Add new file for the new definition of the hard direction's simplification.
Merging Operation is deleted
All definitions are done. Proof still undone.
theory MyhillNerode
imports "Main" "List_Prefix"
begin
text {* sequential composition of languages *}
definition
Seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
where
"L1 ;; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
inductive_set
Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
for L :: "string set"
where
start[intro]: "[] \<in> L\<star>"
| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
theorem ardens_revised:
assumes nemp: "[] \<notin> A"
shows "(X = X ;; A \<union> B) \<longleftrightarrow> (X = B ;; A\<star>)"
proof
assume eq: "X = B ;; A\<star>"
have "A\<star> = {[]} \<union> A\<star> ;; A" sorry
then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)" unfolding Seq_def by simp
also have "\<dots> = B \<union> B ;; (A\<star> ;; A)" unfolding Seq_def by auto
also have "\<dots> = B \<union> (B ;; A\<star>) ;; A" unfolding Seq_def
by (auto) (metis append_assoc)+
finally show "X = X ;; A \<union> B" using eq by auto
next
assume "X = X ;; A \<union> B"
then have "B \<subseteq> X" "X ;; A \<subseteq> X" by auto
show "X = B ;; A\<star>" sorry
qed
datatype rexp =
NULL
| EMPTY
| CHAR char
| SEQ rexp rexp
| ALT rexp rexp
| STAR rexp
consts L:: "'a \<Rightarrow> string set"
overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> string set"
begin
fun
L_rexp :: "rexp \<Rightarrow> string set"
where
"L_rexp (NULL) = {}"
| "L_rexp (EMPTY) = {[]}"
| "L_rexp (CHAR c) = {[c]}"
| "L_rexp (SEQ r1 r2) = (L_rexp r1) ;; (L_rexp r2)"
| "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
| "L_rexp (STAR r) = (L_rexp r)\<star>"
end
definition
folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
where
"folds f z S \<equiv> SOME x. fold_graph f z S x"
lemma folds_simp_null [simp]:
"finite rs \<Longrightarrow> x \<in> L (folds ALT NULL rs) \<longleftrightarrow> (\<exists>r \<in> rs. x \<in> L r)"
apply (simp add: folds_def)
apply (rule someI2_ex)
apply (erule finite_imp_fold_graph)
apply (erule fold_graph.induct)
apply (auto)
done
lemma [simp]:
shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
by simp
definition
str_eq ("_ \<approx>_ _")
where
"x \<approx>Lang y \<equiv> (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)"
definition
str_eq_rel ("\<approx>_")
where
"\<approx>Lang \<equiv> {(x, y). x \<approx>Lang y}"
section {* finite \<Rightarrow> regular *}
definition
transitions :: "string set \<Rightarrow> string set \<Rightarrow> rexp set" ("_\<Turnstile>\<Rightarrow>_")
where
"Y \<Turnstile>\<Rightarrow> X \<equiv> {CHAR c | c. Y ;; {[c]} \<subseteq> X}"
definition
transitions_rexp ("_ \<turnstile>\<rightarrow> _")
where
"Y \<turnstile>\<rightarrow> X \<equiv> folds ALT NULL (Y \<Turnstile>\<Rightarrow>X)"
definition
"init_rhs CS X \<equiv> if X = {[]}
then {({[]}, EMPTY)}
else if ([] \<in> X)
then insert ({[]}, EMPTY) {(Y, Y \<turnstile>\<rightarrow>X) | Y. Y \<in> CS}
else {(Y, Y \<turnstile>\<rightarrow>X) | Y. Y \<in> CS}"
overloading L_rhs \<equiv> "L:: (string set \<times> rexp) set \<Rightarrow> string set"
begin
fun L_rhs:: "(string set \<times> rexp) set \<Rightarrow> string set"
where
"L_rhs rhs = \<Union> {(Y;; L r) | Y r . (Y, r) \<in> rhs}"
end
definition
"eqs CS \<equiv> (\<Union>X \<in> CS. {(X, init_rhs CS X)})"
lemma [simp]:
shows "finite (Y \<Turnstile>\<Rightarrow> X)"
unfolding transitions_def
by auto
lemma defined_by_str:
assumes "s \<in> X"
and "X \<in> UNIV // (\<approx>Lang)"
shows "X = (\<approx>Lang) `` {s}"
using assms
unfolding quotient_def Image_def
unfolding str_eq_rel_def str_eq_def
by auto
(************ arden's lemma variation ********************)
definition
"rexp_of rhs X \<equiv> folds ALT NULL {r. (X, r) \<in> rhs}"
definition
"arden_variate X rhs \<equiv> {(Y, SEQ r (STAR (rexp_of rhs X)))| Y r. (Y, r) \<in> rhs \<and> Y \<noteq> X}"
(************* rhs/equations property **************)
definition
"distinct_equas ES \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
(*********** substitution of ES *************)
text {* rhs_subst rhs X xrhs: substitude all occurence of X in rhs with xrhs *}
definition
"rhs_subst rhs X xrhs \<equiv> {(Y, r) | Y r. Y \<noteq> X \<and> (Y, r) \<in> rhs} \<union>
{(X, SEQ r\<^isub>1 r\<^isub>2 ) | r\<^isub>1 r\<^isub>2. (X, r\<^isub>1) \<in> xrhs \<and> (X, r\<^isub>2) \<in> rhs}"
definition
"eqs_subst ES X xrhs \<equiv> {(Y, rhs_subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
text {*
Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
*}
definition
"ardenable ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> ([] \<notin> L (rexp_of rhs X)) \<and> X = L rhs"
definition
"non_empty ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> X \<noteq> {}"
definition
"self_contained ES \<equiv> \<forall> X xrhs. (X, xrhs) \<in> ES
\<longrightarrow> (\<forall> Y r.(Y, r) \<in> xrhs \<and> Y \<noteq> {[]} \<longrightarrow> (\<exists> yrhs. (Y, yrhs) \<in> ES))"
definition
"Inv ES \<equiv> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and> non_empty ES \<and> self_contained ES"
lemma wf_iter [rule_format]:
fixes f
assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
proof(induct e rule: wf_induct
[OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
fix x
assume h [rule_format]:
"\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
and px: "P x"
show "\<exists>e'. P e' \<and> Q e'"
proof(cases "Q x")
assume "Q x" with px show ?thesis by blast
next
assume nq: "\<not> Q x"
from step [OF px nq]
obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
show ?thesis
proof(rule h)
from ltf show "(e', x) \<in> inv_image less_than f"
by (simp add:inv_image_def)
next
from pe' show "P e'" .
qed
qed
qed
text {* ******BEGIN: proving the initial equation-system satisfies Inv ****** *}
lemma init_ES_satisfy_Inv:
assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
and X_in_eq_cls: "X \<in> (UNIV // (\<approx>Lang))"
shows "Inv (eqs (UNIV // (\<approx>Lang)))"
proof -
have "finite (eqs (UNIV // (\<approx>Lang)))" using finite_CS
by (auto simp add:eqs_def)
moreover have "distinct_equas (eqs (UNIV // (\<approx>Lang)))"
by (auto simp:distinct_equas_def eqs_def)
moreover have "ardenable (eqs (UNIV // (\<approx>Lang)))"
proof-
have "\<And> X rhs. (X, rhs) \<in> (eqs (UNIV // (\<approx>Lang))) \<Longrightarrow> ([] \<notin> L (rexp_of rhs X))"
proof
apply (auto simp:eqs_def rexp_of_def)
sorry
moreover have "\<forall> X rhs. (X, rhs) \<in> (eqs (UNIV // (\<approx>Lang))) \<longrightarrow> X = L rhs"
sorry
ultimately show ?thesis by (simp add:ardenable_def)
qed
moreover have "non_empty (eqs (UNIV // (\<approx>Lang)))"
by (auto simp:non_empty_def eqs_def quotient_def Image_def str_eq_rel_def str_eq_def)
moreover have "self_contained (eqs (UNIV // (\<approx>Lang)))"
by (auto simp:self_contained_def eqs_def init_rhs_def)
ultimately show ?thesis by (simp add:Inv_def)
qed
text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
lemma iteration_step:
assumes Inv_ES: "Inv ES"
and X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES"
and not_T: "card ES > 1"
shows "(\<exists> ES' xrhs'. Inv ES' \<and> (card ES', card ES) \<in> less_than \<and> (X, xrhs') \<in> ES')"
proof -
lemma distinct_rhs_equations:
"(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> distinct_rhs xrhs"
by (auto simp: equations_def equation_rhs_def distinct_rhs_def empty_rhs_def dest:no_two_cls_inters)
lemma every_nonempty_eqclass_has_strings:
"\<lbrakk>X \<in> (UNIV Quo Lang); X \<noteq> {[]}\<rbrakk> \<Longrightarrow> \<exists> clist. clist \<in> X \<and> clist \<noteq> []"
by (auto simp:quot_def equiv_class_def equiv_str_def)
lemma every_eqclass_is_derived_from_empty:
assumes not_empty: "X \<noteq> {[]}"
shows "X \<in> (UNIV Quo Lang) \<Longrightarrow> \<exists> clist. {[]};{clist} \<subseteq> X \<and> clist \<noteq> []"
using not_empty
apply (drule_tac every_nonempty_eqclass_has_strings, simp)
by (auto intro:exI[where x = clist] simp:lang_seq_def)
lemma equiv_str_in_CS:
"\<lbrakk>clist\<rbrakk>Lang \<in> (UNIV Quo Lang)"
by (auto simp:quot_def)
lemma has_str_imp_defined_by_str:
"\<lbrakk>str \<in> X; X \<in> UNIV Quo Lang\<rbrakk> \<Longrightarrow> X = \<lbrakk>str\<rbrakk>Lang"
by (auto simp:quot_def equiv_class_def equiv_str_def)
lemma every_eqclass_has_ascendent:
assumes has_str: "clist @ [c] \<in> X"
and in_CS: "X \<in> UNIV Quo Lang"
shows "\<exists> Y. Y \<in> UNIV Quo Lang \<and> Y-c\<rightarrow>X \<and> clist \<in> Y" (is "\<exists> Y. ?P Y")
proof -
have "?P (\<lbrakk>clist\<rbrakk>Lang)"
proof -
have "\<lbrakk>clist\<rbrakk>Lang \<in> UNIV Quo Lang"
by (simp add:quot_def, rule_tac x = clist in exI, simp)
moreover have "\<lbrakk>clist\<rbrakk>Lang-c\<rightarrow>X"
proof -
have "X = \<lbrakk>(clist @ [c])\<rbrakk>Lang" using has_str in_CS
by (auto intro!:has_str_imp_defined_by_str)
moreover have "\<forall> sl. sl \<in> \<lbrakk>clist\<rbrakk>Lang \<longrightarrow> sl @ [c] \<in> \<lbrakk>(clist @ [c])\<rbrakk>Lang"
by (auto simp:equiv_class_def equiv_str_def)
ultimately show ?thesis unfolding CT_def lang_seq_def
by auto
qed
moreover have "clist \<in> \<lbrakk>clist\<rbrakk>Lang"
by (auto simp:equiv_str_def equiv_class_def)
ultimately show "?P (\<lbrakk>clist\<rbrakk>Lang)" by simp
qed
thus ?thesis by blast
qed
lemma finite_charset_rS:
"finite {CHAR c |c. Y-c\<rightarrow>X}"
by (rule_tac A = UNIV and f = CHAR in finite_surj, auto)
lemma l_eq_r_in_equations:
assumes X_in_equas: "(X, xrhs) \<in> equations (UNIV Quo Lang)"
shows "X = L xrhs"
proof (cases "X = {[]}")
case True
thus ?thesis using X_in_equas
by (simp add:equations_def equation_rhs_def lang_seq_def)
next
case False
show ?thesis
proof
show "X \<subseteq> L xrhs"
proof
fix x
assume "(1)": "x \<in> X"
show "x \<in> L xrhs"
proof (cases "x = []")
assume empty: "x = []"
hence "x \<in> L (empty_rhs X)" using "(1)"
by (auto simp:empty_rhs_def lang_seq_def)
thus ?thesis using X_in_equas False empty "(1)"
unfolding equations_def equation_rhs_def by auto
next
assume not_empty: "x \<noteq> []"
hence "\<exists> clist c. x = clist @ [c]" by (case_tac x rule:rev_cases, auto)
then obtain clist c where decom: "x = clist @ [c]" by blast
moreover have "\<And> Y. \<lbrakk>Y \<in> UNIV Quo Lang; Y-c\<rightarrow>X; clist \<in> Y\<rbrakk>
\<Longrightarrow> [c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
proof -
fix Y
assume Y_is_eq_cl: "Y \<in> UNIV Quo Lang"
and Y_CT_X: "Y-c\<rightarrow>X"
and clist_in_Y: "clist \<in> Y"
with finite_charset_rS
show "[c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
by (auto simp :fold_alt_null_eqs)
qed
hence "\<exists>Xa. Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})"
using X_in_equas False not_empty "(1)" decom
by (auto dest!:every_eqclass_has_ascendent simp:equations_def equation_rhs_def lang_seq_def)
then obtain Xa where
"Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})" by blast
hence "x \<in> L {(S, folds ALT NULL {CHAR c |c. S-c\<rightarrow>X}) |S. S \<in> UNIV Quo Lang}"
using X_in_equas "(1)" decom
by (auto simp add:equations_def equation_rhs_def intro!:exI[where x = Xa])
thus "x \<in> L xrhs" using X_in_equas False not_empty unfolding equations_def equation_rhs_def
by auto
qed
qed
next
show "L xrhs \<subseteq> X"
proof
fix x
assume "(2)": "x \<in> L xrhs"
have "(2_1)": "\<And> s1 s2 r Xa. \<lbrakk>s1 \<in> Xa; s2 \<in> L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
using finite_charset_rS
by (auto simp:CT_def lang_seq_def fold_alt_null_eqs)
have "(2_2)": "\<And> s1 s2 Xa r.\<lbrakk>s1 \<in> Xa; s2 \<in> L r; (Xa, r) \<in> empty_rhs X\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
by (simp add:empty_rhs_def split:if_splits)
show "x \<in> X" using X_in_equas False "(2)"
by (auto intro:"(2_1)" "(2_2)" simp:equations_def equation_rhs_def lang_seq_def)
qed
qed
qed
lemma no_EMPTY_equations:
"(X, xrhs) \<in> equations CS \<Longrightarrow> no_EMPTY_rhs xrhs"
apply (clarsimp simp add:equations_def equation_rhs_def)
apply (simp add:no_EMPTY_rhs_def empty_rhs_def, auto)
apply (subgoal_tac "finite {CHAR c |c. Xa-c\<rightarrow>X}", drule_tac x = "[]" in fold_alt_null_eqs, clarsimp, rule finite_charset_rS)+
done
lemma init_ES_satisfy_ardenable:
"(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> ardenable (X, xrhs)"
unfolding ardenable_def
by (auto intro:distinct_rhs_equations no_EMPTY_equations simp:l_eq_r_in_equations simp del:L_rhs.simps)
lemma init_ES_satisfy_Inv:
assumes finite_CS: "finite (UNIV Quo Lang)"
and X_in_eq_cls: "X \<in> UNIV Quo Lang"
shows "Inv X (equations (UNIV Quo Lang))"
proof -
have "finite (equations (UNIV Quo Lang))" using finite_CS
by (auto simp:equations_def)
moreover have "\<exists>rhs. (X, rhs) \<in> equations (UNIV Quo Lang)" using X_in_eq_cls
by (simp add:equations_def)
moreover have "distinct_equas (equations (UNIV Quo Lang))"
by (auto simp:distinct_equas_def equations_def)
moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow>
rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equations (UNIV Quo Lang)))"
apply (simp add:left_eq_cls_def equations_def rhs_eq_cls_def equation_rhs_def)
by (auto simp:empty_rhs_def split:if_splits)
moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> X \<noteq> {}"
by (clarsimp simp:equations_def empty_notin_CS intro:classical)
moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> ardenable (X, xrhs)"
by (auto intro!:init_ES_satisfy_ardenable)
ultimately show ?thesis by (simp add:Inv_def)
qed
text {* *********** END: proving the initial equation-system satisfies Inv ******* *}
text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
lemma not_T_aux: "\<lbrakk>\<not> TCon (insert a A); x = a\<rbrakk>
\<Longrightarrow> \<exists>y. x \<noteq> y \<and> y \<in> insert a A "
apply (case_tac "insert a A = {a}")
by (auto simp:TCon_def)
lemma not_T_atleast_2[rule_format]:
"finite S \<Longrightarrow> \<forall> x. x \<in> S \<and> (\<not> TCon S)\<longrightarrow> (\<exists> y. x \<noteq> y \<and> y \<in> S)"
apply (erule finite.induct, simp)
apply (clarify, case_tac "x = a")
by (erule not_T_aux, auto)
lemma exist_another_equa:
"\<lbrakk>\<not> TCon ES; finite ES; distinct_equas ES; (X, rhl) \<in> ES\<rbrakk> \<Longrightarrow> \<exists> Y yrhl. (Y, yrhl) \<in> ES \<and> X \<noteq> Y"
apply (drule not_T_atleast_2, simp)
apply (clarsimp simp:distinct_equas_def)
apply (drule_tac x= X in spec, drule_tac x = rhl in spec, drule_tac x = b in spec)
by auto
lemma Inv_mono_with_lambda:
assumes Inv_ES: "Inv X ES"
and X_noteq_Y: "X \<noteq> {[]}"
shows "Inv X (ES - {({[]}, yrhs)})"
proof -
have "finite (ES - {({[]}, yrhs)})" using Inv_ES
by (simp add:Inv_def)
moreover have "\<exists>rhs. (X, rhs) \<in> ES - {({[]}, yrhs)}" using Inv_ES X_noteq_Y
by (simp add:Inv_def)
moreover have "distinct_equas (ES - {({[]}, yrhs)})" using Inv_ES X_noteq_Y
apply (clarsimp simp:Inv_def distinct_equas_def)
by (drule_tac x = Xa in spec, simp)
moreover have "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
ardenable (X, xrhs) \<and> X \<noteq> {}" using Inv_ES
by (clarify, simp add:Inv_def)
moreover
have "insert {[]} (left_eq_cls (ES - {({[]}, yrhs)})) = insert {[]} (left_eq_cls ES)"
by (auto simp:left_eq_cls_def)
hence "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (ES - {({[]}, yrhs)}))"
using Inv_ES by (auto simp:Inv_def)
ultimately show ?thesis by (simp add:Inv_def)
qed
lemma non_empty_card_prop:
"finite ES \<Longrightarrow> \<forall>e. e \<in> ES \<longrightarrow> card ES - Suc 0 < card ES"
apply (erule finite.induct, simp)
apply (case_tac[!] "a \<in> A")
by (auto simp:insert_absorb)
lemma ardenable_prop:
assumes not_lambda: "Y \<noteq> {[]}"
and ardable: "ardenable (Y, yrhs)"
shows "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" (is "\<exists> yrhs'. ?P yrhs'")
proof (cases "(\<exists> reg. (Y, reg) \<in> yrhs)")
case True
thus ?thesis
proof
fix reg
assume self_contained: "(Y, reg) \<in> yrhs"
show ?thesis
proof -
have "?P (arden_variate Y reg yrhs)"
proof -
have "Y = L (arden_variate Y reg yrhs)"
using self_contained not_lambda ardable
by (rule_tac arden_variate_valid, simp_all add:ardenable_def)
moreover have "distinct_rhs (arden_variate Y reg yrhs)"
using ardable
by (auto simp:distinct_rhs_def arden_variate_def seq_rhs_r_def del_x_paired_def ardenable_def)
moreover have "rhs_eq_cls (arden_variate Y reg yrhs) = rhs_eq_cls yrhs - {Y}"
proof -
have "\<And> rhs r. rhs_eq_cls (seq_rhs_r rhs r) = rhs_eq_cls rhs"
apply (auto simp:rhs_eq_cls_def seq_rhs_r_def image_def)
by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "(x, ra)" in bexI, simp+)
moreover have "\<And> rhs X. rhs_eq_cls (del_x_paired rhs X) = rhs_eq_cls rhs - {X}"
by (auto simp:rhs_eq_cls_def del_x_paired_def)
ultimately show ?thesis by (simp add:arden_variate_def)
qed
ultimately show ?thesis by simp
qed
thus ?thesis by (rule_tac x= "arden_variate Y reg yrhs" in exI, simp)
qed
qed
next
case False
hence "(2)": "rhs_eq_cls yrhs - {Y} = rhs_eq_cls yrhs"
by (auto simp:rhs_eq_cls_def)
show ?thesis
proof -
have "?P yrhs" using False ardable "(2)"
by (simp add:ardenable_def)
thus ?thesis by blast
qed
qed
lemma equas_subst_f_del_no_other:
assumes self_contained: "(Y, rhs) \<in> ES"
shows "\<exists> rhs'. (Y, rhs') \<in> (equas_subst_f X xrhs ` ES)" (is "\<exists> rhs'. ?P rhs'")
proof -
have "\<exists> rhs'. equas_subst_f X xrhs (Y, rhs) = (Y, rhs')"
by (auto simp:equas_subst_f_def)
then obtain rhs' where "equas_subst_f X xrhs (Y, rhs) = (Y, rhs')" by blast
hence "?P rhs'" unfolding image_def using self_contained
by (auto intro:bexI[where x = "(Y, rhs)"])
thus ?thesis by blast
qed
lemma del_x_paired_del_only_x:
"\<lbrakk>X \<noteq> Y; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> (X, rhs) \<in> del_x_paired ES Y"
by (auto simp:del_x_paired_def)
lemma equas_subst_del_no_other:
"\<lbrakk>(X, rhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> (\<exists>rhs. (X, rhs) \<in> equas_subst ES Y rhs')"
unfolding equas_subst_def
apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other)
by (erule exE, drule del_x_paired_del_only_x, auto)
lemma equas_subst_holds_distinct:
"distinct_equas ES \<Longrightarrow> distinct_equas (equas_subst ES Y rhs')"
apply (clarsimp simp add:equas_subst_def distinct_equas_def del_x_paired_def equas_subst_f_def)
by (auto split:if_splits)
lemma del_x_paired_dels:
"(X, rhs) \<in> ES \<Longrightarrow> {Y. Y \<in> ES \<and> fst Y = X} \<inter> ES \<noteq> {}"
by (auto)
lemma del_x_paired_subset:
"(X, rhs) \<in> ES \<Longrightarrow> ES - {Y. Y \<in> ES \<and> fst Y = X} \<subset> ES"
apply (drule del_x_paired_dels)
by auto
lemma del_x_paired_card_less:
"\<lbrakk>finite ES; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> card (del_x_paired ES X) < card ES"
apply (simp add:del_x_paired_def)
apply (drule del_x_paired_subset)
by (auto intro:psubset_card_mono)
lemma equas_subst_card_less:
"\<lbrakk>finite ES; (Y, yrhs) \<in> ES\<rbrakk> \<Longrightarrow> card (equas_subst ES Y rhs') < card ES"
apply (simp add:equas_subst_def)
apply (frule_tac h = "equas_subst_f Y rhs'" in finite_imageI)
apply (drule_tac f = "equas_subst_f Y rhs'" in Finite_Set.card_image_le)
apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other,erule exE)
by (drule del_x_paired_card_less, auto)
lemma equas_subst_holds_distinct_rhs:
assumes dist': "distinct_rhs yrhs'"
and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
shows "distinct_rhs xrhs"
using X_in history
apply (clarsimp simp:equas_subst_def del_x_paired_def)
apply (drule_tac x = a in spec, drule_tac x = b in spec)
apply (simp add:ardenable_def equas_subst_f_def)
by (auto intro:rhs_subst_holds_distinct_rhs simp:dist' split:if_splits)
lemma r_no_EMPTY_imp_seq_rhs_r_no_EMPTY:
"[] \<notin> L r \<Longrightarrow> no_EMPTY_rhs (seq_rhs_r rhs r)"
by (auto simp:no_EMPTY_rhs_def seq_rhs_r_def lang_seq_def)
lemma del_x_paired_holds_no_EMPTY:
"no_EMPTY_rhs yrhs \<Longrightarrow> no_EMPTY_rhs (del_x_paired yrhs Y)"
by (auto simp:no_EMPTY_rhs_def del_x_paired_def)
lemma rhs_subst_holds_no_EMPTY:
"\<lbrakk>no_EMPTY_rhs yrhs; (Y, r) \<in> yrhs; Y \<noteq> {[]}\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (rhs_subst yrhs Y rhs' r)"
apply (auto simp:rhs_subst_def intro!:no_EMPTY_rhss_imp_merge_no_EMPTY r_no_EMPTY_imp_seq_rhs_r_no_EMPTY del_x_paired_holds_no_EMPTY)
by (auto simp:no_EMPTY_rhs_def)
lemma equas_subst_holds_no_EMPTY:
assumes substor: "Y \<noteq> {[]}"
and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
shows "no_EMPTY_rhs xrhs"
proof-
from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
by (auto simp add:equas_subst_def del_x_paired_def)
then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
hence dist_zrhs: "distinct_rhs zrhs" using history
by (auto simp:ardenable_def)
show ?thesis
proof (cases "\<exists> r. (Y, r) \<in> zrhs")
case True
then obtain r where Y_in_zrhs: "(Y, r) \<in> zrhs" ..
hence some: "(SOME r. (Y, r) \<in> zrhs) = r" using Z_in dist_zrhs
by (auto simp:distinct_rhs_def)
hence "no_EMPTY_rhs (rhs_subst zrhs Y yrhs' r)"
using substor Y_in_zrhs history Z_in
by (rule_tac rhs_subst_holds_no_EMPTY, auto simp:ardenable_def)
thus ?thesis using X_Z True some
by (simp add:equas_subst_def equas_subst_f_def)
next
case False
hence "(X, xrhs) = (Z, zrhs)" using Z_in X_Z
by (simp add:equas_subst_f_def)
thus ?thesis using history Z_in
by (auto simp:ardenable_def)
qed
qed
lemma equas_subst_f_holds_left_eq_right:
assumes substor: "Y = L rhs'"
and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> distinct_rhs xrhs \<and> X = L xrhs"
and subst: "(X, xrhs) = equas_subst_f Y rhs' (Z, zrhs)"
and self_contained: "(Z, zrhs) \<in> ES"
shows "X = L xrhs"
proof (cases "\<exists> r. (Y, r) \<in> zrhs")
case True
from True obtain r where "(1)":"(Y, r) \<in> zrhs" ..
show ?thesis
proof -
from history self_contained
have dist: "distinct_rhs zrhs" by auto
hence "(SOME r. (Y, r) \<in> zrhs) = r" using self_contained "(1)"
using distinct_rhs_def by (auto intro:some_equality)
moreover have "L zrhs = L (rhs_subst zrhs Y rhs' r)" using substor dist "(1)" self_contained
by (rule_tac rhs_subst_prop1, auto simp:distinct_equas_rhs_def)
ultimately show ?thesis using subst history self_contained
by (auto simp:equas_subst_f_def split:if_splits)
qed
next
case False
thus ?thesis using history subst self_contained
by (auto simp:equas_subst_f_def)
qed
lemma equas_subst_holds_left_eq_right:
assumes history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
and substor: "Y = L rhs'"
and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
shows "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y rhs' \<longrightarrow> X = L xrhs"
apply (clarsimp simp add:equas_subst_def del_x_paired_def)
using substor
apply (drule_tac equas_subst_f_holds_left_eq_right)
using history
by (auto simp:ardenable_def)
lemma equas_subst_holds_ardenable:
assumes substor: "Y = L yrhs'"
and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
and dist': "distinct_rhs yrhs'"
and not_lambda: "Y \<noteq> {[]}"
shows "ardenable (X, xrhs)"
proof -
have "distinct_rhs xrhs" using history X_in dist'
by (auto dest:equas_subst_holds_distinct_rhs)
moreover have "no_EMPTY_rhs xrhs" using history X_in not_lambda
by (auto intro:equas_subst_holds_no_EMPTY)
moreover have "X = L xrhs" using history substor X_in
by (auto dest: equas_subst_holds_left_eq_right simp del:L_rhs.simps)
ultimately show ?thesis using ardenable_def by simp
qed
lemma equas_subst_holds_cls_defined:
assumes X_in: "(X, xrhs) \<in> equas_subst ES Y yrhs'"
and Inv_ES: "Inv X' ES"
and subst: "(Y, yrhs) \<in> ES"
and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
shows "rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
proof-
have tac: "\<lbrakk> A \<subseteq> B; C \<subseteq> D; E \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> E \<subseteq> B \<union> D" by auto
from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
by (auto simp add:equas_subst_def del_x_paired_def)
then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
hence "rhs_eq_cls zrhs \<subseteq> insert {[]} (left_eq_cls ES)" using Inv_ES
by (auto simp:Inv_def)
moreover have "rhs_eq_cls yrhs' \<subseteq> insert {[]} (left_eq_cls ES) - {Y}"
using Inv_ES subst cls_holds_but_Y
by (auto simp:Inv_def)
moreover have "rhs_eq_cls xrhs \<subseteq> rhs_eq_cls zrhs \<union> rhs_eq_cls yrhs' - {Y}"
using X_Z cls_holds_but_Y
apply (clarsimp simp add:equas_subst_f_def rhs_subst_def split:if_splits)
by (auto simp:rhs_eq_cls_def merge_rhs_def del_x_paired_def seq_rhs_r_def)
moreover have "left_eq_cls (equas_subst ES Y yrhs') = left_eq_cls ES - {Y}" using subst
by (auto simp: left_eq_cls_def equas_subst_def del_x_paired_def equas_subst_f_def
dest: equas_subst_f_del_no_other
split: if_splits)
ultimately show ?thesis by blast
qed
lemma iteration_step:
assumes Inv_ES: "Inv X ES"
and not_T: "\<not> TCon ES"
shows "(\<exists> ES'. Inv X ES' \<and> (card ES', card ES) \<in> less_than)"
proof -
from Inv_ES not_T have another: "\<exists>Y yrhs. (Y, yrhs) \<in> ES \<and> X \<noteq> Y" unfolding Inv_def
by (clarify, rule_tac exist_another_equa[where X = X], auto)
then obtain Y yrhs where subst: "(Y, yrhs) \<in> ES" and not_X: " X \<noteq> Y" by blast
show ?thesis (is "\<exists> ES'. ?P ES'")
proof (cases "Y = {[]}")
case True
--"in this situation, we pick a \"\<lambda>\" equation, thus directly remove it from the equation-system"
have "?P (ES - {(Y, yrhs)})"
proof
show "Inv X (ES - {(Y, yrhs)})" using True not_X
by (simp add:Inv_ES Inv_mono_with_lambda)
next
show "(card (ES - {(Y, yrhs)}), card ES) \<in> less_than" using Inv_ES subst
by (auto elim:non_empty_card_prop[rule_format] simp:Inv_def)
qed
thus ?thesis by blast
next
case False
--"in this situation, we pick a equation and using ardenable to get a
rhs without itself in it, then use equas_subst to form a new equation-system"
hence "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
using subst Inv_ES
by (auto intro:ardenable_prop simp add:Inv_def simp del:L_rhs.simps)
then obtain yrhs' where Y'_l_eq_r: "Y = L yrhs'"
and dist_Y': "distinct_rhs yrhs'"
and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" by blast
hence "?P (equas_subst ES Y yrhs')"
proof -
have finite_del: "\<And> S x. finite S \<Longrightarrow> finite (del_x_paired S x)"
apply (rule_tac A = "del_x_paired S x" in finite_subset)
by (auto simp:del_x_paired_def)
have "finite (equas_subst ES Y yrhs')" using Inv_ES
by (auto intro!:finite_del simp:equas_subst_def Inv_def)
moreover have "\<exists>rhs. (X, rhs) \<in> equas_subst ES Y yrhs'" using Inv_ES not_X
by (auto intro:equas_subst_del_no_other simp:Inv_def)
moreover have "distinct_equas (equas_subst ES Y yrhs')" using Inv_ES
by (auto intro:equas_subst_holds_distinct simp:Inv_def)
moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> ardenable (X, xrhs)"
using Inv_ES dist_Y' False Y'_l_eq_r
apply (clarsimp simp:Inv_def)
by (rule equas_subst_holds_ardenable, simp_all)
moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> X \<noteq> {}" using Inv_ES
by (clarsimp simp:equas_subst_def Inv_def del_x_paired_def equas_subst_f_def split:if_splits, auto)
moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow>
rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
using Inv_ES subst cls_holds_but_Y
apply (rule_tac impI | rule_tac allI)+
by (erule equas_subst_holds_cls_defined, auto)
moreover have "(card (equas_subst ES Y yrhs'), card ES) \<in> less_than"using Inv_ES subst
by (simp add:equas_subst_card_less Inv_def)
ultimately show "?P (equas_subst ES Y yrhs')" by (auto simp:Inv_def)
qed
thus ?thesis by blast
qed
qed
text {* ***** END: proving every equation-system's iteration step satisfies Inv ************** *}
lemma iteration_conc:
assumes history: "Inv X ES"
shows "\<exists> ES'. Inv X ES' \<and> TCon ES'" (is "\<exists> ES'. ?P ES'")
proof (cases "TCon ES")
case True
hence "?P ES" using history by simp
thus ?thesis by blast
next
case False
thus ?thesis using history iteration_step
by (rule_tac f = card in wf_iter, simp_all)
qed
lemma eqset_imp_iff': "A = B \<Longrightarrow> \<forall> x. x \<in> A \<longleftrightarrow> x \<in> B"
apply (auto simp:mem_def)
done
lemma set_cases2:
"\<lbrakk>(A = {} \<Longrightarrow> R A); \<And> x. (A = {x}) \<Longrightarrow> R A; \<And> x y. \<lbrakk>x \<noteq> y; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> R A\<rbrakk> \<Longrightarrow> R A"
apply (case_tac "A = {}", simp)
by (case_tac "\<exists> x. A = {x}", clarsimp, blast)
lemma rhs_aux:"\<lbrakk>distinct_rhs rhs; {Y. \<exists>r. (Y, r) \<in> rhs} = {X}\<rbrakk> \<Longrightarrow> (\<exists>r. rhs = {(X, r)})"
apply (rule_tac A = rhs in set_cases2, simp)
apply (drule_tac x = X in eqset_imp_iff, clarsimp)
apply (drule eqset_imp_iff',clarsimp)
apply (frule_tac x = a in spec, drule_tac x = aa in spec)
by (auto simp:distinct_rhs_def)
lemma every_eqcl_has_reg:
assumes finite_CS: "finite (UNIV Quo Lang)"
and X_in_CS: "X \<in> (UNIV Quo Lang)"
shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
proof-
have "\<exists>ES'. Inv X ES' \<and> TCon ES'" using finite_CS X_in_CS
by (auto intro:init_ES_satisfy_Inv iteration_conc)
then obtain ES' where Inv_ES': "Inv X ES'"
and TCon_ES': "TCon ES'" by blast
from Inv_ES' TCon_ES'
have "\<exists> rhs. ES' = {(X, rhs)}"
apply (clarsimp simp:Inv_def TCon_def)
apply (rule_tac x = rhs in exI)
by (auto dest!:card_Suc_Diff1 simp:card_eq_0_iff)
then obtain rhs where ES'_single_equa: "ES' = {(X, rhs)}" ..
hence X_ardenable: "ardenable (X, rhs)" using Inv_ES'
by (simp add:Inv_def)
from X_ardenable have X_l_eq_r: "X = L rhs"
by (simp add:ardenable_def)
hence rhs_not_empty: "rhs \<noteq> {}" using Inv_ES' ES'_single_equa
by (auto simp:Inv_def ardenable_def)
have rhs_eq_cls: "rhs_eq_cls rhs \<subseteq> {X, {[]}}"
using Inv_ES' ES'_single_equa
by (auto simp:Inv_def ardenable_def left_eq_cls_def)
have X_not_empty: "X \<noteq> {}" using Inv_ES' ES'_single_equa
by (auto simp:Inv_def)
show ?thesis
proof (cases "X = {[]}")
case True
hence "?E EMPTY" by (simp)
thus ?thesis by blast
next
case False with X_ardenable
have "\<exists> rhs'. X = L rhs' \<and> rhs_eq_cls rhs' = rhs_eq_cls rhs - {X} \<and> distinct_rhs rhs'"
by (drule_tac ardenable_prop, auto)
then obtain rhs' where X_eq_rhs': "X = L rhs'"
and rhs'_eq_cls: "rhs_eq_cls rhs' = rhs_eq_cls rhs - {X}"
and rhs'_dist : "distinct_rhs rhs'" by blast
have "rhs_eq_cls rhs' \<subseteq> {{[]}}" using rhs_eq_cls False rhs'_eq_cls rhs_not_empty
by blast
hence "rhs_eq_cls rhs' = {{[]}}" using X_not_empty X_eq_rhs'
by (auto simp:rhs_eq_cls_def)
hence "\<exists> r. rhs' = {({[]}, r)}" using rhs'_dist
by (auto intro:rhs_aux simp:rhs_eq_cls_def)
then obtain r where "rhs' = {({[]}, r)}" ..
hence "?E r" using X_eq_rhs' by (auto simp add:lang_seq_def)
thus ?thesis by blast
qed
qed
text {* definition of a regular language *}
abbreviation
reg :: "string set => bool"
where
"reg L1 \<equiv> (\<exists>r::rexp. L r = L1)"
theorem myhill_nerode:
assumes finite_CS: "finite (UNIV Quo Lang)"
shows "\<exists> (reg::rexp). Lang = L reg" (is "\<exists> r. ?P r")
proof -
have has_r_each: "\<forall>C\<in>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists>(r::rexp). C = L r" using finite_CS
by (auto dest:every_eqcl_has_reg)
have "\<exists> (rS::rexp set). finite rS \<and>
(\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> rS. C = L r) \<and>
(\<forall> r \<in> rS. \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r)"
(is "\<exists> rS. ?Q rS")
proof-
have "\<And> C. C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<Longrightarrow> C = L (SOME (ra::rexp). C = L ra)"
using has_r_each
apply (erule_tac x = C in ballE, erule_tac exE)
by (rule_tac a = r in someI2, simp+)
hence "?Q ((\<lambda> C. SOME r. C = L r) ` {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang})" using has_r_each
using finite_CS by auto
thus ?thesis by blast
qed
then obtain rS where finite_rS : "finite rS"
and has_r_each': "\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> (rS::rexp set). C = L r"
and has_cl_each: "\<forall> r \<in> (rS::rexp set). \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r" by blast
have "?P (folds ALT NULL rS)"
proof
show "Lang \<subseteq> L (folds ALT NULL rS)" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_r_each'
apply (clarsimp simp:fold_alt_null_eqs) by blast
next
show "L (folds ALT NULL rS) \<subseteq> Lang" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_cl_each
by (clarsimp simp:fold_alt_null_eqs)
qed
thus ?thesis by blast
qed
text {* tests by Christian *}
(* Alternative definition for Quo *)
definition
QUOT :: "string set \<Rightarrow> (string set) set"
where
"QUOT Lang \<equiv> (\<Union>x. {\<lbrakk>x\<rbrakk>Lang})"
lemma test:
"UNIV Quo Lang = QUOT Lang"
by (auto simp add: quot_def QUOT_def)
lemma test1:
assumes finite_CS: "finite (QUOT Lang)"
shows "reg Lang"
using finite_CS
unfolding test[symmetric]
by (auto dest: myhill_nerode)
lemma cons_one: "x @ y \<in> {z} \<Longrightarrow> x @ y = z"
by simp
lemma quot_lambda: "QUOT {[]} = {{[]}, UNIV - {[]}}"
proof
show "QUOT {[]} \<subseteq> {{[]}, UNIV - {[]}}"
proof
fix x
assume in_quot: "x \<in> QUOT {[]}"
show "x \<in> {{[]}, UNIV - {[]}}"
proof -
from in_quot
have "x = {[]} \<or> x = UNIV - {[]}"
unfolding QUOT_def equiv_class_def
proof
fix xa
assume in_univ: "xa \<in> UNIV"
and in_eqiv: "x \<in> {{y. xa \<equiv>{[]}\<equiv> y}}"
show "x = {[]} \<or> x = UNIV - {[]}"
proof(cases "xa = []")
case True
hence "{y. xa \<equiv>{[]}\<equiv> y} = {[]}" using in_eqiv
by (auto simp add:equiv_str_def)
thus ?thesis using in_eqiv by (rule_tac disjI1, simp)
next
case False
hence "{y. xa \<equiv>{[]}\<equiv> y} = UNIV - {[]}" using in_eqiv
by (auto simp:equiv_str_def)
thus ?thesis using in_eqiv by (rule_tac disjI2, simp)
qed
qed
thus ?thesis by simp
qed
qed
next
show "{{[]}, UNIV - {[]}} \<subseteq> QUOT {[]}"
proof
fix x
assume in_res: "x \<in> {{[]}, (UNIV::string set) - {[]}}"
show "x \<in> (QUOT {[]})"
proof -
have "x = {[]} \<Longrightarrow> x \<in> QUOT {[]}"
apply (simp add:QUOT_def equiv_class_def equiv_str_def)
by (rule_tac x = "[]" in exI, auto)
moreover have "x = UNIV - {[]} \<Longrightarrow> x \<in> QUOT {[]}"
apply (simp add:QUOT_def equiv_class_def equiv_str_def)
by (rule_tac x = "''a''" in exI, auto)
ultimately show ?thesis using in_res by blast
qed
qed
qed
lemma quot_single_aux: "\<lbrakk>x \<noteq> []; x \<noteq> [c]\<rbrakk> \<Longrightarrow> x @ z \<noteq> [c]"
by (induct x, auto)
lemma quot_single: "\<And> (c::char). QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
proof -
fix c::"char"
have exist_another: "\<exists> a. a \<noteq> c"
apply (case_tac "c = CHR ''a''")
apply (rule_tac x = "CHR ''b''" in exI, simp)
by (rule_tac x = "CHR ''a''" in exI, simp)
show "QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
proof
show "QUOT {[c]} \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
proof
fix x
assume in_quot: "x \<in> QUOT {[c]}"
show "x \<in> {{[]}, {[c]}, UNIV - {[],[c]}}"
proof -
from in_quot
have "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[],[c]}"
unfolding QUOT_def equiv_class_def
proof
fix xa
assume in_eqiv: "x \<in> {{y. xa \<equiv>{[c]}\<equiv> y}}"
show "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[], [c]}"
proof-
have "xa = [] \<Longrightarrow> x = {[]}" using in_eqiv
by (auto simp add:equiv_str_def)
moreover have "xa = [c] \<Longrightarrow> x = {[c]}"
proof -
have "xa = [c] \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = {[c]}" using in_eqiv
apply (simp add:equiv_str_def)
apply (rule set_ext, rule iffI, simp)
apply (drule_tac x = "[]" in spec, auto)
done
thus "xa = [c] \<Longrightarrow> x = {[c]}" using in_eqiv by simp
qed
moreover have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}"
proof -
have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = UNIV - {[],[c]}"
apply (clarsimp simp add:equiv_str_def)
apply (rule set_ext, rule iffI, simp)
apply (rule conjI)
apply (drule_tac x = "[c]" in spec, simp)
apply (drule_tac x = "[]" in spec, simp)
by (auto dest:quot_single_aux)
thus "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}" using in_eqiv by simp
qed
ultimately show ?thesis by blast
qed
qed
thus ?thesis by simp
qed
qed
next
show "{{[]}, {[c]}, UNIV - {[],[c]}} \<subseteq> QUOT {[c]}"
proof
fix x
assume in_res: "x \<in> {{[]},{[c]}, (UNIV::string set) - {[],[c]}}"
show "x \<in> (QUOT {[c]})"
proof -
have "x = {[]} \<Longrightarrow> x \<in> QUOT {[c]}"
apply (simp add:QUOT_def equiv_class_def equiv_str_def)
by (rule_tac x = "[]" in exI, auto)
moreover have "x = {[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
apply (simp add:QUOT_def equiv_class_def equiv_str_def)
apply (rule_tac x = "[c]" in exI, simp)
apply (rule set_ext, rule iffI, simp+)
by (drule_tac x = "[]" in spec, simp)
moreover have "x = UNIV - {[],[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
using exist_another
apply (clarsimp simp add:QUOT_def equiv_class_def equiv_str_def)
apply (rule_tac x = "[a]" in exI, simp)
apply (rule set_ext, rule iffI, simp)
apply (clarsimp simp:quot_single_aux, simp)
apply (rule conjI)
apply (drule_tac x = "[c]" in spec, simp)
by (drule_tac x = "[]" in spec, simp)
ultimately show ?thesis using in_res by blast
qed
qed
qed
qed
lemma eq_class_imp_eq_str:
"\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang \<Longrightarrow> x \<equiv>lang\<equiv> y"
by (auto simp:equiv_class_def equiv_str_def)
lemma finite_tag_image:
"finite (range tag) \<Longrightarrow> finite (((op `) tag) ` S)"
apply (rule_tac B = "Pow (tag ` UNIV)" in finite_subset)
by (auto simp add:image_def Pow_def)
lemma str_inj_imps:
assumes str_inj: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<equiv>lang\<equiv> n"
shows "inj_on ((op `) tag) (QUOT lang)"
proof (clarsimp simp add:inj_on_def QUOT_def)
fix x y
assume eq_tag: "tag ` \<lbrakk>x\<rbrakk>lang = tag ` \<lbrakk>y\<rbrakk>lang"
show "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang"
proof -
have aux1:"\<And>a b. a \<in> (\<lbrakk>b\<rbrakk>lang) \<Longrightarrow> (a \<equiv>lang\<equiv> b)"
by (simp add:equiv_class_def equiv_str_def)
have aux2: "\<And> A B f. \<lbrakk>f ` A = f ` B; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists> a b. a \<in> A \<and> b \<in> B \<and> f a = f b"
by auto
have aux3: "\<And> a l. \<lbrakk>a\<rbrakk>l \<noteq> {}"
by (auto simp:equiv_class_def equiv_str_def)
show ?thesis using eq_tag
apply (drule_tac aux2, simp add:aux3, clarsimp)
apply (drule_tac str_inj, (drule_tac aux1)+)
by (simp add:equiv_str_def equiv_class_def)
qed
qed
definition tag_str_ALT :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
where
"tag_str_ALT L\<^isub>1 L\<^isub>2 x \<equiv> (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)"
lemma tag_str_alt_range_finite:
assumes finite1: "finite (QUOT L\<^isub>1)"
and finite2: "finite (QUOT L\<^isub>2)"
shows "finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))"
proof -
have "{y. \<exists>x. y = (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)} \<subseteq> (QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"
by (auto simp:QUOT_def)
thus ?thesis using finite1 finite2
by (auto simp: image_def tag_str_ALT_def UNION_def
intro: finite_subset[where B = "(QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"])
qed
lemma tag_str_alt_inj:
"tag_str_ALT L\<^isub>1 L\<^isub>2 x = tag_str_ALT L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<equiv>(L\<^isub>1 \<union> L\<^isub>2)\<equiv> y"
apply (simp add:tag_str_ALT_def equiv_class_def equiv_str_def)
by blast
lemma quot_alt:
assumes finite1: "finite (QUOT L\<^isub>1)"
and finite2: "finite (QUOT L\<^isub>2)"
shows "finite (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
proof (rule_tac f = "(op `) (tag_str_ALT L\<^isub>1 L\<^isub>2)" in finite_imageD)
show "finite (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 \<union> L\<^isub>2))"
using finite_tag_image tag_str_alt_range_finite finite1 finite2
by auto
next
show "inj_on (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
apply (rule_tac str_inj_imps)
by (erule_tac tag_str_alt_inj)
qed
(* list_diff:: list substract, once different return tailer *)
fun list_diff :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "-" 51)
where
"list_diff [] xs = []" |
"list_diff (x#xs) [] = x#xs" |
"list_diff (x#xs) (y#ys) = (if x = y then list_diff xs ys else (x#xs))"
lemma [simp]: "(x @ y) - x = y"
apply (induct x)
by (case_tac y, simp+)
lemma [simp]: "x - x = []"
by (induct x, auto)
lemma [simp]: "x = xa @ y \<Longrightarrow> x - xa = y "
by (induct x, auto)
lemma [simp]: "x - [] = x"
by (induct x, auto)
lemma [simp]: "xa \<le> x \<Longrightarrow> (x @ y) - xa = (x - xa) @ y"
by (auto elim:prefixE)
definition tag_str_SEQ:: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set set)"
where
"tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv> if (\<exists> xa \<le> x. xa \<in> L\<^isub>1)
then (\<lbrakk>x\<rbrakk>L\<^isub>1, {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1})
else (\<lbrakk>x\<rbrakk>L\<^isub>1, {})"
lemma tag_seq_eq_E:
"tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y \<Longrightarrow>
((\<exists> xa \<le> x. xa \<in> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1 \<and>
{\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1} ) \<or>
((\<forall> xa \<le> x. xa \<notin> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1)"
by (simp add:tag_str_SEQ_def split:if_splits, blast)
lemma tag_str_seq_range_finite:
assumes finite1: "finite (QUOT L\<^isub>1)"
and finite2: "finite (QUOT L\<^isub>2)"
shows "finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
proof -
have "(range (tag_str_SEQ L\<^isub>1 L\<^isub>2)) \<subseteq> (QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))"
by (auto simp:image_def tag_str_SEQ_def QUOT_def)
thus ?thesis using finite1 finite2
by (rule_tac B = "(QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))" in finite_subset, auto)
qed
lemma app_in_seq_decom[rule_format]:
"\<forall> x. x @ z \<in> L\<^isub>1 ; L\<^isub>2 \<longrightarrow> (\<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2) \<or>
(\<exists> za \<le> z. (x @ za) \<in> L\<^isub>1 \<and> (z - za) \<in> L\<^isub>2)"
apply (induct z)
apply (simp, rule allI, rule impI, rule disjI1)
apply (clarsimp simp add:lang_seq_def)
apply (rule_tac x = s1 in exI, simp)
apply (rule allI | rule impI)+
apply (drule_tac x = "x @ [a]" in spec, simp)
apply (erule exE | erule conjE | erule disjE)+
apply (rule disjI2, rule_tac x = "[a]" in exI, simp)
apply (rule disjI1, rule_tac x = xa in exI, simp)
apply (erule exE | erule conjE)+
apply (rule disjI2, rule_tac x = "a # za" in exI, simp)
done
lemma tag_str_seq_inj:
assumes tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
shows "(x::string) \<equiv>(L\<^isub>1 ; L\<^isub>2)\<equiv> y"
proof -
have aux: "\<And> x y z. \<lbrakk>tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y; x @ z \<in> L\<^isub>1 ; L\<^isub>2\<rbrakk>
\<Longrightarrow> y @ z \<in> L\<^isub>1 ; L\<^isub>2"
proof (drule app_in_seq_decom, erule disjE)
fix x y z
assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
and x_gets_l2: "\<exists>xa\<le>x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2"
from x_gets_l2 have "\<exists> xa \<le> x. xa \<in> L\<^isub>1" by blast
hence xy_l2:"{\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1}"
using tag_eq tag_seq_eq_E by blast
from x_gets_l2 obtain xa where xa_le_x: "xa \<le> x"
and xa_in_l1: "xa \<in> L\<^isub>1"
and rest_in_l2: "(x - xa) @ z \<in> L\<^isub>2" by blast
hence "\<exists> ya. \<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 \<and> ya \<le> y \<and> ya \<in> L\<^isub>1" using xy_l2 by auto
then obtain ya where ya_le_x: "ya \<le> y"
and ya_in_l1: "ya \<in> L\<^isub>1"
and rest_eq: "\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2" by blast
from rest_eq rest_in_l2 have "(y - ya) @ z \<in> L\<^isub>2"
by (auto simp:equiv_class_def equiv_str_def)
hence "ya @ ((y - ya) @ z) \<in> L\<^isub>1 ; L\<^isub>2" using ya_in_l1
by (auto simp:lang_seq_def)
thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using ya_le_x
by (erule_tac prefixE, simp)
next
fix x y z
assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
and x_gets_l1: "\<exists>za\<le>z. x @ za \<in> L\<^isub>1 \<and> z - za \<in> L\<^isub>2"
from tag_eq tag_seq_eq_E have x_y_eq: "\<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1" by blast
from x_gets_l1 obtain za where za_le_z: "za \<le> z"
and x_za_in_l1: "(x @ za) \<in> L\<^isub>1"
and rest_in_l2: "z - za \<in> L\<^isub>2" by blast
from x_y_eq x_za_in_l1 have y_za_in_l1: "y @ za \<in> L\<^isub>1"
by (auto simp:equiv_class_def equiv_str_def)
hence "(y @ za) @ (z - za) \<in> L\<^isub>1 ; L\<^isub>2" using rest_in_l2
apply (simp add:lang_seq_def)
by (rule_tac x = "y @ za" in exI, rule_tac x = "z - za" in exI, simp)
thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using za_le_z
by (erule_tac prefixE, simp)
qed
show ?thesis using tag_eq
apply (simp add:equiv_str_def)
by (auto intro:aux)
qed
lemma quot_seq:
assumes finite1: "finite (QUOT L\<^isub>1)"
and finite2: "finite (QUOT L\<^isub>2)"
shows "finite (QUOT (L\<^isub>1;L\<^isub>2))"
proof (rule_tac f = "(op `) (tag_str_SEQ L\<^isub>1 L\<^isub>2)" in finite_imageD)
show "finite (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 ; L\<^isub>2))"
using finite_tag_image tag_str_seq_range_finite finite1 finite2
by auto
next
show "inj_on (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 ; L\<^isub>2))"
apply (rule_tac str_inj_imps)
by (erule_tac tag_str_seq_inj)
qed
(****************** the STAR case ************************)
lemma app_eq_elim[rule_format]:
"\<And> a. \<forall> b x y. a @ b = x @ y \<longrightarrow> (\<exists> aa ab. a = aa @ ab \<and> x = aa \<and> y = ab @ b) \<or>
(\<exists> ba bb. b = ba @ bb \<and> x = a @ ba \<and> y = bb \<and> ba \<noteq> [])"
apply (induct_tac a rule:List.induct, simp)
apply (rule allI | rule impI)+
by (case_tac x, auto)
definition tag_str_STAR:: "string set \<Rightarrow> string \<Rightarrow> string set set"
where
"tag_str_STAR L\<^isub>1 x \<equiv> if (x = [])
then {}
else {\<lbrakk>x\<^isub>2\<rbrakk>L\<^isub>1 | x\<^isub>1 x\<^isub>2. x = x\<^isub>1 @ x\<^isub>2 \<and> x\<^isub>1 \<in> L\<^isub>1\<star> \<and> x\<^isub>2 \<noteq> []}"
lemma tag_str_star_range_finite:
assumes finite1: "finite (QUOT L\<^isub>1)"
shows "finite (range (tag_str_STAR L\<^isub>1))"
proof -
have "range (tag_str_STAR L\<^isub>1) \<subseteq> Pow (QUOT L\<^isub>1)"
by (auto simp:image_def tag_str_STAR_def QUOT_def)
thus ?thesis using finite1
by (rule_tac B = "Pow (QUOT L\<^isub>1)" in finite_subset, auto)
qed
lemma star_prop[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
by (erule Star.induct, auto)
lemma star_prop2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
by (drule step[of y lang "[]"], auto simp:start)
lemma star_prop3[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
by (erule Star.induct, auto intro:star_prop2)
lemma postfix_prop: "y >>= (x @ y) \<Longrightarrow> x = []"
by (erule postfixE, induct x arbitrary:y, auto)
lemma inj_aux:
"\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
\<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>"
proof-
have "\<And>s. s \<in> L\<^isub>1\<star> \<Longrightarrow> \<forall> m z yb. (s = m @ z \<and> m \<equiv>L\<^isub>1\<equiv> yb \<and> x = xa @ m \<and> xa \<in> L\<^isub>1\<star> \<and> m \<noteq> [] \<and>
(\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m) \<longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>)"
apply (erule Star.induct, simp)
apply (rule allI | rule impI | erule conjE)+
apply (drule app_eq_elim)
apply (erule disjE | erule exE | erule conjE)+
apply simp
apply (simp (no_asm) only:append_assoc[THEN sym])
apply (rule step)
apply (simp add:equiv_str_def)
apply simp
apply (erule exE | erule conjE)+
apply (rotate_tac 3)
apply (frule_tac x = "xa @ s1" in spec)
apply (rotate_tac 12)
apply (drule_tac x = ba in spec)
apply (erule impE)
apply ( simp add:star_prop3)
apply (simp)
apply (drule postfix_prop)
apply simp
done
thus "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
\<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>" by blast
qed
lemma min_postfix_exists[rule_format]:
"finite A \<Longrightarrow> A \<noteq> {} \<and> (\<forall> a \<in> A. \<forall> b \<in> A. ((b >>= a) \<or> (a >>= b)))
\<longrightarrow> (\<exists> min. (min \<in> A \<and> (\<forall> a \<in> A. a >>= min)))"
apply (erule finite.induct)
apply simp
apply simp
apply (case_tac "A = {}")
apply simp
apply clarsimp
apply (case_tac "a >>= min")
apply (rule_tac x = min in exI, simp)
apply (rule_tac x = a in exI, simp)
apply clarify
apply (rotate_tac 5)
apply (erule_tac x = aa in ballE) defer apply simp
apply (erule_tac ys = min in postfix_trans)
apply (erule_tac x = min in ballE)
by simp+
lemma tag_str_star_inj:
"tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
proof -
have aux: "\<And> x y z. \<lbrakk>tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y; x @ z \<in> L\<^isub>1\<star>\<rbrakk> \<Longrightarrow> y @ z \<in> L\<^isub>1\<star>"
proof-
fix x y z
assume tag_eq: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
and x_z: "x @ z \<in> L\<^isub>1\<star>"
show "y @ z \<in> L\<^isub>1\<star>"
proof (cases "x = []")
case True
with tag_eq have "y = []" by (simp add:tag_str_STAR_def split:if_splits, blast)
thus ?thesis using x_z True by simp
next
case False
hence not_empty: "{xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>} \<noteq> {}" using x_z
by (simp, rule_tac x = x in exI, rule_tac x = "[]" in exI, simp add:start)
have finite_set: "finite {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}"
apply (rule_tac B = "{xb. \<exists> xa. x = xa @ xb}" in finite_subset)
apply auto
apply (induct x, simp)
apply (subgoal_tac "{xb. \<exists>xa. a # x = xa @ xb} = insert (a # x) {xb. \<exists>xa. x = xa @ xb}")
apply auto
by (case_tac xaa, simp+)
have comparable: "\<forall> a \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
\<forall> b \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
((b >>= a) \<or> (a >>= b))"
by (auto simp:postfix_def, drule app_eq_elim, blast)
hence "\<exists> min. min \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}
\<and> (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min)"
using finite_set not_empty comparable
apply (drule_tac min_postfix_exists, simp)
by (erule exE, rule_tac x = min in exI, auto)
then obtain min xa where x_decom: "x = xa @ min \<and> xa \<in> L\<^isub>1\<star>"
and min_not_empty: "min \<noteq> []"
and min_z_in_star: "min @ z \<in> L\<^isub>1\<star>"
and is_min: "\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min" by blast
from x_decom min_not_empty have "\<lbrakk>min\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 x" by (auto simp:tag_str_STAR_def)
hence "\<exists> yb. \<lbrakk>yb\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 y \<and> \<lbrakk>min\<rbrakk>L\<^isub>1 = \<lbrakk>yb\<rbrakk>L\<^isub>1" using tag_eq by auto
hence "\<exists> ya yb. y = ya @ yb \<and> ya \<in> L\<^isub>1\<star> \<and> min \<equiv>L\<^isub>1\<equiv> yb \<and> yb \<noteq> [] "
by (simp add:tag_str_STAR_def equiv_class_def equiv_str_def split:if_splits, blast)
then obtain ya yb where y_decom: "y = ya @ yb"
and ya_in_star: "ya \<in> L\<^isub>1\<star>"
and yb_not_empty: "yb \<noteq> []"
and min_yb_eq: "min \<equiv>L\<^isub>1\<equiv> yb" by blast
from min_z_in_star min_yb_eq min_not_empty is_min x_decom
have "yb @ z \<in> L\<^isub>1\<star>"
by (rule_tac x = x and xa = xa in inj_aux, simp+)
thus ?thesis using ya_in_star y_decom
by (auto dest:star_prop)
qed
qed
show "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
by (auto intro:aux simp:equiv_str_def)
qed
lemma quot_star:
assumes finite1: "finite (QUOT L\<^isub>1)"
shows "finite (QUOT (L\<^isub>1\<star>))"
proof (rule_tac f = "(op `) (tag_str_STAR L\<^isub>1)" in finite_imageD)
show "finite (op ` (tag_str_STAR L\<^isub>1) ` QUOT (L\<^isub>1\<star>))"
using finite_tag_image tag_str_star_range_finite finite1
by auto
next
show "inj_on (op ` (tag_str_STAR L\<^isub>1)) (QUOT (L\<^isub>1\<star>))"
apply (rule_tac str_inj_imps)
by (erule_tac tag_str_star_inj)
qed
lemma other_direction:
"Lang = L (r::rexp) \<Longrightarrow> finite (QUOT Lang)"
apply (induct arbitrary:Lang rule:rexp.induct)
apply (simp add:QUOT_def equiv_class_def equiv_str_def)
by (simp_all add:quot_lambda quot_single quot_seq quot_alt quot_star)
end