diff -r b04cc5e4e84c -r 7743d2ad71d1 Myhill_1.thy
--- a/Myhill_1.thy	Tue May 31 20:32:49 2011 +0000
+++ b/Myhill_1.thy	Thu Jun 02 16:44:35 2011 +0000
@@ -1,5 +1,5 @@
 theory Myhill_1
-imports Main Folds Regular
+imports Regular
         "~~/src/HOL/Library/While_Combinator" 
 begin
 
@@ -37,43 +37,37 @@
 
 text {* The two kinds of terms in the rhs of equations. *}
 
-datatype rhs_trm = 
+datatype trm = 
    Lam "rexp"            (* Lambda-marker *)
  | Trn "lang" "rexp"     (* Transition *)
 
+fun 
+  L_trm::"trm \<Rightarrow> lang"
+where
+  "L_trm (Lam r) = L_rexp r" 
+| "L_trm (Trn X r) = X \<cdot> L_rexp r"
 
-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
+fun 
+  L_rhs::"trm set \<Rightarrow> lang"
+where 
+  "L_rhs rhs = \<Union> (L_trm ` rhs)"
 
 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)
+  shows "L_rhs {Trn X r | r. P r} = \<Union>{L_trm (Trn X r) | r. P r}"
+by (auto)
 
 lemma L_rhs_union_distrib:
-  fixes A B::"rhs_trm set"
-  shows "L A \<union> L B = L (A \<union> B)"
+  fixes A B::"trm set"
+  shows "L_rhs A \<union> L_rhs B = L_rhs (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"
+  "Y \<Turnstile>c\<Rightarrow> X \<equiv> Y \<cdot> {[c]} \<subseteq> X"
 
 text {* Initial equational system *}
 
@@ -91,7 +85,7 @@
 section {* Arden Operation on equations *}
 
 fun 
-  Append_rexp :: "rexp \<Rightarrow> rhs_trm \<Rightarrow> rhs_trm"
+  Append_rexp :: "rexp \<Rightarrow> trm \<Rightarrow> trm"
 where
   "Append_rexp r (Lam rexp)   = Lam (SEQ rexp r)"
 | "Append_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"
@@ -112,7 +106,7 @@
         (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"
+  Subst_all :: "(lang \<times> trm set) set \<Rightarrow> lang \<Rightarrow> trm set \<Rightarrow> (lang \<times> trm set) set"
 where
   "Subst_all ES X xrhs \<equiv> {(Y, Subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
 
@@ -149,10 +143,10 @@
      \<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"
+  "soundness ES \<equiv> \<forall>(X, rhs) \<in> ES. X = L_rhs rhs"
 
 definition 
-  "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
+  "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L_rexp r)"
 
 definition 
   "ardenable_all ES \<equiv> \<forall>(X, rhs) \<in> ES. ardenable rhs"
@@ -227,29 +221,28 @@
     done
 qed
 
-lemma rhs_trm_soundness:
+lemma 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}))"
+  shows "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<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)
+  then show "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<Uplus>{r. Trn X r \<in> rhs}))"
+    by (simp only: L_rhs_set L_trm.simps) (auto simp add: Seq_def)
 qed
 
 lemma lang_of_append_rexp:
-  "L (Append_rexp r rhs_trm) = L rhs_trm ;; L r"
+  "L_trm (Append_rexp r trm) = L_trm trm \<cdot> L_rexp 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"
+  "L_rhs (Append_rexp_rhs rhs r) = L_rhs rhs \<cdot> L_rexp r"
 unfolding Append_rexp_rhs_def
 by (auto simp add: Seq_def lang_of_append_rexp)
 
 
-
-subsubsection {* Intialization *}
+subsubsection {* Intial Equational System *}
 
 lemma defined_by_str:
   assumes "s \<in> X" "X \<in> UNIV // \<approx>A" 
@@ -261,7 +254,7 @@
 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"
+  obtains Y where "Y \<in> UNIV // \<approx>A" and "Y \<cdot> {[c]} \<subseteq> X" and "s \<in> Y"
 proof -
   def Y \<equiv> "\<approx>A `` {s}"
   have "Y \<in> UNIV // \<approx>A" 
@@ -269,7 +262,7 @@
   moreover
   have "X = \<approx>A `` {s @ [c]}" 
     using has_str in_CS defined_by_str by blast
-  then have "Y ;; {[c]} \<subseteq> X" 
+  then have "Y \<cdot> {[c]} \<subseteq> X" 
     unfolding Y_def Image_def Seq_def
     unfolding str_eq_rel_def
     by clarsimp
@@ -281,14 +274,14 @@
 
 lemma l_eq_r_in_eqs:
   assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
-  shows "X = L rhs"
+  shows "X = L_rhs rhs"
 proof 
-  show "X \<subseteq> L rhs"
+  show "X \<subseteq> L_rhs 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
+      then have "x \<in> L_rhs rhs" using X_in_eqs in_X
 	unfolding Init_def Init_rhs_def
         by auto
     }
@@ -297,40 +290,40 @@
       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"
+      then obtain Y where "Y \<in> UNIV // \<approx>A" "Y \<cdot> {[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}"
+      then have "x \<in> L_rhs {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
+      then have "x \<in> L_rhs rhs" using X_in_eqs in_X
 	unfolding Init_def Init_rhs_def by simp
     }
-    ultimately show "x \<in> L rhs" by blast
+    ultimately show "x \<in> L_rhs rhs" by blast
   qed
 next
-  show "L rhs \<subseteq> X" using X_in_eqs
+  show "L_rhs 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)"
+  shows "X = \<Union> (L_trm `  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 S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y \<cdot> {[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"
+  moreover have "{Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y \<cdot> {[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
+  have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}" by auto
   then show "finite (Init_rhs CS X)" unfolding Init_rhs_def transition_def by simp
 qed
 
@@ -359,32 +352,32 @@
 subsubsection {* Interation step *}
 
 lemma Arden_keeps_eq:
-  assumes l_eq_r: "X = L rhs"
+  assumes l_eq_r: "X = L_rhs rhs"
   and not_empty: "ardenable rhs"
   and finite: "finite rhs"
-  shows "X = L (Arden X rhs)"
+  shows "X = L_rhs (Arden X rhs)"
 proof -
-  def A \<equiv> "L (\<Uplus>{r. Trn X r \<in> rhs})"
+  def A \<equiv> "L_rexp (\<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)"
+  def B \<equiv> "L_rhs (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"
+  have "X = L_rhs rhs" using l_eq_r by simp
+  also have "\<dots> = L_rhs (b \<union> (rhs - b))" unfolding b_def by auto
+  also have "\<dots> = L_rhs b \<union> B" unfolding B_def by (simp only: L_rhs_union_distrib)
+  also have "\<dots> = X \<cdot> A \<union> B"
     unfolding b_def
-    unfolding rhs_trm_soundness[OF finite]
+    unfolding trm_soundness[OF finite]
     unfolding A_def
     by blast
-  finally have "X = X ;; A \<union> B" . 
-  then have "X = B ;; A\<star>"
+  finally have "X = X \<cdot> A \<union> B" . 
+  then have "X = B \<cdot> A\<star>"
     by (simp add: arden[OF not_empty2])
-  also have "\<dots> = L (Arden X rhs)"
+  also have "\<dots> = L_rhs (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
+  finally show "X = L_rhs (Arden X rhs)" by simp
 qed 
 
 lemma Append_keeps_finite:
@@ -418,16 +411,16 @@
 by (simp only: Subst_def Append_keeps_nonempty nonempty_set_union nonempty_set_sub)
 
 lemma Subst_keeps_eq:
-  assumes substor: "X = L xrhs"
+  assumes substor: "X = L_rhs xrhs"
   and finite: "finite rhs"
-  shows "L (Subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
+  shows "L_rhs (Subst rhs X xrhs) = L_rhs 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}))"
+  def A \<equiv> "L_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs})"
+  have "?Left = A \<union> L_rhs (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})"
+  moreover have "?Right = A \<union> L_rhs {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 
@@ -435,8 +428,8 @@
       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)
+  moreover have "L_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = L_rhs {Trn X r | r. Trn X r \<in> rhs}" 
+    using finite substor by (simp only: lang_of_append_rexp_rhs trm_soundness)
   ultimately show ?thesis by simp
 qed
 
@@ -519,7 +512,7 @@
   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" 
+  have Y_eq_yrhs: "Y = L_rhs 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)
@@ -527,7 +520,7 @@
     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)" 
+    have "Y = L_rhs (Arden Y yrhs)" 
       using Y_eq_yrhs invariant_ES finite_yrhs
       using finite_Trn[OF finite_yrhs]
       apply(rule_tac Arden_keeps_eq)
@@ -723,7 +716,7 @@
 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" 
+  shows "\<exists>r. X = L_rexp r" 
 proof -
   from finite_CS X_in_CS 
   obtain xrhs where Inv_ES: "invariant {(X, xrhs)}"
@@ -742,14 +735,14 @@
     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
+  have "X = L_rhs xrhs" using Inv_ES unfolding invariant_def soundness_def
     by simp
-  then have "X = L A" using Inv_ES 
+  then have "X = L_rhs 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
+  then have "X = L_rhs {Lam r | r. Lam r \<in> A}" using eq by simp
+  then have "X = L_rexp (\<Uplus>{r. Lam r \<in> A})" using fin by auto
+  then show "\<exists>r. X = L_rexp r" by blast
 qed
 
 lemma bchoice_finite_set:
@@ -764,19 +757,19 @@
 
 theorem Myhill_Nerode1:
   assumes finite_CS: "finite (UNIV // \<approx>A)"
-  shows   "\<exists>r::rexp. A = L r"
+  shows   "\<exists>r. A = L_rexp 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" 
+  have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r. X = L_rexp r" 
     using finite_CS every_eqcl_has_reg by blast
-  then have a: "\<forall>X \<in> finals A. \<exists>r::rexp. X = L r"
+  then have a: "\<forall>X \<in> finals A. \<exists>r. X = L_rexp r"
     using finals_in_partitions by auto
-  then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L ` rs)" "finite rs"
+  then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L_rexp ` rs)" "finite rs"
     using fin by (auto dest: bchoice_finite_set)
-  then have "A = L (\<Uplus>rs)" 
+  then have "A = L_rexp (\<Uplus>rs)" 
     unfolding lang_is_union_of_finals[symmetric] by simp
-  then show "\<exists>r::rexp. A = L r" by blast
+  then show "\<exists>r. A = L_rexp r" by blast
 qed