--- 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