(* Title: HOLCF/stream.ML
ID: $Id$
Author: Franz Regensburger
Copyright 1993 Technische Universitaet Muenchen
Lemmas for stream.thy
*)
open Stream;
(* ------------------------------------------------------------------------*)
(* The isomorphisms stream_rep_iso and stream_abs_iso are strict *)
(* ------------------------------------------------------------------------*)
val stream_iso_strict= stream_rep_iso RS (stream_abs_iso RS
(allI RSN (2,allI RS iso_strict)));
val stream_rews = [stream_iso_strict RS conjunct1,
stream_iso_strict RS conjunct2];
(* ------------------------------------------------------------------------*)
(* Properties of stream_copy *)
(* ------------------------------------------------------------------------*)
fun prover defs thm = prove_goalw Stream.thy defs thm
(fn prems =>
[
(cut_facts_tac prems 1),
(asm_simp_tac (HOLCF_ss addsimps
(stream_rews @ [stream_abs_iso,stream_rep_iso])) 1)
]);
val stream_copy =
[
prover [stream_copy_def] "stream_copy[f][UU]=UU",
prover [stream_copy_def,scons_def]
"x~=UU ==> stream_copy[f][scons[x][xs]]= scons[x][f[xs]]"
];
val stream_rews = stream_copy @ stream_rews;
(* ------------------------------------------------------------------------*)
(* Exhaustion and elimination for streams *)
(* ------------------------------------------------------------------------*)
val Exh_stream = prove_goalw Stream.thy [scons_def]
"s = UU | (? x xs. x~=UU & s = scons[x][xs])"
(fn prems =>
[
(simp_tac HOLCF_ss 1),
(rtac (stream_rep_iso RS subst) 1),
(res_inst_tac [("p","stream_rep[s]")] sprodE 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(asm_simp_tac HOLCF_ss 1),
(res_inst_tac [("p","y")] liftE1 1),
(contr_tac 1),
(rtac disjI2 1),
(rtac exI 1),
(rtac exI 1),
(etac conjI 1),
(asm_simp_tac HOLCF_ss 1)
]);
val streamE = prove_goal Stream.thy
"[| s=UU ==> Q; !!x xs.[|s=scons[x][xs];x~=UU|]==>Q|]==>Q"
(fn prems =>
[
(rtac (Exh_stream RS disjE) 1),
(eresolve_tac prems 1),
(etac exE 1),
(etac exE 1),
(resolve_tac prems 1),
(fast_tac HOL_cs 1),
(fast_tac HOL_cs 1)
]);
(* ------------------------------------------------------------------------*)
(* Properties of stream_when *)
(* ------------------------------------------------------------------------*)
fun prover defs thm = prove_goalw Stream.thy defs thm
(fn prems =>
[
(cut_facts_tac prems 1),
(asm_simp_tac (HOLCF_ss addsimps
(stream_rews @ [stream_abs_iso,stream_rep_iso])) 1)
]);
val stream_when = [
prover [stream_when_def] "stream_when[f][UU]=UU",
prover [stream_when_def,scons_def]
"x~=UU ==> stream_when[f][scons[x][xs]]= f[x][xs]"
];
val stream_rews = stream_when @ stream_rews;
(* ------------------------------------------------------------------------*)
(* Rewrites for discriminators and selectors *)
(* ------------------------------------------------------------------------*)
fun prover defs thm = prove_goalw Stream.thy defs thm
(fn prems =>
[
(simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val stream_discsel = [
prover [is_scons_def] "is_scons[UU]=UU",
prover [shd_def] "shd[UU]=UU",
prover [stl_def] "stl[UU]=UU"
];
fun prover defs thm = prove_goalw Stream.thy defs thm
(fn prems =>
[
(cut_facts_tac prems 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val stream_discsel = [
prover [is_scons_def,shd_def,stl_def] "x~=UU ==> is_scons[scons[x][xs]]=TT",
prover [is_scons_def,shd_def,stl_def] "x~=UU ==> shd[scons[x][xs]]=x",
prover [is_scons_def,shd_def,stl_def] "x~=UU ==> stl[scons[x][xs]]=xs"
] @ stream_discsel;
val stream_rews = stream_discsel @ stream_rews;
(* ------------------------------------------------------------------------*)
(* Definedness and strictness *)
(* ------------------------------------------------------------------------*)
fun prover contr thm = prove_goal Stream.thy thm
(fn prems =>
[
(res_inst_tac [("P1",contr)] classical3 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(dtac sym 1),
(asm_simp_tac HOLCF_ss 1),
(simp_tac (HOLCF_ss addsimps (prems @ stream_rews)) 1)
]);
val stream_constrdef = [
prover "is_scons[UU] ~= UU" "x~=UU ==> scons[x][xs]~=UU"
];
fun prover defs thm = prove_goalw Stream.thy defs thm
(fn prems =>
[
(simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val stream_constrdef = [
prover [scons_def] "scons[UU][xs]=UU"
] @ stream_constrdef;
val stream_rews = stream_constrdef @ stream_rews;
(* ------------------------------------------------------------------------*)
(* Distinctness wrt. << and = *)
(* ------------------------------------------------------------------------*)
(* ------------------------------------------------------------------------*)
(* Invertibility *)
(* ------------------------------------------------------------------------*)
val stream_invert =
[
prove_goal Stream.thy "[|x1~=UU; y1~=UU;\
\ scons[x1][x2] << scons[y1][y2]|] ==> x1<< y1 & x2 << y2"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac conjI 1),
(dres_inst_tac [("fo5","stream_when[LAM x l.x]")] monofun_cfun_arg 1),
(etac box_less 1),
(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
(dres_inst_tac [("fo5","stream_when[LAM x l.l]")] monofun_cfun_arg 1),
(etac box_less 1),
(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_when) 1)
])
];
(* ------------------------------------------------------------------------*)
(* Injectivity *)
(* ------------------------------------------------------------------------*)
val stream_inject =
[
prove_goal Stream.thy "[|x1~=UU; y1~=UU;\
\ scons[x1][x2] = scons[y1][y2]|] ==> x1= y1 & x2 = y2"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac conjI 1),
(dres_inst_tac [("f","stream_when[LAM x l.x]")] cfun_arg_cong 1),
(etac box_equals 1),
(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
(dres_inst_tac [("f","stream_when[LAM x l.l]")] cfun_arg_cong 1),
(etac box_equals 1),
(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_when) 1)
])
];
(* ------------------------------------------------------------------------*)
(* definedness for discriminators and selectors *)
(* ------------------------------------------------------------------------*)
fun prover thm = prove_goal Stream.thy thm
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac streamE 1),
(contr_tac 1),
(REPEAT (asm_simp_tac (HOLCF_ss addsimps stream_discsel) 1))
]);
val stream_discsel_def =
[
prover "s~=UU ==> is_scons[s]~=UU",
prover "s~=UU ==> shd[s]~=UU"
];
val stream_rews = stream_discsel_def @ stream_rews;
(* ------------------------------------------------------------------------*)
(* Properties stream_take *)
(* ------------------------------------------------------------------------*)
val stream_take =
[
prove_goalw Stream.thy [stream_take_def] "stream_take(n)[UU]=UU"
(fn prems =>
[
(res_inst_tac [("n","n")] natE 1),
(asm_simp_tac iterate_ss 1),
(asm_simp_tac iterate_ss 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1)
]),
prove_goalw Stream.thy [stream_take_def] "stream_take(0)[xs]=UU"
(fn prems =>
[
(asm_simp_tac iterate_ss 1)
])];
fun prover thm = prove_goalw Stream.thy [stream_take_def] thm
(fn prems =>
[
(cut_facts_tac prems 1),
(simp_tac iterate_ss 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val stream_take = [
prover
"x~=UU ==> stream_take(Suc(n))[scons[x][xs]]=scons[x][stream_take(n)[xs]]"
] @ stream_take;
val stream_rews = stream_take @ stream_rews;
(* ------------------------------------------------------------------------*)
(* enhance the simplifier *)
(* ------------------------------------------------------------------------*)
val stream_copy2 = prove_goal Stream.thy
"stream_copy[f][scons[x][xs]]= scons[x][f[xs]]"
(fn prems =>
[
(res_inst_tac [("Q","x=UU")] classical2 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val shd2 = prove_goal Stream.thy "shd[scons[x][xs]]=x"
(fn prems =>
[
(res_inst_tac [("Q","x=UU")] classical2 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val stream_take2 = prove_goal Stream.thy
"stream_take(Suc(n))[scons[x][xs]]=scons[x][stream_take(n)[xs]]"
(fn prems =>
[
(res_inst_tac [("Q","x=UU")] classical2 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val stream_rews = [stream_iso_strict RS conjunct1,
stream_iso_strict RS conjunct2,
hd stream_copy, stream_copy2]
@ stream_when
@ [hd stream_discsel,shd2] @ (tl (tl stream_discsel))
@ stream_constrdef
@ stream_discsel_def
@ [ stream_take2] @ (tl stream_take);
(* ------------------------------------------------------------------------*)
(* take lemma for streams *)
(* ------------------------------------------------------------------------*)
fun prover reach defs thm = prove_goalw Stream.thy defs thm
(fn prems =>
[
(res_inst_tac [("t","s1")] (reach RS subst) 1),
(res_inst_tac [("t","s2")] (reach RS subst) 1),
(rtac (fix_def2 RS ssubst) 1),
(rtac (contlub_cfun_fun RS ssubst) 1),
(rtac is_chain_iterate 1),
(rtac (contlub_cfun_fun RS ssubst) 1),
(rtac is_chain_iterate 1),
(rtac lub_equal 1),
(rtac (is_chain_iterate RS ch2ch_fappL) 1),
(rtac (is_chain_iterate RS ch2ch_fappL) 1),
(rtac allI 1),
(resolve_tac prems 1)
]);
val stream_take_lemma = prover stream_reach [stream_take_def]
"(!!n.stream_take(n)[s1]=stream_take(n)[s2]) ==> s1=s2";
(* ------------------------------------------------------------------------*)
(* Co -induction for streams *)
(* ------------------------------------------------------------------------*)
val stream_coind_lemma = prove_goalw Stream.thy [stream_bisim_def]
"stream_bisim(R) ==> ! p q.R(p,q) --> stream_take(n)[p]=stream_take(n)[q]"
(fn prems =>
[
(cut_facts_tac prems 1),
(nat_ind_tac "n" 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(strip_tac 1),
((etac allE 1) THEN (etac allE 1) THEN (etac (mp RS disjE) 1)),
(atac 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(etac exE 1),
(etac exE 1),
(etac exE 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(REPEAT (etac conjE 1)),
(rtac cfun_arg_cong 1),
(fast_tac HOL_cs 1)
]);
val stream_coind = prove_goal Stream.thy "[|stream_bisim(R);R(p,q)|] ==> p = q"
(fn prems =>
[
(rtac stream_take_lemma 1),
(rtac (stream_coind_lemma RS spec RS spec RS mp) 1),
(resolve_tac prems 1),
(resolve_tac prems 1)
]);
(* ------------------------------------------------------------------------*)
(* structural induction for admissible predicates *)
(* ------------------------------------------------------------------------*)
val stream_finite_ind = prove_goal Stream.thy
"[|P(UU);\
\ !! x s1.[|x~=UU;P(s1)|] ==> P(scons[x][s1])\
\ |] ==> !s.P(stream_take(n)[s])"
(fn prems =>
[
(nat_ind_tac "n" 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(resolve_tac prems 1),
(rtac allI 1),
(res_inst_tac [("s","s")] streamE 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(resolve_tac prems 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(resolve_tac prems 1),
(atac 1),
(etac spec 1)
]);
val stream_finite_ind2 = prove_goalw Stream.thy [stream_finite_def]
"(!!n.P(stream_take(n)[s])) ==> stream_finite(s) -->P(s)"
(fn prems =>
[
(strip_tac 1),
(etac exE 1),
(etac subst 1),
(resolve_tac prems 1)
]);
val stream_finite_ind3 = prove_goal Stream.thy
"[|P(UU);\
\ !! x s1.[|x~=UU;P(s1)|] ==> P(scons[x][s1])\
\ |] ==> stream_finite(s) --> P(s)"
(fn prems =>
[
(rtac stream_finite_ind2 1),
(rtac (stream_finite_ind RS spec) 1),
(REPEAT (resolve_tac prems 1)),
(REPEAT (atac 1))
]);
val stream_ind = prove_goal Stream.thy
"[|adm(P);\
\ P(UU);\
\ !! x s1.[|x~=UU;P(s1)|] ==> P(scons[x][s1])\
\ |] ==> P(s)"
(fn prems =>
[
(rtac (stream_reach RS subst) 1),
(res_inst_tac [("x","s")] spec 1),
(rtac wfix_ind 1),
(rtac adm_impl_admw 1),
(REPEAT (resolve_tac adm_thms 1)),
(rtac adm_subst 1),
(contX_tacR 1),
(resolve_tac prems 1),
(rtac allI 1),
(rtac (rewrite_rule [stream_take_def] stream_finite_ind) 1),
(REPEAT (resolve_tac prems 1)),
(REPEAT (atac 1))
]);
(* ------------------------------------------------------------------------*)
(* simplify use of Co-induction *)
(* ------------------------------------------------------------------------*)
val surjectiv_scons = prove_goal Stream.thy "scons[shd[s]][stl[s]]=s"
(fn prems =>
[
(res_inst_tac [("s","s")] streamE 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val stream_coind_lemma2 = prove_goalw Stream.thy [stream_bisim_def]
"!s1 s2. R(s1, s2)-->shd[s1]=shd[s2] & R(stl[s1],stl[s2]) ==>stream_bisim(R)"
(fn prems =>
[
(cut_facts_tac prems 1),
(strip_tac 1),
(etac allE 1),
(etac allE 1),
(dtac mp 1),
(atac 1),
(etac conjE 1),
(res_inst_tac [("Q","s1 = UU & s2 = UU")] classical2 1),
(rtac disjI1 1),
(fast_tac HOL_cs 1),
(rtac disjI2 1),
(rtac disjE 1),
(etac (de_morgan2 RS ssubst) 1),
(res_inst_tac [("x","shd[s1]")] exI 1),
(res_inst_tac [("x","stl[s1]")] exI 1),
(res_inst_tac [("x","stl[s2]")] exI 1),
(rtac conjI 1),
(eresolve_tac stream_discsel_def 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1),
(eres_inst_tac [("s","shd[s1]"),("t","shd[s2]")] subst 1),
(simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1),
(res_inst_tac [("x","shd[s2]")] exI 1),
(res_inst_tac [("x","stl[s1]")] exI 1),
(res_inst_tac [("x","stl[s2]")] exI 1),
(rtac conjI 1),
(eresolve_tac stream_discsel_def 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1),
(res_inst_tac [("s","shd[s1]"),("t","shd[s2]")] ssubst 1),
(etac sym 1),
(simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1)
]);
(* ------------------------------------------------------------------------*)
(* theorems about finite and infinite streams *)
(* ------------------------------------------------------------------------*)
(* ----------------------------------------------------------------------- *)
(* 2 lemmas about stream_finite *)
(* ----------------------------------------------------------------------- *)
val stream_finite_UU = prove_goalw Stream.thy [stream_finite_def]
"stream_finite(UU)"
(fn prems =>
[
(rtac exI 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val inf_stream_not_UU = prove_goal Stream.thy "~stream_finite(s) ==> s ~= UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(etac swap 1),
(dtac notnotD 1),
(hyp_subst_tac 1),
(rtac stream_finite_UU 1)
]);
(* ----------------------------------------------------------------------- *)
(* a lemma about shd *)
(* ----------------------------------------------------------------------- *)
val stream_shd_lemma1 = prove_goal Stream.thy "shd[s]=UU --> s=UU"
(fn prems =>
[
(res_inst_tac [("s","s")] streamE 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(hyp_subst_tac 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
(* ----------------------------------------------------------------------- *)
(* lemmas about stream_take *)
(* ----------------------------------------------------------------------- *)
val stream_take_lemma1 = prove_goal Stream.thy
"!x xs.x~=UU --> \
\ stream_take(Suc(n))[scons[x][xs]] = scons[x][xs] --> stream_take(n)[xs]=xs"
(fn prems =>
[
(rtac allI 1),
(rtac allI 1),
(rtac impI 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(strip_tac 1),
(rtac ((hd stream_inject) RS conjunct2) 1),
(atac 1),
(atac 1),
(atac 1)
]);
val stream_take_lemma2 = prove_goal Stream.thy
"! s2. stream_take(n)[s2] = s2 --> stream_take(Suc(n))[s2]=s2"
(fn prems =>
[
(nat_ind_tac "n" 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(strip_tac 1 ),
(hyp_subst_tac 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(rtac allI 1),
(res_inst_tac [("s","s2")] streamE 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(strip_tac 1 ),
(subgoal_tac "stream_take(n1)[xs] = xs" 1),
(rtac ((hd stream_inject) RS conjunct2) 2),
(atac 4),
(atac 2),
(atac 2),
(rtac cfun_arg_cong 1),
(fast_tac HOL_cs 1)
]);
val stream_take_lemma3 = prove_goal Stream.thy
"!x xs.x~=UU --> \
\ stream_take(n)[scons[x][xs]] = scons[x][xs] --> stream_take(n)[xs]=xs"
(fn prems =>
[
(nat_ind_tac "n" 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(strip_tac 1 ),
(res_inst_tac [("P","scons[x][xs]=UU")] notE 1),
(eresolve_tac stream_constrdef 1),
(etac sym 1),
(strip_tac 1 ),
(rtac (stream_take_lemma2 RS spec RS mp) 1),
(res_inst_tac [("x1.1","x")] ((hd stream_inject) RS conjunct2) 1),
(atac 1),
(atac 1),
(etac (stream_take2 RS subst) 1)
]);
val stream_take_lemma4 = prove_goal Stream.thy
"!x xs.\
\stream_take(n)[xs]=xs --> stream_take(Suc(n))[scons[x][xs]] = scons[x][xs]"
(fn prems =>
[
(nat_ind_tac "n" 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
(* ---- *)
val stream_take_lemma5 = prove_goal Stream.thy
"!s. stream_take(n)[s]=s --> iterate(n,stl,s)=UU"
(fn prems =>
[
(nat_ind_tac "n" 1),
(simp_tac iterate_ss 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(strip_tac 1),
(res_inst_tac [("s","s")] streamE 1),
(hyp_subst_tac 1),
(rtac (iterate_Suc2 RS ssubst) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(rtac (iterate_Suc2 RS ssubst) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(etac allE 1),
(etac mp 1),
(hyp_subst_tac 1),
(etac (stream_take_lemma1 RS spec RS spec RS mp RS mp) 1),
(atac 1)
]);
val stream_take_lemma6 = prove_goal Stream.thy
"!s.iterate(n,stl,s)=UU --> stream_take(n)[s]=s"
(fn prems =>
[
(nat_ind_tac "n" 1),
(simp_tac iterate_ss 1),
(strip_tac 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(rtac allI 1),
(res_inst_tac [("s","s")] streamE 1),
(hyp_subst_tac 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
(hyp_subst_tac 1),
(rtac (iterate_Suc2 RS ssubst) 1),
(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
]);
val stream_take_lemma7 = prove_goal Stream.thy
"(iterate(n,stl,s)=UU) = (stream_take(n)[s]=s)"
(fn prems =>
[
(rtac iffI 1),
(etac (stream_take_lemma6 RS spec RS mp) 1),
(etac (stream_take_lemma5 RS spec RS mp) 1)
]);
(* ----------------------------------------------------------------------- *)
(* lemmas stream_finite *)
(* ----------------------------------------------------------------------- *)
val stream_finite_lemma1 = prove_goalw Stream.thy [stream_finite_def]
"stream_finite(xs) ==> stream_finite(scons[x][xs])"
(fn prems =>
[
(cut_facts_tac prems 1),
(etac exE 1),
(rtac exI 1),
(etac (stream_take_lemma4 RS spec RS spec RS mp) 1)
]);
val stream_finite_lemma2 = prove_goalw Stream.thy [stream_finite_def]
"[|x~=UU; stream_finite(scons[x][xs])|] ==> stream_finite(xs)"
(fn prems =>
[
(cut_facts_tac prems 1),
(etac exE 1),
(rtac exI 1),
(etac (stream_take_lemma3 RS spec RS spec RS mp RS mp) 1),
(atac 1)
]);
val stream_finite_lemma3 = prove_goal Stream.thy
"x~=UU ==> stream_finite(scons[x][xs]) = stream_finite(xs)"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac iffI 1),
(etac stream_finite_lemma2 1),
(atac 1),
(etac stream_finite_lemma1 1)
]);
val stream_finite_lemma5 = prove_goalw Stream.thy [stream_finite_def]
"(!n. s1 << s2 --> stream_take(n)[s2] = s2 --> stream_finite(s1))\
\=(s1 << s2 --> stream_finite(s2) --> stream_finite(s1))"
(fn prems =>
[
(rtac iffI 1),
(fast_tac HOL_cs 1),
(fast_tac HOL_cs 1)
]);
val stream_finite_lemma6 = prove_goal Stream.thy
"!s1 s2. s1 << s2 --> stream_take(n)[s2] = s2 --> stream_finite(s1)"
(fn prems =>
[
(nat_ind_tac "n" 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(strip_tac 1 ),
(hyp_subst_tac 1),
(dtac UU_I 1),
(hyp_subst_tac 1),
(rtac stream_finite_UU 1),
(rtac allI 1),
(rtac allI 1),
(res_inst_tac [("s","s1")] streamE 1),
(hyp_subst_tac 1),
(strip_tac 1 ),
(rtac stream_finite_UU 1),
(hyp_subst_tac 1),
(res_inst_tac [("s","s2")] streamE 1),
(hyp_subst_tac 1),
(strip_tac 1 ),
(dtac UU_I 1),
(asm_simp_tac(HOLCF_ss addsimps (stream_rews @ [stream_finite_UU])) 1),
(hyp_subst_tac 1),
(simp_tac (HOLCF_ss addsimps stream_rews) 1),
(strip_tac 1 ),
(rtac stream_finite_lemma1 1),
(subgoal_tac "xs << xsa" 1),
(subgoal_tac "stream_take(n1)[xsa] = xsa" 1),
(fast_tac HOL_cs 1),
(res_inst_tac [("x1.1","xa"),("y1.1","xa")]
((hd stream_inject) RS conjunct2) 1),
(atac 1),
(atac 1),
(atac 1),
(res_inst_tac [("x1.1","x"),("y1.1","xa")]
((hd stream_invert) RS conjunct2) 1),
(atac 1),
(atac 1),
(atac 1)
]);
val stream_finite_lemma7 = prove_goal Stream.thy
"s1 << s2 --> stream_finite(s2) --> stream_finite(s1)"
(fn prems =>
[
(rtac (stream_finite_lemma5 RS iffD1) 1),
(rtac allI 1),
(rtac (stream_finite_lemma6 RS spec RS spec) 1)
]);
val stream_finite_lemma8 = prove_goalw Stream.thy [stream_finite_def]
"stream_finite(s) = (? n. iterate(n,stl,s)=UU)"
(fn prems =>
[
(simp_tac (HOL_ss addsimps [stream_take_lemma7]) 1)
]);
(* ----------------------------------------------------------------------- *)
(* admissibility of ~stream_finite *)
(* ----------------------------------------------------------------------- *)
val adm_not_stream_finite = prove_goalw Stream.thy [adm_def]
"adm(%s. ~ stream_finite(s))"
(fn prems =>
[
(strip_tac 1 ),
(res_inst_tac [("P1","!i. ~ stream_finite(Y(i))")] classical3 1),
(atac 2),
(subgoal_tac "!i.stream_finite(Y(i))" 1),
(fast_tac HOL_cs 1),
(rtac allI 1),
(rtac (stream_finite_lemma7 RS mp RS mp) 1),
(etac is_ub_thelub 1),
(atac 1)
]);
(* ----------------------------------------------------------------------- *)
(* alternative prove for admissibility of ~stream_finite *)
(* show that stream_finite(s) = (? n. iterate(n, stl, s) = UU) *)
(* and prove adm. of ~(? n. iterate(n, stl, s) = UU) *)
(* proof uses theorems stream_take_lemma5-7; stream_finite_lemma8 *)
(* ----------------------------------------------------------------------- *)
val adm_not_stream_finite2=prove_goal Stream.thy "adm(%s. ~ stream_finite(s))"
(fn prems =>
[
(subgoal_tac "(!s.(~stream_finite(s))=(!n.iterate(n,stl,s)~=UU))" 1),
(etac (adm_cong RS iffD2)1),
(REPEAT(resolve_tac adm_thms 1)),
(rtac contX_iterate2 1),
(rtac allI 1),
(rtac (stream_finite_lemma8 RS ssubst) 1),
(fast_tac HOL_cs 1)
]);