src/HOL/Codatatype/Tools/bnf_gfp.ML
author blanchet
Tue Sep 04 13:05:01 2012 +0200 (2012-09-04)
changeset 49121 9e0acaa470ab
parent 49109 0e5b859e1c91
child 49123 263b0e330d8b
permissions -rw-r--r--
more work on FP sugar
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(*  Title:      HOL/Codatatype/Tools/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 bnf_gfp: binding list -> typ list list -> BNF_Def.BNF list -> Proof.context ->
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    term list * Proof.context
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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_FP_Util
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open BNF_GFP_Util
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open BNF_GFP_Tactics
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datatype wit_tree = Leaf of int | 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], 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, 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_fld_wit vars _ _ (Leaf j) = ([j], nth vars j)
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  | tree_to_fld_wit vars flds witss (Node ((i, nwit, I), subtrees)) =
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     (I, nth flds i $ (Term.list_comb (snd (nth (nth witss i) nwit),
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       map (snd o tree_to_fld_wit vars flds witss) subtrees)));
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fun tree_to_coind_wits _ (Leaf j) = []
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  | tree_to_coind_wits lwitss (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 bnf_gfp bs Dss_insts bnfs lthy =
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  let
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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 b = Binding.name (fold_rev (fn b => fn s => Binding.name_of b ^ s) bs "");
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    (* TODO: check if m, n etc are sane *)
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    val Dss = map (fn Ds => map TFree (fold Term.add_tfreesT Ds [])) Dss_insts;
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    val deads = distinct (op =) (flat Dss);
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    val names_lthy = fold Variable.declare_typ deads lthy;
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    (* tvars *)
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    val ((((((((passiveAs, activeAs), allAs)), (passiveBs, activeBs)),
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      (passiveCs, activeCs)), passiveXs), passiveYs), idxT) = names_lthy
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      |> mk_TFrees live
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      |> apfst (`(chop m))
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      ||> mk_TFrees live
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      ||>> apfst (chop m)
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      ||> mk_TFrees live
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      ||>> apfst (chop m)
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      ||>> mk_TFrees m
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      ||>> mk_TFrees m
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      ||> fst o mk_TFrees 1
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      ||> the_single;
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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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    (* typs *)
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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) (deads @ 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 all_fTs = map2 (fn T => fn U => T --> U) allAs allBs;
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    val self_fTs = map (fn T => T --> T) activeAs;
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    val gTs = map2 (fn T => fn U => T --> U) activeBs activeCs;
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    val all_gTs = map2 (fn T => fn U => T --> U) allBs allCs';
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    val RTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeBs;
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    val sRTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeAs;
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    val R'Ts = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeBs activeCs;
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    val setsRTs = map HOLogic.mk_setT sRTs;
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    val setRTs = map HOLogic.mk_setT RTs;
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    val all_sbisT = HOLogic.mk_tupleT setsRTs;
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    val setR'Ts = map HOLogic.mk_setT R'Ts;
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    val FRTs = mk_FTs (passiveAs @ RTs);
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    val sumBsAs = map2 (curry mk_sumT) activeBs activeAs;
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    val sumFTs = mk_FTs (passiveAs @ sumBsAs);
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    val sum_sTs = map2 (fn T => fn U => T --> U) activeAs sumFTs;
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    (* terms *)
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    val mapsAsAs = map4 mk_map_of_bnf Dss Ass Ass bnfs;
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    val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;
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    val mapsBsCs' = map4 mk_map_of_bnf Dss Bss Css' bnfs;
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    val mapsAsCs' = map4 mk_map_of_bnf Dss Ass Css' bnfs;
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    val map_Inls = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ sumBsAs)) bnfs;
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    val map_Inls_rev = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ sumBsAs)) Bss bnfs;
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    val map_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs;
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    val map_snds = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs;
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    fun mk_setss Ts = map3 mk_sets_of_bnf (map (replicate live) Dss)
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      (map (replicate live) (replicate n Ts)) bnfs;
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    val setssAs = mk_setss allAs;
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    val setssAs' = transpose setssAs;
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    val bis_setss = mk_setss (passiveAs @ RTs);
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    val relsAsBs = map4 mk_rel_of_bnf Dss Ass Bss bnfs;
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    val bds = map3 mk_bd_of_bnf Dss Ass bnfs;
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    val sum_bd = Library.foldr1 (uncurry mk_csum) bds;
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    val sum_bdT = fst (dest_relT (fastype_of sum_bd));
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    val witss = map wits_of_bnf bnfs;
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    val emptys = map (fn T => HOLogic.mk_set T []) passiveAs;
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    val Zeros = map (fn empty =>
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     HOLogic.mk_tuple (map (fn U => absdummy U empty) activeAs)) emptys;
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    val hrecTs = map fastype_of Zeros;
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    val hsetTs = map (fn hrecT => Library.foldr (op -->) (sTs, HOLogic.natT --> hrecT)) hrecTs;
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    val (((((((((((((((((((((((((((((((((((zs, zs'), zs_copy), zs_copy2),
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      z's), As), As_copy), Bs), Bs_copy), B's), B''s), ss), sum_ss), s's), s''s), fs), fs_copy),
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      self_fs), all_fs), gs), all_gs), xFs), xFs_copy), RFs), (Rtuple, Rtuple')), (hrecs, hrecs')),
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      (nat, nat')), Rs), Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), Kss),
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      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 "A" ATs
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      ||>> mk_Frees "A" ATs
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      ||>> mk_Frees "B" BTs
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      ||>> mk_Frees "B" BTs
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      ||>> mk_Frees "B'" B'Ts
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      ||>> mk_Frees "B''" B''Ts
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      ||>> mk_Frees "s" sTs
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      ||>> mk_Frees "sums" sum_sTs
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      ||>> mk_Frees "s'" s'Ts
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      ||>> mk_Frees "s''" s''Ts
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      ||>> mk_Frees "f" fTs
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      ||>> mk_Frees "f" fTs
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      ||>> mk_Frees "f" self_fTs
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      ||>> mk_Frees "f" all_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 "x" FTsAs
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      ||>> mk_Frees "x" FRTs
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      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Rtuple") all_sbisT
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      ||>> mk_Frees' "rec" hrecTs
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      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT
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      ||>> mk_Frees "R" setRTs
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      ||>> mk_Frees "R" setRTs
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      ||>> mk_Frees "R'" setR'Ts
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      ||>> mk_Frees "R" setsRTs
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      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") idxT
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      ||>> yield_singleton (mk_Frees "I") (HOLogic.mk_setT idxT)
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      ||>> mk_Frees "Ri" (map (fn T => idxT --> T) setRTs)
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      ||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) activeAs) ATs);
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    val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
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    val passive_diags = map mk_diag As;
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    val active_UNIVs = map HOLogic.mk_UNIV activeAs;
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    val sum_UNIVs = map HOLogic.mk_UNIV sumBsAs;
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    val passive_ids = map HOLogic.id_const passiveAs;
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    val active_ids = map HOLogic.id_const activeAs;
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    val Inls = map2 Inl_const activeBs activeAs;
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    val fsts = map fst_const RTs;
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    val snds = map snd_const RTs;
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    (* thms *)
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    val bd_card_orders = map bd_card_order_of_bnf bnfs;
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    val bd_card_order = hd bd_card_orders
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    val bd_Card_orders = map bd_Card_order_of_bnf bnfs;
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    val bd_Card_order = hd bd_Card_orders;
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    val bd_Cinfinites = map bd_Cinfinite_of_bnf bnfs;
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    val bd_Cinfinite = hd bd_Cinfinites;
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    val bd_Cnotzeros = map bd_Cnotzero_of_bnf bnfs;
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    val bd_Cnotzero = hd bd_Cnotzeros;
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    val in_bds = map in_bd_of_bnf bnfs;
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    val in_monos = map in_mono_of_bnf bnfs;
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    val map_comps = map map_comp_of_bnf bnfs;
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    val map_comp's = map map_comp'_of_bnf bnfs;
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    val map_congs = map map_cong_of_bnf bnfs;
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    val map_id's = map map_id'_of_bnf bnfs;
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    val pred_defs = map pred_def_of_bnf bnfs;
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    val rel_congs = map rel_cong_of_bnf bnfs;
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    val rel_converses = map rel_converse_of_bnf bnfs;
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    val rel_defs = map rel_def_of_bnf bnfs;
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    val rel_Grs = map rel_Gr_of_bnf bnfs;
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    val rel_Ids = map rel_Id_of_bnf bnfs;
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    val rel_monos = map rel_mono_of_bnf bnfs;
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    val rel_Os = map rel_O_of_bnf bnfs;
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    val map_wpulls = map map_wpull_of_bnf bnfs;
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    val set_bdss = map set_bd_of_bnf bnfs;
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    val set_natural'ss = map set_natural'_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_comp =
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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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        Skip_Proof.prove lthy [] []
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          (fold_rev Logic.all (x :: fs @ all_gs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))))
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          (K (mk_map_comp_id_tac map_comp))
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        |> Thm.close_derivation
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      end;
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    val map_comp_id_thms = map5 mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comp's;
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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_congL x mapAsAs sets map_cong map_id' =
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      let
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        fun mk_prem set f z z' =
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          HOLogic.mk_Trueprop
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            (mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z))));
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        val prems = map4 mk_prem (drop m sets) self_fs zs zs';
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        val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq
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         (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x))
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      in
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        Skip_Proof.prove lthy [] []
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          (fold_rev Logic.all (x :: self_fs) (Logic.list_implies (prems, goal)))
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          (K (mk_map_congL_tac m map_cong map_id'))
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        |> Thm.close_derivation
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      end;
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    val map_congL_thms = map5 mk_map_congL xFs mapsAsAs setssAs map_congs map_id's;
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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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    val in_cong'_thms = map (fn bnf => in_cong_of_bnf bnf OF (replicate m refl)) bnfs;
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    val map_arg_cong_thms =
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   270
      let
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   271
        val prems = map2 (fn x => fn y =>
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   272
          HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))) xFs xFs_copy;
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   273
        val maps = map (fn map => Term.list_comb (map, all_fs)) mapsAsBs;
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   274
        val concls = map3 (fn x => fn y => fn map =>
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   275
          HOLogic.mk_Trueprop (HOLogic.mk_eq (map $ x, map $ y))) xFs xFs_copy maps;
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   276
        val goals =
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   277
          map4 (fn prem => fn concl => fn x => fn y =>
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   278
            fold_rev Logic.all (x :: y :: all_fs) (Logic.mk_implies (prem, concl)))
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   279
          prems concls xFs xFs_copy;
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   280
      in
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   281
        map (fn goal => Skip_Proof.prove lthy [] [] goal
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   282
          (K ((hyp_subst_tac THEN' rtac refl) 1)) |> Thm.close_derivation) goals
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   283
      end;
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   284
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   285
    val timer = time (timer "Derived simple theorems");
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   286
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   287
    (* coalgebra *)
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   288
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   289
    val coalg_bind = Binding.suffix_name ("_" ^ coN ^ algN) b;
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   290
    val coalg_name = Binding.name_of coalg_bind;
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   291
    val coalg_def_bind = (Thm.def_binding coalg_bind, []);
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   292
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   293
    (*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in A1 .. Am B1 ... Bn)*)
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   294
    val coalg_spec =
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   295
      let
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   296
        val coalgT = Library.foldr (op -->) (ATs @ BTs @ sTs, HOLogic.boolT);
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   297
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   298
        val ins = map3 mk_in (replicate n (As @ Bs)) setssAs FTsAs;
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   299
        fun mk_coalg_conjunct B s X z z' =
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   300
          mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X)));
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   301
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   302
        val lhs = Term.list_comb (Free (coalg_name, coalgT), As @ Bs @ ss);
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   303
        val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_coalg_conjunct Bs ss ins zs zs')
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   304
      in
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   305
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
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   306
      end;
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   307
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   308
    val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) =
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   309
        lthy
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   310
        |> Specification.definition (SOME (coalg_bind, NONE, NoSyn), (coalg_def_bind, coalg_spec))
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   311
        ||> `Local_Theory.restore;
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   312
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   313
    (*transforms defined frees into consts*)
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   314
    val phi = Proof_Context.export_morphism lthy_old lthy;
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   315
    val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free));
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   316
    val coalg_def = Morphism.thm phi coalg_def_free;
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   317
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   318
    fun mk_coalg As Bs ss =
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   319
      let
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   320
        val args = As @ Bs @ ss;
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   321
        val Ts = map fastype_of args;
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   322
        val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT);
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   323
      in
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   324
        Term.list_comb (Const (coalg, coalgT), args)
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   325
      end;
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   326
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   327
    val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
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   328
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   329
    val coalg_in_thms = map (fn i =>
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   330
      coalg_def RS @{thm subst[of _ _ "%x. x"]} RS mk_conjunctN n i RS bspec) ks
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   331
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   332
    val coalg_set_thmss =
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   333
      let
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   334
        val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
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   335
        fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));
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   336
        fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) B);
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   337
        val prems = map2 mk_prem zs Bs;
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   338
        val conclss = map3 (fn s => fn x => fn sets => map2 (mk_concl s x) (As @ Bs) sets)
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   339
          ss zs setssAs;
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   340
        val goalss = map3 (fn x => fn prem => fn concls => map (fn concl =>
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   341
          fold_rev Logic.all (x :: As @ Bs @ ss)
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   342
            (Logic.list_implies (coalg_prem :: [prem], concl))) concls) zs prems conclss;
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   343
      in
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   344
        map (fn goals => map (fn goal => Skip_Proof.prove lthy [] [] goal
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   345
          (K (mk_coalg_set_tac coalg_def)) |> Thm.close_derivation) goals) goalss
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   346
      end;
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   347
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   348
    val coalg_set_thmss' = transpose coalg_set_thmss;
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   349
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   350
    fun mk_tcoalg ATs BTs = mk_coalg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);
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   351
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   352
    val tcoalg_thm =
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   353
      let
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   354
        val goal = fold_rev Logic.all ss
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   355
          (HOLogic.mk_Trueprop (mk_tcoalg passiveAs activeAs ss))
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   356
      in
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   357
        Skip_Proof.prove lthy [] [] goal
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   358
          (K (stac coalg_def 1 THEN CONJ_WRAP
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   359
            (K (EVERY' [rtac ballI, rtac CollectI,
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   360
              CONJ_WRAP' (K (EVERY' [rtac @{thm subset_UNIV}])) allAs] 1)) ss))
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   361
        |> Thm.close_derivation
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   362
      end;
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   363
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   364
    val timer = time (timer "Coalgebra definition & thms");
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   365
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   366
    (* morphism *)
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   367
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   368
    val mor_bind = Binding.suffix_name ("_" ^ morN) b;
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   369
    val mor_name = Binding.name_of mor_bind;
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   370
    val mor_def_bind = (Thm.def_binding mor_bind, []);
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   371
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   372
    (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*)
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   373
    (*mor) forall i = 1 ... n: (\<forall>x \<in> Bi.
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   374
       Fi_map id ... id f1 ... fn (si x) = si' (fi x)*)
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   375
    val mor_spec =
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   376
      let
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   377
        val morT = Library.foldr (op -->) (BTs @ sTs @ B'Ts @ s'Ts @ fTs, HOLogic.boolT);
blanchet@48975
   378
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   379
        fun mk_fbetw f B1 B2 z z' =
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   380
          mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));
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   381
        fun mk_mor B mapAsBs f s s' z z' =
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   382
          mk_Ball B (Term.absfree z' (HOLogic.mk_eq
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   383
            (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z))));
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   384
        val lhs = Term.list_comb (Free (mor_name, morT), Bs @ ss @ B's @ s's @ fs);
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   385
        val rhs = HOLogic.mk_conj
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   386
          (Library.foldr1 HOLogic.mk_conj (map5 mk_fbetw fs Bs B's zs zs'),
blanchet@48975
   387
           Library.foldr1 HOLogic.mk_conj (map7 mk_mor Bs mapsAsBs fs ss s's zs zs'))
blanchet@48975
   388
      in
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   389
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
   390
      end;
blanchet@48975
   391
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   392
    val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
blanchet@48975
   393
        lthy
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   394
        |> Specification.definition (SOME (mor_bind, NONE, NoSyn), (mor_def_bind, mor_spec))
blanchet@48975
   395
        ||> `Local_Theory.restore;
blanchet@48975
   396
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   397
    (*transforms defined frees into consts*)
blanchet@48975
   398
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   399
    val mor = fst (Term.dest_Const (Morphism.term phi mor_free));
blanchet@48975
   400
    val mor_def = Morphism.thm phi mor_def_free;
blanchet@48975
   401
blanchet@48975
   402
    fun mk_mor Bs1 ss1 Bs2 ss2 fs =
blanchet@48975
   403
      let
blanchet@48975
   404
        val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;
blanchet@48975
   405
        val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);
blanchet@48975
   406
        val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   407
      in
blanchet@48975
   408
        Term.list_comb (Const (mor, morT), args)
blanchet@48975
   409
      end;
blanchet@48975
   410
blanchet@48975
   411
    val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
blanchet@48975
   412
blanchet@48975
   413
    val (mor_image_thms, morE_thms) =
blanchet@48975
   414
      let
blanchet@48975
   415
        val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
blanchet@48975
   416
        fun mk_image_goal f B1 B2 = fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs)
blanchet@48975
   417
          (Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_subset (mk_image f $ B1) B2)));
blanchet@48975
   418
        val image_goals = map3 mk_image_goal fs Bs B's;
blanchet@48975
   419
        fun mk_elim_goal B mapAsBs f s s' x =
blanchet@48975
   420
          fold_rev Logic.all (x :: Bs @ ss @ B's @ s's @ fs)
blanchet@48975
   421
            (Logic.list_implies ([prem, HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B))],
blanchet@48975
   422
              HOLogic.mk_Trueprop (HOLogic.mk_eq
blanchet@48975
   423
               (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x)))));
blanchet@48975
   424
        val elim_goals = map6 mk_elim_goal Bs mapsAsBs fs ss s's zs;
blanchet@48975
   425
        fun prove goal =
traytel@49109
   426
          Skip_Proof.prove lthy [] [] goal (K (mk_mor_elim_tac mor_def))
traytel@49109
   427
          |> Thm.close_derivation;
blanchet@48975
   428
      in
blanchet@48975
   429
        (map prove image_goals, map prove elim_goals)
blanchet@48975
   430
      end;
blanchet@48975
   431
blanchet@48975
   432
    val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms;
blanchet@48975
   433
blanchet@48975
   434
    val mor_incl_thm =
blanchet@48975
   435
      let
blanchet@48975
   436
        val prems = map2 (HOLogic.mk_Trueprop oo mk_subset) Bs Bs_copy;
blanchet@48975
   437
        val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
blanchet@48975
   438
      in
blanchet@48975
   439
        Skip_Proof.prove lthy [] []
blanchet@48975
   440
          (fold_rev Logic.all (Bs @ ss @ Bs_copy) (Logic.list_implies (prems, concl)))
blanchet@48975
   441
          (K (mk_mor_incl_tac mor_def map_id's))
traytel@49109
   442
        |> Thm.close_derivation
blanchet@48975
   443
      end;
blanchet@48975
   444
blanchet@48975
   445
    val mor_id_thm = mor_incl_thm OF (replicate n subset_refl);
blanchet@48975
   446
blanchet@48975
   447
    val mor_comp_thm =
blanchet@48975
   448
      let
blanchet@48975
   449
        val prems =
blanchet@48975
   450
          [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),
blanchet@48975
   451
           HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];
blanchet@48975
   452
        val concl =
blanchet@48975
   453
          HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));
blanchet@48975
   454
      in
blanchet@48975
   455
        Skip_Proof.prove lthy [] []
blanchet@48975
   456
          (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)
blanchet@48975
   457
            (Logic.list_implies (prems, concl)))
blanchet@48975
   458
          (K (mk_mor_comp_tac mor_def mor_image'_thms morE_thms map_comp_id_thms))
traytel@49109
   459
        |> Thm.close_derivation
blanchet@48975
   460
      end;
blanchet@48975
   461
blanchet@48975
   462
    val mor_cong_thm =
blanchet@48975
   463
      let
blanchet@48975
   464
        val prems = map HOLogic.mk_Trueprop
blanchet@48975
   465
         (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])
blanchet@48975
   466
        val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);
blanchet@48975
   467
      in
blanchet@48975
   468
        Skip_Proof.prove lthy [] []
blanchet@48975
   469
          (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ fs_copy)
blanchet@48975
   470
            (Logic.list_implies (prems, concl)))
blanchet@48975
   471
          (K ((hyp_subst_tac THEN' atac) 1))
traytel@49109
   472
        |> Thm.close_derivation
blanchet@48975
   473
      end;
blanchet@48975
   474
blanchet@48975
   475
    val mor_UNIV_thm =
blanchet@48975
   476
      let
blanchet@48975
   477
        fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
blanchet@48975
   478
            (HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s),
blanchet@48975
   479
            HOLogic.mk_comp (s', f));
blanchet@48975
   480
        val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;
blanchet@48975
   481
        val rhs = Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct mapsAsBs fs ss s's);
blanchet@48975
   482
      in
blanchet@48975
   483
        Skip_Proof.prove lthy [] []
blanchet@48975
   484
          (fold_rev Logic.all (ss @ s's @ fs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))))
blanchet@48975
   485
          (K (mk_mor_UNIV_tac morE_thms mor_def))
traytel@49109
   486
        |> Thm.close_derivation
blanchet@48975
   487
      end;
blanchet@48975
   488
blanchet@48975
   489
    val mor_str_thm =
blanchet@48975
   490
      let
blanchet@48975
   491
        val maps = map2 (fn Ds => fn bnf => Term.list_comb
blanchet@48975
   492
          (mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs;
blanchet@48975
   493
      in
blanchet@48975
   494
        Skip_Proof.prove lthy [] []
blanchet@48975
   495
          (fold_rev Logic.all ss (HOLogic.mk_Trueprop
blanchet@48975
   496
            (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss)))
blanchet@48975
   497
          (K (mk_mor_str_tac ks mor_UNIV_thm))
traytel@49109
   498
        |> Thm.close_derivation
blanchet@48975
   499
      end;
blanchet@48975
   500
blanchet@48975
   501
    val mor_sum_case_thm =
blanchet@48975
   502
      let
blanchet@48975
   503
        val maps = map3 (fn s => fn sum_s => fn map =>
blanchet@48975
   504
          mk_sum_case (HOLogic.mk_comp (Term.list_comb (map, passive_ids @ Inls), s)) sum_s)
blanchet@48975
   505
          s's sum_ss map_Inls;
blanchet@48975
   506
      in
blanchet@48975
   507
        Skip_Proof.prove lthy [] []
blanchet@48975
   508
          (fold_rev Logic.all (s's @ sum_ss) (HOLogic.mk_Trueprop
blanchet@48975
   509
            (mk_mor (map HOLogic.mk_UNIV activeBs) s's sum_UNIVs maps Inls)))
blanchet@48975
   510
          (K (mk_mor_sum_case_tac ks mor_UNIV_thm))
traytel@49109
   511
        |> Thm.close_derivation
blanchet@48975
   512
      end;
blanchet@48975
   513
blanchet@48975
   514
    val timer = time (timer "Morphism definition & thms");
blanchet@48975
   515
blanchet@48975
   516
    fun hset_rec_bind j = Binding.suffix_name ("_" ^ hset_recN ^ (if m = 1 then "" else
blanchet@48975
   517
      string_of_int j)) b;
blanchet@48975
   518
    val hset_rec_name = Binding.name_of o hset_rec_bind;
blanchet@48975
   519
    val hset_rec_def_bind = rpair [] o Thm.def_binding o hset_rec_bind;
blanchet@48975
   520
blanchet@48975
   521
    fun hset_rec_spec j Zero hsetT hrec hrec' =
blanchet@48975
   522
      let
blanchet@48975
   523
        fun mk_Suc s setsAs z z' =
blanchet@48975
   524
          let
blanchet@48975
   525
            val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m setsAs);
blanchet@48975
   526
            fun mk_UN set k = mk_UNION (set $ (s $ z)) (mk_nthN n hrec k);
blanchet@48975
   527
          in
blanchet@48975
   528
            Term.absfree z'
blanchet@48975
   529
              (mk_union (set $ (s $ z), Library.foldl1 mk_union (map2 mk_UN sets ks)))
blanchet@48975
   530
          end;
blanchet@48975
   531
blanchet@48975
   532
        val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec'
blanchet@48975
   533
          (HOLogic.mk_tuple (map4 mk_Suc ss setssAs zs zs')));
blanchet@48975
   534
blanchet@48975
   535
        val lhs = Term.list_comb (Free (hset_rec_name j, hsetT), ss);
blanchet@48975
   536
        val rhs = mk_nat_rec Zero Suc;
blanchet@48975
   537
      in
blanchet@48975
   538
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
   539
      end;
blanchet@48975
   540
blanchet@48975
   541
    val ((hset_rec_frees, (_, hset_rec_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   542
      lthy
blanchet@48975
   543
      |> fold_map5 (fn j => fn Zero => fn hsetT => fn hrec => fn hrec' => Specification.definition
blanchet@48975
   544
        (SOME (hset_rec_bind j, NONE, NoSyn),
blanchet@48975
   545
          (hset_rec_def_bind j, hset_rec_spec j Zero hsetT hrec hrec')))
blanchet@48975
   546
        ls Zeros hsetTs hrecs hrecs'
blanchet@48975
   547
      |>> apsnd split_list o split_list
blanchet@48975
   548
      ||> `Local_Theory.restore;
blanchet@48975
   549
blanchet@48975
   550
    (*transforms defined frees into consts*)
blanchet@48975
   551
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   552
blanchet@48975
   553
    val hset_rec_defs = map (Morphism.thm phi) hset_rec_def_frees;
blanchet@48975
   554
    val hset_recs = map (fst o Term.dest_Const o Morphism.term phi) hset_rec_frees;
blanchet@48975
   555
blanchet@48975
   556
    fun mk_hset_rec ss nat i j T =
blanchet@48975
   557
      let
blanchet@48975
   558
        val args = ss @ [nat];
blanchet@48975
   559
        val Ts = map fastype_of ss;
blanchet@48975
   560
        val bTs = map domain_type Ts;
blanchet@48975
   561
        val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) bTs)
blanchet@48975
   562
        val hset_recT = Library.foldr (op -->) (Ts, HOLogic.natT --> hrecT);
blanchet@48975
   563
      in
blanchet@48975
   564
        mk_nthN n (Term.list_comb (Const (nth hset_recs (j - 1), hset_recT), args)) i
blanchet@48975
   565
      end;
blanchet@48975
   566
blanchet@48975
   567
    val hset_rec_0ss = mk_rec_simps n @{thm nat_rec_0} hset_rec_defs;
blanchet@48975
   568
    val hset_rec_Sucss = mk_rec_simps n @{thm nat_rec_Suc} hset_rec_defs;
blanchet@48975
   569
    val hset_rec_0ss' = transpose hset_rec_0ss;
blanchet@48975
   570
    val hset_rec_Sucss' = transpose hset_rec_Sucss;
blanchet@48975
   571
blanchet@48975
   572
    fun hset_bind i j = Binding.suffix_name ("_" ^ hsetN ^
blanchet@48975
   573
      (if m = 1 then "" else string_of_int j)) (nth bs (i - 1));
blanchet@48975
   574
    val hset_name = Binding.name_of oo hset_bind;
blanchet@48975
   575
    val hset_def_bind = rpair [] o Thm.def_binding oo hset_bind;
blanchet@48975
   576
blanchet@48975
   577
    fun hset_spec i j =
blanchet@48975
   578
      let
blanchet@48975
   579
        val U = nth activeAs (i - 1);
blanchet@48975
   580
        val z = nth zs (i - 1);
blanchet@48975
   581
        val T = nth passiveAs (j - 1);
blanchet@48975
   582
        val setT = HOLogic.mk_setT T;
blanchet@48975
   583
        val hsetT = Library.foldr (op -->) (sTs, U --> setT);
blanchet@48975
   584
blanchet@48975
   585
        val lhs = Term.list_comb (Free (hset_name i j, hsetT), ss @ [z]);
blanchet@48975
   586
        val rhs = mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
blanchet@48975
   587
          (Term.absfree nat' (mk_hset_rec ss nat i j T $ z));
blanchet@48975
   588
      in
blanchet@48975
   589
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
   590
      end;
blanchet@48975
   591
blanchet@48975
   592
    val ((hset_frees, (_, hset_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   593
      lthy
blanchet@48975
   594
      |> fold_map (fn i => fold_map (fn j => Specification.definition
blanchet@48975
   595
        (SOME (hset_bind i j, NONE, NoSyn), (hset_def_bind i j, hset_spec i j))) ls) ks
blanchet@48975
   596
      |>> map (apsnd split_list o split_list)
blanchet@48975
   597
      |>> apsnd split_list o split_list
blanchet@48975
   598
      ||> `Local_Theory.restore;
blanchet@48975
   599
blanchet@48975
   600
    (*transforms defined frees into consts*)
blanchet@48975
   601
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   602
blanchet@48975
   603
    val hset_defss = map (map (Morphism.thm phi)) hset_def_frees;
blanchet@48975
   604
    val hset_defss' = transpose hset_defss;
blanchet@48975
   605
    val hset_namess = map (map (fst o Term.dest_Const o Morphism.term phi)) hset_frees;
blanchet@48975
   606
blanchet@48975
   607
    fun mk_hset ss i j T =
blanchet@48975
   608
      let
blanchet@48975
   609
        val Ts = map fastype_of ss;
blanchet@48975
   610
        val bTs = map domain_type Ts;
blanchet@48975
   611
        val hsetT = Library.foldr (op -->) (Ts, nth bTs (i - 1) --> HOLogic.mk_setT T);
blanchet@48975
   612
      in
blanchet@48975
   613
        Term.list_comb (Const (nth (nth hset_namess (i - 1)) (j - 1), hsetT), ss)
blanchet@48975
   614
      end;
blanchet@48975
   615
blanchet@48975
   616
    val hsetssAs = map (fn i => map2 (mk_hset ss i) ls passiveAs) ks;
blanchet@48975
   617
blanchet@48975
   618
    val (set_incl_hset_thmss, set_hset_incl_hset_thmsss) =
blanchet@48975
   619
      let
blanchet@48975
   620
        fun mk_set_incl_hset s x set hset = fold_rev Logic.all (x :: ss)
blanchet@48975
   621
          (HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (hset $ x)));
blanchet@48975
   622
blanchet@48975
   623
        fun mk_set_hset_incl_hset s x y set hset1 hset2 =
blanchet@48975
   624
          fold_rev Logic.all (x :: y :: ss)
blanchet@48975
   625
            (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x, set $ (s $ y))),
blanchet@48975
   626
            HOLogic.mk_Trueprop (mk_subset (hset1 $ x) (hset2 $ y))));
blanchet@48975
   627
blanchet@48975
   628
        val set_incl_hset_goalss =
blanchet@48975
   629
          map4 (fn s => fn x => fn sets => fn hsets =>
blanchet@48975
   630
            map2 (mk_set_incl_hset s x) (take m sets) hsets)
blanchet@48975
   631
          ss zs setssAs hsetssAs;
blanchet@48975
   632
blanchet@48975
   633
        (*xk : F(i)set(m+k) (si yi) ==> F(k)_hset(j) s1 ... sn xk <= F(i)_hset(j) s1 ... sn yi*)
blanchet@48975
   634
        val set_hset_incl_hset_goalsss =
blanchet@48975
   635
          map4 (fn si => fn yi => fn sets => fn hsetsi =>
blanchet@48975
   636
            map3 (fn xk => fn set => fn hsetsk =>
blanchet@48975
   637
              map2 (mk_set_hset_incl_hset si xk yi set) hsetsk hsetsi)
blanchet@48975
   638
            zs_copy (drop m sets) hsetssAs)
blanchet@48975
   639
          ss zs setssAs hsetssAs;
blanchet@48975
   640
      in
blanchet@48975
   641
        (map3 (fn goals => fn defs => fn rec_Sucs =>
blanchet@48975
   642
          map3 (fn goal => fn def => fn rec_Suc =>
traytel@49109
   643
            Skip_Proof.prove lthy [] [] goal (K (mk_set_incl_hset_tac def rec_Suc))
traytel@49109
   644
            |> Thm.close_derivation)
blanchet@48975
   645
          goals defs rec_Sucs)
blanchet@48975
   646
        set_incl_hset_goalss hset_defss hset_rec_Sucss,
blanchet@48975
   647
        map3 (fn goalss => fn defsi => fn rec_Sucs =>
blanchet@48975
   648
          map3 (fn k => fn goals => fn defsk =>
blanchet@48975
   649
            map4 (fn goal => fn defk => fn defi => fn rec_Suc =>
blanchet@48975
   650
              Skip_Proof.prove lthy [] [] goal
traytel@49109
   651
                (K (mk_set_hset_incl_hset_tac n [defk, defi] rec_Suc k))
traytel@49109
   652
              |> Thm.close_derivation)
blanchet@48975
   653
            goals defsk defsi rec_Sucs)
blanchet@48975
   654
          ks goalss hset_defss)
blanchet@48975
   655
        set_hset_incl_hset_goalsss hset_defss hset_rec_Sucss)
blanchet@48975
   656
      end;
blanchet@48975
   657
blanchet@48975
   658
    val set_incl_hset_thmss' = transpose set_incl_hset_thmss;
blanchet@48975
   659
    val set_hset_incl_hset_thmsss' = transpose (map transpose set_hset_incl_hset_thmsss);
blanchet@48975
   660
    val set_hset_incl_hset_thmsss'' = map transpose set_hset_incl_hset_thmsss';
blanchet@48975
   661
    val set_hset_thmss = map (map (fn thm => thm RS @{thm set_mp})) set_incl_hset_thmss;
blanchet@48975
   662
    val set_hset_hset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp})))
blanchet@48975
   663
      set_hset_incl_hset_thmsss;
blanchet@48975
   664
    val set_hset_thmss' = transpose set_hset_thmss;
blanchet@48975
   665
    val set_hset_hset_thmsss' = transpose (map transpose set_hset_hset_thmsss);
blanchet@48975
   666
blanchet@48975
   667
    val set_incl_hin_thmss =
blanchet@48975
   668
      let
blanchet@48975
   669
        fun mk_set_incl_hin s x hsets1 set hsets2 T =
blanchet@48975
   670
          fold_rev Logic.all (x :: ss @ As)
blanchet@48975
   671
            (Logic.list_implies
blanchet@48975
   672
              (map2 (fn hset => fn A => HOLogic.mk_Trueprop (mk_subset (hset $ x) A)) hsets1 As,
blanchet@48975
   673
              HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (mk_in As hsets2 T))));
blanchet@48975
   674
blanchet@48975
   675
        val set_incl_hin_goalss =
blanchet@48975
   676
          map4 (fn s => fn x => fn sets => fn hsets =>
blanchet@48975
   677
            map3 (mk_set_incl_hin s x hsets) (drop m sets) hsetssAs activeAs)
blanchet@48975
   678
          ss zs setssAs hsetssAs;
blanchet@48975
   679
      in
blanchet@48975
   680
        map2 (map2 (fn goal => fn thms =>
traytel@49109
   681
          Skip_Proof.prove lthy [] [] goal (K (mk_set_incl_hin_tac thms))
traytel@49109
   682
          |> Thm.close_derivation))
blanchet@48975
   683
        set_incl_hin_goalss set_hset_incl_hset_thmsss
blanchet@48975
   684
      end;
blanchet@48975
   685
blanchet@48975
   686
    val hset_minimal_thms =
blanchet@48975
   687
      let
blanchet@48975
   688
        fun mk_passive_prem set s x K =
blanchet@48975
   689
          Logic.all x (HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (K $ x)));
blanchet@48975
   690
blanchet@48975
   691
        fun mk_active_prem s x1 K1 set x2 K2 =
blanchet@48975
   692
          fold_rev Logic.all [x1, x2]
blanchet@48975
   693
            (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x2, set $ (s $ x1))),
blanchet@48975
   694
              HOLogic.mk_Trueprop (mk_subset (K2 $ x2) (K1 $ x1))));
blanchet@48975
   695
blanchet@48975
   696
        val premss = map2 (fn j => fn Ks =>
blanchet@48975
   697
          map4 mk_passive_prem (map (fn xs => nth xs (j - 1)) setssAs) ss zs Ks @
blanchet@48975
   698
            flat (map4 (fn sets => fn s => fn x1 => fn K1 =>
blanchet@48975
   699
              map3 (mk_active_prem s x1 K1) (drop m sets) zs_copy Ks) setssAs ss zs Ks))
blanchet@48975
   700
          ls Kss;
blanchet@48975
   701
blanchet@48975
   702
        val hset_rec_minimal_thms =
blanchet@48975
   703
          let
blanchet@48975
   704
            fun mk_conjunct j T i K x = mk_subset (mk_hset_rec ss nat i j T $ x) (K $ x);
blanchet@48975
   705
            fun mk_concl j T Ks = list_all_free zs
blanchet@48975
   706
              (Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs));
blanchet@48975
   707
            val concls = map3 mk_concl ls passiveAs Kss;
blanchet@48975
   708
blanchet@48975
   709
            val goals = map2 (fn prems => fn concl =>
blanchet@48975
   710
              Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls
blanchet@48975
   711
blanchet@48975
   712
            val ctss =
blanchet@48975
   713
              map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
blanchet@48975
   714
          in
blanchet@48975
   715
            map4 (fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>
blanchet@48975
   716
              singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   717
                (Skip_Proof.prove lthy [] [] goal
traytel@49109
   718
                  (mk_hset_rec_minimal_tac m cts hset_rec_0s hset_rec_Sucs))
traytel@49109
   719
              |> Thm.close_derivation)
blanchet@48975
   720
            goals ctss hset_rec_0ss' hset_rec_Sucss'
blanchet@48975
   721
          end;
blanchet@48975
   722
blanchet@48975
   723
        fun mk_conjunct j T i K x = mk_subset (mk_hset ss i j T $ x) (K $ x);
blanchet@48975
   724
        fun mk_concl j T Ks = Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs);
blanchet@48975
   725
        val concls = map3 mk_concl ls passiveAs Kss;
blanchet@48975
   726
blanchet@48975
   727
        val goals = map3 (fn Ks => fn prems => fn concl =>
blanchet@48975
   728
          fold_rev Logic.all (Ks @ ss @ zs)
blanchet@48975
   729
            (Logic.list_implies (prems, HOLogic.mk_Trueprop concl))) Kss premss concls;
blanchet@48975
   730
      in
blanchet@48975
   731
        map3 (fn goal => fn hset_defs => fn hset_rec_minimal =>
blanchet@48975
   732
          Skip_Proof.prove lthy [] [] goal
traytel@49109
   733
            (mk_hset_minimal_tac n hset_defs hset_rec_minimal)
traytel@49109
   734
          |> Thm.close_derivation)
blanchet@48975
   735
        goals hset_defss' hset_rec_minimal_thms
blanchet@48975
   736
      end;
blanchet@48975
   737
blanchet@48975
   738
    val mor_hset_thmss =
blanchet@48975
   739
      let
blanchet@48975
   740
        val mor_hset_rec_thms =
blanchet@48975
   741
          let
blanchet@48975
   742
            fun mk_conjunct j T i f x B =
blanchet@48975
   743
              HOLogic.mk_imp (HOLogic.mk_mem (x, B), HOLogic.mk_eq
blanchet@48975
   744
               (mk_hset_rec s's nat i j T $ (f $ x), mk_hset_rec ss nat i j T $ x));
blanchet@48975
   745
blanchet@48975
   746
            fun mk_concl j T = list_all_free zs
blanchet@48975
   747
              (Library.foldr1 HOLogic.mk_conj (map4 (mk_conjunct j T) ks fs zs Bs));
blanchet@48975
   748
            val concls = map2 mk_concl ls passiveAs;
blanchet@48975
   749
blanchet@48975
   750
            val ctss =
blanchet@48975
   751
              map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
blanchet@48975
   752
blanchet@48975
   753
            val goals = map (fn concl =>
blanchet@48975
   754
              Logic.list_implies ([coalg_prem, mor_prem], HOLogic.mk_Trueprop concl)) concls;
blanchet@48975
   755
          in
blanchet@48975
   756
            map5 (fn j => fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>
blanchet@48975
   757
              singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
   758
                (Skip_Proof.prove lthy [] [] goal
blanchet@48975
   759
                  (K (mk_mor_hset_rec_tac m n cts j hset_rec_0s hset_rec_Sucs
traytel@49109
   760
                    morE_thms set_natural'ss coalg_set_thmss)))
traytel@49109
   761
              |> Thm.close_derivation)
blanchet@48975
   762
            ls goals ctss hset_rec_0ss' hset_rec_Sucss'
blanchet@48975
   763
          end;
blanchet@48975
   764
blanchet@48975
   765
        val mor_hset_rec_thmss = map (fn thm => map (fn i =>
blanchet@48975
   766
          mk_specN n thm RS mk_conjunctN n i RS mp) ks) mor_hset_rec_thms;
blanchet@48975
   767
blanchet@48975
   768
        fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));
blanchet@48975
   769
blanchet@48975
   770
        fun mk_concl j T i f x = HOLogic.mk_Trueprop (HOLogic.mk_eq
blanchet@48975
   771
          (mk_hset s's i j T $ (f $ x), mk_hset ss i j T $ x));
blanchet@48975
   772
blanchet@48975
   773
        val goalss = map2 (fn j => fn T => map4 (fn i => fn f => fn x => fn B =>
blanchet@48975
   774
          fold_rev Logic.all (x :: As @ Bs @ ss @ B's @ s's @ fs)
blanchet@48975
   775
            (Logic.list_implies ([coalg_prem, mor_prem,
blanchet@48975
   776
              mk_prem x B], mk_concl j T i f x))) ks fs zs Bs) ls passiveAs;
blanchet@48975
   777
      in
blanchet@48975
   778
        map3 (map3 (fn goal => fn hset_def => fn mor_hset_rec =>
blanchet@48975
   779
          Skip_Proof.prove lthy [] [] goal
traytel@49109
   780
            (K (mk_mor_hset_tac hset_def mor_hset_rec))
traytel@49109
   781
          |> Thm.close_derivation))
blanchet@48975
   782
        goalss hset_defss' mor_hset_rec_thmss
blanchet@48975
   783
      end;
blanchet@48975
   784
blanchet@48975
   785
    val timer = time (timer "Hereditary sets");
blanchet@48975
   786
blanchet@48975
   787
    (* bisimulation *)
blanchet@48975
   788
blanchet@48975
   789
    val bis_bind = Binding.suffix_name ("_" ^ bisN) b;
blanchet@48975
   790
    val bis_name = Binding.name_of bis_bind;
blanchet@48975
   791
    val bis_def_bind = (Thm.def_binding bis_bind, []);
blanchet@48975
   792
blanchet@48975
   793
    fun mk_bis_le_conjunct R B1 B2 = mk_subset R (mk_Times (B1, B2));
blanchet@48975
   794
    val bis_le = Library.foldr1 HOLogic.mk_conj (map3 mk_bis_le_conjunct Rs Bs B's)
blanchet@48975
   795
blanchet@48975
   796
    val bis_spec =
blanchet@48975
   797
      let
blanchet@48975
   798
        val bisT = Library.foldr (op -->) (ATs @ BTs @ sTs @ B'Ts @ s'Ts @ setRTs, HOLogic.boolT);
blanchet@48975
   799
blanchet@48975
   800
        val fst_args = passive_ids @ fsts;
blanchet@48975
   801
        val snd_args = passive_ids @ snds;
blanchet@48975
   802
        fun mk_bis R s s' b1 b2 RF map1 map2 sets =
blanchet@48975
   803
          list_all_free [b1, b2] (HOLogic.mk_imp
blanchet@48975
   804
            (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
blanchet@48975
   805
            mk_Bex (mk_in (As @ Rs) sets (snd (dest_Free RF))) (Term.absfree (dest_Free RF)
blanchet@48975
   806
              (HOLogic.mk_conj
blanchet@48975
   807
                (HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1),
blanchet@48975
   808
                HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2))))));
blanchet@48975
   809
blanchet@48975
   810
        val lhs = Term.list_comb (Free (bis_name, bisT), As @ Bs @ ss @ B's @ s's @ Rs);
blanchet@48975
   811
        val rhs = HOLogic.mk_conj
blanchet@48975
   812
          (bis_le, Library.foldr1 HOLogic.mk_conj
blanchet@48975
   813
            (map9 mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss))
blanchet@48975
   814
      in
blanchet@48975
   815
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
   816
      end;
blanchet@48975
   817
blanchet@48975
   818
    val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) =
blanchet@48975
   819
        lthy
blanchet@48975
   820
        |> Specification.definition (SOME (bis_bind, NONE, NoSyn), (bis_def_bind, bis_spec))
blanchet@48975
   821
        ||> `Local_Theory.restore;
blanchet@48975
   822
blanchet@48975
   823
    (*transforms defined frees into consts*)
blanchet@48975
   824
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   825
    val bis = fst (Term.dest_Const (Morphism.term phi bis_free));
blanchet@48975
   826
    val bis_def = Morphism.thm phi bis_def_free;
blanchet@48975
   827
blanchet@48975
   828
    fun mk_bis As Bs1 ss1 Bs2 ss2 Rs =
blanchet@48975
   829
      let
blanchet@48975
   830
        val args = As @ Bs1 @ ss1 @ Bs2 @ ss2 @ Rs;
blanchet@48975
   831
        val Ts = map fastype_of args;
blanchet@48975
   832
        val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   833
      in
blanchet@48975
   834
        Term.list_comb (Const (bis, bisT), args)
blanchet@48975
   835
      end;
blanchet@48975
   836
blanchet@48975
   837
    val bis_cong_thm =
blanchet@48975
   838
      let
blanchet@48975
   839
        val prems = map HOLogic.mk_Trueprop
blanchet@48975
   840
         (mk_bis As Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs)
blanchet@48975
   841
        val concl = HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs_copy);
blanchet@48975
   842
      in
blanchet@48975
   843
        Skip_Proof.prove lthy [] []
blanchet@48975
   844
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs @ Rs_copy)
blanchet@48975
   845
            (Logic.list_implies (prems, concl)))
blanchet@48975
   846
          (K ((hyp_subst_tac THEN' atac) 1))
traytel@49109
   847
        |> Thm.close_derivation
blanchet@48975
   848
      end;
blanchet@48975
   849
blanchet@48975
   850
    val bis_rel_thm =
blanchet@48975
   851
      let
blanchet@48975
   852
        fun mk_conjunct R s s' b1 b2 rel =
blanchet@48975
   853
          list_all_free [b1, b2] (HOLogic.mk_imp
blanchet@48975
   854
            (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
blanchet@48975
   855
            HOLogic.mk_mem (HOLogic.mk_prod (s $ b1, s' $ b2),
blanchet@48975
   856
              Term.list_comb (rel, passive_diags @ Rs))));
blanchet@48975
   857
blanchet@48975
   858
        val rhs = HOLogic.mk_conj
blanchet@48975
   859
          (bis_le, Library.foldr1 HOLogic.mk_conj
blanchet@48975
   860
            (map6 mk_conjunct Rs ss s's zs z's relsAsBs))
blanchet@48975
   861
      in
blanchet@48975
   862
        Skip_Proof.prove lthy [] []
blanchet@48975
   863
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
blanchet@48975
   864
            (HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_bis As Bs ss B's s's Rs, rhs))))
blanchet@48975
   865
          (K (mk_bis_rel_tac m bis_def rel_defs map_comp's map_congs set_natural'ss))
traytel@49109
   866
        |> Thm.close_derivation
blanchet@48975
   867
      end;
blanchet@48975
   868
blanchet@48975
   869
    val bis_converse_thm =
blanchet@48975
   870
      Skip_Proof.prove lthy [] []
blanchet@48975
   871
        (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
blanchet@48975
   872
          (Logic.mk_implies
blanchet@48975
   873
            (HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
blanchet@48975
   874
            HOLogic.mk_Trueprop (mk_bis As B's s's Bs ss (map mk_converse Rs)))))
traytel@49109
   875
        (K (mk_bis_converse_tac m bis_rel_thm rel_congs rel_converses))
traytel@49109
   876
      |> Thm.close_derivation;
blanchet@48975
   877
blanchet@48975
   878
    val bis_O_thm =
blanchet@48975
   879
      let
blanchet@48975
   880
        val prems =
blanchet@48975
   881
          [HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
blanchet@48975
   882
           HOLogic.mk_Trueprop (mk_bis As B's s's B''s s''s R's)];
blanchet@48975
   883
        val concl =
blanchet@48975
   884
          HOLogic.mk_Trueprop (mk_bis As Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's));
blanchet@48975
   885
      in
blanchet@48975
   886
        Skip_Proof.prove lthy [] []
blanchet@48975
   887
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ B''s @ s''s @ Rs @ R's)
blanchet@48975
   888
            (Logic.list_implies (prems, concl)))
blanchet@48975
   889
          (K (mk_bis_O_tac m bis_rel_thm rel_congs rel_Os))
traytel@49109
   890
        |> Thm.close_derivation
blanchet@48975
   891
      end;
blanchet@48975
   892
blanchet@48975
   893
    val bis_Gr_thm =
blanchet@48975
   894
      let
blanchet@48975
   895
        val concl =
blanchet@48975
   896
          HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map2 mk_Gr Bs fs));
blanchet@48975
   897
      in
blanchet@48975
   898
        Skip_Proof.prove lthy [] []
blanchet@48975
   899
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ fs)
blanchet@48975
   900
            (Logic.list_implies ([coalg_prem, mor_prem], concl)))
blanchet@48975
   901
          (mk_bis_Gr_tac bis_rel_thm rel_Grs mor_image_thms morE_thms coalg_in_thms)
traytel@49109
   902
        |> Thm.close_derivation
blanchet@48975
   903
      end;
blanchet@48975
   904
blanchet@48975
   905
    val bis_image2_thm = bis_cong_thm OF
blanchet@48975
   906
      ((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) ::
blanchet@48975
   907
      replicate n @{thm image2_Gr});
blanchet@48975
   908
blanchet@48975
   909
    val bis_diag_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) ::
blanchet@48975
   910
      replicate n @{thm diag_Gr});
blanchet@48975
   911
blanchet@48975
   912
    val bis_Union_thm =
blanchet@48975
   913
      let
blanchet@48975
   914
        val prem =
blanchet@48975
   915
          HOLogic.mk_Trueprop (mk_Ball Idx
blanchet@48975
   916
            (Term.absfree idx' (mk_bis As Bs ss B's s's (map (fn R => R $ idx) Ris))));
blanchet@48975
   917
        val concl =
blanchet@48975
   918
          HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map (mk_UNION Idx) Ris));
blanchet@48975
   919
      in
blanchet@48975
   920
        Skip_Proof.prove lthy [] []
blanchet@48975
   921
          (fold_rev Logic.all (Idx :: As @ Bs @ ss @ B's @ s's @ Ris)
blanchet@48975
   922
            (Logic.mk_implies (prem, concl)))
blanchet@48975
   923
          (mk_bis_Union_tac bis_def in_mono'_thms)
traytel@49109
   924
        |> Thm.close_derivation
blanchet@48975
   925
      end;
blanchet@48975
   926
blanchet@48975
   927
    (* self-bisimulation *)
blanchet@48975
   928
blanchet@48975
   929
    fun mk_sbis As Bs ss Rs = mk_bis As Bs ss Bs ss Rs;
blanchet@48975
   930
blanchet@48975
   931
    val sbis_prem = HOLogic.mk_Trueprop (mk_sbis As Bs ss sRs);
blanchet@48975
   932
blanchet@48975
   933
    (* largest self-bisimulation *)
blanchet@48975
   934
blanchet@48975
   935
    fun lsbis_bind i = Binding.suffix_name ("_" ^ lsbisN ^ (if n = 1 then "" else
blanchet@48975
   936
      string_of_int i)) b;
blanchet@48975
   937
    val lsbis_name = Binding.name_of o lsbis_bind;
blanchet@48975
   938
    val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind;
blanchet@48975
   939
blanchet@48975
   940
    val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs
blanchet@48975
   941
      (HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis As Bs ss sRs)));
blanchet@48975
   942
blanchet@48975
   943
    fun lsbis_spec i RT =
blanchet@48975
   944
      let
blanchet@48975
   945
        fun mk_lsbisT RT =
blanchet@48975
   946
          Library.foldr (op -->) (map fastype_of (As @ Bs @ ss), RT);
blanchet@48975
   947
        val lhs = Term.list_comb (Free (lsbis_name i, mk_lsbisT RT), As @ Bs @ ss);
blanchet@48975
   948
        val rhs = mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i));
blanchet@48975
   949
      in
blanchet@48975
   950
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
   951
      end;
blanchet@48975
   952
blanchet@48975
   953
    val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   954
      lthy
blanchet@48975
   955
      |> fold_map2 (fn i => fn RT => Specification.definition
blanchet@48975
   956
        (SOME (lsbis_bind i, NONE, NoSyn), (lsbis_def_bind i, lsbis_spec i RT))) ks setsRTs
blanchet@48975
   957
      |>> apsnd split_list o split_list
blanchet@48975
   958
      ||> `Local_Theory.restore;
blanchet@48975
   959
blanchet@48975
   960
    (*transforms defined frees into consts*)
blanchet@48975
   961
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   962
blanchet@48975
   963
    val lsbis_defs = map (Morphism.thm phi) lsbis_def_frees;
blanchet@48975
   964
    val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees;
blanchet@48975
   965
blanchet@48975
   966
    fun mk_lsbis As Bs ss i =
blanchet@48975
   967
      let
blanchet@48975
   968
        val args = As @ Bs @ ss;
blanchet@48975
   969
        val Ts = map fastype_of args;
blanchet@48975
   970
        val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1)))));
blanchet@48975
   971
        val lsbisT = Library.foldr (op -->) (Ts, RT);
blanchet@48975
   972
      in
blanchet@48975
   973
        Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args)
blanchet@48975
   974
      end;
blanchet@48975
   975
blanchet@48975
   976
    val sbis_lsbis_thm =
blanchet@48975
   977
      Skip_Proof.prove lthy [] []
blanchet@48975
   978
        (fold_rev Logic.all (As @ Bs @ ss)
blanchet@48975
   979
          (HOLogic.mk_Trueprop (mk_sbis As Bs ss (map (mk_lsbis As Bs ss) ks))))
traytel@49109
   980
        (K (mk_sbis_lsbis_tac lsbis_defs bis_Union_thm bis_cong_thm))
traytel@49109
   981
      |> Thm.close_derivation;
blanchet@48975
   982
blanchet@48975
   983
    val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS
blanchet@48975
   984
      (bis_def RS @{thm subst[of _ _ "%x. x"]} RS conjunct1 RS mk_conjunctN n i)) ks;
blanchet@48975
   985
    val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS
blanchet@48975
   986
      (bis_def RS @{thm subst[of _ _ "%x. x"]} RS conjunct2 RS mk_conjunctN n i))) RS mp) ks;
blanchet@48975
   987
blanchet@48975
   988
    val incl_lsbis_thms =
blanchet@48975
   989
      let
blanchet@48975
   990
        fun mk_concl i R = HOLogic.mk_Trueprop (mk_subset R (mk_lsbis As Bs ss i));
blanchet@48975
   991
        val goals = map2 (fn i => fn R => fold_rev Logic.all (As @ Bs @ ss @ sRs)
blanchet@48975
   992
          (Logic.mk_implies (sbis_prem, mk_concl i R))) ks sRs;
blanchet@48975
   993
      in
blanchet@48975
   994
        map3 (fn goal => fn i => fn def => Skip_Proof.prove lthy [] [] goal
traytel@49109
   995
          (K (mk_incl_lsbis_tac n i def)) |> Thm.close_derivation) goals ks lsbis_defs
blanchet@48975
   996
      end;
blanchet@48975
   997
blanchet@48975
   998
    val equiv_lsbis_thms =
blanchet@48975
   999
      let
blanchet@48975
  1000
        fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis As Bs ss i));
blanchet@48975
  1001
        val goals = map2 (fn i => fn B => fold_rev Logic.all (As @ Bs @ ss)
blanchet@48975
  1002
          (Logic.mk_implies (coalg_prem, mk_concl i B))) ks Bs;
blanchet@48975
  1003
      in
blanchet@48975
  1004
        map3 (fn goal => fn l_incl => fn incl_l =>
blanchet@48975
  1005
          Skip_Proof.prove lthy [] [] goal
blanchet@48975
  1006
            (K (mk_equiv_lsbis_tac sbis_lsbis_thm l_incl incl_l
traytel@49109
  1007
              bis_diag_thm bis_converse_thm bis_O_thm))
traytel@49109
  1008
          |> Thm.close_derivation)
blanchet@48975
  1009
        goals lsbis_incl_thms incl_lsbis_thms
blanchet@48975
  1010
      end;
blanchet@48975
  1011
blanchet@48975
  1012
    val timer = time (timer "Bisimulations");
blanchet@48975
  1013
blanchet@48975
  1014
    (* bounds *)
blanchet@48975
  1015
blanchet@48975
  1016
    val (lthy, sbd, sbdT,
blanchet@48975
  1017
      sbd_card_order, sbd_Cinfinite, sbd_Cnotzero, sbd_Card_order, set_sbdss, in_sbds) =
blanchet@48975
  1018
      if n = 1
blanchet@48975
  1019
      then (lthy, sum_bd, sum_bdT,
blanchet@48975
  1020
        bd_card_order, bd_Cinfinite, bd_Cnotzero, bd_Card_order, set_bdss, in_bds)
blanchet@48975
  1021
      else
blanchet@48975
  1022
        let
blanchet@48975
  1023
          val sbdT_bind = Binding.suffix_name ("_" ^ sum_bdTN) b;
blanchet@48975
  1024
blanchet@48975
  1025
          val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) =
blanchet@48975
  1026
            typedef true NONE (sbdT_bind, params, NoSyn)
blanchet@48975
  1027
              (HOLogic.mk_UNIV sum_bdT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
blanchet@48975
  1028
blanchet@48975
  1029
          val sbdT = Type (sbdT_name, params');
blanchet@48975
  1030
          val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT);
blanchet@48975
  1031
blanchet@48975
  1032
          val sbd_bind = Binding.suffix_name ("_" ^ sum_bdN) b;
blanchet@48975
  1033
          val sbd_name = Binding.name_of sbd_bind;
blanchet@48975
  1034
          val sbd_def_bind = (Thm.def_binding sbd_bind, []);
blanchet@48975
  1035
blanchet@48975
  1036
          val sbd_spec = HOLogic.mk_Trueprop
blanchet@48975
  1037
            (HOLogic.mk_eq (Free (sbd_name, mk_relT (`I sbdT)), mk_dir_image sum_bd Abs_sbdT));
blanchet@48975
  1038
blanchet@48975
  1039
          val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) =
blanchet@48975
  1040
            lthy
blanchet@48975
  1041
            |> Specification.definition (SOME (sbd_bind, NONE, NoSyn), (sbd_def_bind, sbd_spec))
blanchet@48975
  1042
            ||> `Local_Theory.restore;
blanchet@48975
  1043
blanchet@48975
  1044
          (*transforms defined frees into consts*)
blanchet@48975
  1045
          val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1046
blanchet@48975
  1047
          val sbd_def = Morphism.thm phi sbd_def_free;
blanchet@48975
  1048
          val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT));
blanchet@48975
  1049
blanchet@48975
  1050
          val sbdT_set_def = the (#set_def sbdT_loc_info);
blanchet@48975
  1051
          val sbdT_Abs_inject = #Abs_inject sbdT_loc_info;
blanchet@48975
  1052
          val sbdT_Abs_cases = #Abs_cases sbdT_loc_info;
blanchet@48975
  1053
blanchet@48975
  1054
          val Abs_sbdT_inj = mk_Abs_inj_thm sbdT_set_def sbdT_Abs_inject;
blanchet@48975
  1055
          val Abs_sbdT_bij = mk_Abs_bij_thm lthy sbdT_set_def sbdT_Abs_inject sbdT_Abs_cases;
blanchet@48975
  1056
blanchet@48975
  1057
          fun mk_sum_Cinfinite [thm] = thm
blanchet@48975
  1058
            | mk_sum_Cinfinite (thm :: thms) =
blanchet@48975
  1059
              @{thm Cinfinite_csum_strong} OF [thm, mk_sum_Cinfinite thms];
blanchet@48975
  1060
blanchet@48975
  1061
          val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;
blanchet@48975
  1062
          val sum_Card_order = sum_Cinfinite RS conjunct2;
blanchet@48975
  1063
blanchet@48975
  1064
          fun mk_sum_card_order [thm] = thm
blanchet@48975
  1065
            | mk_sum_card_order (thm :: thms) =
blanchet@48975
  1066
              @{thm card_order_csum} OF [thm, mk_sum_card_order thms];
blanchet@48975
  1067
blanchet@48975
  1068
          val sum_card_order = mk_sum_card_order bd_card_orders;
blanchet@48975
  1069
blanchet@48975
  1070
          val sbd_ordIso = Local_Defs.fold lthy [sbd_def]
blanchet@48975
  1071
            (@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order]);
blanchet@48975
  1072
          val sbd_card_order =  Local_Defs.fold lthy [sbd_def]
blanchet@48975
  1073
            (@{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]);
blanchet@48975
  1074
          val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite];
blanchet@48975
  1075
          val sbd_Cnotzero = sbd_Cinfinite RS @{thm Cinfinite_Cnotzero};
blanchet@48975
  1076
          val sbd_Card_order = sbd_Cinfinite RS conjunct2;
blanchet@48975
  1077
blanchet@48975
  1078
          fun mk_set_sbd i bd_Card_order bds =
blanchet@48975
  1079
            map (fn thm => @{thm ordLeq_ordIso_trans} OF
blanchet@48975
  1080
              [bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds;
blanchet@48975
  1081
          val set_sbdss = map3 mk_set_sbd ks bd_Card_orders set_bdss;
blanchet@48975
  1082
blanchet@48975
  1083
          fun mk_in_sbd i Co Cnz bd =
blanchet@48975
  1084
            Cnz RS ((@{thm ordLeq_ordIso_trans} OF
blanchet@48975
  1085
              [(Co RS mk_ordLeq_csum n i (Co RS @{thm ordLeq_refl})), sbd_ordIso]) RS
blanchet@48975
  1086
              (bd RS @{thm ordLeq_transitive[OF _
blanchet@48975
  1087
                cexp_mono2_Cnotzero[OF _ csum_Cnotzero2[OF ctwo_Cnotzero]]]}));
blanchet@48975
  1088
          val in_sbds = map4 mk_in_sbd ks bd_Card_orders bd_Cnotzeros in_bds;
blanchet@48975
  1089
       in
blanchet@48975
  1090
         (lthy, sbd, sbdT,
blanchet@48975
  1091
           sbd_card_order, sbd_Cinfinite, sbd_Cnotzero, sbd_Card_order, set_sbdss, in_sbds)
blanchet@48975
  1092
       end;
blanchet@48975
  1093
blanchet@48975
  1094
    fun mk_sbd_sbd 1 = sbd_Card_order RS @{thm ordIso_refl}
blanchet@48975
  1095
      | mk_sbd_sbd n = @{thm csum_absorb1} OF
blanchet@48975
  1096
          [sbd_Cinfinite, mk_sbd_sbd (n - 1) RS @{thm ordIso_imp_ordLeq}];
blanchet@48975
  1097
blanchet@48975
  1098
    val sbd_sbd_thm = mk_sbd_sbd n;
blanchet@48975
  1099
blanchet@48975
  1100
    val sbdTs = replicate n sbdT;
blanchet@48975
  1101
    val sum_sbd = Library.foldr1 (uncurry mk_csum) (replicate n sbd);
blanchet@48975
  1102
    val sum_sbdT = mk_sumTN sbdTs;
blanchet@48975
  1103
    val sum_sbd_listT = HOLogic.listT sum_sbdT;
blanchet@48975
  1104
    val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT;
blanchet@48975
  1105
    val bdTs = passiveAs @ replicate n sbdT;
blanchet@48975
  1106
    val to_sbd_maps = map4 mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs;
blanchet@48975
  1107
    val bdFTs = mk_FTs bdTs;
blanchet@48975
  1108
    val sbdFT = mk_sumTN bdFTs;
blanchet@48975
  1109
    val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT);
blanchet@48975
  1110
    val treeQT = HOLogic.mk_setT treeT;
blanchet@48975
  1111
    val treeTs = passiveAs @ replicate n treeT;
blanchet@48975
  1112
    val treeQTs = passiveAs @ replicate n treeQT;
blanchet@48975
  1113
    val treeFTs = mk_FTs treeTs;
blanchet@48975
  1114
    val tree_maps = map4 mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs;
blanchet@48975
  1115
    val final_maps = map4 mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs;
blanchet@48975
  1116
    val tree_setss = mk_setss treeTs;
blanchet@48975
  1117
    val isNode_setss = mk_setss (passiveAs @ replicate n sbdT);
blanchet@48975
  1118
blanchet@48975
  1119
    val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []];
blanchet@48975
  1120
    val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs);
blanchet@48975
  1121
    val Lev_recT = fastype_of Zero;
blanchet@48975
  1122
    val LevT = Library.foldr (op -->) (sTs, HOLogic.natT --> Lev_recT);
blanchet@48975
  1123
blanchet@48975
  1124
    val Nil = HOLogic.mk_tuple (map3 (fn i => fn z => fn z'=>
blanchet@48975
  1125
      Term.absfree z' (mk_InN activeAs z i)) ks zs zs');
blanchet@48975
  1126
    val rv_recT = fastype_of Nil;
blanchet@48975
  1127
    val rvT = Library.foldr (op -->) (sTs, sum_sbd_listT --> rv_recT);
blanchet@48975
  1128
blanchet@48975
  1129
    val (((((((((((sumx, sumx'), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')),
blanchet@48975
  1130
      (lab, lab')), (Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')),
blanchet@48975
  1131
      names_lthy) = names_lthy
blanchet@48975
  1132
      |> yield_singleton (apfst (op ~~) oo mk_Frees' "sumx") sum_sbdT
blanchet@48975
  1133
      ||>> mk_Frees' "k" sbdTs
blanchet@48975
  1134
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
blanchet@48975
  1135
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
blanchet@48975
  1136
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl") sum_sbd_list_setT
blanchet@48975
  1137
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT)
blanchet@48975
  1138
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl_lab") treeT
blanchet@48975
  1139
      ||>> mk_Frees "x" bdFTs
blanchet@48975
  1140
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") Lev_recT
blanchet@48975
  1141
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") rv_recT;
blanchet@48975
  1142
blanchet@48975
  1143
    val (k, k') = (hd kks, hd kks')
blanchet@48975
  1144
blanchet@48975
  1145
    val timer = time (timer "Bounds");
blanchet@48975
  1146
blanchet@48975
  1147
    (* tree coalgebra *)
blanchet@48975
  1148
blanchet@48975
  1149
    fun isNode_bind i = Binding.suffix_name ("_" ^ isNodeN ^ (if n = 1 then "" else
blanchet@48975
  1150
      string_of_int i)) b;
blanchet@48975
  1151
    val isNode_name = Binding.name_of o isNode_bind;
blanchet@48975
  1152
    val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind;
blanchet@48975
  1153
blanchet@48975
  1154
    val isNodeT =
blanchet@48975
  1155
      Library.foldr (op -->) (map fastype_of (As @ [Kl, lab, kl]), HOLogic.boolT);
blanchet@48975
  1156
blanchet@48975
  1157
    val Succs = map3 (fn i => fn k => fn k' =>
blanchet@48975
  1158
      HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl)))
blanchet@48975
  1159
      ks kks kks';
blanchet@48975
  1160
blanchet@48975
  1161
    fun isNode_spec sets x i =
blanchet@48975
  1162
      let
blanchet@48975
  1163
        val (passive_sets, active_sets) = chop m (map (fn set => set $ x) sets);
blanchet@48975
  1164
        val lhs = Term.list_comb (Free (isNode_name i, isNodeT), As @ [Kl, lab, kl]);
blanchet@48975
  1165
        val rhs = list_exists_free [x]
blanchet@48975
  1166
          (Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) ::
blanchet@48975
  1167
          map2 mk_subset passive_sets As @ map2 (curry HOLogic.mk_eq) active_sets Succs));
blanchet@48975
  1168
      in
blanchet@48975
  1169
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  1170
      end;
blanchet@48975
  1171
blanchet@48975
  1172
    val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1173
      lthy
blanchet@48975
  1174
      |> fold_map3 (fn i => fn x => fn sets => Specification.definition
blanchet@48975
  1175
        (SOME (isNode_bind i, NONE, NoSyn), (isNode_def_bind i, isNode_spec sets x i)))
blanchet@48975
  1176
        ks xs isNode_setss
blanchet@48975
  1177
      |>> apsnd split_list o split_list
blanchet@48975
  1178
      ||> `Local_Theory.restore;
blanchet@48975
  1179
blanchet@48975
  1180
    (*transforms defined frees into consts*)
blanchet@48975
  1181
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1182
blanchet@48975
  1183
    val isNode_defs = map (Morphism.thm phi) isNode_def_frees;
blanchet@48975
  1184
    val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees;
blanchet@48975
  1185
blanchet@48975
  1186
    fun mk_isNode As kl i =
blanchet@48975
  1187
      Term.list_comb (Const (nth isNodes (i - 1), isNodeT), As @ [Kl, lab, kl]);
blanchet@48975
  1188
blanchet@48975
  1189
    val isTree =
blanchet@48975
  1190
      let
blanchet@48975
  1191
        val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl);
blanchet@48975
  1192
        val Field = mk_subset Kl (mk_Field (mk_clists sum_sbd));
blanchet@48975
  1193
        val prefCl = mk_prefCl Kl;
blanchet@48975
  1194
blanchet@48975
  1195
        val tree = mk_Ball Kl (Term.absfree kl'
blanchet@48975
  1196
          (HOLogic.mk_conj
blanchet@48975
  1197
            (Library.foldr1 HOLogic.mk_disj (map (mk_isNode As kl) ks),
blanchet@48975
  1198
            Library.foldr1 HOLogic.mk_conj (map4 (fn Succ => fn i => fn k => fn k' =>
blanchet@48975
  1199
              mk_Ball Succ (Term.absfree k' (mk_isNode As
blanchet@48975
  1200
                (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i)))
blanchet@48975
  1201
            Succs ks kks kks'))));
blanchet@48975
  1202
blanchet@48975
  1203
        val undef = list_all_free [kl] (HOLogic.mk_imp
blanchet@48975
  1204
          (HOLogic.mk_not (HOLogic.mk_mem (kl, Kl)),
blanchet@48975
  1205
          HOLogic.mk_eq (lab $ kl, mk_undefined sbdFT)));
blanchet@48975
  1206
      in
blanchet@48975
  1207
        Library.foldr1 HOLogic.mk_conj [empty, Field, prefCl, tree, undef]
blanchet@48975
  1208
      end;
blanchet@48975
  1209
blanchet@48975
  1210
    fun carT_bind i = Binding.suffix_name ("_" ^ carTN ^ (if n = 1 then "" else
blanchet@48975
  1211
      string_of_int i)) b;
blanchet@48975
  1212
    val carT_name = Binding.name_of o carT_bind;
blanchet@48975
  1213
    val carT_def_bind = rpair [] o Thm.def_binding o carT_bind;
blanchet@48975
  1214
blanchet@48975
  1215
    fun carT_spec i =
blanchet@48975
  1216
      let
blanchet@48975
  1217
        val carTT = Library.foldr (op -->) (ATs, HOLogic.mk_setT treeT);
blanchet@48975
  1218
blanchet@48975
  1219
        val lhs = Term.list_comb (Free (carT_name i, carTT), As);
blanchet@48975
  1220
        val rhs = HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
blanchet@48975
  1221
          (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)),
blanchet@48975
  1222
            HOLogic.mk_conj (isTree, mk_isNode As (HOLogic.mk_list sum_sbdT []) i))));
blanchet@48975
  1223
      in
blanchet@48975
  1224
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  1225
      end;
blanchet@48975
  1226
blanchet@48975
  1227
    val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1228
      lthy
blanchet@48975
  1229
      |> fold_map (fn i => Specification.definition
blanchet@48975
  1230
        (SOME (carT_bind i, NONE, NoSyn), (carT_def_bind i, carT_spec i))) ks
blanchet@48975
  1231
      |>> apsnd split_list o split_list
blanchet@48975
  1232
      ||> `Local_Theory.restore;
blanchet@48975
  1233
blanchet@48975
  1234
    (*transforms defined frees into consts*)
blanchet@48975
  1235
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1236
blanchet@48975
  1237
    val carT_defs = map (Morphism.thm phi) carT_def_frees;
blanchet@48975
  1238
    val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees;
blanchet@48975
  1239
blanchet@48975
  1240
    fun mk_carT As i = Term.list_comb
blanchet@48975
  1241
      (Const (nth carTs (i - 1),
blanchet@48975
  1242
         Library.foldr (op -->) (map fastype_of As, HOLogic.mk_setT treeT)), As);
blanchet@48975
  1243
blanchet@48975
  1244
    fun strT_bind i = Binding.suffix_name ("_" ^ strTN ^ (if n = 1 then "" else
blanchet@48975
  1245
      string_of_int i)) b;
blanchet@48975
  1246
    val strT_name = Binding.name_of o strT_bind;
blanchet@48975
  1247
    val strT_def_bind = rpair [] o Thm.def_binding o strT_bind;
blanchet@48975
  1248
blanchet@48975
  1249
    fun strT_spec mapFT FT i =
blanchet@48975
  1250
      let
blanchet@48975
  1251
        val strTT = treeT --> FT;
blanchet@48975
  1252
blanchet@48975
  1253
        fun mk_f i k k' =
blanchet@48975
  1254
          let val in_k = mk_InN sbdTs k i;
blanchet@48975
  1255
          in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end;
blanchet@48975
  1256
blanchet@48975
  1257
        val f = Term.list_comb (mapFT, passive_ids @ map3 mk_f ks kks kks');
blanchet@48975
  1258
        val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs));
blanchet@48975
  1259
        val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2);
blanchet@48975
  1260
        val lhs = Free (strT_name i, strTT);
blanchet@48975
  1261
        val rhs = HOLogic.mk_split (Term.absfree Kl' (Term.absfree lab'
blanchet@48975
  1262
          (mk_sum_caseN fs $ (lab $ HOLogic.mk_list sum_sbdT []))));
blanchet@48975
  1263
      in
blanchet@48975
  1264
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  1265
      end;
blanchet@48975
  1266
blanchet@48975
  1267
    val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1268
      lthy
blanchet@48975
  1269
      |> fold_map3 (fn i => fn mapFT => fn FT => Specification.definition
blanchet@48975
  1270
        (SOME (strT_bind i, NONE, NoSyn), (strT_def_bind i, strT_spec mapFT FT i)))
blanchet@48975
  1271
        ks tree_maps treeFTs
blanchet@48975
  1272
      |>> apsnd split_list o split_list
blanchet@48975
  1273
      ||> `Local_Theory.restore;
blanchet@48975
  1274
blanchet@48975
  1275
    (*transforms defined frees into consts*)
blanchet@48975
  1276
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1277
blanchet@48975
  1278
    val strT_defs = map ((fn def => trans OF [def RS fun_cong, @{thm prod.cases}]) o
blanchet@48975
  1279
      Morphism.thm phi) strT_def_frees;
blanchet@48975
  1280
    val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees;
blanchet@48975
  1281
blanchet@48975
  1282
    fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT);
blanchet@48975
  1283
blanchet@48975
  1284
    val carTAs = map (mk_carT As) ks;
blanchet@48975
  1285
    val carTAs_copy = map (mk_carT As_copy) ks;
blanchet@48975
  1286
    val strTAs = map2 mk_strT treeFTs ks;
blanchet@48975
  1287
    val hset_strTss = map (fn i => map2 (mk_hset strTAs i) ls passiveAs) ks;
blanchet@48975
  1288
blanchet@48975
  1289
    val coalgT_thm =
blanchet@48975
  1290
      Skip_Proof.prove lthy [] []
blanchet@48975
  1291
        (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_coalg As carTAs strTAs)))
traytel@49109
  1292
        (mk_coalgT_tac m (coalg_def :: isNode_defs @ carT_defs) strT_defs set_natural'ss)
traytel@49109
  1293
      |> Thm.close_derivation;
blanchet@48975
  1294
blanchet@48975
  1295
    val card_of_carT_thms =
blanchet@48975
  1296
      let
blanchet@48975
  1297
        val lhs = mk_card_of
blanchet@48975
  1298
          (HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
blanchet@48975
  1299
            (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)), isTree))));
blanchet@48975
  1300
        val rhs = mk_cexp
blanchet@48975
  1301
          (if m = 0 then ctwo else
blanchet@48975
  1302
            (mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo))
blanchet@48975
  1303
            (mk_cexp sbd sbd);
blanchet@48975
  1304
        val card_of_carT =
blanchet@48975
  1305
          Skip_Proof.prove lthy [] []
blanchet@48975
  1306
            (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_ordLeq lhs rhs)))
blanchet@48975
  1307
            (K (mk_card_of_carT_tac m isNode_defs sbd_sbd_thm
blanchet@48975
  1308
              sbd_card_order sbd_Card_order sbd_Cinfinite sbd_Cnotzero in_sbds))
traytel@49109
  1309
          |> Thm.close_derivation
blanchet@48975
  1310
      in
blanchet@48975
  1311
        map (fn def => @{thm ordLeq_transitive[OF
blanchet@48975
  1312
          card_of_mono1[OF ord_eq_le_trans[OF _ Collect_restrict']]]} OF [def, card_of_carT])
blanchet@48975
  1313
        carT_defs
blanchet@48975
  1314
      end;
blanchet@48975
  1315
blanchet@48975
  1316
    val carT_set_thmss =
blanchet@48975
  1317
      let
blanchet@48975
  1318
        val Kl_lab = HOLogic.mk_prod (Kl, lab);
blanchet@48975
  1319
        fun mk_goal carT strT set k i =
blanchet@48975
  1320
          fold_rev Logic.all (sumx :: Kl :: lab :: k :: kl :: As)
blanchet@48975
  1321
            (Logic.list_implies (map HOLogic.mk_Trueprop
blanchet@48975
  1322
              [HOLogic.mk_mem (Kl_lab, carT), HOLogic.mk_mem (mk_Cons sumx kl, Kl),
blanchet@48975
  1323
              HOLogic.mk_eq (sumx, mk_InN sbdTs k i)],
blanchet@48975
  1324
            HOLogic.mk_Trueprop (HOLogic.mk_mem
blanchet@48975
  1325
              (HOLogic.mk_prod (mk_Shift Kl sumx, mk_shift lab sumx),
blanchet@48975
  1326
              set $ (strT $ Kl_lab)))));
blanchet@48975
  1327
blanchet@48975
  1328
        val goalss = map3 (fn carT => fn strT => fn sets =>
blanchet@48975
  1329
          map3 (mk_goal carT strT) (drop m sets) kks ks) carTAs strTAs tree_setss;
blanchet@48975
  1330
      in
blanchet@48975
  1331
        map6 (fn i => fn goals =>
blanchet@48975
  1332
            fn carT_def => fn strT_def => fn isNode_def => fn set_naturals =>
blanchet@48975
  1333
          map2 (fn goal => fn set_natural =>
blanchet@48975
  1334
            Skip_Proof.prove lthy [] [] goal
traytel@49109
  1335
              (mk_carT_set_tac n i carT_def strT_def isNode_def set_natural)
traytel@49109
  1336
            |> Thm.close_derivation)
blanchet@48975
  1337
          goals (drop m set_naturals))
blanchet@48975
  1338
        ks goalss carT_defs strT_defs isNode_defs set_natural'ss
blanchet@48975
  1339
      end;
blanchet@48975
  1340
blanchet@48975
  1341
    val carT_set_thmss' = transpose carT_set_thmss;
blanchet@48975
  1342
blanchet@48975
  1343
    val isNode_hset_thmss =
blanchet@48975
  1344
      let
blanchet@48975
  1345
        val Kl_lab = HOLogic.mk_prod (Kl, lab);
blanchet@48975
  1346
        fun mk_Kl_lab carT = HOLogic.mk_mem (Kl_lab, carT);
blanchet@48975
  1347
blanchet@48975
  1348
        val strT_hset_thmsss =
blanchet@48975
  1349
          let
blanchet@48975
  1350
            val strT_hset_thms =
blanchet@48975
  1351
              let
blanchet@48975
  1352
                fun mk_lab_kl i x = HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i);
blanchet@48975
  1353
blanchet@48975
  1354
                fun mk_inner_conjunct j T i x set i' carT =
blanchet@48975
  1355
                  HOLogic.mk_imp (HOLogic.mk_conj (mk_Kl_lab carT, mk_lab_kl i x),
blanchet@48975
  1356
                    mk_subset (set $ x) (mk_hset strTAs i' j T $ Kl_lab));
blanchet@48975
  1357
blanchet@48975
  1358
                fun mk_conjunct j T i x set =
blanchet@48975
  1359
                  Library.foldr1 HOLogic.mk_conj (map2 (mk_inner_conjunct j T i x set) ks carTAs);
blanchet@48975
  1360
blanchet@48975
  1361
                fun mk_concl j T = list_all_free (Kl :: lab :: xs @ As)
blanchet@48975
  1362
                  (HOLogic.mk_imp (HOLogic.mk_mem (kl, Kl),
blanchet@48975
  1363
                    Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T)
blanchet@48975
  1364
                      ks xs (map (fn xs => nth xs (j - 1)) isNode_setss))));
blanchet@48975
  1365
                val concls = map2 mk_concl ls passiveAs;
blanchet@48975
  1366
blanchet@48975
  1367
                val cTs = [SOME (certifyT lthy sum_sbdT)];
blanchet@48975
  1368
                val arg_cong_cTs = map (SOME o certifyT lthy) treeFTs;
blanchet@48975
  1369
                val ctss =
blanchet@48975
  1370
                  map (fn phi => map (SOME o certify lthy) [Term.absfree kl' phi, kl]) concls;
blanchet@48975
  1371
blanchet@48975
  1372
                val goals = map HOLogic.mk_Trueprop concls;
blanchet@48975
  1373
              in
blanchet@48975
  1374
                map5 (fn j => fn goal => fn cts => fn set_incl_hsets => fn set_hset_incl_hsetss =>
blanchet@48975
  1375
                  singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1376
                    (Skip_Proof.prove lthy [] [] goal
blanchet@48975
  1377
                      (K (mk_strT_hset_tac n m j arg_cong_cTs cTs cts
blanchet@48975
  1378
                        carT_defs strT_defs isNode_defs
blanchet@48975
  1379
                        set_incl_hsets set_hset_incl_hsetss coalg_set_thmss' carT_set_thmss'
traytel@49109
  1380
                        coalgT_thm set_natural'ss)))
traytel@49109
  1381
                  |> Thm.close_derivation)
blanchet@48975
  1382
                ls goals ctss set_incl_hset_thmss' set_hset_incl_hset_thmsss''
blanchet@48975
  1383
              end;
blanchet@48975
  1384
blanchet@48975
  1385
            val strT_hset'_thms = map (fn thm => mk_specN (2 + n + m) thm RS mp) strT_hset_thms;
blanchet@48975
  1386
          in
blanchet@48975
  1387
            map (fn thm => map (fn i => map (fn i' =>
blanchet@48975
  1388
              thm RS mk_conjunctN n i RS mk_conjunctN n i' RS mp) ks) ks) strT_hset'_thms
blanchet@48975
  1389
          end;
blanchet@48975
  1390
blanchet@48975
  1391
        val carT_prems = map (fn carT =>
blanchet@48975
  1392
          HOLogic.mk_Trueprop (HOLogic.mk_mem (Kl_lab, carT))) carTAs_copy;
blanchet@48975
  1393
        val prem = HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, Kl));
blanchet@48975
  1394
        val in_prems = map (fn hsets =>
blanchet@48975
  1395
          HOLogic.mk_Trueprop (HOLogic.mk_mem (Kl_lab, mk_in As hsets treeT))) hset_strTss;
blanchet@48975
  1396
        val isNode_premss = replicate n (map (HOLogic.mk_Trueprop o mk_isNode As_copy kl) ks);
blanchet@48975
  1397
        val conclss = replicate n (map (HOLogic.mk_Trueprop o mk_isNode As kl) ks);
blanchet@48975
  1398
      in
blanchet@48975
  1399
        map5 (fn carT_prem => fn isNode_prems => fn in_prem => fn concls => fn strT_hset_thmss =>
blanchet@48975
  1400
          map4 (fn isNode_prem => fn concl => fn isNode_def => fn strT_hset_thms =>
blanchet@48975
  1401
            Skip_Proof.prove lthy [] []
traytel@49109
  1402
              (fold_rev Logic.all (Kl :: lab :: kl :: As @ As_copy)
traytel@49109
  1403
                (Logic.list_implies ([carT_prem, prem, isNode_prem, in_prem], concl)))
traytel@49109
  1404
              (mk_isNode_hset_tac n isNode_def strT_hset_thms)
traytel@49109
  1405
            |> Thm.close_derivation)
blanchet@48975
  1406
          isNode_prems concls isNode_defs
blanchet@48975
  1407
          (if m = 0 then replicate n [] else transpose strT_hset_thmss))
blanchet@48975
  1408
        carT_prems isNode_premss in_prems conclss
blanchet@48975
  1409
        (if m = 0 then replicate n [] else transpose (map transpose strT_hset_thmsss))
blanchet@48975
  1410
      end;
blanchet@48975
  1411
blanchet@48975
  1412
    val timer = time (timer "Tree coalgebra");
blanchet@48975
  1413
blanchet@48975
  1414
    fun mk_to_sbd s x i i' =
blanchet@48975
  1415
      mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
blanchet@48975
  1416
    fun mk_from_sbd s x i i' =
blanchet@48975
  1417
      mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
blanchet@48975
  1418
blanchet@48975
  1419
    fun mk_to_sbd_thmss thm = map (map (fn set_sbd =>
blanchet@48975
  1420
      thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss;
blanchet@48975
  1421
blanchet@48975
  1422
    val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj};
blanchet@48975
  1423
    val to_sbd_thmss = mk_to_sbd_thmss @{thm toCard};
blanchet@48975
  1424
    val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard};
blanchet@48975
  1425
blanchet@48975
  1426
    val Lev_bind = Binding.suffix_name ("_" ^ LevN) b;
blanchet@48975
  1427
    val Lev_name = Binding.name_of Lev_bind;
blanchet@48975
  1428
    val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind);
blanchet@48975
  1429
blanchet@48975
  1430
    val Lev_spec =
blanchet@48975
  1431
      let
blanchet@48975
  1432
        fun mk_Suc i s setsAs a a' =
blanchet@48975
  1433
          let
blanchet@48975
  1434
            val sets = drop m setsAs;
blanchet@48975
  1435
            fun mk_set i' set b =
blanchet@48975
  1436
              let
blanchet@48975
  1437
                val Cons = HOLogic.mk_eq (kl_copy,
blanchet@48975
  1438
                  mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl)
blanchet@48975
  1439
                val b_set = HOLogic.mk_mem (b, set $ (s $ a));
blanchet@48975
  1440
                val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b);
blanchet@48975
  1441
              in
blanchet@48975
  1442
                HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl]
blanchet@48975
  1443
                  (HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec))))
blanchet@48975
  1444
              end;
blanchet@48975
  1445
          in
blanchet@48975
  1446
            Term.absfree a' (Library.foldl1 mk_union (map3 mk_set ks sets zs_copy))
blanchet@48975
  1447
          end;
blanchet@48975
  1448
blanchet@48975
  1449
        val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec'
blanchet@48975
  1450
          (HOLogic.mk_tuple (map5 mk_Suc ks ss setssAs zs zs')));
blanchet@48975
  1451
blanchet@48975
  1452
        val lhs = Term.list_comb (Free (Lev_name, LevT), ss);
blanchet@48975
  1453
        val rhs = mk_nat_rec Zero Suc;
blanchet@48975
  1454
      in
blanchet@48975
  1455
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  1456
      end;
blanchet@48975
  1457
blanchet@48975
  1458
    val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) =
blanchet@48975
  1459
      lthy
blanchet@48975
  1460
      |> Specification.definition (SOME (Lev_bind, NONE, NoSyn), (Lev_def_bind, Lev_spec))
blanchet@48975
  1461
      ||> `Local_Theory.restore;
blanchet@48975
  1462
blanchet@48975
  1463
    (*transforms defined frees into consts*)
blanchet@48975
  1464
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1465
blanchet@48975
  1466
    val Lev_def = Morphism.thm phi Lev_def_free;
blanchet@48975
  1467
    val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free));
blanchet@48975
  1468
blanchet@48975
  1469
    fun mk_Lev ss nat i =
blanchet@48975
  1470
      let
blanchet@48975
  1471
        val Ts = map fastype_of ss;
blanchet@48975
  1472
        val LevT = Library.foldr (op -->) (Ts, HOLogic.natT -->
blanchet@48975
  1473
          HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts));
blanchet@48975
  1474
      in
blanchet@48975
  1475
        mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i
blanchet@48975
  1476
      end;
blanchet@48975
  1477
blanchet@48975
  1478
    val Lev_0s = flat (mk_rec_simps n @{thm nat_rec_0} [Lev_def]);
blanchet@48975
  1479
    val Lev_Sucs = flat (mk_rec_simps n @{thm nat_rec_Suc} [Lev_def]);
blanchet@48975
  1480
blanchet@48975
  1481
    val rv_bind = Binding.suffix_name ("_" ^ rvN) b;
blanchet@48975
  1482
    val rv_name = Binding.name_of rv_bind;
blanchet@48975
  1483
    val rv_def_bind = rpair [] (Thm.def_binding rv_bind);
blanchet@48975
  1484
blanchet@48975
  1485
    val rv_spec =
blanchet@48975
  1486
      let
blanchet@48975
  1487
        fun mk_Cons i s b b' =
blanchet@48975
  1488
          let
blanchet@48975
  1489
            fun mk_case i' =
blanchet@48975
  1490
              Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k));
blanchet@48975
  1491
          in
blanchet@48975
  1492
            Term.absfree b' (mk_sum_caseN (map mk_case ks) $ sumx)
blanchet@48975
  1493
          end;
blanchet@48975
  1494
blanchet@48975
  1495
        val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec'
blanchet@48975
  1496
          (HOLogic.mk_tuple (map4 mk_Cons ks ss zs zs'))));
blanchet@48975
  1497
blanchet@48975
  1498
        val lhs = Term.list_comb (Free (rv_name, rvT), ss);
blanchet@48975
  1499
        val rhs = mk_list_rec Nil Cons;
blanchet@48975
  1500
      in
blanchet@48975
  1501
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  1502
      end;
blanchet@48975
  1503
blanchet@48975
  1504
    val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) =
blanchet@48975
  1505
      lthy
blanchet@48975
  1506
      |> Specification.definition (SOME (rv_bind, NONE, NoSyn), (rv_def_bind, rv_spec))
blanchet@48975
  1507
      ||> `Local_Theory.restore;
blanchet@48975
  1508
blanchet@48975
  1509
    (*transforms defined frees into consts*)
blanchet@48975
  1510
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1511
blanchet@48975
  1512
    val rv_def = Morphism.thm phi rv_def_free;
blanchet@48975
  1513
    val rv = fst (Term.dest_Const (Morphism.term phi rv_free));
blanchet@48975
  1514
blanchet@48975
  1515
    fun mk_rv ss kl i =
blanchet@48975
  1516
      let
blanchet@48975
  1517
        val Ts = map fastype_of ss;
blanchet@48975
  1518
        val As = map domain_type Ts;
blanchet@48975
  1519
        val rvT = Library.foldr (op -->) (Ts, fastype_of kl -->
blanchet@48975
  1520
          HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As));
blanchet@48975
  1521
      in
blanchet@48975
  1522
        mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i
blanchet@48975
  1523
      end;
blanchet@48975
  1524
blanchet@48975
  1525
    val rv_Nils = flat (mk_rec_simps n @{thm list_rec_Nil} [rv_def]);
blanchet@48975
  1526
    val rv_Conss = flat (mk_rec_simps n @{thm list_rec_Cons} [rv_def]);
blanchet@48975
  1527
blanchet@48975
  1528
    fun beh_bind i = Binding.suffix_name ("_" ^ behN ^ (if n = 1 then "" else
blanchet@48975
  1529
      string_of_int i)) b;
blanchet@48975
  1530
    val beh_name = Binding.name_of o beh_bind;
blanchet@48975
  1531
    val beh_def_bind = rpair [] o Thm.def_binding o beh_bind;
blanchet@48975
  1532
blanchet@48975
  1533
    fun beh_spec i z =
blanchet@48975
  1534
      let
blanchet@48975
  1535
        val mk_behT = Library.foldr (op -->) (map fastype_of (ss @ [z]), treeT);
blanchet@48975
  1536
blanchet@48975
  1537
        fun mk_case i to_sbd_map s k k' =
blanchet@48975
  1538
          Term.absfree k' (mk_InN bdFTs
blanchet@48975
  1539
            (Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i);
blanchet@48975
  1540
blanchet@48975
  1541
        val Lab = Term.absfree kl' (mk_If
blanchet@48975
  1542
          (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))
blanchet@48975
  1543
          (mk_sum_caseN (map5 mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z))
blanchet@48975
  1544
          (mk_undefined sbdFT));
blanchet@48975
  1545
blanchet@48975
  1546
        val lhs = Term.list_comb (Free (beh_name i, mk_behT), ss) $ z;
blanchet@48975
  1547
        val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
blanchet@48975
  1548
          (Term.absfree nat' (mk_Lev ss nat i $ z)), Lab);
blanchet@48975
  1549
      in
blanchet@48975
  1550
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  1551
      end;
blanchet@48975
  1552
blanchet@48975
  1553
    val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1554
      lthy
blanchet@48975
  1555
      |> fold_map2 (fn i => fn z => Specification.definition
blanchet@48975
  1556
        (SOME (beh_bind i, NONE, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs
blanchet@48975
  1557
      |>> apsnd split_list o split_list
blanchet@48975
  1558
      ||> `Local_Theory.restore;
blanchet@48975
  1559
blanchet@48975
  1560
    (*transforms defined frees into consts*)
blanchet@48975
  1561
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1562
blanchet@48975
  1563
    val beh_defs = map (Morphism.thm phi) beh_def_frees;
blanchet@48975
  1564
    val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees;
blanchet@48975
  1565
blanchet@48975
  1566
    fun mk_beh ss i =
blanchet@48975
  1567
      let
blanchet@48975
  1568
        val Ts = map fastype_of ss;
blanchet@48975
  1569
        val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT);
blanchet@48975
  1570
      in
blanchet@48975
  1571
        Term.list_comb (Const (nth behs (i - 1), behT), ss)
blanchet@48975
  1572
      end;
blanchet@48975
  1573
blanchet@48975
  1574
    val Lev_sbd_thms =
blanchet@48975
  1575
      let
blanchet@48975
  1576
        fun mk_conjunct i z = mk_subset (mk_Lev ss nat i $ z) (mk_Field (mk_clists sum_sbd));
blanchet@48975
  1577
        val goal = list_all_free zs
blanchet@48975
  1578
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1579
blanchet@48975
  1580
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1581
blanchet@48975
  1582
        val Lev_sbd = singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1583
          (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@49109
  1584
            (K (mk_Lev_sbd_tac cts Lev_0s Lev_Sucs to_sbd_thmss))
traytel@49109
  1585
          |> Thm.close_derivation);
blanchet@48975
  1586
blanchet@48975
  1587
        val Lev_sbd' = mk_specN n Lev_sbd;
blanchet@48975
  1588
      in
blanchet@48975
  1589
        map (fn i => Lev_sbd' RS mk_conjunctN n i) ks
blanchet@48975
  1590
      end;
blanchet@48975
  1591
blanchet@48975
  1592
    val (length_Lev_thms, length_Lev'_thms) =
blanchet@48975
  1593
      let
blanchet@48975
  1594
        fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1595
          HOLogic.mk_eq (mk_size kl, nat));
blanchet@48975
  1596
        val goal = list_all_free (kl :: zs)
blanchet@48975
  1597
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1598
blanchet@48975
  1599
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1600
blanchet@48975
  1601
        val length_Lev = singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1602
          (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@49109
  1603
            (K (mk_length_Lev_tac cts Lev_0s Lev_Sucs))
traytel@49109
  1604
          |> Thm.close_derivation);
blanchet@48975
  1605
blanchet@48975
  1606
        val length_Lev' = mk_specN (n + 1) length_Lev;
blanchet@48975
  1607
        val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks;
blanchet@48975
  1608
blanchet@48975
  1609
        fun mk_goal i z = fold_rev Logic.all (z :: kl :: nat :: ss) (Logic.mk_implies
blanchet@48975
  1610
            (HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z)),
blanchet@48975
  1611
            HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))));
blanchet@48975
  1612
        val goals = map2 mk_goal ks zs;
blanchet@48975
  1613
blanchet@48975
  1614
        val length_Levs' = map2 (fn goal => fn length_Lev =>
traytel@49109
  1615
          Skip_Proof.prove lthy [] [] goal (K (mk_length_Lev'_tac length_Lev))
traytel@49109
  1616
          |> Thm.close_derivation) goals length_Levs;
blanchet@48975
  1617
      in
blanchet@48975
  1618
        (length_Levs, length_Levs')
blanchet@48975
  1619
      end;
blanchet@48975
  1620
blanchet@48975
  1621
    val prefCl_Lev_thms =
blanchet@48975
  1622
      let
blanchet@48975
  1623
        fun mk_conjunct i z = HOLogic.mk_imp
blanchet@48975
  1624
          (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), mk_subset kl_copy kl),
blanchet@48975
  1625
          HOLogic.mk_mem (kl_copy, mk_Lev ss (mk_size kl_copy) i $ z));
blanchet@48975
  1626
        val goal = list_all_free (kl :: kl_copy :: zs)
blanchet@48975
  1627
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1628
blanchet@48975
  1629
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1630
blanchet@48975
  1631
        val prefCl_Lev = singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1632
          (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@49109
  1633
            (K (mk_prefCl_Lev_tac cts Lev_0s Lev_Sucs)))
traytel@49109
  1634
          |> Thm.close_derivation;
blanchet@48975
  1635
blanchet@48975
  1636
        val prefCl_Lev' = mk_specN (n + 2) prefCl_Lev;
blanchet@48975
  1637
      in
blanchet@48975
  1638
        map (fn i => prefCl_Lev' RS mk_conjunctN n i RS mp) ks
blanchet@48975
  1639
      end;
blanchet@48975
  1640
blanchet@48975
  1641
    val rv_last_thmss =
blanchet@48975
  1642
      let
blanchet@48975
  1643
        fun mk_conjunct i z i' z_copy = list_exists_free [z_copy]
blanchet@48975
  1644
          (HOLogic.mk_eq
blanchet@48975
  1645
            (mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z,
blanchet@48975
  1646
            mk_InN activeAs z_copy i'));
blanchet@48975
  1647
        val goal = list_all_free (k :: zs)
blanchet@48975
  1648
          (Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z =>
blanchet@48975
  1649
            Library.foldr1 HOLogic.mk_conj
blanchet@48975
  1650
              (map2 (mk_conjunct i z) ks zs_copy)) ks zs));
blanchet@48975
  1651
blanchet@48975
  1652
        val cTs = [SOME (certifyT lthy sum_sbdT)];
blanchet@48975
  1653
        val cts = map (SOME o certify lthy) [Term.absfree kl' goal, kl];
blanchet@48975
  1654
blanchet@48975
  1655
        val rv_last = singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1656
          (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@49109
  1657
            (K (mk_rv_last_tac cTs cts rv_Nils rv_Conss)))
traytel@49109
  1658
          |> Thm.close_derivation;
blanchet@48975
  1659
blanchet@48975
  1660
        val rv_last' = mk_specN (n + 1) rv_last;
blanchet@48975
  1661
      in
blanchet@48975
  1662
        map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks
blanchet@48975
  1663
      end;
blanchet@48975
  1664
blanchet@48975
  1665
    val set_rv_Lev_thmsss = if m = 0 then replicate n (replicate n []) else
blanchet@48975
  1666
      let
blanchet@48975
  1667
        fun mk_case s sets z z_free = Term.absfree z_free (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  1668
          (map2 (fn set => fn A => mk_subset (set $ (s $ z)) A) (take m sets) As));
blanchet@48975
  1669
blanchet@48975
  1670
        fun mk_conjunct i z B = HOLogic.mk_imp
blanchet@48975
  1671
          (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), HOLogic.mk_mem (z, B)),
blanchet@48975
  1672
          mk_sum_caseN (map4 mk_case ss setssAs zs zs') $ (mk_rv ss kl i $ z));
blanchet@48975
  1673
blanchet@48975
  1674
        val goal = list_all_free (kl :: zs)
blanchet@48975
  1675
          (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct ks zs Bs));
blanchet@48975
  1676
blanchet@48975
  1677
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1678
blanchet@48975
  1679
        val set_rv_Lev = singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1680
          (Skip_Proof.prove lthy [] []
blanchet@48975
  1681
            (Logic.mk_implies (coalg_prem, HOLogic.mk_Trueprop goal))
blanchet@48975
  1682
            (K (mk_set_rv_Lev_tac m cts Lev_0s Lev_Sucs rv_Nils rv_Conss
traytel@49109
  1683
              coalg_set_thmss from_to_sbd_thmss)))
traytel@49109
  1684
          |> Thm.close_derivation;
blanchet@48975
  1685
blanchet@48975
  1686
        val set_rv_Lev' = mk_specN (n + 1) set_rv_Lev;
blanchet@48975
  1687
      in
blanchet@48975
  1688
        map (fn i => map (fn i' =>
blanchet@48975
  1689
          split_conj_thm (if n = 1 then set_rv_Lev' RS mk_conjunctN n i RS mp
blanchet@48975
  1690
            else set_rv_Lev' RS mk_conjunctN n i RS mp RSN
blanchet@48975
  1691
              (2, @{thm sum_case_cong} RS @{thm subst[of _ _ "%x. x"]}) RS
blanchet@48975
  1692
              (mk_sum_casesN n i' RS @{thm subst[of _ _ "%x. x"]}))) ks) ks
blanchet@48975
  1693
      end;
blanchet@48975
  1694
blanchet@48975
  1695
    val set_Lev_thmsss =
blanchet@48975
  1696
      let
blanchet@48975
  1697
        fun mk_conjunct i z =
blanchet@48975
  1698
          let
blanchet@48975
  1699
            fun mk_conjunct' i' sets s z' =
blanchet@48975
  1700
              let
blanchet@48975
  1701
                fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp
blanchet@48975
  1702
                  (HOLogic.mk_mem (z'', set $ (s $ z')),
blanchet@48975
  1703
                    HOLogic.mk_mem (mk_append (kl,
blanchet@48975
  1704
                      HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']),
blanchet@48975
  1705
                      mk_Lev ss (HOLogic.mk_Suc nat) i $ z));
blanchet@48975
  1706
              in
blanchet@48975
  1707
                HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'),
blanchet@48975
  1708
                  (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct'' ks (drop m sets) zs_copy2)))
blanchet@48975
  1709
              end;
blanchet@48975
  1710
          in
blanchet@48975
  1711
            HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1712
              Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct' ks setssAs ss zs_copy))
blanchet@48975
  1713
          end;
blanchet@48975
  1714
blanchet@48975
  1715
        val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2)
blanchet@48975
  1716
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1717
blanchet@48975
  1718
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1719
blanchet@48975
  1720
        val set_Lev = singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1721
          (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@49109
  1722
            (K (mk_set_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss)))
traytel@49109
  1723
          |> Thm.close_derivation;
blanchet@48975
  1724
blanchet@48975
  1725
        val set_Lev' = mk_specN (3 * n + 1) set_Lev;
blanchet@48975
  1726
      in
blanchet@48975
  1727
        map (fn i => map (fn i' => map (fn i'' => set_Lev' RS
blanchet@48975
  1728
          mk_conjunctN n i RS mp RS
blanchet@48975
  1729
          mk_conjunctN n i' RS mp RS
blanchet@48975
  1730
          mk_conjunctN n i'' RS mp) ks) ks) ks
blanchet@48975
  1731
      end;
blanchet@48975
  1732
blanchet@48975
  1733
    val set_image_Lev_thmsss =
blanchet@48975
  1734
      let
blanchet@48975
  1735
        fun mk_conjunct i z =
blanchet@48975
  1736
          let
blanchet@48975
  1737
            fun mk_conjunct' i' sets =
blanchet@48975
  1738
              let
blanchet@48975
  1739
                fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp
blanchet@48975
  1740
                  (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''),
blanchet@48975
  1741
                  HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z''))));
blanchet@48975
  1742
              in
blanchet@48975
  1743
                HOLogic.mk_imp (HOLogic.mk_mem
blanchet@48975
  1744
                  (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']),
blanchet@48975
  1745
                    mk_Lev ss (HOLogic.mk_Suc nat) i $ z),
blanchet@48975
  1746
                  (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct'' ks sets ss zs_copy)))
blanchet@48975
  1747
              end;
blanchet@48975
  1748
          in
blanchet@48975
  1749
            HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1750
              Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs')))
blanchet@48975
  1751
          end;
blanchet@48975
  1752
blanchet@48975
  1753
        val goal = list_all_free (kl :: k :: zs @ zs_copy)
blanchet@48975
  1754
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1755
blanchet@48975
  1756
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1757
blanchet@48975
  1758
        val set_image_Lev = singleton (Proof_Context.export names_lthy lthy)
blanchet@48975
  1759
          (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
blanchet@48975
  1760
            (K (mk_set_image_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss
traytel@49109
  1761
              from_to_sbd_thmss to_sbd_inj_thmss)))
traytel@49109
  1762
          |> Thm.close_derivation;
blanchet@48975
  1763
blanchet@48975
  1764
        val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev;
blanchet@48975
  1765
      in
blanchet@48975
  1766
        map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS
blanchet@48975
  1767
          mk_conjunctN n i RS mp RS
blanchet@48975
  1768
          mk_conjunctN n i'' RS mp RS
blanchet@48975
  1769
          mk_conjunctN n i' RS mp) ks) ks) ks
blanchet@48975
  1770
      end;
blanchet@48975
  1771
blanchet@48975
  1772
    val mor_beh_thm =
blanchet@48975
  1773
      Skip_Proof.prove lthy [] []
blanchet@48975
  1774
        (fold_rev Logic.all (As @ Bs @ ss) (Logic.mk_implies (coalg_prem,
blanchet@48975
  1775
          HOLogic.mk_Trueprop (mk_mor Bs ss carTAs strTAs (map (mk_beh ss) ks)))))
blanchet@48975
  1776
        (mk_mor_beh_tac m mor_def mor_cong_thm
blanchet@48975
  1777
          beh_defs carT_defs strT_defs isNode_defs
blanchet@48975
  1778
          to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbd_thms
blanchet@48975
  1779
          length_Lev_thms length_Lev'_thms prefCl_Lev_thms rv_last_thmss
blanchet@48975
  1780
          set_rv_Lev_thmsss set_Lev_thmsss set_image_Lev_thmsss
traytel@49109
  1781
          set_natural'ss coalg_set_thmss map_comp_id_thms map_congs map_arg_cong_thms)
traytel@49109
  1782
      |> Thm.close_derivation;
blanchet@48975
  1783
blanchet@48975
  1784
    val timer = time (timer "Behavioral morphism");
blanchet@48975
  1785
blanchet@48975
  1786
    fun mk_LSBIS As i = mk_lsbis As (map (mk_carT As) ks) strTAs i;
blanchet@48975
  1787
    fun mk_car_final As i =
blanchet@48975
  1788
      mk_quotient (mk_carT As i) (mk_LSBIS As i);
blanchet@48975
  1789
    fun mk_str_final As i =
blanchet@48975
  1790
      mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1),
blanchet@48975
  1791
        passive_ids @ map (mk_proj o mk_LSBIS As) ks), nth strTAs (i - 1)));
blanchet@48975
  1792
blanchet@48975
  1793
    val car_finalAs = map (mk_car_final As) ks;
blanchet@48975
  1794
    val str_finalAs = map (mk_str_final As) ks;
blanchet@48975
  1795
    val car_finals = map (mk_car_final passive_UNIVs) ks;
blanchet@48975
  1796
    val str_finals = map (mk_str_final passive_UNIVs) ks;
blanchet@48975
  1797
blanchet@48975
  1798
    val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss;
blanchet@48975
  1799
    val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms;
blanchet@48975
  1800
blanchet@48975
  1801
    val congruent_str_final_thms =
blanchet@48975
  1802
      let
blanchet@48975
  1803
        fun mk_goal R final_map strT =
blanchet@48975
  1804
          fold_rev Logic.all As (HOLogic.mk_Trueprop
blanchet@48975
  1805
            (mk_congruent R (HOLogic.mk_comp
blanchet@48975
  1806
              (Term.list_comb (final_map, passive_ids @ map (mk_proj o mk_LSBIS As) ks), strT))));
blanchet@48975
  1807
blanchet@48975
  1808
        val goals = map3 mk_goal (map (mk_LSBIS As) ks) final_maps strTAs;
blanchet@48975
  1809
      in
blanchet@48975
  1810
        map4 (fn goal => fn lsbisE => fn map_comp_id => fn map_cong =>
blanchet@48975
  1811
          Skip_Proof.prove lthy [] [] goal
traytel@49109
  1812
            (K (mk_congruent_str_final_tac m lsbisE map_comp_id map_cong equiv_LSBIS_thms))
traytel@49109
  1813
          |> Thm.close_derivation)
blanchet@48975
  1814
        goals lsbisE_thms map_comp_id_thms map_congs
blanchet@48975
  1815
      end;
blanchet@48975
  1816
blanchet@48975
  1817
    val coalg_final_thm = Skip_Proof.prove lthy [] [] (fold_rev Logic.all As
blanchet@48975
  1818
      (HOLogic.mk_Trueprop (mk_coalg As car_finalAs str_finalAs)))
blanchet@48975
  1819
      (K (mk_coalg_final_tac m coalg_def congruent_str_final_thms equiv_LSBIS_thms
traytel@49109
  1820
        set_natural'ss coalgT_set_thmss))
traytel@49109
  1821
      |> Thm.close_derivation;
blanchet@48975
  1822
blanchet@48975
  1823
    val mor_T_final_thm = Skip_Proof.prove lthy [] [] (fold_rev Logic.all As
blanchet@48975
  1824
      (HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finalAs str_finalAs
blanchet@48975
  1825
        (map (mk_proj o mk_LSBIS As) ks))))
traytel@49109
  1826
      (K (mk_mor_T_final_tac mor_def congruent_str_final_thms equiv_LSBIS_thms))
traytel@49109
  1827
      |> Thm.close_derivation;
blanchet@48975
  1828
blanchet@48975
  1829
    val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm];
blanchet@48975
  1830
    val in_car_final_thms = map (fn mor_image' => mor_image' OF
blanchet@48975
  1831
      [tcoalg_thm RS mor_final_thm, UNIV_I]) mor_image'_thms;
blanchet@48975
  1832
blanchet@48975
  1833
    val timer = time (timer "Final coalgebra");
blanchet@48975
  1834
blanchet@48975
  1835
    val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
blanchet@48975
  1836
      lthy
blanchet@48975
  1837
      |> fold_map3 (fn b => fn car_final => fn in_car_final =>
blanchet@48975
  1838
        typedef false NONE (b, params, NoSyn) car_final NONE
blanchet@48975
  1839
          (EVERY' [rtac exI, rtac in_car_final] 1)) bs car_finals in_car_final_thms
blanchet@48975
  1840
      |>> apsnd split_list o split_list;
blanchet@48975
  1841
blanchet@48975
  1842
    val Ts = map (fn name => Type (name, params')) T_names;
blanchet@48975
  1843
    fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;
blanchet@48975
  1844
    val Ts' = mk_Ts passiveBs;
blanchet@48975
  1845
    val Ts'' = mk_Ts passiveCs;
blanchet@48975
  1846
    val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts;
blanchet@48975
  1847
    val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts;
blanchet@48975
  1848
blanchet@48975
  1849
    val Reps = map #Rep T_loc_infos;
blanchet@48975
  1850
    val Rep_injects = map #Rep_inject T_loc_infos;
blanchet@48975
  1851
    val Rep_inverses = map #Rep_inverse T_loc_infos;
blanchet@48975
  1852
    val Abs_inverses = map #Abs_inverse T_loc_infos;
blanchet@48975
  1853
blanchet@48975
  1854
    val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");
blanchet@48975
  1855
blanchet@48975
  1856
    val UNIVs = map HOLogic.mk_UNIV Ts;
blanchet@48975
  1857
    val FTs = mk_FTs (passiveAs @ Ts);
blanchet@48975
  1858
    val FTs' = mk_FTs (passiveBs @ Ts);
blanchet@48975
  1859
    val prodTs = map (HOLogic.mk_prodT o `I) Ts;
blanchet@48975
  1860
    val prodFTs = mk_FTs (passiveAs @ prodTs);
blanchet@48975
  1861
    val FTs_setss = mk_setss (passiveAs @ Ts);
blanchet@48975
  1862
    val FTs'_setss = mk_setss (passiveBs @ Ts);
blanchet@48975
  1863
    val prodFT_setss = mk_setss (passiveAs @ prodTs);
blanchet@48975
  1864
    val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs;
blanchet@48975
  1865
    val map_FT_nths = map2 (fn Ds =>
blanchet@48975
  1866
      mk_map_of_bnf Ds (passiveAs @ prodTs) (passiveAs @ Ts)) Dss bnfs;
blanchet@48975
  1867
    val fstsTs = map fst_const prodTs;
blanchet@48975
  1868
    val sndsTs = map snd_const prodTs;
blanchet@48975
  1869
    val unfTs = map2 (curry (op -->)) Ts FTs;
blanchet@48975
  1870
    val fldTs = map2 (curry (op -->)) FTs Ts;
blanchet@48975
  1871
    val coiter_fTs = map2 (curry op -->) activeAs Ts;
blanchet@48975
  1872
    val corec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) sum_sTs;
blanchet@48975
  1873
    val corec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls;
blanchet@48975
  1874
    val corec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls_rev;
blanchet@48975
  1875
    val corec_Inls = map (Term.subst_atomic_types (activeBs ~~ Ts)) Inls;
blanchet@48975
  1876
blanchet@48975
  1877
    val (((((((((((((Jzs, Jzs'), (Jz's, Jz's')), Jzs_copy), Jzs1), Jzs2), Jpairs),
blanchet@48975
  1878
      FJzs), TRs), coiter_fs), coiter_fs_copy), corec_ss), phis), names_lthy) = names_lthy
blanchet@48975
  1879
      |> mk_Frees' "z" Ts
blanchet@48975
  1880
      ||>> mk_Frees' "z" Ts'
blanchet@48975
  1881
      ||>> mk_Frees "z" Ts
blanchet@48975
  1882
      ||>> mk_Frees "z1" Ts
blanchet@48975
  1883
      ||>> mk_Frees "z2" Ts
blanchet@48975
  1884
      ||>> mk_Frees "j" (map2 (curry HOLogic.mk_prodT) Ts Ts')
blanchet@48975
  1885
      ||>> mk_Frees "x" prodFTs
blanchet@48975
  1886
      ||>> mk_Frees "R" (map (mk_relT o `I) Ts)
blanchet@48975
  1887
      ||>> mk_Frees "f" coiter_fTs
blanchet@48975
  1888
      ||>> mk_Frees "g" coiter_fTs
blanchet@48975
  1889
      ||>> mk_Frees "s" corec_sTs
blanchet@48975
  1890
      ||>> mk_Frees "phi" (map (fn T => T --> T --> HOLogic.boolT) Ts);
blanchet@48975
  1891
blanchet@48975
  1892
    fun unf_bind i = Binding.suffix_name ("_" ^ unfN) (nth bs (i - 1));
blanchet@48975
  1893
    val unf_name = Binding.name_of o unf_bind;
blanchet@48975
  1894
    val unf_def_bind = rpair [] o Thm.def_binding o unf_bind;
blanchet@48975
  1895
blanchet@48975
  1896
    fun unf_spec i rep str map_FT unfT Jz Jz' =
blanchet@48975
  1897
      let
blanchet@48975
  1898
        val lhs = Free (unf_name i, unfT);
blanchet@48975
  1899
        val rhs = Term.absfree Jz'
blanchet@48975
  1900
          (Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $
blanchet@48975
  1901
            (str $ (rep $ Jz)));
blanchet@48975
  1902
      in
blanchet@48975
  1903
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  1904
      end;
blanchet@48975
  1905
blanchet@48975
  1906
    val ((unf_frees, (_, unf_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1907
      lthy
blanchet@48975
  1908
      |> fold_map7 (fn i => fn rep => fn str => fn map => fn unfT => fn Jz => fn Jz' =>
blanchet@48975
  1909
        Specification.definition
blanchet@48975
  1910
          (SOME (unf_bind i, NONE, NoSyn), (unf_def_bind i, unf_spec i rep str map unfT Jz Jz')))
blanchet@48975
  1911
          ks Rep_Ts str_finals map_FTs unfTs Jzs Jzs'
blanchet@48975
  1912
      |>> apsnd split_list o split_list
blanchet@48975
  1913
      ||> `Local_Theory.restore;
blanchet@48975
  1914
blanchet@48975
  1915
    (*transforms defined frees into consts*)
blanchet@48975
  1916
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1917
    fun mk_unfs passive =
blanchet@48975
  1918
      map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (deads @ passive)) o
blanchet@48975
  1919
        Morphism.term phi) unf_frees;
blanchet@48975
  1920
    val unfs = mk_unfs passiveAs;
blanchet@48975
  1921
    val unf's = mk_unfs passiveBs;
blanchet@48975
  1922
    val unf_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) unf_def_frees;
blanchet@48975
  1923
blanchet@48975
  1924
    val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss;
blanchet@48975
  1925
    val (mor_Rep_thm, mor_Abs_thm) =
blanchet@48975
  1926
      let
blanchet@48975
  1927
        val mor_Rep =
blanchet@48975
  1928
          Skip_Proof.prove lthy [] []
blanchet@48975
  1929
            (HOLogic.mk_Trueprop (mk_mor UNIVs unfs car_finals str_finals Rep_Ts))
blanchet@48975
  1930
            (mk_mor_Rep_tac m (mor_def :: unf_defs) Reps Abs_inverses coalg_final_set_thmss
traytel@49109
  1931
              map_comp_id_thms map_congL_thms)
traytel@49109
  1932
          |> Thm.close_derivation;
blanchet@48975
  1933
blanchet@48975
  1934
        val mor_Abs =
blanchet@48975
  1935
          Skip_Proof.prove lthy [] []
blanchet@48975
  1936
            (HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs unfs Abs_Ts))
traytel@49109
  1937
            (mk_mor_Abs_tac (mor_def :: unf_defs) Abs_inverses)
traytel@49109
  1938
          |> Thm.close_derivation;
blanchet@48975
  1939
      in
blanchet@48975
  1940
        (mor_Rep, mor_Abs)
blanchet@48975
  1941
      end;
blanchet@48975
  1942
blanchet@48975
  1943
    val timer = time (timer "unf definitions & thms");
blanchet@48975
  1944
blanchet@48975
  1945
    fun coiter_bind i = Binding.suffix_name ("_" ^ coN ^ iterN) (nth bs (i - 1));
blanchet@48975
  1946
    val coiter_name = Binding.name_of o coiter_bind;
blanchet@48975
  1947
    val coiter_def_bind = rpair [] o Thm.def_binding o coiter_bind;
blanchet@48975
  1948
blanchet@48975
  1949
    fun coiter_spec i T AT abs f z z' =
blanchet@48975
  1950
      let
blanchet@48975
  1951
        val coiterT = Library.foldr (op -->) (sTs, AT --> T);
blanchet@48975
  1952
blanchet@48975
  1953
        val lhs = Term.list_comb (Free (coiter_name i, coiterT), ss);
blanchet@48975
  1954
        val rhs = Term.absfree z' (abs $ (f $ z));
blanchet@48975
  1955
      in
blanchet@48975
  1956
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  1957
      end;
blanchet@48975
  1958
blanchet@48975
  1959
    val ((coiter_frees, (_, coiter_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1960
      lthy
blanchet@48975
  1961
      |> fold_map7 (fn i => fn T => fn AT => fn abs => fn f => fn z => fn z' =>
blanchet@48975
  1962
        Specification.definition
blanchet@48975
  1963
          (SOME (coiter_bind i, NONE, NoSyn), (coiter_def_bind i, coiter_spec i T AT abs f z z')))
blanchet@48975
  1964
          ks Ts activeAs Abs_Ts (map (fn i => HOLogic.mk_comp
blanchet@48975
  1965
            (mk_proj (mk_LSBIS passive_UNIVs i), mk_beh ss i)) ks) zs zs'
blanchet@48975
  1966
      |>> apsnd split_list o split_list
blanchet@48975
  1967
      ||> `Local_Theory.restore;
blanchet@48975
  1968
blanchet@48975
  1969
    (*transforms defined frees into consts*)
blanchet@48975
  1970
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1971
    val coiters = map (fst o dest_Const o Morphism.term phi) coiter_frees;
blanchet@48975
  1972
    fun mk_coiter Ts ss i = Term.list_comb (Const (nth coiters (i - 1), Library.foldr (op -->)
blanchet@48975
  1973
      (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
blanchet@48975
  1974
    val coiter_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) coiter_def_frees;
blanchet@48975
  1975
blanchet@48975
  1976
    val mor_coiter_thm =
blanchet@48975
  1977
      let
blanchet@48975
  1978
        val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses;
blanchet@48975
  1979
        val morEs' = map (fn thm =>
blanchet@48975
  1980
          (thm OF [tcoalg_thm RS mor_final_thm, UNIV_I]) RS sym) morE_thms;
blanchet@48975
  1981
      in
blanchet@48975
  1982
        Skip_Proof.prove lthy [] []
blanchet@48975
  1983
          (fold_rev Logic.all ss
blanchet@48975
  1984
            (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs unfs (map (mk_coiter Ts ss) ks))))
blanchet@48975
  1985
          (K (mk_mor_coiter_tac m mor_UNIV_thm unf_defs coiter_defs Abs_inverses' morEs'
blanchet@48975
  1986
            map_comp_id_thms map_congs))
traytel@49109
  1987
        |> Thm.close_derivation
blanchet@48975
  1988
      end;
blanchet@48975
  1989
    val coiter_thms = map (fn thm => (thm OF [mor_coiter_thm, UNIV_I]) RS sym) morE_thms;
blanchet@48975
  1990
blanchet@48975
  1991
    val (raw_coind_thms, raw_coind_thm) =
blanchet@48975
  1992
      let
blanchet@48975
  1993
        val prem = HOLogic.mk_Trueprop (mk_sbis passive_UNIVs UNIVs unfs TRs);
blanchet@48975
  1994
        val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  1995
          (map2 (fn R => fn T => mk_subset R (Id_const T)) TRs Ts));
blanchet@48975
  1996
        val goal = fold_rev Logic.all TRs (Logic.mk_implies (prem, concl));
blanchet@48975
  1997
      in
blanchet@48975
  1998
        `split_conj_thm (Skip_Proof.prove lthy [] [] goal
blanchet@48975
  1999
          (K (mk_raw_coind_tac bis_def bis_cong_thm bis_O_thm bis_converse_thm bis_Gr_thm
blanchet@48975
  2000
            tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm
traytel@49109
  2001
            lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects))
traytel@49109
  2002
          |> Thm.close_derivation)
blanchet@48975
  2003
      end;
blanchet@48975
  2004
blanchet@48975
  2005
    val unique_mor_thms =
blanchet@48975
  2006
      let
blanchet@48975
  2007
        val prems = [HOLogic.mk_Trueprop (mk_coalg passive_UNIVs Bs ss), HOLogic.mk_Trueprop
blanchet@48975
  2008
          (HOLogic.mk_conj (mk_mor Bs ss UNIVs unfs coiter_fs,
blanchet@48975
  2009
            mk_mor Bs ss UNIVs unfs coiter_fs_copy))];
blanchet@48975
  2010
        fun mk_fun_eq B f g z = HOLogic.mk_imp
blanchet@48975
  2011
          (HOLogic.mk_mem (z, B), HOLogic.mk_eq (f $ z, g $ z));
blanchet@48975
  2012
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  2013
          (map4 mk_fun_eq Bs coiter_fs coiter_fs_copy zs));
blanchet@48975
  2014
blanchet@48975
  2015
        val unique_mor = Skip_Proof.prove lthy [] []
blanchet@48975
  2016
          (fold_rev Logic.all (Bs @ ss @ coiter_fs @ coiter_fs_copy @ zs)
blanchet@48975
  2017
            (Logic.list_implies (prems, unique)))
traytel@49109
  2018
          (K (mk_unique_mor_tac raw_coind_thms bis_image2_thm))
traytel@49109
  2019
          |> Thm.close_derivation;
blanchet@48975
  2020
      in
blanchet@48975
  2021
        map (fn thm => conjI RSN (2, thm RS mp)) (split_conj_thm unique_mor)
blanchet@48975
  2022
      end;
blanchet@48975
  2023
blanchet@48975
  2024
    val (coiter_unique_mor_thms, coiter_unique_mor_thm) =
blanchet@48975
  2025
      let
blanchet@48975
  2026
        val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs unfs coiter_fs);
blanchet@48975
  2027
        fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_coiter Ts ss i);
blanchet@48975
  2028
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  2029
          (map2 mk_fun_eq coiter_fs ks));
blanchet@48975
  2030
blanchet@48975
  2031
        val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm);
blanchet@48975
  2032
        val mor_thm = mor_comp_thm OF [tcoalg_thm RS mor_final_thm, mor_Abs_thm];
blanchet@48975
  2033
blanchet@48975
  2034
        val unique_mor = Skip_Proof.prove lthy [] []
blanchet@48975
  2035
          (fold_rev Logic.all (ss @ coiter_fs) (Logic.mk_implies (prem, unique)))
traytel@49109
  2036
          (K (mk_coiter_unique_mor_tac raw_coind_thms bis_thm mor_thm coiter_defs))
traytel@49109
  2037
          |> Thm.close_derivation;
blanchet@48975
  2038
      in
blanchet@48975
  2039
        `split_conj_thm unique_mor
blanchet@48975
  2040
      end;
blanchet@48975
  2041
blanchet@48975
  2042
    val (coiter_unique_thms, coiter_unique_thm) = `split_conj_thm (split_conj_prems n
blanchet@48975
  2043
      (mor_UNIV_thm RS @{thm ssubst[of _ _ "%x. x"]} RS coiter_unique_mor_thm));
blanchet@48975
  2044
blanchet@48975
  2045
    val coiter_unf_thms = map (fn thm => mor_id_thm RS thm RS sym) coiter_unique_mor_thms;
blanchet@48975
  2046
blanchet@48975
  2047
    val coiter_o_unf_thms =
blanchet@48975
  2048
      let
blanchet@48975
  2049
        val mor = mor_comp_thm OF [mor_str_thm, mor_coiter_thm];
blanchet@48975
  2050
      in
blanchet@48975
  2051
        map2 (fn unique => fn coiter_fld =>
blanchet@48975
  2052
          trans OF [mor RS unique, coiter_fld]) coiter_unique_mor_thms coiter_unf_thms
blanchet@48975
  2053
      end;
blanchet@48975
  2054
blanchet@48975
  2055
    val timer = time (timer "coiter definitions & thms");
blanchet@48975
  2056
blanchet@48975
  2057
    val map_unfs = map2 (fn Ds => fn bnf =>
blanchet@48975
  2058
      Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,
blanchet@48975
  2059
        map HOLogic.id_const passiveAs @ unfs)) Dss bnfs;
blanchet@48975
  2060
blanchet@48975
  2061
    fun fld_bind i = Binding.suffix_name ("_" ^ fldN) (nth bs (i - 1));
blanchet@48975
  2062
    val fld_name = Binding.name_of o fld_bind;
blanchet@48975
  2063
    val fld_def_bind = rpair [] o Thm.def_binding o fld_bind;
blanchet@48975
  2064
blanchet@48975
  2065
    fun fld_spec i fldT =
blanchet@48975
  2066
      let
blanchet@48975
  2067
        val lhs = Free (fld_name i, fldT);
blanchet@48975
  2068
        val rhs = mk_coiter Ts map_unfs i;
blanchet@48975
  2069
      in
blanchet@48975
  2070
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  2071
      end;
blanchet@48975
  2072
blanchet@48975
  2073
    val ((fld_frees, (_, fld_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  2074
        lthy
blanchet@48975
  2075
        |> fold_map2 (fn i => fn fldT =>
blanchet@48975
  2076
          Specification.definition
blanchet@48975
  2077
            (SOME (fld_bind i, NONE, NoSyn), (fld_def_bind i, fld_spec i fldT))) ks fldTs
blanchet@48975
  2078
        |>> apsnd split_list o split_list
blanchet@48975
  2079
        ||> `Local_Theory.restore;
blanchet@48975
  2080
blanchet@48975
  2081
    (*transforms defined frees into consts*)
blanchet@48975
  2082
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  2083
    fun mk_flds params =
blanchet@48975
  2084
      map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
blanchet@48975
  2085
        fld_frees;
blanchet@48975
  2086
    val flds = mk_flds params';
blanchet@48975
  2087
    val fld_defs = map (Morphism.thm phi) fld_def_frees;
blanchet@48975
  2088
blanchet@48975
  2089
    val fld_o_unf_thms = map2 (Local_Defs.fold lthy o single) fld_defs coiter_o_unf_thms;
blanchet@48975
  2090
blanchet@48975
  2091
    val unf_o_fld_thms =
blanchet@48975
  2092
      let
blanchet@48975
  2093
        fun mk_goal unf fld FT =
blanchet@48975
  2094
          HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (unf, fld), HOLogic.id_const FT));
blanchet@48975
  2095
        val goals = map3 mk_goal unfs flds FTs;
blanchet@48975
  2096
      in
blanchet@48975
  2097
        map5 (fn goal => fn fld_def => fn coiter => fn map_comp_id => fn map_congL =>
blanchet@48975
  2098
          Skip_Proof.prove lthy [] [] goal
traytel@49109
  2099
            (mk_unf_o_fld_tac fld_def coiter map_comp_id map_congL coiter_o_unf_thms)
traytel@49109
  2100
          |> Thm.close_derivation)
blanchet@48975
  2101
          goals fld_defs coiter_thms map_comp_id_thms map_congL_thms
blanchet@48975
  2102
      end;
blanchet@48975
  2103
blanchet@48975
  2104
    val unf_fld_thms = map (fn thm => thm RS @{thm pointfree_idE}) unf_o_fld_thms;
blanchet@48975
  2105
    val fld_unf_thms = map (fn thm => thm RS @{thm pointfree_idE}) fld_o_unf_thms;
blanchet@48975
  2106
blanchet@48975
  2107
    val bij_unf_thms =
blanchet@48975
  2108
      map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) fld_o_unf_thms unf_o_fld_thms;
blanchet@48975
  2109
    val inj_unf_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_unf_thms;
blanchet@48975
  2110
    val surj_unf_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_unf_thms;
blanchet@48975
  2111
    val unf_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_unf_thms;
blanchet@48975
  2112
    val unf_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_unf_thms;
blanchet@49029
  2113
    val unf_exhaust_thms = map (fn thm => thm RS exE) unf_nchotomy_thms;
blanchet@48975
  2114
blanchet@48975
  2115
    val bij_fld_thms =
blanchet@48975
  2116
      map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) unf_o_fld_thms fld_o_unf_thms;
blanchet@48975
  2117
    val inj_fld_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_fld_thms;
blanchet@48975
  2118
    val surj_fld_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_fld_thms;
blanchet@48975
  2119
    val fld_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_fld_thms;
blanchet@48975
  2120
    val fld_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_fld_thms;
blanchet@49029
  2121
    val fld_exhaust_thms = map (fn thm => thm RS exE) fld_nchotomy_thms;
blanchet@48975
  2122
blanchet@48975
  2123
    val fld_coiter_thms = map3 (fn unf_inject => fn coiter => fn unf_fld =>
blanchet@48975
  2124
      iffD1 OF [unf_inject, trans  OF [coiter, unf_fld RS sym]])
blanchet@48975
  2125
      unf_inject_thms coiter_thms unf_fld_thms;
blanchet@48975
  2126
blanchet@48975
  2127
    val timer = time (timer "fld definitions & thms");
blanchet@48975
  2128
blanchet@48975
  2129
    val corec_Inl_sum_thms =
blanchet@48975
  2130
      let
blanchet@48975
  2131
        val mor = mor_comp_thm OF [mor_sum_case_thm, mor_coiter_thm];
blanchet@48975
  2132
      in
blanchet@48975
  2133
        map2 (fn unique => fn coiter_unf =>
blanchet@48975
  2134
          trans OF [mor RS unique, coiter_unf]) coiter_unique_mor_thms coiter_unf_thms
blanchet@48975
  2135
      end;
blanchet@48975
  2136
blanchet@48975
  2137
    fun corec_bind i = Binding.suffix_name ("_" ^ coN ^ recN) (nth bs (i - 1));
blanchet@48975
  2138
    val corec_name = Binding.name_of o corec_bind;
blanchet@48975
  2139
    val corec_def_bind = rpair [] o Thm.def_binding o corec_bind;
blanchet@48975
  2140
blanchet@48975
  2141
    fun corec_spec i T AT =
blanchet@48975
  2142
      let
blanchet@48975
  2143
        val corecT = Library.foldr (op -->) (corec_sTs, AT --> T);
blanchet@48975
  2144
        val maps = map3 (fn unf => fn sum_s => fn map => mk_sum_case
blanchet@48975
  2145
            (HOLogic.mk_comp (Term.list_comb (map, passive_ids @ corec_Inls), unf)) sum_s)
blanchet@48975
  2146
          unfs corec_ss corec_maps;
blanchet@48975
  2147
blanchet@48975
  2148
        val lhs = Term.list_comb (Free (corec_name i, corecT), corec_ss);
blanchet@48975
  2149
        val rhs = HOLogic.mk_comp (mk_coiter Ts maps i, Inr_const T AT);
blanchet@48975
  2150
      in
blanchet@48975
  2151
        HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
blanchet@48975
  2152
      end;
blanchet@48975
  2153
blanchet@48975
  2154
    val ((corec_frees, (_, corec_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  2155
        lthy
blanchet@48975
  2156
        |> fold_map3 (fn i => fn T => fn AT =>
blanchet@48975
  2157
          Specification.definition
blanchet@48975
  2158
            (SOME (corec_bind i, NONE, NoSyn), (corec_def_bind i, corec_spec i T AT)))
blanchet@48975
  2159
            ks Ts activeAs
blanchet@48975
  2160
        |>> apsnd split_list o split_list
blanchet@48975
  2161
        ||> `Local_Theory.restore;
blanchet@48975
  2162
blanchet@48975
  2163
    (*transforms defined frees into consts*)
blanchet@48975
  2164
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  2165
    val corecs = map (fst o dest_Const o Morphism.term phi) corec_frees;
blanchet@48975
  2166
    fun mk_corec ss i = Term.list_comb (Const (nth corecs (i - 1), Library.foldr (op -->)
blanchet@48975
  2167
      (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
blanchet@48975
  2168
    val corec_defs = map (Morphism.thm phi) corec_def_frees;
blanchet@48975
  2169
blanchet@48975
  2170
    val sum_cases =
blanchet@48975
  2171
      map2 (fn T => fn i => mk_sum_case (HOLogic.id_const T) (mk_corec corec_ss i)) Ts ks;
blanchet@48975
  2172
    val corec_thms =
blanchet@48975
  2173
      let
blanchet@48975
  2174
        fun mk_goal i corec_s corec_map unf z =
blanchet@48975
  2175
          let
blanchet@48975
  2176
            val lhs = unf $ (mk_corec corec_ss i $ z);
blanchet@48975
  2177
            val rhs = Term.list_comb (corec_map, passive_ids @ sum_cases) $ (corec_s $ z);
blanchet@48975
  2178
          in
blanchet@48975
  2179
            fold_rev Logic.all (z :: corec_ss) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
blanchet@48975
  2180
          end;
blanchet@48975
  2181
        val goals = map5 mk_goal ks corec_ss corec_maps_rev unfs zs;
blanchet@48975
  2182
      in
blanchet@48975
  2183
        map3 (fn goal => fn coiter => fn map_cong =>
blanchet@48975
  2184
          Skip_Proof.prove lthy [] [] goal
traytel@49109
  2185
            (mk_corec_tac m corec_defs coiter map_cong corec_Inl_sum_thms)
traytel@49109
  2186
          |> Thm.close_derivation)
traytel@49109
  2187
        goals coiter_thms map_congs
blanchet@48975
  2188
      end;
blanchet@48975
  2189
blanchet@48975
  2190
    val timer = time (timer "corec definitions & thms");
blanchet@48975
  2191
blanchet@48975
  2192
    val (unf_coinduct_thm, coinduct_params, rel_coinduct_thm, pred_coinduct_thm,
blanchet@48975
  2193
         unf_coinduct_upto_thm, rel_coinduct_upto_thm, pred_coinduct_upto_thm) =
blanchet@48975
  2194
      let
blanchet@48975
  2195
        val zs = Jzs1 @ Jzs2;
blanchet@48975
  2196
        val frees = phis @ zs;
blanchet@48975
  2197
blanchet@48975
  2198
        fun mk_Ids Id = if Id then map Id_const passiveAs else map mk_diag passive_UNIVs;
blanchet@48975
  2199
blanchet@48975
  2200
        fun mk_phi upto_eq phi z1 z2 = if upto_eq
blanchet@48975
  2201
          then Term.absfree (dest_Free z1) (Term.absfree (dest_Free z2)
blanchet@48975
  2202
            (HOLogic.mk_disj (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2))))
blanchet@48975
  2203
          else phi;
blanchet@48975
  2204
blanchet@48975
  2205
        fun phi_rels upto_eq = map4 (fn phi => fn T => fn z1 => fn z2 =>
blanchet@48975
  2206
          HOLogic.Collect_const (HOLogic.mk_prodT (T, T)) $
blanchet@48975
  2207
            HOLogic.mk_split (mk_phi upto_eq phi z1 z2)) phis Ts Jzs1 Jzs2;
blanchet@48975
  2208
blanchet@48975
  2209
        val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs;
blanchet@48975
  2210
blanchet@48975
  2211
        fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2));
blanchet@48975
  2212
        val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  2213
          (map3 mk_concl phis Jzs1 Jzs2));
blanchet@48975
  2214
blanchet@48975
  2215
        fun mk_rel_prem upto_eq phi unf rel Jz Jz_copy =
blanchet@48975
  2216
          let
blanchet@48975
  2217
            val concl = HOLogic.mk_mem (HOLogic.mk_tuple [unf $ Jz, unf $ Jz_copy],
blanchet@48975
  2218
              Term.list_comb (rel, mk_Ids upto_eq @ phi_rels upto_eq));
blanchet@48975
  2219
          in