src/HOL/Tools/BNF/bnf_gfp.ML
author blanchet
Mon Jan 20 18:24:56 2014 +0100 (2014-01-20)
changeset 55061 a0adf838e2d1
parent 55060 3105434fb02f
child 55062 6d3fad6f01c9
permissions -rw-r--r--
adjusted comments
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(*  Title:      HOL/Tools/BNF/bnf_gfp.ML
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Andrei Popescu, TU Muenchen
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012
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Codatatype construction.
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*)
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signature BNF_GFP =
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sig
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  val construct_gfp: mixfix list -> binding list -> binding list -> binding list list ->
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    binding list -> (string * sort) list -> typ list * typ list list -> BNF_Def.bnf list ->
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    local_theory -> BNF_FP_Util.fp_result * local_theory
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end;
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structure BNF_GFP : BNF_GFP =
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struct
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open BNF_Def
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open BNF_Util
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open BNF_Tactics
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open BNF_Comp
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open BNF_FP_Util
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open BNF_FP_Def_Sugar
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open BNF_GFP_Rec_Sugar
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open BNF_GFP_Util
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open BNF_GFP_Tactics
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datatype wit_tree = Wit_Leaf of int | Wit_Node of (int * int * int list) * wit_tree list;
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fun mk_tree_args (I, T) (I', Ts) = (sort_distinct int_ord (I @ I'), T :: Ts);
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fun finish Iss m seen i (nwit, I) =
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  let
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    val treess = map (fn j =>
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        if j < m orelse member (op =) seen j then [([j], Wit_Leaf j)]
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        else
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          map_index (finish Iss m (insert (op =) j seen) j) (nth Iss (j - m))
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          |> flat
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          |> minimize_wits)
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      I;
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  in
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    map (fn (I, t) => (I, Wit_Node ((i - m, nwit, filter (fn i => i < m) I), t)))
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      (fold_rev (map_product mk_tree_args) treess [([], [])])
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    |> minimize_wits
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  end;
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fun tree_to_ctor_wit vars _ _ (Wit_Leaf j) = ([j], nth vars j)
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  | tree_to_ctor_wit vars ctors witss (Wit_Node ((i, nwit, I), subtrees)) =
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     (I, nth ctors i $ (Term.list_comb (snd (nth (nth witss i) nwit),
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       map (snd o tree_to_ctor_wit vars ctors witss) subtrees)));
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fun tree_to_coind_wits _ (Wit_Leaf _) = []
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  | tree_to_coind_wits lwitss (Wit_Node ((i, nwit, I), subtrees)) =
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     ((i, I), nth (nth lwitss i) nwit) :: maps (tree_to_coind_wits lwitss) subtrees;
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(*all BNFs have the same lives*)
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fun construct_gfp mixfixes map_bs rel_bs set_bss0 bs resBs (resDs, Dss) bnfs lthy =
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  let
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    val time = time lthy;
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    val timer = time (Timer.startRealTimer ());
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    val live = live_of_bnf (hd bnfs);
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    val n = length bnfs; (*active*)
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    val ks = 1 upto n;
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    val m = live - n; (*passive, if 0 don't generate a new BNF*)
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    val ls = 1 upto m;
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    val note_all = Config.get lthy bnf_note_all;
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    val b_names = map Binding.name_of bs;
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    val b_name = mk_common_name b_names;
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    val b = Binding.name b_name;
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    val mk_internal_b = Binding.name #> Binding.prefix true b_name #> Binding.conceal;
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    fun mk_internal_bs name =
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      map (fn b =>
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        Binding.prefix true b_name (Binding.prefix_name (name ^ "_") b) |> Binding.conceal) bs;
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    val external_bs = map2 (Binding.prefix false) b_names bs
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      |> note_all = false ? map Binding.conceal;
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    (* TODO: check if m, n, etc., are sane *)
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    val deads = fold (union (op =)) Dss resDs;
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    val names_lthy = fold Variable.declare_typ deads lthy;
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    val passives = map fst (subtract (op = o apsnd TFree) deads resBs);
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    (* tvars *)
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    val ((((((passiveAs, activeAs), passiveBs), activeBs), passiveCs), activeCs), idxT) = names_lthy
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      |> variant_tfrees passives
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      ||>> mk_TFrees n
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      ||>> variant_tfrees passives
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      ||>> mk_TFrees n
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      ||>> mk_TFrees m
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      ||>> mk_TFrees n
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      ||> fst o mk_TFrees 1
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      ||> the_single;
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    val allAs = passiveAs @ activeAs;
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    val allBs' = passiveBs @ activeBs;
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    val Ass = replicate n allAs;
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    val allBs = passiveAs @ activeBs;
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    val Bss = replicate n allBs;
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    val allCs = passiveAs @ activeCs;
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    val allCs' = passiveBs @ activeCs;
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    val Css' = replicate n allCs';
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    (* types *)
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    val dead_poss =
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      map (fn x => if member (op =) deads (TFree x) then SOME (TFree x) else NONE) resBs;
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    fun mk_param NONE passive = (hd passive, tl passive)
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      | mk_param (SOME a) passive = (a, passive);
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    val mk_params = fold_map mk_param dead_poss #> fst;
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    fun mk_FTs Ts = map2 (fn Ds => mk_T_of_bnf Ds Ts) Dss bnfs;
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    val (params, params') = `(map Term.dest_TFree) (mk_params passiveAs);
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    val (dead_params, dead_params') = `(map Term.dest_TFree) (subtract (op =) passiveAs params');
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    val FTsAs = mk_FTs allAs;
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    val FTsBs = mk_FTs allBs;
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    val FTsCs = mk_FTs allCs;
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    val ATs = map HOLogic.mk_setT passiveAs;
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    val BTs = map HOLogic.mk_setT activeAs;
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    val B'Ts = map HOLogic.mk_setT activeBs;
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    val B''Ts = map HOLogic.mk_setT activeCs;
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    val sTs = map2 (fn T => fn U => T --> U) activeAs FTsAs;
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    val s'Ts = map2 (fn T => fn U => T --> U) activeBs FTsBs;
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    val s''Ts = map2 (fn T => fn U => T --> U) activeCs FTsCs;
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    val fTs = map2 (fn T => fn U => T --> U) activeAs activeBs;
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    val self_fTs = map (fn T => T --> T) activeAs;
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    val gTs = map2 (fn T => fn U => T --> U) activeBs activeCs;
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    val all_gTs = map2 (fn T => fn U => T --> U) allBs allCs';
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    val RTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeBs;
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    val sRTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeAs;
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    val R'Ts = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeBs activeCs;
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    val setsRTs = map HOLogic.mk_setT sRTs;
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    val setRTs = map HOLogic.mk_setT RTs;
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    val all_sbisT = HOLogic.mk_tupleT setsRTs;
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    val setR'Ts = map HOLogic.mk_setT R'Ts;
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    val FRTs = mk_FTs (passiveAs @ RTs);
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    val sumBsAs = map2 (curry mk_sumT) activeBs activeAs;
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    val sumFTs = mk_FTs (passiveAs @ sumBsAs);
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    val sum_sTs = map2 (fn T => fn U => T --> U) activeAs sumFTs;
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    (* terms *)
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    val mapsAsAs = map4 mk_map_of_bnf Dss Ass Ass bnfs;
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    val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;
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    val mapsBsCs' = map4 mk_map_of_bnf Dss Bss Css' bnfs;
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    val mapsAsCs' = map4 mk_map_of_bnf Dss Ass Css' bnfs;
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    val map_Inls = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ sumBsAs)) bnfs;
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    val map_Inls_rev = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ sumBsAs)) Bss bnfs;
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    val map_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs;
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    val map_snds = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs;
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    fun mk_setss Ts = map3 mk_sets_of_bnf (map (replicate live) Dss)
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      (map (replicate live) (replicate n Ts)) bnfs;
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    val setssAs = mk_setss allAs;
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    val setssAs' = transpose setssAs;
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    val bis_setss = mk_setss (passiveAs @ RTs);
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    val relsAsBs = map4 mk_rel_of_bnf Dss Ass Bss bnfs;
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    val bds = map3 mk_bd_of_bnf Dss Ass bnfs;
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    val sum_bd = Library.foldr1 (uncurry mk_csum) bds;
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    val sum_bdT = fst (dest_relT (fastype_of sum_bd));
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    val emptys = map (fn T => HOLogic.mk_set T []) passiveAs;
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    val Zeros = map (fn empty =>
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     HOLogic.mk_tuple (map (fn U => absdummy U empty) activeAs)) emptys;
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    val hrecTs = map fastype_of Zeros;
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    val hsetTs = map (fn hrecT => Library.foldr (op -->) (sTs, HOLogic.natT --> hrecT)) hrecTs;
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    val (((((((((((((((((((((((((((((((((((zs, zs'), zs_copy), zs_copy2), z's), (ys, ys')),
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      As), Bs), Bs_copy), B's), B''s), ss), sum_ss), s's), s''s), fs), fs_copy),
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      self_fs), gs), all_gs), xFs), yFs), yFs_copy), RFs), (Rtuple, Rtuple')), (hrecs, hrecs')),
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      (nat, nat')), Rs), Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), Kss), names_lthy) = lthy
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      |> mk_Frees' "b" activeAs
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      ||>> mk_Frees "b" activeAs
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      ||>> mk_Frees "b" activeAs
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      ||>> mk_Frees "b" activeBs
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      ||>> mk_Frees' "y" passiveAs
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      ||>> mk_Frees "A" ATs
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      ||>> mk_Frees "B" BTs
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      ||>> mk_Frees "B" BTs
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      ||>> mk_Frees "B'" B'Ts
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      ||>> mk_Frees "B''" B''Ts
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      ||>> mk_Frees "s" sTs
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      ||>> mk_Frees "sums" sum_sTs
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      ||>> mk_Frees "s'" s'Ts
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      ||>> mk_Frees "s''" s''Ts
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      ||>> mk_Frees "f" fTs
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      ||>> mk_Frees "f" fTs
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      ||>> mk_Frees "f" self_fTs
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      ||>> mk_Frees "g" gTs
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      ||>> mk_Frees "g" all_gTs
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      ||>> mk_Frees "x" FTsAs
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      ||>> mk_Frees "y" FTsBs
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      ||>> mk_Frees "y" FTsBs
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      ||>> mk_Frees "x" FRTs
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      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Rtuple") all_sbisT
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      ||>> mk_Frees' "rec" hrecTs
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      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT
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      ||>> mk_Frees "R" setRTs
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      ||>> mk_Frees "R" setRTs
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      ||>> mk_Frees "R'" setR'Ts
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      ||>> mk_Frees "R" setsRTs
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      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") idxT
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      ||>> yield_singleton (mk_Frees "I") (HOLogic.mk_setT idxT)
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      ||>> mk_Frees "Ri" (map (fn T => idxT --> T) setRTs)
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      ||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) activeAs) ATs);
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    val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
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    val passive_Id_ons = map mk_Id_on As;
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    val active_UNIVs = map HOLogic.mk_UNIV activeAs;
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    val sum_UNIVs = map HOLogic.mk_UNIV sumBsAs;
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    val passive_ids = map HOLogic.id_const passiveAs;
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    val active_ids = map HOLogic.id_const activeAs;
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    val Inls = map2 Inl_const activeBs activeAs;
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    val fsts = map fst_const RTs;
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    val snds = map snd_const RTs;
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    (* thms *)
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    val bd_card_orders = map bd_card_order_of_bnf bnfs;
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    val bd_card_order = hd bd_card_orders
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    val bd_Card_orders = map bd_Card_order_of_bnf bnfs;
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    val bd_Card_order = hd bd_Card_orders;
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    val bd_Cinfinites = map bd_Cinfinite_of_bnf bnfs;
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    val bd_Cinfinite = hd bd_Cinfinites;
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    val in_monos = map in_mono_of_bnf bnfs;
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    val map_comp0s = map map_comp0_of_bnf bnfs;
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    val sym_map_comps = map mk_sym map_comp0s;
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    val map_comps = map map_comp_of_bnf bnfs;
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    val map_cong0s = map map_cong0_of_bnf bnfs;
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    val map_id0s = map map_id0_of_bnf bnfs;
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    val map_ids = map map_id_of_bnf bnfs;
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    val set_bdss = map set_bd_of_bnf bnfs;
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    val set_mapss = map set_map_of_bnf bnfs;
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    val rel_congs = map rel_cong_of_bnf bnfs;
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    val rel_converseps = map rel_conversep_of_bnf bnfs;
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    val rel_Grps = map rel_Grp_of_bnf bnfs;
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    val rel_OOs = map rel_OO_of_bnf bnfs;
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    val rel_OO_Grps = map rel_OO_Grp_of_bnf bnfs;
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    val timer = time (timer "Extracted terms & thms");
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    (* derived thms *)
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    (*map g1 ... gm g(m+1) ... g(m+n) (map id ... id f(m+1) ... f(m+n) x) =
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      map g1 ... gm (g(m+1) o f(m+1)) ... (g(m+n) o f(m+n)) x*)
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    fun mk_map_comp_id x mapAsBs mapBsCs mapAsCs map_comp0 =
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      let
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        val lhs = Term.list_comb (mapBsCs, all_gs) $
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          (Term.list_comb (mapAsBs, passive_ids @ fs) $ x);
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        val rhs =
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          Term.list_comb (mapAsCs, take m all_gs @ map HOLogic.mk_comp (drop m all_gs ~~ fs)) $ x;
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      in
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        Goal.prove_sorry lthy [] []
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          (fold_rev Logic.all (x :: fs @ all_gs) (mk_Trueprop_eq (lhs, rhs)))
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          (K (mk_map_comp_id_tac map_comp0))
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        |> Thm.close_derivation
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      end;
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    val map_comp_id_thms = map5 mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comps;
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    (*forall a : set(m+1) x. f(m+1) a = a; ...; forall a : set(m+n) x. f(m+n) a = a ==>
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      map id ... id f(m+1) ... f(m+n) x = x*)
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    fun mk_map_cong0L x mapAsAs sets map_cong0 map_id =
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      let
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        fun mk_prem set f z z' =
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          HOLogic.mk_Trueprop
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            (mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z))));
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        val prems = map4 mk_prem (drop m sets) self_fs zs zs';
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        val goal = mk_Trueprop_eq (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x);
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      in
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        Goal.prove_sorry lthy [] []
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   271
          (fold_rev Logic.all (x :: self_fs) (Logic.list_implies (prems, goal)))
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   272
          (K (mk_map_cong0L_tac m map_cong0 map_id))
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   273
        |> Thm.close_derivation
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   274
      end;
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   275
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   276
    val map_cong0L_thms = map5 mk_map_cong0L xFs mapsAsAs setssAs map_cong0s map_ids;
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   277
    val in_mono'_thms = map (fn thm =>
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   278
      (thm OF (replicate m subset_refl)) RS @{thm set_mp}) in_monos;
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   279
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   280
    val map_arg_cong_thms =
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   281
      let
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   282
        val prems = map2 (curry mk_Trueprop_eq) yFs yFs_copy;
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   283
        val maps = map (fn mapx => Term.list_comb (mapx, all_gs)) mapsBsCs';
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   284
        val concls =
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   285
          map3 (fn x => fn y => fn mapx => mk_Trueprop_eq (mapx $ x, mapx $ y)) yFs yFs_copy maps;
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   286
        val goals =
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   287
          map4 (fn prem => fn concl => fn x => fn y =>
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   288
            fold_rev Logic.all (x :: y :: all_gs) (Logic.mk_implies (prem, concl)))
traytel@51925
   289
          prems concls yFs yFs_copy;
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   290
      in
wenzelm@51551
   291
        map (fn goal => Goal.prove_sorry lthy [] [] goal
wenzelm@51798
   292
          (K ((hyp_subst_tac lthy THEN' rtac refl) 1)) |> Thm.close_derivation) goals
blanchet@48975
   293
      end;
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   294
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   295
    val timer = time (timer "Derived simple theorems");
blanchet@48975
   296
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   297
    (* coalgebra *)
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   298
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   299
    val coalg_bind = mk_internal_b (coN ^ algN) ;
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   300
    val coalg_name = Binding.name_of coalg_bind;
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   301
    val coalg_def_bind = (Thm.def_binding coalg_bind, []);
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   302
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   303
    (*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in A1 .. Am B1 ... Bn)*)
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   304
    val coalg_spec =
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   305
      let
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   306
        val coalgT = Library.foldr (op -->) (ATs @ BTs @ sTs, HOLogic.boolT);
blanchet@48975
   307
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   308
        val ins = map3 mk_in (replicate n (As @ Bs)) setssAs FTsAs;
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   309
        fun mk_coalg_conjunct B s X z z' =
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   310
          mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X)));
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   311
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   312
        val lhs = Term.list_comb (Free (coalg_name, coalgT), As @ Bs @ ss);
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   313
        val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_coalg_conjunct Bs ss ins zs zs')
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   314
      in
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   315
        mk_Trueprop_eq (lhs, rhs)
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   316
      end;
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   317
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   318
    val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) =
blanchet@49311
   319
      lthy
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   320
      |> Specification.definition (SOME (coalg_bind, NONE, NoSyn), (coalg_def_bind, coalg_spec))
blanchet@49311
   321
      ||> `Local_Theory.restore;
blanchet@49311
   322
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   323
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   324
    val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free));
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   325
    val coalg_def = Morphism.thm phi coalg_def_free;
blanchet@48975
   326
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   327
    fun mk_coalg As Bs ss =
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   328
      let
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   329
        val args = As @ Bs @ ss;
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   330
        val Ts = map fastype_of args;
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   331
        val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   332
      in
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   333
        Term.list_comb (Const (coalg, coalgT), args)
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   334
      end;
blanchet@48975
   335
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   336
    val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
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   337
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   338
    val coalg_in_thms = map (fn i =>
traytel@52904
   339
      coalg_def RS iffD1 RS mk_conjunctN n i RS bspec) ks
blanchet@48975
   340
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   341
    val coalg_set_thmss =
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   342
      let
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   343
        val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
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   344
        fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));
traytel@51893
   345
        fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) B);
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   346
        val prems = map2 mk_prem zs Bs;
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   347
        val conclss = map3 (fn s => fn x => fn sets => map2 (mk_concl s x) (As @ Bs) sets)
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   348
          ss zs setssAs;
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   349
        val goalss = map3 (fn x => fn prem => fn concls => map (fn concl =>
blanchet@48975
   350
          fold_rev Logic.all (x :: As @ Bs @ ss)
blanchet@48975
   351
            (Logic.list_implies (coalg_prem :: [prem], concl))) concls) zs prems conclss;
blanchet@48975
   352
      in
wenzelm@51551
   353
        map (fn goals => map (fn goal => Goal.prove_sorry lthy [] [] goal
traytel@49109
   354
          (K (mk_coalg_set_tac coalg_def)) |> Thm.close_derivation) goals) goalss
blanchet@48975
   355
      end;
blanchet@48975
   356
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   357
    fun mk_tcoalg ATs BTs = mk_coalg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);
blanchet@48975
   358
blanchet@48975
   359
    val tcoalg_thm =
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   360
      let
blanchet@48975
   361
        val goal = fold_rev Logic.all ss
blanchet@48975
   362
          (HOLogic.mk_Trueprop (mk_tcoalg passiveAs activeAs ss))
blanchet@48975
   363
      in
wenzelm@51551
   364
        Goal.prove_sorry lthy [] [] goal
blanchet@48975
   365
          (K (stac coalg_def 1 THEN CONJ_WRAP
blanchet@48975
   366
            (K (EVERY' [rtac ballI, rtac CollectI,
blanchet@48975
   367
              CONJ_WRAP' (K (EVERY' [rtac @{thm subset_UNIV}])) allAs] 1)) ss))
traytel@49109
   368
        |> Thm.close_derivation
blanchet@48975
   369
      end;
blanchet@48975
   370
blanchet@48975
   371
    val timer = time (timer "Coalgebra definition & thms");
blanchet@48975
   372
blanchet@48975
   373
    (* morphism *)
blanchet@48975
   374
traytel@53566
   375
    val mor_bind = mk_internal_b morN;
blanchet@48975
   376
    val mor_name = Binding.name_of mor_bind;
blanchet@48975
   377
    val mor_def_bind = (Thm.def_binding mor_bind, []);
blanchet@48975
   378
blanchet@48975
   379
    (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*)
blanchet@48975
   380
    (*mor) forall i = 1 ... n: (\<forall>x \<in> Bi.
blanchet@48975
   381
       Fi_map id ... id f1 ... fn (si x) = si' (fi x)*)
blanchet@48975
   382
    val mor_spec =
blanchet@48975
   383
      let
blanchet@48975
   384
        val morT = Library.foldr (op -->) (BTs @ sTs @ B'Ts @ s'Ts @ fTs, HOLogic.boolT);
blanchet@48975
   385
blanchet@48975
   386
        fun mk_fbetw f B1 B2 z z' =
blanchet@48975
   387
          mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));
blanchet@48975
   388
        fun mk_mor B mapAsBs f s s' z z' =
blanchet@48975
   389
          mk_Ball B (Term.absfree z' (HOLogic.mk_eq
blanchet@48975
   390
            (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z))));
blanchet@48975
   391
        val lhs = Term.list_comb (Free (mor_name, morT), Bs @ ss @ B's @ s's @ fs);
blanchet@48975
   392
        val rhs = HOLogic.mk_conj
blanchet@48975
   393
          (Library.foldr1 HOLogic.mk_conj (map5 mk_fbetw fs Bs B's zs zs'),
blanchet@48975
   394
           Library.foldr1 HOLogic.mk_conj (map7 mk_mor Bs mapsAsBs fs ss s's zs zs'))
blanchet@48975
   395
      in
blanchet@49123
   396
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   397
      end;
blanchet@48975
   398
blanchet@48975
   399
    val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
blanchet@49311
   400
      lthy
blanchet@49311
   401
      |> Specification.definition (SOME (mor_bind, NONE, NoSyn), (mor_def_bind, mor_spec))
blanchet@49311
   402
      ||> `Local_Theory.restore;
blanchet@49311
   403
blanchet@48975
   404
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   405
    val mor = fst (Term.dest_Const (Morphism.term phi mor_free));
blanchet@48975
   406
    val mor_def = Morphism.thm phi mor_def_free;
blanchet@48975
   407
blanchet@48975
   408
    fun mk_mor Bs1 ss1 Bs2 ss2 fs =
blanchet@48975
   409
      let
blanchet@48975
   410
        val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;
blanchet@48975
   411
        val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);
blanchet@48975
   412
        val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   413
      in
blanchet@48975
   414
        Term.list_comb (Const (mor, morT), args)
blanchet@48975
   415
      end;
blanchet@48975
   416
blanchet@48975
   417
    val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
blanchet@48975
   418
blanchet@48975
   419
    val (mor_image_thms, morE_thms) =
blanchet@48975
   420
      let
blanchet@48975
   421
        val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
blanchet@48975
   422
        fun mk_image_goal f B1 B2 = fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs)
traytel@51893
   423
          (Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_leq (mk_image f $ B1) B2)));
blanchet@48975
   424
        val image_goals = map3 mk_image_goal fs Bs B's;
blanchet@48975
   425
        fun mk_elim_goal B mapAsBs f s s' x =
blanchet@48975
   426
          fold_rev Logic.all (x :: Bs @ ss @ B's @ s's @ fs)
blanchet@48975
   427
            (Logic.list_implies ([prem, HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B))],
blanchet@49123
   428
              mk_Trueprop_eq (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x))));
blanchet@48975
   429
        val elim_goals = map6 mk_elim_goal Bs mapsAsBs fs ss s's zs;
blanchet@48975
   430
        fun prove goal =
wenzelm@51551
   431
          Goal.prove_sorry lthy [] [] goal (K (mk_mor_elim_tac mor_def))
traytel@49109
   432
          |> Thm.close_derivation;
blanchet@48975
   433
      in
blanchet@48975
   434
        (map prove image_goals, map prove elim_goals)
blanchet@48975
   435
      end;
blanchet@48975
   436
blanchet@48975
   437
    val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms;
blanchet@48975
   438
blanchet@48975
   439
    val mor_incl_thm =
blanchet@48975
   440
      let
traytel@51893
   441
        val prems = map2 (HOLogic.mk_Trueprop oo mk_leq) Bs Bs_copy;
blanchet@48975
   442
        val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
blanchet@48975
   443
      in
wenzelm@51551
   444
        Goal.prove_sorry lthy [] []
blanchet@48975
   445
          (fold_rev Logic.all (Bs @ ss @ Bs_copy) (Logic.list_implies (prems, concl)))
blanchet@53285
   446
          (K (mk_mor_incl_tac mor_def map_ids))
traytel@49109
   447
        |> Thm.close_derivation
blanchet@48975
   448
      end;
blanchet@48975
   449
blanchet@48975
   450
    val mor_id_thm = mor_incl_thm OF (replicate n subset_refl);
blanchet@48975
   451
blanchet@48975
   452
    val mor_comp_thm =
blanchet@48975
   453
      let
blanchet@48975
   454
        val prems =
blanchet@48975
   455
          [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),
blanchet@48975
   456
           HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];
blanchet@48975
   457
        val concl =
blanchet@48975
   458
          HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));
blanchet@48975
   459
      in
wenzelm@51551
   460
        Goal.prove_sorry lthy [] []
blanchet@48975
   461
          (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)
blanchet@48975
   462
            (Logic.list_implies (prems, concl)))
blanchet@48975
   463
          (K (mk_mor_comp_tac mor_def mor_image'_thms morE_thms map_comp_id_thms))
traytel@49109
   464
        |> Thm.close_derivation
blanchet@48975
   465
      end;
blanchet@48975
   466
blanchet@48975
   467
    val mor_cong_thm =
blanchet@48975
   468
      let
blanchet@48975
   469
        val prems = map HOLogic.mk_Trueprop
blanchet@48975
   470
         (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])
blanchet@48975
   471
        val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);
blanchet@48975
   472
      in
wenzelm@51551
   473
        Goal.prove_sorry lthy [] []
blanchet@48975
   474
          (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ fs_copy)
blanchet@48975
   475
            (Logic.list_implies (prems, concl)))
wenzelm@51798
   476
          (K ((hyp_subst_tac lthy THEN' atac) 1))
traytel@49109
   477
        |> Thm.close_derivation
blanchet@48975
   478
      end;
blanchet@48975
   479
blanchet@48975
   480
    val mor_UNIV_thm =
blanchet@48975
   481
      let
blanchet@48975
   482
        fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
blanchet@48975
   483
            (HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s),
blanchet@48975
   484
            HOLogic.mk_comp (s', f));
blanchet@48975
   485
        val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;
blanchet@48975
   486
        val rhs = Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct mapsAsBs fs ss s's);
blanchet@48975
   487
      in
wenzelm@51551
   488
        Goal.prove_sorry lthy [] [] (fold_rev Logic.all (ss @ s's @ fs) (mk_Trueprop_eq (lhs, rhs)))
blanchet@48975
   489
          (K (mk_mor_UNIV_tac morE_thms mor_def))
traytel@49109
   490
        |> Thm.close_derivation
blanchet@48975
   491
      end;
blanchet@48975
   492
blanchet@48975
   493
    val mor_str_thm =
blanchet@48975
   494
      let
blanchet@48975
   495
        val maps = map2 (fn Ds => fn bnf => Term.list_comb
blanchet@48975
   496
          (mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs;
blanchet@48975
   497
      in
wenzelm@51551
   498
        Goal.prove_sorry lthy [] []
blanchet@48975
   499
          (fold_rev Logic.all ss (HOLogic.mk_Trueprop
blanchet@48975
   500
            (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss)))
blanchet@48975
   501
          (K (mk_mor_str_tac ks mor_UNIV_thm))
traytel@49109
   502
        |> Thm.close_derivation
blanchet@48975
   503
      end;
blanchet@48975
   504
blanchet@48975
   505
    val mor_sum_case_thm =
blanchet@48975
   506
      let
blanchet@49458
   507
        val maps = map3 (fn s => fn sum_s => fn mapx =>
blanchet@49458
   508
          mk_sum_case (HOLogic.mk_comp (Term.list_comb (mapx, passive_ids @ Inls), s), sum_s))
blanchet@48975
   509
          s's sum_ss map_Inls;
blanchet@48975
   510
      in
wenzelm@51551
   511
        Goal.prove_sorry lthy [] []
blanchet@48975
   512
          (fold_rev Logic.all (s's @ sum_ss) (HOLogic.mk_Trueprop
blanchet@48975
   513
            (mk_mor (map HOLogic.mk_UNIV activeBs) s's sum_UNIVs maps Inls)))
blanchet@48975
   514
          (K (mk_mor_sum_case_tac ks mor_UNIV_thm))
traytel@49109
   515
        |> Thm.close_derivation
blanchet@48975
   516
      end;
blanchet@48975
   517
blanchet@48975
   518
    val timer = time (timer "Morphism definition & thms");
blanchet@48975
   519
traytel@53566
   520
    fun hset_rec_bind j = mk_internal_b (hset_recN ^ (if m = 1 then "" else string_of_int j));
blanchet@48975
   521
    val hset_rec_name = Binding.name_of o hset_rec_bind;
blanchet@48975
   522
    val hset_rec_def_bind = rpair [] o Thm.def_binding o hset_rec_bind;
blanchet@48975
   523
blanchet@48975
   524
    fun hset_rec_spec j Zero hsetT hrec hrec' =
blanchet@48975
   525
      let
blanchet@48975
   526
        fun mk_Suc s setsAs z z' =
blanchet@48975
   527
          let
blanchet@48975
   528
            val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m setsAs);
blanchet@48975
   529
            fun mk_UN set k = mk_UNION (set $ (s $ z)) (mk_nthN n hrec k);
blanchet@48975
   530
          in
blanchet@48975
   531
            Term.absfree z'
blanchet@48975
   532
              (mk_union (set $ (s $ z), Library.foldl1 mk_union (map2 mk_UN sets ks)))
blanchet@48975
   533
          end;
blanchet@48975
   534
blanchet@48975
   535
        val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec'
blanchet@48975
   536
          (HOLogic.mk_tuple (map4 mk_Suc ss setssAs zs zs')));
blanchet@48975
   537
blanchet@48975
   538
        val lhs = Term.list_comb (Free (hset_rec_name j, hsetT), ss);
blanchet@48975
   539
        val rhs = mk_nat_rec Zero Suc;
blanchet@48975
   540
      in
blanchet@49123
   541
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   542
      end;
blanchet@48975
   543
blanchet@48975
   544
    val ((hset_rec_frees, (_, hset_rec_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   545
      lthy
blanchet@48975
   546
      |> fold_map5 (fn j => fn Zero => fn hsetT => fn hrec => fn hrec' => Specification.definition
blanchet@48975
   547
        (SOME (hset_rec_bind j, NONE, NoSyn),
blanchet@48975
   548
          (hset_rec_def_bind j, hset_rec_spec j Zero hsetT hrec hrec')))
blanchet@48975
   549
        ls Zeros hsetTs hrecs hrecs'
blanchet@48975
   550
      |>> apsnd split_list o split_list
blanchet@48975
   551
      ||> `Local_Theory.restore;
blanchet@48975
   552
blanchet@48975
   553
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   554
blanchet@48975
   555
    val hset_rec_defs = map (Morphism.thm phi) hset_rec_def_frees;
blanchet@48975
   556
    val hset_recs = map (fst o Term.dest_Const o Morphism.term phi) hset_rec_frees;
blanchet@48975
   557
blanchet@48975
   558
    fun mk_hset_rec ss nat i j T =
blanchet@48975
   559
      let
blanchet@48975
   560
        val args = ss @ [nat];
blanchet@48975
   561
        val Ts = map fastype_of ss;
blanchet@48975
   562
        val bTs = map domain_type Ts;
blanchet@48975
   563
        val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) bTs)
blanchet@48975
   564
        val hset_recT = Library.foldr (op -->) (Ts, HOLogic.natT --> hrecT);
blanchet@48975
   565
      in
blanchet@48975
   566
        mk_nthN n (Term.list_comb (Const (nth hset_recs (j - 1), hset_recT), args)) i
blanchet@48975
   567
      end;
blanchet@48975
   568
blanchet@48975
   569
    val hset_rec_0ss = mk_rec_simps n @{thm nat_rec_0} hset_rec_defs;
blanchet@48975
   570
    val hset_rec_Sucss = mk_rec_simps n @{thm nat_rec_Suc} hset_rec_defs;
blanchet@48975
   571
    val hset_rec_0ss' = transpose hset_rec_0ss;
blanchet@48975
   572
    val hset_rec_Sucss' = transpose hset_rec_Sucss;
blanchet@48975
   573
traytel@53566
   574
    fun hset_binds j = mk_internal_bs (hsetN ^ (if m = 1 then "" else string_of_int j))
traytel@53566
   575
    fun hset_bind i j = nth (hset_binds j) (i - 1);
blanchet@48975
   576
    val hset_name = Binding.name_of oo hset_bind;
blanchet@48975
   577
    val hset_def_bind = rpair [] o Thm.def_binding oo hset_bind;
blanchet@48975
   578
blanchet@48975
   579
    fun hset_spec i j =
blanchet@48975
   580
      let
blanchet@48975
   581
        val U = nth activeAs (i - 1);
blanchet@48975
   582
        val z = nth zs (i - 1);
blanchet@48975
   583
        val T = nth passiveAs (j - 1);
blanchet@48975
   584
        val setT = HOLogic.mk_setT T;
blanchet@48975
   585
        val hsetT = Library.foldr (op -->) (sTs, U --> setT);
blanchet@48975
   586
blanchet@48975
   587
        val lhs = Term.list_comb (Free (hset_name i j, hsetT), ss @ [z]);
blanchet@48975
   588
        val rhs = mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
blanchet@48975
   589
          (Term.absfree nat' (mk_hset_rec ss nat i j T $ z));
blanchet@48975
   590
      in
blanchet@49123
   591
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   592
      end;
blanchet@48975
   593
blanchet@48975
   594
    val ((hset_frees, (_, hset_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   595
      lthy
blanchet@48975
   596
      |> fold_map (fn i => fold_map (fn j => Specification.definition
blanchet@48975
   597
        (SOME (hset_bind i j, NONE, NoSyn), (hset_def_bind i j, hset_spec i j))) ls) ks
blanchet@48975
   598
      |>> map (apsnd split_list o split_list)
blanchet@48975
   599
      |>> apsnd split_list o split_list
blanchet@48975
   600
      ||> `Local_Theory.restore;
blanchet@48975
   601
blanchet@48975
   602
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   603
blanchet@48975
   604
    val hset_defss = map (map (Morphism.thm phi)) hset_def_frees;
blanchet@48975
   605
    val hset_defss' = transpose hset_defss;
blanchet@48975
   606
    val hset_namess = map (map (fst o Term.dest_Const o Morphism.term phi)) hset_frees;
blanchet@48975
   607
blanchet@48975
   608
    fun mk_hset ss i j T =
blanchet@48975
   609
      let
blanchet@48975
   610
        val Ts = map fastype_of ss;
blanchet@48975
   611
        val bTs = map domain_type Ts;
blanchet@48975
   612
        val hsetT = Library.foldr (op -->) (Ts, nth bTs (i - 1) --> HOLogic.mk_setT T);
blanchet@48975
   613
      in
blanchet@48975
   614
        Term.list_comb (Const (nth (nth hset_namess (i - 1)) (j - 1), hsetT), ss)
blanchet@48975
   615
      end;
blanchet@48975
   616
blanchet@48975
   617
    val hsetssAs = map (fn i => map2 (mk_hset ss i) ls passiveAs) ks;
blanchet@48975
   618
blanchet@48975
   619
    val (set_incl_hset_thmss, set_hset_incl_hset_thmsss) =
blanchet@48975
   620
      let
blanchet@48975
   621
        fun mk_set_incl_hset s x set hset = fold_rev Logic.all (x :: ss)
traytel@51893
   622
          (HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) (hset $ x)));
blanchet@48975
   623
blanchet@48975
   624
        fun mk_set_hset_incl_hset s x y set hset1 hset2 =
blanchet@48975
   625
          fold_rev Logic.all (x :: y :: ss)
blanchet@48975
   626
            (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x, set $ (s $ y))),
traytel@51893
   627
            HOLogic.mk_Trueprop (mk_leq (hset1 $ x) (hset2 $ y))));
blanchet@48975
   628
blanchet@48975
   629
        val set_incl_hset_goalss =
blanchet@48975
   630
          map4 (fn s => fn x => fn sets => fn hsets =>
blanchet@48975
   631
            map2 (mk_set_incl_hset s x) (take m sets) hsets)
blanchet@48975
   632
          ss zs setssAs hsetssAs;
blanchet@48975
   633
blanchet@48975
   634
        (*xk : F(i)set(m+k) (si yi) ==> F(k)_hset(j) s1 ... sn xk <= F(i)_hset(j) s1 ... sn yi*)
blanchet@48975
   635
        val set_hset_incl_hset_goalsss =
blanchet@48975
   636
          map4 (fn si => fn yi => fn sets => fn hsetsi =>
blanchet@48975
   637
            map3 (fn xk => fn set => fn hsetsk =>
blanchet@48975
   638
              map2 (mk_set_hset_incl_hset si xk yi set) hsetsk hsetsi)
blanchet@48975
   639
            zs_copy (drop m sets) hsetssAs)
blanchet@48975
   640
          ss zs setssAs hsetssAs;
blanchet@48975
   641
      in
blanchet@48975
   642
        (map3 (fn goals => fn defs => fn rec_Sucs =>
blanchet@48975
   643
          map3 (fn goal => fn def => fn rec_Suc =>
wenzelm@51551
   644
            Goal.prove_sorry lthy [] [] goal (K (mk_set_incl_hset_tac def rec_Suc))
traytel@49109
   645
            |> Thm.close_derivation)
blanchet@48975
   646
          goals defs rec_Sucs)
blanchet@48975
   647
        set_incl_hset_goalss hset_defss hset_rec_Sucss,
blanchet@48975
   648
        map3 (fn goalss => fn defsi => fn rec_Sucs =>
blanchet@48975
   649
          map3 (fn k => fn goals => fn defsk =>
blanchet@48975
   650
            map4 (fn goal => fn defk => fn defi => fn rec_Suc =>
wenzelm@51551
   651
              Goal.prove_sorry lthy [] [] goal
traytel@49109
   652
                (K (mk_set_hset_incl_hset_tac n [defk, defi] rec_Suc k))
traytel@49109
   653
              |> Thm.close_derivation)
blanchet@48975
   654
            goals defsk defsi rec_Sucs)
blanchet@48975
   655
          ks goalss hset_defss)
blanchet@48975
   656
        set_hset_incl_hset_goalsss hset_defss hset_rec_Sucss)
blanchet@48975
   657
      end;
blanchet@48975
   658
blanchet@48975
   659
    val set_incl_hset_thmss' = transpose set_incl_hset_thmss;
blanchet@48975
   660
    val set_hset_incl_hset_thmsss' = transpose (map transpose set_hset_incl_hset_thmsss);
blanchet@48975
   661
    val set_hset_thmss = map (map (fn thm => thm RS @{thm set_mp})) set_incl_hset_thmss;
blanchet@48975
   662
    val set_hset_hset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp})))
blanchet@48975
   663
      set_hset_incl_hset_thmsss;
blanchet@48975
   664
    val set_hset_thmss' = transpose set_hset_thmss;
blanchet@48975
   665
    val set_hset_hset_thmsss' = transpose (map transpose set_hset_hset_thmsss);
blanchet@48975
   666
blanchet@48975
   667
    val hset_minimal_thms =
blanchet@48975
   668
      let
blanchet@48975
   669
        fun mk_passive_prem set s x K =
traytel@51893
   670
          Logic.all x (HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) (K $ x)));
blanchet@48975
   671
blanchet@48975
   672
        fun mk_active_prem s x1 K1 set x2 K2 =
blanchet@48975
   673
          fold_rev Logic.all [x1, x2]
blanchet@48975
   674
            (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x2, set $ (s $ x1))),
traytel@51893
   675
              HOLogic.mk_Trueprop (mk_leq (K2 $ x2) (K1 $ x1))));
blanchet@48975
   676
blanchet@48975
   677
        val premss = map2 (fn j => fn Ks =>
blanchet@48975
   678
          map4 mk_passive_prem (map (fn xs => nth xs (j - 1)) setssAs) ss zs Ks @
blanchet@48975
   679
            flat (map4 (fn sets => fn s => fn x1 => fn K1 =>
blanchet@48975
   680
              map3 (mk_active_prem s x1 K1) (drop m sets) zs_copy Ks) setssAs ss zs Ks))
blanchet@48975
   681
          ls Kss;
blanchet@48975
   682
blanchet@48975
   683
        val hset_rec_minimal_thms =
blanchet@48975
   684
          let
traytel@51893
   685
            fun mk_conjunct j T i K x = mk_leq (mk_hset_rec ss nat i j T $ x) (K $ x);
blanchet@48975
   686
            fun mk_concl j T Ks = list_all_free zs
blanchet@48975
   687
              (Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs));
blanchet@48975
   688
            val concls = map3 mk_concl ls passiveAs Kss;
blanchet@48975
   689
blanchet@48975
   690
            val goals = map2 (fn prems => fn concl =>
blanchet@48975
   691
              Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls
blanchet@48975
   692
blanchet@48975
   693
            val ctss =
blanchet@48975
   694
              map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
blanchet@48975
   695
          in
blanchet@48975
   696
            map4 (fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>
blanchet@48975
   697
              singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
   698
                (Goal.prove_sorry lthy [] [] goal
traytel@49109
   699
                  (mk_hset_rec_minimal_tac m cts hset_rec_0s hset_rec_Sucs))
traytel@49109
   700
              |> Thm.close_derivation)
blanchet@48975
   701
            goals ctss hset_rec_0ss' hset_rec_Sucss'
blanchet@48975
   702
          end;
blanchet@48975
   703
traytel@51893
   704
        fun mk_conjunct j T i K x = mk_leq (mk_hset ss i j T $ x) (K $ x);
blanchet@48975
   705
        fun mk_concl j T Ks = Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs);
blanchet@48975
   706
        val concls = map3 mk_concl ls passiveAs Kss;
blanchet@48975
   707
blanchet@48975
   708
        val goals = map3 (fn Ks => fn prems => fn concl =>
blanchet@48975
   709
          fold_rev Logic.all (Ks @ ss @ zs)
blanchet@48975
   710
            (Logic.list_implies (prems, HOLogic.mk_Trueprop concl))) Kss premss concls;
blanchet@48975
   711
      in
blanchet@48975
   712
        map3 (fn goal => fn hset_defs => fn hset_rec_minimal =>
wenzelm@51551
   713
          Goal.prove_sorry lthy [] [] goal
traytel@49109
   714
            (mk_hset_minimal_tac n hset_defs hset_rec_minimal)
traytel@49109
   715
          |> Thm.close_derivation)
blanchet@48975
   716
        goals hset_defss' hset_rec_minimal_thms
blanchet@48975
   717
      end;
blanchet@48975
   718
blanchet@48975
   719
    val timer = time (timer "Hereditary sets");
blanchet@48975
   720
blanchet@48975
   721
    (* bisimulation *)
blanchet@48975
   722
traytel@53566
   723
    val bis_bind = mk_internal_b bisN;
blanchet@48975
   724
    val bis_name = Binding.name_of bis_bind;
blanchet@48975
   725
    val bis_def_bind = (Thm.def_binding bis_bind, []);
blanchet@48975
   726
traytel@51893
   727
    fun mk_bis_le_conjunct R B1 B2 = mk_leq R (mk_Times (B1, B2));
blanchet@48975
   728
    val bis_le = Library.foldr1 HOLogic.mk_conj (map3 mk_bis_le_conjunct Rs Bs B's)
blanchet@48975
   729
blanchet@48975
   730
    val bis_spec =
blanchet@48975
   731
      let
blanchet@48975
   732
        val bisT = Library.foldr (op -->) (ATs @ BTs @ sTs @ B'Ts @ s'Ts @ setRTs, HOLogic.boolT);
blanchet@48975
   733
blanchet@48975
   734
        val fst_args = passive_ids @ fsts;
blanchet@48975
   735
        val snd_args = passive_ids @ snds;
blanchet@48975
   736
        fun mk_bis R s s' b1 b2 RF map1 map2 sets =
blanchet@48975
   737
          list_all_free [b1, b2] (HOLogic.mk_imp
blanchet@48975
   738
            (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
blanchet@48975
   739
            mk_Bex (mk_in (As @ Rs) sets (snd (dest_Free RF))) (Term.absfree (dest_Free RF)
blanchet@48975
   740
              (HOLogic.mk_conj
blanchet@48975
   741
                (HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1),
blanchet@48975
   742
                HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2))))));
blanchet@48975
   743
blanchet@48975
   744
        val lhs = Term.list_comb (Free (bis_name, bisT), As @ Bs @ ss @ B's @ s's @ Rs);
blanchet@48975
   745
        val rhs = HOLogic.mk_conj
blanchet@48975
   746
          (bis_le, Library.foldr1 HOLogic.mk_conj
blanchet@48975
   747
            (map9 mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss))
blanchet@48975
   748
      in
blanchet@49123
   749
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   750
      end;
blanchet@48975
   751
blanchet@48975
   752
    val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) =
blanchet@49311
   753
      lthy
blanchet@49311
   754
      |> Specification.definition (SOME (bis_bind, NONE, NoSyn), (bis_def_bind, bis_spec))
blanchet@49311
   755
      ||> `Local_Theory.restore;
blanchet@49311
   756
blanchet@48975
   757
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   758
    val bis = fst (Term.dest_Const (Morphism.term phi bis_free));
blanchet@48975
   759
    val bis_def = Morphism.thm phi bis_def_free;
blanchet@48975
   760
blanchet@48975
   761
    fun mk_bis As Bs1 ss1 Bs2 ss2 Rs =
blanchet@48975
   762
      let
blanchet@48975
   763
        val args = As @ Bs1 @ ss1 @ Bs2 @ ss2 @ Rs;
blanchet@48975
   764
        val Ts = map fastype_of args;
blanchet@48975
   765
        val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   766
      in
blanchet@48975
   767
        Term.list_comb (Const (bis, bisT), args)
blanchet@48975
   768
      end;
blanchet@48975
   769
blanchet@48975
   770
    val bis_cong_thm =
blanchet@48975
   771
      let
blanchet@48975
   772
        val prems = map HOLogic.mk_Trueprop
blanchet@48975
   773
         (mk_bis As Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs)
blanchet@48975
   774
        val concl = HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs_copy);
blanchet@48975
   775
      in
wenzelm@51551
   776
        Goal.prove_sorry lthy [] []
blanchet@48975
   777
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs @ Rs_copy)
blanchet@48975
   778
            (Logic.list_implies (prems, concl)))
wenzelm@51798
   779
          (K ((hyp_subst_tac lthy THEN' atac) 1))
traytel@49109
   780
        |> Thm.close_derivation
blanchet@48975
   781
      end;
blanchet@48975
   782
traytel@51893
   783
    val bis_rel_thm =
blanchet@48975
   784
      let
traytel@51893
   785
        fun mk_conjunct R s s' b1 b2 rel =
blanchet@48975
   786
          list_all_free [b1, b2] (HOLogic.mk_imp
blanchet@48975
   787
            (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
traytel@51893
   788
            Term.list_comb (rel, map mk_in_rel (passive_Id_ons @ Rs)) $ (s $ b1) $ (s' $ b2)));
blanchet@48975
   789
blanchet@48975
   790
        val rhs = HOLogic.mk_conj
blanchet@48975
   791
          (bis_le, Library.foldr1 HOLogic.mk_conj
blanchet@48975
   792
            (map6 mk_conjunct Rs ss s's zs z's relsAsBs))
blanchet@48975
   793
      in
wenzelm@51551
   794
        Goal.prove_sorry lthy [] []
blanchet@48975
   795
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
blanchet@49123
   796
            (mk_Trueprop_eq (mk_bis As Bs ss B's s's Rs, rhs)))
blanchet@53290
   797
          (K (mk_bis_rel_tac lthy m bis_def rel_OO_Grps map_comps map_cong0s set_mapss))
traytel@49109
   798
        |> Thm.close_derivation
blanchet@48975
   799
      end;
blanchet@48975
   800
blanchet@48975
   801
    val bis_converse_thm =
wenzelm@51551
   802
      Goal.prove_sorry lthy [] []
blanchet@48975
   803
        (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
blanchet@48975
   804
          (Logic.mk_implies
blanchet@48975
   805
            (HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
blanchet@48975
   806
            HOLogic.mk_Trueprop (mk_bis As B's s's Bs ss (map mk_converse Rs)))))
traytel@51893
   807
        (K (mk_bis_converse_tac m bis_rel_thm rel_congs rel_converseps))
traytel@49109
   808
      |> Thm.close_derivation;
blanchet@48975
   809
blanchet@48975
   810
    val bis_O_thm =
blanchet@48975
   811
      let
blanchet@48975
   812
        val prems =
blanchet@48975
   813
          [HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
blanchet@48975
   814
           HOLogic.mk_Trueprop (mk_bis As B's s's B''s s''s R's)];
blanchet@48975
   815
        val concl =
blanchet@48975
   816
          HOLogic.mk_Trueprop (mk_bis As Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's));
blanchet@48975
   817
      in
wenzelm@51551
   818
        Goal.prove_sorry lthy [] []
blanchet@48975
   819
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ B''s @ s''s @ Rs @ R's)
blanchet@48975
   820
            (Logic.list_implies (prems, concl)))
traytel@51893
   821
          (K (mk_bis_O_tac lthy m bis_rel_thm rel_congs rel_OOs))
traytel@49109
   822
        |> Thm.close_derivation
blanchet@48975
   823
      end;
blanchet@48975
   824
blanchet@48975
   825
    val bis_Gr_thm =
blanchet@48975
   826
      let
blanchet@48975
   827
        val concl =
blanchet@48975
   828
          HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map2 mk_Gr Bs fs));
blanchet@48975
   829
      in
wenzelm@51551
   830
        Goal.prove_sorry lthy [] []
blanchet@48975
   831
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ fs)
blanchet@48975
   832
            (Logic.list_implies ([coalg_prem, mor_prem], concl)))
traytel@51893
   833
          (mk_bis_Gr_tac bis_rel_thm rel_Grps mor_image_thms morE_thms coalg_in_thms)
traytel@49109
   834
        |> Thm.close_derivation
blanchet@48975
   835
      end;
blanchet@48975
   836
blanchet@48975
   837
    val bis_image2_thm = bis_cong_thm OF
blanchet@48975
   838
      ((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) ::
blanchet@48975
   839
      replicate n @{thm image2_Gr});
blanchet@48975
   840
traytel@51447
   841
    val bis_Id_on_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) ::
traytel@51447
   842
      replicate n @{thm Id_on_Gr});
blanchet@48975
   843
blanchet@48975
   844
    val bis_Union_thm =
blanchet@48975
   845
      let
blanchet@48975
   846
        val prem =
blanchet@48975
   847
          HOLogic.mk_Trueprop (mk_Ball Idx
blanchet@48975
   848
            (Term.absfree idx' (mk_bis As Bs ss B's s's (map (fn R => R $ idx) Ris))));
blanchet@48975
   849
        val concl =
blanchet@48975
   850
          HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map (mk_UNION Idx) Ris));
blanchet@48975
   851
      in
wenzelm@51551
   852
        Goal.prove_sorry lthy [] []
blanchet@48975
   853
          (fold_rev Logic.all (Idx :: As @ Bs @ ss @ B's @ s's @ Ris)
blanchet@48975
   854
            (Logic.mk_implies (prem, concl)))
blanchet@48975
   855
          (mk_bis_Union_tac bis_def in_mono'_thms)
traytel@49109
   856
        |> Thm.close_derivation
blanchet@48975
   857
      end;
blanchet@48975
   858
blanchet@48975
   859
    (* self-bisimulation *)
blanchet@48975
   860
blanchet@48975
   861
    fun mk_sbis As Bs ss Rs = mk_bis As Bs ss Bs ss Rs;
blanchet@48975
   862
blanchet@48975
   863
    val sbis_prem = HOLogic.mk_Trueprop (mk_sbis As Bs ss sRs);
blanchet@48975
   864
blanchet@48975
   865
    (* largest self-bisimulation *)
blanchet@48975
   866
traytel@53566
   867
    val lsbis_binds = mk_internal_bs lsbisN;
traytel@53566
   868
    fun lsbis_bind i = nth lsbis_binds (i - 1);
blanchet@48975
   869
    val lsbis_name = Binding.name_of o lsbis_bind;
blanchet@48975
   870
    val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind;
blanchet@48975
   871
blanchet@48975
   872
    val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs
blanchet@48975
   873
      (HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis As Bs ss sRs)));
blanchet@48975
   874
blanchet@48975
   875
    fun lsbis_spec i RT =
blanchet@48975
   876
      let
blanchet@48975
   877
        fun mk_lsbisT RT =
blanchet@48975
   878
          Library.foldr (op -->) (map fastype_of (As @ Bs @ ss), RT);
blanchet@48975
   879
        val lhs = Term.list_comb (Free (lsbis_name i, mk_lsbisT RT), As @ Bs @ ss);
blanchet@48975
   880
        val rhs = mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i));
blanchet@48975
   881
      in
blanchet@49123
   882
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   883
      end;
blanchet@48975
   884
blanchet@48975
   885
    val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   886
      lthy
blanchet@48975
   887
      |> fold_map2 (fn i => fn RT => Specification.definition
blanchet@48975
   888
        (SOME (lsbis_bind i, NONE, NoSyn), (lsbis_def_bind i, lsbis_spec i RT))) ks setsRTs
blanchet@48975
   889
      |>> apsnd split_list o split_list
blanchet@48975
   890
      ||> `Local_Theory.restore;
blanchet@48975
   891
blanchet@48975
   892
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   893
blanchet@48975
   894
    val lsbis_defs = map (Morphism.thm phi) lsbis_def_frees;
blanchet@48975
   895
    val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees;
blanchet@48975
   896
blanchet@48975
   897
    fun mk_lsbis As Bs ss i =
blanchet@48975
   898
      let
blanchet@48975
   899
        val args = As @ Bs @ ss;
blanchet@48975
   900
        val Ts = map fastype_of args;
blanchet@48975
   901
        val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1)))));
blanchet@48975
   902
        val lsbisT = Library.foldr (op -->) (Ts, RT);
blanchet@48975
   903
      in
blanchet@48975
   904
        Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args)
blanchet@48975
   905
      end;
blanchet@48975
   906
blanchet@48975
   907
    val sbis_lsbis_thm =
wenzelm@51551
   908
      Goal.prove_sorry lthy [] []
blanchet@48975
   909
        (fold_rev Logic.all (As @ Bs @ ss)
blanchet@48975
   910
          (HOLogic.mk_Trueprop (mk_sbis As Bs ss (map (mk_lsbis As Bs ss) ks))))
wenzelm@51798
   911
        (K (mk_sbis_lsbis_tac lthy lsbis_defs bis_Union_thm bis_cong_thm))
traytel@49109
   912
      |> Thm.close_derivation;
blanchet@48975
   913
blanchet@48975
   914
    val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS
traytel@52904
   915
      (bis_def RS iffD1 RS conjunct1 RS mk_conjunctN n i)) ks;
blanchet@48975
   916
    val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS
traytel@52904
   917
      (bis_def RS iffD1 RS conjunct2 RS mk_conjunctN n i))) RS mp) ks;
blanchet@48975
   918
blanchet@48975
   919
    val incl_lsbis_thms =
blanchet@48975
   920
      let
traytel@51893
   921
        fun mk_concl i R = HOLogic.mk_Trueprop (mk_leq R (mk_lsbis As Bs ss i));
blanchet@48975
   922
        val goals = map2 (fn i => fn R => fold_rev Logic.all (As @ Bs @ ss @ sRs)
blanchet@48975
   923
          (Logic.mk_implies (sbis_prem, mk_concl i R))) ks sRs;
blanchet@48975
   924
      in
wenzelm@51551
   925
        map3 (fn goal => fn i => fn def => Goal.prove_sorry lthy [] [] goal
traytel@49109
   926
          (K (mk_incl_lsbis_tac n i def)) |> Thm.close_derivation) goals ks lsbis_defs
blanchet@48975
   927
      end;
blanchet@48975
   928
blanchet@48975
   929
    val equiv_lsbis_thms =
blanchet@48975
   930
      let
blanchet@48975
   931
        fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis As Bs ss i));
blanchet@48975
   932
        val goals = map2 (fn i => fn B => fold_rev Logic.all (As @ Bs @ ss)
blanchet@48975
   933
          (Logic.mk_implies (coalg_prem, mk_concl i B))) ks Bs;
blanchet@48975
   934
      in
blanchet@48975
   935
        map3 (fn goal => fn l_incl => fn incl_l =>
wenzelm@51551
   936
          Goal.prove_sorry lthy [] [] goal
blanchet@48975
   937
            (K (mk_equiv_lsbis_tac sbis_lsbis_thm l_incl incl_l
traytel@51447
   938
              bis_Id_on_thm bis_converse_thm bis_O_thm))
traytel@49109
   939
          |> Thm.close_derivation)
blanchet@48975
   940
        goals lsbis_incl_thms incl_lsbis_thms
blanchet@48975
   941
      end;
blanchet@48975
   942
blanchet@48975
   943
    val timer = time (timer "Bisimulations");
blanchet@48975
   944
blanchet@48975
   945
    (* bounds *)
blanchet@48975
   946
blanchet@48975
   947
    val (lthy, sbd, sbdT,
traytel@52635
   948
      sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss) =
blanchet@48975
   949
      if n = 1
traytel@52635
   950
      then (lthy, sum_bd, sum_bdT, bd_card_order, bd_Cinfinite, bd_Card_order, set_bdss)
blanchet@48975
   951
      else
blanchet@48975
   952
        let
traytel@53566
   953
          val sbdT_bind = mk_internal_b sum_bdTN;
blanchet@48975
   954
blanchet@48975
   955
          val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) =
wenzelm@49835
   956
            typedef (sbdT_bind, dead_params, NoSyn)
blanchet@48975
   957
              (HOLogic.mk_UNIV sum_bdT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
blanchet@48975
   958
traytel@49620
   959
          val sbdT = Type (sbdT_name, dead_params');
blanchet@48975
   960
          val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT);
blanchet@48975
   961
traytel@53566
   962
          val sbd_bind = mk_internal_b sum_bdN;
blanchet@48975
   963
          val sbd_name = Binding.name_of sbd_bind;
blanchet@48975
   964
          val sbd_def_bind = (Thm.def_binding sbd_bind, []);
blanchet@48975
   965
blanchet@48975
   966
          val sbd_spec = HOLogic.mk_Trueprop
blanchet@48975
   967
            (HOLogic.mk_eq (Free (sbd_name, mk_relT (`I sbdT)), mk_dir_image sum_bd Abs_sbdT));
blanchet@48975
   968
blanchet@48975
   969
          val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) =
blanchet@48975
   970
            lthy
blanchet@48975
   971
            |> Specification.definition (SOME (sbd_bind, NONE, NoSyn), (sbd_def_bind, sbd_spec))
blanchet@48975
   972
            ||> `Local_Theory.restore;
blanchet@48975
   973
blanchet@48975
   974
          val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   975
blanchet@48975
   976
          val sbd_def = Morphism.thm phi sbd_def_free;
blanchet@48975
   977
          val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT));
blanchet@48975
   978
traytel@49228
   979
          val Abs_sbdT_inj = mk_Abs_inj_thm (#Abs_inject sbdT_loc_info);
traytel@49228
   980
          val Abs_sbdT_bij = mk_Abs_bij_thm lthy Abs_sbdT_inj (#Abs_cases sbdT_loc_info);
blanchet@48975
   981
blanchet@48975
   982
          fun mk_sum_Cinfinite [thm] = thm
blanchet@48975
   983
            | mk_sum_Cinfinite (thm :: thms) =
blanchet@48975
   984
              @{thm Cinfinite_csum_strong} OF [thm, mk_sum_Cinfinite thms];
blanchet@48975
   985
blanchet@48975
   986
          val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;
blanchet@48975
   987
          val sum_Card_order = sum_Cinfinite RS conjunct2;
blanchet@48975
   988
blanchet@48975
   989
          fun mk_sum_card_order [thm] = thm
blanchet@48975
   990
            | mk_sum_card_order (thm :: thms) =
blanchet@48975
   991
              @{thm card_order_csum} OF [thm, mk_sum_card_order thms];
blanchet@48975
   992
blanchet@48975
   993
          val sum_card_order = mk_sum_card_order bd_card_orders;
blanchet@48975
   994
blanchet@49504
   995
          val sbd_ordIso = fold_thms lthy [sbd_def]
blanchet@48975
   996
            (@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order]);
blanchet@49504
   997
          val sbd_card_order =  fold_thms lthy [sbd_def]
blanchet@48975
   998
            (@{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]);
blanchet@48975
   999
          val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite];
blanchet@48975
  1000
          val sbd_Card_order = sbd_Cinfinite RS conjunct2;
blanchet@48975
  1001
blanchet@48975
  1002
          fun mk_set_sbd i bd_Card_order bds =
blanchet@48975
  1003
            map (fn thm => @{thm ordLeq_ordIso_trans} OF
blanchet@48975
  1004
              [bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds;
blanchet@48975
  1005
          val set_sbdss = map3 mk_set_sbd ks bd_Card_orders set_bdss;
blanchet@48975
  1006
       in
traytel@52635
  1007
         (lthy, sbd, sbdT, sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss)
blanchet@48975
  1008
       end;
blanchet@48975
  1009
blanchet@48975
  1010
    val sbdTs = replicate n sbdT;
blanchet@48975
  1011
    val sum_sbd = Library.foldr1 (uncurry mk_csum) (replicate n sbd);
blanchet@48975
  1012
    val sum_sbdT = mk_sumTN sbdTs;
blanchet@48975
  1013
    val sum_sbd_listT = HOLogic.listT sum_sbdT;
blanchet@48975
  1014
    val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT;
blanchet@48975
  1015
    val bdTs = passiveAs @ replicate n sbdT;
blanchet@48975
  1016
    val to_sbd_maps = map4 mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs;
blanchet@48975
  1017
    val bdFTs = mk_FTs bdTs;
blanchet@48975
  1018
    val sbdFT = mk_sumTN bdFTs;
blanchet@48975
  1019
    val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT);
blanchet@48975
  1020
    val treeQT = HOLogic.mk_setT treeT;
blanchet@48975
  1021
    val treeTs = passiveAs @ replicate n treeT;
blanchet@48975
  1022
    val treeQTs = passiveAs @ replicate n treeQT;
blanchet@48975
  1023
    val treeFTs = mk_FTs treeTs;
blanchet@48975
  1024
    val tree_maps = map4 mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs;
blanchet@48975
  1025
    val final_maps = map4 mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs;
blanchet@48975
  1026
    val isNode_setss = mk_setss (passiveAs @ replicate n sbdT);
blanchet@48975
  1027
blanchet@48975
  1028
    val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []];
blanchet@48975
  1029
    val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs);
blanchet@48975
  1030
    val Lev_recT = fastype_of Zero;
blanchet@48975
  1031
    val LevT = Library.foldr (op -->) (sTs, HOLogic.natT --> Lev_recT);
blanchet@48975
  1032
blanchet@48975
  1033
    val Nil = HOLogic.mk_tuple (map3 (fn i => fn z => fn z'=>
blanchet@48975
  1034
      Term.absfree z' (mk_InN activeAs z i)) ks zs zs');
blanchet@48975
  1035
    val rv_recT = fastype_of Nil;
blanchet@48975
  1036
    val rvT = Library.foldr (op -->) (sTs, sum_sbd_listT --> rv_recT);
blanchet@48975
  1037
blanchet@48975
  1038
    val (((((((((((sumx, sumx'), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')),
blanchet@48975
  1039
      (lab, lab')), (Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')),
blanchet@48975
  1040
      names_lthy) = names_lthy
blanchet@48975
  1041
      |> yield_singleton (apfst (op ~~) oo mk_Frees' "sumx") sum_sbdT
blanchet@48975
  1042
      ||>> mk_Frees' "k" sbdTs
blanchet@48975
  1043
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
blanchet@48975
  1044
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
blanchet@48975
  1045
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl") sum_sbd_list_setT
blanchet@48975
  1046
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT)
blanchet@48975
  1047
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl_lab") treeT
blanchet@48975
  1048
      ||>> mk_Frees "x" bdFTs
blanchet@48975
  1049
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") Lev_recT
blanchet@48975
  1050
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") rv_recT;
blanchet@48975
  1051
blanchet@48975
  1052
    val (k, k') = (hd kks, hd kks')
blanchet@48975
  1053
blanchet@48975
  1054
    val timer = time (timer "Bounds");
blanchet@48975
  1055
blanchet@48975
  1056
    (* tree coalgebra *)
blanchet@48975
  1057
traytel@53566
  1058
    val isNode_binds = mk_internal_bs isNodeN;
traytel@53566
  1059
    fun isNode_bind i = nth isNode_binds (i - 1);
blanchet@48975
  1060
    val isNode_name = Binding.name_of o isNode_bind;
blanchet@48975
  1061
    val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind;
blanchet@48975
  1062
blanchet@48975
  1063
    val isNodeT =
blanchet@48975
  1064
      Library.foldr (op -->) (map fastype_of (As @ [Kl, lab, kl]), HOLogic.boolT);
blanchet@48975
  1065
blanchet@48975
  1066
    val Succs = map3 (fn i => fn k => fn k' =>
blanchet@48975
  1067
      HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl)))
blanchet@48975
  1068
      ks kks kks';
blanchet@48975
  1069
blanchet@48975
  1070
    fun isNode_spec sets x i =
blanchet@48975
  1071
      let
blanchet@48975
  1072
        val (passive_sets, active_sets) = chop m (map (fn set => set $ x) sets);
blanchet@48975
  1073
        val lhs = Term.list_comb (Free (isNode_name i, isNodeT), As @ [Kl, lab, kl]);
blanchet@48975
  1074
        val rhs = list_exists_free [x]
blanchet@48975
  1075
          (Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) ::
traytel@51893
  1076
          map2 mk_leq passive_sets As @ map2 (curry HOLogic.mk_eq) active_sets Succs));
blanchet@48975
  1077
      in
blanchet@49123
  1078
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1079
      end;
blanchet@48975
  1080
blanchet@48975
  1081
    val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1082
      lthy
blanchet@48975
  1083
      |> fold_map3 (fn i => fn x => fn sets => Specification.definition
blanchet@48975
  1084
        (SOME (isNode_bind i, NONE, NoSyn), (isNode_def_bind i, isNode_spec sets x i)))
blanchet@48975
  1085
        ks xs isNode_setss
blanchet@48975
  1086
      |>> apsnd split_list o split_list
blanchet@48975
  1087
      ||> `Local_Theory.restore;
blanchet@48975
  1088
blanchet@48975
  1089
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1090
blanchet@48975
  1091
    val isNode_defs = map (Morphism.thm phi) isNode_def_frees;
blanchet@48975
  1092
    val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees;
blanchet@48975
  1093
blanchet@48975
  1094
    fun mk_isNode As kl i =
blanchet@48975
  1095
      Term.list_comb (Const (nth isNodes (i - 1), isNodeT), As @ [Kl, lab, kl]);
blanchet@48975
  1096
blanchet@48975
  1097
    val isTree =
blanchet@48975
  1098
      let
blanchet@48975
  1099
        val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl);
traytel@51893
  1100
        val Field = mk_leq Kl (mk_Field (mk_clists sum_sbd));
blanchet@48975
  1101
        val prefCl = mk_prefCl Kl;
blanchet@48975
  1102
blanchet@48975
  1103
        val tree = mk_Ball Kl (Term.absfree kl'
blanchet@48975
  1104
          (HOLogic.mk_conj
blanchet@48975
  1105
            (Library.foldr1 HOLogic.mk_disj (map (mk_isNode As kl) ks),
blanchet@48975
  1106
            Library.foldr1 HOLogic.mk_conj (map4 (fn Succ => fn i => fn k => fn k' =>
blanchet@48975
  1107
              mk_Ball Succ (Term.absfree k' (mk_isNode As
blanchet@48975
  1108
                (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i)))
blanchet@48975
  1109
            Succs ks kks kks'))));
blanchet@48975
  1110
blanchet@48975
  1111
        val undef = list_all_free [kl] (HOLogic.mk_imp
blanchet@48975
  1112
          (HOLogic.mk_not (HOLogic.mk_mem (kl, Kl)),
blanchet@48975
  1113
          HOLogic.mk_eq (lab $ kl, mk_undefined sbdFT)));
blanchet@48975
  1114
      in
blanchet@48975
  1115
        Library.foldr1 HOLogic.mk_conj [empty, Field, prefCl, tree, undef]
blanchet@48975
  1116
      end;
blanchet@48975
  1117
traytel@53566
  1118
    val carT_binds = mk_internal_bs carTN;
traytel@53566
  1119
    fun carT_bind i = nth carT_binds (i - 1);
blanchet@48975
  1120
    val carT_name = Binding.name_of o carT_bind;
blanchet@48975
  1121
    val carT_def_bind = rpair [] o Thm.def_binding o carT_bind;
blanchet@48975
  1122
blanchet@48975
  1123
    fun carT_spec i =
blanchet@48975
  1124
      let
blanchet@48975
  1125
        val carTT = Library.foldr (op -->) (ATs, HOLogic.mk_setT treeT);
blanchet@48975
  1126
blanchet@48975
  1127
        val lhs = Term.list_comb (Free (carT_name i, carTT), As);
blanchet@48975
  1128
        val rhs = HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
blanchet@48975
  1129
          (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)),
blanchet@48975
  1130
            HOLogic.mk_conj (isTree, mk_isNode As (HOLogic.mk_list sum_sbdT []) i))));
blanchet@48975
  1131
      in
blanchet@49123
  1132
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1133
      end;
blanchet@48975
  1134
blanchet@48975
  1135
    val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1136
      lthy
blanchet@48975
  1137
      |> fold_map (fn i => Specification.definition
blanchet@48975
  1138
        (SOME (carT_bind i, NONE, NoSyn), (carT_def_bind i, carT_spec i))) ks
blanchet@48975
  1139
      |>> apsnd split_list o split_list
blanchet@48975
  1140
      ||> `Local_Theory.restore;
blanchet@48975
  1141
blanchet@48975
  1142
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1143
blanchet@48975
  1144
    val carT_defs = map (Morphism.thm phi) carT_def_frees;
blanchet@48975
  1145
    val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees;
blanchet@48975
  1146
blanchet@48975
  1147
    fun mk_carT As i = Term.list_comb
blanchet@48975
  1148
      (Const (nth carTs (i - 1),
blanchet@48975
  1149
         Library.foldr (op -->) (map fastype_of As, HOLogic.mk_setT treeT)), As);
blanchet@48975
  1150
traytel@53566
  1151
    val strT_binds = mk_internal_bs strTN;
traytel@53566
  1152
    fun strT_bind i = nth strT_binds (i - 1);
blanchet@48975
  1153
    val strT_name = Binding.name_of o strT_bind;
blanchet@48975
  1154
    val strT_def_bind = rpair [] o Thm.def_binding o strT_bind;
blanchet@48975
  1155
blanchet@48975
  1156
    fun strT_spec mapFT FT i =
blanchet@48975
  1157
      let
blanchet@48975
  1158
        val strTT = treeT --> FT;
blanchet@48975
  1159
blanchet@48975
  1160
        fun mk_f i k k' =
blanchet@48975
  1161
          let val in_k = mk_InN sbdTs k i;
blanchet@48975
  1162
          in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end;
blanchet@48975
  1163
blanchet@48975
  1164
        val f = Term.list_comb (mapFT, passive_ids @ map3 mk_f ks kks kks');
blanchet@48975
  1165
        val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs));
blanchet@48975
  1166
        val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2);
blanchet@48975
  1167
        val lhs = Free (strT_name i, strTT);
blanchet@48975
  1168
        val rhs = HOLogic.mk_split (Term.absfree Kl' (Term.absfree lab'
blanchet@48975
  1169
          (mk_sum_caseN fs $ (lab $ HOLogic.mk_list sum_sbdT []))));
blanchet@48975
  1170
      in
blanchet@49123
  1171
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1172
      end;
blanchet@48975
  1173
blanchet@48975
  1174
    val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1175
      lthy
blanchet@48975
  1176
      |> fold_map3 (fn i => fn mapFT => fn FT => Specification.definition
blanchet@48975
  1177
        (SOME (strT_bind i, NONE, NoSyn), (strT_def_bind i, strT_spec mapFT FT i)))
blanchet@48975
  1178
        ks tree_maps treeFTs
blanchet@48975
  1179
      |>> apsnd split_list o split_list
blanchet@48975
  1180
      ||> `Local_Theory.restore;
blanchet@48975
  1181
blanchet@48975
  1182
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1183
blanchet@48975
  1184
    val strT_defs = map ((fn def => trans OF [def RS fun_cong, @{thm prod.cases}]) o
blanchet@48975
  1185
      Morphism.thm phi) strT_def_frees;
blanchet@48975
  1186
    val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees;
blanchet@48975
  1187
blanchet@48975
  1188
    fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT);
blanchet@48975
  1189
blanchet@48975
  1190
    val carTAs = map (mk_carT As) ks;
blanchet@48975
  1191
    val strTAs = map2 mk_strT treeFTs ks;
blanchet@48975
  1192
blanchet@48975
  1193
    val coalgT_thm =
wenzelm@51551
  1194
      Goal.prove_sorry lthy [] []
blanchet@48975
  1195
        (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_coalg As carTAs strTAs)))
blanchet@53290
  1196
        (mk_coalgT_tac m (coalg_def :: isNode_defs @ carT_defs) strT_defs set_mapss)
traytel@49109
  1197
      |> Thm.close_derivation;
blanchet@48975
  1198
blanchet@48975
  1199
    val timer = time (timer "Tree coalgebra");
blanchet@48975
  1200
blanchet@48975
  1201
    fun mk_to_sbd s x i i' =
blanchet@48975
  1202
      mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
blanchet@48975
  1203
    fun mk_from_sbd s x i i' =
blanchet@48975
  1204
      mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
blanchet@48975
  1205
blanchet@48975
  1206
    fun mk_to_sbd_thmss thm = map (map (fn set_sbd =>
blanchet@48975
  1207
      thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss;
blanchet@48975
  1208
blanchet@48975
  1209
    val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj};
blanchet@48975
  1210
    val to_sbd_thmss = mk_to_sbd_thmss @{thm toCard};
blanchet@48975
  1211
    val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard};
blanchet@48975
  1212
traytel@53566
  1213
    val Lev_bind = mk_internal_b LevN;
blanchet@48975
  1214
    val Lev_name = Binding.name_of Lev_bind;
blanchet@48975
  1215
    val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind);
blanchet@48975
  1216
blanchet@48975
  1217
    val Lev_spec =
blanchet@48975
  1218
      let
blanchet@48975
  1219
        fun mk_Suc i s setsAs a a' =
blanchet@48975
  1220
          let
blanchet@48975
  1221
            val sets = drop m setsAs;
blanchet@48975
  1222
            fun mk_set i' set b =
blanchet@48975
  1223
              let
blanchet@48975
  1224
                val Cons = HOLogic.mk_eq (kl_copy,
blanchet@48975
  1225
                  mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl)
blanchet@48975
  1226
                val b_set = HOLogic.mk_mem (b, set $ (s $ a));
blanchet@48975
  1227
                val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b);
blanchet@48975
  1228
              in
blanchet@48975
  1229
                HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl]
blanchet@48975
  1230
                  (HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec))))
blanchet@48975
  1231
              end;
blanchet@48975
  1232
          in
blanchet@48975
  1233
            Term.absfree a' (Library.foldl1 mk_union (map3 mk_set ks sets zs_copy))
blanchet@48975
  1234
          end;
blanchet@48975
  1235
blanchet@48975
  1236
        val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec'
blanchet@48975
  1237
          (HOLogic.mk_tuple (map5 mk_Suc ks ss setssAs zs zs')));
blanchet@48975
  1238
blanchet@48975
  1239
        val lhs = Term.list_comb (Free (Lev_name, LevT), ss);
blanchet@48975
  1240
        val rhs = mk_nat_rec Zero Suc;
blanchet@48975
  1241
      in
blanchet@49123
  1242
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1243
      end;
blanchet@48975
  1244
blanchet@48975
  1245
    val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) =
blanchet@48975
  1246
      lthy
blanchet@48975
  1247
      |> Specification.definition (SOME (Lev_bind, NONE, NoSyn), (Lev_def_bind, Lev_spec))
blanchet@48975
  1248
      ||> `Local_Theory.restore;
blanchet@48975
  1249
blanchet@48975
  1250
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1251
blanchet@48975
  1252
    val Lev_def = Morphism.thm phi Lev_def_free;
blanchet@48975
  1253
    val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free));
blanchet@48975
  1254
blanchet@48975
  1255
    fun mk_Lev ss nat i =
blanchet@48975
  1256
      let
blanchet@48975
  1257
        val Ts = map fastype_of ss;
blanchet@48975
  1258
        val LevT = Library.foldr (op -->) (Ts, HOLogic.natT -->
blanchet@48975
  1259
          HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts));
blanchet@48975
  1260
      in
blanchet@48975
  1261
        mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i
blanchet@48975
  1262
      end;
blanchet@48975
  1263
blanchet@48975
  1264
    val Lev_0s = flat (mk_rec_simps n @{thm nat_rec_0} [Lev_def]);
blanchet@48975
  1265
    val Lev_Sucs = flat (mk_rec_simps n @{thm nat_rec_Suc} [Lev_def]);
blanchet@48975
  1266
traytel@53566
  1267
    val rv_bind = mk_internal_b rvN;
blanchet@48975
  1268
    val rv_name = Binding.name_of rv_bind;
blanchet@48975
  1269
    val rv_def_bind = rpair [] (Thm.def_binding rv_bind);
blanchet@48975
  1270
blanchet@48975
  1271
    val rv_spec =
blanchet@48975
  1272
      let
blanchet@48975
  1273
        fun mk_Cons i s b b' =
blanchet@48975
  1274
          let
blanchet@48975
  1275
            fun mk_case i' =
blanchet@48975
  1276
              Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k));
blanchet@48975
  1277
          in
blanchet@48975
  1278
            Term.absfree b' (mk_sum_caseN (map mk_case ks) $ sumx)
blanchet@48975
  1279
          end;
blanchet@48975
  1280
blanchet@48975
  1281
        val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec'
blanchet@48975
  1282
          (HOLogic.mk_tuple (map4 mk_Cons ks ss zs zs'))));
blanchet@48975
  1283
blanchet@48975
  1284
        val lhs = Term.list_comb (Free (rv_name, rvT), ss);
blanchet@48975
  1285
        val rhs = mk_list_rec Nil Cons;
blanchet@48975
  1286
      in
blanchet@49123
  1287
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1288
      end;
blanchet@48975
  1289
blanchet@48975
  1290
    val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) =
blanchet@48975
  1291
      lthy
blanchet@48975
  1292
      |> Specification.definition (SOME (rv_bind, NONE, NoSyn), (rv_def_bind, rv_spec))
blanchet@48975
  1293
      ||> `Local_Theory.restore;
blanchet@48975
  1294
blanchet@48975
  1295
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1296
blanchet@48975
  1297
    val rv_def = Morphism.thm phi rv_def_free;
blanchet@48975
  1298
    val rv = fst (Term.dest_Const (Morphism.term phi rv_free));
blanchet@48975
  1299
blanchet@48975
  1300
    fun mk_rv ss kl i =
blanchet@48975
  1301
      let
blanchet@48975
  1302
        val Ts = map fastype_of ss;
blanchet@48975
  1303
        val As = map domain_type Ts;
blanchet@48975
  1304
        val rvT = Library.foldr (op -->) (Ts, fastype_of kl -->
blanchet@48975
  1305
          HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As));
blanchet@48975
  1306
      in
blanchet@48975
  1307
        mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i
blanchet@48975
  1308
      end;
blanchet@48975
  1309
blanchet@48975
  1310
    val rv_Nils = flat (mk_rec_simps n @{thm list_rec_Nil} [rv_def]);
blanchet@48975
  1311
    val rv_Conss = flat (mk_rec_simps n @{thm list_rec_Cons} [rv_def]);
blanchet@48975
  1312
traytel@53566
  1313
    val beh_binds = mk_internal_bs behN;
traytel@53566
  1314
    fun beh_bind i = nth beh_binds (i - 1);
blanchet@48975
  1315
    val beh_name = Binding.name_of o beh_bind;
blanchet@48975
  1316
    val beh_def_bind = rpair [] o Thm.def_binding o beh_bind;
blanchet@48975
  1317
blanchet@48975
  1318
    fun beh_spec i z =
blanchet@48975
  1319
      let
blanchet@48975
  1320
        val mk_behT = Library.foldr (op -->) (map fastype_of (ss @ [z]), treeT);
blanchet@48975
  1321
blanchet@48975
  1322
        fun mk_case i to_sbd_map s k k' =
blanchet@48975
  1323
          Term.absfree k' (mk_InN bdFTs
blanchet@48975
  1324
            (Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i);
blanchet@48975
  1325
blanchet@48975
  1326
        val Lab = Term.absfree kl' (mk_If
blanchet@48975
  1327
          (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))
blanchet@48975
  1328
          (mk_sum_caseN (map5 mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z))
blanchet@48975
  1329
          (mk_undefined sbdFT));
blanchet@48975
  1330
blanchet@48975
  1331
        val lhs = Term.list_comb (Free (beh_name i, mk_behT), ss) $ z;
blanchet@48975
  1332
        val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
blanchet@48975
  1333
          (Term.absfree nat' (mk_Lev ss nat i $ z)), Lab);
blanchet@48975
  1334
      in
blanchet@49123
  1335
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1336
      end;
blanchet@48975
  1337
blanchet@48975
  1338
    val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1339
      lthy
blanchet@48975
  1340
      |> fold_map2 (fn i => fn z => Specification.definition
blanchet@48975
  1341
        (SOME (beh_bind i, NONE, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs
blanchet@48975
  1342
      |>> apsnd split_list o split_list
blanchet@48975
  1343
      ||> `Local_Theory.restore;
blanchet@48975
  1344
blanchet@48975
  1345
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1346
blanchet@48975
  1347
    val beh_defs = map (Morphism.thm phi) beh_def_frees;
blanchet@48975
  1348
    val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees;
blanchet@48975
  1349
blanchet@48975
  1350
    fun mk_beh ss i =
blanchet@48975
  1351
      let
blanchet@48975
  1352
        val Ts = map fastype_of ss;
blanchet@48975
  1353
        val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT);
blanchet@48975
  1354
      in
blanchet@48975
  1355
        Term.list_comb (Const (nth behs (i - 1), behT), ss)
blanchet@48975
  1356
      end;
blanchet@48975
  1357
blanchet@48975
  1358
    val Lev_sbd_thms =
blanchet@48975
  1359
      let
traytel@51893
  1360
        fun mk_conjunct i z = mk_leq (mk_Lev ss nat i $ z) (mk_Field (mk_clists sum_sbd));
blanchet@48975
  1361
        val goal = list_all_free zs
blanchet@48975
  1362
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1363
blanchet@48975
  1364
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1365
blanchet@48975
  1366
        val Lev_sbd = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1367
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1368
            (K (mk_Lev_sbd_tac lthy cts Lev_0s Lev_Sucs to_sbd_thmss))
traytel@49109
  1369
          |> Thm.close_derivation);
blanchet@48975
  1370
blanchet@48975
  1371
        val Lev_sbd' = mk_specN n Lev_sbd;
blanchet@48975
  1372
      in
blanchet@48975
  1373
        map (fn i => Lev_sbd' RS mk_conjunctN n i) ks
blanchet@48975
  1374
      end;
blanchet@48975
  1375
blanchet@48975
  1376
    val (length_Lev_thms, length_Lev'_thms) =
blanchet@48975
  1377
      let
blanchet@48975
  1378
        fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1379
          HOLogic.mk_eq (mk_size kl, nat));
blanchet@48975
  1380
        val goal = list_all_free (kl :: zs)
blanchet@48975
  1381
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1382
blanchet@48975
  1383
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1384
blanchet@48975
  1385
        val length_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1386
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1387
            (K (mk_length_Lev_tac lthy cts Lev_0s Lev_Sucs))
traytel@49109
  1388
          |> Thm.close_derivation);
blanchet@48975
  1389
blanchet@48975
  1390
        val length_Lev' = mk_specN (n + 1) length_Lev;
blanchet@48975
  1391
        val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks;
blanchet@48975
  1392
blanchet@48975
  1393
        fun mk_goal i z = fold_rev Logic.all (z :: kl :: nat :: ss) (Logic.mk_implies
blanchet@48975
  1394
            (HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z)),
blanchet@48975
  1395
            HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))));
blanchet@48975
  1396
        val goals = map2 mk_goal ks zs;
blanchet@48975
  1397
blanchet@48975
  1398
        val length_Levs' = map2 (fn goal => fn length_Lev =>
wenzelm@51551
  1399
          Goal.prove_sorry lthy [] [] goal (K (mk_length_Lev'_tac length_Lev))
traytel@49109
  1400
          |> Thm.close_derivation) goals length_Levs;
blanchet@48975
  1401
      in
blanchet@48975
  1402
        (length_Levs, length_Levs')
blanchet@48975
  1403
      end;
blanchet@48975
  1404
blanchet@48975
  1405
    val prefCl_Lev_thms =
blanchet@48975
  1406
      let
blanchet@48975
  1407
        fun mk_conjunct i z = HOLogic.mk_imp
traytel@50058
  1408
          (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), mk_prefixeq kl_copy kl),
blanchet@48975
  1409
          HOLogic.mk_mem (kl_copy, mk_Lev ss (mk_size kl_copy) i $ z));
blanchet@48975
  1410
        val goal = list_all_free (kl :: kl_copy :: zs)
blanchet@48975
  1411
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1412
blanchet@48975
  1413
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1414
blanchet@48975
  1415
        val prefCl_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1416
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1417
            (K (mk_prefCl_Lev_tac lthy cts Lev_0s Lev_Sucs)))
traytel@49109
  1418
          |> Thm.close_derivation;
blanchet@48975
  1419
blanchet@48975
  1420
        val prefCl_Lev' = mk_specN (n + 2) prefCl_Lev;
blanchet@48975
  1421
      in
blanchet@48975
  1422
        map (fn i => prefCl_Lev' RS mk_conjunctN n i RS mp) ks
blanchet@48975
  1423
      end;
blanchet@48975
  1424
blanchet@48975
  1425
    val rv_last_thmss =
blanchet@48975
  1426
      let
blanchet@48975
  1427
        fun mk_conjunct i z i' z_copy = list_exists_free [z_copy]
blanchet@48975
  1428
          (HOLogic.mk_eq
blanchet@48975
  1429
            (mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z,
blanchet@48975
  1430
            mk_InN activeAs z_copy i'));
blanchet@48975
  1431
        val goal = list_all_free (k :: zs)
blanchet@48975
  1432
          (Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z =>
blanchet@48975
  1433
            Library.foldr1 HOLogic.mk_conj
blanchet@48975
  1434
              (map2 (mk_conjunct i z) ks zs_copy)) ks zs));
blanchet@48975
  1435
blanchet@48975
  1436
        val cTs = [SOME (certifyT lthy sum_sbdT)];
blanchet@48975
  1437
        val cts = map (SOME o certify lthy) [Term.absfree kl' goal, kl];
blanchet@48975
  1438
blanchet@48975
  1439
        val rv_last = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1440
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@49109
  1441
            (K (mk_rv_last_tac cTs cts rv_Nils rv_Conss)))
traytel@49109
  1442
          |> Thm.close_derivation;
blanchet@48975
  1443
blanchet@48975
  1444
        val rv_last' = mk_specN (n + 1) rv_last;
blanchet@48975
  1445
      in
blanchet@48975
  1446
        map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks
blanchet@48975
  1447
      end;
blanchet@48975
  1448
blanchet@48975
  1449
    val set_rv_Lev_thmsss = if m = 0 then replicate n (replicate n []) else
blanchet@48975
  1450
      let
blanchet@48975
  1451
        fun mk_case s sets z z_free = Term.absfree z_free (Library.foldr1 HOLogic.mk_conj
traytel@51893
  1452
          (map2 (fn set => fn A => mk_leq (set $ (s $ z)) A) (take m sets) As));
blanchet@48975
  1453
blanchet@48975
  1454
        fun mk_conjunct i z B = HOLogic.mk_imp
blanchet@48975
  1455
          (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), HOLogic.mk_mem (z, B)),
blanchet@48975
  1456
          mk_sum_caseN (map4 mk_case ss setssAs zs zs') $ (mk_rv ss kl i $ z));
blanchet@48975
  1457
blanchet@48975
  1458
        val goal = list_all_free (kl :: zs)
blanchet@48975
  1459
          (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct ks zs Bs));
blanchet@48975
  1460
blanchet@48975
  1461
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1462
blanchet@48975
  1463
        val set_rv_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1464
          (Goal.prove_sorry lthy [] []
blanchet@48975
  1465
            (Logic.mk_implies (coalg_prem, HOLogic.mk_Trueprop goal))
wenzelm@51798
  1466
            (K (mk_set_rv_Lev_tac lthy m cts Lev_0s Lev_Sucs rv_Nils rv_Conss
traytel@49109
  1467
              coalg_set_thmss from_to_sbd_thmss)))
traytel@49109
  1468
          |> Thm.close_derivation;
blanchet@48975
  1469
blanchet@48975
  1470
        val set_rv_Lev' = mk_specN (n + 1) set_rv_Lev;
blanchet@48975
  1471
      in
blanchet@48975
  1472
        map (fn i => map (fn i' =>
blanchet@48975
  1473
          split_conj_thm (if n = 1 then set_rv_Lev' RS mk_conjunctN n i RS mp
blanchet@48975
  1474
            else set_rv_Lev' RS mk_conjunctN n i RS mp RSN
traytel@52904
  1475
              (2, @{thm sum_case_weak_cong} RS iffD1) RS
traytel@52904
  1476
              (mk_sum_casesN n i' RS iffD1))) ks) ks
blanchet@48975
  1477
      end;
blanchet@48975
  1478
blanchet@48975
  1479
    val set_Lev_thmsss =
blanchet@48975
  1480
      let
blanchet@48975
  1481
        fun mk_conjunct i z =
blanchet@48975
  1482
          let
blanchet@48975
  1483
            fun mk_conjunct' i' sets s z' =
blanchet@48975
  1484
              let
blanchet@48975
  1485
                fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp
blanchet@48975
  1486
                  (HOLogic.mk_mem (z'', set $ (s $ z')),
blanchet@48975
  1487
                    HOLogic.mk_mem (mk_append (kl,
blanchet@48975
  1488
                      HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']),
blanchet@48975
  1489
                      mk_Lev ss (HOLogic.mk_Suc nat) i $ z));
blanchet@48975
  1490
              in
blanchet@48975
  1491
                HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'),
blanchet@48975
  1492
                  (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct'' ks (drop m sets) zs_copy2)))
blanchet@48975
  1493
              end;
blanchet@48975
  1494
          in
blanchet@48975
  1495
            HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1496
              Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct' ks setssAs ss zs_copy))
blanchet@48975
  1497
          end;
blanchet@48975
  1498
blanchet@48975
  1499
        val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2)
blanchet@48975
  1500
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1501
blanchet@48975
  1502
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1503
blanchet@48975
  1504
        val set_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1505
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1506
            (K (mk_set_Lev_tac lthy cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss)))
traytel@49109
  1507
          |> Thm.close_derivation;
blanchet@48975
  1508
blanchet@48975
  1509
        val set_Lev' = mk_specN (3 * n + 1) set_Lev;
blanchet@48975
  1510
      in
blanchet@48975
  1511
        map (fn i => map (fn i' => map (fn i'' => set_Lev' RS
blanchet@48975
  1512
          mk_conjunctN n i RS mp RS
blanchet@48975
  1513
          mk_conjunctN n i' RS mp RS
blanchet@48975
  1514
          mk_conjunctN n i'' RS mp) ks) ks) ks
blanchet@48975
  1515
      end;
blanchet@48975
  1516
blanchet@48975
  1517
    val set_image_Lev_thmsss =
blanchet@48975
  1518
      let
blanchet@48975
  1519
        fun mk_conjunct i z =
blanchet@48975
  1520
          let
blanchet@48975
  1521
            fun mk_conjunct' i' sets =
blanchet@48975
  1522
              let
blanchet@48975
  1523
                fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp
blanchet@48975
  1524
                  (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''),
blanchet@48975
  1525
                  HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z''))));
blanchet@48975
  1526
              in
blanchet@48975
  1527
                HOLogic.mk_imp (HOLogic.mk_mem
blanchet@48975
  1528
                  (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']),
blanchet@48975
  1529
                    mk_Lev ss (HOLogic.mk_Suc nat) i $ z),
blanchet@48975
  1530
                  (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct'' ks sets ss zs_copy)))
blanchet@48975
  1531
              end;
blanchet@48975
  1532
          in
blanchet@48975
  1533
            HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1534
              Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs')))
blanchet@48975
  1535
          end;
blanchet@48975
  1536
blanchet@48975
  1537
        val goal = list_all_free (kl :: k :: zs @ zs_copy)
blanchet@48975
  1538
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1539
blanchet@48975
  1540
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1541
blanchet@48975
  1542
        val set_image_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1543
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1544
            (K (mk_set_image_Lev_tac lthy cts Lev_0s Lev_Sucs rv_Nils rv_Conss
traytel@49109
  1545
              from_to_sbd_thmss to_sbd_inj_thmss)))
traytel@49109
  1546
          |> Thm.close_derivation;
blanchet@48975
  1547
blanchet@48975
  1548
        val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev;
blanchet@48975
  1549
      in
blanchet@48975
  1550
        map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS
blanchet@48975
  1551
          mk_conjunctN n i RS mp RS
blanchet@48975
  1552
          mk_conjunctN n i'' RS mp RS
blanchet@48975
  1553
          mk_conjunctN n i' RS mp) ks) ks) ks
blanchet@48975
  1554
      end;
blanchet@48975
  1555
blanchet@48975
  1556
    val mor_beh_thm =
wenzelm@51551
  1557
      Goal.prove_sorry lthy [] []
blanchet@48975
  1558
        (fold_rev Logic.all (As @ Bs @ ss) (Logic.mk_implies (coalg_prem,
blanchet@48975
  1559
          HOLogic.mk_Trueprop (mk_mor Bs ss carTAs strTAs (map (mk_beh ss) ks)))))
blanchet@48975
  1560
        (mk_mor_beh_tac m mor_def mor_cong_thm
blanchet@48975
  1561
          beh_defs carT_defs strT_defs isNode_defs
blanchet@48975
  1562
          to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbd_thms
blanchet@48975
  1563
          length_Lev_thms length_Lev'_thms prefCl_Lev_thms rv_last_thmss
blanchet@48975
  1564
          set_rv_Lev_thmsss set_Lev_thmsss set_image_Lev_thmsss
blanchet@53290
  1565
          set_mapss coalg_set_thmss map_comp_id_thms map_cong0s map_arg_cong_thms)
traytel@49109
  1566
      |> Thm.close_derivation;
blanchet@48975
  1567
blanchet@48975
  1568
    val timer = time (timer "Behavioral morphism");
blanchet@48975
  1569
blanchet@48975
  1570
    fun mk_LSBIS As i = mk_lsbis As (map (mk_carT As) ks) strTAs i;
blanchet@48975
  1571
    fun mk_car_final As i =
blanchet@48975
  1572
      mk_quotient (mk_carT As i) (mk_LSBIS As i);
blanchet@48975
  1573
    fun mk_str_final As i =
blanchet@48975
  1574
      mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1),
blanchet@48975
  1575
        passive_ids @ map (mk_proj o mk_LSBIS As) ks), nth strTAs (i - 1)));
blanchet@48975
  1576
blanchet@48975
  1577
    val car_finalAs = map (mk_car_final As) ks;
blanchet@48975
  1578
    val str_finalAs = map (mk_str_final As) ks;
blanchet@48975
  1579
    val car_finals = map (mk_car_final passive_UNIVs) ks;
blanchet@48975
  1580
    val str_finals = map (mk_str_final passive_UNIVs) ks;
blanchet@48975
  1581
blanchet@48975
  1582
    val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss;
blanchet@48975
  1583
    val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms;
blanchet@48975
  1584
blanchet@48975
  1585
    val congruent_str_final_thms =
blanchet@48975
  1586
      let
blanchet@48975
  1587
        fun mk_goal R final_map strT =
blanchet@48975
  1588
          fold_rev Logic.all As (HOLogic.mk_Trueprop
blanchet@48975
  1589
            (mk_congruent R (HOLogic.mk_comp
blanchet@48975
  1590
              (Term.list_comb (final_map, passive_ids @ map (mk_proj o mk_LSBIS As) ks), strT))));
blanchet@48975
  1591
blanchet@48975
  1592
        val goals = map3 mk_goal (map (mk_LSBIS As) ks) final_maps strTAs;
blanchet@48975
  1593
      in
blanchet@51761
  1594
        map4 (fn goal => fn lsbisE => fn map_comp_id => fn map_cong0 =>
wenzelm@51551
  1595
          Goal.prove_sorry lthy [] [] goal
blanchet@51761
  1596
            (K (mk_congruent_str_final_tac m lsbisE map_comp_id map_cong0 equiv_LSBIS_thms))
traytel@49109
  1597
          |> Thm.close_derivation)
blanchet@51761
  1598
        goals lsbisE_thms map_comp_id_thms map_cong0s
blanchet@48975
  1599
      end;
blanchet@48975
  1600
wenzelm@51551
  1601
    val coalg_final_thm = Goal.prove_sorry lthy [] [] (fold_rev Logic.all As
blanchet@48975
  1602
      (HOLogic.mk_Trueprop (mk_coalg As car_finalAs str_finalAs)))
blanchet@48975
  1603
      (K (mk_coalg_final_tac m coalg_def congruent_str_final_thms equiv_LSBIS_thms
blanchet@53290
  1604
        set_mapss coalgT_set_thmss))
traytel@49109
  1605
      |> Thm.close_derivation;
blanchet@48975
  1606
wenzelm@51551
  1607
    val mor_T_final_thm = Goal.prove_sorry lthy [] [] (fold_rev Logic.all As
blanchet@48975
  1608
      (HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finalAs str_finalAs
blanchet@48975
  1609
        (map (mk_proj o mk_LSBIS As) ks))))
traytel@49109
  1610
      (K (mk_mor_T_final_tac mor_def congruent_str_final_thms equiv_LSBIS_thms))
traytel@49109
  1611
      |> Thm.close_derivation;
blanchet@48975
  1612
blanchet@48975
  1613
    val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm];
blanchet@48975
  1614
    val in_car_final_thms = map (fn mor_image' => mor_image' OF
blanchet@48975
  1615
      [tcoalg_thm RS mor_final_thm, UNIV_I]) mor_image'_thms;
blanchet@48975
  1616
blanchet@48975
  1617
    val timer = time (timer "Final coalgebra");
blanchet@48975
  1618
blanchet@48975
  1619
    val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
blanchet@48975
  1620
      lthy
blanchet@49169
  1621
      |> fold_map4 (fn b => fn mx => fn car_final => fn in_car_final =>
traytel@53566
  1622
        typedef (Binding.conceal b, params, mx) car_final NONE
blanchet@49169
  1623
          (EVERY' [rtac exI, rtac in_car_final] 1)) bs mixfixes car_finals in_car_final_thms
blanchet@48975
  1624
      |>> apsnd split_list o split_list;
blanchet@48975
  1625
blanchet@48975
  1626
    val Ts = map (fn name => Type (name, params')) T_names;
blanchet@48975
  1627
    fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;
blanchet@48975
  1628
    val Ts' = mk_Ts passiveBs;
blanchet@48975
  1629
    val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts;
blanchet@48975
  1630
    val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts;
blanchet@48975
  1631
blanchet@48975
  1632
    val Reps = map #Rep T_loc_infos;
blanchet@48975
  1633
    val Rep_injects = map #Rep_inject T_loc_infos;
blanchet@48975
  1634
    val Abs_inverses = map #Abs_inverse T_loc_infos;
blanchet@48975
  1635
blanchet@48975
  1636
    val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");
blanchet@48975
  1637
blanchet@48975
  1638
    val UNIVs = map HOLogic.mk_UNIV Ts;
blanchet@48975
  1639
    val FTs = mk_FTs (passiveAs @ Ts);
blanchet@48975
  1640
    val FTs' = mk_FTs (passiveBs @ Ts);
blanchet@48975
  1641
    val prodTs = map (HOLogic.mk_prodT o `I) Ts;
blanchet@48975
  1642
    val prodFTs = mk_FTs (passiveAs @ prodTs);
blanchet@48975
  1643
    val FTs_setss = mk_setss (passiveAs @ Ts);
blanchet@48975
  1644
    val prodFT_setss = mk_setss (passiveAs @ prodTs);
blanchet@48975
  1645
    val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs;
blanchet@48975
  1646
    val map_FT_nths = map2 (fn Ds =>
blanchet@48975
  1647
      mk_map_of_bnf Ds (passiveAs @ prodTs) (passiveAs @ Ts)) Dss bnfs;
blanchet@48975
  1648
    val fstsTs = map fst_const prodTs;
blanchet@48975
  1649
    val sndsTs = map snd_const prodTs;
traytel@52923
  1650
    val dtorTs = map2 (curry op -->) Ts FTs;
traytel@52923
  1651
    val ctorTs = map2 (curry op -->) FTs Ts;
blanchet@49504
  1652
    val unfold_fTs = map2 (curry op -->) activeAs Ts;
blanchet@48975
  1653
    val corec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) sum_sTs;
blanchet@48975
  1654
    val corec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls;
blanchet@48975
  1655
    val corec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls_rev;
blanchet@48975
  1656
    val corec_Inls = map (Term.subst_atomic_types (activeBs ~~ Ts)) Inls;
traytel@51739
  1657
    val corec_UNIVs = map2 (HOLogic.mk_UNIV oo curry mk_sumT) Ts activeAs;
blanchet@48975
  1658
traytel@54841
  1659
    val (((((((((((((Jzs, Jzs'), Jz's), Jzs_copy), Jz's_copy), Jzs1), Jzs2),
traytel@54841
  1660
      FJzs), TRs), unfold_fs), corec_ss), phis), dtor_set_induct_phiss),
traytel@54841
  1661
      names_lthy) = names_lthy
blanchet@48975
  1662
      |> mk_Frees' "z" Ts
traytel@54841
  1663
      ||>> mk_Frees "y" Ts'
traytel@51925
  1664
      ||>> mk_Frees "z'" Ts
traytel@51925
  1665
      ||>> mk_Frees "y'" Ts'
blanchet@48975
  1666
      ||>> mk_Frees "z1" Ts
blanchet@48975
  1667
      ||>> mk_Frees "z2" Ts
blanchet@48975
  1668
      ||>> mk_Frees "x" prodFTs
blanchet@49591
  1669
      ||>> mk_Frees "r" (map (mk_relT o `I) Ts)
blanchet@49504
  1670
      ||>> mk_Frees "f" unfold_fTs
blanchet@48975
  1671
      ||>> mk_Frees "s" corec_sTs
traytel@54841
  1672
      ||>> mk_Frees "P" (map2 mk_pred2T Ts Ts)
traytel@54841
  1673
      ||>> mk_Freess "P" (map (fn A => map (mk_pred2T A) Ts) passiveAs);
blanchet@48975
  1674
blanchet@54492
  1675
    fun dtor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtorN ^ "_");
blanchet@49501
  1676
    val dtor_name = Binding.name_of o dtor_bind;
traytel@53566
  1677
    val dtor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o dtor_bind;
blanchet@49501
  1678
blanchet@49501
  1679
    fun dtor_spec i rep str map_FT dtorT Jz Jz' =
blanchet@48975
  1680
      let
blanchet@49501
  1681
        val lhs = Free (dtor_name i, dtorT);
blanchet@48975
  1682
        val rhs = Term.absfree Jz'
blanchet@48975
  1683
          (Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $
blanchet@48975
  1684
            (str $ (rep $ Jz)));
blanchet@48975
  1685
      in
blanchet@49123
  1686
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1687
      end;
blanchet@48975
  1688
blanchet@49501
  1689
    val ((dtor_frees, (_, dtor_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1690
      lthy
blanchet@49501
  1691
      |> fold_map7 (fn i => fn rep => fn str => fn mapx => fn dtorT => fn Jz => fn Jz' =>
blanchet@49501
  1692
        Specification.definition (SOME (dtor_bind i, NONE, NoSyn),
blanchet@49501
  1693
          (dtor_def_bind i, dtor_spec i rep str mapx dtorT Jz Jz')))
blanchet@49501
  1694
        ks Rep_Ts str_finals map_FTs dtorTs Jzs Jzs'
blanchet@48975
  1695
      |>> apsnd split_list o split_list
blanchet@48975
  1696
      ||> `Local_Theory.restore;
blanchet@48975
  1697
blanchet@48975
  1698
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49501
  1699
    fun mk_dtors passive =
traytel@49185
  1700
      map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (mk_params passive)) o
blanchet@49501
  1701
        Morphism.term phi) dtor_frees;
blanchet@49501
  1702
    val dtors = mk_dtors passiveAs;
blanchet@49501
  1703
    val dtor's = mk_dtors passiveBs;
blanchet@49501
  1704
    val dtor_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) dtor_def_frees;
blanchet@48975
  1705
blanchet@48975
  1706
    val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss;
blanchet@48975
  1707
    val (mor_Rep_thm, mor_Abs_thm) =
blanchet@48975
  1708
      let
blanchet@48975
  1709
        val mor_Rep =
wenzelm@51551
  1710
          Goal.prove_sorry lthy [] []
blanchet@49501
  1711
            (HOLogic.mk_Trueprop (mk_mor UNIVs dtors car_finals str_finals Rep_Ts))
blanchet@49501
  1712
            (mk_mor_Rep_tac m (mor_def :: dtor_defs) Reps Abs_inverses coalg_final_set_thmss
blanchet@51761
  1713
              map_comp_id_thms map_cong0L_thms)
traytel@49109
  1714
          |> Thm.close_derivation;
blanchet@48975
  1715
blanchet@48975
  1716
        val mor_Abs =
wenzelm@51551
  1717
          Goal.prove_sorry lthy [] []
blanchet@49501
  1718
            (HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs dtors Abs_Ts))
blanchet@49501
  1719
            (mk_mor_Abs_tac (mor_def :: dtor_defs) Abs_inverses)
traytel@49109
  1720
          |> Thm.close_derivation;
blanchet@48975
  1721
      in
blanchet@48975
  1722
        (mor_Rep, mor_Abs)
blanchet@48975
  1723
      end;
blanchet@48975
  1724
blanchet@49501
  1725
    val timer = time (timer "dtor definitions & thms");
blanchet@49501
  1726
blanchet@54492
  1727
    fun unfold_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_unfoldN ^ "_");
blanchet@49504
  1728
    val unfold_name = Binding.name_of o unfold_bind;
traytel@53566
  1729
    val unfold_def_bind = rpair [] o Binding.conceal o Thm.def_binding o unfold_bind;
blanchet@49504
  1730
blanchet@49504
  1731
    fun unfold_spec i T AT abs f z z' =
blanchet@48975
  1732
      let
blanchet@49504
  1733
        val unfoldT = Library.foldr (op -->) (sTs, AT --> T);
blanchet@49504
  1734
blanchet@49504
  1735
        val lhs = Term.list_comb (Free (unfold_name i, unfoldT), ss);
blanchet@48975
  1736
        val rhs = Term.absfree z' (abs $ (f $ z));
blanchet@48975
  1737
      in
blanchet@49123
  1738
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1739
      end;
blanchet@48975
  1740
blanchet@49504
  1741
    val ((unfold_frees, (_, unfold_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1742
      lthy
blanchet@48975
  1743
      |> fold_map7 (fn i => fn T => fn AT => fn abs => fn f => fn z => fn z' =>
blanchet@48975
  1744
        Specification.definition
blanchet@49504
  1745
          (SOME (unfold_bind i, NONE, NoSyn), (unfold_def_bind i, unfold_spec i T AT abs f z z')))
blanchet@48975
  1746
          ks Ts activeAs Abs_Ts (map (fn i => HOLogic.mk_comp
blanchet@48975
  1747
            (mk_proj (mk_LSBIS passive_UNIVs i), mk_beh ss i)) ks) zs zs'
blanchet@48975
  1748
      |>> apsnd split_list o split_list
blanchet@48975
  1749
      ||> `Local_Theory.restore;
blanchet@48975
  1750
blanchet@48975
  1751
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49504
  1752
    val unfolds = map (Morphism.term phi) unfold_frees;
blanchet@49504
  1753
    val unfold_names = map (fst o dest_Const) unfolds;
traytel@52731
  1754
    fun mk_unfolds passives actives =
traytel@52731
  1755
      map3 (fn name => fn T => fn active =>
traytel@52731
  1756
        Const (name, Library.foldr (op -->)
traytel@52923
  1757
          (map2 (curry op -->) actives (mk_FTs (passives @ actives)), active --> T)))
traytel@52731
  1758
      unfold_names (mk_Ts passives) actives;
blanchet@49504
  1759
    fun mk_unfold Ts ss i = Term.list_comb (Const (nth unfold_names (i - 1), Library.foldr (op -->)
blanchet@48975
  1760
      (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
blanchet@49504
  1761
    val unfold_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) unfold_def_frees;
blanchet@49504
  1762
blanchet@49504
  1763
    val mor_unfold_thm =
blanchet@48975
  1764
      let
blanchet@48975
  1765
        val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses;
blanchet@48975
  1766
        val morEs' = map (fn thm =>
blanchet@48975
  1767
          (thm OF [tcoalg_thm RS mor_final_thm, UNIV_I]) RS sym) morE_thms;
blanchet@48975
  1768
      in
wenzelm@51551
  1769
        Goal.prove_sorry lthy [] []
blanchet@48975
  1770
          (fold_rev Logic.all ss
blanchet@49504
  1771
            (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors (map (mk_unfold Ts ss) ks))))
blanchet@49504
  1772
          (K (mk_mor_unfold_tac m mor_UNIV_thm dtor_defs unfold_defs Abs_inverses' morEs'
blanchet@51761
  1773
            map_comp_id_thms map_cong0s))
traytel@49109
  1774
        |> Thm.close_derivation
blanchet@48975
  1775
      end;
blanchet@49504
  1776
    val dtor_unfold_thms = map (fn thm => (thm OF [mor_unfold_thm, UNIV_I]) RS sym) morE_thms;
blanchet@48975
  1777
blanchet@48975
  1778
    val (raw_coind_thms, raw_coind_thm) =
blanchet@48975
  1779
      let
blanchet@49501
  1780
        val prem = HOLogic.mk_Trueprop (mk_sbis passive_UNIVs UNIVs dtors TRs);
blanchet@48975
  1781
        val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
traytel@51893
  1782
          (map2 (fn R => fn T => mk_leq R (Id_const T)) TRs Ts));
blanchet@48975
  1783
        val goal = fold_rev Logic.all TRs (Logic.mk_implies (prem, concl));
blanchet@48975
  1784
      in
wenzelm@51551
  1785
        `split_conj_thm (Goal.prove_sorry lthy [] [] goal
blanchet@48975
  1786
          (K (mk_raw_coind_tac bis_def bis_cong_thm bis_O_thm bis_converse_thm bis_Gr_thm
blanchet@48975
  1787
            tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm
traytel@49109
  1788
            lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects))
traytel@49109
  1789
          |> Thm.close_derivation)
blanchet@48975
  1790
      end;
blanchet@48975
  1791
blanchet@49504
  1792
    val (unfold_unique_mor_thms, unfold_unique_mor_thm) =
blanchet@48975
  1793
      let
blanchet@49504
  1794
        val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors unfold_fs);
blanchet@49504
  1795
        fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_unfold Ts ss i);
blanchet@48975
  1796
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@49504
  1797
          (map2 mk_fun_eq unfold_fs ks));
blanchet@48975
  1798
blanchet@48975
  1799
        val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm);
blanchet@48975
  1800
        val mor_thm = mor_comp_thm OF [tcoalg_thm RS mor_final_thm, mor_Abs_thm];
blanchet@48975
  1801
wenzelm@51551
  1802
        val unique_mor = Goal.prove_sorry lthy [] []
blanchet@49504
  1803
          (fold_rev Logic.all (ss @ unfold_fs) (Logic.mk_implies (prem, unique)))
blanchet@49504
  1804
          (K (mk_unfold_unique_mor_tac raw_coind_thms bis_thm mor_thm unfold_defs))
traytel@49109
  1805
          |> Thm.close_derivation;
blanchet@48975
  1806
      in
blanchet@48975
  1807
        `split_conj_thm unique_mor
blanchet@48975
  1808
      end;
blanchet@48975
  1809
blanchet@49504
  1810
    val (dtor_unfold_unique_thms, dtor_unfold_unique_thm) = `split_conj_thm (split_conj_prems n
traytel@52904
  1811
      (mor_UNIV_thm RS iffD2 RS unfold_unique_mor_thm));
blanchet@49504
  1812
blanchet@49504
  1813
    val unfold_dtor_thms = map (fn thm => mor_id_thm RS thm RS sym) unfold_unique_mor_thms;
blanchet@49504
  1814
blanchet@49504
  1815
    val unfold_o_dtor_thms =
blanchet@48975
  1816
      let
blanchet@49504
  1817
        val mor = mor_comp_thm OF [mor_str_thm, mor_unfold_thm];
blanchet@48975
  1818
      in
blanchet@49504
  1819
        map2 (fn unique => fn unfold_ctor =>
blanchet@49504
  1820
          trans OF [mor RS unique, unfold_ctor]) unfold_unique_mor_thms unfold_dtor_thms
blanchet@48975
  1821
      end;
blanchet@48975
  1822
blanchet@49504
  1823
    val timer = time (timer "unfold definitions & thms");
blanchet@48975
  1824
blanchet@49501
  1825
    val map_dtors = map2 (fn Ds => fn bnf =>
blanchet@48975
  1826
      Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,
blanchet@49501
  1827
        map HOLogic.id_const passiveAs @ dtors)) Dss bnfs;
blanchet@49501
  1828
blanchet@54492
  1829
    fun ctor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctorN ^ "_");
blanchet@49501
  1830
    val ctor_name = Binding.name_of o ctor_bind;
traytel@53566
  1831
    val ctor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o ctor_bind;
blanchet@49501
  1832
blanchet@49501
  1833
    fun ctor_spec i ctorT =
blanchet@48975
  1834
      let
blanchet@49501
  1835
        val lhs = Free (ctor_name i, ctorT);
blanchet@49504
  1836
        val rhs = mk_unfold Ts map_dtors i;
blanchet@48975
  1837
      in
blanchet@49123
  1838
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1839
      end;
blanchet@48975
  1840
blanchet@49501
  1841
    val ((ctor_frees, (_, ctor_def_frees)), (lthy, lthy_old)) =
blanchet@49311
  1842
      lthy
blanchet@49501
  1843
      |> fold_map2 (fn i => fn ctorT =>
blanchet@49311
  1844
        Specification.definition
blanchet@49501
  1845
          (SOME (ctor_bind i, NONE, NoSyn), (ctor_def_bind i, ctor_spec i ctorT))) ks ctorTs
blanchet@49311
  1846
      |>> apsnd split_list o split_list
blanchet@49311
  1847
      ||> `Local_Theory.restore;
blanchet@49311
  1848
blanchet@48975
  1849
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49501
  1850
    fun mk_ctors params =
blanchet@48975
  1851
      map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
blanchet@49501
  1852
        ctor_frees;
blanchet@49501
  1853
    val ctors = mk_ctors params';
blanchet@49501
  1854
    val ctor_defs = map (Morphism.thm phi) ctor_def_frees;
blanchet@49501
  1855
blanchet@49504
  1856
    val ctor_o_dtor_thms = map2 (fold_thms lthy o single) ctor_defs unfold_o_dtor_thms;
blanchet@49501
  1857
blanchet@49501
  1858
    val dtor_o_ctor_thms =
blanchet@48975
  1859
      let
blanchet@49501
  1860
        fun mk_goal dtor ctor FT =
blanchet@49501
  1861
         mk_Trueprop_eq (HOLogic.mk_comp (dtor, ctor), HOLogic.id_const FT);
blanchet@49501
  1862
        val goals = map3 mk_goal dtors ctors FTs;
blanchet@48975
  1863
      in
blanchet@51761
  1864
        map5 (fn goal => fn ctor_def => fn unfold => fn map_comp_id => fn map_cong0L =>
wenzelm@51551
  1865
          Goal.prove_sorry lthy [] [] goal
blanchet@51761
  1866
            (mk_dtor_o_ctor_tac ctor_def unfold map_comp_id map_cong0L unfold_o_dtor_thms)
traytel@49109
  1867
          |> Thm.close_derivation)
blanchet@51761
  1868
          goals ctor_defs dtor_unfold_thms map_comp_id_thms map_cong0L_thms
blanchet@48975
  1869
      end;
blanchet@48975
  1870
blanchet@49501
  1871
    val dtor_ctor_thms = map (fn thm => thm RS @{thm pointfree_idE}) dtor_o_ctor_thms;
blanchet@49501
  1872
    val ctor_dtor_thms = map (fn thm => thm RS @{thm pointfree_idE}) ctor_o_dtor_thms;
blanchet@49501
  1873
blanchet@49501
  1874
    val bij_dtor_thms =
blanchet@49501
  1875
      map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) ctor_o_dtor_thms dtor_o_ctor_thms;
blanchet@49501
  1876
    val inj_dtor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_dtor_thms;
blanchet@49501
  1877
    val surj_dtor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_dtor_thms;
blanchet@49501
  1878
    val dtor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_dtor_thms;
blanchet@49501
  1879
    val dtor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_dtor_thms;
blanchet@49501
  1880
    val dtor_exhaust_thms = map (fn thm => thm RS exE) dtor_nchotomy_thms;
blanchet@49501
  1881
blanchet@49501
  1882
    val bij_ctor_thms =
blanchet@49501
  1883
      map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) dtor_o_ctor_thms ctor_o_dtor_thms;
blanchet@49501
  1884
    val inj_ctor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_ctor_thms;
blanchet@49501
  1885
    val surj_ctor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_ctor_thms;
blanchet@49501
  1886
    val ctor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_ctor_thms;
blanchet@49501
  1887
    val ctor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_ctor_thms;
blanchet@49501
  1888
    val ctor_exhaust_thms = map (fn thm => thm RS exE) ctor_nchotomy_thms;
blanchet@49501
  1889
blanchet@49501
  1890
    val timer = time (timer "ctor definitions & thms");
blanchet@48975
  1891
blanchet@48975
  1892
    val corec_Inl_sum_thms =
blanchet@48975
  1893
      let
blanchet@49504
  1894
        val mor = mor_comp_thm OF [mor_sum_case_thm, mor_unfold_thm];
blanchet@48975
  1895
      in
blanchet@49504
  1896
        map2 (fn unique => fn unfold_dtor =>
blanchet@49504
  1897
          trans OF [mor RS unique, unfold_dtor]) unfold_unique_mor_thms unfold_dtor_thms
blanchet@48975
  1898
      end;
blanchet@48975
  1899
blanchet@54492
  1900
    fun corec_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_corecN ^ "_");
blanchet@48975
  1901
    val corec_name = Binding.name_of o corec_bind;
traytel@53566
  1902
    val corec_def_bind = rpair [] o Binding.conceal o Thm.def_binding o corec_bind;
blanchet@48975
  1903
traytel@51739
  1904
    val corec_strs =
traytel@51739
  1905
      map3 (fn dtor => fn sum_s => fn mapx =>
traytel@51739
  1906
        mk_sum_case
traytel@51739
  1907
          (HOLogic.mk_comp (Term.list_comb (mapx, passive_ids @ corec_Inls), dtor), sum_s))
traytel@51739
  1908
      dtors corec_ss corec_maps;
traytel@51739
  1909
blanchet@48975
  1910
    fun corec_spec i T AT =
blanchet@48975
  1911
      let
blanchet@48975
  1912
        val corecT = Library.foldr (op -->) (corec_sTs, AT --> T);
blanchet@48975
  1913
blanchet@48975
  1914
        val lhs = Term.list_comb (Free (corec_name i, corecT), corec_ss);
traytel@51739
  1915
        val rhs = HOLogic.mk_comp (mk_unfold Ts corec_strs i, Inr_const T AT);
blanchet@48975
  1916
      in
blanchet@49123
  1917
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1918
      end;
blanchet@48975
  1919
blanchet@48975
  1920
    val ((corec_frees, (_, corec_def_frees)), (lthy, lthy_old)) =
blanchet@49311
  1921
      lthy
blanchet@49311
  1922
      |> fold_map3 (fn i => fn T => fn AT =>
blanchet@49311
  1923
        Specification.definition
blanchet@49311
  1924
          (SOME (corec_bind i, NONE, NoSyn), (corec_def_bind i, corec_spec i T AT)))
blanchet@49311
  1925
          ks Ts activeAs
blanchet@49311
  1926
      |>> apsnd split_list o split_list
blanchet@49311
  1927
      ||> `Local_Theory.restore;
blanchet@49311
  1928
blanchet@48975
  1929
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49176
  1930
    val corecs = map (Morphism.term phi) corec_frees;
blanchet@49176
  1931
    val corec_names = map (fst o dest_Const) corecs;
blanchet@49176
  1932
    fun mk_corec ss i = Term.list_comb (Const (nth corec_names (i - 1), Library.foldr (op -->)
blanchet@48975
  1933
      (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
blanchet@48975
  1934
    val corec_defs = map (Morphism.thm phi) corec_def_frees;
blanchet@48975
  1935
blanchet@48975
  1936
    val sum_cases =
blanchet@49255
  1937
      map2 (fn T => fn i => mk_sum_case (HOLogic.id_const T, mk_corec corec_ss i)) Ts ks;
blanchet@49504
  1938
    val dtor_corec_thms =
blanchet@48975
  1939
      let
blanchet@49501
  1940
        fun mk_goal i corec_s corec_map dtor z =
blanchet@48975
  1941
          let
blanchet@49501
  1942
            val lhs = dtor $ (mk_corec corec_ss i $ z);
blanchet@48975
  1943
            val rhs = Term.list_comb (corec_map, passive_ids @ sum_cases) $ (corec_s $ z);
blanchet@48975
  1944
          in
blanchet@49123
  1945
            fold_rev Logic.all (z :: corec_ss) (mk_Trueprop_eq (lhs, rhs))
blanchet@48975
  1946
          end;
blanchet@49501
  1947
        val goals = map5 mk_goal ks corec_ss corec_maps_rev dtors zs;
blanchet@48975
  1948
      in
blanchet@51761
  1949
        map3 (fn goal => fn unfold => fn map_cong0 =>
wenzelm@51551
  1950
          Goal.prove_sorry lthy [] [] goal
blanchet@51761
  1951
            (mk_corec_tac m corec_defs unfold map_cong0 corec_Inl_sum_thms)
traytel@49109
  1952
          |> Thm.close_derivation)
blanchet@51761
  1953
        goals dtor_unfold_thms map_cong0s
blanchet@48975
  1954
      end;
blanchet@48975
  1955
traytel@51739
  1956
    val corec_unique_mor_thm =
traytel@51739
  1957
      let
traytel@51739
  1958
        val id_fs = map2 (fn T => fn f => mk_sum_case (HOLogic.id_const T, f)) Ts unfold_fs;
traytel@51739
  1959
        val prem = HOLogic.mk_Trueprop (mk_mor corec_UNIVs corec_strs UNIVs dtors id_fs);
traytel@51739
  1960
        fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_corec corec_ss i);
traytel@51739
  1961
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
traytel@51739
  1962
          (map2 mk_fun_eq unfold_fs ks));
traytel@51739
  1963
      in
traytel@51739
  1964
        Goal.prove_sorry lthy [] []
traytel@51739
  1965
          (fold_rev Logic.all (corec_ss @ unfold_fs) (Logic.mk_implies (prem, unique)))
traytel@51739
  1966
          (mk_corec_unique_mor_tac corec_defs corec_Inl_sum_thms unfold_unique_mor_thm)
traytel@51739
  1967
          |> Thm.close_derivation
traytel@51739
  1968
      end;
traytel@51739
  1969
traytel@53696
  1970
    val map_id0s_o_id =
traytel@53696
  1971
      map (fn thm =>
traytel@53696
  1972
        mk_trans (thm RS @{thm arg_cong2[of _ _ _ _ "op o", OF _ refl]}) @{thm id_o})
traytel@53696
  1973
      map_id0s;
traytel@53696
  1974
traytel@52913
  1975
    val (dtor_corec_unique_thms, dtor_corec_unique_thm) =
traytel@52913
  1976
      `split_conj_thm (split_conj_prems n
traytel@52904
  1977
        (mor_UNIV_thm RS iffD2 RS corec_unique_mor_thm)
traytel@53696
  1978
        |> Local_Defs.unfold lthy (@{thms o_sum_case o_id id_o o_assoc sum_case_o_inj(1)} @
traytel@53696
  1979
           map_id0s_o_id @ sym_map_comps)
traytel@53696
  1980
        OF replicate n @{thm arg_cong2[of _ _ _ _ sum_case, OF refl]});
traytel@51739
  1981
blanchet@48975
  1982
    val timer = time (timer "corec definitions & thms");
blanchet@48975
  1983
traytel@53105
  1984
    val (dtor_map_coinduct_thm, coinduct_params, dtor_coinduct_thm) =
blanchet@48975
  1985
      let
blanchet@48975
  1986
        val zs = Jzs1 @ Jzs2;
blanchet@48975
  1987
        val frees = phis @ zs;
blanchet@48975
  1988
traytel@51893
  1989
        val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs;
blanchet@48975
  1990
blanchet@48975
  1991
        fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2));
blanchet@48975
  1992
        val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  1993
          (map3 mk_concl phis Jzs1 Jzs2));
blanchet@48975
  1994
traytel@53105
  1995
        fun mk_rel_prem phi dtor rel Jz Jz_copy =
blanchet@48975
  1996
          let
traytel@53105
  1997
            val concl = Term.list_comb (rel, map HOLogic.eq_const passiveAs @ phis) $
traytel@51893
  1998
              (dtor $ Jz) $ (dtor $ Jz_copy);
blanchet@48975
  1999
          in
blanchet@48975
  2000
            HOLogic.mk_Trueprop
blanchet@48975
  2001
              (list_all_free [Jz, Jz_copy] (HOLogic.mk_imp (phi $ Jz $ Jz_copy, concl)))
blanchet@48975
  2002
          end;
blanchet@48975
  2003
traytel@53105
  2004
        val rel_prems = map5 mk_rel_prem phis dtors rels Jzs Jzs_copy;
traytel@51893
  2005
        val dtor_coinduct_goal =
traytel@51893
  2006
          fold_rev Logic.all frees (Logic.list_implies (rel_prems, concl));
traytel@51893
  2007
traytel@51893
  2008
        val dtor_coinduct =
traytel@51893
  2009
          Goal.prove_sorry lthy [] [] dtor_coinduct_goal
traytel@51893
  2010
            (K (mk_dtor_coinduct_tac m raw_coind_thm bis_rel_thm rel_congs))
traytel@51893
  2011
          |> Thm.close_derivation;
blanchet@48975
  2012
traytel@53105
  2013
        fun mk_prem phi dtor map_nth sets Jz Jz_copy FJz =
blanchet@48975
  2014
          let
blanchet@48975
  2015
            val xs = [Jz, Jz_copy];
blanchet@48975
  2016
blanchet@48975
  2017
            fun mk_map_conjunct nths x =
blanchet@49501
  2018
              HOLogic.mk_eq (Term.list_comb (map_nth, passive_ids @ nths) $ FJz, dtor $ x);
blanchet@48975
  2019
blanchet@48975
  2020
            fun mk_set_conjunct set phi z1 z2 =
blanchet@48975
  2021
              list_all_free [z1, z2]
blanchet@48975
  2022
                (HOLogic.mk_imp (HOLogic.mk_mem (HOLogic.mk_prod (z1, z2), set $ FJz),
traytel@53105
  2023
                  phi $ z1 $ z2));
blanchet@48975
  2024
blanchet@48975
  2025
            val concl = list_exists_free [FJz] (HOLogic.mk_conj
blanchet@48975
  2026
              (Library.foldr1 HOLogic.mk_conj (map2 mk_map_conjunct [fstsTs, sndsTs] xs),
blanchet@48975
  2027
              Library.foldr1 HOLogic.mk_conj
blanchet@48975
  2028
                (map4 mk_set_conjunct (drop m sets) phis Jzs1 Jzs2)));
blanchet@48975
  2029
          in
blanchet@48975
  2030
            fold_rev Logic.all xs (Logic.mk_implies
blanchet@48975
  2031
              (HOLogic.mk_Trueprop (Term.list_comb (phi, xs)), HOLogic.mk_Trueprop concl))
blanchet@48975
  2032
          end;
blanchet@48975
  2033
traytel@53105
  2034
        val prems = map7 mk_prem phis dtors map_FT_nths prodFT_setss Jzs Jzs_copy FJzs;
blanchet@49591
  2035
blanchet@49591
  2036
        val dtor_map_coinduct_goal = fold_rev Logic.all frees (Logic.list_implies (prems, concl));
traytel@51893
  2037
        val dtor_map_coinduct =
traytel@51893
  2038
          Goal.prove_sorry lthy [] [] dtor_map_coinduct_goal
traytel@51893
  2039
            (K (mk_dtor_map_coinduct_tac m ks raw_coind_thm bis_def))
traytel@49109
  2040
          |> Thm.close_derivation;
blanchet@48975
  2041
      in
traytel@53105
  2042
        (dtor_map_coinduct, rev (Term.add_tfrees dtor_map_coinduct_goal []), dtor_coinduct)
blanchet@48975
  2043
      end;
blanchet@48975
  2044
blanchet@48975
  2045
    val timer = time (timer "coinduction");
blanchet@48975
  2046
traytel@54841
  2047
    val setss_by_bnf = map (fn i => map2 (mk_hset dtors i) ls passiveAs) ks;
traytel@54841
  2048
    val setss_by_range = transpose setss_by_bnf;
traytel@54841
  2049
traytel@54841
  2050
    val (hset_dtor_incl_thmss, hset_hset_dtor_incl_thmsss, dtor_hset_induct_thms) =
traytel@54841
  2051
      let
traytel@54841
  2052
        fun tinst_of dtor =
traytel@54841
  2053
          map (SOME o certify lthy) (dtor :: remove (op =) dtor dtors);
traytel@54841
  2054
        fun tinst_of' dtor = case tinst_of dtor of t :: ts => t :: NONE :: ts;
traytel@54841
  2055
        val Tinst = map (pairself (certifyT lthy))
traytel@54841
  2056
          (map Logic.varifyT_global (deads @ allAs) ~~ (deads @ passiveAs @ Ts));
traytel@54841
  2057
        val set_incl_thmss =
traytel@54841
  2058
          map2 (fn dtor => map (singleton (Proof_Context.export names_lthy lthy) o
traytel@54841
  2059
            Drule.instantiate' [] (tinst_of' dtor) o
traytel@54841
  2060
            Thm.instantiate (Tinst, []) o Drule.zero_var_indexes))
traytel@54841
  2061
          dtors set_incl_hset_thmss;
traytel@54841
  2062
traytel@54841
  2063
        val tinst = splice (map (SOME o certify lthy) dtors) (replicate n NONE)
traytel@54841
  2064
        val set_minimal_thms =
traytel@54841
  2065
          map (Drule.instantiate' [] tinst o Thm.instantiate (Tinst, []) o
traytel@54841
  2066
            Drule.zero_var_indexes)
traytel@54841
  2067
          hset_minimal_thms;
traytel@54841
  2068
traytel@54841
  2069
        val set_set_incl_thmsss =
traytel@54841
  2070
          map2 (fn dtor => map (map (singleton (Proof_Context.export names_lthy lthy) o
traytel@54841
  2071
            Drule.instantiate' [] (NONE :: tinst_of' dtor) o
traytel@54841
  2072
            Thm.instantiate (Tinst, []) o Drule.zero_var_indexes)))
traytel@54841
  2073
          dtors set_hset_incl_hset_thmsss;
traytel@54841
  2074
traytel@54841
  2075
        val set_set_incl_thmsss' = transpose (map transpose set_set_incl_thmsss);
traytel@54841
  2076
traytel@54841
  2077
        val incls =
traytel@54841
  2078
          maps (map (fn thm => thm RS @{thm subset_Collect_iff})) set_incl_thmss @
traytel@54841
  2079
            @{thms subset_Collect_iff[OF subset_refl]};
traytel@54841
  2080
traytel@54841
  2081
        fun mk_induct_tinst phis jsets y y' =
traytel@54841
  2082
          map4 (fn phi => fn jset => fn Jz => fn Jz' =>
traytel@54841
  2083
            SOME (certify lthy (Term.absfree Jz' (HOLogic.mk_Collect (fst y', snd y',
traytel@54841
  2084
              HOLogic.mk_conj (HOLogic.mk_mem (y, jset $ Jz), phi $ y $ Jz))))))
traytel@54841
  2085
          phis jsets Jzs Jzs';
traytel@54841
  2086
        val dtor_set_induct_thms =
traytel@54841
  2087
          map6 (fn set_minimal => fn set_set_inclss => fn jsets => fn y => fn y' => fn phis =>
traytel@54841
  2088
            ((set_minimal
traytel@54841
  2089
              |> Drule.instantiate' [] (mk_induct_tinst phis jsets y y')
traytel@54841
  2090
              |> unfold_thms lthy incls) OF
traytel@54841
  2091
              (replicate n ballI @
traytel@54841
  2092
                maps (map (fn thm => thm RS @{thm subset_CollectI})) set_set_inclss))
traytel@54841
  2093
            |> singleton (Proof_Context.export names_lthy lthy)
traytel@54841
  2094
            |> rule_by_tactic lthy (ALLGOALS (TRY o etac asm_rl)))
traytel@54841
  2095
          set_minimal_thms set_set_incl_thmsss' setss_by_range ys ys' dtor_set_induct_phiss
traytel@54841
  2096
      in
traytel@54841
  2097
        (set_incl_thmss, set_set_incl_thmsss, dtor_set_induct_thms)
traytel@54841
  2098
      end;
traytel@54841
  2099
blanchet@53270
  2100
    fun mk_dtor_map_DEADID_thm dtor_inject map_id0 =
blanchet@53270
  2101
      trans OF [iffD2 OF [dtor_inject, id_apply], map_id0 RS sym];
traytel@51925
  2102
traytel@51925
  2103
    fun mk_dtor_Jrel_DEADID_thm dtor_inject bnf =
traytel@51917
  2104
      trans OF [rel_eq_of_bnf bnf RS @{thm predicate2_eqD}, dtor_inject] RS sym;
traytel@51917
  2105
traytel@51925
  2106
    val JphiTs = map2 mk_pred2T passiveAs passiveBs;
traytel@54841
  2107
    val Jpsi1Ts = map2 mk_pred2T passiveAs passiveCs;
traytel@54841
  2108
    val Jpsi2Ts = map2 mk_pred2T passiveCs passiveBs;
traytel@51925
  2109
    val prodTsTs' = map2 (curry HOLogic.mk_prodT) Ts Ts';
traytel@51925
  2110
    val fstsTsTs' = map fst_const prodTsTs';
traytel@51925
  2111
    val sndsTsTs' = map snd_const prodTsTs';
traytel@52731
  2112
    val activephiTs = map2 mk_pred2T activeAs activeBs;
traytel@51925
  2113
    val activeJphiTs = map2 mk_pred2T Ts Ts';
traytel@54841
  2114
    val (((((Jphis, Jpsi1s), Jpsi2s), activephis), activeJphis), names_lthy) = names_lthy
traytel@51925
  2115
      |> mk_Frees "R" JphiTs
traytel@54841
  2116
      ||>> mk_Frees "R" Jpsi1Ts
traytel@54841
  2117
      ||>> mk_Frees "Q" Jpsi2Ts
traytel@52731
  2118
      ||>> mk_Frees "S" activephiTs
traytel@51925
  2119
      ||>> mk_Frees "JR" activeJphiTs;
traytel@51925
  2120
    val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
traytel@51925
  2121
    val in_rels = map in_rel_of_bnf bnfs;
traytel@51925
  2122
traytel@54841
  2123
    fun mk_Jrel_DEADID_coinduct_thm () = 
traytel@54841
  2124
      mk_rel_xtor_co_induct_thm Greatest_FP rels activeJphis (map HOLogic.eq_const Ts) Jphis
traytel@54841
  2125
        Jzs Jz's dtors dtor's (fn {context = ctxt, prems} =>
traytel@54841
  2126
          (unfold_thms_tac ctxt @{thms le_fun_def le_bool_def all_simps(1,2)[symmetric]} THEN
traytel@54841
  2127
          REPEAT_DETERM (rtac allI 1) THEN rtac (dtor_coinduct_thm OF prems) 1)) lthy;
traytel@54841
  2128
blanchet@48975
  2129
    (*register new codatatypes as BNFs*)
traytel@54841
  2130
    val (timer, Jbnfs, (dtor_Jmap_o_thms, dtor_Jmap_thms), dtor_Jset_thmss',
traytel@54841
  2131
      dtor_Jrel_thms, Jrel_coinduct_thm, Jbnf_notes, lthy) =
blanchet@49585
  2132
      if m = 0 then
traytel@52913
  2133
        (timer, replicate n DEADID_bnf,
blanchet@53285
  2134
        map_split (`(mk_pointfree lthy)) (map2 mk_dtor_map_DEADID_thm dtor_inject_thms map_ids),
traytel@54841
  2135
        replicate n [], map2 mk_dtor_Jrel_DEADID_thm dtor_inject_thms bnfs,
traytel@54841
  2136
        mk_Jrel_DEADID_coinduct_thm (), [], lthy)
blanchet@49585
  2137
      else let
blanchet@48975
  2138
        val fTs = map2 (curry op -->) passiveAs passiveBs;
blanchet@48975
  2139
        val gTs = map2 (curry op -->) passiveBs passiveCs;
blanchet@48975
  2140
        val uTs = map2 (curry op -->) Ts Ts';
traytel@54841
  2141
traytel@54841
  2142
        val ((((((((fs, fs'), fs_copy), gs), us), (Jys, Jys')), (Jys_copy, Jys'_copy)),
traytel@54841
  2143
          (ys_copy, ys'_copy)), names_lthy) = names_lthy
blanchet@48975
  2144
          |> mk_Frees' "f" fTs
blanchet@49308
  2145
          ||>> mk_Frees "f" fTs
blanchet@49308
  2146
          ||>> mk_Frees "g" gTs
blanchet@48975
  2147
          ||>> mk_Frees "u" uTs
blanchet@48975
  2148
          ||>> mk_Frees' "b" Ts'
blanchet@48975
  2149
          ||>> mk_Frees' "b" Ts'
blanchet@48975
  2150
          ||>> mk_Frees' "y" passiveAs;
blanchet@48975
  2151
blanchet@48975
  2152
        val map_FTFT's = map2 (fn Ds =>
blanchet@48975
  2153
          mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
blanchet@48975
  2154
blanchet@48975
  2155
        fun mk_maps ATs BTs Ts mk_T =
blanchet@48975
  2156
          map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ map mk_T Ts)) Dss bnfs;
blanchet@48975
  2157
        fun mk_Fmap mk_const fs Ts Fmap = Term.list_comb (Fmap, fs @ map mk_const Ts);
blanchet@49501
  2158
        fun mk_map mk_const mk_T Ts fs Ts' dtors mk_maps =
blanchet@49504
  2159
          mk_unfold Ts' (map2 (fn dtor => fn Fmap =>
blanchet@49501
  2160
            HOLogic.mk_comp (mk_Fmap mk_const fs Ts Fmap, dtor)) dtors (mk_maps Ts mk_T));
blanchet@48975
  2161
        val mk_map_id = mk_map HOLogic.id_const I;
blanchet@48975
  2162
        val mk_mapsAB = mk_maps passiveAs passiveBs;
blanchet@49501
  2163
        val fs_maps = map (mk_map_id Ts fs Ts' dtors mk_mapsAB) ks;
traytel@54841
  2164
traytel@54841
  2165
        val set_bss =
traytel@54841
  2166
          map (flat o map2 (fn B => fn b =>
traytel@54841
  2167
            if member (op =) resDs (TFree B) then [] else [b]) resBs) set_bss0;
blanchet@48975
  2168
traytel@49104
  2169
        fun close_wit I wit = (I, fold_rev Term.absfree (map (nth ys') I) wit);
traytel@49104
  2170
traytel@49104
  2171
        val all_unitTs = replicate live HOLogic.unitT;
traytel@49104
  2172
        val unitTs = replicate n HOLogic.unitT;
traytel@49104
  2173
        val unit_funs = replicate n (Term.absdummy HOLogic.unitT HOLogic.unit);
traytel@49104
  2174
        fun mk_map_args I =
traytel@49104
  2175
          map (fn i =>
traytel@49104
  2176
            if member (op =) I i then Term.absdummy HOLogic.unitT (nth ys i)
traytel@49104
  2177
            else mk_undefined (HOLogic.unitT --> nth passiveAs i))
traytel@49104
  2178
          (0 upto (m - 1));
traytel@49104
  2179
traytel@49104
  2180
        fun mk_nat_wit Ds bnf (I, wit) () =
traytel@49104
  2181
          let
traytel@49104
  2182
            val passiveI = filter (fn i => i < m) I;
traytel@49104
  2183
            val map_args = mk_map_args passiveI;
traytel@49104
  2184
          in
traytel@49104
  2185
            Term.absdummy HOLogic.unitT (Term.list_comb
traytel@49104
  2186
              (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $ wit)
traytel@49104
  2187
          end;
traytel@49104
  2188
traytel@49104
  2189
        fun mk_dummy_wit Ds bnf I =
traytel@49104
  2190
          let
traytel@49104
  2191
            val map_args = mk_map_args I;
traytel@49104
  2192
          in
traytel@49104
  2193
            Term.absdummy HOLogic.unitT (Term.list_comb
traytel@49104
  2194
              (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $
traytel@49104
  2195
              mk_undefined (mk_T_of_bnf Ds all_unitTs bnf))
traytel@49104
  2196
          end;
traytel@49104
  2197
traytel@49104
  2198
        val nat_witss =
blanchet@49308
  2199
          map2 (fn Ds => fn bnf => mk_wits_of_bnf (replicate (nwits_of_bnf bnf) Ds)
traytel@49104
  2200
            (replicate (nwits_of_bnf bnf) (replicate live HOLogic.unitT)) bnf
traytel@49104
  2201
            |> map (fn (I, wit) =>
traytel@49104
  2202
              (I, Lazy.lazy (mk_nat_wit Ds bnf (I, Term.list_comb (wit, map (K HOLogic.unit) I))))))
blanchet@49308
  2203
          Dss bnfs;
traytel@49104
  2204
traytel@49104
  2205
        val nat_wit_thmss = map2 (curry op ~~) nat_witss (map wit_thmss_of_bnf bnfs)
traytel@49104
  2206
traytel@49104
  2207
        val Iss = map (map fst) nat_witss;
traytel@49104
  2208
traytel@49104
  2209
        fun filter_wits (I, wit) =
traytel@49104
  2210
          let val J = filter (fn i => i < m) I;
traytel@49104
  2211
          in (J, (length J < length I, wit)) end;
traytel@49104
  2212
traytel@49104
  2213
        val wit_treess = map_index (fn (i, Is) =>
traytel@49104
  2214
          map_index (finish Iss m [i+m] (i+m)) Is) Iss
traytel@49104
  2215
          |> map (minimize_wits o map filter_wits o minimize_wits o flat);
traytel@49104
  2216
traytel@49104
  2217
        val coind_wit_argsss =
traytel@49104
  2218
          map (map (tree_to_coind_wits nat_wit_thmss o snd o snd) o filter (fst o snd)) wit_treess;
traytel@49104
  2219
traytel@49104
  2220
        val nonredundant_coind_wit_argsss =
traytel@49104
  2221
          fold (fn i => fn argsss =>
traytel@49104
  2222
            nth_map (i - 1) (filter_out (fn xs =>
traytel@49104
  2223
              exists (fn ys =>
traytel@49104
  2224
                let
traytel@49104
  2225
                  val xs' = (map (fst o fst) xs, snd (fst (hd xs)));
traytel@49104
  2226
                  val ys' = (map (fst o fst) ys, snd (fst (hd ys)));
traytel@49104
  2227
                in
traytel@49104
  2228
                  eq_pair (subset (op =)) (eq_set (op =)) (xs', ys') andalso not (fst xs' = fst ys')
traytel@49104
  2229
                end)
traytel@49104
  2230
              (flat argsss)))
traytel@49104
  2231
            argsss)
traytel@49104
  2232
          ks coind_wit_argsss;
traytel@49104
  2233
traytel@49104
  2234
        fun prepare_args args =
traytel@49104
  2235
          let
traytel@49104
  2236
            val I = snd (fst (hd args));
traytel@49104
  2237
            val (dummys, args') =
traytel@49104
  2238
              map_split (fn i =>
traytel@49104
  2239
                (case find_first (fn arg => fst (fst arg) = i - 1) args of
traytel@49104
  2240
                  SOME (_, ((_, wit), thms)) => (NONE, (Lazy.force wit, thms))
traytel@49104
  2241
                | NONE =>
traytel@49104
  2242
                  (SOME (i - 1), (mk_dummy_wit (nth Dss (i - 1)) (nth bnfs (i - 1)) I, []))))
traytel@49104
  2243
              ks;
traytel@49104
  2244
          in
traytel@49104
  2245
            ((I, dummys), apsnd flat (split_list args'))
traytel@49104
  2246
          end;
traytel@49104
  2247
traytel@49104
  2248
        fun mk_coind_wits ((I, dummys), (args, thms)) =
blanchet@49504
  2249
          ((I, dummys), (map (fn i => mk_unfold Ts args i $ HOLogic.unit) ks, thms));
traytel@49104
  2250
traytel@49104
  2251
        val coind_witss =
traytel@49104
  2252
          maps (map (mk_coind_wits o prepare_args)) nonredundant_coind_wit_argsss;
traytel@49104
  2253
traytel@54841
  2254
        val witss = map2 (fn Ds => fn bnf => mk_wits_of_bnf
traytel@54841
  2255
          (replicate (nwits_of_bnf bnf) Ds)
traytel@54841
  2256
          (replicate (nwits_of_bnf bnf) (passiveAs @ Ts)) bnf) Dss bnfs;
traytel@54841
  2257
traytel@54841
  2258
        val ctor_witss =
traytel@54841
  2259
          map (map (uncurry close_wit o tree_to_ctor_wit ys ctors witss o snd o snd) o
traytel@54841
  2260
            filter_out (fst o snd)) wit_treess;
traytel@54841
  2261
traytel@49104
  2262
        fun mk_coind_wit_thms ((I, dummys), (wits, wit_thms)) =
traytel@49104
  2263
          let
traytel@49104
  2264
            fun mk_goal sets y y_copy y'_copy j =
traytel@49104
  2265
              let
traytel@49104
  2266
                fun mk_conjunct set z dummy wit =
traytel@49104
  2267
                  mk_Ball (set $ z) (Term.absfree y'_copy
traytel@49104
  2268
                    (if dummy = NONE orelse member (op =) I (j - 1) then
traytel@49104
  2269
                      HOLogic.mk_imp (HOLogic.mk_eq (z, wit),
traytel@49104
  2270
                        if member (op =) I (j - 1) then HOLogic.mk_eq (y_copy, y)
traytel@49104
  2271
                        else @{term False})
traytel@49104
  2272
                    else @{term True}));
traytel@49104
  2273
              in
traytel@49104
  2274
                fold_rev Logic.all (map (nth ys) I @ Jzs) (HOLogic.mk_Trueprop
traytel@49104
  2275
                  (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct sets Jzs dummys wits)))
traytel@49104
  2276
              end;
traytel@49104
  2277
            val goals = map5 mk_goal setss_by_range ys ys_copy ys'_copy ls;
traytel@49104
  2278
          in
traytel@49104
  2279
            map2 (fn goal => fn induct =>