src/HOLCF/Tools/Domain/domain_theorems.ML
author huffman
Sat Feb 27 14:04:46 2010 -0800 (2010-02-27)
changeset 35459 3d8acfae6fb8
parent 35458 deaf221c4a59
child 35460 8cb42aa19358
permissions -rw-r--r--
move proofs of when_rews intro domain_constructors.ML
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(*  Title:      HOLCF/Tools/Domain/domain_theorems.ML
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    Author:     David von Oheimb
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    Author:     Brian Huffman
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Proof generator for domain command.
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*)
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val HOLCF_ss = @{simpset};
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signature DOMAIN_THEOREMS =
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sig
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  val theorems:
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    Domain_Library.eq * Domain_Library.eq list
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    -> typ * (binding * (bool * binding option * typ) list * mixfix) list
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    -> theory -> thm list * theory;
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  val comp_theorems: bstring * Domain_Library.eq list -> theory -> thm list * theory;
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  val quiet_mode: bool Unsynchronized.ref;
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  val trace_domain: bool Unsynchronized.ref;
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end;
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structure Domain_Theorems :> DOMAIN_THEOREMS =
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struct
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val quiet_mode = Unsynchronized.ref false;
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val trace_domain = Unsynchronized.ref false;
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fun message s = if !quiet_mode then () else writeln s;
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fun trace s = if !trace_domain then tracing s else ();
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val adm_impl_admw = @{thm adm_impl_admw};
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val adm_all = @{thm adm_all};
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val adm_conj = @{thm adm_conj};
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val adm_subst = @{thm adm_subst};
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val antisym_less_inverse = @{thm below_antisym_inverse};
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val beta_cfun = @{thm beta_cfun};
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val cfun_arg_cong = @{thm cfun_arg_cong};
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val ch2ch_fst = @{thm ch2ch_fst};
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val ch2ch_snd = @{thm ch2ch_snd};
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val ch2ch_Rep_CFunL = @{thm ch2ch_Rep_CFunL};
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val ch2ch_Rep_CFunR = @{thm ch2ch_Rep_CFunR};
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val chain_iterate = @{thm chain_iterate};
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val compact_ONE = @{thm compact_ONE};
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val compact_sinl = @{thm compact_sinl};
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val compact_sinr = @{thm compact_sinr};
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val compact_spair = @{thm compact_spair};
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val compact_up = @{thm compact_up};
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val contlub_cfun_arg = @{thm contlub_cfun_arg};
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val contlub_cfun_fun = @{thm contlub_cfun_fun};
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val contlub_fst = @{thm contlub_fst};
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val contlub_snd = @{thm contlub_snd};
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val contlubE = @{thm contlubE};
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val cont_const = @{thm cont_const};
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val cont_id = @{thm cont_id};
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val cont2cont_fst = @{thm cont2cont_fst};
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val cont2cont_snd = @{thm cont2cont_snd};
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val cont2cont_Rep_CFun = @{thm cont2cont_Rep_CFun};
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val fix_def2 = @{thm fix_def2};
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val injection_eq = @{thm injection_eq};
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val injection_less = @{thm injection_below};
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val lub_equal = @{thm lub_equal};
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val monofun_cfun_arg = @{thm monofun_cfun_arg};
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val retraction_strict = @{thm retraction_strict};
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val spair_eq = @{thm spair_eq};
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val spair_less = @{thm spair_below};
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val sscase1 = @{thm sscase1};
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val ssplit1 = @{thm ssplit1};
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val strictify1 = @{thm strictify1};
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val wfix_ind = @{thm wfix_ind};
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val iso_intro       = @{thm iso.intro};
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val iso_abs_iso     = @{thm iso.abs_iso};
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val iso_rep_iso     = @{thm iso.rep_iso};
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val iso_abs_strict  = @{thm iso.abs_strict};
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val iso_rep_strict  = @{thm iso.rep_strict};
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val iso_abs_defined = @{thm iso.abs_defined};
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val iso_rep_defined = @{thm iso.rep_defined};
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val iso_compact_abs = @{thm iso.compact_abs};
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val iso_compact_rep = @{thm iso.compact_rep};
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val iso_iso_swap    = @{thm iso.iso_swap};
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val exh_start = @{thm exh_start};
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val ex_defined_iffs = @{thms ex_defined_iffs};
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val exh_casedist0 = @{thm exh_casedist0};
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val exh_casedists = @{thms exh_casedists};
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open Domain_Library;
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infixr 0 ===>;
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infixr 0 ==>;
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infix 0 == ; 
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infix 1 ===;
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infix 1 ~= ;
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infix 1 <<;
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infix 1 ~<<;
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infix 9 `   ;
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infix 9 `% ;
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infix 9 `%%;
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infixr 9 oo;
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(* ----- general proof facilities ------------------------------------------- *)
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fun legacy_infer_term thy t =
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  let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)
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  in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;
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fun pg'' thy defs t tacs =
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  let
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    val t' = legacy_infer_term thy t;
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    val asms = Logic.strip_imp_prems t';
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    val prop = Logic.strip_imp_concl t';
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    fun tac {prems, context} =
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      rewrite_goals_tac defs THEN
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      EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
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  in Goal.prove_global thy [] asms prop tac end;
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fun pg' thy defs t tacsf =
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  let
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    fun tacs {prems, context} =
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      if null prems then tacsf context
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      else cut_facts_tac prems 1 :: tacsf context;
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  in pg'' thy defs t tacs end;
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(* FIXME!!!!!!!!! *)
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(* We should NEVER re-parse variable names as strings! *)
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(* The names can conflict with existing constants or other syntax! *)
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fun case_UU_tac ctxt rews i v =
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  InductTacs.case_tac ctxt (v^"=UU") i THEN
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  asm_simp_tac (HOLCF_ss addsimps rews) i;
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val chain_tac =
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  REPEAT_DETERM o resolve_tac 
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    [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL, ch2ch_fst, ch2ch_snd];
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(* ----- general proofs ----------------------------------------------------- *)
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val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}
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val dist_eqI = @{lemma "!!x::'a::po. ~ x << y ==> x ~= y" by (blast dest!: below_antisym_inverse)}
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fun theorems
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    (((dname, _), cons) : eq, eqs : eq list)
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    (dom_eqn : typ * (binding * (bool * binding option * typ) list * mixfix) list)
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    (thy : theory) =
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let
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val _ = message ("Proving isomorphism properties of domain "^dname^" ...");
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val map_tab = Domain_Isomorphism.get_map_tab thy;
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(* ----- getting the axioms and definitions --------------------------------- *)
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local
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  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
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in
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  val ax_abs_iso  = ga "abs_iso"  dname;
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  val ax_rep_iso  = ga "rep_iso"  dname;
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  val ax_when_def = ga "when_def" dname;
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  fun get_def mk_name (con, _, _) = ga (mk_name con^"_def") dname;
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  val axs_dis_def = map (get_def dis_name) cons;
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  val axs_mat_def = map (get_def mat_name) cons;
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  val axs_pat_def = map (get_def pat_name) cons;
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  val ax_copy_def = ga "copy_def" dname;
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end; (* local *)
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(* ----- define constructors ------------------------------------------------ *)
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val lhsT = fst dom_eqn;
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val rhsT =
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  let
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    fun mk_arg_typ (lazy, sel, T) = if lazy then mk_uT T else T;
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    fun mk_con_typ (bind, args, mx) =
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        if null args then oneT else foldr1 mk_sprodT (map mk_arg_typ args);
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    fun mk_eq_typ (_, cons) = foldr1 mk_ssumT (map mk_con_typ cons);
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  in
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    mk_eq_typ dom_eqn
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  end;
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val rep_const = Const(dname^"_rep", lhsT ->> rhsT);
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val abs_const = Const(dname^"_abs", rhsT ->> lhsT);
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val (result, thy) =
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  Domain_Constructors.add_domain_constructors
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    (Long_Name.base_name dname) dom_eqn
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    (rep_const, abs_const) (ax_rep_iso, ax_abs_iso) ax_when_def thy;
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val con_appls = #con_betas result;
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val {exhaust, casedist, ...} = result;
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val {con_compacts, con_rews, inverts, injects, dist_les, dist_eqs, ...} = result;
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val {sel_rews, ...} = result;
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val when_rews = #cases result;
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val when_strict = hd when_rews;
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(* ----- theorems concerning the isomorphism -------------------------------- *)
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val pg = pg' thy;
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val dc_abs  = %%:(dname^"_abs");
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val dc_rep  = %%:(dname^"_rep");
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val dc_copy = %%:(dname^"_copy");
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val x_name = "x";
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val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso];
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val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
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val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
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val iso_rews = map Drule.export_without_context [ax_abs_iso, ax_rep_iso, abs_strict, rep_strict];
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(* ----- theorems concerning the constructors, discriminators and selectors - *)
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local
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  fun dis_strict (con, _, _) =
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    let
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      val goal = mk_trp (strict (%%:(dis_name con)));
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    in pg axs_dis_def goal (K [rtac when_strict 1]) end;
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  fun dis_app c (con, _, args) =
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    let
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      val lhs = %%:(dis_name c) ` con_app con args;
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      val rhs = if con = c then TT else FF;
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      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
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      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
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    in pg axs_dis_def goal (K tacs) end;
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  val _ = trace " Proving dis_apps...";
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  val dis_apps = maps (fn (c,_,_) => map (dis_app c) cons) cons;
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  fun dis_defin (con, _, args) =
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    let
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      val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name);
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      val tacs =
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        [rtac casedist 1,
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         contr_tac 1,
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         DETERM_UNTIL_SOLVED (CHANGED
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          (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))];
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    in pg [] goal (K tacs) end;
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  val _ = trace " Proving dis_stricts...";
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  val dis_stricts = map dis_strict cons;
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  val _ = trace " Proving dis_defins...";
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  val dis_defins = map dis_defin cons;
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in
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  val dis_rews = dis_stricts @ dis_defins @ dis_apps;
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end;
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local
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  fun mat_strict (con, _, _) =
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    let
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      val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU);
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      val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1];
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    in pg axs_mat_def goal (K tacs) end;
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  val _ = trace " Proving mat_stricts...";
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  val mat_stricts = map mat_strict cons;
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  fun one_mat c (con, _, args) =
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    let
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      val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs";
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      val rhs =
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        if con = c
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        then list_ccomb (%:"rhs", map %# args)
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        else mk_fail;
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      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
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      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
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    in pg axs_mat_def goal (K tacs) end;
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  val _ = trace " Proving mat_apps...";
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  val mat_apps =
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    maps (fn (c,_,_) => map (one_mat c) cons) cons;
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in
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  val mat_rews = mat_stricts @ mat_apps;
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end;
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local
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  fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
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  fun pat_lhs (con,_,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
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  fun pat_rhs (con,_,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
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    | pat_rhs (con,_,args) =
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        (mk_branch (mk_ctuple_pat (ps args)))
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          `(%:"rhs")`(mk_ctuple (map %# args));
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  fun pat_strict c =
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    let
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      val axs = @{thm branch_def} :: axs_pat_def;
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      val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
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      val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
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    in pg axs goal (K tacs) end;
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  fun pat_app c (con, _, args) =
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    let
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      val axs = @{thm branch_def} :: axs_pat_def;
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      val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
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      val rhs = if con = first c then pat_rhs c else mk_fail;
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      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
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      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
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    in pg axs goal (K tacs) end;
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  val _ = trace " Proving pat_stricts...";
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  val pat_stricts = map pat_strict cons;
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  val _ = trace " Proving pat_apps...";
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  val pat_apps = maps (fn c => map (pat_app c) cons) cons;
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in
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  val pat_rews = pat_stricts @ pat_apps;
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end;
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(* ----- theorems concerning one induction step ----------------------------- *)
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val copy_strict =
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  let
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    val _ = trace " Proving copy_strict...";
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    val goal = mk_trp (strict (dc_copy `% "f"));
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    val rules = [abs_strict, rep_strict] @ @{thms domain_map_stricts};
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    val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
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  in
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    SOME (pg [ax_copy_def] goal (K tacs))
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    handle
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      THM (s, _, _) => (trace s; NONE)
huffman@35058
   320
    | ERROR s => (trace s; NONE)
huffman@35058
   321
  end;
wenzelm@23152
   322
wenzelm@23152
   323
local
huffman@35288
   324
  fun copy_app (con, _, args) =
wenzelm@23152
   325
    let
wenzelm@23152
   326
      val lhs = dc_copy`%"f"`(con_app con args);
huffman@31232
   327
      fun one_rhs arg =
haftmann@33971
   328
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@33801
   329
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@33801
   330
                 (proj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
huffman@31232
   331
          else (%# arg);
huffman@31232
   332
      val rhs = con_app2 con one_rhs args;
huffman@35059
   333
      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
huffman@35059
   334
      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
huffman@35059
   335
      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
wenzelm@23152
   336
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
wenzelm@33317
   337
      val args' = filter_out (fn a => is_rec a orelse is_lazy a) args;
huffman@33504
   338
      val stricts = abs_strict :: rep_strict :: @{thms domain_map_stricts};
huffman@35443
   339
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   340
      fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
huffman@33504
   341
      val rules = [ax_abs_iso] @ @{thms domain_map_simps};
huffman@31232
   342
      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
huffman@31232
   343
    in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
wenzelm@23152
   344
in
huffman@29402
   345
  val _ = trace " Proving copy_apps...";
wenzelm@23152
   346
  val copy_apps = map copy_app cons;
wenzelm@23152
   347
end;
wenzelm@23152
   348
wenzelm@23152
   349
local
huffman@35288
   350
  fun one_strict (con, _, args) = 
wenzelm@23152
   351
    let
wenzelm@23152
   352
      val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
huffman@35058
   353
      val rews = the_list copy_strict @ copy_apps @ con_rews;
huffman@35443
   354
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   355
      fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
wenzelm@23152
   356
        [asm_simp_tac (HOLCF_ss addsimps rews) 1];
huffman@35058
   357
    in
huffman@35058
   358
      SOME (pg [] goal tacs)
huffman@35058
   359
      handle
huffman@35058
   360
        THM (s, _, _) => (trace s; NONE)
huffman@35058
   361
      | ERROR s => (trace s; NONE)
huffman@35058
   362
    end;
wenzelm@23152
   363
huffman@35288
   364
  fun has_nonlazy_rec (_, _, args) = exists is_nonlazy_rec args;
wenzelm@23152
   365
in
huffman@29402
   366
  val _ = trace " Proving copy_stricts...";
huffman@35058
   367
  val copy_stricts = map_filter one_strict (filter has_nonlazy_rec cons);
wenzelm@23152
   368
end;
wenzelm@23152
   369
huffman@35058
   370
val copy_rews = the_list copy_strict @ copy_apps @ copy_stricts;
wenzelm@23152
   371
wenzelm@23152
   372
in
wenzelm@23152
   373
  thy
wenzelm@30364
   374
    |> Sign.add_path (Long_Name.base_name dname)
huffman@31004
   375
    |> snd o PureThy.add_thmss [
huffman@31004
   376
        ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
huffman@31004
   377
        ((Binding.name "exhaust"   , [exhaust]   ), []),
huffman@31004
   378
        ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
huffman@31004
   379
        ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
huffman@31004
   380
        ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
huffman@33427
   381
        ((Binding.name "con_rews"  , con_rews    ),
huffman@33427
   382
         [Simplifier.simp_add, Fixrec.fixrec_simp_add]),
huffman@31004
   383
        ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
huffman@31004
   384
        ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
huffman@31004
   385
        ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
huffman@31004
   386
        ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
huffman@31004
   387
        ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
huffman@31004
   388
        ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
huffman@31004
   389
        ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
huffman@31004
   390
        ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
huffman@33427
   391
        ((Binding.name "match_rews", mat_rews    ),
huffman@33427
   392
         [Simplifier.simp_add, Fixrec.fixrec_simp_add])]
wenzelm@24712
   393
    |> Sign.parent_path
haftmann@28536
   394
    |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
wenzelm@23152
   395
        pat_rews @ dist_les @ dist_eqs @ copy_rews)
wenzelm@23152
   396
end; (* let *)
wenzelm@23152
   397
wenzelm@23152
   398
fun comp_theorems (comp_dnam, eqs: eq list) thy =
wenzelm@23152
   399
let
wenzelm@27232
   400
val global_ctxt = ProofContext.init thy;
huffman@33801
   401
val map_tab = Domain_Isomorphism.get_map_tab thy;
wenzelm@27232
   402
wenzelm@23152
   403
val dnames = map (fst o fst) eqs;
wenzelm@23152
   404
val conss  = map  snd        eqs;
haftmann@28965
   405
val comp_dname = Sign.full_bname thy comp_dnam;
wenzelm@23152
   406
huffman@29402
   407
val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
wenzelm@23152
   408
val pg = pg' thy;
wenzelm@23152
   409
wenzelm@23152
   410
(* ----- getting the composite axiom and definitions ------------------------ *)
wenzelm@23152
   411
wenzelm@23152
   412
local
wenzelm@26343
   413
  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
wenzelm@23152
   414
in
wenzelm@23152
   415
  val axs_reach      = map (ga "reach"     ) dnames;
wenzelm@23152
   416
  val axs_take_def   = map (ga "take_def"  ) dnames;
wenzelm@23152
   417
  val axs_finite_def = map (ga "finite_def") dnames;
wenzelm@23152
   418
  val ax_copy2_def   =      ga "copy_def"  comp_dnam;
huffman@35444
   419
(* TEMPORARILY DISABLED
wenzelm@23152
   420
  val ax_bisim_def   =      ga "bisim_def" comp_dnam;
huffman@35444
   421
TEMPORARILY DISABLED *)
wenzelm@23152
   422
end;
wenzelm@23152
   423
wenzelm@23152
   424
local
wenzelm@26343
   425
  fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
wenzelm@26343
   426
  fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
wenzelm@23152
   427
in
wenzelm@23152
   428
  val cases = map (gt  "casedist" ) dnames;
wenzelm@26336
   429
  val con_rews  = maps (gts "con_rews" ) dnames;
wenzelm@26336
   430
  val copy_rews = maps (gts "copy_rews") dnames;
wenzelm@23152
   431
end;
wenzelm@23152
   432
wenzelm@23152
   433
fun dc_take dn = %%:(dn^"_take");
wenzelm@23152
   434
val x_name = idx_name dnames "x"; 
wenzelm@23152
   435
val P_name = idx_name dnames "P";
wenzelm@23152
   436
val n_eqs = length eqs;
wenzelm@23152
   437
wenzelm@23152
   438
(* ----- theorems concerning finite approximation and finite induction ------ *)
wenzelm@23152
   439
wenzelm@23152
   440
local
wenzelm@32149
   441
  val iterate_Cprod_ss = global_simpset_of @{theory Fix};
wenzelm@23152
   442
  val copy_con_rews  = copy_rews @ con_rews;
wenzelm@23152
   443
  val copy_take_defs =
wenzelm@23152
   444
    (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
huffman@29402
   445
  val _ = trace " Proving take_stricts...";
huffman@35057
   446
  fun one_take_strict ((dn, args), _) =
wenzelm@23152
   447
    let
huffman@35057
   448
      val goal = mk_trp (strict (dc_take dn $ %:"n"));
huffman@35057
   449
      val rules = [
huffman@35057
   450
        @{thm monofun_fst [THEN monofunE]},
huffman@35057
   451
        @{thm monofun_snd [THEN monofunE]}];
huffman@35057
   452
      val tacs = [
huffman@35057
   453
        rtac @{thm UU_I} 1,
huffman@35057
   454
        rtac @{thm below_eq_trans} 1,
huffman@35057
   455
        resolve_tac axs_reach 2,
huffman@35057
   456
        rtac @{thm monofun_cfun_fun} 1,
huffman@35057
   457
        REPEAT (resolve_tac rules 1),
huffman@35057
   458
        rtac @{thm iterate_below_fix} 1];
huffman@35057
   459
    in pg axs_take_def goal (K tacs) end;
huffman@35057
   460
  val take_stricts = map one_take_strict eqs;
wenzelm@23152
   461
  fun take_0 n dn =
wenzelm@23152
   462
    let
huffman@35058
   463
      val goal = mk_trp ((dc_take dn $ @{term "0::nat"}) `% x_name n === UU);
wenzelm@27208
   464
    in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
wenzelm@23152
   465
  val take_0s = mapn take_0 1 dnames;
huffman@29402
   466
  val _ = trace " Proving take_apps...";
huffman@35288
   467
  fun one_take_app dn (con, _, args) =
wenzelm@23152
   468
    let
huffman@35058
   469
      fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
huffman@35058
   470
      fun one_rhs arg =
huffman@35058
   471
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@35058
   472
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@35058
   473
                 mk_take (dtyp_of arg) ` (%# arg)
huffman@35058
   474
          else (%# arg);
huffman@35058
   475
      val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
huffman@35058
   476
      val rhs = con_app2 con one_rhs args;
huffman@35059
   477
      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
huffman@35059
   478
      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
huffman@35059
   479
      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
huffman@35059
   480
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
huffman@35059
   481
      val tacs = [asm_simp_tac (HOLCF_ss addsimps copy_con_rews) 1];
huffman@35059
   482
    in pg copy_take_defs goal (K tacs) end;
huffman@35058
   483
  fun one_take_apps ((dn, _), cons) = map (one_take_app dn) cons;
huffman@35058
   484
  val take_apps = maps one_take_apps eqs;
wenzelm@23152
   485
in
wenzelm@35021
   486
  val take_rews = map Drule.export_without_context
huffman@35058
   487
    (take_stricts @ take_0s @ take_apps);
wenzelm@23152
   488
end; (* local *)
wenzelm@23152
   489
wenzelm@23152
   490
local
huffman@35288
   491
  fun one_con p (con, _, args) =
wenzelm@23152
   492
    let
huffman@35443
   493
      val P_names = map P_name (1 upto (length dnames));
huffman@35443
   494
      val vns = Name.variant_list P_names (map vname args);
huffman@35443
   495
      val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns));
wenzelm@23152
   496
      fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
wenzelm@23152
   497
      val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
wenzelm@33317
   498
      val t2 = lift ind_hyp (filter is_rec args, t1);
huffman@35443
   499
      val t3 = lift_defined (bound_arg vns) (nonlazy_vns, t2);
huffman@35443
   500
    in Library.foldr mk_All (vns, t3) end;
wenzelm@23152
   501
wenzelm@23152
   502
  fun one_eq ((p, cons), concl) =
wenzelm@23152
   503
    mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
wenzelm@23152
   504
wenzelm@23152
   505
  fun ind_term concf = Library.foldr one_eq
wenzelm@23152
   506
    (mapn (fn n => fn x => (P_name n, x)) 1 conss,
wenzelm@23152
   507
     mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
wenzelm@23152
   508
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@27208
   509
  fun quant_tac ctxt i = EVERY
wenzelm@27239
   510
    (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
wenzelm@23152
   511
wenzelm@23152
   512
  fun ind_prems_tac prems = EVERY
wenzelm@26336
   513
    (maps (fn cons =>
wenzelm@23152
   514
      (resolve_tac prems 1 ::
huffman@35288
   515
        maps (fn (_,_,args) => 
wenzelm@23152
   516
          resolve_tac prems 1 ::
wenzelm@23152
   517
          map (K(atac 1)) (nonlazy args) @
wenzelm@33317
   518
          map (K(atac 1)) (filter is_rec args))
wenzelm@26336
   519
        cons))
wenzelm@26336
   520
      conss);
wenzelm@23152
   521
  local 
wenzelm@23152
   522
    (* check whether every/exists constructor of the n-th part of the equation:
wenzelm@23152
   523
       it has a possibly indirectly recursive argument that isn't/is possibly 
wenzelm@23152
   524
       indirectly lazy *)
wenzelm@23152
   525
    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
wenzelm@23152
   526
          is_rec arg andalso not(rec_of arg mem ns) andalso
wenzelm@23152
   527
          ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
wenzelm@23152
   528
            rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
wenzelm@23152
   529
              (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
huffman@35288
   530
          ) o third) cons;
wenzelm@23152
   531
    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
wenzelm@23152
   532
    fun warn (n,cons) =
wenzelm@23152
   533
      if all_rec_to [] false (n,cons)
wenzelm@23152
   534
      then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
wenzelm@23152
   535
      else false;
wenzelm@23152
   536
    fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
wenzelm@23152
   537
wenzelm@23152
   538
  in
wenzelm@23152
   539
    val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
wenzelm@23152
   540
    val is_emptys = map warn n__eqs;
wenzelm@23152
   541
    val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
wenzelm@23152
   542
  end;
wenzelm@23152
   543
in (* local *)
huffman@29402
   544
  val _ = trace " Proving finite_ind...";
wenzelm@23152
   545
  val finite_ind =
wenzelm@23152
   546
    let
wenzelm@23152
   547
      fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
wenzelm@23152
   548
      val goal = ind_term concf;
wenzelm@23152
   549
wenzelm@27208
   550
      fun tacf {prems, context} =
wenzelm@23152
   551
        let
wenzelm@23152
   552
          val tacs1 = [
wenzelm@27208
   553
            quant_tac context 1,
wenzelm@23152
   554
            simp_tac HOL_ss 1,
wenzelm@27208
   555
            InductTacs.induct_tac context [[SOME "n"]] 1,
wenzelm@23152
   556
            simp_tac (take_ss addsimps prems) 1,
wenzelm@23152
   557
            TRY (safe_tac HOL_cs)];
wenzelm@23152
   558
          fun arg_tac arg =
huffman@35443
   559
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   560
            case_UU_tac context (prems @ con_rews) 1
wenzelm@23152
   561
              (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
huffman@35288
   562
          fun con_tacs (con, _, args) = 
wenzelm@23152
   563
            asm_simp_tac take_ss 1 ::
wenzelm@33317
   564
            map arg_tac (filter is_nonlazy_rec args) @
wenzelm@23152
   565
            [resolve_tac prems 1] @
wenzelm@33317
   566
            map (K (atac 1)) (nonlazy args) @
wenzelm@33317
   567
            map (K (etac spec 1)) (filter is_rec args);
wenzelm@23152
   568
          fun cases_tacs (cons, cases) =
wenzelm@27239
   569
            res_inst_tac context [(("x", 0), "x")] cases 1 ::
wenzelm@23152
   570
            asm_simp_tac (take_ss addsimps prems) 1 ::
wenzelm@26336
   571
            maps con_tacs cons;
wenzelm@23152
   572
        in
wenzelm@26336
   573
          tacs1 @ maps cases_tacs (conss ~~ cases)
wenzelm@23152
   574
        end;
huffman@31232
   575
    in pg'' thy [] goal tacf
huffman@31232
   576
       handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
huffman@31232
   577
    end;
wenzelm@23152
   578
huffman@29402
   579
  val _ = trace " Proving take_lemmas...";
wenzelm@23152
   580
  val take_lemmas =
wenzelm@23152
   581
    let
wenzelm@23152
   582
      fun take_lemma n (dn, ax_reach) =
wenzelm@23152
   583
        let
wenzelm@23152
   584
          val lhs = dc_take dn $ Bound 0 `%(x_name n);
wenzelm@23152
   585
          val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
wenzelm@23152
   586
          val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
wenzelm@23152
   587
          val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
huffman@33396
   588
          val rules = [contlub_fst RS contlubE RS ssubst,
huffman@33396
   589
                       contlub_snd RS contlubE RS ssubst];
wenzelm@27208
   590
          fun tacf {prems, context} = [
wenzelm@27239
   591
            res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
wenzelm@27239
   592
            res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
wenzelm@23152
   593
            stac fix_def2 1,
wenzelm@23152
   594
            REPEAT (CHANGED
huffman@33396
   595
              (resolve_tac rules 1 THEN chain_tac 1)),
wenzelm@23152
   596
            stac contlub_cfun_fun 1,
wenzelm@23152
   597
            stac contlub_cfun_fun 2,
wenzelm@23152
   598
            rtac lub_equal 3,
wenzelm@23152
   599
            chain_tac 1,
wenzelm@23152
   600
            rtac allI 1,
wenzelm@23152
   601
            resolve_tac prems 1];
wenzelm@23152
   602
        in pg'' thy axs_take_def goal tacf end;
wenzelm@23152
   603
    in mapn take_lemma 1 (dnames ~~ axs_reach) end;
wenzelm@23152
   604
wenzelm@23152
   605
(* ----- theorems concerning finiteness and induction ----------------------- *)
wenzelm@23152
   606
huffman@29402
   607
  val _ = trace " Proving finites, ind...";
wenzelm@23152
   608
  val (finites, ind) =
huffman@31232
   609
  (
wenzelm@23152
   610
    if is_finite
wenzelm@23152
   611
    then (* finite case *)
wenzelm@23152
   612
      let 
wenzelm@23152
   613
        fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
wenzelm@23152
   614
        fun dname_lemma dn =
wenzelm@23152
   615
          let
wenzelm@23152
   616
            val prem1 = mk_trp (defined (%:"x"));
wenzelm@23152
   617
            val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
wenzelm@23152
   618
            val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
wenzelm@23152
   619
            val concl = mk_trp (take_enough dn);
wenzelm@23152
   620
            val goal = prem1 ===> prem2 ===> concl;
wenzelm@23152
   621
            val tacs = [
wenzelm@23152
   622
              etac disjE 1,
wenzelm@23152
   623
              etac notE 1,
wenzelm@23152
   624
              resolve_tac take_lemmas 1,
wenzelm@23152
   625
              asm_simp_tac take_ss 1,
wenzelm@23152
   626
              atac 1];
wenzelm@27208
   627
          in pg [] goal (K tacs) end;
huffman@31232
   628
        val _ = trace " Proving finite_lemmas1a";
wenzelm@23152
   629
        val finite_lemmas1a = map dname_lemma dnames;
wenzelm@23152
   630
 
huffman@31232
   631
        val _ = trace " Proving finite_lemma1b";
wenzelm@23152
   632
        val finite_lemma1b =
wenzelm@23152
   633
          let
wenzelm@23152
   634
            fun mk_eqn n ((dn, args), _) =
wenzelm@23152
   635
              let
wenzelm@23152
   636
                val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
wenzelm@23152
   637
                val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
wenzelm@23152
   638
              in
wenzelm@23152
   639
                mk_constrainall
wenzelm@23152
   640
                  (x_name n, Type (dn,args), mk_disj (disj1, disj2))
wenzelm@23152
   641
              end;
wenzelm@23152
   642
            val goal =
wenzelm@23152
   643
              mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
wenzelm@27208
   644
            fun arg_tacs ctxt vn = [
wenzelm@27239
   645
              eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
wenzelm@23152
   646
              etac disjE 1,
wenzelm@23152
   647
              asm_simp_tac (HOL_ss addsimps con_rews) 1,
wenzelm@23152
   648
              asm_simp_tac take_ss 1];
huffman@35288
   649
            fun con_tacs ctxt (con, _, args) =
wenzelm@23152
   650
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   651
              maps (arg_tacs ctxt) (nonlazy_rec args);
wenzelm@27208
   652
            fun foo_tacs ctxt n (cons, cases) =
wenzelm@23152
   653
              simp_tac take_ss 1 ::
wenzelm@23152
   654
              rtac allI 1 ::
wenzelm@27239
   655
              res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
wenzelm@23152
   656
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   657
              maps (con_tacs ctxt) cons;
wenzelm@27208
   658
            fun tacs ctxt =
wenzelm@23152
   659
              rtac allI 1 ::
wenzelm@27208
   660
              InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
wenzelm@23152
   661
              simp_tac take_ss 1 ::
wenzelm@23152
   662
              TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
wenzelm@27208
   663
              flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
wenzelm@23152
   664
          in pg [] goal tacs end;
wenzelm@23152
   665
wenzelm@23152
   666
        fun one_finite (dn, l1b) =
wenzelm@23152
   667
          let
wenzelm@23152
   668
            val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
wenzelm@27208
   669
            fun tacs ctxt = [
huffman@35443
   670
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   671
              case_UU_tac ctxt take_rews 1 "x",
wenzelm@23152
   672
              eresolve_tac finite_lemmas1a 1,
wenzelm@23152
   673
              step_tac HOL_cs 1,
wenzelm@23152
   674
              step_tac HOL_cs 1,
wenzelm@23152
   675
              cut_facts_tac [l1b] 1,
wenzelm@23152
   676
              fast_tac HOL_cs 1];
wenzelm@23152
   677
          in pg axs_finite_def goal tacs end;
wenzelm@23152
   678
huffman@31232
   679
        val _ = trace " Proving finites";
wenzelm@27232
   680
        val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
huffman@31232
   681
        val _ = trace " Proving ind";
wenzelm@23152
   682
        val ind =
wenzelm@23152
   683
          let
wenzelm@23152
   684
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@27208
   685
            fun tacf {prems, context} =
wenzelm@23152
   686
              let
wenzelm@23152
   687
                fun finite_tacs (finite, fin_ind) = [
wenzelm@23152
   688
                  rtac(rewrite_rule axs_finite_def finite RS exE)1,
wenzelm@23152
   689
                  etac subst 1,
wenzelm@23152
   690
                  rtac fin_ind 1,
wenzelm@23152
   691
                  ind_prems_tac prems];
wenzelm@23152
   692
              in
wenzelm@23152
   693
                TRY (safe_tac HOL_cs) ::
wenzelm@27232
   694
                maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
wenzelm@23152
   695
              end;
wenzelm@23152
   696
          in pg'' thy [] (ind_term concf) tacf end;
wenzelm@23152
   697
      in (finites, ind) end (* let *)
wenzelm@23152
   698
wenzelm@23152
   699
    else (* infinite case *)
wenzelm@23152
   700
      let
wenzelm@23152
   701
        fun one_finite n dn =
wenzelm@27239
   702
          read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
wenzelm@23152
   703
        val finites = mapn one_finite 1 dnames;
wenzelm@23152
   704
wenzelm@23152
   705
        val goal =
wenzelm@23152
   706
          let
huffman@26012
   707
            fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
wenzelm@23152
   708
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@23152
   709
          in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
huffman@33396
   710
        val cont_rules =
huffman@33396
   711
            [cont_id, cont_const, cont2cont_Rep_CFun,
huffman@33396
   712
             cont2cont_fst, cont2cont_snd];
wenzelm@27208
   713
        fun tacf {prems, context} =
wenzelm@23152
   714
          map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
wenzelm@27208
   715
          quant_tac context 1,
wenzelm@23152
   716
          rtac (adm_impl_admw RS wfix_ind) 1,
huffman@25895
   717
          REPEAT_DETERM (rtac adm_all 1),
wenzelm@23152
   718
          REPEAT_DETERM (
wenzelm@23152
   719
            TRY (rtac adm_conj 1) THEN 
wenzelm@23152
   720
            rtac adm_subst 1 THEN 
huffman@33396
   721
            REPEAT (resolve_tac cont_rules 1) THEN
huffman@33396
   722
            resolve_tac prems 1),
wenzelm@23152
   723
          strip_tac 1,
wenzelm@23152
   724
          rtac (rewrite_rule axs_take_def finite_ind) 1,
wenzelm@23152
   725
          ind_prems_tac prems];
wenzelm@23152
   726
        val ind = (pg'' thy [] goal tacf
wenzelm@23152
   727
          handle ERROR _ =>
huffman@33396
   728
            (warning "Cannot prove infinite induction rule"; TrueI));
huffman@31232
   729
      in (finites, ind) end
huffman@31232
   730
  )
huffman@31232
   731
      handle THM _ =>
huffman@31232
   732
             (warning "Induction proofs failed (THM raised)."; ([], TrueI))
huffman@31232
   733
           | ERROR _ =>
huffman@33810
   734
             (warning "Cannot prove induction rule"; ([], TrueI));
huffman@31232
   735
huffman@31232
   736
wenzelm@23152
   737
end; (* local *)
wenzelm@23152
   738
wenzelm@23152
   739
(* ----- theorem concerning coinduction ------------------------------------- *)
wenzelm@23152
   740
huffman@35444
   741
(* COINDUCTION TEMPORARILY DISABLED
wenzelm@23152
   742
local
wenzelm@23152
   743
  val xs = mapn (fn n => K (x_name n)) 1 dnames;
wenzelm@23152
   744
  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
wenzelm@23152
   745
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@23152
   746
  val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
huffman@29402
   747
  val _ = trace " Proving coind_lemma...";
wenzelm@23152
   748
  val coind_lemma =
wenzelm@23152
   749
    let
wenzelm@23152
   750
      fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
wenzelm@23152
   751
      fun mk_eqn n dn =
wenzelm@23152
   752
        (dc_take dn $ %:"n" ` bnd_arg n 0) ===
wenzelm@23152
   753
        (dc_take dn $ %:"n" ` bnd_arg n 1);
wenzelm@23152
   754
      fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
wenzelm@23152
   755
      val goal =
wenzelm@23152
   756
        mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
wenzelm@23152
   757
          Library.foldr mk_all2 (xs,
wenzelm@23152
   758
            Library.foldr mk_imp (mapn mk_prj 0 dnames,
wenzelm@23152
   759
              foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
wenzelm@27208
   760
      fun x_tacs ctxt n x = [
wenzelm@23152
   761
        rotate_tac (n+1) 1,
wenzelm@23152
   762
        etac all2E 1,
wenzelm@27239
   763
        eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
wenzelm@23152
   764
        TRY (safe_tac HOL_cs),
wenzelm@23152
   765
        REPEAT (CHANGED (asm_simp_tac take_ss 1))];
wenzelm@27208
   766
      fun tacs ctxt = [
wenzelm@23152
   767
        rtac impI 1,
wenzelm@27208
   768
        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
wenzelm@23152
   769
        simp_tac take_ss 1,
wenzelm@23152
   770
        safe_tac HOL_cs] @
wenzelm@27208
   771
        flat (mapn (x_tacs ctxt) 0 xs);
wenzelm@23152
   772
    in pg [ax_bisim_def] goal tacs end;
wenzelm@23152
   773
in
huffman@29402
   774
  val _ = trace " Proving coind...";
wenzelm@23152
   775
  val coind = 
wenzelm@23152
   776
    let
wenzelm@23152
   777
      fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
wenzelm@23152
   778
      fun mk_eqn x = %:x === %:(x^"'");
wenzelm@23152
   779
      val goal =
wenzelm@23152
   780
        mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
wenzelm@23152
   781
          Logic.list_implies (mapn mk_prj 0 xs,
wenzelm@23152
   782
            mk_trp (foldr1 mk_conj (map mk_eqn xs)));
wenzelm@23152
   783
      val tacs =
wenzelm@23152
   784
        TRY (safe_tac HOL_cs) ::
wenzelm@26336
   785
        maps (fn take_lemma => [
wenzelm@23152
   786
          rtac take_lemma 1,
wenzelm@23152
   787
          cut_facts_tac [coind_lemma] 1,
wenzelm@23152
   788
          fast_tac HOL_cs 1])
wenzelm@26336
   789
        take_lemmas;
wenzelm@27208
   790
    in pg [] goal (K tacs) end;
wenzelm@23152
   791
end; (* local *)
huffman@35444
   792
COINDUCTION TEMPORARILY DISABLED *)
wenzelm@23152
   793
wenzelm@32172
   794
val inducts = Project_Rule.projections (ProofContext.init thy) ind;
huffman@30829
   795
fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
huffman@31232
   796
val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
huffman@30829
   797
wenzelm@24712
   798
in thy |> Sign.add_path comp_dnam
huffman@31004
   799
       |> snd o PureThy.add_thmss [
huffman@31004
   800
           ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
huffman@31004
   801
           ((Binding.name "take_lemmas", take_lemmas ), []),
huffman@31004
   802
           ((Binding.name "finites"    , finites     ), []),
huffman@31004
   803
           ((Binding.name "finite_ind" , [finite_ind]), []),
huffman@35444
   804
           ((Binding.name "ind"        , [ind]       ), [])(*,
huffman@35444
   805
           ((Binding.name "coind"      , [coind]     ), [])*)]
huffman@31232
   806
       |> (if induct_failed then I
huffman@31232
   807
           else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
haftmann@28536
   808
       |> Sign.parent_path |> pair take_rews
wenzelm@23152
   809
end; (* let *)
wenzelm@23152
   810
end; (* struct *)