(*  Title:      HOL/BNF/Tools/bnf_gfp.ML

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Andrei Popescu, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012

 Codatatype construction.

 *)

 signature BNF_GFP =

 sig

   val construct_gfp: mixfix list -> binding list -> binding list -> binding list list ->

     binding list -> (string * sort) list -> typ list * typ list list -> BNF_Def.bnf list ->

     local_theory -> BNF_FP_Util.fp_result * local_theory

 end;

 structure BNF_GFP : BNF_GFP =

 struct

 open BNF_Def

 open BNF_Util

 open BNF_Tactics

 open BNF_Comp

 open BNF_FP_Util

 open BNF_FP_Def_Sugar

 open BNF_GFP_Rec_Sugar

 open BNF_GFP_Util

 open BNF_GFP_Tactics

 datatype wit_tree = Wit_Leaf of int | Wit_Node of (int * int * int list) * wit_tree list;

 fun mk_tree_args (I, T) (I', Ts) = (sort_distinct int_ord (I @ I'), T :: Ts);

 fun finish Iss m seen i (nwit, I) =

   let

     val treess = map (fn j =>

         if j < m orelse member (op =) seen j then [([j], Wit_Leaf j)]

         else

           map_index (finish Iss m (insert (op =) j seen) j) (nth Iss (j - m))

           |> flat

           |> minimize_wits)

       I;

   in

     map (fn (I, t) => (I, Wit_Node ((i - m, nwit, filter (fn i => i < m) I), t)))

       (fold_rev (map_product mk_tree_args) treess [([], [])])

     |> minimize_wits

   end;

 fun tree_to_ctor_wit vars _ _ (Wit_Leaf j) = ([j], nth vars j)

   | tree_to_ctor_wit vars ctors witss (Wit_Node ((i, nwit, I), subtrees)) =

      (I, nth ctors i $ (Term.list_comb (snd (nth (nth witss i) nwit),

        map (snd o tree_to_ctor_wit vars ctors witss) subtrees)));

 fun tree_to_coind_wits _ (Wit_Leaf _) = []

   | tree_to_coind_wits lwitss (Wit_Node ((i, nwit, I), subtrees)) =

      ((i, I), nth (nth lwitss i) nwit) :: maps (tree_to_coind_wits lwitss) subtrees;

 (*all BNFs have the same lives*)

 fun construct_gfp mixfixes map_bs rel_bs set_bss0 bs resBs (resDs, Dss) bnfs lthy =

   let

     val time = time lthy;

     val timer = time (Timer.startRealTimer ());

     val live = live_of_bnf (hd bnfs);

     val n = length bnfs; (*active*)

     val ks = 1 upto n;

     val m = live - n; (*passive, if 0 don't generate a new BNF*)

     val ls = 1 upto m;

     val note_all = Config.get lthy bnf_note_all;

     val b_names = map Binding.name_of bs;

     val b_name = mk_common_name b_names;

     val b = Binding.name b_name;

     val mk_internal_b = Binding.name #> Binding.prefix true b_name #> Binding.conceal;

     fun mk_internal_bs name =

       map (fn b =>

         Binding.prefix true b_name (Binding.prefix_name (name ^ "_") b) |> Binding.conceal) bs;

     val external_bs = map2 (Binding.prefix false) b_names bs

       |> note_all = false ? map Binding.conceal;

     (* TODO: check if m, n, etc., are sane *)

     val deads = fold (union (op =)) Dss resDs;

     val names_lthy = fold Variable.declare_typ deads lthy;

     val passives = map fst (subtract (op = o apsnd TFree) deads resBs);

     (* tvars *)

     val ((((((((passiveAs, activeAs), passiveBs), activeBs), passiveCs), activeCs), passiveXs),

         passiveYs), idxT) = names_lthy

       |> variant_tfrees passives

       ||>> mk_TFrees n

       ||>> variant_tfrees passives

       ||>> mk_TFrees n

       ||>> mk_TFrees m

       ||>> mk_TFrees n

       ||>> mk_TFrees m

       ||>> mk_TFrees m

       ||> fst o mk_TFrees 1

       ||> the_single;

     val allAs = passiveAs @ activeAs;

     val allBs' = passiveBs @ activeBs;

     val Ass = replicate n allAs;

     val allBs = passiveAs @ activeBs;

     val Bss = replicate n allBs;

     val allCs = passiveAs @ activeCs;

     val allCs' = passiveBs @ activeCs;

     val Css' = replicate n allCs';

     (* types *)

     val dead_poss =

       map (fn x => if member (op =) deads (TFree x) then SOME (TFree x) else NONE) resBs;

     fun mk_param NONE passive = (hd passive, tl passive)

       | mk_param (SOME a) passive = (a, passive);

     val mk_params = fold_map mk_param dead_poss #> fst;

     fun mk_FTs Ts = map2 (fn Ds => mk_T_of_bnf Ds Ts) Dss bnfs;

     val (params, params') = `(map Term.dest_TFree) (mk_params passiveAs);

     val (dead_params, dead_params') = `(map Term.dest_TFree) (subtract (op =) passiveAs params');

     val FTsAs = mk_FTs allAs;

     val FTsBs = mk_FTs allBs;

     val FTsCs = mk_FTs allCs;

     val ATs = map HOLogic.mk_setT passiveAs;

     val BTs = map HOLogic.mk_setT activeAs;

     val B'Ts = map HOLogic.mk_setT activeBs;

     val B''Ts = map HOLogic.mk_setT activeCs;

     val sTs = map2 (fn T => fn U => T --> U) activeAs FTsAs;

     val s'Ts = map2 (fn T => fn U => T --> U) activeBs FTsBs;

     val s''Ts = map2 (fn T => fn U => T --> U) activeCs FTsCs;

     val fTs = map2 (fn T => fn U => T --> U) activeAs activeBs;

     val self_fTs = map (fn T => T --> T) activeAs;

     val gTs = map2 (fn T => fn U => T --> U) activeBs activeCs;

     val all_gTs = map2 (fn T => fn U => T --> U) allBs allCs';

     val RTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeBs;

     val sRTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeAs;

     val R'Ts = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeBs activeCs;

     val setsRTs = map HOLogic.mk_setT sRTs;

     val setRTs = map HOLogic.mk_setT RTs;

     val all_sbisT = HOLogic.mk_tupleT setsRTs;

     val setR'Ts = map HOLogic.mk_setT R'Ts;

     val FRTs = mk_FTs (passiveAs @ RTs);

     val sumBsAs = map2 (curry mk_sumT) activeBs activeAs;

     val sumFTs = mk_FTs (passiveAs @ sumBsAs);

     val sum_sTs = map2 (fn T => fn U => T --> U) activeAs sumFTs;

     (* terms *)

     val mapsAsAs = map4 mk_map_of_bnf Dss Ass Ass bnfs;

     val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;

     val mapsBsCs' = map4 mk_map_of_bnf Dss Bss Css' bnfs;

     val mapsAsCs' = map4 mk_map_of_bnf Dss Ass Css' bnfs;

     val map_Inls = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ sumBsAs)) bnfs;

     val map_Inls_rev = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ sumBsAs)) Bss bnfs;

     val map_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs;

     val map_snds = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs;

     fun mk_setss Ts = map3 mk_sets_of_bnf (map (replicate live) Dss)

       (map (replicate live) (replicate n Ts)) bnfs;

     val setssAs = mk_setss allAs;

     val setssAs' = transpose setssAs;

     val bis_setss = mk_setss (passiveAs @ RTs);

     val relsAsBs = map4 mk_rel_of_bnf Dss Ass Bss bnfs;

     val bds = map3 mk_bd_of_bnf Dss Ass bnfs;

     val sum_bd = Library.foldr1 (uncurry mk_csum) bds;

     val sum_bdT = fst (dest_relT (fastype_of sum_bd));

     val emptys = map (fn T => HOLogic.mk_set T []) passiveAs;

     val Zeros = map (fn empty =>

      HOLogic.mk_tuple (map (fn U => absdummy U empty) activeAs)) emptys;

     val hrecTs = map fastype_of Zeros;

     val hsetTs = map (fn hrecT => Library.foldr (op -->) (sTs, HOLogic.natT --> hrecT)) hrecTs;

     val ((((((((((((((((((((((((((((((((((zs, zs'), zs_copy), zs_copy2),

       z's), As), Bs), Bs_copy), B's), B''s), ss), sum_ss), s's), s''s), fs), fs_copy),

       self_fs), gs), all_gs), xFs), yFs), yFs_copy), RFs), (Rtuple, Rtuple')), (hrecs, hrecs')),

       (nat, nat')), Rs), Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), Kss), names_lthy) = lthy

       |> mk_Frees' "b" activeAs

       ||>> mk_Frees "b" activeAs

       ||>> mk_Frees "b" activeAs

       ||>> mk_Frees "b" activeBs

       ||>> mk_Frees "A" ATs

       ||>> mk_Frees "B" BTs

       ||>> mk_Frees "B" BTs

       ||>> mk_Frees "B'" B'Ts

       ||>> mk_Frees "B''" B''Ts

       ||>> mk_Frees "s" sTs

       ||>> mk_Frees "sums" sum_sTs

       ||>> mk_Frees "s'" s'Ts

       ||>> mk_Frees "s''" s''Ts

       ||>> mk_Frees "f" fTs

       ||>> mk_Frees "f" fTs

       ||>> mk_Frees "f" self_fTs

       ||>> mk_Frees "g" gTs

       ||>> mk_Frees "g" all_gTs

       ||>> mk_Frees "x" FTsAs

       ||>> mk_Frees "y" FTsBs

       ||>> mk_Frees "y" FTsBs

       ||>> mk_Frees "x" FRTs

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Rtuple") all_sbisT

       ||>> mk_Frees' "rec" hrecTs

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT

       ||>> mk_Frees "R" setRTs

       ||>> mk_Frees "R" setRTs

       ||>> mk_Frees "R'" setR'Ts

       ||>> mk_Frees "R" setsRTs

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") idxT

       ||>> yield_singleton (mk_Frees "I") (HOLogic.mk_setT idxT)

       ||>> mk_Frees "Ri" (map (fn T => idxT --> T) setRTs)

       ||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) activeAs) ATs);

     val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;

     val passive_Id_ons = map mk_Id_on As;

     val active_UNIVs = map HOLogic.mk_UNIV activeAs;

     val sum_UNIVs = map HOLogic.mk_UNIV sumBsAs;

     val passive_ids = map HOLogic.id_const passiveAs;

     val active_ids = map HOLogic.id_const activeAs;

     val Inls = map2 Inl_const activeBs activeAs;

     val fsts = map fst_const RTs;

     val snds = map snd_const RTs;

     (* thms *)

     val bd_card_orders = map bd_card_order_of_bnf bnfs;

     val bd_card_order = hd bd_card_orders

     val bd_Card_orders = map bd_Card_order_of_bnf bnfs;

     val bd_Card_order = hd bd_Card_orders;

     val bd_Cinfinites = map bd_Cinfinite_of_bnf bnfs;

     val bd_Cinfinite = hd bd_Cinfinites;

     val in_monos = map in_mono_of_bnf bnfs;

     val map_comp0s = map map_comp0_of_bnf bnfs;

     val sym_map_comps = map mk_sym map_comp0s;

     val map_comps = map map_comp_of_bnf bnfs;

     val map_cong0s = map map_cong0_of_bnf bnfs;

     val map_id0s = map map_id0_of_bnf bnfs;

     val map_ids = map map_id_of_bnf bnfs;

     val map_wpulls = map map_wpull_of_bnf bnfs;

     val set_bdss = map set_bd_of_bnf bnfs;

     val set_mapss = map set_map_of_bnf bnfs;

     val rel_congs = map rel_cong_of_bnf bnfs;

     val rel_converseps = map rel_conversep_of_bnf bnfs;

     val rel_Grps = map rel_Grp_of_bnf bnfs;

     val rel_OOs = map rel_OO_of_bnf bnfs;

     val rel_OO_Grps = map rel_OO_Grp_of_bnf bnfs;

     val timer = time (timer "Extracted terms & thms");

     (* derived thms *)

     (*map g1 ... gm g(m+1) ... g(m+n) (map id ... id f(m+1) ... f(m+n) x) =

       map g1 ... gm (g(m+1) o f(m+1)) ... (g(m+n) o f(m+n)) x*)

     fun mk_map_comp_id x mapAsBs mapBsCs mapAsCs map_comp0 =

       let

         val lhs = Term.list_comb (mapBsCs, all_gs) $

           (Term.list_comb (mapAsBs, passive_ids @ fs) $ x);

         val rhs =

           Term.list_comb (mapAsCs, take m all_gs @ map HOLogic.mk_comp (drop m all_gs ~~ fs)) $ x;

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (x :: fs @ all_gs) (mk_Trueprop_eq (lhs, rhs)))

           (K (mk_map_comp_id_tac map_comp0))

         |> Thm.close_derivation

       end;

     val map_comp_id_thms = map5 mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comps;

     (*forall a : set(m+1) x. f(m+1) a = a; ...; forall a : set(m+n) x. f(m+n) a = a ==>

       map id ... id f(m+1) ... f(m+n) x = x*)

     fun mk_map_cong0L x mapAsAs sets map_cong0 map_id =

       let

         fun mk_prem set f z z' =

           HOLogic.mk_Trueprop

             (mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z))));

         val prems = map4 mk_prem (drop m sets) self_fs zs zs';

         val goal = mk_Trueprop_eq (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x);

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (x :: self_fs) (Logic.list_implies (prems, goal)))

           (K (mk_map_cong0L_tac m map_cong0 map_id))

         |> Thm.close_derivation

       end;

     val map_cong0L_thms = map5 mk_map_cong0L xFs mapsAsAs setssAs map_cong0s map_ids;

     val in_mono'_thms = map (fn thm =>

       (thm OF (replicate m subset_refl)) RS @{thm set_mp}) in_monos;

     val map_arg_cong_thms =

       let

         val prems = map2 (curry mk_Trueprop_eq) yFs yFs_copy;

         val maps = map (fn mapx => Term.list_comb (mapx, all_gs)) mapsBsCs';

         val concls =

           map3 (fn x => fn y => fn mapx => mk_Trueprop_eq (mapx $ x, mapx $ y)) yFs yFs_copy maps;

         val goals =

           map4 (fn prem => fn concl => fn x => fn y =>

             fold_rev Logic.all (x :: y :: all_gs) (Logic.mk_implies (prem, concl)))

           prems concls yFs yFs_copy;

       in

         map (fn goal => Goal.prove_sorry lthy [] [] goal

           (K ((hyp_subst_tac lthy THEN' rtac refl) 1)) |> Thm.close_derivation) goals

       end;

     val timer = time (timer "Derived simple theorems");

     (* coalgebra *)

     val coalg_bind = mk_internal_b (coN ^ algN) ;

     val coalg_name = Binding.name_of coalg_bind;

     val coalg_def_bind = (Thm.def_binding coalg_bind, []);

     (*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in A1 .. Am B1 ... Bn)*)

     val coalg_spec =

       let

         val coalgT = Library.foldr (op -->) (ATs @ BTs @ sTs, HOLogic.boolT);

         val ins = map3 mk_in (replicate n (As @ Bs)) setssAs FTsAs;

         fun mk_coalg_conjunct B s X z z' =

           mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X)));

         val lhs = Term.list_comb (Free (coalg_name, coalgT), As @ Bs @ ss);

         val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_coalg_conjunct Bs ss ins zs zs')

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) =

       lthy

       |> Specification.definition (SOME (coalg_bind, NONE, NoSyn), (coalg_def_bind, coalg_spec))

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free));

     val coalg_def = Morphism.thm phi coalg_def_free;

     fun mk_coalg As Bs ss =

       let

         val args = As @ Bs @ ss;

         val Ts = map fastype_of args;

         val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT);

       in

         Term.list_comb (Const (coalg, coalgT), args)

       end;

     val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);

     val coalg_in_thms = map (fn i =>

       coalg_def RS iffD1 RS mk_conjunctN n i RS bspec) ks

     val coalg_set_thmss =

       let

         val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);

         fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));

         fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) B);

         val prems = map2 mk_prem zs Bs;

         val conclss = map3 (fn s => fn x => fn sets => map2 (mk_concl s x) (As @ Bs) sets)

           ss zs setssAs;

         val goalss = map3 (fn x => fn prem => fn concls => map (fn concl =>

           fold_rev Logic.all (x :: As @ Bs @ ss)

             (Logic.list_implies (coalg_prem :: [prem], concl))) concls) zs prems conclss;

       in

         map (fn goals => map (fn goal => Goal.prove_sorry lthy [] [] goal

           (K (mk_coalg_set_tac coalg_def)) |> Thm.close_derivation) goals) goalss

       end;

     fun mk_tcoalg ATs BTs = mk_coalg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);

     val tcoalg_thm =

       let

         val goal = fold_rev Logic.all ss

           (HOLogic.mk_Trueprop (mk_tcoalg passiveAs activeAs ss))

       in

         Goal.prove_sorry lthy [] [] goal

           (K (stac coalg_def 1 THEN CONJ_WRAP

             (K (EVERY' [rtac ballI, rtac CollectI,

               CONJ_WRAP' (K (EVERY' [rtac @{thm subset_UNIV}])) allAs] 1)) ss))

         |> Thm.close_derivation

       end;

     val timer = time (timer "Coalgebra definition & thms");

     (* morphism *)

     val mor_bind = mk_internal_b morN;

     val mor_name = Binding.name_of mor_bind;

     val mor_def_bind = (Thm.def_binding mor_bind, []);

     (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*)

     (*mor) forall i = 1 ... n: (\<forall>x \<in> Bi.

        Fi_map id ... id f1 ... fn (si x) = si' (fi x)*)

     val mor_spec =

       let

         val morT = Library.foldr (op -->) (BTs @ sTs @ B'Ts @ s'Ts @ fTs, HOLogic.boolT);

         fun mk_fbetw f B1 B2 z z' =

           mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));

         fun mk_mor B mapAsBs f s s' z z' =

           mk_Ball B (Term.absfree z' (HOLogic.mk_eq

             (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z))));

         val lhs = Term.list_comb (Free (mor_name, morT), Bs @ ss @ B's @ s's @ fs);

         val rhs = HOLogic.mk_conj

           (Library.foldr1 HOLogic.mk_conj (map5 mk_fbetw fs Bs B's zs zs'),

            Library.foldr1 HOLogic.mk_conj (map7 mk_mor Bs mapsAsBs fs ss s's zs zs'))

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =

       lthy

       |> Specification.definition (SOME (mor_bind, NONE, NoSyn), (mor_def_bind, mor_spec))

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val mor = fst (Term.dest_Const (Morphism.term phi mor_free));

     val mor_def = Morphism.thm phi mor_def_free;

     fun mk_mor Bs1 ss1 Bs2 ss2 fs =

       let

         val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;

         val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);

         val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);

       in

         Term.list_comb (Const (mor, morT), args)

       end;

     val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);

     val (mor_image_thms, morE_thms) =

       let

         val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);

         fun mk_image_goal f B1 B2 = fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs)

           (Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_leq (mk_image f $ B1) B2)));

         val image_goals = map3 mk_image_goal fs Bs B's;

         fun mk_elim_goal B mapAsBs f s s' x =

           fold_rev Logic.all (x :: Bs @ ss @ B's @ s's @ fs)

             (Logic.list_implies ([prem, HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B))],

               mk_Trueprop_eq (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x))));

         val elim_goals = map6 mk_elim_goal Bs mapsAsBs fs ss s's zs;

         fun prove goal =

           Goal.prove_sorry lthy [] [] goal (K (mk_mor_elim_tac mor_def))

           |> Thm.close_derivation;

       in

         (map prove image_goals, map prove elim_goals)

       end;

     val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms;

     val mor_incl_thm =

       let

         val prems = map2 (HOLogic.mk_Trueprop oo mk_leq) Bs Bs_copy;

         val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (Bs @ ss @ Bs_copy) (Logic.list_implies (prems, concl)))

           (K (mk_mor_incl_tac mor_def map_ids))

         |> Thm.close_derivation

       end;

     val mor_id_thm = mor_incl_thm OF (replicate n subset_refl);

     val mor_comp_thm =

       let

         val prems =

           [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),

            HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];

         val concl =

           HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)

             (Logic.list_implies (prems, concl)))

           (K (mk_mor_comp_tac mor_def mor_image'_thms morE_thms map_comp_id_thms))

         |> Thm.close_derivation

       end;

     val mor_cong_thm =

       let

         val prems = map HOLogic.mk_Trueprop

          (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])

         val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ fs_copy)

             (Logic.list_implies (prems, concl)))

           (K ((hyp_subst_tac lthy THEN' atac) 1))

         |> Thm.close_derivation

       end;

     val mor_UNIV_thm =

       let

         fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq

             (HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s),

             HOLogic.mk_comp (s', f));

         val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;

         val rhs = Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct mapsAsBs fs ss s's);

       in

         Goal.prove_sorry lthy [] [] (fold_rev Logic.all (ss @ s's @ fs) (mk_Trueprop_eq (lhs, rhs)))

           (K (mk_mor_UNIV_tac morE_thms mor_def))

         |> Thm.close_derivation

       end;

     val mor_str_thm =

       let

         val maps = map2 (fn Ds => fn bnf => Term.list_comb

           (mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs;

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all ss (HOLogic.mk_Trueprop

             (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss)))

           (K (mk_mor_str_tac ks mor_UNIV_thm))

         |> Thm.close_derivation

       end;

     val mor_sum_case_thm =

       let

         val maps = map3 (fn s => fn sum_s => fn mapx =>

           mk_sum_case (HOLogic.mk_comp (Term.list_comb (mapx, passive_ids @ Inls), s), sum_s))

           s's sum_ss map_Inls;

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (s's @ sum_ss) (HOLogic.mk_Trueprop

             (mk_mor (map HOLogic.mk_UNIV activeBs) s's sum_UNIVs maps Inls)))

           (K (mk_mor_sum_case_tac ks mor_UNIV_thm))

         |> Thm.close_derivation

       end;

     val timer = time (timer "Morphism definition & thms");

     fun hset_rec_bind j = mk_internal_b (hset_recN ^ (if m = 1 then "" else string_of_int j));

     val hset_rec_name = Binding.name_of o hset_rec_bind;

     val hset_rec_def_bind = rpair [] o Thm.def_binding o hset_rec_bind;

     fun hset_rec_spec j Zero hsetT hrec hrec' =

       let

         fun mk_Suc s setsAs z z' =

           let

             val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m setsAs);

             fun mk_UN set k = mk_UNION (set $ (s $ z)) (mk_nthN n hrec k);

           in

             Term.absfree z'

               (mk_union (set $ (s $ z), Library.foldl1 mk_union (map2 mk_UN sets ks)))

           end;

         val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec'

           (HOLogic.mk_tuple (map4 mk_Suc ss setssAs zs zs')));

         val lhs = Term.list_comb (Free (hset_rec_name j, hsetT), ss);

         val rhs = mk_nat_rec Zero Suc;

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((hset_rec_frees, (_, hset_rec_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map5 (fn j => fn Zero => fn hsetT => fn hrec => fn hrec' => Specification.definition

         (SOME (hset_rec_bind j, NONE, NoSyn),

           (hset_rec_def_bind j, hset_rec_spec j Zero hsetT hrec hrec')))

         ls Zeros hsetTs hrecs hrecs'

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val hset_rec_defs = map (Morphism.thm phi) hset_rec_def_frees;

     val hset_recs = map (fst o Term.dest_Const o Morphism.term phi) hset_rec_frees;

     fun mk_hset_rec ss nat i j T =

       let

         val args = ss @ [nat];

         val Ts = map fastype_of ss;

         val bTs = map domain_type Ts;

         val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) bTs)

         val hset_recT = Library.foldr (op -->) (Ts, HOLogic.natT --> hrecT);

       in

         mk_nthN n (Term.list_comb (Const (nth hset_recs (j - 1), hset_recT), args)) i

       end;

     val hset_rec_0ss = mk_rec_simps n @{thm nat_rec_0} hset_rec_defs;

     val hset_rec_Sucss = mk_rec_simps n @{thm nat_rec_Suc} hset_rec_defs;

     val hset_rec_0ss' = transpose hset_rec_0ss;

     val hset_rec_Sucss' = transpose hset_rec_Sucss;

     fun hset_binds j = mk_internal_bs (hsetN ^ (if m = 1 then "" else string_of_int j))

     fun hset_bind i j = nth (hset_binds j) (i - 1);

     val hset_name = Binding.name_of oo hset_bind;

     val hset_def_bind = rpair [] o Thm.def_binding oo hset_bind;

     fun hset_spec i j =

       let

         val U = nth activeAs (i - 1);

         val z = nth zs (i - 1);

         val T = nth passiveAs (j - 1);

         val setT = HOLogic.mk_setT T;

         val hsetT = Library.foldr (op -->) (sTs, U --> setT);

         val lhs = Term.list_comb (Free (hset_name i j, hsetT), ss @ [z]);

         val rhs = mk_UNION (HOLogic.mk_UNIV HOLogic.natT)

           (Term.absfree nat' (mk_hset_rec ss nat i j T $ z));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((hset_frees, (_, hset_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map (fn i => fold_map (fn j => Specification.definition

         (SOME (hset_bind i j, NONE, NoSyn), (hset_def_bind i j, hset_spec i j))) ls) ks

       |>> map (apsnd split_list o split_list)

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val hset_defss = map (map (Morphism.thm phi)) hset_def_frees;

     val hset_defss' = transpose hset_defss;

     val hset_namess = map (map (fst o Term.dest_Const o Morphism.term phi)) hset_frees;

     fun mk_hset ss i j T =

       let

         val Ts = map fastype_of ss;

         val bTs = map domain_type Ts;

         val hsetT = Library.foldr (op -->) (Ts, nth bTs (i - 1) --> HOLogic.mk_setT T);

       in

         Term.list_comb (Const (nth (nth hset_namess (i - 1)) (j - 1), hsetT), ss)

       end;

     val hsetssAs = map (fn i => map2 (mk_hset ss i) ls passiveAs) ks;

     val (set_incl_hset_thmss, set_hset_incl_hset_thmsss) =

       let

         fun mk_set_incl_hset s x set hset = fold_rev Logic.all (x :: ss)

           (HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) (hset $ x)));

         fun mk_set_hset_incl_hset s x y set hset1 hset2 =

           fold_rev Logic.all (x :: y :: ss)

             (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x, set $ (s $ y))),

             HOLogic.mk_Trueprop (mk_leq (hset1 $ x) (hset2 $ y))));

         val set_incl_hset_goalss =

           map4 (fn s => fn x => fn sets => fn hsets =>

             map2 (mk_set_incl_hset s x) (take m sets) hsets)

           ss zs setssAs hsetssAs;

         (*xk : F(i)set(m+k) (si yi) ==> F(k)_hset(j) s1 ... sn xk <= F(i)_hset(j) s1 ... sn yi*)

         val set_hset_incl_hset_goalsss =

           map4 (fn si => fn yi => fn sets => fn hsetsi =>

             map3 (fn xk => fn set => fn hsetsk =>

               map2 (mk_set_hset_incl_hset si xk yi set) hsetsk hsetsi)

             zs_copy (drop m sets) hsetssAs)

           ss zs setssAs hsetssAs;

       in

         (map3 (fn goals => fn defs => fn rec_Sucs =>

           map3 (fn goal => fn def => fn rec_Suc =>

             Goal.prove_sorry lthy [] [] goal (K (mk_set_incl_hset_tac def rec_Suc))

             |> Thm.close_derivation)

           goals defs rec_Sucs)

         set_incl_hset_goalss hset_defss hset_rec_Sucss,

         map3 (fn goalss => fn defsi => fn rec_Sucs =>

           map3 (fn k => fn goals => fn defsk =>

             map4 (fn goal => fn defk => fn defi => fn rec_Suc =>

               Goal.prove_sorry lthy [] [] goal

                 (K (mk_set_hset_incl_hset_tac n [defk, defi] rec_Suc k))

               |> Thm.close_derivation)

             goals defsk defsi rec_Sucs)

           ks goalss hset_defss)

         set_hset_incl_hset_goalsss hset_defss hset_rec_Sucss)

       end;

     val set_incl_hset_thmss' = transpose set_incl_hset_thmss;

     val set_hset_incl_hset_thmsss' = transpose (map transpose set_hset_incl_hset_thmsss);

     val set_hset_thmss = map (map (fn thm => thm RS @{thm set_mp})) set_incl_hset_thmss;

     val set_hset_hset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp})))

       set_hset_incl_hset_thmsss;

     val set_hset_thmss' = transpose set_hset_thmss;

     val set_hset_hset_thmsss' = transpose (map transpose set_hset_hset_thmsss);

     val set_incl_hin_thmss =

       let

         fun mk_set_incl_hin s x hsets1 set hsets2 T =

           fold_rev Logic.all (x :: ss @ As)

             (Logic.list_implies

               (map2 (fn hset => fn A => HOLogic.mk_Trueprop (mk_leq (hset $ x) A)) hsets1 As,

               HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) (mk_in As hsets2 T))));

         val set_incl_hin_goalss =

           map4 (fn s => fn x => fn sets => fn hsets =>

             map3 (mk_set_incl_hin s x hsets) (drop m sets) hsetssAs activeAs)

           ss zs setssAs hsetssAs;

       in

         map2 (map2 (fn goal => fn thms =>

           Goal.prove_sorry lthy [] [] goal (K (mk_set_incl_hin_tac thms))

           |> Thm.close_derivation))

         set_incl_hin_goalss set_hset_incl_hset_thmsss

       end;

     val hset_minimal_thms =

       let

         fun mk_passive_prem set s x K =

           Logic.all x (HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) (K $ x)));

         fun mk_active_prem s x1 K1 set x2 K2 =

           fold_rev Logic.all [x1, x2]

             (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x2, set $ (s $ x1))),

               HOLogic.mk_Trueprop (mk_leq (K2 $ x2) (K1 $ x1))));

         val premss = map2 (fn j => fn Ks =>

           map4 mk_passive_prem (map (fn xs => nth xs (j - 1)) setssAs) ss zs Ks @

             flat (map4 (fn sets => fn s => fn x1 => fn K1 =>

               map3 (mk_active_prem s x1 K1) (drop m sets) zs_copy Ks) setssAs ss zs Ks))

           ls Kss;

         val hset_rec_minimal_thms =

           let

             fun mk_conjunct j T i K x = mk_leq (mk_hset_rec ss nat i j T $ x) (K $ x);

             fun mk_concl j T Ks = list_all_free zs

               (Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs));

             val concls = map3 mk_concl ls passiveAs Kss;

             val goals = map2 (fn prems => fn concl =>

               Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls

             val ctss =

               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;

           in

             map4 (fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>

               singleton (Proof_Context.export names_lthy lthy)

                 (Goal.prove_sorry lthy [] [] goal

                   (mk_hset_rec_minimal_tac m cts hset_rec_0s hset_rec_Sucs))

               |> Thm.close_derivation)

             goals ctss hset_rec_0ss' hset_rec_Sucss'

           end;

         fun mk_conjunct j T i K x = mk_leq (mk_hset ss i j T $ x) (K $ x);

         fun mk_concl j T Ks = Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs);

         val concls = map3 mk_concl ls passiveAs Kss;

         val goals = map3 (fn Ks => fn prems => fn concl =>

           fold_rev Logic.all (Ks @ ss @ zs)

             (Logic.list_implies (prems, HOLogic.mk_Trueprop concl))) Kss premss concls;

       in

         map3 (fn goal => fn hset_defs => fn hset_rec_minimal =>

           Goal.prove_sorry lthy [] [] goal

             (mk_hset_minimal_tac n hset_defs hset_rec_minimal)

           |> Thm.close_derivation)

         goals hset_defss' hset_rec_minimal_thms

       end;

     val timer = time (timer "Hereditary sets");

     (* bisimulation *)

     val bis_bind = mk_internal_b bisN;

     val bis_name = Binding.name_of bis_bind;

     val bis_def_bind = (Thm.def_binding bis_bind, []);

     fun mk_bis_le_conjunct R B1 B2 = mk_leq R (mk_Times (B1, B2));

     val bis_le = Library.foldr1 HOLogic.mk_conj (map3 mk_bis_le_conjunct Rs Bs B's)

     val bis_spec =

       let

         val bisT = Library.foldr (op -->) (ATs @ BTs @ sTs @ B'Ts @ s'Ts @ setRTs, HOLogic.boolT);

         val fst_args = passive_ids @ fsts;

         val snd_args = passive_ids @ snds;

         fun mk_bis R s s' b1 b2 RF map1 map2 sets =

           list_all_free [b1, b2] (HOLogic.mk_imp

             (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),

             mk_Bex (mk_in (As @ Rs) sets (snd (dest_Free RF))) (Term.absfree (dest_Free RF)

               (HOLogic.mk_conj

                 (HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1),

                 HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2))))));

         val lhs = Term.list_comb (Free (bis_name, bisT), As @ Bs @ ss @ B's @ s's @ Rs);

         val rhs = HOLogic.mk_conj

           (bis_le, Library.foldr1 HOLogic.mk_conj

             (map9 mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss))

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) =

       lthy

       |> Specification.definition (SOME (bis_bind, NONE, NoSyn), (bis_def_bind, bis_spec))

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val bis = fst (Term.dest_Const (Morphism.term phi bis_free));

     val bis_def = Morphism.thm phi bis_def_free;

     fun mk_bis As Bs1 ss1 Bs2 ss2 Rs =

       let

         val args = As @ Bs1 @ ss1 @ Bs2 @ ss2 @ Rs;

         val Ts = map fastype_of args;

         val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT);

       in

         Term.list_comb (Const (bis, bisT), args)

       end;

     val bis_cong_thm =

       let

         val prems = map HOLogic.mk_Trueprop

          (mk_bis As Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs)

         val concl = HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs_copy);

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs @ Rs_copy)

             (Logic.list_implies (prems, concl)))

           (K ((hyp_subst_tac lthy THEN' atac) 1))

         |> Thm.close_derivation

       end;

     val bis_rel_thm =

       let

         fun mk_conjunct R s s' b1 b2 rel =

           list_all_free [b1, b2] (HOLogic.mk_imp

             (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),

             Term.list_comb (rel, map mk_in_rel (passive_Id_ons @ Rs)) $ (s $ b1) $ (s' $ b2)));

         val rhs = HOLogic.mk_conj

           (bis_le, Library.foldr1 HOLogic.mk_conj

             (map6 mk_conjunct Rs ss s's zs z's relsAsBs))

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)

             (mk_Trueprop_eq (mk_bis As Bs ss B's s's Rs, rhs)))

           (K (mk_bis_rel_tac lthy m bis_def rel_OO_Grps map_comps map_cong0s set_mapss))

         |> Thm.close_derivation

       end;

     val bis_converse_thm =

       Goal.prove_sorry lthy [] []

         (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)

           (Logic.mk_implies

             (HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),

             HOLogic.mk_Trueprop (mk_bis As B's s's Bs ss (map mk_converse Rs)))))

         (K (mk_bis_converse_tac m bis_rel_thm rel_congs rel_converseps))

       |> Thm.close_derivation;

     val bis_O_thm =

       let

         val prems =

           [HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),

            HOLogic.mk_Trueprop (mk_bis As B's s's B''s s''s R's)];

         val concl =

           HOLogic.mk_Trueprop (mk_bis As Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ B''s @ s''s @ Rs @ R's)

             (Logic.list_implies (prems, concl)))

           (K (mk_bis_O_tac lthy m bis_rel_thm rel_congs rel_OOs))

         |> Thm.close_derivation

       end;

     val bis_Gr_thm =

       let

         val concl =

           HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map2 mk_Gr Bs fs));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ fs)

             (Logic.list_implies ([coalg_prem, mor_prem], concl)))

           (mk_bis_Gr_tac bis_rel_thm rel_Grps mor_image_thms morE_thms coalg_in_thms)

         |> Thm.close_derivation

       end;

     val bis_image2_thm = bis_cong_thm OF

       ((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) ::

       replicate n @{thm image2_Gr});

     val bis_Id_on_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) ::

       replicate n @{thm Id_on_Gr});

     val bis_Union_thm =

       let

         val prem =

           HOLogic.mk_Trueprop (mk_Ball Idx

             (Term.absfree idx' (mk_bis As Bs ss B's s's (map (fn R => R $ idx) Ris))));

         val concl =

           HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map (mk_UNION Idx) Ris));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (Idx :: As @ Bs @ ss @ B's @ s's @ Ris)

             (Logic.mk_implies (prem, concl)))

           (mk_bis_Union_tac bis_def in_mono'_thms)

         |> Thm.close_derivation

       end;

     (* self-bisimulation *)

     fun mk_sbis As Bs ss Rs = mk_bis As Bs ss Bs ss Rs;

     val sbis_prem = HOLogic.mk_Trueprop (mk_sbis As Bs ss sRs);

     (* largest self-bisimulation *)

     val lsbis_binds = mk_internal_bs lsbisN;

     fun lsbis_bind i = nth lsbis_binds (i - 1);

     val lsbis_name = Binding.name_of o lsbis_bind;

     val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind;

     val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs

       (HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis As Bs ss sRs)));

     fun lsbis_spec i RT =

       let

         fun mk_lsbisT RT =

           Library.foldr (op -->) (map fastype_of (As @ Bs @ ss), RT);

         val lhs = Term.list_comb (Free (lsbis_name i, mk_lsbisT RT), As @ Bs @ ss);

         val rhs = mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map2 (fn i => fn RT => Specification.definition

         (SOME (lsbis_bind i, NONE, NoSyn), (lsbis_def_bind i, lsbis_spec i RT))) ks setsRTs

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val lsbis_defs = map (Morphism.thm phi) lsbis_def_frees;

     val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees;

     fun mk_lsbis As Bs ss i =

       let

         val args = As @ Bs @ ss;

         val Ts = map fastype_of args;

         val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1)))));

         val lsbisT = Library.foldr (op -->) (Ts, RT);

       in

         Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args)

       end;

     val sbis_lsbis_thm =

       Goal.prove_sorry lthy [] []

         (fold_rev Logic.all (As @ Bs @ ss)

           (HOLogic.mk_Trueprop (mk_sbis As Bs ss (map (mk_lsbis As Bs ss) ks))))

         (K (mk_sbis_lsbis_tac lthy lsbis_defs bis_Union_thm bis_cong_thm))

       |> Thm.close_derivation;

     val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS

       (bis_def RS iffD1 RS conjunct1 RS mk_conjunctN n i)) ks;

     val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS

       (bis_def RS iffD1 RS conjunct2 RS mk_conjunctN n i))) RS mp) ks;

     val incl_lsbis_thms =

       let

         fun mk_concl i R = HOLogic.mk_Trueprop (mk_leq R (mk_lsbis As Bs ss i));

         val goals = map2 (fn i => fn R => fold_rev Logic.all (As @ Bs @ ss @ sRs)

           (Logic.mk_implies (sbis_prem, mk_concl i R))) ks sRs;

       in

         map3 (fn goal => fn i => fn def => Goal.prove_sorry lthy [] [] goal

           (K (mk_incl_lsbis_tac n i def)) |> Thm.close_derivation) goals ks lsbis_defs

       end;

     val equiv_lsbis_thms =

       let

         fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis As Bs ss i));

         val goals = map2 (fn i => fn B => fold_rev Logic.all (As @ Bs @ ss)

           (Logic.mk_implies (coalg_prem, mk_concl i B))) ks Bs;

       in

         map3 (fn goal => fn l_incl => fn incl_l =>

           Goal.prove_sorry lthy [] [] goal

             (K (mk_equiv_lsbis_tac sbis_lsbis_thm l_incl incl_l

               bis_Id_on_thm bis_converse_thm bis_O_thm))

           |> Thm.close_derivation)

         goals lsbis_incl_thms incl_lsbis_thms

       end;

     val timer = time (timer "Bisimulations");

     (* bounds *)

     val (lthy, sbd, sbdT,

       sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss) =

       if n = 1

       then (lthy, sum_bd, sum_bdT, bd_card_order, bd_Cinfinite, bd_Card_order, set_bdss)

       else

         let

           val sbdT_bind = mk_internal_b sum_bdTN;

           val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) =

             typedef (sbdT_bind, dead_params, NoSyn)

               (HOLogic.mk_UNIV sum_bdT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1) lthy;

           val sbdT = Type (sbdT_name, dead_params');

           val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT);

           val sbd_bind = mk_internal_b sum_bdN;

           val sbd_name = Binding.name_of sbd_bind;

           val sbd_def_bind = (Thm.def_binding sbd_bind, []);

           val sbd_spec = HOLogic.mk_Trueprop

             (HOLogic.mk_eq (Free (sbd_name, mk_relT (`I sbdT)), mk_dir_image sum_bd Abs_sbdT));

           val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) =

             lthy

             |> Specification.definition (SOME (sbd_bind, NONE, NoSyn), (sbd_def_bind, sbd_spec))

             ||> `Local_Theory.restore;

           val phi = Proof_Context.export_morphism lthy_old lthy;

           val sbd_def = Morphism.thm phi sbd_def_free;

           val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT));

           val Abs_sbdT_inj = mk_Abs_inj_thm (#Abs_inject sbdT_loc_info);

           val Abs_sbdT_bij = mk_Abs_bij_thm lthy Abs_sbdT_inj (#Abs_cases sbdT_loc_info);

           fun mk_sum_Cinfinite [thm] = thm

             | mk_sum_Cinfinite (thm :: thms) =

               @{thm Cinfinite_csum_strong} OF [thm, mk_sum_Cinfinite thms];

           val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;

           val sum_Card_order = sum_Cinfinite RS conjunct2;

           fun mk_sum_card_order [thm] = thm

             | mk_sum_card_order (thm :: thms) =

               @{thm card_order_csum} OF [thm, mk_sum_card_order thms];

           val sum_card_order = mk_sum_card_order bd_card_orders;

           val sbd_ordIso = fold_thms lthy [sbd_def]

             (@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order]);

           val sbd_card_order =  fold_thms lthy [sbd_def]

             (@{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]);

           val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite];

           val sbd_Card_order = sbd_Cinfinite RS conjunct2;

           fun mk_set_sbd i bd_Card_order bds =

             map (fn thm => @{thm ordLeq_ordIso_trans} OF

               [bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds;

           val set_sbdss = map3 mk_set_sbd ks bd_Card_orders set_bdss;

        in

          (lthy, sbd, sbdT, sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss)

        end;

     val sbdTs = replicate n sbdT;

     val sum_sbd = Library.foldr1 (uncurry mk_csum) (replicate n sbd);

     val sum_sbdT = mk_sumTN sbdTs;

     val sum_sbd_listT = HOLogic.listT sum_sbdT;

     val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT;

     val bdTs = passiveAs @ replicate n sbdT;

     val to_sbd_maps = map4 mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs;

     val bdFTs = mk_FTs bdTs;

     val sbdFT = mk_sumTN bdFTs;

     val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT);

     val treeQT = HOLogic.mk_setT treeT;

     val treeTs = passiveAs @ replicate n treeT;

     val treeQTs = passiveAs @ replicate n treeQT;

     val treeFTs = mk_FTs treeTs;

     val tree_maps = map4 mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs;

     val final_maps = map4 mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs;

     val isNode_setss = mk_setss (passiveAs @ replicate n sbdT);

     val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []];

     val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs);

     val Lev_recT = fastype_of Zero;

     val LevT = Library.foldr (op -->) (sTs, HOLogic.natT --> Lev_recT);

     val Nil = HOLogic.mk_tuple (map3 (fn i => fn z => fn z'=>

       Term.absfree z' (mk_InN activeAs z i)) ks zs zs');

     val rv_recT = fastype_of Nil;

     val rvT = Library.foldr (op -->) (sTs, sum_sbd_listT --> rv_recT);

     val (((((((((((sumx, sumx'), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')),

       (lab, lab')), (Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')),

       names_lthy) = names_lthy

       |> yield_singleton (apfst (op ~~) oo mk_Frees' "sumx") sum_sbdT

       ||>> mk_Frees' "k" sbdTs

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl") sum_sbd_list_setT

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT)

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl_lab") treeT

       ||>> mk_Frees "x" bdFTs

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") Lev_recT

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") rv_recT;

     val (k, k') = (hd kks, hd kks')

     val timer = time (timer "Bounds");

     (* tree coalgebra *)

     val isNode_binds = mk_internal_bs isNodeN;

     fun isNode_bind i = nth isNode_binds (i - 1);

     val isNode_name = Binding.name_of o isNode_bind;

     val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind;

     val isNodeT =

       Library.foldr (op -->) (map fastype_of (As @ [Kl, lab, kl]), HOLogic.boolT);

     val Succs = map3 (fn i => fn k => fn k' =>

       HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl)))

       ks kks kks';

     fun isNode_spec sets x i =

       let

         val (passive_sets, active_sets) = chop m (map (fn set => set $ x) sets);

         val lhs = Term.list_comb (Free (isNode_name i, isNodeT), As @ [Kl, lab, kl]);

         val rhs = list_exists_free [x]

           (Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) ::

           map2 mk_leq passive_sets As @ map2 (curry HOLogic.mk_eq) active_sets Succs));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map3 (fn i => fn x => fn sets => Specification.definition

         (SOME (isNode_bind i, NONE, NoSyn), (isNode_def_bind i, isNode_spec sets x i)))

         ks xs isNode_setss

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val isNode_defs = map (Morphism.thm phi) isNode_def_frees;

     val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees;

     fun mk_isNode As kl i =

       Term.list_comb (Const (nth isNodes (i - 1), isNodeT), As @ [Kl, lab, kl]);

     val isTree =

       let

         val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl);

         val Field = mk_leq Kl (mk_Field (mk_clists sum_sbd));

         val prefCl = mk_prefCl Kl;

         val tree = mk_Ball Kl (Term.absfree kl'

           (HOLogic.mk_conj

             (Library.foldr1 HOLogic.mk_disj (map (mk_isNode As kl) ks),

             Library.foldr1 HOLogic.mk_conj (map4 (fn Succ => fn i => fn k => fn k' =>

               mk_Ball Succ (Term.absfree k' (mk_isNode As

                 (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i)))

             Succs ks kks kks'))));

         val undef = list_all_free [kl] (HOLogic.mk_imp

           (HOLogic.mk_not (HOLogic.mk_mem (kl, Kl)),

           HOLogic.mk_eq (lab $ kl, mk_undefined sbdFT)));

       in

         Library.foldr1 HOLogic.mk_conj [empty, Field, prefCl, tree, undef]

       end;

     val carT_binds = mk_internal_bs carTN;

     fun carT_bind i = nth carT_binds (i - 1);

     val carT_name = Binding.name_of o carT_bind;

     val carT_def_bind = rpair [] o Thm.def_binding o carT_bind;

     fun carT_spec i =

       let

         val carTT = Library.foldr (op -->) (ATs, HOLogic.mk_setT treeT);

         val lhs = Term.list_comb (Free (carT_name i, carTT), As);

         val rhs = HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]

           (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)),

             HOLogic.mk_conj (isTree, mk_isNode As (HOLogic.mk_list sum_sbdT []) i))));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map (fn i => Specification.definition

         (SOME (carT_bind i, NONE, NoSyn), (carT_def_bind i, carT_spec i))) ks

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val carT_defs = map (Morphism.thm phi) carT_def_frees;

     val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees;

     fun mk_carT As i = Term.list_comb

       (Const (nth carTs (i - 1),

          Library.foldr (op -->) (map fastype_of As, HOLogic.mk_setT treeT)), As);

     val strT_binds = mk_internal_bs strTN;

     fun strT_bind i = nth strT_binds (i - 1);

     val strT_name = Binding.name_of o strT_bind;

     val strT_def_bind = rpair [] o Thm.def_binding o strT_bind;

     fun strT_spec mapFT FT i =

       let

         val strTT = treeT --> FT;

         fun mk_f i k k' =

           let val in_k = mk_InN sbdTs k i;

           in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end;

         val f = Term.list_comb (mapFT, passive_ids @ map3 mk_f ks kks kks');

         val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs));

         val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2);

         val lhs = Free (strT_name i, strTT);

         val rhs = HOLogic.mk_split (Term.absfree Kl' (Term.absfree lab'

           (mk_sum_caseN fs $ (lab $ HOLogic.mk_list sum_sbdT []))));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map3 (fn i => fn mapFT => fn FT => Specification.definition

         (SOME (strT_bind i, NONE, NoSyn), (strT_def_bind i, strT_spec mapFT FT i)))

         ks tree_maps treeFTs

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val strT_defs = map ((fn def => trans OF [def RS fun_cong, @{thm prod.cases}]) o

       Morphism.thm phi) strT_def_frees;

     val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees;

     fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT);

     val carTAs = map (mk_carT As) ks;

     val strTAs = map2 mk_strT treeFTs ks;

     val coalgT_thm =

       Goal.prove_sorry lthy [] []

         (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_coalg As carTAs strTAs)))

         (mk_coalgT_tac m (coalg_def :: isNode_defs @ carT_defs) strT_defs set_mapss)

       |> Thm.close_derivation;

     val timer = time (timer "Tree coalgebra");

     fun mk_to_sbd s x i i' =

       mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;

     fun mk_from_sbd s x i i' =

       mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;

     fun mk_to_sbd_thmss thm = map (map (fn set_sbd =>

       thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss;

     val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj};

     val to_sbd_thmss = mk_to_sbd_thmss @{thm toCard};

     val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard};

     val Lev_bind = mk_internal_b LevN;

     val Lev_name = Binding.name_of Lev_bind;

     val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind);

     val Lev_spec =

       let

         fun mk_Suc i s setsAs a a' =

           let

             val sets = drop m setsAs;

             fun mk_set i' set b =

               let

                 val Cons = HOLogic.mk_eq (kl_copy,

                   mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl)

                 val b_set = HOLogic.mk_mem (b, set $ (s $ a));

                 val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b);

               in

                 HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl]

                   (HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec))))

               end;

           in

             Term.absfree a' (Library.foldl1 mk_union (map3 mk_set ks sets zs_copy))

           end;

         val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec'

           (HOLogic.mk_tuple (map5 mk_Suc ks ss setssAs zs zs')));

         val lhs = Term.list_comb (Free (Lev_name, LevT), ss);

         val rhs = mk_nat_rec Zero Suc;

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) =

       lthy

       |> Specification.definition (SOME (Lev_bind, NONE, NoSyn), (Lev_def_bind, Lev_spec))

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val Lev_def = Morphism.thm phi Lev_def_free;

     val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free));

     fun mk_Lev ss nat i =

       let

         val Ts = map fastype_of ss;

         val LevT = Library.foldr (op -->) (Ts, HOLogic.natT -->

           HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts));

       in

         mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i

       end;

     val Lev_0s = flat (mk_rec_simps n @{thm nat_rec_0} [Lev_def]);

     val Lev_Sucs = flat (mk_rec_simps n @{thm nat_rec_Suc} [Lev_def]);

     val rv_bind = mk_internal_b rvN;

     val rv_name = Binding.name_of rv_bind;

     val rv_def_bind = rpair [] (Thm.def_binding rv_bind);

     val rv_spec =

       let

         fun mk_Cons i s b b' =

           let

             fun mk_case i' =

               Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k));

           in

             Term.absfree b' (mk_sum_caseN (map mk_case ks) $ sumx)

           end;

         val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec'

           (HOLogic.mk_tuple (map4 mk_Cons ks ss zs zs'))));

         val lhs = Term.list_comb (Free (rv_name, rvT), ss);

         val rhs = mk_list_rec Nil Cons;

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) =

       lthy

       |> Specification.definition (SOME (rv_bind, NONE, NoSyn), (rv_def_bind, rv_spec))

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val rv_def = Morphism.thm phi rv_def_free;

     val rv = fst (Term.dest_Const (Morphism.term phi rv_free));

     fun mk_rv ss kl i =

       let

         val Ts = map fastype_of ss;

         val As = map domain_type Ts;

         val rvT = Library.foldr (op -->) (Ts, fastype_of kl -->

           HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As));

       in

         mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i

       end;

     val rv_Nils = flat (mk_rec_simps n @{thm list_rec_Nil} [rv_def]);

     val rv_Conss = flat (mk_rec_simps n @{thm list_rec_Cons} [rv_def]);

     val beh_binds = mk_internal_bs behN;

     fun beh_bind i = nth beh_binds (i - 1);

     val beh_name = Binding.name_of o beh_bind;

     val beh_def_bind = rpair [] o Thm.def_binding o beh_bind;

     fun beh_spec i z =

       let

         val mk_behT = Library.foldr (op -->) (map fastype_of (ss @ [z]), treeT);

         fun mk_case i to_sbd_map s k k' =

           Term.absfree k' (mk_InN bdFTs

             (Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i);

         val Lab = Term.absfree kl' (mk_If

           (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))

           (mk_sum_caseN (map5 mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z))

           (mk_undefined sbdFT));

         val lhs = Term.list_comb (Free (beh_name i, mk_behT), ss) $ z;

         val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT)

           (Term.absfree nat' (mk_Lev ss nat i $ z)), Lab);

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map2 (fn i => fn z => Specification.definition

         (SOME (beh_bind i, NONE, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val beh_defs = map (Morphism.thm phi) beh_def_frees;

     val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees;

     fun mk_beh ss i =

       let

         val Ts = map fastype_of ss;

         val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT);

       in

         Term.list_comb (Const (nth behs (i - 1), behT), ss)

       end;

     val Lev_sbd_thms =

       let

         fun mk_conjunct i z = mk_leq (mk_Lev ss nat i $ z) (mk_Field (mk_clists sum_sbd));

         val goal = list_all_free zs

           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));

         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];

         val Lev_sbd = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)

             (K (mk_Lev_sbd_tac lthy cts Lev_0s Lev_Sucs to_sbd_thmss))

           |> Thm.close_derivation);

         val Lev_sbd' = mk_specN n Lev_sbd;

       in

         map (fn i => Lev_sbd' RS mk_conjunctN n i) ks

       end;

     val (length_Lev_thms, length_Lev'_thms) =

       let

         fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),

           HOLogic.mk_eq (mk_size kl, nat));

         val goal = list_all_free (kl :: zs)

           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));

         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];

         val length_Lev = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)

             (K (mk_length_Lev_tac lthy cts Lev_0s Lev_Sucs))

           |> Thm.close_derivation);

         val length_Lev' = mk_specN (n + 1) length_Lev;

         val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks;

         fun mk_goal i z = fold_rev Logic.all (z :: kl :: nat :: ss) (Logic.mk_implies

             (HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z)),

             HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))));

         val goals = map2 mk_goal ks zs;

         val length_Levs' = map2 (fn goal => fn length_Lev =>

           Goal.prove_sorry lthy [] [] goal (K (mk_length_Lev'_tac length_Lev))

           |> Thm.close_derivation) goals length_Levs;

       in

         (length_Levs, length_Levs')

       end;

     val prefCl_Lev_thms =

       let

         fun mk_conjunct i z = HOLogic.mk_imp

           (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), mk_prefixeq kl_copy kl),

           HOLogic.mk_mem (kl_copy, mk_Lev ss (mk_size kl_copy) i $ z));

         val goal = list_all_free (kl :: kl_copy :: zs)

           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));

         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];

         val prefCl_Lev = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)

             (K (mk_prefCl_Lev_tac lthy cts Lev_0s Lev_Sucs)))

           |> Thm.close_derivation;

         val prefCl_Lev' = mk_specN (n + 2) prefCl_Lev;

       in

         map (fn i => prefCl_Lev' RS mk_conjunctN n i RS mp) ks

       end;

     val rv_last_thmss =

       let

         fun mk_conjunct i z i' z_copy = list_exists_free [z_copy]

           (HOLogic.mk_eq

             (mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z,

             mk_InN activeAs z_copy i'));

         val goal = list_all_free (k :: zs)

           (Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z =>

             Library.foldr1 HOLogic.mk_conj

               (map2 (mk_conjunct i z) ks zs_copy)) ks zs));

         val cTs = [SOME (certifyT lthy sum_sbdT)];

         val cts = map (SOME o certify lthy) [Term.absfree kl' goal, kl];

         val rv_last = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)

             (K (mk_rv_last_tac cTs cts rv_Nils rv_Conss)))

           |> Thm.close_derivation;

         val rv_last' = mk_specN (n + 1) rv_last;

       in

         map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks

       end;

     val set_rv_Lev_thmsss = if m = 0 then replicate n (replicate n []) else

       let

         fun mk_case s sets z z_free = Term.absfree z_free (Library.foldr1 HOLogic.mk_conj

           (map2 (fn set => fn A => mk_leq (set $ (s $ z)) A) (take m sets) As));

         fun mk_conjunct i z B = HOLogic.mk_imp

           (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), HOLogic.mk_mem (z, B)),

           mk_sum_caseN (map4 mk_case ss setssAs zs zs') $ (mk_rv ss kl i $ z));

         val goal = list_all_free (kl :: zs)

           (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct ks zs Bs));

         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];

         val set_rv_Lev = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] []

             (Logic.mk_implies (coalg_prem, HOLogic.mk_Trueprop goal))

             (K (mk_set_rv_Lev_tac lthy m cts Lev_0s Lev_Sucs rv_Nils rv_Conss

               coalg_set_thmss from_to_sbd_thmss)))

           |> Thm.close_derivation;

         val set_rv_Lev' = mk_specN (n + 1) set_rv_Lev;

       in

         map (fn i => map (fn i' =>

           split_conj_thm (if n = 1 then set_rv_Lev' RS mk_conjunctN n i RS mp

             else set_rv_Lev' RS mk_conjunctN n i RS mp RSN

               (2, @{thm sum_case_weak_cong} RS iffD1) RS

               (mk_sum_casesN n i' RS iffD1))) ks) ks

       end;

     val set_Lev_thmsss =

       let

         fun mk_conjunct i z =

           let

             fun mk_conjunct' i' sets s z' =

               let

                 fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp

                   (HOLogic.mk_mem (z'', set $ (s $ z')),

                     HOLogic.mk_mem (mk_append (kl,

                       HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']),

                       mk_Lev ss (HOLogic.mk_Suc nat) i $ z));

               in

                 HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'),

                   (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct'' ks (drop m sets) zs_copy2)))

               end;

           in

             HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),

               Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct' ks setssAs ss zs_copy))

           end;

         val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2)

           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));

         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];

         val set_Lev = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)

             (K (mk_set_Lev_tac lthy cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss)))

           |> Thm.close_derivation;

         val set_Lev' = mk_specN (3 * n + 1) set_Lev;

       in

         map (fn i => map (fn i' => map (fn i'' => set_Lev' RS

           mk_conjunctN n i RS mp RS

           mk_conjunctN n i' RS mp RS

           mk_conjunctN n i'' RS mp) ks) ks) ks

       end;

     val set_image_Lev_thmsss =

       let

         fun mk_conjunct i z =

           let

             fun mk_conjunct' i' sets =

               let

                 fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp

                   (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''),

                   HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z''))));

               in

                 HOLogic.mk_imp (HOLogic.mk_mem

                   (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']),

                     mk_Lev ss (HOLogic.mk_Suc nat) i $ z),

                   (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct'' ks sets ss zs_copy)))

               end;

           in

             HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),

               Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs')))

           end;

         val goal = list_all_free (kl :: k :: zs @ zs_copy)

           (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));

         val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];

         val set_image_Lev = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)

             (K (mk_set_image_Lev_tac lthy cts Lev_0s Lev_Sucs rv_Nils rv_Conss

               from_to_sbd_thmss to_sbd_inj_thmss)))

           |> Thm.close_derivation;

         val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev;

       in

         map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS

           mk_conjunctN n i RS mp RS

           mk_conjunctN n i'' RS mp RS

           mk_conjunctN n i' RS mp) ks) ks) ks

       end;

     val mor_beh_thm =

       Goal.prove_sorry lthy [] []

         (fold_rev Logic.all (As @ Bs @ ss) (Logic.mk_implies (coalg_prem,

           HOLogic.mk_Trueprop (mk_mor Bs ss carTAs strTAs (map (mk_beh ss) ks)))))

         (mk_mor_beh_tac m mor_def mor_cong_thm

           beh_defs carT_defs strT_defs isNode_defs

           to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbd_thms

           length_Lev_thms length_Lev'_thms prefCl_Lev_thms rv_last_thmss

           set_rv_Lev_thmsss set_Lev_thmsss set_image_Lev_thmsss

           set_mapss coalg_set_thmss map_comp_id_thms map_cong0s map_arg_cong_thms)

       |> Thm.close_derivation;

     val timer = time (timer "Behavioral morphism");

     fun mk_LSBIS As i = mk_lsbis As (map (mk_carT As) ks) strTAs i;

     fun mk_car_final As i =

       mk_quotient (mk_carT As i) (mk_LSBIS As i);

     fun mk_str_final As i =

       mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1),

         passive_ids @ map (mk_proj o mk_LSBIS As) ks), nth strTAs (i - 1)));

     val car_finalAs = map (mk_car_final As) ks;

     val str_finalAs = map (mk_str_final As) ks;

     val car_finals = map (mk_car_final passive_UNIVs) ks;

     val str_finals = map (mk_str_final passive_UNIVs) ks;

     val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss;

     val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms;

     val congruent_str_final_thms =

       let

         fun mk_goal R final_map strT =

           fold_rev Logic.all As (HOLogic.mk_Trueprop

             (mk_congruent R (HOLogic.mk_comp

               (Term.list_comb (final_map, passive_ids @ map (mk_proj o mk_LSBIS As) ks), strT))));

         val goals = map3 mk_goal (map (mk_LSBIS As) ks) final_maps strTAs;

       in

         map4 (fn goal => fn lsbisE => fn map_comp_id => fn map_cong0 =>

           Goal.prove_sorry lthy [] [] goal

             (K (mk_congruent_str_final_tac m lsbisE map_comp_id map_cong0 equiv_LSBIS_thms))

           |> Thm.close_derivation)

         goals lsbisE_thms map_comp_id_thms map_cong0s

       end;

     val coalg_final_thm = Goal.prove_sorry lthy [] [] (fold_rev Logic.all As

       (HOLogic.mk_Trueprop (mk_coalg As car_finalAs str_finalAs)))

       (K (mk_coalg_final_tac m coalg_def congruent_str_final_thms equiv_LSBIS_thms

         set_mapss coalgT_set_thmss))

       |> Thm.close_derivation;

     val mor_T_final_thm = Goal.prove_sorry lthy [] [] (fold_rev Logic.all As

       (HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finalAs str_finalAs

         (map (mk_proj o mk_LSBIS As) ks))))

       (K (mk_mor_T_final_tac mor_def congruent_str_final_thms equiv_LSBIS_thms))

       |> Thm.close_derivation;

     val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm];

     val in_car_final_thms = map (fn mor_image' => mor_image' OF

       [tcoalg_thm RS mor_final_thm, UNIV_I]) mor_image'_thms;

     val timer = time (timer "Final coalgebra");

     val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =

       lthy

       |> fold_map4 (fn b => fn mx => fn car_final => fn in_car_final =>

         typedef (Binding.conceal b, params, mx) car_final NONE

           (EVERY' [rtac exI, rtac in_car_final] 1)) bs mixfixes car_finals in_car_final_thms

       |>> apsnd split_list o split_list;

     val Ts = map (fn name => Type (name, params')) T_names;

     fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;

     val Ts' = mk_Ts passiveBs;

     val Ts'' = mk_Ts passiveCs;

     val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts;

     val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts;

     val Reps = map #Rep T_loc_infos;

     val Rep_injects = map #Rep_inject T_loc_infos;

     val Abs_inverses = map #Abs_inverse T_loc_infos;

     val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");

     val UNIVs = map HOLogic.mk_UNIV Ts;

     val FTs = mk_FTs (passiveAs @ Ts);

     val FTs' = mk_FTs (passiveBs @ Ts);

     val prodTs = map (HOLogic.mk_prodT o `I) Ts;

     val prodFTs = mk_FTs (passiveAs @ prodTs);

     val FTs_setss = mk_setss (passiveAs @ Ts);

     val prodFT_setss = mk_setss (passiveAs @ prodTs);

     val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs;

     val map_FT_nths = map2 (fn Ds =>

       mk_map_of_bnf Ds (passiveAs @ prodTs) (passiveAs @ Ts)) Dss bnfs;

     val fstsTs = map fst_const prodTs;

     val sndsTs = map snd_const prodTs;

     val dtorTs = map2 (curry op -->) Ts FTs;

     val ctorTs = map2 (curry op -->) FTs Ts;

     val unfold_fTs = map2 (curry op -->) activeAs Ts;

     val corec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) sum_sTs;

     val corec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls;

     val corec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls_rev;

     val corec_Inls = map (Term.subst_atomic_types (activeBs ~~ Ts)) Inls;

     val corec_UNIVs = map2 (HOLogic.mk_UNIV oo curry mk_sumT) Ts activeAs;

     val ((((((((((((((Jzs, Jzs'), (Jz's, Jz's')), Jzs_copy), Jz's_copy), Jzs1), Jzs2), Jpairs),

       FJzs), TRs), unfold_fs), unfold_fs_copy), corec_ss), phis), names_lthy) = names_lthy

       |> mk_Frees' "z" Ts

       ||>> mk_Frees' "y" Ts'

       ||>> mk_Frees "z'" Ts

       ||>> mk_Frees "y'" Ts'

       ||>> mk_Frees "z1" Ts

       ||>> mk_Frees "z2" Ts

       ||>> mk_Frees "j" (map2 (curry HOLogic.mk_prodT) Ts Ts')

       ||>> mk_Frees "x" prodFTs

       ||>> mk_Frees "r" (map (mk_relT o `I) Ts)

       ||>> mk_Frees "f" unfold_fTs

       ||>> mk_Frees "g" unfold_fTs

       ||>> mk_Frees "s" corec_sTs

       ||>> mk_Frees "P" (map2 mk_pred2T Ts Ts);

     fun dtor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtorN ^ "_");

     val dtor_name = Binding.name_of o dtor_bind;

     val dtor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o dtor_bind;

     fun dtor_spec i rep str map_FT dtorT Jz Jz' =

       let

         val lhs = Free (dtor_name i, dtorT);

         val rhs = Term.absfree Jz'

           (Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $

             (str $ (rep $ Jz)));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((dtor_frees, (_, dtor_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map7 (fn i => fn rep => fn str => fn mapx => fn dtorT => fn Jz => fn Jz' =>

         Specification.definition (SOME (dtor_bind i, NONE, NoSyn),

           (dtor_def_bind i, dtor_spec i rep str mapx dtorT Jz Jz')))

         ks Rep_Ts str_finals map_FTs dtorTs Jzs Jzs'

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     fun mk_dtors passive =

       map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (mk_params passive)) o

         Morphism.term phi) dtor_frees;

     val dtors = mk_dtors passiveAs;

     val dtor's = mk_dtors passiveBs;

     val dtor_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) dtor_def_frees;

     val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss;

     val (mor_Rep_thm, mor_Abs_thm) =

       let

         val mor_Rep =

           Goal.prove_sorry lthy [] []

             (HOLogic.mk_Trueprop (mk_mor UNIVs dtors car_finals str_finals Rep_Ts))

             (mk_mor_Rep_tac m (mor_def :: dtor_defs) Reps Abs_inverses coalg_final_set_thmss

               map_comp_id_thms map_cong0L_thms)

           |> Thm.close_derivation;

         val mor_Abs =

           Goal.prove_sorry lthy [] []

             (HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs dtors Abs_Ts))

             (mk_mor_Abs_tac (mor_def :: dtor_defs) Abs_inverses)

           |> Thm.close_derivation;

       in

         (mor_Rep, mor_Abs)

       end;

     val timer = time (timer "dtor definitions & thms");

     fun unfold_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_unfoldN ^ "_");

     val unfold_name = Binding.name_of o unfold_bind;

     val unfold_def_bind = rpair [] o Binding.conceal o Thm.def_binding o unfold_bind;

     fun unfold_spec i T AT abs f z z' =

       let

         val unfoldT = Library.foldr (op -->) (sTs, AT --> T);

         val lhs = Term.list_comb (Free (unfold_name i, unfoldT), ss);

         val rhs = Term.absfree z' (abs $ (f $ z));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((unfold_frees, (_, unfold_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map7 (fn i => fn T => fn AT => fn abs => fn f => fn z => fn z' =>

         Specification.definition

           (SOME (unfold_bind i, NONE, NoSyn), (unfold_def_bind i, unfold_spec i T AT abs f z z')))

           ks Ts activeAs Abs_Ts (map (fn i => HOLogic.mk_comp

             (mk_proj (mk_LSBIS passive_UNIVs i), mk_beh ss i)) ks) zs zs'

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val unfolds = map (Morphism.term phi) unfold_frees;

     val unfold_names = map (fst o dest_Const) unfolds;

     fun mk_unfolds passives actives =

       map3 (fn name => fn T => fn active =>

         Const (name, Library.foldr (op -->)

           (map2 (curry op -->) actives (mk_FTs (passives @ actives)), active --> T)))

       unfold_names (mk_Ts passives) actives;

     fun mk_unfold Ts ss i = Term.list_comb (Const (nth unfold_names (i - 1), Library.foldr (op -->)

       (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);

     val unfold_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) unfold_def_frees;

     val mor_unfold_thm =

       let

         val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses;

         val morEs' = map (fn thm =>

           (thm OF [tcoalg_thm RS mor_final_thm, UNIV_I]) RS sym) morE_thms;

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all ss

             (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors (map (mk_unfold Ts ss) ks))))

           (K (mk_mor_unfold_tac m mor_UNIV_thm dtor_defs unfold_defs Abs_inverses' morEs'

             map_comp_id_thms map_cong0s))

         |> Thm.close_derivation

       end;

     val dtor_unfold_thms = map (fn thm => (thm OF [mor_unfold_thm, UNIV_I]) RS sym) morE_thms;

     val (raw_coind_thms, raw_coind_thm) =

       let

         val prem = HOLogic.mk_Trueprop (mk_sbis passive_UNIVs UNIVs dtors TRs);

         val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

           (map2 (fn R => fn T => mk_leq R (Id_const T)) TRs Ts));

         val goal = fold_rev Logic.all TRs (Logic.mk_implies (prem, concl));

       in

         `split_conj_thm (Goal.prove_sorry lthy [] [] goal

           (K (mk_raw_coind_tac bis_def bis_cong_thm bis_O_thm bis_converse_thm bis_Gr_thm

             tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm

             lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects))

           |> Thm.close_derivation)

       end;

     val unique_mor_thms =

       let

         val prems = [HOLogic.mk_Trueprop (mk_coalg passive_UNIVs Bs ss), HOLogic.mk_Trueprop

           (HOLogic.mk_conj (mk_mor Bs ss UNIVs dtors unfold_fs,

             mk_mor Bs ss UNIVs dtors unfold_fs_copy))];

         fun mk_fun_eq B f g z = HOLogic.mk_imp

           (HOLogic.mk_mem (z, B), HOLogic.mk_eq (f $ z, g $ z));

         val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

           (map4 mk_fun_eq Bs unfold_fs unfold_fs_copy zs));

         val unique_mor = Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (Bs @ ss @ unfold_fs @ unfold_fs_copy @ zs)

             (Logic.list_implies (prems, unique)))

           (K (mk_unique_mor_tac raw_coind_thms bis_image2_thm))

           |> Thm.close_derivation;

       in

         map (fn thm => conjI RSN (2, thm RS mp)) (split_conj_thm unique_mor)

       end;

     val (unfold_unique_mor_thms, unfold_unique_mor_thm) =

       let

         val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors unfold_fs);

         fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_unfold Ts ss i);

         val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

           (map2 mk_fun_eq unfold_fs ks));

         val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm);

         val mor_thm = mor_comp_thm OF [tcoalg_thm RS mor_final_thm, mor_Abs_thm];

         val unique_mor = Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (ss @ unfold_fs) (Logic.mk_implies (prem, unique)))

           (K (mk_unfold_unique_mor_tac raw_coind_thms bis_thm mor_thm unfold_defs))

           |> Thm.close_derivation;

       in

         `split_conj_thm unique_mor

       end;

     val (dtor_unfold_unique_thms, dtor_unfold_unique_thm) = `split_conj_thm (split_conj_prems n

       (mor_UNIV_thm RS iffD2 RS unfold_unique_mor_thm));

     val unfold_dtor_thms = map (fn thm => mor_id_thm RS thm RS sym) unfold_unique_mor_thms;

     val unfold_o_dtor_thms =

       let

         val mor = mor_comp_thm OF [mor_str_thm, mor_unfold_thm];

       in

         map2 (fn unique => fn unfold_ctor =>

           trans OF [mor RS unique, unfold_ctor]) unfold_unique_mor_thms unfold_dtor_thms

       end;

     val timer = time (timer "unfold definitions & thms");

     val map_dtors = map2 (fn Ds => fn bnf =>

       Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,

         map HOLogic.id_const passiveAs @ dtors)) Dss bnfs;

     fun ctor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctorN ^ "_");

     val ctor_name = Binding.name_of o ctor_bind;

     val ctor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o ctor_bind;

     fun ctor_spec i ctorT =

       let

         val lhs = Free (ctor_name i, ctorT);

         val rhs = mk_unfold Ts map_dtors i;

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((ctor_frees, (_, ctor_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map2 (fn i => fn ctorT =>

         Specification.definition

           (SOME (ctor_bind i, NONE, NoSyn), (ctor_def_bind i, ctor_spec i ctorT))) ks ctorTs

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     fun mk_ctors params =

       map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)

         ctor_frees;

     val ctors = mk_ctors params';

     val ctor_defs = map (Morphism.thm phi) ctor_def_frees;

     val ctor_o_dtor_thms = map2 (fold_thms lthy o single) ctor_defs unfold_o_dtor_thms;

     val dtor_o_ctor_thms =

       let

         fun mk_goal dtor ctor FT =

          mk_Trueprop_eq (HOLogic.mk_comp (dtor, ctor), HOLogic.id_const FT);

         val goals = map3 mk_goal dtors ctors FTs;

       in

         map5 (fn goal => fn ctor_def => fn unfold => fn map_comp_id => fn map_cong0L =>

           Goal.prove_sorry lthy [] [] goal

             (mk_dtor_o_ctor_tac ctor_def unfold map_comp_id map_cong0L unfold_o_dtor_thms)

           |> Thm.close_derivation)

           goals ctor_defs dtor_unfold_thms map_comp_id_thms map_cong0L_thms

       end;

     val dtor_ctor_thms = map (fn thm => thm RS @{thm pointfree_idE}) dtor_o_ctor_thms;

     val ctor_dtor_thms = map (fn thm => thm RS @{thm pointfree_idE}) ctor_o_dtor_thms;

     val bij_dtor_thms =

       map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) ctor_o_dtor_thms dtor_o_ctor_thms;

     val inj_dtor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_dtor_thms;

     val surj_dtor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_dtor_thms;

     val dtor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_dtor_thms;

     val dtor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_dtor_thms;

     val dtor_exhaust_thms = map (fn thm => thm RS exE) dtor_nchotomy_thms;

     val bij_ctor_thms =

       map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) dtor_o_ctor_thms ctor_o_dtor_thms;

     val inj_ctor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_ctor_thms;

     val surj_ctor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_ctor_thms;

     val ctor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_ctor_thms;

     val ctor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_ctor_thms;

     val ctor_exhaust_thms = map (fn thm => thm RS exE) ctor_nchotomy_thms;

     val timer = time (timer "ctor definitions & thms");

     val corec_Inl_sum_thms =

       let

         val mor = mor_comp_thm OF [mor_sum_case_thm, mor_unfold_thm];

       in

         map2 (fn unique => fn unfold_dtor =>

           trans OF [mor RS unique, unfold_dtor]) unfold_unique_mor_thms unfold_dtor_thms

       end;

     fun corec_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_corecN ^ "_");

     val corec_name = Binding.name_of o corec_bind;

     val corec_def_bind = rpair [] o Binding.conceal o Thm.def_binding o corec_bind;

     val corec_strs =

       map3 (fn dtor => fn sum_s => fn mapx =>

         mk_sum_case

           (HOLogic.mk_comp (Term.list_comb (mapx, passive_ids @ corec_Inls), dtor), sum_s))

       dtors corec_ss corec_maps;

     fun corec_spec i T AT =

       let

         val corecT = Library.foldr (op -->) (corec_sTs, AT --> T);

         val lhs = Term.list_comb (Free (corec_name i, corecT), corec_ss);

         val rhs = HOLogic.mk_comp (mk_unfold Ts corec_strs i, Inr_const T AT);

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((corec_frees, (_, corec_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map3 (fn i => fn T => fn AT =>

         Specification.definition

           (SOME (corec_bind i, NONE, NoSyn), (corec_def_bind i, corec_spec i T AT)))

           ks Ts activeAs

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val corecs = map (Morphism.term phi) corec_frees;

     val corec_names = map (fst o dest_Const) corecs;

     fun mk_corec ss i = Term.list_comb (Const (nth corec_names (i - 1), Library.foldr (op -->)

       (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);

     val corec_defs = map (Morphism.thm phi) corec_def_frees;

     val sum_cases =

       map2 (fn T => fn i => mk_sum_case (HOLogic.id_const T, mk_corec corec_ss i)) Ts ks;

     val dtor_corec_thms =

       let

         fun mk_goal i corec_s corec_map dtor z =

           let

             val lhs = dtor $ (mk_corec corec_ss i $ z);

             val rhs = Term.list_comb (corec_map, passive_ids @ sum_cases) $ (corec_s $ z);

           in

             fold_rev Logic.all (z :: corec_ss) (mk_Trueprop_eq (lhs, rhs))

           end;

         val goals = map5 mk_goal ks corec_ss corec_maps_rev dtors zs;

       in

         map3 (fn goal => fn unfold => fn map_cong0 =>

           Goal.prove_sorry lthy [] [] goal

             (mk_corec_tac m corec_defs unfold map_cong0 corec_Inl_sum_thms)

           |> Thm.close_derivation)

         goals dtor_unfold_thms map_cong0s

       end;

     val corec_unique_mor_thm =

       let

         val id_fs = map2 (fn T => fn f => mk_sum_case (HOLogic.id_const T, f)) Ts unfold_fs;

         val prem = HOLogic.mk_Trueprop (mk_mor corec_UNIVs corec_strs UNIVs dtors id_fs);

         fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_corec corec_ss i);

         val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

           (map2 mk_fun_eq unfold_fs ks));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (corec_ss @ unfold_fs) (Logic.mk_implies (prem, unique)))

           (mk_corec_unique_mor_tac corec_defs corec_Inl_sum_thms unfold_unique_mor_thm)

           |> Thm.close_derivation

       end;

     val map_id0s_o_id =

       map (fn thm =>

         mk_trans (thm RS @{thm arg_cong2[of _ _ _ _ "op o", OF _ refl]}) @{thm id_o})

       map_id0s;

     val (dtor_corec_unique_thms, dtor_corec_unique_thm) =

       `split_conj_thm (split_conj_prems n

         (mor_UNIV_thm RS iffD2 RS corec_unique_mor_thm)

         |> Local_Defs.unfold lthy (@{thms o_sum_case o_id id_o o_assoc sum_case_o_inj(1)} @

            map_id0s_o_id @ sym_map_comps)

         OF replicate n @{thm arg_cong2[of _ _ _ _ sum_case, OF refl]});

     val timer = time (timer "corec definitions & thms");

     val (dtor_map_coinduct_thm, coinduct_params, dtor_coinduct_thm) =

       let

         val zs = Jzs1 @ Jzs2;

         val frees = phis @ zs;

         val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs;

         fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2));

         val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

           (map3 mk_concl phis Jzs1 Jzs2));

         fun mk_rel_prem phi dtor rel Jz Jz_copy =

           let

             val concl = Term.list_comb (rel, map HOLogic.eq_const passiveAs @ phis) $

               (dtor $ Jz) $ (dtor $ Jz_copy);

           in

             HOLogic.mk_Trueprop

               (list_all_free [Jz, Jz_copy] (HOLogic.mk_imp (phi $ Jz $ Jz_copy, concl)))

           end;

         val rel_prems = map5 mk_rel_prem phis dtors rels Jzs Jzs_copy;

         val dtor_coinduct_goal =

           fold_rev Logic.all frees (Logic.list_implies (rel_prems, concl));

         val dtor_coinduct =

           Goal.prove_sorry lthy [] [] dtor_coinduct_goal

             (K (mk_dtor_coinduct_tac m raw_coind_thm bis_rel_thm rel_congs))

           |> Thm.close_derivation;

         fun mk_prem phi dtor map_nth sets Jz Jz_copy FJz =

           let

             val xs = [Jz, Jz_copy];

             fun mk_map_conjunct nths x =

               HOLogic.mk_eq (Term.list_comb (map_nth, passive_ids @ nths) $ FJz, dtor $ x);

             fun mk_set_conjunct set phi z1 z2 =

               list_all_free [z1, z2]

                 (HOLogic.mk_imp (HOLogic.mk_mem (HOLogic.mk_prod (z1, z2), set $ FJz),

                   phi $ z1 $ z2));

             val concl = list_exists_free [FJz] (HOLogic.mk_conj

               (Library.foldr1 HOLogic.mk_conj (map2 mk_map_conjunct [fstsTs, sndsTs] xs),

               Library.foldr1 HOLogic.mk_conj

                 (map4 mk_set_conjunct (drop m sets) phis Jzs1 Jzs2)));

           in

             fold_rev Logic.all xs (Logic.mk_implies

               (HOLogic.mk_Trueprop (Term.list_comb (phi, xs)), HOLogic.mk_Trueprop concl))

           end;

         val prems = map7 mk_prem phis dtors map_FT_nths prodFT_setss Jzs Jzs_copy FJzs;

         val dtor_map_coinduct_goal = fold_rev Logic.all frees (Logic.list_implies (prems, concl));

         val dtor_map_coinduct =

           Goal.prove_sorry lthy [] [] dtor_map_coinduct_goal

             (K (mk_dtor_map_coinduct_tac m ks raw_coind_thm bis_def))

           |> Thm.close_derivation;

       in

         (dtor_map_coinduct, rev (Term.add_tfrees dtor_map_coinduct_goal []), dtor_coinduct)

       end;

     val timer = time (timer "coinduction");

     fun mk_dtor_map_DEADID_thm dtor_inject map_id0 =

       trans OF [iffD2 OF [dtor_inject, id_apply], map_id0 RS sym];

     fun mk_dtor_Jrel_DEADID_thm dtor_inject bnf =

       trans OF [rel_eq_of_bnf bnf RS @{thm predicate2_eqD}, dtor_inject] RS sym;

     val JphiTs = map2 mk_pred2T passiveAs passiveBs;

     val prodTsTs' = map2 (curry HOLogic.mk_prodT) Ts Ts';

     val fstsTsTs' = map fst_const prodTsTs';

     val sndsTsTs' = map snd_const prodTsTs';

     val activephiTs = map2 mk_pred2T activeAs activeBs;

     val activeJphiTs = map2 mk_pred2T Ts Ts';

     val (((Jphis, activephis), activeJphis), names_lthy) = names_lthy

       |> mk_Frees "R" JphiTs

       ||>> mk_Frees "S" activephiTs

       ||>> mk_Frees "JR" activeJphiTs;

     val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;

     val in_rels = map in_rel_of_bnf bnfs;

     (*register new codatatypes as BNFs*)

     val (timer, Jbnfs, (folded_dtor_map_o_thms, folded_dtor_map_thms), folded_dtor_set_thmss',

       dtor_set_induct_thms, dtor_Jrel_thms, Jbnf_notes, lthy) =

       if m = 0 then

         (timer, replicate n DEADID_bnf,

         map_split (`(mk_pointfree lthy)) (map2 mk_dtor_map_DEADID_thm dtor_inject_thms map_ids),

         replicate n [], [], map2 mk_dtor_Jrel_DEADID_thm dtor_inject_thms bnfs, [], lthy)

       else let

         val fTs = map2 (curry op -->) passiveAs passiveBs;

         val gTs = map2 (curry op -->) passiveBs passiveCs;

         val f1Ts = map2 (curry op -->) passiveAs passiveYs;

         val f2Ts = map2 (curry op -->) passiveBs passiveYs;

         val p1Ts = map2 (curry op -->) passiveXs passiveAs;

         val p2Ts = map2 (curry op -->) passiveXs passiveBs;

         val pTs = map2 (curry op -->) passiveXs passiveCs;

         val uTs = map2 (curry op -->) Ts Ts';

         val B1Ts = map HOLogic.mk_setT passiveAs;

         val B2Ts = map HOLogic.mk_setT passiveBs;

         val AXTs = map HOLogic.mk_setT passiveXs;

         val XTs = mk_Ts passiveXs;

         val YTs = mk_Ts passiveYs;

         val ((((((((((((((((((fs, fs'), fs_copy), gs), us),

           (Jys, Jys')), (Jys_copy, Jys'_copy)), dtor_set_induct_phiss),

           B1s), B2s), AXs), f1s), f2s), p1s), p2s), ps), (ys, ys')), (ys_copy, ys'_copy)),

           names_lthy) = names_lthy

           |> mk_Frees' "f" fTs

           ||>> mk_Frees "f" fTs

           ||>> mk_Frees "g" gTs

           ||>> mk_Frees "u" uTs

           ||>> mk_Frees' "b" Ts'

           ||>> mk_Frees' "b" Ts'

           ||>> mk_Freess "P" (map (fn A => map (mk_pred2T A) Ts) passiveAs)

           ||>> mk_Frees "B1" B1Ts

           ||>> mk_Frees "B2" B2Ts

           ||>> mk_Frees "A" AXTs

           ||>> mk_Frees "f1" f1Ts

           ||>> mk_Frees "f2" f2Ts

           ||>> mk_Frees "p1" p1Ts

           ||>> mk_Frees "p2" p2Ts

           ||>> mk_Frees "p" pTs

           ||>> mk_Frees' "y" passiveAs

           ||>> mk_Frees' "y" passiveAs;

         val map_FTFT's = map2 (fn Ds =>

           mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;

         fun mk_maps ATs BTs Ts mk_T =

           map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ map mk_T Ts)) Dss bnfs;

         fun mk_Fmap mk_const fs Ts Fmap = Term.list_comb (Fmap, fs @ map mk_const Ts);

         fun mk_map mk_const mk_T Ts fs Ts' dtors mk_maps =

           mk_unfold Ts' (map2 (fn dtor => fn Fmap =>

             HOLogic.mk_comp (mk_Fmap mk_const fs Ts Fmap, dtor)) dtors (mk_maps Ts mk_T));

         val mk_map_id = mk_map HOLogic.id_const I;

         val mk_mapsAB = mk_maps passiveAs passiveBs;

         val mk_mapsBC = mk_maps passiveBs passiveCs;

         val mk_mapsAC = mk_maps passiveAs passiveCs;

         val mk_mapsAY = mk_maps passiveAs passiveYs;

         val mk_mapsBY = mk_maps passiveBs passiveYs;

         val mk_mapsXA = mk_maps passiveXs passiveAs;

         val mk_mapsXB = mk_maps passiveXs passiveBs;

         val mk_mapsXC = mk_maps passiveXs passiveCs;

         val fs_maps = map (mk_map_id Ts fs Ts' dtors mk_mapsAB) ks;

         val fs_copy_maps = map (mk_map_id Ts fs_copy Ts' dtors mk_mapsAB) ks;

         val gs_maps = map (mk_map_id Ts' gs Ts'' dtor's mk_mapsBC) ks;

         val fgs_maps =

           map (mk_map_id Ts (map2 (curry HOLogic.mk_comp) gs fs) Ts'' dtors mk_mapsAC) ks;

         val Xdtors = mk_dtors passiveXs;

         val UNIV's = map HOLogic.mk_UNIV Ts';

         val CUNIVs = map HOLogic.mk_UNIV passiveCs;

         val UNIV''s = map HOLogic.mk_UNIV Ts'';

         val dtor''s = mk_dtors passiveCs;

         val f1s_maps = map (mk_map_id Ts f1s YTs dtors mk_mapsAY) ks;

         val f2s_maps = map (mk_map_id Ts' f2s YTs dtor's mk_mapsBY) ks;

         val pid_maps = map (mk_map_id XTs ps Ts'' Xdtors mk_mapsXC) ks;

         val pfst_Fmaps =

           map (mk_Fmap fst_const p1s prodTsTs') (mk_mapsXA prodTsTs' (fst o HOLogic.dest_prodT));

         val psnd_Fmaps =

           map (mk_Fmap snd_const p2s prodTsTs') (mk_mapsXB prodTsTs' (snd o HOLogic.dest_prodT));

         val p1id_Fmaps = map (mk_Fmap HOLogic.id_const p1s prodTsTs') (mk_mapsXA prodTsTs' I);

         val p2id_Fmaps = map (mk_Fmap HOLogic.id_const p2s prodTsTs') (mk_mapsXB prodTsTs' I);

         val pid_Fmaps = map (mk_Fmap HOLogic.id_const ps prodTsTs') (mk_mapsXC prodTsTs' I);

         val (dtor_map_thms, map_thms) =

           let

             fun mk_goal fs_map map dtor dtor' = fold_rev Logic.all fs

               (mk_Trueprop_eq (HOLogic.mk_comp (dtor', fs_map),

                 HOLogic.mk_comp (Term.list_comb (map, fs @ fs_maps), dtor)));

             val goals = map4 mk_goal fs_maps map_FTFT's dtors dtor's;

             val cTs = map (SOME o certifyT lthy) FTs';

             val maps =

               map5 (fn goal => fn cT => fn unfold => fn map_comp => fn map_cong0 =>

                 Goal.prove_sorry lthy [] [] goal

                   (K (mk_map_tac m n cT unfold map_comp map_cong0))

                 |> Thm.close_derivation)

               goals cTs dtor_unfold_thms map_comps map_cong0s;

           in

             map_split (fn thm => (thm RS @{thm comp_eq_dest}, thm)) maps

           end;

         val map_comp0_thms =

           let

             val goal = fold_rev Logic.all (fs @ gs)

               (HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                 (map3 (fn fmap => fn gmap => fn fgmap =>

                    HOLogic.mk_eq (HOLogic.mk_comp (gmap, fmap), fgmap))

                 fs_maps gs_maps fgs_maps)))

           in

             split_conj_thm (Goal.prove_sorry lthy [] [] goal

               (K (mk_map_comp0_tac m n map_thms map_comp0s map_cong0s dtor_unfold_unique_thm))

               |> Thm.close_derivation)

           end;

         val dtor_map_unique_thm =

           let

             fun mk_prem u map dtor dtor' =

               mk_Trueprop_eq (HOLogic.mk_comp (dtor', u),

                 HOLogic.mk_comp (Term.list_comb (map, fs @ us), dtor));

             val prems = map4 mk_prem us map_FTFT's dtors dtor's;

             val goal =

               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                 (map2 (curry HOLogic.mk_eq) us fs_maps));

           in

             Goal.prove_sorry lthy [] []

               (fold_rev Logic.all (us @ fs) (Logic.list_implies (prems, goal)))

               (mk_dtor_map_unique_tac dtor_unfold_unique_thm sym_map_comps)

               |> Thm.close_derivation

           end;

         val timer = time (timer "map functions for the new codatatypes");

         val bd = mk_cexp sbd sbd;

         val timer = time (timer "bounds for the new codatatypes");

         val setss_by_bnf = map (fn i => map2 (mk_hset dtors i) ls passiveAs) ks;

         val setss_by_bnf' = map (fn i => map2 (mk_hset dtor's i) ls passiveBs) ks;

         val setss_by_range = transpose setss_by_bnf;

         val dtor_set_thmss =

           let

             fun mk_simp_goal relate pas_set act_sets sets dtor z set =

               relate (set $ z, mk_union (pas_set $ (dtor $ z),

                  Library.foldl1 mk_union

                    (map2 (fn X => mk_UNION (X $ (dtor $ z))) act_sets sets)));

             fun mk_goals eq =

               map2 (fn i => fn sets =>

                 map4 (fn Fsets =>

                   mk_simp_goal eq (nth Fsets (i - 1)) (drop m Fsets) sets)

                 FTs_setss dtors Jzs sets)

               ls setss_by_range;

             val le_goals = map

               (fold_rev Logic.all Jzs o HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj)

               (mk_goals (uncurry mk_leq));

             val set_le_thmss = map split_conj_thm

               (map4 (fn goal => fn hset_minimal => fn set_hsets => fn set_hset_hsetss =>

                 Goal.prove_sorry lthy [] [] goal

                   (K (mk_set_le_tac n hset_minimal set_hsets set_hset_hsetss))

                 |> Thm.close_derivation)

               le_goals hset_minimal_thms set_hset_thmss' set_hset_hset_thmsss');

             val simp_goalss = map (map2 (fn z => fn goal =>

                 Logic.all z (HOLogic.mk_Trueprop goal)) Jzs)

               (mk_goals HOLogic.mk_eq);

           in

             map4 (map4 (fn goal => fn set_le => fn set_incl_hset => fn set_hset_incl_hsets =>

               Goal.prove_sorry lthy [] [] goal

                 (K (mk_dtor_set_tac n set_le set_incl_hset set_hset_incl_hsets))

               |> Thm.close_derivation))

             simp_goalss set_le_thmss set_incl_hset_thmss' set_hset_incl_hset_thmsss'

           end;

         val timer = time (timer "set functions for the new codatatypes");

         val colss = map2 (fn j => fn T =>

           map (fn i => mk_hset_rec dtors nat i j T) ks) ls passiveAs;

         val colss' = map2 (fn j => fn T =>

           map (fn i => mk_hset_rec dtor's nat i j T) ks) ls passiveBs;

         val Xcolss = map2 (fn j => fn T =>

           map (fn i => mk_hset_rec Xdtors nat i j T) ks) ls passiveXs;

         val col_natural_thmss =

           let

             fun mk_col_natural f map z col col' =

               HOLogic.mk_eq (mk_image f $ (col $ z), col' $ (map $ z));

             fun mk_goal f cols cols' = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj

               (map4 (mk_col_natural f) fs_maps Jzs cols cols'));

             val goals = map3 mk_goal fs colss colss';

             val ctss =

               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) goals;

             val thms =

               map4 (fn goal => fn cts => fn rec_0s => fn rec_Sucs =>

                 singleton (Proof_Context.export names_lthy lthy)

                   (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)

                     (mk_col_natural_tac cts rec_0s rec_Sucs dtor_map_thms set_mapss))

                 |> Thm.close_derivation)

               goals ctss hset_rec_0ss' hset_rec_Sucss';

           in

             map (split_conj_thm o mk_specN n) thms

           end;

         val col_bd_thmss =

           let

             fun mk_col_bd z col = mk_ordLeq (mk_card_of (col $ z)) sbd;

             fun mk_goal cols = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj

               (map2 mk_col_bd Jzs cols));

             val goals = map mk_goal colss;

             val ctss =

               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) goals;

             val thms =

               map5 (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>

                 singleton (Proof_Context.export names_lthy lthy)

                   (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)

                     (K (mk_col_bd_tac m j cts rec_0s rec_Sucs

                       sbd_Card_order sbd_Cinfinite set_sbdss)))

                 |> Thm.close_derivation)

               ls goals ctss hset_rec_0ss' hset_rec_Sucss';

           in

             map (split_conj_thm o mk_specN n) thms

           end;

         val map_cong0_thms =

           let

             val cTs = map (SOME o certifyT lthy o

               Term.typ_subst_atomic (passiveAs ~~ passiveBs) o TFree) coinduct_params;

             fun mk_prem z set f g y y' =

               mk_Ball (set $ z) (Term.absfree y' (HOLogic.mk_eq (f $ y, g $ y)));

             fun mk_prems sets z =

               Library.foldr1 HOLogic.mk_conj (map5 (mk_prem z) sets fs fs_copy ys ys')

             fun mk_map_cong0 sets z fmap gmap =

               HOLogic.mk_imp (mk_prems sets z, HOLogic.mk_eq (fmap $ z, gmap $ z));

             fun mk_coind_body sets (x, T) z fmap gmap y y_copy =

               HOLogic.mk_conj

                 (HOLogic.mk_mem (z, HOLogic.mk_Collect (x, T, mk_prems sets z)),

                   HOLogic.mk_conj (HOLogic.mk_eq (y, fmap $ z),

                     HOLogic.mk_eq (y_copy, gmap $ z)))

             fun mk_cphi sets (z' as (x, T)) z fmap gmap y' y y'_copy y_copy =

               HOLogic.mk_exists (x, T, mk_coind_body sets z' z fmap gmap y y_copy)

               |> Term.absfree y'_copy

               |> Term.absfree y'

               |> certify lthy;

             val cphis =

               map9 mk_cphi setss_by_bnf Jzs' Jzs fs_maps fs_copy_maps Jys' Jys Jys'_copy Jys_copy;

             val coinduct = Drule.instantiate' cTs (map SOME cphis) dtor_map_coinduct_thm;

             val goal =

               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                 (map4 mk_map_cong0 setss_by_bnf Jzs fs_maps fs_copy_maps));

             val thm = singleton (Proof_Context.export names_lthy lthy)

               (Goal.prove_sorry lthy [] [] goal

               (K (mk_mcong_tac lthy m (rtac coinduct) map_comps dtor_map_thms map_cong0s set_mapss

               set_hset_thmss set_hset_hset_thmsss)))

               |> Thm.close_derivation

           in

             split_conj_thm thm

           end;

         val B1_ins = map2 (mk_in B1s) setss_by_bnf Ts;

         val B2_ins = map2 (mk_in B2s) setss_by_bnf' Ts';

         val thePulls = map4 mk_thePull B1_ins B2_ins f1s_maps f2s_maps;

         val thePullTs = passiveXs @ map2 (curry HOLogic.mk_prodT) Ts Ts';

         val thePull_ins = map2 (mk_in (AXs @ thePulls)) (mk_setss thePullTs) (mk_FTs thePullTs);

         val pickFs = map5 mk_pickWP thePull_ins pfst_Fmaps psnd_Fmaps

           (map2 (curry op $) dtors Jzs) (map2 (curry op $) dtor's Jz's);

         val pickF_ss = map3 (fn pickF => fn z => fn z' =>

           HOLogic.mk_split (Term.absfree z (Term.absfree z' pickF))) pickFs Jzs' Jz's';

         val picks = map (mk_unfold XTs pickF_ss) ks;

         val wpull_prem = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

           (map8 mk_wpull AXs B1s B2s f1s f2s (replicate m NONE) p1s p2s));

         val map_eq_thms = map2 (fn simp => fn diff => box_equals OF [diff RS iffD2, simp, simp])

           dtor_map_thms dtor_inject_thms;

         val map_wpull_thms = map (fn thm => thm OF

           (replicate m asm_rl @ replicate n @{thm wpull_thePull})) map_wpulls;

         val pickWP_assms_tacs =

           map3 mk_pickWP_assms_tac set_incl_hset_thmss set_incl_hin_thmss map_eq_thms;

         val coalg_thePull_thm =

           let

             val coalg = HOLogic.mk_Trueprop

               (mk_coalg CUNIVs thePulls (map2 (curry HOLogic.mk_comp) pid_Fmaps pickF_ss));

             val goal = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s @ ps)

               (Logic.mk_implies (wpull_prem, coalg));

           in

             Goal.prove_sorry lthy [] [] goal (mk_coalg_thePull_tac m coalg_def map_wpull_thms

               set_mapss pickWP_assms_tacs)

             |> Thm.close_derivation

           end;

         val (mor_thePull_fst_thm, mor_thePull_snd_thm, mor_thePull_pick_thm) =

           let

             val mor_fst = HOLogic.mk_Trueprop

               (mk_mor thePulls (map2 (curry HOLogic.mk_comp) p1id_Fmaps pickF_ss)

                 UNIVs dtors fstsTsTs');

             val mor_snd = HOLogic.mk_Trueprop

               (mk_mor thePulls (map2 (curry HOLogic.mk_comp) p2id_Fmaps pickF_ss)

                 UNIV's dtor's sndsTsTs');

             val mor_pick = HOLogic.mk_Trueprop

               (mk_mor thePulls (map2 (curry HOLogic.mk_comp) pid_Fmaps pickF_ss)

                 UNIV''s dtor''s (map2 (curry HOLogic.mk_comp) pid_maps picks));

             val fst_goal = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s)

               (Logic.mk_implies (wpull_prem, mor_fst));

             val snd_goal = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s)

               (Logic.mk_implies (wpull_prem, mor_snd));

             val pick_goal = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s @ ps)

               (Logic.mk_implies (wpull_prem, mor_pick));

           in

             (Goal.prove_sorry lthy [] [] fst_goal (mk_mor_thePull_fst_tac m mor_def map_wpull_thms

               map_comps pickWP_assms_tacs) |> Thm.close_derivation,

             Goal.prove_sorry lthy [] [] snd_goal (mk_mor_thePull_snd_tac m mor_def map_wpull_thms

               map_comps pickWP_assms_tacs) |> Thm.close_derivation,

             Goal.prove_sorry lthy [] [] pick_goal (mk_mor_thePull_pick_tac mor_def dtor_unfold_thms

               map_comps) |> Thm.close_derivation)

           end;

         val pick_col_thmss =

           let

             fun mk_conjunct AX Jpair pick thePull col =

               HOLogic.mk_imp (HOLogic.mk_mem (Jpair, thePull), mk_leq (col $ (pick $ Jpair)) AX);

             fun mk_concl AX cols =

               list_all_free Jpairs (Library.foldr1 HOLogic.mk_conj

                 (map4 (mk_conjunct AX) Jpairs picks thePulls cols));

             val concls = map2 mk_concl AXs Xcolss;

             val ctss =

               map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;

             val goals =

               map (fn concl => Logic.mk_implies (wpull_prem, HOLogic.mk_Trueprop concl)) concls;

             val thms =

               map5 (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>

                 singleton (Proof_Context.export names_lthy lthy) (Goal.prove_sorry lthy [] [] goal

                   (mk_pick_col_tac m j cts rec_0s rec_Sucs dtor_unfold_thms set_mapss

                     map_wpull_thms pickWP_assms_tacs))

                 |> Thm.close_derivation)

               ls goals ctss hset_rec_0ss' hset_rec_Sucss';

           in

             map (map (fn thm => thm RS mp) o split_conj_thm o mk_specN n) thms

           end;

         val timer = time (timer "helpers for BNF properties");

         val map_id0_tacs =

           map2 (K oo mk_map_id0_tac map_thms) dtor_unfold_unique_thms unfold_dtor_thms;

         val map_comp0_tacs = map (fn thm => K (rtac (thm RS sym) 1)) map_comp0_thms;

         val map_cong0_tacs = map (mk_map_cong0_tac m) map_cong0_thms;

         val set_nat_tacss =

           map2 (map2 (K oo mk_set_map0_tac)) hset_defss (transpose col_natural_thmss);

         val bd_co_tacs = replicate n (K (mk_bd_card_order_tac sbd_card_order));

         val bd_cinf_tacs = replicate n (K (mk_bd_cinfinite_tac sbd_Cinfinite));

         val set_bd_tacss =

           map2 (map2 (K oo mk_set_bd_tac sbd_Cinfinite)) hset_defss (transpose col_bd_thmss);

         val map_wpull_tacs =

           map3 (K ooo mk_wpull_tac m coalg_thePull_thm mor_thePull_fst_thm mor_thePull_snd_thm

             mor_thePull_pick_thm) unique_mor_thms (transpose pick_col_thmss) hset_defss;

         val rel_OO_Grp_tacs = replicate n (K (rtac refl 1));

         val tacss = map9 zip_axioms map_id0_tacs map_comp0_tacs map_cong0_tacs set_nat_tacss

           bd_co_tacs bd_cinf_tacs set_bd_tacss map_wpull_tacs rel_OO_Grp_tacs;

         val (hset_dtor_incl_thmss, hset_hset_dtor_incl_thmsss, dtor_hset_induct_thms) =

           let

             fun tinst_of dtor =

               map (SOME o certify lthy) (dtor :: remove (op =) dtor dtors);

             fun tinst_of' dtor = case tinst_of dtor of t :: ts => t :: NONE :: ts;

             val Tinst = map (pairself (certifyT lthy))

               (map Logic.varifyT_global (deads @ allAs) ~~ (deads @ passiveAs @ Ts));

             val set_incl_thmss =

               map2 (fn dtor => map (singleton (Proof_Context.export names_lthy lthy) o

                 Drule.instantiate' [] (tinst_of' dtor) o

                 Thm.instantiate (Tinst, []) o Drule.zero_var_indexes))

               dtors set_incl_hset_thmss;

             val tinst = splice (map (SOME o certify lthy) dtors) (replicate n NONE)

             val set_minimal_thms =

               map (Drule.instantiate' [] tinst o Thm.instantiate (Tinst, []) o

                 Drule.zero_var_indexes)

               hset_minimal_thms;

             val set_set_incl_thmsss =

               map2 (fn dtor => map (map (singleton (Proof_Context.export names_lthy lthy) o

                 Drule.instantiate' [] (NONE :: tinst_of' dtor) o

                 Thm.instantiate (Tinst, []) o Drule.zero_var_indexes)))

               dtors set_hset_incl_hset_thmsss;

             val set_set_incl_thmsss' = transpose (map transpose set_set_incl_thmsss);

             val incls =

               maps (map (fn thm => thm RS @{thm subset_Collect_iff})) set_incl_thmss @

                 @{thms subset_Collect_iff[OF subset_refl]};

             fun mk_induct_tinst phis jsets y y' =

               map4 (fn phi => fn jset => fn Jz => fn Jz' =>

                 SOME (certify lthy (Term.absfree Jz' (HOLogic.mk_Collect (fst y', snd y',

                   HOLogic.mk_conj (HOLogic.mk_mem (y, jset $ Jz), phi $ y $ Jz))))))

               phis jsets Jzs Jzs';

             val dtor_set_induct_thms =

               map6 (fn set_minimal => fn set_set_inclss => fn jsets => fn y => fn y' => fn phis =>

                 ((set_minimal

                   |> Drule.instantiate' [] (mk_induct_tinst phis jsets y y')

                   |> unfold_thms lthy incls) OF

                   (replicate n ballI @

                     maps (map (fn thm => thm RS @{thm subset_CollectI})) set_set_inclss))

                 |> singleton (Proof_Context.export names_lthy lthy)

                 |> rule_by_tactic lthy (ALLGOALS (TRY o etac asm_rl)))

               set_minimal_thms set_set_incl_thmsss' setss_by_range ys ys' dtor_set_induct_phiss

           in

             (set_incl_thmss, set_set_incl_thmsss, dtor_set_induct_thms)

           end;

         fun close_wit I wit = (I, fold_rev Term.absfree (map (nth ys') I) wit);

         val all_unitTs = replicate live HOLogic.unitT;

         val unitTs = replicate n HOLogic.unitT;

         val unit_funs = replicate n (Term.absdummy HOLogic.unitT HOLogic.unit);

         fun mk_map_args I =

           map (fn i =>

             if member (op =) I i then Term.absdummy HOLogic.unitT (nth ys i)

             else mk_undefined (HOLogic.unitT --> nth passiveAs i))

           (0 upto (m - 1));

         fun mk_nat_wit Ds bnf (I, wit) () =

           let

             val passiveI = filter (fn i => i < m) I;

             val map_args = mk_map_args passiveI;

           in

             Term.absdummy HOLogic.unitT (Term.list_comb

               (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $ wit)

           end;

         fun mk_dummy_wit Ds bnf I =

           let

             val map_args = mk_map_args I;

           in

             Term.absdummy HOLogic.unitT (Term.list_comb

               (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $

               mk_undefined (mk_T_of_bnf Ds all_unitTs bnf))

           end;

         val nat_witss =

           map2 (fn Ds => fn bnf => mk_wits_of_bnf (replicate (nwits_of_bnf bnf) Ds)

             (replicate (nwits_of_bnf bnf) (replicate live HOLogic.unitT)) bnf

             |> map (fn (I, wit) =>

               (I, Lazy.lazy (mk_nat_wit Ds bnf (I, Term.list_comb (wit, map (K HOLogic.unit) I))))))

           Dss bnfs;

         val nat_wit_thmss = map2 (curry op ~~) nat_witss (map wit_thmss_of_bnf bnfs)

         val Iss = map (map fst) nat_witss;

         fun filter_wits (I, wit) =

           let val J = filter (fn i => i < m) I;

           in (J, (length J < length I, wit)) end;

         val wit_treess = map_index (fn (i, Is) =>

           map_index (finish Iss m [i+m] (i+m)) Is) Iss

           |> map (minimize_wits o map filter_wits o minimize_wits o flat);

         val coind_wit_argsss =

           map (map (tree_to_coind_wits nat_wit_thmss o snd o snd) o filter (fst o snd)) wit_treess;

         val nonredundant_coind_wit_argsss =

           fold (fn i => fn argsss =>

             nth_map (i - 1) (filter_out (fn xs =>

               exists (fn ys =>

                 let

                   val xs' = (map (fst o fst) xs, snd (fst (hd xs)));

                   val ys' = (map (fst o fst) ys, snd (fst (hd ys)));

                 in

                   eq_pair (subset (op =)) (eq_set (op =)) (xs', ys') andalso not (fst xs' = fst ys')

                 end)

               (flat argsss)))

             argsss)

           ks coind_wit_argsss;

         fun prepare_args args =

           let

             val I = snd (fst (hd args));

             val (dummys, args') =

               map_split (fn i =>

                 (case find_first (fn arg => fst (fst arg) = i - 1) args of

                   SOME (_, ((_, wit), thms)) => (NONE, (Lazy.force wit, thms))

                 | NONE =>

                   (SOME (i - 1), (mk_dummy_wit (nth Dss (i - 1)) (nth bnfs (i - 1)) I, []))))

               ks;

           in

             ((I, dummys), apsnd flat (split_list args'))

           end;

         fun mk_coind_wits ((I, dummys), (args, thms)) =

           ((I, dummys), (map (fn i => mk_unfold Ts args i $ HOLogic.unit) ks, thms));

         val coind_witss =

           maps (map (mk_coind_wits o prepare_args)) nonredundant_coind_wit_argsss;

         fun mk_coind_wit_thms ((I, dummys), (wits, wit_thms)) =

           let

             fun mk_goal sets y y_copy y'_copy j =

               let

                 fun mk_conjunct set z dummy wit =

                   mk_Ball (set $ z) (Term.absfree y'_copy

                     (if dummy = NONE orelse member (op =) I (j - 1) then

                       HOLogic.mk_imp (HOLogic.mk_eq (z, wit),

                         if member (op =) I (j - 1) then HOLogic.mk_eq (y_copy, y)

                         else @{term False})

                     else @{term True}));

               in

                 fold_rev Logic.all (map (nth ys) I @ Jzs) (HOLogic.mk_Trueprop

                   (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct sets Jzs dummys wits)))

               end;

             val goals = map5 mk_goal setss_by_range ys ys_copy ys'_copy ls;

           in

             map2 (fn goal => fn induct =>

               Goal.prove_sorry lthy [] [] goal

                 (mk_coind_wit_tac induct dtor_unfold_thms (flat set_mapss) wit_thms)

               |> Thm.close_derivation)