
   634
         (indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
berghofe@5177
   635
         EVERY (map (fn (prem, r) => (EVERY
berghofe@13641
   636
           [REPEAT (eresolve_tac Abs_inverse_thms 1),
berghofe@5177
   637
            simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
berghofe@13641
   638
            DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
wenzelm@20046
   639
                (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
berghofe@5177
   640
haftmann@18377
   641
    val ([dt_induct'], thy7) =
haftmann@18377
   642
      thy6
haftmann@18377
   643
      |> Theory.add_path big_name
haftmann@18377
   644
      |> PureThy.add_thms [(("induct", dt_induct), [case_names_induct])]
haftmann@18377
   645
      ||> Theory.parent_path;
berghofe@5177
   646
haftmann@18314
   647
  in
haftmann@18314
   648
    ((constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct'), thy7)
berghofe@5177
   649
  end;
berghofe@5177
   650
berghofe@5177
   651
end;