File theorems.ML (Isabelle2005: October 2005)

# File theorems.ML

    (*  Title:      HOLCF/domain/theorems.ML
        ID:         $Id: theorems.ML,v 1.55 2005/09/17 16:11:25 wenzelm Exp $
        Author:     David von Oheimb
                    New proofs/tactics by Brian Huffman

    Proof generator for domain section.
    *)

    val HOLCF_ss = simpset();

    structure Domain_Theorems = struct

    local

    open Domain_Library;
    infixr 0 ===>;infixr 0 ==>;infix 0 == ;
    infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
    infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;

    (* ----- general proof facilities ------------------------------------------- *)

    fun inferT sg pre_tm =
      #1 (Sign.infer_types (Sign.pp sg) sg (K NONE) (K NONE) [] true ([pre_tm],propT));

    fun pg'' thy defs t = let val sg = sign_of thy;
                              val ct = Thm.cterm_of sg (inferT sg t);
                          in prove_goalw_cterm defs ct end;
    fun pg'  thy defs t tacsf=pg'' thy defs t (fn []   => tacsf
                                    | prems=> (cut_facts_tac prems 1)::tacsf);

    fun case_UU_tac rews i v =      case_tac (v^"=UU") i THEN
                                    asm_simp_tac (HOLCF_ss addsimps rews) i;

    val chain_tac = REPEAT_DETERM o resolve_tac
                    [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL];

    (* ----- general proofs ----------------------------------------------------- *)

    val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R"
     (fn prems =>[
                                    resolve_tac prems 1,
                                    cut_facts_tac prems 1,
                                    fast_tac HOL_cs 1]);

    val dist_eqI = prove_goal Porder.thy "!!x::'a::po. ~ x << y ==> x ~= y"
                 (fn prems => [
                   (blast_tac (claset() addDs [antisym_less_inverse]) 1)]);
    (*
    infixr 0 y;
    val b = 0;
    fun _ y t = by t;
    fun g defs t = let val sg = sign_of thy;
                         val ct = Thm.cterm_of sg (inferT sg t);
                     in goalw_cterm defs ct end;
    *)

    in

    fun theorems (((dname,_),cons) : eq, eqs : eq list) thy =
    let

    val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ...");
    val pg = pg' thy;

    (* ----- getting the axioms and definitions --------------------------------- *)

    local fun ga s dn = get_thm thy (Name (dn ^ "." ^ s)) in
    val ax_abs_iso    = ga "abs_iso"  dname;
    val ax_rep_iso    = ga "rep_iso"  dname;
    val ax_when_def   = ga "when_def" dname;
    val axs_con_def   = map (fn (con,_) => ga (extern_name con^"_def") dname) cons;
    val axs_dis_def   = map (fn (con,_) => ga (   dis_name con^"_def") dname) cons;
    val axs_mat_def   = map (fn (con,_) => ga (   mat_name con^"_def") dname) cons;
    val axs_sel_def   = List.concat(map (fn (_,args) => List.mapPartial (fn arg =>
                     Option.map (fn sel => ga (sel^"_def") dname) (sel_of arg)) args)
    									  cons);
    val ax_copy_def   = ga "copy_def" dname;
    end; (* local *)

    (* ----- theorems concerning the isomorphism -------------------------------- *)

    val dc_abs  = %%:(dname^"_abs");
    val dc_rep  = %%:(dname^"_rep");
    val dc_copy = %%:(dname^"_copy");
    val x_name = "x";

    val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso];
    val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
    val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
    val abs_defin' = iso_locale RS iso_abs_defin';
    val rep_defin' = iso_locale RS iso_rep_defin';
    val iso_rews = map standard [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];

    (* ----- generating beta reduction rules from definitions-------------------- *)

    local
      fun arglist (Const _ $ Abs (s,_,t)) = let
            val (vars,body) = arglist t
            in  (s :: vars, body) end
      |   arglist t = ([],t);
      fun bind_fun vars t = Library.foldr mk_All (vars,t);
      fun bound_vars 0 = [] | bound_vars i = (Bound (i-1) :: bound_vars (i-1));
    in
      fun appl_of_def def = let
            val (_ $ con $ lam) = concl_of def;
            val (vars, rhs) = arglist lam;
            val lhs = mk_cRep_CFun (con, bound_vars (length vars));
            val appl = bind_fun vars (lhs == rhs);
            val cs = ContProc.cont_thms lam;
            val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs;
            in pg (def::betas) appl [rtac reflexive_thm 1] end;
    end;

    val when_appl = appl_of_def ax_when_def;
    val con_appls = map appl_of_def axs_con_def;

    local
      fun arg2typ n arg = let val t = TVar (("'a",n),pcpoS)
                          in (n+1, if is_lazy arg then mk_uT t else t) end;
      fun args2typ n [] = (n,oneT)
      |   args2typ n [arg] = arg2typ n arg
      |   args2typ n (arg::args) = let val (n1,t1) = arg2typ n arg;
                                       val (n2,t2) = args2typ n1 args
    			       in  (n2, mk_sprodT (t1, t2)) end;
      fun cons2typ n [] = (n,oneT)
      |   cons2typ n [con] = args2typ n (snd con)
      |   cons2typ n (con::cons) = let val (n1,t1) = args2typ n (snd con);
                                       val (n2,t2) = cons2typ n1 cons
    			       in  (n2, mk_ssumT (t1, t2)) end;
    in
      fun cons2ctyp cons = ctyp_of (sign_of thy) (snd (cons2typ 1 cons));
    end;

    local
      val iso_swap = iso_locale RS iso_iso_swap;
      fun one_con (con,args) = let val vns = map vname args in
        Library.foldr mk_ex (vns, foldr' mk_conj ((%:x_name === con_app2 con %: vns)::
                                  map (defined o %:) (nonlazy args))) end;
      val exh = foldr' mk_disj ((%:x_name===UU)::map one_con cons);
      val my_ctyp = cons2ctyp cons;
      val thm1 = instantiate' [SOME my_ctyp] [] exh_start;
      val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1;
      val thm3 = rewrite_rule [mk_meta_eq conj_assoc] thm2;
    in
    val exhaust = pg con_appls (mk_trp exh)[
    (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *)
    			rtac disjE 1,
    			etac (rep_defin' RS disjI1) 2,
    			etac disjI2 2,
    			rewrite_goals_tac [mk_meta_eq iso_swap],
    			rtac thm3 1];
    val casedist = standard (rewrite_rule exh_casedists (exhaust RS exh_casedist0));
    end;

    local
      fun bind_fun t = Library.foldr mk_All (when_funs cons,t);
      fun bound_fun i _ = Bound (length cons - i);
      val when_app  = Library.foldl (op `) (%%:(dname^"_when"), mapn bound_fun 1 cons);
    in
    val when_strict = pg [when_appl, mk_meta_eq rep_strict]
    			(bind_fun(mk_trp(strict when_app)))
    			[resolve_tac [sscase1,ssplit1,strictify1] 1];
    val when_apps = let fun one_when n (con,args) = pg (when_appl :: con_appls)
                    (bind_fun (lift_defined %: (nonlazy args,
                    mk_trp(when_app`(con_app con args) ===
                           mk_cRep_CFun(bound_fun n 0,map %# args)))))[
                    asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1];
            in mapn one_when 1 cons end;
    end;
    val when_rews = when_strict::when_apps;

    (* ----- theorems concerning the constructors, discriminators and selectors - *)

    val dis_rews = let
      val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
                                 strict(%%:(dis_name con)))) [
                                    rtac when_strict 1]) cons;
      val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def
                       (lift_defined %: (nonlazy args,
                            (mk_trp((%%:(dis_name c))`(con_app con args) ===
                                  %%:(if con=c then TT_N else FF_N))))) [
                                    asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
            in List.concat(map (fn (c,_) => map (one_dis c) cons) cons) end;
      val dis_defins = map (fn (con,args) => pg [] (defined(%:x_name) ==>
                          defined(%%:(dis_name con)`%x_name)) [
                                    rtac casedist 1,
                                    contr_tac 1,
                                    DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac
                                            (HOLCF_ss addsimps dis_apps) 1))]) cons;
    in dis_stricts @ dis_defins @ dis_apps end;

    val mat_rews = let
      val mat_stricts = map (fn (con,_) => pg axs_mat_def (mk_trp(
                                 strict(%%:(mat_name con)))) [
                                    rtac when_strict 1]) cons;
      val mat_apps = let fun one_mat c (con,args)= pg axs_mat_def
                       (lift_defined %: (nonlazy args,
                            (mk_trp((%%:(mat_name c))`(con_app con args) ===
                                  (if con=c
                                      then %%:returnN`(mk_ctuple (map %# args))
                                      else %%:failN)))))
                       [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
            in List.concat(map (fn (c,_) => map (one_mat c) cons) cons) end;
    in mat_stricts @ mat_apps end;

    val con_stricts = List.concat(map (fn (con,args) => map (fn vn =>
                            pg con_appls
                               (mk_trp(con_app2 con (fn arg => if vname arg = vn
                                            then UU else %# arg) args === UU))[
                                    asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
                            ) (nonlazy args)) cons);
    val con_defins = map (fn (con,args) => pg []
                            (lift_defined %: (nonlazy args,
                                    mk_trp(defined(con_app con args)))) ([
                              rtac rev_contrapos 1,
                              eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
                              asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons;
    val con_rews = con_stricts @ con_defins;

    val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%:sel))) [
                                    simp_tac (HOLCF_ss addsimps when_rews) 1];
    in List.concat(map (fn (_,args) => List.mapPartial (fn arg => Option.map one_sel (sel_of arg)) args) cons) end;
    val sel_apps = let fun one_sel c n sel = map (fn (con,args) =>
                    let val nlas = nonlazy args;
                        val vns  = map vname args;
                    in pg axs_sel_def (lift_defined %:
                       (List.filter (fn v => con=c andalso (v<>List.nth(vns,n))) nlas,
                                    mk_trp((%%:sel)`(con_app con args) ===
                                    (if con=c then %:(List.nth(vns,n)) else UU))))
                                ( (if con=c then []
                           else map(case_UU_tac(when_rews@con_stricts)1) nlas)
                         @(if con=c andalso ((List.nth(vns,n)) mem nlas)
                                     then[case_UU_tac (when_rews @ con_stricts) 1
                                                      (List.nth(vns,n))] else [])
                         @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
    in List.concat(map  (fn (c,args) =>
         List.concat(List.mapPartial I (mapn (fn n => fn arg => Option.map (one_sel c n) (sel_of arg)) 0 args))) cons) end;
    val sel_defins = if length cons=1 then List.mapPartial (fn arg => Option.map (fn sel => pg [](defined(%:x_name)==>
                            defined(%%:sel`%x_name)) [
                                    rtac casedist 1,
                                    contr_tac 1,
                                    DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac
                                                 (HOLCF_ss addsimps sel_apps) 1))])(sel_of arg))
                     (filter_out is_lazy (snd(hd cons))) else [];
    val sel_rews = sel_stricts @ sel_defins @ sel_apps;

    val distincts_le = let
        fun dist (con1, args1) (con2, args2) = pg []
                  (lift_defined %: ((nonlazy args1),
                            (mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
                            rtac rev_contrapos 1,
                            eres_inst_tac[("f",dis_name con1)] monofun_cfun_arg 1]
                          @map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2)
                          @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
        fun distinct (con1,args1) (con2,args2) =
            let val arg1 = (con1, args1)
                val arg2 = (con2,
    			ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
                            (args2, variantlist(map vname args2,map vname args1)))
            in [dist arg1 arg2, dist arg2 arg1] end;
        fun distincts []      = []
        |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
    in distincts cons end;
    val dist_les = List.concat (List.concat distincts_le);
    val dist_eqs = let
        fun distinct (_,args1) ((_,args2),leqs) = let
            val (le1,le2) = (hd leqs, hd(tl leqs));
            val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
            if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
            if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
                                            [eq1, eq2] end;
        fun distincts []      = []
        |   distincts ((c,leqs)::cs) = List.concat
    	            (ListPair.map (distinct c) ((map #1 cs),leqs)) @
    		    distincts cs;
        in map standard (distincts (cons~~distincts_le)) end;

    local
      fun pgterm rel con args =
        let
          fun append s = upd_vname(fn v => v^s);
          val (largs,rargs) = (args, map (append "'") args);
          val concl = mk_trp (foldr' mk_conj (ListPair.map rel (map %# largs, map %# rargs)));
          val prem = mk_trp (rel(con_app con largs,con_app con rargs));
          val prop = prem ===> lift_defined %: (nonlazy largs, concl);
        in pg con_appls prop end;
      val cons' = List.filter (fn (_,args) => args<>[]) cons;
    in
    val inverts =
      let
        val abs_less = ax_abs_iso RS (allI RS injection_less) RS iffD1;
        val tacs = [
          dtac abs_less 1,
          REPEAT (dresolve_tac [sinl_less RS iffD1, sinr_less RS iffD1] 1),
          asm_full_simp_tac (HOLCF_ss addsimps [spair_less]) 1];
      in map (fn (con,args) => pgterm (op <<) con args tacs) cons' end;
    val injects =
      let
        val abs_eq = ax_abs_iso RS (allI RS injection_eq) RS iffD1;
        val tacs = [
          dtac abs_eq 1,
          REPEAT (dresolve_tac [sinl_inject, sinr_inject] 1),
          asm_full_simp_tac (HOLCF_ss addsimps [spair_eq]) 1];
      in map (fn (con,args) => pgterm (op ===) con args tacs) cons' end;
    end;

    (* ----- theorems concerning one induction step ----------------------------- *)

    val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
                       asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict]) 1];
    val copy_apps = map (fn (con,args) => pg [ax_copy_def]
                        (lift_defined %: (nonlazy_rec args,
                            mk_trp(dc_copy`%"f"`(con_app con args) ===
                    (con_app2 con (app_rec_arg (cproj (%:"f") eqs)) args))))
                            (map (case_UU_tac (abs_strict::when_strict::con_stricts)
                                     1 o vname)
                             (List.filter (fn a => not (is_rec a orelse is_lazy a)) args)
                            @[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
                              simp_tac (HOLCF_ss addsimps con_appls) 1]))cons;
    val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
                                            (con_app con args) ===UU))
         (let val rews = copy_strict::copy_apps@con_rews
                             in map (case_UU_tac rews 1) (nonlazy args) @ [
                                 asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
                            (List.filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
    val copy_rews = copy_strict::copy_apps @ copy_stricts;
    in thy |> Theory.add_path (Sign.base_name dname)
           |> (#1 o (PureThy.add_thmss (map Thm.no_attributes [
    		("iso_rews" , iso_rews  ),
    		("exhaust"  , [exhaust] ),
    		("casedist" , [casedist]),
    		("when_rews", when_rews ),
    		("con_rews", con_rews),
    		("sel_rews", sel_rews),
    		("dis_rews", dis_rews),
    		("dist_les", dist_les),
    		("dist_eqs", dist_eqs),
    		("inverts" , inverts ),
    		("injects" , injects ),
    		("copy_rews", copy_rews)])))
           |> (#1 o PureThy.add_thmss [(("match_rews", mat_rews), [Simplifier.simp_add_global])])
           |> Theory.parent_path |> rpair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
                     dist_les @ dist_eqs @ copy_rews)
    end; (* let *)

    fun comp_theorems (comp_dnam, eqs: eq list) thy =
    let
    val dnames = map (fst o fst) eqs;
    val conss  = map  snd        eqs;
    val comp_dname = Sign.full_name (sign_of thy) comp_dnam;

    val d = writeln("Proving induction properties of domain "^comp_dname^" ...");
    val pg = pg' thy;

    (* ----- getting the composite axiom and definitions ------------------------ *)

    local fun ga s dn = get_thm thy (Name (dn ^ "." ^ s)) in
    val axs_reach      = map (ga "reach"     ) dnames;
    val axs_take_def   = map (ga "take_def"  ) dnames;
    val axs_finite_def = map (ga "finite_def") dnames;
    val ax_copy2_def   =      ga "copy_def"  comp_dnam;
    val ax_bisim_def   =      ga "bisim_def" comp_dnam;
    end; (* local *)

    local fun gt  s dn = get_thm  thy (Name (dn ^ "." ^ s));
          fun gts s dn = get_thms thy (Name (dn ^ "." ^ s)) in
    val cases     =       map (gt  "casedist" ) dnames;
    val con_rews  = List.concat (map (gts "con_rews" ) dnames);
    val copy_rews = List.concat (map (gts "copy_rews") dnames);
    end; (* local *)

    fun dc_take dn = %%:(dn^"_take");
    val x_name = idx_name dnames "x";
    val P_name = idx_name dnames "P";
    val n_eqs = length eqs;

    (* ----- theorems concerning finite approximation and finite induction ------ *)

    local
      val iterate_Cprod_ss = simpset_of Fix.thy;
      val copy_con_rews  = copy_rews @ con_rews;
      val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
      val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
                strict(dc_take dn $ %:"n")) eqs))) ([
                            induct_tac "n" 1,
                             simp_tac iterate_Cprod_ss 1,
                            asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]);
      val take_stricts' = rewrite_rule copy_take_defs take_stricts;
      val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%:"0")
                                                            `%x_name n === UU))[
                                    simp_tac iterate_Cprod_ss 1]) 1 dnames;
      val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
      val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj
                (List.concat(map (fn ((dn,_),cons) => map (fn (con,args) => Library.foldr mk_all
            (map vname args,(dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args) ===
             con_app2 con (app_rec_arg (fn n=>dc_take (List.nth(dnames,n))$ %:"n"))
                                  args)) cons) eqs)))) ([
                                    simp_tac iterate_Cprod_ss 1,
                                    induct_tac "n" 1,
                                simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
                                    asm_full_simp_tac (HOLCF_ss addsimps
                                          (List.filter (has_fewer_prems 1) copy_rews)) 1,
                                    TRY(safe_tac HOL_cs)] @
                            (List.concat(map (fn ((dn,_),cons) => map (fn (con,args) =>
                                    if nonlazy_rec args = [] then all_tac else
                                    EVERY(map c_UU_tac (nonlazy_rec args)) THEN
                                    asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1
                                                               ) cons) eqs)));
    in
    val take_rews = map standard (atomize take_stricts @ take_0s @ atomize take_apps);
    end; (* local *)

    local
      fun one_con p (con,args) = Library.foldr mk_All (map vname args,
            lift_defined (bound_arg (map vname args)) (nonlazy args,
            lift (fn arg => %:(P_name (1+rec_of arg)) $ bound_arg args arg)
             (List.filter is_rec args,mk_trp(%:p $ con_app2 con (bound_arg args) args))));
      fun one_eq ((p,cons),concl) = (mk_trp(%:p $ UU) ===>
                               Library.foldr (op ===>) (map (one_con p) cons,concl));
      fun ind_term concf = Library.foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
                            mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
      val take_ss = HOL_ss addsimps take_rews;
      fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
                                   1 dnames);
      fun ind_prems_tac prems = EVERY(List.concat (map (fn cons => (
                                         resolve_tac prems 1 ::
                                         List.concat (map (fn (_,args) =>
                                           resolve_tac prems 1 ::
                                           map (K(atac 1)) (nonlazy args) @
                                           map (K(atac 1)) (List.filter is_rec args))
                                         cons))) conss));
      local
        (* check whether every/exists constructor of the n-th part of the equation:
           it has a possibly indirectly recursive argument that isn't/is possibly
           indirectly lazy *)
        fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg =>
              is_rec arg andalso not(rec_of arg mem ns) andalso
              ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse
                rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns)
                  (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
              ) o snd) cons;
        fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
        fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (warning
            ("domain "^List.nth(dnames,n)^" is empty!"); true) else false;
        fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;

      in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
         val is_emptys = map warn n__eqs;
         val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
      end;
    in (* local *)
    val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %:(P_name n)$
                                 (dc_take dn $ %:"n" `%(x_name n)))) (fn prems => [
                                    quant_tac 1,
                                    simp_tac HOL_ss 1,
                                    induct_tac "n" 1,
                                    simp_tac (take_ss addsimps prems) 1,
                                    TRY(safe_tac HOL_cs)]
                                    @ List.concat(map (fn (cons,cases) => [
                                     res_inst_tac [("x","x")] cases 1,
                                     asm_simp_tac (take_ss addsimps prems) 1]
                                     @ List.concat(map (fn (con,args) =>
                                      asm_simp_tac take_ss 1 ::
                                      map (fn arg =>
                                       case_UU_tac (prems@con_rews) 1 (
                               List.nth(dnames,rec_of arg)^"_take n$"^vname arg))
                                      (List.filter is_nonlazy_rec args) @ [
                                      resolve_tac prems 1] @
                                      map (K (atac 1))      (nonlazy args) @
                                      map (K (etac spec 1)) (List.filter is_rec args))
                                     cons))
                                    (conss~~cases)));

    val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
                    mk_trp(dc_take dn $ Bound 0 `%(x_name n) ===
                           dc_take dn $ Bound 0 `%(x_name n^"'")))
               ===> mk_trp(%:(x_name n) === %:(x_name n^"'"))) (fn prems => [
                            res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
                            res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
                                    stac fix_def2 1,
                                    REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
                                                   THEN chain_tac 1)),
                                    stac contlub_cfun_fun 1,
                                    stac contlub_cfun_fun 2,
                                    rtac lub_equal 3,
                                    chain_tac 1,
                                    rtac allI 1,
                                    resolve_tac prems 1])) 1 (dnames~~axs_reach);

    (* ----- theorems concerning finiteness and induction ----------------------- *)

    val (finites,ind) = if is_finite then
      let
        fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
        val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%:"x")) ===>
            mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %:"x" === UU),
            take_enough dn)) ===> mk_trp(take_enough dn)) [
                                    etac disjE 1,
                                    etac notE 1,
                                    resolve_tac take_lemmas 1,
                                    asm_simp_tac take_ss 1,
                                    atac 1]) dnames;
        val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn
            (fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
             mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU,
                     dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
                                    rtac allI 1,
                                    induct_tac "n" 1,
                                    simp_tac take_ss 1,
                            TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
                                    List.concat(mapn (fn n => fn (cons,cases) => [
                                      simp_tac take_ss 1,
                                      rtac allI 1,
                                      res_inst_tac [("x",x_name n)] cases 1,
                                      asm_simp_tac take_ss 1] @
                                      List.concat(map (fn (con,args) =>
                                        asm_simp_tac take_ss 1 ::
                                        List.concat(map (fn vn => [
                                          eres_inst_tac [("x",vn)] all_dupE 1,
                                          etac disjE 1,
                                          asm_simp_tac (HOL_ss addsimps con_rews) 1,
                                          asm_simp_tac take_ss 1])
                                        (nonlazy_rec args)))
                                      cons))
                                    1 (conss~~cases)));
        val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
                                                    %%:(dn^"_finite") $ %:"x"))[
                                    case_UU_tac take_rews 1 "x",
                                    eresolve_tac finite_lemmas1a 1,
                                    step_tac HOL_cs 1,
                                    step_tac HOL_cs 1,
                                    cut_facts_tac [l1b] 1,
                            fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
      in
      (finites,
       pg'' thy[](ind_term (fn n => fn dn => %:(P_name n) $ %:(x_name n)))(fn prems =>
                                    TRY(safe_tac HOL_cs) ::
                             List.concat (map (fn (finite,fin_ind) => [
                                   rtac(rewrite_rule axs_finite_def finite RS exE)1,
                                    etac subst 1,
                                    rtac fin_ind 1,
                                    ind_prems_tac prems])
                                       (finites~~(atomize finite_ind)) ))
    ) end (* let *) else
      (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy)
                        [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
       pg'' thy [] (Library.foldr (op ===>) (mapn (fn n => K(mk_trp(%%:admN $ %:(P_name n))))
                   1 dnames, ind_term (fn n => fn dn => %:(P_name n) $ %:(x_name n))))
                       (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1)
                                        axs_reach @ [
                                    quant_tac 1,
                                    rtac (adm_impl_admw RS wfix_ind) 1,
                                     REPEAT_DETERM(rtac adm_all2 1),
                                     REPEAT_DETERM(TRY(rtac adm_conj 1) THEN
                                                       rtac adm_subst 1 THEN
                                            cont_tacR 1 THEN resolve_tac prems 1),
                                    strip_tac 1,
                                    rtac (rewrite_rule axs_take_def finite_ind) 1,
                                    ind_prems_tac prems])
      handle ERROR => (warning "Cannot prove infinite induction rule"; refl))
    end; (* local *)

    (* ----- theorem concerning coinduction ------------------------------------- *)

    local
      val xs = mapn (fn n => K (x_name n)) 1 dnames;
      fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
      val take_ss = HOL_ss addsimps take_rews;
      val sproj   = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
      val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%:(comp_dname^"_bisim") $ %:"R",
                    Library.foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
                      Library.foldr mk_imp (mapn (fn n => K(proj (%:"R") eqs n $
                                          bnd_arg n 0 $ bnd_arg n 1)) 0 dnames,
                        foldr' mk_conj (mapn (fn n => fn dn =>
                                    (dc_take dn $ %:"n" `bnd_arg n 0 ===
                                    (dc_take dn $ %:"n" `bnd_arg n 1)))0 dnames))))))
                                 ([ rtac impI 1,
                                    induct_tac "n" 1,
                                    simp_tac take_ss 1,
                                    safe_tac HOL_cs] @
                                    List.concat(mapn (fn n => fn x => [
                                      rotate_tac (n+1) 1,
                                      etac all2E 1,
                                      eres_inst_tac [("P1", sproj "R" eqs n^
                                            " "^x^" "^x^"'")](mp RS disjE) 1,
                                      TRY(safe_tac HOL_cs),
                                      REPEAT(CHANGED(asm_simp_tac take_ss 1))])
                                    0 xs));
    in
    val coind = pg [] (mk_trp(%%:(comp_dname^"_bisim") $ %:"R") ===>
                    Library.foldr (op ===>) (mapn (fn n => fn x =>
                      mk_trp(proj (%:"R") eqs n $ %:x $ %:(x^"'"))) 0 xs,
                      mk_trp(foldr' mk_conj (map (fn x => %:x === %:(x^"'")) xs)))) ([
                                    TRY(safe_tac HOL_cs)] @
                                    List.concat(map (fn take_lemma => [
                                      rtac take_lemma 1,
                                      cut_facts_tac [coind_lemma] 1,
                                      fast_tac HOL_cs 1])
                                    take_lemmas));
    end; (* local *)

    in thy |> Theory.add_path comp_dnam
           |> (#1 o (PureThy.add_thmss (map Thm.no_attributes [
    		("take_rews"  , take_rews  ),
    		("take_lemmas", take_lemmas),
    		("finites"    , finites    ),
    		("finite_ind", [finite_ind]),
    		("ind"       , [ind       ]),
    		("coind"     , [coind     ])])))
           |> Theory.parent_path |> rpair take_rews
    end; (* let *)
    end; (* local *)
    end; (* struct *)