(* theorems.ML
ID: $Id$
Author : David von Oheimb
Created: 06-Jun-95
Updated: 08-Jun-95 first proof from cterms
Updated: 26-Jun-95 proofs for exhaustion thms
Updated: 27-Jun-95 proofs for discriminators, constructors and selectors
Updated: 06-Jul-95 proofs for distinctness, invertibility and injectivity
Updated: 17-Jul-95 proofs for induction rules
Updated: 19-Jul-95 proof for co-induction rule
Updated: 28-Aug-95 definedness theorems for selectors (completion)
Updated: 05-Sep-95 simultaneous domain equations (main part)
Updated: 11-Sep-95 simultaneous domain equations (coding finished)
Updated: 13-Sep-95 simultaneous domain equations (debugging)
Copyright 1995 TU Muenchen
*)
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 = #2(Sign.infer_types sg (K None)(K None)[]true([pre_tm],propT));
(*
infix 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;
*)
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 REPEAT_DETERM_UNTIL p tac =
let fun drep st = if p st then Sequence.single st
else (case Sequence.pull(tapply(tac,st)) of
None => Sequence.null
| Some(st',_) => drep st')
in Tactic drep end;
val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);
local val trueI2 = prove_goal HOL.thy "f~=x ==> True" (fn prems => [rtac TrueI 1]) in
val kill_neq_tac = dtac trueI2 end;
fun case_UU_tac rews i v = res_inst_tac [("Q",v^"=UU")] classical2 i THEN
asm_simp_tac (HOLCF_ss addsimps rews) i;
val chain_tac = REPEAT_DETERM o resolve_tac
[is_chain_iterate, ch2ch_fappR, ch2ch_fappL];
(* ----- general proofs ----------------------------------------------------------- *)
val swap3 = prove_goal HOL.thy "[| Q ==> P; ~P |] ==> ~Q" (fn prems => [
cut_facts_tac prems 1,
etac swap 1,
dtac notnotD 1,
etac (hd prems) 1]);
val dist_eqI = prove_goal Porder0.thy "~ x << y ==> x ~= y" (fn prems => [
cut_facts_tac prems 1,
etac swap 1,
dtac notnotD 1,
asm_simp_tac HOLCF_ss 1]);
val cfst_strict = prove_goal Cprod3.thy "cfst`UU = UU" (fn _ => [
(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
val csnd_strict = prove_goal Cprod3.thy "csnd`UU = UU" (fn _ => [
(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
in
fun theorems thy (((dname,_),cons) : eq, eqs :eq list) =
let
val dummy = writeln ("Proving isomorphism properties of domain "^dname^"...");
val pg = pg' thy;
(* ----- getting the axioms and definitions --------------------------------------- *)
local val ga = get_axiom thy in
val ax_abs_iso = ga (dname^"_abs_iso" );
val ax_rep_iso = ga (dname^"_rep_iso" );
val ax_when_def = ga (dname^"_when_def" );
val axs_con_def = map (fn (con,_) => ga (extern_name con ^"_def")) cons;
val axs_dis_def = map (fn (con,_) => ga ( dis_name con ^"_def")) cons;
val axs_sel_def = flat(map (fn (_,args) =>
map (fn arg => ga (sel_of arg ^"_def")) args) cons);
val ax_copy_def = ga (dname^"_copy_def" );
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 (rep_strict, abs_strict) = let
val r = ax_rep_iso RS (ax_abs_iso RS (allI RSN(2,allI RS iso_strict)))
in (r RS conjunct1, r RS conjunct2) end;
val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
etac ssubst 1,
rtac rep_strict 1];
val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
etac ssubst 1,
rtac abs_strict 1];
val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
local
val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
etac (ax_rep_iso RS subst) 1];
fun exh foldr1 cn quant foldr2 var = let
fun one_con (con,args) = let val vns = map vname args in
foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
map (defined o (var vns)) (nonlazy args))) end
in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
in
val cases = let
fun common_tac thm = rtac thm 1 THEN contr_tac 1;
fun unit_tac true = common_tac liftE1
| unit_tac _ = all_tac;
fun prod_tac [] = common_tac oneE
| prod_tac [arg] = unit_tac (is_lazy arg)
| prod_tac (arg::args) =
common_tac sprodE THEN
kill_neq_tac 1 THEN
unit_tac (is_lazy arg) THEN
prod_tac args;
fun sum_one_tac p = SELECT_GOAL(EVERY[
rtac p 1,
rewrite_goals_tac axs_con_def,
dtac iso_swap 1,
simp_tac HOLCF_ss 1,
UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
fun sum_tac [(_,args)] [p] =
prod_tac args THEN sum_one_tac p
| sum_tac ((_,args)::cons') (p::prems) = DETERM(
common_tac ssumE THEN
kill_neq_tac 1 THEN kill_neq_tac 2 THEN
prod_tac args THEN sum_one_tac p) THEN
sum_tac cons' prems
| sum_tac _ _ = Imposs "theorems:sum_tac";
in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
(fn T => T ==> %"P") mk_All
(fn l => foldr (op ===>) (map mk_trp l,mk_trp(%"P")))
bound_arg)
(fn prems => [
cut_facts_tac [excluded_middle] 1,
etac disjE 1,
rtac (hd prems) 2,
etac rep_defin' 2,
if is_one_con_one_arg (not o is_lazy) cons
then rtac (hd (tl prems)) 1 THEN atac 2 THEN
rewrite_goals_tac axs_con_def THEN
simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
else sum_tac cons (tl prems)])end;
val exhaust = pg [] (mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %))) [
rtac cases 1,
UNTIL_SOLVED(fast_tac HOL_cs 1)];
end;
local
val when_app = foldl (op `) (%%(dname^"_when"), map % (when_funs cons));
val when_appl = pg [ax_when_def] (mk_trp(when_app`%x_name===when_body cons
(fn (_,n) => %(nth_elem(n-1,when_funs cons)))`(dc_rep`%x_name))) [
simp_tac HOLCF_ss 1];
in
val when_strict = pg [] ((if is_one_con_one_arg (K true) cons
then fn t => mk_trp(strict(%"f")) ===> t else Id)(mk_trp(strict when_app))) [
simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
val when_apps = let fun one_when n (con,args) = pg axs_con_def
(lift_defined % (nonlazy args, mk_trp(when_app`(con_app con args) ===
mk_cfapp(%(nth_elem(n,when_funs cons)),map %# args))))[
asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
in mapn one_when 0 cons end;
end;
val when_rews = when_strict::when_apps;
(* ----- theorems concerning the constructors, discriminators and selectors ------- *)
val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
(if is_one_con_one_arg (K true) cons then mk_not else Id)
(strict(%%(dis_name con))))) [
simp_tac (HOLCF_ss addsimps (if is_one_con_one_arg (K true) cons
then [ax_when_def] else when_rews)) 1]) cons;
val dis_apps = let fun one_dis c (con,args)= pg (axs_dis_def)
(lift_defined % (nonlazy args, (*(if is_one_con_one_arg is_lazy cons
then curry (lift_defined %#) args else Id)
#################*)
(mk_trp((%%(dis_name c))`(con_app con args) ===
%%(if con=c then "TT" else "FF"))))) [
asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
in flat(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 cases 1,
contr_tac 1,
UNTIL_SOLVED (CHANGED(asm_simp_tac
(HOLCF_ss addsimps dis_apps) 1))]) cons;
val dis_rews = dis_stricts @ dis_defins @ dis_apps;
val con_stricts = flat(map (fn (con,args) => map (fn vn =>
pg (axs_con_def)
(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 swap3 1] @ (if is_one_con_one_arg (K true) cons
then [
if is_lazy (hd args) then rtac defined_up 2
else atac 2,
rtac abs_defin' 1,
asm_full_simp_tac (HOLCF_ss addsimps axs_con_def) 1]
else [
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 flat(map (fn (_,args) => map (fn arg => 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 %
(filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
mk_trp((%%sel)`(con_app con args) === (if con=c then %(nth_elem(n,vns)) else UU))))
( (if con=c then []
else map(case_UU_tac(when_rews@con_stricts)1) nlas)
@(if con=c andalso ((nth_elem(n,vns)) mem nlas)
then[case_UU_tac (when_rews @ con_stricts) 1
(nth_elem(n,vns))] else [])
@ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
in flat(map (fn (c,args) =>
flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
val sel_defins = if length cons = 1 then map (fn arg => pg [] (defined(%x_name) ==>
defined(%%(sel_of arg)`%x_name)) [
rtac cases 1,
contr_tac 1,
UNTIL_SOLVED (CHANGED(asm_simp_tac
(HOLCF_ss addsimps sel_apps) 1))])
(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 swap3 1,
eres_inst_tac [("fo5",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, (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 dists_le = flat (flat distincts_le);
val dists_eq = 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(map (distinct c) ((map fst cs)~~leqs)) @
distincts cs;
in 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);
in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
mk_trp (foldr' mk_conj
(map rel (map %# largs ~~ map %# rargs)))))) end;
val cons' = filter (fn (_,args) => args<>[]) cons;
in
val inverts = map (fn (con,args) =>
pgterm (op <<) con args (flat(map (fn arg => [
TRY(rtac conjI 1),
dres_inst_tac [("fo5",sel_of arg)] monofun_cfun_arg 1,
asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
) args))) cons';
val injects = map (fn ((con,args),inv_thm) =>
pgterm (op ===) con args [
etac (antisym_less_inverse RS conjE) 1,
dtac inv_thm 1, REPEAT(atac 1),
dtac inv_thm 1, REPEAT(atac 1),
TRY(safe_tac HOL_cs),
REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
(cons'~~inverts);
end;
(* ----- theorems concerning one induction step ----------------------------------- *)
val copy_strict = pg [ax_copy_def] ((if is_one_con_one_arg (K true) cons then fn t =>
mk_trp(strict(cproj (%"f") eqs (rec_of (hd(snd(hd cons)))))) ===> t
else Id) (mk_trp(strict(dc_copy`%"f")))) [
asm_simp_tac(HOLCF_ss addsimps [abs_strict,rep_strict,
cfst_strict,csnd_strict]) 1];
val copy_apps = map (fn (con,args) => pg (ax_copy_def::axs_con_def)
(lift_defined %# (filter is_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 [ax_abs_iso] 1 o vname)
(filter(fn a=>not(is_rec a orelse is_lazy a))args)@
[asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1])
)cons;
val copy_stricts = map(fn(con,args)=>pg[](mk_trp(dc_copy`UU`(con_app con args) ===UU))
(let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
in map (case_UU_tac rews 1) (nonlazy args) @ [
asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
(filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
val copy_rews = copy_strict::copy_apps @ copy_stricts;
in (iso_rews, exhaust, cases, when_rews,
con_rews, sel_rews, dis_rews, dists_eq, dists_le, inverts, injects,
copy_rews)
end; (* let *)
fun comp_theorems thy (comp_dname, eqs: eq list, casess, con_rews, copy_rews) =
let
val dummy = writeln ("Proving induction properties of domain "^comp_dname^"...");
val pg = pg' thy;
val dnames = map (fst o fst) eqs;
val conss = map snd eqs;
(* ----- getting the composite axiom and definitions ------------------------------ *)
local val ga = get_axiom thy in
val axs_reach = map (fn dn => ga (dn ^ "_reach" )) dnames;
val axs_take_def = map (fn dn => ga (dn ^ "_take_def")) dnames;
val axs_finite_def = map (fn dn => ga (dn ^"_finite_def")) dnames;
val ax_copy2_def = ga (comp_dname^ "_copy_def");
val ax_bisim_def = ga (comp_dname^"_bisim_def");
end; (* local *)
(* ----- theorems concerning finiteness and induction ----------------------------- *)
fun dc_take dn = %%(dn^"_take");
val x_name = idx_name dnames "x";
val P_name = idx_name dnames "P";
local
val iterate_ss = simpset_of "Fix";
val iterate_Cprod_strict_ss = iterate_ss addsimps [cfst_strict, csnd_strict];
val iterate_Cprod_ss = iterate_ss addsimps [cfst2,csnd2,csplit2];
val copy_con_rews = copy_rews @ con_rews;
val copy_take_defs = (if length dnames=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),_)=>
(dc_take dn $ %"n")`UU === mk_constrain(Type(dn,args),UU)) eqs)))([
nat_ind_tac "n" 1,
simp_tac iterate_ss 1,
simp_tac iterate_Cprod_strict_ss 1,
asm_simp_tac iterate_Cprod_ss 1,
TRY(safe_tac HOL_cs)] @
map(K(asm_simp_tac (HOL_ss addsimps copy_rews)1))dnames);
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_strict_ss 1]) 1 dnames;
val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj
(flat(map (fn ((dn,_),cons) => map (fn (con,args) => 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 (nth_elem(n,dnames))$ %"n"))
args)) cons) eqs)))) ([
nat_ind_tac "n" 1,
simp_tac iterate_Cprod_strict_ss 1,
simp_tac (HOLCF_ss addsimps copy_con_rews) 1,
TRY(safe_tac HOL_cs)] @
(flat(map (fn ((dn,_),cons) => map (fn (con,args) => EVERY (
asm_full_simp_tac iterate_Cprod_ss 1::
map (case_UU_tac (take_stricts'::copy_con_rews) 1)
(nonlazy args) @[
asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1])
) cons) eqs)));
in
val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
end; (* local *)
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,
rtac (fix_def2 RS ssubst) 1,
REPEAT(CHANGED(rtac (contlub_cfun_arg RS ssubst) 1
THEN chain_tac 1)),
rtac (contlub_cfun_fun RS ssubst) 1,
rtac (contlub_cfun_fun RS ssubst) 2,
rtac lub_equal 3,
chain_tac 1,
rtac allI 1,
resolve_tac prems 1])) 1 (dnames~~axs_reach);
local
fun one_con p (con,args) = 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)
(filter is_rec args,mk_trp(%p $ con_app2 con (bound_arg args) args))));
fun one_eq ((p,cons),concl) = (mk_trp(%p $ UU) ===>
foldr (op ===>) (map (one_con p) cons,concl));
fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x)) 1 conss,
mk_trp(foldr' mk_conj (mapn (fn n => concf (P_name n,x_name n)) 1 dnames)));
val take_ss = HOL_ss addsimps take_rews;
fun ind_tacs tacsf thms1 thms2 prems = TRY(safe_tac HOL_cs)::
flat (mapn (fn n => fn (thm1,thm2) =>
tacsf (n,prems) (thm1,thm2) @
flat (map (fn cons =>
(resolve_tac prems 1 ::
flat (map (fn (_,args) =>
resolve_tac prems 1::
map (K(atac 1)) (nonlazy args) @
map (K(atac 1)) (filter is_rec args))
cons)))
conss))
0 (thms1~~thms2));
local
fun all_rec_to ns lazy_rec (n,cons) = forall (exists (fn arg =>
is_rec arg andalso not(rec_of arg mem ns) andalso
((rec_of arg = n andalso not(lazy_rec orelse is_lazy arg)) orelse
rec_of arg <> n andalso all_rec_to (rec_of arg::ns)
(lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
) o snd) cons;
fun warn (n,cons) = if all_rec_to [] false (n,cons) then (writeln
("WARNING: domain "^nth_elem(n,dnames)^" is empty!"); true)
else false;
fun lazy_rec_to ns lazy_rec (n,cons) = exists (exists (fn arg =>
is_rec arg andalso not(rec_of arg mem ns) andalso
((rec_of arg = n andalso (lazy_rec orelse is_lazy arg)) orelse
rec_of arg <> n andalso lazy_rec_to (rec_of arg::ns)
(lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
) o snd) cons;
in val is_emptys = map warn (mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs);
val is_finite = forall (not o lazy_rec_to [] false)
(mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs)
end;
in
val finite_ind = pg'' thy [] (ind_term (fn (P,x) => fn dn =>
mk_all(x,%P $ (dc_take dn $ %"n" `Bound 0)))) (fn prems=> [
nat_ind_tac "n" 1,
simp_tac (take_ss addsimps prems) 1,
TRY(safe_tac HOL_cs)]
@ flat(mapn (fn n => fn (cons,cases) => [
res_inst_tac [("x",x_name n)] cases 1,
asm_simp_tac (take_ss addsimps prems) 1]
@ flat(map (fn (con,args) =>
asm_simp_tac take_ss 1 ::
map (fn arg =>
case_UU_tac (prems@con_rews) 1 (
nth_elem(rec_of arg,dnames)^"_take n1`"^vname arg))
(filter is_nonlazy_rec args) @ [
resolve_tac prems 1] @
map (K (atac 1)) (nonlazy args) @
map (K (etac spec 1)) (filter is_rec args))
cons))
1 (conss~~casess)));
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,
nat_ind_tac "n" 1,
simp_tac take_ss 1,
TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
flat(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] @
flat(map (fn (con,args) =>
asm_simp_tac take_ss 1 ::
flat(map (fn arg => [
eres_inst_tac [("x",vname arg)] all_dupE 1,
etac disjE 1,
asm_simp_tac (HOL_ss addsimps con_rews) 1,
asm_simp_tac take_ss 1])
(filter is_nonlazy_rec args)))
cons))
1 (conss~~casess))) handle ERROR => raise ERROR;
val all_finite=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
(all_finite,
pg'' thy [] (ind_term (fn (P,x) => fn dn => %P $ %x))
(ind_tacs (fn _ => fn (all_fin,finite_ind) => [
rtac (rewrite_rule axs_finite_def all_fin RS exE) 1,
etac subst 1,
rtac finite_ind 1]) all_finite (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 [] (foldr (op ===>) (mapn (fn n =>K(mk_trp(%%"adm" $ %(P_name n))))1
dnames,ind_term (fn(P,x)=>fn dn=> %P $ %x)))
(ind_tacs (fn (n,prems) => fn (ax_reach,finite_ind) =>[
rtac (ax_reach RS subst) 1,
res_inst_tac [("x",x_name n)] spec 1,
rtac wfix_ind 1,
rtac adm_impl_admw 1,
resolve_tac adm_thms 1,
rtac adm_subst 1,
cont_tacR 1,
resolve_tac prems 1,
strip_tac 1,
rtac(rewrite_rule axs_take_def finite_ind) 1])
axs_reach (atomize finite_ind))
)
end; (* local *)
local
val xs = mapn (fn n => K (x_name n)) 1 dnames;
fun bnd_arg n i = Bound(2*(length dnames - n)-i-1);
val take_ss = HOL_ss addsimps take_rews;
val sproj = bin_branchr (fn s => "fst("^s^")") (fn s => "snd("^s^")");
val coind_lemma = pg [ax_bisim_def] (mk_trp(mk_imp(%%(comp_dname^"_bisim") $ %"R",
foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
foldr mk_imp (mapn (fn n => K(proj (%"R") dnames 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,
nat_ind_tac "n" 1,
simp_tac take_ss 1,
safe_tac HOL_cs] @
flat(mapn (fn n => fn x => [
etac allE 1, etac allE 1,
eres_inst_tac [("P1",sproj "R" dnames 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") ===>
foldr (op ===>) (mapn (fn n => fn x =>
mk_trp(proj (%"R") dnames n $ %x $ %(x^"'"))) 0 xs,
mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
TRY(safe_tac HOL_cs)] @
flat(map (fn take_lemma => [
rtac take_lemma 1,
cut_facts_tac [coind_lemma] 1,
fast_tac HOL_cs 1])
take_lemmas));
end; (* local *)
in (take_rews, take_lemmas, finites, finite_ind, ind, coind)
end; (* let *)
end; (* local *)
end; (* struct *)