src/HOLCF/Tools/Domain/domain_theorems.ML
author huffman
Sat Feb 27 18:45:06 2010 -0800 (2010-02-27)
changeset 35464 2ae3362ba8ee
parent 35462 f5461b02d754
child 35466 9fcfd5763181
permissions -rw-r--r--
removed dead code
     1 (*  Title:      HOLCF/Tools/Domain/domain_theorems.ML
     2     Author:     David von Oheimb
     3     Author:     Brian Huffman
     4 
     5 Proof generator for domain command.
     6 *)
     7 
     8 val HOLCF_ss = @{simpset};
     9 
    10 signature DOMAIN_THEOREMS =
    11 sig
    12   val theorems:
    13     Domain_Library.eq * Domain_Library.eq list
    14     -> typ * (binding * (bool * binding option * typ) list * mixfix) list
    15     -> theory -> thm list * theory;
    16 
    17   val comp_theorems: bstring * Domain_Library.eq list -> theory -> thm list * theory;
    18   val quiet_mode: bool Unsynchronized.ref;
    19   val trace_domain: bool Unsynchronized.ref;
    20 end;
    21 
    22 structure Domain_Theorems :> DOMAIN_THEOREMS =
    23 struct
    24 
    25 val quiet_mode = Unsynchronized.ref false;
    26 val trace_domain = Unsynchronized.ref false;
    27 
    28 fun message s = if !quiet_mode then () else writeln s;
    29 fun trace s = if !trace_domain then tracing s else ();
    30 
    31 val adm_impl_admw = @{thm adm_impl_admw};
    32 val adm_all = @{thm adm_all};
    33 val adm_conj = @{thm adm_conj};
    34 val adm_subst = @{thm adm_subst};
    35 val ch2ch_fst = @{thm ch2ch_fst};
    36 val ch2ch_snd = @{thm ch2ch_snd};
    37 val ch2ch_Rep_CFunL = @{thm ch2ch_Rep_CFunL};
    38 val ch2ch_Rep_CFunR = @{thm ch2ch_Rep_CFunR};
    39 val chain_iterate = @{thm chain_iterate};
    40 val contlub_cfun_fun = @{thm contlub_cfun_fun};
    41 val contlub_fst = @{thm contlub_fst};
    42 val contlub_snd = @{thm contlub_snd};
    43 val contlubE = @{thm contlubE};
    44 val cont_const = @{thm cont_const};
    45 val cont_id = @{thm cont_id};
    46 val cont2cont_fst = @{thm cont2cont_fst};
    47 val cont2cont_snd = @{thm cont2cont_snd};
    48 val cont2cont_Rep_CFun = @{thm cont2cont_Rep_CFun};
    49 val fix_def2 = @{thm fix_def2};
    50 val lub_equal = @{thm lub_equal};
    51 val retraction_strict = @{thm retraction_strict};
    52 val wfix_ind = @{thm wfix_ind};
    53 val iso_intro = @{thm iso.intro};
    54 
    55 open Domain_Library;
    56 infixr 0 ===>;
    57 infixr 0 ==>;
    58 infix 0 == ; 
    59 infix 1 ===;
    60 infix 1 ~= ;
    61 infix 1 <<;
    62 infix 1 ~<<;
    63 infix 9 `   ;
    64 infix 9 `% ;
    65 infix 9 `%%;
    66 infixr 9 oo;
    67 
    68 (* ----- general proof facilities ------------------------------------------- *)
    69 
    70 fun legacy_infer_term thy t =
    71   let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)
    72   in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;
    73 
    74 fun pg'' thy defs t tacs =
    75   let
    76     val t' = legacy_infer_term thy t;
    77     val asms = Logic.strip_imp_prems t';
    78     val prop = Logic.strip_imp_concl t';
    79     fun tac {prems, context} =
    80       rewrite_goals_tac defs THEN
    81       EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
    82   in Goal.prove_global thy [] asms prop tac end;
    83 
    84 fun pg' thy defs t tacsf =
    85   let
    86     fun tacs {prems, context} =
    87       if null prems then tacsf context
    88       else cut_facts_tac prems 1 :: tacsf context;
    89   in pg'' thy defs t tacs end;
    90 
    91 (* FIXME!!!!!!!!! *)
    92 (* We should NEVER re-parse variable names as strings! *)
    93 (* The names can conflict with existing constants or other syntax! *)
    94 fun case_UU_tac ctxt rews i v =
    95   InductTacs.case_tac ctxt (v^"=UU") i THEN
    96   asm_simp_tac (HOLCF_ss addsimps rews) i;
    97 
    98 val chain_tac =
    99   REPEAT_DETERM o resolve_tac 
   100     [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL, ch2ch_fst, ch2ch_snd];
   101 
   102 (* ----- general proofs ----------------------------------------------------- *)
   103 
   104 val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}
   105 
   106 fun theorems
   107     (((dname, _), cons) : eq, eqs : eq list)
   108     (dom_eqn : typ * (binding * (bool * binding option * typ) list * mixfix) list)
   109     (thy : theory) =
   110 let
   111 
   112 val _ = message ("Proving isomorphism properties of domain "^dname^" ...");
   113 val map_tab = Domain_Isomorphism.get_map_tab thy;
   114 
   115 
   116 (* ----- getting the axioms and definitions --------------------------------- *)
   117 
   118 local
   119   fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
   120 in
   121   val ax_abs_iso  = ga "abs_iso"  dname;
   122   val ax_rep_iso  = ga "rep_iso"  dname;
   123   val ax_when_def = ga "when_def" dname;
   124   fun get_def mk_name (con, _, _) = ga (mk_name con^"_def") dname;
   125   val axs_pat_def = map (get_def pat_name) cons;
   126   val ax_copy_def = ga "copy_def" dname;
   127 end; (* local *)
   128 
   129 (* ----- define constructors ------------------------------------------------ *)
   130 
   131 val lhsT = fst dom_eqn;
   132 
   133 val rhsT =
   134   let
   135     fun mk_arg_typ (lazy, sel, T) = if lazy then mk_uT T else T;
   136     fun mk_con_typ (bind, args, mx) =
   137         if null args then oneT else foldr1 mk_sprodT (map mk_arg_typ args);
   138     fun mk_eq_typ (_, cons) = foldr1 mk_ssumT (map mk_con_typ cons);
   139   in
   140     mk_eq_typ dom_eqn
   141   end;
   142 
   143 val rep_const = Const(dname^"_rep", lhsT ->> rhsT);
   144 
   145 val abs_const = Const(dname^"_abs", rhsT ->> lhsT);
   146 
   147 val (result, thy) =
   148   Domain_Constructors.add_domain_constructors
   149     (Long_Name.base_name dname) dom_eqn
   150     (rep_const, abs_const) (ax_rep_iso, ax_abs_iso) ax_when_def thy;
   151 
   152 val con_appls = #con_betas result;
   153 val {exhaust, casedist, ...} = result;
   154 val {con_compacts, con_rews, inverts, injects, dist_les, dist_eqs, ...} = result;
   155 val {sel_rews, ...} = result;
   156 val when_rews = #cases result;
   157 val when_strict = hd when_rews;
   158 val dis_rews = #dis_rews result;
   159 val axs_mat_def = #match_rews result;
   160 
   161 (* ----- theorems concerning the isomorphism -------------------------------- *)
   162 
   163 val pg = pg' thy;
   164 
   165 val dc_copy = %%:(dname^"_copy");
   166 
   167 val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
   168 val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
   169 val iso_rews = map Drule.export_without_context [ax_abs_iso, ax_rep_iso, abs_strict, rep_strict];
   170 
   171 (* ----- theorems concerning the constructors, discriminators and selectors - *)
   172 
   173 local
   174   fun mat_strict (con, _, _) =
   175     let
   176       val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU);
   177       val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1];
   178     in pg axs_mat_def goal (K tacs) end;
   179 
   180   val _ = trace " Proving mat_stricts...";
   181   val mat_stricts = map mat_strict cons;
   182 
   183   fun one_mat c (con, _, args) =
   184     let
   185       val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs";
   186       val rhs =
   187         if con = c
   188         then list_ccomb (%:"rhs", map %# args)
   189         else mk_fail;
   190       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   191       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   192     in pg axs_mat_def goal (K tacs) end;
   193 
   194   val _ = trace " Proving mat_apps...";
   195   val mat_apps =
   196     maps (fn (c,_,_) => map (one_mat c) cons) cons;
   197 in
   198   val mat_rews = mat_stricts @ mat_apps;
   199 end;
   200 
   201 local
   202   fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
   203 
   204   fun pat_lhs (con,_,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
   205 
   206   fun pat_rhs (con,_,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
   207     | pat_rhs (con,_,args) =
   208         (mk_branch (mk_ctuple_pat (ps args)))
   209           `(%:"rhs")`(mk_ctuple (map %# args));
   210 
   211   fun pat_strict c =
   212     let
   213       val axs = @{thm branch_def} :: axs_pat_def;
   214       val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
   215       val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
   216     in pg axs goal (K tacs) end;
   217 
   218   fun pat_app c (con, _, args) =
   219     let
   220       val axs = @{thm branch_def} :: axs_pat_def;
   221       val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
   222       val rhs = if con = first c then pat_rhs c else mk_fail;
   223       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   224       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   225     in pg axs goal (K tacs) end;
   226 
   227   val _ = trace " Proving pat_stricts...";
   228   val pat_stricts = map pat_strict cons;
   229   val _ = trace " Proving pat_apps...";
   230   val pat_apps = maps (fn c => map (pat_app c) cons) cons;
   231 in
   232   val pat_rews = pat_stricts @ pat_apps;
   233 end;
   234 
   235 (* ----- theorems concerning one induction step ----------------------------- *)
   236 
   237 val copy_strict =
   238   let
   239     val _ = trace " Proving copy_strict...";
   240     val goal = mk_trp (strict (dc_copy `% "f"));
   241     val rules = [abs_strict, rep_strict] @ @{thms domain_map_stricts};
   242     val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
   243   in
   244     SOME (pg [ax_copy_def] goal (K tacs))
   245     handle
   246       THM (s, _, _) => (trace s; NONE)
   247     | ERROR s => (trace s; NONE)
   248   end;
   249 
   250 local
   251   fun copy_app (con, _, args) =
   252     let
   253       val lhs = dc_copy`%"f"`(con_app con args);
   254       fun one_rhs arg =
   255           if Datatype_Aux.is_rec_type (dtyp_of arg)
   256           then Domain_Axioms.copy_of_dtyp map_tab
   257                  (proj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
   258           else (%# arg);
   259       val rhs = con_app2 con one_rhs args;
   260       fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
   261       fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
   262       fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
   263       val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
   264       val args' = filter_out (fn a => is_rec a orelse is_lazy a) args;
   265       val stricts = abs_strict :: rep_strict :: @{thms domain_map_stricts};
   266                         (* FIXME! case_UU_tac *)
   267       fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
   268       val rules = [ax_abs_iso] @ @{thms domain_map_simps};
   269       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
   270     in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
   271 in
   272   val _ = trace " Proving copy_apps...";
   273   val copy_apps = map copy_app cons;
   274 end;
   275 
   276 local
   277   fun one_strict (con, _, args) = 
   278     let
   279       val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
   280       val rews = the_list copy_strict @ copy_apps @ con_rews;
   281                         (* FIXME! case_UU_tac *)
   282       fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
   283         [asm_simp_tac (HOLCF_ss addsimps rews) 1];
   284     in
   285       SOME (pg [] goal tacs)
   286       handle
   287         THM (s, _, _) => (trace s; NONE)
   288       | ERROR s => (trace s; NONE)
   289     end;
   290 
   291   fun has_nonlazy_rec (_, _, args) = exists is_nonlazy_rec args;
   292 in
   293   val _ = trace " Proving copy_stricts...";
   294   val copy_stricts = map_filter one_strict (filter has_nonlazy_rec cons);
   295 end;
   296 
   297 val copy_rews = the_list copy_strict @ copy_apps @ copy_stricts;
   298 
   299 in
   300   thy
   301     |> Sign.add_path (Long_Name.base_name dname)
   302     |> snd o PureThy.add_thmss [
   303         ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
   304         ((Binding.name "exhaust"   , [exhaust]   ), []),
   305         ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
   306         ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
   307         ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
   308         ((Binding.name "con_rews"  , con_rews    ),
   309          [Simplifier.simp_add, Fixrec.fixrec_simp_add]),
   310         ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
   311         ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
   312         ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
   313         ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
   314         ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
   315         ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
   316         ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
   317         ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
   318         ((Binding.name "match_rews", mat_rews    ),
   319          [Simplifier.simp_add, Fixrec.fixrec_simp_add])]
   320     |> Sign.parent_path
   321     |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
   322         pat_rews @ dist_les @ dist_eqs @ copy_rews)
   323 end; (* let *)
   324 
   325 fun comp_theorems (comp_dnam, eqs: eq list) thy =
   326 let
   327 val global_ctxt = ProofContext.init thy;
   328 val map_tab = Domain_Isomorphism.get_map_tab thy;
   329 
   330 val dnames = map (fst o fst) eqs;
   331 val conss  = map  snd        eqs;
   332 val comp_dname = Sign.full_bname thy comp_dnam;
   333 
   334 val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
   335 val pg = pg' thy;
   336 
   337 (* ----- getting the composite axiom and definitions ------------------------ *)
   338 
   339 local
   340   fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
   341 in
   342   val axs_reach      = map (ga "reach"     ) dnames;
   343   val axs_take_def   = map (ga "take_def"  ) dnames;
   344   val axs_finite_def = map (ga "finite_def") dnames;
   345   val ax_copy2_def   =      ga "copy_def"  comp_dnam;
   346 (* TEMPORARILY DISABLED
   347   val ax_bisim_def   =      ga "bisim_def" comp_dnam;
   348 TEMPORARILY DISABLED *)
   349 end;
   350 
   351 local
   352   fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
   353   fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
   354 in
   355   val cases = map (gt  "casedist" ) dnames;
   356   val con_rews  = maps (gts "con_rews" ) dnames;
   357   val copy_rews = maps (gts "copy_rews") dnames;
   358 end;
   359 
   360 fun dc_take dn = %%:(dn^"_take");
   361 val x_name = idx_name dnames "x"; 
   362 val P_name = idx_name dnames "P";
   363 val n_eqs = length eqs;
   364 
   365 (* ----- theorems concerning finite approximation and finite induction ------ *)
   366 
   367 local
   368   val iterate_Cprod_ss = global_simpset_of @{theory Fix};
   369   val copy_con_rews  = copy_rews @ con_rews;
   370   val copy_take_defs =
   371     (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
   372   val _ = trace " Proving take_stricts...";
   373   fun one_take_strict ((dn, args), _) =
   374     let
   375       val goal = mk_trp (strict (dc_take dn $ %:"n"));
   376       val rules = [
   377         @{thm monofun_fst [THEN monofunE]},
   378         @{thm monofun_snd [THEN monofunE]}];
   379       val tacs = [
   380         rtac @{thm UU_I} 1,
   381         rtac @{thm below_eq_trans} 1,
   382         resolve_tac axs_reach 2,
   383         rtac @{thm monofun_cfun_fun} 1,
   384         REPEAT (resolve_tac rules 1),
   385         rtac @{thm iterate_below_fix} 1];
   386     in pg axs_take_def goal (K tacs) end;
   387   val take_stricts = map one_take_strict eqs;
   388   fun take_0 n dn =
   389     let
   390       val goal = mk_trp ((dc_take dn $ @{term "0::nat"}) `% x_name n === UU);
   391     in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
   392   val take_0s = mapn take_0 1 dnames;
   393   val _ = trace " Proving take_apps...";
   394   fun one_take_app dn (con, _, args) =
   395     let
   396       fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
   397       fun one_rhs arg =
   398           if Datatype_Aux.is_rec_type (dtyp_of arg)
   399           then Domain_Axioms.copy_of_dtyp map_tab
   400                  mk_take (dtyp_of arg) ` (%# arg)
   401           else (%# arg);
   402       val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
   403       val rhs = con_app2 con one_rhs args;
   404       fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
   405       fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
   406       fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
   407       val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
   408       val tacs = [asm_simp_tac (HOLCF_ss addsimps copy_con_rews) 1];
   409     in pg copy_take_defs goal (K tacs) end;
   410   fun one_take_apps ((dn, _), cons) = map (one_take_app dn) cons;
   411   val take_apps = maps one_take_apps eqs;
   412 in
   413   val take_rews = map Drule.export_without_context
   414     (take_stricts @ take_0s @ take_apps);
   415 end; (* local *)
   416 
   417 local
   418   fun one_con p (con, _, args) =
   419     let
   420       val P_names = map P_name (1 upto (length dnames));
   421       val vns = Name.variant_list P_names (map vname args);
   422       val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns));
   423       fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
   424       val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
   425       val t2 = lift ind_hyp (filter is_rec args, t1);
   426       val t3 = lift_defined (bound_arg vns) (nonlazy_vns, t2);
   427     in Library.foldr mk_All (vns, t3) end;
   428 
   429   fun one_eq ((p, cons), concl) =
   430     mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
   431 
   432   fun ind_term concf = Library.foldr one_eq
   433     (mapn (fn n => fn x => (P_name n, x)) 1 conss,
   434      mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
   435   val take_ss = HOL_ss addsimps take_rews;
   436   fun quant_tac ctxt i = EVERY
   437     (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
   438 
   439   fun ind_prems_tac prems = EVERY
   440     (maps (fn cons =>
   441       (resolve_tac prems 1 ::
   442         maps (fn (_,_,args) => 
   443           resolve_tac prems 1 ::
   444           map (K(atac 1)) (nonlazy args) @
   445           map (K(atac 1)) (filter is_rec args))
   446         cons))
   447       conss);
   448   local 
   449     (* check whether every/exists constructor of the n-th part of the equation:
   450        it has a possibly indirectly recursive argument that isn't/is possibly 
   451        indirectly lazy *)
   452     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
   453           is_rec arg andalso not(rec_of arg mem ns) andalso
   454           ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
   455             rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
   456               (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
   457           ) o third) cons;
   458     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
   459     fun warn (n,cons) =
   460       if all_rec_to [] false (n,cons)
   461       then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
   462       else false;
   463     fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
   464 
   465   in
   466     val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
   467     val is_emptys = map warn n__eqs;
   468     val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   469   end;
   470 in (* local *)
   471   val _ = trace " Proving finite_ind...";
   472   val finite_ind =
   473     let
   474       fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
   475       val goal = ind_term concf;
   476 
   477       fun tacf {prems, context} =
   478         let
   479           val tacs1 = [
   480             quant_tac context 1,
   481             simp_tac HOL_ss 1,
   482             InductTacs.induct_tac context [[SOME "n"]] 1,
   483             simp_tac (take_ss addsimps prems) 1,
   484             TRY (safe_tac HOL_cs)];
   485           fun arg_tac arg =
   486                         (* FIXME! case_UU_tac *)
   487             case_UU_tac context (prems @ con_rews) 1
   488               (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
   489           fun con_tacs (con, _, args) = 
   490             asm_simp_tac take_ss 1 ::
   491             map arg_tac (filter is_nonlazy_rec args) @
   492             [resolve_tac prems 1] @
   493             map (K (atac 1)) (nonlazy args) @
   494             map (K (etac spec 1)) (filter is_rec args);
   495           fun cases_tacs (cons, cases) =
   496             res_inst_tac context [(("x", 0), "x")] cases 1 ::
   497             asm_simp_tac (take_ss addsimps prems) 1 ::
   498             maps con_tacs cons;
   499         in
   500           tacs1 @ maps cases_tacs (conss ~~ cases)
   501         end;
   502     in pg'' thy [] goal tacf
   503        handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
   504     end;
   505 
   506   val _ = trace " Proving take_lemmas...";
   507   val take_lemmas =
   508     let
   509       fun take_lemma n (dn, ax_reach) =
   510         let
   511           val lhs = dc_take dn $ Bound 0 `%(x_name n);
   512           val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
   513           val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
   514           val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
   515           val rules = [contlub_fst RS contlubE RS ssubst,
   516                        contlub_snd RS contlubE RS ssubst];
   517           fun tacf {prems, context} = [
   518             res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
   519             res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
   520             stac fix_def2 1,
   521             REPEAT (CHANGED
   522               (resolve_tac rules 1 THEN chain_tac 1)),
   523             stac contlub_cfun_fun 1,
   524             stac contlub_cfun_fun 2,
   525             rtac lub_equal 3,
   526             chain_tac 1,
   527             rtac allI 1,
   528             resolve_tac prems 1];
   529         in pg'' thy axs_take_def goal tacf end;
   530     in mapn take_lemma 1 (dnames ~~ axs_reach) end;
   531 
   532 (* ----- theorems concerning finiteness and induction ----------------------- *)
   533 
   534   val _ = trace " Proving finites, ind...";
   535   val (finites, ind) =
   536   (
   537     if is_finite
   538     then (* finite case *)
   539       let 
   540         fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
   541         fun dname_lemma dn =
   542           let
   543             val prem1 = mk_trp (defined (%:"x"));
   544             val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
   545             val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
   546             val concl = mk_trp (take_enough dn);
   547             val goal = prem1 ===> prem2 ===> concl;
   548             val tacs = [
   549               etac disjE 1,
   550               etac notE 1,
   551               resolve_tac take_lemmas 1,
   552               asm_simp_tac take_ss 1,
   553               atac 1];
   554           in pg [] goal (K tacs) end;
   555         val _ = trace " Proving finite_lemmas1a";
   556         val finite_lemmas1a = map dname_lemma dnames;
   557  
   558         val _ = trace " Proving finite_lemma1b";
   559         val finite_lemma1b =
   560           let
   561             fun mk_eqn n ((dn, args), _) =
   562               let
   563                 val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
   564                 val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
   565               in
   566                 mk_constrainall
   567                   (x_name n, Type (dn,args), mk_disj (disj1, disj2))
   568               end;
   569             val goal =
   570               mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
   571             fun arg_tacs ctxt vn = [
   572               eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
   573               etac disjE 1,
   574               asm_simp_tac (HOL_ss addsimps con_rews) 1,
   575               asm_simp_tac take_ss 1];
   576             fun con_tacs ctxt (con, _, args) =
   577               asm_simp_tac take_ss 1 ::
   578               maps (arg_tacs ctxt) (nonlazy_rec args);
   579             fun foo_tacs ctxt n (cons, cases) =
   580               simp_tac take_ss 1 ::
   581               rtac allI 1 ::
   582               res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
   583               asm_simp_tac take_ss 1 ::
   584               maps (con_tacs ctxt) cons;
   585             fun tacs ctxt =
   586               rtac allI 1 ::
   587               InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
   588               simp_tac take_ss 1 ::
   589               TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
   590               flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
   591           in pg [] goal tacs end;
   592 
   593         fun one_finite (dn, l1b) =
   594           let
   595             val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
   596             fun tacs ctxt = [
   597                         (* FIXME! case_UU_tac *)
   598               case_UU_tac ctxt take_rews 1 "x",
   599               eresolve_tac finite_lemmas1a 1,
   600               step_tac HOL_cs 1,
   601               step_tac HOL_cs 1,
   602               cut_facts_tac [l1b] 1,
   603               fast_tac HOL_cs 1];
   604           in pg axs_finite_def goal tacs end;
   605 
   606         val _ = trace " Proving finites";
   607         val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
   608         val _ = trace " Proving ind";
   609         val ind =
   610           let
   611             fun concf n dn = %:(P_name n) $ %:(x_name n);
   612             fun tacf {prems, context} =
   613               let
   614                 fun finite_tacs (finite, fin_ind) = [
   615                   rtac(rewrite_rule axs_finite_def finite RS exE)1,
   616                   etac subst 1,
   617                   rtac fin_ind 1,
   618                   ind_prems_tac prems];
   619               in
   620                 TRY (safe_tac HOL_cs) ::
   621                 maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
   622               end;
   623           in pg'' thy [] (ind_term concf) tacf end;
   624       in (finites, ind) end (* let *)
   625 
   626     else (* infinite case *)
   627       let
   628         fun one_finite n dn =
   629           read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
   630         val finites = mapn one_finite 1 dnames;
   631 
   632         val goal =
   633           let
   634             fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
   635             fun concf n dn = %:(P_name n) $ %:(x_name n);
   636           in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
   637         val cont_rules =
   638             [cont_id, cont_const, cont2cont_Rep_CFun,
   639              cont2cont_fst, cont2cont_snd];
   640         fun tacf {prems, context} =
   641           map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
   642           quant_tac context 1,
   643           rtac (adm_impl_admw RS wfix_ind) 1,
   644           REPEAT_DETERM (rtac adm_all 1),
   645           REPEAT_DETERM (
   646             TRY (rtac adm_conj 1) THEN 
   647             rtac adm_subst 1 THEN 
   648             REPEAT (resolve_tac cont_rules 1) THEN
   649             resolve_tac prems 1),
   650           strip_tac 1,
   651           rtac (rewrite_rule axs_take_def finite_ind) 1,
   652           ind_prems_tac prems];
   653         val ind = (pg'' thy [] goal tacf
   654           handle ERROR _ =>
   655             (warning "Cannot prove infinite induction rule"; TrueI));
   656       in (finites, ind) end
   657   )
   658       handle THM _ =>
   659              (warning "Induction proofs failed (THM raised)."; ([], TrueI))
   660            | ERROR _ =>
   661              (warning "Cannot prove induction rule"; ([], TrueI));
   662 
   663 
   664 end; (* local *)
   665 
   666 (* ----- theorem concerning coinduction ------------------------------------- *)
   667 
   668 (* COINDUCTION TEMPORARILY DISABLED
   669 local
   670   val xs = mapn (fn n => K (x_name n)) 1 dnames;
   671   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
   672   val take_ss = HOL_ss addsimps take_rews;
   673   val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
   674   val _ = trace " Proving coind_lemma...";
   675   val coind_lemma =
   676     let
   677       fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
   678       fun mk_eqn n dn =
   679         (dc_take dn $ %:"n" ` bnd_arg n 0) ===
   680         (dc_take dn $ %:"n" ` bnd_arg n 1);
   681       fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
   682       val goal =
   683         mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
   684           Library.foldr mk_all2 (xs,
   685             Library.foldr mk_imp (mapn mk_prj 0 dnames,
   686               foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
   687       fun x_tacs ctxt n x = [
   688         rotate_tac (n+1) 1,
   689         etac all2E 1,
   690         eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
   691         TRY (safe_tac HOL_cs),
   692         REPEAT (CHANGED (asm_simp_tac take_ss 1))];
   693       fun tacs ctxt = [
   694         rtac impI 1,
   695         InductTacs.induct_tac ctxt [[SOME "n"]] 1,
   696         simp_tac take_ss 1,
   697         safe_tac HOL_cs] @
   698         flat (mapn (x_tacs ctxt) 0 xs);
   699     in pg [ax_bisim_def] goal tacs end;
   700 in
   701   val _ = trace " Proving coind...";
   702   val coind = 
   703     let
   704       fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
   705       fun mk_eqn x = %:x === %:(x^"'");
   706       val goal =
   707         mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
   708           Logic.list_implies (mapn mk_prj 0 xs,
   709             mk_trp (foldr1 mk_conj (map mk_eqn xs)));
   710       val tacs =
   711         TRY (safe_tac HOL_cs) ::
   712         maps (fn take_lemma => [
   713           rtac take_lemma 1,
   714           cut_facts_tac [coind_lemma] 1,
   715           fast_tac HOL_cs 1])
   716         take_lemmas;
   717     in pg [] goal (K tacs) end;
   718 end; (* local *)
   719 COINDUCTION TEMPORARILY DISABLED *)
   720 
   721 val inducts = Project_Rule.projections (ProofContext.init thy) ind;
   722 fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
   723 val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
   724 
   725 in thy |> Sign.add_path comp_dnam
   726        |> snd o PureThy.add_thmss [
   727            ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
   728            ((Binding.name "take_lemmas", take_lemmas ), []),
   729            ((Binding.name "finites"    , finites     ), []),
   730            ((Binding.name "finite_ind" , [finite_ind]), []),
   731            ((Binding.name "ind"        , [ind]       ), [])(*,
   732            ((Binding.name "coind"      , [coind]     ), [])*)]
   733        |> (if induct_failed then I
   734            else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
   735        |> Sign.parent_path |> pair take_rews
   736 end; (* let *)
   737 end; (* struct *)