src/HOL/BNF/Tools/bnf_gfp.ML
author traytel
Thu Sep 12 16:31:42 2013 +0200 (2013-09-12)
changeset 53566 5ff3a2d112d7
parent 53553 d4191bf88529
child 53567 7f84e5e7a49b
permissions -rw-r--r--
conceal internal bindings
blanchet@49509
     1
(*  Title:      HOL/BNF/Tools/bnf_gfp.ML
blanchet@48975
     2
    Author:     Dmitriy Traytel, TU Muenchen
blanchet@48975
     3
    Author:     Andrei Popescu, TU Muenchen
blanchet@48975
     4
    Author:     Jasmin Blanchette, TU Muenchen
blanchet@48975
     5
    Copyright   2012
blanchet@48975
     6
blanchet@48975
     7
Codatatype construction.
blanchet@48975
     8
*)
blanchet@48975
     9
blanchet@48975
    10
signature BNF_GFP =
blanchet@48975
    11
sig
blanchet@51867
    12
  val construct_gfp: mixfix list -> binding list -> binding list -> binding list list ->
blanchet@51867
    13
    binding list -> (string * sort) list -> typ list * typ list list -> BNF_Def.bnf list ->
blanchet@51866
    14
    local_theory -> BNF_FP_Util.fp_result * local_theory
blanchet@48975
    15
end;
blanchet@48975
    16
blanchet@48975
    17
structure BNF_GFP : BNF_GFP =
blanchet@48975
    18
struct
blanchet@48975
    19
blanchet@48975
    20
open BNF_Def
blanchet@48975
    21
open BNF_Util
blanchet@48975
    22
open BNF_Tactics
blanchet@49585
    23
open BNF_Comp
blanchet@51850
    24
open BNF_FP_Util
blanchet@49636
    25
open BNF_FP_Def_Sugar
blanchet@53310
    26
open BNF_FP_Rec_Sugar
blanchet@48975
    27
open BNF_GFP_Util
blanchet@48975
    28
open BNF_GFP_Tactics
blanchet@48975
    29
blanchet@49460
    30
datatype wit_tree = Wit_Leaf of int | Wit_Node of (int * int * int list) * wit_tree list;
traytel@49104
    31
traytel@49104
    32
fun mk_tree_args (I, T) (I', Ts) = (sort_distinct int_ord (I @ I'), T :: Ts);
traytel@49104
    33
traytel@49104
    34
fun finish Iss m seen i (nwit, I) =
traytel@49104
    35
  let
traytel@49104
    36
    val treess = map (fn j =>
blanchet@49460
    37
        if j < m orelse member (op =) seen j then [([j], Wit_Leaf j)]
traytel@49104
    38
        else
traytel@49104
    39
          map_index (finish Iss m (insert (op =) j seen) j) (nth Iss (j - m))
traytel@49104
    40
          |> flat
traytel@49104
    41
          |> minimize_wits)
traytel@49104
    42
      I;
traytel@49104
    43
  in
blanchet@49460
    44
    map (fn (I, t) => (I, Wit_Node ((i - m, nwit, filter (fn i => i < m) I), t)))
traytel@49104
    45
      (fold_rev (map_product mk_tree_args) treess [([], [])])
traytel@49104
    46
    |> minimize_wits
traytel@49104
    47
  end;
traytel@49104
    48
blanchet@49501
    49
fun tree_to_ctor_wit vars _ _ (Wit_Leaf j) = ([j], nth vars j)
blanchet@49501
    50
  | tree_to_ctor_wit vars ctors witss (Wit_Node ((i, nwit, I), subtrees)) =
blanchet@49501
    51
     (I, nth ctors i $ (Term.list_comb (snd (nth (nth witss i) nwit),
blanchet@49501
    52
       map (snd o tree_to_ctor_wit vars ctors witss) subtrees)));
traytel@49104
    53
blanchet@49460
    54
fun tree_to_coind_wits _ (Wit_Leaf _) = []
blanchet@49460
    55
  | tree_to_coind_wits lwitss (Wit_Node ((i, nwit, I), subtrees)) =
traytel@49104
    56
     ((i, I), nth (nth lwitss i) nwit) :: maps (tree_to_coind_wits lwitss) subtrees;
traytel@49104
    57
blanchet@49460
    58
(*all BNFs have the same lives*)
blanchet@51867
    59
fun construct_gfp mixfixes map_bs rel_bs set_bss bs resBs (resDs, Dss) bnfs lthy =
blanchet@48975
    60
  let
traytel@53143
    61
    val time = time lthy;
blanchet@48975
    62
    val timer = time (Timer.startRealTimer ());
blanchet@48975
    63
blanchet@48975
    64
    val live = live_of_bnf (hd bnfs);
blanchet@48975
    65
    val n = length bnfs; (*active*)
blanchet@48975
    66
    val ks = 1 upto n;
blanchet@52956
    67
    val m = live - n; (*passive, if 0 don't generate a new BNF*)
blanchet@48975
    68
    val ls = 1 upto m;
traytel@53566
    69
traytel@53566
    70
    val note_all = Config.get lthy bnf_note_all;
blanchet@52956
    71
    val b_names = map Binding.name_of bs;
traytel@53566
    72
    val b_name = mk_common_name b_names;
traytel@53566
    73
    val b = Binding.name b_name;
traytel@53566
    74
    val mk_internal_b = Binding.name #> Binding.prefix true b_name #> Binding.conceal;
traytel@53566
    75
    fun mk_internal_bs name =
traytel@53566
    76
      map (fn b =>
traytel@53566
    77
        Binding.prefix true b_name (Binding.suffix_name ("_" ^ name) b) |> Binding.conceal) bs;
traytel@53566
    78
    val external_bs = map2 (Binding.prefix false) b_names bs
traytel@53566
    79
      |> note_all = false ? map Binding.conceal;
blanchet@48975
    80
blanchet@49460
    81
    (* TODO: check if m, n, etc., are sane *)
blanchet@48975
    82
traytel@49185
    83
    val deads = fold (union (op =)) Dss resDs;
blanchet@48975
    84
    val names_lthy = fold Variable.declare_typ deads lthy;
blanchet@52956
    85
    val passives = map fst (subtract (op = o apsnd TFree) deads resBs);
blanchet@48975
    86
blanchet@48975
    87
    (* tvars *)
blanchet@52938
    88
    val ((((((((passiveAs, activeAs), passiveBs), activeBs), passiveCs), activeCs), passiveXs),
blanchet@52938
    89
        passiveYs), idxT) = names_lthy
blanchet@52956
    90
      |> variant_tfrees passives
blanchet@52938
    91
      ||>> mk_TFrees n
blanchet@52956
    92
      ||>> variant_tfrees passives
blanchet@52938
    93
      ||>> mk_TFrees n
blanchet@52938
    94
      ||>> mk_TFrees m
blanchet@52938
    95
      ||>> mk_TFrees n
blanchet@48975
    96
      ||>> mk_TFrees m
blanchet@48975
    97
      ||>> mk_TFrees m
blanchet@48975
    98
      ||> fst o mk_TFrees 1
blanchet@48975
    99
      ||> the_single;
blanchet@48975
   100
blanchet@52938
   101
    val allAs = passiveAs @ activeAs;
blanchet@52938
   102
    val allBs' = passiveBs @ activeBs;
blanchet@48975
   103
    val Ass = replicate n allAs;
blanchet@48975
   104
    val allBs = passiveAs @ activeBs;
blanchet@48975
   105
    val Bss = replicate n allBs;
blanchet@48975
   106
    val allCs = passiveAs @ activeCs;
blanchet@48975
   107
    val allCs' = passiveBs @ activeCs;
blanchet@48975
   108
    val Css' = replicate n allCs';
blanchet@48975
   109
blanchet@51866
   110
    (* types *)
traytel@49185
   111
    val dead_poss =
blanchet@52956
   112
      map (fn x => if member (op =) deads (TFree x) then SOME (TFree x) else NONE) resBs;
traytel@49185
   113
    fun mk_param NONE passive = (hd passive, tl passive)
traytel@49185
   114
      | mk_param (SOME a) passive = (a, passive);
traytel@49185
   115
    val mk_params = fold_map mk_param dead_poss #> fst;
traytel@49185
   116
blanchet@48975
   117
    fun mk_FTs Ts = map2 (fn Ds => mk_T_of_bnf Ds Ts) Dss bnfs;
traytel@49185
   118
    val (params, params') = `(map Term.dest_TFree) (mk_params passiveAs);
traytel@49620
   119
    val (dead_params, dead_params') = `(map Term.dest_TFree) (subtract (op =) passiveAs params');
blanchet@48975
   120
    val FTsAs = mk_FTs allAs;
blanchet@48975
   121
    val FTsBs = mk_FTs allBs;
blanchet@48975
   122
    val FTsCs = mk_FTs allCs;
blanchet@48975
   123
    val ATs = map HOLogic.mk_setT passiveAs;
blanchet@48975
   124
    val BTs = map HOLogic.mk_setT activeAs;
blanchet@48975
   125
    val B'Ts = map HOLogic.mk_setT activeBs;
blanchet@48975
   126
    val B''Ts = map HOLogic.mk_setT activeCs;
blanchet@48975
   127
    val sTs = map2 (fn T => fn U => T --> U) activeAs FTsAs;
blanchet@48975
   128
    val s'Ts = map2 (fn T => fn U => T --> U) activeBs FTsBs;
blanchet@48975
   129
    val s''Ts = map2 (fn T => fn U => T --> U) activeCs FTsCs;
blanchet@48975
   130
    val fTs = map2 (fn T => fn U => T --> U) activeAs activeBs;
blanchet@48975
   131
    val self_fTs = map (fn T => T --> T) activeAs;
blanchet@48975
   132
    val gTs = map2 (fn T => fn U => T --> U) activeBs activeCs;
blanchet@48975
   133
    val all_gTs = map2 (fn T => fn U => T --> U) allBs allCs';
blanchet@48975
   134
    val RTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeBs;
blanchet@48975
   135
    val sRTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeAs;
blanchet@48975
   136
    val R'Ts = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeBs activeCs;
blanchet@48975
   137
    val setsRTs = map HOLogic.mk_setT sRTs;
blanchet@48975
   138
    val setRTs = map HOLogic.mk_setT RTs;
blanchet@48975
   139
    val all_sbisT = HOLogic.mk_tupleT setsRTs;
blanchet@48975
   140
    val setR'Ts = map HOLogic.mk_setT R'Ts;
blanchet@48975
   141
    val FRTs = mk_FTs (passiveAs @ RTs);
blanchet@48975
   142
    val sumBsAs = map2 (curry mk_sumT) activeBs activeAs;
blanchet@48975
   143
    val sumFTs = mk_FTs (passiveAs @ sumBsAs);
blanchet@48975
   144
    val sum_sTs = map2 (fn T => fn U => T --> U) activeAs sumFTs;
blanchet@48975
   145
blanchet@48975
   146
    (* terms *)
blanchet@48975
   147
    val mapsAsAs = map4 mk_map_of_bnf Dss Ass Ass bnfs;
blanchet@48975
   148
    val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;
blanchet@48975
   149
    val mapsBsCs' = map4 mk_map_of_bnf Dss Bss Css' bnfs;
blanchet@48975
   150
    val mapsAsCs' = map4 mk_map_of_bnf Dss Ass Css' bnfs;
blanchet@48975
   151
    val map_Inls = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ sumBsAs)) bnfs;
blanchet@48975
   152
    val map_Inls_rev = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ sumBsAs)) Bss bnfs;
blanchet@48975
   153
    val map_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs;
blanchet@48975
   154
    val map_snds = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs;
blanchet@48975
   155
    fun mk_setss Ts = map3 mk_sets_of_bnf (map (replicate live) Dss)
blanchet@48975
   156
      (map (replicate live) (replicate n Ts)) bnfs;
blanchet@48975
   157
    val setssAs = mk_setss allAs;
blanchet@48975
   158
    val setssAs' = transpose setssAs;
blanchet@48975
   159
    val bis_setss = mk_setss (passiveAs @ RTs);
traytel@51893
   160
    val relsAsBs = map4 mk_rel_of_bnf Dss Ass Bss bnfs;
blanchet@48975
   161
    val bds = map3 mk_bd_of_bnf Dss Ass bnfs;
blanchet@48975
   162
    val sum_bd = Library.foldr1 (uncurry mk_csum) bds;
blanchet@48975
   163
    val sum_bdT = fst (dest_relT (fastype_of sum_bd));
blanchet@48975
   164
blanchet@48975
   165
    val emptys = map (fn T => HOLogic.mk_set T []) passiveAs;
blanchet@48975
   166
    val Zeros = map (fn empty =>
blanchet@48975
   167
     HOLogic.mk_tuple (map (fn U => absdummy U empty) activeAs)) emptys;
blanchet@48975
   168
    val hrecTs = map fastype_of Zeros;
blanchet@48975
   169
    val hsetTs = map (fn hrecT => Library.foldr (op -->) (sTs, HOLogic.natT --> hrecT)) hrecTs;
blanchet@48975
   170
traytel@52635
   171
    val ((((((((((((((((((((((((((((((((((zs, zs'), zs_copy), zs_copy2),
traytel@52635
   172
      z's), As), Bs), Bs_copy), B's), B''s), ss), sum_ss), s's), s''s), fs), fs_copy),
blanchet@52298
   173
      self_fs), gs), all_gs), xFs), yFs), yFs_copy), RFs), (Rtuple, Rtuple')), (hrecs, hrecs')),
blanchet@52298
   174
      (nat, nat')), Rs), Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), Kss), names_lthy) = lthy
blanchet@48975
   175
      |> mk_Frees' "b" activeAs
blanchet@48975
   176
      ||>> mk_Frees "b" activeAs
blanchet@48975
   177
      ||>> mk_Frees "b" activeAs
blanchet@48975
   178
      ||>> mk_Frees "b" activeBs
blanchet@48975
   179
      ||>> mk_Frees "A" ATs
blanchet@48975
   180
      ||>> mk_Frees "B" BTs
blanchet@48975
   181
      ||>> mk_Frees "B" BTs
blanchet@48975
   182
      ||>> mk_Frees "B'" B'Ts
blanchet@48975
   183
      ||>> mk_Frees "B''" B''Ts
blanchet@48975
   184
      ||>> mk_Frees "s" sTs
blanchet@48975
   185
      ||>> mk_Frees "sums" sum_sTs
blanchet@48975
   186
      ||>> mk_Frees "s'" s'Ts
blanchet@48975
   187
      ||>> mk_Frees "s''" s''Ts
blanchet@48975
   188
      ||>> mk_Frees "f" fTs
blanchet@48975
   189
      ||>> mk_Frees "f" fTs
blanchet@48975
   190
      ||>> mk_Frees "f" self_fTs
blanchet@48975
   191
      ||>> mk_Frees "g" gTs
blanchet@48975
   192
      ||>> mk_Frees "g" all_gTs
blanchet@48975
   193
      ||>> mk_Frees "x" FTsAs
traytel@51925
   194
      ||>> mk_Frees "y" FTsBs
traytel@51925
   195
      ||>> mk_Frees "y" FTsBs
blanchet@48975
   196
      ||>> mk_Frees "x" FRTs
blanchet@48975
   197
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Rtuple") all_sbisT
blanchet@48975
   198
      ||>> mk_Frees' "rec" hrecTs
blanchet@48975
   199
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT
blanchet@48975
   200
      ||>> mk_Frees "R" setRTs
blanchet@48975
   201
      ||>> mk_Frees "R" setRTs
blanchet@48975
   202
      ||>> mk_Frees "R'" setR'Ts
blanchet@48975
   203
      ||>> mk_Frees "R" setsRTs
blanchet@48975
   204
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") idxT
blanchet@48975
   205
      ||>> yield_singleton (mk_Frees "I") (HOLogic.mk_setT idxT)
blanchet@48975
   206
      ||>> mk_Frees "Ri" (map (fn T => idxT --> T) setRTs)
blanchet@48975
   207
      ||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) activeAs) ATs);
blanchet@48975
   208
blanchet@48975
   209
    val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
traytel@51447
   210
    val passive_Id_ons = map mk_Id_on As;
blanchet@48975
   211
    val active_UNIVs = map HOLogic.mk_UNIV activeAs;
blanchet@48975
   212
    val sum_UNIVs = map HOLogic.mk_UNIV sumBsAs;
blanchet@48975
   213
    val passive_ids = map HOLogic.id_const passiveAs;
blanchet@48975
   214
    val active_ids = map HOLogic.id_const activeAs;
blanchet@48975
   215
    val Inls = map2 Inl_const activeBs activeAs;
blanchet@48975
   216
    val fsts = map fst_const RTs;
blanchet@48975
   217
    val snds = map snd_const RTs;
blanchet@48975
   218
blanchet@48975
   219
    (* thms *)
blanchet@48975
   220
    val bd_card_orders = map bd_card_order_of_bnf bnfs;
blanchet@48975
   221
    val bd_card_order = hd bd_card_orders
blanchet@48975
   222
    val bd_Card_orders = map bd_Card_order_of_bnf bnfs;
blanchet@48975
   223
    val bd_Card_order = hd bd_Card_orders;
blanchet@48975
   224
    val bd_Cinfinites = map bd_Cinfinite_of_bnf bnfs;
blanchet@48975
   225
    val bd_Cinfinite = hd bd_Cinfinites;
blanchet@48975
   226
    val in_monos = map in_mono_of_bnf bnfs;
blanchet@53287
   227
    val map_comp0s = map map_comp0_of_bnf bnfs;
blanchet@53287
   228
    val sym_map_comps = map mk_sym map_comp0s;
blanchet@53288
   229
    val map_comps = map map_comp_of_bnf bnfs;
blanchet@51761
   230
    val map_cong0s = map map_cong0_of_bnf bnfs;
blanchet@53270
   231
    val map_id0s = map map_id0_of_bnf bnfs;
blanchet@53285
   232
    val map_ids = map map_id_of_bnf bnfs;
blanchet@48975
   233
    val map_wpulls = map map_wpull_of_bnf bnfs;
blanchet@48975
   234
    val set_bdss = map set_bd_of_bnf bnfs;
blanchet@53290
   235
    val set_mapss = map set_map_of_bnf bnfs;
traytel@51893
   236
    val rel_congs = map rel_cong_of_bnf bnfs;
traytel@51893
   237
    val rel_converseps = map rel_conversep_of_bnf bnfs;
traytel@51893
   238
    val rel_Grps = map rel_Grp_of_bnf bnfs;
traytel@51893
   239
    val rel_eqs = map rel_eq_of_bnf bnfs;
traytel@51893
   240
    val rel_monos = map rel_mono_of_bnf bnfs;
traytel@51893
   241
    val rel_OOs = map rel_OO_of_bnf bnfs;
traytel@51893
   242
    val rel_OO_Grps = map rel_OO_Grp_of_bnf bnfs;
blanchet@48975
   243
blanchet@48975
   244
    val timer = time (timer "Extracted terms & thms");
blanchet@48975
   245
blanchet@48975
   246
    (* derived thms *)
blanchet@48975
   247
blanchet@52956
   248
    (*map g1 ... gm g(m+1) ... g(m+n) (map id ... id f(m+1) ... f(m+n) x) =
blanchet@48975
   249
      map g1 ... gm (g(m+1) o f(m+1)) ... (g(m+n) o f(m+n)) x*)
blanchet@53287
   250
    fun mk_map_comp_id x mapAsBs mapBsCs mapAsCs map_comp0 =
blanchet@48975
   251
      let
blanchet@48975
   252
        val lhs = Term.list_comb (mapBsCs, all_gs) $
blanchet@48975
   253
          (Term.list_comb (mapAsBs, passive_ids @ fs) $ x);
blanchet@48975
   254
        val rhs =
blanchet@48975
   255
          Term.list_comb (mapAsCs, take m all_gs @ map HOLogic.mk_comp (drop m all_gs ~~ fs)) $ x;
blanchet@48975
   256
      in
wenzelm@51551
   257
        Goal.prove_sorry lthy [] []
blanchet@49123
   258
          (fold_rev Logic.all (x :: fs @ all_gs) (mk_Trueprop_eq (lhs, rhs)))
blanchet@53287
   259
          (K (mk_map_comp_id_tac map_comp0))
traytel@49109
   260
        |> Thm.close_derivation
blanchet@48975
   261
      end;
blanchet@48975
   262
blanchet@53288
   263
    val map_comp_id_thms = map5 mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comps;
blanchet@48975
   264
blanchet@48975
   265
    (*forall a : set(m+1) x. f(m+1) a = a; ...; forall a : set(m+n) x. f(m+n) a = a ==>
blanchet@48975
   266
      map id ... id f(m+1) ... f(m+n) x = x*)
blanchet@53285
   267
    fun mk_map_cong0L x mapAsAs sets map_cong0 map_id =
blanchet@48975
   268
      let
blanchet@48975
   269
        fun mk_prem set f z z' =
blanchet@48975
   270
          HOLogic.mk_Trueprop
blanchet@48975
   271
            (mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z))));
blanchet@48975
   272
        val prems = map4 mk_prem (drop m sets) self_fs zs zs';
blanchet@49123
   273
        val goal = mk_Trueprop_eq (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x);
blanchet@48975
   274
      in
wenzelm@51551
   275
        Goal.prove_sorry lthy [] []
blanchet@48975
   276
          (fold_rev Logic.all (x :: self_fs) (Logic.list_implies (prems, goal)))
blanchet@53285
   277
          (K (mk_map_cong0L_tac m map_cong0 map_id))
traytel@49109
   278
        |> Thm.close_derivation
blanchet@48975
   279
      end;
blanchet@48975
   280
blanchet@53285
   281
    val map_cong0L_thms = map5 mk_map_cong0L xFs mapsAsAs setssAs map_cong0s map_ids;
blanchet@48975
   282
    val in_mono'_thms = map (fn thm =>
blanchet@48975
   283
      (thm OF (replicate m subset_refl)) RS @{thm set_mp}) in_monos;
blanchet@48975
   284
blanchet@48975
   285
    val map_arg_cong_thms =
blanchet@48975
   286
      let
traytel@51925
   287
        val prems = map2 (curry mk_Trueprop_eq) yFs yFs_copy;
traytel@51925
   288
        val maps = map (fn mapx => Term.list_comb (mapx, all_gs)) mapsBsCs';
blanchet@49123
   289
        val concls =
traytel@51925
   290
          map3 (fn x => fn y => fn mapx => mk_Trueprop_eq (mapx $ x, mapx $ y)) yFs yFs_copy maps;
blanchet@48975
   291
        val goals =
blanchet@48975
   292
          map4 (fn prem => fn concl => fn x => fn y =>
traytel@51925
   293
            fold_rev Logic.all (x :: y :: all_gs) (Logic.mk_implies (prem, concl)))
traytel@51925
   294
          prems concls yFs yFs_copy;
blanchet@48975
   295
      in
wenzelm@51551
   296
        map (fn goal => Goal.prove_sorry lthy [] [] goal
wenzelm@51798
   297
          (K ((hyp_subst_tac lthy THEN' rtac refl) 1)) |> Thm.close_derivation) goals
blanchet@48975
   298
      end;
blanchet@48975
   299
blanchet@48975
   300
    val timer = time (timer "Derived simple theorems");
blanchet@48975
   301
blanchet@48975
   302
    (* coalgebra *)
blanchet@48975
   303
traytel@53566
   304
    val coalg_bind = mk_internal_b (coN ^ algN) ;
blanchet@48975
   305
    val coalg_name = Binding.name_of coalg_bind;
blanchet@48975
   306
    val coalg_def_bind = (Thm.def_binding coalg_bind, []);
blanchet@48975
   307
blanchet@48975
   308
    (*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in A1 .. Am B1 ... Bn)*)
blanchet@48975
   309
    val coalg_spec =
blanchet@48975
   310
      let
blanchet@48975
   311
        val coalgT = Library.foldr (op -->) (ATs @ BTs @ sTs, HOLogic.boolT);
blanchet@48975
   312
blanchet@48975
   313
        val ins = map3 mk_in (replicate n (As @ Bs)) setssAs FTsAs;
blanchet@48975
   314
        fun mk_coalg_conjunct B s X z z' =
blanchet@48975
   315
          mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X)));
blanchet@48975
   316
blanchet@48975
   317
        val lhs = Term.list_comb (Free (coalg_name, coalgT), As @ Bs @ ss);
blanchet@48975
   318
        val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_coalg_conjunct Bs ss ins zs zs')
blanchet@48975
   319
      in
blanchet@49123
   320
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   321
      end;
blanchet@48975
   322
blanchet@48975
   323
    val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) =
blanchet@49311
   324
      lthy
blanchet@49311
   325
      |> Specification.definition (SOME (coalg_bind, NONE, NoSyn), (coalg_def_bind, coalg_spec))
blanchet@49311
   326
      ||> `Local_Theory.restore;
blanchet@49311
   327
blanchet@48975
   328
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   329
    val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free));
blanchet@48975
   330
    val coalg_def = Morphism.thm phi coalg_def_free;
blanchet@48975
   331
blanchet@48975
   332
    fun mk_coalg As Bs ss =
blanchet@48975
   333
      let
blanchet@48975
   334
        val args = As @ Bs @ ss;
blanchet@48975
   335
        val Ts = map fastype_of args;
blanchet@48975
   336
        val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   337
      in
blanchet@48975
   338
        Term.list_comb (Const (coalg, coalgT), args)
blanchet@48975
   339
      end;
blanchet@48975
   340
blanchet@48975
   341
    val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
blanchet@48975
   342
blanchet@48975
   343
    val coalg_in_thms = map (fn i =>
traytel@52904
   344
      coalg_def RS iffD1 RS mk_conjunctN n i RS bspec) ks
blanchet@48975
   345
blanchet@48975
   346
    val coalg_set_thmss =
blanchet@48975
   347
      let
blanchet@48975
   348
        val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
blanchet@48975
   349
        fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));
traytel@51893
   350
        fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) B);
blanchet@48975
   351
        val prems = map2 mk_prem zs Bs;
blanchet@48975
   352
        val conclss = map3 (fn s => fn x => fn sets => map2 (mk_concl s x) (As @ Bs) sets)
blanchet@48975
   353
          ss zs setssAs;
blanchet@48975
   354
        val goalss = map3 (fn x => fn prem => fn concls => map (fn concl =>
blanchet@48975
   355
          fold_rev Logic.all (x :: As @ Bs @ ss)
blanchet@48975
   356
            (Logic.list_implies (coalg_prem :: [prem], concl))) concls) zs prems conclss;
blanchet@48975
   357
      in
wenzelm@51551
   358
        map (fn goals => map (fn goal => Goal.prove_sorry lthy [] [] goal
traytel@49109
   359
          (K (mk_coalg_set_tac coalg_def)) |> Thm.close_derivation) goals) goalss
blanchet@48975
   360
      end;
blanchet@48975
   361
blanchet@48975
   362
    fun mk_tcoalg ATs BTs = mk_coalg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);
blanchet@48975
   363
blanchet@48975
   364
    val tcoalg_thm =
blanchet@48975
   365
      let
blanchet@48975
   366
        val goal = fold_rev Logic.all ss
blanchet@48975
   367
          (HOLogic.mk_Trueprop (mk_tcoalg passiveAs activeAs ss))
blanchet@48975
   368
      in
wenzelm@51551
   369
        Goal.prove_sorry lthy [] [] goal
blanchet@48975
   370
          (K (stac coalg_def 1 THEN CONJ_WRAP
blanchet@48975
   371
            (K (EVERY' [rtac ballI, rtac CollectI,
blanchet@48975
   372
              CONJ_WRAP' (K (EVERY' [rtac @{thm subset_UNIV}])) allAs] 1)) ss))
traytel@49109
   373
        |> Thm.close_derivation
blanchet@48975
   374
      end;
blanchet@48975
   375
blanchet@48975
   376
    val timer = time (timer "Coalgebra definition & thms");
blanchet@48975
   377
blanchet@48975
   378
    (* morphism *)
blanchet@48975
   379
traytel@53566
   380
    val mor_bind = mk_internal_b morN;
blanchet@48975
   381
    val mor_name = Binding.name_of mor_bind;
blanchet@48975
   382
    val mor_def_bind = (Thm.def_binding mor_bind, []);
blanchet@48975
   383
blanchet@48975
   384
    (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*)
blanchet@48975
   385
    (*mor) forall i = 1 ... n: (\<forall>x \<in> Bi.
blanchet@48975
   386
       Fi_map id ... id f1 ... fn (si x) = si' (fi x)*)
blanchet@48975
   387
    val mor_spec =
blanchet@48975
   388
      let
blanchet@48975
   389
        val morT = Library.foldr (op -->) (BTs @ sTs @ B'Ts @ s'Ts @ fTs, HOLogic.boolT);
blanchet@48975
   390
blanchet@48975
   391
        fun mk_fbetw f B1 B2 z z' =
blanchet@48975
   392
          mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));
blanchet@48975
   393
        fun mk_mor B mapAsBs f s s' z z' =
blanchet@48975
   394
          mk_Ball B (Term.absfree z' (HOLogic.mk_eq
blanchet@48975
   395
            (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z))));
blanchet@48975
   396
        val lhs = Term.list_comb (Free (mor_name, morT), Bs @ ss @ B's @ s's @ fs);
blanchet@48975
   397
        val rhs = HOLogic.mk_conj
blanchet@48975
   398
          (Library.foldr1 HOLogic.mk_conj (map5 mk_fbetw fs Bs B's zs zs'),
blanchet@48975
   399
           Library.foldr1 HOLogic.mk_conj (map7 mk_mor Bs mapsAsBs fs ss s's zs zs'))
blanchet@48975
   400
      in
blanchet@49123
   401
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   402
      end;
blanchet@48975
   403
blanchet@48975
   404
    val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
blanchet@49311
   405
      lthy
blanchet@49311
   406
      |> Specification.definition (SOME (mor_bind, NONE, NoSyn), (mor_def_bind, mor_spec))
blanchet@49311
   407
      ||> `Local_Theory.restore;
blanchet@49311
   408
blanchet@48975
   409
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   410
    val mor = fst (Term.dest_Const (Morphism.term phi mor_free));
blanchet@48975
   411
    val mor_def = Morphism.thm phi mor_def_free;
blanchet@48975
   412
blanchet@48975
   413
    fun mk_mor Bs1 ss1 Bs2 ss2 fs =
blanchet@48975
   414
      let
blanchet@48975
   415
        val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;
blanchet@48975
   416
        val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);
blanchet@48975
   417
        val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   418
      in
blanchet@48975
   419
        Term.list_comb (Const (mor, morT), args)
blanchet@48975
   420
      end;
blanchet@48975
   421
blanchet@48975
   422
    val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
blanchet@48975
   423
blanchet@48975
   424
    val (mor_image_thms, morE_thms) =
blanchet@48975
   425
      let
blanchet@48975
   426
        val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
blanchet@48975
   427
        fun mk_image_goal f B1 B2 = fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs)
traytel@51893
   428
          (Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_leq (mk_image f $ B1) B2)));
blanchet@48975
   429
        val image_goals = map3 mk_image_goal fs Bs B's;
blanchet@48975
   430
        fun mk_elim_goal B mapAsBs f s s' x =
blanchet@48975
   431
          fold_rev Logic.all (x :: Bs @ ss @ B's @ s's @ fs)
blanchet@48975
   432
            (Logic.list_implies ([prem, HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B))],
blanchet@49123
   433
              mk_Trueprop_eq (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x))));
blanchet@48975
   434
        val elim_goals = map6 mk_elim_goal Bs mapsAsBs fs ss s's zs;
blanchet@48975
   435
        fun prove goal =
wenzelm@51551
   436
          Goal.prove_sorry lthy [] [] goal (K (mk_mor_elim_tac mor_def))
traytel@49109
   437
          |> Thm.close_derivation;
blanchet@48975
   438
      in
blanchet@48975
   439
        (map prove image_goals, map prove elim_goals)
blanchet@48975
   440
      end;
blanchet@48975
   441
blanchet@48975
   442
    val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms;
blanchet@48975
   443
blanchet@48975
   444
    val mor_incl_thm =
blanchet@48975
   445
      let
traytel@51893
   446
        val prems = map2 (HOLogic.mk_Trueprop oo mk_leq) Bs Bs_copy;
blanchet@48975
   447
        val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
blanchet@48975
   448
      in
wenzelm@51551
   449
        Goal.prove_sorry lthy [] []
blanchet@48975
   450
          (fold_rev Logic.all (Bs @ ss @ Bs_copy) (Logic.list_implies (prems, concl)))
blanchet@53285
   451
          (K (mk_mor_incl_tac mor_def map_ids))
traytel@49109
   452
        |> Thm.close_derivation
blanchet@48975
   453
      end;
blanchet@48975
   454
blanchet@48975
   455
    val mor_id_thm = mor_incl_thm OF (replicate n subset_refl);
blanchet@48975
   456
blanchet@48975
   457
    val mor_comp_thm =
blanchet@48975
   458
      let
blanchet@48975
   459
        val prems =
blanchet@48975
   460
          [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),
blanchet@48975
   461
           HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];
blanchet@48975
   462
        val concl =
blanchet@48975
   463
          HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));
blanchet@48975
   464
      in
wenzelm@51551
   465
        Goal.prove_sorry lthy [] []
blanchet@48975
   466
          (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)
blanchet@48975
   467
            (Logic.list_implies (prems, concl)))
blanchet@48975
   468
          (K (mk_mor_comp_tac mor_def mor_image'_thms morE_thms map_comp_id_thms))
traytel@49109
   469
        |> Thm.close_derivation
blanchet@48975
   470
      end;
blanchet@48975
   471
blanchet@48975
   472
    val mor_cong_thm =
blanchet@48975
   473
      let
blanchet@48975
   474
        val prems = map HOLogic.mk_Trueprop
blanchet@48975
   475
         (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])
blanchet@48975
   476
        val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);
blanchet@48975
   477
      in
wenzelm@51551
   478
        Goal.prove_sorry lthy [] []
blanchet@48975
   479
          (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ fs_copy)
blanchet@48975
   480
            (Logic.list_implies (prems, concl)))
wenzelm@51798
   481
          (K ((hyp_subst_tac lthy THEN' atac) 1))
traytel@49109
   482
        |> Thm.close_derivation
blanchet@48975
   483
      end;
blanchet@48975
   484
blanchet@48975
   485
    val mor_UNIV_thm =
blanchet@48975
   486
      let
blanchet@48975
   487
        fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
blanchet@48975
   488
            (HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s),
blanchet@48975
   489
            HOLogic.mk_comp (s', f));
blanchet@48975
   490
        val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;
blanchet@48975
   491
        val rhs = Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct mapsAsBs fs ss s's);
blanchet@48975
   492
      in
wenzelm@51551
   493
        Goal.prove_sorry lthy [] [] (fold_rev Logic.all (ss @ s's @ fs) (mk_Trueprop_eq (lhs, rhs)))
blanchet@48975
   494
          (K (mk_mor_UNIV_tac morE_thms mor_def))
traytel@49109
   495
        |> Thm.close_derivation
blanchet@48975
   496
      end;
blanchet@48975
   497
blanchet@48975
   498
    val mor_str_thm =
blanchet@48975
   499
      let
blanchet@48975
   500
        val maps = map2 (fn Ds => fn bnf => Term.list_comb
blanchet@48975
   501
          (mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs;
blanchet@48975
   502
      in
wenzelm@51551
   503
        Goal.prove_sorry lthy [] []
blanchet@48975
   504
          (fold_rev Logic.all ss (HOLogic.mk_Trueprop
blanchet@48975
   505
            (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss)))
blanchet@48975
   506
          (K (mk_mor_str_tac ks mor_UNIV_thm))
traytel@49109
   507
        |> Thm.close_derivation
blanchet@48975
   508
      end;
blanchet@48975
   509
blanchet@48975
   510
    val mor_sum_case_thm =
blanchet@48975
   511
      let
blanchet@49458
   512
        val maps = map3 (fn s => fn sum_s => fn mapx =>
blanchet@49458
   513
          mk_sum_case (HOLogic.mk_comp (Term.list_comb (mapx, passive_ids @ Inls), s), sum_s))
blanchet@48975
   514
          s's sum_ss map_Inls;
blanchet@48975
   515
      in
wenzelm@51551
   516
        Goal.prove_sorry lthy [] []
blanchet@48975
   517
          (fold_rev Logic.all (s's @ sum_ss) (HOLogic.mk_Trueprop
blanchet@48975
   518
            (mk_mor (map HOLogic.mk_UNIV activeBs) s's sum_UNIVs maps Inls)))
blanchet@48975
   519
          (K (mk_mor_sum_case_tac ks mor_UNIV_thm))
traytel@49109
   520
        |> Thm.close_derivation
blanchet@48975
   521
      end;
blanchet@48975
   522
blanchet@48975
   523
    val timer = time (timer "Morphism definition & thms");
blanchet@48975
   524
traytel@53566
   525
    fun hset_rec_bind j = mk_internal_b (hset_recN ^ (if m = 1 then "" else string_of_int j));
blanchet@48975
   526
    val hset_rec_name = Binding.name_of o hset_rec_bind;
blanchet@48975
   527
    val hset_rec_def_bind = rpair [] o Thm.def_binding o hset_rec_bind;
blanchet@48975
   528
blanchet@48975
   529
    fun hset_rec_spec j Zero hsetT hrec hrec' =
blanchet@48975
   530
      let
blanchet@48975
   531
        fun mk_Suc s setsAs z z' =
blanchet@48975
   532
          let
blanchet@48975
   533
            val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m setsAs);
blanchet@48975
   534
            fun mk_UN set k = mk_UNION (set $ (s $ z)) (mk_nthN n hrec k);
blanchet@48975
   535
          in
blanchet@48975
   536
            Term.absfree z'
blanchet@48975
   537
              (mk_union (set $ (s $ z), Library.foldl1 mk_union (map2 mk_UN sets ks)))
blanchet@48975
   538
          end;
blanchet@48975
   539
blanchet@48975
   540
        val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec'
blanchet@48975
   541
          (HOLogic.mk_tuple (map4 mk_Suc ss setssAs zs zs')));
blanchet@48975
   542
blanchet@48975
   543
        val lhs = Term.list_comb (Free (hset_rec_name j, hsetT), ss);
blanchet@48975
   544
        val rhs = mk_nat_rec Zero Suc;
blanchet@48975
   545
      in
blanchet@49123
   546
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   547
      end;
blanchet@48975
   548
blanchet@48975
   549
    val ((hset_rec_frees, (_, hset_rec_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   550
      lthy
blanchet@48975
   551
      |> fold_map5 (fn j => fn Zero => fn hsetT => fn hrec => fn hrec' => Specification.definition
blanchet@48975
   552
        (SOME (hset_rec_bind j, NONE, NoSyn),
blanchet@48975
   553
          (hset_rec_def_bind j, hset_rec_spec j Zero hsetT hrec hrec')))
blanchet@48975
   554
        ls Zeros hsetTs hrecs hrecs'
blanchet@48975
   555
      |>> apsnd split_list o split_list
blanchet@48975
   556
      ||> `Local_Theory.restore;
blanchet@48975
   557
blanchet@48975
   558
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   559
blanchet@48975
   560
    val hset_rec_defs = map (Morphism.thm phi) hset_rec_def_frees;
blanchet@48975
   561
    val hset_recs = map (fst o Term.dest_Const o Morphism.term phi) hset_rec_frees;
blanchet@48975
   562
blanchet@48975
   563
    fun mk_hset_rec ss nat i j T =
blanchet@48975
   564
      let
blanchet@48975
   565
        val args = ss @ [nat];
blanchet@48975
   566
        val Ts = map fastype_of ss;
blanchet@48975
   567
        val bTs = map domain_type Ts;
blanchet@48975
   568
        val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) bTs)
blanchet@48975
   569
        val hset_recT = Library.foldr (op -->) (Ts, HOLogic.natT --> hrecT);
blanchet@48975
   570
      in
blanchet@48975
   571
        mk_nthN n (Term.list_comb (Const (nth hset_recs (j - 1), hset_recT), args)) i
blanchet@48975
   572
      end;
blanchet@48975
   573
blanchet@48975
   574
    val hset_rec_0ss = mk_rec_simps n @{thm nat_rec_0} hset_rec_defs;
blanchet@48975
   575
    val hset_rec_Sucss = mk_rec_simps n @{thm nat_rec_Suc} hset_rec_defs;
blanchet@48975
   576
    val hset_rec_0ss' = transpose hset_rec_0ss;
blanchet@48975
   577
    val hset_rec_Sucss' = transpose hset_rec_Sucss;
blanchet@48975
   578
traytel@53566
   579
    fun hset_binds j = mk_internal_bs (hsetN ^ (if m = 1 then "" else string_of_int j))
traytel@53566
   580
    fun hset_bind i j = nth (hset_binds j) (i - 1);
blanchet@48975
   581
    val hset_name = Binding.name_of oo hset_bind;
blanchet@48975
   582
    val hset_def_bind = rpair [] o Thm.def_binding oo hset_bind;
blanchet@48975
   583
blanchet@48975
   584
    fun hset_spec i j =
blanchet@48975
   585
      let
blanchet@48975
   586
        val U = nth activeAs (i - 1);
blanchet@48975
   587
        val z = nth zs (i - 1);
blanchet@48975
   588
        val T = nth passiveAs (j - 1);
blanchet@48975
   589
        val setT = HOLogic.mk_setT T;
blanchet@48975
   590
        val hsetT = Library.foldr (op -->) (sTs, U --> setT);
blanchet@48975
   591
blanchet@48975
   592
        val lhs = Term.list_comb (Free (hset_name i j, hsetT), ss @ [z]);
blanchet@48975
   593
        val rhs = mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
blanchet@48975
   594
          (Term.absfree nat' (mk_hset_rec ss nat i j T $ z));
blanchet@48975
   595
      in
blanchet@49123
   596
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   597
      end;
blanchet@48975
   598
blanchet@48975
   599
    val ((hset_frees, (_, hset_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   600
      lthy
blanchet@48975
   601
      |> fold_map (fn i => fold_map (fn j => Specification.definition
blanchet@48975
   602
        (SOME (hset_bind i j, NONE, NoSyn), (hset_def_bind i j, hset_spec i j))) ls) ks
blanchet@48975
   603
      |>> map (apsnd split_list o split_list)
blanchet@48975
   604
      |>> apsnd split_list o split_list
blanchet@48975
   605
      ||> `Local_Theory.restore;
blanchet@48975
   606
blanchet@48975
   607
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   608
blanchet@48975
   609
    val hset_defss = map (map (Morphism.thm phi)) hset_def_frees;
blanchet@48975
   610
    val hset_defss' = transpose hset_defss;
blanchet@48975
   611
    val hset_namess = map (map (fst o Term.dest_Const o Morphism.term phi)) hset_frees;
blanchet@48975
   612
blanchet@48975
   613
    fun mk_hset ss i j T =
blanchet@48975
   614
      let
blanchet@48975
   615
        val Ts = map fastype_of ss;
blanchet@48975
   616
        val bTs = map domain_type Ts;
blanchet@48975
   617
        val hsetT = Library.foldr (op -->) (Ts, nth bTs (i - 1) --> HOLogic.mk_setT T);
blanchet@48975
   618
      in
blanchet@48975
   619
        Term.list_comb (Const (nth (nth hset_namess (i - 1)) (j - 1), hsetT), ss)
blanchet@48975
   620
      end;
blanchet@48975
   621
blanchet@48975
   622
    val hsetssAs = map (fn i => map2 (mk_hset ss i) ls passiveAs) ks;
blanchet@48975
   623
blanchet@48975
   624
    val (set_incl_hset_thmss, set_hset_incl_hset_thmsss) =
blanchet@48975
   625
      let
blanchet@48975
   626
        fun mk_set_incl_hset s x set hset = fold_rev Logic.all (x :: ss)
traytel@51893
   627
          (HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) (hset $ x)));
blanchet@48975
   628
blanchet@48975
   629
        fun mk_set_hset_incl_hset s x y set hset1 hset2 =
blanchet@48975
   630
          fold_rev Logic.all (x :: y :: ss)
blanchet@48975
   631
            (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x, set $ (s $ y))),
traytel@51893
   632
            HOLogic.mk_Trueprop (mk_leq (hset1 $ x) (hset2 $ y))));
blanchet@48975
   633
blanchet@48975
   634
        val set_incl_hset_goalss =
blanchet@48975
   635
          map4 (fn s => fn x => fn sets => fn hsets =>
blanchet@48975
   636
            map2 (mk_set_incl_hset s x) (take m sets) hsets)
blanchet@48975
   637
          ss zs setssAs hsetssAs;
blanchet@48975
   638
blanchet@48975
   639
        (*xk : F(i)set(m+k) (si yi) ==> F(k)_hset(j) s1 ... sn xk <= F(i)_hset(j) s1 ... sn yi*)
blanchet@48975
   640
        val set_hset_incl_hset_goalsss =
blanchet@48975
   641
          map4 (fn si => fn yi => fn sets => fn hsetsi =>
blanchet@48975
   642
            map3 (fn xk => fn set => fn hsetsk =>
blanchet@48975
   643
              map2 (mk_set_hset_incl_hset si xk yi set) hsetsk hsetsi)
blanchet@48975
   644
            zs_copy (drop m sets) hsetssAs)
blanchet@48975
   645
          ss zs setssAs hsetssAs;
blanchet@48975
   646
      in
blanchet@48975
   647
        (map3 (fn goals => fn defs => fn rec_Sucs =>
blanchet@48975
   648
          map3 (fn goal => fn def => fn rec_Suc =>
wenzelm@51551
   649
            Goal.prove_sorry lthy [] [] goal (K (mk_set_incl_hset_tac def rec_Suc))
traytel@49109
   650
            |> Thm.close_derivation)
blanchet@48975
   651
          goals defs rec_Sucs)
blanchet@48975
   652
        set_incl_hset_goalss hset_defss hset_rec_Sucss,
blanchet@48975
   653
        map3 (fn goalss => fn defsi => fn rec_Sucs =>
blanchet@48975
   654
          map3 (fn k => fn goals => fn defsk =>
blanchet@48975
   655
            map4 (fn goal => fn defk => fn defi => fn rec_Suc =>
wenzelm@51551
   656
              Goal.prove_sorry lthy [] [] goal
traytel@49109
   657
                (K (mk_set_hset_incl_hset_tac n [defk, defi] rec_Suc k))
traytel@49109
   658
              |> Thm.close_derivation)
blanchet@48975
   659
            goals defsk defsi rec_Sucs)
blanchet@48975
   660
          ks goalss hset_defss)
blanchet@48975
   661
        set_hset_incl_hset_goalsss hset_defss hset_rec_Sucss)
blanchet@48975
   662
      end;
blanchet@48975
   663
blanchet@48975
   664
    val set_incl_hset_thmss' = transpose set_incl_hset_thmss;
blanchet@48975
   665
    val set_hset_incl_hset_thmsss' = transpose (map transpose set_hset_incl_hset_thmsss);
blanchet@48975
   666
    val set_hset_thmss = map (map (fn thm => thm RS @{thm set_mp})) set_incl_hset_thmss;
blanchet@48975
   667
    val set_hset_hset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp})))
blanchet@48975
   668
      set_hset_incl_hset_thmsss;
blanchet@48975
   669
    val set_hset_thmss' = transpose set_hset_thmss;
blanchet@48975
   670
    val set_hset_hset_thmsss' = transpose (map transpose set_hset_hset_thmsss);
blanchet@48975
   671
blanchet@48975
   672
    val set_incl_hin_thmss =
blanchet@48975
   673
      let
blanchet@48975
   674
        fun mk_set_incl_hin s x hsets1 set hsets2 T =
blanchet@48975
   675
          fold_rev Logic.all (x :: ss @ As)
blanchet@48975
   676
            (Logic.list_implies
traytel@51893
   677
              (map2 (fn hset => fn A => HOLogic.mk_Trueprop (mk_leq (hset $ x) A)) hsets1 As,
traytel@51893
   678
              HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) (mk_in As hsets2 T))));
blanchet@48975
   679
blanchet@48975
   680
        val set_incl_hin_goalss =
blanchet@48975
   681
          map4 (fn s => fn x => fn sets => fn hsets =>
blanchet@48975
   682
            map3 (mk_set_incl_hin s x hsets) (drop m sets) hsetssAs activeAs)
blanchet@48975
   683
          ss zs setssAs hsetssAs;
blanchet@48975
   684
      in
blanchet@48975
   685
        map2 (map2 (fn goal => fn thms =>
wenzelm@51551
   686
          Goal.prove_sorry lthy [] [] goal (K (mk_set_incl_hin_tac thms))
traytel@49109
   687
          |> Thm.close_derivation))
blanchet@48975
   688
        set_incl_hin_goalss set_hset_incl_hset_thmsss
blanchet@48975
   689
      end;
blanchet@48975
   690
blanchet@48975
   691
    val hset_minimal_thms =
blanchet@48975
   692
      let
blanchet@48975
   693
        fun mk_passive_prem set s x K =
traytel@51893
   694
          Logic.all x (HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) (K $ x)));
blanchet@48975
   695
blanchet@48975
   696
        fun mk_active_prem s x1 K1 set x2 K2 =
blanchet@48975
   697
          fold_rev Logic.all [x1, x2]
blanchet@48975
   698
            (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x2, set $ (s $ x1))),
traytel@51893
   699
              HOLogic.mk_Trueprop (mk_leq (K2 $ x2) (K1 $ x1))));
blanchet@48975
   700
blanchet@48975
   701
        val premss = map2 (fn j => fn Ks =>
blanchet@48975
   702
          map4 mk_passive_prem (map (fn xs => nth xs (j - 1)) setssAs) ss zs Ks @
blanchet@48975
   703
            flat (map4 (fn sets => fn s => fn x1 => fn K1 =>
blanchet@48975
   704
              map3 (mk_active_prem s x1 K1) (drop m sets) zs_copy Ks) setssAs ss zs Ks))
blanchet@48975
   705
          ls Kss;
blanchet@48975
   706
blanchet@48975
   707
        val hset_rec_minimal_thms =
blanchet@48975
   708
          let
traytel@51893
   709
            fun mk_conjunct j T i K x = mk_leq (mk_hset_rec ss nat i j T $ x) (K $ x);
blanchet@48975
   710
            fun mk_concl j T Ks = list_all_free zs
blanchet@48975
   711
              (Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs));
blanchet@48975
   712
            val concls = map3 mk_concl ls passiveAs Kss;
blanchet@48975
   713
blanchet@48975
   714
            val goals = map2 (fn prems => fn concl =>
blanchet@48975
   715
              Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls
blanchet@48975
   716
blanchet@48975
   717
            val ctss =
blanchet@48975
   718
              map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
blanchet@48975
   719
          in
blanchet@48975
   720
            map4 (fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>
blanchet@48975
   721
              singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
   722
                (Goal.prove_sorry lthy [] [] goal
traytel@49109
   723
                  (mk_hset_rec_minimal_tac m cts hset_rec_0s hset_rec_Sucs))
traytel@49109
   724
              |> Thm.close_derivation)
blanchet@48975
   725
            goals ctss hset_rec_0ss' hset_rec_Sucss'
blanchet@48975
   726
          end;
blanchet@48975
   727
traytel@51893
   728
        fun mk_conjunct j T i K x = mk_leq (mk_hset ss i j T $ x) (K $ x);
blanchet@48975
   729
        fun mk_concl j T Ks = Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs);
blanchet@48975
   730
        val concls = map3 mk_concl ls passiveAs Kss;
blanchet@48975
   731
blanchet@48975
   732
        val goals = map3 (fn Ks => fn prems => fn concl =>
blanchet@48975
   733
          fold_rev Logic.all (Ks @ ss @ zs)
blanchet@48975
   734
            (Logic.list_implies (prems, HOLogic.mk_Trueprop concl))) Kss premss concls;
blanchet@48975
   735
      in
blanchet@48975
   736
        map3 (fn goal => fn hset_defs => fn hset_rec_minimal =>
wenzelm@51551
   737
          Goal.prove_sorry lthy [] [] goal
traytel@49109
   738
            (mk_hset_minimal_tac n hset_defs hset_rec_minimal)
traytel@49109
   739
          |> Thm.close_derivation)
blanchet@48975
   740
        goals hset_defss' hset_rec_minimal_thms
blanchet@48975
   741
      end;
blanchet@48975
   742
blanchet@48975
   743
    val timer = time (timer "Hereditary sets");
blanchet@48975
   744
blanchet@48975
   745
    (* bisimulation *)
blanchet@48975
   746
traytel@53566
   747
    val bis_bind = mk_internal_b bisN;
blanchet@48975
   748
    val bis_name = Binding.name_of bis_bind;
blanchet@48975
   749
    val bis_def_bind = (Thm.def_binding bis_bind, []);
blanchet@48975
   750
traytel@51893
   751
    fun mk_bis_le_conjunct R B1 B2 = mk_leq R (mk_Times (B1, B2));
blanchet@48975
   752
    val bis_le = Library.foldr1 HOLogic.mk_conj (map3 mk_bis_le_conjunct Rs Bs B's)
blanchet@48975
   753
blanchet@48975
   754
    val bis_spec =
blanchet@48975
   755
      let
blanchet@48975
   756
        val bisT = Library.foldr (op -->) (ATs @ BTs @ sTs @ B'Ts @ s'Ts @ setRTs, HOLogic.boolT);
blanchet@48975
   757
blanchet@48975
   758
        val fst_args = passive_ids @ fsts;
blanchet@48975
   759
        val snd_args = passive_ids @ snds;
blanchet@48975
   760
        fun mk_bis R s s' b1 b2 RF map1 map2 sets =
blanchet@48975
   761
          list_all_free [b1, b2] (HOLogic.mk_imp
blanchet@48975
   762
            (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
blanchet@48975
   763
            mk_Bex (mk_in (As @ Rs) sets (snd (dest_Free RF))) (Term.absfree (dest_Free RF)
blanchet@48975
   764
              (HOLogic.mk_conj
blanchet@48975
   765
                (HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1),
blanchet@48975
   766
                HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2))))));
blanchet@48975
   767
blanchet@48975
   768
        val lhs = Term.list_comb (Free (bis_name, bisT), As @ Bs @ ss @ B's @ s's @ Rs);
blanchet@48975
   769
        val rhs = HOLogic.mk_conj
blanchet@48975
   770
          (bis_le, Library.foldr1 HOLogic.mk_conj
blanchet@48975
   771
            (map9 mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss))
blanchet@48975
   772
      in
blanchet@49123
   773
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   774
      end;
blanchet@48975
   775
blanchet@48975
   776
    val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) =
blanchet@49311
   777
      lthy
blanchet@49311
   778
      |> Specification.definition (SOME (bis_bind, NONE, NoSyn), (bis_def_bind, bis_spec))
blanchet@49311
   779
      ||> `Local_Theory.restore;
blanchet@49311
   780
blanchet@48975
   781
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   782
    val bis = fst (Term.dest_Const (Morphism.term phi bis_free));
blanchet@48975
   783
    val bis_def = Morphism.thm phi bis_def_free;
blanchet@48975
   784
blanchet@48975
   785
    fun mk_bis As Bs1 ss1 Bs2 ss2 Rs =
blanchet@48975
   786
      let
blanchet@48975
   787
        val args = As @ Bs1 @ ss1 @ Bs2 @ ss2 @ Rs;
blanchet@48975
   788
        val Ts = map fastype_of args;
blanchet@48975
   789
        val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT);
blanchet@48975
   790
      in
blanchet@48975
   791
        Term.list_comb (Const (bis, bisT), args)
blanchet@48975
   792
      end;
blanchet@48975
   793
blanchet@48975
   794
    val bis_cong_thm =
blanchet@48975
   795
      let
blanchet@48975
   796
        val prems = map HOLogic.mk_Trueprop
blanchet@48975
   797
         (mk_bis As Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs)
blanchet@48975
   798
        val concl = HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs_copy);
blanchet@48975
   799
      in
wenzelm@51551
   800
        Goal.prove_sorry lthy [] []
blanchet@48975
   801
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs @ Rs_copy)
blanchet@48975
   802
            (Logic.list_implies (prems, concl)))
wenzelm@51798
   803
          (K ((hyp_subst_tac lthy THEN' atac) 1))
traytel@49109
   804
        |> Thm.close_derivation
blanchet@48975
   805
      end;
blanchet@48975
   806
traytel@51893
   807
    val bis_rel_thm =
blanchet@48975
   808
      let
traytel@51893
   809
        fun mk_conjunct R s s' b1 b2 rel =
blanchet@48975
   810
          list_all_free [b1, b2] (HOLogic.mk_imp
blanchet@48975
   811
            (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
traytel@51893
   812
            Term.list_comb (rel, map mk_in_rel (passive_Id_ons @ Rs)) $ (s $ b1) $ (s' $ b2)));
blanchet@48975
   813
blanchet@48975
   814
        val rhs = HOLogic.mk_conj
blanchet@48975
   815
          (bis_le, Library.foldr1 HOLogic.mk_conj
blanchet@48975
   816
            (map6 mk_conjunct Rs ss s's zs z's relsAsBs))
blanchet@48975
   817
      in
wenzelm@51551
   818
        Goal.prove_sorry lthy [] []
blanchet@48975
   819
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
blanchet@49123
   820
            (mk_Trueprop_eq (mk_bis As Bs ss B's s's Rs, rhs)))
blanchet@53290
   821
          (K (mk_bis_rel_tac lthy m bis_def rel_OO_Grps map_comps map_cong0s set_mapss))
traytel@49109
   822
        |> Thm.close_derivation
blanchet@48975
   823
      end;
blanchet@48975
   824
blanchet@48975
   825
    val bis_converse_thm =
wenzelm@51551
   826
      Goal.prove_sorry lthy [] []
blanchet@48975
   827
        (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
blanchet@48975
   828
          (Logic.mk_implies
blanchet@48975
   829
            (HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
blanchet@48975
   830
            HOLogic.mk_Trueprop (mk_bis As B's s's Bs ss (map mk_converse Rs)))))
traytel@51893
   831
        (K (mk_bis_converse_tac m bis_rel_thm rel_congs rel_converseps))
traytel@49109
   832
      |> Thm.close_derivation;
blanchet@48975
   833
blanchet@48975
   834
    val bis_O_thm =
blanchet@48975
   835
      let
blanchet@48975
   836
        val prems =
blanchet@48975
   837
          [HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
blanchet@48975
   838
           HOLogic.mk_Trueprop (mk_bis As B's s's B''s s''s R's)];
blanchet@48975
   839
        val concl =
blanchet@48975
   840
          HOLogic.mk_Trueprop (mk_bis As Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's));
blanchet@48975
   841
      in
wenzelm@51551
   842
        Goal.prove_sorry lthy [] []
blanchet@48975
   843
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ B''s @ s''s @ Rs @ R's)
blanchet@48975
   844
            (Logic.list_implies (prems, concl)))
traytel@51893
   845
          (K (mk_bis_O_tac lthy m bis_rel_thm rel_congs rel_OOs))
traytel@49109
   846
        |> Thm.close_derivation
blanchet@48975
   847
      end;
blanchet@48975
   848
blanchet@48975
   849
    val bis_Gr_thm =
blanchet@48975
   850
      let
blanchet@48975
   851
        val concl =
blanchet@48975
   852
          HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map2 mk_Gr Bs fs));
blanchet@48975
   853
      in
wenzelm@51551
   854
        Goal.prove_sorry lthy [] []
blanchet@48975
   855
          (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ fs)
blanchet@48975
   856
            (Logic.list_implies ([coalg_prem, mor_prem], concl)))
traytel@51893
   857
          (mk_bis_Gr_tac bis_rel_thm rel_Grps mor_image_thms morE_thms coalg_in_thms)
traytel@49109
   858
        |> Thm.close_derivation
blanchet@48975
   859
      end;
blanchet@48975
   860
blanchet@48975
   861
    val bis_image2_thm = bis_cong_thm OF
blanchet@48975
   862
      ((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) ::
blanchet@48975
   863
      replicate n @{thm image2_Gr});
blanchet@48975
   864
traytel@51447
   865
    val bis_Id_on_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) ::
traytel@51447
   866
      replicate n @{thm Id_on_Gr});
blanchet@48975
   867
blanchet@48975
   868
    val bis_Union_thm =
blanchet@48975
   869
      let
blanchet@48975
   870
        val prem =
blanchet@48975
   871
          HOLogic.mk_Trueprop (mk_Ball Idx
blanchet@48975
   872
            (Term.absfree idx' (mk_bis As Bs ss B's s's (map (fn R => R $ idx) Ris))));
blanchet@48975
   873
        val concl =
blanchet@48975
   874
          HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map (mk_UNION Idx) Ris));
blanchet@48975
   875
      in
wenzelm@51551
   876
        Goal.prove_sorry lthy [] []
blanchet@48975
   877
          (fold_rev Logic.all (Idx :: As @ Bs @ ss @ B's @ s's @ Ris)
blanchet@48975
   878
            (Logic.mk_implies (prem, concl)))
blanchet@48975
   879
          (mk_bis_Union_tac bis_def in_mono'_thms)
traytel@49109
   880
        |> Thm.close_derivation
blanchet@48975
   881
      end;
blanchet@48975
   882
blanchet@48975
   883
    (* self-bisimulation *)
blanchet@48975
   884
blanchet@48975
   885
    fun mk_sbis As Bs ss Rs = mk_bis As Bs ss Bs ss Rs;
blanchet@48975
   886
blanchet@48975
   887
    val sbis_prem = HOLogic.mk_Trueprop (mk_sbis As Bs ss sRs);
blanchet@48975
   888
blanchet@48975
   889
    (* largest self-bisimulation *)
blanchet@48975
   890
traytel@53566
   891
    val lsbis_binds = mk_internal_bs lsbisN;
traytel@53566
   892
    fun lsbis_bind i = nth lsbis_binds (i - 1);
blanchet@48975
   893
    val lsbis_name = Binding.name_of o lsbis_bind;
blanchet@48975
   894
    val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind;
blanchet@48975
   895
blanchet@48975
   896
    val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs
blanchet@48975
   897
      (HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis As Bs ss sRs)));
blanchet@48975
   898
blanchet@48975
   899
    fun lsbis_spec i RT =
blanchet@48975
   900
      let
blanchet@48975
   901
        fun mk_lsbisT RT =
blanchet@48975
   902
          Library.foldr (op -->) (map fastype_of (As @ Bs @ ss), RT);
blanchet@48975
   903
        val lhs = Term.list_comb (Free (lsbis_name i, mk_lsbisT RT), As @ Bs @ ss);
blanchet@48975
   904
        val rhs = mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i));
blanchet@48975
   905
      in
blanchet@49123
   906
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
   907
      end;
blanchet@48975
   908
blanchet@48975
   909
    val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) =
blanchet@48975
   910
      lthy
blanchet@48975
   911
      |> fold_map2 (fn i => fn RT => Specification.definition
blanchet@48975
   912
        (SOME (lsbis_bind i, NONE, NoSyn), (lsbis_def_bind i, lsbis_spec i RT))) ks setsRTs
blanchet@48975
   913
      |>> apsnd split_list o split_list
blanchet@48975
   914
      ||> `Local_Theory.restore;
blanchet@48975
   915
blanchet@48975
   916
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   917
blanchet@48975
   918
    val lsbis_defs = map (Morphism.thm phi) lsbis_def_frees;
blanchet@48975
   919
    val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees;
blanchet@48975
   920
blanchet@48975
   921
    fun mk_lsbis As Bs ss i =
blanchet@48975
   922
      let
blanchet@48975
   923
        val args = As @ Bs @ ss;
blanchet@48975
   924
        val Ts = map fastype_of args;
blanchet@48975
   925
        val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1)))));
blanchet@48975
   926
        val lsbisT = Library.foldr (op -->) (Ts, RT);
blanchet@48975
   927
      in
blanchet@48975
   928
        Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args)
blanchet@48975
   929
      end;
blanchet@48975
   930
blanchet@48975
   931
    val sbis_lsbis_thm =
wenzelm@51551
   932
      Goal.prove_sorry lthy [] []
blanchet@48975
   933
        (fold_rev Logic.all (As @ Bs @ ss)
blanchet@48975
   934
          (HOLogic.mk_Trueprop (mk_sbis As Bs ss (map (mk_lsbis As Bs ss) ks))))
wenzelm@51798
   935
        (K (mk_sbis_lsbis_tac lthy lsbis_defs bis_Union_thm bis_cong_thm))
traytel@49109
   936
      |> Thm.close_derivation;
blanchet@48975
   937
blanchet@48975
   938
    val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS
traytel@52904
   939
      (bis_def RS iffD1 RS conjunct1 RS mk_conjunctN n i)) ks;
blanchet@48975
   940
    val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS
traytel@52904
   941
      (bis_def RS iffD1 RS conjunct2 RS mk_conjunctN n i))) RS mp) ks;
blanchet@48975
   942
blanchet@48975
   943
    val incl_lsbis_thms =
blanchet@48975
   944
      let
traytel@51893
   945
        fun mk_concl i R = HOLogic.mk_Trueprop (mk_leq R (mk_lsbis As Bs ss i));
blanchet@48975
   946
        val goals = map2 (fn i => fn R => fold_rev Logic.all (As @ Bs @ ss @ sRs)
blanchet@48975
   947
          (Logic.mk_implies (sbis_prem, mk_concl i R))) ks sRs;
blanchet@48975
   948
      in
wenzelm@51551
   949
        map3 (fn goal => fn i => fn def => Goal.prove_sorry lthy [] [] goal
traytel@49109
   950
          (K (mk_incl_lsbis_tac n i def)) |> Thm.close_derivation) goals ks lsbis_defs
blanchet@48975
   951
      end;
blanchet@48975
   952
blanchet@48975
   953
    val equiv_lsbis_thms =
blanchet@48975
   954
      let
blanchet@48975
   955
        fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis As Bs ss i));
blanchet@48975
   956
        val goals = map2 (fn i => fn B => fold_rev Logic.all (As @ Bs @ ss)
blanchet@48975
   957
          (Logic.mk_implies (coalg_prem, mk_concl i B))) ks Bs;
blanchet@48975
   958
      in
blanchet@48975
   959
        map3 (fn goal => fn l_incl => fn incl_l =>
wenzelm@51551
   960
          Goal.prove_sorry lthy [] [] goal
blanchet@48975
   961
            (K (mk_equiv_lsbis_tac sbis_lsbis_thm l_incl incl_l
traytel@51447
   962
              bis_Id_on_thm bis_converse_thm bis_O_thm))
traytel@49109
   963
          |> Thm.close_derivation)
blanchet@48975
   964
        goals lsbis_incl_thms incl_lsbis_thms
blanchet@48975
   965
      end;
blanchet@48975
   966
blanchet@48975
   967
    val timer = time (timer "Bisimulations");
blanchet@48975
   968
blanchet@48975
   969
    (* bounds *)
blanchet@48975
   970
blanchet@48975
   971
    val (lthy, sbd, sbdT,
traytel@52635
   972
      sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss) =
blanchet@48975
   973
      if n = 1
traytel@52635
   974
      then (lthy, sum_bd, sum_bdT, bd_card_order, bd_Cinfinite, bd_Card_order, set_bdss)
blanchet@48975
   975
      else
blanchet@48975
   976
        let
traytel@53566
   977
          val sbdT_bind = mk_internal_b sum_bdTN;
blanchet@48975
   978
blanchet@48975
   979
          val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) =
wenzelm@49835
   980
            typedef (sbdT_bind, dead_params, NoSyn)
blanchet@48975
   981
              (HOLogic.mk_UNIV sum_bdT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
blanchet@48975
   982
traytel@49620
   983
          val sbdT = Type (sbdT_name, dead_params');
blanchet@48975
   984
          val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT);
blanchet@48975
   985
traytel@53566
   986
          val sbd_bind = mk_internal_b sum_bdN;
blanchet@48975
   987
          val sbd_name = Binding.name_of sbd_bind;
blanchet@48975
   988
          val sbd_def_bind = (Thm.def_binding sbd_bind, []);
blanchet@48975
   989
blanchet@48975
   990
          val sbd_spec = HOLogic.mk_Trueprop
blanchet@48975
   991
            (HOLogic.mk_eq (Free (sbd_name, mk_relT (`I sbdT)), mk_dir_image sum_bd Abs_sbdT));
blanchet@48975
   992
blanchet@48975
   993
          val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) =
blanchet@48975
   994
            lthy
blanchet@48975
   995
            |> Specification.definition (SOME (sbd_bind, NONE, NoSyn), (sbd_def_bind, sbd_spec))
blanchet@48975
   996
            ||> `Local_Theory.restore;
blanchet@48975
   997
blanchet@48975
   998
          val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
   999
blanchet@48975
  1000
          val sbd_def = Morphism.thm phi sbd_def_free;
blanchet@48975
  1001
          val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT));
blanchet@48975
  1002
traytel@49228
  1003
          val Abs_sbdT_inj = mk_Abs_inj_thm (#Abs_inject sbdT_loc_info);
traytel@49228
  1004
          val Abs_sbdT_bij = mk_Abs_bij_thm lthy Abs_sbdT_inj (#Abs_cases sbdT_loc_info);
blanchet@48975
  1005
blanchet@48975
  1006
          fun mk_sum_Cinfinite [thm] = thm
blanchet@48975
  1007
            | mk_sum_Cinfinite (thm :: thms) =
blanchet@48975
  1008
              @{thm Cinfinite_csum_strong} OF [thm, mk_sum_Cinfinite thms];
blanchet@48975
  1009
blanchet@48975
  1010
          val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;
blanchet@48975
  1011
          val sum_Card_order = sum_Cinfinite RS conjunct2;
blanchet@48975
  1012
blanchet@48975
  1013
          fun mk_sum_card_order [thm] = thm
blanchet@48975
  1014
            | mk_sum_card_order (thm :: thms) =
blanchet@48975
  1015
              @{thm card_order_csum} OF [thm, mk_sum_card_order thms];
blanchet@48975
  1016
blanchet@48975
  1017
          val sum_card_order = mk_sum_card_order bd_card_orders;
blanchet@48975
  1018
blanchet@49504
  1019
          val sbd_ordIso = fold_thms lthy [sbd_def]
blanchet@48975
  1020
            (@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order]);
blanchet@49504
  1021
          val sbd_card_order =  fold_thms lthy [sbd_def]
blanchet@48975
  1022
            (@{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]);
blanchet@48975
  1023
          val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite];
blanchet@48975
  1024
          val sbd_Card_order = sbd_Cinfinite RS conjunct2;
blanchet@48975
  1025
blanchet@48975
  1026
          fun mk_set_sbd i bd_Card_order bds =
blanchet@48975
  1027
            map (fn thm => @{thm ordLeq_ordIso_trans} OF
blanchet@48975
  1028
              [bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds;
blanchet@48975
  1029
          val set_sbdss = map3 mk_set_sbd ks bd_Card_orders set_bdss;
blanchet@48975
  1030
       in
traytel@52635
  1031
         (lthy, sbd, sbdT, sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss)
blanchet@48975
  1032
       end;
blanchet@48975
  1033
blanchet@48975
  1034
    val sbdTs = replicate n sbdT;
blanchet@48975
  1035
    val sum_sbd = Library.foldr1 (uncurry mk_csum) (replicate n sbd);
blanchet@48975
  1036
    val sum_sbdT = mk_sumTN sbdTs;
blanchet@48975
  1037
    val sum_sbd_listT = HOLogic.listT sum_sbdT;
blanchet@48975
  1038
    val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT;
blanchet@48975
  1039
    val bdTs = passiveAs @ replicate n sbdT;
blanchet@48975
  1040
    val to_sbd_maps = map4 mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs;
blanchet@48975
  1041
    val bdFTs = mk_FTs bdTs;
blanchet@48975
  1042
    val sbdFT = mk_sumTN bdFTs;
blanchet@48975
  1043
    val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT);
blanchet@48975
  1044
    val treeQT = HOLogic.mk_setT treeT;
blanchet@48975
  1045
    val treeTs = passiveAs @ replicate n treeT;
blanchet@48975
  1046
    val treeQTs = passiveAs @ replicate n treeQT;
blanchet@48975
  1047
    val treeFTs = mk_FTs treeTs;
blanchet@48975
  1048
    val tree_maps = map4 mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs;
blanchet@48975
  1049
    val final_maps = map4 mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs;
blanchet@48975
  1050
    val isNode_setss = mk_setss (passiveAs @ replicate n sbdT);
blanchet@48975
  1051
blanchet@48975
  1052
    val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []];
blanchet@48975
  1053
    val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs);
blanchet@48975
  1054
    val Lev_recT = fastype_of Zero;
blanchet@48975
  1055
    val LevT = Library.foldr (op -->) (sTs, HOLogic.natT --> Lev_recT);
blanchet@48975
  1056
blanchet@48975
  1057
    val Nil = HOLogic.mk_tuple (map3 (fn i => fn z => fn z'=>
blanchet@48975
  1058
      Term.absfree z' (mk_InN activeAs z i)) ks zs zs');
blanchet@48975
  1059
    val rv_recT = fastype_of Nil;
blanchet@48975
  1060
    val rvT = Library.foldr (op -->) (sTs, sum_sbd_listT --> rv_recT);
blanchet@48975
  1061
blanchet@48975
  1062
    val (((((((((((sumx, sumx'), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')),
blanchet@48975
  1063
      (lab, lab')), (Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')),
blanchet@48975
  1064
      names_lthy) = names_lthy
blanchet@48975
  1065
      |> yield_singleton (apfst (op ~~) oo mk_Frees' "sumx") sum_sbdT
blanchet@48975
  1066
      ||>> mk_Frees' "k" sbdTs
blanchet@48975
  1067
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
blanchet@48975
  1068
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
blanchet@48975
  1069
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl") sum_sbd_list_setT
blanchet@48975
  1070
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT)
blanchet@48975
  1071
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl_lab") treeT
blanchet@48975
  1072
      ||>> mk_Frees "x" bdFTs
blanchet@48975
  1073
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") Lev_recT
blanchet@48975
  1074
      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") rv_recT;
blanchet@48975
  1075
blanchet@48975
  1076
    val (k, k') = (hd kks, hd kks')
blanchet@48975
  1077
blanchet@48975
  1078
    val timer = time (timer "Bounds");
blanchet@48975
  1079
blanchet@48975
  1080
    (* tree coalgebra *)
blanchet@48975
  1081
traytel@53566
  1082
    val isNode_binds = mk_internal_bs isNodeN;
traytel@53566
  1083
    fun isNode_bind i = nth isNode_binds (i - 1);
blanchet@48975
  1084
    val isNode_name = Binding.name_of o isNode_bind;
blanchet@48975
  1085
    val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind;
blanchet@48975
  1086
blanchet@48975
  1087
    val isNodeT =
blanchet@48975
  1088
      Library.foldr (op -->) (map fastype_of (As @ [Kl, lab, kl]), HOLogic.boolT);
blanchet@48975
  1089
blanchet@48975
  1090
    val Succs = map3 (fn i => fn k => fn k' =>
blanchet@48975
  1091
      HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl)))
blanchet@48975
  1092
      ks kks kks';
blanchet@48975
  1093
blanchet@48975
  1094
    fun isNode_spec sets x i =
blanchet@48975
  1095
      let
blanchet@48975
  1096
        val (passive_sets, active_sets) = chop m (map (fn set => set $ x) sets);
blanchet@48975
  1097
        val lhs = Term.list_comb (Free (isNode_name i, isNodeT), As @ [Kl, lab, kl]);
blanchet@48975
  1098
        val rhs = list_exists_free [x]
blanchet@48975
  1099
          (Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) ::
traytel@51893
  1100
          map2 mk_leq passive_sets As @ map2 (curry HOLogic.mk_eq) active_sets Succs));
blanchet@48975
  1101
      in
blanchet@49123
  1102
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1103
      end;
blanchet@48975
  1104
blanchet@48975
  1105
    val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1106
      lthy
blanchet@48975
  1107
      |> fold_map3 (fn i => fn x => fn sets => Specification.definition
blanchet@48975
  1108
        (SOME (isNode_bind i, NONE, NoSyn), (isNode_def_bind i, isNode_spec sets x i)))
blanchet@48975
  1109
        ks xs isNode_setss
blanchet@48975
  1110
      |>> apsnd split_list o split_list
blanchet@48975
  1111
      ||> `Local_Theory.restore;
blanchet@48975
  1112
blanchet@48975
  1113
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1114
blanchet@48975
  1115
    val isNode_defs = map (Morphism.thm phi) isNode_def_frees;
blanchet@48975
  1116
    val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees;
blanchet@48975
  1117
blanchet@48975
  1118
    fun mk_isNode As kl i =
blanchet@48975
  1119
      Term.list_comb (Const (nth isNodes (i - 1), isNodeT), As @ [Kl, lab, kl]);
blanchet@48975
  1120
blanchet@48975
  1121
    val isTree =
blanchet@48975
  1122
      let
blanchet@48975
  1123
        val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl);
traytel@51893
  1124
        val Field = mk_leq Kl (mk_Field (mk_clists sum_sbd));
blanchet@48975
  1125
        val prefCl = mk_prefCl Kl;
blanchet@48975
  1126
blanchet@48975
  1127
        val tree = mk_Ball Kl (Term.absfree kl'
blanchet@48975
  1128
          (HOLogic.mk_conj
blanchet@48975
  1129
            (Library.foldr1 HOLogic.mk_disj (map (mk_isNode As kl) ks),
blanchet@48975
  1130
            Library.foldr1 HOLogic.mk_conj (map4 (fn Succ => fn i => fn k => fn k' =>
blanchet@48975
  1131
              mk_Ball Succ (Term.absfree k' (mk_isNode As
blanchet@48975
  1132
                (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i)))
blanchet@48975
  1133
            Succs ks kks kks'))));
blanchet@48975
  1134
blanchet@48975
  1135
        val undef = list_all_free [kl] (HOLogic.mk_imp
blanchet@48975
  1136
          (HOLogic.mk_not (HOLogic.mk_mem (kl, Kl)),
blanchet@48975
  1137
          HOLogic.mk_eq (lab $ kl, mk_undefined sbdFT)));
blanchet@48975
  1138
      in
blanchet@48975
  1139
        Library.foldr1 HOLogic.mk_conj [empty, Field, prefCl, tree, undef]
blanchet@48975
  1140
      end;
blanchet@48975
  1141
traytel@53566
  1142
    val carT_binds = mk_internal_bs carTN;
traytel@53566
  1143
    fun carT_bind i = nth carT_binds (i - 1);
blanchet@48975
  1144
    val carT_name = Binding.name_of o carT_bind;
blanchet@48975
  1145
    val carT_def_bind = rpair [] o Thm.def_binding o carT_bind;
blanchet@48975
  1146
blanchet@48975
  1147
    fun carT_spec i =
blanchet@48975
  1148
      let
blanchet@48975
  1149
        val carTT = Library.foldr (op -->) (ATs, HOLogic.mk_setT treeT);
blanchet@48975
  1150
blanchet@48975
  1151
        val lhs = Term.list_comb (Free (carT_name i, carTT), As);
blanchet@48975
  1152
        val rhs = HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
blanchet@48975
  1153
          (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)),
blanchet@48975
  1154
            HOLogic.mk_conj (isTree, mk_isNode As (HOLogic.mk_list sum_sbdT []) i))));
blanchet@48975
  1155
      in
blanchet@49123
  1156
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1157
      end;
blanchet@48975
  1158
blanchet@48975
  1159
    val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1160
      lthy
blanchet@48975
  1161
      |> fold_map (fn i => Specification.definition
blanchet@48975
  1162
        (SOME (carT_bind i, NONE, NoSyn), (carT_def_bind i, carT_spec i))) ks
blanchet@48975
  1163
      |>> apsnd split_list o split_list
blanchet@48975
  1164
      ||> `Local_Theory.restore;
blanchet@48975
  1165
blanchet@48975
  1166
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1167
blanchet@48975
  1168
    val carT_defs = map (Morphism.thm phi) carT_def_frees;
blanchet@48975
  1169
    val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees;
blanchet@48975
  1170
blanchet@48975
  1171
    fun mk_carT As i = Term.list_comb
blanchet@48975
  1172
      (Const (nth carTs (i - 1),
blanchet@48975
  1173
         Library.foldr (op -->) (map fastype_of As, HOLogic.mk_setT treeT)), As);
blanchet@48975
  1174
traytel@53566
  1175
    val strT_binds = mk_internal_bs strTN;
traytel@53566
  1176
    fun strT_bind i = nth strT_binds (i - 1);
blanchet@48975
  1177
    val strT_name = Binding.name_of o strT_bind;
blanchet@48975
  1178
    val strT_def_bind = rpair [] o Thm.def_binding o strT_bind;
blanchet@48975
  1179
blanchet@48975
  1180
    fun strT_spec mapFT FT i =
blanchet@48975
  1181
      let
blanchet@48975
  1182
        val strTT = treeT --> FT;
blanchet@48975
  1183
blanchet@48975
  1184
        fun mk_f i k k' =
blanchet@48975
  1185
          let val in_k = mk_InN sbdTs k i;
blanchet@48975
  1186
          in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end;
blanchet@48975
  1187
blanchet@48975
  1188
        val f = Term.list_comb (mapFT, passive_ids @ map3 mk_f ks kks kks');
blanchet@48975
  1189
        val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs));
blanchet@48975
  1190
        val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2);
blanchet@48975
  1191
        val lhs = Free (strT_name i, strTT);
blanchet@48975
  1192
        val rhs = HOLogic.mk_split (Term.absfree Kl' (Term.absfree lab'
blanchet@48975
  1193
          (mk_sum_caseN fs $ (lab $ HOLogic.mk_list sum_sbdT []))));
blanchet@48975
  1194
      in
blanchet@49123
  1195
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1196
      end;
blanchet@48975
  1197
blanchet@48975
  1198
    val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1199
      lthy
blanchet@48975
  1200
      |> fold_map3 (fn i => fn mapFT => fn FT => Specification.definition
blanchet@48975
  1201
        (SOME (strT_bind i, NONE, NoSyn), (strT_def_bind i, strT_spec mapFT FT i)))
blanchet@48975
  1202
        ks tree_maps treeFTs
blanchet@48975
  1203
      |>> apsnd split_list o split_list
blanchet@48975
  1204
      ||> `Local_Theory.restore;
blanchet@48975
  1205
blanchet@48975
  1206
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1207
blanchet@48975
  1208
    val strT_defs = map ((fn def => trans OF [def RS fun_cong, @{thm prod.cases}]) o
blanchet@48975
  1209
      Morphism.thm phi) strT_def_frees;
blanchet@48975
  1210
    val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees;
blanchet@48975
  1211
blanchet@48975
  1212
    fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT);
blanchet@48975
  1213
blanchet@48975
  1214
    val carTAs = map (mk_carT As) ks;
blanchet@48975
  1215
    val strTAs = map2 mk_strT treeFTs ks;
blanchet@48975
  1216
blanchet@48975
  1217
    val coalgT_thm =
wenzelm@51551
  1218
      Goal.prove_sorry lthy [] []
blanchet@48975
  1219
        (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_coalg As carTAs strTAs)))
blanchet@53290
  1220
        (mk_coalgT_tac m (coalg_def :: isNode_defs @ carT_defs) strT_defs set_mapss)
traytel@49109
  1221
      |> Thm.close_derivation;
blanchet@48975
  1222
blanchet@48975
  1223
    val timer = time (timer "Tree coalgebra");
blanchet@48975
  1224
blanchet@48975
  1225
    fun mk_to_sbd s x i i' =
blanchet@48975
  1226
      mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
blanchet@48975
  1227
    fun mk_from_sbd s x i i' =
blanchet@48975
  1228
      mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
blanchet@48975
  1229
blanchet@48975
  1230
    fun mk_to_sbd_thmss thm = map (map (fn set_sbd =>
blanchet@48975
  1231
      thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss;
blanchet@48975
  1232
blanchet@48975
  1233
    val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj};
blanchet@48975
  1234
    val to_sbd_thmss = mk_to_sbd_thmss @{thm toCard};
blanchet@48975
  1235
    val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard};
blanchet@48975
  1236
traytel@53566
  1237
    val Lev_bind = mk_internal_b LevN;
blanchet@48975
  1238
    val Lev_name = Binding.name_of Lev_bind;
blanchet@48975
  1239
    val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind);
blanchet@48975
  1240
blanchet@48975
  1241
    val Lev_spec =
blanchet@48975
  1242
      let
blanchet@48975
  1243
        fun mk_Suc i s setsAs a a' =
blanchet@48975
  1244
          let
blanchet@48975
  1245
            val sets = drop m setsAs;
blanchet@48975
  1246
            fun mk_set i' set b =
blanchet@48975
  1247
              let
blanchet@48975
  1248
                val Cons = HOLogic.mk_eq (kl_copy,
blanchet@48975
  1249
                  mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl)
blanchet@48975
  1250
                val b_set = HOLogic.mk_mem (b, set $ (s $ a));
blanchet@48975
  1251
                val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b);
blanchet@48975
  1252
              in
blanchet@48975
  1253
                HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl]
blanchet@48975
  1254
                  (HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec))))
blanchet@48975
  1255
              end;
blanchet@48975
  1256
          in
blanchet@48975
  1257
            Term.absfree a' (Library.foldl1 mk_union (map3 mk_set ks sets zs_copy))
blanchet@48975
  1258
          end;
blanchet@48975
  1259
blanchet@48975
  1260
        val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec'
blanchet@48975
  1261
          (HOLogic.mk_tuple (map5 mk_Suc ks ss setssAs zs zs')));
blanchet@48975
  1262
blanchet@48975
  1263
        val lhs = Term.list_comb (Free (Lev_name, LevT), ss);
blanchet@48975
  1264
        val rhs = mk_nat_rec Zero Suc;
blanchet@48975
  1265
      in
blanchet@49123
  1266
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1267
      end;
blanchet@48975
  1268
blanchet@48975
  1269
    val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) =
blanchet@48975
  1270
      lthy
blanchet@48975
  1271
      |> Specification.definition (SOME (Lev_bind, NONE, NoSyn), (Lev_def_bind, Lev_spec))
blanchet@48975
  1272
      ||> `Local_Theory.restore;
blanchet@48975
  1273
blanchet@48975
  1274
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1275
blanchet@48975
  1276
    val Lev_def = Morphism.thm phi Lev_def_free;
blanchet@48975
  1277
    val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free));
blanchet@48975
  1278
blanchet@48975
  1279
    fun mk_Lev ss nat i =
blanchet@48975
  1280
      let
blanchet@48975
  1281
        val Ts = map fastype_of ss;
blanchet@48975
  1282
        val LevT = Library.foldr (op -->) (Ts, HOLogic.natT -->
blanchet@48975
  1283
          HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts));
blanchet@48975
  1284
      in
blanchet@48975
  1285
        mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i
blanchet@48975
  1286
      end;
blanchet@48975
  1287
blanchet@48975
  1288
    val Lev_0s = flat (mk_rec_simps n @{thm nat_rec_0} [Lev_def]);
blanchet@48975
  1289
    val Lev_Sucs = flat (mk_rec_simps n @{thm nat_rec_Suc} [Lev_def]);
blanchet@48975
  1290
traytel@53566
  1291
    val rv_bind = mk_internal_b rvN;
blanchet@48975
  1292
    val rv_name = Binding.name_of rv_bind;
blanchet@48975
  1293
    val rv_def_bind = rpair [] (Thm.def_binding rv_bind);
blanchet@48975
  1294
blanchet@48975
  1295
    val rv_spec =
blanchet@48975
  1296
      let
blanchet@48975
  1297
        fun mk_Cons i s b b' =
blanchet@48975
  1298
          let
blanchet@48975
  1299
            fun mk_case i' =
blanchet@48975
  1300
              Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k));
blanchet@48975
  1301
          in
blanchet@48975
  1302
            Term.absfree b' (mk_sum_caseN (map mk_case ks) $ sumx)
blanchet@48975
  1303
          end;
blanchet@48975
  1304
blanchet@48975
  1305
        val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec'
blanchet@48975
  1306
          (HOLogic.mk_tuple (map4 mk_Cons ks ss zs zs'))));
blanchet@48975
  1307
blanchet@48975
  1308
        val lhs = Term.list_comb (Free (rv_name, rvT), ss);
blanchet@48975
  1309
        val rhs = mk_list_rec Nil Cons;
blanchet@48975
  1310
      in
blanchet@49123
  1311
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1312
      end;
blanchet@48975
  1313
blanchet@48975
  1314
    val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) =
blanchet@48975
  1315
      lthy
blanchet@48975
  1316
      |> Specification.definition (SOME (rv_bind, NONE, NoSyn), (rv_def_bind, rv_spec))
blanchet@48975
  1317
      ||> `Local_Theory.restore;
blanchet@48975
  1318
blanchet@48975
  1319
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1320
blanchet@48975
  1321
    val rv_def = Morphism.thm phi rv_def_free;
blanchet@48975
  1322
    val rv = fst (Term.dest_Const (Morphism.term phi rv_free));
blanchet@48975
  1323
blanchet@48975
  1324
    fun mk_rv ss kl i =
blanchet@48975
  1325
      let
blanchet@48975
  1326
        val Ts = map fastype_of ss;
blanchet@48975
  1327
        val As = map domain_type Ts;
blanchet@48975
  1328
        val rvT = Library.foldr (op -->) (Ts, fastype_of kl -->
blanchet@48975
  1329
          HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As));
blanchet@48975
  1330
      in
blanchet@48975
  1331
        mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i
blanchet@48975
  1332
      end;
blanchet@48975
  1333
blanchet@48975
  1334
    val rv_Nils = flat (mk_rec_simps n @{thm list_rec_Nil} [rv_def]);
blanchet@48975
  1335
    val rv_Conss = flat (mk_rec_simps n @{thm list_rec_Cons} [rv_def]);
blanchet@48975
  1336
traytel@53566
  1337
    val beh_binds = mk_internal_bs behN;
traytel@53566
  1338
    fun beh_bind i = nth beh_binds (i - 1);
blanchet@48975
  1339
    val beh_name = Binding.name_of o beh_bind;
blanchet@48975
  1340
    val beh_def_bind = rpair [] o Thm.def_binding o beh_bind;
blanchet@48975
  1341
blanchet@48975
  1342
    fun beh_spec i z =
blanchet@48975
  1343
      let
blanchet@48975
  1344
        val mk_behT = Library.foldr (op -->) (map fastype_of (ss @ [z]), treeT);
blanchet@48975
  1345
blanchet@48975
  1346
        fun mk_case i to_sbd_map s k k' =
blanchet@48975
  1347
          Term.absfree k' (mk_InN bdFTs
blanchet@48975
  1348
            (Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i);
blanchet@48975
  1349
blanchet@48975
  1350
        val Lab = Term.absfree kl' (mk_If
blanchet@48975
  1351
          (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))
blanchet@48975
  1352
          (mk_sum_caseN (map5 mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z))
blanchet@48975
  1353
          (mk_undefined sbdFT));
blanchet@48975
  1354
blanchet@48975
  1355
        val lhs = Term.list_comb (Free (beh_name i, mk_behT), ss) $ z;
blanchet@48975
  1356
        val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
blanchet@48975
  1357
          (Term.absfree nat' (mk_Lev ss nat i $ z)), Lab);
blanchet@48975
  1358
      in
blanchet@49123
  1359
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1360
      end;
blanchet@48975
  1361
blanchet@48975
  1362
    val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1363
      lthy
blanchet@48975
  1364
      |> fold_map2 (fn i => fn z => Specification.definition
blanchet@48975
  1365
        (SOME (beh_bind i, NONE, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs
blanchet@48975
  1366
      |>> apsnd split_list o split_list
blanchet@48975
  1367
      ||> `Local_Theory.restore;
blanchet@48975
  1368
blanchet@48975
  1369
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@48975
  1370
blanchet@48975
  1371
    val beh_defs = map (Morphism.thm phi) beh_def_frees;
blanchet@48975
  1372
    val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees;
blanchet@48975
  1373
blanchet@48975
  1374
    fun mk_beh ss i =
blanchet@48975
  1375
      let
blanchet@48975
  1376
        val Ts = map fastype_of ss;
blanchet@48975
  1377
        val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT);
blanchet@48975
  1378
      in
blanchet@48975
  1379
        Term.list_comb (Const (nth behs (i - 1), behT), ss)
blanchet@48975
  1380
      end;
blanchet@48975
  1381
blanchet@48975
  1382
    val Lev_sbd_thms =
blanchet@48975
  1383
      let
traytel@51893
  1384
        fun mk_conjunct i z = mk_leq (mk_Lev ss nat i $ z) (mk_Field (mk_clists sum_sbd));
blanchet@48975
  1385
        val goal = list_all_free zs
blanchet@48975
  1386
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1387
blanchet@48975
  1388
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1389
blanchet@48975
  1390
        val Lev_sbd = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1391
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1392
            (K (mk_Lev_sbd_tac lthy cts Lev_0s Lev_Sucs to_sbd_thmss))
traytel@49109
  1393
          |> Thm.close_derivation);
blanchet@48975
  1394
blanchet@48975
  1395
        val Lev_sbd' = mk_specN n Lev_sbd;
blanchet@48975
  1396
      in
blanchet@48975
  1397
        map (fn i => Lev_sbd' RS mk_conjunctN n i) ks
blanchet@48975
  1398
      end;
blanchet@48975
  1399
blanchet@48975
  1400
    val (length_Lev_thms, length_Lev'_thms) =
blanchet@48975
  1401
      let
blanchet@48975
  1402
        fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1403
          HOLogic.mk_eq (mk_size kl, nat));
blanchet@48975
  1404
        val goal = list_all_free (kl :: zs)
blanchet@48975
  1405
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1406
blanchet@48975
  1407
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1408
blanchet@48975
  1409
        val length_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1410
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1411
            (K (mk_length_Lev_tac lthy cts Lev_0s Lev_Sucs))
traytel@49109
  1412
          |> Thm.close_derivation);
blanchet@48975
  1413
blanchet@48975
  1414
        val length_Lev' = mk_specN (n + 1) length_Lev;
blanchet@48975
  1415
        val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks;
blanchet@48975
  1416
blanchet@48975
  1417
        fun mk_goal i z = fold_rev Logic.all (z :: kl :: nat :: ss) (Logic.mk_implies
blanchet@48975
  1418
            (HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z)),
blanchet@48975
  1419
            HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))));
blanchet@48975
  1420
        val goals = map2 mk_goal ks zs;
blanchet@48975
  1421
blanchet@48975
  1422
        val length_Levs' = map2 (fn goal => fn length_Lev =>
wenzelm@51551
  1423
          Goal.prove_sorry lthy [] [] goal (K (mk_length_Lev'_tac length_Lev))
traytel@49109
  1424
          |> Thm.close_derivation) goals length_Levs;
blanchet@48975
  1425
      in
blanchet@48975
  1426
        (length_Levs, length_Levs')
blanchet@48975
  1427
      end;
blanchet@48975
  1428
blanchet@48975
  1429
    val prefCl_Lev_thms =
blanchet@48975
  1430
      let
blanchet@48975
  1431
        fun mk_conjunct i z = HOLogic.mk_imp
traytel@50058
  1432
          (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), mk_prefixeq kl_copy kl),
blanchet@48975
  1433
          HOLogic.mk_mem (kl_copy, mk_Lev ss (mk_size kl_copy) i $ z));
blanchet@48975
  1434
        val goal = list_all_free (kl :: kl_copy :: zs)
blanchet@48975
  1435
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1436
blanchet@48975
  1437
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1438
blanchet@48975
  1439
        val prefCl_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1440
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1441
            (K (mk_prefCl_Lev_tac lthy cts Lev_0s Lev_Sucs)))
traytel@49109
  1442
          |> Thm.close_derivation;
blanchet@48975
  1443
blanchet@48975
  1444
        val prefCl_Lev' = mk_specN (n + 2) prefCl_Lev;
blanchet@48975
  1445
      in
blanchet@48975
  1446
        map (fn i => prefCl_Lev' RS mk_conjunctN n i RS mp) ks
blanchet@48975
  1447
      end;
blanchet@48975
  1448
blanchet@48975
  1449
    val rv_last_thmss =
blanchet@48975
  1450
      let
blanchet@48975
  1451
        fun mk_conjunct i z i' z_copy = list_exists_free [z_copy]
blanchet@48975
  1452
          (HOLogic.mk_eq
blanchet@48975
  1453
            (mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z,
blanchet@48975
  1454
            mk_InN activeAs z_copy i'));
blanchet@48975
  1455
        val goal = list_all_free (k :: zs)
blanchet@48975
  1456
          (Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z =>
blanchet@48975
  1457
            Library.foldr1 HOLogic.mk_conj
blanchet@48975
  1458
              (map2 (mk_conjunct i z) ks zs_copy)) ks zs));
blanchet@48975
  1459
blanchet@48975
  1460
        val cTs = [SOME (certifyT lthy sum_sbdT)];
blanchet@48975
  1461
        val cts = map (SOME o certify lthy) [Term.absfree kl' goal, kl];
blanchet@48975
  1462
blanchet@48975
  1463
        val rv_last = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1464
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
traytel@49109
  1465
            (K (mk_rv_last_tac cTs cts rv_Nils rv_Conss)))
traytel@49109
  1466
          |> Thm.close_derivation;
blanchet@48975
  1467
blanchet@48975
  1468
        val rv_last' = mk_specN (n + 1) rv_last;
blanchet@48975
  1469
      in
blanchet@48975
  1470
        map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks
blanchet@48975
  1471
      end;
blanchet@48975
  1472
blanchet@48975
  1473
    val set_rv_Lev_thmsss = if m = 0 then replicate n (replicate n []) else
blanchet@48975
  1474
      let
blanchet@48975
  1475
        fun mk_case s sets z z_free = Term.absfree z_free (Library.foldr1 HOLogic.mk_conj
traytel@51893
  1476
          (map2 (fn set => fn A => mk_leq (set $ (s $ z)) A) (take m sets) As));
blanchet@48975
  1477
blanchet@48975
  1478
        fun mk_conjunct i z B = HOLogic.mk_imp
blanchet@48975
  1479
          (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), HOLogic.mk_mem (z, B)),
blanchet@48975
  1480
          mk_sum_caseN (map4 mk_case ss setssAs zs zs') $ (mk_rv ss kl i $ z));
blanchet@48975
  1481
blanchet@48975
  1482
        val goal = list_all_free (kl :: zs)
blanchet@48975
  1483
          (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct ks zs Bs));
blanchet@48975
  1484
blanchet@48975
  1485
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1486
blanchet@48975
  1487
        val set_rv_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1488
          (Goal.prove_sorry lthy [] []
blanchet@48975
  1489
            (Logic.mk_implies (coalg_prem, HOLogic.mk_Trueprop goal))
wenzelm@51798
  1490
            (K (mk_set_rv_Lev_tac lthy m cts Lev_0s Lev_Sucs rv_Nils rv_Conss
traytel@49109
  1491
              coalg_set_thmss from_to_sbd_thmss)))
traytel@49109
  1492
          |> Thm.close_derivation;
blanchet@48975
  1493
blanchet@48975
  1494
        val set_rv_Lev' = mk_specN (n + 1) set_rv_Lev;
blanchet@48975
  1495
      in
blanchet@48975
  1496
        map (fn i => map (fn i' =>
blanchet@48975
  1497
          split_conj_thm (if n = 1 then set_rv_Lev' RS mk_conjunctN n i RS mp
blanchet@48975
  1498
            else set_rv_Lev' RS mk_conjunctN n i RS mp RSN
traytel@52904
  1499
              (2, @{thm sum_case_weak_cong} RS iffD1) RS
traytel@52904
  1500
              (mk_sum_casesN n i' RS iffD1))) ks) ks
blanchet@48975
  1501
      end;
blanchet@48975
  1502
blanchet@48975
  1503
    val set_Lev_thmsss =
blanchet@48975
  1504
      let
blanchet@48975
  1505
        fun mk_conjunct i z =
blanchet@48975
  1506
          let
blanchet@48975
  1507
            fun mk_conjunct' i' sets s z' =
blanchet@48975
  1508
              let
blanchet@48975
  1509
                fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp
blanchet@48975
  1510
                  (HOLogic.mk_mem (z'', set $ (s $ z')),
blanchet@48975
  1511
                    HOLogic.mk_mem (mk_append (kl,
blanchet@48975
  1512
                      HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']),
blanchet@48975
  1513
                      mk_Lev ss (HOLogic.mk_Suc nat) i $ z));
blanchet@48975
  1514
              in
blanchet@48975
  1515
                HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'),
blanchet@48975
  1516
                  (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct'' ks (drop m sets) zs_copy2)))
blanchet@48975
  1517
              end;
blanchet@48975
  1518
          in
blanchet@48975
  1519
            HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1520
              Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct' ks setssAs ss zs_copy))
blanchet@48975
  1521
          end;
blanchet@48975
  1522
blanchet@48975
  1523
        val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2)
blanchet@48975
  1524
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1525
blanchet@48975
  1526
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1527
blanchet@48975
  1528
        val set_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1529
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1530
            (K (mk_set_Lev_tac lthy cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss)))
traytel@49109
  1531
          |> Thm.close_derivation;
blanchet@48975
  1532
blanchet@48975
  1533
        val set_Lev' = mk_specN (3 * n + 1) set_Lev;
blanchet@48975
  1534
      in
blanchet@48975
  1535
        map (fn i => map (fn i' => map (fn i'' => set_Lev' RS
blanchet@48975
  1536
          mk_conjunctN n i RS mp RS
blanchet@48975
  1537
          mk_conjunctN n i' RS mp RS
blanchet@48975
  1538
          mk_conjunctN n i'' RS mp) ks) ks) ks
blanchet@48975
  1539
      end;
blanchet@48975
  1540
blanchet@48975
  1541
    val set_image_Lev_thmsss =
blanchet@48975
  1542
      let
blanchet@48975
  1543
        fun mk_conjunct i z =
blanchet@48975
  1544
          let
blanchet@48975
  1545
            fun mk_conjunct' i' sets =
blanchet@48975
  1546
              let
blanchet@48975
  1547
                fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp
blanchet@48975
  1548
                  (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''),
blanchet@48975
  1549
                  HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z''))));
blanchet@48975
  1550
              in
blanchet@48975
  1551
                HOLogic.mk_imp (HOLogic.mk_mem
blanchet@48975
  1552
                  (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']),
blanchet@48975
  1553
                    mk_Lev ss (HOLogic.mk_Suc nat) i $ z),
blanchet@48975
  1554
                  (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct'' ks sets ss zs_copy)))
blanchet@48975
  1555
              end;
blanchet@48975
  1556
          in
blanchet@48975
  1557
            HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
blanchet@48975
  1558
              Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs')))
blanchet@48975
  1559
          end;
blanchet@48975
  1560
blanchet@48975
  1561
        val goal = list_all_free (kl :: k :: zs @ zs_copy)
blanchet@48975
  1562
          (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
blanchet@48975
  1563
blanchet@48975
  1564
        val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
blanchet@48975
  1565
blanchet@48975
  1566
        val set_image_Lev = singleton (Proof_Context.export names_lthy lthy)
wenzelm@51551
  1567
          (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51798
  1568
            (K (mk_set_image_Lev_tac lthy cts Lev_0s Lev_Sucs rv_Nils rv_Conss
traytel@49109
  1569
              from_to_sbd_thmss to_sbd_inj_thmss)))
traytel@49109
  1570
          |> Thm.close_derivation;
blanchet@48975
  1571
blanchet@48975
  1572
        val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev;
blanchet@48975
  1573
      in
blanchet@48975
  1574
        map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS
blanchet@48975
  1575
          mk_conjunctN n i RS mp RS
blanchet@48975
  1576
          mk_conjunctN n i'' RS mp RS
blanchet@48975
  1577
          mk_conjunctN n i' RS mp) ks) ks) ks
blanchet@48975
  1578
      end;
blanchet@48975
  1579
blanchet@48975
  1580
    val mor_beh_thm =
wenzelm@51551
  1581
      Goal.prove_sorry lthy [] []
blanchet@48975
  1582
        (fold_rev Logic.all (As @ Bs @ ss) (Logic.mk_implies (coalg_prem,
blanchet@48975
  1583
          HOLogic.mk_Trueprop (mk_mor Bs ss carTAs strTAs (map (mk_beh ss) ks)))))
blanchet@48975
  1584
        (mk_mor_beh_tac m mor_def mor_cong_thm
blanchet@48975
  1585
          beh_defs carT_defs strT_defs isNode_defs
blanchet@48975
  1586
          to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbd_thms
blanchet@48975
  1587
          length_Lev_thms length_Lev'_thms prefCl_Lev_thms rv_last_thmss
blanchet@48975
  1588
          set_rv_Lev_thmsss set_Lev_thmsss set_image_Lev_thmsss
blanchet@53290
  1589
          set_mapss coalg_set_thmss map_comp_id_thms map_cong0s map_arg_cong_thms)
traytel@49109
  1590
      |> Thm.close_derivation;
blanchet@48975
  1591
blanchet@48975
  1592
    val timer = time (timer "Behavioral morphism");
blanchet@48975
  1593
blanchet@48975
  1594
    fun mk_LSBIS As i = mk_lsbis As (map (mk_carT As) ks) strTAs i;
blanchet@48975
  1595
    fun mk_car_final As i =
blanchet@48975
  1596
      mk_quotient (mk_carT As i) (mk_LSBIS As i);
blanchet@48975
  1597
    fun mk_str_final As i =
blanchet@48975
  1598
      mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1),
blanchet@48975
  1599
        passive_ids @ map (mk_proj o mk_LSBIS As) ks), nth strTAs (i - 1)));
blanchet@48975
  1600
blanchet@48975
  1601
    val car_finalAs = map (mk_car_final As) ks;
blanchet@48975
  1602
    val str_finalAs = map (mk_str_final As) ks;
blanchet@48975
  1603
    val car_finals = map (mk_car_final passive_UNIVs) ks;
blanchet@48975
  1604
    val str_finals = map (mk_str_final passive_UNIVs) ks;
blanchet@48975
  1605
blanchet@48975
  1606
    val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss;
blanchet@48975
  1607
    val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms;
blanchet@48975
  1608
blanchet@48975
  1609
    val congruent_str_final_thms =
blanchet@48975
  1610
      let
blanchet@48975
  1611
        fun mk_goal R final_map strT =
blanchet@48975
  1612
          fold_rev Logic.all As (HOLogic.mk_Trueprop
blanchet@48975
  1613
            (mk_congruent R (HOLogic.mk_comp
blanchet@48975
  1614
              (Term.list_comb (final_map, passive_ids @ map (mk_proj o mk_LSBIS As) ks), strT))));
blanchet@48975
  1615
blanchet@48975
  1616
        val goals = map3 mk_goal (map (mk_LSBIS As) ks) final_maps strTAs;
blanchet@48975
  1617
      in
blanchet@51761
  1618
        map4 (fn goal => fn lsbisE => fn map_comp_id => fn map_cong0 =>
wenzelm@51551
  1619
          Goal.prove_sorry lthy [] [] goal
blanchet@51761
  1620
            (K (mk_congruent_str_final_tac m lsbisE map_comp_id map_cong0 equiv_LSBIS_thms))
traytel@49109
  1621
          |> Thm.close_derivation)
blanchet@51761
  1622
        goals lsbisE_thms map_comp_id_thms map_cong0s
blanchet@48975
  1623
      end;
blanchet@48975
  1624
wenzelm@51551
  1625
    val coalg_final_thm = Goal.prove_sorry lthy [] [] (fold_rev Logic.all As
blanchet@48975
  1626
      (HOLogic.mk_Trueprop (mk_coalg As car_finalAs str_finalAs)))
blanchet@48975
  1627
      (K (mk_coalg_final_tac m coalg_def congruent_str_final_thms equiv_LSBIS_thms
blanchet@53290
  1628
        set_mapss coalgT_set_thmss))
traytel@49109
  1629
      |> Thm.close_derivation;
blanchet@48975
  1630
wenzelm@51551
  1631
    val mor_T_final_thm = Goal.prove_sorry lthy [] [] (fold_rev Logic.all As
blanchet@48975
  1632
      (HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finalAs str_finalAs
blanchet@48975
  1633
        (map (mk_proj o mk_LSBIS As) ks))))
traytel@49109
  1634
      (K (mk_mor_T_final_tac mor_def congruent_str_final_thms equiv_LSBIS_thms))
traytel@49109
  1635
      |> Thm.close_derivation;
blanchet@48975
  1636
blanchet@48975
  1637
    val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm];
blanchet@48975
  1638
    val in_car_final_thms = map (fn mor_image' => mor_image' OF
blanchet@48975
  1639
      [tcoalg_thm RS mor_final_thm, UNIV_I]) mor_image'_thms;
blanchet@48975
  1640
blanchet@48975
  1641
    val timer = time (timer "Final coalgebra");
blanchet@48975
  1642
blanchet@48975
  1643
    val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
blanchet@48975
  1644
      lthy
blanchet@49169
  1645
      |> fold_map4 (fn b => fn mx => fn car_final => fn in_car_final =>
traytel@53566
  1646
        typedef (Binding.conceal b, params, mx) car_final NONE
blanchet@49169
  1647
          (EVERY' [rtac exI, rtac in_car_final] 1)) bs mixfixes car_finals in_car_final_thms
blanchet@48975
  1648
      |>> apsnd split_list o split_list;
blanchet@48975
  1649
blanchet@48975
  1650
    val Ts = map (fn name => Type (name, params')) T_names;
blanchet@48975
  1651
    fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;
blanchet@48975
  1652
    val Ts' = mk_Ts passiveBs;
blanchet@48975
  1653
    val Ts'' = mk_Ts passiveCs;
blanchet@48975
  1654
    val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts;
blanchet@48975
  1655
    val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts;
blanchet@48975
  1656
blanchet@48975
  1657
    val Reps = map #Rep T_loc_infos;
blanchet@48975
  1658
    val Rep_injects = map #Rep_inject T_loc_infos;
blanchet@48975
  1659
    val Abs_inverses = map #Abs_inverse T_loc_infos;
blanchet@48975
  1660
blanchet@48975
  1661
    val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");
blanchet@48975
  1662
blanchet@48975
  1663
    val UNIVs = map HOLogic.mk_UNIV Ts;
blanchet@48975
  1664
    val FTs = mk_FTs (passiveAs @ Ts);
blanchet@48975
  1665
    val FTs' = mk_FTs (passiveBs @ Ts);
blanchet@48975
  1666
    val prodTs = map (HOLogic.mk_prodT o `I) Ts;
blanchet@48975
  1667
    val prodFTs = mk_FTs (passiveAs @ prodTs);
blanchet@48975
  1668
    val FTs_setss = mk_setss (passiveAs @ Ts);
blanchet@48975
  1669
    val prodFT_setss = mk_setss (passiveAs @ prodTs);
blanchet@48975
  1670
    val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs;
blanchet@48975
  1671
    val map_FT_nths = map2 (fn Ds =>
blanchet@48975
  1672
      mk_map_of_bnf Ds (passiveAs @ prodTs) (passiveAs @ Ts)) Dss bnfs;
blanchet@48975
  1673
    val fstsTs = map fst_const prodTs;
blanchet@48975
  1674
    val sndsTs = map snd_const prodTs;
traytel@52923
  1675
    val dtorTs = map2 (curry op -->) Ts FTs;
traytel@52923
  1676
    val ctorTs = map2 (curry op -->) FTs Ts;
blanchet@49504
  1677
    val unfold_fTs = map2 (curry op -->) activeAs Ts;
blanchet@48975
  1678
    val corec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) sum_sTs;
blanchet@48975
  1679
    val corec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls;
blanchet@48975
  1680
    val corec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls_rev;
blanchet@48975
  1681
    val corec_Inls = map (Term.subst_atomic_types (activeBs ~~ Ts)) Inls;
traytel@51739
  1682
    val corec_UNIVs = map2 (HOLogic.mk_UNIV oo curry mk_sumT) Ts activeAs;
blanchet@48975
  1683
traytel@51925
  1684
    val ((((((((((((((Jzs, Jzs'), (Jz's, Jz's')), Jzs_copy), Jz's_copy), Jzs1), Jzs2), Jpairs),
blanchet@49504
  1685
      FJzs), TRs), unfold_fs), unfold_fs_copy), corec_ss), phis), names_lthy) = names_lthy
blanchet@48975
  1686
      |> mk_Frees' "z" Ts
traytel@51925
  1687
      ||>> mk_Frees' "y" Ts'
traytel@51925
  1688
      ||>> mk_Frees "z'" Ts
traytel@51925
  1689
      ||>> mk_Frees "y'" Ts'
blanchet@48975
  1690
      ||>> mk_Frees "z1" Ts
blanchet@48975
  1691
      ||>> mk_Frees "z2" Ts
blanchet@48975
  1692
      ||>> mk_Frees "j" (map2 (curry HOLogic.mk_prodT) Ts Ts')
blanchet@48975
  1693
      ||>> mk_Frees "x" prodFTs
blanchet@49591
  1694
      ||>> mk_Frees "r" (map (mk_relT o `I) Ts)
blanchet@49504
  1695
      ||>> mk_Frees "f" unfold_fTs
blanchet@49504
  1696
      ||>> mk_Frees "g" unfold_fTs
blanchet@48975
  1697
      ||>> mk_Frees "s" corec_sTs
blanchet@49592
  1698
      ||>> mk_Frees "P" (map2 mk_pred2T Ts Ts);
blanchet@48975
  1699
traytel@53300
  1700
    fun dtor_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ dtorN);
blanchet@49501
  1701
    val dtor_name = Binding.name_of o dtor_bind;
traytel@53566
  1702
    val dtor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o dtor_bind;
blanchet@49501
  1703
blanchet@49501
  1704
    fun dtor_spec i rep str map_FT dtorT Jz Jz' =
blanchet@48975
  1705
      let
blanchet@49501
  1706
        val lhs = Free (dtor_name i, dtorT);
blanchet@48975
  1707
        val rhs = Term.absfree Jz'
blanchet@48975
  1708
          (Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $
blanchet@48975
  1709
            (str $ (rep $ Jz)));
blanchet@48975
  1710
      in
blanchet@49123
  1711
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1712
      end;
blanchet@48975
  1713
blanchet@49501
  1714
    val ((dtor_frees, (_, dtor_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1715
      lthy
blanchet@49501
  1716
      |> fold_map7 (fn i => fn rep => fn str => fn mapx => fn dtorT => fn Jz => fn Jz' =>
blanchet@49501
  1717
        Specification.definition (SOME (dtor_bind i, NONE, NoSyn),
blanchet@49501
  1718
          (dtor_def_bind i, dtor_spec i rep str mapx dtorT Jz Jz')))
blanchet@49501
  1719
        ks Rep_Ts str_finals map_FTs dtorTs Jzs Jzs'
blanchet@48975
  1720
      |>> apsnd split_list o split_list
blanchet@48975
  1721
      ||> `Local_Theory.restore;
blanchet@48975
  1722
blanchet@48975
  1723
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49501
  1724
    fun mk_dtors passive =
traytel@49185
  1725
      map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (mk_params passive)) o
blanchet@49501
  1726
        Morphism.term phi) dtor_frees;
blanchet@49501
  1727
    val dtors = mk_dtors passiveAs;
blanchet@49501
  1728
    val dtor's = mk_dtors passiveBs;
blanchet@49501
  1729
    val dtor_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) dtor_def_frees;
blanchet@48975
  1730
blanchet@48975
  1731
    val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss;
blanchet@48975
  1732
    val (mor_Rep_thm, mor_Abs_thm) =
blanchet@48975
  1733
      let
blanchet@48975
  1734
        val mor_Rep =
wenzelm@51551
  1735
          Goal.prove_sorry lthy [] []
blanchet@49501
  1736
            (HOLogic.mk_Trueprop (mk_mor UNIVs dtors car_finals str_finals Rep_Ts))
blanchet@49501
  1737
            (mk_mor_Rep_tac m (mor_def :: dtor_defs) Reps Abs_inverses coalg_final_set_thmss
blanchet@51761
  1738
              map_comp_id_thms map_cong0L_thms)
traytel@49109
  1739
          |> Thm.close_derivation;
blanchet@48975
  1740
blanchet@48975
  1741
        val mor_Abs =
wenzelm@51551
  1742
          Goal.prove_sorry lthy [] []
blanchet@49501
  1743
            (HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs dtors Abs_Ts))
blanchet@49501
  1744
            (mk_mor_Abs_tac (mor_def :: dtor_defs) Abs_inverses)
traytel@49109
  1745
          |> Thm.close_derivation;
blanchet@48975
  1746
      in
blanchet@48975
  1747
        (mor_Rep, mor_Abs)
blanchet@48975
  1748
      end;
blanchet@48975
  1749
blanchet@49501
  1750
    val timer = time (timer "dtor definitions & thms");
blanchet@49501
  1751
traytel@53300
  1752
    fun unfold_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ dtor_unfoldN);
blanchet@49504
  1753
    val unfold_name = Binding.name_of o unfold_bind;
traytel@53566
  1754
    val unfold_def_bind = rpair [] o Binding.conceal o Thm.def_binding o unfold_bind;
blanchet@49504
  1755
blanchet@49504
  1756
    fun unfold_spec i T AT abs f z z' =
blanchet@48975
  1757
      let
blanchet@49504
  1758
        val unfoldT = Library.foldr (op -->) (sTs, AT --> T);
blanchet@49504
  1759
blanchet@49504
  1760
        val lhs = Term.list_comb (Free (unfold_name i, unfoldT), ss);
blanchet@48975
  1761
        val rhs = Term.absfree z' (abs $ (f $ z));
blanchet@48975
  1762
      in
blanchet@49123
  1763
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1764
      end;
blanchet@48975
  1765
blanchet@49504
  1766
    val ((unfold_frees, (_, unfold_def_frees)), (lthy, lthy_old)) =
blanchet@48975
  1767
      lthy
blanchet@48975
  1768
      |> fold_map7 (fn i => fn T => fn AT => fn abs => fn f => fn z => fn z' =>
blanchet@48975
  1769
        Specification.definition
blanchet@49504
  1770
          (SOME (unfold_bind i, NONE, NoSyn), (unfold_def_bind i, unfold_spec i T AT abs f z z')))
blanchet@48975
  1771
          ks Ts activeAs Abs_Ts (map (fn i => HOLogic.mk_comp
blanchet@48975
  1772
            (mk_proj (mk_LSBIS passive_UNIVs i), mk_beh ss i)) ks) zs zs'
blanchet@48975
  1773
      |>> apsnd split_list o split_list
blanchet@48975
  1774
      ||> `Local_Theory.restore;
blanchet@48975
  1775
blanchet@48975
  1776
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49504
  1777
    val unfolds = map (Morphism.term phi) unfold_frees;
blanchet@49504
  1778
    val unfold_names = map (fst o dest_Const) unfolds;
traytel@52731
  1779
    fun mk_unfolds passives actives =
traytel@52731
  1780
      map3 (fn name => fn T => fn active =>
traytel@52731
  1781
        Const (name, Library.foldr (op -->)
traytel@52923
  1782
          (map2 (curry op -->) actives (mk_FTs (passives @ actives)), active --> T)))
traytel@52731
  1783
      unfold_names (mk_Ts passives) actives;
blanchet@49504
  1784
    fun mk_unfold Ts ss i = Term.list_comb (Const (nth unfold_names (i - 1), Library.foldr (op -->)
blanchet@48975
  1785
      (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
blanchet@49504
  1786
    val unfold_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) unfold_def_frees;
blanchet@49504
  1787
blanchet@49504
  1788
    val mor_unfold_thm =
blanchet@48975
  1789
      let
blanchet@48975
  1790
        val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses;
blanchet@48975
  1791
        val morEs' = map (fn thm =>
blanchet@48975
  1792
          (thm OF [tcoalg_thm RS mor_final_thm, UNIV_I]) RS sym) morE_thms;
blanchet@48975
  1793
      in
wenzelm@51551
  1794
        Goal.prove_sorry lthy [] []
blanchet@48975
  1795
          (fold_rev Logic.all ss
blanchet@49504
  1796
            (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors (map (mk_unfold Ts ss) ks))))
blanchet@49504
  1797
          (K (mk_mor_unfold_tac m mor_UNIV_thm dtor_defs unfold_defs Abs_inverses' morEs'
blanchet@51761
  1798
            map_comp_id_thms map_cong0s))
traytel@49109
  1799
        |> Thm.close_derivation
blanchet@48975
  1800
      end;
blanchet@49504
  1801
    val dtor_unfold_thms = map (fn thm => (thm OF [mor_unfold_thm, UNIV_I]) RS sym) morE_thms;
blanchet@48975
  1802
blanchet@48975
  1803
    val (raw_coind_thms, raw_coind_thm) =
blanchet@48975
  1804
      let
blanchet@49501
  1805
        val prem = HOLogic.mk_Trueprop (mk_sbis passive_UNIVs UNIVs dtors TRs);
blanchet@48975
  1806
        val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
traytel@51893
  1807
          (map2 (fn R => fn T => mk_leq R (Id_const T)) TRs Ts));
blanchet@48975
  1808
        val goal = fold_rev Logic.all TRs (Logic.mk_implies (prem, concl));
blanchet@48975
  1809
      in
wenzelm@51551
  1810
        `split_conj_thm (Goal.prove_sorry lthy [] [] goal
blanchet@48975
  1811
          (K (mk_raw_coind_tac bis_def bis_cong_thm bis_O_thm bis_converse_thm bis_Gr_thm
blanchet@48975
  1812
            tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm
traytel@49109
  1813
            lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects))
traytel@49109
  1814
          |> Thm.close_derivation)
blanchet@48975
  1815
      end;
blanchet@48975
  1816
blanchet@48975
  1817
    val unique_mor_thms =
blanchet@48975
  1818
      let
blanchet@48975
  1819
        val prems = [HOLogic.mk_Trueprop (mk_coalg passive_UNIVs Bs ss), HOLogic.mk_Trueprop
blanchet@49504
  1820
          (HOLogic.mk_conj (mk_mor Bs ss UNIVs dtors unfold_fs,
blanchet@49504
  1821
            mk_mor Bs ss UNIVs dtors unfold_fs_copy))];
blanchet@48975
  1822
        fun mk_fun_eq B f g z = HOLogic.mk_imp
blanchet@48975
  1823
          (HOLogic.mk_mem (z, B), HOLogic.mk_eq (f $ z, g $ z));
blanchet@48975
  1824
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@49504
  1825
          (map4 mk_fun_eq Bs unfold_fs unfold_fs_copy zs));
blanchet@48975
  1826
wenzelm@51551
  1827
        val unique_mor = Goal.prove_sorry lthy [] []
blanchet@49504
  1828
          (fold_rev Logic.all (Bs @ ss @ unfold_fs @ unfold_fs_copy @ zs)
blanchet@48975
  1829
            (Logic.list_implies (prems, unique)))
traytel@49109
  1830
          (K (mk_unique_mor_tac raw_coind_thms bis_image2_thm))
traytel@49109
  1831
          |> Thm.close_derivation;
blanchet@48975
  1832
      in
blanchet@48975
  1833
        map (fn thm => conjI RSN (2, thm RS mp)) (split_conj_thm unique_mor)
blanchet@48975
  1834
      end;
blanchet@48975
  1835
blanchet@49504
  1836
    val (unfold_unique_mor_thms, unfold_unique_mor_thm) =
blanchet@48975
  1837
      let
blanchet@49504
  1838
        val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors unfold_fs);
blanchet@49504
  1839
        fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_unfold Ts ss i);
blanchet@48975
  1840
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@49504
  1841
          (map2 mk_fun_eq unfold_fs ks));
blanchet@48975
  1842
blanchet@48975
  1843
        val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm);
blanchet@48975
  1844
        val mor_thm = mor_comp_thm OF [tcoalg_thm RS mor_final_thm, mor_Abs_thm];
blanchet@48975
  1845
wenzelm@51551
  1846
        val unique_mor = Goal.prove_sorry lthy [] []
blanchet@49504
  1847
          (fold_rev Logic.all (ss @ unfold_fs) (Logic.mk_implies (prem, unique)))
blanchet@49504
  1848
          (K (mk_unfold_unique_mor_tac raw_coind_thms bis_thm mor_thm unfold_defs))
traytel@49109
  1849
          |> Thm.close_derivation;
blanchet@48975
  1850
      in
blanchet@48975
  1851
        `split_conj_thm unique_mor
blanchet@48975
  1852
      end;
blanchet@48975
  1853
blanchet@49504
  1854
    val (dtor_unfold_unique_thms, dtor_unfold_unique_thm) = `split_conj_thm (split_conj_prems n
traytel@52904
  1855
      (mor_UNIV_thm RS iffD2 RS unfold_unique_mor_thm));
blanchet@49504
  1856
blanchet@49504
  1857
    val unfold_dtor_thms = map (fn thm => mor_id_thm RS thm RS sym) unfold_unique_mor_thms;
blanchet@49504
  1858
blanchet@49504
  1859
    val unfold_o_dtor_thms =
blanchet@48975
  1860
      let
blanchet@49504
  1861
        val mor = mor_comp_thm OF [mor_str_thm, mor_unfold_thm];
blanchet@48975
  1862
      in
blanchet@49504
  1863
        map2 (fn unique => fn unfold_ctor =>
blanchet@49504
  1864
          trans OF [mor RS unique, unfold_ctor]) unfold_unique_mor_thms unfold_dtor_thms
blanchet@48975
  1865
      end;
blanchet@48975
  1866
blanchet@49504
  1867
    val timer = time (timer "unfold definitions & thms");
blanchet@48975
  1868
blanchet@49501
  1869
    val map_dtors = map2 (fn Ds => fn bnf =>
blanchet@48975
  1870
      Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,
blanchet@49501
  1871
        map HOLogic.id_const passiveAs @ dtors)) Dss bnfs;
blanchet@49501
  1872
traytel@53300
  1873
    fun ctor_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ ctorN);
blanchet@49501
  1874
    val ctor_name = Binding.name_of o ctor_bind;
traytel@53566
  1875
    val ctor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o ctor_bind;
blanchet@49501
  1876
blanchet@49501
  1877
    fun ctor_spec i ctorT =
blanchet@48975
  1878
      let
blanchet@49501
  1879
        val lhs = Free (ctor_name i, ctorT);
blanchet@49504
  1880
        val rhs = mk_unfold Ts map_dtors i;
blanchet@48975
  1881
      in
blanchet@49123
  1882
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1883
      end;
blanchet@48975
  1884
blanchet@49501
  1885
    val ((ctor_frees, (_, ctor_def_frees)), (lthy, lthy_old)) =
blanchet@49311
  1886
      lthy
blanchet@49501
  1887
      |> fold_map2 (fn i => fn ctorT =>
blanchet@49311
  1888
        Specification.definition
blanchet@49501
  1889
          (SOME (ctor_bind i, NONE, NoSyn), (ctor_def_bind i, ctor_spec i ctorT))) ks ctorTs
blanchet@49311
  1890
      |>> apsnd split_list o split_list
blanchet@49311
  1891
      ||> `Local_Theory.restore;
blanchet@49311
  1892
blanchet@48975
  1893
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49501
  1894
    fun mk_ctors params =
blanchet@48975
  1895
      map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
blanchet@49501
  1896
        ctor_frees;
blanchet@49501
  1897
    val ctors = mk_ctors params';
blanchet@49501
  1898
    val ctor_defs = map (Morphism.thm phi) ctor_def_frees;
blanchet@49501
  1899
blanchet@49504
  1900
    val ctor_o_dtor_thms = map2 (fold_thms lthy o single) ctor_defs unfold_o_dtor_thms;
blanchet@49501
  1901
blanchet@49501
  1902
    val dtor_o_ctor_thms =
blanchet@48975
  1903
      let
blanchet@49501
  1904
        fun mk_goal dtor ctor FT =
blanchet@49501
  1905
         mk_Trueprop_eq (HOLogic.mk_comp (dtor, ctor), HOLogic.id_const FT);
blanchet@49501
  1906
        val goals = map3 mk_goal dtors ctors FTs;
blanchet@48975
  1907
      in
blanchet@51761
  1908
        map5 (fn goal => fn ctor_def => fn unfold => fn map_comp_id => fn map_cong0L =>
wenzelm@51551
  1909
          Goal.prove_sorry lthy [] [] goal
blanchet@51761
  1910
            (mk_dtor_o_ctor_tac ctor_def unfold map_comp_id map_cong0L unfold_o_dtor_thms)
traytel@49109
  1911
          |> Thm.close_derivation)
blanchet@51761
  1912
          goals ctor_defs dtor_unfold_thms map_comp_id_thms map_cong0L_thms
blanchet@48975
  1913
      end;
blanchet@48975
  1914
blanchet@49501
  1915
    val dtor_ctor_thms = map (fn thm => thm RS @{thm pointfree_idE}) dtor_o_ctor_thms;
blanchet@49501
  1916
    val ctor_dtor_thms = map (fn thm => thm RS @{thm pointfree_idE}) ctor_o_dtor_thms;
blanchet@49501
  1917
blanchet@49501
  1918
    val bij_dtor_thms =
blanchet@49501
  1919
      map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) ctor_o_dtor_thms dtor_o_ctor_thms;
blanchet@49501
  1920
    val inj_dtor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_dtor_thms;
blanchet@49501
  1921
    val surj_dtor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_dtor_thms;
blanchet@49501
  1922
    val dtor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_dtor_thms;
blanchet@49501
  1923
    val dtor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_dtor_thms;
blanchet@49501
  1924
    val dtor_exhaust_thms = map (fn thm => thm RS exE) dtor_nchotomy_thms;
blanchet@49501
  1925
blanchet@49501
  1926
    val bij_ctor_thms =
blanchet@49501
  1927
      map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) dtor_o_ctor_thms ctor_o_dtor_thms;
blanchet@49501
  1928
    val inj_ctor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_ctor_thms;
blanchet@49501
  1929
    val surj_ctor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_ctor_thms;
blanchet@49501
  1930
    val ctor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_ctor_thms;
blanchet@49501
  1931
    val ctor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_ctor_thms;
blanchet@49501
  1932
    val ctor_exhaust_thms = map (fn thm => thm RS exE) ctor_nchotomy_thms;
blanchet@49501
  1933
blanchet@49501
  1934
    val timer = time (timer "ctor definitions & thms");
blanchet@48975
  1935
blanchet@48975
  1936
    val corec_Inl_sum_thms =
blanchet@48975
  1937
      let
blanchet@49504
  1938
        val mor = mor_comp_thm OF [mor_sum_case_thm, mor_unfold_thm];
blanchet@48975
  1939
      in
blanchet@49504
  1940
        map2 (fn unique => fn unfold_dtor =>
blanchet@49504
  1941
          trans OF [mor RS unique, unfold_dtor]) unfold_unique_mor_thms unfold_dtor_thms
blanchet@48975
  1942
      end;
blanchet@48975
  1943
traytel@53300
  1944
    fun corec_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ dtor_corecN);
blanchet@48975
  1945
    val corec_name = Binding.name_of o corec_bind;
traytel@53566
  1946
    val corec_def_bind = rpair [] o Binding.conceal o Thm.def_binding o corec_bind;
blanchet@48975
  1947
traytel@51739
  1948
    val corec_strs =
traytel@51739
  1949
      map3 (fn dtor => fn sum_s => fn mapx =>
traytel@51739
  1950
        mk_sum_case
traytel@51739
  1951
          (HOLogic.mk_comp (Term.list_comb (mapx, passive_ids @ corec_Inls), dtor), sum_s))
traytel@51739
  1952
      dtors corec_ss corec_maps;
traytel@51739
  1953
blanchet@48975
  1954
    fun corec_spec i T AT =
blanchet@48975
  1955
      let
blanchet@48975
  1956
        val corecT = Library.foldr (op -->) (corec_sTs, AT --> T);
blanchet@48975
  1957
blanchet@48975
  1958
        val lhs = Term.list_comb (Free (corec_name i, corecT), corec_ss);
traytel@51739
  1959
        val rhs = HOLogic.mk_comp (mk_unfold Ts corec_strs i, Inr_const T AT);
blanchet@48975
  1960
      in
blanchet@49123
  1961
        mk_Trueprop_eq (lhs, rhs)
blanchet@48975
  1962
      end;
blanchet@48975
  1963
blanchet@48975
  1964
    val ((corec_frees, (_, corec_def_frees)), (lthy, lthy_old)) =
blanchet@49311
  1965
      lthy
blanchet@49311
  1966
      |> fold_map3 (fn i => fn T => fn AT =>
blanchet@49311
  1967
        Specification.definition
blanchet@49311
  1968
          (SOME (corec_bind i, NONE, NoSyn), (corec_def_bind i, corec_spec i T AT)))
blanchet@49311
  1969
          ks Ts activeAs
blanchet@49311
  1970
      |>> apsnd split_list o split_list
blanchet@49311
  1971
      ||> `Local_Theory.restore;
blanchet@49311
  1972
blanchet@48975
  1973
    val phi = Proof_Context.export_morphism lthy_old lthy;
blanchet@49176
  1974
    val corecs = map (Morphism.term phi) corec_frees;
blanchet@49176
  1975
    val corec_names = map (fst o dest_Const) corecs;
blanchet@49176
  1976
    fun mk_corec ss i = Term.list_comb (Const (nth corec_names (i - 1), Library.foldr (op -->)
blanchet@48975
  1977
      (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
blanchet@48975
  1978
    val corec_defs = map (Morphism.thm phi) corec_def_frees;
blanchet@48975
  1979
blanchet@48975
  1980
    val sum_cases =
blanchet@49255
  1981
      map2 (fn T => fn i => mk_sum_case (HOLogic.id_const T, mk_corec corec_ss i)) Ts ks;
blanchet@49504
  1982
    val dtor_corec_thms =
blanchet@48975
  1983
      let
blanchet@49501
  1984
        fun mk_goal i corec_s corec_map dtor z =
blanchet@48975
  1985
          let
blanchet@49501
  1986
            val lhs = dtor $ (mk_corec corec_ss i $ z);
blanchet@48975
  1987
            val rhs = Term.list_comb (corec_map, passive_ids @ sum_cases) $ (corec_s $ z);
blanchet@48975
  1988
          in
blanchet@49123
  1989
            fold_rev Logic.all (z :: corec_ss) (mk_Trueprop_eq (lhs, rhs))
blanchet@48975
  1990
          end;
blanchet@49501
  1991
        val goals = map5 mk_goal ks corec_ss corec_maps_rev dtors zs;
blanchet@48975
  1992
      in
blanchet@51761
  1993
        map3 (fn goal => fn unfold => fn map_cong0 =>
wenzelm@51551
  1994
          Goal.prove_sorry lthy [] [] goal
blanchet@51761
  1995
            (mk_corec_tac m corec_defs unfold map_cong0 corec_Inl_sum_thms)
traytel@49109
  1996
          |> Thm.close_derivation)
blanchet@51761
  1997
        goals dtor_unfold_thms map_cong0s
blanchet@48975
  1998
      end;
blanchet@48975
  1999
traytel@51739
  2000
    val corec_unique_mor_thm =
traytel@51739
  2001
      let
traytel@51739
  2002
        val id_fs = map2 (fn T => fn f => mk_sum_case (HOLogic.id_const T, f)) Ts unfold_fs;
traytel@51739
  2003
        val prem = HOLogic.mk_Trueprop (mk_mor corec_UNIVs corec_strs UNIVs dtors id_fs);
traytel@51739
  2004
        fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_corec corec_ss i);
traytel@51739
  2005
        val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
traytel@51739
  2006
          (map2 mk_fun_eq unfold_fs ks));
traytel@51739
  2007
      in
traytel@51739
  2008
        Goal.prove_sorry lthy [] []
traytel@51739
  2009
          (fold_rev Logic.all (corec_ss @ unfold_fs) (Logic.mk_implies (prem, unique)))
traytel@51739
  2010
          (mk_corec_unique_mor_tac corec_defs corec_Inl_sum_thms unfold_unique_mor_thm)
traytel@51739
  2011
          |> Thm.close_derivation
traytel@51739
  2012
      end;
traytel@51739
  2013
traytel@52913
  2014
    val (dtor_corec_unique_thms, dtor_corec_unique_thm) =
traytel@52913
  2015
      `split_conj_thm (split_conj_prems n
traytel@52904
  2016
        (mor_UNIV_thm RS iffD2 RS corec_unique_mor_thm)
traytel@52634
  2017
        |> Local_Defs.unfold lthy (@{thms o_sum_case o_id id_o o_assoc sum_case_o_inj(1)} @
blanchet@53270
  2018
           map_id0s @ sym_map_comps) OF replicate n @{thm arg_cong2[of _ _ _ _ sum_case, OF refl]});
traytel@51739
  2019
blanchet@48975
  2020
    val timer = time (timer "corec definitions & thms");
blanchet@48975
  2021
traytel@53105
  2022
    val (dtor_map_coinduct_thm, coinduct_params, dtor_coinduct_thm) =
blanchet@48975
  2023
      let
blanchet@48975
  2024
        val zs = Jzs1 @ Jzs2;
blanchet@48975
  2025
        val frees = phis @ zs;
blanchet@48975
  2026
traytel@51893
  2027
        val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs;
blanchet@48975
  2028
blanchet@48975
  2029
        fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2));
blanchet@48975
  2030
        val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  2031
          (map3 mk_concl phis Jzs1 Jzs2));
blanchet@48975
  2032
traytel@53105
  2033
        fun mk_rel_prem phi dtor rel Jz Jz_copy =
blanchet@48975
  2034
          let
traytel@53105
  2035
            val concl = Term.list_comb (rel, map HOLogic.eq_const passiveAs @ phis) $
traytel@51893
  2036
              (dtor $ Jz) $ (dtor $ Jz_copy);
blanchet@48975
  2037
          in
blanchet@48975
  2038
            HOLogic.mk_Trueprop
blanchet@48975
  2039
              (list_all_free [Jz, Jz_copy] (HOLogic.mk_imp (phi $ Jz $ Jz_copy, concl)))
blanchet@48975
  2040
          end;
blanchet@48975
  2041
traytel@53105
  2042
        val rel_prems = map5 mk_rel_prem phis dtors rels Jzs Jzs_copy;
traytel@51893
  2043
        val dtor_coinduct_goal =
traytel@51893
  2044
          fold_rev Logic.all frees (Logic.list_implies (rel_prems, concl));
traytel@51893
  2045
traytel@51893
  2046
        val dtor_coinduct =
traytel@51893
  2047
          Goal.prove_sorry lthy [] [] dtor_coinduct_goal
traytel@51893
  2048
            (K (mk_dtor_coinduct_tac m raw_coind_thm bis_rel_thm rel_congs))
traytel@51893
  2049
          |> Thm.close_derivation;
blanchet@48975
  2050
traytel@53105
  2051
        fun mk_prem phi dtor map_nth sets Jz Jz_copy FJz =
blanchet@48975
  2052
          let
blanchet@48975
  2053
            val xs = [Jz, Jz_copy];
blanchet@48975
  2054
blanchet@48975
  2055
            fun mk_map_conjunct nths x =
blanchet@49501
  2056
              HOLogic.mk_eq (Term.list_comb (map_nth, passive_ids @ nths) $ FJz, dtor $ x);
blanchet@48975
  2057
blanchet@48975
  2058
            fun mk_set_conjunct set phi z1 z2 =
blanchet@48975
  2059
              list_all_free [z1, z2]
blanchet@48975
  2060
                (HOLogic.mk_imp (HOLogic.mk_mem (HOLogic.mk_prod (z1, z2), set $ FJz),
traytel@53105
  2061
                  phi $ z1 $ z2));
blanchet@48975
  2062
blanchet@48975
  2063
            val concl = list_exists_free [FJz] (HOLogic.mk_conj
blanchet@48975
  2064
              (Library.foldr1 HOLogic.mk_conj (map2 mk_map_conjunct [fstsTs, sndsTs] xs),
blanchet@48975
  2065
              Library.foldr1 HOLogic.mk_conj
blanchet@48975
  2066
                (map4 mk_set_conjunct (drop m sets) phis Jzs1 Jzs2)));
blanchet@48975
  2067
          in
blanchet@48975
  2068
            fold_rev Logic.all xs (Logic.mk_implies
blanchet@48975
  2069
              (HOLogic.mk_Trueprop (Term.list_comb (phi, xs)), HOLogic.mk_Trueprop concl))
blanchet@48975
  2070
          end;
blanchet@48975
  2071
traytel@53105
  2072
        val prems = map7 mk_prem phis dtors map_FT_nths prodFT_setss Jzs Jzs_copy FJzs;
blanchet@49591
  2073
blanchet@49591
  2074
        val dtor_map_coinduct_goal = fold_rev Logic.all frees (Logic.list_implies (prems, concl));
traytel@51893
  2075
        val dtor_map_coinduct =
traytel@51893
  2076
          Goal.prove_sorry lthy [] [] dtor_map_coinduct_goal
traytel@51893
  2077
            (K (mk_dtor_map_coinduct_tac m ks raw_coind_thm bis_def))
traytel@49109
  2078
          |> Thm.close_derivation;
blanchet@48975
  2079
      in
traytel@53105
  2080
        (dtor_map_coinduct, rev (Term.add_tfrees dtor_map_coinduct_goal []), dtor_coinduct)
blanchet@48975
  2081
      end;
blanchet@48975
  2082
blanchet@48975
  2083
    val timer = time (timer "coinduction");
blanchet@48975
  2084
blanchet@53270
  2085
    fun mk_dtor_map_DEADID_thm dtor_inject map_id0 =
blanchet@53270
  2086
      trans OF [iffD2 OF [dtor_inject, id_apply], map_id0 RS sym];
traytel@51925
  2087
traytel@51925
  2088
    fun mk_dtor_Jrel_DEADID_thm dtor_inject bnf =
traytel@51917
  2089
      trans OF [rel_eq_of_bnf bnf RS @{thm predicate2_eqD}, dtor_inject] RS sym;
traytel@51917
  2090
traytel@51925
  2091
    val JphiTs = map2 mk_pred2T passiveAs passiveBs;
traytel@51925
  2092
    val prodTsTs' = map2 (curry HOLogic.mk_prodT) Ts Ts';
traytel@51925
  2093
    val fstsTsTs' = map fst_const prodTsTs';
traytel@51925
  2094
    val sndsTsTs' = map snd_const prodTsTs';
traytel@52731
  2095
    val activephiTs = map2 mk_pred2T activeAs activeBs;
traytel@51925
  2096
    val activeJphiTs = map2 mk_pred2T Ts Ts';
traytel@52731
  2097
    val (((Jphis, activephis), activeJphis), names_lthy) = names_lthy
traytel@51925
  2098
      |> mk_Frees "R" JphiTs
traytel@52731
  2099
      ||>> mk_Frees "S" activephiTs
traytel@51925
  2100
      ||>> mk_Frees "JR" activeJphiTs;
traytel@51925
  2101
    val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
traytel@51925
  2102
    val in_rels = map in_rel_of_bnf bnfs;
traytel@51925
  2103
blanchet@48975
  2104
    (*register new codatatypes as BNFs*)
traytel@52913
  2105
    val (timer, Jbnfs, (folded_dtor_map_o_thms, folded_dtor_map_thms), folded_dtor_set_thmss',
traytel@52913
  2106
      dtor_set_induct_thms, dtor_Jrel_thms, lthy) =
blanchet@49585
  2107
      if m = 0 then
traytel@52913
  2108
        (timer, replicate n DEADID_bnf,
blanchet@53285
  2109
        map_split (`(mk_pointfree lthy)) (map2 mk_dtor_map_DEADID_thm dtor_inject_thms map_ids),
traytel@51925
  2110
        replicate n [], [], map2 mk_dtor_Jrel_DEADID_thm dtor_inject_thms bnfs, lthy)
blanchet@49585
  2111
      else let
blanchet@48975
  2112
        val fTs = map2 (curry op -->) passiveAs passiveBs;
blanchet@48975
  2113
        val gTs = map2 (curry op -->) passiveBs passiveCs;
blanchet@48975
  2114
        val f1Ts = map2 (curry op -->) passiveAs passiveYs;
blanchet@48975
  2115
        val f2Ts = map2 (curry op -->) passiveBs passiveYs;
blanchet@48975
  2116
        val p1Ts = map2 (curry op -->) passiveXs passiveAs;
blanchet@48975
  2117
        val p2Ts = map2 (curry op -->) passiveXs passiveBs;
blanchet@48975
  2118
        val pTs = map2 (curry op -->) passiveXs passiveCs;
blanchet@48975
  2119
        val uTs = map2 (curry op -->) Ts Ts';
blanchet@48975
  2120
        val B1Ts = map HOLogic.mk_setT passiveAs;
blanchet@48975
  2121
        val B2Ts = map HOLogic.mk_setT passiveBs;
blanchet@48975
  2122
        val AXTs = map HOLogic.mk_setT passiveXs;
blanchet@48975
  2123
        val XTs = mk_Ts passiveXs;
blanchet@48975
  2124
        val YTs = mk_Ts passiveYs;
blanchet@48975
  2125
traytel@51925
  2126
        val ((((((((((((((((((fs, fs'), fs_copy), gs), us),
traytel@51925
  2127
          (Jys, Jys')), (Jys_copy, Jys'_copy)), dtor_set_induct_phiss),
blanchet@49308
  2128
          B1s), B2s), AXs), f1s), f2s), p1s), p2s), ps), (ys, ys')), (ys_copy, ys'_copy)),
traytel@49104
  2129
          names_lthy) = names_lthy
blanchet@48975
  2130
          |> mk_Frees' "f" fTs
blanchet@49308
  2131
          ||>> mk_Frees "f" fTs
blanchet@49308
  2132
          ||>> mk_Frees "g" gTs
blanchet@48975
  2133
          ||>> mk_Frees "u" uTs
blanchet@48975
  2134
          ||>> mk_Frees' "b" Ts'
blanchet@48975
  2135
          ||>> mk_Frees' "b" Ts'
blanchet@49592
  2136
          ||>> mk_Freess "P" (map (fn A => map (mk_pred2T A) Ts) passiveAs)
blanchet@48975
  2137
          ||>> mk_Frees "B1" B1Ts
blanchet@48975
  2138
          ||>> mk_Frees "B2" B2Ts
blanchet@48975
  2139
          ||>> mk_Frees "A" AXTs
blanchet@48975
  2140
          ||>> mk_Frees "f1" f1Ts
blanchet@48975
  2141
          ||>> mk_Frees "f2" f2Ts
blanchet@48975
  2142
          ||>> mk_Frees "p1" p1Ts
blanchet@48975
  2143
          ||>> mk_Frees "p2" p2Ts
blanchet@48975
  2144
          ||>> mk_Frees "p" pTs
traytel@49104
  2145
          ||>> mk_Frees' "y" passiveAs
blanchet@48975
  2146
          ||>> mk_Frees' "y" passiveAs;
blanchet@48975
  2147
blanchet@48975
  2148
        val map_FTFT's = map2 (fn Ds =>
blanchet@48975
  2149
          mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
blanchet@48975
  2150
blanchet@48975
  2151
        fun mk_maps ATs BTs Ts mk_T =
blanchet@48975
  2152
          map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ map mk_T Ts)) Dss bnfs;
blanchet@48975
  2153
        fun mk_Fmap mk_const fs Ts Fmap = Term.list_comb (Fmap, fs @ map mk_const Ts);
blanchet@49501
  2154
        fun mk_map mk_const mk_T Ts fs Ts' dtors mk_maps =
blanchet@49504
  2155
          mk_unfold Ts' (map2 (fn dtor => fn Fmap =>
blanchet@49501
  2156
            HOLogic.mk_comp (mk_Fmap mk_const fs Ts Fmap, dtor)) dtors (mk_maps Ts mk_T));
blanchet@48975
  2157
        val mk_map_id = mk_map HOLogic.id_const I;
blanchet@48975
  2158
        val mk_mapsAB = mk_maps passiveAs passiveBs;
blanchet@48975
  2159
        val mk_mapsBC = mk_maps passiveBs passiveCs;
blanchet@48975
  2160
        val mk_mapsAC = mk_maps passiveAs passiveCs;
blanchet@48975
  2161
        val mk_mapsAY = mk_maps passiveAs passiveYs;
blanchet@48975
  2162
        val mk_mapsBY = mk_maps passiveBs passiveYs;
blanchet@48975
  2163
        val mk_mapsXA = mk_maps passiveXs passiveAs;
blanchet@48975
  2164
        val mk_mapsXB = mk_maps passiveXs passiveBs;
blanchet@48975
  2165
        val mk_mapsXC = mk_maps passiveXs passiveCs;
blanchet@49501
  2166
        val fs_maps = map (mk_map_id Ts fs Ts' dtors mk_mapsAB) ks;
blanchet@49501
  2167
        val fs_copy_maps = map (mk_map_id Ts fs_copy Ts' dtors mk_mapsAB) ks;
blanchet@49501
  2168
        val gs_maps = map (mk_map_id Ts' gs Ts'' dtor's mk_mapsBC) ks;
blanchet@48975
  2169
        val fgs_maps =
blanchet@49501
  2170
          map (mk_map_id Ts (map2 (curry HOLogic.mk_comp) gs fs) Ts'' dtors mk_mapsAC) ks;
blanchet@49501
  2171
        val Xdtors = mk_dtors passiveXs;
blanchet@48975
  2172
        val UNIV's = map HOLogic.mk_UNIV Ts';
blanchet@48975
  2173
        val CUNIVs = map HOLogic.mk_UNIV passiveCs;
blanchet@48975
  2174
        val UNIV''s = map HOLogic.mk_UNIV Ts'';
blanchet@49501
  2175
        val dtor''s = mk_dtors passiveCs;
blanchet@49501
  2176
        val f1s_maps = map (mk_map_id Ts f1s YTs dtors mk_mapsAY) ks;
blanchet@49501
  2177
        val f2s_maps = map (mk_map_id Ts' f2s YTs dtor's mk_mapsBY) ks;
blanchet@49501
  2178
        val pid_maps = map (mk_map_id XTs ps Ts'' Xdtors mk_mapsXC) ks;
blanchet@48975
  2179
        val pfst_Fmaps =
traytel@51925
  2180
          map (mk_Fmap fst_const p1s prodTsTs') (mk_mapsXA prodTsTs' (fst o HOLogic.dest_prodT));
blanchet@48975
  2181
        val psnd_Fmaps =
traytel@51925
  2182
          map (mk_Fmap snd_const p2s prodTsTs') (mk_mapsXB prodTsTs' (snd o HOLogic.dest_prodT));
traytel@51925
  2183
        val p1id_Fmaps = map (mk_Fmap HOLogic.id_const p1s prodTsTs') (mk_mapsXA prodTsTs' I);
traytel@51925
  2184
        val p2id_Fmaps = map (mk_Fmap HOLogic.id_const p2s prodTsTs') (mk_mapsXB prodTsTs' I);
traytel@51925
  2185
        val pid_Fmaps = map (mk_Fmap HOLogic.id_const ps prodTsTs') (mk_mapsXC prodTsTs' I);
blanchet@48975
  2186
blanchet@49541
  2187
        val (dtor_map_thms, map_thms) =
blanchet@48975
  2188
          let
blanchet@49501
  2189
            fun mk_goal fs_map map dtor dtor' = fold_rev Logic.all fs
blanchet@49501
  2190
              (mk_Trueprop_eq (HOLogic.mk_comp (dtor', fs_map),
blanchet@49501
  2191
                HOLogic.mk_comp (Term.list_comb (map, fs @ fs_maps), dtor)));
blanchet@49501
  2192
            val goals = map4 mk_goal fs_maps map_FTFT's dtors dtor's;
blanchet@48975
  2193
            val cTs = map (SOME o certifyT lthy) FTs';
traytel@49109
  2194
            val maps =
blanchet@53288
  2195
              map5 (fn goal => fn cT => fn unfold => fn map_comp => fn map_cong0 =>
wenzelm@51551
  2196
                Goal.prove_sorry lthy [] [] goal
blanchet@53288
  2197
                  (K (mk_map_tac m n cT unfold map_comp map_cong0))
traytel@49109
  2198
                |> Thm.close_derivation)
blanchet@53288
  2199
              goals cTs dtor_unfold_thms map_comps map_cong0s;
blanchet@48975
  2200
          in
traytel@52659
  2201
            map_split (fn thm => (thm RS @{thm comp_eq_dest}, thm)) maps
blanchet@48975
  2202
          end;
blanchet@48975
  2203
blanchet@53287
  2204
        val map_comp0_thms =
blanchet@48975
  2205
          let
blanchet@48975
  2206
            val goal = fold_rev Logic.all (fs @ gs)
blanchet@48975
  2207
              (HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  2208
                (map3 (fn fmap => fn gmap => fn fgmap =>
blanchet@48975
  2209
                   HOLogic.mk_eq (HOLogic.mk_comp (gmap, fmap), fgmap))
blanchet@48975
  2210
                fs_maps gs_maps fgs_maps)))
blanchet@48975
  2211
          in
wenzelm@51551
  2212
            split_conj_thm (Goal.prove_sorry lthy [] [] goal
blanchet@53287
  2213
              (K (mk_map_comp0_tac m n map_thms map_comp0s map_cong0s dtor_unfold_unique_thm))
traytel@49109
  2214
              |> Thm.close_derivation)
blanchet@48975
  2215
          end;
blanchet@48975
  2216
blanchet@49543
  2217
        val dtor_map_unique_thm =
blanchet@48975
  2218
          let
blanchet@49501
  2219
            fun mk_prem u map dtor dtor' =
blanchet@49501
  2220
              mk_Trueprop_eq (HOLogic.mk_comp (dtor', u),
blanchet@49501
  2221
                HOLogic.mk_comp (Term.list_comb (map, fs @ us), dtor));
blanchet@49501
  2222
            val prems = map4 mk_prem us map_FTFT's dtors dtor's;
blanchet@48975
  2223
            val goal =
blanchet@48975
  2224
              HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
blanchet@48975
  2225
                (map2 (curry HOLogic.mk_eq) us fs_maps));
blanchet@49308
  2226
          in
wenzelm@51551
  2227
            Goal.prove_sorry lthy [] []
blanchet@48975
  2228
              (fold_rev Logic.all (us @ fs) (Logic.list_implies (prems, goal)))
traytel@52911
  2229
              (mk_dtor_map_unique_tac dtor_unfold_unique_thm sym_map_comps)
blanchet@49308
  2230
              |> Thm.close_derivation
blanchet@48975
  2231
          end;
blanchet@48975
  2232
blanchet@48975
  2233
        val timer = time (timer "map functions for the new codatatypes");
blanchet@48975
  2234
traytel@52545
  2235
        val bd = mk_cexp sbd sbd;
blanchet@48975
  2236
blanchet@48975
  2237
        val timer = time (timer "bounds for the new codatatypes");
blanchet@48975
  2238
blanchet@49501
  2239
        val setss_by_bnf = map (fn i => map2 (mk_hset dtors i) ls passiveAs) ks;
blanchet@49501
  2240
        val setss_by_bnf' = map (fn i => map2 (mk_hset dtor's i) ls passiveBs) ks;
blanchet@48975
  2241
        val setss_by_range = transpose setss_by_bnf;
blanchet@48975
  2242
blanchet@49585
  2243
        val dtor_set_thmss =
blanchet@48975
  2244
          let
blanchet@49501
  2245
            fun mk_simp_goal relate pas_set act_sets sets dtor z set =
blanchet@49501
  2246
              relate (set $ z, mk_union (pas_set $ (dtor $ z),
blanchet@48975
  2247
                 Library.foldl1 mk_union
blanchet@49501
  2248
                   (map2 (fn X => mk_UNION (X $ (dtor $ z))) act_sets sets)));
blanchet@48975
  2249
            fun mk_goals eq =
blanchet@48975
  2250
              map2 (fn i => fn sets =>
blanchet@48975
  2251
                map4 (fn Fsets =>
blanchet@48975
  2252
                  mk_simp_goal eq (nth Fsets (i - 1)) (drop m Fsets) sets)
blanchet@49501
  2253
                FTs_setss dtors Jzs sets)
blanchet@48975
  2254
              ls setss_by_range;
blanchet@48975
  2255
blanchet@48975
  2256
            val le_goals = map
blanchet@48975
  2257
              (fold_rev Logic.all Jzs o HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj)
traytel@51893
  2258
              (mk_goals (uncurry mk_leq));
blanchet@48975
  2259
            val set_le_thmss = map split_conj_thm
blanchet@48975
  2260
              (map4 (fn goal => fn hset_minimal => fn set_hsets => fn set_hset_hsetss =>
wenzelm@51551
  2261
                Goal.prove_sorry lthy [] [] goal
traytel@49109
  2262
                  (K (mk_set_le_tac n hset_minimal set_hsets set_hset_hsetss))
traytel@49109
  2263
                |> Thm.close_derivation)
blanchet@48975
  2264
              le_goals hset_minimal_thms set_hset_thmss' set_hset_hset_thmsss');
blanchet@48975
  2265