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

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Andrei Popescu, TU Muenchen

     Copyright   2012

 Datatype construction.

 *)

 signature BNF_LFP =

 sig

   val construct_lfp: 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_LFP : BNF_LFP =

 struct

 open BNF_Def

 open BNF_Util

 open BNF_Tactics

 open BNF_Comp

 open BNF_FP_Util

 open BNF_FP_Def_Sugar

 open BNF_LFP_Rec_Sugar

 open BNF_LFP_Util

 open BNF_LFP_Tactics

 (*all BNFs have the same lives*)

 fun construct_lfp mixfixes map_bs rel_bs set_bss 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 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.suffix_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), activeCs), passiveXs), passiveYs) =

       names_lthy

       |> variant_tfrees passives

       ||>> mk_TFrees n

       ||>> variant_tfrees passives

       ||>> mk_TFrees n

       ||>> mk_TFrees n

       ||>> mk_TFrees m

       ||> fst o mk_TFrees m;

     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 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 (curry op -->) FTsAs activeAs;

     val s'Ts = map2 (curry op -->) FTsBs activeBs;

     val s''Ts = map2 (curry op -->) FTsCs activeCs;

     val fTs = map2 (curry op -->) activeAs activeBs;

     val inv_fTs = map2 (curry op -->) activeBs activeAs;

     val self_fTs = map2 (curry op -->) activeAs activeAs;

     val gTs = map2 (curry op -->) activeBs activeCs;

     val all_gTs = map2 (curry op -->) allBs allCs';

     val prodBsAs = map2 (curry HOLogic.mk_prodT) activeBs activeAs;

     val prodFTs = mk_FTs (passiveAs @ prodBsAs);

     val prod_sTs = map2 (curry op -->) prodFTs activeAs;

     (* terms *)

     val mapsAsAs = map4 mk_map_of_bnf Dss Ass Ass bnfs;

     val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;

     val mapsBsAs = map4 mk_map_of_bnf Dss Bss Ass 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_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ prodBsAs)) Bss bnfs;

     val map_fsts_rev = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ prodBsAs)) 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 bd0s = map3 mk_bd_of_bnf Dss Ass bnfs;

     val bds =

       map3 (fn bd0 => fn Ds => fn bnf => mk_csum bd0

         (mk_card_of (HOLogic.mk_UNIV

           (mk_T_of_bnf Ds (replicate live (fst (dest_relT (fastype_of bd0)))) bnf))))

       bd0s Dss bnfs;

     val witss = map wits_of_bnf bnfs;

     val (((((((((((((((((((zs, zs'), As), Bs), Bs_copy), B's), B''s), ss), prod_ss), s's), s''s),

       fs), fs_copy), inv_fs), self_fs), gs), all_gs), (xFs, xFs')), (yFs, yFs')),

       names_lthy) = lthy

       |> mk_Frees' "z" activeAs

       ||>> 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 "prods" prod_sTs

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

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

       ||>> mk_Frees "f" fTs

       ||>> mk_Frees "f" fTs

       ||>> mk_Frees "f" inv_fTs

       ||>> mk_Frees "f" self_fTs

       ||>> mk_Frees "g" gTs

       ||>> mk_Frees "g" all_gTs

       ||>> mk_Frees' "x" FTsAs

       ||>> mk_Frees' "y" FTsBs;

     val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;

     val active_UNIVs = map HOLogic.mk_UNIV activeAs;

     val prod_UNIVs = map HOLogic.mk_UNIV prodBsAs;

     val passive_ids = map HOLogic.id_const passiveAs;

     val active_ids = map HOLogic.id_const activeAs;

     val fsts = map fst_const prodBsAs;

     (* thms *)

     val bd0_card_orders = map bd_card_order_of_bnf bnfs;

     val bd0_Card_orders = map bd_Card_order_of_bnf bnfs;

     val bd0_Cinfinites = map bd_Cinfinite_of_bnf bnfs;

     val set_bd0ss = map set_bd_of_bnf bnfs;

     val bd_card_orders =

       map (fn thm => @{thm card_order_csum} OF [thm, @{thm card_of_card_order_on}]) bd0_card_orders;

     val bd_Card_order = @{thm Card_order_csum};

     val bd_Card_orders = replicate n bd_Card_order;

     val bd_Cinfinites = map (fn thm => thm RS @{thm Cinfinite_csum1}) bd0_Cinfinites;

     val bd_Cnotzeros = map (fn thm => thm RS @{thm Cinfinite_Cnotzero}) bd_Cinfinites;

     val bd_Cinfinite = hd bd_Cinfinites;

     val bd_Cnotzero = hd bd_Cnotzeros;

     val set_bdss =

       map2 (fn set_bd0s => fn bd0_Card_order =>

         map (fn thm => ctrans OF [thm, bd0_Card_order RS @{thm ordLeq_csum1}]) set_bd0s)

       set_bd0ss bd0_Card_orders;

     val in_bds = map in_bd_of_bnf bnfs;

     val sym_map_comps = map (fn bnf => map_comp0_of_bnf bnf RS sym) bnfs;

     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_mapss = map set_map_of_bnf bnfs;

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

     (* nonemptiness check *)

     fun new_wit X (wit: nonemptiness_witness) = subset (op =) (#I wit, (0 upto m - 1) @ map snd X);

     val all = m upto m + n - 1;

     fun enrich X = map_filter (fn i =>

       (case find_first (fn (_, i') => i = i') X of

         NONE =>

           (case find_index (new_wit X) (nth witss (i - m)) of

             ~1 => NONE

           | j => SOME (j, i))

       | SOME ji => SOME ji)) all;

     val reachable = fixpoint (op =) enrich [];

     val _ = (case subtract (op =) (map snd reachable) all of

         [] => ()

       | i :: _ => error ("Cannot define empty datatype " ^ quote (Binding.name_of (nth bs (i - m)))));

     val wit_thms = flat (map2 (fn bnf => fn (j, _) => nth (wit_thmss_of_bnf bnf) j) bnfs reachable);

     val timer = time (timer "Checked nonemptiness");

     (* 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 bnf => in_mono_of_bnf bnf OF (replicate m subset_refl)) bnfs;

     val in_cong'_thms = map (fn bnf => in_cong_of_bnf bnf OF (replicate m refl)) bnfs;

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

     (* algebra *)

     val alg_bind = mk_internal_b algN;

     val alg_name = Binding.name_of alg_bind;

     val alg_def_bind = (Thm.def_binding alg_bind, []);

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

     val alg_spec =

       let

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

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

         fun mk_alg_conjunct B s X x x' =

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

         val lhs = Term.list_comb (Free (alg_name, algT), As @ Bs @ ss);

         val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_alg_conjunct Bs ss ins xFs xFs')

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((alg_free, (_, alg_def_free)), (lthy, lthy_old)) =

         lthy

         |> Specification.definition (SOME (alg_bind, NONE, NoSyn), (alg_def_bind, alg_spec))

         ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val alg = fst (Term.dest_Const (Morphism.term phi alg_free));

     val alg_def = Morphism.thm phi alg_def_free;

     fun mk_alg As Bs ss =

       let

         val args = As @ Bs @ ss;

         val Ts = map fastype_of args;

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

       in

         Term.list_comb (Const (alg, algT), args)

       end;

     val alg_set_thms =

       let

         val alg_prem = HOLogic.mk_Trueprop (mk_alg As Bs ss);

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

         fun mk_concl s x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (s $ x, B));

         val premss = map2 ((fn x => fn sets =>  map2 (mk_prem x) sets (As @ Bs))) xFs setssAs;

         val concls = map3 mk_concl ss xFs Bs;

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

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

             (Logic.list_implies (alg_prem :: prems, concl))) xFs premss concls;

       in

         map (fn goal =>

           Goal.prove_sorry lthy [] [] goal (K (mk_alg_set_tac alg_def)) |> Thm.close_derivation)

         goals

       end;

     fun mk_talg ATs BTs = mk_alg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);

     val talg_thm =

       let

         val goal = fold_rev Logic.all ss

           (HOLogic.mk_Trueprop (mk_talg passiveAs activeAs ss))

       in

         Goal.prove_sorry lthy [] [] goal

           (K (stac alg_def 1 THEN CONJ_WRAP (K (EVERY' [rtac ballI, rtac UNIV_I] 1)) ss))

         |> Thm.close_derivation

       end;

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

     val alg_not_empty_thms =

       let

         val alg_prem =

           HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);

         val concls = map (HOLogic.mk_Trueprop o mk_not_empty) Bs;

         val goals =

           map (fn concl =>

             fold_rev Logic.all (Bs @ ss) (Logic.mk_implies (alg_prem, concl))) concls;

       in

         map2 (fn goal => fn alg_set =>

           Goal.prove_sorry lthy [] []

             goal (K (mk_alg_not_empty_tac lthy alg_set alg_set_thms wit_thms))

           |> Thm.close_derivation)

         goals alg_set_thms

       end;

     val timer = time (timer "Proved nonemptiness");

     (* 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. f x \<in> B'i)*)

     (*mor) forall i = 1 ... n: (\<forall>x \<in> Fi_in UNIV ... UNIV B1 ... Bn.

        f (s1 x) = s1' (Fi_map id ... id f1 ... fn 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 sets mapAsBs f s s' T x x' =

           mk_Ball (mk_in (passive_UNIVs @ Bs) sets T)

             (Term.absfree x' (HOLogic.mk_eq (f $ (s $ x), s' $

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

         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

             (map8 mk_mor setssAs mapsAsBs fs ss s's FTsAs xFs xFs'))

       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_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_prem sets x T = HOLogic.mk_Trueprop

           (HOLogic.mk_mem (x, mk_in (passive_UNIVs @ Bs) sets T));

         fun mk_elim_goal sets mapAsBs f s s' x T =

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

             (Logic.list_implies ([prem, mk_elim_prem sets x T],

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

         val elim_goals = map7 mk_elim_goal setssAs mapsAsBs fs ss s's xFs FTsAs;

         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_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_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 set_mapss map_comp_id_thms))

         |> Thm.close_derivation

       end;

     val mor_inv_thm =

       let

         fun mk_inv_prem f inv_f B B' = HOLogic.mk_conj (mk_leq (mk_image inv_f $ B') B,

           HOLogic.mk_conj (mk_inver inv_f f B, mk_inver f inv_f B'));

         val prems = map HOLogic.mk_Trueprop

           ([mk_mor Bs ss B's s's fs,

           mk_alg passive_UNIVs Bs ss,

           mk_alg passive_UNIVs B's s's] @

           map4 mk_inv_prem fs inv_fs Bs B's);

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

       in

         Goal.prove_sorry lthy [] []

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

             (Logic.list_implies (prems, concl)))

           (K (mk_mor_inv_tac alg_def mor_def set_mapss morE_thms map_comp_id_thms map_cong0L_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_str_thm =

       let

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

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

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all ss (HOLogic.mk_Trueprop

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

           (K (mk_mor_str_tac ks mor_def))

         |> Thm.close_derivation

       end;

     val mor_convol_thm =

       let

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

           mk_convol (HOLogic.mk_comp (s, Term.list_comb (mapx, passive_ids @ fsts)), prod_s))

           s's prod_ss map_fsts;

       in

         Goal.prove_sorry lthy [] []

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

             (mk_mor prod_UNIVs maps (map HOLogic.mk_UNIV activeBs) s's fsts)))

           (K (mk_mor_convol_tac ks mor_def))

         |> Thm.close_derivation

       end;

     val mor_UNIV_thm =

       let

         fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq

             (HOLogic.mk_comp (f, s),

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

         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 m morE_thms mor_def))

         |> Thm.close_derivation

       end;

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

     (* isomorphism *)

     (*mor Bs1 ss1 Bs2 ss2 fs \<and> (\<exists>gs. mor Bs2 ss2 Bs1 ss1 fs \<and>

        forall i = 1 ... n. (inver gs[i] fs[i] Bs1[i] \<and> inver fs[i] gs[i] Bs2[i]))*)

     fun mk_iso Bs1 ss1 Bs2 ss2 fs gs =

       let

         val ex_inv_mor = list_exists_free gs

           (HOLogic.mk_conj (mk_mor Bs2 ss2 Bs1 ss1 gs,

             Library.foldr1 HOLogic.mk_conj (map2 (curry HOLogic.mk_conj)

               (map3 mk_inver gs fs Bs1) (map3 mk_inver fs gs Bs2))));

       in

         HOLogic.mk_conj (mk_mor Bs1 ss1 Bs2 ss2 fs, ex_inv_mor)

       end;

     val iso_alt_thm =

       let

         val prems = map HOLogic.mk_Trueprop

          [mk_alg passive_UNIVs Bs ss,

          mk_alg passive_UNIVs B's s's]

         val concl = mk_Trueprop_eq (mk_iso Bs ss B's s's fs inv_fs,

           HOLogic.mk_conj (mk_mor Bs ss B's s's fs,

             Library.foldr1 HOLogic.mk_conj (map3 mk_bij_betw fs Bs B's)));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs) (Logic.list_implies (prems, concl)))

           (K (mk_iso_alt_tac mor_image_thms mor_inv_thm))

         |> Thm.close_derivation

       end;

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

     (* algebra copies *)

     val (copy_alg_thm, ex_copy_alg_thm) =

       let

         val prems = map HOLogic.mk_Trueprop

          (mk_alg passive_UNIVs Bs ss :: map3 mk_bij_betw inv_fs B's Bs);

         val inver_prems = map HOLogic.mk_Trueprop

           (map3 mk_inver inv_fs fs Bs @ map3 mk_inver fs inv_fs B's);

         val all_prems = prems @ inver_prems;

         fun mk_s f s mapT y y' = Term.absfree y' (f $ (s $

           (Term.list_comb (mapT, passive_ids @ inv_fs) $ y)));

         val alg = HOLogic.mk_Trueprop

           (mk_alg passive_UNIVs B's (map5 mk_s fs ss mapsBsAs yFs yFs'));

         val copy_str_thm = Goal.prove_sorry lthy [] []

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

             (Logic.list_implies (all_prems, alg)))

           (K (mk_copy_str_tac set_mapss alg_def alg_set_thms))

           |> Thm.close_derivation;

         val iso = HOLogic.mk_Trueprop

           (mk_iso B's (map5 mk_s fs ss mapsBsAs yFs yFs') Bs ss inv_fs fs_copy);

         val copy_alg_thm = Goal.prove_sorry lthy [] []

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

             (Logic.list_implies (all_prems, iso)))

           (K (mk_copy_alg_tac set_mapss alg_set_thms mor_def iso_alt_thm copy_str_thm))

           |> Thm.close_derivation;

         val ex = HOLogic.mk_Trueprop

           (list_exists_free s's

             (HOLogic.mk_conj (mk_alg passive_UNIVs B's s's,

               mk_iso B's s's Bs ss inv_fs fs_copy)));

         val ex_copy_alg_thm = Goal.prove_sorry lthy [] []

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

              (Logic.list_implies (prems, ex)))

           (K (mk_ex_copy_alg_tac n copy_str_thm copy_alg_thm))

           |> Thm.close_derivation;

       in

         (copy_alg_thm, ex_copy_alg_thm)

       end;

     val timer = time (timer "Copy thms");

     (* bounds *)

     val sum_Card_order = if n = 1 then bd_Card_order else @{thm Card_order_csum};

     val sum_Cnotzero = if n = 1 then bd_Cnotzero else bd_Cnotzero RS @{thm csum_Cnotzero1};

     val sum_Cinfinite = if n = 1 then bd_Cinfinite else bd_Cinfinite RS @{thm Cinfinite_csum1};

     fun mk_set_bd_sums i bd_Card_order bds =

       if n = 1 then bds

       else map (fn thm => bd_Card_order RS mk_ordLeq_csum n i thm) bds;

     val set_bd_sumss = map3 mk_set_bd_sums ks bd_Card_orders set_bdss;

     fun mk_in_bd_sum i Co Cnz bd =

       if n = 1 then bd

       else Cnz RS ((Co RS mk_ordLeq_csum n i (Co RS @{thm ordLeq_refl})) RS

         (bd RS @{thm ordLeq_transitive[OF _ cexp_mono2_Cnotzero[OF _ Card_order_csum]]}));

     val in_bd_sums = map4 mk_in_bd_sum ks bd_Card_orders bd_Cnotzeros in_bds;

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

     val suc_bd = mk_cardSuc sum_bd;

     val field_suc_bd = mk_Field suc_bd;

     val suc_bdT = fst (dest_relT (fastype_of suc_bd));

     fun mk_Asuc_bd [] = mk_cexp ctwo suc_bd

       | mk_Asuc_bd As =

         mk_cexp (mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo) suc_bd;

     val suc_bd_Card_order = if n = 1 then bd_Card_order RS @{thm cardSuc_Card_order}

       else @{thm cardSuc_Card_order[OF Card_order_csum]};

     val suc_bd_Cinfinite = if n = 1 then bd_Cinfinite RS @{thm Cinfinite_cardSuc}

       else bd_Cinfinite RS @{thm Cinfinite_cardSuc[OF Cinfinite_csum1]};

     val suc_bd_Cnotzero = suc_bd_Cinfinite RS @{thm Cinfinite_Cnotzero};

     val suc_bd_worel = suc_bd_Card_order RS @{thm Card_order_wo_rel}

     val basis_Asuc = if m = 0 then @{thm ordLeq_refl[OF Card_order_ctwo]}

         else @{thm ordLeq_csum2[OF Card_order_ctwo]};

     val Asuc_bd_Cinfinite = suc_bd_Cinfinite RS (basis_Asuc RS @{thm Cinfinite_cexp});

     val suc_bd_Asuc_bd = @{thm ordLess_ordLeq_trans[OF ordLess_ctwo_cexp cexp_mono1]} OF

       [suc_bd_Card_order, basis_Asuc, suc_bd_Card_order];

     val Asuc_bdT = fst (dest_relT (fastype_of (mk_Asuc_bd As)));

     val II_BTs = replicate n (HOLogic.mk_setT Asuc_bdT);

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

       mk_T_of_bnf Ds (passiveAs @ replicate n Asuc_bdT) bnf --> Asuc_bdT) Dss bnfs;

     val (((((((idxs, Asi_name), (idx, idx')), (jdx, jdx')), II_Bs), II_ss), Asuc_fs),

       names_lthy) = names_lthy

       |> mk_Frees "i" (replicate n suc_bdT)

       ||>> (fn ctxt => apfst the_single (mk_fresh_names ctxt 1 "Asi"))

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

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "j") suc_bdT

       ||>> mk_Frees "IIB" II_BTs

       ||>> mk_Frees "IIs" II_sTs

       ||>> mk_Frees "f" (map (fn T => Asuc_bdT --> T) activeAs);

     val suc_bd_limit_thm =

       let

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

           (map (fn idx => HOLogic.mk_mem (idx, field_suc_bd)) idxs));

         fun mk_conjunct idx = HOLogic.mk_conj (mk_not_eq idx jdx,

           HOLogic.mk_mem (HOLogic.mk_prod (idx, jdx), suc_bd));

         val concl = HOLogic.mk_Trueprop (mk_Bex field_suc_bd

           (Term.absfree jdx' (Library.foldr1 HOLogic.mk_conj (map mk_conjunct idxs))));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all idxs (Logic.list_implies ([prem], concl)))

           (K (mk_bd_limit_tac n suc_bd_Cinfinite))

         |> Thm.close_derivation

       end;

     val timer = time (timer "Bounds");

     (* minimal algebra *)

     fun mk_minG Asi i k = mk_UNION (mk_underS suc_bd $ i)

       (Term.absfree jdx' (mk_nthN n (Asi $ jdx) k));

     fun mk_minH_component As Asi i sets Ts s k =

       HOLogic.mk_binop @{const_name "sup"}

       (mk_minG Asi i k, mk_image s $ mk_in (As @ map (mk_minG Asi i) ks) sets Ts);

     fun mk_min_algs As ss =

       let

         val BTs = map (range_type o fastype_of) ss;

         val Ts = map (HOLogic.dest_setT o fastype_of) As @ BTs;

         val (Asi, Asi') = `Free (Asi_name, suc_bdT -->

           Library.foldr1 HOLogic.mk_prodT (map HOLogic.mk_setT BTs));

       in

          mk_worec suc_bd (Term.absfree Asi' (Term.absfree idx' (HOLogic.mk_tuple

            (map4 (mk_minH_component As Asi idx) (mk_setss Ts) (mk_FTs Ts) ss ks))))

       end;

     val (min_algs_thms, min_algs_mono_thms, card_of_min_algs_thm, least_min_algs_thm) =

       let

         val i_field = HOLogic.mk_mem (idx, field_suc_bd);

         val min_algs = mk_min_algs As ss;

         val min_algss = map (fn k => mk_nthN n (min_algs $ idx) k) ks;

         val concl = HOLogic.mk_Trueprop

           (HOLogic.mk_eq (min_algs $ idx, HOLogic.mk_tuple

             (map4 (mk_minH_component As min_algs idx) setssAs FTsAs ss ks)));

         val goal = fold_rev Logic.all (idx :: As @ ss)

           (Logic.mk_implies (HOLogic.mk_Trueprop i_field, concl));

         val min_algs_thm = Goal.prove_sorry lthy [] [] goal

           (K (mk_min_algs_tac suc_bd_worel in_cong'_thms))

           |> Thm.close_derivation;

         val min_algs_thms = map (fn k => min_algs_thm RS mk_nthI n k) ks;

         fun mk_mono_goal min_alg =

           fold_rev Logic.all (As @ ss) (HOLogic.mk_Trueprop (mk_relChain suc_bd

             (Term.absfree idx' min_alg)));

         val monos =

           map2 (fn goal => fn min_algs =>

             Goal.prove_sorry lthy [] [] goal (K (mk_min_algs_mono_tac lthy min_algs))

             |> Thm.close_derivation)

           (map mk_mono_goal min_algss) min_algs_thms;

         val Asuc_bd = mk_Asuc_bd As;

         fun mk_card_conjunct min_alg = mk_ordLeq (mk_card_of min_alg) Asuc_bd;

         val card_conjunction = Library.foldr1 HOLogic.mk_conj (map mk_card_conjunct min_algss);

         val card_cT = certifyT lthy suc_bdT;

         val card_ct = certify lthy (Term.absfree idx' card_conjunction);

         val card_of = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] []

             (HOLogic.mk_Trueprop (HOLogic.mk_imp (i_field, card_conjunction)))

             (K (mk_min_algs_card_of_tac card_cT card_ct

               m suc_bd_worel min_algs_thms in_bd_sums

               sum_Card_order sum_Cnotzero suc_bd_Card_order suc_bd_Cinfinite suc_bd_Cnotzero

               suc_bd_Asuc_bd Asuc_bd_Cinfinite)))

           |> Thm.close_derivation;

         val least_prem = HOLogic.mk_Trueprop (mk_alg As Bs ss);

         val least_conjunction = Library.foldr1 HOLogic.mk_conj (map2 mk_leq min_algss Bs);

         val least_cT = certifyT lthy suc_bdT;

         val least_ct = certify lthy (Term.absfree idx' least_conjunction);

         val least = singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] []

             (Logic.mk_implies (least_prem,

               HOLogic.mk_Trueprop (HOLogic.mk_imp (i_field, least_conjunction))))

             (K (mk_min_algs_least_tac least_cT least_ct

               suc_bd_worel min_algs_thms alg_set_thms)))

           |> Thm.close_derivation;

       in

         (min_algs_thms, monos, card_of, least)

       end;

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

     val min_alg_binds = mk_internal_bs min_algN;

     fun min_alg_bind i = nth min_alg_binds (i - 1);

     fun min_alg_name i = Binding.name_of (min_alg_bind i);

     val min_alg_def_bind = rpair [] o Thm.def_binding o min_alg_bind;

     fun min_alg_spec i =

       let

         val min_algT =

           Library.foldr (op -->) (ATs @ sTs, HOLogic.mk_setT (nth activeAs (i - 1)));

         val lhs = Term.list_comb (Free (min_alg_name i, min_algT), As @ ss);

         val rhs = mk_UNION (field_suc_bd)

           (Term.absfree idx' (mk_nthN n (mk_min_algs As ss $ idx) i));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((min_alg_frees, (_, min_alg_def_frees)), (lthy, lthy_old)) =

         lthy

         |> fold_map (fn i => Specification.definition

           (SOME (min_alg_bind i, NONE, NoSyn), (min_alg_def_bind i, min_alg_spec i))) ks

         |>> apsnd split_list o split_list

         ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val min_algs = map (fst o Term.dest_Const o Morphism.term phi) min_alg_frees;

     val min_alg_defs = map (Morphism.thm phi) min_alg_def_frees;

     fun mk_min_alg As ss i =

       let

         val T = HOLogic.mk_setT (range_type (fastype_of (nth ss (i - 1))))

         val args = As @ ss;

         val Ts = map fastype_of args;

         val min_algT = Library.foldr (op -->) (Ts, T);

       in

         Term.list_comb (Const (nth min_algs (i - 1), min_algT), args)

       end;

     val (alg_min_alg_thm, card_of_min_alg_thms, least_min_alg_thms, mor_incl_min_alg_thm) =

       let

         val min_algs = map (mk_min_alg As ss) ks;

         val goal = fold_rev Logic.all (As @ ss) (HOLogic.mk_Trueprop (mk_alg As min_algs ss));

         val alg_min_alg = Goal.prove_sorry lthy [] [] goal

           (K (mk_alg_min_alg_tac m alg_def min_alg_defs suc_bd_limit_thm sum_Cinfinite

             set_bd_sumss min_algs_thms min_algs_mono_thms))

           |> Thm.close_derivation;

         val Asuc_bd = mk_Asuc_bd As;

         fun mk_card_of_thm min_alg def = Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (As @ ss)

             (HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of min_alg) Asuc_bd)))

           (K (mk_card_of_min_alg_tac def card_of_min_algs_thm

             suc_bd_Card_order suc_bd_Asuc_bd Asuc_bd_Cinfinite))

           |> Thm.close_derivation;

         val least_prem = HOLogic.mk_Trueprop (mk_alg As Bs ss);

         fun mk_least_thm min_alg B def = Goal.prove_sorry lthy [] []

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

             (Logic.mk_implies (least_prem, HOLogic.mk_Trueprop (mk_leq min_alg B))))

           (K (mk_least_min_alg_tac def least_min_algs_thm))

           |> Thm.close_derivation;

         val leasts = map3 mk_least_thm min_algs Bs min_alg_defs;

         val incl_prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);

         val incl_min_algs = map (mk_min_alg passive_UNIVs ss) ks;

         val incl = Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (Bs @ ss)

             (Logic.mk_implies (incl_prem,

               HOLogic.mk_Trueprop (mk_mor incl_min_algs ss Bs ss active_ids))))

           (K (EVERY' (rtac mor_incl_thm :: map etac leasts) 1))

           |> Thm.close_derivation;

       in

         (alg_min_alg, map2 mk_card_of_thm min_algs min_alg_defs, leasts, incl)

       end;

     val timer = time (timer "Minimal algebra definition & thms");

     val II_repT = HOLogic.mk_prodT (HOLogic.mk_tupleT II_BTs, HOLogic.mk_tupleT II_sTs);

     val IIT_bind = mk_internal_b IITN;

     val ((IIT_name, (IIT_glob_info, IIT_loc_info)), lthy) =

       typedef (IIT_bind, params, NoSyn)

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

     val IIT = Type (IIT_name, params');

     val Abs_IIT = Const (#Abs_name IIT_glob_info, II_repT --> IIT);

     val Rep_IIT = Const (#Rep_name IIT_glob_info, IIT --> II_repT);

     val Abs_IIT_inverse_thm = UNIV_I RS #Abs_inverse IIT_loc_info;

     val initT = IIT --> Asuc_bdT;

     val active_initTs = replicate n initT;

     val init_FTs = map2 (fn Ds => mk_T_of_bnf Ds (passiveAs @ active_initTs)) Dss bnfs;

     val init_fTs = map (fn T => initT --> T) activeAs;

     val (((((((iidx, iidx'), init_xs), (init_xFs, init_xFs')),

       init_fs), init_fs_copy), init_phis), names_lthy) = names_lthy

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

       ||>> mk_Frees "ix" active_initTs

       ||>> mk_Frees' "x" init_FTs

       ||>> mk_Frees "f" init_fTs

       ||>> mk_Frees "f" init_fTs

       ||>> mk_Frees "P" (replicate n (mk_pred1T initT));

     val II = HOLogic.mk_Collect (fst iidx', IIT, list_exists_free (II_Bs @ II_ss)

       (HOLogic.mk_conj (HOLogic.mk_eq (iidx,

         Abs_IIT $ (HOLogic.mk_prod (HOLogic.mk_tuple II_Bs, HOLogic.mk_tuple II_ss))),

         mk_alg passive_UNIVs II_Bs II_ss)));

     val select_Bs = map (mk_nthN n (HOLogic.mk_fst (Rep_IIT $ iidx))) ks;

     val select_ss = map (mk_nthN n (HOLogic.mk_snd (Rep_IIT $ iidx))) ks;

     val str_init_binds = mk_internal_bs str_initN;

     fun str_init_bind i = nth str_init_binds (i - 1);

     val str_init_name = Binding.name_of o str_init_bind;

     val str_init_def_bind = rpair [] o Thm.def_binding o str_init_bind;

     fun str_init_spec i =

       let

         val T = nth init_FTs (i - 1);

         val init_xF = nth init_xFs (i - 1)

         val select_s = nth select_ss (i - 1);

         val map = mk_map_of_bnf (nth Dss (i - 1))

           (passiveAs @ active_initTs) (passiveAs @ replicate n Asuc_bdT)

           (nth bnfs (i - 1));

         val map_args = passive_ids @ replicate n (mk_rapp iidx Asuc_bdT);

         val str_initT = T --> IIT --> Asuc_bdT;

         val lhs = Term.list_comb (Free (str_init_name i, str_initT), [init_xF, iidx]);

         val rhs = select_s $ (Term.list_comb (map, map_args) $ init_xF);

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((str_init_frees, (_, str_init_def_frees)), (lthy, lthy_old)) =

       lthy

       |> fold_map (fn i => Specification.definition

         (SOME (str_init_bind i, NONE, NoSyn), (str_init_def_bind i, str_init_spec i))) ks

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val str_inits =

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

         str_init_frees;

     val str_init_defs = map (Morphism.thm phi) str_init_def_frees;

     val car_inits = map (mk_min_alg passive_UNIVs str_inits) ks;

     (*TODO: replace with instantiate? (problem: figure out right type instantiation)*)

     val alg_init_thm = Goal.prove_sorry lthy [] []

       (HOLogic.mk_Trueprop (mk_alg passive_UNIVs car_inits str_inits))

       (K (rtac alg_min_alg_thm 1))

       |> Thm.close_derivation;

     val alg_select_thm = Goal.prove_sorry lthy [] []

       (HOLogic.mk_Trueprop (mk_Ball II

         (Term.absfree iidx' (mk_alg passive_UNIVs select_Bs select_ss))))

       (mk_alg_select_tac Abs_IIT_inverse_thm)

       |> Thm.close_derivation;

     val mor_select_thm =

       let

         val alg_prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);

         val i_prem = HOLogic.mk_Trueprop (HOLogic.mk_mem (iidx, II));

         val mor_prem = HOLogic.mk_Trueprop (mk_mor select_Bs select_ss Bs ss Asuc_fs);

         val prems = [alg_prem, i_prem, mor_prem];

         val concl = HOLogic.mk_Trueprop

           (mk_mor car_inits str_inits Bs ss

             (map (fn f => HOLogic.mk_comp (f, mk_rapp iidx Asuc_bdT)) Asuc_fs));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (iidx :: Bs @ ss @ Asuc_fs) (Logic.list_implies (prems, concl)))

           (K (mk_mor_select_tac mor_def mor_cong_thm mor_comp_thm mor_incl_min_alg_thm alg_def

             alg_select_thm alg_set_thms set_mapss str_init_defs))

         |> Thm.close_derivation

       end;

     val (init_ex_mor_thm, init_unique_mor_thms) =

       let

         val prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);

         val concl = HOLogic.mk_Trueprop

           (list_exists_free init_fs (mk_mor car_inits str_inits Bs ss init_fs));

         val ex_mor = Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (Bs @ ss) (Logic.mk_implies (prem, concl)))

           (mk_init_ex_mor_tac Abs_IIT_inverse_thm ex_copy_alg_thm alg_min_alg_thm

             card_of_min_alg_thms mor_comp_thm mor_select_thm mor_incl_min_alg_thm)

           |> Thm.close_derivation;

         val prems = map2 (HOLogic.mk_Trueprop oo curry HOLogic.mk_mem) init_xs car_inits

         val mor_prems = map HOLogic.mk_Trueprop

           [mk_mor car_inits str_inits Bs ss init_fs,

           mk_mor car_inits str_inits Bs ss init_fs_copy];

         fun mk_fun_eq f g x = HOLogic.mk_eq (f $ x, g $ x);

         val unique = HOLogic.mk_Trueprop

           (Library.foldr1 HOLogic.mk_conj (map3 mk_fun_eq init_fs init_fs_copy init_xs));

         val unique_mor = Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (init_xs @ Bs @ ss @ init_fs @ init_fs_copy)

             (Logic.list_implies (prems @ mor_prems, unique)))

           (K (mk_init_unique_mor_tac m alg_def alg_init_thm least_min_alg_thms

             in_mono'_thms alg_set_thms morE_thms map_cong0s))

           |> Thm.close_derivation;

       in

         (ex_mor, split_conj_thm unique_mor)

       end;

     val init_setss = mk_setss (passiveAs @ active_initTs);

     val active_init_setss = map (drop m) init_setss;

     val init_ins = map2 (fn sets => mk_in (passive_UNIVs @ car_inits) sets) init_setss init_FTs;

     fun mk_closed phis =

       let

         fun mk_conjunct phi str_init init_sets init_in x x' =

           let

             val prem = Library.foldr1 HOLogic.mk_conj

               (map2 (fn set => mk_Ball (set $ x)) init_sets phis);

             val concl = phi $ (str_init $ x);

           in

             mk_Ball init_in (Term.absfree x' (HOLogic.mk_imp (prem, concl)))

           end;

       in

         Library.foldr1 HOLogic.mk_conj

           (map6 mk_conjunct phis str_inits active_init_setss init_ins init_xFs init_xFs')

       end;

     val init_induct_thm =

       let

         val prem = HOLogic.mk_Trueprop (mk_closed init_phis);

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

           (map2 mk_Ball car_inits init_phis));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all init_phis (Logic.mk_implies (prem, concl)))

           (K (mk_init_induct_tac m alg_def alg_init_thm least_min_alg_thms alg_set_thms))

         |> Thm.close_derivation

       end;

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

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

       lthy

       |> fold_map3 (fn b => fn mx => fn car_init =>

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

           (EVERY' [rtac ssubst, rtac @{thm ex_in_conv}, resolve_tac alg_not_empty_thms,

             rtac alg_init_thm] 1)) bs mixfixes car_inits

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

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

     val type_defs = map #type_definition T_loc_infos;

     val Reps = map #Rep T_loc_infos;

     val Rep_casess = map #Rep_cases T_loc_infos;

     val Rep_injects = map #Rep_inject T_loc_infos;

     val Rep_inverses = map #Rep_inverse T_loc_infos;

     val Abs_inverses = map #Abs_inverse T_loc_infos;

     fun mk_inver_thm mk_tac rep abs X thm =

       Goal.prove_sorry lthy [] []

         (HOLogic.mk_Trueprop (mk_inver rep abs X))

         (K (EVERY' [rtac ssubst, rtac @{thm inver_def}, rtac ballI, mk_tac thm] 1))

       |> Thm.close_derivation;

     val inver_Reps = map4 (mk_inver_thm rtac) Abs_Ts Rep_Ts (map HOLogic.mk_UNIV Ts) Rep_inverses;

     val inver_Abss = map4 (mk_inver_thm etac) Rep_Ts Abs_Ts car_inits Abs_inverses;

     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');

     fun mk_set_Ts T = passiveAs @ replicate n (HOLogic.mk_setT T);

     val setFTss = map (mk_FTs o mk_set_Ts) passiveAs;

     val FTs_setss = mk_setss (passiveAs @ Ts);

     val FTs'_setss = mk_setss (passiveBs @ Ts');

     val map_FT_inits = map2 (fn Ds =>

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

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

     val foldT = Library.foldr1 HOLogic.mk_prodT (map2 (curry op -->) Ts activeAs);

     val rec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) prod_sTs;

     val rec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_fsts;

     val rec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_fsts_rev;

     val rec_fsts = map (Term.subst_atomic_types (activeBs ~~ Ts)) fsts;

     val rec_UNIVs = map2 (HOLogic.mk_UNIV oo curry HOLogic.mk_prodT) Ts activeAs;

     val (((((((((Izs1, Izs1'), (Izs2, Izs2')), (xFs, xFs')), yFs), (AFss, AFss')),

       (fold_f, fold_f')), fs), rec_ss), names_lthy) = names_lthy

       |> mk_Frees' "z1" Ts

       ||>> mk_Frees' "z2" Ts'

       ||>> mk_Frees' "x" FTs

       ||>> mk_Frees "y" FTs'

       ||>> mk_Freess' "z" setFTss

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "f") foldT

       ||>> mk_Frees "f" fTs

       ||>> mk_Frees "s" rec_sTs;

     val Izs = map2 retype_free Ts zs;

     val phis = map2 retype_free (map mk_pred1T Ts) init_phis;

     val phi2s = map2 retype_free (map2 mk_pred2T Ts Ts') init_phis;

     fun ctor_bind i = nth external_bs (i - 1) |> Binding.suffix_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 abs str map_FT_init x x' =

       let

         val ctorT = nth FTs (i - 1) --> nth Ts (i - 1);

         val lhs = Free (ctor_name i, ctorT);

         val rhs = Term.absfree x' (abs $ (str $

           (Term.list_comb (map_FT_init, map HOLogic.id_const passiveAs @ Rep_Ts) $ x)));

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

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

       lthy

       |> fold_map6 (fn i => fn abs => fn str => fn mapx => fn x => fn x' =>

         Specification.definition

           (SOME (ctor_bind i, NONE, NoSyn), (ctor_def_bind i, ctor_spec i abs str mapx x x')))

           ks Abs_Ts str_inits map_FT_inits xFs xFs'

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     fun mk_ctors passive =

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

         Morphism.term phi) ctor_frees;

     val ctors = mk_ctors passiveAs;

     val ctor's = mk_ctors passiveBs;

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

     val (mor_Rep_thm, mor_Abs_thm) =

       let

         val copy = alg_init_thm RS copy_alg_thm;

         fun mk_bij inj Rep cases = @{thm bij_betwI'} OF [inj, Rep, cases];

         val bijs = map3 mk_bij Rep_injects Reps Rep_casess;

         val mor_Rep =

           Goal.prove_sorry lthy [] []

             (HOLogic.mk_Trueprop (mk_mor UNIVs ctors car_inits str_inits Rep_Ts))

             (mk_mor_Rep_tac ctor_defs copy bijs inver_Abss inver_Reps)

           |> Thm.close_derivation;

         val inv = mor_inv_thm OF [mor_Rep, talg_thm, alg_init_thm];

         val mor_Abs =

           Goal.prove_sorry lthy [] []

             (HOLogic.mk_Trueprop (mk_mor car_inits str_inits UNIVs ctors Abs_Ts))

             (K (mk_mor_Abs_tac inv inver_Abss inver_Reps))

           |> Thm.close_derivation;

       in

         (mor_Rep, mor_Abs)

       end;

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

     val fold_fun = Term.absfree fold_f'

       (mk_mor UNIVs ctors active_UNIVs ss (map (mk_nthN n fold_f) ks));

     val foldx = HOLogic.choice_const foldT $ fold_fun;

     fun fold_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ ctor_foldN);

     val fold_name = Binding.name_of o fold_bind;

     val fold_def_bind = rpair [] o Binding.conceal o Thm.def_binding o fold_bind;

     fun fold_spec i T AT =

       let

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

         val lhs = Term.list_comb (Free (fold_name i, foldT), ss);

         val rhs = mk_nthN n foldx i;

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((fold_frees, (_, fold_def_frees)), (lthy, lthy_old)) =

       lthy

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

         Specification.definition

           (SOME (fold_bind i, NONE, NoSyn), (fold_def_bind i, fold_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 folds = map (Morphism.term phi) fold_frees;

     val fold_names = map (fst o dest_Const) folds;

     fun mk_folds passives actives =

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

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

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

       fold_names (mk_Ts passives) actives;

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

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

     val fold_defs = map (Morphism.thm phi) fold_def_frees;

     val mor_fold_thm =

       let

         val ex_mor = talg_thm RS init_ex_mor_thm;

         val mor_cong = mor_cong_thm OF (map (mk_nth_conv n) ks);

         val mor_comp = mor_Rep_thm RS mor_comp_thm;

         val cT = certifyT lthy foldT;

         val ct = certify lthy fold_fun

       in

         singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] []

             (HOLogic.mk_Trueprop (mk_mor UNIVs ctors active_UNIVs ss (map (mk_fold Ts ss) ks)))

             (K (mk_mor_fold_tac cT ct fold_defs ex_mor (mor_comp RS mor_cong))))

         |> Thm.close_derivation

       end;

     val ctor_fold_thms = map (fn morE => rule_by_tactic lthy

       ((rtac CollectI THEN' CONJ_WRAP' (K (rtac @{thm subset_UNIV})) (1 upto m + n)) 1)

       (mor_fold_thm RS morE)) morE_thms;

     val (fold_unique_mor_thms, fold_unique_mor_thm) =

       let

         val prem = HOLogic.mk_Trueprop (mk_mor UNIVs ctors active_UNIVs ss fs);

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

         val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_fun_eq fs ks));

         val unique_mor = Goal.prove_sorry lthy [] []

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

           (K (mk_fold_unique_mor_tac type_defs init_unique_mor_thms Reps

             mor_comp_thm mor_Abs_thm mor_fold_thm))

           |> Thm.close_derivation;

       in

         `split_conj_thm unique_mor

       end;

     val (ctor_fold_unique_thms, ctor_fold_unique_thm) =

       `split_conj_thm (mk_conjIN n RS

         (mor_UNIV_thm RS iffD2 RS fold_unique_mor_thm))

     val fold_ctor_thms =

       map (fn thm => (mor_incl_thm OF replicate n @{thm subset_UNIV}) RS thm RS sym)

         fold_unique_mor_thms;

     val ctor_o_fold_thms =

       let

         val mor = mor_comp_thm OF [mor_fold_thm, mor_str_thm];

       in

         map2 (fn unique => fn fold_ctor =>

           trans OF [mor RS unique, fold_ctor]) fold_unique_mor_thms fold_ctor_thms

       end;

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

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

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

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

     fun dtor_bind i = nth external_bs (i - 1) |> Binding.suffix_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 FT T =

       let

         val dtorT = T --> FT;

         val lhs = Free (dtor_name i, dtorT);

         val rhs = mk_fold Ts map_ctors i;

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

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

       lthy

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

         Specification.definition

           (SOME (dtor_bind i, NONE, NoSyn), (dtor_def_bind i, dtor_spec i FT T))) ks FTs Ts

       |>> apsnd split_list o split_list

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy_old lthy;

     fun mk_dtors params =

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

         dtor_frees;

     val dtors = mk_dtors params';

     val dtor_defs = map (Morphism.thm phi) dtor_def_frees;

     val ctor_o_dtor_thms = map2 (fold_thms lthy o single) dtor_defs ctor_o_fold_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 dtor_def => fn foldx => fn map_comp_id => fn map_cong0L =>

           Goal.prove_sorry lthy [] [] goal

             (K (mk_dtor_o_ctor_tac dtor_def foldx map_comp_id map_cong0L ctor_o_fold_thms))

           |> Thm.close_derivation)

         goals dtor_defs ctor_fold_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 "dtor definitions & thms");

     val fst_rec_pair_thms =

       let

         val mor = mor_comp_thm OF [mor_fold_thm, mor_convol_thm];

       in

         map2 (fn unique => fn fold_ctor =>

           trans OF [mor RS unique, fold_ctor]) fold_unique_mor_thms fold_ctor_thms

       end;

     fun rec_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ ctor_recN);

     val rec_name = Binding.name_of o rec_bind;

     val rec_def_bind = rpair [] o Binding.conceal o Thm.def_binding o rec_bind;

     val rec_strs =

       map3 (fn ctor => fn prod_s => fn mapx =>

         mk_convol (HOLogic.mk_comp (ctor, Term.list_comb (mapx, passive_ids @ rec_fsts)), prod_s))

       ctors rec_ss rec_maps;

     fun rec_spec i T AT =

       let

         val recT = Library.foldr (op -->) (rec_sTs, T --> AT);

         val lhs = Term.list_comb (Free (rec_name i, recT), rec_ss);

         val rhs = HOLogic.mk_comp (snd_const (HOLogic.mk_prodT (T, AT)), mk_fold Ts rec_strs i);

       in

         mk_Trueprop_eq (lhs, rhs)

       end;

     val ((rec_frees, (_, rec_def_frees)), (lthy, lthy_old)) =

       lthy

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

         Specification.definition

           (SOME (rec_bind i, NONE, NoSyn), (rec_def_bind i, rec_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 recs = map (Morphism.term phi) rec_frees;

     val rec_names = map (fst o dest_Const) recs;

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

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

     val rec_defs = map (Morphism.thm phi) rec_def_frees;

     val convols = map2 (fn T => fn i => mk_convol (HOLogic.id_const T, mk_rec rec_ss i)) Ts ks;

     val ctor_rec_thms =

       let

         fun mk_goal i rec_s rec_map ctor x =

           let

             val lhs = mk_rec rec_ss i $ (ctor $ x);

             val rhs = rec_s $ (Term.list_comb (rec_map, passive_ids @ convols) $ x);

           in

             fold_rev Logic.all (x :: rec_ss) (mk_Trueprop_eq (lhs, rhs))

           end;

         val goals = map5 mk_goal ks rec_ss rec_maps_rev ctors xFs;

       in

         map2 (fn goal => fn foldx =>

           Goal.prove_sorry lthy [] [] goal (mk_rec_tac rec_defs foldx fst_rec_pair_thms)

           |> Thm.close_derivation)

         goals ctor_fold_thms

       end;

     val rec_unique_mor_thm =

       let

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

         val prem = HOLogic.mk_Trueprop (mk_mor UNIVs ctors rec_UNIVs rec_strs id_fs);

         fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_rec rec_ss i);

         val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_fun_eq fs ks));

       in

         Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (rec_ss @ fs) (Logic.mk_implies (prem, unique)))

           (mk_rec_unique_mor_tac rec_defs fst_rec_pair_thms fold_unique_mor_thm)

           |> Thm.close_derivation

       end;

     val (ctor_rec_unique_thms, ctor_rec_unique_thm) =

       `split_conj_thm (split_conj_prems n

         (mor_UNIV_thm RS iffD2 RS rec_unique_mor_thm)

         |> Local_Defs.unfold lthy (@{thms convol_o o_id id_o o_assoc[symmetric] fst_convol} @

            map_id0s @ sym_map_comps) OF replicate n @{thm arg_cong2[of _ _ _ _ convol, OF refl]});

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

     val (ctor_induct_thm, induct_params) =

       let

         fun mk_prem phi ctor sets x =

           let

             fun mk_IH phi set z =

               let

                 val prem = HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set $ x));

                 val concl = HOLogic.mk_Trueprop (phi $ z);

               in

                 Logic.all z (Logic.mk_implies (prem, concl))

               end;

             val IHs = map3 mk_IH phis (drop m sets) Izs;

             val concl = HOLogic.mk_Trueprop (phi $ (ctor $ x));

           in

             Logic.all x (Logic.list_implies (IHs, concl))

           end;

         val prems = map4 mk_prem phis ctors FTs_setss xFs;

         fun mk_concl phi z = phi $ z;

         val concl =

           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_concl phis Izs));

         val goal = Logic.list_implies (prems, concl);

       in

         (Goal.prove_sorry lthy [] []

           (fold_rev Logic.all (phis @ Izs) goal)

           (K (mk_ctor_induct_tac lthy m set_mapss init_induct_thm morE_thms mor_Abs_thm

             Rep_inverses Abs_inverses Reps))

         |> Thm.close_derivation,

         rev (Term.add_tfrees goal []))

       end;

     val cTs = map (SOME o certifyT lthy o TFree) induct_params;

     val weak_ctor_induct_thms =

       let fun insts i = (replicate (i - 1) TrueI) @ (@{thm asm_rl} :: replicate (n - i) TrueI);

       in map (fn i => (ctor_induct_thm OF insts i) RS mk_conjunctN n i) ks end;

     val (ctor_induct2_thm, induct2_params) =

       let

         fun mk_prem phi ctor ctor' sets sets' x y =

           let

             fun mk_IH phi set set' z1 z2 =

               let

                 val prem1 = HOLogic.mk_Trueprop (HOLogic.mk_mem (z1, (set $ x)));

                 val prem2 = HOLogic.mk_Trueprop (HOLogic.mk_mem (z2, (set' $ y)));

                 val concl = HOLogic.mk_Trueprop (phi $ z1 $ z2);

               in

                 fold_rev Logic.all [z1, z2] (Logic.list_implies ([prem1, prem2], concl))

               end;

             val IHs = map5 mk_IH phi2s (drop m sets) (drop m sets') Izs1 Izs2;

             val concl = HOLogic.mk_Trueprop (phi $ (ctor $ x) $ (ctor' $ y));

           in

             fold_rev Logic.all [x, y] (Logic.list_implies (IHs, concl))

           end;

         val prems = map7 mk_prem phi2s ctors ctor's FTs_setss FTs'_setss xFs yFs;

         fun mk_concl phi z1 z2 = phi $ z1 $ z2;

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

           (map3 mk_concl phi2s Izs1 Izs2));

         fun mk_t phi (z1, z1') (z2, z2') =

           Term.absfree z1' (HOLogic.mk_all (fst z2', snd z2', phi $ z1 $ z2));

         val cts = map3 (SOME o certify lthy ooo mk_t) phi2s (Izs1 ~~ Izs1') (Izs2 ~~ Izs2');

         val goal = Logic.list_implies (prems, concl);

       in

         (singleton (Proof_Context.export names_lthy lthy)

           (Goal.prove_sorry lthy [] [] goal

             (mk_ctor_induct2_tac cTs cts ctor_induct_thm weak_ctor_induct_thms))

           |> Thm.close_derivation,

         rev (Term.add_tfrees goal []))

       end;

     val timer = time (timer "induction");

     fun mk_ctor_map_DEADID_thm ctor_inject map_id0 =

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

     fun mk_ctor_Irel_DEADID_thm ctor_inject bnf =

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

     val IphiTs = map2 mk_pred2T passiveAs passiveBs;

     val activephiTs = map2 mk_pred2T activeAs activeBs;

     val activeIphiTs = map2 mk_pred2T Ts Ts';

     val (((Iphis, activephis), activeIphis), names_lthy) = names_lthy

       |> mk_Frees "R" IphiTs

       ||>> mk_Frees "S" activephiTs

       ||>> mk_Frees "IR" activeIphiTs;

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

     (*register new datatypes as BNFs*)

     val (timer, Ibnfs, (folded_ctor_map_o_thms, folded_ctor_map_thms), folded_ctor_set_thmss',

         ctor_Irel_thms, Ibnf_notes, lthy) =

       if m = 0 then

         (timer, replicate n DEADID_bnf,

         map_split (`(mk_pointfree lthy)) (map2 mk_ctor_map_DEADID_thm ctor_inject_thms map_ids),

         replicate n [], map2 mk_ctor_Irel_DEADID_thm ctor_inject_thms bnfs, [], lthy)

       else let

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

         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 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), us),

           B1s), B2s), AXs), (xs, xs')), f1s), f2s), p1s), p2s), (ys, ys')),

           names_lthy) = names_lthy

           |> mk_Frees' "f" fTs

           ||>> mk_Frees "f" fTs

           ||>> mk_Frees "u" uTs

           ||>> mk_Frees "B1" B1Ts

           ||>> mk_Frees "B2" B2Ts

           ||>> mk_Frees "A" AXTs

           ||>> mk_Frees' "x" XTs

           ||>> mk_Frees "f1" f1Ts

           ||>> mk_Frees "f2" f2Ts

           ||>> mk_Frees "p1" p1Ts

           ||>> mk_Frees "p2" p2Ts

           ||>> mk_Frees' "y" passiveAs;

         val map_FTFT's = map2 (fn Ds =>

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

         fun mk_passive_maps ATs BTs Ts =

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

         fun mk_map_fold_arg fs Ts ctor fmap =

           HOLogic.mk_comp (ctor, Term.list_comb (fmap, fs @ map HOLogic.id_const Ts));

         fun mk_map Ts fs Ts' ctors mk_maps =

           mk_fold Ts (map2 (mk_map_fold_arg fs Ts') ctors (mk_maps Ts'));

         val pmapsABT' = mk_passive_maps passiveAs passiveBs;

         val fs_maps = map (mk_map Ts fs Ts' ctor's pmapsABT') ks;

         val fs_copy_maps = map (mk_map Ts fs_copy Ts' ctor's pmapsABT') ks;

         val Yctors = mk_ctors passiveYs;

         val f1s_maps = map (mk_map Ts f1s YTs Yctors (mk_passive_maps passiveAs passiveYs)) ks;

         val f2s_maps = map (mk_map Ts' f2s YTs Yctors (mk_passive_maps passiveBs passiveYs)) ks;

         val p1s_maps = map (mk_map XTs p1s Ts ctors (mk_passive_maps passiveXs passiveAs)) ks;

         val p2s_maps = map (mk_map XTs p2s Ts' ctor's (mk_passive_maps passiveXs passiveBs)) ks;

         val (ctor_map_thms, ctor_map_o_thms) =

           let

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

               (mk_Trueprop_eq (HOLogic.mk_comp (fs_map, ctor),

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

             val goals = map4 mk_goal fs_maps map_FTFT's ctors ctor's;

             val maps =

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

                 Goal.prove_sorry lthy [] [] goal (K (mk_map_tac m n foldx map_comp_id map_cong0))

                 |> Thm.close_derivation)

               goals ctor_fold_thms map_comp_id_thms map_cong0s;

           in

             `(map (fn thm => thm RS @{thm comp_eq_dest})) maps

           end;

         val (ctor_map_unique_thms, ctor_map_unique_thm) =

           let

             fun mk_prem u map ctor ctor' =

               mk_Trueprop_eq (HOLogic.mk_comp (u, ctor),

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

             val prems = map4 mk_prem us map_FTFT's ctors ctor's;

             val goal =

               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

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

             val unique = Goal.prove_sorry lthy [] []

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

               (mk_ctor_map_unique_tac ctor_fold_unique_thm sym_map_comps)

               |> Thm.close_derivation;

           in

             `split_conj_thm unique

           end;

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

         val bd = mk_cpow sum_bd;

         val bd_Cinfinite = sum_Cinfinite RS @{thm Cinfinite_cpow};

         fun mk_cpow_bd thm = @{thm ordLeq_transitive} OF

           [thm, sum_Card_order RS @{thm cpow_greater_eq}];

         val set_bd_cpowss = map (map mk_cpow_bd) set_bd_sumss;

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

         val ls = 1 upto m;

         val setsss = map (mk_setss o mk_set_Ts) passiveAs;

         val map_setss = map (fn T => map2 (fn Ds =>

           mk_map_of_bnf Ds (passiveAs @ Ts) (mk_set_Ts T)) Dss bnfs) passiveAs;

         fun mk_col l T z z' sets =

           let

             fun mk_UN set = mk_Union T $ (set $ z);

           in

             Term.absfree z'

               (mk_union (nth sets (l - 1) $ z,

                 Library.foldl1 mk_union (map mk_UN (drop m sets))))

           end;

         val colss = map5 (fn l => fn T => map3 (mk_col l T)) ls passiveAs AFss AFss' setsss;

         val setss_by_range = map (fn cols => map (mk_fold Ts cols) ks) colss;

         val setss_by_bnf = transpose setss_by_range;

         val ctor_set_thmss =

           let

             fun mk_goal sets ctor set col map =

               mk_Trueprop_eq (HOLogic.mk_comp (set, ctor),

                 HOLogic.mk_comp (col, Term.list_comb (map, passive_ids @ sets)));

             val goalss =

               map3 (fn sets => map4 (mk_goal sets) ctors sets) setss_by_range colss map_setss;

             val setss = map (map2 (fn foldx => fn goal =>

               Goal.prove_sorry lthy [] [] goal (K (mk_set_tac foldx)) |> Thm.close_derivation)

               ctor_fold_thms) goalss;

             fun mk_simp_goal pas_set act_sets sets ctor z set =

               Logic.all z (mk_Trueprop_eq (set $ (ctor $ z),

                 mk_union (pas_set $ z,

                   Library.foldl1 mk_union (map2 (fn X => mk_UNION (X $ z)) act_sets sets))));

             val simp_goalss =

               map2 (fn i => fn sets =>

                 map4 (fn Fsets => mk_simp_goal (nth Fsets (i - 1)) (drop m Fsets) sets)

                   FTs_setss ctors xFs sets)

                 ls setss_by_range;

             val ctor_setss = map3 (fn i => map3 (fn set_nats => fn goal => fn set =>

                 Goal.prove_sorry lthy [] [] goal

                   (K (mk_ctor_set_tac set (nth set_nats (i - 1)) (drop m set_nats)))

                 |> Thm.close_derivation)

               set_mapss) ls simp_goalss setss;

           in

             ctor_setss

           end;

         fun mk_set_thms ctor_set = (@{thm xt1(3)} OF [ctor_set, @{thm Un_upper1}]) ::

           map (fn i => (@{thm xt1(3)} OF [ctor_set, @{thm Un_upper2}]) RS

             (mk_Un_upper n i RS subset_trans) RSN

             (2, @{thm UN_upper} RS subset_trans))

             (1 upto n);

         val Fset_set_thmsss = transpose (map (map mk_set_thms) ctor_set_thmss);

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

         val cxs = map (SOME o certify lthy) Izs;

         val setss_by_bnf' =

           map (map (Term.subst_atomic_types (passiveAs ~~ passiveBs))) setss_by_bnf;

         val setss_by_range' = transpose setss_by_bnf';

         val set_map0_thmss =

           let

             fun mk_set_map0 f map z set set' =

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

             fun mk_cphi f map z set set' = certify lthy

               (Term.absfree (dest_Free z) (mk_set_map0 f map z set set'));

             val csetss = map (map (certify lthy)) setss_by_range';

             val cphiss = map3 (fn f => fn sets => fn sets' =>

               (map4 (mk_cphi f) fs_maps Izs sets sets')) fs setss_by_range setss_by_range';

             val inducts = map (fn cphis =>

               Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm) cphiss;

             val goals =

               map3 (fn f => fn sets => fn sets' =>

                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                   (map4 (mk_set_map0 f) fs_maps Izs sets sets')))

                   fs setss_by_range setss_by_range';

             fun mk_tac induct = mk_set_nat_tac m (rtac induct) set_mapss ctor_map_thms;

             val thms =

               map5 (fn goal => fn csets => fn ctor_sets => fn induct => fn i =>

                 singleton (Proof_Context.export names_lthy lthy)

                   (Goal.prove_sorry lthy [] [] goal (mk_tac induct csets ctor_sets i))

                 |> Thm.close_derivation)

               goals csetss ctor_set_thmss inducts ls;

           in

             map split_conj_thm thms

           end;

         val set_bd_thmss =

           let

             fun mk_set_bd z set = mk_ordLeq (mk_card_of (set $ z)) bd;

             fun mk_cphi z set = certify lthy (Term.absfree (dest_Free z) (mk_set_bd z set));

             val cphiss = map (map2 mk_cphi Izs) setss_by_range;

             val inducts = map (fn cphis =>

               Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm) cphiss;

             val goals =

               map (fn sets =>

                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                   (map2 mk_set_bd Izs sets))) setss_by_range;

             fun mk_tac induct = mk_set_bd_tac m (rtac induct) bd_Cinfinite set_bd_cpowss;

             val thms =

               map4 (fn goal => fn ctor_sets => fn induct => fn i =>

                 singleton (Proof_Context.export names_lthy lthy)

                   (Goal.prove_sorry lthy [] [] goal (mk_tac induct ctor_sets i))

                 |> Thm.close_derivation)

               goals ctor_set_thmss inducts ls;

           in

             map split_conj_thm thms

           end;

         val map_cong0_thms =

           let

             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_map_cong0 sets z fmap gmap =

               HOLogic.mk_imp

                 (Library.foldr1 HOLogic.mk_conj (map5 (mk_prem z) sets fs fs_copy ys ys'),

                 HOLogic.mk_eq (fmap $ z, gmap $ z));

             fun mk_cphi sets z fmap gmap =

               certify lthy (Term.absfree (dest_Free z) (mk_map_cong0 sets z fmap gmap));

             val cphis = map4 mk_cphi setss_by_bnf Izs fs_maps fs_copy_maps;

             val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm;

             val goal =

               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                 (map4 mk_map_cong0 setss_by_bnf Izs fs_maps fs_copy_maps));

             val thm = singleton (Proof_Context.export names_lthy lthy)

               (Goal.prove_sorry lthy [] [] goal

               (mk_mcong_tac (rtac induct) Fset_set_thmsss map_cong0s ctor_map_thms))

               |> Thm.close_derivation;

           in

             split_conj_thm thm

           end;

         val Xsetss = map (map (Term.subst_atomic_types (passiveAs ~~ passiveXs))) setss_by_bnf;

         val map_wpull_thms =

           let

             val cTs = map (SOME o certifyT lthy o TFree) induct2_params;

             val cxs = map (SOME o certify lthy) (splice Izs1 Izs2);

             fun mk_prem z1 z2 sets1 sets2 map1 map2 =

               HOLogic.mk_conj

                 (HOLogic.mk_mem (z1, mk_in B1s sets1 (fastype_of z1)),

                 HOLogic.mk_conj

                   (HOLogic.mk_mem (z2, mk_in B2s sets2 (fastype_of z2)),

                   HOLogic.mk_eq (map1 $ z1, map2 $ z2)));

             val prems = map6 mk_prem Izs1 Izs2 setss_by_bnf setss_by_bnf' f1s_maps f2s_maps;

             fun mk_concl z1 z2 sets map1 map2 T x x' =

               mk_Bex (mk_in AXs sets T) (Term.absfree x'

                 (HOLogic.mk_conj (HOLogic.mk_eq (map1 $ x, z1), HOLogic.mk_eq (map2 $ x, z2))));

             val concls = map8 mk_concl Izs1 Izs2 Xsetss p1s_maps p2s_maps XTs xs xs';

             val goals = map2 (curry HOLogic.mk_imp) prems concls;

             fun mk_cphi z1 z2 goal = certify lthy (Term.absfree z1 (Term.absfree z2 goal));

             val cphis = map3 mk_cphi Izs1' Izs2' goals;

             val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct2_thm;

             val goal = Logic.list_implies (map HOLogic.mk_Trueprop

                 (map8 mk_wpull AXs B1s B2s f1s f2s (replicate m NONE) p1s p2s),

               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj goals));

             val thm = singleton (Proof_Context.export names_lthy lthy)

               (Goal.prove_sorry lthy [] [] goal

               (K (mk_lfp_map_wpull_tac lthy m (rtac induct) map_wpulls ctor_map_thms

                 (transpose ctor_set_thmss) Fset_set_thmsss ctor_inject_thms)))

               |> Thm.close_derivation;

           in

             split_conj_thm thm

           end;

         val timer = time (timer "helpers for BNF properties");

         val map_id0_tacs = map (K o mk_map_id0_tac map_id0s) ctor_map_unique_thms;

         val map_comp0_tacs =

           map2 (K oo mk_map_comp0_tac map_comps ctor_map_thms) ctor_map_unique_thms ks;

         val map_cong0_tacs = map (mk_map_cong0_tac m) map_cong0_thms;

         val set_nat_tacss = map (map (K o mk_set_map0_tac)) (transpose set_map0_thmss);

         val bd_co_tacs = replicate n (K (mk_bd_card_order_tac bd_card_orders));

         val bd_cinf_tacs = replicate n (K (rtac (bd_Cinfinite RS conjunct1) 1));

         val set_bd_tacss = map (map (fn thm => K (rtac thm 1))) (transpose set_bd_thmss);

         val map_wpull_tacs = map (K o mk_wpull_tac) map_wpull_thms;

         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 ctor_witss =

           let

             val witss = map2 (fn Ds => fn bnf => mk_wits_of_bnf

               (replicate (nwits_of_bnf bnf) Ds)

               (replicate (nwits_of_bnf bnf) (passiveAs @ Ts)) bnf) Dss bnfs;

             fun close_wit (I, wit) = fold_rev Term.absfree (map (nth ys') I) wit;

             fun wit_apply (arg_I, arg_wit) (fun_I, fun_wit) =

               (union (op =) arg_I fun_I, fun_wit $ arg_wit);

             fun gen_arg support i =

               if i < m then [([i], nth ys i)]

               else maps (mk_wit support (nth ctors (i - m)) (i - m)) (nth support (i - m))

             and mk_wit support ctor i (I, wit) =

               let val args = map (gen_arg (nth_map i (remove (op =) (I, wit)) support)) I;

               in

                 (args, [([], wit)])

                 |-> fold (map_product wit_apply)

                 |> map (apsnd (fn t => ctor $ t))

                 |> minimize_wits

               end;

           in

             map3 (fn ctor => fn i => map close_wit o minimize_wits o maps (mk_wit witss ctor i))

               ctors (0 upto n - 1) witss

           end;

         fun wit_tac {context = ctxt, prems = _} =

           mk_wit_tac ctxt n (flat ctor_set_thmss) (maps wit_thms_of_bnf bnfs);

         val (Ibnfs, lthy) =

           fold_map9 (fn tacs => fn b => fn map_b => fn rel_b => fn set_bs => fn mapx => fn sets =>

               fn T => fn wits => fn lthy =>

             bnf_def Dont_Inline (user_policy Note_Some) I tacs wit_tac (SOME deads)

               map_b rel_b set_bs

               (((((b, fold_rev Term.absfree fs' mapx), sets), absdummy T bd), wits), NONE)

               lthy

             |> register_bnf (Local_Theory.full_name lthy b))

           tacss bs map_bs rel_bs set_bss fs_maps setss_by_bnf Ts ctor_witss lthy;

         val fold_maps = fold_thms lthy (map (fn bnf =>

           mk_unabs_def m (map_def_of_bnf bnf RS meta_eq_to_obj_eq)) Ibnfs);

         val fold_sets = fold_thms lthy (maps (fn bnf =>

           map (fn thm => thm RS meta_eq_to_obj_eq) (set_defs_of_bnf bnf)) Ibnfs);

         val timer = time (timer "registered new datatypes as BNFs");

         val Irels = map (mk_rel_of_bnf deads passiveAs passiveBs) Ibnfs;

         val Irelphis = map (fn Irel => Term.list_comb (Irel, Iphis)) Irels;

         val relphis = map (fn rel => Term.list_comb (rel, Iphis @ Irelphis)) rels;

         val in_rels = map in_rel_of_bnf bnfs;

         val in_Irels = map in_rel_of_bnf Ibnfs;

         val ctor_set_incl_thmss = map (map (fold_sets o hd)) Fset_set_thmsss;

         val ctor_set_set_incl_thmsss = map (transpose o map (map fold_sets o tl)) Fset_set_thmsss;

         val folded_ctor_map_thms = map fold_maps ctor_map_thms;

         val folded_ctor_map_o_thms = map fold_maps ctor_map_o_thms;

         val folded_ctor_set_thmss = map (map fold_sets) ctor_set_thmss;

         val folded_ctor_set_thmss' = transpose folded_ctor_set_thmss;

         val ctor_Irel_thms =

           let

             fun mk_goal xF yF ctor ctor' Irelphi relphi = fold_rev Logic.all (xF :: yF :: Iphis)

               (mk_Trueprop_eq (Irelphi $ (ctor $ xF) $ (ctor' $ yF), relphi $ xF $ yF));

             val goals = map6 mk_goal xFs yFs ctors ctor's Irelphis relphis;

           in

             map12 (fn i => fn goal => fn in_rel => fn map_comp0 => fn map_cong0 =>

               fn ctor_map => fn ctor_sets => fn ctor_inject => fn ctor_dtor =>

               fn set_map0s => fn ctor_set_incls => fn ctor_set_set_inclss =>

               Goal.prove_sorry lthy [] [] goal

                (K (mk_ctor_rel_tac lthy in_Irels i in_rel map_comp0 map_cong0 ctor_map ctor_sets

                  ctor_inject ctor_dtor set_map0s ctor_set_incls ctor_set_set_inclss))

               |> Thm.close_derivation)

             ks goals in_rels map_comps map_cong0s folded_ctor_map_thms folded_ctor_set_thmss'

               ctor_inject_thms ctor_dtor_thms set_mapss ctor_set_incl_thmss

               ctor_set_set_incl_thmsss

           end;

         val timer = time (timer "additional properties");

         val ls' = if m = 1 then [0] else ls

         val Ibnf_common_notes =

           [(ctor_map_uniqueN, [fold_maps ctor_map_unique_thm])]

           |> map (fn (thmN, thms) =>

             ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));

         val Ibnf_notes =

           [(ctor_mapN, map single folded_ctor_map_thms),

           (ctor_relN, map single ctor_Irel_thms),

           (ctor_set_inclN, ctor_set_incl_thmss),

           (ctor_set_set_inclN, map flat ctor_set_set_incl_thmsss)] @

           map2 (fn i => fn thms => (mk_ctor_setN i, map single thms)) ls' folded_ctor_set_thmss

           |> maps (fn (thmN, thmss) =>

             map2 (fn b => fn thms =>

               ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))

             bs thmss)

       in

         (timer, Ibnfs, (folded_ctor_map_o_thms, folded_ctor_map_thms), folded_ctor_set_thmss',

           ctor_Irel_thms, Ibnf_common_notes @ Ibnf_notes, lthy)

       end;

       val ctor_fold_o_map_thms = mk_xtor_un_fold_o_map_thms Least_FP false m ctor_fold_unique_thm

         folded_ctor_map_o_thms (map (mk_pointfree lthy) ctor_fold_thms) sym_map_comps map_cong0s;

       val ctor_rec_o_map_thms = mk_xtor_un_fold_o_map_thms Least_FP true m ctor_rec_unique_thm

         folded_ctor_map_o_thms (map (mk_pointfree lthy) ctor_rec_thms) sym_map_comps map_cong0s;

       val Irels = if m = 0 then map HOLogic.eq_const Ts

         else map (mk_rel_of_bnf deads passiveAs passiveBs) Ibnfs;

       val Irel_induct_thm =

         mk_rel_xtor_co_induct_thm Least_FP rels activeIphis Irels Iphis xFs yFs ctors ctor's

           (mk_rel_induct_tac m ctor_induct2_thm ks ctor_Irel_thms (map rel_mono_strong_of_bnf bnfs))

           lthy;

       val rels = map2 (fn Ds => mk_rel_of_bnf Ds allAs allBs') Dss bnfs;

       val ctor_fold_transfer_thms =

         mk_un_fold_transfer_thms Least_FP rels activephis Irels Iphis

           (mk_folds passiveAs activeAs) (mk_folds passiveBs activeBs)

           (mk_fold_transfer_tac m Irel_induct_thm (map map_transfer_of_bnf bnfs) ctor_fold_thms)

           lthy;

       val timer = time (timer "relator induction");

       val common_notes =

         [(ctor_inductN, [ctor_induct_thm]),

         (ctor_induct2N, [ctor_induct2_thm]),

         (rel_inductN, [Irel_induct_thm]),

         (ctor_fold_transferN, ctor_fold_transfer_thms)]

         |> map (fn (thmN, thms) =>

           ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));

       val notes =

         [(ctor_dtorN, ctor_dtor_thms),

         (ctor_exhaustN, ctor_exhaust_thms),

         (ctor_foldN, ctor_fold_thms),

         (ctor_fold_uniqueN, ctor_fold_unique_thms),

         (ctor_rec_uniqueN, ctor_rec_unique_thms),

         (ctor_fold_o_mapN, ctor_fold_o_map_thms),

         (ctor_rec_o_mapN, ctor_rec_o_map_thms),

         (ctor_injectN, ctor_inject_thms),

         (ctor_recN, ctor_rec_thms),

         (dtor_ctorN, dtor_ctor_thms),

         (dtor_exhaustN, dtor_exhaust_thms),

         (dtor_injectN, dtor_inject_thms)]

         |> map (apsnd (map single))

         |> maps (fn (thmN, thmss) =>

           map2 (fn b => fn thms =>

             ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))

           bs thmss);

     (*FIXME: once the package exports all the necessary high-level characteristic theorems,

        those should not only be concealed but rather not noted at all*)

     val maybe_conceal_notes = note_all = false ? map (apfst (apfst Binding.conceal));

   in

     timer;

     ({Ts = Ts, bnfs = Ibnfs, ctors = ctors, dtors = dtors, xtor_co_iterss = transpose [folds, recs],

       xtor_co_induct = ctor_induct_thm, dtor_ctors = dtor_ctor_thms, ctor_dtors = ctor_dtor_thms,

       ctor_injects = ctor_inject_thms, dtor_injects = dtor_inject_thms,

       xtor_map_thms = folded_ctor_map_thms, xtor_set_thmss = folded_ctor_set_thmss',

       xtor_rel_thms = ctor_Irel_thms,

       xtor_co_iter_thmss = transpose [ctor_fold_thms, ctor_rec_thms],

       xtor_co_iter_o_map_thmss = transpose [ctor_fold_o_map_thms, ctor_rec_o_map_thms],

       rel_xtor_co_induct_thm = Irel_induct_thm},

      lthy |> Local_Theory.notes (maybe_conceal_notes (common_notes @ notes @ Ibnf_notes)) |> snd)

   end;

 val _ =

   Outer_Syntax.local_theory @{command_spec "datatype_new"} "define new-style inductive datatypes"

     (parse_co_datatype_cmd Least_FP construct_lfp);

 val _ = Outer_Syntax.local_theory @{command_spec "primrec_new"}

   "define primitive recursive functions"

   (Parse.fixes -- Parse_Spec.where_alt_specs >> (snd oo uncurry add_primrec_cmd));

 end;