(*  Title:      HOLCF/Tools/domain/domain_theorems.ML

     ID:         $Id$

     Author:     David von Oheimb

                 New proofs/tactics by Brian Huffman

 Proof generator for domain command.

 *)

 val HOLCF_ss = @{simpset};

 structure Domain_Theorems = struct

 val quiet_mode = ref false;

 val trace_domain = ref false;

 fun message s = if !quiet_mode then () else writeln s;

 fun trace s = if !trace_domain then tracing s else ();

 local

 val adm_impl_admw = @{thm adm_impl_admw};

 val adm_all = @{thm adm_all};

 val adm_conj = @{thm adm_conj};

 val adm_subst = @{thm adm_subst};

 val antisym_less_inverse = @{thm antisym_less_inverse};

 val beta_cfun = @{thm beta_cfun};

 val cfun_arg_cong = @{thm cfun_arg_cong};

 val ch2ch_Rep_CFunL = @{thm ch2ch_Rep_CFunL};

 val ch2ch_Rep_CFunR = @{thm ch2ch_Rep_CFunR};

 val chain_iterate = @{thm chain_iterate};

 val compact_ONE = @{thm compact_ONE};

 val compact_sinl = @{thm compact_sinl};

 val compact_sinr = @{thm compact_sinr};

 val compact_spair = @{thm compact_spair};

 val compact_up = @{thm compact_up};

 val contlub_cfun_arg = @{thm contlub_cfun_arg};

 val contlub_cfun_fun = @{thm contlub_cfun_fun};

 val fix_def2 = @{thm fix_def2};

 val injection_eq = @{thm injection_eq};

 val injection_less = @{thm injection_less};

 val lub_equal = @{thm lub_equal};

 val monofun_cfun_arg = @{thm monofun_cfun_arg};

 val retraction_strict = @{thm retraction_strict};

 val spair_eq = @{thm spair_eq};

 val spair_less = @{thm spair_less};

 val sscase1 = @{thm sscase1};

 val ssplit1 = @{thm ssplit1};

 val strictify1 = @{thm strictify1};

 val wfix_ind = @{thm wfix_ind};

 val iso_intro       = @{thm iso.intro};

 val iso_abs_iso     = @{thm iso.abs_iso};

 val iso_rep_iso     = @{thm iso.rep_iso};

 val iso_abs_strict  = @{thm iso.abs_strict};

 val iso_rep_strict  = @{thm iso.rep_strict};

 val iso_abs_defin'  = @{thm iso.abs_defin'};

 val iso_rep_defin'  = @{thm iso.rep_defin'};

 val iso_abs_defined = @{thm iso.abs_defined};

 val iso_rep_defined = @{thm iso.rep_defined};

 val iso_compact_abs = @{thm iso.compact_abs};

 val iso_compact_rep = @{thm iso.compact_rep};

 val iso_iso_swap    = @{thm iso.iso_swap};

 val exh_start = @{thm exh_start};

 val ex_defined_iffs = @{thms ex_defined_iffs};

 val exh_casedist0 = @{thm exh_casedist0};

 val exh_casedists = @{thms exh_casedists};

 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 legacy_infer_term thy t =

   let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)

   in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;

 fun pg'' thy defs t tacs =

   let

     val t' = legacy_infer_term thy t;

     val asms = Logic.strip_imp_prems t';

     val prop = Logic.strip_imp_concl t';

     fun tac {prems, context} =

       rewrite_goals_tac defs THEN

       EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})

   in Goal.prove_global thy [] asms prop tac end;

 fun pg' thy defs t tacsf =

   let

     fun tacs {prems, context} =

       if null prems then tacsf context

       else cut_facts_tac prems 1 :: tacsf context;

   in pg'' thy defs t tacs end;

 fun case_UU_tac ctxt rews i v =

   InductTacs.case_tac ctxt (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 = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}

 val dist_eqI = @{lemma "!!x::'a::po. ~ x << y ==> x ~= y" by (blast dest!: antisym_less_inverse)}

 in

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

 let

 val _ = message ("Proving isomorphism properties of domain "^dname^" ...");

 val pg = pg' thy;

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

 local

   fun ga s dn = PureThy.get_thm thy (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;

   fun get_def mk_name (con,_) = ga (mk_name con^"_def") dname;

   val axs_con_def = map (get_def extern_name) cons;

   val axs_dis_def = map (get_def dis_name) cons;

   val axs_mat_def = map (get_def mat_name) cons;

   val axs_pat_def = map (get_def pat_name) cons;

   val axs_sel_def =

     let

       fun def_of_sel sel = ga (sel^"_def") dname;

       fun def_of_arg arg = Option.map def_of_sel (sel_of arg);

       fun defs_of_con (_, args) = List.mapPartial def_of_arg args;

     in

       maps defs_of_con cons

     end;

   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-------------------- *)

 val _ = trace " Proving beta reduction rules...";

 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 = list_ccomb (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 (K [rtac reflexive_thm 1]) end;

 end;

 val _ = trace "Proving when_appl...";

 val when_appl = appl_of_def ax_when_def;

 val _ = trace "Proving con_appls...";

 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 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;

       val eqn = %:x_name === con_app2 con %: vns;

       val conj = foldr1 mk_conj (eqn :: map (defined o %:) (nonlazy args));

     in Library.foldr mk_ex (vns, conj) end;

   val conj_assoc = @{thm conj_assoc};

   val exh = foldr1 mk_disj ((%:x_name === UU) :: map one_con cons);

   val thm1 = instantiate' [SOME (cons2ctyp cons)] [] exh_start;

   val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1;

   val thm3 = rewrite_rule [mk_meta_eq @{thm conj_assoc}] thm2;

   (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *)

   val tacs = [

     rtac disjE 1,

     etac (rep_defin' RS disjI1) 2,

     etac disjI2 2,

     rewrite_goals_tac [mk_meta_eq iso_swap],

     rtac thm3 1];

 in

   val _ = trace " Proving exhaust...";

   val exhaust = pg con_appls (mk_trp exh) (K tacs);

   val _ = trace " Proving casedist...";

   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 = list_ccomb (%%:(dname^"_when"), mapn bound_fun 1 cons);

 in

   val _ = trace " Proving when_strict...";

   val when_strict =

     let

       val axs = [when_appl, mk_meta_eq rep_strict];

       val goal = bind_fun (mk_trp (strict when_app));

       val tacs = [resolve_tac [sscase1, ssplit1, strictify1] 1];

     in pg axs goal (K tacs) end;

   val _ = trace " Proving when_apps...";

   val when_apps =

     let

       fun one_when n (con,args) =

         let

           val axs = when_appl :: con_appls;

           val goal = bind_fun (lift_defined %: (nonlazy args, 

                 mk_trp (when_app`(con_app con args) ===

                        list_ccomb (bound_fun n 0, map %# args))));

           val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1];

         in pg axs goal (K tacs) end;

     in mapn one_when 1 cons end;

 end;

 val when_rews = when_strict :: when_apps;

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

 local

   fun dis_strict (con, _) =

     let

       val goal = mk_trp (strict (%%:(dis_name con)));

     in pg axs_dis_def goal (K [rtac when_strict 1]) end;

   fun dis_app c (con, args) =

     let

       val lhs = %%:(dis_name c) ` con_app con args;

       val rhs = if con = c then TT else FF;

       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));

       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];

     in pg axs_dis_def goal (K tacs) end;

   val _ = trace " Proving dis_apps...";

   val dis_apps = maps (fn (c,_) => map (dis_app c) cons) cons;

   fun dis_defin (con, args) =

     let

       val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name);

       val tacs =

         [rtac casedist 1,

          contr_tac 1,

          DETERM_UNTIL_SOLVED (CHANGED

           (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))];

     in pg [] goal (K tacs) end;

   val _ = trace " Proving dis_stricts...";

   val dis_stricts = map dis_strict cons;

   val _ = trace " Proving dis_defins...";

   val dis_defins = map dis_defin cons;

 in

   val dis_rews = dis_stricts @ dis_defins @ dis_apps;

 end;

 local

   fun mat_strict (con, _) =

     let

       val goal = mk_trp (strict (%%:(mat_name con)));

       val tacs = [rtac when_strict 1];

     in pg axs_mat_def goal (K tacs) end;

   val _ = trace " Proving mat_stricts...";

   val mat_stricts = map mat_strict cons;

   fun one_mat c (con, args) =

     let

       val lhs = %%:(mat_name c) ` con_app con args;

       val rhs =

         if con = c

         then mk_return (mk_ctuple (map %# args))

         else mk_fail;

       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));

       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];

     in pg axs_mat_def goal (K tacs) end;

   val _ = trace " Proving mat_apps...";

   val mat_apps =

     maps (fn (c,_) => map (one_mat c) cons) cons;

 in

   val mat_rews = mat_stricts @ mat_apps;

 end;

 local

   fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;

   fun pat_lhs (con,args) = mk_branch (list_comb (%%:(pat_name con), ps args));

   fun pat_rhs (con,[]) = mk_return ((%:"rhs") ` HOLogic.unit)

     | pat_rhs (con,args) =

         (mk_branch (mk_ctuple_pat (ps args)))

           `(%:"rhs")`(mk_ctuple (map %# args));

   fun pat_strict c =

     let

       val axs = @{thm branch_def} :: axs_pat_def;

       val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));

       val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];

     in pg axs goal (K tacs) end;

   fun pat_app c (con, args) =

     let

       val axs = @{thm branch_def} :: axs_pat_def;

       val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);

       val rhs = if con = fst c then pat_rhs c else mk_fail;

       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));

       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];

     in pg axs goal (K tacs) end;

   val _ = trace " Proving pat_stricts...";

   val pat_stricts = map pat_strict cons;

   val _ = trace " Proving pat_apps...";

   val pat_apps = maps (fn c => map (pat_app c) cons) cons;

 in

   val pat_rews = pat_stricts @ pat_apps;

 end;

 local

   val rev_contrapos = @{thm rev_contrapos};

   fun con_strict (con, args) = 

     let

       fun one_strict vn =

         let

           fun f arg = if vname arg = vn then UU else %# arg;

           val goal = mk_trp (con_app2 con f args === UU);

           val tacs = [asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1];

         in pg con_appls goal (K tacs) end;

     in map one_strict (nonlazy args) end;

   fun con_defin (con, args) =

     let

       val concl = mk_trp (defined (con_app con args));

       val goal = lift_defined %: (nonlazy args, concl);

       fun tacs ctxt = [

         rtac @{thm rev_contrapos} 1,

         eres_inst_tac ctxt [(("f", 0), dis_name con)] cfun_arg_cong 1,

         asm_simp_tac (HOLCF_ss addsimps dis_rews) 1];

     in pg [] goal tacs end;

 in

   val _ = trace " Proving con_stricts...";

   val con_stricts = maps con_strict cons;

   val _ = trace " Proving pat_defins...";

   val con_defins = map con_defin cons;

   val con_rews = con_stricts @ con_defins;

 end;

 local

   val rules =

     [compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE];

   fun con_compact (con, args) =

     let

       val concl = mk_trp (mk_compact (con_app con args));

       val goal = lift (fn x => mk_compact (%#x)) (args, concl);

       val tacs = [

         rtac (iso_locale RS iso_compact_abs) 1,

         REPEAT (resolve_tac rules 1 ORELSE atac 1)];

     in pg con_appls goal (K tacs) end;

 in

   val _ = trace " Proving con_compacts...";

   val con_compacts = map con_compact cons;

 end;

 local

   fun one_sel sel =

     pg axs_sel_def (mk_trp (strict (%%:sel)))

       (K [simp_tac (HOLCF_ss addsimps when_rews) 1]);

   fun sel_strict (_, args) =

     List.mapPartial (Option.map one_sel o sel_of) args;

 in

   val _ = trace " Proving sel_stricts...";

   val sel_stricts = maps sel_strict cons;

 end;

 local

   fun sel_app_same c n sel (con, args) =

     let

       val nlas = nonlazy args;

       val vns = map vname args;

       val vnn = List.nth (vns, n);

       val nlas' = List.filter (fn v => v <> vnn) nlas;

       val lhs = (%%:sel)`(con_app con args);

       val goal = lift_defined %: (nlas', mk_trp (lhs === %:vnn));

       fun tacs1 ctxt =

         if vnn mem nlas

         then [case_UU_tac ctxt (when_rews @ con_stricts) 1 vnn]

         else [];

       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];

     in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;

   fun sel_app_diff c n sel (con, args) =

     let

       val nlas = nonlazy args;

       val goal = mk_trp (%%:sel ` con_app con args === UU);

       fun tacs1 ctxt = map (case_UU_tac ctxt (when_rews @ con_stricts) 1) nlas;

       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];

     in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;

   fun sel_app c n sel (con, args) =

     if con = c

     then sel_app_same c n sel (con, args)

     else sel_app_diff c n sel (con, args);

   fun one_sel c n sel = map (sel_app c n sel) cons;

   fun one_sel' c n arg = Option.map (one_sel c n) (sel_of arg);

   fun one_con (c, args) =

     flat (map_filter I (mapn (one_sel' c) 0 args));

 in

   val _ = trace " Proving sel_apps...";

   val sel_apps = maps one_con cons;

 end;

 local

   fun sel_defin sel =

     let

       val goal = defined (%:x_name) ==> defined (%%:sel`%x_name);

       val tacs = [

         rtac casedist 1,

         contr_tac 1,

         DETERM_UNTIL_SOLVED (CHANGED

           (asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))];

     in pg [] goal (K tacs) end;

 in

   val _ = trace " Proving sel_defins...";

   val sel_defins =

     if length cons = 1

     then List.mapPartial (fn arg => Option.map sel_defin (sel_of arg))

                  (filter_out is_lazy (snd (hd cons)))

     else [];

 end;

 val sel_rews = sel_stricts @ sel_defins @ sel_apps;

 val rev_contrapos = @{thm rev_contrapos};

 val _ = trace " Proving dist_les...";

 val distincts_le =

   let

     fun dist (con1, args1) (con2, args2) =

       let

         val goal = lift_defined %: (nonlazy args1,

                         mk_trp (con_app con1 args1 ~<< con_app con2 args2));

         fun tacs ctxt = [

           rtac @{thm rev_contrapos} 1,

           eres_inst_tac ctxt [(("f", 0), dis_name con1)] monofun_cfun_arg 1]

           @ map (case_UU_tac ctxt (con_stricts @ dis_rews) 1) (nonlazy args2)

           @ [asm_simp_tac (HOLCF_ss addsimps dis_rews) 1];

       in pg [] goal tacs end;

     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, Name.variant_list (map vname args1) (map vname args2)));

         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 = flat (flat distincts_le);

 val _ = trace " Proving dist_eqs...";

 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) = flat

 	            (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 =

         foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs));

       val prem = rel (con_app con largs, con_app con rargs);

       val sargs = case largs of [_] => [] | _ => nonlazy args;

       val prop = lift_defined %: (sargs, mk_trp (prem === concl));

     in pg con_appls prop end;

   val cons' = List.filter (fn (_,args) => args<>[]) cons;

 in

   val _ = trace " Proving inverts...";

   val inverts =

     let

       val abs_less = ax_abs_iso RS (allI RS injection_less);

       val tacs =

         [asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1];

     in map (fn (con, args) => pgterm (op <<) con args (K tacs)) cons' end;

   val _ = trace " Proving injects...";

   val injects =

     let

       val abs_eq = ax_abs_iso RS (allI RS injection_eq);

       val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1];

     in map (fn (con, args) => pgterm (op ===) con args (K tacs)) cons' end;

 end;

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

 val copy_strict =

   let

     val goal = mk_trp (strict (dc_copy `% "f"));

     val tacs = [asm_simp_tac (HOLCF_ss addsimps [abs_strict, when_strict]) 1];

   in pg [ax_copy_def] goal (K tacs) end;

 local

   fun copy_app (con, args) =

     let

       val lhs = dc_copy`%"f"`(con_app con args);

       val rhs = con_app2 con (app_rec_arg (cproj (%:"f") eqs)) args;

       val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));

       val args' = List.filter (fn a => not (is_rec a orelse is_lazy a)) args;

       val stricts = abs_strict::when_strict::con_stricts;

       fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';

       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_apps) 1];

     in pg [ax_copy_def] goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;

 in

   val _ = trace " Proving copy_apps...";

   val copy_apps = map copy_app cons;

 end;

 local

   fun one_strict (con, args) = 

     let

       val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);

       val rews = copy_strict :: copy_apps @ con_rews;

       fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @

         [asm_simp_tac (HOLCF_ss addsimps rews) 1];

     in pg [] goal tacs end;

   fun has_nonlazy_rec (_, args) = exists is_nonlazy_rec args;

 in

   val _ = trace " Proving copy_stricts...";

   val copy_stricts = map one_strict (List.filter has_nonlazy_rec cons);

 end;

 val copy_rews = copy_strict :: copy_apps @ copy_stricts;

 in

   thy

     |> Sign.add_path (Long_Name.base_name dname)

     |> (snd o PureThy.add_thmss [

         ((Binding.name "iso_rews" , iso_rews  ), [Simplifier.simp_add]),

         ((Binding.name "exhaust"  , [exhaust] ), []),

         ((Binding.name "casedist" , [casedist]), [Induct.cases_type dname]),

         ((Binding.name "when_rews", when_rews ), [Simplifier.simp_add]),

         ((Binding.name "compacts", con_compacts), [Simplifier.simp_add]),

         ((Binding.name "con_rews", con_rews), [Simplifier.simp_add]),

         ((Binding.name "sel_rews", sel_rews), [Simplifier.simp_add]),

         ((Binding.name "dis_rews", dis_rews), [Simplifier.simp_add]),

         ((Binding.name "pat_rews", pat_rews), [Simplifier.simp_add]),

         ((Binding.name "dist_les", dist_les), [Simplifier.simp_add]),

         ((Binding.name "dist_eqs", dist_eqs), [Simplifier.simp_add]),

         ((Binding.name "inverts" , inverts ), [Simplifier.simp_add]),

         ((Binding.name "injects" , injects ), [Simplifier.simp_add]),

         ((Binding.name "copy_rews", copy_rews), [Simplifier.simp_add]),

         ((Binding.name "match_rews", mat_rews), [Simplifier.simp_add])

        ])

     |> Sign.parent_path

     |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @

         pat_rews @ dist_les @ dist_eqs @ copy_rews)

 end; (* let *)

 fun comp_theorems (comp_dnam, eqs: eq list) thy =

 let

 val global_ctxt = ProofContext.init thy;

 val dnames = map (fst o fst) eqs;

 val conss  = map  snd        eqs;

 val comp_dname = Sign.full_bname thy comp_dnam;

 val _ = message ("Proving induction properties of domain "^comp_dname^" ...");

 val pg = pg' thy;

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

 local

   fun ga s dn = PureThy.get_thm thy (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

   fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);

   fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);

 in

   val cases = map (gt  "casedist" ) dnames;

   val con_rews  = maps (gts "con_rews" ) dnames;

   val copy_rews = maps (gts "copy_rews") dnames;

 end;

 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 @{theory Fix};

   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 _ = trace " Proving take_stricts...";

   val take_stricts =

     let

       fun one_eq ((dn, args), _) = strict (dc_take dn $ %:"n");

       val goal = mk_trp (foldr1 mk_conj (map one_eq eqs));

       fun tacs ctxt = [

         InductTacs.induct_tac ctxt [[SOME "n"]] 1,

         simp_tac iterate_Cprod_ss 1,

         asm_simp_tac (iterate_Cprod_ss addsimps copy_rews) 1];

     in pg copy_take_defs goal tacs end;

   val take_stricts' = rewrite_rule copy_take_defs take_stricts;

   fun take_0 n dn =

     let

       val goal = mk_trp ((dc_take dn $ %%:"HOL.zero") `% x_name n === UU);

     in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;

   val take_0s = mapn take_0 1 dnames;

   fun c_UU_tac ctxt = case_UU_tac ctxt (take_stricts'::copy_con_rews) 1;

   val _ = trace " Proving take_apps...";

   val take_apps =

     let

       fun mk_eqn dn (con, args) =

         let

           fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";

           val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);

           val rhs = con_app2 con (app_rec_arg mk_take) args;

         in Library.foldr mk_all (map vname args, lhs === rhs) end;

       fun mk_eqns ((dn, _), cons) = map (mk_eqn dn) cons;

       val goal = mk_trp (foldr1 mk_conj (maps mk_eqns eqs));

       val simps = List.filter (has_fewer_prems 1) copy_rews;

       fun con_tac ctxt (con, args) =

         if nonlazy_rec args = []

         then all_tac

         else EVERY (map (c_UU_tac ctxt) (nonlazy_rec args)) THEN

           asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1;

       fun eq_tacs ctxt ((dn, _), cons) = map (con_tac ctxt) cons;

       fun tacs ctxt =

         simp_tac iterate_Cprod_ss 1 ::

         InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::

         simp_tac (iterate_Cprod_ss addsimps copy_con_rews) 1 ::

         asm_full_simp_tac (HOLCF_ss addsimps simps) 1 ::

         TRY (safe_tac HOL_cs) ::

         maps (eq_tacs ctxt) eqs;

     in pg copy_take_defs goal tacs end;

 in

   val take_rews = map standard

     (atomize global_ctxt take_stricts @ take_0s @ atomize global_ctxt take_apps);

 end; (* local *)

 local

   fun one_con p (con,args) =

     let

       fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;

       val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);

       val t2 = lift ind_hyp (List.filter is_rec args, t1);

       val t3 = lift_defined (bound_arg (map vname args)) (nonlazy args, t2);

     in Library.foldr mk_All (map vname args, t3) end;

   fun one_eq ((p, cons), concl) =

     mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);

   fun ind_term concf = Library.foldr one_eq

     (mapn (fn n => fn x => (P_name n, x)) 1 conss,

      mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));

   val take_ss = HOL_ss addsimps take_rews;

   fun quant_tac ctxt i = EVERY

     (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);

   fun ind_prems_tac prems = EVERY

     (maps (fn cons =>

       (resolve_tac prems 1 ::

         maps (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 _ = trace " Proving finite_ind...";

   val finite_ind =

     let

       fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));

       val goal = ind_term concf;

       fun tacf {prems, context} =

         let

           val tacs1 = [

             quant_tac context 1,

             simp_tac HOL_ss 1,

             InductTacs.induct_tac context [[SOME "n"]] 1,

             simp_tac (take_ss addsimps prems) 1,

             TRY (safe_tac HOL_cs)];

           fun arg_tac arg =

             case_UU_tac context (prems @ con_rews) 1

               (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);

           fun con_tacs (con, args) = 

             asm_simp_tac take_ss 1 ::

             map arg_tac (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);

           fun cases_tacs (cons, cases) =

             res_inst_tac context [(("x", 0), "x")] cases 1 ::

             asm_simp_tac (take_ss addsimps prems) 1 ::

             maps con_tacs cons;

         in

           tacs1 @ maps cases_tacs (conss ~~ cases)

         end;

     in pg'' thy [] goal tacf end;

   val _ = trace " Proving take_lemmas...";

   val take_lemmas =

     let

       fun take_lemma n (dn, ax_reach) =

         let

           val lhs = dc_take dn $ Bound 0 `%(x_name n);

           val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");

           val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));

           val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;

           fun tacf {prems, context} = [

             res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,

             res_inst_tac context [(("t", 0), 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];

         in pg'' thy axs_take_def goal tacf end;

     in mapn take_lemma 1 (dnames ~~ axs_reach) end;

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

   val _ = trace " Proving finites, ind...";

   val (finites, ind) =

     if is_finite

     then (* finite case *)

       let 

         fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");

         fun dname_lemma dn =

           let

             val prem1 = mk_trp (defined (%:"x"));

             val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);

             val prem2 = mk_trp (mk_disj (disj1, take_enough dn));

             val concl = mk_trp (take_enough dn);

             val goal = prem1 ===> prem2 ===> concl;

             val tacs = [

               etac disjE 1,

               etac notE 1,

               resolve_tac take_lemmas 1,

               asm_simp_tac take_ss 1,

               atac 1];

           in pg [] goal (K tacs) end;

         val finite_lemmas1a = map dname_lemma dnames;

         val finite_lemma1b =

           let

             fun mk_eqn n ((dn, args), _) =

               let

                 val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;

                 val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;

               in

                 mk_constrainall

                   (x_name n, Type (dn,args), mk_disj (disj1, disj2))

               end;

             val goal =

               mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));

             fun arg_tacs ctxt vn = [

               eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,

               etac disjE 1,

               asm_simp_tac (HOL_ss addsimps con_rews) 1,

               asm_simp_tac take_ss 1];

             fun con_tacs ctxt (con, args) =

               asm_simp_tac take_ss 1 ::

               maps (arg_tacs ctxt) (nonlazy_rec args);

             fun foo_tacs ctxt n (cons, cases) =

               simp_tac take_ss 1 ::

               rtac allI 1 ::

               res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::

               asm_simp_tac take_ss 1 ::

               maps (con_tacs ctxt) cons;

             fun tacs ctxt =

               rtac allI 1 ::

               InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::

               simp_tac take_ss 1 ::

               TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::

               flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));

           in pg [] goal tacs end;

         fun one_finite (dn, l1b) =

           let

             val goal = mk_trp (%%:(dn^"_finite") $ %:"x");

             fun tacs ctxt = [

               case_UU_tac ctxt 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];

           in pg axs_finite_def goal tacs end;

         val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);

         val ind =

           let

             fun concf n dn = %:(P_name n) $ %:(x_name n);

             fun tacf {prems, context} =

               let

                 fun finite_tacs (finite, fin_ind) = [

                   rtac(rewrite_rule axs_finite_def finite RS exE)1,

                   etac subst 1,

                   rtac fin_ind 1,

                   ind_prems_tac prems];

               in

                 TRY (safe_tac HOL_cs) ::

                 maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)

               end;

           in pg'' thy [] (ind_term concf) tacf end;

       in (finites, ind) end (* let *)

     else (* infinite case *)

       let

         fun one_finite n dn =

           read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;

         val finites = mapn one_finite 1 dnames;

         val goal =

           let

             fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));

             fun concf n dn = %:(P_name n) $ %:(x_name n);

           in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;

         fun tacf {prems, context} =

           map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [

           quant_tac context 1,

           rtac (adm_impl_admw RS wfix_ind) 1,

           REPEAT_DETERM (rtac adm_all 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];

         val ind = (pg'' thy [] goal tacf

           handle ERROR _ =>

             (warning "Cannot prove infinite induction rule"; refl));

       in (finites, ind) end;

 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 _ = trace " Proving coind_lemma...";

   val coind_lemma =

     let

       fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;

       fun mk_eqn n dn =

         (dc_take dn $ %:"n" ` bnd_arg n 0) ===

         (dc_take dn $ %:"n" ` bnd_arg n 1);

       fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));

       val goal =

         mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",

           Library.foldr mk_all2 (xs,

             Library.foldr mk_imp (mapn mk_prj 0 dnames,

               foldr1 mk_conj (mapn mk_eqn 0 dnames)))));

       fun x_tacs ctxt n x = [

         rotate_tac (n+1) 1,

         etac all2E 1,

         eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,

         TRY (safe_tac HOL_cs),

         REPEAT (CHANGED (asm_simp_tac take_ss 1))];

       fun tacs ctxt = [

         rtac impI 1,

         InductTacs.induct_tac ctxt [[SOME "n"]] 1,

         simp_tac take_ss 1,

         safe_tac HOL_cs] @

         flat (mapn (x_tacs ctxt) 0 xs);

     in pg [ax_bisim_def] goal tacs end;

 in

   val _ = trace " Proving coind...";

   val coind = 

     let

       fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));

       fun mk_eqn x = %:x === %:(x^"'");

       val goal =

         mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>

           Logic.list_implies (mapn mk_prj 0 xs,

             mk_trp (foldr1 mk_conj (map mk_eqn xs)));

       val tacs =

         TRY (safe_tac HOL_cs) ::

         maps (fn take_lemma => [

           rtac take_lemma 1,

           cut_facts_tac [coind_lemma] 1,

           fast_tac HOL_cs 1])

         take_lemmas;

     in pg [] goal (K tacs) end;

 end; (* local *)

 val inducts = ProjectRule.projections (ProofContext.init thy) ind;

 fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);

 in thy |> Sign.add_path comp_dnam

        |> (snd o (PureThy.add_thmss (map (Thm.no_attributes o apfst Binding.name) [

 		("take_rews"  , take_rews  ),

 		("take_lemmas", take_lemmas),

 		("finites"    , finites    ),

 		("finite_ind", [finite_ind]),

 		("ind"       , [ind       ]),

 		("coind"     , [coind     ])])))

        |> (snd o (PureThy.add_thmss (map ind_rule (dnames ~~ inducts))))

        |> Sign.parent_path |> pair take_rews

 end; (* let *)

 end; (* local *)

 end; (* struct *)