src/HOLCF/domain/theorems.ML
 author wenzelm Wed, 02 Nov 2005 14:46:51 +0100 changeset 18057 ad97e231bf8a parent 17985 d5d576b72371 child 18083 cf7669049df5 permissions -rw-r--r--
dist_eqI: the_context();
```
(*  Title:      HOLCF/domain/theorems.ML
ID:         \$Id\$
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 tacs =
let val t' = inferT thy t in
standard (Goal.prove thy [] (Logic.strip_imp_prems t') (Logic.strip_imp_concl t')
(fn prems => rewrite_goals_tac defs THEN EVERY (tacs (map (rewrite_rule defs) prems))))
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

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 (the_context ()) "!!x::'a::po. ~ x << y ==> x ~= y"
(fn prems => [
(*
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, foldr1 mk_conj ((%:x_name === con_app2 con %: vns)::
map (defined o %:) (nonlazy args))) end;
val exh = foldr1 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)))))[
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))))) [
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
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)))))
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))[
) (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 con_compacts =
let
val rules = [compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE];
fun one_compact (con,args) = pg con_appls
(lift (fn x => %%:compactN \$ %:(vname x)) (args, mk_trp(%%:compactN \$ (con_app con args))))
[rtac (iso_locale RS iso_compact_abs) 1, REPEAT (resolve_tac rules 1 ORELSE atac 1)];
in map one_compact cons end;

val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%:sel))) [
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 [])
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
(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)
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 (foldr1 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),
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),
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"))) [
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)
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 ),
("compacts", con_compacts),
("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)])))
|> 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(foldr1 mk_conj(map(fn((dn,args),_)=>
strict(dc_take dn \$ %:"n")) eqs))) ([
induct_tac "n" 1,
simp_tac iterate_Cprod_ss 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(foldr1 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,
(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
) 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(foldr1 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,
TRY(safe_tac HOL_cs)]
@ List.concat(map (fn (cons,cases) => [
res_inst_tac [("x","x")] cases 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",foldr1 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,
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 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,
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,
foldr1 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(foldr1 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 *)

|> (#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 *)
```