src/HOL/Codatatype/Tools/bnf_gfp.ML
author traytel
Tue Sep 04 12:10:19 2012 +0200 (2012-09-04)
changeset 49109 0e5b859e1c91
parent 49105 a426099dc343
child 49121 9e0acaa470ab
permissions -rw-r--r--
no more aliases for Local_Theory.note; use Thm.close_derivation in internal theorems;
     1 (*  Title:      HOL/Codatatype/Tools/bnf_gfp.ML
     2     Author:     Dmitriy Traytel, TU Muenchen
     3     Author:     Andrei Popescu, TU Muenchen
     4     Author:     Jasmin Blanchette, TU Muenchen
     5     Copyright   2012
     6 
     7 Codatatype construction.
     8 *)
     9 
    10 signature BNF_GFP =
    11 sig
    12   val bnf_gfp: binding list -> typ list list -> BNF_Def.BNF list -> Proof.context -> Proof.context
    13 end;
    14 
    15 structure BNF_GFP : BNF_GFP =
    16 struct
    17 
    18 open BNF_Def
    19 open BNF_Util
    20 open BNF_Tactics
    21 open BNF_FP_Util
    22 open BNF_GFP_Util
    23 open BNF_GFP_Tactics
    24 
    25 datatype wit_tree = Leaf of int | Node of (int * int * int list) * wit_tree list;
    26 
    27 fun mk_tree_args (I, T) (I', Ts) = (sort_distinct int_ord (I @ I'), T :: Ts);
    28 
    29 fun finish Iss m seen i (nwit, I) =
    30   let
    31     val treess = map (fn j =>
    32         if j < m orelse member (op =) seen j then [([j], Leaf j)]
    33         else
    34           map_index (finish Iss m (insert (op =) j seen) j) (nth Iss (j - m))
    35           |> flat
    36           |> minimize_wits)
    37       I;
    38   in
    39     map (fn (I, t) => (I, Node ((i - m, nwit, filter (fn i => i < m) I), t)))
    40       (fold_rev (map_product mk_tree_args) treess [([], [])])
    41     |> minimize_wits
    42   end;
    43 
    44 fun tree_to_fld_wit vars _ _ (Leaf j) = ([j], nth vars j)
    45   | tree_to_fld_wit vars flds witss (Node ((i, nwit, I), subtrees)) =
    46      (I, nth flds i $ (Term.list_comb (snd (nth (nth witss i) nwit),
    47        map (snd o tree_to_fld_wit vars flds witss) subtrees)));
    48 
    49 fun tree_to_coind_wits _ (Leaf j) = []
    50   | tree_to_coind_wits lwitss (Node ((i, nwit, I), subtrees)) =
    51      ((i, I), nth (nth lwitss i) nwit) :: maps (tree_to_coind_wits lwitss) subtrees;
    52 
    53 (*all bnfs have the same lives*)
    54 fun bnf_gfp bs Dss_insts bnfs lthy =
    55   let
    56     val timer = time (Timer.startRealTimer ());
    57 
    58     val live = live_of_bnf (hd bnfs);
    59     val n = length bnfs; (*active*)
    60     val ks = 1 upto n;
    61     val m = live - n (*passive, if 0 don't generate a new bnf*);
    62     val ls = 1 upto m;
    63     val b = Binding.name (fold_rev (fn b => fn s => Binding.name_of b ^ s) bs "");
    64 
    65     (* TODO: check if m, n etc are sane *)
    66 
    67     val Dss = map (fn Ds => map TFree (fold Term.add_tfreesT Ds [])) Dss_insts;
    68     val deads = distinct (op =) (flat Dss);
    69     val names_lthy = fold Variable.declare_typ deads lthy;
    70 
    71     (* tvars *)
    72     val ((((((((passiveAs, activeAs), allAs)), (passiveBs, activeBs)),
    73       (passiveCs, activeCs)), passiveXs), passiveYs), idxT) = names_lthy
    74       |> mk_TFrees live
    75       |> apfst (`(chop m))
    76       ||> mk_TFrees live
    77       ||>> apfst (chop m)
    78       ||> mk_TFrees live
    79       ||>> apfst (chop m)
    80       ||>> mk_TFrees m
    81       ||>> mk_TFrees m
    82       ||> fst o mk_TFrees 1
    83       ||> the_single;
    84 
    85     val Ass = replicate n allAs;
    86     val allBs = passiveAs @ activeBs;
    87     val Bss = replicate n allBs;
    88     val allCs = passiveAs @ activeCs;
    89     val allCs' = passiveBs @ activeCs;
    90     val Css' = replicate n allCs';
    91 
    92     (* typs *)
    93     fun mk_FTs Ts = map2 (fn Ds => mk_T_of_bnf Ds Ts) Dss bnfs;
    94     val (params, params') = `(map dest_TFree) (deads @ passiveAs);
    95     val FTsAs = mk_FTs allAs;
    96     val FTsBs = mk_FTs allBs;
    97     val FTsCs = mk_FTs allCs;
    98     val ATs = map HOLogic.mk_setT passiveAs;
    99     val BTs = map HOLogic.mk_setT activeAs;
   100     val B'Ts = map HOLogic.mk_setT activeBs;
   101     val B''Ts = map HOLogic.mk_setT activeCs;
   102     val sTs = map2 (fn T => fn U => T --> U) activeAs FTsAs;
   103     val s'Ts = map2 (fn T => fn U => T --> U) activeBs FTsBs;
   104     val s''Ts = map2 (fn T => fn U => T --> U) activeCs FTsCs;
   105     val fTs = map2 (fn T => fn U => T --> U) activeAs activeBs;
   106     val all_fTs = map2 (fn T => fn U => T --> U) allAs allBs;
   107     val self_fTs = map (fn T => T --> T) activeAs;
   108     val gTs = map2 (fn T => fn U => T --> U) activeBs activeCs;
   109     val all_gTs = map2 (fn T => fn U => T --> U) allBs allCs';
   110     val RTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeBs;
   111     val sRTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeAs;
   112     val R'Ts = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeBs activeCs;
   113     val setsRTs = map HOLogic.mk_setT sRTs;
   114     val setRTs = map HOLogic.mk_setT RTs;
   115     val all_sbisT = HOLogic.mk_tupleT setsRTs;
   116     val setR'Ts = map HOLogic.mk_setT R'Ts;
   117     val FRTs = mk_FTs (passiveAs @ RTs);
   118     val sumBsAs = map2 (curry mk_sumT) activeBs activeAs;
   119     val sumFTs = mk_FTs (passiveAs @ sumBsAs);
   120     val sum_sTs = map2 (fn T => fn U => T --> U) activeAs sumFTs;
   121 
   122     (* terms *)
   123     val mapsAsAs = map4 mk_map_of_bnf Dss Ass Ass bnfs;
   124     val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;
   125     val mapsBsCs' = map4 mk_map_of_bnf Dss Bss Css' bnfs;
   126     val mapsAsCs' = map4 mk_map_of_bnf Dss Ass Css' bnfs;
   127     val map_Inls = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ sumBsAs)) bnfs;
   128     val map_Inls_rev = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ sumBsAs)) Bss bnfs;
   129     val map_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs;
   130     val map_snds = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs;
   131     fun mk_setss Ts = map3 mk_sets_of_bnf (map (replicate live) Dss)
   132       (map (replicate live) (replicate n Ts)) bnfs;
   133     val setssAs = mk_setss allAs;
   134     val setssAs' = transpose setssAs;
   135     val bis_setss = mk_setss (passiveAs @ RTs);
   136     val relsAsBs = map4 mk_rel_of_bnf Dss Ass Bss bnfs;
   137     val bds = map3 mk_bd_of_bnf Dss Ass bnfs;
   138     val sum_bd = Library.foldr1 (uncurry mk_csum) bds;
   139     val sum_bdT = fst (dest_relT (fastype_of sum_bd));
   140     val witss = map wits_of_bnf bnfs;
   141 
   142     val emptys = map (fn T => HOLogic.mk_set T []) passiveAs;
   143     val Zeros = map (fn empty =>
   144      HOLogic.mk_tuple (map (fn U => absdummy U empty) activeAs)) emptys;
   145     val hrecTs = map fastype_of Zeros;
   146     val hsetTs = map (fn hrecT => Library.foldr (op -->) (sTs, HOLogic.natT --> hrecT)) hrecTs;
   147 
   148     val (((((((((((((((((((((((((((((((((((zs, zs'), zs_copy), zs_copy2),
   149       z's), As), As_copy), Bs), Bs_copy), B's), B''s), ss), sum_ss), s's), s''s), fs), fs_copy),
   150       self_fs), all_fs), gs), all_gs), xFs), xFs_copy), RFs), (Rtuple, Rtuple')), (hrecs, hrecs')),
   151       (nat, nat')), Rs), Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), Kss),
   152       names_lthy) = lthy
   153       |> mk_Frees' "b" activeAs
   154       ||>> mk_Frees "b" activeAs
   155       ||>> mk_Frees "b" activeAs
   156       ||>> mk_Frees "b" activeBs
   157       ||>> mk_Frees "A" ATs
   158       ||>> mk_Frees "A" ATs
   159       ||>> mk_Frees "B" BTs
   160       ||>> mk_Frees "B" BTs
   161       ||>> mk_Frees "B'" B'Ts
   162       ||>> mk_Frees "B''" B''Ts
   163       ||>> mk_Frees "s" sTs
   164       ||>> mk_Frees "sums" sum_sTs
   165       ||>> mk_Frees "s'" s'Ts
   166       ||>> mk_Frees "s''" s''Ts
   167       ||>> mk_Frees "f" fTs
   168       ||>> mk_Frees "f" fTs
   169       ||>> mk_Frees "f" self_fTs
   170       ||>> mk_Frees "f" all_fTs
   171       ||>> mk_Frees "g" gTs
   172       ||>> mk_Frees "g" all_gTs
   173       ||>> mk_Frees "x" FTsAs
   174       ||>> mk_Frees "x" FTsAs
   175       ||>> mk_Frees "x" FRTs
   176       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Rtuple") all_sbisT
   177       ||>> mk_Frees' "rec" hrecTs
   178       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT
   179       ||>> mk_Frees "R" setRTs
   180       ||>> mk_Frees "R" setRTs
   181       ||>> mk_Frees "R'" setR'Ts
   182       ||>> mk_Frees "R" setsRTs
   183       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") idxT
   184       ||>> yield_singleton (mk_Frees "I") (HOLogic.mk_setT idxT)
   185       ||>> mk_Frees "Ri" (map (fn T => idxT --> T) setRTs)
   186       ||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) activeAs) ATs);
   187 
   188     val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
   189     val passive_diags = map mk_diag As;
   190     val active_UNIVs = map HOLogic.mk_UNIV activeAs;
   191     val sum_UNIVs = map HOLogic.mk_UNIV sumBsAs;
   192     val passive_ids = map HOLogic.id_const passiveAs;
   193     val active_ids = map HOLogic.id_const activeAs;
   194     val Inls = map2 Inl_const activeBs activeAs;
   195     val fsts = map fst_const RTs;
   196     val snds = map snd_const RTs;
   197 
   198     (* thms *)
   199     val bd_card_orders = map bd_card_order_of_bnf bnfs;
   200     val bd_card_order = hd bd_card_orders
   201     val bd_Card_orders = map bd_Card_order_of_bnf bnfs;
   202     val bd_Card_order = hd bd_Card_orders;
   203     val bd_Cinfinites = map bd_Cinfinite_of_bnf bnfs;
   204     val bd_Cinfinite = hd bd_Cinfinites;
   205     val bd_Cnotzeros = map bd_Cnotzero_of_bnf bnfs;
   206     val bd_Cnotzero = hd bd_Cnotzeros;
   207     val in_bds = map in_bd_of_bnf bnfs;
   208     val in_monos = map in_mono_of_bnf bnfs;
   209     val map_comps = map map_comp_of_bnf bnfs;
   210     val map_comp's = map map_comp'_of_bnf bnfs;
   211     val map_congs = map map_cong_of_bnf bnfs;
   212     val map_id's = map map_id'_of_bnf bnfs;
   213     val pred_defs = map pred_def_of_bnf bnfs;
   214     val rel_congs = map rel_cong_of_bnf bnfs;
   215     val rel_converses = map rel_converse_of_bnf bnfs;
   216     val rel_defs = map rel_def_of_bnf bnfs;
   217     val rel_Grs = map rel_Gr_of_bnf bnfs;
   218     val rel_Ids = map rel_Id_of_bnf bnfs;
   219     val rel_monos = map rel_mono_of_bnf bnfs;
   220     val rel_Os = map rel_O_of_bnf bnfs;
   221     val map_wpulls = map map_wpull_of_bnf bnfs;
   222     val set_bdss = map set_bd_of_bnf bnfs;
   223     val set_natural'ss = map set_natural'_of_bnf bnfs;
   224 
   225     val timer = time (timer "Extracted terms & thms");
   226 
   227     (* derived thms *)
   228 
   229     (*map g1 ... gm g(m+1) ... g(m+n) (map id ... id f(m+1) ... f(m+n) x)=
   230       map g1 ... gm (g(m+1) o f(m+1)) ... (g(m+n) o f(m+n)) x*)
   231     fun mk_map_comp_id x mapAsBs mapBsCs mapAsCs map_comp =
   232       let
   233         val lhs = Term.list_comb (mapBsCs, all_gs) $
   234           (Term.list_comb (mapAsBs, passive_ids @ fs) $ x);
   235         val rhs =
   236           Term.list_comb (mapAsCs, take m all_gs @ map HOLogic.mk_comp (drop m all_gs ~~ fs)) $ x;
   237       in
   238         Skip_Proof.prove lthy [] []
   239           (fold_rev Logic.all (x :: fs @ all_gs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))))
   240           (K (mk_map_comp_id_tac map_comp))
   241         |> Thm.close_derivation
   242       end;
   243 
   244     val map_comp_id_thms = map5 mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comp's;
   245 
   246     (*forall a : set(m+1) x. f(m+1) a = a; ...; forall a : set(m+n) x. f(m+n) a = a ==>
   247       map id ... id f(m+1) ... f(m+n) x = x*)
   248     fun mk_map_congL x mapAsAs sets map_cong map_id' =
   249       let
   250         fun mk_prem set f z z' =
   251           HOLogic.mk_Trueprop
   252             (mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z))));
   253         val prems = map4 mk_prem (drop m sets) self_fs zs zs';
   254         val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq
   255          (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x))
   256       in
   257         Skip_Proof.prove lthy [] []
   258           (fold_rev Logic.all (x :: self_fs) (Logic.list_implies (prems, goal)))
   259           (K (mk_map_congL_tac m map_cong map_id'))
   260         |> Thm.close_derivation
   261       end;
   262 
   263     val map_congL_thms = map5 mk_map_congL xFs mapsAsAs setssAs map_congs map_id's;
   264     val in_mono'_thms = map (fn thm =>
   265       (thm OF (replicate m subset_refl)) RS @{thm set_mp}) in_monos;
   266     val in_cong'_thms = map (fn bnf => in_cong_of_bnf bnf OF (replicate m refl)) bnfs;
   267 
   268     val map_arg_cong_thms =
   269       let
   270         val prems = map2 (fn x => fn y =>
   271           HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))) xFs xFs_copy;
   272         val maps = map (fn map => Term.list_comb (map, all_fs)) mapsAsBs;
   273         val concls = map3 (fn x => fn y => fn map =>
   274           HOLogic.mk_Trueprop (HOLogic.mk_eq (map $ x, map $ y))) xFs xFs_copy maps;
   275         val goals =
   276           map4 (fn prem => fn concl => fn x => fn y =>
   277             fold_rev Logic.all (x :: y :: all_fs) (Logic.mk_implies (prem, concl)))
   278           prems concls xFs xFs_copy;
   279       in
   280         map (fn goal => Skip_Proof.prove lthy [] [] goal
   281           (K ((hyp_subst_tac THEN' rtac refl) 1)) |> Thm.close_derivation) goals
   282       end;
   283 
   284     val timer = time (timer "Derived simple theorems");
   285 
   286     (* coalgebra *)
   287 
   288     val coalg_bind = Binding.suffix_name ("_" ^ coN ^ algN) b;
   289     val coalg_name = Binding.name_of coalg_bind;
   290     val coalg_def_bind = (Thm.def_binding coalg_bind, []);
   291 
   292     (*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in A1 .. Am B1 ... Bn)*)
   293     val coalg_spec =
   294       let
   295         val coalgT = Library.foldr (op -->) (ATs @ BTs @ sTs, HOLogic.boolT);
   296 
   297         val ins = map3 mk_in (replicate n (As @ Bs)) setssAs FTsAs;
   298         fun mk_coalg_conjunct B s X z z' =
   299           mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X)));
   300 
   301         val lhs = Term.list_comb (Free (coalg_name, coalgT), As @ Bs @ ss);
   302         val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_coalg_conjunct Bs ss ins zs zs')
   303       in
   304         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
   305       end;
   306 
   307     val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) =
   308         lthy
   309         |> Specification.definition (SOME (coalg_bind, NONE, NoSyn), (coalg_def_bind, coalg_spec))
   310         ||> `Local_Theory.restore;
   311 
   312     (*transforms defined frees into consts*)
   313     val phi = Proof_Context.export_morphism lthy_old lthy;
   314     val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free));
   315     val coalg_def = Morphism.thm phi coalg_def_free;
   316 
   317     fun mk_coalg As Bs ss =
   318       let
   319         val args = As @ Bs @ ss;
   320         val Ts = map fastype_of args;
   321         val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT);
   322       in
   323         Term.list_comb (Const (coalg, coalgT), args)
   324       end;
   325 
   326     val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
   327 
   328     val coalg_in_thms = map (fn i =>
   329       coalg_def RS @{thm subst[of _ _ "%x. x"]} RS mk_conjunctN n i RS bspec) ks
   330 
   331     val coalg_set_thmss =
   332       let
   333         val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
   334         fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));
   335         fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) B);
   336         val prems = map2 mk_prem zs Bs;
   337         val conclss = map3 (fn s => fn x => fn sets => map2 (mk_concl s x) (As @ Bs) sets)
   338           ss zs setssAs;
   339         val goalss = map3 (fn x => fn prem => fn concls => map (fn concl =>
   340           fold_rev Logic.all (x :: As @ Bs @ ss)
   341             (Logic.list_implies (coalg_prem :: [prem], concl))) concls) zs prems conclss;
   342       in
   343         map (fn goals => map (fn goal => Skip_Proof.prove lthy [] [] goal
   344           (K (mk_coalg_set_tac coalg_def)) |> Thm.close_derivation) goals) goalss
   345       end;
   346 
   347     val coalg_set_thmss' = transpose coalg_set_thmss;
   348 
   349     fun mk_tcoalg ATs BTs = mk_coalg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);
   350 
   351     val tcoalg_thm =
   352       let
   353         val goal = fold_rev Logic.all ss
   354           (HOLogic.mk_Trueprop (mk_tcoalg passiveAs activeAs ss))
   355       in
   356         Skip_Proof.prove lthy [] [] goal
   357           (K (stac coalg_def 1 THEN CONJ_WRAP
   358             (K (EVERY' [rtac ballI, rtac CollectI,
   359               CONJ_WRAP' (K (EVERY' [rtac @{thm subset_UNIV}])) allAs] 1)) ss))
   360         |> Thm.close_derivation
   361       end;
   362 
   363     val timer = time (timer "Coalgebra definition & thms");
   364 
   365     (* morphism *)
   366 
   367     val mor_bind = Binding.suffix_name ("_" ^ morN) b;
   368     val mor_name = Binding.name_of mor_bind;
   369     val mor_def_bind = (Thm.def_binding mor_bind, []);
   370 
   371     (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*)
   372     (*mor) forall i = 1 ... n: (\<forall>x \<in> Bi.
   373        Fi_map id ... id f1 ... fn (si x) = si' (fi x)*)
   374     val mor_spec =
   375       let
   376         val morT = Library.foldr (op -->) (BTs @ sTs @ B'Ts @ s'Ts @ fTs, HOLogic.boolT);
   377 
   378         fun mk_fbetw f B1 B2 z z' =
   379           mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));
   380         fun mk_mor B mapAsBs f s s' z z' =
   381           mk_Ball B (Term.absfree z' (HOLogic.mk_eq
   382             (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z))));
   383         val lhs = Term.list_comb (Free (mor_name, morT), Bs @ ss @ B's @ s's @ fs);
   384         val rhs = HOLogic.mk_conj
   385           (Library.foldr1 HOLogic.mk_conj (map5 mk_fbetw fs Bs B's zs zs'),
   386            Library.foldr1 HOLogic.mk_conj (map7 mk_mor Bs mapsAsBs fs ss s's zs zs'))
   387       in
   388         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
   389       end;
   390 
   391     val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
   392         lthy
   393         |> Specification.definition (SOME (mor_bind, NONE, NoSyn), (mor_def_bind, mor_spec))
   394         ||> `Local_Theory.restore;
   395 
   396     (*transforms defined frees into consts*)
   397     val phi = Proof_Context.export_morphism lthy_old lthy;
   398     val mor = fst (Term.dest_Const (Morphism.term phi mor_free));
   399     val mor_def = Morphism.thm phi mor_def_free;
   400 
   401     fun mk_mor Bs1 ss1 Bs2 ss2 fs =
   402       let
   403         val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;
   404         val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);
   405         val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);
   406       in
   407         Term.list_comb (Const (mor, morT), args)
   408       end;
   409 
   410     val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
   411 
   412     val (mor_image_thms, morE_thms) =
   413       let
   414         val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
   415         fun mk_image_goal f B1 B2 = fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs)
   416           (Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_subset (mk_image f $ B1) B2)));
   417         val image_goals = map3 mk_image_goal fs Bs B's;
   418         fun mk_elim_goal B mapAsBs f s s' x =
   419           fold_rev Logic.all (x :: Bs @ ss @ B's @ s's @ fs)
   420             (Logic.list_implies ([prem, HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B))],
   421               HOLogic.mk_Trueprop (HOLogic.mk_eq
   422                (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x)))));
   423         val elim_goals = map6 mk_elim_goal Bs mapsAsBs fs ss s's zs;
   424         fun prove goal =
   425           Skip_Proof.prove lthy [] [] goal (K (mk_mor_elim_tac mor_def))
   426           |> Thm.close_derivation;
   427       in
   428         (map prove image_goals, map prove elim_goals)
   429       end;
   430 
   431     val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms;
   432 
   433     val mor_incl_thm =
   434       let
   435         val prems = map2 (HOLogic.mk_Trueprop oo mk_subset) Bs Bs_copy;
   436         val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
   437       in
   438         Skip_Proof.prove lthy [] []
   439           (fold_rev Logic.all (Bs @ ss @ Bs_copy) (Logic.list_implies (prems, concl)))
   440           (K (mk_mor_incl_tac mor_def map_id's))
   441         |> Thm.close_derivation
   442       end;
   443 
   444     val mor_id_thm = mor_incl_thm OF (replicate n subset_refl);
   445 
   446     val mor_comp_thm =
   447       let
   448         val prems =
   449           [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),
   450            HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];
   451         val concl =
   452           HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));
   453       in
   454         Skip_Proof.prove lthy [] []
   455           (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)
   456             (Logic.list_implies (prems, concl)))
   457           (K (mk_mor_comp_tac mor_def mor_image'_thms morE_thms map_comp_id_thms))
   458         |> Thm.close_derivation
   459       end;
   460 
   461     val mor_cong_thm =
   462       let
   463         val prems = map HOLogic.mk_Trueprop
   464          (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])
   465         val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);
   466       in
   467         Skip_Proof.prove lthy [] []
   468           (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ fs_copy)
   469             (Logic.list_implies (prems, concl)))
   470           (K ((hyp_subst_tac THEN' atac) 1))
   471         |> Thm.close_derivation
   472       end;
   473 
   474     val mor_UNIV_thm =
   475       let
   476         fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
   477             (HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s),
   478             HOLogic.mk_comp (s', f));
   479         val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;
   480         val rhs = Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct mapsAsBs fs ss s's);
   481       in
   482         Skip_Proof.prove lthy [] []
   483           (fold_rev Logic.all (ss @ s's @ fs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))))
   484           (K (mk_mor_UNIV_tac morE_thms mor_def))
   485         |> Thm.close_derivation
   486       end;
   487 
   488     val mor_str_thm =
   489       let
   490         val maps = map2 (fn Ds => fn bnf => Term.list_comb
   491           (mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs;
   492       in
   493         Skip_Proof.prove lthy [] []
   494           (fold_rev Logic.all ss (HOLogic.mk_Trueprop
   495             (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss)))
   496           (K (mk_mor_str_tac ks mor_UNIV_thm))
   497         |> Thm.close_derivation
   498       end;
   499 
   500     val mor_sum_case_thm =
   501       let
   502         val maps = map3 (fn s => fn sum_s => fn map =>
   503           mk_sum_case (HOLogic.mk_comp (Term.list_comb (map, passive_ids @ Inls), s)) sum_s)
   504           s's sum_ss map_Inls;
   505       in
   506         Skip_Proof.prove lthy [] []
   507           (fold_rev Logic.all (s's @ sum_ss) (HOLogic.mk_Trueprop
   508             (mk_mor (map HOLogic.mk_UNIV activeBs) s's sum_UNIVs maps Inls)))
   509           (K (mk_mor_sum_case_tac ks mor_UNIV_thm))
   510         |> Thm.close_derivation
   511       end;
   512 
   513     val timer = time (timer "Morphism definition & thms");
   514 
   515     fun hset_rec_bind j = Binding.suffix_name ("_" ^ hset_recN ^ (if m = 1 then "" else
   516       string_of_int j)) b;
   517     val hset_rec_name = Binding.name_of o hset_rec_bind;
   518     val hset_rec_def_bind = rpair [] o Thm.def_binding o hset_rec_bind;
   519 
   520     fun hset_rec_spec j Zero hsetT hrec hrec' =
   521       let
   522         fun mk_Suc s setsAs z z' =
   523           let
   524             val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m setsAs);
   525             fun mk_UN set k = mk_UNION (set $ (s $ z)) (mk_nthN n hrec k);
   526           in
   527             Term.absfree z'
   528               (mk_union (set $ (s $ z), Library.foldl1 mk_union (map2 mk_UN sets ks)))
   529           end;
   530 
   531         val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec'
   532           (HOLogic.mk_tuple (map4 mk_Suc ss setssAs zs zs')));
   533 
   534         val lhs = Term.list_comb (Free (hset_rec_name j, hsetT), ss);
   535         val rhs = mk_nat_rec Zero Suc;
   536       in
   537         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
   538       end;
   539 
   540     val ((hset_rec_frees, (_, hset_rec_def_frees)), (lthy, lthy_old)) =
   541       lthy
   542       |> fold_map5 (fn j => fn Zero => fn hsetT => fn hrec => fn hrec' => Specification.definition
   543         (SOME (hset_rec_bind j, NONE, NoSyn),
   544           (hset_rec_def_bind j, hset_rec_spec j Zero hsetT hrec hrec')))
   545         ls Zeros hsetTs hrecs hrecs'
   546       |>> apsnd split_list o split_list
   547       ||> `Local_Theory.restore;
   548 
   549     (*transforms defined frees into consts*)
   550     val phi = Proof_Context.export_morphism lthy_old lthy;
   551 
   552     val hset_rec_defs = map (Morphism.thm phi) hset_rec_def_frees;
   553     val hset_recs = map (fst o Term.dest_Const o Morphism.term phi) hset_rec_frees;
   554 
   555     fun mk_hset_rec ss nat i j T =
   556       let
   557         val args = ss @ [nat];
   558         val Ts = map fastype_of ss;
   559         val bTs = map domain_type Ts;
   560         val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) bTs)
   561         val hset_recT = Library.foldr (op -->) (Ts, HOLogic.natT --> hrecT);
   562       in
   563         mk_nthN n (Term.list_comb (Const (nth hset_recs (j - 1), hset_recT), args)) i
   564       end;
   565 
   566     val hset_rec_0ss = mk_rec_simps n @{thm nat_rec_0} hset_rec_defs;
   567     val hset_rec_Sucss = mk_rec_simps n @{thm nat_rec_Suc} hset_rec_defs;
   568     val hset_rec_0ss' = transpose hset_rec_0ss;
   569     val hset_rec_Sucss' = transpose hset_rec_Sucss;
   570 
   571     fun hset_bind i j = Binding.suffix_name ("_" ^ hsetN ^
   572       (if m = 1 then "" else string_of_int j)) (nth bs (i - 1));
   573     val hset_name = Binding.name_of oo hset_bind;
   574     val hset_def_bind = rpair [] o Thm.def_binding oo hset_bind;
   575 
   576     fun hset_spec i j =
   577       let
   578         val U = nth activeAs (i - 1);
   579         val z = nth zs (i - 1);
   580         val T = nth passiveAs (j - 1);
   581         val setT = HOLogic.mk_setT T;
   582         val hsetT = Library.foldr (op -->) (sTs, U --> setT);
   583 
   584         val lhs = Term.list_comb (Free (hset_name i j, hsetT), ss @ [z]);
   585         val rhs = mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
   586           (Term.absfree nat' (mk_hset_rec ss nat i j T $ z));
   587       in
   588         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
   589       end;
   590 
   591     val ((hset_frees, (_, hset_def_frees)), (lthy, lthy_old)) =
   592       lthy
   593       |> fold_map (fn i => fold_map (fn j => Specification.definition
   594         (SOME (hset_bind i j, NONE, NoSyn), (hset_def_bind i j, hset_spec i j))) ls) ks
   595       |>> map (apsnd split_list o split_list)
   596       |>> apsnd split_list o split_list
   597       ||> `Local_Theory.restore;
   598 
   599     (*transforms defined frees into consts*)
   600     val phi = Proof_Context.export_morphism lthy_old lthy;
   601 
   602     val hset_defss = map (map (Morphism.thm phi)) hset_def_frees;
   603     val hset_defss' = transpose hset_defss;
   604     val hset_namess = map (map (fst o Term.dest_Const o Morphism.term phi)) hset_frees;
   605 
   606     fun mk_hset ss i j T =
   607       let
   608         val Ts = map fastype_of ss;
   609         val bTs = map domain_type Ts;
   610         val hsetT = Library.foldr (op -->) (Ts, nth bTs (i - 1) --> HOLogic.mk_setT T);
   611       in
   612         Term.list_comb (Const (nth (nth hset_namess (i - 1)) (j - 1), hsetT), ss)
   613       end;
   614 
   615     val hsetssAs = map (fn i => map2 (mk_hset ss i) ls passiveAs) ks;
   616 
   617     val (set_incl_hset_thmss, set_hset_incl_hset_thmsss) =
   618       let
   619         fun mk_set_incl_hset s x set hset = fold_rev Logic.all (x :: ss)
   620           (HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (hset $ x)));
   621 
   622         fun mk_set_hset_incl_hset s x y set hset1 hset2 =
   623           fold_rev Logic.all (x :: y :: ss)
   624             (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x, set $ (s $ y))),
   625             HOLogic.mk_Trueprop (mk_subset (hset1 $ x) (hset2 $ y))));
   626 
   627         val set_incl_hset_goalss =
   628           map4 (fn s => fn x => fn sets => fn hsets =>
   629             map2 (mk_set_incl_hset s x) (take m sets) hsets)
   630           ss zs setssAs hsetssAs;
   631 
   632         (*xk : F(i)set(m+k) (si yi) ==> F(k)_hset(j) s1 ... sn xk <= F(i)_hset(j) s1 ... sn yi*)
   633         val set_hset_incl_hset_goalsss =
   634           map4 (fn si => fn yi => fn sets => fn hsetsi =>
   635             map3 (fn xk => fn set => fn hsetsk =>
   636               map2 (mk_set_hset_incl_hset si xk yi set) hsetsk hsetsi)
   637             zs_copy (drop m sets) hsetssAs)
   638           ss zs setssAs hsetssAs;
   639       in
   640         (map3 (fn goals => fn defs => fn rec_Sucs =>
   641           map3 (fn goal => fn def => fn rec_Suc =>
   642             Skip_Proof.prove lthy [] [] goal (K (mk_set_incl_hset_tac def rec_Suc))
   643             |> Thm.close_derivation)
   644           goals defs rec_Sucs)
   645         set_incl_hset_goalss hset_defss hset_rec_Sucss,
   646         map3 (fn goalss => fn defsi => fn rec_Sucs =>
   647           map3 (fn k => fn goals => fn defsk =>
   648             map4 (fn goal => fn defk => fn defi => fn rec_Suc =>
   649               Skip_Proof.prove lthy [] [] goal
   650                 (K (mk_set_hset_incl_hset_tac n [defk, defi] rec_Suc k))
   651               |> Thm.close_derivation)
   652             goals defsk defsi rec_Sucs)
   653           ks goalss hset_defss)
   654         set_hset_incl_hset_goalsss hset_defss hset_rec_Sucss)
   655       end;
   656 
   657     val set_incl_hset_thmss' = transpose set_incl_hset_thmss;
   658     val set_hset_incl_hset_thmsss' = transpose (map transpose set_hset_incl_hset_thmsss);
   659     val set_hset_incl_hset_thmsss'' = map transpose set_hset_incl_hset_thmsss';
   660     val set_hset_thmss = map (map (fn thm => thm RS @{thm set_mp})) set_incl_hset_thmss;
   661     val set_hset_hset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp})))
   662       set_hset_incl_hset_thmsss;
   663     val set_hset_thmss' = transpose set_hset_thmss;
   664     val set_hset_hset_thmsss' = transpose (map transpose set_hset_hset_thmsss);
   665 
   666     val set_incl_hin_thmss =
   667       let
   668         fun mk_set_incl_hin s x hsets1 set hsets2 T =
   669           fold_rev Logic.all (x :: ss @ As)
   670             (Logic.list_implies
   671               (map2 (fn hset => fn A => HOLogic.mk_Trueprop (mk_subset (hset $ x) A)) hsets1 As,
   672               HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (mk_in As hsets2 T))));
   673 
   674         val set_incl_hin_goalss =
   675           map4 (fn s => fn x => fn sets => fn hsets =>
   676             map3 (mk_set_incl_hin s x hsets) (drop m sets) hsetssAs activeAs)
   677           ss zs setssAs hsetssAs;
   678       in
   679         map2 (map2 (fn goal => fn thms =>
   680           Skip_Proof.prove lthy [] [] goal (K (mk_set_incl_hin_tac thms))
   681           |> Thm.close_derivation))
   682         set_incl_hin_goalss set_hset_incl_hset_thmsss
   683       end;
   684 
   685     val hset_minimal_thms =
   686       let
   687         fun mk_passive_prem set s x K =
   688           Logic.all x (HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (K $ x)));
   689 
   690         fun mk_active_prem s x1 K1 set x2 K2 =
   691           fold_rev Logic.all [x1, x2]
   692             (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x2, set $ (s $ x1))),
   693               HOLogic.mk_Trueprop (mk_subset (K2 $ x2) (K1 $ x1))));
   694 
   695         val premss = map2 (fn j => fn Ks =>
   696           map4 mk_passive_prem (map (fn xs => nth xs (j - 1)) setssAs) ss zs Ks @
   697             flat (map4 (fn sets => fn s => fn x1 => fn K1 =>
   698               map3 (mk_active_prem s x1 K1) (drop m sets) zs_copy Ks) setssAs ss zs Ks))
   699           ls Kss;
   700 
   701         val hset_rec_minimal_thms =
   702           let
   703             fun mk_conjunct j T i K x = mk_subset (mk_hset_rec ss nat i j T $ x) (K $ x);
   704             fun mk_concl j T Ks = list_all_free zs
   705               (Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs));
   706             val concls = map3 mk_concl ls passiveAs Kss;
   707 
   708             val goals = map2 (fn prems => fn concl =>
   709               Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls
   710 
   711             val ctss =
   712               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
   713           in
   714             map4 (fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>
   715               singleton (Proof_Context.export names_lthy lthy)
   716                 (Skip_Proof.prove lthy [] [] goal
   717                   (mk_hset_rec_minimal_tac m cts hset_rec_0s hset_rec_Sucs))
   718               |> Thm.close_derivation)
   719             goals ctss hset_rec_0ss' hset_rec_Sucss'
   720           end;
   721 
   722         fun mk_conjunct j T i K x = mk_subset (mk_hset ss i j T $ x) (K $ x);
   723         fun mk_concl j T Ks = Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs);
   724         val concls = map3 mk_concl ls passiveAs Kss;
   725 
   726         val goals = map3 (fn Ks => fn prems => fn concl =>
   727           fold_rev Logic.all (Ks @ ss @ zs)
   728             (Logic.list_implies (prems, HOLogic.mk_Trueprop concl))) Kss premss concls;
   729       in
   730         map3 (fn goal => fn hset_defs => fn hset_rec_minimal =>
   731           Skip_Proof.prove lthy [] [] goal
   732             (mk_hset_minimal_tac n hset_defs hset_rec_minimal)
   733           |> Thm.close_derivation)
   734         goals hset_defss' hset_rec_minimal_thms
   735       end;
   736 
   737     val mor_hset_thmss =
   738       let
   739         val mor_hset_rec_thms =
   740           let
   741             fun mk_conjunct j T i f x B =
   742               HOLogic.mk_imp (HOLogic.mk_mem (x, B), HOLogic.mk_eq
   743                (mk_hset_rec s's nat i j T $ (f $ x), mk_hset_rec ss nat i j T $ x));
   744 
   745             fun mk_concl j T = list_all_free zs
   746               (Library.foldr1 HOLogic.mk_conj (map4 (mk_conjunct j T) ks fs zs Bs));
   747             val concls = map2 mk_concl ls passiveAs;
   748 
   749             val ctss =
   750               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
   751 
   752             val goals = map (fn concl =>
   753               Logic.list_implies ([coalg_prem, mor_prem], HOLogic.mk_Trueprop concl)) concls;
   754           in
   755             map5 (fn j => fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>
   756               singleton (Proof_Context.export names_lthy lthy)
   757                 (Skip_Proof.prove lthy [] [] goal
   758                   (K (mk_mor_hset_rec_tac m n cts j hset_rec_0s hset_rec_Sucs
   759                     morE_thms set_natural'ss coalg_set_thmss)))
   760               |> Thm.close_derivation)
   761             ls goals ctss hset_rec_0ss' hset_rec_Sucss'
   762           end;
   763 
   764         val mor_hset_rec_thmss = map (fn thm => map (fn i =>
   765           mk_specN n thm RS mk_conjunctN n i RS mp) ks) mor_hset_rec_thms;
   766 
   767         fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));
   768 
   769         fun mk_concl j T i f x = HOLogic.mk_Trueprop (HOLogic.mk_eq
   770           (mk_hset s's i j T $ (f $ x), mk_hset ss i j T $ x));
   771 
   772         val goalss = map2 (fn j => fn T => map4 (fn i => fn f => fn x => fn B =>
   773           fold_rev Logic.all (x :: As @ Bs @ ss @ B's @ s's @ fs)
   774             (Logic.list_implies ([coalg_prem, mor_prem,
   775               mk_prem x B], mk_concl j T i f x))) ks fs zs Bs) ls passiveAs;
   776       in
   777         map3 (map3 (fn goal => fn hset_def => fn mor_hset_rec =>
   778           Skip_Proof.prove lthy [] [] goal
   779             (K (mk_mor_hset_tac hset_def mor_hset_rec))
   780           |> Thm.close_derivation))
   781         goalss hset_defss' mor_hset_rec_thmss
   782       end;
   783 
   784     val timer = time (timer "Hereditary sets");
   785 
   786     (* bisimulation *)
   787 
   788     val bis_bind = Binding.suffix_name ("_" ^ bisN) b;
   789     val bis_name = Binding.name_of bis_bind;
   790     val bis_def_bind = (Thm.def_binding bis_bind, []);
   791 
   792     fun mk_bis_le_conjunct R B1 B2 = mk_subset R (mk_Times (B1, B2));
   793     val bis_le = Library.foldr1 HOLogic.mk_conj (map3 mk_bis_le_conjunct Rs Bs B's)
   794 
   795     val bis_spec =
   796       let
   797         val bisT = Library.foldr (op -->) (ATs @ BTs @ sTs @ B'Ts @ s'Ts @ setRTs, HOLogic.boolT);
   798 
   799         val fst_args = passive_ids @ fsts;
   800         val snd_args = passive_ids @ snds;
   801         fun mk_bis R s s' b1 b2 RF map1 map2 sets =
   802           list_all_free [b1, b2] (HOLogic.mk_imp
   803             (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
   804             mk_Bex (mk_in (As @ Rs) sets (snd (dest_Free RF))) (Term.absfree (dest_Free RF)
   805               (HOLogic.mk_conj
   806                 (HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1),
   807                 HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2))))));
   808 
   809         val lhs = Term.list_comb (Free (bis_name, bisT), As @ Bs @ ss @ B's @ s's @ Rs);
   810         val rhs = HOLogic.mk_conj
   811           (bis_le, Library.foldr1 HOLogic.mk_conj
   812             (map9 mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss))
   813       in
   814         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
   815       end;
   816 
   817     val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) =
   818         lthy
   819         |> Specification.definition (SOME (bis_bind, NONE, NoSyn), (bis_def_bind, bis_spec))
   820         ||> `Local_Theory.restore;
   821 
   822     (*transforms defined frees into consts*)
   823     val phi = Proof_Context.export_morphism lthy_old lthy;
   824     val bis = fst (Term.dest_Const (Morphism.term phi bis_free));
   825     val bis_def = Morphism.thm phi bis_def_free;
   826 
   827     fun mk_bis As Bs1 ss1 Bs2 ss2 Rs =
   828       let
   829         val args = As @ Bs1 @ ss1 @ Bs2 @ ss2 @ Rs;
   830         val Ts = map fastype_of args;
   831         val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT);
   832       in
   833         Term.list_comb (Const (bis, bisT), args)
   834       end;
   835 
   836     val bis_cong_thm =
   837       let
   838         val prems = map HOLogic.mk_Trueprop
   839          (mk_bis As Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs)
   840         val concl = HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs_copy);
   841       in
   842         Skip_Proof.prove lthy [] []
   843           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs @ Rs_copy)
   844             (Logic.list_implies (prems, concl)))
   845           (K ((hyp_subst_tac THEN' atac) 1))
   846         |> Thm.close_derivation
   847       end;
   848 
   849     val bis_rel_thm =
   850       let
   851         fun mk_conjunct R s s' b1 b2 rel =
   852           list_all_free [b1, b2] (HOLogic.mk_imp
   853             (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
   854             HOLogic.mk_mem (HOLogic.mk_prod (s $ b1, s' $ b2),
   855               Term.list_comb (rel, passive_diags @ Rs))));
   856 
   857         val rhs = HOLogic.mk_conj
   858           (bis_le, Library.foldr1 HOLogic.mk_conj
   859             (map6 mk_conjunct Rs ss s's zs z's relsAsBs))
   860       in
   861         Skip_Proof.prove lthy [] []
   862           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
   863             (HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_bis As Bs ss B's s's Rs, rhs))))
   864           (K (mk_bis_rel_tac m bis_def rel_defs map_comp's map_congs set_natural'ss))
   865         |> Thm.close_derivation
   866       end;
   867 
   868     val bis_converse_thm =
   869       Skip_Proof.prove lthy [] []
   870         (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
   871           (Logic.mk_implies
   872             (HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
   873             HOLogic.mk_Trueprop (mk_bis As B's s's Bs ss (map mk_converse Rs)))))
   874         (K (mk_bis_converse_tac m bis_rel_thm rel_congs rel_converses))
   875       |> Thm.close_derivation;
   876 
   877     val bis_O_thm =
   878       let
   879         val prems =
   880           [HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
   881            HOLogic.mk_Trueprop (mk_bis As B's s's B''s s''s R's)];
   882         val concl =
   883           HOLogic.mk_Trueprop (mk_bis As Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's));
   884       in
   885         Skip_Proof.prove lthy [] []
   886           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ B''s @ s''s @ Rs @ R's)
   887             (Logic.list_implies (prems, concl)))
   888           (K (mk_bis_O_tac m bis_rel_thm rel_congs rel_Os))
   889         |> Thm.close_derivation
   890       end;
   891 
   892     val bis_Gr_thm =
   893       let
   894         val concl =
   895           HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map2 mk_Gr Bs fs));
   896       in
   897         Skip_Proof.prove lthy [] []
   898           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ fs)
   899             (Logic.list_implies ([coalg_prem, mor_prem], concl)))
   900           (mk_bis_Gr_tac bis_rel_thm rel_Grs mor_image_thms morE_thms coalg_in_thms)
   901         |> Thm.close_derivation
   902       end;
   903 
   904     val bis_image2_thm = bis_cong_thm OF
   905       ((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) ::
   906       replicate n @{thm image2_Gr});
   907 
   908     val bis_diag_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) ::
   909       replicate n @{thm diag_Gr});
   910 
   911     val bis_Union_thm =
   912       let
   913         val prem =
   914           HOLogic.mk_Trueprop (mk_Ball Idx
   915             (Term.absfree idx' (mk_bis As Bs ss B's s's (map (fn R => R $ idx) Ris))));
   916         val concl =
   917           HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map (mk_UNION Idx) Ris));
   918       in
   919         Skip_Proof.prove lthy [] []
   920           (fold_rev Logic.all (Idx :: As @ Bs @ ss @ B's @ s's @ Ris)
   921             (Logic.mk_implies (prem, concl)))
   922           (mk_bis_Union_tac bis_def in_mono'_thms)
   923         |> Thm.close_derivation
   924       end;
   925 
   926     (* self-bisimulation *)
   927 
   928     fun mk_sbis As Bs ss Rs = mk_bis As Bs ss Bs ss Rs;
   929 
   930     val sbis_prem = HOLogic.mk_Trueprop (mk_sbis As Bs ss sRs);
   931 
   932     (* largest self-bisimulation *)
   933 
   934     fun lsbis_bind i = Binding.suffix_name ("_" ^ lsbisN ^ (if n = 1 then "" else
   935       string_of_int i)) b;
   936     val lsbis_name = Binding.name_of o lsbis_bind;
   937     val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind;
   938 
   939     val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs
   940       (HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis As Bs ss sRs)));
   941 
   942     fun lsbis_spec i RT =
   943       let
   944         fun mk_lsbisT RT =
   945           Library.foldr (op -->) (map fastype_of (As @ Bs @ ss), RT);
   946         val lhs = Term.list_comb (Free (lsbis_name i, mk_lsbisT RT), As @ Bs @ ss);
   947         val rhs = mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i));
   948       in
   949         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
   950       end;
   951 
   952     val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) =
   953       lthy
   954       |> fold_map2 (fn i => fn RT => Specification.definition
   955         (SOME (lsbis_bind i, NONE, NoSyn), (lsbis_def_bind i, lsbis_spec i RT))) ks setsRTs
   956       |>> apsnd split_list o split_list
   957       ||> `Local_Theory.restore;
   958 
   959     (*transforms defined frees into consts*)
   960     val phi = Proof_Context.export_morphism lthy_old lthy;
   961 
   962     val lsbis_defs = map (Morphism.thm phi) lsbis_def_frees;
   963     val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees;
   964 
   965     fun mk_lsbis As Bs ss i =
   966       let
   967         val args = As @ Bs @ ss;
   968         val Ts = map fastype_of args;
   969         val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1)))));
   970         val lsbisT = Library.foldr (op -->) (Ts, RT);
   971       in
   972         Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args)
   973       end;
   974 
   975     val sbis_lsbis_thm =
   976       Skip_Proof.prove lthy [] []
   977         (fold_rev Logic.all (As @ Bs @ ss)
   978           (HOLogic.mk_Trueprop (mk_sbis As Bs ss (map (mk_lsbis As Bs ss) ks))))
   979         (K (mk_sbis_lsbis_tac lsbis_defs bis_Union_thm bis_cong_thm))
   980       |> Thm.close_derivation;
   981 
   982     val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS
   983       (bis_def RS @{thm subst[of _ _ "%x. x"]} RS conjunct1 RS mk_conjunctN n i)) ks;
   984     val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS
   985       (bis_def RS @{thm subst[of _ _ "%x. x"]} RS conjunct2 RS mk_conjunctN n i))) RS mp) ks;
   986 
   987     val incl_lsbis_thms =
   988       let
   989         fun mk_concl i R = HOLogic.mk_Trueprop (mk_subset R (mk_lsbis As Bs ss i));
   990         val goals = map2 (fn i => fn R => fold_rev Logic.all (As @ Bs @ ss @ sRs)
   991           (Logic.mk_implies (sbis_prem, mk_concl i R))) ks sRs;
   992       in
   993         map3 (fn goal => fn i => fn def => Skip_Proof.prove lthy [] [] goal
   994           (K (mk_incl_lsbis_tac n i def)) |> Thm.close_derivation) goals ks lsbis_defs
   995       end;
   996 
   997     val equiv_lsbis_thms =
   998       let
   999         fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis As Bs ss i));
  1000         val goals = map2 (fn i => fn B => fold_rev Logic.all (As @ Bs @ ss)
  1001           (Logic.mk_implies (coalg_prem, mk_concl i B))) ks Bs;
  1002       in
  1003         map3 (fn goal => fn l_incl => fn incl_l =>
  1004           Skip_Proof.prove lthy [] [] goal
  1005             (K (mk_equiv_lsbis_tac sbis_lsbis_thm l_incl incl_l
  1006               bis_diag_thm bis_converse_thm bis_O_thm))
  1007           |> Thm.close_derivation)
  1008         goals lsbis_incl_thms incl_lsbis_thms
  1009       end;
  1010 
  1011     val timer = time (timer "Bisimulations");
  1012 
  1013     (* bounds *)
  1014 
  1015     val (lthy, sbd, sbdT,
  1016       sbd_card_order, sbd_Cinfinite, sbd_Cnotzero, sbd_Card_order, set_sbdss, in_sbds) =
  1017       if n = 1
  1018       then (lthy, sum_bd, sum_bdT,
  1019         bd_card_order, bd_Cinfinite, bd_Cnotzero, bd_Card_order, set_bdss, in_bds)
  1020       else
  1021         let
  1022           val sbdT_bind = Binding.suffix_name ("_" ^ sum_bdTN) b;
  1023 
  1024           val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) =
  1025             typedef true NONE (sbdT_bind, params, NoSyn)
  1026               (HOLogic.mk_UNIV sum_bdT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
  1027 
  1028           val sbdT = Type (sbdT_name, params');
  1029           val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT);
  1030 
  1031           val sbd_bind = Binding.suffix_name ("_" ^ sum_bdN) b;
  1032           val sbd_name = Binding.name_of sbd_bind;
  1033           val sbd_def_bind = (Thm.def_binding sbd_bind, []);
  1034 
  1035           val sbd_spec = HOLogic.mk_Trueprop
  1036             (HOLogic.mk_eq (Free (sbd_name, mk_relT (`I sbdT)), mk_dir_image sum_bd Abs_sbdT));
  1037 
  1038           val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) =
  1039             lthy
  1040             |> Specification.definition (SOME (sbd_bind, NONE, NoSyn), (sbd_def_bind, sbd_spec))
  1041             ||> `Local_Theory.restore;
  1042 
  1043           (*transforms defined frees into consts*)
  1044           val phi = Proof_Context.export_morphism lthy_old lthy;
  1045 
  1046           val sbd_def = Morphism.thm phi sbd_def_free;
  1047           val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT));
  1048 
  1049           val sbdT_set_def = the (#set_def sbdT_loc_info);
  1050           val sbdT_Abs_inject = #Abs_inject sbdT_loc_info;
  1051           val sbdT_Abs_cases = #Abs_cases sbdT_loc_info;
  1052 
  1053           val Abs_sbdT_inj = mk_Abs_inj_thm sbdT_set_def sbdT_Abs_inject;
  1054           val Abs_sbdT_bij = mk_Abs_bij_thm lthy sbdT_set_def sbdT_Abs_inject sbdT_Abs_cases;
  1055 
  1056           fun mk_sum_Cinfinite [thm] = thm
  1057             | mk_sum_Cinfinite (thm :: thms) =
  1058               @{thm Cinfinite_csum_strong} OF [thm, mk_sum_Cinfinite thms];
  1059 
  1060           val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;
  1061           val sum_Card_order = sum_Cinfinite RS conjunct2;
  1062 
  1063           fun mk_sum_card_order [thm] = thm
  1064             | mk_sum_card_order (thm :: thms) =
  1065               @{thm card_order_csum} OF [thm, mk_sum_card_order thms];
  1066 
  1067           val sum_card_order = mk_sum_card_order bd_card_orders;
  1068 
  1069           val sbd_ordIso = Local_Defs.fold lthy [sbd_def]
  1070             (@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order]);
  1071           val sbd_card_order =  Local_Defs.fold lthy [sbd_def]
  1072             (@{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]);
  1073           val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite];
  1074           val sbd_Cnotzero = sbd_Cinfinite RS @{thm Cinfinite_Cnotzero};
  1075           val sbd_Card_order = sbd_Cinfinite RS conjunct2;
  1076 
  1077           fun mk_set_sbd i bd_Card_order bds =
  1078             map (fn thm => @{thm ordLeq_ordIso_trans} OF
  1079               [bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds;
  1080           val set_sbdss = map3 mk_set_sbd ks bd_Card_orders set_bdss;
  1081 
  1082           fun mk_in_sbd i Co Cnz bd =
  1083             Cnz RS ((@{thm ordLeq_ordIso_trans} OF
  1084               [(Co RS mk_ordLeq_csum n i (Co RS @{thm ordLeq_refl})), sbd_ordIso]) RS
  1085               (bd RS @{thm ordLeq_transitive[OF _
  1086                 cexp_mono2_Cnotzero[OF _ csum_Cnotzero2[OF ctwo_Cnotzero]]]}));
  1087           val in_sbds = map4 mk_in_sbd ks bd_Card_orders bd_Cnotzeros in_bds;
  1088        in
  1089          (lthy, sbd, sbdT,
  1090            sbd_card_order, sbd_Cinfinite, sbd_Cnotzero, sbd_Card_order, set_sbdss, in_sbds)
  1091        end;
  1092 
  1093     fun mk_sbd_sbd 1 = sbd_Card_order RS @{thm ordIso_refl}
  1094       | mk_sbd_sbd n = @{thm csum_absorb1} OF
  1095           [sbd_Cinfinite, mk_sbd_sbd (n - 1) RS @{thm ordIso_imp_ordLeq}];
  1096 
  1097     val sbd_sbd_thm = mk_sbd_sbd n;
  1098 
  1099     val sbdTs = replicate n sbdT;
  1100     val sum_sbd = Library.foldr1 (uncurry mk_csum) (replicate n sbd);
  1101     val sum_sbdT = mk_sumTN sbdTs;
  1102     val sum_sbd_listT = HOLogic.listT sum_sbdT;
  1103     val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT;
  1104     val bdTs = passiveAs @ replicate n sbdT;
  1105     val to_sbd_maps = map4 mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs;
  1106     val bdFTs = mk_FTs bdTs;
  1107     val sbdFT = mk_sumTN bdFTs;
  1108     val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT);
  1109     val treeQT = HOLogic.mk_setT treeT;
  1110     val treeTs = passiveAs @ replicate n treeT;
  1111     val treeQTs = passiveAs @ replicate n treeQT;
  1112     val treeFTs = mk_FTs treeTs;
  1113     val tree_maps = map4 mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs;
  1114     val final_maps = map4 mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs;
  1115     val tree_setss = mk_setss treeTs;
  1116     val isNode_setss = mk_setss (passiveAs @ replicate n sbdT);
  1117 
  1118     val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []];
  1119     val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs);
  1120     val Lev_recT = fastype_of Zero;
  1121     val LevT = Library.foldr (op -->) (sTs, HOLogic.natT --> Lev_recT);
  1122 
  1123     val Nil = HOLogic.mk_tuple (map3 (fn i => fn z => fn z'=>
  1124       Term.absfree z' (mk_InN activeAs z i)) ks zs zs');
  1125     val rv_recT = fastype_of Nil;
  1126     val rvT = Library.foldr (op -->) (sTs, sum_sbd_listT --> rv_recT);
  1127 
  1128     val (((((((((((sumx, sumx'), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')),
  1129       (lab, lab')), (Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')),
  1130       names_lthy) = names_lthy
  1131       |> yield_singleton (apfst (op ~~) oo mk_Frees' "sumx") sum_sbdT
  1132       ||>> mk_Frees' "k" sbdTs
  1133       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
  1134       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
  1135       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl") sum_sbd_list_setT
  1136       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT)
  1137       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl_lab") treeT
  1138       ||>> mk_Frees "x" bdFTs
  1139       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") Lev_recT
  1140       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") rv_recT;
  1141 
  1142     val (k, k') = (hd kks, hd kks')
  1143 
  1144     val timer = time (timer "Bounds");
  1145 
  1146     (* tree coalgebra *)
  1147 
  1148     fun isNode_bind i = Binding.suffix_name ("_" ^ isNodeN ^ (if n = 1 then "" else
  1149       string_of_int i)) b;
  1150     val isNode_name = Binding.name_of o isNode_bind;
  1151     val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind;
  1152 
  1153     val isNodeT =
  1154       Library.foldr (op -->) (map fastype_of (As @ [Kl, lab, kl]), HOLogic.boolT);
  1155 
  1156     val Succs = map3 (fn i => fn k => fn k' =>
  1157       HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl)))
  1158       ks kks kks';
  1159 
  1160     fun isNode_spec sets x i =
  1161       let
  1162         val (passive_sets, active_sets) = chop m (map (fn set => set $ x) sets);
  1163         val lhs = Term.list_comb (Free (isNode_name i, isNodeT), As @ [Kl, lab, kl]);
  1164         val rhs = list_exists_free [x]
  1165           (Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) ::
  1166           map2 mk_subset passive_sets As @ map2 (curry HOLogic.mk_eq) active_sets Succs));
  1167       in
  1168         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  1169       end;
  1170 
  1171     val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) =
  1172       lthy
  1173       |> fold_map3 (fn i => fn x => fn sets => Specification.definition
  1174         (SOME (isNode_bind i, NONE, NoSyn), (isNode_def_bind i, isNode_spec sets x i)))
  1175         ks xs isNode_setss
  1176       |>> apsnd split_list o split_list
  1177       ||> `Local_Theory.restore;
  1178 
  1179     (*transforms defined frees into consts*)
  1180     val phi = Proof_Context.export_morphism lthy_old lthy;
  1181 
  1182     val isNode_defs = map (Morphism.thm phi) isNode_def_frees;
  1183     val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees;
  1184 
  1185     fun mk_isNode As kl i =
  1186       Term.list_comb (Const (nth isNodes (i - 1), isNodeT), As @ [Kl, lab, kl]);
  1187 
  1188     val isTree =
  1189       let
  1190         val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl);
  1191         val Field = mk_subset Kl (mk_Field (mk_clists sum_sbd));
  1192         val prefCl = mk_prefCl Kl;
  1193 
  1194         val tree = mk_Ball Kl (Term.absfree kl'
  1195           (HOLogic.mk_conj
  1196             (Library.foldr1 HOLogic.mk_disj (map (mk_isNode As kl) ks),
  1197             Library.foldr1 HOLogic.mk_conj (map4 (fn Succ => fn i => fn k => fn k' =>
  1198               mk_Ball Succ (Term.absfree k' (mk_isNode As
  1199                 (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i)))
  1200             Succs ks kks kks'))));
  1201 
  1202         val undef = list_all_free [kl] (HOLogic.mk_imp
  1203           (HOLogic.mk_not (HOLogic.mk_mem (kl, Kl)),
  1204           HOLogic.mk_eq (lab $ kl, mk_undefined sbdFT)));
  1205       in
  1206         Library.foldr1 HOLogic.mk_conj [empty, Field, prefCl, tree, undef]
  1207       end;
  1208 
  1209     fun carT_bind i = Binding.suffix_name ("_" ^ carTN ^ (if n = 1 then "" else
  1210       string_of_int i)) b;
  1211     val carT_name = Binding.name_of o carT_bind;
  1212     val carT_def_bind = rpair [] o Thm.def_binding o carT_bind;
  1213 
  1214     fun carT_spec i =
  1215       let
  1216         val carTT = Library.foldr (op -->) (ATs, HOLogic.mk_setT treeT);
  1217 
  1218         val lhs = Term.list_comb (Free (carT_name i, carTT), As);
  1219         val rhs = HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
  1220           (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)),
  1221             HOLogic.mk_conj (isTree, mk_isNode As (HOLogic.mk_list sum_sbdT []) i))));
  1222       in
  1223         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  1224       end;
  1225 
  1226     val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) =
  1227       lthy
  1228       |> fold_map (fn i => Specification.definition
  1229         (SOME (carT_bind i, NONE, NoSyn), (carT_def_bind i, carT_spec i))) ks
  1230       |>> apsnd split_list o split_list
  1231       ||> `Local_Theory.restore;
  1232 
  1233     (*transforms defined frees into consts*)
  1234     val phi = Proof_Context.export_morphism lthy_old lthy;
  1235 
  1236     val carT_defs = map (Morphism.thm phi) carT_def_frees;
  1237     val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees;
  1238 
  1239     fun mk_carT As i = Term.list_comb
  1240       (Const (nth carTs (i - 1),
  1241          Library.foldr (op -->) (map fastype_of As, HOLogic.mk_setT treeT)), As);
  1242 
  1243     fun strT_bind i = Binding.suffix_name ("_" ^ strTN ^ (if n = 1 then "" else
  1244       string_of_int i)) b;
  1245     val strT_name = Binding.name_of o strT_bind;
  1246     val strT_def_bind = rpair [] o Thm.def_binding o strT_bind;
  1247 
  1248     fun strT_spec mapFT FT i =
  1249       let
  1250         val strTT = treeT --> FT;
  1251 
  1252         fun mk_f i k k' =
  1253           let val in_k = mk_InN sbdTs k i;
  1254           in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end;
  1255 
  1256         val f = Term.list_comb (mapFT, passive_ids @ map3 mk_f ks kks kks');
  1257         val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs));
  1258         val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2);
  1259         val lhs = Free (strT_name i, strTT);
  1260         val rhs = HOLogic.mk_split (Term.absfree Kl' (Term.absfree lab'
  1261           (mk_sum_caseN fs $ (lab $ HOLogic.mk_list sum_sbdT []))));
  1262       in
  1263         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  1264       end;
  1265 
  1266     val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) =
  1267       lthy
  1268       |> fold_map3 (fn i => fn mapFT => fn FT => Specification.definition
  1269         (SOME (strT_bind i, NONE, NoSyn), (strT_def_bind i, strT_spec mapFT FT i)))
  1270         ks tree_maps treeFTs
  1271       |>> apsnd split_list o split_list
  1272       ||> `Local_Theory.restore;
  1273 
  1274     (*transforms defined frees into consts*)
  1275     val phi = Proof_Context.export_morphism lthy_old lthy;
  1276 
  1277     val strT_defs = map ((fn def => trans OF [def RS fun_cong, @{thm prod.cases}]) o
  1278       Morphism.thm phi) strT_def_frees;
  1279     val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees;
  1280 
  1281     fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT);
  1282 
  1283     val carTAs = map (mk_carT As) ks;
  1284     val carTAs_copy = map (mk_carT As_copy) ks;
  1285     val strTAs = map2 mk_strT treeFTs ks;
  1286     val hset_strTss = map (fn i => map2 (mk_hset strTAs i) ls passiveAs) ks;
  1287 
  1288     val coalgT_thm =
  1289       Skip_Proof.prove lthy [] []
  1290         (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_coalg As carTAs strTAs)))
  1291         (mk_coalgT_tac m (coalg_def :: isNode_defs @ carT_defs) strT_defs set_natural'ss)
  1292       |> Thm.close_derivation;
  1293 
  1294     val card_of_carT_thms =
  1295       let
  1296         val lhs = mk_card_of
  1297           (HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
  1298             (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)), isTree))));
  1299         val rhs = mk_cexp
  1300           (if m = 0 then ctwo else
  1301             (mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo))
  1302             (mk_cexp sbd sbd);
  1303         val card_of_carT =
  1304           Skip_Proof.prove lthy [] []
  1305             (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_ordLeq lhs rhs)))
  1306             (K (mk_card_of_carT_tac m isNode_defs sbd_sbd_thm
  1307               sbd_card_order sbd_Card_order sbd_Cinfinite sbd_Cnotzero in_sbds))
  1308           |> Thm.close_derivation
  1309       in
  1310         map (fn def => @{thm ordLeq_transitive[OF
  1311           card_of_mono1[OF ord_eq_le_trans[OF _ Collect_restrict']]]} OF [def, card_of_carT])
  1312         carT_defs
  1313       end;
  1314 
  1315     val carT_set_thmss =
  1316       let
  1317         val Kl_lab = HOLogic.mk_prod (Kl, lab);
  1318         fun mk_goal carT strT set k i =
  1319           fold_rev Logic.all (sumx :: Kl :: lab :: k :: kl :: As)
  1320             (Logic.list_implies (map HOLogic.mk_Trueprop
  1321               [HOLogic.mk_mem (Kl_lab, carT), HOLogic.mk_mem (mk_Cons sumx kl, Kl),
  1322               HOLogic.mk_eq (sumx, mk_InN sbdTs k i)],
  1323             HOLogic.mk_Trueprop (HOLogic.mk_mem
  1324               (HOLogic.mk_prod (mk_Shift Kl sumx, mk_shift lab sumx),
  1325               set $ (strT $ Kl_lab)))));
  1326 
  1327         val goalss = map3 (fn carT => fn strT => fn sets =>
  1328           map3 (mk_goal carT strT) (drop m sets) kks ks) carTAs strTAs tree_setss;
  1329       in
  1330         map6 (fn i => fn goals =>
  1331             fn carT_def => fn strT_def => fn isNode_def => fn set_naturals =>
  1332           map2 (fn goal => fn set_natural =>
  1333             Skip_Proof.prove lthy [] [] goal
  1334               (mk_carT_set_tac n i carT_def strT_def isNode_def set_natural)
  1335             |> Thm.close_derivation)
  1336           goals (drop m set_naturals))
  1337         ks goalss carT_defs strT_defs isNode_defs set_natural'ss
  1338       end;
  1339 
  1340     val carT_set_thmss' = transpose carT_set_thmss;
  1341 
  1342     val isNode_hset_thmss =
  1343       let
  1344         val Kl_lab = HOLogic.mk_prod (Kl, lab);
  1345         fun mk_Kl_lab carT = HOLogic.mk_mem (Kl_lab, carT);
  1346 
  1347         val strT_hset_thmsss =
  1348           let
  1349             val strT_hset_thms =
  1350               let
  1351                 fun mk_lab_kl i x = HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i);
  1352 
  1353                 fun mk_inner_conjunct j T i x set i' carT =
  1354                   HOLogic.mk_imp (HOLogic.mk_conj (mk_Kl_lab carT, mk_lab_kl i x),
  1355                     mk_subset (set $ x) (mk_hset strTAs i' j T $ Kl_lab));
  1356 
  1357                 fun mk_conjunct j T i x set =
  1358                   Library.foldr1 HOLogic.mk_conj (map2 (mk_inner_conjunct j T i x set) ks carTAs);
  1359 
  1360                 fun mk_concl j T = list_all_free (Kl :: lab :: xs @ As)
  1361                   (HOLogic.mk_imp (HOLogic.mk_mem (kl, Kl),
  1362                     Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T)
  1363                       ks xs (map (fn xs => nth xs (j - 1)) isNode_setss))));
  1364                 val concls = map2 mk_concl ls passiveAs;
  1365 
  1366                 val cTs = [SOME (certifyT lthy sum_sbdT)];
  1367                 val arg_cong_cTs = map (SOME o certifyT lthy) treeFTs;
  1368                 val ctss =
  1369                   map (fn phi => map (SOME o certify lthy) [Term.absfree kl' phi, kl]) concls;
  1370 
  1371                 val goals = map HOLogic.mk_Trueprop concls;
  1372               in
  1373                 map5 (fn j => fn goal => fn cts => fn set_incl_hsets => fn set_hset_incl_hsetss =>
  1374                   singleton (Proof_Context.export names_lthy lthy)
  1375                     (Skip_Proof.prove lthy [] [] goal
  1376                       (K (mk_strT_hset_tac n m j arg_cong_cTs cTs cts
  1377                         carT_defs strT_defs isNode_defs
  1378                         set_incl_hsets set_hset_incl_hsetss coalg_set_thmss' carT_set_thmss'
  1379                         coalgT_thm set_natural'ss)))
  1380                   |> Thm.close_derivation)
  1381                 ls goals ctss set_incl_hset_thmss' set_hset_incl_hset_thmsss''
  1382               end;
  1383 
  1384             val strT_hset'_thms = map (fn thm => mk_specN (2 + n + m) thm RS mp) strT_hset_thms;
  1385           in
  1386             map (fn thm => map (fn i => map (fn i' =>
  1387               thm RS mk_conjunctN n i RS mk_conjunctN n i' RS mp) ks) ks) strT_hset'_thms
  1388           end;
  1389 
  1390         val carT_prems = map (fn carT =>
  1391           HOLogic.mk_Trueprop (HOLogic.mk_mem (Kl_lab, carT))) carTAs_copy;
  1392         val prem = HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, Kl));
  1393         val in_prems = map (fn hsets =>
  1394           HOLogic.mk_Trueprop (HOLogic.mk_mem (Kl_lab, mk_in As hsets treeT))) hset_strTss;
  1395         val isNode_premss = replicate n (map (HOLogic.mk_Trueprop o mk_isNode As_copy kl) ks);
  1396         val conclss = replicate n (map (HOLogic.mk_Trueprop o mk_isNode As kl) ks);
  1397       in
  1398         map5 (fn carT_prem => fn isNode_prems => fn in_prem => fn concls => fn strT_hset_thmss =>
  1399           map4 (fn isNode_prem => fn concl => fn isNode_def => fn strT_hset_thms =>
  1400             Skip_Proof.prove lthy [] []
  1401               (fold_rev Logic.all (Kl :: lab :: kl :: As @ As_copy)
  1402                 (Logic.list_implies ([carT_prem, prem, isNode_prem, in_prem], concl)))
  1403               (mk_isNode_hset_tac n isNode_def strT_hset_thms)
  1404             |> Thm.close_derivation)
  1405           isNode_prems concls isNode_defs
  1406           (if m = 0 then replicate n [] else transpose strT_hset_thmss))
  1407         carT_prems isNode_premss in_prems conclss
  1408         (if m = 0 then replicate n [] else transpose (map transpose strT_hset_thmsss))
  1409       end;
  1410 
  1411     val timer = time (timer "Tree coalgebra");
  1412 
  1413     fun mk_to_sbd s x i i' =
  1414       mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
  1415     fun mk_from_sbd s x i i' =
  1416       mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
  1417 
  1418     fun mk_to_sbd_thmss thm = map (map (fn set_sbd =>
  1419       thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss;
  1420 
  1421     val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj};
  1422     val to_sbd_thmss = mk_to_sbd_thmss @{thm toCard};
  1423     val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard};
  1424 
  1425     val Lev_bind = Binding.suffix_name ("_" ^ LevN) b;
  1426     val Lev_name = Binding.name_of Lev_bind;
  1427     val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind);
  1428 
  1429     val Lev_spec =
  1430       let
  1431         fun mk_Suc i s setsAs a a' =
  1432           let
  1433             val sets = drop m setsAs;
  1434             fun mk_set i' set b =
  1435               let
  1436                 val Cons = HOLogic.mk_eq (kl_copy,
  1437                   mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl)
  1438                 val b_set = HOLogic.mk_mem (b, set $ (s $ a));
  1439                 val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b);
  1440               in
  1441                 HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl]
  1442                   (HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec))))
  1443               end;
  1444           in
  1445             Term.absfree a' (Library.foldl1 mk_union (map3 mk_set ks sets zs_copy))
  1446           end;
  1447 
  1448         val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec'
  1449           (HOLogic.mk_tuple (map5 mk_Suc ks ss setssAs zs zs')));
  1450 
  1451         val lhs = Term.list_comb (Free (Lev_name, LevT), ss);
  1452         val rhs = mk_nat_rec Zero Suc;
  1453       in
  1454         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  1455       end;
  1456 
  1457     val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) =
  1458       lthy
  1459       |> Specification.definition (SOME (Lev_bind, NONE, NoSyn), (Lev_def_bind, Lev_spec))
  1460       ||> `Local_Theory.restore;
  1461 
  1462     (*transforms defined frees into consts*)
  1463     val phi = Proof_Context.export_morphism lthy_old lthy;
  1464 
  1465     val Lev_def = Morphism.thm phi Lev_def_free;
  1466     val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free));
  1467 
  1468     fun mk_Lev ss nat i =
  1469       let
  1470         val Ts = map fastype_of ss;
  1471         val LevT = Library.foldr (op -->) (Ts, HOLogic.natT -->
  1472           HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts));
  1473       in
  1474         mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i
  1475       end;
  1476 
  1477     val Lev_0s = flat (mk_rec_simps n @{thm nat_rec_0} [Lev_def]);
  1478     val Lev_Sucs = flat (mk_rec_simps n @{thm nat_rec_Suc} [Lev_def]);
  1479 
  1480     val rv_bind = Binding.suffix_name ("_" ^ rvN) b;
  1481     val rv_name = Binding.name_of rv_bind;
  1482     val rv_def_bind = rpair [] (Thm.def_binding rv_bind);
  1483 
  1484     val rv_spec =
  1485       let
  1486         fun mk_Cons i s b b' =
  1487           let
  1488             fun mk_case i' =
  1489               Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k));
  1490           in
  1491             Term.absfree b' (mk_sum_caseN (map mk_case ks) $ sumx)
  1492           end;
  1493 
  1494         val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec'
  1495           (HOLogic.mk_tuple (map4 mk_Cons ks ss zs zs'))));
  1496 
  1497         val lhs = Term.list_comb (Free (rv_name, rvT), ss);
  1498         val rhs = mk_list_rec Nil Cons;
  1499       in
  1500         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  1501       end;
  1502 
  1503     val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) =
  1504       lthy
  1505       |> Specification.definition (SOME (rv_bind, NONE, NoSyn), (rv_def_bind, rv_spec))
  1506       ||> `Local_Theory.restore;
  1507 
  1508     (*transforms defined frees into consts*)
  1509     val phi = Proof_Context.export_morphism lthy_old lthy;
  1510 
  1511     val rv_def = Morphism.thm phi rv_def_free;
  1512     val rv = fst (Term.dest_Const (Morphism.term phi rv_free));
  1513 
  1514     fun mk_rv ss kl i =
  1515       let
  1516         val Ts = map fastype_of ss;
  1517         val As = map domain_type Ts;
  1518         val rvT = Library.foldr (op -->) (Ts, fastype_of kl -->
  1519           HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As));
  1520       in
  1521         mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i
  1522       end;
  1523 
  1524     val rv_Nils = flat (mk_rec_simps n @{thm list_rec_Nil} [rv_def]);
  1525     val rv_Conss = flat (mk_rec_simps n @{thm list_rec_Cons} [rv_def]);
  1526 
  1527     fun beh_bind i = Binding.suffix_name ("_" ^ behN ^ (if n = 1 then "" else
  1528       string_of_int i)) b;
  1529     val beh_name = Binding.name_of o beh_bind;
  1530     val beh_def_bind = rpair [] o Thm.def_binding o beh_bind;
  1531 
  1532     fun beh_spec i z =
  1533       let
  1534         val mk_behT = Library.foldr (op -->) (map fastype_of (ss @ [z]), treeT);
  1535 
  1536         fun mk_case i to_sbd_map s k k' =
  1537           Term.absfree k' (mk_InN bdFTs
  1538             (Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i);
  1539 
  1540         val Lab = Term.absfree kl' (mk_If
  1541           (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))
  1542           (mk_sum_caseN (map5 mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z))
  1543           (mk_undefined sbdFT));
  1544 
  1545         val lhs = Term.list_comb (Free (beh_name i, mk_behT), ss) $ z;
  1546         val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
  1547           (Term.absfree nat' (mk_Lev ss nat i $ z)), Lab);
  1548       in
  1549         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  1550       end;
  1551 
  1552     val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) =
  1553       lthy
  1554       |> fold_map2 (fn i => fn z => Specification.definition
  1555         (SOME (beh_bind i, NONE, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs
  1556       |>> apsnd split_list o split_list
  1557       ||> `Local_Theory.restore;
  1558 
  1559     (*transforms defined frees into consts*)
  1560     val phi = Proof_Context.export_morphism lthy_old lthy;
  1561 
  1562     val beh_defs = map (Morphism.thm phi) beh_def_frees;
  1563     val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees;
  1564 
  1565     fun mk_beh ss i =
  1566       let
  1567         val Ts = map fastype_of ss;
  1568         val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT);
  1569       in
  1570         Term.list_comb (Const (nth behs (i - 1), behT), ss)
  1571       end;
  1572 
  1573     val Lev_sbd_thms =
  1574       let
  1575         fun mk_conjunct i z = mk_subset (mk_Lev ss nat i $ z) (mk_Field (mk_clists sum_sbd));
  1576         val goal = list_all_free zs
  1577           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
  1578 
  1579         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
  1580 
  1581         val Lev_sbd = singleton (Proof_Context.export names_lthy lthy)
  1582           (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
  1583             (K (mk_Lev_sbd_tac cts Lev_0s Lev_Sucs to_sbd_thmss))
  1584           |> Thm.close_derivation);
  1585 
  1586         val Lev_sbd' = mk_specN n Lev_sbd;
  1587       in
  1588         map (fn i => Lev_sbd' RS mk_conjunctN n i) ks
  1589       end;
  1590 
  1591     val (length_Lev_thms, length_Lev'_thms) =
  1592       let
  1593         fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
  1594           HOLogic.mk_eq (mk_size kl, nat));
  1595         val goal = list_all_free (kl :: zs)
  1596           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
  1597 
  1598         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
  1599 
  1600         val length_Lev = singleton (Proof_Context.export names_lthy lthy)
  1601           (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
  1602             (K (mk_length_Lev_tac cts Lev_0s Lev_Sucs))
  1603           |> Thm.close_derivation);
  1604 
  1605         val length_Lev' = mk_specN (n + 1) length_Lev;
  1606         val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks;
  1607 
  1608         fun mk_goal i z = fold_rev Logic.all (z :: kl :: nat :: ss) (Logic.mk_implies
  1609             (HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z)),
  1610             HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))));
  1611         val goals = map2 mk_goal ks zs;
  1612 
  1613         val length_Levs' = map2 (fn goal => fn length_Lev =>
  1614           Skip_Proof.prove lthy [] [] goal (K (mk_length_Lev'_tac length_Lev))
  1615           |> Thm.close_derivation) goals length_Levs;
  1616       in
  1617         (length_Levs, length_Levs')
  1618       end;
  1619 
  1620     val prefCl_Lev_thms =
  1621       let
  1622         fun mk_conjunct i z = HOLogic.mk_imp
  1623           (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), mk_subset kl_copy kl),
  1624           HOLogic.mk_mem (kl_copy, mk_Lev ss (mk_size kl_copy) i $ z));
  1625         val goal = list_all_free (kl :: kl_copy :: zs)
  1626           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
  1627 
  1628         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
  1629 
  1630         val prefCl_Lev = singleton (Proof_Context.export names_lthy lthy)
  1631           (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
  1632             (K (mk_prefCl_Lev_tac cts Lev_0s Lev_Sucs)))
  1633           |> Thm.close_derivation;
  1634 
  1635         val prefCl_Lev' = mk_specN (n + 2) prefCl_Lev;
  1636       in
  1637         map (fn i => prefCl_Lev' RS mk_conjunctN n i RS mp) ks
  1638       end;
  1639 
  1640     val rv_last_thmss =
  1641       let
  1642         fun mk_conjunct i z i' z_copy = list_exists_free [z_copy]
  1643           (HOLogic.mk_eq
  1644             (mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z,
  1645             mk_InN activeAs z_copy i'));
  1646         val goal = list_all_free (k :: zs)
  1647           (Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z =>
  1648             Library.foldr1 HOLogic.mk_conj
  1649               (map2 (mk_conjunct i z) ks zs_copy)) ks zs));
  1650 
  1651         val cTs = [SOME (certifyT lthy sum_sbdT)];
  1652         val cts = map (SOME o certify lthy) [Term.absfree kl' goal, kl];
  1653 
  1654         val rv_last = singleton (Proof_Context.export names_lthy lthy)
  1655           (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
  1656             (K (mk_rv_last_tac cTs cts rv_Nils rv_Conss)))
  1657           |> Thm.close_derivation;
  1658 
  1659         val rv_last' = mk_specN (n + 1) rv_last;
  1660       in
  1661         map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks
  1662       end;
  1663 
  1664     val set_rv_Lev_thmsss = if m = 0 then replicate n (replicate n []) else
  1665       let
  1666         fun mk_case s sets z z_free = Term.absfree z_free (Library.foldr1 HOLogic.mk_conj
  1667           (map2 (fn set => fn A => mk_subset (set $ (s $ z)) A) (take m sets) As));
  1668 
  1669         fun mk_conjunct i z B = HOLogic.mk_imp
  1670           (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), HOLogic.mk_mem (z, B)),
  1671           mk_sum_caseN (map4 mk_case ss setssAs zs zs') $ (mk_rv ss kl i $ z));
  1672 
  1673         val goal = list_all_free (kl :: zs)
  1674           (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct ks zs Bs));
  1675 
  1676         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
  1677 
  1678         val set_rv_Lev = singleton (Proof_Context.export names_lthy lthy)
  1679           (Skip_Proof.prove lthy [] []
  1680             (Logic.mk_implies (coalg_prem, HOLogic.mk_Trueprop goal))
  1681             (K (mk_set_rv_Lev_tac m cts Lev_0s Lev_Sucs rv_Nils rv_Conss
  1682               coalg_set_thmss from_to_sbd_thmss)))
  1683           |> Thm.close_derivation;
  1684 
  1685         val set_rv_Lev' = mk_specN (n + 1) set_rv_Lev;
  1686       in
  1687         map (fn i => map (fn i' =>
  1688           split_conj_thm (if n = 1 then set_rv_Lev' RS mk_conjunctN n i RS mp
  1689             else set_rv_Lev' RS mk_conjunctN n i RS mp RSN
  1690               (2, @{thm sum_case_cong} RS @{thm subst[of _ _ "%x. x"]}) RS
  1691               (mk_sum_casesN n i' RS @{thm subst[of _ _ "%x. x"]}))) ks) ks
  1692       end;
  1693 
  1694     val set_Lev_thmsss =
  1695       let
  1696         fun mk_conjunct i z =
  1697           let
  1698             fun mk_conjunct' i' sets s z' =
  1699               let
  1700                 fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp
  1701                   (HOLogic.mk_mem (z'', set $ (s $ z')),
  1702                     HOLogic.mk_mem (mk_append (kl,
  1703                       HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']),
  1704                       mk_Lev ss (HOLogic.mk_Suc nat) i $ z));
  1705               in
  1706                 HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'),
  1707                   (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct'' ks (drop m sets) zs_copy2)))
  1708               end;
  1709           in
  1710             HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
  1711               Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct' ks setssAs ss zs_copy))
  1712           end;
  1713 
  1714         val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2)
  1715           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
  1716 
  1717         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
  1718 
  1719         val set_Lev = singleton (Proof_Context.export names_lthy lthy)
  1720           (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
  1721             (K (mk_set_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss)))
  1722           |> Thm.close_derivation;
  1723 
  1724         val set_Lev' = mk_specN (3 * n + 1) set_Lev;
  1725       in
  1726         map (fn i => map (fn i' => map (fn i'' => set_Lev' RS
  1727           mk_conjunctN n i RS mp RS
  1728           mk_conjunctN n i' RS mp RS
  1729           mk_conjunctN n i'' RS mp) ks) ks) ks
  1730       end;
  1731 
  1732     val set_image_Lev_thmsss =
  1733       let
  1734         fun mk_conjunct i z =
  1735           let
  1736             fun mk_conjunct' i' sets =
  1737               let
  1738                 fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp
  1739                   (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''),
  1740                   HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z''))));
  1741               in
  1742                 HOLogic.mk_imp (HOLogic.mk_mem
  1743                   (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']),
  1744                     mk_Lev ss (HOLogic.mk_Suc nat) i $ z),
  1745                   (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct'' ks sets ss zs_copy)))
  1746               end;
  1747           in
  1748             HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
  1749               Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs')))
  1750           end;
  1751 
  1752         val goal = list_all_free (kl :: k :: zs @ zs_copy)
  1753           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
  1754 
  1755         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
  1756 
  1757         val set_image_Lev = singleton (Proof_Context.export names_lthy lthy)
  1758           (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
  1759             (K (mk_set_image_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss
  1760               from_to_sbd_thmss to_sbd_inj_thmss)))
  1761           |> Thm.close_derivation;
  1762 
  1763         val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev;
  1764       in
  1765         map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS
  1766           mk_conjunctN n i RS mp RS
  1767           mk_conjunctN n i'' RS mp RS
  1768           mk_conjunctN n i' RS mp) ks) ks) ks
  1769       end;
  1770 
  1771     val mor_beh_thm =
  1772       Skip_Proof.prove lthy [] []
  1773         (fold_rev Logic.all (As @ Bs @ ss) (Logic.mk_implies (coalg_prem,
  1774           HOLogic.mk_Trueprop (mk_mor Bs ss carTAs strTAs (map (mk_beh ss) ks)))))
  1775         (mk_mor_beh_tac m mor_def mor_cong_thm
  1776           beh_defs carT_defs strT_defs isNode_defs
  1777           to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbd_thms
  1778           length_Lev_thms length_Lev'_thms prefCl_Lev_thms rv_last_thmss
  1779           set_rv_Lev_thmsss set_Lev_thmsss set_image_Lev_thmsss
  1780           set_natural'ss coalg_set_thmss map_comp_id_thms map_congs map_arg_cong_thms)
  1781       |> Thm.close_derivation;
  1782 
  1783     val timer = time (timer "Behavioral morphism");
  1784 
  1785     fun mk_LSBIS As i = mk_lsbis As (map (mk_carT As) ks) strTAs i;
  1786     fun mk_car_final As i =
  1787       mk_quotient (mk_carT As i) (mk_LSBIS As i);
  1788     fun mk_str_final As i =
  1789       mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1),
  1790         passive_ids @ map (mk_proj o mk_LSBIS As) ks), nth strTAs (i - 1)));
  1791 
  1792     val car_finalAs = map (mk_car_final As) ks;
  1793     val str_finalAs = map (mk_str_final As) ks;
  1794     val car_finals = map (mk_car_final passive_UNIVs) ks;
  1795     val str_finals = map (mk_str_final passive_UNIVs) ks;
  1796 
  1797     val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss;
  1798     val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms;
  1799 
  1800     val congruent_str_final_thms =
  1801       let
  1802         fun mk_goal R final_map strT =
  1803           fold_rev Logic.all As (HOLogic.mk_Trueprop
  1804             (mk_congruent R (HOLogic.mk_comp
  1805               (Term.list_comb (final_map, passive_ids @ map (mk_proj o mk_LSBIS As) ks), strT))));
  1806 
  1807         val goals = map3 mk_goal (map (mk_LSBIS As) ks) final_maps strTAs;
  1808       in
  1809         map4 (fn goal => fn lsbisE => fn map_comp_id => fn map_cong =>
  1810           Skip_Proof.prove lthy [] [] goal
  1811             (K (mk_congruent_str_final_tac m lsbisE map_comp_id map_cong equiv_LSBIS_thms))
  1812           |> Thm.close_derivation)
  1813         goals lsbisE_thms map_comp_id_thms map_congs
  1814       end;
  1815 
  1816     val coalg_final_thm = Skip_Proof.prove lthy [] [] (fold_rev Logic.all As
  1817       (HOLogic.mk_Trueprop (mk_coalg As car_finalAs str_finalAs)))
  1818       (K (mk_coalg_final_tac m coalg_def congruent_str_final_thms equiv_LSBIS_thms
  1819         set_natural'ss coalgT_set_thmss))
  1820       |> Thm.close_derivation;
  1821 
  1822     val mor_T_final_thm = Skip_Proof.prove lthy [] [] (fold_rev Logic.all As
  1823       (HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finalAs str_finalAs
  1824         (map (mk_proj o mk_LSBIS As) ks))))
  1825       (K (mk_mor_T_final_tac mor_def congruent_str_final_thms equiv_LSBIS_thms))
  1826       |> Thm.close_derivation;
  1827 
  1828     val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm];
  1829     val in_car_final_thms = map (fn mor_image' => mor_image' OF
  1830       [tcoalg_thm RS mor_final_thm, UNIV_I]) mor_image'_thms;
  1831 
  1832     val timer = time (timer "Final coalgebra");
  1833 
  1834     val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
  1835       lthy
  1836       |> fold_map3 (fn b => fn car_final => fn in_car_final =>
  1837         typedef false NONE (b, params, NoSyn) car_final NONE
  1838           (EVERY' [rtac exI, rtac in_car_final] 1)) bs car_finals in_car_final_thms
  1839       |>> apsnd split_list o split_list;
  1840 
  1841     val Ts = map (fn name => Type (name, params')) T_names;
  1842     fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;
  1843     val Ts' = mk_Ts passiveBs;
  1844     val Ts'' = mk_Ts passiveCs;
  1845     val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts;
  1846     val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts;
  1847 
  1848     val Reps = map #Rep T_loc_infos;
  1849     val Rep_injects = map #Rep_inject T_loc_infos;
  1850     val Rep_inverses = map #Rep_inverse T_loc_infos;
  1851     val Abs_inverses = map #Abs_inverse T_loc_infos;
  1852 
  1853     val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");
  1854 
  1855     val UNIVs = map HOLogic.mk_UNIV Ts;
  1856     val FTs = mk_FTs (passiveAs @ Ts);
  1857     val FTs' = mk_FTs (passiveBs @ Ts);
  1858     val prodTs = map (HOLogic.mk_prodT o `I) Ts;
  1859     val prodFTs = mk_FTs (passiveAs @ prodTs);
  1860     val FTs_setss = mk_setss (passiveAs @ Ts);
  1861     val FTs'_setss = mk_setss (passiveBs @ Ts);
  1862     val prodFT_setss = mk_setss (passiveAs @ prodTs);
  1863     val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs;
  1864     val map_FT_nths = map2 (fn Ds =>
  1865       mk_map_of_bnf Ds (passiveAs @ prodTs) (passiveAs @ Ts)) Dss bnfs;
  1866     val fstsTs = map fst_const prodTs;
  1867     val sndsTs = map snd_const prodTs;
  1868     val unfTs = map2 (curry (op -->)) Ts FTs;
  1869     val fldTs = map2 (curry (op -->)) FTs Ts;
  1870     val coiter_fTs = map2 (curry op -->) activeAs Ts;
  1871     val corec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) sum_sTs;
  1872     val corec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls;
  1873     val corec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls_rev;
  1874     val corec_Inls = map (Term.subst_atomic_types (activeBs ~~ Ts)) Inls;
  1875 
  1876     val (((((((((((((Jzs, Jzs'), (Jz's, Jz's')), Jzs_copy), Jzs1), Jzs2), Jpairs),
  1877       FJzs), TRs), coiter_fs), coiter_fs_copy), corec_ss), phis), names_lthy) = names_lthy
  1878       |> mk_Frees' "z" Ts
  1879       ||>> mk_Frees' "z" Ts'
  1880       ||>> mk_Frees "z" Ts
  1881       ||>> mk_Frees "z1" Ts
  1882       ||>> mk_Frees "z2" Ts
  1883       ||>> mk_Frees "j" (map2 (curry HOLogic.mk_prodT) Ts Ts')
  1884       ||>> mk_Frees "x" prodFTs
  1885       ||>> mk_Frees "R" (map (mk_relT o `I) Ts)
  1886       ||>> mk_Frees "f" coiter_fTs
  1887       ||>> mk_Frees "g" coiter_fTs
  1888       ||>> mk_Frees "s" corec_sTs
  1889       ||>> mk_Frees "phi" (map (fn T => T --> T --> HOLogic.boolT) Ts);
  1890 
  1891     fun unf_bind i = Binding.suffix_name ("_" ^ unfN) (nth bs (i - 1));
  1892     val unf_name = Binding.name_of o unf_bind;
  1893     val unf_def_bind = rpair [] o Thm.def_binding o unf_bind;
  1894 
  1895     fun unf_spec i rep str map_FT unfT Jz Jz' =
  1896       let
  1897         val lhs = Free (unf_name i, unfT);
  1898         val rhs = Term.absfree Jz'
  1899           (Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $
  1900             (str $ (rep $ Jz)));
  1901       in
  1902         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  1903       end;
  1904 
  1905     val ((unf_frees, (_, unf_def_frees)), (lthy, lthy_old)) =
  1906       lthy
  1907       |> fold_map7 (fn i => fn rep => fn str => fn map => fn unfT => fn Jz => fn Jz' =>
  1908         Specification.definition
  1909           (SOME (unf_bind i, NONE, NoSyn), (unf_def_bind i, unf_spec i rep str map unfT Jz Jz')))
  1910           ks Rep_Ts str_finals map_FTs unfTs Jzs Jzs'
  1911       |>> apsnd split_list o split_list
  1912       ||> `Local_Theory.restore;
  1913 
  1914     (*transforms defined frees into consts*)
  1915     val phi = Proof_Context.export_morphism lthy_old lthy;
  1916     fun mk_unfs passive =
  1917       map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (deads @ passive)) o
  1918         Morphism.term phi) unf_frees;
  1919     val unfs = mk_unfs passiveAs;
  1920     val unf's = mk_unfs passiveBs;
  1921     val unf_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) unf_def_frees;
  1922 
  1923     val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss;
  1924     val (mor_Rep_thm, mor_Abs_thm) =
  1925       let
  1926         val mor_Rep =
  1927           Skip_Proof.prove lthy [] []
  1928             (HOLogic.mk_Trueprop (mk_mor UNIVs unfs car_finals str_finals Rep_Ts))
  1929             (mk_mor_Rep_tac m (mor_def :: unf_defs) Reps Abs_inverses coalg_final_set_thmss
  1930               map_comp_id_thms map_congL_thms)
  1931           |> Thm.close_derivation;
  1932 
  1933         val mor_Abs =
  1934           Skip_Proof.prove lthy [] []
  1935             (HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs unfs Abs_Ts))
  1936             (mk_mor_Abs_tac (mor_def :: unf_defs) Abs_inverses)
  1937           |> Thm.close_derivation;
  1938       in
  1939         (mor_Rep, mor_Abs)
  1940       end;
  1941 
  1942     val timer = time (timer "unf definitions & thms");
  1943 
  1944     fun coiter_bind i = Binding.suffix_name ("_" ^ coN ^ iterN) (nth bs (i - 1));
  1945     val coiter_name = Binding.name_of o coiter_bind;
  1946     val coiter_def_bind = rpair [] o Thm.def_binding o coiter_bind;
  1947 
  1948     fun coiter_spec i T AT abs f z z' =
  1949       let
  1950         val coiterT = Library.foldr (op -->) (sTs, AT --> T);
  1951 
  1952         val lhs = Term.list_comb (Free (coiter_name i, coiterT), ss);
  1953         val rhs = Term.absfree z' (abs $ (f $ z));
  1954       in
  1955         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  1956       end;
  1957 
  1958     val ((coiter_frees, (_, coiter_def_frees)), (lthy, lthy_old)) =
  1959       lthy
  1960       |> fold_map7 (fn i => fn T => fn AT => fn abs => fn f => fn z => fn z' =>
  1961         Specification.definition
  1962           (SOME (coiter_bind i, NONE, NoSyn), (coiter_def_bind i, coiter_spec i T AT abs f z z')))
  1963           ks Ts activeAs Abs_Ts (map (fn i => HOLogic.mk_comp
  1964             (mk_proj (mk_LSBIS passive_UNIVs i), mk_beh ss i)) ks) zs zs'
  1965       |>> apsnd split_list o split_list
  1966       ||> `Local_Theory.restore;
  1967 
  1968     (*transforms defined frees into consts*)
  1969     val phi = Proof_Context.export_morphism lthy_old lthy;
  1970     val coiters = map (fst o dest_Const o Morphism.term phi) coiter_frees;
  1971     fun mk_coiter Ts ss i = Term.list_comb (Const (nth coiters (i - 1), Library.foldr (op -->)
  1972       (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
  1973     val coiter_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) coiter_def_frees;
  1974 
  1975     val mor_coiter_thm =
  1976       let
  1977         val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses;
  1978         val morEs' = map (fn thm =>
  1979           (thm OF [tcoalg_thm RS mor_final_thm, UNIV_I]) RS sym) morE_thms;
  1980       in
  1981         Skip_Proof.prove lthy [] []
  1982           (fold_rev Logic.all ss
  1983             (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs unfs (map (mk_coiter Ts ss) ks))))
  1984           (K (mk_mor_coiter_tac m mor_UNIV_thm unf_defs coiter_defs Abs_inverses' morEs'
  1985             map_comp_id_thms map_congs))
  1986         |> Thm.close_derivation
  1987       end;
  1988     val coiter_thms = map (fn thm => (thm OF [mor_coiter_thm, UNIV_I]) RS sym) morE_thms;
  1989 
  1990     val (raw_coind_thms, raw_coind_thm) =
  1991       let
  1992         val prem = HOLogic.mk_Trueprop (mk_sbis passive_UNIVs UNIVs unfs TRs);
  1993         val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
  1994           (map2 (fn R => fn T => mk_subset R (Id_const T)) TRs Ts));
  1995         val goal = fold_rev Logic.all TRs (Logic.mk_implies (prem, concl));
  1996       in
  1997         `split_conj_thm (Skip_Proof.prove lthy [] [] goal
  1998           (K (mk_raw_coind_tac bis_def bis_cong_thm bis_O_thm bis_converse_thm bis_Gr_thm
  1999             tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm
  2000             lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects))
  2001           |> Thm.close_derivation)
  2002       end;
  2003 
  2004     val unique_mor_thms =
  2005       let
  2006         val prems = [HOLogic.mk_Trueprop (mk_coalg passive_UNIVs Bs ss), HOLogic.mk_Trueprop
  2007           (HOLogic.mk_conj (mk_mor Bs ss UNIVs unfs coiter_fs,
  2008             mk_mor Bs ss UNIVs unfs coiter_fs_copy))];
  2009         fun mk_fun_eq B f g z = HOLogic.mk_imp
  2010           (HOLogic.mk_mem (z, B), HOLogic.mk_eq (f $ z, g $ z));
  2011         val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
  2012           (map4 mk_fun_eq Bs coiter_fs coiter_fs_copy zs));
  2013 
  2014         val unique_mor = Skip_Proof.prove lthy [] []
  2015           (fold_rev Logic.all (Bs @ ss @ coiter_fs @ coiter_fs_copy @ zs)
  2016             (Logic.list_implies (prems, unique)))
  2017           (K (mk_unique_mor_tac raw_coind_thms bis_image2_thm))
  2018           |> Thm.close_derivation;
  2019       in
  2020         map (fn thm => conjI RSN (2, thm RS mp)) (split_conj_thm unique_mor)
  2021       end;
  2022 
  2023     val (coiter_unique_mor_thms, coiter_unique_mor_thm) =
  2024       let
  2025         val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs unfs coiter_fs);
  2026         fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_coiter Ts ss i);
  2027         val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
  2028           (map2 mk_fun_eq coiter_fs ks));
  2029 
  2030         val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm);
  2031         val mor_thm = mor_comp_thm OF [tcoalg_thm RS mor_final_thm, mor_Abs_thm];
  2032 
  2033         val unique_mor = Skip_Proof.prove lthy [] []
  2034           (fold_rev Logic.all (ss @ coiter_fs) (Logic.mk_implies (prem, unique)))
  2035           (K (mk_coiter_unique_mor_tac raw_coind_thms bis_thm mor_thm coiter_defs))
  2036           |> Thm.close_derivation;
  2037       in
  2038         `split_conj_thm unique_mor
  2039       end;
  2040 
  2041     val (coiter_unique_thms, coiter_unique_thm) = `split_conj_thm (split_conj_prems n
  2042       (mor_UNIV_thm RS @{thm ssubst[of _ _ "%x. x"]} RS coiter_unique_mor_thm));
  2043 
  2044     val coiter_unf_thms = map (fn thm => mor_id_thm RS thm RS sym) coiter_unique_mor_thms;
  2045 
  2046     val coiter_o_unf_thms =
  2047       let
  2048         val mor = mor_comp_thm OF [mor_str_thm, mor_coiter_thm];
  2049       in
  2050         map2 (fn unique => fn coiter_fld =>
  2051           trans OF [mor RS unique, coiter_fld]) coiter_unique_mor_thms coiter_unf_thms
  2052       end;
  2053 
  2054     val timer = time (timer "coiter definitions & thms");
  2055 
  2056     val map_unfs = map2 (fn Ds => fn bnf =>
  2057       Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,
  2058         map HOLogic.id_const passiveAs @ unfs)) Dss bnfs;
  2059 
  2060     fun fld_bind i = Binding.suffix_name ("_" ^ fldN) (nth bs (i - 1));
  2061     val fld_name = Binding.name_of o fld_bind;
  2062     val fld_def_bind = rpair [] o Thm.def_binding o fld_bind;
  2063 
  2064     fun fld_spec i fldT =
  2065       let
  2066         val lhs = Free (fld_name i, fldT);
  2067         val rhs = mk_coiter Ts map_unfs i;
  2068       in
  2069         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  2070       end;
  2071 
  2072     val ((fld_frees, (_, fld_def_frees)), (lthy, lthy_old)) =
  2073         lthy
  2074         |> fold_map2 (fn i => fn fldT =>
  2075           Specification.definition
  2076             (SOME (fld_bind i, NONE, NoSyn), (fld_def_bind i, fld_spec i fldT))) ks fldTs
  2077         |>> apsnd split_list o split_list
  2078         ||> `Local_Theory.restore;
  2079 
  2080     (*transforms defined frees into consts*)
  2081     val phi = Proof_Context.export_morphism lthy_old lthy;
  2082     fun mk_flds params =
  2083       map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
  2084         fld_frees;
  2085     val flds = mk_flds params';
  2086     val fld_defs = map (Morphism.thm phi) fld_def_frees;
  2087 
  2088     val fld_o_unf_thms = map2 (Local_Defs.fold lthy o single) fld_defs coiter_o_unf_thms;
  2089 
  2090     val unf_o_fld_thms =
  2091       let
  2092         fun mk_goal unf fld FT =
  2093           HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (unf, fld), HOLogic.id_const FT));
  2094         val goals = map3 mk_goal unfs flds FTs;
  2095       in
  2096         map5 (fn goal => fn fld_def => fn coiter => fn map_comp_id => fn map_congL =>
  2097           Skip_Proof.prove lthy [] [] goal
  2098             (mk_unf_o_fld_tac fld_def coiter map_comp_id map_congL coiter_o_unf_thms)
  2099           |> Thm.close_derivation)
  2100           goals fld_defs coiter_thms map_comp_id_thms map_congL_thms
  2101       end;
  2102 
  2103     val unf_fld_thms = map (fn thm => thm RS @{thm pointfree_idE}) unf_o_fld_thms;
  2104     val fld_unf_thms = map (fn thm => thm RS @{thm pointfree_idE}) fld_o_unf_thms;
  2105 
  2106     val bij_unf_thms =
  2107       map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) fld_o_unf_thms unf_o_fld_thms;
  2108     val inj_unf_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_unf_thms;
  2109     val surj_unf_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_unf_thms;
  2110     val unf_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_unf_thms;
  2111     val unf_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_unf_thms;
  2112     val unf_exhaust_thms = map (fn thm => thm RS exE) unf_nchotomy_thms;
  2113 
  2114     val bij_fld_thms =
  2115       map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) unf_o_fld_thms fld_o_unf_thms;
  2116     val inj_fld_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_fld_thms;
  2117     val surj_fld_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_fld_thms;
  2118     val fld_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_fld_thms;
  2119     val fld_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_fld_thms;
  2120     val fld_exhaust_thms = map (fn thm => thm RS exE) fld_nchotomy_thms;
  2121 
  2122     val fld_coiter_thms = map3 (fn unf_inject => fn coiter => fn unf_fld =>
  2123       iffD1 OF [unf_inject, trans  OF [coiter, unf_fld RS sym]])
  2124       unf_inject_thms coiter_thms unf_fld_thms;
  2125 
  2126     val timer = time (timer "fld definitions & thms");
  2127 
  2128     val corec_Inl_sum_thms =
  2129       let
  2130         val mor = mor_comp_thm OF [mor_sum_case_thm, mor_coiter_thm];
  2131       in
  2132         map2 (fn unique => fn coiter_unf =>
  2133           trans OF [mor RS unique, coiter_unf]) coiter_unique_mor_thms coiter_unf_thms
  2134       end;
  2135 
  2136     fun corec_bind i = Binding.suffix_name ("_" ^ coN ^ recN) (nth bs (i - 1));
  2137     val corec_name = Binding.name_of o corec_bind;
  2138     val corec_def_bind = rpair [] o Thm.def_binding o corec_bind;
  2139 
  2140     fun corec_spec i T AT =
  2141       let
  2142         val corecT = Library.foldr (op -->) (corec_sTs, AT --> T);
  2143         val maps = map3 (fn unf => fn sum_s => fn map => mk_sum_case
  2144             (HOLogic.mk_comp (Term.list_comb (map, passive_ids @ corec_Inls), unf)) sum_s)
  2145           unfs corec_ss corec_maps;
  2146 
  2147         val lhs = Term.list_comb (Free (corec_name i, corecT), corec_ss);
  2148         val rhs = HOLogic.mk_comp (mk_coiter Ts maps i, Inr_const T AT);
  2149       in
  2150         HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
  2151       end;
  2152 
  2153     val ((corec_frees, (_, corec_def_frees)), (lthy, lthy_old)) =
  2154         lthy
  2155         |> fold_map3 (fn i => fn T => fn AT =>
  2156           Specification.definition
  2157             (SOME (corec_bind i, NONE, NoSyn), (corec_def_bind i, corec_spec i T AT)))
  2158             ks Ts activeAs
  2159         |>> apsnd split_list o split_list
  2160         ||> `Local_Theory.restore;
  2161 
  2162     (*transforms defined frees into consts*)
  2163     val phi = Proof_Context.export_morphism lthy_old lthy;
  2164     val corecs = map (fst o dest_Const o Morphism.term phi) corec_frees;
  2165     fun mk_corec ss i = Term.list_comb (Const (nth corecs (i - 1), Library.foldr (op -->)
  2166       (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
  2167     val corec_defs = map (Morphism.thm phi) corec_def_frees;
  2168 
  2169     val sum_cases =
  2170       map2 (fn T => fn i => mk_sum_case (HOLogic.id_const T) (mk_corec corec_ss i)) Ts ks;
  2171     val corec_thms =
  2172       let
  2173         fun mk_goal i corec_s corec_map unf z =
  2174           let
  2175             val lhs = unf $ (mk_corec corec_ss i $ z);
  2176             val rhs = Term.list_comb (corec_map, passive_ids @ sum_cases) $ (corec_s $ z);
  2177           in
  2178             fold_rev Logic.all (z :: corec_ss) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
  2179           end;
  2180         val goals = map5 mk_goal ks corec_ss corec_maps_rev unfs zs;
  2181       in
  2182         map3 (fn goal => fn coiter => fn map_cong =>
  2183           Skip_Proof.prove lthy [] [] goal
  2184             (mk_corec_tac m corec_defs coiter map_cong corec_Inl_sum_thms)
  2185           |> Thm.close_derivation)
  2186         goals coiter_thms map_congs
  2187       end;
  2188 
  2189     val timer = time (timer "corec definitions & thms");
  2190 
  2191     val (unf_coinduct_thm, coinduct_params, rel_coinduct_thm, pred_coinduct_thm,
  2192          unf_coinduct_upto_thm, rel_coinduct_upto_thm, pred_coinduct_upto_thm) =
  2193       let
  2194         val zs = Jzs1 @ Jzs2;
  2195         val frees = phis @ zs;
  2196 
  2197         fun mk_Ids Id = if Id then map Id_const passiveAs else map mk_diag passive_UNIVs;
  2198 
  2199         fun mk_phi upto_eq phi z1 z2 = if upto_eq
  2200           then Term.absfree (dest_Free z1) (Term.absfree (dest_Free z2)
  2201             (HOLogic.mk_disj (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2))))
  2202           else phi;
  2203 
  2204         fun phi_rels upto_eq = map4 (fn phi => fn T => fn z1 => fn z2 =>
  2205           HOLogic.Collect_const (HOLogic.mk_prodT (T, T)) $
  2206             HOLogic.mk_split (mk_phi upto_eq phi z1 z2)) phis Ts Jzs1 Jzs2;
  2207 
  2208         val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs;
  2209 
  2210         fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2));
  2211         val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
  2212           (map3 mk_concl phis Jzs1 Jzs2));
  2213 
  2214         fun mk_rel_prem upto_eq phi unf rel Jz Jz_copy =
  2215           let
  2216             val concl = HOLogic.mk_mem (HOLogic.mk_tuple [unf $ Jz, unf $ Jz_copy],
  2217               Term.list_comb (rel, mk_Ids upto_eq @ phi_rels upto_eq));
  2218           in
  2219             HOLogic.mk_Trueprop
  2220               (list_all_free [Jz, Jz_copy] (HOLogic.mk_imp (phi $ Jz $ Jz_copy, concl)))
  2221           end;
  2222 
  2223         val rel_prems = map5 (mk_rel_prem false) phis unfs rels Jzs Jzs_copy;
  2224         val rel_upto_prems = map5 (mk_rel_prem true) phis unfs rels Jzs Jzs_copy;
  2225 
  2226         val rel_coinduct_goal = fold_rev Logic.all frees (Logic.list_implies (rel_prems, concl));
  2227         val coinduct_params = rev (Term.add_tfrees rel_coinduct_goal []);
  2228 
  2229         val rel_coinduct = Local_Defs.unfold lthy @{thms diag_UNIV}
  2230           (Skip_Proof.prove lthy [] [] rel_coinduct_goal
  2231             (K (mk_rel_coinduct_tac ks raw_coind_thm bis_rel_thm))
  2232           |> Thm.close_derivation);
  2233 
  2234         fun mk_unf_prem upto_eq phi unf map_nth sets Jz Jz_copy FJz =
  2235           let
  2236             val xs = [Jz, Jz_copy];
  2237 
  2238             fun mk_map_conjunct nths x =
  2239               HOLogic.mk_eq (Term.list_comb (map_nth, passive_ids @ nths) $ FJz, unf $ x);
  2240 
  2241             fun mk_set_conjunct set phi z1 z2 =
  2242               list_all_free [z1, z2]
  2243                 (HOLogic.mk_imp (HOLogic.mk_mem (HOLogic.mk_prod (z1, z2), set $ FJz),
  2244                   mk_phi upto_eq phi z1 z2 $ z1 $ z2));
  2245 
  2246             val concl = list_exists_free [FJz] (HOLogic.mk_conj
  2247               (Library.foldr1 HOLogic.mk_conj (map2 mk_map_conjunct [fstsTs, sndsTs] xs),
  2248               Library.foldr1 HOLogic.mk_conj
  2249                 (map4 mk_set_conjunct (drop m sets) phis Jzs1 Jzs2)));
  2250           in
  2251             fold_rev Logic.all xs (Logic.mk_implies
  2252               (HOLogic.mk_Trueprop (Term.list_comb (phi, xs)), HOLogic.mk_Trueprop concl))
  2253           end;
  2254 
  2255         fun mk_unf_prems upto_eq =
  2256           map7 (mk_unf_prem upto_eq) phis unfs map_FT_nths prodFT_setss Jzs Jzs_copy FJzs
  2257 
  2258         val unf_prems = mk_unf_prems false;
  2259         val unf_upto_prems = mk_unf_prems true;
  2260 
  2261         val unf_coinduct_goal = fold_rev Logic.all frees (Logic.list_implies (unf_prems, concl));
  2262         val unf_coinduct = Skip_Proof.prove lthy [] [] unf_coinduct_goal
  2263           (K (mk_unf_coinduct_tac m ks raw_coind_thm bis_def))
  2264           |> Thm.close_derivation;
  2265 
  2266         val cTs = map (SOME o certifyT lthy o TFree) coinduct_params;
  2267         val cts = map3 (SOME o certify lthy ooo mk_phi true) phis Jzs1 Jzs2;
  2268 
  2269         val rel_coinduct_upto = singleton (Proof_Context.export names_lthy lthy)
  2270           (Skip_Proof.prove lthy [] []
  2271             (fold_rev Logic.all zs (Logic.list_implies (rel_upto_prems, concl)))
  2272             (K (mk_rel_coinduct_upto_tac m cTs cts rel_coinduct rel_monos rel_Ids)))
  2273           |> Thm.close_derivation;
  2274 
  2275         val unf_coinduct_upto = singleton (Proof_Context.export names_lthy lthy)
  2276           (Skip_Proof.prove lthy [] []
  2277             (fold_rev Logic.all zs (Logic.list_implies (unf_upto_prems, concl)))
  2278             (K (mk_unf_coinduct_upto_tac ks cTs cts unf_coinduct bis_def
  2279               (tcoalg_thm RS bis_diag_thm))))
  2280           |> Thm.close_derivation;
  2281 
  2282         val pred_coinduct = rel_coinduct
  2283           |> Local_Defs.unfold lthy @{thms Id_def'}
  2284           |> Local_Defs.fold lthy pred_defs;
  2285         val pred_coinduct_upto = rel_coinduct_upto
  2286           |> Local_Defs.unfold lthy @{thms Id_def'}
  2287           |> Local_Defs.fold lthy pred_defs;
  2288       in
  2289         (unf_coinduct, rev (Term.add_tfrees unf_coinduct_goal []), rel_coinduct, pred_coinduct,
  2290          unf_coinduct_upto, rel_coinduct_upto, pred_coinduct_upto)
  2291       end;
  2292 
  2293     val timer = time (timer "coinduction");
  2294 
  2295     (*register new codatatypes as BNFs*)
  2296     val lthy = if m = 0 then lthy else
  2297       let
  2298         val fTs = map2 (curry op -->) passiveAs passiveBs;
  2299         val gTs = map2 (curry op -->) passiveBs passiveCs;
  2300         val f1Ts = map2 (curry op -->) passiveAs passiveYs;
  2301         val f2Ts = map2 (curry op -->) passiveBs passiveYs;
  2302         val p1Ts = map2 (curry op -->) passiveXs passiveAs;
  2303         val p2Ts = map2 (curry op -->) passiveXs passiveBs;
  2304         val pTs = map2 (curry op -->) passiveXs passiveCs;
  2305         val uTs = map2 (curry op -->) Ts Ts';
  2306         val JRTs = map2 (curry mk_relT) passiveAs passiveBs;
  2307         val JphiTs = map2 (fn T => fn U => T --> U --> HOLogic.boolT) passiveAs passiveBs;
  2308         val prodTs = map2 (curry HOLogic.mk_prodT) Ts Ts';
  2309         val B1Ts = map HOLogic.mk_setT passiveAs;
  2310         val B2Ts = map HOLogic.mk_setT passiveBs;
  2311         val AXTs = map HOLogic.mk_setT passiveXs;
  2312         val XTs = mk_Ts passiveXs;
  2313         val YTs = mk_Ts passiveYs;
  2314 
  2315         val (((((((((((((((((((((fs, fs'), (fs_copy, fs'_copy)), (gs, gs')), us),
  2316           (Jys, Jys')), (Jys_copy, Jys'_copy)), set_induct_phiss), JRs), Jphis),
  2317           B1s), B2s), AXs), Xs), f1s), f2s), p1s), p2s), ps), (ys, ys')), (ys_copy, ys'_copy)),
  2318           names_lthy) = names_lthy
  2319           |> mk_Frees' "f" fTs
  2320           ||>> mk_Frees' "f" fTs
  2321           ||>> mk_Frees' "g" gTs
  2322           ||>> mk_Frees "u" uTs
  2323           ||>> mk_Frees' "b" Ts'
  2324           ||>> mk_Frees' "b" Ts'
  2325           ||>> mk_Freess "phi" (map (fn T => map (fn U => T --> U --> HOLogic.boolT) Ts) passiveAs)
  2326           ||>> mk_Frees "R" JRTs
  2327           ||>> mk_Frees "phi" JphiTs
  2328           ||>> mk_Frees "B1" B1Ts
  2329           ||>> mk_Frees "B2" B2Ts
  2330           ||>> mk_Frees "A" AXTs
  2331           ||>> mk_Frees "x" XTs
  2332           ||>> mk_Frees "f1" f1Ts
  2333           ||>> mk_Frees "f2" f2Ts
  2334           ||>> mk_Frees "p1" p1Ts
  2335           ||>> mk_Frees "p2" p2Ts
  2336           ||>> mk_Frees "p" pTs
  2337           ||>> mk_Frees' "y" passiveAs
  2338           ||>> mk_Frees' "y" passiveAs;
  2339 
  2340         val map_FTFT's = map2 (fn Ds =>
  2341           mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
  2342 
  2343         fun mk_maps ATs BTs Ts mk_T =
  2344           map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ map mk_T Ts)) Dss bnfs;
  2345         fun mk_Fmap mk_const fs Ts Fmap = Term.list_comb (Fmap, fs @ map mk_const Ts);
  2346         fun mk_map mk_const mk_T Ts fs Ts' unfs mk_maps =
  2347           mk_coiter Ts' (map2 (fn unf => fn Fmap =>
  2348             HOLogic.mk_comp (mk_Fmap mk_const fs Ts Fmap, unf)) unfs (mk_maps Ts mk_T));
  2349         val mk_map_id = mk_map HOLogic.id_const I;
  2350         val mk_mapsAB = mk_maps passiveAs passiveBs;
  2351         val mk_mapsBC = mk_maps passiveBs passiveCs;
  2352         val mk_mapsAC = mk_maps passiveAs passiveCs;
  2353         val mk_mapsAY = mk_maps passiveAs passiveYs;
  2354         val mk_mapsBY = mk_maps passiveBs passiveYs;
  2355         val mk_mapsXA = mk_maps passiveXs passiveAs;
  2356         val mk_mapsXB = mk_maps passiveXs passiveBs;
  2357         val mk_mapsXC = mk_maps passiveXs passiveCs;
  2358         val fs_maps = map (mk_map_id Ts fs Ts' unfs mk_mapsAB) ks;
  2359         val fs_copy_maps = map (mk_map_id Ts fs_copy Ts' unfs mk_mapsAB) ks;
  2360         val gs_maps = map (mk_map_id Ts' gs Ts'' unf's mk_mapsBC) ks;
  2361         val fgs_maps =
  2362           map (mk_map_id Ts (map2 (curry HOLogic.mk_comp) gs fs) Ts'' unfs mk_mapsAC) ks;
  2363         val Xunfs = mk_unfs passiveXs;
  2364         val UNIV's = map HOLogic.mk_UNIV Ts';
  2365         val CUNIVs = map HOLogic.mk_UNIV passiveCs;
  2366         val UNIV''s = map HOLogic.mk_UNIV Ts'';
  2367         val fstsTsTs' = map fst_const prodTs;
  2368         val sndsTsTs' = map snd_const prodTs;
  2369         val unf''s = mk_unfs passiveCs;
  2370         val f1s_maps = map (mk_map_id Ts f1s YTs unfs mk_mapsAY) ks;
  2371         val f2s_maps = map (mk_map_id Ts' f2s YTs unf's mk_mapsBY) ks;
  2372         val pid_maps = map (mk_map_id XTs ps Ts'' Xunfs mk_mapsXC) ks;
  2373         val pfst_Fmaps =
  2374           map (mk_Fmap fst_const p1s prodTs) (mk_mapsXA prodTs (fst o HOLogic.dest_prodT));
  2375         val psnd_Fmaps =
  2376           map (mk_Fmap snd_const p2s prodTs) (mk_mapsXB prodTs (snd o HOLogic.dest_prodT));
  2377         val p1id_Fmaps = map (mk_Fmap HOLogic.id_const p1s prodTs) (mk_mapsXA prodTs I);
  2378         val p2id_Fmaps = map (mk_Fmap HOLogic.id_const p2s prodTs) (mk_mapsXB prodTs I);
  2379         val pid_Fmaps = map (mk_Fmap HOLogic.id_const ps prodTs) (mk_mapsXC prodTs I);
  2380 
  2381         val (map_simp_thms, map_thms) =
  2382           let
  2383             fun mk_goal fs_map map unf unf' = fold_rev Logic.all fs
  2384               (HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (unf', fs_map),
  2385                 HOLogic.mk_comp (Term.list_comb (map, fs @ fs_maps), unf))));
  2386             val goals = map4 mk_goal fs_maps map_FTFT's unfs unf's;
  2387             val cTs = map (SOME o certifyT lthy) FTs';
  2388             val maps =
  2389               map5 (fn goal => fn cT => fn coiter => fn map_comp' => fn map_cong =>
  2390                 Skip_Proof.prove lthy [] [] goal
  2391                   (K (mk_map_tac m n cT coiter map_comp' map_cong))
  2392                 |> Thm.close_derivation)
  2393               goals cTs coiter_thms map_comp's map_congs;
  2394           in
  2395             map_split (fn thm => (thm RS @{thm pointfreeE}, thm)) maps
  2396           end;
  2397 
  2398         val map_comp_thms =
  2399           let
  2400             val goal = fold_rev Logic.all (fs @ gs)
  2401               (HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
  2402                 (map3 (fn fmap => fn gmap => fn fgmap =>
  2403                    HOLogic.mk_eq (HOLogic.mk_comp (gmap, fmap), fgmap))
  2404                 fs_maps gs_maps fgs_maps)))
  2405           in
  2406             split_conj_thm (Skip_Proof.prove lthy [] [] goal
  2407               (K (mk_map_comp_tac m n map_thms map_comps map_congs coiter_unique_thm))
  2408               |> Thm.close_derivation)
  2409           end;
  2410 
  2411         val (map_unique_thms, map_unique_thm) =
  2412           let
  2413             fun mk_prem u map unf unf' =
  2414               HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (unf', u),
  2415                 HOLogic.mk_comp (Term.list_comb (map, fs @ us), unf)));
  2416             val prems = map4 mk_prem us map_FTFT's unfs unf's;
  2417             val goal =
  2418               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
  2419                 (map2 (curry HOLogic.mk_eq) us fs_maps));
  2420             val unique = Skip_Proof.prove lthy [] []
  2421               (fold_rev Logic.all (us @ fs) (Logic.list_implies (prems, goal)))
  2422               (mk_map_unique_tac coiter_unique_thm map_comps)
  2423               |> Thm.close_derivation;
  2424           in
  2425             `split_conj_thm unique
  2426           end;
  2427 
  2428         val timer = time (timer "map functions for the new codatatypes");
  2429 
  2430         val bd = mk_ccexp sbd sbd;
  2431 
  2432         val timer = time (timer "bounds for the new codatatypes");
  2433 
  2434         fun mk_set_Ts T = passiveAs @ replicate n (HOLogic.mk_setT T);
  2435         val setsss = map (mk_setss o mk_set_Ts) passiveAs;
  2436         val map_setss = map (fn T => map2 (fn Ds =>
  2437           mk_map_of_bnf Ds (passiveAs @ Ts) (mk_set_Ts T)) Dss bnfs) passiveAs;
  2438 
  2439         val setss_by_bnf = map (fn i => map2 (mk_hset unfs i) ls passiveAs) ks;
  2440         val setss_by_bnf' = map (fn i => map2 (mk_hset unf's i) ls passiveBs) ks;
  2441         val setss_by_range = transpose setss_by_bnf;
  2442 
  2443         val set_simp_thmss =
  2444           let
  2445             fun mk_simp_goal relate pas_set act_sets sets unf z set =
  2446               relate (set $ z, mk_union (pas_set $ (unf $ z),
  2447                  Library.foldl1 mk_union
  2448                    (map2 (fn X => mk_UNION (X $ (unf $ z))) act_sets sets)));
  2449             fun mk_goals eq =
  2450               map2 (fn i => fn sets =>
  2451                 map4 (fn Fsets =>
  2452                   mk_simp_goal eq (nth Fsets (i - 1)) (drop m Fsets) sets)
  2453                 FTs_setss unfs Jzs sets)
  2454               ls setss_by_range;
  2455 
  2456             val le_goals = map
  2457               (fold_rev Logic.all Jzs o HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj)
  2458               (mk_goals (uncurry mk_subset));
  2459             val set_le_thmss = map split_conj_thm
  2460               (map4 (fn goal => fn hset_minimal => fn set_hsets => fn set_hset_hsetss =>
  2461                 Skip_Proof.prove lthy [] [] goal
  2462                   (K (mk_set_le_tac n hset_minimal set_hsets set_hset_hsetss))
  2463                 |> Thm.close_derivation)
  2464               le_goals hset_minimal_thms set_hset_thmss' set_hset_hset_thmsss');
  2465 
  2466             val simp_goalss = map (map2 (fn z => fn goal =>
  2467                 Logic.all z (HOLogic.mk_Trueprop goal)) Jzs)
  2468               (mk_goals HOLogic.mk_eq);
  2469           in
  2470             map4 (map4 (fn goal => fn set_le => fn set_incl_hset => fn set_hset_incl_hsets =>
  2471               Skip_Proof.prove lthy [] [] goal
  2472                 (K (mk_set_simp_tac n set_le set_incl_hset set_hset_incl_hsets))
  2473               |> Thm.close_derivation))
  2474             simp_goalss set_le_thmss set_incl_hset_thmss' set_hset_incl_hset_thmsss'
  2475           end;
  2476 
  2477         val timer = time (timer "set functions for the new codatatypes");
  2478 
  2479         val colss = map2 (fn j => fn T =>
  2480           map (fn i => mk_hset_rec unfs nat i j T) ks) ls passiveAs;
  2481         val colss' = map2 (fn j => fn T =>
  2482           map (fn i => mk_hset_rec unf's nat i j T) ks) ls passiveBs;
  2483         val Xcolss = map2 (fn j => fn T =>
  2484           map (fn i => mk_hset_rec Xunfs nat i j T) ks) ls passiveXs;
  2485 
  2486         val col_natural_thmss =
  2487           let
  2488             fun mk_col_natural f map z col col' =
  2489               HOLogic.mk_eq (mk_image f $ (col $ z), col' $ (map $ z));
  2490 
  2491             fun mk_goal f cols cols' = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj
  2492               (map4 (mk_col_natural f) fs_maps Jzs cols cols'));
  2493 
  2494             val goals = map3 mk_goal fs colss colss';
  2495 
  2496             val ctss =
  2497               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) goals;
  2498 
  2499             val thms =
  2500               map4 (fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
  2501                 singleton (Proof_Context.export names_lthy lthy)
  2502                   (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
  2503                     (mk_col_natural_tac cts rec_0s rec_Sucs map_simp_thms set_natural'ss))
  2504                 |> Thm.close_derivation)
  2505               goals ctss hset_rec_0ss' hset_rec_Sucss';
  2506           in
  2507             map (split_conj_thm o mk_specN n) thms
  2508           end;
  2509 
  2510         val col_bd_thmss =
  2511           let
  2512             fun mk_col_bd z col = mk_ordLeq (mk_card_of (col $ z)) sbd;
  2513 
  2514             fun mk_goal cols = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj
  2515               (map2 mk_col_bd Jzs cols));
  2516 
  2517             val goals = map mk_goal colss;
  2518 
  2519             val ctss =
  2520               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) goals;
  2521 
  2522             val thms =
  2523               map5 (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
  2524                 singleton (Proof_Context.export names_lthy lthy)
  2525                   (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
  2526                     (K (mk_col_bd_tac m j cts rec_0s rec_Sucs
  2527                       sbd_Card_order sbd_Cinfinite set_sbdss)))
  2528                 |> Thm.close_derivation)
  2529               ls goals ctss hset_rec_0ss' hset_rec_Sucss';
  2530           in
  2531             map (split_conj_thm o mk_specN n) thms
  2532           end;
  2533 
  2534         val map_cong_thms =
  2535           let
  2536             val cTs = map (SOME o certifyT lthy o
  2537               Term.typ_subst_atomic (passiveAs ~~ passiveBs) o TFree) coinduct_params;
  2538 
  2539             fun mk_prem z set f g y y' =
  2540               mk_Ball (set $ z) (Term.absfree y' (HOLogic.mk_eq (f $ y, g $ y)));
  2541 
  2542             fun mk_prems sets z =
  2543               Library.foldr1 HOLogic.mk_conj (map5 (mk_prem z) sets fs fs_copy ys ys')
  2544 
  2545             fun mk_map_cong sets z fmap gmap =
  2546               HOLogic.mk_imp (mk_prems sets z, HOLogic.mk_eq (fmap $ z, gmap $ z));
  2547 
  2548             fun mk_coind_body sets (x, T) z fmap gmap y y_copy =
  2549               HOLogic.mk_conj
  2550                 (HOLogic.mk_mem (z, HOLogic.mk_Collect (x, T, mk_prems sets z)),
  2551                   HOLogic.mk_conj (HOLogic.mk_eq (y, fmap $ z),
  2552                     HOLogic.mk_eq (y_copy, gmap $ z)))
  2553 
  2554             fun mk_cphi sets (z' as (x, T)) z fmap gmap y' y y'_copy y_copy =
  2555               HOLogic.mk_exists (x, T, mk_coind_body sets z' z fmap gmap y y_copy)
  2556               |> Term.absfree y'_copy
  2557               |> Term.absfree y'
  2558               |> certify lthy;
  2559 
  2560             val cphis =
  2561               map9 mk_cphi setss_by_bnf Jzs' Jzs fs_maps fs_copy_maps Jys' Jys Jys'_copy Jys_copy;
  2562 
  2563             val coinduct = Drule.instantiate' cTs (map SOME cphis) unf_coinduct_thm;
  2564 
  2565             val goal =
  2566               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
  2567                 (map4 mk_map_cong setss_by_bnf Jzs fs_maps fs_copy_maps));
  2568 
  2569             val thm = singleton (Proof_Context.export names_lthy lthy)
  2570               (Skip_Proof.prove lthy [] [] goal
  2571               (K (mk_mcong_tac m (rtac coinduct) map_comp's map_simp_thms map_congs set_natural'ss
  2572               set_hset_thmss set_hset_hset_thmsss)))
  2573               |> Thm.close_derivation
  2574           in
  2575             split_conj_thm thm
  2576           end;
  2577 
  2578         val B1_ins = map2 (mk_in B1s) setss_by_bnf Ts;
  2579         val B2_ins = map2 (mk_in B2s) setss_by_bnf' Ts';
  2580         val thePulls = map4 mk_thePull B1_ins B2_ins f1s_maps f2s_maps;
  2581         val thePullTs = passiveXs @ map2 (curry HOLogic.mk_prodT) Ts Ts';
  2582         val thePull_ins = map2 (mk_in (AXs @ thePulls)) (mk_setss thePullTs) (mk_FTs thePullTs);
  2583         val pickFs = map5 mk_pickWP thePull_ins pfst_Fmaps psnd_Fmaps
  2584           (map2 (curry (op $)) unfs Jzs) (map2 (curry (op $)) unf's Jz's);
  2585         val pickF_ss = map3 (fn pickF => fn z => fn z' =>
  2586           HOLogic.mk_split (Term.absfree z (Term.absfree z' pickF))) pickFs Jzs' Jz's';
  2587         val picks = map (mk_coiter XTs pickF_ss) ks;
  2588 
  2589         val wpull_prem = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
  2590           (map8 mk_wpull AXs B1s B2s f1s f2s (replicate m NONE) p1s p2s));
  2591 
  2592         val map_eq_thms = map2 (fn simp => fn diff => box_equals OF [diff RS iffD2, simp, simp])
  2593           map_simp_thms unf_inject_thms;
  2594         val map_wpull_thms = map (fn thm => thm OF
  2595           (replicate m asm_rl @ replicate n @{thm wpull_thePull})) map_wpulls;
  2596         val pickWP_assms_tacs =
  2597           map3 mk_pickWP_assms_tac set_incl_hset_thmss set_incl_hin_thmss map_eq_thms;
  2598 
  2599         val coalg_thePull_thm =
  2600           let
  2601             val coalg = HOLogic.mk_Trueprop
  2602               (mk_coalg CUNIVs thePulls (map2 (curry HOLogic.mk_comp) pid_Fmaps pickF_ss));
  2603             val goal = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s @ ps)
  2604               (Logic.mk_implies (wpull_prem, coalg));
  2605           in
  2606             Skip_Proof.prove lthy [] [] goal (mk_coalg_thePull_tac m coalg_def map_wpull_thms
  2607               set_natural'ss pickWP_assms_tacs)
  2608             |> Thm.close_derivation
  2609           end;
  2610 
  2611         val (mor_thePull_fst_thm, mor_thePull_snd_thm, mor_thePull_pick_thm) =
  2612           let
  2613             val mor_fst = HOLogic.mk_Trueprop
  2614               (mk_mor thePulls (map2 (curry HOLogic.mk_comp) p1id_Fmaps pickF_ss)
  2615                 UNIVs unfs fstsTsTs');
  2616             val mor_snd = HOLogic.mk_Trueprop
  2617               (mk_mor thePulls (map2 (curry HOLogic.mk_comp) p2id_Fmaps pickF_ss)
  2618                 UNIV's unf's sndsTsTs');
  2619             val mor_pick = HOLogic.mk_Trueprop
  2620               (mk_mor thePulls (map2 (curry HOLogic.mk_comp) pid_Fmaps pickF_ss)
  2621                 UNIV''s unf''s (map2 (curry HOLogic.mk_comp) pid_maps picks));
  2622 
  2623             val goal_fst = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s)
  2624               (Logic.mk_implies (wpull_prem, mor_fst));
  2625             val goal_snd = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s)
  2626               (Logic.mk_implies (wpull_prem, mor_snd));
  2627             val goal_pick = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s @ ps)
  2628               (Logic.mk_implies (wpull_prem, mor_pick));
  2629           in
  2630             (Skip_Proof.prove lthy [] [] goal_fst (mk_mor_thePull_fst_tac m mor_def map_wpull_thms
  2631               map_comp's pickWP_assms_tacs) |> Thm.close_derivation,
  2632             Skip_Proof.prove lthy [] [] goal_snd (mk_mor_thePull_snd_tac m mor_def map_wpull_thms
  2633               map_comp's pickWP_assms_tacs) |> Thm.close_derivation,
  2634             Skip_Proof.prove lthy [] [] goal_pick (mk_mor_thePull_pick_tac mor_def coiter_thms
  2635               map_comp's) |> Thm.close_derivation)
  2636           end;
  2637 
  2638         val pick_col_thmss =
  2639           let
  2640             fun mk_conjunct AX Jpair pick thePull col =
  2641               HOLogic.mk_imp (HOLogic.mk_mem (Jpair, thePull), mk_subset (col $ (pick $ Jpair)) AX);
  2642 
  2643             fun mk_concl AX cols =
  2644               list_all_free Jpairs (Library.foldr1 HOLogic.mk_conj
  2645                 (map4 (mk_conjunct AX) Jpairs picks thePulls cols));
  2646 
  2647             val concls = map2 mk_concl AXs Xcolss;
  2648 
  2649             val ctss =
  2650               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
  2651 
  2652             val goals =
  2653               map (fn concl => Logic.mk_implies (wpull_prem, HOLogic.mk_Trueprop concl)) concls;
  2654 
  2655             val thms =
  2656               map5 (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
  2657                 singleton (Proof_Context.export names_lthy lthy) (Skip_Proof.prove lthy [] [] goal
  2658                   (mk_pick_col_tac m j cts rec_0s rec_Sucs coiter_thms set_natural'ss map_wpull_thms
  2659                     pickWP_assms_tacs))
  2660                 |> Thm.close_derivation)
  2661               ls goals ctss hset_rec_0ss' hset_rec_Sucss';
  2662           in
  2663             map (map (fn thm => thm RS mp) o split_conj_thm o mk_specN n) thms
  2664           end;
  2665 
  2666         val timer = time (timer "helpers for BNF properties");
  2667 
  2668         val map_id_tacs = map2 (K oo mk_map_id_tac map_thms) coiter_unique_thms coiter_unf_thms;
  2669         val map_comp_tacs = map (fn thm => K (rtac (thm RS sym) 1)) map_comp_thms;
  2670         val map_cong_tacs = map (mk_map_cong_tac m) map_cong_thms;
  2671         val set_nat_tacss =
  2672           map2 (map2 (K oo mk_set_natural_tac)) hset_defss (transpose col_natural_thmss);
  2673 
  2674         val bd_co_tacs = replicate n (K (mk_bd_card_order_tac sbd_card_order));
  2675         val bd_cinf_tacs = replicate n (K (mk_bd_cinfinite_tac sbd_Cinfinite));
  2676 
  2677         val set_bd_tacss =
  2678           map2 (map2 (K oo mk_set_bd_tac sbd_Cinfinite)) hset_defss (transpose col_bd_thmss);
  2679 
  2680         val in_bd_tacs = map7 (fn i => fn isNode_hsets => fn carT_def =>
  2681             fn card_of_carT => fn mor_image => fn Rep_inverse => fn mor_hsets =>
  2682           K (mk_in_bd_tac (nth isNode_hsets (i - 1)) isNode_hsets carT_def
  2683             card_of_carT mor_image Rep_inverse mor_hsets
  2684             sbd_Cnotzero sbd_Card_order mor_Rep_thm coalgT_thm mor_T_final_thm tcoalg_thm))
  2685           ks isNode_hset_thmss carT_defs card_of_carT_thms
  2686           mor_image'_thms Rep_inverses (transpose mor_hset_thmss);
  2687 
  2688         val map_wpull_tacs =
  2689           map3 (K ooo mk_wpull_tac m coalg_thePull_thm mor_thePull_fst_thm mor_thePull_snd_thm
  2690             mor_thePull_pick_thm) unique_mor_thms (transpose pick_col_thmss) hset_defss;
  2691 
  2692         val tacss = map9 mk_tactics map_id_tacs map_comp_tacs map_cong_tacs set_nat_tacss bd_co_tacs
  2693           bd_cinf_tacs set_bd_tacss in_bd_tacs map_wpull_tacs;
  2694 
  2695         val (hset_unf_incl_thmss, hset_hset_unf_incl_thmsss, hset_induct_thms) =
  2696           let
  2697             fun tinst_of unf =
  2698               map (SOME o certify lthy) (unf :: remove (op =) unf unfs);
  2699             fun tinst_of' unf = case tinst_of unf of t :: ts => t :: NONE :: ts;
  2700             val Tinst = map (pairself (certifyT lthy))
  2701               (map Logic.varifyT_global (deads @ allAs) ~~ (deads @ passiveAs @ Ts));
  2702             val set_incl_thmss =
  2703               map2 (fn unf => map (singleton (Proof_Context.export names_lthy lthy) o
  2704                 Drule.instantiate' [] (tinst_of' unf) o
  2705                 Thm.instantiate (Tinst, []) o Drule.zero_var_indexes))
  2706               unfs set_incl_hset_thmss;
  2707 
  2708             val tinst = interleave (map (SOME o certify lthy) unfs) (replicate n NONE)
  2709             val set_minimal_thms =
  2710               map (Drule.instantiate' [] tinst o Thm.instantiate (Tinst, []) o
  2711                 Drule.zero_var_indexes)
  2712               hset_minimal_thms;
  2713 
  2714             val set_set_incl_thmsss =
  2715               map2 (fn unf => map (map (singleton (Proof_Context.export names_lthy lthy) o
  2716                 Drule.instantiate' [] (NONE :: tinst_of' unf) o
  2717                 Thm.instantiate (Tinst, []) o Drule.zero_var_indexes)))
  2718               unfs set_hset_incl_hset_thmsss;
  2719 
  2720             val set_set_incl_thmsss' = transpose (map transpose set_set_incl_thmsss);
  2721 
  2722             val incls =
  2723               maps (map (fn thm => thm RS @{thm subset_Collect_iff})) set_incl_thmss @
  2724                 @{thms subset_Collect_iff[OF subset_refl]};
  2725 
  2726             fun mk_induct_tinst phis jsets y y' =
  2727               map4 (fn phi => fn jset => fn Jz => fn Jz' =>
  2728                 SOME (certify lthy (Term.absfree Jz' (HOLogic.mk_Collect (fst y', snd y',
  2729                   HOLogic.mk_conj (HOLogic.mk_mem (y, jset $ Jz), phi $ y $ Jz))))))
  2730               phis jsets Jzs Jzs';
  2731             val set_induct_thms =
  2732               map6 (fn set_minimal => fn set_set_inclss => fn jsets => fn y => fn y' => fn phis =>
  2733                 ((set_minimal
  2734                   |> Drule.instantiate' [] (mk_induct_tinst phis jsets y y')
  2735                   |> Local_Defs.unfold lthy incls) OF
  2736                   (replicate n ballI @
  2737                     maps (map (fn thm => thm RS @{thm subset_CollectI})) set_set_inclss))
  2738                 |> singleton (Proof_Context.export names_lthy lthy)
  2739                 |> rule_by_tactic lthy (ALLGOALS (TRY o etac asm_rl)))
  2740               set_minimal_thms set_set_incl_thmsss' setss_by_range ys ys' set_induct_phiss
  2741           in
  2742             (set_incl_thmss, set_set_incl_thmsss, set_induct_thms)
  2743           end;
  2744 
  2745         fun close_wit I wit = (I, fold_rev Term.absfree (map (nth ys') I) wit);
  2746 
  2747         val all_unitTs = replicate live HOLogic.unitT;
  2748         val unitTs = replicate n HOLogic.unitT;
  2749         val unit_funs = replicate n (Term.absdummy HOLogic.unitT HOLogic.unit);
  2750         fun mk_map_args I =
  2751           map (fn i =>
  2752             if member (op =) I i then Term.absdummy HOLogic.unitT (nth ys i)
  2753             else mk_undefined (HOLogic.unitT --> nth passiveAs i))
  2754           (0 upto (m - 1));
  2755 
  2756         fun mk_nat_wit Ds bnf (I, wit) () =
  2757           let
  2758             val passiveI = filter (fn i => i < m) I;
  2759             val map_args = mk_map_args passiveI;
  2760           in
  2761             Term.absdummy HOLogic.unitT (Term.list_comb
  2762               (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $ wit)
  2763           end;
  2764 
  2765         fun mk_dummy_wit Ds bnf I =
  2766           let
  2767             val map_args = mk_map_args I;
  2768           in
  2769             Term.absdummy HOLogic.unitT (Term.list_comb
  2770               (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $
  2771               mk_undefined (mk_T_of_bnf Ds all_unitTs bnf))
  2772           end;
  2773 
  2774         val nat_witss =
  2775           map3 (fn i => fn Ds => fn bnf => mk_wits_of_bnf (replicate (nwits_of_bnf bnf) Ds)
  2776             (replicate (nwits_of_bnf bnf) (replicate live HOLogic.unitT)) bnf
  2777             |> map (fn (I, wit) =>
  2778               (I, Lazy.lazy (mk_nat_wit Ds bnf (I, Term.list_comb (wit, map (K HOLogic.unit) I))))))
  2779           ks Dss bnfs;
  2780 
  2781         val nat_wit_thmss = map2 (curry op ~~) nat_witss (map wit_thmss_of_bnf bnfs)
  2782 
  2783         val Iss = map (map fst) nat_witss;
  2784 
  2785         fun filter_wits (I, wit) =
  2786           let val J = filter (fn i => i < m) I;
  2787           in (J, (length J < length I, wit)) end;
  2788 
  2789         val wit_treess = map_index (fn (i, Is) =>
  2790           map_index (finish Iss m [i+m] (i+m)) Is) Iss
  2791           |> map (minimize_wits o map filter_wits o minimize_wits o flat);
  2792 
  2793         val coind_wit_argsss =
  2794           map (map (tree_to_coind_wits nat_wit_thmss o snd o snd) o filter (fst o snd)) wit_treess;
  2795 
  2796         val nonredundant_coind_wit_argsss =
  2797           fold (fn i => fn argsss =>
  2798             nth_map (i - 1) (filter_out (fn xs =>
  2799               exists (fn ys =>
  2800                 let
  2801                   val xs' = (map (fst o fst) xs, snd (fst (hd xs)));
  2802                   val ys' = (map (fst o fst) ys, snd (fst (hd ys)));
  2803                 in
  2804                   eq_pair (subset (op =)) (eq_set (op =)) (xs', ys') andalso not (fst xs' = fst ys')
  2805                 end)
  2806               (flat argsss)))
  2807             argsss)
  2808           ks coind_wit_argsss;
  2809 
  2810         fun prepare_args args =
  2811           let
  2812             val I = snd (fst (hd args));
  2813             val (dummys, args') =
  2814               map_split (fn i =>
  2815                 (case find_first (fn arg => fst (fst arg) = i - 1) args of
  2816                   SOME (_, ((_, wit), thms)) => (NONE, (Lazy.force wit, thms))
  2817                 | NONE =>
  2818                   (SOME (i - 1), (mk_dummy_wit (nth Dss (i - 1)) (nth bnfs (i - 1)) I, []))))
  2819               ks;
  2820           in
  2821             ((I, dummys), apsnd flat (split_list args'))
  2822           end;
  2823 
  2824         fun mk_coind_wits ((I, dummys), (args, thms)) =
  2825           ((I, dummys), (map (fn i => mk_coiter Ts args i $ HOLogic.unit) ks, thms));
  2826 
  2827         val coind_witss =
  2828           maps (map (mk_coind_wits o prepare_args)) nonredundant_coind_wit_argsss;
  2829 
  2830         fun mk_coind_wit_thms ((I, dummys), (wits, wit_thms)) =
  2831           let
  2832             fun mk_goal sets y y_copy y'_copy j =
  2833               let
  2834                 fun mk_conjunct set z dummy wit =
  2835                   mk_Ball (set $ z) (Term.absfree y'_copy
  2836                     (if dummy = NONE orelse member (op =) I (j - 1) then
  2837                       HOLogic.mk_imp (HOLogic.mk_eq (z, wit),
  2838                         if member (op =) I (j - 1) then HOLogic.mk_eq (y_copy, y)
  2839                         else @{term False})
  2840                     else @{term True}));
  2841               in
  2842                 fold_rev Logic.all (map (nth ys) I @ Jzs) (HOLogic.mk_Trueprop
  2843                   (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct sets Jzs dummys wits)))
  2844               end;
  2845             val goals = map5 mk_goal setss_by_range ys ys_copy ys'_copy ls;
  2846           in
  2847             map2 (fn goal => fn induct =>
  2848               Skip_Proof.prove lthy [] [] goal
  2849                 (mk_coind_wit_tac induct coiter_thms (flat set_natural'ss) wit_thms)
  2850               |> Thm.close_derivation)
  2851             goals hset_induct_thms
  2852             |> map split_conj_thm
  2853             |> transpose
  2854             |> map (map_filter (try (fn thm => thm RS bspec RS mp)))
  2855             |> curry op ~~ (map_index Library.I (map (close_wit I) wits))
  2856             |> filter (fn (_, thms) => length thms = m)
  2857           end;
  2858 
  2859         val coind_wit_thms = maps mk_coind_wit_thms coind_witss;
  2860 
  2861         val witss = map2 (fn Ds => fn bnf => mk_wits_of_bnf
  2862           (replicate (nwits_of_bnf bnf) Ds)
  2863           (replicate (nwits_of_bnf bnf) (passiveAs @ Ts)) bnf) Dss bnfs;
  2864 
  2865         val fld_witss =
  2866           map (map (uncurry close_wit o tree_to_fld_wit ys flds witss o snd o snd) o
  2867             filter_out (fst o snd)) wit_treess;
  2868 
  2869         val all_witss =
  2870           fold (fn ((i, wit), thms) => fn witss =>
  2871             nth_map i (fn (thms', wits) => (thms @ thms', wit :: wits)) witss)
  2872           coind_wit_thms (map (pair []) fld_witss)
  2873           |> map (apsnd (map snd o minimize_wits));
  2874 
  2875         val wit_tac = mk_wit_tac n unf_fld_thms (flat set_simp_thmss) (maps wit_thms_of_bnf bnfs);
  2876 
  2877         val (Jbnfs, lthy) =
  2878           fold_map6 (fn tacs => fn b => fn map => fn sets => fn T => fn (thms, wits) =>
  2879             bnf_def Dont_Inline user_policy I tacs (wit_tac thms) (SOME deads)
  2880               ((((b, fold_rev Term.absfree fs' map), sets), absdummy T bd), wits))
  2881           tacss bs fs_maps setss_by_bnf Ts all_witss lthy;
  2882 
  2883         val fold_maps = Local_Defs.fold lthy (map (fn bnf =>
  2884           mk_unabs_def m (map_def_of_bnf bnf RS @{thm meta_eq_to_obj_eq})) Jbnfs);
  2885 
  2886         val fold_sets = Local_Defs.fold lthy (maps (fn bnf =>
  2887          map (fn thm => thm RS @{thm meta_eq_to_obj_eq}) (set_defs_of_bnf bnf)) Jbnfs);
  2888 
  2889         val timer = time (timer "registered new codatatypes as BNFs");
  2890 
  2891         val set_incl_thmss = map (map fold_sets) hset_unf_incl_thmss;
  2892         val set_set_incl_thmsss = map (map (map fold_sets)) hset_hset_unf_incl_thmsss;
  2893         val set_induct_thms = map fold_sets hset_induct_thms;
  2894 
  2895         val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
  2896         val Jrels = map (mk_rel_of_bnf deads passiveAs passiveBs) Jbnfs;
  2897         val preds = map2 (fn Ds => mk_pred_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
  2898         val Jpreds = map (mk_pred_of_bnf deads passiveAs passiveBs) Jbnfs;
  2899 
  2900         val JrelRs = map (fn Jrel => Term.list_comb (Jrel, JRs)) Jrels;
  2901         val relRs = map (fn rel => Term.list_comb (rel, JRs @ JrelRs)) rels;
  2902         val Jpredphis = map (fn Jrel => Term.list_comb (Jrel, Jphis)) Jpreds;
  2903         val predphis = map (fn rel => Term.list_comb (rel, Jphis @ Jpredphis)) preds;
  2904 
  2905         val in_rels = map in_rel_of_bnf bnfs;
  2906         val in_Jrels = map in_rel_of_bnf Jbnfs;
  2907         val Jpred_defs =
  2908           map (Drule.abs_def o (fn thm => thm RS @{thm eq_reflection}) o pred_def_of_bnf) Jbnfs;
  2909 
  2910         val folded_map_simp_thms = map fold_maps map_simp_thms;
  2911         val folded_set_simp_thmss = map (map fold_sets) set_simp_thmss;
  2912         val folded_set_simp_thmss' = transpose folded_set_simp_thmss;
  2913 
  2914         val Jrel_unfold_thms =
  2915           let
  2916             fun mk_goal Jz Jz' unf unf' JrelR relR = fold_rev Logic.all (Jz :: Jz' :: JRs)
  2917               (HOLogic.mk_Trueprop (HOLogic.mk_eq
  2918                 (HOLogic.mk_mem (HOLogic.mk_prod (Jz, Jz'), JrelR),
  2919                   HOLogic.mk_mem (HOLogic.mk_prod (unf $ Jz, unf' $ Jz'), relR))));
  2920             val goals = map6 mk_goal Jzs Jz's unfs unf's JrelRs relRs;
  2921           in
  2922             map12 (fn i => fn goal => fn in_rel => fn map_comp => fn map_cong =>
  2923               fn map_simp => fn set_simps => fn unf_inject => fn unf_fld =>
  2924               fn set_naturals => fn set_incls => fn set_set_inclss =>
  2925               Skip_Proof.prove lthy [] [] goal
  2926                 (K (mk_rel_unfold_tac in_Jrels i in_rel map_comp map_cong map_simp set_simps
  2927                   unf_inject unf_fld set_naturals set_incls set_set_inclss))
  2928               |> Thm.close_derivation)
  2929             ks goals in_rels map_comp's map_congs folded_map_simp_thms folded_set_simp_thmss'
  2930               unf_inject_thms unf_fld_thms set_natural'ss set_incl_thmss set_set_incl_thmsss
  2931           end;
  2932 
  2933         val Jpred_unfold_thms =
  2934           let
  2935             fun mk_goal Jz Jz' unf unf' Jpredphi predphi = fold_rev Logic.all (Jz :: Jz' :: Jphis)
  2936               (HOLogic.mk_Trueprop (HOLogic.mk_eq
  2937                 (Jpredphi $ Jz $ Jz', predphi $ (unf $ Jz) $ (unf' $ Jz'))));
  2938             val goals = map6 mk_goal Jzs Jz's unfs unf's Jpredphis predphis;
  2939           in
  2940             map3 (fn goal => fn pred_def => fn Jrel_unfold =>
  2941               Skip_Proof.prove lthy [] [] goal (mk_pred_unfold_tac pred_def Jpred_defs Jrel_unfold)
  2942               |> Thm.close_derivation)
  2943             goals pred_defs Jrel_unfold_thms
  2944           end;
  2945 
  2946         val timer = time (timer "additional properties");
  2947 
  2948         val ls' = if m = 1 then [0] else ls;
  2949 
  2950         val Jbnf_common_notes =
  2951           [(map_uniqueN, [fold_maps map_unique_thm])] @
  2952           map2 (fn i => fn thm => (mk_set_inductN i, [thm])) ls' set_induct_thms
  2953           |> map (fn (thmN, thms) =>
  2954             ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));
  2955 
  2956         val Jbnf_notes =
  2957           [(map_simpsN, map single folded_map_simp_thms),
  2958           (set_inclN, set_incl_thmss),
  2959           (set_set_inclN, map flat set_set_incl_thmsss), (* nicer names? *)
  2960           (rel_unfoldN, map single Jrel_unfold_thms),
  2961           (pred_unfoldN, map single Jpred_unfold_thms)] @
  2962           map2 (fn i => fn thms => (mk_set_simpsN i, map single thms)) ls' folded_set_simp_thmss
  2963           |> maps (fn (thmN, thmss) =>
  2964             map2 (fn b => fn thms =>
  2965               ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))
  2966             bs thmss)
  2967       in
  2968         lthy |> Local_Theory.notes (Jbnf_common_notes @ Jbnf_notes) |> snd
  2969       end;
  2970 
  2971       val common_notes =
  2972         [(unf_coinductN, [unf_coinduct_thm]),
  2973         (rel_coinductN, [rel_coinduct_thm]),
  2974         (pred_coinductN, [pred_coinduct_thm]),
  2975         (unf_coinduct_uptoN, [unf_coinduct_upto_thm]),
  2976         (rel_coinduct_uptoN, [rel_coinduct_upto_thm]),
  2977         (pred_coinduct_uptoN, [pred_coinduct_upto_thm])]
  2978         |> map (fn (thmN, thms) =>
  2979           ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));
  2980 
  2981       val notes =
  2982         [(coiterN, coiter_thms),
  2983         (coiter_uniqueN, coiter_unique_thms),
  2984         (corecN, corec_thms),
  2985         (unf_fldN, unf_fld_thms),
  2986         (fld_unfN, fld_unf_thms),
  2987         (unf_injectN, unf_inject_thms),
  2988         (unf_exhaustN, unf_exhaust_thms),
  2989         (fld_injectN, fld_inject_thms),
  2990         (fld_exhaustN, fld_exhaust_thms),
  2991         (fld_coiterN, fld_coiter_thms)]
  2992         |> map (apsnd (map single))
  2993         |> maps (fn (thmN, thmss) =>
  2994           map2 (fn b => fn thms =>
  2995             ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))
  2996           bs thmss)
  2997   in
  2998     lthy |> Local_Theory.notes (common_notes @ notes) |> snd
  2999   end;
  3000 
  3001 val _ =
  3002   Outer_Syntax.local_theory @{command_spec "codata_raw"} "greatest fixed points for BNF equations"
  3003     (Parse.and_list1
  3004       ((Parse.binding --| Parse.$$$ ":") -- (Parse.typ --| Parse.$$$ "=" -- Parse.typ)) >>
  3005       (fp_bnf_cmd bnf_gfp o apsnd split_list o split_list));
  3006 
  3007 end;