--- a/MyhillNerode.thy Tue Oct 19 11:51:05 2010 +0000
+++ b/MyhillNerode.thy Wed Oct 20 14:11:14 2010 +0000
@@ -1,1826 +1,1806 @@
-theory MyhilNerode
- imports "Main"
-begin
-
-text {* sequential composition of languages *}
-
-definition
- lang_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}"
-
-lemma lang_seq_empty:
- shows "{[]} ; L = L"
- and "L ; {[]} = L"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_null:
- shows "{} ; L = {}"
- and "L ; {} = {}"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_append:
- assumes a: "s1 \<in> L1"
- and b: "s2 \<in> L2"
- shows "s1@s2 \<in> L1 ; L2"
-unfolding lang_seq_def
-using a b by auto
-
-lemma lang_seq_union:
- shows "(L1 \<union> L2); L3 = (L1; L3) \<union> (L2; L3)"
- and "L1; (L2 \<union> L3) = (L1; L2) \<union> (L1; L3)"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_assoc:
- shows "(L1 ; L2) ; L3 = L1 ; (L2 ; L3)"
-by (simp add: lang_seq_def Collect_def mem_def expand_fun_eq)
- (metis append_assoc)
-
-lemma lang_seq_minus:
- shows "(L1; L2) - {[]} =
- (if [] \<in> L1 then L2 - {[]} else {}) \<union>
- (if [] \<in> L2 then L1 - {[]} else {}) \<union> ((L1 - {[]}); (L2 - {[]}))"
-apply(auto simp add: lang_seq_def)
-apply(metis mem_def self_append_conv)
-apply(metis mem_def self_append_conv2)
-apply(metis mem_def self_append_conv2)
-apply(metis mem_def self_append_conv)
-done
-
-section {* Kleene star for languages defined as least fixed point *}
-
-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 lang_star_empty:
- shows "{}\<star> = {[]}"
-by (auto elim: Star.cases)
-
-lemma lang_star_cases:
- shows "L\<star> = {[]} \<union> L ; L\<star>"
-proof
- { fix x
- have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L ; L\<star>"
- unfolding lang_seq_def
- by (induct rule: Star.induct) (auto)
- }
- then show "L\<star> \<subseteq> {[]} \<union> L ; L\<star>" by auto
-next
- show "{[]} \<union> L ; L\<star> \<subseteq> L\<star>"
- unfolding lang_seq_def by auto
-qed
-
-lemma lang_star_cases':
- shows "L\<star> = {[]} \<union> L\<star> ; L"
-proof
- { fix x
- have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L\<star> ; L"
- unfolding lang_seq_def
- apply (induct rule: Star.induct)
- apply simp
- apply simp
- apply (erule disjE)
- apply (auto)[1]
- apply (erule exE | erule conjE)+
- apply (rule disjI2)
- apply (rule_tac x = "s1 @ s1a" in exI)
- by auto
- }
- then show "L\<star> \<subseteq> {[]} \<union> L\<star> ; L" by auto
-next
- show "{[]} \<union> L\<star> ; L \<subseteq> L\<star>"
- unfolding lang_seq_def
- apply auto
- apply (erule Star.induct)
- apply auto
- apply (drule step[of _ _ "[]"])
- by (auto intro:start)
-qed
-
-lemma lang_star_simple:
- shows "L \<subseteq> L\<star>"
-by (subst lang_star_cases)
- (auto simp only: lang_seq_def)
-
-lemma lang_star_prop0_aux:
- "s2 \<in> L\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L \<longrightarrow> (\<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4)"
-apply (erule Star.induct)
-apply (clarify, rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
-apply (clarify, drule_tac x = s1 in spec)
-apply (drule mp, simp, clarify)
-apply (rule_tac x = "s1a @ s3" in exI, rule_tac x = s4 in exI)
-by auto
-
-lemma lang_star_prop0:
- "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> \<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4"
-by (auto dest:lang_star_prop0_aux)
-
-lemma lang_star_prop1:
- assumes asm: "L1; L2 \<subseteq> L2"
- shows "L1\<star>; L2 \<subseteq> L2"
-proof -
- { fix s1 s2
- assume minor: "s2 \<in> L2"
- assume major: "s1 \<in> L1\<star>"
- then have "s1@s2 \<in> L2"
- proof(induct rule: Star.induct)
- case start
- show "[]@s2 \<in> L2" using minor by simp
- next
- case (step s1 s1')
- have "s1 \<in> L1" by fact
- moreover
- have "s1'@s2 \<in> L2" by fact
- ultimately have "s1@(s1'@s2) \<in> L1; L2" by (auto simp add: lang_seq_def)
- with asm have "s1@(s1'@s2) \<in> L2" by auto
- then show "(s1@s1')@s2 \<in> L2" by simp
- qed
- }
- then show "L1\<star>; L2 \<subseteq> L2" by (auto simp add: lang_seq_def)
-qed
-
-lemma lang_star_prop2_aux:
- "s2 \<in> L2\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L1 \<and> L1 ; L2 \<subseteq> L1 \<longrightarrow> s1 @ s2 \<in> L1"
-apply (erule Star.induct, simp)
-apply (clarify, drule_tac x = "s1a @ s1" in spec)
-by (auto simp:lang_seq_def)
-
-lemma lang_star_prop2:
- "L1; L2 \<subseteq> L1 \<Longrightarrow> L1 ; L2\<star> \<subseteq> L1"
-by (auto dest!:lang_star_prop2_aux simp:lang_seq_def)
-
-lemma lang_star_seq_subseteq:
- shows "L ; L\<star> \<subseteq> L\<star>"
-using lang_star_cases by blast
-
-lemma lang_star_double:
- shows "L\<star>; L\<star> = L\<star>"
-proof
- show "L\<star> ; L\<star> \<subseteq> L\<star>"
- using lang_star_prop1 lang_star_seq_subseteq by blast
-next
- have "L\<star> \<subseteq> L\<star> \<union> L\<star>; (L; L\<star>)" by auto
- also have "\<dots> = L\<star>;{[]} \<union> L\<star>; (L; L\<star>)" by (simp add: lang_seq_empty)
- also have "\<dots> = L\<star>; ({[]} \<union> L; L\<star>)" by (simp only: lang_seq_union)
- also have "\<dots> = L\<star>; L\<star>" using lang_star_cases by simp
- finally show "L\<star> \<subseteq> L\<star> ; L\<star>" by simp
-qed
-
-lemma lang_star_seq_subseteq':
- shows "L\<star>; L \<subseteq> L\<star>"
-proof -
- have "L \<subseteq> L\<star>" by (rule lang_star_simple)
- then have "L\<star>; L \<subseteq> L\<star>; L\<star>" by (auto simp add: lang_seq_def)
- then show "L\<star>; L \<subseteq> L\<star>" using lang_star_double by blast
-qed
-
-lemma
- shows "L\<star> \<subseteq> L\<star>\<star>"
-by (rule lang_star_simple)
-
-section {* tricky section *}
-
-lemma k1:
- assumes b: "s \<in> L\<star>"
- and a: "s \<noteq> []"
- shows "s \<in> (L - {[]}); L\<star>"
-using b a
-apply(induct rule: Star.induct)
-apply(simp)
-apply(case_tac "s1=[]")
-apply(simp)
-apply(simp add: lang_seq_def)
-apply(blast)
-done
-
-section {* (relies on lemma k1) *}
-
-lemma zzz:
- shows "{s. c#s \<in> L1\<star>} = {s. c#s \<in> L1} ; (L1\<star>)"
-apply(auto simp add: lang_seq_def Cons_eq_append_conv)
-apply(drule k1)
-apply(auto)[1]
-apply(auto simp add: lang_seq_def)[1]
-apply(rule_tac x="tl s1" in exI)
-apply(rule_tac x="s2" in exI)
-apply(auto)[1]
-apply(auto simp add: Cons_eq_append_conv)[2]
-apply(drule lang_seq_append)
-apply(assumption)
-apply(rotate_tac 1)
-apply(drule rev_subsetD)
-apply(rule lang_star_seq_subseteq)
-apply(simp)
-done
-
-
-
-section {* regular expressions *}
-
-datatype rexp =
- NULL
-| EMPTY
-| CHAR char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-
-
-consts L:: "'a \<Rightarrow> string set"
-
-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>"
-
-defs (overloaded)
- l_rexp_abs: "L rexp \<equiv> L_rexp rexp"
-
-declare L_rexp.simps [simp del] L_rexp.simps [folded l_rexp_abs, simp add]
-
-definition
- Ls :: "rexp set \<Rightarrow> string set"
-where
- "Ls R = (\<Union>r\<in>R. (L r))"
-
-lemma Ls_union:
- "Ls (R1 \<union> R2) = (Ls R1) \<union> (Ls R2)"
-unfolding Ls_def by auto
-
-text {* helper function for termination proofs *}
-fun
- Left :: "rexp \<Rightarrow> rexp"
-where
- "Left (SEQ r1 r2) = r1"
-
-text {* dagger function *}
-
-function
- dagger :: "rexp \<Rightarrow> char \<Rightarrow> rexp list" ("_ \<dagger> _")
-where
- c1: "(NULL \<dagger> c) = []"
-| c2: "(EMPTY) \<dagger> c = []"
-| c3: "(CHAR c') \<dagger> c = (if c = c' then [EMPTY] else [])"
-| c4: "(ALT r1 r2) \<dagger> c = r1 \<dagger> c @ r2 \<dagger> c"
-| c5: "(SEQ NULL r2) \<dagger> c = []"
-| c6: "(SEQ EMPTY r2) \<dagger> c = r2 \<dagger> c"
-| c7: "(SEQ (CHAR c') r2) \<dagger> c = (if c = c' then [r2] else [])"
-| c8: "(SEQ (SEQ r11 r12) r2) \<dagger> c = (SEQ r11 (SEQ r12 r2)) \<dagger> c"
-| c9: "(SEQ (ALT r11 r12) r2) \<dagger> c = (SEQ r11 r2) \<dagger> c @ (SEQ r12 r2) \<dagger> c"
-| c10: "(SEQ (STAR r1) r2) \<dagger> c = r2 \<dagger> c @ [SEQ r' (SEQ (STAR r1) r2). r' \<leftarrow> r1 \<dagger> c]"
-| c11: "(STAR r) \<dagger> c = [SEQ r' (STAR r) . r' \<leftarrow> r \<dagger> c]"
-by (pat_completeness) (auto)
-
-termination dagger
- by (relation "measures [\<lambda>(r, c). size r, \<lambda>(r, c). size (Left r)]") (simp_all)
-
-lemma dagger_correctness:
- "Ls (set r \<dagger> c) = {s. c#s \<in> L r}"
-proof (induct rule: dagger.induct)
- case (1 c)
- show "Ls (set NULL \<dagger> c) = {s. c#s \<in> L NULL}" by (simp add: Ls_def)
-next
- case (2 c)
- show "Ls (set EMPTY \<dagger> c) = {s. c#s \<in> L EMPTY}" by (simp add: Ls_def)
-next
- case (3 c' c)
- show "Ls (set CHAR c' \<dagger> c) = {s. c#s \<in> L (CHAR c')}" by (simp add: Ls_def)
-next
- case (4 r1 r2 c)
- have ih1: "Ls (set r1 \<dagger> c) = {s. c#s \<in> L r1}" by fact
- have ih2: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
- show "Ls (set ALT r1 r2 \<dagger> c) = {s. c#s \<in> L (ALT r1 r2)}"
- by (simp add: Ls_union ih1 ih2 Collect_disj_eq)
-next
- case (5 r2 c)
- show "Ls (set SEQ NULL r2 \<dagger> c) = {s. c#s \<in> L (SEQ NULL r2)}" by (simp add: Ls_def lang_seq_null)
-next
- case (6 r2 c)
- have ih: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
- show "Ls (set SEQ EMPTY r2 \<dagger> c) = {s. c#s \<in> L (SEQ EMPTY r2)}"
- by (simp add: ih lang_seq_empty)
-next
- case (7 c' r2 c)
- show "Ls (set SEQ (CHAR c') r2 \<dagger> c) = {s. c#s \<in> L (SEQ (CHAR c') r2)}"
- by (simp add: Ls_def lang_seq_def)
-next
- case (8 r11 r12 r2 c)
- have ih: "Ls (set SEQ r11 (SEQ r12 r2) \<dagger> c) = {s. c#s \<in> L (SEQ r11 (SEQ r12 r2))}" by fact
- show "Ls (set SEQ (SEQ r11 r12) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (SEQ r11 r12) r2)}"
- by (simp add: ih lang_seq_assoc)
-next
- case (9 r11 r12 r2 c)
- have ih1: "Ls (set SEQ r11 r2 \<dagger> c) = {s. c#s \<in> L (SEQ r11 r2)}" by fact
- have ih2: "Ls (set SEQ r12 r2 \<dagger> c) = {s. c#s \<in> L (SEQ r12 r2)}" by fact
- show "Ls (set SEQ (ALT r11 r12) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (ALT r11 r12) r2)}"
- by (simp add: Ls_union ih1 ih2 lang_seq_union Collect_disj_eq)
-next
- case (10 r1 r2 c)
- have ih2: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
- have ih1: "Ls (set r1 \<dagger> c) = {s. c#s \<in> L r1}" by fact
- have "Ls (set SEQ (STAR r1) r2 \<dagger> c) = Ls (set r2 \<dagger> c) \<union> (Ls (set r1 \<dagger> c); ((L r1)\<star> ; L r2))"
- by (auto simp add: lang_seq_def Ls_def)
- also have "\<dots> = {s. c#s \<in> L r2} \<union> ({s. c#s \<in> L r1} ; ((L r1)\<star> ; L r2))" using ih1 ih2 by simp
- also have "\<dots> = {s. c#s \<in> L r2} \<union> ({s. c#s \<in> L r1} ; (L r1)\<star>) ; L r2" by (simp add: lang_seq_assoc)
- also have "\<dots> = {s. c#s \<in> L r2} \<union> {s. c#s \<in> (L r1)\<star>} ; L r2" by (simp add: zzz)
- also have "\<dots> = {s. c#s \<in> L r2} \<union> {s. c#s \<in> (L r1)\<star> ; L r2}"
- by (auto simp add: lang_seq_def Cons_eq_append_conv)
- also have "\<dots> = {s. c#s \<in> (L r1)\<star> ; L r2}"
- by (force simp add: lang_seq_def)
- finally show "Ls (set SEQ (STAR r1) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (STAR r1) r2)}" by simp
-next
- case (11 r c)
- have ih: "Ls (set r \<dagger> c) = {s. c#s \<in> L r}" by fact
- have "Ls (set (STAR r) \<dagger> c) = Ls (set r \<dagger> c) ; (L r)\<star>"
- by (auto simp add: lang_seq_def Ls_def)
- also have "\<dots> = {s. c#s \<in> L r} ; (L r)\<star>" using ih by simp
- also have "\<dots> = {s. c#s \<in> (L r)\<star>}" using zzz by simp
- finally show "Ls (set (STAR r) \<dagger> c) = {s. c#s \<in> L (STAR r)}" by simp
-qed
-
-
-text {* matcher function (based on the "list"-dagger function) *}
-fun
- first_True :: "bool list \<Rightarrow> bool"
-where
- "first_True [] = False"
-| "first_True (x#xs) = (if x then True else first_True xs)"
-
-lemma not_first_True[simp]:
- shows "(\<not>(first_True xs)) = (\<forall>x \<in> set xs. \<not>x)"
-by (induct xs) (auto)
-
-lemma first_True:
- shows "(first_True xs) = (\<exists>x \<in> set xs. x)"
-by (induct xs) (auto)
-
-text {* matcher function *}
-
-function
- matcher :: "rexp \<Rightarrow> string \<Rightarrow> bool" ("_ ! _")
-where
- "NULL ! s = False"
-| "EMPTY ! s = (s =[])"
-| "CHAR c ! s = (s = [c])"
-| "ALT r1 r2 ! s = (r1 ! s \<or> r2 ! s)"
-| "STAR r ! [] = True"
-| "STAR r ! c#s = first_True [SEQ (r') (STAR r) ! s. r' \<leftarrow> r \<dagger> c]"
-| "SEQ r1 r2 ! [] = (r1 ! [] \<and> r2 ! [])"
-| "SEQ NULL r2 ! (c#s) = False"
-| "SEQ EMPTY r2 ! (c#s) = (r2 ! c#s)"
-| "SEQ (CHAR c') r2 ! (c#s) = (if c'=c then r2 ! s else False)"
-| "SEQ (SEQ r11 r12) r2 ! (c#s) = (SEQ r11 (SEQ r12 r2) ! c#s)"
-| "SEQ (ALT r11 r12) r2 ! (c#s) = ((SEQ r11 r2) ! (c#s) \<or> (SEQ r12 r2) ! (c#s))"
-| "SEQ (STAR r1) r2 ! (c#s) = (r2 ! (c#s) \<or> first_True [SEQ r' (SEQ (STAR r1) r2) ! s. r' \<leftarrow> r1 \<dagger> c])"
-by (pat_completeness) (auto)
-
-termination matcher
- by(relation "measures [\<lambda>(r,s). length s, \<lambda>(r,s). size r, \<lambda>(r,s). size (Left r)]") (simp_all)
-
-text {* positive correctness of the matcher *}
-lemma matcher1:
- shows "r ! s \<Longrightarrow> s \<in> L r"
-proof (induct rule: matcher.induct)
- case (1 s)
- have "NULL ! s" by fact
- then show "s \<in> L NULL" by simp
-next
- case (2 s)
- have "EMPTY ! s" by fact
- then show "s \<in> L EMPTY" by simp
-next
- case (3 c s)
- have "CHAR c ! s" by fact
- then show "s \<in> L (CHAR c)" by simp
-next
- case (4 r1 r2 s)
- have ih1: "r1 ! s \<Longrightarrow> s \<in> L r1" by fact
- have ih2: "r2 ! s \<Longrightarrow> s \<in> L r2" by fact
- have "ALT r1 r2 ! s" by fact
- with ih1 ih2 show "s \<in> L (ALT r1 r2)" by auto
-next
- case (5 r)
- have "STAR r ! []" by fact
- then show "[] \<in> L (STAR r)" by auto
-next
- case (6 r c s)
- have ih1: "\<And>rx. \<lbrakk>rx \<in> set r \<dagger> c; SEQ rx (STAR r) ! s\<rbrakk> \<Longrightarrow> s \<in> L (SEQ rx (STAR r))" by fact
- have as: "STAR r ! c#s" by fact
- then obtain r' where imp1: "r' \<in> set r \<dagger> c" and imp2: "SEQ r' (STAR r) ! s"
- by (auto simp add: first_True)
- from imp2 imp1 have "s \<in> L (SEQ r' (STAR r))" using ih1 by simp
- then have "s \<in> L r' ; (L r)\<star>" by simp
- then have "s \<in> Ls (set r \<dagger> c) ; (L r)\<star>" using imp1 by (auto simp add: Ls_def lang_seq_def)
- then have "s \<in> {s. c#s \<in> L r} ; (L r)\<star>" by (auto simp add: dagger_correctness)
- then have "s \<in> {s. c#s \<in> (L r)\<star>}" by (simp add: zzz)
- then have "c#s \<in> {[c]}; {s. c#s \<in> (L r)\<star>}" by (auto simp add: lang_seq_def)
- then have "c#s \<in> (L r)\<star>" by (auto simp add: lang_seq_def)
- then show "c#s \<in> L (STAR r)" by simp
-next
- case (7 r1 r2)
- have ih1: "r1 ! [] \<Longrightarrow> [] \<in> L r1" by fact
- have ih2: "r2 ! [] \<Longrightarrow> [] \<in> L r2" by fact
- have as: "SEQ r1 r2 ! []" by fact
- then have "r1 ! [] \<and> r2 ! []" by simp
- then show "[] \<in> L (SEQ r1 r2)" using ih1 ih2 by (simp add: lang_seq_def)
-next
- case (8 r2 c s)
- have "SEQ NULL r2 ! c#s" by fact
- then show "c#s \<in> L (SEQ NULL r2)" by simp
-next
- case (9 r2 c s)
- have ih1: "r2 ! c#s \<Longrightarrow> c#s \<in> L r2" by fact
- have "SEQ EMPTY r2 ! c#s" by fact
- then show "c#s \<in> L (SEQ EMPTY r2)" using ih1 by (simp add: lang_seq_def)
-next
- case (10 c' r2 c s)
- have ih1: "\<lbrakk>c' = c; r2 ! s\<rbrakk> \<Longrightarrow> s \<in> L r2" by fact
- have "SEQ (CHAR c') r2 ! c#s" by fact
- then show "c#s \<in> L (SEQ (CHAR c') r2)"
- using ih1 by (auto simp add: lang_seq_def split: if_splits)
-next
- case (11 r11 r12 r2 c s)
- have ih1: "SEQ r11 (SEQ r12 r2) ! c#s \<Longrightarrow> c#s \<in> L (SEQ r11 (SEQ r12 r2))" by fact
- have "SEQ (SEQ r11 r12) r2 ! c#s" by fact
- then have "c#s \<in> L (SEQ r11 (SEQ r12 r2))" using ih1 by simp
- then show "c#s \<in> L (SEQ (SEQ r11 r12) r2)" by (simp add: lang_seq_assoc)
-next
- case (12 r11 r12 r2 c s)
- have ih1: "SEQ r11 r2 ! c#s \<Longrightarrow> c#s \<in> L (SEQ r11 r2)" by fact
- have ih2: "SEQ r12 r2 ! c#s \<Longrightarrow> c#s \<in> L (SEQ r12 r2)" by fact
- have "SEQ (ALT r11 r12) r2 ! c#s" by fact
- then show "c#s \<in> L (SEQ (ALT r11 r12) r2)"
- using ih1 ih2 by (auto simp add: lang_seq_union)
-next
- case (13 r1 r2 c s)
- have ih1: "r2 ! c#s \<Longrightarrow> c#s \<in> L r2" by fact
- have ih2: "\<And>r'. \<lbrakk>r' \<in> set r1 \<dagger> c; SEQ r' (SEQ (STAR r1) r2) ! s\<rbrakk> \<Longrightarrow>
- s \<in> L (SEQ r' (SEQ (STAR r1) r2))" by fact
- have "SEQ (STAR r1) r2 ! c#s" by fact
- then have "(r2 ! c#s) \<or> (\<exists>r' \<in> set r1 \<dagger> c. SEQ r' (SEQ (STAR r1) r2) ! s)" by (auto simp add: first_True)
- moreover
- { assume "r2 ! c#s"
- with ih1 have "c#s \<in> L r2" by simp
- then have "c # s \<in> L r1\<star> ; L r2"
- by (auto simp add: lang_seq_def)
- then have "c#s \<in> L (SEQ (STAR r1) r2)" by simp
- }
- moreover
- { assume "\<exists>r' \<in> set r1 \<dagger> c. SEQ r' (SEQ (STAR r1) r2) ! s"
- then obtain r' where imp1: "r' \<in> set r1 \<dagger> c" and imp2: "SEQ r' (SEQ (STAR r1) r2) ! s" by blast
- from imp2 imp1 have "s \<in> L (SEQ r' (SEQ (STAR r1) r2))" using ih2 by simp
- then have "s \<in> L r' ; ((L r1)\<star> ; L r2)" by simp
- then have "s \<in> Ls (set r1 \<dagger> c) ; ((L r1)\<star> ; L r2)" using imp1 by (auto simp add: Ls_def lang_seq_def)
- then have "s \<in> {s. c#s \<in> L r1} ; ((L r1)\<star> ; L r2)" by (simp add: dagger_correctness)
- then have "s \<in> ({s. c#s \<in> L r1} ; (L r1)\<star>) ; L r2" by (simp add: lang_seq_assoc)
- then have "s \<in> {s. c#s \<in> (L r1)\<star>} ; L r2" by (simp add: zzz)
- then have "c#s \<in> {[c]}; ({s. c#s \<in> (L r1)\<star>}; L r2)" by (auto simp add: lang_seq_def)
- then have "c#s \<in> ({[c]}; {s. c#s \<in> (L r1)\<star>}) ; L r2" by (simp add: lang_seq_assoc)
- then have "c#s \<in> (L r1)\<star>; L r2" by (auto simp add: lang_seq_def)
- then have "c#s \<in> L (SEQ (STAR r1) r2)" by simp
- }
- ultimately show "c#s \<in> L (SEQ (STAR r1) r2)" by blast
-qed
-
-text {* negative correctness of the matcher *}
-lemma matcher2:
- shows "\<not> r ! s \<Longrightarrow> s \<notin> L r"
-proof (induct rule: matcher.induct)
- case (1 s)
- have "\<not> NULL ! s" by fact
- then show "s \<notin> L NULL" by simp
-next
- case (2 s)
- have "\<not> EMPTY ! s" by fact
- then show "s \<notin> L EMPTY" by simp
-next
- case (3 c s)
- have "\<not> CHAR c ! s" by fact
- then show "s \<notin> L (CHAR c)" by simp
-next
- case (4 r1 r2 s)
- have ih2: "\<not> r1 ! s \<Longrightarrow> s \<notin> L r1" by fact
- have ih4: "\<not> r2 ! s \<Longrightarrow> s \<notin> L r2" by fact
- have "\<not> ALT r1 r2 ! s" by fact
- then show "s \<notin> L (ALT r1 r2)" by (simp add: ih2 ih4)
-next
- case (5 r)
- have "\<not> STAR r ! []" by fact
- then show "[] \<notin> L (STAR r)" by simp
-next
- case (6 r c s)
- have ih: "\<And>rx. \<lbrakk>rx \<in> set r \<dagger> c; \<not>SEQ rx (STAR r) ! s\<rbrakk> \<Longrightarrow> s \<notin> L (SEQ rx (STAR r))" by fact
- have as: "\<not> STAR r ! c#s" by fact
- then have "\<forall>r'\<in> set r \<dagger> c. \<not> (SEQ r' (STAR r) ! s)" by simp
- then have "\<forall>r'\<in> set r \<dagger> c. s \<notin> L (SEQ r' (STAR r))" using ih by auto
- then have "\<forall>r'\<in> set r \<dagger> c. s \<notin> L r' ; ((L r)\<star>)" by simp
- then have "s \<notin> (Ls (set r \<dagger> c)) ; ((L r)\<star>)" by (auto simp add: Ls_def lang_seq_def)
- then have "s \<notin> {s. c#s \<in> L r} ; ((L r)\<star>)" by (simp add: dagger_correctness)
- then have "s \<notin> {s. c#s \<in> (L r)\<star>}" by (simp add: zzz)
- then have "c#s \<notin> {[c]} ; {s. c#s \<in> (L r)\<star>}" by (auto simp add: lang_seq_assoc lang_seq_def)
- then have "c#s \<notin> (L r)\<star>" by (simp add: lang_seq_def)
- then show "c#s \<notin> L (STAR r)" by simp
-next
- case (7 r1 r2)
- have ih2: "\<not> r1 ! [] \<Longrightarrow> [] \<notin> L r1" by fact
- have ih4: "\<not> r2 ! [] \<Longrightarrow> [] \<notin> L r2" by fact
- have "\<not> SEQ r1 r2 ! []" by fact
- then have "\<not> r1 ! [] \<or> \<not> r2 ! []" by simp
- then show "[] \<notin> L (SEQ r1 r2)" using ih2 ih4
- by (auto simp add: lang_seq_def)
-next
- case (8 r2 c s)
- have "\<not> SEQ NULL r2 ! c#s" by fact
- then show "c#s \<notin> L (SEQ NULL r2)" by (simp add: lang_seq_null)
-next
- case (9 r2 c s)
- have ih1: "\<not> r2 ! c#s \<Longrightarrow> c#s \<notin> L r2" by fact
- have "\<not> SEQ EMPTY r2 ! c#s" by fact
- then show "c#s \<notin> L (SEQ EMPTY r2)"
- using ih1 by (simp add: lang_seq_def)
-next
- case (10 c' r2 c s)
- have ih2: "\<lbrakk>c' = c; \<not>r2 ! s\<rbrakk> \<Longrightarrow> s \<notin> L r2" by fact
- have "\<not> SEQ (CHAR c') r2 ! c#s" by fact
- then show "c#s \<notin> L (SEQ (CHAR c') r2)"
- using ih2 by (auto simp add: lang_seq_def)
-next
- case (11 r11 r12 r2 c s)
- have ih2: "\<not> SEQ r11 (SEQ r12 r2) ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r11 (SEQ r12 r2))" by fact
- have "\<not> SEQ (SEQ r11 r12) r2 ! c#s" by fact
- then show "c#s \<notin> L (SEQ (SEQ r11 r12) r2)"
- using ih2 by (auto simp add: lang_seq_def)
-next
- case (12 r11 r12 r2 c s)
- have ih2: "\<not> SEQ r11 r2 ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r11 r2)" by fact
- have ih4: "\<not> SEQ r12 r2 ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r12 r2)" by fact
- have "\<not> SEQ (ALT r11 r12) r2 ! c#s" by fact
- then show " c#s \<notin> L (SEQ (ALT r11 r12) r2)"
- using ih2 ih4 by (simp add: lang_seq_union)
-next
- case (13 r1 r2 c s)
- have ih1: "\<not>r2 ! c#s \<Longrightarrow> c#s \<notin> L r2" by fact
- have ih2: "\<And>rx. \<lbrakk>rx \<in> set r1 \<dagger> c; \<not> SEQ rx (SEQ (STAR r1) r2) ! s\<rbrakk>
- \<Longrightarrow> s \<notin> L (SEQ rx (SEQ (STAR r1) r2))" by fact
- have as: "\<not> SEQ (STAR r1) r2 ! c#s" by fact
- then have as1: "\<not>r2 ! c#s" and as2: "\<forall>r1'\<in>set r1 \<dagger> c. \<not> SEQ r1' (SEQ (STAR r1) r2) ! s" by simp_all
- from as1 have bs1: "c#s \<notin> L r2" using ih1 by simp
- from as2 have "\<forall>r1'\<in>set r1 \<dagger> c. \<not> SEQ r1' (SEQ (STAR r1) r2) ! s" by simp
- then have "\<forall>r1'\<in>set r1 \<dagger> c. s \<notin> L (SEQ r1' (SEQ (STAR r1) r2))" using ih2 by simp
- then have "\<forall>r1'\<in>set r1 \<dagger> c. s \<notin> L r1'; ((L r1)\<star>; L r2)" by simp
- then have "s \<notin> (Ls (set r1 \<dagger> c)) ; ((L r1)\<star>; L r2)" by (auto simp add: Ls_def lang_seq_def)
- then have "s \<notin> {s. c#s \<in> L r1} ; ((L r1)\<star>; L r2)" by (simp add: dagger_correctness)
- then have "s \<notin> ({s. c#s \<in> L r1} ; (L r1)\<star>); L r2" by (simp add: lang_seq_assoc)
- then have "s \<notin> {s. c#s \<in> (L r1)\<star>}; L r2" by (simp add: zzz)
- then have "c#s \<notin> {[c]}; ({s. c#s \<in> (L r1)\<star>}; L r2)" by (auto simp add: lang_seq_def)
- then have "c#s \<notin> (L r1)\<star>; L r2" using bs1 by (auto simp add: lang_seq_def Cons_eq_append_conv)
- then show "c#s \<notin> L (SEQ (STAR r1) r2)" by simp
-qed
-
-section {* Questions *}
-
-text {*
- - Why was the key lemma k1 omitted; is there an easy, non-induction
- way for obtaining this property?
- - Why was False included in the definition of the STAR-clause in
- the matcher? Has this something to do with executing the code?
-
-*}
-
-section {* Code *}
-
-export_code dagger in SML module_name Dagger file -
-export_code matcher in SML module_name Dagger file -
-
-section {* Examples *}
-
-text {* since now everything is based on lists, the evaluation is quite fast *}
-
-value "NULL ! []"
-value "(CHAR (CHR ''a'')) ! [CHR ''a'']"
-value "((CHAR a) ! [a,a])"
-value "(STAR (CHAR a)) ! []"
-value "(STAR (CHAR a)) ! [a,a]"
-value "(SEQ (CHAR (CHR ''a'')) (SEQ (STAR (CHAR (CHR ''b''))) (CHAR (CHR ''c'')))) ! ''abbbbc''"
-value "(SEQ (CHAR (CHR ''a'')) (SEQ (STAR (CHAR (CHR ''b''))) (CHAR (CHR ''c'')))) ! ''abbcbbc''"
-
-section {* Slind et al's matcher based on derivatives *}
-
-fun
- nullable :: "rexp \<Rightarrow> bool"
-where
- "nullable (NULL) = False"
-| "nullable (EMPTY) = True"
-| "nullable (CHAR c) = False"
-| "nullable (ALT r1 r2) = ((nullable r1) \<or> (nullable r2))"
-| "nullable (SEQ r1 r2) = ((nullable r1) \<and> (nullable r2))"
-| "nullable (STAR r) = True"
-
-lemma nullable:
- shows "([] \<in> L r) = nullable r"
-by (induct r)
- (auto simp add: lang_seq_def)
-
-fun
- der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
-where
- "der c (NULL) = NULL"
-| "der c (EMPTY) = NULL"
-| "der c (CHAR c') = (if c=c' then EMPTY else NULL)"
-| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
-| "der c (SEQ r1 r2) = ALT (SEQ (der c r1) r2) (if nullable r1 then der c r2 else NULL)"
-| "der c (STAR r) = SEQ (der c r) (STAR r)"
-
-lemma k2:
- assumes b: "s \<in> L1\<star>"
- and a: "s \<noteq> []"
- shows "s \<in> (L1; (L1\<star>))"
-using b a
-apply(induct rule: Star.induct)
-apply(simp)
-apply(case_tac "s1=[]")
-apply(simp)
-apply(simp add: lang_seq_def)
-apply(blast)
-done
-
-
-lemma der_correctness:
- shows "(s \<in> L (der c r)) = ((c#s) \<in> L r)"
-proof (induct r arbitrary: s)
- case (NULL s)
- show "(s \<in> L (der c NULL)) = (c#s \<in> L NULL)" by simp
-next
- case (EMPTY s)
- show "(s \<in> L (der c EMPTY)) = (c#s \<in> L EMPTY)" by simp
-next
- case (CHAR c' s)
- show "(s \<in> L (der c (CHAR c'))) = (c#s \<in> L (CHAR c'))" by simp
-next
- case (SEQ r1 r2 s)
- have ih1: "\<And>s. (s \<in> L (der c r1)) = (c#s \<in> L r1)" by fact
- have ih2: "\<And>s. (s \<in> L (der c r2)) = (c#s \<in> L r2)" by fact
- show "(s \<in> L (der c (SEQ r1 r2))) = (c#s \<in> L (SEQ r1 r2))"
- using ih1 ih2
- by (auto simp add: nullable[symmetric] lang_seq_def Cons_eq_append_conv)
-next
- case (ALT r1 r2 s)
- have ih1: "\<And>s. (s \<in> L (der c r1)) = (c#s \<in> L r1)" by fact
- have ih2: "\<And>s. (s \<in> L (der c r2)) = (c#s \<in> L r2)" by fact
- show "(s \<in> L (der c (ALT r1 r2))) = (c#s \<in> L (ALT r1 r2))"
- using ih1 ih2 by (auto simp add: lang_seq_def)
-next
- case (STAR r s)
- have ih1: "\<And>s. (s \<in> L (der c r)) = (c#s \<in> L r)" by fact
- show "(s \<in> L (der c (STAR r))) = (c#s \<in> L (STAR r))"
- using ih1
- apply(simp)
- apply(auto simp add: lang_seq_def)
- apply(drule lang_seq_append)
- apply(assumption)
- apply(simp)
- apply(subst lang_star_cases)
- apply(simp)
- thm k1
- apply(drule k2)
- apply(simp)
- apply(simp add: lang_seq_def)
- apply(erule exE)+
- apply(erule conjE)+
- apply(auto simp add: lang_seq_def Cons_eq_append_conv)
- apply(drule k1)
- apply(simp)
- apply(simp add: lang_seq_def)
- apply(erule exE)+
- apply(erule conjE)+
- apply(auto simp add: lang_seq_def Cons_eq_append_conv)
- done
-qed
-
-fun
- derivative :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
-where
- "derivative [] r = r"
-| "derivative (c#s) r = derivative s (der c r)"
-
-fun
- slind_matcher :: "rexp \<Rightarrow> string \<Rightarrow> bool"
-where
- "slind_matcher r s = nullable (derivative s r)"
-
-lemma slind_matcher:
- shows "slind_matcher r s = (s \<in> L r)"
-by (induct s arbitrary: r)
- (auto simp add: nullable der_correctness)
-
-export_code slind_matcher in SML module_name Slind file -
-
-
-(* ******************************************** now is the codes writen by chunhan ************************************* *)
-
-section {* Arden's Lemma revised *}
-
-lemma arden_aux1:
- assumes a: "X \<subseteq> X ; A \<union> B"
- and b: "[] \<notin> A"
- shows "x \<in> X \<Longrightarrow> x \<in> B ; A\<star>"
-apply (induct x taking:length rule:measure_induct)
-apply (subgoal_tac "x \<in> X ; A \<union> B")
-defer
-using a
-apply (auto)[1]
-apply simp
-apply (erule disjE)
-defer
-apply (auto simp add:lang_seq_def) [1]
-apply (subgoal_tac "\<exists> x1 x2. x = x1 @ x2 \<and> x1 \<in> X \<and> x2 \<in> A")
-defer
-apply (auto simp add:lang_seq_def) [1]
-apply (erule exE | erule conjE)+
-apply simp
-apply (drule_tac x = x1 in spec)
-apply (simp)
-using b
-apply -
-apply (auto)[1]
-apply (subgoal_tac "x1 @ x2 \<in> (B ; A\<star>) ; A")
-defer
-apply (auto simp add:lang_seq_def)[1]
-by (metis Un_absorb1 lang_seq_assoc lang_seq_union(2) lang_star_double lang_star_simple mem_def sup1CI)
-
-theorem ardens_revised:
- assumes nemp: "[] \<notin> A"
- shows "(X = X ; A \<union> B) \<longleftrightarrow> (X = B ; A\<star>)"
-apply(rule iffI)
-defer
-apply(simp)
-apply(subst lang_star_cases')
-apply(subst lang_seq_union)
-apply(simp add: lang_seq_empty)
-apply(simp add: lang_seq_assoc)
-apply(auto)[1]
-proof -
- assume "X = X ; A \<union> B"
- then have as1: "X ; A \<union> B \<subseteq> X" and as2: "X \<subseteq> X ; A \<union> B" by simp_all
- from as1 have a: "X ; A \<subseteq> X" and b: "B \<subseteq> X" by simp_all
- from b have "B; A\<star> \<subseteq> X ; A\<star>" by (auto simp add: lang_seq_def)
- moreover
- from a have "X ; A\<star> \<subseteq> X"
-
-by (rule lang_star_prop2)
- ultimately have f1: "B ; A\<star> \<subseteq> X" by simp
- from as2 nemp
- have f2: "X \<subseteq> B; A\<star>" using arden_aux1 by auto
- from f1 f2 show "X = B; A\<star>" by auto
-qed
-
-section {* equiv class' definition *}
-
-definition
- equiv_str :: "string \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> bool" ("_ \<equiv>_\<equiv> _" [100, 100, 100] 100)
-where
- "x \<equiv>L'\<equiv> y \<longleftrightarrow> (\<forall>z. x@z \<in> L' \<longleftrightarrow> y@z \<in> L')"
-
-definition
- equiv_class :: "string \<Rightarrow> (string set) \<Rightarrow> string set" ("\<lbrakk>_\<rbrakk>_" [100, 100] 100)
-where
- "\<lbrakk>x\<rbrakk>L' \<equiv> {y. x \<equiv>L'\<equiv> y}"
-
-text {* Chunhan modifies Q to Quo *}
-definition
- quot :: "string set \<Rightarrow> (string set) \<Rightarrow> (string set) set" ("_ Quo _" [100, 100] 100)
-where
- "L' Quo R \<equiv> { \<lbrakk>x\<rbrakk>R | x. x \<in> L'}"
-
-lemma lang_eqs_union_of_eqcls:
- "Lang = \<Union> {X. X \<in> (UNIV Quo Lang) \<and> (\<forall> x \<in> X. x \<in> Lang)}"
-proof
- show "Lang \<subseteq> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
- proof
- fix x
- assume "x \<in> Lang"
- thus "x \<in> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
- proof (simp add:quot_def)
- assume "(1)": "x \<in> Lang"
- show "\<exists>xa. (\<exists>x. xa = \<lbrakk>x\<rbrakk>Lang) \<and> (\<forall>x\<in>xa. x \<in> Lang) \<and> x \<in> xa" (is "\<exists>xa.?P xa")
- proof -
- have "?P (\<lbrakk>x\<rbrakk>Lang)" using "(1)"
- by (auto simp:equiv_class_def equiv_str_def dest: spec[where x = "[]"])
- thus ?thesis by blast
- qed
- qed
- qed
-next
- show "\<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<subseteq> Lang"
- by auto
-qed
-
-lemma empty_notin_CS: "{} \<notin> UNIV Quo Lang"
-apply (clarsimp simp:quot_def equiv_class_def)
-by (rule_tac x = x in exI, auto simp:equiv_str_def)
-
-lemma no_two_cls_inters:
- "\<lbrakk>X \<in> UNIV Quo Lang; Y \<in> UNIV Quo Lang; X \<noteq> Y\<rbrakk> \<Longrightarrow> X \<inter> Y = {}"
-by (auto simp:quot_def equiv_class_def equiv_str_def)
-
-text {* equiv_class transition *}
-definition
- CT :: "string set \<Rightarrow> char \<Rightarrow> string set \<Rightarrow> bool" ("_-_\<rightarrow>_" [99,99]99)
-where
- "X-c\<rightarrow>Y \<equiv> ((X;{[c]}) \<subseteq> Y)"
-
-types t_equa_rhs = "(string set \<times> rexp) set"
-
-types t_equa = "(string set \<times> t_equa_rhs)"
-
-types t_equas = "t_equa set"
-
-text {* "empty_rhs" generates "\<lambda>" for init-state, just like "\<lambda>" for final states in Brzozowski method.
- But if the init-state is "{[]}" ("\<lambda>" itself) then empty set is returned, see definition of "equation_rhs" *}
-definition
- empty_rhs :: "string set \<Rightarrow> t_equa_rhs"
-where
- "empty_rhs X \<equiv> if ([] \<in> X) then {({[]}, EMPTY)} else {}"
-
-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"
-
-definition
- equation_rhs :: "(string set) set \<Rightarrow> string set \<Rightarrow> t_equa_rhs"
-where
- "equation_rhs CS X \<equiv> if (X = {[]}) then {({[]}, EMPTY)}
- else {(S, folds ALT NULL {CHAR c| c. S-c\<rightarrow>X} )| S. S \<in> CS} \<union> empty_rhs X"
-
-definition
- equations :: "(string set) set \<Rightarrow> t_equas"
-where
- "equations CS \<equiv> {(X, equation_rhs CS X) | X. X \<in> CS}"
-
-definition
- L_rhs :: "t_equa_rhs \<Rightarrow> string set"
-where
- "L_rhs rhs \<equiv> {x. \<exists> X r. (X, r) \<in> rhs \<and> x \<in> X;(L r)}"
-
-defs (overloaded)
- l_rhs_abs: "L rhs \<equiv> L_rhs rhs"
-
-lemmas L_def = L_rhs_def [folded l_rhs_abs] L_rexp.simps (* ??? is this OK ?? *)
-
-definition
- distinct_rhs :: "t_equa_rhs \<Rightarrow> bool"
-where
- "distinct_rhs rhs \<equiv> \<forall> X reg\<^isub>1 reg\<^isub>2. (X, reg\<^isub>1) \<in> rhs \<and> (X, reg\<^isub>2) \<in> rhs \<longrightarrow> reg\<^isub>1 = reg\<^isub>2"
-
-definition
- distinct_equas_rhs :: "t_equas \<Rightarrow> bool"
-where
- "distinct_equas_rhs equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> distinct_rhs rhs"
-
-definition
- distinct_equas :: "t_equas \<Rightarrow> bool"
-where
- "distinct_equas equas \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> equas \<and> (X, rhs') \<in> equas \<longrightarrow> rhs = rhs'"
-
-definition
- seq_rhs_r :: "t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
-where
- "seq_rhs_r rhs r \<equiv> (\<lambda>(X, reg). (X, SEQ reg r)) ` rhs"
-
-definition
- del_x_paired :: "('a \<times> 'b) set \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'b) set"
-where
- "del_x_paired S x \<equiv> S - {X. X \<in> S \<and> fst X = x}"
-
-definition
- arden_variate :: "string set \<Rightarrow> rexp \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
-where
- "arden_variate X r rhs \<equiv> seq_rhs_r (del_x_paired rhs X) (STAR r)"
-
-definition
- no_EMPTY_rhs :: "t_equa_rhs \<Rightarrow> bool"
-where
- "no_EMPTY_rhs rhs \<equiv> \<forall> X r. (X, r) \<in> rhs \<and> X \<noteq> {[]} \<longrightarrow> [] \<notin> L r"
-
-definition
- no_EMPTY_equas :: "t_equas \<Rightarrow> bool"
-where
- "no_EMPTY_equas equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> no_EMPTY_rhs rhs"
-
-lemma fold_alt_null_eqs:
- "finite rS \<Longrightarrow> x \<in> L (folds ALT NULL rS) = (\<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)
-by auto (*??? how do this be in Isar ?? *)
-
-lemma seq_rhs_r_prop1:
- "L (seq_rhs_r rhs r) = (L rhs);(L r)"
-apply (rule set_ext, rule iffI)
-apply (auto simp:L_def seq_rhs_r_def image_def lang_seq_def)
-apply (rule_tac x = "s1 @ s1a" in exI, rule_tac x = "s2a" in exI, simp)
-apply (rule_tac x = a in exI, rule_tac x = b in exI, simp)
-apply (rule_tac x = s1 in exI, rule_tac x = s1a in exI, simp)
-apply (rule_tac x = X in exI, rule_tac x = "SEQ ra r" in exI, simp)
-apply (rule conjI)
-apply (rule_tac x = "(X, ra)" in bexI, simp+)
-apply (rule_tac x = s1a in exI, rule_tac x = "s2a @ s2" in exI, simp)
-apply (simp add:lang_seq_def)
-by (rule_tac x = s2a in exI, rule_tac x = s2 in exI, simp)
-
-lemma del_x_paired_prop1:
- "\<lbrakk>distinct_rhs rhs; (X, r) \<in> rhs\<rbrakk> \<Longrightarrow> X ; L r \<union> L (del_x_paired rhs X) = L rhs"
-apply (simp add:L_def del_x_paired_def)
-apply (rule set_ext, rule iffI, simp)
-apply (erule disjE, rule_tac x = X in exI, rule_tac x = r in exI, simp)
-apply (clarify, rule_tac x = Xa in exI, rule_tac x = ra in exI, simp)
-apply (clarsimp, drule_tac x = Xa in spec, drule_tac x = ra in spec)
-apply (auto simp:distinct_rhs_def)
-done
-
-lemma arden_variate_prop:
- assumes "(X, rx) \<in> rhs"
- shows "(\<forall> Y. Y \<noteq> X \<longrightarrow> (\<exists> r. (Y, r) \<in> rhs) = (\<exists> r. (Y, r) \<in> (arden_variate X rx rhs)))"
-proof (rule allI, rule impI)
- fix Y
- assume "(1)": "Y \<noteq> X"
- show "(\<exists>r. (Y, r) \<in> rhs) = (\<exists>r. (Y, r) \<in> arden_variate X rx rhs)"
- proof
- assume "(1_1)": "\<exists>r. (Y, r) \<in> rhs"
- show "\<exists>r. (Y, r) \<in> arden_variate X rx rhs" (is "\<exists>r. ?P r")
- proof -
- from "(1_1)" obtain r where "(Y, r) \<in> rhs" ..
- hence "?P (SEQ r (STAR rx))"
- proof (simp add:arden_variate_def image_def)
- have "(Y, r) \<in> rhs \<Longrightarrow> (Y, r) \<in> del_x_paired rhs X"
- by (auto simp:del_x_paired_def "(1)")
- thus "(Y, r) \<in> rhs \<Longrightarrow> (Y, SEQ r (STAR rx)) \<in> seq_rhs_r (del_x_paired rhs X) (STAR rx)"
- by (auto simp:seq_rhs_r_def)
- qed
- thus ?thesis by blast
- qed
- next
- assume "(2_1)": "\<exists>r. (Y, r) \<in> arden_variate X rx rhs"
- thus "\<exists>r. (Y, r) \<in> rhs" (is "\<exists> r. ?P r")
- by (auto simp:arden_variate_def del_x_paired_def seq_rhs_r_def image_def)
- qed
-qed
-
-text {*
- arden_variate_valid: proves variation from "X = X;r + Y;ry + \<dots>" to "X = Y;(SEQ ry (STAR r)) + \<dots>" holds the law of "language of left equiv language of right"
-*}
-lemma arden_variate_valid:
- assumes X_not_empty: "X \<noteq> {[]}"
- and l_eq_r: "X = L rhs"
- and dist: "distinct_rhs rhs"
- and no_empty: "no_EMPTY_rhs rhs"
- and self_contained: "(X, r) \<in> rhs"
- shows "X = L (arden_variate X r rhs)"
-proof -
- have "[] \<notin> L r" using no_empty X_not_empty self_contained
- by (auto simp:no_EMPTY_rhs_def)
- hence ardens: "X = X;(L r) \<union> (L (del_x_paired rhs X)) \<longleftrightarrow> X = (L (del_x_paired rhs X)) ; (L r)\<star>"
- by (rule ardens_revised)
- have del_x: "X = X ; L r \<union> L (del_x_paired rhs X) \<longleftrightarrow> X = L rhs" using dist l_eq_r self_contained
- by (auto dest:del_x_paired_prop1)
- show ?thesis
- proof
- show "X \<subseteq> L (arden_variate X r rhs)"
- proof
- fix x
- assume "(1_1)": "x \<in> X" with l_eq_r ardens del_x
- show "x \<in> L (arden_variate X r rhs)"
- by (simp add:arden_variate_def seq_rhs_r_prop1)
- qed
- next
- show "L (arden_variate X r rhs) \<subseteq> X"
- proof
- fix x
- assume "(2_1)": "x \<in> L (arden_variate X r rhs)" with ardens del_x l_eq_r
- show "x \<in> X"
- by (simp add:arden_variate_def seq_rhs_r_prop1)
- qed
- qed
-qed
-
-text {* merge_rhs {(X1, r1), (x2, r2}, (x4, r4), \<dots>} {(x1, r1'), (x3, r3'), \<dots>} = {(x1, ALT r1 r1'}, (x2, r2), (x3, r3'), (x4, r4), \<dots>} *}
-definition
- merge_rhs :: "t_equa_rhs \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
-where
- "merge_rhs rhs rhs' \<equiv> {(X, r). (\<exists> r1 r2. ((X,r1) \<in> rhs \<and> (X, r2) \<in> rhs') \<and> r = ALT r1 r2) \<or>
- (\<exists> r1. (X, r1) \<in> rhs \<and> (\<not> (\<exists> r2. (X, r2) \<in> rhs')) \<and> r = r1) \<or>
- (\<exists> r2. (X, r2) \<in> rhs' \<and> (\<not> (\<exists> r1. (X, r1) \<in> rhs)) \<and> r = r2) }"
-
-
-text {* rhs_subst rhs X=xrhs r: substitude all occurence of X in rhs((X,r) \<in> rhs) with xrhs *}
-definition
- rhs_subst :: "t_equa_rhs \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
-where
- "rhs_subst rhs X xrhs r \<equiv> merge_rhs (del_x_paired rhs X) (seq_rhs_r xrhs r)"
-
-definition
- equas_subst_f :: "string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa \<Rightarrow> t_equa"
-where
- "equas_subst_f X xrhs equa \<equiv> let (Y, rhs) = equa in
- if (\<exists> r. (X, r) \<in> rhs)
- then (Y, rhs_subst rhs X xrhs (SOME r. (X, r) \<in> rhs))
- else equa"
-
-definition
- equas_subst :: "t_equas \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equas"
-where
- "equas_subst ES X xrhs \<equiv> del_x_paired (equas_subst_f X xrhs ` ES) X"
-
-lemma lang_seq_prop1:
- "x \<in> X ; L r \<Longrightarrow> x \<in> X ; (L r \<union> L r')"
-by (auto simp:lang_seq_def)
-
-lemma lang_seq_prop1':
- "x \<in> X; L r \<Longrightarrow> x \<in> X ; (L r' \<union> L r)"
-by (auto simp:lang_seq_def)
-
-lemma lang_seq_prop2:
- "x \<in> X; (L r \<union> L r') \<Longrightarrow> x \<in> X;L r \<or> x \<in> X;L r'"
-by (auto simp:lang_seq_def)
-
-lemma merge_rhs_prop1:
- shows "L (merge_rhs rhs rhs') = L rhs \<union> L rhs' "
-apply (auto simp add:merge_rhs_def L_def dest!:lang_seq_prop2 intro:lang_seq_prop1)
-apply (rule_tac x = X in exI, rule_tac x = r1 in exI, simp)
-apply (case_tac "\<exists> r2. (X, r2) \<in> rhs'")
-apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r r2" in exI, simp add:lang_seq_prop1)
-apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
-apply (case_tac "\<exists> r1. (X, r1) \<in> rhs")
-apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r1 r" in exI, simp add:lang_seq_prop1')
-apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
-done
-
-lemma no_EMPTY_rhss_imp_merge_no_EMPTY:
- "\<lbrakk>no_EMPTY_rhs rhs; no_EMPTY_rhs rhs'\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (merge_rhs rhs rhs')"
-apply (simp add:no_EMPTY_rhs_def merge_rhs_def)
-apply (clarify, (erule conjE | erule exE | erule disjE)+)
-by auto
-
-lemma distinct_rhs_prop:
- "\<lbrakk>distinct_rhs rhs; (X, r1) \<in> rhs; (X, r2) \<in> rhs\<rbrakk> \<Longrightarrow> r1 = r2"
-by (auto simp:distinct_rhs_def)
-
-lemma merge_rhs_prop2:
- assumes dist_rhs: "distinct_rhs rhs"
- and dist_rhs':"distinct_rhs rhs'"
- shows "distinct_rhs (merge_rhs rhs rhs')"
-apply (auto simp:merge_rhs_def distinct_rhs_def)
-using dist_rhs
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs'
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs'
-apply (drule distinct_rhs_prop, simp+)
-done
-
-lemma seq_rhs_r_holds_distinct:
- "distinct_rhs rhs \<Longrightarrow> distinct_rhs (seq_rhs_r rhs r)"
-by (auto simp:distinct_rhs_def seq_rhs_r_def)
-
-lemma seq_rhs_r_prop0:
- assumes l_eq_r: "X = L xrhs"
- shows "L (seq_rhs_r xrhs r) = X ; L r "
-using l_eq_r
-by (auto simp:seq_rhs_r_prop1)
-
-lemma rhs_subst_prop1:
- assumes l_eq_r: "X = L xrhs"
- and dist: "distinct_rhs rhs"
- shows "(X, r) \<in> rhs \<Longrightarrow> L rhs = L (rhs_subst rhs X xrhs r)"
-apply (simp add:rhs_subst_def merge_rhs_prop1)
-using l_eq_r
-apply (drule_tac r = r in seq_rhs_r_prop0, simp)
-using dist
-apply (auto dest:del_x_paired_prop1)
-done
-
-lemma del_x_paired_holds_distinct_rhs:
- "distinct_rhs rhs \<Longrightarrow> distinct_rhs (del_x_paired rhs x)"
-by (auto simp:distinct_rhs_def del_x_paired_def)
-
-lemma rhs_subst_holds_distinct_rhs:
- "\<lbrakk>distinct_rhs rhs; distinct_rhs xrhs\<rbrakk> \<Longrightarrow> distinct_rhs (rhs_subst rhs X xrhs r)"
-apply (drule_tac r = r and rhs = xrhs in seq_rhs_r_holds_distinct)
-apply (drule_tac x = X in del_x_paired_holds_distinct_rhs)
-by (auto dest:merge_rhs_prop2[where rhs = "del_x_paired rhs X"] simp:rhs_subst_def)
-
-section {* myhill-nerode theorem *}
-
-definition left_eq_cls :: "t_equas \<Rightarrow> (string set) set"
-where
- "left_eq_cls ES \<equiv> {X. \<exists> rhs. (X, rhs) \<in> ES} "
-
-definition right_eq_cls :: "t_equas \<Rightarrow> (string set) set"
-where
- "right_eq_cls ES \<equiv> {Y. \<exists> X rhs r. (X, rhs) \<in> ES \<and> (Y, r) \<in> rhs }"
-
-definition rhs_eq_cls :: "t_equa_rhs \<Rightarrow> (string set) set"
-where
- "rhs_eq_cls rhs \<equiv> {Y. \<exists> r. (Y, r) \<in> rhs}"
-
-definition ardenable :: "t_equa \<Rightarrow> bool"
-where
- "ardenable equa \<equiv> let (X, rhs) = equa in
- distinct_rhs rhs \<and> no_EMPTY_rhs rhs \<and> X = L rhs"
-
-text {*
- Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
-*}
-definition Inv :: "string set \<Rightarrow> t_equas \<Rightarrow> bool"
-where
- "Inv X ES \<equiv> finite ES \<and> (\<exists> rhs. (X, rhs) \<in> ES) \<and> distinct_equas ES \<and>
- (\<forall> X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs) \<and> X \<noteq> {} \<and> rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls ES))"
-
-text {*
- TCon: Termination Condition of the equation-system decreasion.
-*}
-definition TCon:: "'a set \<Rightarrow> bool"
-where
- "TCon ES \<equiv> card ES = 1"
-
-
-text {*
- The following is a iteration principle, and is the main framework for the proof:
- 1: We can form the initial equation-system using "equations" defined above, and prove it has invariance Inv by lemma "init_ES_satisfy_Inv";
- 2: We can decrease the number of the equation-system using ardens_lemma_revised and substitution ("equas_subst", defined above),
- and prove it holds the property "step" in "wf_iter" by lemma "iteration_step"
- and finally using property P and Q to prove the myhill-nerode theorem
-
-*}
-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
-
-
-(* ********************************* BEGIN: proving the initial equation-system satisfies Inv **************************************** *)
-
-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 L_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 L_def lang_seq_def)
- thus ?thesis using X_in_equas False empty "(1)"
- unfolding equations_def equation_rhs_def by (auto simp:L_def)
- 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 L_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:L_def 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 simp:L_def)
- 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 L_def lang_seq_def)
- qed
- qed
-qed
-
-lemma finite_CT_chars:
- "finite {CHAR c |c. Xa-c\<rightarrow>X}"
-by (auto)
-
-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_CT_chars)+
-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)
-
-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
-
-
-(* ********************************* END: proving the initial equation-system satisfies Inv **************************************** *)
-
-
-(* ***************************** 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 del_x_paired_del_only_x':
- "(X, rhs) \<in> del_x_paired ES Y \<Longrightarrow> X \<noteq> Y \<and> (X, rhs) \<in> ES"
-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)
- 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:Inv_def)
- 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
-
-(* ****************************** 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 L_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 add:L_def)
- 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:L_def 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:L_def lang_seq_def)
- thus ?thesis by blast
- qed
-qed
-
-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
-
-end
-
-
-
-
-
-
-
-
-
-
+theory MyhillNerode
+ imports "Main"
+begin
+
+text {* sequential composition of languages *}
+
+definition
+ lang_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}"
+
+lemma lang_seq_empty:
+ shows "{[]} ; L = L"
+ and "L ; {[]} = L"
+unfolding lang_seq_def by auto
+
+lemma lang_seq_null:
+ shows "{} ; L = {}"
+ and "L ; {} = {}"
+unfolding lang_seq_def by auto
+
+lemma lang_seq_append:
+ assumes a: "s1 \<in> L1"
+ and b: "s2 \<in> L2"
+ shows "s1@s2 \<in> L1 ; L2"
+unfolding lang_seq_def
+using a b by auto
+
+lemma lang_seq_union:
+ shows "(L1 \<union> L2); L3 = (L1; L3) \<union> (L2; L3)"
+ and "L1; (L2 \<union> L3) = (L1; L2) \<union> (L1; L3)"
+unfolding lang_seq_def by auto
+
+lemma lang_seq_assoc:
+ shows "(L1 ; L2) ; L3 = L1 ; (L2 ; L3)"
+by (simp add: lang_seq_def Collect_def mem_def expand_fun_eq)
+ (metis append_assoc)
+
+lemma lang_seq_minus:
+ shows "(L1; L2) - {[]} =
+ (if [] \<in> L1 then L2 - {[]} else {}) \<union>
+ (if [] \<in> L2 then L1 - {[]} else {}) \<union> ((L1 - {[]}); (L2 - {[]}))"
+apply(auto simp add: lang_seq_def)
+apply(metis mem_def self_append_conv)
+apply(metis mem_def self_append_conv2)
+apply(metis mem_def self_append_conv2)
+apply(metis mem_def self_append_conv)
+done
+
+section {* Kleene star for languages defined as least fixed point *}
+
+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 lang_star_empty:
+ shows "{}\<star> = {[]}"
+by (auto elim: Star.cases)
+
+lemma lang_star_cases:
+ shows "L\<star> = {[]} \<union> L ; L\<star>"
+proof
+ { fix x
+ have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L ; L\<star>"
+ unfolding lang_seq_def
+ by (induct rule: Star.induct) (auto)
+ }
+ then show "L\<star> \<subseteq> {[]} \<union> L ; L\<star>" by auto
+next
+ show "{[]} \<union> L ; L\<star> \<subseteq> L\<star>"
+ unfolding lang_seq_def by auto
+qed
+
+lemma lang_star_cases':
+ shows "L\<star> = {[]} \<union> L\<star> ; L"
+proof
+ { fix x
+ have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L\<star> ; L"
+ unfolding lang_seq_def
+ apply (induct rule: Star.induct)
+ apply simp
+ apply simp
+ apply (erule disjE)
+ apply (auto)[1]
+ apply (erule exE | erule conjE)+
+ apply (rule disjI2)
+ apply (rule_tac x = "s1 @ s1a" in exI)
+ by auto
+ }
+ then show "L\<star> \<subseteq> {[]} \<union> L\<star> ; L" by auto
+next
+ show "{[]} \<union> L\<star> ; L \<subseteq> L\<star>"
+ unfolding lang_seq_def
+ apply auto
+ apply (erule Star.induct)
+ apply auto
+ apply (drule step[of _ _ "[]"])
+ by (auto intro:start)
+qed
+
+lemma lang_star_simple:
+ shows "L \<subseteq> L\<star>"
+by (subst lang_star_cases)
+ (auto simp only: lang_seq_def)
+
+lemma lang_star_prop0_aux:
+ "s2 \<in> L\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L \<longrightarrow> (\<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4)"
+apply (erule Star.induct)
+apply (clarify, rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
+apply (clarify, drule_tac x = s1 in spec)
+apply (drule mp, simp, clarify)
+apply (rule_tac x = "s1a @ s3" in exI, rule_tac x = s4 in exI)
+by auto
+
+lemma lang_star_prop0:
+ "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> \<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4"
+by (auto dest:lang_star_prop0_aux)
+
+lemma lang_star_prop1:
+ assumes asm: "L1; L2 \<subseteq> L2"
+ shows "L1\<star>; L2 \<subseteq> L2"
+proof -
+ { fix s1 s2
+ assume minor: "s2 \<in> L2"
+ assume major: "s1 \<in> L1\<star>"
+ then have "s1@s2 \<in> L2"
+ proof(induct rule: Star.induct)
+ case start
+ show "[]@s2 \<in> L2" using minor by simp
+ next
+ case (step s1 s1')
+ have "s1 \<in> L1" by fact
+ moreover
+ have "s1'@s2 \<in> L2" by fact
+ ultimately have "s1@(s1'@s2) \<in> L1; L2" by (auto simp add: lang_seq_def)
+ with asm have "s1@(s1'@s2) \<in> L2" by auto
+ then show "(s1@s1')@s2 \<in> L2" by simp
+ qed
+ }
+ then show "L1\<star>; L2 \<subseteq> L2" by (auto simp add: lang_seq_def)
+qed
+
+lemma lang_star_prop2_aux:
+ "s2 \<in> L2\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L1 \<and> L1 ; L2 \<subseteq> L1 \<longrightarrow> s1 @ s2 \<in> L1"
+apply (erule Star.induct, simp)
+apply (clarify, drule_tac x = "s1a @ s1" in spec)
+by (auto simp:lang_seq_def)
+
+lemma lang_star_prop2:
+ "L1; L2 \<subseteq> L1 \<Longrightarrow> L1 ; L2\<star> \<subseteq> L1"
+by (auto dest!:lang_star_prop2_aux simp:lang_seq_def)
+
+lemma lang_star_seq_subseteq:
+ shows "L ; L\<star> \<subseteq> L\<star>"
+using lang_star_cases by blast
+
+lemma lang_star_double:
+ shows "L\<star>; L\<star> = L\<star>"
+proof
+ show "L\<star> ; L\<star> \<subseteq> L\<star>"
+ using lang_star_prop1 lang_star_seq_subseteq by blast
+next
+ have "L\<star> \<subseteq> L\<star> \<union> L\<star>; (L; L\<star>)" by auto
+ also have "\<dots> = L\<star>;{[]} \<union> L\<star>; (L; L\<star>)" by (simp add: lang_seq_empty)
+ also have "\<dots> = L\<star>; ({[]} \<union> L; L\<star>)" by (simp only: lang_seq_union)
+ also have "\<dots> = L\<star>; L\<star>" using lang_star_cases by simp
+ finally show "L\<star> \<subseteq> L\<star> ; L\<star>" by simp
+qed
+
+lemma lang_star_seq_subseteq':
+ shows "L\<star>; L \<subseteq> L\<star>"
+proof -
+ have "L \<subseteq> L\<star>" by (rule lang_star_simple)
+ then have "L\<star>; L \<subseteq> L\<star>; L\<star>" by (auto simp add: lang_seq_def)
+ then show "L\<star>; L \<subseteq> L\<star>" using lang_star_double by blast
+qed
+
+lemma
+ shows "L\<star> \<subseteq> L\<star>\<star>"
+by (rule lang_star_simple)
+
+section {* tricky section *}
+
+lemma k1:
+ assumes b: "s \<in> L\<star>"
+ and a: "s \<noteq> []"
+ shows "s \<in> (L - {[]}); L\<star>"
+using b a
+apply(induct rule: Star.induct)
+apply(simp)
+apply(case_tac "s1=[]")
+apply(simp)
+apply(simp add: lang_seq_def)
+apply(blast)
+done
+
+section {* (relies on lemma k1) *}
+
+lemma zzz:
+ shows "{s. c#s \<in> L1\<star>} = {s. c#s \<in> L1} ; (L1\<star>)"
+apply(auto simp add: lang_seq_def Cons_eq_append_conv)
+apply(drule k1)
+apply(auto)[1]
+apply(auto simp add: lang_seq_def)[1]
+apply(rule_tac x="tl s1" in exI)
+apply(rule_tac x="s2" in exI)
+apply(auto)[1]
+apply(auto simp add: Cons_eq_append_conv)[2]
+apply(drule lang_seq_append)
+apply(assumption)
+apply(rotate_tac 1)
+apply(drule rev_subsetD)
+apply(rule lang_star_seq_subseteq)
+apply(simp)
+done
+
+
+
+section {* regular expressions *}
+
+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
+ Ls :: "rexp set \<Rightarrow> string set"
+where
+ "Ls R = (\<Union>r\<in>R. (L r))"
+
+lemma Ls_union:
+ "Ls (R1 \<union> R2) = (Ls R1) \<union> (Ls R2)"
+unfolding Ls_def by auto
+
+text {* helper function for termination proofs *}
+fun
+ Left :: "rexp \<Rightarrow> rexp"
+where
+ "Left (SEQ r1 r2) = r1"
+
+text {* dagger function *}
+
+function
+ dagger :: "rexp \<Rightarrow> char \<Rightarrow> rexp list" ("_ \<dagger> _")
+where
+ c1: "(NULL \<dagger> c) = []"
+| c2: "(EMPTY) \<dagger> c = []"
+| c3: "(CHAR c') \<dagger> c = (if c = c' then [EMPTY] else [])"
+| c4: "(ALT r1 r2) \<dagger> c = r1 \<dagger> c @ r2 \<dagger> c"
+| c5: "(SEQ NULL r2) \<dagger> c = []"
+| c6: "(SEQ EMPTY r2) \<dagger> c = r2 \<dagger> c"
+| c7: "(SEQ (CHAR c') r2) \<dagger> c = (if c = c' then [r2] else [])"
+| c8: "(SEQ (SEQ r11 r12) r2) \<dagger> c = (SEQ r11 (SEQ r12 r2)) \<dagger> c"
+| c9: "(SEQ (ALT r11 r12) r2) \<dagger> c = (SEQ r11 r2) \<dagger> c @ (SEQ r12 r2) \<dagger> c"
+| c10: "(SEQ (STAR r1) r2) \<dagger> c = r2 \<dagger> c @ [SEQ r' (SEQ (STAR r1) r2). r' \<leftarrow> r1 \<dagger> c]"
+| c11: "(STAR r) \<dagger> c = [SEQ r' (STAR r) . r' \<leftarrow> r \<dagger> c]"
+by (pat_completeness) (auto)
+
+termination dagger
+ by (relation "measures [\<lambda>(r, c). size r, \<lambda>(r, c). size (Left r)]") (simp_all)
+
+lemma dagger_correctness:
+ "Ls (set r \<dagger> c) = {s. c#s \<in> L r}"
+proof (induct rule: dagger.induct)
+ case (1 c)
+ show "Ls (set NULL \<dagger> c) = {s. c#s \<in> L NULL}" by (simp add: Ls_def)
+next
+ case (2 c)
+ show "Ls (set EMPTY \<dagger> c) = {s. c#s \<in> L EMPTY}" by (simp add: Ls_def)
+next
+ case (3 c' c)
+ show "Ls (set CHAR c' \<dagger> c) = {s. c#s \<in> L (CHAR c')}" by (simp add: Ls_def)
+next
+ case (4 r1 r2 c)
+ have ih1: "Ls (set r1 \<dagger> c) = {s. c#s \<in> L r1}" by fact
+ have ih2: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
+ show "Ls (set ALT r1 r2 \<dagger> c) = {s. c#s \<in> L (ALT r1 r2)}"
+ by (simp add: Ls_union ih1 ih2 Collect_disj_eq)
+next
+ case (5 r2 c)
+ show "Ls (set SEQ NULL r2 \<dagger> c) = {s. c#s \<in> L (SEQ NULL r2)}" by (simp add: Ls_def lang_seq_null)
+next
+ case (6 r2 c)
+ have ih: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
+ show "Ls (set SEQ EMPTY r2 \<dagger> c) = {s. c#s \<in> L (SEQ EMPTY r2)}"
+ by (simp add: ih lang_seq_empty)
+next
+ case (7 c' r2 c)
+ show "Ls (set SEQ (CHAR c') r2 \<dagger> c) = {s. c#s \<in> L (SEQ (CHAR c') r2)}"
+ by (simp add: Ls_def lang_seq_def)
+next
+ case (8 r11 r12 r2 c)
+ have ih: "Ls (set SEQ r11 (SEQ r12 r2) \<dagger> c) = {s. c#s \<in> L (SEQ r11 (SEQ r12 r2))}" by fact
+ show "Ls (set SEQ (SEQ r11 r12) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (SEQ r11 r12) r2)}"
+ by (simp add: ih lang_seq_assoc)
+next
+ case (9 r11 r12 r2 c)
+ have ih1: "Ls (set SEQ r11 r2 \<dagger> c) = {s. c#s \<in> L (SEQ r11 r2)}" by fact
+ have ih2: "Ls (set SEQ r12 r2 \<dagger> c) = {s. c#s \<in> L (SEQ r12 r2)}" by fact
+ show "Ls (set SEQ (ALT r11 r12) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (ALT r11 r12) r2)}"
+ by (simp add: Ls_union ih1 ih2 lang_seq_union Collect_disj_eq)
+next
+ case (10 r1 r2 c)
+ have ih2: "Ls (set r2 \<dagger> c) = {s. c#s \<in> L r2}" by fact
+ have ih1: "Ls (set r1 \<dagger> c) = {s. c#s \<in> L r1}" by fact
+ have "Ls (set SEQ (STAR r1) r2 \<dagger> c) = Ls (set r2 \<dagger> c) \<union> (Ls (set r1 \<dagger> c); ((L r1)\<star> ; L r2))"
+ by (auto simp add: lang_seq_def Ls_def)
+ also have "\<dots> = {s. c#s \<in> L r2} \<union> ({s. c#s \<in> L r1} ; ((L r1)\<star> ; L r2))" using ih1 ih2 by simp
+ also have "\<dots> = {s. c#s \<in> L r2} \<union> ({s. c#s \<in> L r1} ; (L r1)\<star>) ; L r2" by (simp add: lang_seq_assoc)
+ also have "\<dots> = {s. c#s \<in> L r2} \<union> {s. c#s \<in> (L r1)\<star>} ; L r2" by (simp add: zzz)
+ also have "\<dots> = {s. c#s \<in> L r2} \<union> {s. c#s \<in> (L r1)\<star> ; L r2}"
+ by (auto simp add: lang_seq_def Cons_eq_append_conv)
+ also have "\<dots> = {s. c#s \<in> (L r1)\<star> ; L r2}"
+ by (force simp add: lang_seq_def)
+ finally show "Ls (set SEQ (STAR r1) r2 \<dagger> c) = {s. c#s \<in> L (SEQ (STAR r1) r2)}" by simp
+next
+ case (11 r c)
+ have ih: "Ls (set r \<dagger> c) = {s. c#s \<in> L r}" by fact
+ have "Ls (set (STAR r) \<dagger> c) = Ls (set r \<dagger> c) ; (L r)\<star>"
+ by (auto simp add: lang_seq_def Ls_def)
+ also have "\<dots> = {s. c#s \<in> L r} ; (L r)\<star>" using ih by simp
+ also have "\<dots> = {s. c#s \<in> (L r)\<star>}" using zzz by simp
+ finally show "Ls (set (STAR r) \<dagger> c) = {s. c#s \<in> L (STAR r)}" by simp
+qed
+
+
+text {* matcher function (based on the "list"-dagger function) *}
+fun
+ first_True :: "bool list \<Rightarrow> bool"
+where
+ "first_True [] = False"
+| "first_True (x#xs) = (if x then True else first_True xs)"
+
+lemma not_first_True[simp]:
+ shows "(\<not>(first_True xs)) = (\<forall>x \<in> set xs. \<not>x)"
+by (induct xs) (auto)
+
+lemma first_True:
+ shows "(first_True xs) = (\<exists>x \<in> set xs. x)"
+by (induct xs) (auto)
+
+text {* matcher function *}
+
+function
+ matcher :: "rexp \<Rightarrow> string \<Rightarrow> bool" ("_ ! _")
+where
+ "NULL ! s = False"
+| "EMPTY ! s = (s =[])"
+| "CHAR c ! s = (s = [c])"
+| "ALT r1 r2 ! s = (r1 ! s \<or> r2 ! s)"
+| "STAR r ! [] = True"
+| "STAR r ! c#s = first_True [SEQ (r') (STAR r) ! s. r' \<leftarrow> r \<dagger> c]"
+| "SEQ r1 r2 ! [] = (r1 ! [] \<and> r2 ! [])"
+| "SEQ NULL r2 ! (c#s) = False"
+| "SEQ EMPTY r2 ! (c#s) = (r2 ! c#s)"
+| "SEQ (CHAR c') r2 ! (c#s) = (if c'=c then r2 ! s else False)"
+| "SEQ (SEQ r11 r12) r2 ! (c#s) = (SEQ r11 (SEQ r12 r2) ! c#s)"
+| "SEQ (ALT r11 r12) r2 ! (c#s) = ((SEQ r11 r2) ! (c#s) \<or> (SEQ r12 r2) ! (c#s))"
+| "SEQ (STAR r1) r2 ! (c#s) = (r2 ! (c#s) \<or> first_True [SEQ r' (SEQ (STAR r1) r2) ! s. r' \<leftarrow> r1 \<dagger> c])"
+by (pat_completeness) (auto)
+
+termination matcher
+ by(relation "measures [\<lambda>(r,s). length s, \<lambda>(r,s). size r, \<lambda>(r,s). size (Left r)]") (simp_all)
+
+text {* positive correctness of the matcher *}
+lemma matcher1:
+ shows "r ! s \<Longrightarrow> s \<in> L r"
+proof (induct rule: matcher.induct)
+ case (1 s)
+ have "NULL ! s" by fact
+ then show "s \<in> L NULL" by simp
+next
+ case (2 s)
+ have "EMPTY ! s" by fact
+ then show "s \<in> L EMPTY" by simp
+next
+ case (3 c s)
+ have "CHAR c ! s" by fact
+ then show "s \<in> L (CHAR c)" by simp
+next
+ case (4 r1 r2 s)
+ have ih1: "r1 ! s \<Longrightarrow> s \<in> L r1" by fact
+ have ih2: "r2 ! s \<Longrightarrow> s \<in> L r2" by fact
+ have "ALT r1 r2 ! s" by fact
+ with ih1 ih2 show "s \<in> L (ALT r1 r2)" by auto
+next
+ case (5 r)
+ have "STAR r ! []" by fact
+ then show "[] \<in> L (STAR r)" by auto
+next
+ case (6 r c s)
+ have ih1: "\<And>rx. \<lbrakk>rx \<in> set r \<dagger> c; SEQ rx (STAR r) ! s\<rbrakk> \<Longrightarrow> s \<in> L (SEQ rx (STAR r))" by fact
+ have as: "STAR r ! c#s" by fact
+ then obtain r' where imp1: "r' \<in> set r \<dagger> c" and imp2: "SEQ r' (STAR r) ! s"
+ by (auto simp add: first_True)
+ from imp2 imp1 have "s \<in> L (SEQ r' (STAR r))" using ih1 by simp
+ then have "s \<in> L r' ; (L r)\<star>" by simp
+ then have "s \<in> Ls (set r \<dagger> c) ; (L r)\<star>" using imp1 by (auto simp add: Ls_def lang_seq_def)
+ then have "s \<in> {s. c#s \<in> L r} ; (L r)\<star>" by (auto simp add: dagger_correctness)
+ then have "s \<in> {s. c#s \<in> (L r)\<star>}" by (simp add: zzz)
+ then have "c#s \<in> {[c]}; {s. c#s \<in> (L r)\<star>}" by (auto simp add: lang_seq_def)
+ then have "c#s \<in> (L r)\<star>" by (auto simp add: lang_seq_def)
+ then show "c#s \<in> L (STAR r)" by simp
+next
+ case (7 r1 r2)
+ have ih1: "r1 ! [] \<Longrightarrow> [] \<in> L r1" by fact
+ have ih2: "r2 ! [] \<Longrightarrow> [] \<in> L r2" by fact
+ have as: "SEQ r1 r2 ! []" by fact
+ then have "r1 ! [] \<and> r2 ! []" by simp
+ then show "[] \<in> L (SEQ r1 r2)" using ih1 ih2 by (simp add: lang_seq_def)
+next
+ case (8 r2 c s)
+ have "SEQ NULL r2 ! c#s" by fact
+ then show "c#s \<in> L (SEQ NULL r2)" by simp
+next
+ case (9 r2 c s)
+ have ih1: "r2 ! c#s \<Longrightarrow> c#s \<in> L r2" by fact
+ have "SEQ EMPTY r2 ! c#s" by fact
+ then show "c#s \<in> L (SEQ EMPTY r2)" using ih1 by (simp add: lang_seq_def)
+next
+ case (10 c' r2 c s)
+ have ih1: "\<lbrakk>c' = c; r2 ! s\<rbrakk> \<Longrightarrow> s \<in> L r2" by fact
+ have "SEQ (CHAR c') r2 ! c#s" by fact
+ then show "c#s \<in> L (SEQ (CHAR c') r2)"
+ using ih1 by (auto simp add: lang_seq_def split: if_splits)
+next
+ case (11 r11 r12 r2 c s)
+ have ih1: "SEQ r11 (SEQ r12 r2) ! c#s \<Longrightarrow> c#s \<in> L (SEQ r11 (SEQ r12 r2))" by fact
+ have "SEQ (SEQ r11 r12) r2 ! c#s" by fact
+ then have "c#s \<in> L (SEQ r11 (SEQ r12 r2))" using ih1 by simp
+ then show "c#s \<in> L (SEQ (SEQ r11 r12) r2)" by (simp add: lang_seq_assoc)
+next
+ case (12 r11 r12 r2 c s)
+ have ih1: "SEQ r11 r2 ! c#s \<Longrightarrow> c#s \<in> L (SEQ r11 r2)" by fact
+ have ih2: "SEQ r12 r2 ! c#s \<Longrightarrow> c#s \<in> L (SEQ r12 r2)" by fact
+ have "SEQ (ALT r11 r12) r2 ! c#s" by fact
+ then show "c#s \<in> L (SEQ (ALT r11 r12) r2)"
+ using ih1 ih2 by (auto simp add: lang_seq_union)
+next
+ case (13 r1 r2 c s)
+ have ih1: "r2 ! c#s \<Longrightarrow> c#s \<in> L r2" by fact
+ have ih2: "\<And>r'. \<lbrakk>r' \<in> set r1 \<dagger> c; SEQ r' (SEQ (STAR r1) r2) ! s\<rbrakk> \<Longrightarrow>
+ s \<in> L (SEQ r' (SEQ (STAR r1) r2))" by fact
+ have "SEQ (STAR r1) r2 ! c#s" by fact
+ then have "(r2 ! c#s) \<or> (\<exists>r' \<in> set r1 \<dagger> c. SEQ r' (SEQ (STAR r1) r2) ! s)" by (auto simp add: first_True)
+ moreover
+ { assume "r2 ! c#s"
+ with ih1 have "c#s \<in> L r2" by simp
+ then have "c # s \<in> L r1\<star> ; L r2"
+ by (auto simp add: lang_seq_def)
+ then have "c#s \<in> L (SEQ (STAR r1) r2)" by simp
+ }
+ moreover
+ { assume "\<exists>r' \<in> set r1 \<dagger> c. SEQ r' (SEQ (STAR r1) r2) ! s"
+ then obtain r' where imp1: "r' \<in> set r1 \<dagger> c" and imp2: "SEQ r' (SEQ (STAR r1) r2) ! s" by blast
+ from imp2 imp1 have "s \<in> L (SEQ r' (SEQ (STAR r1) r2))" using ih2 by simp
+ then have "s \<in> L r' ; ((L r1)\<star> ; L r2)" by simp
+ then have "s \<in> Ls (set r1 \<dagger> c) ; ((L r1)\<star> ; L r2)" using imp1 by (auto simp add: Ls_def lang_seq_def)
+ then have "s \<in> {s. c#s \<in> L r1} ; ((L r1)\<star> ; L r2)" by (simp add: dagger_correctness)
+ then have "s \<in> ({s. c#s \<in> L r1} ; (L r1)\<star>) ; L r2" by (simp add: lang_seq_assoc)
+ then have "s \<in> {s. c#s \<in> (L r1)\<star>} ; L r2" by (simp add: zzz)
+ then have "c#s \<in> {[c]}; ({s. c#s \<in> (L r1)\<star>}; L r2)" by (auto simp add: lang_seq_def)
+ then have "c#s \<in> ({[c]}; {s. c#s \<in> (L r1)\<star>}) ; L r2" by (simp add: lang_seq_assoc)
+ then have "c#s \<in> (L r1)\<star>; L r2" by (auto simp add: lang_seq_def)
+ then have "c#s \<in> L (SEQ (STAR r1) r2)" by simp
+ }
+ ultimately show "c#s \<in> L (SEQ (STAR r1) r2)" by blast
+qed
+
+text {* negative correctness of the matcher *}
+lemma matcher2:
+ shows "\<not> r ! s \<Longrightarrow> s \<notin> L r"
+proof (induct rule: matcher.induct)
+ case (1 s)
+ have "\<not> NULL ! s" by fact
+ then show "s \<notin> L NULL" by simp
+next
+ case (2 s)
+ have "\<not> EMPTY ! s" by fact
+ then show "s \<notin> L EMPTY" by simp
+next
+ case (3 c s)
+ have "\<not> CHAR c ! s" by fact
+ then show "s \<notin> L (CHAR c)" by simp
+next
+ case (4 r1 r2 s)
+ have ih2: "\<not> r1 ! s \<Longrightarrow> s \<notin> L r1" by fact
+ have ih4: "\<not> r2 ! s \<Longrightarrow> s \<notin> L r2" by fact
+ have "\<not> ALT r1 r2 ! s" by fact
+ then show "s \<notin> L (ALT r1 r2)" by (simp add: ih2 ih4)
+next
+ case (5 r)
+ have "\<not> STAR r ! []" by fact
+ then show "[] \<notin> L (STAR r)" by simp
+next
+ case (6 r c s)
+ have ih: "\<And>rx. \<lbrakk>rx \<in> set r \<dagger> c; \<not>SEQ rx (STAR r) ! s\<rbrakk> \<Longrightarrow> s \<notin> L (SEQ rx (STAR r))" by fact
+ have as: "\<not> STAR r ! c#s" by fact
+ then have "\<forall>r'\<in> set r \<dagger> c. \<not> (SEQ r' (STAR r) ! s)" by simp
+ then have "\<forall>r'\<in> set r \<dagger> c. s \<notin> L (SEQ r' (STAR r))" using ih by auto
+ then have "\<forall>r'\<in> set r \<dagger> c. s \<notin> L r' ; ((L r)\<star>)" by simp
+ then have "s \<notin> (Ls (set r \<dagger> c)) ; ((L r)\<star>)" by (auto simp add: Ls_def lang_seq_def)
+ then have "s \<notin> {s. c#s \<in> L r} ; ((L r)\<star>)" by (simp add: dagger_correctness)
+ then have "s \<notin> {s. c#s \<in> (L r)\<star>}" by (simp add: zzz)
+ then have "c#s \<notin> {[c]} ; {s. c#s \<in> (L r)\<star>}" by (auto simp add: lang_seq_assoc lang_seq_def)
+ then have "c#s \<notin> (L r)\<star>" by (simp add: lang_seq_def)
+ then show "c#s \<notin> L (STAR r)" by simp
+next
+ case (7 r1 r2)
+ have ih2: "\<not> r1 ! [] \<Longrightarrow> [] \<notin> L r1" by fact
+ have ih4: "\<not> r2 ! [] \<Longrightarrow> [] \<notin> L r2" by fact
+ have "\<not> SEQ r1 r2 ! []" by fact
+ then have "\<not> r1 ! [] \<or> \<not> r2 ! []" by simp
+ then show "[] \<notin> L (SEQ r1 r2)" using ih2 ih4
+ by (auto simp add: lang_seq_def)
+next
+ case (8 r2 c s)
+ have "\<not> SEQ NULL r2 ! c#s" by fact
+ then show "c#s \<notin> L (SEQ NULL r2)" by (simp add: lang_seq_null)
+next
+ case (9 r2 c s)
+ have ih1: "\<not> r2 ! c#s \<Longrightarrow> c#s \<notin> L r2" by fact
+ have "\<not> SEQ EMPTY r2 ! c#s" by fact
+ then show "c#s \<notin> L (SEQ EMPTY r2)"
+ using ih1 by (simp add: lang_seq_def)
+next
+ case (10 c' r2 c s)
+ have ih2: "\<lbrakk>c' = c; \<not>r2 ! s\<rbrakk> \<Longrightarrow> s \<notin> L r2" by fact
+ have "\<not> SEQ (CHAR c') r2 ! c#s" by fact
+ then show "c#s \<notin> L (SEQ (CHAR c') r2)"
+ using ih2 by (auto simp add: lang_seq_def)
+next
+ case (11 r11 r12 r2 c s)
+ have ih2: "\<not> SEQ r11 (SEQ r12 r2) ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r11 (SEQ r12 r2))" by fact
+ have "\<not> SEQ (SEQ r11 r12) r2 ! c#s" by fact
+ then show "c#s \<notin> L (SEQ (SEQ r11 r12) r2)"
+ using ih2 by (auto simp add: lang_seq_def)
+next
+ case (12 r11 r12 r2 c s)
+ have ih2: "\<not> SEQ r11 r2 ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r11 r2)" by fact
+ have ih4: "\<not> SEQ r12 r2 ! c#s \<Longrightarrow> c#s \<notin> L (SEQ r12 r2)" by fact
+ have "\<not> SEQ (ALT r11 r12) r2 ! c#s" by fact
+ then show " c#s \<notin> L (SEQ (ALT r11 r12) r2)"
+ using ih2 ih4 by (simp add: lang_seq_union)
+next
+ case (13 r1 r2 c s)
+ have ih1: "\<not>r2 ! c#s \<Longrightarrow> c#s \<notin> L r2" by fact
+ have ih2: "\<And>rx. \<lbrakk>rx \<in> set r1 \<dagger> c; \<not> SEQ rx (SEQ (STAR r1) r2) ! s\<rbrakk>
+ \<Longrightarrow> s \<notin> L (SEQ rx (SEQ (STAR r1) r2))" by fact
+ have as: "\<not> SEQ (STAR r1) r2 ! c#s" by fact
+ then have as1: "\<not>r2 ! c#s" and as2: "\<forall>r1'\<in>set r1 \<dagger> c. \<not> SEQ r1' (SEQ (STAR r1) r2) ! s" by simp_all
+ from as1 have bs1: "c#s \<notin> L r2" using ih1 by simp
+ from as2 have "\<forall>r1'\<in>set r1 \<dagger> c. \<not> SEQ r1' (SEQ (STAR r1) r2) ! s" by simp
+ then have "\<forall>r1'\<in>set r1 \<dagger> c. s \<notin> L (SEQ r1' (SEQ (STAR r1) r2))" using ih2 by simp
+ then have "\<forall>r1'\<in>set r1 \<dagger> c. s \<notin> L r1'; ((L r1)\<star>; L r2)" by simp
+ then have "s \<notin> (Ls (set r1 \<dagger> c)) ; ((L r1)\<star>; L r2)" by (auto simp add: Ls_def lang_seq_def)
+ then have "s \<notin> {s. c#s \<in> L r1} ; ((L r1)\<star>; L r2)" by (simp add: dagger_correctness)
+ then have "s \<notin> ({s. c#s \<in> L r1} ; (L r1)\<star>); L r2" by (simp add: lang_seq_assoc)
+ then have "s \<notin> {s. c#s \<in> (L r1)\<star>}; L r2" by (simp add: zzz)
+ then have "c#s \<notin> {[c]}; ({s. c#s \<in> (L r1)\<star>}; L r2)" by (auto simp add: lang_seq_def)
+ then have "c#s \<notin> (L r1)\<star>; L r2" using bs1 by (auto simp add: lang_seq_def Cons_eq_append_conv)
+ then show "c#s \<notin> L (SEQ (STAR r1) r2)" by simp
+qed
+
+section {* Questions *}
+
+text {*
+ - Why was the key lemma k1 omitted; is there an easy, non-induction
+ way for obtaining this property?
+ - Why was False included in the definition of the STAR-clause in
+ the matcher? Has this something to do with executing the code?
+
+*}
+
+section {* Code *}
+
+export_code dagger in SML module_name Dagger file -
+export_code matcher in SML module_name Dagger file -
+
+section {* Examples *}
+
+text {* since now everything is based on lists, the evaluation is quite fast *}
+
+value "NULL ! []"
+value "(CHAR (CHR ''a'')) ! [CHR ''a'']"
+value "((CHAR a) ! [a,a])"
+value "(STAR (CHAR a)) ! []"
+value "(STAR (CHAR a)) ! [a,a]"
+value "(SEQ (CHAR (CHR ''a'')) (SEQ (STAR (CHAR (CHR ''b''))) (CHAR (CHR ''c'')))) ! ''abbbbc''"
+value "(SEQ (CHAR (CHR ''a'')) (SEQ (STAR (CHAR (CHR ''b''))) (CHAR (CHR ''c'')))) ! ''abbcbbc''"
+
+section {* Slind et al's matcher based on derivatives *}
+
+fun
+ nullable :: "rexp \<Rightarrow> bool"
+where
+ "nullable (NULL) = False"
+| "nullable (EMPTY) = True"
+| "nullable (CHAR c) = False"
+| "nullable (ALT r1 r2) = ((nullable r1) \<or> (nullable r2))"
+| "nullable (SEQ r1 r2) = ((nullable r1) \<and> (nullable r2))"
+| "nullable (STAR r) = True"
+
+lemma nullable:
+ shows "([] \<in> L r) = nullable r"
+by (induct r)
+ (auto simp add: lang_seq_def)
+
+fun
+ der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+ "der c (NULL) = NULL"
+| "der c (EMPTY) = NULL"
+| "der c (CHAR c') = (if c=c' then EMPTY else NULL)"
+| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
+| "der c (SEQ r1 r2) = ALT (SEQ (der c r1) r2) (if nullable r1 then der c r2 else NULL)"
+| "der c (STAR r) = SEQ (der c r) (STAR r)"
+
+lemma k2:
+ assumes b: "s \<in> L1\<star>"
+ and a: "s \<noteq> []"
+ shows "s \<in> (L1; (L1\<star>))"
+using b a
+apply(induct rule: Star.induct)
+apply(simp)
+apply(case_tac "s1=[]")
+apply(simp)
+apply(simp add: lang_seq_def)
+apply(blast)
+done
+
+
+lemma der_correctness:
+ shows "(s \<in> L (der c r)) = ((c#s) \<in> L r)"
+proof (induct r arbitrary: s)
+ case (NULL s)
+ show "(s \<in> L (der c NULL)) = (c#s \<in> L NULL)" by simp
+next
+ case (EMPTY s)
+ show "(s \<in> L (der c EMPTY)) = (c#s \<in> L EMPTY)" by simp
+next
+ case (CHAR c' s)
+ show "(s \<in> L (der c (CHAR c'))) = (c#s \<in> L (CHAR c'))" by simp
+next
+ case (SEQ r1 r2 s)
+ have ih1: "\<And>s. (s \<in> L (der c r1)) = (c#s \<in> L r1)" by fact
+ have ih2: "\<And>s. (s \<in> L (der c r2)) = (c#s \<in> L r2)" by fact
+ show "(s \<in> L (der c (SEQ r1 r2))) = (c#s \<in> L (SEQ r1 r2))"
+ using ih1 ih2
+ by (auto simp add: nullable[symmetric] lang_seq_def Cons_eq_append_conv)
+next
+ case (ALT r1 r2 s)
+ have ih1: "\<And>s. (s \<in> L (der c r1)) = (c#s \<in> L r1)" by fact
+ have ih2: "\<And>s. (s \<in> L (der c r2)) = (c#s \<in> L r2)" by fact
+ show "(s \<in> L (der c (ALT r1 r2))) = (c#s \<in> L (ALT r1 r2))"
+ using ih1 ih2 by (auto simp add: lang_seq_def)
+next
+ case (STAR r s)
+ have ih1: "\<And>s. (s \<in> L (der c r)) = (c#s \<in> L r)" by fact
+ show "(s \<in> L (der c (STAR r))) = (c#s \<in> L (STAR r))"
+ using ih1
+ apply(simp)
+ apply(auto simp add: lang_seq_def)
+ apply(drule lang_seq_append)
+ apply(assumption)
+ apply(simp)
+ apply(subst lang_star_cases)
+ apply(simp)
+ thm k1
+ apply(drule k2)
+ apply(simp)
+ apply(simp add: lang_seq_def)
+ apply(erule exE)+
+ apply(erule conjE)+
+ apply(auto simp add: lang_seq_def Cons_eq_append_conv)
+ apply(drule k1)
+ apply(simp)
+ apply(simp add: lang_seq_def)
+ apply(erule exE)+
+ apply(erule conjE)+
+ apply(auto simp add: lang_seq_def Cons_eq_append_conv)
+ done
+qed
+
+fun
+ derivative :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+ "derivative [] r = r"
+| "derivative (c#s) r = derivative s (der c r)"
+
+fun
+ slind_matcher :: "rexp \<Rightarrow> string \<Rightarrow> bool"
+where
+ "slind_matcher r s = nullable (derivative s r)"
+
+lemma slind_matcher:
+ shows "slind_matcher r s = (s \<in> L r)"
+by (induct s arbitrary: r)
+ (auto simp add: nullable der_correctness)
+
+export_code slind_matcher in SML module_name Slind file -
+
+
+(* ******************************************** now is the codes writen by chunhan ************************************* *)
+
+section {* Arden's Lemma revised *}
+
+lemma arden_aux1:
+ assumes a: "X \<subseteq> X ; A \<union> B"
+ and b: "[] \<notin> A"
+ shows "x \<in> X \<Longrightarrow> x \<in> B ; A\<star>"
+apply (induct x taking:length rule:measure_induct)
+apply (subgoal_tac "x \<in> X ; A \<union> B")
+defer
+using a
+apply (auto)[1]
+apply simp
+apply (erule disjE)
+defer
+apply (auto simp add:lang_seq_def) [1]
+apply (subgoal_tac "\<exists> x1 x2. x = x1 @ x2 \<and> x1 \<in> X \<and> x2 \<in> A")
+defer
+apply (auto simp add:lang_seq_def) [1]
+apply (erule exE | erule conjE)+
+apply simp
+apply (drule_tac x = x1 in spec)
+apply (simp)
+using b
+apply -
+apply (auto)[1]
+apply (subgoal_tac "x1 @ x2 \<in> (B ; A\<star>) ; A")
+defer
+apply (auto simp add:lang_seq_def)[1]
+by (metis Un_absorb1 lang_seq_assoc lang_seq_union(2) lang_star_double lang_star_simple mem_def sup1CI)
+
+theorem ardens_revised:
+ assumes nemp: "[] \<notin> A"
+ shows "(X = X ; A \<union> B) \<longleftrightarrow> (X = B ; A\<star>)"
+apply(rule iffI)
+defer
+apply(simp)
+apply(subst lang_star_cases')
+apply(subst lang_seq_union)
+apply(simp add: lang_seq_empty)
+apply(simp add: lang_seq_assoc)
+apply(auto)[1]
+proof -
+ assume "X = X ; A \<union> B"
+ then have as1: "X ; A \<union> B \<subseteq> X" and as2: "X \<subseteq> X ; A \<union> B" by simp_all
+ from as1 have a: "X ; A \<subseteq> X" and b: "B \<subseteq> X" by simp_all
+ from b have "B; A\<star> \<subseteq> X ; A\<star>" by (auto simp add: lang_seq_def)
+ moreover
+ from a have "X ; A\<star> \<subseteq> X"
+
+by (rule lang_star_prop2)
+ ultimately have f1: "B ; A\<star> \<subseteq> X" by simp
+ from as2 nemp
+ have f2: "X \<subseteq> B; A\<star>" using arden_aux1 by auto
+ from f1 f2 show "X = B; A\<star>" by auto
+qed
+
+section {* equiv class' definition *}
+
+definition
+ equiv_str :: "string \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> bool" ("_ \<equiv>_\<equiv> _" [100, 100, 100] 100)
+where
+ "x \<equiv>L'\<equiv> y \<longleftrightarrow> (\<forall>z. x@z \<in> L' \<longleftrightarrow> y@z \<in> L')"
+
+definition
+ equiv_class :: "string \<Rightarrow> (string set) \<Rightarrow> string set" ("\<lbrakk>_\<rbrakk>_" [100, 100] 100)
+where
+ "\<lbrakk>x\<rbrakk>L' \<equiv> {y. x \<equiv>L'\<equiv> y}"
+
+text {* Chunhan modifies Q to Quo *}
+definition
+ quot :: "string set \<Rightarrow> (string set) \<Rightarrow> (string set) set" ("_ Quo _" [100, 100] 100)
+where
+ "L' Quo R \<equiv> { \<lbrakk>x\<rbrakk>R | x. x \<in> L'}"
+
+lemma lang_eqs_union_of_eqcls:
+ "Lang = \<Union> {X. X \<in> (UNIV Quo Lang) \<and> (\<forall> x \<in> X. x \<in> Lang)}"
+proof
+ show "Lang \<subseteq> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
+ proof
+ fix x
+ assume "x \<in> Lang"
+ thus "x \<in> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
+ proof (simp add:quot_def)
+ assume "(1)": "x \<in> Lang"
+ show "\<exists>xa. (\<exists>x. xa = \<lbrakk>x\<rbrakk>Lang) \<and> (\<forall>x\<in>xa. x \<in> Lang) \<and> x \<in> xa" (is "\<exists>xa.?P xa")
+ proof -
+ have "?P (\<lbrakk>x\<rbrakk>Lang)" using "(1)"
+ by (auto simp:equiv_class_def equiv_str_def dest: spec[where x = "[]"])
+ thus ?thesis by blast
+ qed
+ qed
+ qed
+next
+ show "\<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<subseteq> Lang"
+ by auto
+qed
+
+lemma empty_notin_CS: "{} \<notin> UNIV Quo Lang"
+apply (clarsimp simp:quot_def equiv_class_def)
+by (rule_tac x = x in exI, auto simp:equiv_str_def)
+
+lemma no_two_cls_inters:
+ "\<lbrakk>X \<in> UNIV Quo Lang; Y \<in> UNIV Quo Lang; X \<noteq> Y\<rbrakk> \<Longrightarrow> X \<inter> Y = {}"
+by (auto simp:quot_def equiv_class_def equiv_str_def)
+
+text {* equiv_class transition *}
+definition
+ CT :: "string set \<Rightarrow> char \<Rightarrow> string set \<Rightarrow> bool" ("_-_\<rightarrow>_" [99,99]99)
+where
+ "X-c\<rightarrow>Y \<equiv> ((X;{[c]}) \<subseteq> Y)"
+
+types t_equa_rhs = "(string set \<times> rexp) set"
+
+types t_equa = "(string set \<times> t_equa_rhs)"
+
+types t_equas = "t_equa set"
+
+text {* "empty_rhs" generates "\<lambda>" for init-state, just like "\<lambda>" for final states in Brzozowski method.
+ But if the init-state is "{[]}" ("\<lambda>" itself) then empty set is returned, see definition of "equation_rhs" *}
+definition
+ empty_rhs :: "string set \<Rightarrow> t_equa_rhs"
+where
+ "empty_rhs X \<equiv> if ([] \<in> X) then {({[]}, EMPTY)} else {}"
+
+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"
+
+definition
+ equation_rhs :: "(string set) set \<Rightarrow> string set \<Rightarrow> t_equa_rhs"
+where
+ "equation_rhs CS X \<equiv> if (X = {[]}) then {({[]}, EMPTY)}
+ else {(S, folds ALT NULL {CHAR c| c. S-c\<rightarrow>X} )| S. S \<in> CS} \<union> empty_rhs X"
+
+definition
+ equations :: "(string set) set \<Rightarrow> t_equas"
+where
+ "equations CS \<equiv> {(X, equation_rhs CS X) | X. X \<in> CS}"
+
+overloading L_rhs \<equiv> "L:: t_equa_rhs \<Rightarrow> string set"
+begin
+fun L_rhs:: "t_equa_rhs \<Rightarrow> string set"
+where
+ "L_rhs rhs = {x. \<exists> X r. (X, r) \<in> rhs \<and> x \<in> X;(L r)}"
+end
+
+definition
+ distinct_rhs :: "t_equa_rhs \<Rightarrow> bool"
+where
+ "distinct_rhs rhs \<equiv> \<forall> X reg\<^isub>1 reg\<^isub>2. (X, reg\<^isub>1) \<in> rhs \<and> (X, reg\<^isub>2) \<in> rhs \<longrightarrow> reg\<^isub>1 = reg\<^isub>2"
+
+definition
+ distinct_equas_rhs :: "t_equas \<Rightarrow> bool"
+where
+ "distinct_equas_rhs equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> distinct_rhs rhs"
+
+definition
+ distinct_equas :: "t_equas \<Rightarrow> bool"
+where
+ "distinct_equas equas \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> equas \<and> (X, rhs') \<in> equas \<longrightarrow> rhs = rhs'"
+
+definition
+ seq_rhs_r :: "t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
+where
+ "seq_rhs_r rhs r \<equiv> (\<lambda>(X, reg). (X, SEQ reg r)) ` rhs"
+
+definition
+ del_x_paired :: "('a \<times> 'b) set \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'b) set"
+where
+ "del_x_paired S x \<equiv> S - {X. X \<in> S \<and> fst X = x}"
+
+definition
+ arden_variate :: "string set \<Rightarrow> rexp \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
+where
+ "arden_variate X r rhs \<equiv> seq_rhs_r (del_x_paired rhs X) (STAR r)"
+
+definition
+ no_EMPTY_rhs :: "t_equa_rhs \<Rightarrow> bool"
+where
+ "no_EMPTY_rhs rhs \<equiv> \<forall> X r. (X, r) \<in> rhs \<and> X \<noteq> {[]} \<longrightarrow> [] \<notin> L r"
+
+definition
+ no_EMPTY_equas :: "t_equas \<Rightarrow> bool"
+where
+ "no_EMPTY_equas equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> no_EMPTY_rhs rhs"
+
+lemma fold_alt_null_eqs:
+ "finite rS \<Longrightarrow> x \<in> L (folds ALT NULL rS) = (\<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)
+by auto (*??? how do this be in Isar ?? *)
+
+lemma seq_rhs_r_prop1:
+ "L (seq_rhs_r rhs r) = (L rhs);(L r)"
+apply (rule set_ext, rule iffI)
+apply (auto simp:seq_rhs_r_def image_def lang_seq_def)
+apply (rule_tac x = "s1 @ s1a" in exI, rule_tac x = "s2a" in exI, simp)
+apply (rule_tac x = a in exI, rule_tac x = b in exI, simp)
+apply (rule_tac x = s1 in exI, rule_tac x = s1a in exI, simp)
+apply (rule_tac x = X in exI, rule_tac x = "SEQ ra r" in exI, simp)
+apply (rule conjI)
+apply (rule_tac x = "(X, ra)" in bexI, simp+)
+apply (rule_tac x = s1a in exI, rule_tac x = "s2a @ s2" in exI, simp)
+apply (simp add:lang_seq_def)
+by (rule_tac x = s2a in exI, rule_tac x = s2 in exI, simp)
+
+lemma del_x_paired_prop1:
+ "\<lbrakk>distinct_rhs rhs; (X, r) \<in> rhs\<rbrakk> \<Longrightarrow> X ; L r \<union> L (del_x_paired rhs X) = L rhs"
+apply (simp add:del_x_paired_def)
+apply (rule set_ext, rule iffI, simp)
+apply (erule disjE, rule_tac x = X in exI, rule_tac x = r in exI, simp)
+apply (clarify, rule_tac x = Xa in exI, rule_tac x = ra in exI, simp)
+apply (clarsimp, drule_tac x = Xa in spec, drule_tac x = ra in spec)
+apply (auto simp:distinct_rhs_def)
+done
+
+lemma arden_variate_prop:
+ assumes "(X, rx) \<in> rhs"
+ shows "(\<forall> Y. Y \<noteq> X \<longrightarrow> (\<exists> r. (Y, r) \<in> rhs) = (\<exists> r. (Y, r) \<in> (arden_variate X rx rhs)))"
+proof (rule allI, rule impI)
+ fix Y
+ assume "(1)": "Y \<noteq> X"
+ show "(\<exists>r. (Y, r) \<in> rhs) = (\<exists>r. (Y, r) \<in> arden_variate X rx rhs)"
+ proof
+ assume "(1_1)": "\<exists>r. (Y, r) \<in> rhs"
+ show "\<exists>r. (Y, r) \<in> arden_variate X rx rhs" (is "\<exists>r. ?P r")
+ proof -
+ from "(1_1)" obtain r where "(Y, r) \<in> rhs" ..
+ hence "?P (SEQ r (STAR rx))"
+ proof (simp add:arden_variate_def image_def)
+ have "(Y, r) \<in> rhs \<Longrightarrow> (Y, r) \<in> del_x_paired rhs X"
+ by (auto simp:del_x_paired_def "(1)")
+ thus "(Y, r) \<in> rhs \<Longrightarrow> (Y, SEQ r (STAR rx)) \<in> seq_rhs_r (del_x_paired rhs X) (STAR rx)"
+ by (auto simp:seq_rhs_r_def)
+ qed
+ thus ?thesis by blast
+ qed
+ next
+ assume "(2_1)": "\<exists>r. (Y, r) \<in> arden_variate X rx rhs"
+ thus "\<exists>r. (Y, r) \<in> rhs" (is "\<exists> r. ?P r")
+ by (auto simp:arden_variate_def del_x_paired_def seq_rhs_r_def image_def)
+ qed
+qed
+
+text {*
+ arden_variate_valid: proves variation from "X = X;r + Y;ry + \<dots>" to "X = Y;(SEQ ry (STAR r)) + \<dots>" holds the law of "language of left equiv language of right"
+*}
+lemma arden_variate_valid:
+ assumes X_not_empty: "X \<noteq> {[]}"
+ and l_eq_r: "X = L rhs"
+ and dist: "distinct_rhs rhs"
+ and no_empty: "no_EMPTY_rhs rhs"
+ and self_contained: "(X, r) \<in> rhs"
+ shows "X = L (arden_variate X r rhs)"
+proof -
+ have "[] \<notin> L r" using no_empty X_not_empty self_contained
+ by (auto simp:no_EMPTY_rhs_def)
+ hence ardens: "X = X;(L r) \<union> (L (del_x_paired rhs X)) \<longleftrightarrow> X = (L (del_x_paired rhs X)) ; (L r)\<star>"
+ by (rule ardens_revised)
+ have del_x: "X = X ; L r \<union> L (del_x_paired rhs X) \<longleftrightarrow> X = L rhs" using dist l_eq_r self_contained
+ by (auto dest!:del_x_paired_prop1)
+ show ?thesis
+ proof
+ show "X \<subseteq> L (arden_variate X r rhs)"
+ proof
+ fix x
+ assume "(1_1)": "x \<in> X" with l_eq_r ardens del_x
+ show "x \<in> L (arden_variate X r rhs)"
+ by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
+ qed
+ next
+ show "L (arden_variate X r rhs) \<subseteq> X"
+ proof
+ fix x
+ assume "(2_1)": "x \<in> L (arden_variate X r rhs)" with ardens del_x l_eq_r
+ show "x \<in> X"
+ by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
+ qed
+ qed
+qed
+
+text {* merge_rhs {(X1, r1), (x2, r2}, (x4, r4), \<dots>} {(x1, r1'), (x3, r3'), \<dots>} = {(x1, ALT r1 r1'}, (x2, r2), (x3, r3'), (x4, r4), \<dots>} *}
+definition
+ merge_rhs :: "t_equa_rhs \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
+where
+ "merge_rhs rhs rhs' \<equiv> {(X, r). (\<exists> r1 r2. ((X,r1) \<in> rhs \<and> (X, r2) \<in> rhs') \<and> r = ALT r1 r2) \<or>
+ (\<exists> r1. (X, r1) \<in> rhs \<and> (\<not> (\<exists> r2. (X, r2) \<in> rhs')) \<and> r = r1) \<or>
+ (\<exists> r2. (X, r2) \<in> rhs' \<and> (\<not> (\<exists> r1. (X, r1) \<in> rhs)) \<and> r = r2) }"
+
+
+text {* rhs_subst rhs X=xrhs r: substitude all occurence of X in rhs((X,r) \<in> rhs) with xrhs *}
+definition
+ rhs_subst :: "t_equa_rhs \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
+where
+ "rhs_subst rhs X xrhs r \<equiv> merge_rhs (del_x_paired rhs X) (seq_rhs_r xrhs r)"
+
+definition
+ equas_subst_f :: "string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa \<Rightarrow> t_equa"
+where
+ "equas_subst_f X xrhs equa \<equiv> let (Y, rhs) = equa in
+ if (\<exists> r. (X, r) \<in> rhs)
+ then (Y, rhs_subst rhs X xrhs (SOME r. (X, r) \<in> rhs))
+ else equa"
+
+definition
+ equas_subst :: "t_equas \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equas"
+where
+ "equas_subst ES X xrhs \<equiv> del_x_paired (equas_subst_f X xrhs ` ES) X"
+
+lemma lang_seq_prop1:
+ "x \<in> X ; L r \<Longrightarrow> x \<in> X ; (L r \<union> L r')"
+by (auto simp:lang_seq_def)
+
+lemma lang_seq_prop1':
+ "x \<in> X; L r \<Longrightarrow> x \<in> X ; (L r' \<union> L r)"
+by (auto simp:lang_seq_def)
+
+lemma lang_seq_prop2:
+ "x \<in> X; (L r \<union> L r') \<Longrightarrow> x \<in> X;L r \<or> x \<in> X;L r'"
+by (auto simp:lang_seq_def)
+
+lemma merge_rhs_prop1:
+ shows "L (merge_rhs rhs rhs') = L rhs \<union> L rhs' "
+apply (auto simp add:merge_rhs_def dest!:lang_seq_prop2 intro:lang_seq_prop1)
+apply (rule_tac x = X in exI, rule_tac x = r1 in exI, simp)
+apply (case_tac "\<exists> r2. (X, r2) \<in> rhs'")
+apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r r2" in exI, simp add:lang_seq_prop1)
+apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
+apply (case_tac "\<exists> r1. (X, r1) \<in> rhs")
+apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r1 r" in exI, simp add:lang_seq_prop1')
+apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
+done
+
+lemma no_EMPTY_rhss_imp_merge_no_EMPTY:
+ "\<lbrakk>no_EMPTY_rhs rhs; no_EMPTY_rhs rhs'\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (merge_rhs rhs rhs')"
+apply (simp add:no_EMPTY_rhs_def merge_rhs_def)
+apply (clarify, (erule conjE | erule exE | erule disjE)+)
+by auto
+
+lemma distinct_rhs_prop:
+ "\<lbrakk>distinct_rhs rhs; (X, r1) \<in> rhs; (X, r2) \<in> rhs\<rbrakk> \<Longrightarrow> r1 = r2"
+by (auto simp:distinct_rhs_def)
+
+lemma merge_rhs_prop2:
+ assumes dist_rhs: "distinct_rhs rhs"
+ and dist_rhs':"distinct_rhs rhs'"
+ shows "distinct_rhs (merge_rhs rhs rhs')"
+apply (auto simp:merge_rhs_def distinct_rhs_def)
+using dist_rhs
+apply (drule distinct_rhs_prop, simp+)
+using dist_rhs'
+apply (drule distinct_rhs_prop, simp+)
+using dist_rhs
+apply (drule distinct_rhs_prop, simp+)
+using dist_rhs'
+apply (drule distinct_rhs_prop, simp+)
+done
+
+lemma seq_rhs_r_holds_distinct:
+ "distinct_rhs rhs \<Longrightarrow> distinct_rhs (seq_rhs_r rhs r)"
+by (auto simp:distinct_rhs_def seq_rhs_r_def)
+
+lemma seq_rhs_r_prop0:
+ assumes l_eq_r: "X = L xrhs"
+ shows "L (seq_rhs_r xrhs r) = X ; L r "
+using l_eq_r
+by (auto simp only:seq_rhs_r_prop1)
+
+lemma rhs_subst_prop1:
+ assumes l_eq_r: "X = L xrhs"
+ and dist: "distinct_rhs rhs"
+ shows "(X, r) \<in> rhs \<Longrightarrow> L rhs = L (rhs_subst rhs X xrhs r)"
+apply (simp add:rhs_subst_def merge_rhs_prop1 del:L_rhs.simps)
+using l_eq_r
+apply (drule_tac r = r in seq_rhs_r_prop0, simp del:L_rhs.simps)
+using dist
+by (auto dest!:del_x_paired_prop1 simp del:L_rhs.simps)
+
+lemma del_x_paired_holds_distinct_rhs:
+ "distinct_rhs rhs \<Longrightarrow> distinct_rhs (del_x_paired rhs x)"
+by (auto simp:distinct_rhs_def del_x_paired_def)
+
+lemma rhs_subst_holds_distinct_rhs:
+ "\<lbrakk>distinct_rhs rhs; distinct_rhs xrhs\<rbrakk> \<Longrightarrow> distinct_rhs (rhs_subst rhs X xrhs r)"
+apply (drule_tac r = r and rhs = xrhs in seq_rhs_r_holds_distinct)
+apply (drule_tac x = X in del_x_paired_holds_distinct_rhs)
+by (auto dest:merge_rhs_prop2[where rhs = "del_x_paired rhs X"] simp:rhs_subst_def)
+
+section {* myhill-nerode theorem *}
+
+definition left_eq_cls :: "t_equas \<Rightarrow> (string set) set"
+where
+ "left_eq_cls ES \<equiv> {X. \<exists> rhs. (X, rhs) \<in> ES} "
+
+definition right_eq_cls :: "t_equas \<Rightarrow> (string set) set"
+where
+ "right_eq_cls ES \<equiv> {Y. \<exists> X rhs r. (X, rhs) \<in> ES \<and> (Y, r) \<in> rhs }"
+
+definition rhs_eq_cls :: "t_equa_rhs \<Rightarrow> (string set) set"
+where
+ "rhs_eq_cls rhs \<equiv> {Y. \<exists> r. (Y, r) \<in> rhs}"
+
+definition ardenable :: "t_equa \<Rightarrow> bool"
+where
+ "ardenable equa \<equiv> let (X, rhs) = equa in
+ distinct_rhs rhs \<and> no_EMPTY_rhs rhs \<and> X = L rhs"
+
+text {*
+ Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
+*}
+definition Inv :: "string set \<Rightarrow> t_equas \<Rightarrow> bool"
+where
+ "Inv X ES \<equiv> finite ES \<and> (\<exists> rhs. (X, rhs) \<in> ES) \<and> distinct_equas ES \<and>
+ (\<forall> X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs) \<and> X \<noteq> {} \<and> rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls ES))"
+
+text {*
+ TCon: Termination Condition of the equation-system decreasion.
+*}
+definition TCon:: "'a set \<Rightarrow> bool"
+where
+ "TCon ES \<equiv> card ES = 1"
+
+
+text {*
+ The following is a iteration principle, and is the main framework for the proof:
+ 1: We can form the initial equation-system using "equations" defined above, and prove it has invariance Inv by lemma "init_ES_satisfy_Inv";
+ 2: We can decrease the number of the equation-system using ardens_lemma_revised and substitution ("equas_subst", defined above),
+ and prove it holds the property "step" in "wf_iter" by lemma "iteration_step"
+ and finally using property Inv and TCon to prove the myhill-nerode theorem
+
+*}
+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
+
+
+(* ********************************* BEGIN: proving the initial equation-system satisfies Inv **************************************** *)
+
+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 finite_CT_chars:
+ "finite {CHAR c |c. Xa-c\<rightarrow>X}"
+by (auto)
+
+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_CT_chars)+
+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
+
+
+(* ********************************* END: proving the initial equation-system satisfies Inv **************************************** *)
+
+
+(* ***************************** 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
+
+(* ****************************** 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
+
+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
+
+end
\ No newline at end of file