src/HOL/Tools/Datatype/datatype_rep_proofs.ML
author nipkow
Thu Oct 22 09:27:48 2009 +0200 (2009-10-22)
changeset 33057 764547b68538
parent 33040 cffdb7b28498
child 33171 292970b42770
permissions -rw-r--r--
inv_onto -> inv_into
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(*  Title:      HOL/Tools/datatype_rep_proofs.ML
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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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  include DATATYPE_COMMON
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  val representation_proofs : config -> info Symtab.table ->
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    string list -> descr list -> (string * sort) list ->
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      (binding * mixfix) list -> (binding * mixfix) list list -> attribute
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        -> theory -> (thm list list * thm list 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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val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
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(** theory context references **)
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fun exh_thm_of (dt_info : info Symtab.table) tname =
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  #exhaust (the (Symtab.lookup dt_info tname));
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(******************************************************************************)
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fun representation_proofs (config : config) (dt_info : 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.node";
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    val In0_name = "Datatype.In0";
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    val In1_name = "Datatype.In1";
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    val Scons_name = "Datatype.Scons";
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    val Leaf_name = "Datatype.Leaf";
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    val Numb_name = "Datatype.Numb";
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    val Lim_name = "Datatype.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 (PureThy.get_thm Datatype_thy)
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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' = flat descr;
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    val big_name = space_implode "_" new_type_names;
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    val thy1 = Sign.add_path big_name thy;
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    val big_rec_name = big_name ^ "_rep_set";
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    val rep_set_names' =
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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 rep_set_names = map (Sign.full_bname thy1) rep_set_names';
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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 Balanced_Tree.make (fn (T, U) => Type ("+", [T, U])) branchTs;
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    val arities = remove (op =) 0 (get_arities descr');
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    val unneeded_vars = subtract (op =) (List.foldr OldTerm.add_typ_tfree_names [] (leafTs' @ branchTs)) (hd tyvars);
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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 Balanced_Tree.make (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 UnivT' = Univ_elT --> HOLogic.boolT;
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    val Collect = Const ("Collect", UnivT' --> UnivT);
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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 (fn T'' => T'' = T') leafTs)
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      end;
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    (* make injections for constructors *)
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    fun mk_univ_inj ts = Balanced_Tree.access
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      {left = fn t => In0 $ t,
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        right = fn t => In1 $ t,
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        init =
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          if ts = [] then Const (@{const_name undefined}, Univ_elT)
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          else 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 (fn T'' => T'' = T') branchTs)
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      end;
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    val mk_lim = List.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 config "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
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                    (Free (nth rep_set_names' k, UnivT') $ free_t)) :: 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) = List.foldr mk_prem (1, [], []) cargs;
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        val concl = HOLogic.mk_Trueprop
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          (Free (s, UnivT') $ mk_univ_inj ts n i)
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      in Logic.list_implies (prems, concl)
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      end;
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    val intr_ts = maps (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 ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
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        Inductive.add_inductive_global (serial_string ())
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          {quiet_mode = #quiet config, verbose = false, kind = Thm.internalK,
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           alt_name = Binding.name big_rec_name, coind = false, no_elim = true, no_ind = false,
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           skip_mono = true, fork_mono = false}
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          (map (fn s => ((Binding.name s, UnivT'), NoSyn)) rep_set_names') []
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          (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] thy1;
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    (********************************* typedef ********************************)
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    val (typedefs, thy3) = thy2 |>
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      Sign.parent_path |>
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      fold_map (fn ((((name, mx), tvs), c), name') =>
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          Typedef.add_typedef false (SOME (Binding.name name')) (name, tvs, mx)
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            (Collect $ Const (c, UnivT')) NONE
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            (rtac exI 1 THEN rtac CollectI 1 THEN
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              QUIET_BREADTH_FIRST (has_fewer_prems 1)
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              (resolve_tac rep_intrs 1)))
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                (types_syntax ~~ tyvars ~~
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                  (Library.take (length newTs, rep_set_names)) ~~ new_type_names) ||>
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      Sign.add_path big_name;
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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 thy3) rep_names @
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      map (Sign.full_bname thy3) rep_names';
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    (* isomorphism declarations *)
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    val iso_decls = map (fn (T, s) => (Binding.name 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 (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) = List.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 thy ("Abs_" ^ tname);
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        val rep_name = Sign.intern_const 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 = Logic.mk_equals (lhs, Const (abs_name, Univ_elT --> T) $ rhs);
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        val def_name = Long_Name.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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          |> Sign.add_consts_i [(cname', constrT, mx)]
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          |> (PureThy.add_defs false o map Thm.no_attributes) [(Binding.name 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 rep_const = cterm_of thy
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          (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> Univ_elT));
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        val cong' =
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          Drule.standard (cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong);
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        val dist =
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          Drule.standard (cterm_instantiate [(cterm_of thy 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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          ((Sign.add_path tname thy, defs, [], 1), constrs ~~ constr_syntax)
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      in
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        (Sign.parent_path 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 |> Sign.add_consts_i iso_decls |> Sign.parent_path, [], [], [], []),
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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 config "Proving isomorphism properties ...";
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    val newT_iso_axms = map (fn (_, td) =>
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      (collect_simp (#Abs_inverse td), #Rep_inverse td,
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       collect_simp (#Rep td))) typedefs;
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    val newT_iso_inj_thms = map (fn (_, td) =>
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      (collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;
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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 (Scons (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 (Scons (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 recTs k;
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        val rep_name = nth all_rep_names k;
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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, Ts), dt) =
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          let
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            val T' = typ_of_dtyp descr' sorts dt;
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            val (Us, U) = strip_type T'
berghofe@13641
   298
          in (case strip_dtyp dt of
berghofe@13641
   299
              (_, DtRec j) => if j mem ks' then
skalberg@15574
   300
                  (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds
skalberg@15574
   301
                     (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
berghofe@13641
   302
                   Ts @ [Us ---> Univ_elT])
berghofe@5177
   303
                else
skalberg@15574
   304
                  (i2 + 1, i2', ts @ [mk_lim
haftmann@31949
   305
                     (Const (nth all_rep_names j, U --> Univ_elT) $
skalberg@15574
   306
                        app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)
berghofe@7015
   307
            | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
berghofe@5177
   308
          end;
berghofe@5177
   309
skalberg@15570
   310
        val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);
berghofe@5177
   311
        val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
berghofe@7015
   312
        val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
berghofe@5177
   313
        val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
berghofe@5177
   314
skalberg@15570
   315
        val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);
berghofe@5177
   316
        val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
berghofe@5177
   317
          (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
berghofe@5177
   318
berghofe@5177
   319
      in (fs @ [f], eqns @ [eqn], i + 1) end;
berghofe@5177
   320
berghofe@5177
   321
    (* define isomorphisms for all mutually recursive datatypes in list ds *)
berghofe@5177
   322
berghofe@5177
   323
    fun make_iso_defs (ds, (thy, char_thms)) =
berghofe@5177
   324
      let
berghofe@5177
   325
        val ks = map fst ds;
berghofe@5177
   326
        val (_, (tname, _, _)) = hd ds;
wenzelm@17412
   327
        val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);
berghofe@5177
   328
berghofe@5177
   329
        fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
berghofe@5177
   330
          let
skalberg@15570
   331
            val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))
berghofe@5177
   332
              ((fs, eqns, 1), constrs);
haftmann@31949
   333
            val iso = (nth recTs k, nth all_rep_names k)
berghofe@5177
   334
          in (fs', eqns', isos @ [iso]) end;
berghofe@5177
   335
        
skalberg@15570
   336
        val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);
berghofe@5177
   337
        val fTs = map fastype_of fs;
wenzelm@30364
   338
        val defs = map (fn (rec_name, (T, iso_name)) => (Binding.name (Long_Name.base_name iso_name ^ "_def"),
wenzelm@27330
   339
          Logic.mk_equals (Const (iso_name, T --> Univ_elT),
wenzelm@27330
   340
            list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs)))) (rec_names ~~ isos);
wenzelm@28362
   341
        val (def_thms, thy') =
wenzelm@28362
   342
          apsnd Theory.checkpoint ((PureThy.add_defs false o map Thm.no_attributes) defs thy);
berghofe@5177
   343
berghofe@5177
   344
        (* prove characteristic equations *)
berghofe@5177
   345
oheimb@5553
   346
        val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
wenzelm@32970
   347
        val char_thms' = map (fn eqn => Skip_Proof.prove_global thy' [] [] eqn
wenzelm@20046
   348
          (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
berghofe@5177
   349
berghofe@5177
   350
      in (thy', char_thms' @ char_thms) end;
berghofe@5177
   351
wenzelm@30190
   352
    val (thy5, iso_char_thms) = apfst Theory.checkpoint (List.foldr make_iso_defs
haftmann@32124
   353
      (Sign.add_path big_name thy4, []) (tl descr));
berghofe@5177
   354
berghofe@5177
   355
    (* prove isomorphism properties *)
berghofe@5177
   356
wenzelm@28362
   357
    fun mk_funs_inv thy thm =
berghofe@7015
   358
      let
wenzelm@26626
   359
        val prop = Thm.prop_of thm;
berghofe@21021
   360
        val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
wenzelm@16287
   361
          (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
wenzelm@29270
   362
        val used = OldTerm.add_term_tfree_names (a, []);
berghofe@13641
   363
berghofe@13641
   364
        fun mk_thm i =
berghofe@13641
   365
          let
berghofe@13641
   366
            val Ts = map (TFree o rpair HOLogic.typeS)
wenzelm@20071
   367
              (Name.variant_list used (replicate i "'t"));
berghofe@13641
   368
            val f = Free ("f", Ts ---> U)
wenzelm@32970
   369
          in Skip_Proof.prove_global thy [] [] (Logic.mk_implies
berghofe@13641
   370
            (HOLogic.mk_Trueprop (HOLogic.list_all
berghofe@21021
   371
               (map (pair "x") Ts, S $ app_bnds f i)),
berghofe@13641
   372
             HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
wenzelm@17985
   373
               r $ (a $ app_bnds f i)), f))))
berghofe@26806
   374
            (fn _ => EVERY [REPEAT_DETERM_N i (rtac ext 1),
berghofe@26806
   375
               REPEAT (etac allE 1), rtac thm 1, atac 1])
berghofe@13641
   376
          end
berghofe@13641
   377
      in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
berghofe@7015
   378
berghofe@5177
   379
    (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
berghofe@5177
   380
berghofe@26806
   381
    val fun_congs = map (fn T => make_elim (Drule.instantiate'
berghofe@26806
   382
      [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs;
berghofe@26806
   383
berghofe@5177
   384
    fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
berghofe@5177
   385
      let
berghofe@5177
   386
        val (_, (tname, _, _)) = hd ds;
haftmann@32727
   387
        val induct = (#induct o the o Symtab.lookup dt_info) tname;
berghofe@5177
   388
berghofe@5177
   389
        fun mk_ind_concl (i, _) =
berghofe@5177
   390
          let
haftmann@31949
   391
            val T = nth recTs i;
haftmann@31949
   392
            val Rep_t = Const (nth all_rep_names i, T --> Univ_elT);
haftmann@31949
   393
            val rep_set_name = nth rep_set_names i
berghofe@5177
   394
          in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
berghofe@5177
   395
                HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
berghofe@5177
   396
                  HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
berghofe@21021
   397
              Const (rep_set_name, UnivT') $ (Rep_t $ mk_Free "x" T i))
berghofe@5177
   398
          end;
berghofe@5177
   399
berghofe@5177
   400
        val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
berghofe@5177
   401
oheimb@5553
   402
        val rewrites = map mk_meta_eq iso_char_thms;
berghofe@21021
   403
        val inj_thms' = map snd newT_iso_inj_thms @
haftmann@26359
   404
          map (fn r => r RS @{thm injD}) inj_thms;
berghofe@5177
   405
wenzelm@32970
   406
        val inj_thm = Skip_Proof.prove_global thy5 [] []
wenzelm@17985
   407
          (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY
haftmann@32712
   408
            [(indtac induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
berghofe@5177
   409
             REPEAT (EVERY
berghofe@5177
   410
               [rtac allI 1, rtac impI 1,
berghofe@5177
   411
                exh_tac (exh_thm_of dt_info) 1,
berghofe@5177
   412
                REPEAT (EVERY
berghofe@5177
   413
                  [hyp_subst_tac 1,
berghofe@5177
   414
                   rewrite_goals_tac rewrites,
berghofe@5177
   415
                   REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
berghofe@5177
   416
                   (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
berghofe@5177
   417
                   ORELSE (EVERY
berghofe@13641
   418
                     [REPEAT (eresolve_tac (Scons_inject ::
berghofe@13641
   419
                        map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
berghofe@13641
   420
                      REPEAT (cong_tac 1), rtac refl 1,
berghofe@13641
   421
                      REPEAT (atac 1 ORELSE (EVERY
berghofe@13641
   422
                        [REPEAT (rtac ext 1),
berghofe@13641
   423
                         REPEAT (eresolve_tac (mp :: allE ::
berghofe@13641
   424
                           map make_elim (Suml_inject :: Sumr_inject ::
berghofe@26806
   425
                             Lim_inject :: inj_thms') @ fun_congs) 1),
wenzelm@20046
   426
                         atac 1]))])])])]);
berghofe@5177
   427
haftmann@26359
   428
        val inj_thms'' = map (fn r => r RS @{thm datatype_injI})
paulson@6171
   429
                             (split_conj_thm inj_thm);
berghofe@5177
   430
paulson@6171
   431
        val elem_thm = 
wenzelm@32970
   432
            Skip_Proof.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))
wenzelm@20046
   433
              (fn _ =>
haftmann@32712
   434
               EVERY [(indtac induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
wenzelm@20046
   435
                rewrite_goals_tac rewrites,
wenzelm@20046
   436
                REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
wenzelm@20046
   437
                  ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
berghofe@5177
   438
berghofe@11471
   439
      in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
berghofe@11471
   440
      end;
berghofe@11471
   441
wenzelm@30190
   442
    val (iso_inj_thms_unfolded, iso_elem_thms) = List.foldr prove_iso_thms
skalberg@15574
   443
      ([], map #3 newT_iso_axms) (tl descr);
berghofe@21021
   444
    val iso_inj_thms = map snd newT_iso_inj_thms @
haftmann@26359
   445
      map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded;
berghofe@11471
   446
berghofe@21021
   447
    (* prove  dt_rep_set_i x --> x : range dt_Rep_i *)
berghofe@11471
   448
berghofe@11471
   449
    fun mk_iso_t (((set_name, iso_name), i), T) =
berghofe@11471
   450
      let val isoT = T --> Univ_elT
berghofe@11471
   451
      in HOLogic.imp $ 
berghofe@21021
   452
        (Const (set_name, UnivT') $ mk_Free "x" Univ_elT i) $
haftmann@31643
   453
          (if i < length newTs then HOLogic.true_const
berghofe@11471
   454
           else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
haftmann@31643
   455
             Const (@{const_name image}, isoT --> HOLogic.mk_setT T --> UnivT) $
haftmann@30304
   456
               Const (iso_name, isoT) $ Const (@{const_name UNIV}, HOLogic.mk_setT T)))
berghofe@5177
   457
      end;
berghofe@5177
   458
berghofe@11471
   459
    val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
berghofe@11471
   460
      (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
berghofe@11471
   461
berghofe@11471
   462
    (* all the theorems are proved by one single simultaneous induction *)
berghofe@11471
   463
haftmann@26359
   464
    val range_eqs = map (fn r => mk_meta_eq (r RS @{thm range_ex1_eq}))
berghofe@13641
   465
      iso_inj_thms_unfolded;
berghofe@13641
   466
berghofe@11471
   467
    val iso_thms = if length descr = 1 then [] else
skalberg@15570
   468
      Library.drop (length newTs, split_conj_thm
wenzelm@32970
   469
        (Skip_Proof.prove_global thy5 [] [] iso_t (fn _ => EVERY
berghofe@25678
   470
           [(indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
berghofe@11471
   471
            REPEAT (rtac TrueI 1),
berghofe@13641
   472
            rewrite_goals_tac (mk_meta_eq choice_eq ::
haftmann@26359
   473
              symmetric (mk_meta_eq @{thm expand_fun_eq}) :: range_eqs),
berghofe@13641
   474
            rewrite_goals_tac (map symmetric range_eqs),
berghofe@11471
   475
            REPEAT (EVERY
berghofe@13641
   476
              [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
wenzelm@28362
   477
                 maps (mk_funs_inv thy5 o #1) newT_iso_axms) 1),
berghofe@11471
   478
               TRY (hyp_subst_tac 1),
berghofe@11471
   479
               rtac (sym RS range_eqI) 1,
wenzelm@20046
   480
               resolve_tac iso_char_thms 1])])));
wenzelm@11435
   481
wenzelm@11435
   482
    val Abs_inverse_thms' =
wenzelm@11435
   483
      map #1 newT_iso_axms @
nipkow@33057
   484
      map2 (fn r_inj => fn r => @{thm f_the_inv_into_f} OF [r_inj, r RS mp])
haftmann@18330
   485
        iso_inj_thms_unfolded iso_thms;
wenzelm@11435
   486
wenzelm@28362
   487
    val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms';
berghofe@5177
   488
berghofe@5177
   489
    (******************* freeness theorems for constructors *******************)
berghofe@5177
   490
haftmann@31668
   491
    val _ = message config "Proving freeness of constructors ...";
berghofe@5177
   492
berghofe@5177
   493
    (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
berghofe@5177
   494
    
berghofe@5177
   495
    fun prove_constr_rep_thm eqn =
berghofe@5177
   496
      let
berghofe@21021
   497
        val inj_thms = map fst newT_iso_inj_thms;
haftmann@26359
   498
        val rewrites = @{thm o_def} :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
wenzelm@32970
   499
      in Skip_Proof.prove_global thy5 [] [] eqn (fn _ => EVERY
berghofe@5177
   500
        [resolve_tac inj_thms 1,
berghofe@5177
   501
         rewrite_goals_tac rewrites,
berghofe@21021
   502
         rtac refl 3,
berghofe@5177
   503
         resolve_tac rep_intrs 2,
wenzelm@20046
   504
         REPEAT (resolve_tac iso_elem_thms 1)])
berghofe@5177
   505
      end;
berghofe@5177
   506
berghofe@5177
   507
    (*--------------------------------------------------------------*)
berghofe@5177
   508
    (* constr_rep_thms and rep_congs are used to prove distinctness *)
berghofe@7015
   509
    (* of constructors.                                             *)
berghofe@5177
   510
    (*--------------------------------------------------------------*)
berghofe@5177
   511
berghofe@5177
   512
    val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
berghofe@5177
   513
berghofe@5177
   514
    val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
berghofe@5177
   515
      dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
berghofe@5177
   516
        (constr_rep_thms ~~ dist_lemmas);
berghofe@5177
   517
haftmann@32900
   518
    fun prove_distinct_thms dist_rewrites' (k, ts) =
haftmann@32900
   519
      let
haftmann@32900
   520
        fun prove [] = []
haftmann@32900
   521
          | prove (t :: ts) =
haftmann@32900
   522
              let
wenzelm@32970
   523
                val dist_thm = Skip_Proof.prove_global thy5 [] [] t (fn _ =>
haftmann@32900
   524
                  EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
wenzelm@32957
   525
              in dist_thm :: Drule.standard (dist_thm RS not_sym) :: prove ts end;
haftmann@32900
   526
      in prove ts end;
berghofe@7015
   527
haftmann@32900
   528
    val distinct_thms = map2 (prove_distinct_thms)
haftmann@32900
   529
      dist_rewrites (DatatypeProp.make_distincts descr sorts);
berghofe@7015
   530
berghofe@5177
   531
    (* prove injectivity of constructors *)
berghofe@5177
   532
berghofe@5177
   533
    fun prove_constr_inj_thm rep_thms t =
berghofe@13641
   534
      let val inj_thms = Scons_inject :: (map make_elim
berghofe@21021
   535
        (iso_inj_thms @
berghofe@13641
   536
          [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
berghofe@13641
   537
           Lim_inject, Suml_inject, Sumr_inject]))
wenzelm@32970
   538
      in Skip_Proof.prove_global thy5 [] [] t (fn _ => EVERY
berghofe@5177
   539
        [rtac iffI 1,
berghofe@5177
   540
         REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
berghofe@5177
   541
         dresolve_tac rep_congs 1, dtac box_equals 1,
berghofe@13641
   542
         REPEAT (resolve_tac rep_thms 1),
berghofe@5177
   543
         REPEAT (eresolve_tac inj_thms 1),
berghofe@13641
   544
         REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
berghofe@13641
   545
           REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
wenzelm@20046
   546
           atac 1]))])
berghofe@5177
   547
      end;
berghofe@5177
   548
berghofe@5177
   549
    val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
berghofe@5177
   550
      ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
berghofe@5177
   551
haftmann@18314
   552
    val ((constr_inject', distinct_thms'), thy6) =
haftmann@18314
   553
      thy5
haftmann@32124
   554
      |> Sign.parent_path
haftmann@18314
   555
      |> store_thmss "inject" new_type_names constr_inject
haftmann@18314
   556
      ||>> store_thmss "distinct" new_type_names distinct_thms;
berghofe@5177
   557
berghofe@5177
   558
    (*************************** induction theorem ****************************)
berghofe@5177
   559
haftmann@31668
   560
    val _ = message config "Proving induction rule for datatypes ...";
berghofe@5177
   561
berghofe@5177
   562
    val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
berghofe@32999
   563
      (map (fn r => r RS @{thm the_inv_f_f} RS subst) iso_inj_thms_unfolded);
berghofe@32999
   564
    val Rep_inverse_thms' = map (fn r => r RS @{thm the_inv_f_f}) iso_inj_thms_unfolded;
berghofe@5177
   565
berghofe@5177
   566
    fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
berghofe@5177
   567
      let
haftmann@31949
   568
        val Rep_t = Const (nth all_rep_names i, T --> Univ_elT) $
berghofe@5177
   569
          mk_Free "x" T i;
berghofe@5177
   570
berghofe@5177
   571
        val Abs_t = if i < length newTs then
wenzelm@22578
   572
            Const (Sign.intern_const thy6
haftmann@31949
   573
              ("Abs_" ^ (nth new_type_names i)), Univ_elT --> T)
nipkow@33057
   574
          else Const (@{const_name the_inv_into},
berghofe@32999
   575
              [HOLogic.mk_setT T, T --> Univ_elT, Univ_elT] ---> T) $
berghofe@32999
   576
            HOLogic.mk_UNIV T $ Const (nth all_rep_names i, T --> Univ_elT)
berghofe@5177
   577
berghofe@21021
   578
      in (prems @ [HOLogic.imp $
haftmann@31949
   579
            (Const (nth rep_set_names i, UnivT') $ Rep_t) $
berghofe@5177
   580
              (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
berghofe@5177
   581
          concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
berghofe@5177
   582
      end;
berghofe@5177
   583
berghofe@5177
   584
    val (indrule_lemma_prems, indrule_lemma_concls) =
skalberg@15570
   585
      Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
berghofe@5177
   586
wenzelm@22578
   587
    val cert = cterm_of thy6;
berghofe@5177
   588
wenzelm@32970
   589
    val indrule_lemma = Skip_Proof.prove_global thy6 [] []
berghofe@5177
   590
      (Logic.mk_implies
berghofe@5177
   591
        (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
wenzelm@17985
   592
         HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
wenzelm@17985
   593
           [REPEAT (etac conjE 1),
berghofe@5177
   594
            REPEAT (EVERY
berghofe@5177
   595
              [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
wenzelm@20046
   596
               etac mp 1, resolve_tac iso_elem_thms 1])]);
berghofe@5177
   597
wenzelm@8305
   598
    val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
berghofe@5177
   599
    val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
berghofe@5177
   600
      map (Free o apfst fst o dest_Var) Ps;
berghofe@5177
   601
    val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
berghofe@5177
   602
wenzelm@17985
   603
    val dt_induct_prop = DatatypeProp.make_ind descr sorts;
wenzelm@32970
   604
    val dt_induct = Skip_Proof.prove_global thy6 []
wenzelm@17985
   605
      (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
wenzelm@26711
   606
      (fn {prems, ...} => EVERY
berghofe@13641
   607
        [rtac indrule_lemma' 1,
berghofe@25678
   608
         (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
berghofe@5177
   609
         EVERY (map (fn (prem, r) => (EVERY
berghofe@13641
   610
           [REPEAT (eresolve_tac Abs_inverse_thms 1),
berghofe@5177
   611
            simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
berghofe@13641
   612
            DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
wenzelm@20046
   613
                (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
berghofe@5177
   614
haftmann@18377
   615
    val ([dt_induct'], thy7) =
haftmann@18377
   616
      thy6
wenzelm@24712
   617
      |> Sign.add_path big_name
haftmann@29579
   618
      |> PureThy.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])]
wenzelm@28362
   619
      ||> Sign.parent_path
wenzelm@28362
   620
      ||> Theory.checkpoint;
berghofe@5177
   621
haftmann@18314
   622
  in
haftmann@32907
   623
    ((constr_inject', distinct_thms', dt_induct'), thy7)
berghofe@5177
   624
  end;
berghofe@5177
   625
berghofe@5177
   626
end;