src/HOL/Tools/BNF/bnf_gfp.ML
author wenzelm
Tue Feb 10 14:48:26 2015 +0100 (2015-02-10)
changeset 59498 50b60f501b05
parent 58963 26bf09b95dda
child 59580 cbc38731d42f
permissions -rw-r--r--
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
occasionally clarified use of context;
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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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    BNF_Comp.absT_info list -> 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_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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    fun mk_internal_of_b name =
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      Binding.prefix_name (name ^ "_") #> Binding.prefix true b_name #> Binding.conceal;
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    fun mk_internal_b name = mk_internal_of_b name b;
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    fun mk_internal_bs name = map (mk_internal_of_b name) bs;
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    val external_bs = map2 (Binding.prefix false) b_names bs
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      |> not note_all ? map Binding.conceal;
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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 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 = @{map 4} mk_map_of_bnf Dss Ass Ass bnfs;
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    val mapsAsBs = @{map 4} mk_map_of_bnf Dss Ass Bss bnfs;
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    val mapsBsCs' = @{map 4} mk_map_of_bnf Dss Bss Css' bnfs;
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    val mapsAsCs' = @{map 4} mk_map_of_bnf Dss Ass Css' bnfs;
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    val map_Inls = @{map 4} mk_map_of_bnf Dss Bss (replicate n (passiveAs @ sumBsAs)) bnfs;
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    val map_Inls_rev = @{map 4} mk_map_of_bnf Dss (replicate n (passiveAs @ sumBsAs)) Bss bnfs;
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    val map_fsts = @{map 4} mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs;
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    val map_snds = @{map 4} mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs;
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    fun mk_setss Ts = @{map 3} 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 = @{map 4} mk_rel_of_bnf Dss Ass Bss bnfs;
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    val bds = @{map 3} 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 (sum_bdT_params, sum_bdT_params') = `(map TFree) (Term.add_tfreesT sum_bdT []);
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    val ((((((((((((((((((((((((((((((((zs, zs'), zs_copy), zs_copy2), z's), (ys, ys')),
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      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')),
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      (nat, nat')), Rs), Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), 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 "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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      ||>> 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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    val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
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    val passive_eqs = map HOLogic.eq_const passiveAs;
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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 le_rel_OOs = map le_rel_OO_of_bnf bnfs;
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    val in_rels = map in_rel_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 [] [] (mk_Trueprop_eq (lhs, rhs))
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          (fn {context = ctxt, prems = _} => mk_map_comp_id_tac ctxt map_comp0)
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        |> Thm.close_derivation
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        |> singleton (Proof_Context.export names_lthy lthy)
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      end;
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    val map_comp_id_thms = @{map 5} 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 = @{map 4} 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 [] [] (Logic.list_implies (prems, goal))
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          (K (mk_map_cong0L_tac m map_cong0 map_id))
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        |> Thm.close_derivation
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        |> singleton (Proof_Context.export names_lthy lthy)
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      end;
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    val map_cong0L_thms = @{map 5} mk_map_cong0L xFs mapsAsAs setssAs map_cong0s map_ids;
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    val in_mono'_thms = map (fn thm =>
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      (thm OF (replicate m subset_refl)) RS @{thm set_mp}) in_monos;
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   268
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   269
    val map_arg_cong_thms =
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   270
      let
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   271
        val prems = map2 (curry mk_Trueprop_eq) yFs yFs_copy;
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   272
        val maps = map (fn mapx => Term.list_comb (mapx, all_gs)) mapsBsCs';
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   273
        val concls =
wenzelm@58634
   274
          @{map 3} (fn x => fn y => fn mapx => mk_Trueprop_eq (mapx $ x, mapx $ y))
wenzelm@58634
   275
            yFs yFs_copy maps;
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   276
        val goals = map2 (fn prem => fn concl => Logic.mk_implies (prem, concl)) prems concls;
blanchet@48975
   277
      in
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   278
        map (fn goal =>
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   279
          Goal.prove_sorry lthy [] [] goal (K ((hyp_subst_tac lthy THEN' rtac refl) 1))
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   280
          |> Thm.close_derivation
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   281
          |> singleton (Proof_Context.export names_lthy lthy)) goals
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   282
      end;
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   283
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   284
    val timer = time (timer "Derived simple theorems");
blanchet@48975
   285
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   286
    (* coalgebra *)
blanchet@48975
   287
traytel@53566
   288
    val coalg_bind = mk_internal_b (coN ^ algN) ;
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   289
    val coalg_def_bind = (Thm.def_binding coalg_bind, []);
blanchet@48975
   290
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   291
    (*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in UNIV .. UNIV B1 ... Bn)*)
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   292
    val coalg_spec =
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   293
      let
wenzelm@58634
   294
        val ins = @{map 3} mk_in (replicate n (passive_UNIVs @ Bs)) setssAs FTsAs;
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   295
        fun mk_coalg_conjunct B s X z z' =
blanchet@48975
   296
          mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X)));
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   297
wenzelm@58634
   298
        val rhs = Library.foldr1 HOLogic.mk_conj (@{map 5} mk_coalg_conjunct Bs ss ins zs zs')
blanchet@48975
   299
      in
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   300
        fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss) rhs
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   301
      end;
blanchet@48975
   302
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   303
    val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) =
blanchet@49311
   304
      lthy
traytel@55204
   305
      |> Local_Theory.define ((coalg_bind, NoSyn), (coalg_def_bind, coalg_spec))
blanchet@49311
   306
      ||> `Local_Theory.restore;
blanchet@49311
   307
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   308
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   309
    val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free));
traytel@55541
   310
    val coalg_def = mk_unabs_def (2 * n) (Morphism.thm phi coalg_def_free RS meta_eq_to_obj_eq);
traytel@55541
   311
traytel@55541
   312
    fun mk_coalg Bs ss =
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   313
      let
traytel@55541
   314
        val args = Bs @ ss;
blanchet@48975
   315
        val Ts = map fastype_of args;
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   316
        val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   317
      in
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   318
        Term.list_comb (Const (coalg, coalgT), args)
blanchet@48975
   319
      end;
blanchet@48975
   320
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   321
    val coalg_prem = HOLogic.mk_Trueprop (mk_coalg Bs ss);
blanchet@48975
   322
blanchet@48975
   323
    val coalg_in_thms = map (fn i =>
traytel@52904
   324
      coalg_def RS iffD1 RS mk_conjunctN n i RS bspec) ks
blanchet@48975
   325
blanchet@48975
   326
    val coalg_set_thmss =
blanchet@48975
   327
      let
traytel@55541
   328
        val coalg_prem = HOLogic.mk_Trueprop (mk_coalg Bs ss);
desharna@57567
   329
        fun mk_prem x B = mk_Trueprop_mem (x, B);
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   330
        fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) B);
blanchet@48975
   331
        val prems = map2 mk_prem zs Bs;
wenzelm@58634
   332
        val conclss = @{map 3} (fn s => fn x => fn sets => map2 (mk_concl s x) Bs (drop m sets))
blanchet@48975
   333
          ss zs setssAs;
traytel@56272
   334
        val goalss = map2 (fn prem => fn concls => map (fn concl =>
traytel@56272
   335
          Logic.list_implies (coalg_prem :: [prem], concl)) concls) prems conclss;
blanchet@48975
   336
      in
traytel@56272
   337
        map (fn goals => map (fn goal =>
traytel@56272
   338
          Goal.prove_sorry lthy [] [] goal (K (mk_coalg_set_tac coalg_def))
traytel@56272
   339
          |> Thm.close_derivation
traytel@56272
   340
          |> singleton (Proof_Context.export names_lthy lthy)) goals) goalss
blanchet@48975
   341
      end;
blanchet@48975
   342
traytel@55541
   343
    fun mk_tcoalg BTs = mk_coalg (map HOLogic.mk_UNIV BTs);
blanchet@48975
   344
blanchet@48975
   345
    val tcoalg_thm =
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   346
      let
traytel@56272
   347
        val goal = HOLogic.mk_Trueprop (mk_tcoalg activeAs ss)
blanchet@48975
   348
      in
wenzelm@51551
   349
        Goal.prove_sorry lthy [] [] goal
traytel@56114
   350
          (K (rtac (coalg_def RS iffD2) 1 THEN CONJ_WRAP
blanchet@48975
   351
            (K (EVERY' [rtac ballI, rtac CollectI,
blanchet@48975
   352
              CONJ_WRAP' (K (EVERY' [rtac @{thm subset_UNIV}])) allAs] 1)) ss))
traytel@49109
   353
        |> Thm.close_derivation
traytel@56272
   354
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   355
      end;
blanchet@48975
   356
blanchet@48975
   357
    val timer = time (timer "Coalgebra definition & thms");
blanchet@48975
   358
blanchet@48975
   359
    (* morphism *)
blanchet@48975
   360
traytel@53566
   361
    val mor_bind = mk_internal_b morN;
blanchet@48975
   362
    val mor_def_bind = (Thm.def_binding mor_bind, []);
blanchet@48975
   363
blanchet@48975
   364
    (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*)
blanchet@48975
   365
    (*mor) forall i = 1 ... n: (\<forall>x \<in> Bi.
blanchet@48975
   366
       Fi_map id ... id f1 ... fn (si x) = si' (fi x)*)
blanchet@48975
   367
    val mor_spec =
blanchet@48975
   368
      let
blanchet@48975
   369
        fun mk_fbetw f B1 B2 z z' =
blanchet@48975
   370
          mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));
blanchet@48975
   371
        fun mk_mor B mapAsBs f s s' z z' =
blanchet@48975
   372
          mk_Ball B (Term.absfree z' (HOLogic.mk_eq
blanchet@48975
   373
            (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z))));
blanchet@48975
   374
        val rhs = HOLogic.mk_conj
wenzelm@58634
   375
          (Library.foldr1 HOLogic.mk_conj (@{map 5} mk_fbetw fs Bs B's zs zs'),
wenzelm@58634
   376
           Library.foldr1 HOLogic.mk_conj (@{map 7} mk_mor Bs mapsAsBs fs ss s's zs zs'))
blanchet@48975
   377
      in
traytel@55204
   378
        fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss @ B's @ s's @ fs) rhs
blanchet@48975
   379
      end;
blanchet@48975
   380
blanchet@48975
   381
    val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
blanchet@49311
   382
      lthy
traytel@55204
   383
      |> Local_Theory.define ((mor_bind, NoSyn), (mor_def_bind, mor_spec))
blanchet@49311
   384
      ||> `Local_Theory.restore;
blanchet@49311
   385
blanchet@48975
   386
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   387
    val mor = fst (Term.dest_Const (Morphism.term phi mor_free));
traytel@55204
   388
    val mor_def = mk_unabs_def (5 * n) (Morphism.thm phi mor_def_free RS meta_eq_to_obj_eq);
blanchet@48975
   389
blanchet@48975
   390
    fun mk_mor Bs1 ss1 Bs2 ss2 fs =
blanchet@48975
   391
      let
blanchet@48975
   392
        val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;
blanchet@48975
   393
        val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);
blanchet@48975
   394
        val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   395
      in
blanchet@48975
   396
        Term.list_comb (Const (mor, morT), args)
blanchet@48975
   397
      end;
blanchet@48975
   398
blanchet@48975
   399
    val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
blanchet@48975
   400
blanchet@48975
   401
    val (mor_image_thms, morE_thms) =
blanchet@48975
   402
      let
blanchet@48975
   403
        val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
traytel@56272
   404
        fun mk_image_goal f B1 B2 =
traytel@56272
   405
          Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_leq (mk_image f $ B1) B2));
wenzelm@58634
   406
        val image_goals = @{map 3} mk_image_goal fs Bs B's;
blanchet@48975
   407
        fun mk_elim_goal B mapAsBs f s s' x =
desharna@57567
   408
          Logic.list_implies ([prem, mk_Trueprop_mem (x, B)],
traytel@56272
   409
            mk_Trueprop_eq (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x)));
wenzelm@58634
   410
        val elim_goals = @{map 6} mk_elim_goal Bs mapsAsBs fs ss s's zs;
blanchet@48975
   411
        fun prove goal =
wenzelm@51551
   412
          Goal.prove_sorry lthy [] [] goal (K (mk_mor_elim_tac mor_def))
traytel@56272
   413
          |> Thm.close_derivation
traytel@56272
   414
          |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
   415
      in
blanchet@48975
   416
        (map prove image_goals, map prove elim_goals)
blanchet@48975
   417
      end;
blanchet@48975
   418
blanchet@48975
   419
    val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms;
blanchet@48975
   420
blanchet@48975
   421
    val mor_incl_thm =
blanchet@48975
   422
      let
traytel@51893
   423
        val prems = map2 (HOLogic.mk_Trueprop oo mk_leq) Bs Bs_copy;
blanchet@48975
   424
        val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
blanchet@48975
   425
      in
traytel@56272
   426
        Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
blanchet@53285
   427
          (K (mk_mor_incl_tac mor_def map_ids))
traytel@49109
   428
        |> Thm.close_derivation
traytel@56272
   429
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   430
      end;
blanchet@48975
   431
blanchet@48975
   432
    val mor_id_thm = mor_incl_thm OF (replicate n subset_refl);
blanchet@48975
   433
blanchet@48975
   434
    val mor_comp_thm =
blanchet@48975
   435
      let
blanchet@48975
   436
        val prems =
blanchet@48975
   437
          [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),
blanchet@48975
   438
           HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];
blanchet@48975
   439
        val concl =
blanchet@48975
   440
          HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));
blanchet@48975
   441
      in
traytel@56272
   442
        Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
wenzelm@59498
   443
          (K (mk_mor_comp_tac lthy mor_def mor_image'_thms morE_thms map_comp_id_thms))
traytel@49109
   444
        |> Thm.close_derivation
traytel@56272
   445
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   446
      end;
blanchet@48975
   447
blanchet@48975
   448
    val mor_cong_thm =
blanchet@48975
   449
      let
blanchet@48975
   450
        val prems = map HOLogic.mk_Trueprop
blanchet@48975
   451
         (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])
blanchet@48975
   452
        val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);
blanchet@48975
   453
      in
traytel@56272
   454
        Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
wenzelm@58963
   455
          (K ((hyp_subst_tac lthy THEN' assume_tac lthy) 1))
traytel@49109
   456
        |> Thm.close_derivation
traytel@56272
   457
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   458
      end;
blanchet@48975
   459
blanchet@48975
   460
    val mor_UNIV_thm =
blanchet@48975
   461
      let
blanchet@48975
   462
        fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
blanchet@48975
   463
            (HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s),
blanchet@48975
   464
            HOLogic.mk_comp (s', f));
blanchet@48975
   465
        val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;
wenzelm@58634
   466
        val rhs = Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct mapsAsBs fs ss s's);
blanchet@48975
   467
      in
traytel@56272
   468
        Goal.prove_sorry lthy [] [] (mk_Trueprop_eq (lhs, rhs))
blanchet@48975
   469
          (K (mk_mor_UNIV_tac morE_thms mor_def))
traytel@49109
   470
        |> Thm.close_derivation
traytel@56272
   471
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   472
      end;
blanchet@48975
   473
blanchet@48975
   474
    val mor_str_thm =
blanchet@48975
   475
      let
blanchet@48975
   476
        val maps = map2 (fn Ds => fn bnf => Term.list_comb
blanchet@48975
   477
          (mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs;
blanchet@48975
   478
      in
wenzelm@51551
   479
        Goal.prove_sorry lthy [] []
traytel@56272
   480
          (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss))
blanchet@48975
   481
          (K (mk_mor_str_tac ks mor_UNIV_thm))
traytel@49109
   482
        |> Thm.close_derivation
traytel@56272
   483
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   484
      end;
blanchet@48975
   485
blanchet@55414
   486
    val mor_case_sum_thm =
blanchet@48975
   487
      let
wenzelm@58634
   488
        val maps = @{map 3} (fn s => fn sum_s => fn mapx =>
blanchet@55414
   489
          mk_case_sum (HOLogic.mk_comp (Term.list_comb (mapx, passive_ids @ Inls), s), sum_s))
blanchet@48975
   490
          s's sum_ss map_Inls;
blanchet@48975
   491
      in
wenzelm@51551
   492
        Goal.prove_sorry lthy [] []
traytel@56272
   493
          (HOLogic.mk_Trueprop (mk_mor (map HOLogic.mk_UNIV activeBs) s's sum_UNIVs maps Inls))
blanchet@55414
   494
          (K (mk_mor_case_sum_tac ks mor_UNIV_thm))
traytel@49109
   495
        |> Thm.close_derivation
traytel@56272
   496
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   497
      end;
blanchet@48975
   498
blanchet@48975
   499
    val timer = time (timer "Morphism definition & thms");
blanchet@48975
   500
blanchet@48975
   501
    (* bisimulation *)
blanchet@48975
   502
traytel@53566
   503
    val bis_bind = mk_internal_b bisN;
blanchet@48975
   504
    val bis_def_bind = (Thm.def_binding bis_bind, []);
blanchet@48975
   505
traytel@51893
   506
    fun mk_bis_le_conjunct R B1 B2 = mk_leq R (mk_Times (B1, B2));
wenzelm@58634
   507
    val bis_le = Library.foldr1 HOLogic.mk_conj (@{map 3} mk_bis_le_conjunct Rs Bs B's)
blanchet@48975
   508
blanchet@48975
   509
    val bis_spec =
blanchet@48975
   510
      let
blanchet@48975
   511
        val fst_args = passive_ids @ fsts;
blanchet@48975
   512
        val snd_args = passive_ids @ snds;
blanchet@48975
   513
        fun mk_bis R s s' b1 b2 RF map1 map2 sets =
blanchet@48975
   514
          list_all_free [b1, b2] (HOLogic.mk_imp
blanchet@48975
   515
            (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
traytel@55541
   516
            mk_Bex (mk_in (passive_UNIVs @ Rs) sets (snd (dest_Free RF)))
traytel@55541
   517
              (Term.absfree (dest_Free RF) (HOLogic.mk_conj
blanchet@48975
   518
                (HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1),
blanchet@48975
   519
                HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2))))));
blanchet@48975
   520
blanchet@48975
   521
        val rhs = HOLogic.mk_conj
blanchet@48975
   522
          (bis_le, Library.foldr1 HOLogic.mk_conj
wenzelm@58634
   523
            (@{map 9} mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss))
blanchet@48975
   524
      in
traytel@55541
   525
        fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss @ B's @ s's @ Rs) rhs
blanchet@48975
   526
      end;
blanchet@48975
   527
blanchet@48975
   528
    val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) =
blanchet@49311
   529
      lthy
traytel@55204
   530
      |> Local_Theory.define ((bis_bind, NoSyn), (bis_def_bind, bis_spec))
blanchet@49311
   531
      ||> `Local_Theory.restore;
blanchet@49311
   532
blanchet@48975
   533
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   534
    val bis = fst (Term.dest_Const (Morphism.term phi bis_free));
traytel@55541
   535
    val bis_def = mk_unabs_def (5 * n) (Morphism.thm phi bis_def_free RS meta_eq_to_obj_eq);
traytel@55541
   536
traytel@55541
   537
    fun mk_bis Bs1 ss1 Bs2 ss2 Rs =
blanchet@48975
   538
      let
traytel@55541
   539
        val args = Bs1 @ ss1 @ Bs2 @ ss2 @ Rs;
blanchet@48975
   540
        val Ts = map fastype_of args;
blanchet@48975
   541
        val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   542
      in
blanchet@48975
   543
        Term.list_comb (Const (bis, bisT), args)
blanchet@48975
   544
      end;
blanchet@48975
   545
blanchet@48975
   546
    val bis_cong_thm =
blanchet@48975
   547
      let
blanchet@48975
   548
        val prems = map HOLogic.mk_Trueprop
traytel@55541
   549
         (mk_bis Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs)
traytel@55541
   550
        val concl = HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs_copy);
blanchet@48975
   551
      in
traytel@56272
   552
        Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
wenzelm@58963
   553
          (fn {context = ctxt, prems = _} => (hyp_subst_tac ctxt THEN' assume_tac ctxt) 1)
traytel@49109
   554
        |> Thm.close_derivation
traytel@56272
   555
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   556
      end;
blanchet@48975
   557
traytel@51893
   558
    val bis_rel_thm =
blanchet@48975
   559
      let
traytel@51893
   560
        fun mk_conjunct R s s' b1 b2 rel =
blanchet@48975
   561
          list_all_free [b1, b2] (HOLogic.mk_imp
blanchet@48975
   562
            (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
traytel@55541
   563
            Term.list_comb (rel, passive_eqs @ map mk_in_rel Rs) $ (s $ b1) $ (s' $ b2)));
blanchet@48975
   564
blanchet@48975
   565
        val rhs = HOLogic.mk_conj
blanchet@48975
   566
          (bis_le, Library.foldr1 HOLogic.mk_conj
wenzelm@58634
   567
            (@{map 6} mk_conjunct Rs ss s's zs z's relsAsBs))
blanchet@48975
   568
      in
traytel@56272
   569
        Goal.prove_sorry lthy [] [] (mk_Trueprop_eq (mk_bis Bs ss B's s's Rs, rhs))
traytel@56017
   570
          (K (mk_bis_rel_tac m bis_def in_rels map_comps map_cong0s set_mapss))
traytel@49109
   571
        |> Thm.close_derivation
traytel@56272
   572
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   573
      end;
blanchet@48975
   574
blanchet@48975
   575
    val bis_converse_thm =
wenzelm@51551
   576
      Goal.prove_sorry lthy [] []
traytel@56272
   577
        (Logic.mk_implies (HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs),
traytel@56272
   578
          HOLogic.mk_Trueprop (mk_bis B's s's Bs ss (map mk_converse Rs))))
traytel@51893
   579
        (K (mk_bis_converse_tac m bis_rel_thm rel_congs rel_converseps))
traytel@56272
   580
      |> Thm.close_derivation
traytel@56272
   581
      |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
   582
blanchet@48975
   583
    val bis_O_thm =
blanchet@48975
   584
      let
blanchet@48975
   585
        val prems =
traytel@55541
   586
          [HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs),
traytel@55541
   587
           HOLogic.mk_Trueprop (mk_bis B's s's B''s s''s R's)];
blanchet@48975
   588
        val concl =
traytel@55541
   589
          HOLogic.mk_Trueprop (mk_bis Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's));
blanchet@48975
   590
      in
traytel@56272
   591
        Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
traytel@57726
   592
          (K (mk_bis_O_tac lthy m bis_rel_thm rel_congs le_rel_OOs))
traytel@49109
   593
        |> Thm.close_derivation
traytel@56272
   594
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   595
      end;
blanchet@48975
   596
blanchet@48975
   597
    val bis_Gr_thm =
blanchet@48975
   598
      let
blanchet@48975
   599
        val concl =
traytel@55541
   600
          HOLogic.mk_Trueprop (mk_bis Bs ss B's s's (map2 mk_Gr Bs fs));
blanchet@48975
   601
      in
traytel@56272
   602
        Goal.prove_sorry lthy [] [] (Logic.list_implies ([coalg_prem, mor_prem], concl))
traytel@55197
   603
          (fn {context = ctxt, prems = _} => mk_bis_Gr_tac ctxt bis_rel_thm rel_Grps mor_image_thms
traytel@55197
   604
            morE_thms coalg_in_thms)
traytel@49109
   605
        |> Thm.close_derivation
traytel@56272
   606
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   607
      end;
blanchet@48975
   608
blanchet@48975
   609
    val bis_image2_thm = bis_cong_thm OF
blanchet@48975
   610
      ((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) ::
blanchet@48975
   611
      replicate n @{thm image2_Gr});
blanchet@48975
   612
traytel@51447
   613
    val bis_Id_on_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) ::
traytel@51447
   614
      replicate n @{thm Id_on_Gr});
blanchet@48975
   615
blanchet@48975
   616
    val bis_Union_thm =
blanchet@48975
   617
      let
blanchet@48975
   618
        val prem =
blanchet@48975
   619
          HOLogic.mk_Trueprop (mk_Ball Idx
traytel@55541
   620
            (Term.absfree idx' (mk_bis Bs ss B's s's (map (fn R => R $ idx) Ris))));
blanchet@48975
   621
        val concl =
traytel@55541
   622
          HOLogic.mk_Trueprop (mk_bis Bs ss B's s's (map (mk_UNION Idx) Ris));
blanchet@48975
   623
      in
traytel@56272
   624
        Goal.prove_sorry lthy [] [] (Logic.mk_implies (prem, concl))
traytel@55197
   625
          (fn {context = ctxt, prems = _} => mk_bis_Union_tac ctxt bis_def in_mono'_thms)
traytel@49109
   626
        |> Thm.close_derivation
traytel@56272
   627
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   628
      end;
blanchet@48975
   629
blanchet@48975
   630
    (* self-bisimulation *)
blanchet@48975
   631
traytel@55541
   632
    fun mk_sbis Bs ss Rs = mk_bis Bs ss Bs ss Rs;
traytel@55541
   633
traytel@55541
   634
    val sbis_prem = HOLogic.mk_Trueprop (mk_sbis Bs ss sRs);
blanchet@48975
   635
blanchet@48975
   636
    (* largest self-bisimulation *)
blanchet@48975
   637
traytel@53566
   638
    val lsbis_binds = mk_internal_bs lsbisN;
traytel@53566
   639
    fun lsbis_bind i = nth lsbis_binds (i - 1);
blanchet@48975
   640
    val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind;
blanchet@48975
   641
blanchet@48975
   642
    val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs
traytel@55541
   643
      (HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis Bs ss sRs)));
blanchet@48975
   644
traytel@55204
   645
    fun lsbis_spec i =
traytel@55541
   646
      fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss)
traytel@55204
   647
        (mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i)));
blanchet@48975
   648
blanchet@48975
   649
    val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   650
      lthy
traytel@55204
   651
      |> fold_map (fn i => Local_Theory.define
traytel@55204
   652
        ((lsbis_bind i, NoSyn), (lsbis_def_bind i, lsbis_spec i))) ks
blanchet@48975
   653
      |>> apsnd split_list o split_list
blanchet@48975
   654
      ||> `Local_Theory.restore;
blanchet@48975
   655
blanchet@48975
   656
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   657
traytel@55204
   658
    val lsbis_defs = map (fn def =>
traytel@55541
   659
      mk_unabs_def (2 * n) (Morphism.thm phi def RS meta_eq_to_obj_eq)) lsbis_def_frees;
blanchet@48975
   660
    val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees;
blanchet@48975
   661
traytel@55541
   662
    fun mk_lsbis Bs ss i =
blanchet@48975
   663
      let
traytel@55541
   664
        val args = Bs @ ss;
blanchet@48975
   665
        val Ts = map fastype_of args;
blanchet@48975
   666
        val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1)))));
blanchet@48975
   667
        val lsbisT = Library.foldr (op -->) (Ts, RT);
blanchet@48975
   668
      in
blanchet@48975
   669
        Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args)
blanchet@48975
   670
      end;
blanchet@48975
   671
blanchet@48975
   672
    val sbis_lsbis_thm =
traytel@56272
   673
      Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop (mk_sbis Bs ss (map (mk_lsbis Bs ss) ks)))
wenzelm@51798
   674
        (K (mk_sbis_lsbis_tac lthy lsbis_defs bis_Union_thm bis_cong_thm))
traytel@56272
   675
      |> Thm.close_derivation
traytel@56272
   676
      |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
   677
blanchet@48975
   678
    val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS
traytel@52904
   679
      (bis_def RS iffD1 RS conjunct1 RS mk_conjunctN n i)) ks;
blanchet@48975
   680
    val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS
traytel@52904
   681
      (bis_def RS iffD1 RS conjunct2 RS mk_conjunctN n i))) RS mp) ks;
blanchet@48975
   682
blanchet@48975
   683
    val incl_lsbis_thms =
blanchet@48975
   684
      let
traytel@55541
   685
        fun mk_concl i R = HOLogic.mk_Trueprop (mk_leq R (mk_lsbis Bs ss i));
traytel@56272
   686
        val goals = map2 (fn i => fn R => Logic.mk_implies (sbis_prem, mk_concl i R)) ks sRs;
blanchet@48975
   687
      in
wenzelm@58634
   688
        @{map 3} (fn goal => fn i => fn def =>
traytel@56272
   689
          Goal.prove_sorry lthy [] [] goal (K (mk_incl_lsbis_tac n i def))
traytel@56272
   690
          |> Thm.close_derivation
traytel@56272
   691
          |> singleton (Proof_Context.export names_lthy lthy)) goals ks lsbis_defs
blanchet@48975
   692
      end;
blanchet@48975
   693
blanchet@48975
   694
    val equiv_lsbis_thms =
blanchet@48975
   695
      let
traytel@55541
   696
        fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis Bs ss i));
traytel@56272
   697
        val goals = map2 (fn i => fn B => Logic.mk_implies (coalg_prem, mk_concl i B)) ks Bs;
blanchet@48975
   698
      in
wenzelm@58634
   699
        @{map 3} (fn goal => fn l_incl => fn incl_l =>
wenzelm@51551
   700
          Goal.prove_sorry lthy [] [] goal
blanchet@48975
   701
            (K (mk_equiv_lsbis_tac sbis_lsbis_thm l_incl incl_l
traytel@51447
   702
              bis_Id_on_thm bis_converse_thm bis_O_thm))
traytel@56272
   703
          |> Thm.close_derivation
traytel@56272
   704
          |> singleton (Proof_Context.export names_lthy lthy))
blanchet@48975
   705
        goals lsbis_incl_thms incl_lsbis_thms
blanchet@48975
   706
      end;
blanchet@48975
   707
blanchet@48975
   708
    val timer = time (timer "Bisimulations");
blanchet@48975
   709
blanchet@48975
   710
    (* bounds *)
blanchet@48975
   711
blanchet@48975
   712
    val (lthy, sbd, sbdT,
traytel@52635
   713
      sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss) =
blanchet@48975
   714
      if n = 1
traytel@52635
   715
      then (lthy, sum_bd, sum_bdT, bd_card_order, bd_Cinfinite, bd_Card_order, set_bdss)
blanchet@48975
   716
      else
blanchet@48975
   717
        let
traytel@53566
   718
          val sbdT_bind = mk_internal_b sum_bdTN;
blanchet@48975
   719
blanchet@48975
   720
          val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) =
traytel@56516
   721
            typedef (sbdT_bind, sum_bdT_params', NoSyn)
wenzelm@58959
   722
              (HOLogic.mk_UNIV sum_bdT) NONE (fn _ => EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
blanchet@48975
   723
traytel@56516
   724
          val sbdT = Type (sbdT_name, sum_bdT_params);
blanchet@48975
   725
          val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT);
blanchet@48975
   726
traytel@53566
   727
          val sbd_bind = mk_internal_b sum_bdN;
blanchet@48975
   728
          val sbd_def_bind = (Thm.def_binding sbd_bind, []);
blanchet@48975
   729
traytel@55204
   730
          val sbd_spec = mk_dir_image sum_bd Abs_sbdT;
blanchet@48975
   731
blanchet@48975
   732
          val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) =
blanchet@48975
   733
            lthy
traytel@55204
   734
            |> Local_Theory.define ((sbd_bind, NoSyn), (sbd_def_bind, sbd_spec))
blanchet@48975
   735
            ||> `Local_Theory.restore;
blanchet@48975
   736
blanchet@48975
   737
          val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   738
traytel@55204
   739
          val sbd_def = Morphism.thm phi sbd_def_free RS meta_eq_to_obj_eq;
blanchet@48975
   740
          val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT));
blanchet@48975
   741
traytel@49228
   742
          val Abs_sbdT_inj = mk_Abs_inj_thm (#Abs_inject sbdT_loc_info);
traytel@49228
   743
          val Abs_sbdT_bij = mk_Abs_bij_thm lthy Abs_sbdT_inj (#Abs_cases sbdT_loc_info);
blanchet@48975
   744
blanchet@48975
   745
          val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;
blanchet@48975
   746
          val sum_Card_order = sum_Cinfinite RS conjunct2;
blanchet@48975
   747
          val sum_card_order = mk_sum_card_order bd_card_orders;
blanchet@48975
   748
traytel@55770
   749
          val sbd_ordIso = @{thm ssubst_Pair_rhs} OF
traytel@55770
   750
            [@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order], sbd_def];
traytel@55770
   751
          val sbd_card_order = @{thm iffD2[OF arg_cong[of _ _ card_order]]} OF
traytel@55770
   752
            [sbd_def, @{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]];
blanchet@48975
   753
          val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite];
blanchet@48975
   754
          val sbd_Card_order = sbd_Cinfinite RS conjunct2;
blanchet@48975
   755
blanchet@48975
   756
          fun mk_set_sbd i bd_Card_order bds =
blanchet@48975
   757
            map (fn thm => @{thm ordLeq_ordIso_trans} OF
blanchet@48975
   758
              [bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds;
wenzelm@58634
   759
          val set_sbdss = @{map 3} mk_set_sbd ks bd_Card_orders set_bdss;
blanchet@48975
   760
       in
traytel@52635
   761
         (lthy, sbd, sbdT, sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss)
blanchet@48975
   762
       end;
blanchet@48975
   763
blanchet@48975
   764
    val sbdTs = replicate n sbdT;
blanchet@48975
   765
    val sum_sbdT = mk_sumTN sbdTs;
blanchet@48975
   766
    val sum_sbd_listT = HOLogic.listT sum_sbdT;
blanchet@48975
   767
    val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT;
blanchet@48975
   768
    val bdTs = passiveAs @ replicate n sbdT;
wenzelm@58634
   769
    val to_sbd_maps = @{map 4} mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs;
blanchet@48975
   770
    val bdFTs = mk_FTs bdTs;
blanchet@48975
   771
    val sbdFT = mk_sumTN bdFTs;
blanchet@48975
   772
    val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT);
blanchet@48975
   773
    val treeQT = HOLogic.mk_setT treeT;
blanchet@48975
   774
    val treeTs = passiveAs @ replicate n treeT;
blanchet@48975
   775
    val treeQTs = passiveAs @ replicate n treeQT;
blanchet@48975
   776
    val treeFTs = mk_FTs treeTs;
wenzelm@58634
   777
    val tree_maps = @{map 4} mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs;
wenzelm@58634
   778
    val final_maps = @{map 4} mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs;
blanchet@48975
   779
    val isNode_setss = mk_setss (passiveAs @ replicate n sbdT);
blanchet@48975
   780
blanchet@48975
   781
    val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []];
blanchet@48975
   782
    val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs);
blanchet@48975
   783
    val Lev_recT = fastype_of Zero;
blanchet@48975
   784
wenzelm@58634
   785
    val Nil = HOLogic.mk_tuple (@{map 3} (fn i => fn z => fn z'=>
blanchet@48975
   786
      Term.absfree z' (mk_InN activeAs z i)) ks zs zs');
blanchet@48975
   787
    val rv_recT = fastype_of Nil;
blanchet@48975
   788
blanchet@48975
   789
    val (((((((((((sumx, sumx'), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')),
blanchet@48975
   790
      (lab, lab')), (Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')),
blanchet@48975
   791
      names_lthy) = names_lthy
blanchet@48975
   792
      |> yield_singleton (apfst (op ~~) oo mk_Frees' "sumx") sum_sbdT
blanchet@48975
   793
      ||>> mk_Frees' "k" sbdTs
blanchet@48975
   794
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
blanchet@48975
   795
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
blanchet@48975
   796
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl") sum_sbd_list_setT
blanchet@48975
   797
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT)
blanchet@48975
   798
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl_lab") treeT
blanchet@48975
   799
      ||>> mk_Frees "x" bdFTs
blanchet@48975
   800
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") Lev_recT
blanchet@48975
   801
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") rv_recT;
blanchet@48975
   802
blanchet@48975
   803
    val (k, k') = (hd kks, hd kks')
blanchet@48975
   804
blanchet@48975
   805
    val timer = time (timer "Bounds");
blanchet@48975
   806
blanchet@48975
   807
    (* tree coalgebra *)
blanchet@48975
   808
traytel@53566
   809
    val isNode_binds = mk_internal_bs isNodeN;
traytel@53566
   810
    fun isNode_bind i = nth isNode_binds (i - 1);
blanchet@48975
   811
    val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind;
blanchet@48975
   812
blanchet@48975
   813
    val isNodeT =
traytel@55541
   814
      Library.foldr (op -->) (map fastype_of [Kl, lab, kl], HOLogic.boolT);
blanchet@48975
   815
wenzelm@58634
   816
    val Succs = @{map 3} (fn i => fn k => fn k' =>
blanchet@48975
   817
      HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl)))
blanchet@48975
   818
      ks kks kks';
blanchet@48975
   819
blanchet@48975
   820
    fun isNode_spec sets x i =
blanchet@48975
   821
      let
traytel@55541
   822
        val active_sets = drop m (map (fn set => set $ x) sets);
blanchet@48975
   823
        val rhs = list_exists_free [x]
blanchet@48975
   824
          (Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) ::
traytel@55541
   825
          map2 (curry HOLogic.mk_eq) active_sets Succs));
blanchet@48975
   826
      in
traytel@55541
   827
        fold_rev (Term.absfree o Term.dest_Free) [Kl, lab, kl] rhs
blanchet@48975
   828
      end;
blanchet@48975
   829
blanchet@48975
   830
    val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   831
      lthy
wenzelm@58634
   832
      |> @{fold_map 3} (fn i => fn x => fn sets => Local_Theory.define
traytel@55204
   833
        ((isNode_bind i, NoSyn), (isNode_def_bind i, isNode_spec sets x i)))
blanchet@48975
   834
        ks xs isNode_setss
blanchet@48975
   835
      |>> apsnd split_list o split_list
blanchet@48975
   836
      ||> `Local_Theory.restore;
blanchet@48975
   837
blanchet@48975
   838
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   839
traytel@55204
   840
    val isNode_defs = map (fn def =>
traytel@55541
   841
      mk_unabs_def 3 (Morphism.thm phi def RS meta_eq_to_obj_eq)) isNode_def_frees;
blanchet@48975
   842
    val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees;
blanchet@48975
   843
traytel@55541
   844
    fun mk_isNode kl i =
traytel@55541
   845
      Term.list_comb (Const (nth isNodes (i - 1), isNodeT), [Kl, lab, kl]);
blanchet@48975
   846
blanchet@48975
   847
    val isTree =
blanchet@48975
   848
      let
blanchet@48975
   849
        val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl);
blanchet@48975
   850
blanchet@48975
   851
        val tree = mk_Ball Kl (Term.absfree kl'
wenzelm@58634
   852
          (Library.foldr1 HOLogic.mk_conj (@{map 4} (fn Succ => fn i => fn k => fn k' =>
traytel@55581
   853
            mk_Ball Succ (Term.absfree k' (mk_isNode
traytel@55581
   854
              (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i)))
traytel@55581
   855
          Succs ks kks kks')));
blanchet@48975
   856
      in
traytel@55577
   857
        HOLogic.mk_conj (empty, tree)
blanchet@48975
   858
      end;
blanchet@48975
   859
traytel@53566
   860
    val carT_binds = mk_internal_bs carTN;
traytel@53566
   861
    fun carT_bind i = nth carT_binds (i - 1);
blanchet@48975
   862
    val carT_def_bind = rpair [] o Thm.def_binding o carT_bind;
blanchet@48975
   863
blanchet@48975
   864
    fun carT_spec i =
traytel@55541
   865
      HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
traytel@55541
   866
        (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)),
traytel@55541
   867
          HOLogic.mk_conj (isTree, mk_isNode (HOLogic.mk_list sum_sbdT []) i))));
blanchet@48975
   868
blanchet@48975
   869
    val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   870
      lthy
traytel@55204
   871
      |> fold_map (fn i =>
traytel@55204
   872
        Local_Theory.define ((carT_bind i, NoSyn), (carT_def_bind i, carT_spec i))) ks
blanchet@48975
   873
      |>> apsnd split_list o split_list
blanchet@48975
   874
      ||> `Local_Theory.restore;
blanchet@48975
   875
blanchet@48975
   876
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   877
traytel@55541
   878
    val carT_defs = map (fn def => Morphism.thm phi def RS meta_eq_to_obj_eq) carT_def_frees;
blanchet@48975
   879
    val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees;
blanchet@48975
   880
traytel@55541
   881
    fun mk_carT i = Const (nth carTs (i - 1), HOLogic.mk_setT treeT);
blanchet@48975
   882
traytel@53566
   883
    val strT_binds = mk_internal_bs strTN;
traytel@53566
   884
    fun strT_bind i = nth strT_binds (i - 1);
blanchet@48975
   885
    val strT_def_bind = rpair [] o Thm.def_binding o strT_bind;
blanchet@48975
   886
blanchet@48975
   887
    fun strT_spec mapFT FT i =
blanchet@48975
   888
      let
blanchet@48975
   889
        fun mk_f i k k' =
blanchet@48975
   890
          let val in_k = mk_InN sbdTs k i;
blanchet@48975
   891
          in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end;
blanchet@48975
   892
wenzelm@58634
   893
        val f = Term.list_comb (mapFT, passive_ids @ @{map 3} mk_f ks kks kks');
blanchet@48975
   894
        val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs));
blanchet@48975
   895
        val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2);
blanchet@48975
   896
      in
traytel@55204
   897
        HOLogic.mk_split (Term.absfree Kl' (Term.absfree lab'
blanchet@55414
   898
          (mk_case_sumN fs $ (lab $ HOLogic.mk_list sum_sbdT []))))
blanchet@48975
   899
      end;
blanchet@48975
   900
blanchet@48975
   901
    val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   902
      lthy
wenzelm@58634
   903
      |> @{fold_map 3} (fn i => fn mapFT => fn FT => Local_Theory.define
traytel@55204
   904
        ((strT_bind i, NoSyn), (strT_def_bind i, strT_spec mapFT FT i)))
blanchet@48975
   905
        ks tree_maps treeFTs
blanchet@48975
   906
      |>> apsnd split_list o split_list
blanchet@48975
   907
      ||> `Local_Theory.restore;
blanchet@48975
   908
blanchet@48975
   909
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   910
traytel@55204
   911
    val strT_defs = map (fn def =>
blanchet@55642
   912
        trans OF [Morphism.thm phi def RS meta_eq_to_obj_eq RS fun_cong, @{thm prod.case}])
traytel@55204
   913
      strT_def_frees;
blanchet@48975
   914
    val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees;
blanchet@48975
   915
blanchet@48975
   916
    fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT);
blanchet@48975
   917
traytel@55541
   918
    val carTAs = map mk_carT ks;
blanchet@48975
   919
    val strTAs = map2 mk_strT treeFTs ks;
blanchet@48975
   920
blanchet@48975
   921
    val coalgT_thm =
traytel@55541
   922
      Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop (mk_coalg carTAs strTAs))
traytel@55197
   923
        (fn {context = ctxt, prems = _} => mk_coalgT_tac ctxt m
traytel@55197
   924
          (coalg_def :: isNode_defs @ carT_defs) strT_defs set_mapss)
traytel@49109
   925
      |> Thm.close_derivation;
blanchet@48975
   926
blanchet@48975
   927
    val timer = time (timer "Tree coalgebra");
blanchet@48975
   928
blanchet@48975
   929
    fun mk_to_sbd s x i i' =
blanchet@48975
   930
      mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
blanchet@48975
   931
    fun mk_from_sbd s x i i' =
blanchet@48975
   932
      mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
blanchet@48975
   933
blanchet@48975
   934
    fun mk_to_sbd_thmss thm = map (map (fn set_sbd =>
blanchet@48975
   935
      thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss;
blanchet@48975
   936
blanchet@48975
   937
    val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj};
blanchet@48975
   938
    val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard};
blanchet@48975
   939
traytel@53566
   940
    val Lev_bind = mk_internal_b LevN;
blanchet@48975
   941
    val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind);
blanchet@48975
   942
blanchet@48975
   943
    val Lev_spec =
blanchet@48975
   944
      let
blanchet@48975
   945
        fun mk_Suc i s setsAs a a' =
blanchet@48975
   946
          let
blanchet@48975
   947
            val sets = drop m setsAs;
blanchet@48975
   948
            fun mk_set i' set b =
blanchet@48975
   949
              let
blanchet@48975
   950
                val Cons = HOLogic.mk_eq (kl_copy,
blanchet@48975
   951
                  mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl)
blanchet@48975
   952
                val b_set = HOLogic.mk_mem (b, set $ (s $ a));
blanchet@48975
   953
                val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b);
blanchet@48975
   954
              in
blanchet@48975
   955
                HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl]
blanchet@48975
   956
                  (HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec))))
blanchet@48975
   957
              end;
blanchet@48975
   958
          in
wenzelm@58634
   959
            Term.absfree a' (Library.foldl1 mk_union (@{map 3} mk_set ks sets zs_copy))
blanchet@48975
   960
          end;
blanchet@48975
   961
blanchet@48975
   962
        val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec'
wenzelm@58634
   963
          (HOLogic.mk_tuple (@{map 5} mk_Suc ks ss setssAs zs zs')));
blanchet@48975
   964
blanchet@55415
   965
        val rhs = mk_rec_nat Zero Suc;
blanchet@48975
   966
      in
traytel@55204
   967
        fold_rev (Term.absfree o Term.dest_Free) ss rhs
blanchet@48975
   968
      end;
blanchet@48975
   969
blanchet@48975
   970
    val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) =
blanchet@48975
   971
      lthy
traytel@55204
   972
      |> Local_Theory.define ((Lev_bind, NoSyn), (Lev_def_bind, Lev_spec))
blanchet@48975
   973
      ||> `Local_Theory.restore;
blanchet@48975
   974
blanchet@48975
   975
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   976
traytel@55204
   977
    val Lev_def = mk_unabs_def n (Morphism.thm phi Lev_def_free RS meta_eq_to_obj_eq);
blanchet@48975
   978
    val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free));
blanchet@48975
   979
blanchet@48975
   980
    fun mk_Lev ss nat i =
blanchet@48975
   981
      let
blanchet@48975
   982
        val Ts = map fastype_of ss;
blanchet@48975
   983
        val LevT = Library.foldr (op -->) (Ts, HOLogic.natT -->
blanchet@48975
   984
          HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts));
blanchet@48975
   985
      in
blanchet@48975
   986
        mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i
blanchet@48975
   987
      end;
blanchet@48975
   988
blanchet@55416
   989
    val Lev_0s = flat (mk_rec_simps n @{thm rec_nat_0_imp} [Lev_def]);
blanchet@55416
   990
    val Lev_Sucs = flat (mk_rec_simps n @{thm rec_nat_Suc_imp} [Lev_def]);
blanchet@48975
   991
traytel@53566
   992
    val rv_bind = mk_internal_b rvN;
blanchet@48975
   993
    val rv_def_bind = rpair [] (Thm.def_binding rv_bind);
blanchet@48975
   994
blanchet@48975
   995
    val rv_spec =
blanchet@48975
   996
      let
blanchet@48975
   997
        fun mk_Cons i s b b' =
blanchet@48975
   998
          let
blanchet@48975
   999
            fun mk_case i' =
blanchet@48975
  1000
              Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k));
blanchet@48975
  1001
          in
blanchet@55414
  1002
            Term.absfree b' (mk_case_sumN (map mk_case ks) $ sumx)
blanchet@48975
  1003
          end;
blanchet@48975
  1004
blanchet@48975
  1005
        val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec'
wenzelm@58634
  1006
          (HOLogic.mk_tuple (@{map 4} mk_Cons ks ss zs zs'))));
blanchet@48975
  1007
blanchet@55413
  1008
        val rhs = mk_rec_list Nil Cons;
blanchet@48975
  1009
      in
traytel@55204
  1010
        fold_rev (Term.absfree o Term.dest_Free) ss rhs
blanchet@48975
  1011
      end;
blanchet@48975
  1012
blanchet@48975
  1013
    val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) =
blanchet@48975
  1014
      lthy
traytel@55204
  1015
      |> Local_Theory.define ((rv_bind, NoSyn), (rv_def_bind, rv_spec))
blanchet@48975
  1016
      ||> `Local_Theory.restore;
blanchet@48975
  1017
blanchet@48975
  1018
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1019
traytel@55204
  1020
    val rv_def = mk_unabs_def n (Morphism.thm phi rv_def_free RS meta_eq_to_obj_eq);
blanchet@48975
  1021
    val rv = fst (Term.dest_Const (Morphism.term phi rv_free));
blanchet@48975
  1022
blanchet@48975
  1023
    fun mk_rv ss kl i =
blanchet@48975
  1024
      let
blanchet@48975
  1025
        val Ts = map fastype_of ss;
blanchet@48975
  1026
        val As = map domain_type Ts;
blanchet@48975
  1027
        val rvT = Library.foldr (op -->) (Ts, fastype_of kl -->
blanchet@48975
  1028
          HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As));
blanchet@48975
  1029
      in
blanchet@48975
  1030
        mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i
blanchet@48975
  1031
      end;
blanchet@48975
  1032
blanchet@55413
  1033
    val rv_Nils = flat (mk_rec_simps n @{thm rec_list_Nil_imp} [rv_def]);
blanchet@55413
  1034
    val rv_Conss = flat (mk_rec_simps n @{thm rec_list_Cons_imp} [rv_def]);
blanchet@48975
  1035
traytel@53566
  1036
    val beh_binds = mk_internal_bs behN;
traytel@53566
  1037
    fun beh_bind i = nth beh_binds (i - 1);
blanchet@48975
  1038
    val beh_def_bind = rpair [] o Thm.def_binding o beh_bind;
blanchet@48975
  1039
blanchet@48975
  1040
    fun beh_spec i z =
blanchet@48975
  1041
      let
blanchet@48975
  1042
        fun mk_case i to_sbd_map s k k' =
blanchet@48975
  1043
          Term.absfree k' (mk_InN bdFTs
blanchet@48975
  1044
            (Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i);
blanchet@48975
  1045
traytel@55577
  1046
        val Lab = Term.absfree kl'
wenzelm@58634
  1047
          (mk_case_sumN (@{map 5} mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z));
blanchet@48975
  1048
blanchet@48975
  1049
        val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
blanchet@48975
  1050
          (Term.absfree nat' (mk_Lev ss nat i $ z)), Lab);
blanchet@48975
  1051
      in
traytel@55204
  1052
        fold_rev (Term.absfree o Term.dest_Free) (ss @ [z]) rhs
blanchet@48975
  1053
      end;
blanchet@48975
  1054
blanchet@48975
  1055
    val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1056
      lthy
wenzelm@58634
  1057
      |> @{fold_map 2} (fn i => fn z =>
traytel@55204
  1058
        Local_Theory.define ((beh_bind i, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs
blanchet@48975
  1059
      |>> apsnd split_list o split_list
blanchet@48975
  1060
      ||> `Local_Theory.restore;
blanchet@48975
  1061
blanchet@48975
  1062
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1063
traytel@55204
  1064
    val beh_defs = map (fn def =>
traytel@55204
  1065
      mk_unabs_def (n + 1) (Morphism.thm phi def RS meta_eq_to_obj_eq)) beh_def_frees;
blanchet@48975
  1066
    val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees;
blanchet@48975
  1067
blanchet@48975
  1068
    fun mk_beh ss i =
blanchet@48975
  1069
      let
blanchet@48975
  1070
        val Ts = map fastype_of ss;
blanchet@48975
  1071
        val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT);
blanchet@48975
  1072
      in
blanchet@48975
  1073
        Term.list_comb (Const (nth behs (i - 1), behT), ss)
blanchet@48975
  1074
      end;
blanchet@48975
  1075
blanchet@48975
  1076
    val (length_Lev_thms, length_Lev'_thms) =
blanchet@48975
  1077
      let
blanchet@48975
  1078
        fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1079
          HOLogic.mk_eq (mk_size kl, nat));
blanchet@48975
  1080
        val goal = list_all_free (kl :: zs)
blanchet@48975
  1081
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1082
blanchet@48975
  1083
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1084
traytel@56272
  1085
        val length_Lev =
traytel@56272
  1086
          Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1087
            (K (mk_length_Lev_tac lthy cts Lev_0s Lev_Sucs))
traytel@56272
  1088
          |> Thm.close_derivation
traytel@56272
  1089
          |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
  1090
blanchet@48975
  1091
        val length_Lev' = mk_specN (n + 1) length_Lev;
blanchet@48975
  1092
        val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks;
blanchet@48975
  1093
traytel@56272
  1094
        fun mk_goal i z = Logic.mk_implies
desharna@57567
  1095
            (mk_Trueprop_mem (kl, mk_Lev ss nat i $ z),
desharna@57567
  1096
            mk_Trueprop_mem (kl, mk_Lev ss (mk_size kl) i $ z));
blanchet@48975
  1097
        val goals = map2 mk_goal ks zs;
blanchet@48975
  1098
blanchet@48975
  1099
        val length_Levs' = map2 (fn goal => fn length_Lev =>
wenzelm@51551
  1100
          Goal.prove_sorry lthy [] [] goal (K (mk_length_Lev'_tac length_Lev))
traytel@56272
  1101
          |> Thm.close_derivation
traytel@56272
  1102
          |> singleton (Proof_Context.export names_lthy lthy)) goals length_Levs;
blanchet@48975
  1103
      in
blanchet@48975
  1104
        (length_Levs, length_Levs')
blanchet@48975
  1105
      end;
blanchet@48975
  1106
blanchet@48975
  1107
    val rv_last_thmss =
blanchet@48975
  1108
      let
blanchet@48975
  1109
        fun mk_conjunct i z i' z_copy = list_exists_free [z_copy]
blanchet@48975
  1110
          (HOLogic.mk_eq
blanchet@48975
  1111
            (mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z,
blanchet@48975
  1112
            mk_InN activeAs z_copy i'));
blanchet@48975
  1113
        val goal = list_all_free (k :: zs)
blanchet@48975
  1114
          (Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z =>
blanchet@48975
  1115
            Library.foldr1 HOLogic.mk_conj
blanchet@48975
  1116
              (map2 (mk_conjunct i z) ks zs_copy)) ks zs));
blanchet@48975
  1117
blanchet@48975
  1118
        val cTs = [SOME (certifyT lthy sum_sbdT)];
blanchet@48975
  1119
        val cts = map (SOME o certify lthy) [Term.absfree kl' goal, kl];
blanchet@48975
  1120
traytel@56272
  1121
        val rv_last =
traytel@56272
  1122
          Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@56272
  1123
            (K (mk_rv_last_tac cTs cts rv_Nils rv_Conss))
traytel@56272
  1124
          |> Thm.close_derivation
traytel@56272
  1125
          |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
  1126
blanchet@48975
  1127
        val rv_last' = mk_specN (n + 1) rv_last;
blanchet@48975
  1128
      in
blanchet@48975
  1129
        map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks
blanchet@48975
  1130
      end;
blanchet@48975
  1131
blanchet@48975
  1132
    val set_Lev_thmsss =
blanchet@48975
  1133
      let
blanchet@48975
  1134
        fun mk_conjunct i z =
blanchet@48975
  1135
          let
blanchet@48975
  1136
            fun mk_conjunct' i' sets s z' =
blanchet@48975
  1137
              let
blanchet@48975
  1138
                fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp
blanchet@48975
  1139
                  (HOLogic.mk_mem (z'', set $ (s $ z')),
blanchet@48975
  1140
                    HOLogic.mk_mem (mk_append (kl,
blanchet@48975
  1141
                      HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']),
blanchet@48975
  1142
                      mk_Lev ss (HOLogic.mk_Suc nat) i $ z));
blanchet@48975
  1143
              in
blanchet@48975
  1144
                HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'),
wenzelm@58634
  1145
                  (Library.foldr1 HOLogic.mk_conj
wenzelm@58634
  1146
                    (@{map 3} mk_conjunct'' ks (drop m sets) zs_copy2)))
blanchet@48975
  1147
              end;
blanchet@48975
  1148
          in
blanchet@48975
  1149
            HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
wenzelm@58634
  1150
              Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct' ks setssAs ss zs_copy))
blanchet@48975
  1151
          end;
blanchet@48975
  1152
blanchet@48975
  1153
        val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2)
blanchet@48975
  1154
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1155
blanchet@48975
  1156
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1157
traytel@56272
  1158
        val set_Lev =
traytel@56272
  1159
          Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@56272
  1160
            (K (mk_set_Lev_tac lthy cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss))
traytel@56272
  1161
          |> Thm.close_derivation
traytel@56272
  1162
          |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
  1163
blanchet@48975
  1164
        val set_Lev' = mk_specN (3 * n + 1) set_Lev;
blanchet@48975
  1165
      in
blanchet@48975
  1166
        map (fn i => map (fn i' => map (fn i'' => set_Lev' RS
blanchet@48975
  1167
          mk_conjunctN n i RS mp RS
blanchet@48975
  1168
          mk_conjunctN n i' RS mp RS
blanchet@48975
  1169
          mk_conjunctN n i'' RS mp) ks) ks) ks
blanchet@48975
  1170
      end;
blanchet@48975
  1171
blanchet@48975
  1172
    val set_image_Lev_thmsss =
blanchet@48975
  1173
      let
blanchet@48975
  1174
        fun mk_conjunct i z =
blanchet@48975
  1175
          let
blanchet@48975
  1176
            fun mk_conjunct' i' sets =
blanchet@48975
  1177
              let
blanchet@48975
  1178
                fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp
blanchet@48975
  1179
                  (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''),
blanchet@48975
  1180
                  HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z''))));
blanchet@48975
  1181
              in
blanchet@48975
  1182
                HOLogic.mk_imp (HOLogic.mk_mem
blanchet@48975
  1183
                  (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']),
blanchet@48975
  1184
                    mk_Lev ss (HOLogic.mk_Suc nat) i $ z),
wenzelm@58634
  1185
                  (Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct'' ks sets ss zs_copy)))
blanchet@48975
  1186
              end;
blanchet@48975
  1187
          in
blanchet@48975
  1188
            HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1189
              Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs')))
blanchet@48975
  1190
          end;
blanchet@48975
  1191
blanchet@48975
  1192
        val goal = list_all_free (kl :: k :: zs @ zs_copy)
blanchet@48975
  1193
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1194
blanchet@48975
  1195
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1196
traytel@56272
  1197
        val set_image_Lev =
traytel@56272
  1198
          Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1199
            (K (mk_set_image_Lev_tac lthy cts Lev_0s Lev_Sucs rv_Nils rv_Conss
traytel@56272
  1200
              from_to_sbd_thmss to_sbd_inj_thmss))
traytel@56272
  1201
          |> Thm.close_derivation
traytel@56272
  1202
          |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
  1203
blanchet@48975
  1204
        val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev;
blanchet@48975
  1205
      in
blanchet@48975
  1206
        map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS
blanchet@48975
  1207
          mk_conjunctN n i RS mp RS
blanchet@48975
  1208
          mk_conjunctN n i'' RS mp RS
blanchet@48975
  1209
          mk_conjunctN n i' RS mp) ks) ks) ks
blanchet@48975
  1210
      end;
blanchet@48975
  1211
blanchet@48975
  1212
    val mor_beh_thm =
wenzelm@51551
  1213
      Goal.prove_sorry lthy [] []
traytel@56272
  1214
        (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss carTAs strTAs (map (mk_beh ss) ks)))
traytel@55197
  1215
        (fn {context = ctxt, prems = _} => mk_mor_beh_tac ctxt m mor_def mor_cong_thm
blanchet@48975
  1216
          beh_defs carT_defs strT_defs isNode_defs
traytel@55577
  1217
          to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss
traytel@55577
  1218
          length_Lev_thms length_Lev'_thms rv_last_thmss set_Lev_thmsss
traytel@55577
  1219
          set_image_Lev_thmsss set_mapss map_comp_id_thms map_cong0s)
traytel@56272
  1220
      |> Thm.close_derivation
traytel@56272
  1221
      |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
  1222
blanchet@48975
  1223
    val timer = time (timer "Behavioral morphism");
blanchet@48975
  1224
traytel@55541
  1225
    val lsbisAs = map (mk_lsbis carTAs strTAs) ks;
traytel@55541
  1226
traytel@55541
  1227
    fun mk_str_final i =
blanchet@48975
  1228
      mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1),
traytel@55541
  1229
        passive_ids @ map mk_proj lsbisAs), nth strTAs (i - 1)));
traytel@55541
  1230
traytel@55541
  1231
    val car_finals = map2 mk_quotient carTAs lsbisAs;
traytel@55541
  1232
    val str_finals = map mk_str_final ks;
blanchet@48975
  1233
blanchet@48975
  1234
    val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss;
blanchet@48975
  1235
    val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms;
blanchet@48975
  1236
blanchet@48975
  1237
    val congruent_str_final_thms =
blanchet@48975
  1238
      let
blanchet@48975
  1239
        fun mk_goal R final_map strT =
traytel@55541
  1240
          HOLogic.mk_Trueprop (mk_congruent R (HOLogic.mk_comp
traytel@55541
  1241
            (Term.list_comb (final_map, passive_ids @ map mk_proj lsbisAs), strT)));
traytel@55541
  1242
wenzelm@58634
  1243
        val goals = @{map 3} mk_goal lsbisAs final_maps strTAs;
blanchet@48975
  1244
      in
wenzelm@58634
  1245
        @{map 4} (fn goal => fn lsbisE => fn map_comp_id => fn map_cong0 =>
wenzelm@51551
  1246
          Goal.prove_sorry lthy [] [] goal
blanchet@51761
  1247
            (K (mk_congruent_str_final_tac m lsbisE map_comp_id map_cong0 equiv_LSBIS_thms))
traytel@49109
  1248
          |> Thm.close_derivation)
blanchet@51761
  1249
        goals lsbisE_thms map_comp_id_thms map_cong0s
blanchet@48975
  1250
      end;
blanchet@48975
  1251
traytel@55541
  1252
    val coalg_final_thm = Goal.prove_sorry lthy [] []
traytel@55541
  1253
      (HOLogic.mk_Trueprop (mk_coalg car_finals str_finals))
blanchet@48975
  1254
      (K (mk_coalg_final_tac m coalg_def congruent_str_final_thms equiv_LSBIS_thms
blanchet@53290
  1255
        set_mapss coalgT_set_thmss))
traytel@49109
  1256
      |> Thm.close_derivation;
blanchet@48975
  1257
traytel@55541
  1258
    val mor_T_final_thm = Goal.prove_sorry lthy [] []
traytel@55541
  1259
      (HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finals str_finals (map mk_proj lsbisAs)))
traytel@49109
  1260
      (K (mk_mor_T_final_tac mor_def congruent_str_final_thms equiv_LSBIS_thms))
traytel@49109
  1261
      |> Thm.close_derivation;
blanchet@48975
  1262
blanchet@48975
  1263
    val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm];
traytel@55541
  1264
    val in_car_final_thms = map (fn thm => thm OF [mor_final_thm, UNIV_I]) mor_image'_thms;
blanchet@48975
  1265
blanchet@48975
  1266
    val timer = time (timer "Final coalgebra");
blanchet@48975
  1267
blanchet@48975
  1268
    val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
blanchet@48975
  1269
      lthy
wenzelm@58634
  1270
      |> @{fold_map 4} (fn b => fn mx => fn car_final => fn in_car_final =>
blanchet@57093
  1271
        typedef (b, params, mx) car_final NONE
wenzelm@58959
  1272
          (fn _ => EVERY' [rtac exI, rtac in_car_final] 1)) bs mixfixes car_finals in_car_final_thms
blanchet@48975
  1273
      |>> apsnd split_list o split_list;
blanchet@48975
  1274
blanchet@48975
  1275
    val Ts = map (fn name => Type (name, params')) T_names;
blanchet@48975
  1276
    fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;
blanchet@48975
  1277
    val Ts' = mk_Ts passiveBs;
blanchet@48975
  1278
    val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts;
blanchet@48975
  1279
    val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts;
blanchet@48975
  1280
blanchet@48975
  1281
    val Reps = map #Rep T_loc_infos;
blanchet@48975
  1282
    val Rep_injects = map #Rep_inject T_loc_infos;
blanchet@48975
  1283
    val Abs_inverses = map #Abs_inverse T_loc_infos;
blanchet@48975
  1284
blanchet@48975
  1285
    val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");
blanchet@48975
  1286
blanchet@48975
  1287
    val UNIVs = map HOLogic.mk_UNIV Ts;
blanchet@48975
  1288
    val FTs = mk_FTs (passiveAs @ Ts);
blanchet@48975
  1289
    val FTs_setss = mk_setss (passiveAs @ Ts);
blanchet@48975
  1290
    val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs;
blanchet@49504
  1291
    val unfold_fTs = map2 (curry op -->) activeAs Ts;
blanchet@48975
  1292
    val corec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) sum_sTs;
blanchet@48975
  1293
    val corec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls;
blanchet@48975
  1294
    val corec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls_rev;
blanchet@48975
  1295
    val corec_Inls = map (Term.subst_atomic_types (activeBs ~~ Ts)) Inls;
traytel@51739
  1296
    val corec_UNIVs = map2 (HOLogic.mk_UNIV oo curry mk_sumT) Ts activeAs;
blanchet@48975
  1297
traytel@56113
  1298
    val emptys = map (fn T => HOLogic.mk_set T []) passiveAs;
traytel@56113
  1299
    val Zeros = map (fn empty =>
traytel@56113
  1300
     HOLogic.mk_tuple (map (fn U => absdummy U empty) Ts)) emptys;
traytel@56113
  1301
    val hrecTs = map fastype_of Zeros;
traytel@56113
  1302
traytel@56113
  1303
    val (((((((((((((Jzs, Jzs'), Jz's), Jzs_copy), Jz's_copy), Jzs1), Jzs2),
traytel@56113
  1304
      TRs), unfold_fs), corec_ss), phis), dtor_set_induct_phiss), (hrecs, hrecs')),
traytel@54841
  1305
      names_lthy) = names_lthy
blanchet@48975
  1306
      |> mk_Frees' "z" Ts
traytel@54841
  1307
      ||>> mk_Frees "y" Ts'
traytel@51925
  1308
      ||>> mk_Frees "z'" Ts
traytel@51925
  1309
      ||>> mk_Frees "y'" Ts'
blanchet@48975
  1310
      ||>> mk_Frees "z1" Ts
blanchet@48975
  1311
      ||>> mk_Frees "z2" Ts
blanchet@49591
  1312
      ||>> mk_Frees "r" (map (mk_relT o `I) Ts)
blanchet@49504
  1313
      ||>> mk_Frees "f" unfold_fTs
blanchet@48975
  1314
      ||>> mk_Frees "s" corec_sTs
traytel@54841
  1315
      ||>> mk_Frees "P" (map2 mk_pred2T Ts Ts)
traytel@56113
  1316
      ||>> mk_Freess "P" (map (fn A => map (mk_pred2T A) Ts) passiveAs)
traytel@56113
  1317
      ||>> mk_Frees' "rec" hrecTs;
blanchet@48975
  1318
blanchet@54492
  1319
    fun dtor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtorN ^ "_");
traytel@53566
  1320
    val dtor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o dtor_bind;
blanchet@49501
  1321
traytel@55204
  1322
    fun dtor_spec rep str map_FT Jz Jz' =
traytel@55204
  1323
      Term.absfree Jz'
traytel@55204
  1324
        (Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $ (str $ (rep $ Jz)));
blanchet@48975
  1325
blanchet@49501
  1326
    val ((dtor_frees, (_, dtor_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1327
      lthy
wenzelm@58634
  1328
      |> @{fold_map 6} (fn i => fn rep => fn str => fn mapx => fn Jz => fn Jz' =>
traytel@55204
  1329
        Local_Theory.define ((dtor_bind i, NoSyn),
traytel@55204
  1330
          (dtor_def_bind i, dtor_spec rep str mapx Jz Jz')))
traytel@55204
  1331
        ks Rep_Ts str_finals map_FTs Jzs Jzs'
blanchet@48975
  1332
      |>> apsnd split_list o split_list
blanchet@48975
  1333
      ||> `Local_Theory.restore;
blanchet@48975
  1334
blanchet@48975
  1335
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49501
  1336
    fun mk_dtors passive =
traytel@49185
  1337
      map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (mk_params passive)) o
blanchet@49501
  1338
        Morphism.term phi) dtor_frees;
blanchet@49501
  1339
    val dtors = mk_dtors passiveAs;
blanchet@49501
  1340
    val dtor's = mk_dtors passiveBs;
traytel@55204
  1341
    val dtor_defs = map (fn def =>
traytel@55204
  1342
      Morphism.thm phi def RS meta_eq_to_obj_eq RS fun_cong) dtor_def_frees;
blanchet@48975
  1343
blanchet@48975
  1344
    val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss;
blanchet@48975
  1345
    val (mor_Rep_thm, mor_Abs_thm) =
blanchet@48975
  1346
      let
blanchet@48975
  1347
        val mor_Rep =
wenzelm@51551
  1348
          Goal.prove_sorry lthy [] []
blanchet@49501
  1349
            (HOLogic.mk_Trueprop (mk_mor UNIVs dtors car_finals str_finals Rep_Ts))
traytel@55541
  1350
            (fn {context = ctxt, prems = _} => mk_mor_Rep_tac ctxt (mor_def :: dtor_defs) Reps
traytel@55197
  1351
              Abs_inverses coalg_final_set_thmss map_comp_id_thms map_cong0L_thms)
traytel@49109
  1352
          |> Thm.close_derivation;
blanchet@48975
  1353
blanchet@48975
  1354
        val mor_Abs =
wenzelm@51551
  1355
          Goal.prove_sorry lthy [] []
blanchet@49501
  1356
            (HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs dtors Abs_Ts))
traytel@55197
  1357
            (fn {context = ctxt, prems = _} => mk_mor_Abs_tac ctxt (mor_def :: dtor_defs)
traytel@55197
  1358
              Abs_inverses)
traytel@49109
  1359
          |> Thm.close_derivation;
blanchet@48975
  1360
      in
blanchet@48975
  1361
        (mor_Rep, mor_Abs)
blanchet@48975
  1362
      end;
blanchet@48975
  1363
blanchet@49501
  1364
    val timer = time (timer "dtor definitions & thms");
blanchet@49501
  1365
blanchet@54492
  1366
    fun unfold_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_unfoldN ^ "_");
traytel@53566
  1367
    val unfold_def_bind = rpair [] o Binding.conceal o Thm.def_binding o unfold_bind;
blanchet@49504
  1368
traytel@55204
  1369
    fun unfold_spec abs f z = fold_rev (Term.absfree o Term.dest_Free) (ss @ [z]) (abs $ (f $ z));
blanchet@48975
  1370
blanchet@49504
  1371
    val ((unfold_frees, (_, unfold_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1372
      lthy
wenzelm@58634
  1373
      |> @{fold_map 4} (fn i => fn abs => fn f => fn z =>
traytel@55204
  1374
        Local_Theory.define ((unfold_bind i, NoSyn), (unfold_def_bind i, unfold_spec abs f z)))
traytel@55204
  1375
          ks Abs_Ts (map (fn i => HOLogic.mk_comp
traytel@55541
  1376
            (mk_proj (nth lsbisAs (i - 1)), mk_beh ss i)) ks) zs
blanchet@48975
  1377
      |>> apsnd split_list o split_list
blanchet@48975
  1378
      ||> `Local_Theory.restore;
blanchet@48975
  1379
blanchet@48975
  1380
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49504
  1381
    val unfolds = map (Morphism.term phi) unfold_frees;
blanchet@49504
  1382
    val unfold_names = map (fst o dest_Const) unfolds;
traytel@52731
  1383
    fun mk_unfolds passives actives =
wenzelm@58634
  1384
      @{map 3} (fn name => fn T => fn active =>
traytel@52731
  1385
        Const (name, Library.foldr (op -->)
traytel@52923
  1386
          (map2 (curry op -->) actives (mk_FTs (passives @ actives)), active --> T)))
traytel@52731
  1387
      unfold_names (mk_Ts passives) actives;
blanchet@49504
  1388
    fun mk_unfold Ts ss i = Term.list_comb (Const (nth unfold_names (i - 1), Library.foldr (op -->)
blanchet@48975
  1389
      (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
traytel@55204
  1390
    val unfold_defs = map (fn def =>
traytel@55204
  1391
      mk_unabs_def (n + 1) (Morphism.thm phi def RS meta_eq_to_obj_eq)) unfold_def_frees;
blanchet@49504
  1392
blanchet@49504
  1393
    val mor_unfold_thm =
blanchet@48975
  1394
      let
blanchet@48975
  1395
        val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses;
traytel@55541
  1396
        val morEs' = map (fn thm => (thm OF [mor_final_thm, UNIV_I]) RS sym) morE_thms;
blanchet@48975
  1397
      in
wenzelm@51551
  1398
        Goal.prove_sorry lthy [] []
traytel@56272
  1399
          (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors (map (mk_unfold Ts ss) ks)))
blanchet@49504
  1400
          (K (mk_mor_unfold_tac m mor_UNIV_thm dtor_defs unfold_defs Abs_inverses' morEs'
blanchet@51761
  1401
            map_comp_id_thms map_cong0s))
traytel@49109
  1402
        |> Thm.close_derivation
traytel@56272
  1403
        |> singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1404
      end;
blanchet@49504
  1405
    val dtor_unfold_thms = map (fn thm => (thm OF [mor_unfold_thm, UNIV_I]) RS sym) morE_thms;
blanchet@48975
  1406
blanchet@48975
  1407
    val (raw_coind_thms, raw_coind_thm) =
blanchet@48975
  1408
      let
traytel@55541
  1409
        val prem = HOLogic.mk_Trueprop (mk_sbis UNIVs dtors TRs);
blanchet@48975
  1410
        val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
traytel@51893
  1411
          (map2 (fn R => fn T => mk_leq R (Id_const T)) TRs Ts));
blanchet@48975
  1412
      in
traytel@56272
  1413
        `split_conj_thm (Goal.prove_sorry lthy [] [] (Logic.mk_implies (prem, concl))
blanchet@48975
  1414
          (K (mk_raw_coind_tac bis_def bis_cong_thm bis_O_thm bis_converse_thm bis_Gr_thm
blanchet@48975
  1415
            tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm
traytel@49109
  1416
            lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects))
traytel@56272
  1417
          |> Thm.close_derivation
traytel@56272
  1418
          |> singleton (Proof_Context.export names_lthy lthy))
blanchet@48975
  1419
      end;
blanchet@48975
  1420
blanchet@49504
  1421
    val (unfold_unique_mor_thms, unfold_unique_mor_thm) =
blanchet@48975
  1422
      let
blanchet@49504
  1423
        val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors unfold_fs);
blanchet@49504
  1424
        fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_unfold Ts ss i);
blanchet@48975
  1425
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@49504
  1426
          (map2 mk_fun_eq unfold_fs ks));
blanchet@48975
  1427
blanchet@48975
  1428
        val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm);
traytel@55541
  1429
        val mor_thm = mor_comp_thm OF [mor_final_thm, mor_Abs_thm];
blanchet@48975
  1430
traytel@56272
  1431
        val unique_mor = Goal.prove_sorry lthy [] [] (Logic.mk_implies (prem, unique))
blanchet@49504
  1432
          (K (mk_unfold_unique_mor_tac raw_coind_thms bis_thm mor_thm unfold_defs))
traytel@56272
  1433
          |> Thm.close_derivation
traytel@56272
  1434
          |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
  1435
      in
blanchet@48975
  1436
        `split_conj_thm unique_mor
blanchet@48975
  1437
      end;
blanchet@48975
  1438
blanchet@49504
  1439
    val (dtor_unfold_unique_thms, dtor_unfold_unique_thm) = `split_conj_thm (split_conj_prems n
traytel@52904
  1440
      (mor_UNIV_thm RS iffD2 RS unfold_unique_mor_thm));
blanchet@49504
  1441
blanchet@49504
  1442
    val unfold_dtor_thms = map (fn thm => mor_id_thm RS thm RS sym) unfold_unique_mor_thms;
blanchet@49504
  1443
blanchet@49504
  1444
    val unfold_o_dtor_thms =
blanchet@48975
  1445
      let
blanchet@49504
  1446
        val mor = mor_comp_thm OF [mor_str_thm, mor_unfold_thm];
blanchet@48975
  1447
      in
blanchet@49504
  1448
        map2 (fn unique => fn unfold_ctor =>
blanchet@49504
  1449
          trans OF [mor RS unique, unfold_ctor]) unfold_unique_mor_thms unfold_dtor_thms
blanchet@48975
  1450
      end;
blanchet@48975
  1451
blanchet@49504
  1452
    val timer = time (timer "unfold definitions & thms");
blanchet@48975
  1453
blanchet@49501
  1454
    val map_dtors = map2 (fn Ds => fn bnf =>
blanchet@48975
  1455
      Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,
blanchet@49501
  1456
        map HOLogic.id_const passiveAs @ dtors)) Dss bnfs;
blanchet@49501
  1457
blanchet@54492
  1458
    fun ctor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctorN ^ "_");
traytel@53566
  1459
    val ctor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o ctor_bind;
blanchet@49501
  1460
traytel@55204
  1461
    fun ctor_spec i = mk_unfold Ts map_dtors i;
blanchet@48975
  1462
blanchet@49501
  1463
    val ((ctor_frees, (_, ctor_def_frees)), (lthy, lthy_old)) =
blanchet@49311
  1464
      lthy
traytel@55204
  1465
      |> fold_map (fn i =>
traytel@55204
  1466
        Local_Theory.define ((ctor_bind i, NoSyn), (ctor_def_bind i, ctor_spec i))) ks
blanchet@49311
  1467
      |>> apsnd split_list o split_list
blanchet@49311
  1468
      ||> `Local_Theory.restore;
blanchet@49311
  1469
blanchet@48975
  1470
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49501
  1471
    fun mk_ctors params =
blanchet@48975
  1472
      map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
blanchet@49501
  1473
        ctor_frees;
blanchet@49501
  1474
    val ctors = mk_ctors params';
traytel@55204
  1475
    val ctor_defs = map (fn def => Morphism.thm phi def RS meta_eq_to_obj_eq) ctor_def_frees;
blanchet@49501
  1476
blanchet@49504
  1477
    val ctor_o_dtor_thms = map2 (fold_thms lthy o single) ctor_defs unfold_o_dtor_thms;
blanchet@49501
  1478
blanchet@49501
  1479
    val dtor_o_ctor_thms =
blanchet@48975
  1480
      let
blanchet@49501
  1481
        fun mk_goal dtor ctor FT =
blanchet@49501
  1482
         mk_Trueprop_eq (HOLogic.mk_comp (dtor, ctor), HOLogic.id_const FT);
wenzelm@58634
  1483
        val goals = @{map 3} mk_goal dtors ctors FTs;
blanchet@48975
  1484
      in
wenzelm@58634
  1485
        @{map 5} (fn goal => fn ctor_def => fn unfold => fn map_comp_id => fn map_cong0L =>
wenzelm@51551
  1486
          Goal.prove_sorry lthy [] [] goal
traytel@55197
  1487
            (fn {context = ctxt, prems = _} => mk_dtor_o_ctor_tac ctxt ctor_def unfold map_comp_id
traytel@55197
  1488
              map_cong0L unfold_o_dtor_thms)
traytel@49109
  1489
          |> Thm.close_derivation)
blanchet@51761
  1490
          goals ctor_defs dtor_unfold_thms map_comp_id_thms map_cong0L_thms
blanchet@48975
  1491
      end;
blanchet@48975
  1492
blanchet@49501
  1493
    val dtor_ctor_thms = map (fn thm => thm RS @{thm pointfree_idE}) dtor_o_ctor_thms;
blanchet@49501
  1494
    val ctor_dtor_thms = map (fn thm => thm RS @{thm pointfree_idE}) ctor_o_dtor_thms;
blanchet@49501
  1495
blanchet@49501
  1496
    val bij_dtor_thms =
blanchet@49501
  1497
      map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) ctor_o_dtor_thms dtor_o_ctor_thms;
blanchet@49501
  1498
    val inj_dtor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_dtor_thms;
blanchet@49501
  1499
    val surj_dtor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_dtor_thms;
blanchet@49501
  1500
    val dtor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_dtor_thms;
blanchet@49501
  1501
    val dtor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_dtor_thms;
blanchet@49501
  1502
    val dtor_exhaust_thms = map (fn thm => thm RS exE) dtor_nchotomy_thms;
blanchet@49501
  1503
blanchet@49501
  1504
    val bij_ctor_thms =
blanchet@49501
  1505
      map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) dtor_o_ctor_thms ctor_o_dtor_thms;
blanchet@49501
  1506
    val inj_ctor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_ctor_thms;
blanchet@49501
  1507
    val surj_ctor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_ctor_thms;
blanchet@49501
  1508
    val ctor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_ctor_thms;
blanchet@49501
  1509
    val ctor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_ctor_thms;
blanchet@49501
  1510
    val ctor_exhaust_thms = map (fn thm => thm RS exE) ctor_nchotomy_thms;
blanchet@49501
  1511
blanchet@49501
  1512
    val timer = time (timer "ctor definitions & thms");
blanchet@48975
  1513
blanchet@48975
  1514
    val corec_Inl_sum_thms =
blanchet@48975
  1515
      let
blanchet@55414
  1516
        val mor = mor_comp_thm OF [mor_case_sum_thm, mor_unfold_thm];
blanchet@48975
  1517
      in
blanchet@49504
  1518
        map2 (fn unique => fn unfold_dtor =>
blanchet@49504
  1519
          trans OF [mor RS unique, unfold_dtor]) unfold_unique_mor_thms unfold_dtor_thms
blanchet@48975
  1520
      end;
blanchet@48975
  1521
blanchet@54492
  1522
    fun corec_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_corecN ^ "_");
traytel@53566
  1523
    val corec_def_bind = rpair [] o Binding.conceal o Thm.def_binding o corec_bind;
blanchet@48975
  1524
traytel@51739
  1525
    val corec_strs =
wenzelm@58634
  1526
      @{map 3} (fn dtor => fn sum_s => fn mapx =>
blanchet@55414
  1527
        mk_case_sum
traytel@51739
  1528
          (HOLogic.mk_comp (Term.list_comb (mapx, passive_ids @ corec_Inls), dtor), sum_s))
traytel@51739
  1529
      dtors corec_ss corec_maps;
traytel@51739
  1530
blanchet@48975
  1531
    fun corec_spec i T AT =
traytel@55204
  1532
      fold_rev (Term.absfree o Term.dest_Free) corec_ss
traytel@55204
  1533
        (HOLogic.mk_comp (mk_unfold Ts corec_strs i, Inr_const T AT));
blanchet@48975
  1534
blanchet@48975
  1535
    val ((corec_frees, (_, corec_def_frees)), (lthy, lthy_old)) =
blanchet@49311
  1536
      lthy
wenzelm@58634
  1537
      |> @{fold_map 3} (fn i => fn T => fn AT =>
traytel@55204
  1538
        Local_Theory.define ((corec_bind i, NoSyn), (corec_def_bind i, corec_spec i T AT)))
blanchet@49311
  1539
          ks Ts activeAs
blanchet@49311
  1540
      |>> apsnd split_list o split_list
blanchet@49311
  1541
      ||> `Local_Theory.restore;
blanchet@49311
  1542
blanchet@48975
  1543
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49176
  1544
    val corecs = map (Morphism.term phi) corec_frees;
blanchet@49176
  1545
    val corec_names = map (fst o dest_Const) corecs;
traytel@58443
  1546
    fun mk_corecs Ts passives actives =
traytel@58443
  1547
      let val Tactives = map2 (curry mk_sumT) Ts actives;
traytel@58443
  1548
      in
wenzelm@58634
  1549
        @{map 3} (fn name => fn T => fn active =>
traytel@58443
  1550
          Const (name, Library.foldr (op -->)
traytel@58443
  1551
            (map2 (curry op -->) actives (mk_FTs (passives @ Tactives)), active --> T)))
traytel@58443
  1552
        corec_names Ts actives
traytel@58443
  1553
      end;
blanchet@49176
  1554
    fun mk_corec ss i = Term.list_comb (Const (nth corec_names (i - 1), Library.foldr (op -->)
blanchet@48975
  1555
      (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
traytel@55204
  1556
    val corec_defs = map (fn def =>
traytel@55204
  1557
      mk_unabs_def n (Morphism.thm phi def RS meta_eq_to_obj_eq)) corec_def_frees;
blanchet@48975
  1558
blanchet@55414
  1559
    val case_sums =
blanchet@55414
  1560
      map2 (fn T => fn i => mk_case_sum (HOLogic.id_const T, mk_corec corec_ss i)) Ts ks;
blanchet@49504
  1561
    val dtor_corec_thms =
blanchet@48975
  1562
      let
blanchet@49501
  1563
        fun mk_goal i corec_s corec_map dtor z =
blanchet@48975
  1564
          let
blanchet@49501
  1565
            val lhs = dtor $ (mk_corec corec_ss i $ z);
blanchet@55414
  1566
            val rhs = Term.list_comb (corec_map, passive_ids @ case_sums) $ (corec_s $ z);
blanchet@48975
  1567
          in
traytel@56272
  1568
            mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1569
          end;
wenzelm@58634
  1570
        val goals = @{map 5} mk_goal ks corec_ss corec_maps_rev dtors zs;
blanchet@48975
  1571
      in
wenzelm@58634
  1572
        @{map 3} (fn goal => fn unfold => fn map_cong0 =>
wenzelm@51551
  1573
          Goal.prove_sorry lthy [] [] goal
traytel@55197
  1574
            (fn {context = ctxt, prems = _} => mk_corec_tac ctxt m corec_defs unfold map_cong0
traytel@55197
  1575
              corec_Inl_sum_thms)
traytel@56272
  1576
          |> Thm.close_derivation
traytel@56272
  1577
          |> singleton (Proof_Context.export names_lthy lthy))
blanchet@51761
  1578
        goals dtor_unfold_thms map_cong0s
blanchet@48975
  1579
      end;
blanchet@48975
  1580
traytel@51739
  1581
    val corec_unique_mor_thm =
traytel@51739
  1582
      let
blanchet@55414
  1583
        val id_fs = map2 (fn T => fn f => mk_case_sum (HOLogic.id_const T, f)) Ts unfold_fs;
traytel@51739
  1584
        val prem = HOLogic.mk_Trueprop (mk_mor corec_UNIVs corec_strs UNIVs dtors id_fs);
traytel@51739
  1585
        fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_corec corec_ss i);
traytel@51739
  1586
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
traytel@51739
  1587
          (map2 mk_fun_eq unfold_fs ks));
traytel@51739
  1588
      in
traytel@56272
  1589
        Goal.prove_sorry lthy [] [] (Logic.mk_implies (prem, unique))
traytel@55197
  1590
          (fn {context = ctxt, prems = _} => mk_corec_unique_mor_tac ctxt corec_defs
traytel@55197
  1591
            corec_Inl_sum_thms unfold_unique_mor_thm)
traytel@51739
  1592
          |> Thm.close_derivation
traytel@56272
  1593
          |> singleton (Proof_Context.export names_lthy lthy)
traytel@51739
  1594
      end;
traytel@51739
  1595
traytel@53696
  1596
    val map_id0s_o_id =
traytel@53696
  1597
      map (fn thm =>
blanchet@55067
  1598
        mk_trans (thm RS @{thm arg_cong2[of _ _ _ _ "op o", OF _ refl]}) @{thm id_comp})
traytel@53696
  1599
      map_id0s;
traytel@53696
  1600
traytel@52913
  1601
    val (dtor_corec_unique_thms, dtor_corec_unique_thm) =
traytel@52913
  1602
      `split_conj_thm (split_conj_prems n
traytel@52904
  1603
        (mor_UNIV_thm RS iffD2 RS corec_unique_mor_thm)
desharna@57932
  1604
        |> unfold_thms lthy (@{thms o_case_sum comp_id id_comp comp_assoc[symmetric]
blanchet@55414
  1605
           case_sum_o_inj(1)} @ map_id0s_o_id @ sym_map_comps)
blanchet@55414
  1606
        OF replicate n @{thm arg_cong2[of _ _ _ _ case_sum, OF refl]});
traytel@51739
  1607
blanchet@48975
  1608
    val timer = time (timer "corec definitions & thms");
blanchet@48975
  1609
traytel@55644
  1610
    val (coinduct_params, dtor_coinduct_thm) =
blanchet@48975
  1611
      let
traytel@51893
  1612
        val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs;
blanchet@48975
  1613
blanchet@48975
  1614
        fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2));
blanchet@48975
  1615
        val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
wenzelm@58634
  1616
          (@{map 3} mk_concl phis Jzs1 Jzs2));
blanchet@48975
  1617
traytel@53105
  1618
        fun mk_rel_prem phi dtor rel Jz Jz_copy =
blanchet@48975
  1619
          let
traytel@55541
  1620
            val concl = Term.list_comb (rel, passive_eqs @ phis) $
traytel@51893
  1621
              (dtor $ Jz) $ (dtor $ Jz_copy);
blanchet@48975
  1622
          in
blanchet@48975
  1623
            HOLogic.mk_Trueprop
blanchet@48975
  1624
              (list_all_free [Jz, Jz_copy] (HOLogic.mk_imp (phi $ Jz $ Jz_copy, concl)))
blanchet@48975
  1625
          end;
blanchet@48975
  1626
wenzelm@58634
  1627
        val rel_prems = @{map 5} mk_rel_prem phis dtors rels Jzs Jzs_copy;
traytel@56272
  1628
        val dtor_coinduct_goal = Logic.list_implies (rel_prems, concl);
traytel@51893
  1629
traytel@51893
  1630
        val dtor_coinduct =
traytel@51893
  1631
          Goal.prove_sorry lthy [] [] dtor_coinduct_goal
traytel@51893
  1632
            (K (mk_dtor_coinduct_tac m raw_coind_thm bis_rel_thm rel_congs))
traytel@56272
  1633
          |> Thm.close_derivation
traytel@56272
  1634
          |> singleton (Proof_Context.export names_lthy lthy);
blanchet@48975
  1635
      in
traytel@55644
  1636
        (rev (Term.add_tfrees dtor_coinduct_goal []), dtor_coinduct)
blanchet@48975
  1637
      end;
blanchet@48975
  1638
blanchet@48975
  1639
    val timer = time (timer "coinduction");
blanchet@48975
  1640
blanchet@53270
  1641
    fun mk_dtor_map_DEADID_thm dtor_inject map_id0 =
blanchet@53270
  1642
      trans OF [iffD2 OF [dtor_inject, id_apply], map_id0 RS sym];
traytel@51925
  1643
traytel@51925
  1644
    fun mk_dtor_Jrel_DEADID_thm dtor_inject bnf =
traytel@51917
  1645
      trans OF [rel_eq_of_bnf bnf RS @{thm predicate2_eqD}, dtor_inject] RS sym;
traytel@51917
  1646
traytel@51925
  1647
    val JphiTs = map2 mk_pred2T passiveAs passiveBs;
traytel@54841
  1648
    val Jpsi1Ts = map2 mk_pred2T passiveAs passiveCs;
traytel@54841
  1649
    val Jpsi2Ts = map2 mk_pred2T passiveCs passiveBs;
traytel@51925
  1650
    val prodTsTs' = map2 (curry HOLogic.mk_prodT) Ts Ts';
traytel@51925
  1651
    val fstsTsTs' = map fst_const prodTsTs';
traytel@51925
  1652
    val sndsTsTs' = map snd_const prodTsTs';
traytel@52731
  1653
    val activephiTs = map2 mk_pred2T activeAs activeBs;
traytel@51925
  1654
    val activeJphiTs = map2 mk_pred2T Ts Ts';
traytel@54841
  1655
    val (((((Jphis, Jpsi1s), Jpsi2s), activephis), activeJphis), names_lthy) = names_lthy
traytel@51925
  1656
      |> mk_Frees "R" JphiTs
traytel@54841
  1657
      ||>> mk_Frees "R" Jpsi1Ts
traytel@54841
  1658
      ||>> mk_Frees "Q" Jpsi2Ts
traytel@52731
  1659
      ||>> mk_Frees "S" activephiTs
traytel@51925
  1660
      ||>> mk_Frees "JR" activeJphiTs;
traytel@51925
  1661
    val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
traytel@51925
  1662
desharna@57567
  1663
    fun mk_Jrel_DEADID_coinduct_thm () =
desharna@58579
  1664
      mk_xtor_rel_co_induct_thm Greatest_FP rels activeJphis (map HOLogic.eq_const Ts) Jphis
traytel@54841
  1665
        Jzs Jz's dtors dtor's (fn {context = ctxt, prems} =>
traytel@54841
  1666
          (unfold_thms_tac ctxt @{thms le_fun_def le_bool_def all_simps(1,2)[symmetric]} THEN
traytel@54841
  1667
          REPEAT_DETERM (rtac allI 1) THEN rtac (dtor_coinduct_thm OF prems) 1)) lthy;
traytel@54841
  1668
blanchet@48975
  1669
    (*register new codatatypes as BNFs*)
traytel@54841
  1670
    val (timer, Jbnfs, (dtor_Jmap_o_thms, dtor_Jmap_thms), dtor_Jset_thmss',
desharna@57700
  1671
      dtor_Jrel_thms, Jrel_coinduct_thm, Jbnf_notes, dtor_Jset_induct_thms, lthy) =
blanchet@49585
  1672
      if m = 0 then
traytel@52913
  1673
        (timer, replicate n DEADID_bnf,
blanchet@53285
  1674
        map_split (`(mk_pointfree lthy)) (map2 mk_dtor_map_DEADID_thm dtor_inject_thms map_ids),
traytel@54841
  1675
        replicate n [], map2 mk_dtor_Jrel_DEADID_thm dtor_inject_thms bnfs,
desharna@57700
  1676
        mk_Jrel_DEADID_coinduct_thm (), [], [], lthy)
blanchet@49585
  1677
      else let
blanchet@48975
  1678
        val fTs = map2 (curry op -->) passiveAs passiveBs;
blanchet@48975
  1679
        val gTs = map2 (curry op -->) passiveBs passiveCs;
blanchet@48975
  1680
        val uTs = map2 (curry op -->) Ts Ts';
traytel@54841
  1681
traytel@56113
  1682
        val (((((((((fs, fs'), fs_copy), gs), us), (Jys, Jys')), (Jys_copy, Jys'_copy)),
traytel@56113
  1683
          (ys_copy, ys'_copy)), Kss), names_lthy) = names_lthy
blanchet@48975
  1684
          |> mk_Frees' "f" fTs
blanchet@49308
  1685
          ||>> mk_Frees "f" fTs
blanchet@49308
  1686
          ||>> mk_Frees "g" gTs
blanchet@48975
  1687
          ||>> mk_Frees "u" uTs
blanchet@48975
  1688
          ||>> mk_Frees' "b" Ts'
blanchet@48975
  1689
          ||>> mk_Frees' "b" Ts'
traytel@56113
  1690
          ||>> mk_Frees' "y" passiveAs
traytel@56113
  1691
          ||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) Ts) ATs);;
blanchet@48975
  1692
blanchet@48975
  1693
        val map_FTFT's = map2 (fn Ds =>
blanchet@48975
  1694
          mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
blanchet@48975
  1695
blanchet@48975
  1696
        fun mk_maps ATs BTs Ts mk_T =
blanchet@48975
  1697
          map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ map mk_T Ts)) Dss bnfs;
blanchet@48975
  1698
        fun mk_Fmap mk_const fs Ts Fmap = Term.list_comb (Fmap, fs @ map mk_const Ts);
blanchet@49501
  1699
        fun mk_map mk_const mk_T Ts fs Ts' dtors mk_maps =
blanchet@49504
  1700
          mk_unfold Ts' (map2 (fn dtor => fn Fmap =>
blanchet@49501
  1701
            HOLogic.mk_comp (mk_Fmap mk_const fs Ts Fmap, dtor)) dtors (mk_maps Ts mk_T));
blanchet@48975
  1702
        val mk_map_id = mk_map HOLogic.id_const I;
blanchet@48975
  1703
        val mk_mapsAB = mk_maps passiveAs passiveBs;
blanchet@49501
  1704
        val fs_maps = map (mk_map_id Ts fs Ts' dtors mk_mapsAB) ks;
traytel@54841
  1705
traytel@54841
  1706
        val set_bss =
traytel@54841
  1707
          map (flat o map2 (fn B => fn b =>
traytel@54841
  1708
            if member (op =) resDs (TFree B) then [] else [b]) resBs) set_bss0;
desharna@57567
  1709
traytel@56113
  1710
        fun col_bind j = mk_internal_b (colN ^ (if m = 1 then "" else string_of_int j));
traytel@56113
  1711
        val col_def_bind = rpair [] o Thm.def_binding o col_bind;
desharna@57567
  1712
traytel@56113
  1713
        fun col_spec j Zero hrec hrec' =
traytel@56113
  1714
          let
traytel@56179
  1715
            fun mk_Suc dtor sets z z' =
traytel@56113
  1716
              let
traytel@56179
  1717
                val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m sets);
traytel@56113
  1718
                fun mk_UN set k = mk_UNION (set $ (dtor $ z)) (mk_nthN n hrec k);
traytel@56113
  1719
              in
traytel@56113
  1720
                Term.absfree z'
traytel@56113
  1721
                  (mk_union (set $ (dtor $ z), Library.foldl1 mk_union (map2 mk_UN sets ks)))
traytel@56113
  1722
              end;
desharna@57567
  1723
traytel@56113
  1724
            val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec'
wenzelm@58634
  1725
              (HOLogic.mk_tuple (@{map 4} mk_Suc dtors FTs_setss Jzs Jzs')));
traytel@56113
  1726
          in
traytel@56113
  1727
            mk_rec_nat Zero Suc
traytel@56113
  1728
          end;
desharna@57567
  1729
traytel@56113
  1730
        val ((col_frees, (_, col_def_frees)), (lthy, lthy_old)) =
traytel@56113
  1731
          lthy
wenzelm@58634
  1732
          |> @{fold_map 4} (fn j => fn Zero => fn hrec => fn hrec' => Local_Theory.define
traytel@56113
  1733
            ((col_bind j, NoSyn), (col_def_bind j, col_spec j Zero hrec hrec')))
traytel@56113
  1734
            ls Zeros hrecs hrecs'
traytel@56113
  1735
          |>> apsnd split_list o split_list
traytel@56113
  1736
          ||> `Local_Theory.restore;
desharna@57567
  1737
traytel@56113
  1738
        val phi = Proof_Context.export_morphism lthy_old lthy;
desharna@57567
  1739
traytel@56113
  1740
        val col_defs = map (fn def => Morphism.thm phi def RS meta_eq_to_obj_eq) col_def_frees;
traytel@56113
  1741
        val cols = map (fst o Term.dest_Const o Morphism.term phi) col_frees;
desharna@57567
  1742
traytel@56113
  1743
        fun mk_col Ts nat i j T =
traytel@56113
  1744
          let
traytel@56113
  1745
            val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) Ts)
traytel@56113
  1746
            val colT = HOLogic.natT --> hrecT;
traytel@56113
  1747
          in
traytel@56113
  1748
            mk_nthN n (Term.list_comb (Const (nth cols (j - 1), colT), [nat])) i
traytel@56113
  1749
          end;
desharna@57567
  1750
traytel@56113
  1751
        val col_0ss = mk_rec_simps n @{thm rec_nat_0_imp} col_defs;
traytel@56113
  1752
        val col_Sucss = mk_rec_simps n @{thm rec_nat_Suc_imp} col_defs;
traytel@56113
  1753
        val col_0ss' = transpose col_0ss;
traytel@56113
  1754
        val col_Sucss' = transpose col_Sucss;
desharna@57567
  1755
traytel@56179
  1756
        fun mk_set Ts i j T =
traytel@56113
  1757
          Abs (Name.uu, nth Ts (i - 1), mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
traytel@56113
  1758
            (Term.absfree nat' (mk_col Ts nat i j T $ Bound 1)));
traytel@56113
  1759
traytel@56179
  1760
        val setss = map (fn i => map2 (mk_set Ts i) ls passiveAs) ks;
traytel@54841
  1761
traytel@54841
  1762
        val (Jbnf_consts, lthy) =
wenzelm@58634
  1763
          @{fold_map 7} (fn b => fn map_b => fn rel_b => fn set_bs => fn mapx => fn sets => fn T =>
traytel@56179
  1764
              fn lthy =>
traytel@56179
  1765
            define_bnf_consts Hardly_Inline (user_policy Note_Some lthy) false (SOME deads)
traytel@54841
  1766
              map_b rel_b set_bs
traytel@56179
  1767
              ((((((b, T), fold_rev Term.absfree fs' mapx), sets), sbd),
traytel@56179
  1768
                [Const (@{const_name undefined}, T)]), NONE) lthy)
traytel@56179
  1769
          bs map_bs rel_bs set_bss fs_maps setss Ts lthy;
traytel@54841
  1770
wenzelm@58634
  1771
        val (_, Jconsts, Jconst_defs, mk_Jconsts) = @{split_list 4} Jbnf_consts;
wenzelm@58634
  1772
        val (_, Jsetss, Jbds_Ds, _, _) = @{split_list 5} Jconsts;
wenzelm@58634
  1773
        val (Jmap_defs, Jset_defss, Jbd_defs, _, Jrel_defs) = @{split_list 5} Jconst_defs;
wenzelm@58634
  1774
        val (mk_Jmaps_Ds, mk_Jt_Ds, _, mk_Jrels_Ds, _) = @{split_list 5} mk_Jconsts;
traytel@54841
  1775
traytel@54841
  1776
        val Jrel_unabs_defs = map (fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) Jrel_defs;
traytel@54841
  1777
        val Jset_defs = flat Jset_defss;
traytel@54841
  1778
traytel@54841
  1779
        fun mk_Jmaps As Bs = map (fn mk => mk deads As Bs) mk_Jmaps_Ds;
traytel@54841
  1780
        fun mk_Jsetss As = map2 (fn mk => fn Jsets => map (mk deads As) Jsets) mk_Jt_Ds Jsetss;
traytel@54841
  1781
        val Jbds = map2 (fn mk => mk deads passiveAs) mk_Jt_Ds Jbds_Ds;
traytel@54841
  1782
        fun mk_Jrels As Bs = map (fn mk => mk deads As Bs) mk_Jrels_Ds;
traytel@54841
  1783
traytel@54841
  1784
        val Jmaps = mk_Jmaps passiveAs passiveBs;
traytel@54841
  1785
        val fs_Jmaps = map (fn m => Term.list_comb (m, fs)) Jmaps;
traytel@54841
  1786
        val fs_copy_Jmaps = map (fn m => Term.list_comb (m, fs_copy)) Jmaps;
traytel@54841
  1787
        val gs_Jmaps = map (fn m => Term.list_comb (m, gs)) (mk_Jmaps passiveBs passiveCs);
traytel@54841
  1788
        val fgs_Jmaps = map (fn m => Term.list_comb (m, map2 (curry HOLogic.mk_comp) gs fs))
traytel@54841
  1789
          (mk_Jmaps passiveAs passiveCs);
traytel@54841
  1790
        val (Jsetss_by_range, Jsetss_by_bnf) = `transpose (mk_Jsetss passiveAs);
traytel@54841
  1791
traytel@54841
  1792
        val timer = time (timer "bnf constants for the new datatypes");
traytel@54841
  1793
traytel@54841
  1794
        val (dtor_Jmap_thms, Jmap_thms) =
traytel@54841
  1795
          let
traytel@56272
  1796
            fun mk_goal fs_Jmap map dtor dtor' = mk_Trueprop_eq (HOLogic.mk_comp (dtor', fs_Jmap),
traytel@56272
  1797
              HOLogic.mk_comp (Term.list_comb (map, fs @ fs_Jmaps), dtor));
wenzelm@58634
  1798
            val goals = @{map 4} mk_goal fs_Jmaps map_FTFT's dtors dtor's;
traytel@54841
  1799
            val maps =
wenzelm@58634
  1800
              @{map 5} (fn goal => fn unfold => fn map_comp => fn map_cong0 => fn map_arg_cong =>
traytel@54841
  1801
                Goal.prove_sorry lthy [] [] goal
traytel@54841
  1802
                  (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jmap_defs THEN
traytel@55602
  1803
                     mk_map_tac m n map_arg_cong unfold map_comp map_cong0)
traytel@56272
  1804
                |> Thm.close_derivation
traytel@56272
  1805
                |> singleton (Proof_Context.export names_lthy lthy))
traytel@55602
  1806
              goals dtor_unfold_thms map_comps map_cong0s map_arg_cong_thms;
traytel@54841
  1807
          in
traytel@54841
  1808
            map_split (fn thm => (thm RS @{thm comp_eq_dest}, thm)) maps
traytel@54841
  1809
          end;
traytel@54841
  1810
traytel@54841
  1811
        val dtor_Jmap_unique_thm =
traytel@54841
  1812
          let
traytel@54841
  1813
            fun mk_prem u map dtor dtor' =
traytel@54841
  1814
              mk_Trueprop_eq (HOLogic.mk_comp (dtor', u),
traytel@54841
  1815
                HOLogic.mk_comp (Term.list_comb (map, fs @ us), dtor));
wenzelm@58634
  1816
            val prems = @{map 4} mk_prem us map_FTFT's dtors dtor's;
traytel@54841
  1817
            val goal =
traytel@54841
  1818
              HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
traytel@54841
  1819
                (map2 (curry HOLogic.mk_eq) us fs_Jmaps));
traytel@54841
  1820
          in
traytel@56272
  1821
            Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, goal))
traytel@56272
  1822
              (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jmap_defs THEN
traytel@56272
  1823
                mk_dtor_map_unique_tac ctxt dtor_unfold_unique_thm sym_map_comps)
traytel@56272
  1824
            |> Thm.close_derivation
traytel@56272
  1825
            |> singleton (Proof_Context.export names_lthy lthy)
traytel@54841
  1826
          end;
traytel@54841
  1827
traytel@54841
  1828
        val Jmap_comp0_thms =
traytel@54841
  1829
          let
traytel@56272
  1830
            val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
wenzelm@58634
  1831
              (@{map 3} (fn fmap => fn gmap => fn fgmap =>
traytel@56272
  1832
                 HOLogic.mk_eq (HOLogic.mk_comp (gmap, fmap), fgmap))
traytel@56272
  1833
              fs_Jmaps gs_Jmaps fgs_Jmaps))
traytel@54841
  1834
          in
traytel@54841
  1835
            split_conj_thm (Goal.prove_sorry lthy [] [] goal
traytel@54841
  1836
              (K (mk_map_comp0_tac Jmap_thms map_comp0s dtor_Jmap_unique_thm))
traytel@56272
  1837
              |> Thm.close_derivation
traytel@56272
  1838
              |> singleton (Proof_Context.export names_lthy lthy))
traytel@54841
  1839
          end;
traytel@54841
  1840
traytel@54841
  1841
        val timer = time (timer "map functions for the new codatatypes");
traytel@54841
  1842
traytel@56179
  1843
        val Jset_minimal_thms =
traytel@56179
  1844
          let
traytel@56179
  1845
            fun mk_passive_prem set dtor x K =
traytel@56179
  1846
              Logic.all x (HOLogic.mk_Trueprop (mk_leq (set $ (dtor $ x)) (K $ x)));
desharna@57567
  1847
traytel@56179
  1848
            fun mk_active_prem dtor x1 K1 set x2 K2 =
traytel@56179
  1849
              fold_rev Logic.all [x1, x2]
desharna@57567
  1850
                (Logic.mk_implies (mk_Trueprop_mem (x2, set $ (dtor $ x1)),
traytel@56179
  1851
                  HOLogic.mk_Trueprop (mk_leq (K2 $ x2) (K1 $ x1))));
desharna@57567
  1852
traytel@56179
  1853
            val premss = map2 (fn j => fn Ks =>
wenzelm@58634
  1854
              @{map 4} mk_passive_prem (map (fn xs => nth xs (j - 1)) FTs_setss) dtors Jzs Ks @
wenzelm@58634
  1855
                flat (@{map 4} (fn sets => fn s => fn x1 => fn K1 =>
wenzelm@58634
  1856
                  @{map 3} (mk_active_prem s x1 K1) (drop m sets) Jzs_copy Ks) FTs_setss dtors Jzs Ks))
traytel@56179
  1857
              ls Kss;
desharna@57567
  1858
traytel@56179
  1859
            val col_minimal_thms =
traytel@56179
  1860
              let
traytel@56179
  1861
                fun mk_conjunct j T i K x = mk_leq (mk_col Ts nat i j T $ x) (K $ x);
traytel@56179
  1862
                fun mk_concl j T Ks = list_all_free Jzs
wenzelm@58634
  1863
                  (Library.foldr1 HOLogic.mk_conj (@{map 3} (mk_conjunct j T) ks Ks Jzs));
wenzelm@58634
  1864
                val concls = @{map 3} mk_concl ls passiveAs Kss;
desharna@57567
  1865
traytel@56179
  1866
                val goals = map2 (fn prems => fn concl =>
traytel@56179
  1867
                  Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls
desharna@57567
  1868
traytel@56179
  1869
                val ctss =
traytel@56179
  1870
                  map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
traytel@56179
  1871
              in
wenzelm@58634
  1872
                @{map 4} (fn goal => fn cts => fn col_0s => fn col_Sucs =>
traytel@56272
  1873
                  Goal.prove_sorry lthy [] [] goal
traytel@56272
  1874
                    (fn {context = ctxt, prems = _} => mk_col_minimal_tac ctxt m cts col_0s
traytel@56272
  1875
                      col_Sucs)
traytel@56272
  1876
                  |> Thm.close_derivation
traytel@56272
  1877
                  |> singleton (Proof_Context.export names_lthy lthy))
traytel@56179
  1878
                goals ctss col_0ss' col_Sucss'
traytel@56179
  1879
              end;
desharna@57567
  1880
traytel@56179
  1881
            fun mk_conjunct set K x = mk_leq (set $ x) (K $ x);
wenzelm@58634
  1882
            fun mk_concl sets Ks = Library.foldr1 HOLogic.mk_conj (@{map 3} mk_conjunct sets Ks Jzs);
traytel@56179
  1883
            val concls = map2 mk_concl Jsetss_by_range Kss;
desharna@57567
  1884
traytel@56272
  1885
            val goals = map2 (fn prems => fn concl =>
traytel@56272
  1886
              Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls;
traytel@56179
  1887
          in
traytel@56179
  1888
            map2 (fn goal => fn col_minimal =>
traytel@56179
  1889
              Goal.prove_sorry lthy [] [] goal
traytel@56179
  1890
                (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN
traytel@56179
  1891
                  mk_Jset_minimal_tac ctxt n col_minimal)
traytel@56272
  1892
              |> Thm.close_derivation
traytel@56272
  1893
              |> singleton (Proof_Context.export names_lthy lthy))
traytel@56179
  1894
            goals col_minimal_thms
traytel@56179
  1895
          end;
traytel@56179
  1896
traytel@56179
  1897
        val (dtor_Jset_incl_thmss, dtor_set_Jset_incl_thmsss) =
traytel@56179
  1898
          let
traytel@56272
  1899
            fun mk_set_incl_Jset dtor x set Jset =
traytel@56272
  1900
              HOLogic.mk_Trueprop (mk_leq (set $ (dtor $ x)) (Jset $ x));
desharna@57567
  1901
traytel@56179
  1902
            fun mk_set_Jset_incl_Jset dtor x y set Jset1 Jset2 =
desharna@57567
  1903
              Logic.mk_implies (mk_Trueprop_mem (x, set $ (dtor $ y)),
traytel@56272
  1904
                HOLogic.mk_Trueprop (mk_leq (Jset1 $ x) (Jset2 $ y)));
desharna@57567
  1905
traytel@56179
  1906
            val set_incl_Jset_goalss =
wenzelm@58634
  1907
              @{map 4} (fn dtor => fn x => fn sets => fn Jsets =>
traytel@56179
  1908
                map2 (mk_set_incl_Jset dtor x) (take m sets) Jsets)
traytel@56179
  1909
              dtors Jzs FTs_setss Jsetss_by_bnf;
desharna@57567
  1910
traytel@56179
  1911
            (*x(k) : F(i)set(m+k) (dtor(i) y(i)) ==> J(k)set(j) x(k) <= J(i)set(j) y(i)*)
traytel@56179
  1912
            val set_Jset_incl_Jset_goalsss =
wenzelm@58634
  1913
              @{map 4} (fn dtori => fn yi => fn sets => fn Jsetsi =>
wenzelm@58634
  1914
                @{map 3} (fn xk => fn set => fn Jsetsk =>
traytel@56179
  1915
                  map2 (mk_set_Jset_incl_Jset dtori xk yi set) Jsetsk Jsetsi)
traytel@56179
  1916
                Jzs_copy (drop m sets) Jsetss_by_bnf)
traytel@56179
  1917
              dtors Jzs FTs_setss Jsetss_by_bnf;
traytel@56179
  1918
          in
traytel@56179
  1919
            (map2 (fn goals => fn rec_Sucs =>
traytel@56179
  1920
              map2 (fn goal => fn rec_Suc =>
traytel@56179
  1921
                Goal.prove_sorry lthy [] [] goal
traytel@56179
  1922
                  (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN
traytel@56179
  1923
                    mk_set_incl_Jset_tac rec_Suc)
traytel@56272
  1924
                |> Thm.close_derivation
traytel@56272
  1925
                |> singleton (Proof_Context.export names_lthy lthy))
traytel@56179
  1926
              goals rec_Sucs)
traytel@56179
  1927
            set_incl_Jset_goalss col_Sucss,
traytel@56179
  1928
            map2 (fn goalss => fn rec_Sucs =>
traytel@56179
  1929
              map2 (fn k => fn goals =>
traytel@56179
  1930
                map2 (fn goal => fn rec_Suc =>
traytel@56179
  1931
                  Goal.prove_sorry lthy [] [] goal
traytel@56179
  1932
                    (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN
traytel@56179
  1933
                      mk_set_Jset_incl_Jset_tac n rec_Suc k)
traytel@56272
  1934
                  |> Thm.close_derivation
traytel@56272
  1935
                  |> singleton (Proof_Context.export names_lthy lthy))
traytel@56179
  1936
                goals rec_Sucs)
traytel@56179
  1937
              ks goalss)
traytel@56179
  1938
            set_Jset_incl_Jset_goalsss col_Sucss)
traytel@56179
  1939
          end;
desharna@57567
  1940
traytel@56179
  1941
        val set_incl_Jset_thmss' = transpose dtor_Jset_incl_thmss;
traytel@56179
  1942
        val set_Jset_incl_Jset_thmsss' = transpose (map transpose dtor_set_Jset_incl_thmsss);
traytel@56179
  1943
        val set_Jset_thmss = map (map (fn thm => thm RS @{thm set_mp})) dtor_Jset_incl_thmss;
traytel@56179
  1944
        val set_Jset_Jset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp})))
traytel@56179
  1945
          dtor_set_Jset_incl_thmsss;
traytel@56179
  1946
        val set_Jset_thmss' = transpose set_Jset_thmss;
traytel@56179
  1947
        val set_Jset_Jset_thmsss' = transpose (map transpose set_Jset_Jset_thmsss);
desharna@57567
  1948
traytel@56179
  1949
        val dtor_Jset_induct_thms =
traytel@56179
  1950
          let
traytel@56179
  1951
            val incls =
traytel@56179
  1952
              maps (map (fn thm => thm RS @{thm subset_Collect_iff})) dtor_Jset_incl_thmss @
traytel@56179
  1953
                @{thms subset_Collect_iff[OF subset_refl]};
traytel@56179
  1954
desharna@57567
  1955
            val cTs = map (SOME o certifyT lthy) params';
traytel@56179
  1956
            fun mk_induct_tinst phis jsets y y' =
wenzelm@58634
  1957
              @{map 4} (fn phi => fn jset => fn Jz => fn Jz' =>
traytel@56179
  1958
                SOME (certify lthy (Term.absfree Jz' (HOLogic.mk_Collect (fst y', snd y',
traytel@56179
  1959
                  HOLogic.mk_conj (HOLogic.mk_mem (y, jset $ Jz), phi $ y $ Jz))))))
traytel@56179
  1960
              phis jsets Jzs Jzs';
traytel@56179
  1961
          in
wenzelm@58634
  1962
            @{map 6} (fn set_minimal => fn set_set_inclss => fn jsets => fn y => fn y' => fn phis =>
traytel@56179
  1963
              ((set_minimal
traytel@56179
  1964
                |> Drule.instantiate' cTs (mk_induct_tinst phis jsets y y')
traytel@56179
  1965
                |> unfold_thms lthy incls) OF
traytel@56179
  1966
                (replicate n ballI @
traytel@56179
  1967
                  maps (map (fn thm => thm RS @{thm subset_CollectI})) set_set_inclss))
traytel@56179
  1968
              |> singleton (Proof_Context.export names_lthy lthy)
traytel@56179
  1969
              |> rule_by_tactic lthy (ALLGOALS (TRY o etac asm_rl)))
traytel@56179
  1970
            Jset_minimal_thms set_Jset_incl_Jset_thmsss' Jsetss_by_range ys ys' dtor_set_induct_phiss
traytel@56179
  1971
          end;
traytel@56179
  1972
traytel@54841
  1973
        val (dtor_Jset_thmss', dtor_Jset_thmss) =
traytel@54841
  1974
          let
traytel@54841
  1975
            fun mk_simp_goal relate pas_set act_sets sets dtor z set =
traytel@54841
  1976
              relate (set $ z, mk_union (pas_set $ (dtor $ z),
traytel@54841
  1977
                 Library.foldl1 mk_union
traytel@54841
  1978
                   (map2 (fn X => mk_UNION (X $ (dtor $ z))) act_sets sets)));
traytel@54841
  1979
            fun mk_goals eq =
traytel@54841
  1980
              map2 (fn i => fn sets =>
wenzelm@58634
  1981
                @{map 4} (fn Fsets =>
traytel@54841
  1982
                  mk_simp_goal eq (nth Fsets (i - 1)) (drop m Fsets) sets)
traytel@54841
  1983
                FTs_setss dtors Jzs sets)
traytel@54841
  1984
              ls Jsetss_by_range;
traytel@54841
  1985
traytel@56272
  1986
            val le_goals = map (HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj)
traytel@54841
  1987
              (mk_goals (uncurry mk_leq));
traytel@54841
  1988
            val set_le_thmss = map split_conj_thm
wenzelm@58634
  1989
              (@{map 4} (fn goal => fn Jset_minimal => fn set_Jsets => fn set_Jset_Jsetss =>
traytel@54841
  1990
                Goal.prove_sorry lthy [] [] goal
traytel@56179
  1991
                  (K (mk_set_le_tac n Jset_minimal set_Jsets set_Jset_Jsetss))
traytel@56272
  1992
                |> Thm.close_derivation
traytel@56272
  1993
                |> singleton (Proof_Context.export names_lthy lthy))
traytel@56179
  1994
              le_goals Jset_minimal_thms set_Jset_thmss' set_Jset_Jset_thmsss');
traytel@54841
  1995
traytel@56272
  1996
            val ge_goalss = map (map HOLogic.mk_Trueprop) (mk_goals (uncurry mk_leq o swap));
desharna@57567
  1997
            val set_ge_thmss =
wenzelm@58634
  1998
              @{map 3} (@{map 3} (fn goal => fn set_incl_Jset => fn set_Jset_incl_Jsets =>
traytel@54841
  1999
                Goal.prove_sorry lthy [] [] goal
traytel@56179
  2000
                  (K (mk_set_ge_tac n set_incl_Jset set_Jset_incl_Jsets))
traytel@56272
  2001
                |> Thm.close_derivation
traytel@56272
  2002
                |> singleton (Proof_Context.export names_lthy lthy)))
traytel@56179
  2003
              ge_goalss set_incl_Jset_thmss' set_Jset_incl_Jset_thmsss'
traytel@54841
  2004
          in
traytel@54841
  2005
            map2 (map2 (fn le => fn ge => equalityI OF [le, ge])) set_le_thmss set_ge_thmss
traytel@54841
  2006
            |> `transpose
traytel@54841
  2007
          end;
traytel@54841
  2008
traytel@54841
  2009
        val timer = time (timer "set functions for the new codatatypes");
traytel@54841
  2010
traytel@54841
  2011
        val colss = map2 (fn j => fn T =>
traytel@56113
  2012
          map (fn i => mk_col Ts nat i j T) ks) ls passiveAs;
traytel@54841
  2013
        val colss' = map2 (fn j => fn T =>
traytel@56113
  2014
          map (fn i => mk_col Ts' nat i j T) ks) ls passiveBs;
traytel@54841
  2015
traytel@54841
  2016
        val col_natural_thmss =
traytel@54841
  2017
          let
traytel@54841
  2018
            fun mk_col_natural f map z col col' =
traytel@54841
  2019
              HOLogic.mk_eq (mk_image f $ (col $ z), col' $ (map $ z));
traytel@54841
  2020
traytel@54841
  2021
            fun mk_goal f cols cols' = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj
wenzelm@58634
  2022
              (@{map 4} (mk_col_natural f) fs_Jmaps Jzs cols cols'));
traytel@54841
  2023
wenzelm@58634
  2024
            val goals = @{map 3} mk_goal fs colss colss';
traytel@54841
  2025
traytel@54841
  2026
            val ctss =
traytel@54841
  2027
              map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) goals;
traytel@54841
  2028
traytel@54841
  2029
            val thms =
wenzelm@58634
  2030
              @{map 4} (fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
traytel@56272
  2031
                Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@56272
  2032
                  (fn {context = ctxt, prems = _} => mk_col_natural_tac ctxt cts rec_0s rec_Sucs
traytel@56272
  2033
                    dtor_Jmap_thms set_mapss)
traytel@56272
  2034
                |> Thm.close_derivation
traytel@56272
  2035
                |> singleton (Proof_Context.export names_lthy lthy))
traytel@56113
  2036
              goals ctss col_0ss' col_Sucss';
traytel@54841
  2037
          in
traytel@54841
  2038
            map (split_conj_thm o mk_specN n) thms
traytel@54841
  2039
          end;
traytel@54841
  2040
traytel@54841
  2041
        val col_bd_thmss =
traytel@54841
  2042
          let
traytel@54841
  2043
            fun mk_col_bd z col bd = mk_ordLeq (mk_card_of (col $ z)) bd;
traytel@54841
  2044
traytel@54841
  2045
            fun mk_goal bds cols = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj
wenzelm@58634
  2046
              (@{map 3} mk_col_bd Jzs cols bds));
traytel@54841
  2047
traytel@54841
  2048
            val goals = map (mk_goal Jbds) colss;
traytel@54841
  2049
traytel@54841
  2050
            val ctss = map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat])
traytel@54841
  2051
              (map (mk_goal (replicate n sbd)) colss);
traytel@54841
  2052
traytel@54841
  2053
            val thms =
wenzelm@58634
  2054
              @{map 5} (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
traytel@56272
  2055
                Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@56272
  2056
                  (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jbd_defs THEN
traytel@56272
  2057
                    mk_col_bd_tac m j cts rec_0s rec_Sucs sbd_Card_order sbd_Cinfinite set_sbdss)
traytel@56272
  2058
                |> Thm.close_derivation
traytel@56272
  2059
                |> singleton (Proof_Context.export names_lthy lthy))
traytel@56113
  2060
              ls goals ctss col_0ss' col_Sucss';
traytel@54841
  2061
          in
traytel@54841
  2062
            map (split_conj_thm o mk_specN n) thms
traytel@54841
  2063
          end;
traytel@54841
  2064
traytel@54841
  2065
        val map_cong0_thms =
traytel@54841
  2066
          let
traytel@54841
  2067
            val cTs = map (SOME o certifyT lthy o
traytel@54841
  2068
              Term.typ_subst_atomic (passiveAs ~~ passiveBs) o TFree) coinduct_params;
traytel@54841
  2069
traytel@54841
  2070
            fun mk_prem z set f g y y' =
traytel@54841
  2071
              mk_Ball (set $ z) (Term.absfree y' (HOLogic.mk_eq (f $ y, g $ y)));
traytel@54841
  2072
traytel@54841
  2073
            fun mk_prems sets z =
wenzelm@58634
  2074
              Library.foldr1 HOLogic.mk_conj (@{map 5} (mk_prem z) sets fs fs_copy ys ys')
traytel@54841
  2075
traytel@54841
  2076
            fun mk_map_cong0 sets z fmap gmap =
traytel@54841
  2077
              HOLogic.mk_imp (mk_prems sets z, HOLogic.mk_eq (fmap $ z, gmap $ z));
traytel@54841
  2078
traytel@54841
  2079
            fun mk_coind_body sets (x, T) z fmap gmap y y_copy =
traytel@54841
  2080
              HOLogic.mk_conj
traytel@54841
  2081
                (HOLogic.mk_mem (z, HOLogic.mk_Collect (x, T, mk_prems sets z)),
traytel@54841
  2082
                  HOLogic.mk_conj (HOLogic.mk_eq (y, fmap $ z),
traytel@54841
  2083
                    HOLogic.mk_eq (y_copy, gmap $ z)))
traytel@54841
  2084
traytel@54841
  2085
            fun mk_cphi sets (z' as (x, T)) z fmap gmap y' y y'_copy y_copy =
traytel@54841
  2086
              HOLogic.mk_exists (x, T, mk_coind_body sets z' z fmap gmap y y_copy)
traytel@54841
  2087
              |> Term.absfree y'_copy
traytel@54841
  2088
              |> Term.absfree y'
traytel@54841
  2089
              |> certify lthy;
traytel@54841
  2090
wenzelm@58634
  2091
            val cphis = @{map 9} mk_cphi
traytel@54841
  2092
              Jsetss_by_bnf Jzs' Jzs fs_Jmaps fs_copy_Jmaps Jys' Jys Jys'_copy Jys_copy;
traytel@54841
  2093
traytel@56179
  2094
            val coinduct = Drule.instantiate' cTs (map SOME cphis) dtor_coinduct_thm;
traytel@54841
  2095
traytel@54841
  2096
            val goal =
traytel@54841
  2097
              HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
wenzelm@58634
  2098
                (@{map 4} mk_map_cong0 Jsetss_by_bnf Jzs fs_Jmaps fs_copy_Jmaps));
traytel@54841
  2099
traytel@56272
  2100
            val thm =
traytel@56272
  2101
              Goal.prove_sorry lthy [] [] goal
traytel@56179
  2102
                (K (mk_mcong_tac lthy m (rtac coinduct) map_comps dtor_Jmap_thms map_cong0s
traytel@56272
  2103
                  set_mapss set_Jset_thmss set_Jset_Jset_thmsss in_rels))
traytel@54841
  2104
              |> Thm.close_derivation
traytel@56272
  2105
              |>  singleton (Proof_Context.export names_lthy lthy);
traytel@54841
  2106
          in
traytel@54841
  2107
            split_conj_thm thm
traytel@54841
  2108
          end;
traytel@54841
  2109
traytel@54841
  2110
        val in_Jrels = map (fn def => trans OF [def, @{thm OO_Grp_alt}] RS @{thm predicate2_eqD})
traytel@54841
  2111
            Jrel_unabs_defs;
traytel@54841
  2112
traytel@54841
  2113
        val Jrels = mk_Jrels passiveAs passiveBs;
traytel@54841
  2114
        val Jrelphis = map (fn rel => Term.list_comb (rel, Jphis)) Jrels;
traytel@51893
  2115
        val relphis = map (fn rel => Term.list_comb (rel, Jphis @ Jrelphis)) rels;
traytel@54841
  2116
        val Jrelpsi1s = map (fn rel => Term.list_comb (rel, Jpsi1s)) (mk_Jrels passiveAs passiveCs);
traytel@54841
  2117
        val Jrelpsi2s = map (fn rel => Term.list_comb (rel, Jpsi2s)) (mk_Jrels passiveCs passiveBs);
traytel@54841
  2118
        val Jrelpsi12s = map (fn rel =>
traytel@54841
  2119
            Term.list_comb (rel, map2 (curry mk_rel_compp) Jpsi1s Jpsi2s)) Jrels;
blanchet@48975
  2120
traytel@51893
  2121
        val dtor_Jrel_thms =
blanchet@48975
  2122
          let
traytel@56272
  2123
            fun mk_goal Jz Jz' dtor dtor' Jrelphi relphi =
traytel@56272
  2124
              mk_Trueprop_eq (Jrelphi $ Jz $ Jz', relphi $ (dtor $ Jz) $ (dtor' $ Jz'));
wenzelm@58634
  2125
            val goals = @{map 6} mk_goal Jzs Jz's dtors dtor's Jrelphis relphis;
blanchet@48975
  2126
          in
wenzelm@58634
  2127
            @{map 12} (fn i => fn goal => fn in_rel => fn map_comp0 => fn map_cong0 =>
blanchet@49542
  2128
              fn dtor_map => fn dtor_sets => fn dtor_inject => fn dtor_ctor =>
blanchet@53289
  2129
              fn set_map0s => fn dtor_set_incls => fn dtor_set_set_inclss =>
wenzelm@51551
  2130
              Goal.prove_sorry lthy [] [] goal
blanchet@53287
  2131
                (K (mk_dtor_rel_tac lthy in_Jrels i in_rel map_comp0 map_cong0 dtor_map dtor_sets
blanchet@53289
  2132
                  dtor_inject dtor_ctor set_map0s dtor_set_incls dtor_set_set_inclss))
traytel@56272
  2133
              |> Thm.close_derivation
traytel@56272
  2134
              |> singleton (Proof_Context.export names_lthy lthy))
traytel@54841
  2135
            ks goals in_rels map_comps map_cong0s dtor_Jmap_thms dtor_Jset_thmss'
traytel@54841
  2136
              dtor_inject_thms dtor_ctor_thms set_mapss dtor_Jset_incl_thmss
traytel@54841
  2137
              dtor_set_Jset_incl_thmsss
blanchet@48975
  2138
          end;
blanchet@48975
  2139
traytel@51925
  2140
      val passiveABs = map2 (curry HOLogic.mk_prodT) passiveAs passiveBs;
traytel@51925
  2141
      val zip_ranTs = passiveABs @ prodTsTs';
traytel@51925
  2142
      val allJphis = Jphis @ activeJphis;
traytel@51925
  2143
      val zipFTs = mk_FTs zip_ranTs;
wenzelm@58634
  2144
      val zipTs = @{map 3} (fn T => fn T' => fn FT => T --> T' --> FT) Ts Ts' zipFTs;
traytel@51925
  2145
      val zip_zTs = mk_Ts passiveABs;
traytel@55602
  2146
      val (((zips, (abs, abs')), (zip_zs, zip_zs')), names_lthy) = names_lthy
traytel@51925
  2147
        |> mk_Frees "zip" zipTs
traytel@51925
  2148
        ||>> mk_Frees' "ab" passiveABs
traytel@55602
  2149
        ||>> mk_Frees' "z" zip_zTs;
traytel@51925
  2150
traytel@51925
  2151
      val Iphi_sets =
traytel@51925
  2152
        map2 (fn phi => fn T => HOLogic.Collect_const T $ HOLogic.mk_split phi) allJphis zip_ranTs;
traytel@51925
  2153
      val in_phis = map2 (mk_in Iphi_sets) (mk_setss zip_ranTs) zipFTs;
traytel@51925
  2154
      val fstABs = map fst_const passiveABs;
traytel@51925
  2155
      val all_fsts = fstABs @ fstsTsTs';
traytel@51925
  2156
      val map_all_fsts = map2 (fn Ds => fn bnf =>
traytel@51925
  2157
        Term.list_comb (mk_map_of_bnf Ds zip_ranTs (passiveAs @ Ts) bnf, all_fsts)) Dss bnfs;
traytel@54841
  2158
      val Jmap_fsts = map2 (fn map => fn T => if m = 0 then HOLogic.id_const T
traytel@54841
  2159
        else Term.list_comb (map, fstABs)) (mk_Jmaps passiveABs passiveAs) Ts;
traytel@51925
  2160
traytel@51925
  2161
      val sndABs = map snd_const passiveABs;
traytel@51925
  2162
      val all_snds = sndABs @ sndsTsTs';
traytel@51925
  2163
      val map_all_snds = map2 (fn Ds => fn bnf =>
traytel@51925
  2164
        Term.list_comb (mk_map_of_bnf Ds zip_ranTs (passiveBs @ Ts') bnf, all_snds)) Dss bnfs;
traytel@54841
  2165
      val Jmap_snds = map2 (fn map => fn T => if m = 0 then HOLogic.id_const T
traytel@54841
  2166
        else Term.list_comb (map, sndABs)) (mk_Jmaps passiveABs passiveBs) Ts;
traytel@51925
  2167
      val zip_unfolds = map (mk_unfold zip_zTs (map HOLogic.mk_split zips)) ks;
traytel@54841
  2168
      val zip_setss = mk_Jsetss passiveABs |> transpose;
traytel@51925
  2169
traytel@55197
  2170
      fun Jrel_coinduct_tac {context = ctxt, prems = CIHs} =
traytel@51925
  2171
        let
traytel@51925
  2172
          fun mk_helper_prem phi in_phi zip x y map map' dtor dtor' =
traytel@51925
  2173
            let
traytel@51925
  2174
              val zipxy = zip $ x $ y;
traytel@51925
  2175
            in
traytel@51925
  2176
              HOLogic.mk_Trueprop (list_all_free [x, y]
traytel@51925
  2177
                (HOLogic.mk_imp (phi $ x $ y, HOLogic.mk_conj (HOLogic.mk_mem (zipxy, in_phi),
traytel@51925
  2178
                  HOLogic.mk_conj (HOLogic.mk_eq (map $ zipxy, dtor $ x),
traytel@51925
  2179
                    HOLogic.mk_eq (map' $ zipxy, dtor' $ y))))))
traytel@51925
  2180
            end;
wenzelm@58634
  2181
          val helper_prems = @{map 9} mk_helper_prem
traytel@51925
  2182
            activeJphis in_phis zips Jzs Jz's map_all_fsts map_all_snds dtors dtor's;
traytel@55602
  2183
          fun mk_helper_coind_phi fst phi x alt y map zip_unfold =
traytel@55602
  2184
            list_exists_free [if fst then y else x] (HOLogic.mk_conj (phi $ x $ y,
traytel@55602
  2185
              HOLogic.mk_eq (alt, map $ (zip_unfold $ HOLogic.mk_prod (x, y)))))
wenzelm@58634
  2186
          val coind1_phis = @{map 6} (mk_helper_coind_phi true)
traytel@55602
  2187
            activeJphis Jzs Jzs_copy Jz's Jmap_fsts zip_unfolds;
wenzelm@58634
  2188
          val coind2_phis = @{map 6} (mk_helper_coind_phi false)
traytel@55602
  2189
              activeJphis Jzs Jz's_copy Jz's Jmap_snds zip_unfolds;
traytel@55602
  2190
          fun mk_cts zs z's phis =
wenzelm@58634
  2191
            @{map 3} (fn z => fn z' => fn phi =>
traytel@55602
  2192
              SOME (certify lthy (fold_rev (Term.absfree o Term.dest_Free) [z', z] phi)))
traytel@55602
  2193
            zs z's phis @
traytel@55602
  2194
            map (SOME o certify lthy) (splice z's zs);
traytel@55602
  2195
          val cts1 = mk_cts Jzs Jzs_copy coind1_phis;
traytel@55602
  2196
          val cts2 = mk_cts Jz's Jz's_copy coind2_phis;
traytel@55602
  2197
traytel@55602
  2198
          fun mk_helper_coind_concl z alt coind_phi =
traytel@55602
  2199
            HOLogic.mk_imp (coind_phi, HOLogic.mk_eq (alt, z));
traytel@51925
  2200
          val helper_coind1_concl =
traytel@51925
  2201
            HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
wenzelm@58634
  2202
              (@{map 3} mk_helper_coind_concl Jzs Jzs_copy coind1_phis));
traytel@51925
  2203
          val helper_coind2_concl =
traytel@51925
  2204
            HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
wenzelm@58634
  2205
              (@{map 3} mk_helper_coind_concl Jz's Jz's_copy coind2_phis));
traytel@55602
  2206
traytel@55644
  2207
          fun mk_helper_coind_thms fst concl cts =
traytel@55602
  2208
            Goal.prove_sorry lthy [] [] (Logic.list_implies (helper_prems, concl))
traytel@55644
  2209
              (fn {context = ctxt, prems = _} => mk_rel_coinduct_coind_tac ctxt fst m
traytel@55644
  2210
                (cterm_instantiate_pos cts dtor_coinduct_thm) ks map_comps map_cong0s
traytel@55644
  2211
                map_arg_cong_thms set_mapss dtor_unfold_thms dtor_Jmap_thms in_rels)
traytel@51925
  2212
            |> Thm.close_derivation
traytel@55602
  2213
            |> split_conj_thm
traytel@55602
  2214
            |> Proof_Context.export names_lthy lthy;
traytel@55602
  2215
traytel@55644
  2216
          val helper_coind1_thms = mk_helper_coind_thms true helper_coind1_concl cts1;
traytel@55644
  2217
          val helper_coind2_thms = mk_helper_coind_thms false helper_coind2_concl cts2;
traytel@55602
  2218
traytel@55602
  2219
          fun mk_helper_ind_phi phi ab fst snd z active_phi x y zip_unfold =
traytel@55602
  2220
            list_all_free [x, y] (HOLogic.mk_imp
traytel@51925
  2221
              (HOLogic.mk_conj (active_phi $ x $ y,
traytel@51925
  2222
                 HOLogic.mk_eq (z, zip_unfold $ HOLogic.mk_prod (x, y))),
traytel@55602
  2223
              phi $ (fst $ ab) $ (snd $ ab)));
traytel@55602
  2224
          val helper_ind_phiss =
wenzelm@58634
  2225
            @{map 4} (fn Jphi => fn ab => fn fst => fn snd =>
wenzelm@58634
  2226
              @{map 5} (mk_helper_ind_phi Jphi ab fst snd)
traytel@55602
  2227
              zip_zs activeJphis Jzs Jz's zip_unfolds)
traytel@55602
  2228
            Jphis abs fstABs sndABs;
traytel@55602
  2229
          val ctss = map2 (fn ab' => fn phis =>
traytel@55602
  2230
              map2 (fn z' => fn phi =>
traytel@55602
  2231
                SOME (certify lthy (Term.absfree ab' (Term.absfree z' phi))))
traytel@55602
  2232
              zip_zs' phis @
traytel@55602
  2233
              map (SOME o certify lthy) zip_zs)
traytel@55602
  2234
            abs' helper_ind_phiss;
traytel@55602
  2235
          fun mk_helper_ind_concl ab' z ind_phi set =
traytel@55602
  2236
            mk_Ball (set $ z) (Term.absfree ab' ind_phi);
desharna@57567
  2237
traytel@51925
  2238
          val mk_helper_ind_concls =
wenzelm@58634
  2239
            @{map 3} (fn ab' => fn ind_phis => fn zip_sets =>
wenzelm@58634
  2240
              @{map 3} (mk_helper_ind_concl ab') zip_zs ind_phis zip_sets)
traytel@55602
  2241
            abs' helper_ind_phiss zip_setss
traytel@51925
  2242
            |> map (HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj);
traytel@55602
  2243
traytel@51925
  2244
          val helper_ind_thmss = if m = 0 then replicate n [] else
wenzelm@58634
  2245
            @{map 4} (fn concl => fn j => fn set_induct => fn cts =>
traytel@55602
  2246
              Goal.prove_sorry lthy [] [] (Logic.list_implies (helper_prems, concl))
traytel@55197
  2247
                (fn {context = ctxt, prems = _} => mk_rel_coinduct_ind_tac ctxt m ks
traytel@55602
  2248
                  dtor_unfold_thms set_mapss j (cterm_instantiate_pos cts set_induct))
traytel@51925
  2249
              |> Thm.close_derivation
traytel@55602
  2250
              |> split_conj_thm
traytel@55602
  2251
              |> Proof_Context.export names_lthy lthy)
traytel@55602
  2252
            mk_helper_ind_concls ls dtor_Jset_induct_thms ctss
traytel@51925
  2253
            |> transpose;
traytel@51925
  2254
        in
traytel@55197
  2255
          mk_rel_coinduct_tac ctxt CIHs in_rels in_Jrels
traytel@52505
  2256
            helper_ind_thmss helper_coind1_thms helper_coind2_thms
traytel@51925
  2257
        end;
traytel@54841
  2258
traytel@52505
  2259
      val Jrel_coinduct_thm =
desharna@58579
  2260
        mk_xtor_rel_co_induct_thm Greatest_FP rels activeJphis Jrels Jphis Jzs Jz's dtors dtor's
traytel@52505
  2261
          Jrel_coinduct_tac lthy;
traytel@51925
  2262
traytel@54841
  2263
        val le_Jrel_OO_thm =
traytel@54841
  2264
          let
traytel@54841
  2265
            fun mk_le_Jrel_OO Jrelpsi1 Jrelpsi2 Jrelpsi12 =
traytel@54841
  2266
              mk_leq (mk_rel_compp (Jrelpsi1, Jrelpsi2)) Jrelpsi12;
wenzelm@58634
  2267
            val goals = @{map 3} mk_le_Jrel_OO Jrelpsi1s Jrelpsi2s Jrelpsi12s;
traytel@54841
  2268
traytel@54841
  2269
            val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj goals);
traytel@54841
  2270
          in
traytel@56272
  2271
            Goal.prove_sorry lthy [] [] goal
traytel@57726
  2272
              (K (mk_le_rel_OO_tac Jrel_coinduct_thm dtor_Jrel_thms le_rel_OOs))
traytel@56272
  2273
            |> Thm.close_derivation
traytel@56272
  2274
            |> singleton (Proof_Context.export names_lthy lthy)
traytel@54841
  2275
          end;
traytel@54841
  2276
traytel@54841
  2277
        val timer = time (timer "helpers for BNF properties");
traytel@54841
  2278
traytel@56179
  2279
        fun close_wit I wit = (I, fold_rev Term.absfree (map (nth ys') I) wit);
traytel@56179
  2280
traytel@56179
  2281
        val all_unitTs = replicate live HOLogic.unitT;
traytel@56179
  2282
        val unitTs = replicate n HOLogic.unitT;
traytel@56179
  2283
        val unit_funs = replicate n (Term.absdummy HOLogic.unitT HOLogic.unit);
traytel@56179
  2284
        fun mk_map_args I =
traytel@56179
  2285
          map (fn i =>
traytel@56179
  2286
            if member (op =) I i then Term.absdummy HOLogic.unitT (nth ys i)
traytel@56179
  2287
            else mk_undefined (HOLogic.unitT --> nth passiveAs i))
traytel@56179
  2288
          (0 upto (m - 1));
traytel@56179
  2289
traytel@56179
  2290
        fun mk_nat_wit Ds bnf (I, wit) () =
traytel@56179
  2291
          let
traytel@56179
  2292
            val passiveI = filter (fn i => i < m) I;
traytel@56179
  2293
            val map_args = mk_map_args passiveI;
traytel@56179
  2294
          in
traytel@56179
  2295
            Term.absdummy HOLogic.unitT (Term.list_comb
traytel@56179
  2296
              (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $ wit)
traytel@56179
  2297
          end;
traytel@56179
  2298
traytel@56179
  2299
        fun mk_dummy_wit Ds bnf I =
traytel@56179
  2300
          let
traytel@56179
  2301
            val map_args = mk_map_args I;
traytel@56179
  2302
          in
traytel@56179
  2303
            Term.absdummy HOLogic.unitT (Term.list_comb
traytel@56179
  2304
              (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $
traytel@56179
  2305
              mk_undefined (mk_T_of_bnf Ds all_unitTs bnf))
traytel@56179
  2306
          end;
traytel@56179
  2307
traytel@56179
  2308
        val nat_witss =
traytel@56179
  2309
          map2 (fn Ds => fn bnf => mk_wits_of_bnf (replicate (nwits_of_bnf bnf) Ds)
traytel@56179
  2310
            (replicate (nwits_of_bnf bnf) (replicate live HOLogic.unitT)) bnf
traytel@56179
  2311
            |> map (fn (I, wit) =>
traytel@56179
  2312
              (I, Lazy.lazy (mk_nat_wit Ds bnf (I, Term.list_comb (wit, map (K HOLogic.unit) I))))))
traytel@56179
  2313
          Dss bnfs;
traytel@56179
  2314
traytel@56179
  2315
        val nat_wit_thmss = map2 (curry op ~~) nat_witss (map wit_thmss_of_bnf bnfs)
traytel@56179
  2316
traytel@56179
  2317
        val Iss = map (map fst) nat_witss;
traytel@56179
  2318
traytel@56179
  2319
        fun filter_wits (I, wit) =
traytel@56179
  2320
          let val J = filter (fn i => i < m) I;
traytel@56179
  2321
          in (J, (length J < length I, wit)) end;
traytel@56179
  2322
traytel@56179
  2323
        val wit_treess = map_index (fn (i, Is) =>
traytel@56179
  2324
          map_index (finish Iss m [i+m] (i+m)) Is) Iss
traytel@56179
  2325
          |> map (minimize_wits o map filter_wits o minimize_wits o flat);
traytel@56179
  2326
traytel@56179
  2327
        val coind_wit_argsss =
traytel@56179
  2328
          map (map (tree_to_coind_wits nat_wit_thmss o snd o snd) o filter (fst o snd)) wit_treess;
traytel@56179
  2329
traytel@56179
  2330
        val nonredundant_coind_wit_argsss =
traytel@56179
  2331
          fold (fn i => fn argsss =>
traytel@56179
  2332
            nth_map (i - 1) (filter_out (fn xs =>
traytel@56179
  2333
              exists (fn ys =>
traytel@56179
  2334
                let
traytel@56179
  2335
                  val xs' = (map (fst o fst) xs, snd (fst (hd xs)));
traytel@56179
  2336
                  val ys' = (map (fst o fst) ys, snd (fst (hd ys)));
traytel@56179
  2337
                in
traytel@56179
  2338
                  eq_pair (subset (op =)) (eq_set