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theory Myhill_1
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imports Main List_Prefix Prefix_subtract Prelude
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begin
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(*
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text {*
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\begin{figure}
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\centering
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\scalebox{0.95}{
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\begin{tikzpicture}[->,>=latex,shorten >=1pt,auto,node distance=1.2cm, semithick]
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\node[state,initial] (n1) {$1$};
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\node[state,accepting] (n2) [right = 10em of n1] {$2$};
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\path (n1) edge [bend left] node {$0$} (n2)
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(n1) edge [loop above] node{$1$} (n1)
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(n2) edge [loop above] node{$0$} (n2)
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(n2) edge [bend left] node {$1$} (n1)
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;
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\end{tikzpicture}}
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\caption{An example automaton (or partition)}\label{fig:example_automata}
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\end{figure}
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*}
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*)
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section {* Preliminary definitions *}
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types lang = "string set"
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text {*
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Sequential composition of two languages @{text "L1"} and @{text "L2"}
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*}
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definition Seq :: "lang \<Rightarrow> lang \<Rightarrow> lang" ("_ ;; _" [100,100] 100)
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where
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"L1 ;; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
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text {* Transitive closure of language @{text "L"}. *}
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inductive_set
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Star :: "lang \<Rightarrow> lang" ("_\<star>" [101] 102)
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for L
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where
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start[intro]: "[] \<in> L\<star>"
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| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
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text {* Some properties of operator @{text ";;"}.*}
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lemma seq_union_distrib:
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"(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"
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by (auto simp:Seq_def)
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lemma seq_intro:
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"\<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x @ y \<in> A ;; B "
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by (auto simp:Seq_def)
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lemma seq_assoc:
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"(A ;; B) ;; C = A ;; (B ;; C)"
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apply(auto simp:Seq_def)
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apply blast
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by (metis append_assoc)
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lemma star_intro1[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
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by (erule Star.induct, auto)
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lemma star_intro2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
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by (drule step[of y lang "[]"], auto simp:start)
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lemma star_intro3[rule_format]:
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"x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
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by (erule Star.induct, auto intro:star_intro2)
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lemma star_decom:
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"\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow>(\<exists> a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> lang \<and> b \<in> lang\<star>)"
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by (induct x rule: Star.induct, simp, blast)
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lemma star_decom':
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"\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow> \<exists>a b. x = a @ b \<and> a \<in> lang\<star> \<and> b \<in> lang"
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apply (induct x rule:Star.induct, simp)
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apply (case_tac "s2 = []")
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apply (rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
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apply (simp, (erule exE| erule conjE)+)
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by (rule_tac x = "s1 @ a" in exI, rule_tac x = b in exI, simp add:step)
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text {* Ardens lemma expressed at the level of language, rather than the level of regular expression. *}
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theorem ardens_revised:
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assumes nemp: "[] \<notin> A"
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shows "(X = X ;; A \<union> B) \<longleftrightarrow> (X = B ;; A\<star>)"
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proof
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assume eq: "X = B ;; A\<star>"
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have "A\<star> = {[]} \<union> A\<star> ;; A"
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by (auto simp:Seq_def star_intro3 star_decom')
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then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)"
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unfolding Seq_def by simp
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also have "\<dots> = B \<union> B ;; (A\<star> ;; A)"
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unfolding Seq_def by auto
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also have "\<dots> = B \<union> (B ;; A\<star>) ;; A"
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by (simp only:seq_assoc)
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finally show "X = X ;; A \<union> B"
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using eq by blast
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next
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assume eq': "X = X ;; A \<union> B"
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hence c1': "\<And> x. x \<in> B \<Longrightarrow> x \<in> X"
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and c2': "\<And> x y. \<lbrakk>x \<in> X; y \<in> A\<rbrakk> \<Longrightarrow> x @ y \<in> X"
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using Seq_def by auto
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show "X = B ;; A\<star>"
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proof
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show "B ;; A\<star> \<subseteq> X"
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proof-
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{ fix x y
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have "\<lbrakk>y \<in> A\<star>; x \<in> X\<rbrakk> \<Longrightarrow> x @ y \<in> X "
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apply (induct arbitrary:x rule:Star.induct, simp)
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by (auto simp only:append_assoc[THEN sym] dest:c2')
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} thus ?thesis using c1' by (auto simp:Seq_def)
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qed
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next
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show "X \<subseteq> B ;; A\<star>"
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proof-
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{ fix x
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have "x \<in> X \<Longrightarrow> x \<in> B ;; A\<star>"
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proof (induct x taking:length rule:measure_induct)
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fix z
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assume hyps:
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"\<forall>y. length y < length z \<longrightarrow> y \<in> X \<longrightarrow> y \<in> B ;; A\<star>"
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and z_in: "z \<in> X"
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show "z \<in> B ;; A\<star>"
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proof (cases "z \<in> B")
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case True thus ?thesis by (auto simp:Seq_def start)
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next
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case False hence "z \<in> X ;; A" using eq' z_in by auto
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then obtain za zb where za_in: "za \<in> X"
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and zab: "z = za @ zb \<and> zb \<in> A" and zbne: "zb \<noteq> []"
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using nemp unfolding Seq_def by blast
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from zbne zab have "length za < length z" by auto
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with za_in hyps have "za \<in> B ;; A\<star>" by blast
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hence "za @ zb \<in> B ;; A\<star>" using zab
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by (clarsimp simp:Seq_def, blast dest:star_intro3)
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thus ?thesis using zab by simp
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qed
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qed
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} thus ?thesis by blast
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qed
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qed
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qed
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text {* The syntax of regular expressions is defined by the datatype @{text "rexp"}. *}
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datatype rexp =
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NULL
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| EMPTY
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| CHAR char
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| SEQ rexp rexp
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| ALT rexp rexp
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| STAR rexp
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text {*
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The following @{text "L"} is an overloaded operator, where @{text "L(x)"} evaluates to
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the language represented by the syntactic object @{text "x"}.
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*}
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consts L:: "'a \<Rightarrow> string set"
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text {*
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The @{text "L(rexp)"} for regular expression @{text "rexp"} is defined by the
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following overloading function @{text "L_rexp"}.
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*}
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overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> string set"
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begin
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fun
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L_rexp :: "rexp \<Rightarrow> string set"
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where
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"L_rexp (NULL) = {}"
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| "L_rexp (EMPTY) = {[]}"
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| "L_rexp (CHAR c) = {[c]}"
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| "L_rexp (SEQ r1 r2) = (L_rexp r1) ;; (L_rexp r2)"
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| "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
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| "L_rexp (STAR r) = (L_rexp r)\<star>"
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end
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text {*
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To obtain equational system out of finite set of equivalent classes, a fold operation
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on finite set @{text "folds"} is defined. The use of @{text "SOME"} makes @{text "fold"}
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more robust than the @{text "fold"} in Isabelle library. The expression @{text "folds f"}
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makes sense when @{text "f"} is not @{text "associative"} and @{text "commutitive"},
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while @{text "fold f"} does not.
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*}
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definition
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folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
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where
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"folds f z S \<equiv> SOME x. fold_graph f z S x"
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text {*
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The following lemma assures that the arbitrary choice made by the @{text "SOME"} in @{text "folds"}
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does not affect the @{text "L"}-value of the resultant regular expression.
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*}
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lemma folds_alt_simp [simp]:
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"finite rs \<Longrightarrow> L (folds ALT NULL rs) = \<Union> (L ` rs)"
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apply (rule set_eq_intro, simp add:folds_def)
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apply (rule someI2_ex, erule finite_imp_fold_graph)
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by (erule fold_graph.induct, auto)
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(* Just a technical lemma. *)
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lemma [simp]:
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shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
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by simp
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text {*
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@{text "\<approx>L"} is an equivalent class defined by language @{text "Lang"}.
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*}
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definition
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str_eq_rel ("\<approx>_" [100] 100)
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where
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"\<approx>Lang \<equiv> {(x, y). (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)}"
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text {*
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Among equivlant clases of @{text "\<approx>Lang"}, the set @{text "finals(Lang)"} singles out
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those which contains strings from @{text "Lang"}.
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*}
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definition
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"finals Lang \<equiv> {\<approx>Lang `` {x} | x . x \<in> Lang}"
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text {*
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The following lemma show the relationshipt between @{text "finals(Lang)"} and @{text "Lang"}.
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*}
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lemma lang_is_union_of_finals:
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"Lang = \<Union> finals(Lang)"
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proof
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show "Lang \<subseteq> \<Union> (finals Lang)"
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proof
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fix x
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assume "x \<in> Lang"
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thus "x \<in> \<Union> (finals Lang)"
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apply (simp add:finals_def, rule_tac x = "(\<approx>Lang) `` {x}" in exI)
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by (auto simp:Image_def str_eq_rel_def)
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qed
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next
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show "\<Union> (finals Lang) \<subseteq> Lang"
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apply (clarsimp simp:finals_def str_eq_rel_def)
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by (drule_tac x = "[]" in spec, auto)
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qed
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section {* Direction @{text "finite partition \<Rightarrow> regular language"}*}
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text {*
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The relationship between equivalent classes can be described by an
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equational system.
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For example, in equational system \eqref{example_eqns}, $X_0, X_1$ are equivalent
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classes. The first equation says every string in $X_0$ is obtained either by
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appending one $b$ to a string in $X_0$ or by appending one $a$ to a string in
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$X_1$ or just be an empty string (represented by the regular expression $\lambda$). Similary,
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the second equation tells how the strings inside $X_1$ are composed.
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\begin{equation}\label{example_eqns}
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\begin{aligned}
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X_0 & = X_0 b + X_1 a + \lambda \\
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X_1 & = X_0 a + X_1 b
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\end{aligned}
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\end{equation}
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The summands on the right hand side is represented by the following data type
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@{text "rhs_item"}, mnemonic for 'right hand side item'.
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Generally, there are two kinds of right hand side items, one kind corresponds to
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pure regular expressions, like the $\lambda$ in \eqref{example_eqns}, the other kind corresponds to
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transitions from one one equivalent class to another, like the $X_0 b, X_1 a$ etc.
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*}
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datatype rhs_item =
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Lam "rexp" (* Lambda *)
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| Trn "(string set)" "rexp" (* Transition *)
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text {*
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In this formalization, pure regular expressions like $\lambda$ is
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repsented by @{text "Lam(EMPTY)"}, while transitions like $X_0 a$ is represented by $Trn~X_0~(CHAR~a)$.
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*}
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text {*
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The functions @{text "the_r"} and @{text "the_Trn"} are used to extract
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subcomponents from right hand side items.
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*}
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fun the_r :: "rhs_item \<Rightarrow> rexp"
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where "the_r (Lam r) = r"
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fun the_Trn:: "rhs_item \<Rightarrow> (string set \<times> rexp)"
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where "the_Trn (Trn Y r) = (Y, r)"
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text {*
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Every right hand side item @{text "itm"} defines a string set given
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@{text "L(itm)"}, defined as:
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*}
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overloading L_rhs_e \<equiv> "L:: rhs_item \<Rightarrow> string set"
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begin
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fun L_rhs_e:: "rhs_item \<Rightarrow> string set"
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where
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"L_rhs_e (Lam r) = L r" |
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"L_rhs_e (Trn X r) = X ;; L r"
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end
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text {*
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The right hand side of every equation is represented by a set of
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items. The string set defined by such a set @{text "itms"} is given
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by @{text "L(itms)"}, defined as:
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*}
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overloading L_rhs \<equiv> "L:: rhs_item set \<Rightarrow> string set"
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begin
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fun L_rhs:: "rhs_item set \<Rightarrow> string set"
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where "L_rhs rhs = \<Union> (L ` rhs)"
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end
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text {*
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Given a set of equivalent classses @{text "CS"} and one equivalent class @{text "X"} among
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@{text "CS"}, the term @{text "init_rhs CS X"} is used to extract the right hand side of
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the equation describing the formation of @{text "X"}. The definition of @{text "init_rhs"}
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is:
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*}
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definition
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"init_rhs CS X \<equiv>
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if ([] \<in> X) then
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{Lam(EMPTY)} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}
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else
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{Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
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text {*
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In the definition of @{text "init_rhs"}, the term
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@{text "{Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"} appearing on both branches
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describes the formation of strings in @{text "X"} out of transitions, while
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the term @{text "{Lam(EMPTY)}"} describes the empty string which is intrinsically contained in
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@{text "X"} rather than by transition. This @{text "{Lam(EMPTY)}"} corresponds to
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the $\lambda$ in \eqref{example_eqns}.
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With the help of @{text "init_rhs"}, the equitional system descrbing the formation of every
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equivalent class inside @{text "CS"} is given by the following @{text "eqs(CS)"}.
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*}
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definition "eqs CS \<equiv> {(X, init_rhs CS X) | X. X \<in> CS}"
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(************ arden's lemma variation ********************)
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text {*
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The following @{text "items_of rhs X"} returns all @{text "X"}-items in @{text "rhs"}.
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*}
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definition
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"items_of rhs X \<equiv> {Trn X r | r. (Trn X r) \<in> rhs}"
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text {*
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The following @{text "rexp_of rhs X"} combines all regular expressions in @{text "X"}-items
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using @{text "ALT"} to form a single regular expression.
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It will be used later to implement @{text "arden_variate"} and @{text "rhs_subst"}.
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*}
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definition
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"rexp_of rhs X \<equiv> folds ALT NULL ((snd o the_Trn) ` items_of rhs X)"
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text {*
+ − 358
The following @{text "lam_of rhs"} returns all pure regular expression items in @{text "rhs"}.
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*}
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definition
+ − 362
"lam_of rhs \<equiv> {Lam r | r. Lam r \<in> rhs}"
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text {*
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The following @{text "rexp_of_lam rhs"} combines pure regular expression items in @{text "rhs"}
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using @{text "ALT"} to form a single regular expression.
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When all variables inside @{text "rhs"} are eliminated, @{text "rexp_of_lam rhs"}
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is used to compute compute the regular expression corresponds to @{text "rhs"}.
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*}
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definition
+ − 372
"rexp_of_lam rhs \<equiv> folds ALT NULL (the_r ` lam_of rhs)"
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text {*
+ − 375
The following @{text "attach_rexp rexp' itm"} attach
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the regular expression @{text "rexp'"} to
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the right of right hand side item @{text "itm"}.
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*}
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fun attach_rexp :: "rexp \<Rightarrow> rhs_item \<Rightarrow> rhs_item"
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where
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"attach_rexp rexp' (Lam rexp) = Lam (SEQ rexp rexp')"
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| "attach_rexp rexp' (Trn X rexp) = Trn X (SEQ rexp rexp')"
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+ − 385
text {*
+ − 386
The following @{text "append_rhs_rexp rhs rexp"} attaches
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@{text "rexp"} to every item in @{text "rhs"}.
+ − 388
*}
+ − 389
+ − 390
definition
+ − 391
"append_rhs_rexp rhs rexp \<equiv> (attach_rexp rexp) ` rhs"
+ − 392
+ − 393
text {*
+ − 394
With the help of the two functions immediately above, Ardens'
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transformation on right hand side @{text "rhs"} is implemented
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by the following function @{text "arden_variate X rhs"}.
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After this transformation, the recursive occurent of @{text "X"}
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in @{text "rhs"} will be eliminated, while the
+ − 399
string set defined by @{text "rhs"} is kept unchanged.
+ − 400
*}
+ − 401
definition
+ − 402
"arden_variate X rhs \<equiv>
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append_rhs_rexp (rhs - items_of rhs X) (STAR (rexp_of rhs X))"
+ − 404
+ − 405
+ − 406
(*********** substitution of ES *************)
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+ − 408
text {*
+ − 409
Suppose the equation defining @{text "X"} is $X = xrhs$,
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the purpose of @{text "rhs_subst"} is to substitute all occurences of @{text "X"} in
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@{text "rhs"} by @{text "xrhs"}.
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A litte thought may reveal that the final result
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should be: first append $(a_1 | a_2 | \ldots | a_n)$ to every item of @{text "xrhs"} and then
+ − 414
union the result with all non-@{text "X"}-items of @{text "rhs"}.
+ − 415
*}
+ − 416
definition
+ − 417
"rhs_subst rhs X xrhs \<equiv>
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(rhs - (items_of rhs X)) \<union> (append_rhs_rexp xrhs (rexp_of rhs X))"
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text {*
+ − 421
Suppose the equation defining @{text "X"} is $X = xrhs$, the follwing
+ − 422
@{text "eqs_subst ES X xrhs"} substitute @{text "xrhs"} into every equation
+ − 423
of the equational system @{text "ES"}.
+ − 424
*}
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+ − 426
definition
+ − 427
"eqs_subst ES X xrhs \<equiv> {(Y, rhs_subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
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+ − 429
text {*
+ − 430
The computation of regular expressions for equivalent classes is accomplished
+ − 431
using a iteration principle given by the following lemma.
+ − 432
*}
+ − 433
+ − 434
lemma wf_iter [rule_format]:
+ − 435
fixes f
+ − 436
assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
+ − 437
shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
+ − 438
proof(induct e rule: wf_induct
+ − 439
[OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
+ − 440
fix x
+ − 441
assume h [rule_format]:
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"\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
+ − 443
and px: "P x"
+ − 444
show "\<exists>e'. P e' \<and> Q e'"
+ − 445
proof(cases "Q x")
+ − 446
assume "Q x" with px show ?thesis by blast
+ − 447
next
+ − 448
assume nq: "\<not> Q x"
+ − 449
from step [OF px nq]
+ − 450
obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
+ − 451
show ?thesis
+ − 452
proof(rule h)
+ − 453
from ltf show "(e', x) \<in> inv_image less_than f"
+ − 454
by (simp add:inv_image_def)
+ − 455
next
+ − 456
from pe' show "P e'" .
+ − 457
qed
+ − 458
qed
+ − 459
qed
+ − 460
+ − 461
text {*
+ − 462
The @{text "P"} in lemma @{text "wf_iter"} is an invaiant kept throughout the iteration procedure.
+ − 463
The particular invariant used to solve our problem is defined by function @{text "Inv(ES)"},
+ − 464
an invariant over equal system @{text "ES"}.
+ − 465
Every definition starting next till @{text "Inv"} stipulates a property to be satisfied by @{text "ES"}.
+ − 466
*}
+ − 467
+ − 468
text {*
+ − 469
Every variable is defined at most onece in @{text "ES"}.
+ − 470
*}
+ − 471
definition
+ − 472
"distinct_equas ES \<equiv>
+ − 473
\<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
+ − 474
text {*
+ − 475
Every equation in @{text "ES"} (represented by @{text "(X, rhs)"}) is valid, i.e. @{text "(X = L rhs)"}.
+ − 476
*}
+ − 477
definition
+ − 478
"valid_eqns ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> (X = L rhs)"
+ − 479
+ − 480
text {*
+ − 481
The following @{text "rhs_nonempty rhs"} requires regular expressions occuring in transitional
+ − 482
items of @{text "rhs"} does not contain empty string. This is necessary for
+ − 483
the application of Arden's transformation to @{text "rhs"}.
+ − 484
*}
+ − 485
definition
+ − 486
"rhs_nonempty rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
+ − 487
+ − 488
text {*
+ − 489
The following @{text "ardenable ES"} requires that Arden's transformation is applicable
+ − 490
to every equation of equational system @{text "ES"}.
+ − 491
*}
+ − 492
definition
+ − 493
"ardenable ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> rhs_nonempty rhs"
+ − 494
+ − 495
(* The following non_empty seems useless. *)
+ − 496
definition
+ − 497
"non_empty ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> X \<noteq> {}"
+ − 498
+ − 499
text {*
+ − 500
The following @{text "finite_rhs ES"} requires every equation in @{text "rhs"} be finite.
+ − 501
*}
+ − 502
definition
+ − 503
"finite_rhs ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs"
+ − 504
+ − 505
text {*
+ − 506
The following @{text "classes_of rhs"} returns all variables (or equivalent classes)
+ − 507
occuring in @{text "rhs"}.
+ − 508
*}
+ − 509
definition
+ − 510
"classes_of rhs \<equiv> {X. \<exists> r. Trn X r \<in> rhs}"
+ − 511
+ − 512
text {*
+ − 513
The following @{text "lefts_of ES"} returns all variables
+ − 514
defined by equational system @{text "ES"}.
+ − 515
*}
+ − 516
definition
+ − 517
"lefts_of ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
+ − 518
+ − 519
text {*
+ − 520
The following @{text "self_contained ES"} requires that every
+ − 521
variable occuring on the right hand side of equations is already defined by some
+ − 522
equation in @{text "ES"}.
+ − 523
*}
+ − 524
definition
+ − 525
"self_contained ES \<equiv> \<forall> (X, xrhs) \<in> ES. classes_of xrhs \<subseteq> lefts_of ES"
+ − 526
+ − 527
+ − 528
text {*
+ − 529
The invariant @{text "Inv(ES)"} is a conjunction of all the previously defined constaints.
+ − 530
*}
+ − 531
definition
+ − 532
"Inv ES \<equiv> valid_eqns ES \<and> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and>
+ − 533
non_empty ES \<and> finite_rhs ES \<and> self_contained ES"
+ − 534
+ − 535
subsection {* The proof of this direction *}
+ − 536
+ − 537
subsubsection {* Basic properties *}
+ − 538
+ − 539
text {*
+ − 540
The following are some basic properties of the above definitions.
+ − 541
*}
+ − 542
+ − 543
lemma L_rhs_union_distrib:
+ − 544
" L (A::rhs_item set) \<union> L B = L (A \<union> B)"
+ − 545
by simp
+ − 546
+ − 547
lemma finite_snd_Trn:
+ − 548
assumes finite:"finite rhs"
+ − 549
shows "finite {r\<^isub>2. Trn Y r\<^isub>2 \<in> rhs}" (is "finite ?B")
+ − 550
proof-
+ − 551
def rhs' \<equiv> "{e \<in> rhs. \<exists> r. e = Trn Y r}"
+ − 552
have "?B = (snd o the_Trn) ` rhs'" using rhs'_def by (auto simp:image_def)
+ − 553
moreover have "finite rhs'" using finite rhs'_def by auto
+ − 554
ultimately show ?thesis by simp
+ − 555
qed
+ − 556
+ − 557
lemma rexp_of_empty:
+ − 558
assumes finite:"finite rhs"
+ − 559
and nonempty:"rhs_nonempty rhs"
+ − 560
shows "[] \<notin> L (rexp_of rhs X)"
+ − 561
using finite nonempty rhs_nonempty_def
+ − 562
by (drule_tac finite_snd_Trn[where Y = X], auto simp:rexp_of_def items_of_def)
+ − 563
+ − 564
lemma [intro!]:
+ − 565
"P (Trn X r) \<Longrightarrow> (\<exists>a. (\<exists>r. a = Trn X r \<and> P a))" by auto
+ − 566
+ − 567
lemma finite_items_of:
+ − 568
"finite rhs \<Longrightarrow> finite (items_of rhs X)"
+ − 569
by (auto simp:items_of_def intro:finite_subset)
+ − 570
+ − 571
lemma lang_of_rexp_of:
+ − 572
assumes finite:"finite rhs"
+ − 573
shows "L (items_of rhs X) = X ;; (L (rexp_of rhs X))"
+ − 574
proof -
+ − 575
have "finite ((snd \<circ> the_Trn) ` items_of rhs X)" using finite_items_of[OF finite] by auto
+ − 576
thus ?thesis
+ − 577
apply (auto simp:rexp_of_def Seq_def items_of_def)
+ − 578
apply (rule_tac x = s1 in exI, rule_tac x = s2 in exI, auto)
+ − 579
by (rule_tac x= "Trn X r" in exI, auto simp:Seq_def)
+ − 580
qed
+ − 581
+ − 582
lemma rexp_of_lam_eq_lam_set:
+ − 583
assumes finite: "finite rhs"
+ − 584
shows "L (rexp_of_lam rhs) = L (lam_of rhs)"
+ − 585
proof -
+ − 586
have "finite (the_r ` {Lam r |r. Lam r \<in> rhs})" using finite
+ − 587
by (rule_tac finite_imageI, auto intro:finite_subset)
+ − 588
thus ?thesis by (auto simp:rexp_of_lam_def lam_of_def)
+ − 589
qed
+ − 590
+ − 591
lemma [simp]:
+ − 592
" L (attach_rexp r xb) = L xb ;; L r"
+ − 593
apply (cases xb, auto simp:Seq_def)
+ − 594
by (rule_tac x = "s1 @ s1a" in exI, rule_tac x = s2a in exI,auto simp:Seq_def)
+ − 595
+ − 596
lemma lang_of_append_rhs:
+ − 597
"L (append_rhs_rexp rhs r) = L rhs ;; L r"
+ − 598
apply (auto simp:append_rhs_rexp_def image_def)
+ − 599
apply (auto simp:Seq_def)
+ − 600
apply (rule_tac x = "L xb ;; L r" in exI, auto simp add:Seq_def)
+ − 601
by (rule_tac x = "attach_rexp r xb" in exI, auto simp:Seq_def)
+ − 602
+ − 603
lemma classes_of_union_distrib:
+ − 604
"classes_of A \<union> classes_of B = classes_of (A \<union> B)"
+ − 605
by (auto simp add:classes_of_def)
+ − 606
+ − 607
lemma lefts_of_union_distrib:
+ − 608
"lefts_of A \<union> lefts_of B = lefts_of (A \<union> B)"
+ − 609
by (auto simp:lefts_of_def)
+ − 610
+ − 611
+ − 612
subsubsection {* Intialization *}
+ − 613
+ − 614
text {*
+ − 615
The following several lemmas until @{text "init_ES_satisfy_Inv"} shows that
+ − 616
the initial equational system satisfies invariant @{text "Inv"}.
+ − 617
*}
+ − 618
+ − 619
lemma defined_by_str:
+ − 620
"\<lbrakk>s \<in> X; X \<in> UNIV // (\<approx>Lang)\<rbrakk> \<Longrightarrow> X = (\<approx>Lang) `` {s}"
+ − 621
by (auto simp:quotient_def Image_def str_eq_rel_def)
+ − 622
+ − 623
lemma every_eqclass_has_transition:
+ − 624
assumes has_str: "s @ [c] \<in> X"
+ − 625
and in_CS: "X \<in> UNIV // (\<approx>Lang)"
+ − 626
obtains Y where "Y \<in> UNIV // (\<approx>Lang)" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
+ − 627
proof -
+ − 628
def Y \<equiv> "(\<approx>Lang) `` {s}"
+ − 629
have "Y \<in> UNIV // (\<approx>Lang)"
+ − 630
unfolding Y_def quotient_def by auto
+ − 631
moreover
+ − 632
have "X = (\<approx>Lang) `` {s @ [c]}"
+ − 633
using has_str in_CS defined_by_str by blast
+ − 634
then have "Y ;; {[c]} \<subseteq> X"
+ − 635
unfolding Y_def Image_def Seq_def
+ − 636
unfolding str_eq_rel_def
+ − 637
by clarsimp
+ − 638
moreover
+ − 639
have "s \<in> Y" unfolding Y_def
+ − 640
unfolding Image_def str_eq_rel_def by simp
+ − 641
ultimately show thesis by (blast intro: that)
+ − 642
qed
+ − 643
+ − 644
lemma l_eq_r_in_eqs:
+ − 645
assumes X_in_eqs: "(X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
+ − 646
shows "X = L xrhs"
+ − 647
proof
+ − 648
show "X \<subseteq> L xrhs"
+ − 649
proof
+ − 650
fix x
+ − 651
assume "(1)": "x \<in> X"
+ − 652
show "x \<in> L xrhs"
+ − 653
proof (cases "x = []")
+ − 654
assume empty: "x = []"
+ − 655
thus ?thesis using X_in_eqs "(1)"
+ − 656
by (auto simp:eqs_def init_rhs_def)
+ − 657
next
+ − 658
assume not_empty: "x \<noteq> []"
+ − 659
then obtain clist c where decom: "x = clist @ [c]"
+ − 660
by (case_tac x rule:rev_cases, auto)
+ − 661
have "X \<in> UNIV // (\<approx>Lang)" using X_in_eqs by (auto simp:eqs_def)
+ − 662
then obtain Y
+ − 663
where "Y \<in> UNIV // (\<approx>Lang)"
+ − 664
and "Y ;; {[c]} \<subseteq> X"
+ − 665
and "clist \<in> Y"
+ − 666
using decom "(1)" every_eqclass_has_transition by blast
+ − 667
hence
+ − 668
"x \<in> L {Trn Y (CHAR c)| Y c. Y \<in> UNIV // (\<approx>Lang) \<and> Y ;; {[c]} \<subseteq> X}"
+ − 669
using "(1)" decom
+ − 670
by (simp, rule_tac x = "Trn Y (CHAR c)" in exI, simp add:Seq_def)
+ − 671
thus ?thesis using X_in_eqs "(1)"
+ − 672
by (simp add:eqs_def init_rhs_def)
+ − 673
qed
+ − 674
qed
+ − 675
next
+ − 676
show "L xrhs \<subseteq> X" using X_in_eqs
+ − 677
by (auto simp:eqs_def init_rhs_def)
+ − 678
qed
+ − 679
+ − 680
lemma finite_init_rhs:
+ − 681
assumes finite: "finite CS"
+ − 682
shows "finite (init_rhs CS X)"
+ − 683
proof-
+ − 684
have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" (is "finite ?A")
+ − 685
proof -
+ − 686
def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
+ − 687
def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
+ − 688
have "finite (CS \<times> (UNIV::char set))" using finite by auto
+ − 689
hence "finite S" using S_def
+ − 690
by (rule_tac B = "CS \<times> UNIV" in finite_subset, auto)
+ − 691
moreover have "?A = h ` S" by (auto simp: S_def h_def image_def)
+ − 692
ultimately show ?thesis
+ − 693
by auto
+ − 694
qed
+ − 695
thus ?thesis by (simp add:init_rhs_def)
+ − 696
qed
+ − 697
+ − 698
lemma init_ES_satisfy_Inv:
+ − 699
assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ − 700
shows "Inv (eqs (UNIV // (\<approx>Lang)))"
+ − 701
proof -
+ − 702
have "finite (eqs (UNIV // (\<approx>Lang)))" using finite_CS
+ − 703
by (simp add:eqs_def)
+ − 704
moreover have "distinct_equas (eqs (UNIV // (\<approx>Lang)))"
+ − 705
by (simp add:distinct_equas_def eqs_def)
+ − 706
moreover have "ardenable (eqs (UNIV // (\<approx>Lang)))"
+ − 707
by (auto simp add:ardenable_def eqs_def init_rhs_def rhs_nonempty_def del:L_rhs.simps)
+ − 708
moreover have "valid_eqns (eqs (UNIV // (\<approx>Lang)))"
+ − 709
using l_eq_r_in_eqs by (simp add:valid_eqns_def)
+ − 710
moreover have "non_empty (eqs (UNIV // (\<approx>Lang)))"
+ − 711
by (auto simp:non_empty_def eqs_def quotient_def Image_def str_eq_rel_def)
+ − 712
moreover have "finite_rhs (eqs (UNIV // (\<approx>Lang)))"
+ − 713
using finite_init_rhs[OF finite_CS]
+ − 714
by (auto simp:finite_rhs_def eqs_def)
+ − 715
moreover have "self_contained (eqs (UNIV // (\<approx>Lang)))"
+ − 716
by (auto simp:self_contained_def eqs_def init_rhs_def classes_of_def lefts_of_def)
+ − 717
ultimately show ?thesis by (simp add:Inv_def)
+ − 718
qed
+ − 719
+ − 720
subsubsection {*
+ − 721
Interation step
+ − 722
*}
+ − 723
+ − 724
text {*
+ − 725
From this point until @{text "iteration_step"}, it is proved
+ − 726
that there exists iteration steps which keep @{text "Inv(ES)"} while
+ − 727
decreasing the size of @{text "ES"}.
+ − 728
*}
+ − 729
lemma arden_variate_keeps_eq:
+ − 730
assumes l_eq_r: "X = L rhs"
+ − 731
and not_empty: "[] \<notin> L (rexp_of rhs X)"
+ − 732
and finite: "finite rhs"
+ − 733
shows "X = L (arden_variate X rhs)"
+ − 734
proof -
+ − 735
def A \<equiv> "L (rexp_of rhs X)"
+ − 736
def b \<equiv> "rhs - items_of rhs X"
+ − 737
def B \<equiv> "L b"
+ − 738
have "X = B ;; A\<star>"
+ − 739
proof-
+ − 740
have "rhs = items_of rhs X \<union> b" by (auto simp:b_def items_of_def)
+ − 741
hence "L rhs = L(items_of rhs X \<union> b)" by simp
+ − 742
hence "L rhs = L(items_of rhs X) \<union> B" by (simp only:L_rhs_union_distrib B_def)
+ − 743
with lang_of_rexp_of
+ − 744
have "L rhs = X ;; A \<union> B " using finite by (simp only:B_def b_def A_def)
+ − 745
thus ?thesis
+ − 746
using l_eq_r not_empty
+ − 747
apply (drule_tac B = B and X = X in ardens_revised)
+ − 748
by (auto simp:A_def simp del:L_rhs.simps)
+ − 749
qed
+ − 750
moreover have "L (arden_variate X rhs) = (B ;; A\<star>)" (is "?L = ?R")
+ − 751
by (simp only:arden_variate_def L_rhs_union_distrib lang_of_append_rhs
+ − 752
B_def A_def b_def L_rexp.simps seq_union_distrib)
+ − 753
ultimately show ?thesis by simp
+ − 754
qed
+ − 755
+ − 756
lemma append_keeps_finite:
+ − 757
"finite rhs \<Longrightarrow> finite (append_rhs_rexp rhs r)"
+ − 758
by (auto simp:append_rhs_rexp_def)
+ − 759
+ − 760
lemma arden_variate_keeps_finite:
+ − 761
"finite rhs \<Longrightarrow> finite (arden_variate X rhs)"
+ − 762
by (auto simp:arden_variate_def append_keeps_finite)
+ − 763
+ − 764
lemma append_keeps_nonempty:
+ − 765
"rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (append_rhs_rexp rhs r)"
+ − 766
apply (auto simp:rhs_nonempty_def append_rhs_rexp_def)
+ − 767
by (case_tac x, auto simp:Seq_def)
+ − 768
+ − 769
lemma nonempty_set_sub:
+ − 770
"rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (rhs - A)"
+ − 771
by (auto simp:rhs_nonempty_def)
+ − 772
+ − 773
lemma nonempty_set_union:
+ − 774
"\<lbrakk>rhs_nonempty rhs; rhs_nonempty rhs'\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs \<union> rhs')"
+ − 775
by (auto simp:rhs_nonempty_def)
+ − 776
+ − 777
lemma arden_variate_keeps_nonempty:
+ − 778
"rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (arden_variate X rhs)"
+ − 779
by (simp only:arden_variate_def append_keeps_nonempty nonempty_set_sub)
+ − 780
+ − 781
+ − 782
lemma rhs_subst_keeps_nonempty:
+ − 783
"\<lbrakk>rhs_nonempty rhs; rhs_nonempty xrhs\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs_subst rhs X xrhs)"
+ − 784
by (simp only:rhs_subst_def append_keeps_nonempty nonempty_set_union nonempty_set_sub)
+ − 785
+ − 786
lemma rhs_subst_keeps_eq:
+ − 787
assumes substor: "X = L xrhs"
+ − 788
and finite: "finite rhs"
+ − 789
shows "L (rhs_subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
+ − 790
proof-
+ − 791
def A \<equiv> "L (rhs - items_of rhs X)"
+ − 792
have "?Left = A \<union> L (append_rhs_rexp xrhs (rexp_of rhs X))"
+ − 793
by (simp only:rhs_subst_def L_rhs_union_distrib A_def)
+ − 794
moreover have "?Right = A \<union> L (items_of rhs X)"
+ − 795
proof-
+ − 796
have "rhs = (rhs - items_of rhs X) \<union> (items_of rhs X)" by (auto simp:items_of_def)
+ − 797
thus ?thesis by (simp only:L_rhs_union_distrib A_def)
+ − 798
qed
+ − 799
moreover have "L (append_rhs_rexp xrhs (rexp_of rhs X)) = L (items_of rhs X)"
+ − 800
using finite substor by (simp only:lang_of_append_rhs lang_of_rexp_of)
+ − 801
ultimately show ?thesis by simp
+ − 802
qed
+ − 803
+ − 804
lemma rhs_subst_keeps_finite_rhs:
+ − 805
"\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (rhs_subst rhs Y yrhs)"
+ − 806
by (auto simp:rhs_subst_def append_keeps_finite)
+ − 807
+ − 808
lemma eqs_subst_keeps_finite:
+ − 809
assumes finite:"finite (ES:: (string set \<times> rhs_item set) set)"
+ − 810
shows "finite (eqs_subst ES Y yrhs)"
+ − 811
proof -
+ − 812
have "finite {(Ya, rhs_subst yrhsa Y yrhs) |Ya yrhsa. (Ya, yrhsa) \<in> ES}"
+ − 813
(is "finite ?A")
+ − 814
proof-
+ − 815
def eqns' \<equiv> "{((Ya::string set), yrhsa)| Ya yrhsa. (Ya, yrhsa) \<in> ES}"
+ − 816
def h \<equiv> "\<lambda> ((Ya::string set), yrhsa). (Ya, rhs_subst yrhsa Y yrhs)"
+ − 817
have "finite (h ` eqns')" using finite h_def eqns'_def by auto
+ − 818
moreover have "?A = h ` eqns'" by (auto simp:h_def eqns'_def)
+ − 819
ultimately show ?thesis by auto
+ − 820
qed
+ − 821
thus ?thesis by (simp add:eqs_subst_def)
+ − 822
qed
+ − 823
+ − 824
lemma eqs_subst_keeps_finite_rhs:
+ − 825
"\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (eqs_subst ES Y yrhs)"
+ − 826
by (auto intro:rhs_subst_keeps_finite_rhs simp add:eqs_subst_def finite_rhs_def)
+ − 827
+ − 828
lemma append_rhs_keeps_cls:
+ − 829
"classes_of (append_rhs_rexp rhs r) = classes_of rhs"
+ − 830
apply (auto simp:classes_of_def append_rhs_rexp_def)
+ − 831
apply (case_tac xa, auto simp:image_def)
+ − 832
by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
+ − 833
+ − 834
lemma arden_variate_removes_cl:
+ − 835
"classes_of (arden_variate Y yrhs) = classes_of yrhs - {Y}"
+ − 836
apply (simp add:arden_variate_def append_rhs_keeps_cls items_of_def)
+ − 837
by (auto simp:classes_of_def)
+ − 838
+ − 839
lemma lefts_of_keeps_cls:
+ − 840
"lefts_of (eqs_subst ES Y yrhs) = lefts_of ES"
+ − 841
by (auto simp:lefts_of_def eqs_subst_def)
+ − 842
+ − 843
lemma rhs_subst_updates_cls:
+ − 844
"X \<notin> classes_of xrhs \<Longrightarrow>
+ − 845
classes_of (rhs_subst rhs X xrhs) = classes_of rhs \<union> classes_of xrhs - {X}"
+ − 846
apply (simp only:rhs_subst_def append_rhs_keeps_cls
+ − 847
classes_of_union_distrib[THEN sym])
+ − 848
by (auto simp:classes_of_def items_of_def)
+ − 849
+ − 850
lemma eqs_subst_keeps_self_contained:
+ − 851
fixes Y
+ − 852
assumes sc: "self_contained (ES \<union> {(Y, yrhs)})" (is "self_contained ?A")
+ − 853
shows "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 854
(is "self_contained ?B")
+ − 855
proof-
+ − 856
{ fix X xrhs'
+ − 857
assume "(X, xrhs') \<in> ?B"
+ − 858
then obtain xrhs
+ − 859
where xrhs_xrhs': "xrhs' = rhs_subst xrhs Y (arden_variate Y yrhs)"
+ − 860
and X_in: "(X, xrhs) \<in> ES" by (simp add:eqs_subst_def, blast)
+ − 861
have "classes_of xrhs' \<subseteq> lefts_of ?B"
+ − 862
proof-
+ − 863
have "lefts_of ?B = lefts_of ES" by (auto simp add:lefts_of_def eqs_subst_def)
+ − 864
moreover have "classes_of xrhs' \<subseteq> lefts_of ES"
+ − 865
proof-
+ − 866
have "classes_of xrhs' \<subseteq>
+ − 867
classes_of xrhs \<union> classes_of (arden_variate Y yrhs) - {Y}"
+ − 868
proof-
+ − 869
have "Y \<notin> classes_of (arden_variate Y yrhs)"
+ − 870
using arden_variate_removes_cl by simp
+ − 871
thus ?thesis using xrhs_xrhs' by (auto simp:rhs_subst_updates_cls)
+ − 872
qed
+ − 873
moreover have "classes_of xrhs \<subseteq> lefts_of ES \<union> {Y}" using X_in sc
+ − 874
apply (simp only:self_contained_def lefts_of_union_distrib[THEN sym])
+ − 875
by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lefts_of_def)
+ − 876
moreover have "classes_of (arden_variate Y yrhs) \<subseteq> lefts_of ES \<union> {Y}"
+ − 877
using sc
+ − 878
by (auto simp add:arden_variate_removes_cl self_contained_def lefts_of_def)
+ − 879
ultimately show ?thesis by auto
+ − 880
qed
+ − 881
ultimately show ?thesis by simp
+ − 882
qed
+ − 883
} thus ?thesis by (auto simp only:eqs_subst_def self_contained_def)
+ − 884
qed
+ − 885
+ − 886
lemma eqs_subst_satisfy_Inv:
+ − 887
assumes Inv_ES: "Inv (ES \<union> {(Y, yrhs)})"
+ − 888
shows "Inv (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 889
proof -
+ − 890
have finite_yrhs: "finite yrhs"
+ − 891
using Inv_ES by (auto simp:Inv_def finite_rhs_def)
+ − 892
have nonempty_yrhs: "rhs_nonempty yrhs"
+ − 893
using Inv_ES by (auto simp:Inv_def ardenable_def)
+ − 894
have Y_eq_yrhs: "Y = L yrhs"
+ − 895
using Inv_ES by (simp only:Inv_def valid_eqns_def, blast)
+ − 896
have "distinct_equas (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 897
using Inv_ES
+ − 898
by (auto simp:distinct_equas_def eqs_subst_def Inv_def)
+ − 899
moreover have "finite (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 900
using Inv_ES by (simp add:Inv_def eqs_subst_keeps_finite)
+ − 901
moreover have "finite_rhs (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 902
proof-
+ − 903
have "finite_rhs ES" using Inv_ES
+ − 904
by (simp add:Inv_def finite_rhs_def)
+ − 905
moreover have "finite (arden_variate Y yrhs)"
+ − 906
proof -
+ − 907
have "finite yrhs" using Inv_ES
+ − 908
by (auto simp:Inv_def finite_rhs_def)
+ − 909
thus ?thesis using arden_variate_keeps_finite by simp
+ − 910
qed
+ − 911
ultimately show ?thesis
+ − 912
by (simp add:eqs_subst_keeps_finite_rhs)
+ − 913
qed
+ − 914
moreover have "ardenable (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 915
proof -
+ − 916
{ fix X rhs
+ − 917
assume "(X, rhs) \<in> ES"
+ − 918
hence "rhs_nonempty rhs" using prems Inv_ES
+ − 919
by (simp add:Inv_def ardenable_def)
+ − 920
with nonempty_yrhs
+ − 921
have "rhs_nonempty (rhs_subst rhs Y (arden_variate Y yrhs))"
+ − 922
by (simp add:nonempty_yrhs
+ − 923
rhs_subst_keeps_nonempty arden_variate_keeps_nonempty)
+ − 924
} thus ?thesis by (auto simp add:ardenable_def eqs_subst_def)
+ − 925
qed
+ − 926
moreover have "valid_eqns (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 927
proof-
+ − 928
have "Y = L (arden_variate Y yrhs)"
+ − 929
using Y_eq_yrhs Inv_ES finite_yrhs nonempty_yrhs
+ − 930
by (rule_tac arden_variate_keeps_eq, (simp add:rexp_of_empty)+)
+ − 931
thus ?thesis using Inv_ES
+ − 932
by (clarsimp simp add:valid_eqns_def
+ − 933
eqs_subst_def rhs_subst_keeps_eq Inv_def finite_rhs_def
+ − 934
simp del:L_rhs.simps)
+ − 935
qed
+ − 936
moreover have
+ − 937
non_empty_subst: "non_empty (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 938
using Inv_ES by (auto simp:Inv_def non_empty_def eqs_subst_def)
+ − 939
moreover
+ − 940
have self_subst: "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
+ − 941
using Inv_ES eqs_subst_keeps_self_contained by (simp add:Inv_def)
+ − 942
ultimately show ?thesis using Inv_ES by (simp add:Inv_def)
+ − 943
qed
+ − 944
+ − 945
lemma eqs_subst_card_le:
+ − 946
assumes finite: "finite (ES::(string set \<times> rhs_item set) set)"
+ − 947
shows "card (eqs_subst ES Y yrhs) <= card ES"
+ − 948
proof-
+ − 949
def f \<equiv> "\<lambda> x. ((fst x)::string set, rhs_subst (snd x) Y yrhs)"
+ − 950
have "eqs_subst ES Y yrhs = f ` ES"
+ − 951
apply (auto simp:eqs_subst_def f_def image_def)
+ − 952
by (rule_tac x = "(Ya, yrhsa)" in bexI, simp+)
+ − 953
thus ?thesis using finite by (auto intro:card_image_le)
+ − 954
qed
+ − 955
+ − 956
lemma eqs_subst_cls_remains:
+ − 957
"(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (eqs_subst ES Y yrhs)"
+ − 958
by (auto simp:eqs_subst_def)
+ − 959
+ − 960
lemma card_noteq_1_has_more:
+ − 961
assumes card:"card S \<noteq> 1"
+ − 962
and e_in: "e \<in> S"
+ − 963
and finite: "finite S"
+ − 964
obtains e' where "e' \<in> S \<and> e \<noteq> e'"
+ − 965
proof-
+ − 966
have "card (S - {e}) > 0"
+ − 967
proof -
+ − 968
have "card S > 1" using card e_in finite
+ − 969
by (case_tac "card S", auto)
+ − 970
thus ?thesis using finite e_in by auto
+ − 971
qed
+ − 972
hence "S - {e} \<noteq> {}" using finite by (rule_tac notI, simp)
+ − 973
thus "(\<And>e'. e' \<in> S \<and> e \<noteq> e' \<Longrightarrow> thesis) \<Longrightarrow> thesis" by auto
+ − 974
qed
+ − 975
+ − 976
lemma iteration_step:
+ − 977
assumes Inv_ES: "Inv ES"
+ − 978
and X_in_ES: "(X, xrhs) \<in> ES"
+ − 979
and not_T: "card ES \<noteq> 1"
+ − 980
shows "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'.(X, xrhs') \<in> ES')) \<and>
+ − 981
(card ES', card ES) \<in> less_than" (is "\<exists> ES'. ?P ES'")
+ − 982
proof -
+ − 983
have finite_ES: "finite ES" using Inv_ES by (simp add:Inv_def)
+ − 984
then obtain Y yrhs
+ − 985
where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
+ − 986
using not_T X_in_ES by (drule_tac card_noteq_1_has_more, auto)
+ − 987
def ES' == "ES - {(Y, yrhs)}"
+ − 988
let ?ES'' = "eqs_subst ES' Y (arden_variate Y yrhs)"
+ − 989
have "?P ?ES''"
+ − 990
proof -
+ − 991
have "Inv ?ES''" using Y_in_ES Inv_ES
+ − 992
by (rule_tac eqs_subst_satisfy_Inv, simp add:ES'_def insert_absorb)
+ − 993
moreover have "\<exists>xrhs'. (X, xrhs') \<in> ?ES''" using not_eq X_in_ES
+ − 994
by (rule_tac ES = ES' in eqs_subst_cls_remains, auto simp add:ES'_def)
+ − 995
moreover have "(card ?ES'', card ES) \<in> less_than"
+ − 996
proof -
+ − 997
have "finite ES'" using finite_ES ES'_def by auto
+ − 998
moreover have "card ES' < card ES" using finite_ES Y_in_ES
+ − 999
by (auto simp:ES'_def card_gt_0_iff intro:diff_Suc_less)
+ − 1000
ultimately show ?thesis
+ − 1001
by (auto dest:eqs_subst_card_le elim:le_less_trans)
+ − 1002
qed
+ − 1003
ultimately show ?thesis by simp
+ − 1004
qed
+ − 1005
thus ?thesis by blast
+ − 1006
qed
+ − 1007
+ − 1008
subsubsection {*
+ − 1009
Conclusion of the proof
+ − 1010
*}
+ − 1011
+ − 1012
text {*
+ − 1013
From this point until @{text "hard_direction"}, the hard direction is proved
+ − 1014
through a simple application of the iteration principle.
+ − 1015
*}
+ − 1016
+ − 1017
lemma iteration_conc:
+ − 1018
assumes history: "Inv ES"
+ − 1019
and X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES"
+ − 1020
shows
+ − 1021
"\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'. (X, xrhs') \<in> ES')) \<and> card ES' = 1"
+ − 1022
(is "\<exists> ES'. ?P ES'")
+ − 1023
proof (cases "card ES = 1")
+ − 1024
case True
+ − 1025
thus ?thesis using history X_in_ES
+ − 1026
by blast
+ − 1027
next
+ − 1028
case False
+ − 1029
thus ?thesis using history iteration_step X_in_ES
+ − 1030
by (rule_tac f = card in wf_iter, auto)
+ − 1031
qed
+ − 1032
+ − 1033
lemma last_cl_exists_rexp:
+ − 1034
assumes ES_single: "ES = {(X, xrhs)}"
+ − 1035
and Inv_ES: "Inv ES"
+ − 1036
shows "\<exists> (r::rexp). L r = X" (is "\<exists> r. ?P r")
+ − 1037
proof-
+ − 1038
let ?A = "arden_variate X xrhs"
+ − 1039
have "?P (rexp_of_lam ?A)"
+ − 1040
proof -
+ − 1041
have "L (rexp_of_lam ?A) = L (lam_of ?A)"
+ − 1042
proof(rule rexp_of_lam_eq_lam_set)
+ − 1043
show "finite (arden_variate X xrhs)" using Inv_ES ES_single
+ − 1044
by (rule_tac arden_variate_keeps_finite,
+ − 1045
auto simp add:Inv_def finite_rhs_def)
+ − 1046
qed
+ − 1047
also have "\<dots> = L ?A"
+ − 1048
proof-
+ − 1049
have "lam_of ?A = ?A"
+ − 1050
proof-
+ − 1051
have "classes_of ?A = {}" using Inv_ES ES_single
+ − 1052
by (simp add:arden_variate_removes_cl
+ − 1053
self_contained_def Inv_def lefts_of_def)
+ − 1054
thus ?thesis
+ − 1055
by (auto simp only:lam_of_def classes_of_def, case_tac x, auto)
+ − 1056
qed
+ − 1057
thus ?thesis by simp
+ − 1058
qed
+ − 1059
also have "\<dots> = X"
+ − 1060
proof(rule arden_variate_keeps_eq [THEN sym])
+ − 1061
show "X = L xrhs" using Inv_ES ES_single
+ − 1062
by (auto simp only:Inv_def valid_eqns_def)
+ − 1063
next
+ − 1064
from Inv_ES ES_single show "[] \<notin> L (rexp_of xrhs X)"
+ − 1065
by(simp add:Inv_def ardenable_def rexp_of_empty finite_rhs_def)
+ − 1066
next
+ − 1067
from Inv_ES ES_single show "finite xrhs"
+ − 1068
by (simp add:Inv_def finite_rhs_def)
+ − 1069
qed
+ − 1070
finally show ?thesis by simp
+ − 1071
qed
+ − 1072
thus ?thesis by auto
+ − 1073
qed
+ − 1074
+ − 1075
lemma every_eqcl_has_reg:
+ − 1076
assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ − 1077
and X_in_CS: "X \<in> (UNIV // (\<approx>Lang))"
+ − 1078
shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
+ − 1079
proof -
+ − 1080
from X_in_CS have "\<exists> xrhs. (X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
+ − 1081
by (auto simp:eqs_def init_rhs_def)
+ − 1082
then obtain ES xrhs where Inv_ES: "Inv ES"
+ − 1083
and X_in_ES: "(X, xrhs) \<in> ES"
+ − 1084
and card_ES: "card ES = 1"
+ − 1085
using finite_CS X_in_CS init_ES_satisfy_Inv iteration_conc
+ − 1086
by blast
+ − 1087
hence ES_single_equa: "ES = {(X, xrhs)}"
+ − 1088
by (auto simp:Inv_def dest!:card_Suc_Diff1 simp:card_eq_0_iff)
+ − 1089
thus ?thesis using Inv_ES
+ − 1090
by (rule last_cl_exists_rexp)
+ − 1091
qed
+ − 1092
+ − 1093
lemma finals_in_partitions:
+ − 1094
"finals Lang \<subseteq> (UNIV // (\<approx>Lang))"
+ − 1095
by (auto simp:finals_def quotient_def)
+ − 1096
+ − 1097
theorem hard_direction:
+ − 1098
assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ − 1099
shows "\<exists> (reg::rexp). Lang = L reg"
+ − 1100
proof -
+ − 1101
have "\<forall> X \<in> (UNIV // (\<approx>Lang)). \<exists> (reg::rexp). X = L reg"
+ − 1102
using finite_CS every_eqcl_has_reg by blast
+ − 1103
then obtain f
+ − 1104
where f_prop: "\<forall> X \<in> (UNIV // (\<approx>Lang)). X = L ((f X)::rexp)"
+ − 1105
by (auto dest:bchoice)
+ − 1106
def rs \<equiv> "f ` (finals Lang)"
+ − 1107
have "Lang = \<Union> (finals Lang)" using lang_is_union_of_finals by auto
+ − 1108
also have "\<dots> = L (folds ALT NULL rs)"
+ − 1109
proof -
+ − 1110
have "finite rs"
+ − 1111
proof -
+ − 1112
have "finite (finals Lang)"
+ − 1113
using finite_CS finals_in_partitions[of "Lang"]
+ − 1114
by (erule_tac finite_subset, simp)
+ − 1115
thus ?thesis using rs_def by auto
+ − 1116
qed
+ − 1117
thus ?thesis
+ − 1118
using f_prop rs_def finals_in_partitions[of "Lang"] by auto
+ − 1119
qed
+ − 1120
finally show ?thesis by blast
+ − 1121
qed
+ − 1122
+ − 1123
end