diff -r 3b7477db3462 -r e122cb146ecc Theories/Myhill_1.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Theories/Myhill_1.thy	Wed Mar 23 12:17:30 2011 +0000
@@ -0,0 +1,783 @@
+theory Myhill_1
+imports Main Folds Regular
+        "~~/src/HOL/Library/While_Combinator" 
+begin
+
+section {* Direction @{text "finite partition \<Rightarrow> regular language"} *}
+
+lemma Pair_Collect[simp]:
+  shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
+by simp
+
+text {* Myhill-Nerode relation *}
+
+definition
+  str_eq_rel :: "lang \<Rightarrow> (string \<times> string) set" ("\<approx>_" [100] 100)
+where
+  "\<approx>A \<equiv> {(x, y).  (\<forall>z. x @ z \<in> A \<longleftrightarrow> y @ z \<in> A)}"
+
+definition 
+  finals :: "lang \<Rightarrow> lang set"
+where
+  "finals A \<equiv> {\<approx>A `` {s} | s . s \<in> A}"
+
+lemma lang_is_union_of_finals: 
+  shows "A = \<Union> finals A"
+unfolding finals_def
+unfolding Image_def
+unfolding str_eq_rel_def
+by (auto) (metis append_Nil2)
+
+lemma finals_in_partitions:
+  shows "finals A \<subseteq> (UNIV // \<approx>A)"
+unfolding finals_def quotient_def
+by auto
+
+section {* Equational systems *}
+
+text {* The two kinds of terms in the rhs of equations. *}
+
+datatype rhs_trm = 
+   Lam "rexp"            (* Lambda-marker *)
+ | Trn "lang" "rexp"     (* Transition *)
+
+
+overloading L_rhs_trm \<equiv> "L:: rhs_trm \<Rightarrow> lang"
+begin
+  fun L_rhs_trm:: "rhs_trm \<Rightarrow> lang"
+  where
+    "L_rhs_trm (Lam r) = L r" 
+  | "L_rhs_trm (Trn X r) = X ;; L r"
+end
+
+overloading L_rhs \<equiv> "L:: rhs_trm set \<Rightarrow> lang"
+begin
+   fun L_rhs:: "rhs_trm set \<Rightarrow> lang"
+   where 
+     "L_rhs rhs = \<Union> (L ` rhs)"
+end
+
+lemma L_rhs_set:
+  shows "L {Trn X r | r. P r} = \<Union>{L (Trn X r) | r. P r}"
+by (auto simp del: L_rhs_trm.simps)
+
+lemma L_rhs_union_distrib:
+  fixes A B::"rhs_trm set"
+  shows "L A \<union> L B = L (A \<union> B)"
+by simp
+
+
+
+text {* Transitions between equivalence classes *}
+
+definition 
+  transition :: "lang \<Rightarrow> char \<Rightarrow> lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)
+where
+  "Y \<Turnstile>c\<Rightarrow> X \<equiv> Y ;; {[c]} \<subseteq> X"
+
+text {* Initial equational system *}
+
+definition
+  "Init_rhs CS X \<equiv>  
+      if ([] \<in> X) then 
+          {Lam EMPTY} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}
+      else 
+          {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}"
+
+definition 
+  "Init CS \<equiv> {(X, Init_rhs CS X) | X.  X \<in> CS}"
+
+
+section {* Arden Operation on equations *}
+
+fun 
+  Append_rexp :: "rexp \<Rightarrow> rhs_trm \<Rightarrow> rhs_trm"
+where
+  "Append_rexp r (Lam rexp)   = Lam (SEQ rexp r)"
+| "Append_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"
+
+
+definition
+  "Append_rexp_rhs rhs rexp \<equiv> (Append_rexp rexp) ` rhs"
+
+definition 
+  "Arden X rhs \<equiv> 
+     Append_rexp_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs}) (STAR (\<Uplus> {r. Trn X r \<in> rhs}))"
+
+
+section {* Substitution Operation on equations *}
+
+definition 
+  "Subst rhs X xrhs \<equiv> 
+        (rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> (Append_rexp_rhs xrhs (\<Uplus> {r. Trn X r \<in> rhs}))"
+
+definition
+  Subst_all :: "(lang \<times> rhs_trm set) set \<Rightarrow> lang \<Rightarrow> rhs_trm set \<Rightarrow> (lang \<times> rhs_trm set) set"
+where
+  "Subst_all ES X xrhs \<equiv> {(Y, Subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
+
+definition
+  "Remove ES X xrhs \<equiv> 
+      Subst_all  (ES - {(X, xrhs)}) X (Arden X xrhs)"
+
+
+section {* While-combinator *}
+
+definition 
+  "Iter X ES \<equiv> (let (Y, yrhs) = SOME (Y, yrhs). (Y, yrhs) \<in> ES \<and> X \<noteq> Y
+                in Remove ES Y yrhs)"
+
+lemma IterI2:
+  assumes "(Y, yrhs) \<in> ES"
+  and     "X \<noteq> Y"
+  and     "\<And>Y yrhs. \<lbrakk>(Y, yrhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> Q (Remove ES Y yrhs)"
+  shows "Q (Iter X ES)"
+unfolding Iter_def using assms
+by (rule_tac a="(Y, yrhs)" in someI2) (auto)
+
+abbreviation
+  "Cond ES \<equiv> card ES \<noteq> 1"
+
+definition 
+  "Solve X ES \<equiv> while Cond (Iter X) ES"
+
+
+section {* Invariants *}
+
+definition 
+  "distinctness ES \<equiv> 
+     \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
+
+definition 
+  "soundness ES \<equiv> \<forall>(X, rhs) \<in> ES. X = L rhs"
+
+definition 
+  "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
+
+definition 
+  "ardenable_all ES \<equiv> \<forall>(X, rhs) \<in> ES. ardenable rhs"
+
+definition
+  "finite_rhs ES \<equiv> \<forall>(X, rhs) \<in> ES. finite rhs"
+
+lemma finite_rhs_def2:
+  "finite_rhs ES = (\<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs)"
+unfolding finite_rhs_def by auto
+
+definition 
+  "rhss rhs \<equiv> {X | X r. Trn X r \<in> rhs}"
+
+definition
+  "lhss ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
+
+definition 
+  "validity ES \<equiv> \<forall>(X, rhs) \<in> ES. rhss rhs \<subseteq> lhss ES"
+
+lemma rhss_union_distrib:
+  shows "rhss (A \<union> B) = rhss A \<union> rhss B"
+by (auto simp add: rhss_def)
+
+lemma lhss_union_distrib:
+  shows "lhss (A \<union> B) = lhss A \<union> lhss B"
+by (auto simp add: lhss_def)
+
+
+definition 
+  "invariant ES \<equiv> finite ES
+                \<and> finite_rhs ES
+                \<and> soundness ES 
+                \<and> distinctness ES 
+                \<and> ardenable_all ES 
+                \<and> validity ES"
+
+
+lemma invariantI:
+  assumes "soundness ES" "finite ES" "distinctness ES" "ardenable_all ES" 
+          "finite_rhs ES" "validity ES"
+  shows "invariant ES"
+using assms by (simp add: invariant_def)
+
+
+subsection {* The proof of this direction *}
+
+lemma finite_Trn:
+  assumes fin: "finite rhs"
+  shows "finite {r. Trn Y r \<in> rhs}"
+proof -
+  have "finite {Trn Y r | Y r. Trn Y r \<in> rhs}"
+    by (rule rev_finite_subset[OF fin]) (auto)
+  then have "finite ((\<lambda>(Y, r). Trn Y r) ` {(Y, r) | Y r. Trn Y r \<in> rhs})"
+    by (simp add: image_Collect)
+  then have "finite {(Y, r) | Y r. Trn Y r \<in> rhs}"
+    by (erule_tac finite_imageD) (simp add: inj_on_def)
+  then show "finite {r. Trn Y r \<in> rhs}"
+    by (erule_tac f="snd" in finite_surj) (auto simp add: image_def)
+qed
+
+lemma finite_Lam:
+  assumes fin: "finite rhs"
+  shows "finite {r. Lam r \<in> rhs}"
+proof -
+  have "finite {Lam r | r. Lam r \<in> rhs}"
+    by (rule rev_finite_subset[OF fin]) (auto)
+  then show "finite {r. Lam r \<in> rhs}"
+    apply(simp add: image_Collect[symmetric])
+    apply(erule finite_imageD)
+    apply(auto simp add: inj_on_def)
+    done
+qed
+
+lemma rhs_trm_soundness:
+  assumes finite:"finite rhs"
+  shows "L ({Trn X r| r. Trn X r \<in> rhs}) = X ;; (L (\<Uplus>{r. Trn X r \<in> rhs}))"
+proof -
+  have "finite {r. Trn X r \<in> rhs}" 
+    by (rule finite_Trn[OF finite]) 
+  then show "L ({Trn X r| r. Trn X r \<in> rhs}) = X ;; (L (\<Uplus>{r. Trn X r \<in> rhs}))"
+    by (simp only: L_rhs_set L_rhs_trm.simps) (auto simp add: Seq_def)
+qed
+
+lemma lang_of_append_rexp:
+  "L (Append_rexp r rhs_trm) = L rhs_trm ;; L r"
+by (induct rule: Append_rexp.induct)
+   (auto simp add: seq_assoc)
+
+lemma lang_of_append_rexp_rhs:
+  "L (Append_rexp_rhs rhs r) = L rhs ;; L r"
+unfolding Append_rexp_rhs_def
+by (auto simp add: Seq_def lang_of_append_rexp)
+
+
+
+subsubsection {* Intialization *}
+
+lemma defined_by_str:
+  assumes "s \<in> X" "X \<in> UNIV // \<approx>A" 
+  shows "X = \<approx>A `` {s}"
+using assms
+unfolding quotient_def Image_def str_eq_rel_def
+by auto
+
+lemma every_eqclass_has_transition:
+  assumes has_str: "s @ [c] \<in> X"
+  and     in_CS:   "X \<in> UNIV // \<approx>A"
+  obtains Y where "Y \<in> UNIV // \<approx>A" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
+proof -
+  def Y \<equiv> "\<approx>A `` {s}"
+  have "Y \<in> UNIV // \<approx>A" 
+    unfolding Y_def quotient_def by auto
+  moreover
+  have "X = \<approx>A `` {s @ [c]}" 
+    using has_str in_CS defined_by_str by blast
+  then have "Y ;; {[c]} \<subseteq> X" 
+    unfolding Y_def Image_def Seq_def
+    unfolding str_eq_rel_def
+    by clarsimp
+  moreover
+  have "s \<in> Y" unfolding Y_def 
+    unfolding Image_def str_eq_rel_def by simp
+  ultimately show thesis using that by blast
+qed
+
+lemma l_eq_r_in_eqs:
+  assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
+  shows "X = L rhs"
+proof 
+  show "X \<subseteq> L rhs"
+  proof
+    fix x
+    assume in_X: "x \<in> X"
+    { assume empty: "x = []"
+      then have "x \<in> L rhs" using X_in_eqs in_X
+	unfolding Init_def Init_rhs_def
+        by auto
+    }
+    moreover
+    { assume not_empty: "x \<noteq> []"
+      then obtain s c where decom: "x = s @ [c]"
+	using rev_cases by blast
+      have "X \<in> UNIV // \<approx>A" using X_in_eqs unfolding Init_def by auto
+      then obtain Y where "Y \<in> UNIV // \<approx>A" "Y ;; {[c]} \<subseteq> X" "s \<in> Y"
+        using decom in_X every_eqclass_has_transition by blast
+      then have "x \<in> L {Trn Y (CHAR c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"
+        unfolding transition_def
+	using decom by (force simp add: Seq_def)
+      then have "x \<in> L rhs" using X_in_eqs in_X
+	unfolding Init_def Init_rhs_def by simp
+    }
+    ultimately show "x \<in> L rhs" by blast
+  qed
+next
+  show "L rhs \<subseteq> X" using X_in_eqs
+    unfolding Init_def Init_rhs_def transition_def
+    by auto 
+qed
+
+lemma test:
+  assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
+  shows "X = \<Union> (L `  rhs)"
+using assms l_eq_r_in_eqs by (simp)
+
+lemma finite_Init_rhs: 
+  assumes finite: "finite CS"
+  shows "finite (Init_rhs CS X)"
+proof-
+  def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" 
+  def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
+  have "finite (CS \<times> (UNIV::char set))" using finite by auto
+  then have "finite S" using S_def 
+    by (rule_tac B = "CS \<times> UNIV" in finite_subset) (auto)
+  moreover have "{Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X} = h ` S"
+    unfolding S_def h_def image_def by auto
+  ultimately
+  have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" by auto
+  then show "finite (Init_rhs CS X)" unfolding Init_rhs_def transition_def by simp
+qed
+
+lemma Init_ES_satisfies_invariant:
+  assumes finite_CS: "finite (UNIV // \<approx>A)"
+  shows "invariant (Init (UNIV // \<approx>A))"
+proof (rule invariantI)
+  show "soundness (Init (UNIV // \<approx>A))"
+    unfolding soundness_def 
+    using l_eq_r_in_eqs by auto
+  show "finite (Init (UNIV // \<approx>A))" using finite_CS
+    unfolding Init_def by simp
+  show "distinctness (Init (UNIV // \<approx>A))"     
+    unfolding distinctness_def Init_def by simp
+  show "ardenable_all (Init (UNIV // \<approx>A))"
+    unfolding ardenable_all_def Init_def Init_rhs_def ardenable_def
+   by auto 
+  show "finite_rhs (Init (UNIV // \<approx>A))"
+    using finite_Init_rhs[OF finite_CS]
+    unfolding finite_rhs_def Init_def by auto
+  show "validity (Init (UNIV // \<approx>A))"
+    unfolding validity_def Init_def Init_rhs_def rhss_def lhss_def
+    by auto
+qed
+
+subsubsection {* Interation step *}
+
+lemma Arden_keeps_eq:
+  assumes l_eq_r: "X = L rhs"
+  and not_empty: "ardenable rhs"
+  and finite: "finite rhs"
+  shows "X = L (Arden X rhs)"
+proof -
+  def A \<equiv> "L (\<Uplus>{r. Trn X r \<in> rhs})"
+  def b \<equiv> "{Trn X r | r. Trn X r \<in> rhs}"
+  def B \<equiv> "L (rhs - b)"
+  have not_empty2: "[] \<notin> A" 
+    using finite_Trn[OF finite] not_empty
+    unfolding A_def ardenable_def by simp
+  have "X = L rhs" using l_eq_r by simp
+  also have "\<dots> = L (b \<union> (rhs - b))" unfolding b_def by auto
+  also have "\<dots> = L b \<union> B" unfolding B_def by (simp only: L_rhs_union_distrib)
+  also have "\<dots> = X ;; A \<union> B"
+    unfolding b_def
+    unfolding rhs_trm_soundness[OF finite]
+    unfolding A_def
+    by blast
+  finally have "X = X ;; A \<union> B" . 
+  then have "X = B ;; A\<star>"
+    by (simp add: arden[OF not_empty2])
+  also have "\<dots> = L (Arden X rhs)"
+    unfolding Arden_def A_def B_def b_def
+    by (simp only: lang_of_append_rexp_rhs L_rexp.simps)
+  finally show "X = L (Arden X rhs)" by simp
+qed 
+
+lemma Append_keeps_finite:
+  "finite rhs \<Longrightarrow> finite (Append_rexp_rhs rhs r)"
+by (auto simp:Append_rexp_rhs_def)
+
+lemma Arden_keeps_finite:
+  "finite rhs \<Longrightarrow> finite (Arden X rhs)"
+by (auto simp:Arden_def Append_keeps_finite)
+
+lemma Append_keeps_nonempty:
+  "ardenable rhs \<Longrightarrow> ardenable (Append_rexp_rhs rhs r)"
+apply (auto simp:ardenable_def Append_rexp_rhs_def)
+by (case_tac x, auto simp:Seq_def)
+
+lemma nonempty_set_sub:
+  "ardenable rhs \<Longrightarrow> ardenable (rhs - A)"
+by (auto simp:ardenable_def)
+
+lemma nonempty_set_union:
+  "\<lbrakk>ardenable rhs; ardenable rhs'\<rbrakk> \<Longrightarrow> ardenable (rhs \<union> rhs')"
+by (auto simp:ardenable_def)
+
+lemma Arden_keeps_nonempty:
+  "ardenable rhs \<Longrightarrow> ardenable (Arden X rhs)"
+by (simp only:Arden_def Append_keeps_nonempty nonempty_set_sub)
+
+
+lemma Subst_keeps_nonempty:
+  "\<lbrakk>ardenable rhs; ardenable xrhs\<rbrakk> \<Longrightarrow> ardenable (Subst rhs X xrhs)"
+by (simp only: Subst_def Append_keeps_nonempty nonempty_set_union nonempty_set_sub)
+
+lemma Subst_keeps_eq:
+  assumes substor: "X = L xrhs"
+  and finite: "finite rhs"
+  shows "L (Subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
+proof-
+  def A \<equiv> "L (rhs - {Trn X r | r. Trn X r \<in> rhs})"
+  have "?Left = A \<union> L (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"
+    unfolding Subst_def
+    unfolding L_rhs_union_distrib[symmetric]
+    by (simp add: A_def)
+  moreover have "?Right = A \<union> L ({Trn X r | r. Trn X r \<in> rhs})"
+  proof-
+    have "rhs = (rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> ({Trn X r | r. Trn X r \<in> rhs})" by auto
+    thus ?thesis 
+      unfolding A_def
+      unfolding L_rhs_union_distrib
+      by simp
+  qed
+  moreover have "L (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = L ({Trn X r | r. Trn X r \<in> rhs})" 
+    using finite substor by (simp only: lang_of_append_rexp_rhs rhs_trm_soundness)
+  ultimately show ?thesis by simp
+qed
+
+lemma Subst_keeps_finite_rhs:
+  "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (Subst rhs Y yrhs)"
+by (auto simp: Subst_def Append_keeps_finite)
+
+lemma Subst_all_keeps_finite:
+  assumes finite: "finite ES"
+  shows "finite (Subst_all ES Y yrhs)"
+proof -
+  def eqns \<equiv> "{(X::lang, rhs) |X rhs. (X, rhs) \<in> ES}"
+  def h \<equiv> "\<lambda>(X::lang, rhs). (X, Subst rhs Y yrhs)"
+  have "finite (h ` eqns)" using finite h_def eqns_def by auto
+  moreover 
+  have "Subst_all ES Y yrhs = h ` eqns" unfolding h_def eqns_def Subst_all_def by auto
+  ultimately
+  show "finite (Subst_all ES Y yrhs)" by simp
+qed
+
+lemma Subst_all_keeps_finite_rhs:
+  "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (Subst_all ES Y yrhs)"
+by (auto intro:Subst_keeps_finite_rhs simp add:Subst_all_def finite_rhs_def)
+
+lemma append_rhs_keeps_cls:
+  "rhss (Append_rexp_rhs rhs r) = rhss rhs"
+apply (auto simp:rhss_def Append_rexp_rhs_def)
+apply (case_tac xa, auto simp:image_def)
+by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
+
+lemma Arden_removes_cl:
+  "rhss (Arden Y yrhs) = rhss yrhs - {Y}"
+apply (simp add:Arden_def append_rhs_keeps_cls)
+by (auto simp:rhss_def)
+
+lemma lhss_keeps_cls:
+  "lhss (Subst_all ES Y yrhs) = lhss ES"
+by (auto simp:lhss_def Subst_all_def)
+
+lemma Subst_updates_cls:
+  "X \<notin> rhss xrhs \<Longrightarrow> 
+      rhss (Subst rhs X xrhs) = rhss rhs \<union> rhss xrhs - {X}"
+apply (simp only:Subst_def append_rhs_keeps_cls rhss_union_distrib)
+by (auto simp:rhss_def)
+
+lemma Subst_all_keeps_validity:
+  assumes sc: "validity (ES \<union> {(Y, yrhs)})"        (is "validity ?A")
+  shows "validity (Subst_all ES Y (Arden Y yrhs))"  (is "validity ?B")
+proof -
+  { fix X xrhs'
+    assume "(X, xrhs') \<in> ?B"
+    then obtain xrhs 
+      where xrhs_xrhs': "xrhs' = Subst xrhs Y (Arden Y yrhs)"
+      and X_in: "(X, xrhs) \<in> ES" by (simp add:Subst_all_def, blast)    
+    have "rhss xrhs' \<subseteq> lhss ?B"
+    proof-
+      have "lhss ?B = lhss ES" by (auto simp add:lhss_def Subst_all_def)
+      moreover have "rhss xrhs' \<subseteq> lhss ES"
+      proof-
+        have "rhss xrhs' \<subseteq>  rhss xrhs \<union> rhss (Arden Y yrhs) - {Y}"
+        proof-
+          have "Y \<notin> rhss (Arden Y yrhs)" 
+            using Arden_removes_cl by simp
+          thus ?thesis using xrhs_xrhs' by (auto simp:Subst_updates_cls)
+        qed
+        moreover have "rhss xrhs \<subseteq> lhss ES \<union> {Y}" using X_in sc
+          apply (simp only:validity_def lhss_union_distrib)
+          by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lhss_def)
+        moreover have "rhss (Arden Y yrhs) \<subseteq> lhss ES \<union> {Y}" 
+          using sc 
+          by (auto simp add:Arden_removes_cl validity_def lhss_def)
+        ultimately show ?thesis by auto
+      qed
+      ultimately show ?thesis by simp
+    qed
+  } thus ?thesis by (auto simp only:Subst_all_def validity_def)
+qed
+
+lemma Subst_all_satisfies_invariant:
+  assumes invariant_ES: "invariant (ES \<union> {(Y, yrhs)})"
+  shows "invariant (Subst_all ES Y (Arden Y yrhs))"
+proof (rule invariantI)
+  have Y_eq_yrhs: "Y = L yrhs" 
+    using invariant_ES by (simp only:invariant_def soundness_def, blast)
+   have finite_yrhs: "finite yrhs" 
+    using invariant_ES by (auto simp:invariant_def finite_rhs_def)
+  have nonempty_yrhs: "ardenable yrhs" 
+    using invariant_ES by (auto simp:invariant_def ardenable_all_def)
+  show "soundness (Subst_all ES Y (Arden Y yrhs))"
+  proof -
+    have "Y = L (Arden Y yrhs)" 
+      using Y_eq_yrhs invariant_ES finite_yrhs
+      using finite_Trn[OF finite_yrhs]
+      apply(rule_tac Arden_keeps_eq)
+      apply(simp_all)
+      unfolding invariant_def ardenable_all_def ardenable_def
+      apply(auto)
+      done
+    thus ?thesis using invariant_ES
+      unfolding invariant_def finite_rhs_def2 soundness_def Subst_all_def
+      by (auto simp add: Subst_keeps_eq simp del: L_rhs.simps)
+  qed
+  show "finite (Subst_all ES Y (Arden Y yrhs))" 
+    using invariant_ES by (simp add:invariant_def Subst_all_keeps_finite)
+  show "distinctness (Subst_all ES Y (Arden Y yrhs))" 
+    using invariant_ES 
+    unfolding distinctness_def Subst_all_def invariant_def by auto
+  show "ardenable_all (Subst_all ES Y (Arden Y yrhs))"
+  proof - 
+    { fix X rhs
+      assume "(X, rhs) \<in> ES"
+      hence "ardenable rhs"  using invariant_ES  
+        by (auto simp add:invariant_def ardenable_all_def)
+      with nonempty_yrhs 
+      have "ardenable (Subst rhs Y (Arden Y yrhs))"
+        by (simp add:nonempty_yrhs 
+               Subst_keeps_nonempty Arden_keeps_nonempty)
+    } thus ?thesis by (auto simp add:ardenable_all_def Subst_all_def)
+  qed
+  show "finite_rhs (Subst_all ES Y (Arden Y yrhs))"
+  proof-
+    have "finite_rhs ES" using invariant_ES 
+      by (simp add:invariant_def finite_rhs_def)
+    moreover have "finite (Arden Y yrhs)"
+    proof -
+      have "finite yrhs" using invariant_ES 
+        by (auto simp:invariant_def finite_rhs_def)
+      thus ?thesis using Arden_keeps_finite by simp
+    qed
+    ultimately show ?thesis 
+      by (simp add:Subst_all_keeps_finite_rhs)
+  qed
+  show "validity (Subst_all ES Y (Arden Y yrhs))"
+    using invariant_ES Subst_all_keeps_validity by (simp add:invariant_def)
+qed
+
+lemma Remove_in_card_measure:
+  assumes finite: "finite ES"
+  and     in_ES: "(X, rhs) \<in> ES"
+  shows "(Remove ES X rhs, ES) \<in> measure card"
+proof -
+  def f \<equiv> "\<lambda> x. ((fst x)::lang, Subst (snd x) X (Arden X rhs))"
+  def ES' \<equiv> "ES - {(X, rhs)}"
+  have "Subst_all ES' X (Arden X rhs) = f ` ES'" 
+    apply (auto simp: Subst_all_def f_def image_def)
+    by (rule_tac x = "(Y, yrhs)" in bexI, simp+)
+  then have "card (Subst_all ES' X (Arden X rhs)) \<le> card ES'"
+    unfolding ES'_def using finite by (auto intro: card_image_le)
+  also have "\<dots> < card ES" unfolding ES'_def 
+    using in_ES finite by (rule_tac card_Diff1_less)
+  finally show "(Remove ES X rhs, ES) \<in> measure card" 
+    unfolding Remove_def ES'_def by simp
+qed
+    
+
+lemma Subst_all_cls_remains: 
+  "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (Subst_all ES Y yrhs)"
+by (auto simp: Subst_all_def)
+
+lemma card_noteq_1_has_more:
+  assumes card:"Cond ES"
+  and e_in: "(X, xrhs) \<in> ES"
+  and finite: "finite ES"
+  shows "\<exists>(Y, yrhs) \<in> ES. (X, xrhs) \<noteq> (Y, yrhs)"
+proof-
+  have "card ES > 1" using card e_in finite 
+    by (cases "card ES") (auto) 
+  then have "card (ES - {(X, xrhs)}) > 0"
+    using finite e_in by auto
+  then have "(ES - {(X, xrhs)}) \<noteq> {}" using finite by (rule_tac notI, simp)
+  then show "\<exists>(Y, yrhs) \<in> ES. (X, xrhs) \<noteq> (Y, yrhs)"
+    by auto
+qed
+
+lemma iteration_step_measure:
+  assumes Inv_ES: "invariant ES"
+  and    X_in_ES: "(X, xrhs) \<in> ES"
+  and    Cnd:     "Cond ES "
+  shows "(Iter X ES, ES) \<in> measure card"
+proof -
+  have fin: "finite ES" using Inv_ES unfolding invariant_def by simp
+  then obtain Y yrhs 
+    where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)" 
+    using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+  then have "(Y, yrhs) \<in> ES " "X \<noteq> Y"  
+    using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+    by auto
+  then show "(Iter X ES, ES) \<in> measure card" 
+  apply(rule IterI2)
+  apply(rule Remove_in_card_measure)
+  apply(simp_all add: fin)
+  done
+qed
+
+lemma iteration_step_invariant:
+  assumes Inv_ES: "invariant ES"
+  and    X_in_ES: "(X, xrhs) \<in> ES"
+  and    Cnd: "Cond ES"
+  shows "invariant (Iter X ES)"
+proof -
+  have finite_ES: "finite ES" using Inv_ES by (simp add: invariant_def)
+  then obtain Y yrhs 
+    where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)" 
+    using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+  then have "(Y, yrhs) \<in> ES" "X \<noteq> Y" 
+    using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+    by auto
+  then show "invariant (Iter X ES)" 
+  proof(rule IterI2)
+    fix Y yrhs
+    assume h: "(Y, yrhs) \<in> ES" "X \<noteq> Y"
+    then have "ES - {(Y, yrhs)} \<union> {(Y, yrhs)} = ES" by auto
+    then show "invariant (Remove ES Y yrhs)" unfolding Remove_def
+      using Inv_ES
+      by (rule_tac Subst_all_satisfies_invariant) (simp) 
+  qed
+qed
+
+lemma iteration_step_ex:
+  assumes Inv_ES: "invariant ES"
+  and    X_in_ES: "(X, xrhs) \<in> ES"
+  and    Cnd: "Cond ES"
+  shows "\<exists>xrhs'. (X, xrhs') \<in> (Iter X ES)"
+proof -
+  have finite_ES: "finite ES" using Inv_ES by (simp add: invariant_def)
+  then obtain Y yrhs 
+    where "(Y, yrhs) \<in> ES" "(X, xrhs) \<noteq> (Y, yrhs)" 
+    using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+  then have "(Y, yrhs) \<in> ES " "X \<noteq> Y"  
+    using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+    by auto
+  then show "\<exists>xrhs'. (X, xrhs') \<in> (Iter X ES)" 
+  apply(rule IterI2)
+  unfolding Remove_def
+  apply(rule Subst_all_cls_remains)
+  using X_in_ES
+  apply(auto)
+  done
+qed
+
+
+subsubsection {* Conclusion of the proof *}
+
+lemma Solve:
+  assumes fin: "finite (UNIV // \<approx>A)"
+  and     X_in: "X \<in> (UNIV // \<approx>A)"
+  shows "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}"
+proof -
+  def Inv \<equiv> "\<lambda>ES. invariant ES \<and> (\<exists>rhs. (X, rhs) \<in> ES)"
+  have "Inv (Init (UNIV // \<approx>A))" unfolding Inv_def
+      using fin X_in by (simp add: Init_ES_satisfies_invariant, simp add: Init_def)
+  moreover
+  { fix ES
+    assume inv: "Inv ES" and crd: "Cond ES"
+    then have "Inv (Iter X ES)"
+      unfolding Inv_def
+      by (auto simp add: iteration_step_invariant iteration_step_ex) }
+  moreover
+  { fix ES
+    assume inv: "Inv ES" and not_crd: "\<not>Cond ES"
+    from inv obtain rhs where "(X, rhs) \<in> ES" unfolding Inv_def by auto
+    moreover
+    from not_crd have "card ES = 1" by simp
+    ultimately 
+    have "ES = {(X, rhs)}" by (auto simp add: card_Suc_eq) 
+    then have "\<exists>rhs'. ES = {(X, rhs')} \<and> invariant {(X, rhs')}" using inv
+      unfolding Inv_def by auto }
+  moreover
+    have "wf (measure card)" by simp
+  moreover
+  { fix ES
+    assume inv: "Inv ES" and crd: "Cond ES"
+    then have "(Iter X ES, ES) \<in> measure card"
+      unfolding Inv_def
+      apply(clarify)
+      apply(rule_tac iteration_step_measure)
+      apply(auto)
+      done }
+  ultimately 
+  show "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}" 
+    unfolding Solve_def by (rule while_rule)
+qed
+
+lemma every_eqcl_has_reg:
+  assumes finite_CS: "finite (UNIV // \<approx>A)"
+  and X_in_CS: "X \<in> (UNIV // \<approx>A)"
+  shows "\<exists>r::rexp. X = L r" 
+proof -
+  from finite_CS X_in_CS 
+  obtain xrhs where Inv_ES: "invariant {(X, xrhs)}"
+    using Solve by metis
+
+  def A \<equiv> "Arden X xrhs"
+  have "rhss xrhs \<subseteq> {X}" using Inv_ES 
+    unfolding validity_def invariant_def rhss_def lhss_def
+    by auto
+  then have "rhss A = {}" unfolding A_def 
+    by (simp add: Arden_removes_cl)
+  then have eq: "{Lam r | r. Lam r \<in> A} = A" unfolding rhss_def
+    by (auto, case_tac x, auto)
+  
+  have "finite A" using Inv_ES unfolding A_def invariant_def finite_rhs_def
+    using Arden_keeps_finite by auto
+  then have fin: "finite {r. Lam r \<in> A}" by (rule finite_Lam)
+
+  have "X = L xrhs" using Inv_ES unfolding invariant_def soundness_def
+    by simp
+  then have "X = L A" using Inv_ES 
+    unfolding A_def invariant_def ardenable_all_def finite_rhs_def 
+    by (rule_tac Arden_keeps_eq) (simp_all add: finite_Trn)
+  then have "X = L {Lam r | r. Lam r \<in> A}" using eq by simp
+  then have "X = L (\<Uplus>{r. Lam r \<in> A})" using fin by auto
+  then show "\<exists>r::rexp. X = L r" by blast
+qed
+
+lemma bchoice_finite_set:
+  assumes a: "\<forall>x \<in> S. \<exists>y. x = f y" 
+  and     b: "finite S"
+  shows "\<exists>ys. (\<Union> S) = \<Union>(f ` ys) \<and> finite ys"
+using bchoice[OF a] b
+apply(erule_tac exE)
+apply(rule_tac x="fa ` S" in exI)
+apply(auto)
+done
+
+theorem Myhill_Nerode1:
+  assumes finite_CS: "finite (UNIV // \<approx>A)"
+  shows   "\<exists>r::rexp. A = L r"
+proof -
+  have fin: "finite (finals A)" 
+    using finals_in_partitions finite_CS by (rule finite_subset)
+  have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r::rexp. X = L r" 
+    using finite_CS every_eqcl_has_reg by blast
+  then have a: "\<forall>X \<in> finals A. \<exists>r::rexp. X = L r"
+    using finals_in_partitions by auto
+  then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L ` rs)" "finite rs"
+    using fin by (auto dest: bchoice_finite_set)
+  then have "A = L (\<Uplus>rs)" 
+    unfolding lang_is_union_of_finals[symmetric] by simp
+  then show "\<exists>r::rexp. A = L r" by blast
+qed 
+
+
+end
\ No newline at end of file