
diff -r a3e0056c228b -r 6457e668dee5 Myhill_1.thy
--- a/Myhill_1.thy	Wed Feb 09 03:52:28 2011 +0000
+++ b/Myhill_1.thy	Wed Feb 09 04:50:18 2011 +0000
@@ -1,33 +1,12 @@
 theory Myhill_1
-  imports Main 
+imports Main Folds
 begin
 
-(*
-text {*
-     \begin{figure}
-    \centering
-    \scalebox{0.95}{
-    \begin{tikzpicture}[->,>=latex,shorten >=1pt,auto,node distance=1.2cm, semithick]
-        \node[state,initial] (n1)                   {$1$};
-        \node[state,accepting] (n2) [right = 10em of n1]   {$2$};
-
-        \path (n1) edge [bend left] node {$0$} (n2)
-            (n1) edge [loop above] node{$1$} (n1)
-            (n2) edge [loop above] node{$0$} (n2)
-            (n2) edge [bend left]  node {$1$} (n1)
-            ;
-    \end{tikzpicture}}
-    \caption{An example automaton (or partition)}\label{fig:example_automata}
-    \end{figure}
-*}
-
-*)
-
-
 section {* Preliminary definitions *}
 
 types lang = "string set"
 
+
 text {*  Sequential composition of two languages *}
 
 definition 
@@ -151,10 +130,7 @@
   assumes a: "x \<in> A\<star>" "x \<noteq> []"
   shows "\<exists>a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> A \<and> b \<in> A\<star>"
 using a
-apply(induct rule: star_induct)
-apply(simp)
-apply(blast)
-done
+by (induct rule: star_induct) (blast)+
 
 lemma
   shows seq_Union_left:  "B ;; (\<Union>n. A \<up> n) = (\<Union>n. B ;; (A \<up> n))"
@@ -167,12 +143,11 @@
 
 lemma seq_star_comm:
   shows "A ;; A\<star> = A\<star> ;; A"
-unfolding Star_def
-unfolding seq_Union_left
-unfolding seq_pow_comm
-unfolding seq_Union_right 
+unfolding Star_def seq_Union_left
+unfolding seq_pow_comm seq_Union_right 
 by simp
 
+
 text {* Two lemmas about the length of strings in @{text "A \<up> n"} *}
 
 lemma pow_length:
@@ -209,12 +184,13 @@
 qed
 
 
-section {* A slightly modified version of Arden's lemma *}
+
+section {* A modified version of Arden's lemma *}
 
 
 text {*  A helper lemma for Arden *}
 
-lemma ardens_helper:
+lemma arden_helper:
   assumes eq: "X = X ;; A \<union> B"
   shows "X = X ;; (A \<up> Suc n) \<union> (\<Union>m\<in>{0..n}. B ;; (A \<up> m))"
 proof (induct n)
@@ -232,7 +208,7 @@
   finally show "X = X ;; (A \<up> Suc (Suc n)) \<union> (\<Union>m\<in>{0..Suc n}. B ;; (A \<up> m))" .
 qed
 
-theorem ardens_revised:
+theorem arden:
   assumes nemp: "[] \<notin> A"
   shows "X = X ;; A \<union> B \<longleftrightarrow> X = B ;; A\<star>"
 proof
@@ -251,10 +227,9 @@
 next
   assume eq: "X = X ;; A \<union> B"
   { fix n::nat
-    have "B ;; (A \<up> n) \<subseteq> X" using ardens_helper[OF eq, of "n"] by auto }
+    have "B ;; (A \<up> n) \<subseteq> X" using arden_helper[OF eq, of "n"] by auto }
   then have "B ;; A\<star> \<subseteq> X" 
-    unfolding Seq_def Star_def UNION_def
-    by auto
+    unfolding Seq_def Star_def UNION_def by auto
   moreover
   { fix s::string
     obtain k where "k = length s" by auto
@@ -262,14 +237,13 @@
       using seq_pow_length[OF nemp] by blast
     assume "s \<in> X"
     then have "s \<in> X ;; (A \<up> Suc k) \<union> (\<Union>m\<in>{0..k}. B ;; (A \<up> m))"
-      using ardens_helper[OF eq, of "k"] by auto
+      using arden_helper[OF eq, of "k"] by auto
     then have "s \<in> (\<Union>m\<in>{0..k}. B ;; (A \<up> m))" using not_in by auto
     moreover
     have "(\<Union>m\<in>{0..k}. B ;; (A \<up> m)) \<subseteq> (\<Union>n. B ;; (A \<up> n))" by auto
     ultimately 
     have "s \<in> B ;; A\<star>" 
-      unfolding seq_Union_left Star_def
-      by auto }
+      unfolding seq_Union_left Star_def by auto }
   then have "X \<subseteq> B ;; A\<star>" by auto
   ultimately 
   show "X = B ;; A\<star>" by simp
@@ -288,14 +262,12 @@
 
 
 text {* 
-  The following @{text "L"} is an overloaded operator, where @{text "L(x)"} evaluates to 
-  the language represented by the syntactic object @{text "x"}.
+  The function @{text L} is overloaded, with the idea that @{text "L x"} 
+  evaluates to the language represented by the object @{text x}.
 *}
 
 consts L:: "'a \<Rightarrow> lang"
 
-text {* The @{text "L (rexp)"} for regular expressions. *}
-
 overloading L_rexp \<equiv> "L::  rexp \<Rightarrow> lang"
 begin
 fun
@@ -309,22 +281,7 @@
   | "L_rexp (STAR r) = (L_rexp r)\<star>"
 end
 
-
-section {* Folds for Sets *}
-
-text {*
-  To obtain equational system out of finite set of equivalence classes, a fold operation
-  on finite sets @{text "folds"} is defined. The use of @{text "SOME"} makes @{text "folds"}
-  more robust than the @{text "fold"} in the Isabelle library. The expression @{text "folds f"}
-  makes sense when @{text "f"} is not @{text "associative"} and @{text "commutitive"},
-  while @{text "fold f"} does not.  
-*}
-
-
-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"
+text {* ALT-combination of a set or regulare expressions *}
 
 abbreviation
   Setalt  ("\<Uplus>_" [1000] 999) 
@@ -332,9 +289,7 @@
   "\<Uplus>A == folds ALT NULL A"
 
 text {* 
-  The following lemma ensures that the arbitrary choice made by the 
-  @{text "SOME"} in @{text "folds"} does not affect the @{text "L"}-value 
-  of the resultant regular expression. 
+  For finite sets, @{term Setalt} is preserved under @{term L}.
 *}
 
 lemma folds_alt_simp [simp]:
@@ -349,23 +304,26 @@
 done
 
 
+
+section {* Direction @{text "finite partition \<Rightarrow> regular language"} *}
+
+
 text {* Just a technical lemma for collections and pairs *}
 
 lemma Pair_Collect[simp]:
   shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
 by simp
 
-text {*
-  @{text "\<approx>A"} is an equivalence class defined by language @{text "A"}.
-*}
+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)}"
 
 text {* 
-  Among the equivalence clases of @{text "\<approx>A"}, the set @{text "finals A"} singles out 
-  those which contains the strings from @{text "A"}.
+  Among the equivalence clases of @{text "\<approx>A"}, the set @{text "finals A"} 
+  singles out those which contains the strings from @{text A}.
 *}
 
 definition 
@@ -373,10 +331,6 @@
 where
   "finals A \<equiv> {\<approx>A `` {x} | x . x \<in> A}"
 
-text {* 
-  The following lemma establishes the relationshipt between 
-  @{text "finals A"} and @{text "A"}.
-*}
 
 lemma lang_is_union_of_finals: 
   shows "A = \<Union> finals A"
@@ -394,64 +348,22 @@
 unfolding quotient_def
 by auto
 
-section {* Direction @{text "finite partition \<Rightarrow> regular language"}*}
 
-text {* 
-  The relationship between equivalent classes can be described by an
-  equational system.  For example, in equational system \eqref{example_eqns},
-  $X_0, X_1$ are equivalent classes. The first equation says every string in
-  $X_0$ is obtained either by appending one $b$ to a string in $X_0$ or by
-  appending one $a$ to a string in $X_1$ or just be an empty string
-  (represented by the regular expression $\lambda$). Similary, the second
-  equation tells how the strings inside $X_1$ are composed.
-
-  \begin{equation}\label{example_eqns}
-    \begin{aligned}
-      X_0 & = X_0 b + X_1 a + \lambda \\
-      X_1 & = X_0 a + X_1 b
-    \end{aligned}
-  \end{equation}
-
-  \noindent
-  The summands on the right hand side is represented by the following data
-  type @{text "rhs_item"}, mnemonic for 'right hand side item'.  Generally,
-  there are two kinds of right hand side items, one kind corresponds to pure
-  regular expressions, like the $\lambda$ in \eqref{example_eqns}, the other
-  kind corresponds to transitions from one one equivalent class to another,
-  like the $X_0 b, X_1 a$ etc.
-
-*}
+section {* Equational systems *}
 
 datatype rhs_item = 
-   Lam "rexp"            (* Lambda *)
+   Lam "rexp"            (* Lambda-marker *)
  | Trn "lang" "rexp"     (* Transition *)
 
 
-text {*
-  In this formalization, pure regular expressions like $\lambda$ is 
-  repsented by @{text "Lam(EMPTY)"}, while transitions like $X_0 a$ is 
-  represented by @{term "Trn X\<^isub>0 (CHAR a)"}.
-*}
-
-text {*
-  Every right-hand side item @{text "itm"} defines a language given 
-  by @{text "L(itm)"}, defined as:
-*}
-
-overloading L_rhs_e \<equiv> "L:: rhs_item \<Rightarrow> lang"
+overloading L_rhs_item \<equiv> "L:: rhs_item \<Rightarrow> lang"
 begin
-  fun L_rhs_e:: "rhs_item \<Rightarrow> lang"
+  fun L_rhs_item:: "rhs_item \<Rightarrow> lang"
   where
-    "L_rhs_e (Lam r) = L r" 
-  | "L_rhs_e (Trn X r) = X ;; L r"
+    "L_rhs_item (Lam r) = L r" 
+  | "L_rhs_item (Trn X r) = X ;; L r"
 end
 
-text {*
-  The right hand side of every equation is represented by a set of
-  items. The string set defined by such a set @{text "itms"} is given
-  by @{text "L(itms)"}, defined as:
-*}
-
 overloading L_rhs \<equiv> "L:: rhs_item set \<Rightarrow> lang"
 begin
    fun L_rhs:: "rhs_item set \<Rightarrow> lang"
@@ -459,18 +371,18 @@
      "L_rhs rhs = \<Union> (L ` rhs)"
 end
 
-text {* 
-  Given a set of equivalence classes @{text "CS"} and one equivalence class @{text "X"} among
-  @{text "CS"}, the term @{text "init_rhs CS X"} is used to extract the right hand side of
-  the equation describing the formation of @{text "X"}. The definition of @{text "init_rhs"}
-  is:
-*}
+definition
+  "trns_of rhs X \<equiv> {Trn X r | r. Trn X r \<in> rhs}"
+
+text {* Transitions between equivalence classes *}
 
 definition 
   transition :: "lang \<Rightarrow> rexp \<Rightarrow> lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)
 where
   "Y \<Turnstile>r\<Rightarrow> X \<equiv> Y ;; (L r) \<subseteq> X"
 
+text {* Initial equational system *}
+
 definition
   "init_rhs CS X \<equiv>  
       if ([] \<in> X) then 
@@ -478,89 +390,54 @@
       else 
           {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y \<Turnstile>(CHAR c)\<Rightarrow> X}"
 
-text {*
-  In the definition of @{text "init_rhs"}, the term 
-  @{text "{Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"} appearing on both branches
-  describes the formation of strings in @{text "X"} out of transitions, while 
-  the term @{text "{Lam(EMPTY)}"} describes the empty string which is intrinsically contained in
-  @{text "X"} rather than by transition. This @{text "{Lam(EMPTY)}"} corresponds to 
-  the $\lambda$ in \eqref{example_eqns}.
-
-  With the help of @{text "init_rhs"}, the equitional system descrbing the formation of every
-  equivalent class inside @{text "CS"} is given by the following @{text "eqs(CS)"}.
-*}
-
-
-definition "eqs CS \<equiv> {(X, init_rhs CS X) | X.  X \<in> CS}"
+definition 
+  "eqs CS \<equiv> {(X, init_rhs CS X) | X.  X \<in> CS}"
 
 
 
-(************ arden's lemma variation ********************)
-
-text {* 
-  The following @{text "trns_of rhs X"} returns all @{text "X"}-items in @{text "rhs"}.
-*}
-
-definition
-  "trns_of rhs X \<equiv> {Trn X r | r. Trn X r \<in> rhs}"
+section {* Arden Operation on equations *}
 
 text {*
-  The following @{text "attach_rexp rexp' itm"} attach 
-  the regular expression @{text "rexp'"} to
-  the right of right hand side item @{text "itm"}.
+  The function @{text "attach_rexp r item"} SEQ-composes @{text r} to the
+  right of every rhs-item.
 *}
 
 fun 
   attach_rexp :: "rexp \<Rightarrow> rhs_item \<Rightarrow> rhs_item"
 where
-  "attach_rexp rexp' (Lam rexp)   = Lam (SEQ rexp rexp')"
-| "attach_rexp rexp' (Trn X rexp) = Trn X (SEQ rexp rexp')"
+  "attach_rexp r (Lam rexp)   = Lam (SEQ rexp r)"
+| "attach_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"
 
-text {* 
-  The following @{text "append_rhs_rexp rhs rexp"} attaches 
-  @{text "rexp"} to every item in @{text "rhs"}.
-*}
 
 definition
   "append_rhs_rexp rhs rexp \<equiv> (attach_rexp rexp) ` rhs"
 
-text {*
-  With the help of the two functions immediately above, Ardens'
-  transformation on right hand side @{text "rhs"} is implemented
-  by the following function @{text "arden_variate X rhs"}.
-  After this transformation, the recursive occurence of @{text "X"}
-  in @{text "rhs"} will be eliminated, while the string set defined 
-  by @{text "rhs"} is kept unchanged.
+definition 
+  "arden_op X rhs \<equiv> 
+     append_rhs_rexp (rhs - trns_of rhs X) (STAR (\<Uplus> {r. Trn X r \<in> rhs}))"
+
+
+section {* Substitution Operation on equations *}
+
+text {* 
+  Suppose and equation @{text "X = xrhs"}, @{text "subst_op"} substitutes 
+  all occurences of @{text "X"} in @{text "rhs"} by @{text "xrhs"}.
 *}
 
 definition 
-  "arden_variate X rhs \<equiv> 
-        append_rhs_rexp (rhs - trns_of rhs X) (STAR (\<Uplus> {r. Trn X r \<in> rhs}))"
-
-
-(*********** substitution of ES *************)
-
-text {* 
-  Suppose the equation defining @{text "X"} is $X = xrhs$,
-  the purpose of @{text "rhs_subst"} is to substitute all occurences of @{text "X"} in
-  @{text "rhs"} by @{text "xrhs"}.
-  A litte thought may reveal that the final result
-  should be: first append $(a_1 | a_2 | \ldots | a_n)$ to every item of @{text "xrhs"} and then
-  union the result with all non-@{text "X"}-items of @{text "rhs"}.
- *}
-
-definition 
-  "rhs_subst rhs X xrhs \<equiv> 
+  "subst_op rhs X xrhs \<equiv> 
         (rhs - (trns_of rhs X)) \<union> (append_rhs_rexp xrhs (\<Uplus> {r. Trn X r \<in> rhs}))"
 
 text {*
-  Suppose the equation defining @{text "X"} is $X = xrhs$, the follwing
-  @{text "eqs_subst ES X xrhs"} substitute @{text "xrhs"} into every equation
-  of the equational system @{text "ES"}.
-  *}
+  @{text "eqs_subst ES X xrhs"} substitutes @{text xrhs} into every 
+  equation of the equational system @{text ES}.
+*}
 
 definition
-  "eqs_subst ES X xrhs \<equiv> {(Y, rhs_subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
+  "subst_op_all ES X xrhs \<equiv> {(Y, subst_op yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
+
+
+section {* Well-founded iteration *}
 
 text {*
   The computation of regular expressions for equivalence classes is accomplished
@@ -601,33 +478,36 @@
   Every definition starting next till @{text "Inv"} stipulates a property to be satisfied by @{text "ES"}.
 *}
 
-text {* 
-  Every variable is defined at most onece in @{text "ES"}.
-  *}
+
+section {* Invariants *}
+
+text {* Every variable is defined at most onece in @{text ES}. *}
 
 definition 
   "distinct_equas ES \<equiv> 
-            \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
+     \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
 
 text {* 
-  Every equation in @{text "ES"} (represented by @{text "(X, rhs)"}) is valid, i.e. @{text "(X = L rhs)"}.
-  *}
+  Every equation in @{text ES} (represented by @{text "(X, rhs)"}) 
+  is valid, i.e. @{text "(X = L rhs)"}.
+*}
+
 definition 
   "valid_eqns ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> (X = L rhs)"
 
 text {*
-  The following @{text "rhs_nonempty rhs"} requires regular expressions occuring in transitional 
-  items of @{text "rhs"} does not contain empty string. This is necessary for
-  the application of Arden's transformation to @{text "rhs"}.
-  *}
+  @{text "rhs_nonempty rhs"} requires regular expressions occuring in 
+  transitional items of @{text "rhs"} do not contain empty string. This is 
+  necessary for the application of Arden's transformation to @{text "rhs"}.
+*}
 
 definition 
   "rhs_nonempty rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
 
 text {*
-  The following @{text "ardenable ES"} requires that Arden's transformation is applicable
-  to every equation of equational system @{text "ES"}.
-  *}
+  The following @{text "ardenable ES"} requires that Arden's transformation 
+  is applicable to every equation of equational system @{text "ES"}.
+*}
 
 definition 
   "ardenable ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> rhs_nonempty rhs"
@@ -636,40 +516,41 @@
 definition 
   "non_empty ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> X \<noteq> {}"
 
-text {*
-  The following @{text "finite_rhs ES"} requires every equation in @{text "rhs"} be finite.
-  *}
+text {* 
+  @{text "finite_rhs ES"} requires every equation in @{text "rhs"} 
+  be finite.
+*}
 definition
   "finite_rhs ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs"
 
 text {*
-  The following @{text "classes_of rhs"} returns all variables (or equivalent classes)
+  @{text "classes_of rhs"} returns all variables (or equivalent classes)
   occuring in @{text "rhs"}.
   *}
+
 definition 
   "classes_of rhs \<equiv> {X. \<exists> r. Trn X r \<in> rhs}"
 
 text {*
-  The following @{text "lefts_of ES"} returns all variables 
-  defined by equational system @{text "ES"}.
-  *}
+  @{text "lefts_of ES"} returns all variables defined by an 
+  equational system @{text "ES"}.
+*}
 definition
   "lefts_of ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
 
 text {*
-  The following @{text "self_contained ES"} requires that every
-  variable occuring on the right hand side of equations is already defined by some
-  equation in @{text "ES"}.
-  *}
+  The following @{text "self_contained ES"} requires that every variable occuring 
+  on the right hand side of equations is already defined by some equation in @{text "ES"}.
+*}
 definition 
   "self_contained ES \<equiv> \<forall> (X, xrhs) \<in> ES. classes_of xrhs \<subseteq> lefts_of ES"
 
 
 text {*
-  The invariant @{text "Inv(ES)"} is a conjunction of all the previously defined constaints.
+  The invariant @{text "invariant(ES)"} is a conjunction of all the previously defined constaints.
   *}
 definition 
-  "Inv ES \<equiv> valid_eqns ES \<and> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and> 
+  "invariant ES \<equiv> valid_eqns ES \<and> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and> 
                 non_empty ES \<and> finite_rhs ES \<and> self_contained ES"
 
 subsection {* The proof of this direction *}
@@ -771,8 +652,8 @@
 subsubsection {* Intialization *}
 
 text {*
-  The following several lemmas until @{text "init_ES_satisfy_Inv"} shows that
-  the initial equational system satisfies invariant @{text "Inv"}.
+  The following several lemmas until @{text "init_ES_satisfy_invariant"} shows that
+  the initial equational system satisfies invariant @{text "invariant"}.
 *}
 
 lemma defined_by_str:
@@ -855,9 +736,9 @@
   thus ?thesis by (simp add:init_rhs_def transition_def)
 qed
 
-lemma init_ES_satisfy_Inv:
+lemma init_ES_satisfy_invariant:
   assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
-  shows "Inv (eqs (UNIV // (\<approx>Lang)))"
+  shows "invariant (eqs (UNIV // (\<approx>Lang)))"
 proof -
   have "finite (eqs (UNIV // (\<approx>Lang)))" using finite_CS
     by (simp add:eqs_def)
@@ -874,7 +755,7 @@
     by (auto simp:finite_rhs_def eqs_def)
   moreover have "self_contained (eqs (UNIV // (\<approx>Lang)))"
     by (auto simp:self_contained_def eqs_def init_rhs_def classes_of_def lefts_of_def)
-  ultimately show ?thesis by (simp add:Inv_def)
+  ultimately show ?thesis by (simp add:invariant_def)
 qed
 
 subsubsection {* 
@@ -883,15 +764,15 @@
 
 text {*
   From this point until @{text "iteration_step"}, it is proved
-  that there exists iteration steps which keep @{text "Inv(ES)"} while
+  that there exists iteration steps which keep @{text "invariant(ES)"} while
   decreasing the size of @{text "ES"}.
 *}
 
-lemma arden_variate_keeps_eq:
+lemma arden_op_keeps_eq:
   assumes l_eq_r: "X = L rhs"
   and not_empty: "[] \<notin> L (\<Uplus>{r. Trn X r \<in> rhs})"
   and finite: "finite rhs"
-  shows "X = L (arden_variate X rhs)"
+  shows "X = L (arden_op X rhs)"
 proof -
   def A \<equiv> "L (\<Uplus>{r. Trn X r \<in> rhs})"
   def b \<equiv> "rhs - trns_of rhs X"
@@ -905,13 +786,13 @@
       by (simp only: lang_of_rexp_of finite B_def A_def)
     finally show ?thesis
       using l_eq_r not_empty
-      apply(rule_tac ardens_revised[THEN iffD1])
+      apply(rule_tac arden[THEN iffD1])
       apply(simp add: A_def)
       apply(simp)
       done
   qed
-  moreover have "L (arden_variate X rhs) = (B ;; A\<star>)"
-    by (simp only:arden_variate_def L_rhs_union_distrib lang_of_append_rhs 
+  moreover have "L (arden_op X rhs) = (B ;; A\<star>)"
+    by (simp only:arden_op_def L_rhs_union_distrib lang_of_append_rhs 
                   B_def A_def b_def L_rexp.simps seq_union_distrib_left)
    ultimately show ?thesis by simp
 qed 
@@ -920,9 +801,9 @@
   "finite rhs \<Longrightarrow> finite (append_rhs_rexp rhs r)"
 by (auto simp:append_rhs_rexp_def)
 
-lemma arden_variate_keeps_finite:
-  "finite rhs \<Longrightarrow> finite (arden_variate X rhs)"
-by (auto simp:arden_variate_def append_keeps_finite)
+lemma arden_op_keeps_finite:
+  "finite rhs \<Longrightarrow> finite (arden_op X rhs)"
+by (auto simp:arden_op_def append_keeps_finite)
 
 lemma append_keeps_nonempty:
   "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (append_rhs_rexp rhs r)"
@@ -937,23 +818,23 @@
   "\<lbrakk>rhs_nonempty rhs; rhs_nonempty rhs'\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs \<union> rhs')"
 by (auto simp:rhs_nonempty_def)
 
-lemma arden_variate_keeps_nonempty:
-  "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (arden_variate X rhs)"
-by (simp only:arden_variate_def append_keeps_nonempty nonempty_set_sub)
+lemma arden_op_keeps_nonempty:
+  "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (arden_op X rhs)"
+by (simp only:arden_op_def append_keeps_nonempty nonempty_set_sub)
 
 
-lemma rhs_subst_keeps_nonempty:
-  "\<lbrakk>rhs_nonempty rhs; rhs_nonempty xrhs\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs_subst rhs X xrhs)"
-by (simp only:rhs_subst_def append_keeps_nonempty  nonempty_set_union nonempty_set_sub)
+lemma subst_op_keeps_nonempty:
+  "\<lbrakk>rhs_nonempty rhs; rhs_nonempty xrhs\<rbrakk> \<Longrightarrow> rhs_nonempty (subst_op rhs X xrhs)"
+by (simp only:subst_op_def append_keeps_nonempty  nonempty_set_union nonempty_set_sub)
 
-lemma rhs_subst_keeps_eq:
+lemma subst_op_keeps_eq:
   assumes substor: "X = L xrhs"
   and finite: "finite rhs"
-  shows "L (rhs_subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
+  shows "L (subst_op rhs X xrhs) = L rhs" (is "?Left = ?Right")
 proof-
   def A \<equiv> "L (rhs - trns_of rhs X)"
   have "?Left = A \<union> L (append_rhs_rexp xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"
-    unfolding rhs_subst_def
+    unfolding subst_op_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})"
@@ -970,29 +851,29 @@
   ultimately show ?thesis by simp
 qed
 
-lemma rhs_subst_keeps_finite_rhs:
-  "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (rhs_subst rhs Y yrhs)"
-by (auto simp:rhs_subst_def append_keeps_finite)
+lemma subst_op_keeps_finite_rhs:
+  "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (subst_op rhs Y yrhs)"
+by (auto simp:subst_op_def append_keeps_finite)
 
-lemma eqs_subst_keeps_finite:
+lemma subst_op_all_keeps_finite:
   assumes finite:"finite (ES:: (string set \<times> rhs_item set) set)"
-  shows "finite (eqs_subst ES Y yrhs)"
+  shows "finite (subst_op_all ES Y yrhs)"
 proof -
-  have "finite {(Ya, rhs_subst yrhsa Y yrhs) |Ya yrhsa. (Ya, yrhsa) \<in> ES}" 
+  have "finite {(Ya, subst_op yrhsa Y yrhs) |Ya yrhsa. (Ya, yrhsa) \<in> ES}" 
                                                                   (is "finite ?A")
   proof-
     def eqns' \<equiv> "{((Ya::string set), yrhsa)| Ya yrhsa. (Ya, yrhsa) \<in> ES}"
-    def h \<equiv> "\<lambda> ((Ya::string set), yrhsa). (Ya, rhs_subst yrhsa Y yrhs)"
+    def h \<equiv> "\<lambda> ((Ya::string set), yrhsa). (Ya, subst_op yrhsa Y yrhs)"
     have "finite (h ` eqns')" using finite h_def eqns'_def by auto
     moreover have "?A = h ` eqns'" by (auto simp:h_def eqns'_def)
     ultimately show ?thesis by auto      
   qed
-  thus ?thesis by (simp add:eqs_subst_def)
+  thus ?thesis by (simp add:subst_op_all_def)
 qed
 
-lemma eqs_subst_keeps_finite_rhs:
-  "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (eqs_subst ES Y yrhs)"
-by (auto intro:rhs_subst_keeps_finite_rhs simp add:eqs_subst_def finite_rhs_def)
+lemma subst_op_all_keeps_finite_rhs:
+  "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (subst_op_all ES Y yrhs)"
+by (auto intro:subst_op_keeps_finite_rhs simp add:subst_op_all_def finite_rhs_def)
 
 lemma append_rhs_keeps_cls:
   "classes_of (append_rhs_rexp rhs r) = classes_of rhs"
@@ -1000,131 +881,131 @@
 apply (case_tac xa, auto simp:image_def)
 by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
 
-lemma arden_variate_removes_cl:
-  "classes_of (arden_variate Y yrhs) = classes_of yrhs - {Y}"
-apply (simp add:arden_variate_def append_rhs_keeps_cls trns_of_def)
+lemma arden_op_removes_cl:
+  "classes_of (arden_op Y yrhs) = classes_of yrhs - {Y}"
+apply (simp add:arden_op_def append_rhs_keeps_cls trns_of_def)
 by (auto simp:classes_of_def)
 
 lemma lefts_of_keeps_cls:
-  "lefts_of (eqs_subst ES Y yrhs) = lefts_of ES"
-by (auto simp:lefts_of_def eqs_subst_def)
+  "lefts_of (subst_op_all ES Y yrhs) = lefts_of ES"
+by (auto simp:lefts_of_def subst_op_all_def)
 
-lemma rhs_subst_updates_cls:
+lemma subst_op_updates_cls:
   "X \<notin> classes_of xrhs \<Longrightarrow> 
-      classes_of (rhs_subst rhs X xrhs) = classes_of rhs \<union> classes_of xrhs - {X}"
-apply (simp only:rhs_subst_def append_rhs_keeps_cls 
+      classes_of (subst_op rhs X xrhs) = classes_of rhs \<union> classes_of xrhs - {X}"
+apply (simp only:subst_op_def append_rhs_keeps_cls 
                               classes_of_union_distrib[THEN sym])
 by (auto simp:classes_of_def trns_of_def)
 
-lemma eqs_subst_keeps_self_contained:
+lemma subst_op_all_keeps_self_contained:
   fixes Y
   assumes sc: "self_contained (ES \<union> {(Y, yrhs)})" (is "self_contained ?A")
-  shows "self_contained (eqs_subst ES Y (arden_variate Y yrhs))" 
+  shows "self_contained (subst_op_all ES Y (arden_op Y yrhs))" 
                                                    (is "self_contained ?B")
 proof-
   { fix X xrhs'
     assume "(X, xrhs') \<in> ?B"
     then obtain xrhs 
-      where xrhs_xrhs': "xrhs' = rhs_subst xrhs Y (arden_variate Y yrhs)"
-      and X_in: "(X, xrhs) \<in> ES" by (simp add:eqs_subst_def, blast)    
+      where xrhs_xrhs': "xrhs' = subst_op xrhs Y (arden_op Y yrhs)"
+      and X_in: "(X, xrhs) \<in> ES" by (simp add:subst_op_all_def, blast)    
     have "classes_of xrhs' \<subseteq> lefts_of ?B"
     proof-
-      have "lefts_of ?B = lefts_of ES" by (auto simp add:lefts_of_def eqs_subst_def)
+      have "lefts_of ?B = lefts_of ES" by (auto simp add:lefts_of_def subst_op_all_def)
       moreover have "classes_of xrhs' \<subseteq> lefts_of ES"
       proof-
         have "classes_of xrhs' \<subseteq> 
-                        classes_of xrhs \<union> classes_of (arden_variate Y yrhs) - {Y}"
+                        classes_of xrhs \<union> classes_of (arden_op Y yrhs) - {Y}"
         proof-
-          have "Y \<notin> classes_of (arden_variate Y yrhs)" 
-            using arden_variate_removes_cl by simp
-          thus ?thesis using xrhs_xrhs' by (auto simp:rhs_subst_updates_cls)
+          have "Y \<notin> classes_of (arden_op Y yrhs)" 
+            using arden_op_removes_cl by simp
+          thus ?thesis using xrhs_xrhs' by (auto simp:subst_op_updates_cls)
         qed
         moreover have "classes_of xrhs \<subseteq> lefts_of ES \<union> {Y}" using X_in sc
           apply (simp only:self_contained_def lefts_of_union_distrib[THEN sym])
           by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lefts_of_def)
-        moreover have "classes_of (arden_variate Y yrhs) \<subseteq> lefts_of ES \<union> {Y}" 
+        moreover have "classes_of (arden_op Y yrhs) \<subseteq> lefts_of ES \<union> {Y}" 
           using sc 
-          by (auto simp add:arden_variate_removes_cl self_contained_def lefts_of_def)
+          by (auto simp add:arden_op_removes_cl self_contained_def lefts_of_def)
         ultimately show ?thesis by auto
       qed
       ultimately show ?thesis by simp
     qed
-  } thus ?thesis by (auto simp only:eqs_subst_def self_contained_def)
+  } thus ?thesis by (auto simp only:subst_op_all_def self_contained_def)
 qed
 
-lemma eqs_subst_satisfy_Inv:
-  assumes Inv_ES: "Inv (ES \<union> {(Y, yrhs)})"
-  shows "Inv (eqs_subst ES Y (arden_variate Y yrhs))"
+lemma subst_op_all_satisfy_invariant:
+  assumes invariant_ES: "invariant (ES \<union> {(Y, yrhs)})"
+  shows "invariant (subst_op_all ES Y (arden_op Y yrhs))"
 proof -  
   have finite_yrhs: "finite yrhs" 
-    using Inv_ES by (auto simp:Inv_def finite_rhs_def)
+    using invariant_ES by (auto simp:invariant_def finite_rhs_def)
   have nonempty_yrhs: "rhs_nonempty yrhs" 
-    using Inv_ES by (auto simp:Inv_def ardenable_def)
+    using invariant_ES by (auto simp:invariant_def ardenable_def)
   have Y_eq_yrhs: "Y = L yrhs" 
-    using Inv_ES by (simp only:Inv_def valid_eqns_def, blast)
-  have "distinct_equas (eqs_subst ES Y (arden_variate Y yrhs))" 
-    using Inv_ES
-    by (auto simp:distinct_equas_def eqs_subst_def Inv_def)
-  moreover have "finite (eqs_subst ES Y (arden_variate Y yrhs))" 
-    using Inv_ES by (simp add:Inv_def eqs_subst_keeps_finite)
-  moreover have "finite_rhs (eqs_subst ES Y (arden_variate Y yrhs))"
+    using invariant_ES by (simp only:invariant_def valid_eqns_def, blast)
+  have "distinct_equas (subst_op_all ES Y (arden_op Y yrhs))" 
+    using invariant_ES
+    by (auto simp:distinct_equas_def subst_op_all_def invariant_def)
+  moreover have "finite (subst_op_all ES Y (arden_op Y yrhs))" 
+    using invariant_ES by (simp add:invariant_def subst_op_all_keeps_finite)
+  moreover have "finite_rhs (subst_op_all ES Y (arden_op Y yrhs))"
   proof-
-    have "finite_rhs ES" using Inv_ES 
-      by (simp add:Inv_def finite_rhs_def)
-    moreover have "finite (arden_variate Y yrhs)"
+    have "finite_rhs ES" using invariant_ES 
+      by (simp add:invariant_def finite_rhs_def)
+    moreover have "finite (arden_op Y yrhs)"
     proof -
-      have "finite yrhs" using Inv_ES 
-        by (auto simp:Inv_def finite_rhs_def)
-      thus ?thesis using arden_variate_keeps_finite by simp
+      have "finite yrhs" using invariant_ES 
+        by (auto simp:invariant_def finite_rhs_def)
+      thus ?thesis using arden_op_keeps_finite by simp
     qed
     ultimately show ?thesis 
-      by (simp add:eqs_subst_keeps_finite_rhs)
+      by (simp add:subst_op_all_keeps_finite_rhs)
   qed
-  moreover have "ardenable (eqs_subst ES Y (arden_variate Y yrhs))"
+  moreover have "ardenable (subst_op_all ES Y (arden_op Y yrhs))"
   proof - 
     { fix X rhs
       assume "(X, rhs) \<in> ES"
-      hence "rhs_nonempty rhs"  using prems Inv_ES  
-        by (simp add:Inv_def ardenable_def)
+      hence "rhs_nonempty rhs"  using prems invariant_ES  
+        by (simp add:invariant_def ardenable_def)
       with nonempty_yrhs 
-      have "rhs_nonempty (rhs_subst rhs Y (arden_variate Y yrhs))"
+      have "rhs_nonempty (subst_op rhs Y (arden_op Y yrhs))"
         by (simp add:nonempty_yrhs 
-               rhs_subst_keeps_nonempty arden_variate_keeps_nonempty)
-    } thus ?thesis by (auto simp add:ardenable_def eqs_subst_def)
+               subst_op_keeps_nonempty arden_op_keeps_nonempty)
+    } thus ?thesis by (auto simp add:ardenable_def subst_op_all_def)
   qed
-  moreover have "valid_eqns (eqs_subst ES Y (arden_variate Y yrhs))"
+  moreover have "valid_eqns (subst_op_all ES Y (arden_op Y yrhs))"
   proof-
-    have "Y = L (arden_variate Y yrhs)" 
-      using Y_eq_yrhs Inv_ES finite_yrhs nonempty_yrhs      
-      by (rule_tac arden_variate_keeps_eq, (simp add:rexp_of_empty)+)
-    thus ?thesis using Inv_ES 
+    have "Y = L (arden_op Y yrhs)" 
+      using Y_eq_yrhs invariant_ES finite_yrhs nonempty_yrhs      
+      by (rule_tac arden_op_keeps_eq, (simp add:rexp_of_empty)+)
+    thus ?thesis using invariant_ES 
       by (clarsimp simp add:valid_eqns_def 
-              eqs_subst_def rhs_subst_keeps_eq Inv_def finite_rhs_def
+              subst_op_all_def subst_op_keeps_eq invariant_def finite_rhs_def
                    simp del:L_rhs.simps)
   qed
   moreover have 
-    non_empty_subst: "non_empty (eqs_subst ES Y (arden_variate Y yrhs))"
-    using Inv_ES by (auto simp:Inv_def non_empty_def eqs_subst_def)
+    non_empty_subst: "non_empty (subst_op_all ES Y (arden_op Y yrhs))"
+    using invariant_ES by (auto simp:invariant_def non_empty_def subst_op_all_def)
   moreover 
-  have self_subst: "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
-    using Inv_ES eqs_subst_keeps_self_contained by (simp add:Inv_def)
-  ultimately show ?thesis using Inv_ES by (simp add:Inv_def)
+  have self_subst: "self_contained (subst_op_all ES Y (arden_op Y yrhs))"
+    using invariant_ES subst_op_all_keeps_self_contained by (simp add:invariant_def)
+  ultimately show ?thesis using invariant_ES by (simp add:invariant_def)
 qed
 
-lemma eqs_subst_card_le: 
+lemma subst_op_all_card_le: 
   assumes finite: "finite (ES::(string set \<times> rhs_item set) set)"
-  shows "card (eqs_subst ES Y yrhs) <= card ES"
+  shows "card (subst_op_all ES Y yrhs) <= card ES"
 proof-
-  def f \<equiv> "\<lambda> x. ((fst x)::string set, rhs_subst (snd x) Y yrhs)"
-  have "eqs_subst ES Y yrhs = f ` ES" 
-    apply (auto simp:eqs_subst_def f_def image_def)
+  def f \<equiv> "\<lambda> x. ((fst x)::string set, subst_op (snd x) Y yrhs)"
+  have "subst_op_all ES Y yrhs = f ` ES" 
+    apply (auto simp:subst_op_all_def f_def image_def)
     by (rule_tac x = "(Ya, yrhsa)" in bexI, simp+)
   thus ?thesis using finite by (auto intro:card_image_le)
 qed
 
-lemma eqs_subst_cls_remains: 
-  "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (eqs_subst ES Y yrhs)"
-by (auto simp:eqs_subst_def)
+lemma subst_op_all_cls_remains: 
+  "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (subst_op_all ES Y yrhs)"
+by (auto simp:subst_op_all_def)
 
 lemma card_noteq_1_has_more:
   assumes card:"card S \<noteq> 1"
@@ -1143,31 +1024,31 @@
 qed
 
 lemma iteration_step: 
-  assumes Inv_ES: "Inv ES"
+  assumes invariant_ES: "invariant ES"
   and    X_in_ES: "(X, xrhs) \<in> ES"
   and    not_T: "card ES \<noteq> 1"
-  shows "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'.(X, xrhs') \<in> ES')) \<and> 
+  shows "\<exists> ES'. (invariant ES' \<and> (\<exists> xrhs'.(X, xrhs') \<in> ES')) \<and> 
                 (card ES', card ES) \<in> less_than" (is "\<exists> ES'. ?P ES'")
 proof -
-  have finite_ES: "finite ES" using Inv_ES by (simp add:Inv_def)
+  have finite_ES: "finite ES" using invariant_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 not_T X_in_ES by (drule_tac card_noteq_1_has_more, auto)
   def ES' == "ES - {(Y, yrhs)}"
-  let ?ES'' = "eqs_subst ES' Y (arden_variate Y yrhs)"
+  let ?ES'' = "subst_op_all ES' Y (arden_op Y yrhs)"
   have "?P ?ES''"
   proof -
-    have "Inv ?ES''" using Y_in_ES Inv_ES
-      by (rule_tac eqs_subst_satisfy_Inv, simp add:ES'_def insert_absorb)
+    have "invariant ?ES''" using Y_in_ES invariant_ES
+      by (rule_tac subst_op_all_satisfy_invariant, simp add:ES'_def insert_absorb)
     moreover have "\<exists>xrhs'. (X, xrhs') \<in> ?ES''"  using not_eq X_in_ES
-      by (rule_tac ES = ES' in eqs_subst_cls_remains, auto simp add:ES'_def)
+      by (rule_tac ES = ES' in subst_op_all_cls_remains, auto simp add:ES'_def)
     moreover have "(card ?ES'', card ES) \<in> less_than" 
     proof -
       have "finite ES'" using finite_ES ES'_def by auto
       moreover have "card ES' < card ES" using finite_ES Y_in_ES
         by (auto simp:ES'_def card_gt_0_iff intro:diff_Suc_less)
       ultimately show ?thesis 
-        by (auto dest:eqs_subst_card_le elim:le_less_trans)
+        by (auto dest:subst_op_all_card_le elim:le_less_trans)
     qed
     ultimately show ?thesis by simp
   qed
@@ -1184,10 +1065,10 @@
 *}
 
 lemma iteration_conc: 
-  assumes history: "Inv ES"
+  assumes history: "invariant ES"
   and    X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES"
   shows 
-  "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'. (X, xrhs') \<in> ES')) \<and> card ES' = 1" 
+  "\<exists> ES'. (invariant ES' \<and> (\<exists> xrhs'. (X, xrhs') \<in> ES')) \<and> card ES' = 1" 
                                                           (is "\<exists> ES'. ?P ES'")
 proof (cases "card ES = 1")
   case True
@@ -1201,28 +1082,28 @@
   
 lemma last_cl_exists_rexp:
   assumes ES_single: "ES = {(X, xrhs)}" 
-  and Inv_ES: "Inv ES"
+  and invariant_ES: "invariant ES"
   shows "\<exists> (r::rexp). L r = X" (is "\<exists> r. ?P r")
 proof-
-  def A \<equiv> "arden_variate X xrhs"
+  def A \<equiv> "arden_op X xrhs"
   have "?P (\<Uplus>{r. Lam r \<in> A})"
   proof -
     have "L (\<Uplus>{r. Lam r \<in> A}) = L ({Lam r | r. Lam r \<in>  A})"
     proof(rule rexp_of_lam_eq_lam_set)
       show "finite A" 
 	unfolding A_def
-	using Inv_ES ES_single 
-        by (rule_tac arden_variate_keeps_finite) 
-           (auto simp add: Inv_def finite_rhs_def)
+	using invariant_ES ES_single 
+        by (rule_tac arden_op_keeps_finite) 
+           (auto simp add: invariant_def finite_rhs_def)
     qed
     also have "\<dots> = L A"
     proof-
       have "{Lam r | r. Lam r \<in> A} = A"
       proof-
-        have "classes_of A = {}" using Inv_ES ES_single
+        have "classes_of A = {}" using invariant_ES ES_single
 	  unfolding A_def
-          by (simp add:arden_variate_removes_cl 
-                       self_contained_def Inv_def lefts_of_def) 
+          by (simp add:arden_op_removes_cl 
+                       self_contained_def invariant_def lefts_of_def) 
         thus ?thesis
 	  unfolding A_def
           by (auto simp only: classes_of_def, case_tac x, auto)
@@ -1231,15 +1112,15 @@
     qed
     also have "\<dots> = X"
     unfolding A_def
-    proof(rule arden_variate_keeps_eq [THEN sym])
-      show "X = L xrhs" using Inv_ES ES_single 
-        by (auto simp only:Inv_def valid_eqns_def)  
+    proof(rule arden_op_keeps_eq [THEN sym])
+      show "X = L xrhs" using invariant_ES ES_single 
+        by (auto simp only:invariant_def valid_eqns_def)  
     next
-      from Inv_ES ES_single show "[] \<notin> L (\<Uplus>{r. Trn X r \<in> xrhs})"
-        by(simp add:Inv_def ardenable_def rexp_of_empty finite_rhs_def)
+      from invariant_ES ES_single show "[] \<notin> L (\<Uplus>{r. Trn X r \<in> xrhs})"
+        by(simp add:invariant_def ardenable_def rexp_of_empty finite_rhs_def)
     next
-      from Inv_ES ES_single show "finite xrhs" 
-        by (simp add:Inv_def finite_rhs_def)
+      from invariant_ES ES_single show "finite xrhs" 
+        by (simp add:invariant_def finite_rhs_def)
     qed
     finally show ?thesis by simp
   qed
@@ -1253,14 +1134,14 @@
 proof -
   from X_in_CS have "\<exists> xrhs. (X, xrhs) \<in> (eqs (UNIV  // (\<approx>Lang)))"
     by (auto simp:eqs_def init_rhs_def)
-  then obtain ES xrhs where Inv_ES: "Inv ES" 
+  then obtain ES xrhs where invariant_ES: "invariant ES" 
     and X_in_ES: "(X, xrhs) \<in> ES"
     and card_ES: "card ES = 1"
-    using finite_CS X_in_CS init_ES_satisfy_Inv iteration_conc
+    using finite_CS X_in_CS init_ES_satisfy_invariant iteration_conc
     by blast
   hence ES_single_equa: "ES = {(X, xrhs)}" 
-    by (auto simp:Inv_def dest!:card_Suc_Diff1 simp:card_eq_0_iff) 
-  thus ?thesis using Inv_ES
+    by (auto simp:invariant_def dest!:card_Suc_Diff1 simp:card_eq_0_iff) 
+  thus ?thesis using invariant_ES
     by (rule last_cl_exists_rexp)
 qed
 




