(* Title: HOLCF/domain/theorems.ML
ID: $Id$
Author : David von Oheimb
Copyright 1995, 1996 TU Muenchen
proof generator for domain section
*)
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));
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(tac st) of
None => Sequence.null
| Some(st',_) => drep st')
in drep end;
val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);
local val trueI2 = prove_goal HOL.thy"f~=x ==> True"(fn _ => [rtac TrueI 1]) in
val kill_neq_tac = dtac trueI2 end;
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
[is_chain_iterate, ch2ch_fappR, ch2ch_fappL];
(* ----- 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) << y ==> x ~= y" (fn prems => [
rtac rev_contrapos 1,
etac (antisym_less_inverse RS conjunct1) 1,
resolve_tac prems 1]);
in
fun theorems (((dname,_),cons) : eq, eqs : eq list) thy =
let
val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ...");
val pg = pg' thy;
(*
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;
*)
(* ----- getting the axioms and definitions --------------------------------- *)
local fun ga s dn = get_axiom 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;
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_sel_def = flat(map (fn (_,args) =>
map (fn arg => ga (sel_of arg ^"_def") dname) 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 (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 casedist = let
fun common_tac thm = rtac thm 1 THEN contr_tac 1;
fun unit_tac true = common_tac upE1
| 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_rest_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_rest_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_rest_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 length cons = 1 andalso
length (snd(hd cons)) = 1 andalso
not(is_lazy(hd(snd(hd 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 casedist 1,
UNTIL_SOLVED(fast_tac HOL_cs 1)];
end;
local
fun bind_fun t = foldr mk_All (when_funs cons,t);
fun bound_fun i _ = Bound (length cons - i);
val when_app = foldl (op `) (%%(dname^"_when"), mapn bound_fun 1 cons);
val when_appl = pg [ax_when_def] (bind_fun(mk_trp(when_app`%x_name ===
when_body cons (fn (m,n)=> bound_fun (n-m) 0)`(dc_rep`%x_name))))[
simp_tac HOLCF_ss 1];
in
val when_strict = pg [] (bind_fun(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
(bind_fun (lift_defined % (nonlazy args,
mk_trp(when_app`(con_app con args) ===
mk_cfapp(bound_fun n 0,map %# args)))))[
asm_simp_tac (HOLCF_ss addsimps [when_appl,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)))) [
simp_tac (HOLCF_ss addsimps when_rews) 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" 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 casedist 1,
contr_tac 1,
UNTIL_SOLVED (CHANGED(asm_simp_tac
(HOLCF_ss addsimps dis_apps) 1))]) cons;
in dis_stricts @ dis_defins @ dis_apps end;
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 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 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 casedist 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 rev_contrapos 1,
eres_inst_tac[("fo",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 = flat (flat 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;
open Basis_Library (*restore original List*)
fun distincts [] = []
| distincts ((c,leqs)::cs) = List.concat
(ListPair.map (distinct c) ((map #1 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
(ListPair.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 [("fo",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](mk_trp(strict(dc_copy`%"f"))) [
asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict,
cfst_strict,csnd_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") (length eqs))) args))))
(map (case_UU_tac (abs_strict::when_strict::con_stricts)
1 o vname)
(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 axs_con_def) 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 thy |> Theory.add_path (Sign.base_name dname)
|> PureThy.store_thmss [
("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)]
|> Theory.add_path ".."
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_axiom 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 *)
local fun gt s dn = get_thm thy (dn^"."^s);
fun gts s dn = get_thms thy (dn^"."^s) in
val cases = map (gt "casedist" ) dnames;
val con_rews = flat (map (gts "con_rews" ) dnames);
val copy_rews = flat (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"
addsimps [cfst_strict, csnd_strict]addsimps Cprod_rews;
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))) ([
nat_ind_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
(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)))) ([
simp_tac iterate_Cprod_ss 1,
nat_ind_tac "n" 1,
simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
asm_full_simp_tac (HOLCF_ss addsimps
(filter (has_fewer_prems 1) copy_rews)) 1,
TRY(safe_tac HOL_cs)] @
(flat(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 = atomize take_stricts @ take_0s @ atomize take_apps;
end; (* local *)
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))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(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));
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, (nth_elem(rec_of arg,conss))))
) 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 "^nth_elem(n,dnames)^" is empty!"); true) else false;
fun lazy_rec_to ns = rec_to exists Id 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,
nat_ind_tac "n" 1,
simp_tac (take_ss addsimps prems) 1,
TRY(safe_tac HOL_cs)]
@ flat(map (fn (cons,cases) => [
res_inst_tac [("x","x")] 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 n`"^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))
(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,
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 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) ::
flat (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 [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%"adm" $ %(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])
)
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 => "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") n_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,
nat_ind_tac "n" 1,
simp_tac take_ss 1,
safe_tac HOL_cs] @
flat(mapn (fn n => fn x => [
rotate_tac (n+1) 1,
etac all2E 1,
eres_inst_tac [("P1", sproj "R" n_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") ===>
foldr (op ===>) (mapn (fn n => fn x =>
mk_trp(proj (%"R") n_eqs 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 thy |> Theory.add_path comp_dnam
|> PureThy.store_thmss [
("take_rews" , take_rews ),
("take_lemmas", take_lemmas),
("finites" , finites ),
("finite_ind", [finite_ind]),
("ind" , [ind ]),
("coind" , [coind ])]
|> Theory.add_path ".."
end; (* let *)
end; (* local *)
end; (* struct *)