(*  Title:      HOL/Tools/datatype_rep_proofs.ML

     ID:         $Id$

     Author:     Stefan Berghofer, TU Muenchen

 Definitional introduction of datatypes

 Proof of characteristic theorems:

  - injectivity of constructors

  - distinctness of constructors

  - induction theorem

 *)

 signature DATATYPE_REP_PROOFS =

 sig

   val representation_proofs : bool -> DatatypeAux.datatype_info Symtab.table ->

     string list -> DatatypeAux.descr list -> (string * sort) list ->

       (string * mixfix) list -> (string * mixfix) list list -> attribute

         -> theory -> (thm list list * thm list list * thm list list *

           DatatypeAux.simproc_dist list * thm) * theory

 end;

 structure DatatypeRepProofs : DATATYPE_REP_PROOFS =

 struct

 open DatatypeAux;

 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);

 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];

 (** theory context references **)

 val f_myinv_f = thm "f_myinv_f";

 val myinv_f_f = thm "myinv_f_f";

 fun exh_thm_of (dt_info : datatype_info Symtab.table) tname =

   #exhaustion (the (Symtab.lookup dt_info tname));

 (******************************************************************************)

 fun representation_proofs flat_names (dt_info : datatype_info Symtab.table)

       new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =

   let

     val Datatype_thy = ThyInfo.the_theory "Datatype" thy;

     val node_name = "Datatype.node";

     val In0_name = "Datatype.In0";

     val In1_name = "Datatype.In1";

     val Scons_name = "Datatype.Scons";

     val Leaf_name = "Datatype.Leaf";

     val Numb_name = "Datatype.Numb";

     val Lim_name = "Datatype.Lim";

     val Suml_name = "Datatype.Suml";

     val Sumr_name = "Datatype.Sumr";

     val [In0_inject, In1_inject, Scons_inject, Leaf_inject,

          In0_eq, In1_eq, In0_not_In1, In1_not_In0,

          Lim_inject, Suml_inject, Sumr_inject] = map (get_thm Datatype_thy o Name)

         ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject",

          "In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0",

          "Lim_inject", "Suml_inject", "Sumr_inject"];

     val descr' = List.concat descr;

     val big_name = space_implode "_" new_type_names;

     val thy1 = add_path flat_names big_name thy;

     val big_rec_name = big_name ^ "_rep_set";

     val rep_set_names' =

       (if length descr' = 1 then [big_rec_name] else

         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)

           (1 upto (length descr'))));

     val rep_set_names = map (Sign.full_name (Theory.sign_of thy1)) rep_set_names';

     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);

     val leafTs' = get_nonrec_types descr' sorts;

     val branchTs = get_branching_types descr' sorts;

     val branchT = if null branchTs then HOLogic.unitT

       else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;

     val arities = get_arities descr' \ 0;

     val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names [] (leafTs' @ branchTs);

     val leafTs = leafTs' @ (map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars);

     val recTs = get_rec_types descr' sorts;

     val newTs = Library.take (length (hd descr), recTs);

     val oldTs = Library.drop (length (hd descr), recTs);

     val sumT = if null leafTs then HOLogic.unitT

       else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;

     val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));

     val UnivT = HOLogic.mk_setT Univ_elT;

     val UnivT' = Univ_elT --> HOLogic.boolT;

     val Collect = Const ("Collect", UnivT' --> UnivT);

     val In0 = Const (In0_name, Univ_elT --> Univ_elT);

     val In1 = Const (In1_name, Univ_elT --> Univ_elT);

     val Leaf = Const (Leaf_name, sumT --> Univ_elT);

     val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);

     (* make injections needed for embedding types in leaves *)

     fun mk_inj T' x =

       let

         fun mk_inj' T n i =

           if n = 1 then x else

           let val n2 = n div 2;

               val Type (_, [T1, T2]) = T

           in

             if i <= n2 then

               Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)

             else

               Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))

           end

       in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)

       end;

     (* make injections for constructors *)

     fun mk_univ_inj ts = access_bal (fn t => In0 $ t, fn t => In1 $ t, if ts = [] then

         Const ("arbitrary", Univ_elT)

       else

         foldr1 (HOLogic.mk_binop Scons_name) ts);

     (* function spaces *)

     fun mk_fun_inj T' x =

       let

         fun mk_inj T n i =

           if n = 1 then x else

           let

             val n2 = n div 2;

             val Type (_, [T1, T2]) = T;

             fun mkT U = (U --> Univ_elT) --> T --> Univ_elT

           in

             if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i

             else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)

           end

       in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)

       end;

     val mk_lim = foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t)));

     (************** generate introduction rules for representing set **********)

     val _ = message "Constructing representing sets ...";

     (* make introduction rule for a single constructor *)

     fun make_intr s n (i, (_, cargs)) =

       let

         fun mk_prem (dt, (j, prems, ts)) = (case strip_dtyp dt of

             (dts, DtRec k) =>

               let

                 val Ts = map (typ_of_dtyp descr' sorts) dts;

                 val free_t =

                   app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)

               in (j + 1, list_all (map (pair "x") Ts,

                   HOLogic.mk_Trueprop

                     (Free (List.nth (rep_set_names', k), UnivT') $ free_t)) :: prems,

                 mk_lim free_t Ts :: ts)

               end

           | _ =>

               let val T = typ_of_dtyp descr' sorts dt

               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)

               end);

         val (_, prems, ts) = foldr mk_prem (1, [], []) cargs;

         val concl = HOLogic.mk_Trueprop

           (Free (s, UnivT') $ mk_univ_inj ts n i)

       in Logic.list_implies (prems, concl)

       end;

     val intr_ts = List.concat (map (fn ((_, (_, _, constrs)), rep_set_name) =>

       map (make_intr rep_set_name (length constrs))

         ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names'));

     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =

       setmp InductivePackage.quiet_mode (!quiet_mode)

         (TheoryTarget.init NONE #>

          InductivePackage.add_inductive_i false big_rec_name false true false

            (map (fn s => (s, SOME UnivT', NoSyn)) rep_set_names') []

            (map (fn x => (("", []), x)) intr_ts) [] #>

          apsnd (ProofContext.theory_of o LocalTheory.exit)) thy1;

     (********************************* typedef ********************************)

     val (typedefs, thy3) = thy2 |>

       parent_path flat_names |>

       fold_map (fn ((((name, mx), tvs), c), name') =>

         setmp TypedefPackage.quiet_mode true

           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx)

             (Collect $ Const (c, UnivT')) NONE

             (rtac exI 1 THEN rtac CollectI 1 THEN

               QUIET_BREADTH_FIRST (has_fewer_prems 1)

               (resolve_tac rep_intrs 1))))

                 (types_syntax ~~ tyvars ~~

                   (Library.take (length newTs, rep_set_names)) ~~ new_type_names) ||>

       add_path flat_names big_name;

     (*********************** definition of constructors ***********************)

     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";

     val rep_names = map (curry op ^ "Rep_") new_type_names;

     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))

       (1 upto (length (List.concat (tl descr))));

     val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @

       map (Sign.full_name (Theory.sign_of thy3)) rep_names';

     (* isomorphism declarations *)

     val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))

       (oldTs ~~ rep_names');

     (* constructor definitions *)

     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =

       let

         fun constr_arg (dt, (j, l_args, r_args)) =

           let val T = typ_of_dtyp descr' sorts dt;

               val free_t = mk_Free "x" T j

           in (case (strip_dtyp dt, strip_type T) of

               ((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim

                 (Const (List.nth (all_rep_names, m), U --> Univ_elT) $

                    app_bnds free_t (length Us)) Us :: r_args)

             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))

           end;

         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;

         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;

         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);

         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);

         val lhs = list_comb (Const (cname, constrT), l_args);

         val rhs = mk_univ_inj r_args n i;

         val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);

         val def_name = (Sign.base_name cname) ^ "_def";

         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq

           (Const (rep_name, T --> Univ_elT) $ lhs, rhs));

         val ([def_thm], thy') =

           thy

           |> Theory.add_consts_i [(cname', constrT, mx)]

           |> (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)];

       in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;

     (* constructor definitions for datatype *)

     fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),

         ((((_, (_, _, constrs)), tname), T), constr_syntax)) =

       let

         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;

         val sg = Theory.sign_of thy;

         val rep_const = cterm_of sg

           (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));

         val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);

         val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);

         val (thy', defs', eqns', _) = Library.foldl ((make_constr_def tname T) (length constrs))

           ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)

       in

         (parent_path flat_names thy', defs', eqns @ [eqns'],

           rep_congs @ [cong'], dist_lemmas @ [dist])

       end;

     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = Library.foldl dt_constr_defs

       ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),

         hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);

     (*********** isomorphisms for new types (introduced by typedef) ***********)

     val _ = message "Proving isomorphism properties ...";

     val newT_iso_axms = map (fn (_, td) =>

       (collect_simp (#Abs_inverse td), #Rep_inverse td,

        collect_simp (#Rep td))) typedefs;

     val newT_iso_inj_thms = map (fn (_, td) =>

       (collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;

     (********* isomorphisms between existing types and "unfolded" types *******)

     (*---------------------------------------------------------------------*)

     (* isomorphisms are defined using primrec-combinators:                 *)

     (* generate appropriate functions for instantiating primrec-combinator *)

     (*                                                                     *)

     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)

     (*                                                                     *)

     (* also generate characteristic equations for isomorphisms             *)

     (*                                                                     *)

     (*   e.g.  dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)

     (*---------------------------------------------------------------------*)

     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =

       let

         val argTs = map (typ_of_dtyp descr' sorts) cargs;

         val T = List.nth (recTs, k);

         val rep_name = List.nth (all_rep_names, k);

         val rep_const = Const (rep_name, T --> Univ_elT);

         val constr = Const (cname, argTs ---> T);

         fun process_arg ks' ((i2, i2', ts, Ts), dt) =

           let

             val T' = typ_of_dtyp descr' sorts dt;

             val (Us, U) = strip_type T'

           in (case strip_dtyp dt of

               (_, DtRec j) => if j mem ks' then

                   (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds

                      (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],

                    Ts @ [Us ---> Univ_elT])

                 else

                   (i2 + 1, i2', ts @ [mk_lim

                      (Const (List.nth (all_rep_names, j), U --> Univ_elT) $

                         app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)

             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))

           end;

         val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);

         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));

         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));

         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);

         val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);

         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq

           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))

       in (fs @ [f], eqns @ [eqn], i + 1) end;

     (* define isomorphisms for all mutually recursive datatypes in list ds *)

     fun make_iso_defs (ds, (thy, char_thms)) =

       let

         val ks = map fst ds;

         val (_, (tname, _, _)) = hd ds;

         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);

         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =

           let

             val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))

               ((fs, eqns, 1), constrs);

             val iso = (List.nth (recTs, k), List.nth (all_rep_names, k))

           in (fs', eqns', isos @ [iso]) end;

         val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);

         val fTs = map fastype_of fs;

         val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",

           equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $

             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);

         val (def_thms, thy') = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;

         (* prove characteristic equations *)

         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);

         val char_thms' = map (fn eqn => Goal.prove_global thy' [] [] eqn

           (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;

       in (thy', char_thms' @ char_thms) end;

     val (thy5, iso_char_thms) = foldr make_iso_defs

       (add_path flat_names big_name thy4, []) (tl descr);

     (* prove isomorphism properties *)

     fun mk_funs_inv thm =

       let

         val {sign, prop, ...} = rep_thm thm;

         val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $

           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;

         val used = add_term_tfree_names (a, []);

         fun mk_thm i =

           let

             val Ts = map (TFree o rpair HOLogic.typeS)

               (Name.variant_list used (replicate i "'t"));

             val f = Free ("f", Ts ---> U)

           in Goal.prove_global sign [] [] (Logic.mk_implies

             (HOLogic.mk_Trueprop (HOLogic.list_all

                (map (pair "x") Ts, S $ app_bnds f i)),

              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,

                r $ (a $ app_bnds f i)), f))))

             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1])

           end

       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;

     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)

     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =

       let

         val (_, (tname, _, _)) = hd ds;

         val {induction, ...} = the (Symtab.lookup dt_info tname);

         fun mk_ind_concl (i, _) =

           let

             val T = List.nth (recTs, i);

             val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT);

             val rep_set_name = List.nth (rep_set_names, i)

           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $

                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $

                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),

               Const (rep_set_name, UnivT') $ (Rep_t $ mk_Free "x" T i))

           end;

         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);

         val rewrites = map mk_meta_eq iso_char_thms;

         val inj_thms' = map snd newT_iso_inj_thms @

           map (fn r => r RS injD) inj_thms;

         val inj_thm = Goal.prove_global thy5 [] []

           (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY

             [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,

              REPEAT (EVERY

                [rtac allI 1, rtac impI 1,

                 exh_tac (exh_thm_of dt_info) 1,

                 REPEAT (EVERY

                   [hyp_subst_tac 1,

                    rewrite_goals_tac rewrites,

                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),

                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)

                    ORELSE (EVERY

                      [REPEAT (eresolve_tac (Scons_inject ::

                         map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),

                       REPEAT (cong_tac 1), rtac refl 1,

                       REPEAT (atac 1 ORELSE (EVERY

                         [REPEAT (rtac ext 1),

                          REPEAT (eresolve_tac (mp :: allE ::

                            map make_elim (Suml_inject :: Sumr_inject ::

                              Lim_inject :: fun_cong :: inj_thms')) 1),

                          atac 1]))])])])]);

         val inj_thms'' = map (fn r => r RS datatype_injI)

                              (split_conj_thm inj_thm);

         val elem_thm = 

             Goal.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))

               (fn _ =>

                EVERY [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,

                 rewrite_goals_tac rewrites,

                 REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW

                   ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);

       in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))

       end;

     val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms

       ([], map #3 newT_iso_axms) (tl descr);

     val iso_inj_thms = map snd newT_iso_inj_thms @

       map (fn r => r RS injD) iso_inj_thms_unfolded;

     (* prove  dt_rep_set_i x --> x : range dt_Rep_i *)

     fun mk_iso_t (((set_name, iso_name), i), T) =

       let val isoT = T --> Univ_elT

       in HOLogic.imp $ 

         (Const (set_name, UnivT') $ mk_Free "x" Univ_elT i) $

           (if i < length newTs then Const ("True", HOLogic.boolT)

            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,

              Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $

                Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))

       end;

     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t

       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));

     (* all the theorems are proved by one single simultaneous induction *)

     val range_eqs = map (fn r => mk_meta_eq (r RS range_ex1_eq))

       iso_inj_thms_unfolded;

     val iso_thms = if length descr = 1 then [] else

       Library.drop (length newTs, split_conj_thm

         (Goal.prove_global thy5 [] [] iso_t (fn _ => EVERY

            [(indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,

             REPEAT (rtac TrueI 1),

             rewrite_goals_tac (mk_meta_eq choice_eq ::

               symmetric (mk_meta_eq expand_fun_eq) :: range_eqs),

             rewrite_goals_tac (map symmetric range_eqs),

             REPEAT (EVERY

               [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @

                  List.concat (map (mk_funs_inv o #1) newT_iso_axms)) 1),

                TRY (hyp_subst_tac 1),

                rtac (sym RS range_eqI) 1,

                resolve_tac iso_char_thms 1])])));

     val Abs_inverse_thms' =

       map #1 newT_iso_axms @

       map2 (fn r_inj => fn r => f_myinv_f OF [r_inj, r RS mp])

         iso_inj_thms_unfolded iso_thms;

     val Abs_inverse_thms = List.concat (map mk_funs_inv Abs_inverse_thms');

     (******************* freeness theorems for constructors *******************)

     val _ = message "Proving freeness of constructors ...";

     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)

     fun prove_constr_rep_thm eqn =

       let

         val inj_thms = map fst newT_iso_inj_thms;

         val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)

       in Goal.prove_global thy5 [] [] eqn (fn _ => EVERY

         [resolve_tac inj_thms 1,

          rewrite_goals_tac rewrites,

          rtac refl 3,

          resolve_tac rep_intrs 2,

          REPEAT (resolve_tac iso_elem_thms 1)])

       end;

     (*--------------------------------------------------------------*)

     (* constr_rep_thms and rep_congs are used to prove distinctness *)

     (* of constructors.                                             *)

     (*--------------------------------------------------------------*)

     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;

     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>

       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))

         (constr_rep_thms ~~ dist_lemmas);

     fun prove_distinct_thms (_, []) = []

       | prove_distinct_thms (dist_rewrites', t::_::ts) =

           let

             val dist_thm = Goal.prove_global thy5 [] [] t (fn _ =>

               EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])

           in dist_thm::(standard (dist_thm RS not_sym))::

             (prove_distinct_thms (dist_rewrites', ts))

           end;

     val distinct_thms = map prove_distinct_thms (dist_rewrites ~~

       DatatypeProp.make_distincts new_type_names descr sorts thy5);

     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>

       if length constrs < !DatatypeProp.dtK then FewConstrs dists

       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~

         constr_rep_thms ~~ rep_congs ~~ distinct_thms);

     (* prove injectivity of constructors *)

     fun prove_constr_inj_thm rep_thms t =

       let val inj_thms = Scons_inject :: (map make_elim

         (iso_inj_thms @

           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,

            Lim_inject, Suml_inject, Sumr_inject]))

       in Goal.prove_global thy5 [] [] t (fn _ => EVERY

         [rtac iffI 1,

          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,

          dresolve_tac rep_congs 1, dtac box_equals 1,

          REPEAT (resolve_tac rep_thms 1),

          REPEAT (eresolve_tac inj_thms 1),

          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),

            REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),

            atac 1]))])

       end;

     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)

       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);

     val ((constr_inject', distinct_thms'), thy6) =

       thy5

       |> parent_path flat_names

       |> store_thmss "inject" new_type_names constr_inject

       ||>> store_thmss "distinct" new_type_names distinct_thms;

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

     val _ = message "Proving induction rule for datatypes ...";

     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @

       (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);

     val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;

     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =

       let

         val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT) $

           mk_Free "x" T i;

         val Abs_t = if i < length newTs then

             Const (Sign.intern_const (Theory.sign_of thy6)

               ("Abs_" ^ (List.nth (new_type_names, i))), Univ_elT --> T)

           else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $

             Const (List.nth (all_rep_names, i), T --> Univ_elT)

       in (prems @ [HOLogic.imp $

             (Const (List.nth (rep_set_names, i), UnivT') $ Rep_t) $

               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],

           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])

       end;

     val (indrule_lemma_prems, indrule_lemma_concls) =

       Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));

     val cert = cterm_of (Theory.sign_of thy6);

     val indrule_lemma = Goal.prove_global thy6 [] []

       (Logic.mk_implies

         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),

          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY

            [REPEAT (etac conjE 1),

             REPEAT (EVERY

               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,

                etac mp 1, resolve_tac iso_elem_thms 1])]);

     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));

     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else

       map (Free o apfst fst o dest_Var) Ps;

     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;

     val dt_induct_prop = DatatypeProp.make_ind descr sorts;

     val dt_induct = Goal.prove_global thy6 []

       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)

       (fn prems => EVERY

         [rtac indrule_lemma' 1,

          (indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,

          EVERY (map (fn (prem, r) => (EVERY

            [REPEAT (eresolve_tac Abs_inverse_thms 1),

             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,

             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))

                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);

     val ([dt_induct'], thy7) =

       thy6

       |> Theory.add_path big_name

       |> PureThy.add_thms [(("induct", dt_induct), [case_names_induct])]

       ||> Theory.parent_path;

   in

     ((constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct'), thy7)

   end;

 end;