Beautifying of the Other Direction is finished.
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>"
lemma seq_union_distrib:
"(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"
by (auto simp:Seq_def)
lemma seq_intro:
"\<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x @ y \<in> A ;; B "
by (auto simp:Seq_def)
lemma seq_assoc:
"(A ;; B) ;; C = A ;; (B ;; C)"
apply(auto simp:Seq_def)
apply blast
by (metis append_assoc)
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
thus "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_alt_simp [simp]:
"finite rs \<Longrightarrow> L (folds ALT NULL rs) = \<Union> (L ` rs)"
apply (rule set_ext, simp add:folds_def)
apply (rule someI2_ex, erule finite_imp_fold_graph)
by (erule fold_graph.induct, auto)
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}"
definition
final :: "string set \<Rightarrow> string set \<Rightarrow> bool"
where
"final X Lang \<equiv> (X \<in> UNIV // \<approx>Lang) \<and> (\<forall>s \<in> X. s \<in> Lang)"
lemma lang_is_union_of_finals:
"Lang = \<Union> {X. final X Lang}"
proof
show "Lang \<subseteq> \<Union> {X. final X Lang}"
proof
fix x
assume "x \<in> Lang"
thus "x \<in> \<Union> {X. final X Lang}"
apply (simp, rule_tac x = "(\<approx>Lang) `` {x}" in exI)
apply (auto simp:final_def quotient_def Image_def str_eq_rel_def str_eq_def)
by (drule_tac x = "[]" in spec, simp)
qed
next
show "\<Union>{X. final X Lang} \<subseteq> Lang"
by (auto simp:final_def)
qed
section {* finite \<Rightarrow> regular *}
datatype rhs_item =
Lam "rexp" (* Lambda *)
| Trn "string set" "rexp" (* Transition *)
fun the_Trn:: "rhs_item \<Rightarrow> (string set \<times> rexp)"
where "the_Trn (Trn Y r) = (Y, r)"
fun the_r :: "rhs_item \<Rightarrow> rexp"
where "the_r (Lam r) = r"
overloading L_rhs_e \<equiv> "L:: rhs_item \<Rightarrow> string set"
begin
fun L_rhs_e:: "rhs_item \<Rightarrow> string set"
where
"L_rhs_e (Lam r) = L r" |
"L_rhs_e (Trn X r) = X ;; L r"
end
overloading L_rhs \<equiv> "L:: rhs_item set \<Rightarrow> string set"
begin
fun L_rhs:: "rhs_item set \<Rightarrow> string set"
where
"L_rhs rhs = \<Union> (L ` rhs)"
end
definition
"init_rhs CS X \<equiv> if ([] \<in> X)
then {Lam EMPTY} \<union> {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}
else {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
definition
"eqs CS \<equiv> {(X, init_rhs CS X)|X. X \<in> CS}"
(************ arden's lemma variation ********************)
definition
"items_of rhs X \<equiv> {Trn X r | r. (Trn X r) \<in> rhs}"
definition
"lam_of rhs \<equiv> {Lam r | r. Lam r \<in> rhs}"
definition
"rexp_of rhs X \<equiv> folds ALT NULL ((snd o the_Trn) ` items_of rhs X)"
definition
"rexp_of_lam rhs \<equiv> folds ALT NULL (the_r ` lam_of rhs)"
fun attach_rexp :: "rexp \<Rightarrow> rhs_item \<Rightarrow> rhs_item"
where
"attach_rexp r' (Lam r) = Lam (SEQ r r')"
| "attach_rexp r' (Trn X r) = Trn X (SEQ r r')"
definition
"append_rhs_rexp rhs r \<equiv> (attach_rexp r) ` rhs"
definition
"arden_variate X rhs \<equiv> append_rhs_rexp (rhs - items_of rhs X) (STAR (rexp_of rhs X))"
(*********** 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> (rhs - (items_of rhs X)) \<union> (append_rhs_rexp xrhs (rexp_of rhs X))"
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
"distinct_equas ES \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
definition
"valid_eqns ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> (X = L rhs)"
definition
"rhs_nonempty rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
definition
"ardenable ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> rhs_nonempty rhs"
definition
"non_empty ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> X \<noteq> {}"
definition
"finite_rhs ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs"
definition
"classes_of rhs \<equiv> {X. \<exists> r. Trn X r \<in> rhs}"
definition
"lefts_of ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
definition
"self_contained ES \<equiv> \<forall> (X, xrhs) \<in> ES. classes_of xrhs \<subseteq> lefts_of ES"
definition
"Inv ES \<equiv> valid_eqns ES \<and> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and>
non_empty ES \<and> finite_rhs 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 {* ************* basic properties of definitions above ************************ *}
lemma L_rhs_union_distrib:
" L (A::rhs_item set) \<union> L B = L (A \<union> B)"
by simp
lemma finite_snd_Trn:
assumes finite:"finite rhs"
shows "finite {r\<^isub>2. Trn Y r\<^isub>2 \<in> rhs}" (is "finite ?B")
proof-
def rhs' \<equiv> "{e \<in> rhs. \<exists> r. e = Trn Y r}"
have "?B = (snd o the_Trn) ` rhs'" using rhs'_def by (auto simp:image_def)
moreover have "finite rhs'" using finite rhs'_def by auto
ultimately show ?thesis by simp
qed
lemma rexp_of_empty:
assumes finite:"finite rhs"
and nonempty:"rhs_nonempty rhs"
shows "[] \<notin> L (rexp_of rhs X)"
using finite nonempty rhs_nonempty_def
by (drule_tac finite_snd_Trn[where Y = X], auto simp:rexp_of_def items_of_def)
lemma [intro!]:
"P (Trn X r) \<Longrightarrow> (\<exists>a. (\<exists>r. a = Trn X r \<and> P a))" by auto
lemma finite_items_of:
"finite rhs \<Longrightarrow> finite (items_of rhs X)"
by (auto simp:items_of_def intro:finite_subset)
lemma lang_of_rexp_of:
assumes finite:"finite rhs"
shows "L (items_of rhs X) = X ;; (L (rexp_of rhs X))"
proof -
have "finite ((snd \<circ> the_Trn) ` items_of rhs X)" using finite_items_of[OF finite] by auto
thus ?thesis
apply (auto simp:rexp_of_def Seq_def items_of_def)
apply (rule_tac x = s1 in exI, rule_tac x = s2 in exI, auto)
by (rule_tac x= "Trn X r" in exI, auto simp:Seq_def)
qed
lemma rexp_of_lam_eq_lam_set:
assumes finite: "finite rhs"
shows "L (rexp_of_lam rhs) = L (lam_of rhs)"
proof -
have "finite (the_r ` {Lam r |r. Lam r \<in> rhs})" using finite
by (rule_tac finite_imageI, auto intro:finite_subset)
thus ?thesis by (auto simp:rexp_of_lam_def lam_of_def)
qed
lemma [simp]:
" L (attach_rexp r xb) = L xb ;; L r"
apply (cases xb, auto simp:Seq_def)
by (rule_tac x = "s1 @ s1a" in exI, rule_tac x = s2a in exI,auto simp:Seq_def)
lemma lang_of_append_rhs:
"L (append_rhs_rexp rhs r) = L rhs ;; L r"
apply (auto simp:append_rhs_rexp_def image_def)
apply (auto simp:Seq_def)
apply (rule_tac x = "L xb ;; L r" in exI, auto simp add:Seq_def)
by (rule_tac x = "attach_rexp r xb" in exI, auto simp:Seq_def)
lemma classes_of_union_distrib:
"classes_of A \<union> classes_of B = classes_of (A \<union> B)"
by (auto simp add:classes_of_def)
lemma lefts_of_union_distrib:
"lefts_of A \<union> lefts_of B = lefts_of (A \<union> B)"
by (auto simp:lefts_of_def)
text {* ******BEGIN: proving the initial equation-system satisfies Inv ****** *}
lemma defined_by_str:
"\<lbrakk>s \<in> X; X \<in> UNIV // (\<approx>Lang)\<rbrakk> \<Longrightarrow> X = (\<approx>Lang) `` {s}"
by (auto simp:quotient_def Image_def str_eq_rel_def str_eq_def)
lemma every_eqclass_has_transition:
assumes has_str: "s @ [c] \<in> X"
and in_CS: "X \<in> UNIV // (\<approx>Lang)"
obtains Y where "Y \<in> UNIV // (\<approx>Lang)" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
proof -
def Y \<equiv> "(\<approx>Lang) `` {s}"
have "Y \<in> UNIV // (\<approx>Lang)"
unfolding Y_def quotient_def by auto
moreover
have "X = (\<approx>Lang) `` {s @ [c]}"
using has_str in_CS defined_by_str by blast
then have "Y ;; {[c]} \<subseteq> X"
unfolding Y_def Image_def Seq_def
unfolding str_eq_rel_def
by (auto) (simp add: str_eq_def)
moreover
have "s \<in> Y" unfolding Y_def
unfolding Image_def str_eq_rel_def str_eq_def by simp
ultimately show thesis by (blast intro: that)
qed
lemma l_eq_r_in_eqs:
assumes X_in_eqs: "(X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
shows "X = L xrhs"
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 = []"
thus ?thesis using X_in_eqs "(1)"
by (auto simp:eqs_def init_rhs_def)
next
assume not_empty: "x \<noteq> []"
then obtain clist c where decom: "x = clist @ [c]"
by (case_tac x rule:rev_cases, auto)
have "X \<in> UNIV // (\<approx>Lang)" using X_in_eqs by (auto simp:eqs_def)
then obtain Y
where "Y \<in> UNIV // (\<approx>Lang)"
and "Y ;; {[c]} \<subseteq> X"
and "clist \<in> Y"
using decom "(1)" every_eqclass_has_transition by blast
hence "x \<in> L {Trn Y (CHAR c)| Y c. Y \<in> UNIV // (\<approx>Lang) \<and> Y ;; {[c]} \<subseteq> X}"
using "(1)" decom
by (simp, rule_tac x = "Trn Y (CHAR c)" in exI, simp add:Seq_def)
thus ?thesis using X_in_eqs "(1)"
by (simp add:eqs_def init_rhs_def)
qed
qed
next
show "L xrhs \<subseteq> X" using X_in_eqs
by (auto simp:eqs_def init_rhs_def)
qed
lemma finite_init_rhs:
assumes finite: "finite CS"
shows "finite (init_rhs CS X)"
proof-
have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" (is "finite ?A")
proof -
def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
have "finite (CS \<times> (UNIV::char set))" using finite by auto
hence "finite S" using S_def
by (rule_tac B = "CS \<times> UNIV" in finite_subset, auto)
moreover have "?A = h ` S" by (auto simp: S_def h_def image_def)
ultimately show ?thesis
by auto
qed
thus ?thesis by (simp add:init_rhs_def)
qed
lemma init_ES_satisfy_Inv:
assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
shows "Inv (eqs (UNIV // (\<approx>Lang)))"
proof -
have "finite (eqs (UNIV // (\<approx>Lang)))" using finite_CS
by (simp add:eqs_def)
moreover have "distinct_equas (eqs (UNIV // (\<approx>Lang)))"
by (simp add:distinct_equas_def eqs_def)
moreover have "ardenable (eqs (UNIV // (\<approx>Lang)))"
by (auto simp add:ardenable_def eqs_def init_rhs_def rhs_nonempty_def del:L_rhs.simps)
moreover have "valid_eqns (eqs (UNIV // (\<approx>Lang)))"
using l_eq_r_in_eqs by (simp add:valid_eqns_def)
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 "finite_rhs (eqs (UNIV // (\<approx>Lang)))"
using finite_init_rhs[OF finite_CS]
by (auto simp:finite_rhs_def eqs_def)
moreover have "self_contained (eqs (UNIV // (\<approx>Lang)))"
by (auto simp:self_contained_def eqs_def init_rhs_def classes_of_def lefts_of_def)
ultimately show ?thesis by (simp add:Inv_def)
qed
text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
lemma arden_variate_keeps_eq:
assumes l_eq_r: "X = L rhs"
and not_empty: "[] \<notin> L (rexp_of rhs X)"
and finite: "finite rhs"
shows "X = L (arden_variate X rhs)"
proof -
def A \<equiv> "L (rexp_of rhs X)"
def b \<equiv> "rhs - items_of rhs X"
def B \<equiv> "L b"
have "X = B ;; A\<star>"
proof-
have "rhs = items_of rhs X \<union> b" by (auto simp:b_def items_of_def)
hence "L rhs = L(items_of rhs X \<union> b)" by simp
hence "L rhs = L(items_of rhs X) \<union> B" by (simp only:L_rhs_union_distrib B_def)
with lang_of_rexp_of
have "L rhs = X ;; A \<union> B " using finite by (simp only:B_def b_def A_def)
thus ?thesis
using l_eq_r not_empty
apply (drule_tac B = B and X = X in ardens_revised)
by (auto simp:A_def simp del:L_rhs.simps)
qed
moreover have "L (arden_variate X rhs) = (B ;; A\<star>)" (is "?L = ?R")
by (simp only:arden_variate_def L_rhs_union_distrib lang_of_append_rhs
B_def A_def b_def L_rexp.simps seq_union_distrib)
ultimately show ?thesis by simp
qed
lemma append_keeps_finite:
"finite rhs \<Longrightarrow> finite (append_rhs_rexp rhs r)"
by (auto simp:append_rhs_rexp_def)
lemma arden_variate_keeps_finite:
"finite rhs \<Longrightarrow> finite (arden_variate X rhs)"
by (auto simp:arden_variate_def append_keeps_finite)
lemma append_keeps_nonempty:
"rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (append_rhs_rexp rhs r)"
apply (auto simp:rhs_nonempty_def append_rhs_rexp_def)
by (case_tac x, auto simp:Seq_def)
lemma nonempty_set_sub:
"rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (rhs - A)"
by (auto simp:rhs_nonempty_def)
lemma nonempty_set_union:
"\<lbrakk>rhs_nonempty rhs; rhs_nonempty rhs'\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs \<union> rhs')"
by (auto simp:rhs_nonempty_def)
lemma arden_variate_keeps_nonempty:
"rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (arden_variate X rhs)"
by (simp only:arden_variate_def append_keeps_nonempty nonempty_set_sub)
lemma rhs_subst_keeps_nonempty:
"\<lbrakk>rhs_nonempty rhs; rhs_nonempty xrhs\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs_subst rhs X xrhs)"
by (simp only:rhs_subst_def append_keeps_nonempty nonempty_set_union nonempty_set_sub)
lemma rhs_subst_keeps_eq:
assumes substor: "X = L xrhs"
and finite: "finite rhs"
shows "L (rhs_subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
proof-
def A \<equiv> "L (rhs - items_of rhs X)"
have "?Left = A \<union> L (append_rhs_rexp xrhs (rexp_of rhs X))"
by (simp only:rhs_subst_def L_rhs_union_distrib A_def)
moreover have "?Right = A \<union> L (items_of rhs X)"
proof-
have "rhs = (rhs - items_of rhs X) \<union> (items_of rhs X)" by (auto simp:items_of_def)
thus ?thesis by (simp only:L_rhs_union_distrib A_def)
qed
moreover have "L (append_rhs_rexp xrhs (rexp_of rhs X)) = L (items_of rhs X)"
using finite substor by (simp only:lang_of_append_rhs lang_of_rexp_of)
ultimately show ?thesis by simp
qed
lemma rhs_subst_keeps_finite_rhs:
"\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (rhs_subst rhs Y yrhs)"
by (auto simp:rhs_subst_def append_keeps_finite)
lemma eqs_subst_keeps_finite:
assumes finite:"finite (ES:: (string set \<times> rhs_item set) set)"
shows "finite (eqs_subst ES Y yrhs)"
proof -
have "finite {(Ya, rhs_subst yrhsa Y yrhs) |Ya yrhsa. (Ya, yrhsa) \<in> ES}" (is "finite ?A")
proof-
def eqns' \<equiv> "{((Ya::string set), yrhsa)| Ya yrhsa. (Ya, yrhsa) \<in> ES}"
def h \<equiv> "\<lambda> ((Ya::string set), yrhsa). (Ya, rhs_subst yrhsa Y yrhs)"
have "finite (h ` eqns')" using finite h_def eqns'_def by auto
moreover have "?A = h ` eqns'" by (auto simp:h_def eqns'_def)
ultimately show ?thesis by auto
qed
thus ?thesis by (simp add:eqs_subst_def)
qed
lemma eqs_subst_keeps_finite_rhs:
"\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (eqs_subst ES Y yrhs)"
by (auto intro:rhs_subst_keeps_finite_rhs simp add:eqs_subst_def finite_rhs_def)
lemma append_rhs_keeps_cls:
"classes_of (append_rhs_rexp rhs r) = classes_of rhs"
apply (auto simp:classes_of_def append_rhs_rexp_def)
apply (case_tac xa, auto simp:image_def)
by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
lemma arden_variate_removes_cl:
"classes_of (arden_variate Y yrhs) = classes_of yrhs - {Y}"
apply (simp add:arden_variate_def append_rhs_keeps_cls items_of_def)
by (auto simp:classes_of_def)
lemma lefts_of_keeps_cls:
"lefts_of (eqs_subst ES Y yrhs) = lefts_of ES"
by (auto simp:lefts_of_def eqs_subst_def)
lemma rhs_subst_updates_cls:
"X \<notin> classes_of xrhs \<Longrightarrow> classes_of (rhs_subst rhs X xrhs) = classes_of rhs \<union> classes_of xrhs - {X}"
apply (simp only:rhs_subst_def append_rhs_keeps_cls classes_of_union_distrib[THEN sym])
by (auto simp:classes_of_def items_of_def)
lemma eqs_subst_keeps_self_contained:
fixes Y
assumes sc: "self_contained (ES \<union> {(Y, yrhs)})" (is "self_contained ?A")
shows "self_contained (eqs_subst ES Y (arden_variate Y yrhs))" (is "self_contained ?B")
proof-
{ fix X xrhs'
assume "(X, xrhs') \<in> ?B"
then obtain xrhs
where xrhs_xrhs': "xrhs' = rhs_subst xrhs Y (arden_variate Y yrhs)"
and X_in: "(X, xrhs) \<in> ES" by (simp add:eqs_subst_def, blast)
have "classes_of xrhs' \<subseteq> lefts_of ?B"
proof-
have "lefts_of ?B = lefts_of ES" by (auto simp add:lefts_of_def eqs_subst_def)
moreover have "classes_of xrhs' \<subseteq> lefts_of ES"
proof-
have "classes_of xrhs' \<subseteq> classes_of xrhs \<union> classes_of (arden_variate Y yrhs) - {Y}"
proof-
have "Y \<notin> classes_of (arden_variate Y yrhs)" using arden_variate_removes_cl by simp
thus ?thesis using xrhs_xrhs' by (auto simp:rhs_subst_updates_cls)
qed
moreover have "classes_of xrhs \<subseteq> lefts_of ES \<union> {Y}" using X_in sc
apply (simp only:self_contained_def lefts_of_union_distrib[THEN sym])
by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lefts_of_def)
moreover have "classes_of (arden_variate Y yrhs) \<subseteq> lefts_of ES \<union> {Y}" using sc
by (auto simp add:arden_variate_removes_cl self_contained_def lefts_of_def)
ultimately show ?thesis by auto
qed
ultimately show ?thesis by simp
qed
} thus ?thesis by (auto simp only:eqs_subst_def self_contained_def)
qed
lemma eqs_subst_satisfy_Inv:
assumes Inv_ES: "Inv (ES \<union> {(Y, yrhs)})"
shows "Inv (eqs_subst ES Y (arden_variate Y yrhs))"
proof -
have finite_yrhs: "finite yrhs" using Inv_ES by (auto simp:Inv_def finite_rhs_def)
have nonempty_yrhs: "rhs_nonempty yrhs" using Inv_ES by (auto simp:Inv_def ardenable_def)
have Y_eq_yrhs: "Y = L yrhs" using Inv_ES by (simp only:Inv_def valid_eqns_def, blast)
have "distinct_equas (eqs_subst ES Y (arden_variate Y yrhs))" using Inv_ES
by (auto simp:distinct_equas_def eqs_subst_def Inv_def)
moreover have "finite (eqs_subst ES Y (arden_variate Y yrhs))" using Inv_ES
by (simp add:Inv_def eqs_subst_keeps_finite)
moreover have "finite_rhs (eqs_subst ES Y (arden_variate Y yrhs))"
proof-
have "finite_rhs ES" using Inv_ES by (simp add:Inv_def finite_rhs_def)
moreover have "finite (arden_variate Y yrhs)"
proof -
have "finite yrhs" using Inv_ES by (auto simp:Inv_def finite_rhs_def)
thus ?thesis using arden_variate_keeps_finite by simp
qed
ultimately show ?thesis by (simp add:eqs_subst_keeps_finite_rhs)
qed
moreover have "ardenable (eqs_subst ES Y (arden_variate Y yrhs))"
proof -
{ fix X rhs
assume "(X, rhs) \<in> ES"
hence "rhs_nonempty rhs" using prems Inv_ES by (simp add:Inv_def ardenable_def)
with nonempty_yrhs have "rhs_nonempty (rhs_subst rhs Y (arden_variate Y yrhs))"
by (simp add:nonempty_yrhs rhs_subst_keeps_nonempty arden_variate_keeps_nonempty)
} thus ?thesis by (auto simp add:ardenable_def eqs_subst_def)
qed
moreover have "valid_eqns (eqs_subst ES Y (arden_variate Y yrhs))"
proof-
have "Y = L (arden_variate Y yrhs)" using Y_eq_yrhs Inv_ES finite_yrhs nonempty_yrhs
by (rule_tac arden_variate_keeps_eq, (simp add:rexp_of_empty)+)
thus ?thesis using Inv_ES
by (clarsimp simp add:valid_eqns_def eqs_subst_def rhs_subst_keeps_eq Inv_def finite_rhs_def
simp del:L_rhs.simps)
qed
moreover have non_empty_subst: "non_empty (eqs_subst ES Y (arden_variate Y yrhs))"
using Inv_ES by (auto simp:Inv_def non_empty_def eqs_subst_def)
moreover have self_subst: "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
using Inv_ES eqs_subst_keeps_self_contained by (simp add:Inv_def)
ultimately show ?thesis using Inv_ES by (simp add:Inv_def)
qed
lemma eqs_subst_card_le:
assumes finite: "finite (ES::(string set \<times> rhs_item set) set)"
shows "card (eqs_subst ES Y yrhs) <= card ES"
proof-
def f \<equiv> "\<lambda> x. ((fst x)::string set, rhs_subst (snd x) Y yrhs)"
have "eqs_subst ES Y yrhs = f ` ES"
apply (auto simp:eqs_subst_def f_def image_def)
by (rule_tac x = "(Ya, yrhsa)" in bexI, simp+)
thus ?thesis using finite by (auto intro:card_image_le)
qed
lemma eqs_subst_cls_remains:
"(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (eqs_subst ES Y yrhs)"
by (auto simp:eqs_subst_def)
lemma card_noteq_1_has_more:
assumes card:"card S \<noteq> 1"
and e_in: "e \<in> S"
and finite: "finite S"
obtains e' where "e' \<in> S \<and> e \<noteq> e'"
proof-
have "card (S - {e}) > 0"
proof -
have "card S > 1" using card e_in finite by (case_tac "card S", auto)
thus ?thesis using finite e_in by auto
qed
hence "S - {e} \<noteq> {}" using finite by (rule_tac notI, simp)
thus "(\<And>e'. e' \<in> S \<and> e \<noteq> e' \<Longrightarrow> thesis) \<Longrightarrow> thesis" by auto
qed
lemma iteration_step:
assumes Inv_ES: "Inv ES"
and X_in_ES: "(X, xrhs) \<in> ES"
and not_T: "card ES \<noteq> 1"
shows "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'.(X, xrhs') \<in> ES')) \<and> (card ES', card ES) \<in> less_than" (is "\<exists> ES'. ?P ES'")
proof -
have finite_ES: "finite ES" using Inv_ES by (simp add:Inv_def)
then obtain Y yrhs where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
using not_T X_in_ES by (drule_tac card_noteq_1_has_more, auto)
def ES' == "ES - {(Y, yrhs)}"
let ?ES'' = "eqs_subst ES' Y (arden_variate Y yrhs)"
have "?P ?ES''"
proof -
have "Inv ?ES''" using Y_in_ES Inv_ES
by (rule_tac eqs_subst_satisfy_Inv, simp add:ES'_def insert_absorb)
moreover have "\<exists>xrhs'. (X, xrhs') \<in> ?ES''" using not_eq X_in_ES
by (rule_tac ES = ES' in eqs_subst_cls_remains, auto simp add:ES'_def)
moreover have "(card ?ES'', card ES) \<in> less_than"
proof -
have "finite ES'" using finite_ES ES'_def by auto
moreover have "card ES' < card ES" using finite_ES Y_in_ES
by (auto simp:ES'_def card_gt_0_iff intro:diff_Suc_less)
ultimately show ?thesis
by (auto dest:eqs_subst_card_le elim:le_less_trans)
qed
ultimately show ?thesis by simp
qed
thus ?thesis by blast
qed
text {* ***** END: proving every equation-system's iteration step satisfies Inv ************** *}
lemma iteration_conc:
assumes history: "Inv ES"
and X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES"
shows "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'. (X, xrhs') \<in> ES')) \<and> card ES' = 1" (is "\<exists> ES'. ?P ES'")
proof (cases "card ES = 1")
case True
thus ?thesis using history X_in_ES
by blast
next
case False
thus ?thesis using history iteration_step X_in_ES
by (rule_tac f = card in wf_iter, auto)
qed
lemma last_cl_exists_rexp:
assumes ES_single: "ES = {(X, xrhs)}"
and Inv_ES: "Inv ES"
shows "\<exists> (r::rexp). L r = X" (is "\<exists> r. ?P r")
proof-
let ?A = "arden_variate X xrhs"
have "?P (rexp_of_lam ?A)"
proof -
have "L (rexp_of_lam ?A) = L (lam_of ?A)"
proof(rule rexp_of_lam_eq_lam_set)
show "finite (arden_variate X xrhs)" using Inv_ES ES_single
by (rule_tac arden_variate_keeps_finite, auto simp add:Inv_def finite_rhs_def)
qed
also have "\<dots> = L ?A"
proof-
have "lam_of ?A = ?A"
proof-
have "classes_of ?A = {}" using Inv_ES ES_single
by (simp add:arden_variate_removes_cl self_contained_def Inv_def lefts_of_def)
thus ?thesis by (auto simp only:lam_of_def classes_of_def, case_tac x, auto)
qed
thus ?thesis by simp
qed
also have "\<dots> = X"
proof(rule arden_variate_keeps_eq [THEN sym])
show "X = L xrhs" using Inv_ES ES_single by (auto simp only:Inv_def valid_eqns_def)
next
from Inv_ES ES_single show "[] \<notin> L (rexp_of xrhs X)"
by(simp add:Inv_def ardenable_def rexp_of_empty finite_rhs_def)
next
from Inv_ES ES_single show "finite xrhs" by (simp add:Inv_def finite_rhs_def)
qed
finally show ?thesis by simp
qed
thus ?thesis by auto
qed
lemma every_eqcl_has_reg:
assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
and X_in_CS: "X \<in> (UNIV // (\<approx>Lang))"
shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
proof -
from X_in_CS have "\<exists> xrhs. (X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
by (auto simp:eqs_def init_rhs_def)
then obtain ES xrhs where Inv_ES: "Inv ES"
and X_in_ES: "(X, xrhs) \<in> ES"
and card_ES: "card ES = 1"
using finite_CS X_in_CS init_ES_satisfy_Inv iteration_conc
by blast
hence ES_single_equa: "ES = {(X, xrhs)}"
by (auto simp:Inv_def dest!:card_Suc_Diff1 simp:card_eq_0_iff)
thus ?thesis using Inv_ES
by (rule last_cl_exists_rexp)
qed
theorem hard_direction:
assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
shows "\<exists> (reg::rexp). Lang = L reg"
proof -
have "\<forall> X \<in> (UNIV // (\<approx>Lang)). \<exists> (reg::rexp). X = L reg"
using finite_CS every_eqcl_has_reg by blast
then obtain f where f_prop: "\<forall> X \<in> (UNIV // (\<approx>Lang)). X = L ((f X)::rexp)"
by (auto dest:bchoice)
def rs \<equiv> "f ` {X. final X Lang}"
have "Lang = \<Union> {X. final X Lang}" using lang_is_union_of_finals by simp
also have "\<dots> = L (folds ALT NULL rs)"
proof -
have "finite {X. final X Lang}" using finite_CS by (auto simp:final_def)
thus ?thesis using f_prop by (auto simp:rs_def final_def)
qed
finally show ?thesis by blast
qed
section {* regular \<Rightarrow> finite*}
lemma quot_empty_subset:
"UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}"
proof
fix x
assume "x \<in> UNIV // \<approx>{[]}"
then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[]}}" unfolding quotient_def Image_def by blast
show "x \<in> {{[]}, UNIV - {[]}}"
proof (cases "y = []")
case True with h
have "x = {[]}" by (auto simp:str_eq_rel_def str_eq_def)
thus ?thesis by simp
next
case False with h
have "x = UNIV - {[]}" by (auto simp:str_eq_rel_def str_eq_def)
thus ?thesis by simp
qed
qed
lemma quot_char_subset:
"UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
proof
fix x
assume "x \<in> UNIV // \<approx>{[c]}"
then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[c]}}" unfolding quotient_def Image_def by blast
show "x \<in> {{[]},{[c]}, UNIV - {[], [c]}}"
proof -
{ assume "y = []" hence "x = {[]}" using h by (auto simp:str_eq_rel_def str_eq_def)
} moreover {
assume "y = [c]" hence "x = {[c]}" using h
by (auto dest!:spec[where x = "[]"] simp:str_eq_rel_def str_eq_def)
} moreover {
assume "y \<noteq> []" and "y \<noteq> [c]"
hence "\<forall> z. (y @ z) \<noteq> [c]" by (case_tac y, auto)
moreover have "\<And> p. (p \<noteq> [] \<and> p \<noteq> [c]) = (\<forall> q. p @ q \<noteq> [c])" by (case_tac p, auto)
ultimately have "x = UNIV - {[],[c]}" using h
by (auto simp add:str_eq_rel_def str_eq_def)
} ultimately show ?thesis by blast
qed
qed
text {* *************** Some common lemmas for following ALT, SEQ & STAR cases ******************* *}
lemma finite_tag_imageI:
"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 eq_class_equalI:
"\<lbrakk>X \<in> UNIV // \<approx>lang; Y \<in> UNIV // \<approx>lang; x \<in> X; y \<in> Y; x \<approx>lang y\<rbrakk> \<Longrightarrow> X = Y"
by (auto simp:quotient_def str_eq_rel_def str_eq_def)
lemma tag_image_injI:
assumes str_inj: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<approx>lang n"
shows "inj_on ((op `) tag) (UNIV // \<approx>lang)"
proof-
{ fix X Y
assume X_in: "X \<in> UNIV // \<approx>lang"
and Y_in: "Y \<in> UNIV // \<approx>lang"
and tag_eq: "tag ` X = tag ` Y"
then obtain x y where "x \<in> X" and "y \<in> Y" and "tag x = tag y"
unfolding quotient_def Image_def str_eq_rel_def str_eq_def image_def
apply simp by blast
with X_in Y_in str_inj
have "X = Y" by (rule_tac eq_class_equalI, simp+)
}
thus ?thesis unfolding inj_on_def by auto
qed
text {* **************** the SEQ case ************************ *}
(* 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]: "(x - y = []) \<Longrightarrow> (x \<le> y)"
proof-
have "\<exists>xa. x = xa @ (x - y) \<and> xa \<le> y"
apply (rule list_diff.induct[of _ x y], simp+)
by (clarsimp, rule_tac x = "y # xa" in exI, simp+)
thus "(x - y = []) \<Longrightarrow> (x \<le> y)" by simp
qed
lemma diff_prefix:
"\<lbrakk>c \<le> a - b; b \<le> a\<rbrakk> \<Longrightarrow> b @ c \<le> a"
by (auto elim:prefixE)
lemma diff_diff_appd:
"\<lbrakk>c < a - b; b < a\<rbrakk> \<Longrightarrow> (a - b) - c = a - (b @ c)"
apply (clarsimp simp:strict_prefix_def)
by (drule diff_prefix, auto elim:prefixE)
lemma app_eq_cases[rule_format]:
"\<forall> x . x @ y = m @ n \<longrightarrow> (x \<le> m \<or> m \<le> x)"
apply (induct y, simp)
apply (clarify, drule_tac x = "x @ [a]" in spec)
by (clarsimp, auto simp:prefix_def)
lemma app_eq_dest:
"x @ y = m @ n \<Longrightarrow> (x \<le> m \<and> (m - x) @ n = y) \<or> (m \<le> x \<and> (x - m) @ y = n)"
by (frule_tac app_eq_cases, auto elim:prefixE)
definition
"tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv> ((\<approx>L\<^isub>1) `` {x}, {(\<approx>L\<^isub>2) `` {x - xa}| xa. xa \<le> x \<and> xa \<in> L\<^isub>1})"
lemma tag_str_seq_range_finite:
"\<lbrakk>finite (UNIV // \<approx>L\<^isub>1); finite (UNIV // \<approx>L\<^isub>2)\<rbrakk> \<Longrightarrow> finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
apply (rule_tac B = "(UNIV // \<approx>L\<^isub>1) \<times> (Pow (UNIV // \<approx>L\<^isub>2))" in finite_subset)
by (auto simp:tag_str_SEQ_def Image_def quotient_def split:if_splits)
lemma append_seq_elim:
assumes "x @ y \<in> L\<^isub>1 ;; L\<^isub>2"
shows "(\<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ y \<in> L\<^isub>2) \<or> (\<exists> ya \<le> y. (x @ ya) \<in> L\<^isub>1 \<and> (y - ya) \<in> L\<^isub>2)"
proof-
from assms obtain s\<^isub>1 s\<^isub>2 where "x @ y = s\<^isub>1 @ s\<^isub>2" and in_seq: "s\<^isub>1 \<in> L\<^isub>1 \<and> s\<^isub>2 \<in> L\<^isub>2"
by (auto simp:Seq_def)
hence "(x \<le> s\<^isub>1 \<and> (s\<^isub>1 - x) @ s\<^isub>2 = y) \<or> (s\<^isub>1 \<le> x \<and> (x - s\<^isub>1) @ y = s\<^isub>2)"
using app_eq_dest by auto
moreover have "\<lbrakk>x \<le> s\<^isub>1; (s\<^isub>1 - x) @ s\<^isub>2 = y\<rbrakk> \<Longrightarrow> \<exists> ya \<le> y. (x @ ya) \<in> L\<^isub>1 \<and> (y - ya) \<in> L\<^isub>2" using in_seq
by (rule_tac x = "s\<^isub>1 - x" in exI, auto elim:prefixE)
moreover have "\<lbrakk>s\<^isub>1 \<le> x; (x - s\<^isub>1) @ y = s\<^isub>2\<rbrakk> \<Longrightarrow> \<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ y \<in> L\<^isub>2" using in_seq
by (rule_tac x = s\<^isub>1 in exI, auto)
ultimately show ?thesis by blast
qed
lemma tag_str_SEQ_injI:
"tag_str_SEQ L\<^isub>1 L\<^isub>2 m = tag_str_SEQ L\<^isub>1 L\<^isub>2 n \<Longrightarrow> m \<approx>(L\<^isub>1 ;; L\<^isub>2) n"
proof-
{ fix x y z
assume xz_in_seq: "x @ z \<in> L\<^isub>1 ;; L\<^isub>2"
and tag_xy: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
have"y @ z \<in> L\<^isub>1 ;; L\<^isub>2"
proof-
have "(\<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)"
using xz_in_seq append_seq_elim by simp
moreover {
fix xa
assume h1: "xa \<le> x" and h2: "xa \<in> L\<^isub>1" and h3: "(x - xa) @ z \<in> L\<^isub>2"
obtain ya where "ya \<le> y" and "ya \<in> L\<^isub>1" and "(y - ya) @ z \<in> L\<^isub>2"
proof -
have "\<exists> ya. ya \<le> y \<and> ya \<in> L\<^isub>1 \<and> (x - xa) \<approx>L\<^isub>2 (y - ya)"
proof -
have "{\<approx>L\<^isub>2 `` {x - xa} |xa. xa \<le> x \<and> xa \<in> L\<^isub>1} =
{\<approx>L\<^isub>2 `` {y - xa} |xa. xa \<le> y \<and> xa \<in> L\<^isub>1}" (is "?Left = ?Right")
using h1 tag_xy by (auto simp:tag_str_SEQ_def)
moreover have "\<approx>L\<^isub>2 `` {x - xa} \<in> ?Left" using h1 h2 by auto
ultimately have "\<approx>L\<^isub>2 `` {x - xa} \<in> ?Right" by simp
thus ?thesis by (auto simp:Image_def str_eq_rel_def str_eq_def)
qed
with prems show ?thesis by (auto simp:str_eq_rel_def str_eq_def)
qed
hence "y @ z \<in> L\<^isub>1 ;; L\<^isub>2" by (erule_tac prefixE, auto simp:Seq_def)
} moreover {
fix za
assume h1: "za \<le> z" and h2: "(x @ za) \<in> L\<^isub>1" and h3: "z - za \<in> L\<^isub>2"
hence "y @ za \<in> L\<^isub>1"
proof-
have "\<approx>L\<^isub>1 `` {x} = \<approx>L\<^isub>1 `` {y}" using h1 tag_xy by (auto simp:tag_str_SEQ_def)
with h2 show ?thesis by (auto simp:Image_def str_eq_rel_def str_eq_def)
qed
with h1 h3 have "y @ z \<in> L\<^isub>1 ;; L\<^isub>2" by (drule_tac A = L\<^isub>1 in seq_intro, auto elim:prefixE)
}
ultimately show ?thesis by blast
qed
} thus "tag_str_SEQ L\<^isub>1 L\<^isub>2 m = tag_str_SEQ L\<^isub>1 L\<^isub>2 n \<Longrightarrow> m \<approx>(L\<^isub>1 ;; L\<^isub>2) n"
by (auto simp add: str_eq_def str_eq_rel_def)
qed
lemma quot_seq_finiteI:
assumes finite1: "finite (UNIV // \<approx>(L\<^isub>1::string set))"
and finite2: "finite (UNIV // \<approx>L\<^isub>2)"
shows "finite (UNIV // \<approx>(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) ` UNIV // \<approx>L\<^isub>1 ;; L\<^isub>2)" using finite1 finite2
by (auto intro:finite_tag_imageI tag_str_seq_range_finite)
next
show "inj_on (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2)) (UNIV // \<approx>L\<^isub>1 ;; L\<^isub>2)"
apply (rule tag_image_injI)
apply (rule tag_str_SEQ_injI)
by (auto intro:tag_image_injI tag_str_SEQ_injI simp:)
qed
text {* **************** the ALT case ************************ *}
definition
"tag_str_ALT L\<^isub>1 L\<^isub>2 (x::string) \<equiv> ((\<approx>L\<^isub>1) `` {x}, (\<approx>L\<^isub>2) `` {x})"
lemma tag_str_alt_range_finite:
"\<lbrakk>finite (UNIV // \<approx>L\<^isub>1); finite (UNIV // \<approx>L\<^isub>2)\<rbrakk> \<Longrightarrow> finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))"
apply (rule_tac B = "(UNIV // \<approx>L\<^isub>1) \<times> (UNIV // \<approx>L\<^isub>2)" in finite_subset)
by (auto simp:tag_str_ALT_def Image_def quotient_def)
lemma quot_union_finiteI:
assumes finite1: "finite (UNIV // \<approx>(L\<^isub>1::string set))"
and finite2: "finite (UNIV // \<approx>L\<^isub>2)"
shows "finite (UNIV // \<approx>(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) ` UNIV // \<approx>L\<^isub>1 \<union> L\<^isub>2)" using finite1 finite2
by (auto intro:finite_tag_imageI tag_str_alt_range_finite)
next
show "inj_on (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2)) (UNIV // \<approx>L\<^isub>1 \<union> L\<^isub>2)"
proof-
have "\<And>m n. tag_str_ALT L\<^isub>1 L\<^isub>2 m = tag_str_ALT L\<^isub>1 L\<^isub>2 n \<Longrightarrow> m \<approx>(L\<^isub>1 \<union> L\<^isub>2) n"
unfolding tag_str_ALT_def str_eq_def Image_def str_eq_rel_def by auto
thus ?thesis by (auto intro:tag_image_injI)
qed
qed
text {* **************** the Star case ****************** *}
lemma finite_set_has_max: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> (\<exists> max \<in> A. \<forall> a \<in> A. f a <= (f max :: nat))"
proof (induct rule:finite.induct)
case emptyI thus ?case by simp
next
case (insertI A a)
show ?case
proof (cases "A = {}")
case True thus ?thesis by (rule_tac x = a in bexI, auto)
next
case False
with prems obtain max where h1: "max \<in> A" and h2: "\<forall>a\<in>A. f a \<le> f max" by blast
show ?thesis
proof (cases "f a \<le> f max")
assume "f a \<le> f max"
with h1 h2 show ?thesis by (rule_tac x = max in bexI, auto)
next
assume "\<not> (f a \<le> f max)"
thus ?thesis using h2 by (rule_tac x = a in bexI, auto)
qed
qed
qed
lemma star_intro1[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_intro2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
by (drule step[of y lang "[]"], auto simp:start)
lemma star_intro3[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_intro2)
lemma star_decom: "\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow>(\<exists> a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> lang \<and> b \<in> lang\<star>)"
by (induct x rule: Star.induct, simp, blast)
lemma finite_strict_prefix_set: "finite {xa. xa < (x::string)}"
apply (induct x rule:rev_induct, simp)
apply (subgoal_tac "{xa. xa < xs @ [x]} = {xa. xa < xs} \<union> {xs}")
by (auto simp:strict_prefix_def)
definition
"tag_str_STAR L\<^isub>1 x \<equiv> {(\<approx>L\<^isub>1) `` {x - xa} | xa. xa < x \<and> xa \<in> L\<^isub>1\<star>}"
lemma tag_str_star_range_finite:
"finite (UNIV // \<approx>L\<^isub>1) \<Longrightarrow> finite (range (tag_str_STAR L\<^isub>1))"
apply (rule_tac B = "Pow (UNIV // \<approx>L\<^isub>1)" in finite_subset)
by (auto simp:tag_str_STAR_def Image_def quotient_def split:if_splits)
lemma tag_str_STAR_injI:
"tag_str_STAR L\<^isub>1 m = tag_str_STAR L\<^isub>1 n \<Longrightarrow> m \<approx>(L\<^isub>1\<star>) n"
proof-
{ fix x y z
assume xz_in_star: "x @ z \<in> L\<^isub>1\<star>"
and tag_xy: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
have "y @ z \<in> L\<^isub>1\<star>"
proof(cases "x = []")
case True
with tag_xy have "y = []" by (auto simp:tag_str_STAR_def strict_prefix_def)
thus ?thesis using xz_in_star True by simp
next
case False
obtain x_max where h1: "x_max < x" and h2: "x_max \<in> L\<^isub>1\<star>" and h3: "(x - x_max) @ z \<in> L\<^isub>1\<star>"
and h4:"\<forall> xa < x. xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star> \<longrightarrow> length xa \<le> length x_max"
proof-
let ?S = "{xa. xa < x \<and> xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>}"
have "finite ?S"
by (rule_tac B = "{xa. xa < x}" in finite_subset, auto simp:finite_strict_prefix_set)
moreover have "?S \<noteq> {}" using False xz_in_star
by (simp, rule_tac x = "[]" in exI, auto simp:strict_prefix_def)
ultimately have "\<exists> max \<in> ?S. \<forall> a \<in> ?S. length a \<le> length max" using finite_set_has_max by blast
with prems show ?thesis by blast
qed
obtain ya where h5: "ya < y" and h6: "ya \<in> L\<^isub>1\<star>" and h7: "(x - x_max) \<approx>L\<^isub>1 (y - ya)"
proof-
from tag_xy have "{\<approx>L\<^isub>1 `` {x - xa} |xa. xa < x \<and> xa \<in> L\<^isub>1\<star>} =
{\<approx>L\<^isub>1 `` {y - xa} |xa. xa < y \<and> xa \<in> L\<^isub>1\<star>}" (is "?left = ?right")
by (auto simp:tag_str_STAR_def)
moreover have "\<approx>L\<^isub>1 `` {x - x_max} \<in> ?left" using h1 h2 by auto
ultimately have "\<approx>L\<^isub>1 `` {x - x_max} \<in> ?right" by simp
with prems show ?thesis apply (simp add:Image_def str_eq_rel_def str_eq_def) by blast
qed
have "(y - ya) @ z \<in> L\<^isub>1\<star>"
proof-
from h3 h1 obtain a b where a_in: "a \<in> L\<^isub>1" and a_neq: "a \<noteq> []" and b_in: "b \<in> L\<^isub>1\<star>"
and ab_max: "(x - x_max) @ z = a @ b"
by (drule_tac star_decom, auto simp:strict_prefix_def elim:prefixE)
have "(x - x_max) \<le> a \<and> (a - (x - x_max)) @ b = z"
proof -
have "((x - x_max) \<le> a \<and> (a - (x - x_max)) @ b = z) \<or> (a < (x - x_max) \<and> ((x - x_max) - a) @ z = b)"
using app_eq_dest[OF ab_max] by (auto simp:strict_prefix_def)
moreover {
assume np: "a < (x - x_max)" and b_eqs: " ((x - x_max) - a) @ z = b"
have "False"
proof -
let ?x_max' = "x_max @ a"
have "?x_max' < x" using np h1 by (clarsimp simp:strict_prefix_def diff_prefix)
moreover have "?x_max' \<in> L\<^isub>1\<star>" using a_in h2 by (simp add:star_intro3)
moreover have "(x - ?x_max') @ z \<in> L\<^isub>1\<star>" using b_eqs b_in np h1 by (simp add:diff_diff_appd)
moreover have "\<not> (length ?x_max' \<le> length x_max)" using a_neq by simp
ultimately show ?thesis using h4 by blast
qed
} ultimately show ?thesis by blast
qed
then obtain za where z_decom: "z = za @ b" and x_za: "(x - x_max) @ za \<in> L\<^isub>1"
using a_in by (auto elim:prefixE)
from x_za h7 have "(y - ya) @ za \<in> L\<^isub>1" by (auto simp:str_eq_def)
with z_decom b_in show ?thesis by (auto dest!:step[of "(y - ya) @ za"])
qed
with h5 h6 show ?thesis by (drule_tac star_intro1, auto simp:strict_prefix_def elim:prefixE)
qed
} thus "tag_str_STAR L\<^isub>1 m = tag_str_STAR L\<^isub>1 n \<Longrightarrow> m \<approx>(L\<^isub>1\<star>) n"
by (auto simp add:str_eq_def str_eq_rel_def)
qed
lemma quot_star_finiteI:
assumes finite: "finite (UNIV // \<approx>(L\<^isub>1::string set))"
shows "finite (UNIV // \<approx>(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) ` UNIV // \<approx>L\<^isub>1\<star>)" using finite
by (auto intro:finite_tag_imageI tag_str_star_range_finite)
next
show "inj_on (op ` (tag_str_STAR L\<^isub>1)) (UNIV // \<approx>L\<^isub>1\<star>)"
by (auto intro:tag_image_injI tag_str_STAR_injI)
qed
text {* **************** the Other Direction ************ *}
lemma other_direction:
"Lang = L (r::rexp) \<Longrightarrow> finite (UNIV // (\<approx>Lang))"
proof (induct arbitrary:Lang rule:rexp.induct)
case NULL
have "UNIV // (\<approx>{}) \<subseteq> {UNIV} "
by (auto simp:quotient_def str_eq_rel_def str_eq_def)
with prems show "?case" by (auto intro:finite_subset)
next
case EMPTY
have "UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}" by (rule quot_empty_subset)
with prems show ?case by (auto intro:finite_subset)
next
case (CHAR c)
have "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}" by (rule quot_char_subset)
with prems show ?case by (auto intro:finite_subset)
next
case (SEQ r\<^isub>1 r\<^isub>2)
have "\<lbrakk>finite (UNIV // \<approx>(L r\<^isub>1)); finite (UNIV // \<approx>(L r\<^isub>2))\<rbrakk> \<Longrightarrow> finite (UNIV // \<approx>(L r\<^isub>1 ;; L r\<^isub>2))"
by (erule quot_seq_finiteI, simp)
with prems show ?case by simp
next
case (ALT r\<^isub>1 r\<^isub>2)
have "\<lbrakk>finite (UNIV // \<approx>(L r\<^isub>1)); finite (UNIV // \<approx>(L r\<^isub>2))\<rbrakk> \<Longrightarrow> finite (UNIV // \<approx>(L r\<^isub>1 \<union> L r\<^isub>2))"
by (erule quot_union_finiteI, simp)
with prems show ?case by simp
next
case (STAR r)
have "finite (UNIV // \<approx>(L r)) \<Longrightarrow> finite (UNIV // \<approx>((L r)\<star>))"
by (erule quot_star_finiteI)
with prems show ?case by simp
qed
end