+
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theory LexerExt+
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  imports Main+
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begin+
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+
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+
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section {* Sequential Composition of Languages *}+
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+
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definition+
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  Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)+
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where +
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  "A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"+
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+
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text {* Two Simple Properties about Sequential Composition *}+
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+
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lemma Sequ_empty [simp]:+
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  shows "A ;; {[]} = A"+
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  and   "{[]} ;; A = A"+
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by (simp_all add: Sequ_def)+
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+
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lemma Sequ_null [simp]:+
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  shows "A ;; {} = {}"+
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  and   "{} ;; A = {}"+
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by (simp_all add: Sequ_def)+
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+
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lemma Sequ_assoc: +
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  shows "A ;; (B ;; C) = (A ;; B) ;; C"+
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apply(auto simp add: Sequ_def)+
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apply(metis append_assoc)+
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apply(metis)+
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done+
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+
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lemma Sequ_UNION: +
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  shows "(\<Union>x\<in>A. B ;; C x) = B ;; (\<Union>x\<in>A. C x)"+
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by (auto simp add: Sequ_def)+
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+
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+
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section {* Semantic Derivative (Left Quotient) of Languages *}+
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+
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definition+
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  Der :: "char \<Rightarrow> string set \<Rightarrow> string set"+
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where+
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  "Der c A \<equiv> {s. c # s \<in> A}"+
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+
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definition+
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  Ders :: "string \<Rightarrow> string set \<Rightarrow> string set"+
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where+
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  "Ders s A \<equiv> {s'. s @ s' \<in> A}"+
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+
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lemma Der_null [simp]:+
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  shows "Der c {} = {}"+
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unfolding Der_def+
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by auto+
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+
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lemma Der_empty [simp]:+
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  shows "Der c {[]} = {}"+
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unfolding Der_def+
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by auto+
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+
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lemma Der_char [simp]:+
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  shows "Der c {[d]} = (if c = d then {[]} else {})"+
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unfolding Der_def+
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by auto+
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+
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lemma Der_Sequ [simp]:+
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  shows "Der c (A ;; B) = (Der c A) ;; B \<union> (if [] \<in> A then Der c B else {})"+
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unfolding Der_def Sequ_def+
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by (auto simp add: Cons_eq_append_conv)+
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+
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lemma Der_union [simp]:+
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  shows "Der c (A \<union> B) = Der c A \<union> Der c B"+
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unfolding Der_def+
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by auto+
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+
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lemma Der_UNION: +
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  shows "Der c (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. Der c (B x))"+
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by (auto simp add: Der_def)+
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+
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+
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section {* Power operation for Sets *}+
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+
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fun +
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  Pow :: "string set \<Rightarrow> nat \<Rightarrow> string set" ("_ \<up> _" [101, 102] 101)+
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where+
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   "A \<up> 0 = {[]}"+
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|  "A \<up> (Suc n) = A ;; (A \<up> n)"+
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+
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lemma Pow_empty [simp]:+
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  shows "[] \<in> A \<up> n \<longleftrightarrow> (n = 0 \<or> [] \<in> A)"+
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by(induct n) (auto simp add: Sequ_def)+
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+
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lemma Der_Pow [simp]:+
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  shows "Der c (A \<up> (Suc n)) = +
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     (Der c A) ;; (A \<up> n) \<union> (if [] \<in> A then Der c (A \<up> n) else {})"+
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unfolding Der_def Sequ_def +
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by(auto simp add: Cons_eq_append_conv Sequ_def)+
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+
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lemma Der_Pow_subseteq:+
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  assumes "[] \<in> A"  +
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  shows "Der c (A \<up> n) \<subseteq> (Der c A) ;; (A \<up> n)"+
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using assms+
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apply(induct n)+
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apply(simp add: Sequ_def Der_def)+
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apply(simp)+
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apply(rule conjI)+
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apply (smt Sequ_def append_Nil2 mem_Collect_eq Sequ_assoc subsetI)+
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apply(subgoal_tac "((Der c A) ;; (A \<up> n)) \<subseteq> ((Der c A) ;; (A ;; (A \<up> n)))")+
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apply(auto)[1]+
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by (smt Sequ_def append_Nil2 mem_Collect_eq Sequ_assoc subsetI)+
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+
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lemma test:+
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  shows "(\<Union>x\<le>Suc n. Der c (A \<up> x)) = (\<Union>x\<le>n. Der c A ;; A \<up> x)"+
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apply(induct n)+
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apply(simp)+
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apply(auto)[1]+
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apply(case_tac xa)+
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apply(simp)+
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apply(simp)+
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apply(auto)[1]+
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apply(case_tac "[] \<in> A")+
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apply(simp)+
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apply(simp)+
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by (smt Der_Pow Der_Pow_subseteq UN_insert atMost_Suc sup.orderE sup_bot.right_neutral)+
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+
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lemma test2:+
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  shows "(\<Union>x\<in>(Suc ` A). Der c (B \<up> x)) = (\<Union>x\<in>A. Der c B ;; B \<up> x)"+
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apply(auto)[1]+
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apply(case_tac "[] \<in> B")+
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apply(simp)+
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using Der_Pow_subseteq apply blast+
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apply(simp)+
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done+
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+
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+
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section {* Kleene Star for Languages *}+
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+
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definition +
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  Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102) where+
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  "A\<star> = (\<Union>n. A \<up> n)"+
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+
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lemma Star_empty [intro]:+
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  shows "[] \<in> A\<star>"+
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unfolding Star_def+
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by auto+
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+
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lemma Star_step [intro]:+
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  assumes "s1 \<in> A" "s2 \<in> A\<star>"+
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  shows "s1 @ s2 \<in> A\<star>"+
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proof -+
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  from assms obtain n where "s1 \<in>A" "s2 \<in> A \<up> n"+
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  unfolding Star_def by auto+
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  then have "s1 @ s2 \<in> A ;; (A \<up> n)"+
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  by (auto simp add: Sequ_def)+
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  then have "s1 @ s2 \<in> A \<up> (Suc n)"+
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  by simp+
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  then show "s1 @ s2 \<in> A\<star>" +
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  unfolding Star_def +
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  by (auto simp del: Pow.simps)+
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qed+
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+
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lemma star_cases:+
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  shows "A\<star> = {[]} \<union> A ;; A\<star>"+
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unfolding Star_def+
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apply(simp add: Sequ_def)+
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apply(auto)+
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apply(case_tac xa)+
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apply(auto simp add: Sequ_def)+
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apply(rule_tac x="Suc xa" in exI)+
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apply(auto simp add: Sequ_def)+
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done+
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+
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lemma Der_Star1:+
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  shows "Der c (A ;; A\<star>) = (Der c A) ;; A\<star>"+
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proof -+
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  have "(Der c A) ;; A\<star> = (Der c A) ;; (\<Union>n. A \<up> n)"+
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  unfolding Star_def by simp+
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  also have "... = (\<Union>n. Der c A ;; A \<up> n)"+
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  unfolding Sequ_UNION by simp+
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  also have "... = (\<Union>x\<in>(Suc ` UNIV). Der c (A \<up> x))"+
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  unfolding test2 by simp+
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  also have "... = (\<Union>n. Der c (A \<up> (Suc n)))"+
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  by (simp)+
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  also have "... = Der c (\<Union>n. A \<up> (Suc n))"+
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  unfolding Der_UNION by simp+
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  also have "... = Der c (A ;; (\<Union>n. A \<up> n))"+
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  by (simp only: Pow.simps Sequ_UNION)+
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  finally show "Der c (A ;; A\<star>) = (Der c A) ;; A\<star>"+
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  unfolding Star_def[symmetric] by blast +
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qed+
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+
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lemma Der_star [simp]:+
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  shows "Der c (A\<star>) = (Der c A) ;; A\<star>"+
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proof -    +
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  have "Der c (A\<star>) = Der c ({[]} \<union> A ;; A\<star>)"  +
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    by (simp only: star_cases[symmetric])+
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  also have "... = Der c (A ;; A\<star>)"+
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    by (simp only: Der_union Der_empty) (simp)+
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  also have "... =  (Der c A) ;; A\<star>"+
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    using Der_Star1 by simp+
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  finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .+
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qed+
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+
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+
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+
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+
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section {* Regular Expressions *}+
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+
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datatype rexp =+
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  ZERO+
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| ONE+
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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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| UPNTIMES rexp nat+
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| NTIMES rexp nat+
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| FROMNTIMES rexp nat+
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| NMTIMES rexp nat nat+
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| PLUS rexp+
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+
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section {* Semantics of Regular Expressions *}+
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 +
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fun+
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  L :: "rexp \<Rightarrow> string set"+
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where+
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  "L (ZERO) = {}"+
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| "L (ONE) = {[]}"+
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| "L (CHAR c) = {[c]}"+
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| "L (SEQ r1 r2) = (L r1) ;; (L r2)"+
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| "L (ALT r1 r2) = (L r1) \<union> (L r2)"+
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| "L (STAR r) = (L r)\<star>"+
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| "L (UPNTIMES r n) = (\<Union>i\<in> {..n} . (L r) \<up> i)"+
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| "L (NTIMES r n) = (L r) \<up> n"+
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| "L (FROMNTIMES r n) = (\<Union>i\<in> {n..} . (L r) \<up> i)"+
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| "L (NMTIMES r n m) = (\<Union>i\<in>{n..m} . (L r) \<up> i)" +
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| "L (PLUS r) = (\<Union>i\<in> {1..} . (L r) \<up> i)"+
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+
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section {* Nullable, Derivatives *}+
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+
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fun+
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 nullable :: "rexp \<Rightarrow> bool"+
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where+
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  "nullable (ZERO) = False"+
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| "nullable (ONE) = True"+
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| "nullable (CHAR c) = False"+
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| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"+
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| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"+
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| "nullable (STAR r) = True"+
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| "nullable (UPNTIMES r n) = True"+
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| "nullable (NTIMES r n) = (if n = 0 then True else nullable r)"+
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| "nullable (FROMNTIMES r n) = (if n = 0 then True else nullable r)"+
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| "nullable (NMTIMES r n m) = (if m < n then False else (if n = 0 then True else nullable r))"+
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| "nullable (PLUS r) = (nullable r)"+
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+
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fun+
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 der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"+
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where+
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  "der c (ZERO) = ZERO"+
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| "der c (ONE) = ZERO"+
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| "der c (CHAR d) = (if c = d then ONE else ZERO)"+
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| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"+
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| "der c (SEQ r1 r2) = +
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     (if nullable r1+
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      then ALT (SEQ (der c r1) r2) (der c r2)+
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      else SEQ (der c r1) r2)"+
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| "der c (STAR r) = SEQ (der c r) (STAR r)"+
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| "der c (UPNTIMES r 0) = ZERO"+
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| "der c (UPNTIMES r (Suc n)) = SEQ (der c r) (UPNTIMES r n)"+
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| "der c (NTIMES r 0) = ZERO"+
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| "der c (NTIMES r (Suc n)) = SEQ (der c r) (NTIMES r n)"+
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| "der c (FROMNTIMES r 0) = SEQ (der c r) (FROMNTIMES r 0)"+
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| "der c (FROMNTIMES r (Suc n)) = SEQ (der c r) (FROMNTIMES r n)"+
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| "der c (NMTIMES r n m) = +
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     (if m < n then ZERO +
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      else (if n = 0 then (if m = 0 then ZERO else +
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                           SEQ (der c r) (UPNTIMES r (m - 1))) else +
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                           SEQ (der c r) (NMTIMES r (n - 1) (m - 1))))" +
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| "der c (PLUS r) = SEQ (der c r) (STAR r)"+
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+
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fun +
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 ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"+
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where+
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  "ders [] r = r"+
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| "ders (c # s) r = ders s (der c r)"+
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+
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+
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lemma nullable_correctness:+
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  shows "nullable r  \<longleftrightarrow> [] \<in> (L r)"+
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apply(induct r) +
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apply(auto simp add: Sequ_def) +
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done+
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+
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lemma Der_Pow_notin:+
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  assumes "[] \<notin> A"+
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  shows "Der c (A \<up> (Suc n)) = (Der c A) ;; (A \<up> n)"+
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using assms+
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by(simp)+
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+
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lemma der_correctness:+
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  shows "L (der c r) = Der c (L r)"+
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apply(induct c r rule: der.induct) +
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apply(simp_all add: nullable_correctness)[7]+
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apply(simp only: der.simps L.simps)+
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apply(simp only: Der_UNION)+
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apply(simp only: Sequ_UNION[symmetric])+
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apply(simp add: test)+
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apply(simp)+
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(* NTIMES *)+
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apply(simp only: L.simps der.simps)+
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apply(simp)+
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apply(rule impI)+
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apply (simp add: Der_Pow_subseteq sup_absorb1)+
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(* FROMNTIMES *)+
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apply(simp only: der.simps)+
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apply(simp only: L.simps)+
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apply(simp)+
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using Der_star Star_def apply auto[1]+
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apply(simp only: der.simps)+
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apply(simp only: L.simps)+
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apply(simp add: Der_UNION)+
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apply(subst Sequ_UNION[symmetric])+
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apply(subst test2[symmetric])+
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apply(subgoal_tac "(Suc ` {n..}) = {Suc n..}")+
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apply simp+
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apply(auto simp add: image_def)[1]+
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using Suc_le_D apply blast+
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(* NMTIMES *)+
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apply(case_tac "n \<le> m")+
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prefer 2+
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apply(simp only: der.simps if_True)+
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apply(simp only: L.simps)+
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apply(simp)+
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apply(auto)+
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apply(subst (asm) Sequ_UNION[symmetric])+
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apply(subst (asm) test[symmetric])+
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apply(auto simp add: Der_UNION)[1]+
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apply(auto simp add: Der_UNION)[1]+
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apply(subst Sequ_UNION[symmetric])+
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apply(subst test[symmetric])+
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apply(auto)[1]+
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apply(subst (asm) Sequ_UNION[symmetric])+
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apply(subst (asm) test2[symmetric])+
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apply(auto simp add: Der_UNION)[1]+
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apply(subst Sequ_UNION[symmetric])+
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apply(subst test2[symmetric])+
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apply(auto simp add: Der_UNION)[1]+
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(* PLUS *)+
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apply(auto simp add: Sequ_def Star_def)[1]+
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apply(simp add: Der_UNION)+
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apply(rule_tac x="Suc xa" in bexI)+
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apply(auto simp add: Sequ_def Der_def)[2]+
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apply (metis append_Cons)+
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apply(simp add: Der_UNION)+
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apply(erule_tac bexE)+
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apply(case_tac xa)+
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apply(simp)+
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apply(simp)+
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apply(auto)+
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apply(auto simp add: Sequ_def Der_def)[1]+
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using Star_def apply auto[1]+
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apply(case_tac "[] \<in> L r")+
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apply(auto)+
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using Der_UNION Der_star Star_def by fastforce+
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+
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lemma ders_correctness:+
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  shows "L (ders s r) = Ders s (L r)"+
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apply(induct s arbitrary: r)+
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apply(simp_all add: Ders_def der_correctness Der_def)+
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done+
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+
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lemma ders_ZERO:+
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  shows "ders s (ZERO) = ZERO"+
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apply(induct s)+
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apply(simp_all)+
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done+
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+
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lemma ders_ONE:+
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  shows "ders s (ONE) = (if s = [] then ONE else ZERO)"+
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apply(induct s)+
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apply(simp_all add: ders_ZERO)+
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done+
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+
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lemma ders_CHAR:+
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  shows "ders s (CHAR c) = (if s = [c] then ONE else +
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                           (if s = [] then (CHAR c) else ZERO))"+
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apply(induct s)+
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apply(simp_all add: ders_ZERO ders_ONE)+
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done+
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+
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lemma  ders_ALT:+
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  shows "ders s (ALT r1 r2) = ALT (ders s r1) (ders s r2)"+
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apply(induct s arbitrary: r1 r2)+
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apply(simp_all)+
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done+
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+
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section {* Values *}+
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+
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datatype val = +
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  Void+
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| Char char+
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| Seq val val+
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| Right val+
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| Left val+
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| Stars "val list"+
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+
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+
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section {* The string behind a value *}+
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+
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fun +
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  flat :: "val \<Rightarrow> string"+
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where+
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  "flat (Void) = []"+
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| "flat (Char c) = [c]"+
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| "flat (Left v) = flat v"+
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| "flat (Right v) = flat v"+
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| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"+
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| "flat (Stars []) = []"+
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| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))" +
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+
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lemma flat_Stars [simp]:+
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 "flat (Stars vs) = concat (map flat vs)"+
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by (induct vs) (auto)+
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+
−
+
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section {* Relation between values and regular expressions *}+
−
+
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inductive +
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  Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100)+
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where+
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 "\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : SEQ r1 r2"+
−
| "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : ALT r1 r2"+
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| "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : ALT r1 r2"+
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| "\<turnstile> Void : ONE"+
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| "\<turnstile> Char c : CHAR c"+
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| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : STAR r"+
−
| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<le> n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : UPNTIMES r n"+
−
| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs = n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : NTIMES r n"+
−
| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : FROMNTIMES r n"+
−
| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> n; length vs \<le> m\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : NMTIMES r n m"+
−
| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> 1\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : PLUS r"+
−
+
−
+
−
inductive_cases Prf_elims:+
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  "\<turnstile> v : ZERO"+
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  "\<turnstile> v : SEQ r1 r2"+
−
  "\<turnstile> v : ALT r1 r2"+
−
  "\<turnstile> v : ONE"+
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  "\<turnstile> v : CHAR c"+
−
(*  "\<turnstile> vs : STAR r"*)+
−
+
−
lemma Prf_STAR:+
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   assumes "\<forall>v\<in>set vs. \<turnstile> v : r \<and> flat v \<in> L r"  +
−
   shows "concat (map flat vs) \<in> L r\<star>"+
−
using assms +
−
apply(induct vs)+
−
apply(auto)+
−
done+
−
+
−
lemma Prf_Pow:+
−
  assumes "\<forall>v\<in>set vs. \<turnstile> v : r \<and> flat v \<in> L r" +
−
  shows "concat (map flat vs) \<in> L r \<up> length vs"+
−
using assms+
−
apply(induct vs)+
−
apply(auto simp add: Sequ_def)+
−
done+
−
+
−
lemma Prf_flat_L:+
−
  assumes "\<turnstile> v : r" shows "flat v \<in> L r"+
−
using assms+
−
apply(induct v r rule: Prf.induct)+
−
apply(auto simp add: Sequ_def)+
−
apply(rule Prf_STAR)+
−
apply(simp)+
−
apply(rule_tac x="length vs" in bexI)+
−
apply(rule Prf_Pow)+
−
apply(simp)+
−
apply(simp)+
−
apply(rule Prf_Pow)+
−
apply(simp)+
−
apply(rule_tac x="length vs" in bexI)+
−
apply(rule Prf_Pow)+
−
apply(simp)+
−
apply(simp)+
−
apply(rule_tac x="length vs" in bexI)+
−
apply(rule Prf_Pow)+
−
apply(simp)+
−
apply(simp)+
−
using Prf_Pow by blast+
−
+
−
lemma Star_Pow:+
−
  assumes "s \<in> A \<up> n"+
−
  shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A) \<and> (length ss = n)"+
−
using assms+
−
apply(induct n arbitrary: s)+
−
apply(auto simp add: Sequ_def)+
−
apply(drule_tac x="s2" in meta_spec)+
−
apply(auto)+
−
apply(rule_tac x="s1#ss" in exI)+
−
apply(simp)+
−
done+
−
+
−
lemma Star_string:+
−
  assumes "s \<in> A\<star>"+
−
  shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A)"+
−
using assms+
−
apply(auto simp add: Star_def)+
−
using Star_Pow by blast+
−
+
−
+
−
lemma Star_val:+
−
  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"+
−
  shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"+
−
using assms+
−
apply(induct ss)+
−
apply(auto)+
−
apply (metis empty_iff list.set(1))+
−
by (metis concat.simps(2) list.simps(9) set_ConsD)+
−
+
−
lemma Star_val_length:+
−
  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"+
−
  shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r) \<and> (length vs) = (length ss)"+
−
using assms+
−
apply(induct ss)+
−
apply(auto)+
−
by (metis List.bind_def bind_simps(2) length_Suc_conv set_ConsD)+
−
+
−
+
−
+
−
+
−
+
−
lemma L_flat_Prf2:+
−
  assumes "s \<in> L r" shows "\<exists>v. \<turnstile> v : r \<and> flat v = s"+
−
using assms+
−
apply(induct r arbitrary: s)+
−
apply(auto simp add: Sequ_def intro: Prf.intros)+
−
using Prf.intros(1) flat.simps(5) apply blast+
−
using Prf.intros(2) flat.simps(3) apply blast+
−
using Prf.intros(3) flat.simps(4) apply blast+
−
apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r)")+
−
apply(auto)[1]+
−
apply(rule_tac x="Stars vs" in exI)+
−
apply(simp)+
−
apply(rule Prf.intros)+
−
apply(simp)+
−
using Star_string Star_val apply force+
−
apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")+
−
apply(auto)[1]+
−
apply(rule_tac x="Stars vs" in exI)+
−
apply(simp)+
−
apply(rule Prf.intros)+
−
apply(simp)+
−
apply(simp)+
−
using Star_Pow Star_val_length apply blast+
−
apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x2)")+
−
apply(auto)[1]+
−
apply(rule_tac x="Stars vs" in exI)+
−
apply(simp)+
−
apply(rule Prf.intros)+
−
apply(simp)+
−
apply(simp)+
−
using Star_Pow Star_val_length apply blast+
−
apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")+
−
apply(auto)[1]+
−
apply(rule_tac x="Stars vs" in exI)+
−
apply(simp)+
−
apply(rule Prf.intros)+
−
apply(simp)+
−
apply(simp)+
−
using Star_Pow Star_val_length apply blast+
−
apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")+
−
apply(auto)[1]+
−
apply(rule_tac x="Stars vs" in exI)+
−
apply(simp)+
−
apply(rule Prf.intros)+
−
apply(simp)+
−
apply(simp)+
−
apply(simp)+
−
using Star_Pow Star_val_length apply blast+
−
apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")+
−
apply(auto)[1]+
−
apply(rule_tac x="Stars vs" in exI)+
−
apply(simp)+
−
apply(rule Prf.intros)+
−
apply(simp)+
−
apply(simp)+
−
using Star_Pow Star_val_length apply blast+
−
done+
−
+
−
lemma L_flat_Prf:+
−
  "L(r) = {flat v | v. \<turnstile> v : r}"+
−
using Prf_flat_L L_flat_Prf2 by blast+
−
+
−
+
−
section {* Sulzmann and Lu functions *}+
−
+
−
fun +
−
  mkeps :: "rexp \<Rightarrow> val"+
−
where+
−
  "mkeps(ONE) = Void"+
−
| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"+
−
| "mkeps(ALT r1 r2) = (if nullable r1 then Left (mkeps r1) else Right (mkeps r2))"+
−
| "mkeps(STAR r) = Stars []"+
−
| "mkeps(UPNTIMES r n) = Stars []"+
−
| "mkeps(NTIMES r n) = Stars (replicate n (mkeps r))"+
−
| "mkeps(FROMNTIMES r n) = Stars (replicate n (mkeps r))"+
−
| "mkeps(NMTIMES r n m) = Stars (replicate n (mkeps r))"+
−
| "mkeps(PLUS r) = Stars ([mkeps r])"+
−
+
−
+
−
fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"+
−
where+
−
  "injval (CHAR d) c Void = Char d"+
−
| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"+
−
| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"+
−
| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"+
−
| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"+
−
| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"+
−
| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" +
−
| "injval (UPNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"+
−
| "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"+
−
| "injval (FROMNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"+
−
| "injval (NMTIMES r n m) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"+
−
| "injval (PLUS r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"+
−
+
−
section {* Mkeps, injval *}+
−
+
−
lemma mkeps_nullable:+
−
  assumes "nullable(r)" +
−
  shows "\<turnstile> mkeps r : r"+
−
using assms+
−
apply(induct r rule: mkeps.induct) +
−
apply(auto intro: Prf.intros)+
−
by (metis Prf.intros(10) in_set_replicate length_replicate not_le order_refl)+
−
+
−
lemma mkeps_flat:+
−
  assumes "nullable(r)" +
−
  shows "flat (mkeps r) = []"+
−
using assms+
−
apply (induct rule: nullable.induct) +
−
apply(auto)+
−
by meson+
−
+
−
+
−
lemma Prf_injval:+
−
  assumes "\<turnstile> v : der c r" +
−
  shows "\<turnstile> (injval r c v) : r"+
−
using assms+
−
apply(induct r arbitrary: c v rule: rexp.induct)+
−
apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)+
−
(* STAR *)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(auto)+
−
using Prf.intros(6) apply auto[1]+
−
(* UPNTIMES *)+
−
apply(case_tac x2)+
−
apply(simp)+
−
using Prf_elims(1) apply auto[1]+
−
apply(simp)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
using Prf.intros(7) apply auto[1]+
−
(* NTIMES *)+
−
apply(case_tac x2)+
−
apply(simp)+
−
using Prf_elims(1) apply auto[1]+
−
apply(simp)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
using Prf.intros(8) apply auto[1]+
−
(* FROMNTIMES *)+
−
apply(case_tac x2)+
−
apply(simp)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
using Prf.intros(9) apply auto[1]+
−
apply(rotate_tac 1)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
using Prf.intros(9) apply auto+
−
(* NMTIMES *)+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
apply(rule Prf.intros)+
−
apply(auto)[2]+
−
apply simp+
−
apply(rotate_tac 4)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(clarify)+
−
apply(rule Prf.intros)+
−
apply(auto)[2]+
−
apply simp+
−
(* PLUS *)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(auto)+
−
using Prf.intros apply auto[1]+
−
done+
−
+
−
+
−
lemma Prf_injval_flat:+
−
  assumes "\<turnstile> v : der c r" +
−
  shows "flat (injval r c v) = c # (flat v)"+
−
using assms+
−
apply(induct arbitrary: v rule: der.induct)+
−
apply(auto elim!: Prf_elims split: if_splits)+
−
apply(metis mkeps_flat)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 4)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 2)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
done+
−
+
−
+
−
section {* Our Alternative Posix definition *}+
−
+
−
inductive +
−
  Posix :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)+
−
where+
−
  Posix_ONE: "[] \<in> ONE \<rightarrow> Void"+
−
| Posix_CHAR: "[c] \<in> (CHAR c) \<rightarrow> (Char c)"+
−
| Posix_ALT1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"+
−
| Posix_ALT2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"+
−
| Posix_SEQ: "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;+
−
    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)\<rbrakk> \<Longrightarrow> +
−
    (s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"+
−
| Posix_STAR1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];+
−
    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>+
−
    \<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"+
−
| Posix_STAR2: "[] \<in> STAR r \<rightarrow> Stars []"+
−
| Posix_UPNTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> UPNTIMES r n \<rightarrow> Stars vs; flat v \<noteq> [];+
−
    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r n))\<rbrakk>+
−
    \<Longrightarrow> (s1 @ s2) \<in> UPNTIMES r (Suc n) \<rightarrow> Stars (v # vs)"+
−
| Posix_UPNTIMES2: "[] \<in> UPNTIMES r n \<rightarrow> Stars []"+
−
| Posix_NTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NTIMES r n \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (Stars vs) = [];+
−
    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r n))\<rbrakk>+
−
    \<Longrightarrow> (s1 @ s2) \<in> NTIMES r (Suc n) \<rightarrow> Stars (v # vs)"+
−
| Posix_NTIMES2: "[] \<in> NTIMES r 0 \<rightarrow> Stars []"+
−
| Posix_FROMNTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> FROMNTIMES r n \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (Stars vs) = [];+
−
    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r n))\<rbrakk>+
−
    \<Longrightarrow> (s1 @ s2) \<in> FROMNTIMES r (Suc n) \<rightarrow> Stars (v # vs)"+
−
| Posix_FROMNTIMES2: "s \<in> STAR r \<rightarrow> Stars vs \<Longrightarrow>  s \<in> FROMNTIMES r 0 \<rightarrow> Stars vs"+
−
| Posix_NMTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NMTIMES r n m \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (Stars vs) = [];+
−
    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (NMTIMES r n m))\<rbrakk>+
−
    \<Longrightarrow> (s1 @ s2) \<in> NMTIMES r (Suc n) (Suc m) \<rightarrow> Stars (v # vs)"+
−
| Posix_NMTIMES2: "s \<in> UPNTIMES r m \<rightarrow> Stars vs \<Longrightarrow>  s \<in> NMTIMES r 0 m \<rightarrow> Stars vs"+
−
| Posix_PLUS1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (Stars vs) = [];+
−
    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>+
−
    \<Longrightarrow> (s1 @ s2) \<in> PLUS r \<rightarrow> Stars (v # vs)"+
−
+
−
inductive_cases Posix_elims:+
−
  "s \<in> ZERO \<rightarrow> v"+
−
  "s \<in> ONE \<rightarrow> v"+
−
  "s \<in> CHAR c \<rightarrow> v"+
−
  "s \<in> ALT r1 r2 \<rightarrow> v"+
−
  "s \<in> SEQ r1 r2 \<rightarrow> v"+
−
  "s \<in> STAR r \<rightarrow> v"+
−
+
−
lemma Posix1:+
−
  assumes "s \<in> r \<rightarrow> v"+
−
  shows "s \<in> L r" "flat v = s"+
−
using assms+
−
apply (induct s r v rule: Posix.induct)+
−
apply(auto simp add: Sequ_def)+
−
apply(rule_tac x="Suc x" in bexI)+
−
apply(auto simp add: Sequ_def)+
−
apply(rule_tac x="Suc x" in bexI)+
−
using Sequ_def apply auto[2]+
−
apply(simp add: Star_def)+
−
apply(rule_tac x="Suc x" in bexI)+
−
apply(auto simp add: Sequ_def)+
−
apply(simp add: Star_def)+
−
apply(clarify)+
−
apply(rule_tac x="Suc x" in bexI)+
−
apply(auto simp add: Sequ_def)+
−
done+
−
+
−
+
−
lemma +
−
  "([] @ []) \<in> PLUS (ONE) \<rightarrow> Stars ([Void])"+
−
apply(rule Posix_PLUS1)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(simp)+
−
apply(simp)+
−
done+
−
+
−
lemma +
−
  assumes "s \<in> r \<rightarrow> v" "flat v \<noteq> []" "\<forall>s' \<in> L r. length s' < length s"+
−
  shows "([] @ (s @ [])) \<in> PLUS (ALT ONE r) \<rightarrow> Stars ([Left Void, Right v])"+
−
using assms+
−
oops+
−
+
−
lemma +
−
  "([] @ ([] @ [])) \<in> FROMNTIMES (ONE) (Suc (Suc 0)) \<rightarrow> Stars ([Void, Void])"+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(auto)+
−
done+
−
+
−
+
−
lemma +
−
  assumes "s \<in> r \<rightarrow> v" "flat v \<noteq> []"+
−
  "s \<in> PLUS (ALT ONE r) \<rightarrow> Stars ([Left(Void), Right(v)])"+
−
  shows "False"+
−
using assms+
−
apply(rotate_tac 2)+
−
apply(erule_tac Posix.cases)+
−
apply(simp_all)+
−
apply(auto)+
−
oops+
−
+
−
+
−
+
−
+
−
+
−
lemma Posix1a:+
−
  assumes "s \<in> r \<rightarrow> v"+
−
  shows "\<turnstile> v : r"+
−
using assms+
−
apply(induct s r v rule: Posix.induct)+
−
apply(auto intro: Prf.intros)+
−
apply(rule Prf.intros)+
−
apply(auto)[1]+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rule Prf.intros)+
−
apply(auto)[1]+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rule Prf.intros)+
−
apply(auto)[1]+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rule Prf.intros)+
−
apply(auto)[1]+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rule Prf.intros)+
−
apply(auto)[2]+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rule Prf.intros)+
−
apply(auto)[3]+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rule Prf.intros)+
−
apply(auto)[3]+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rotate_tac 3)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(rule Prf.intros)+
−
apply(auto)+
−
done+
−
+
−
+
−
lemma  Posix_NTIMES_mkeps:+
−
  assumes "[] \<in> r \<rightarrow> mkeps r"+
−
  shows "[] \<in> NTIMES r n \<rightarrow> Stars (replicate n (mkeps r))"+
−
apply(induct n)+
−
apply(simp)+
−
apply (rule Posix_NTIMES2)+
−
apply(simp)+
−
apply(subst append_Nil[symmetric])+
−
apply (rule Posix_NTIMES1)+
−
apply(auto)+
−
apply(rule assms)+
−
done+
−
+
−
lemma  Posix_FROMNTIMES_mkeps:+
−
  assumes "[] \<in> r \<rightarrow> mkeps r"+
−
  shows "[] \<in> FROMNTIMES r n \<rightarrow> Stars (replicate n (mkeps r))"+
−
apply(induct n)+
−
apply(simp)+
−
apply (rule Posix_FROMNTIMES2)+
−
apply (rule Posix.intros)+
−
apply(simp)+
−
apply(subst append_Nil[symmetric])+
−
apply (rule Posix_FROMNTIMES1)+
−
apply(auto)+
−
apply(rule assms)+
−
done+
−
+
−
lemma  Posix_NMTIMES_mkeps:+
−
  assumes "[] \<in> r \<rightarrow> mkeps r" "n \<le> m"+
−
  shows "[] \<in> NMTIMES r n m \<rightarrow> Stars (replicate n (mkeps r))"+
−
using assms(2)+
−
apply(induct n arbitrary: m)+
−
apply(simp)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(case_tac m)+
−
apply(simp)+
−
apply(simp)+
−
apply(subst append_Nil[symmetric])+
−
apply(rule Posix.intros)+
−
apply(auto)+
−
apply(rule assms)+
−
done+
−
+
−
+
−
+
−
lemma Posix_mkeps:+
−
  assumes "nullable r"+
−
  shows "[] \<in> r \<rightarrow> mkeps r"+
−
using assms+
−
apply(induct r rule: nullable.induct)+
−
apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)+
−
apply(subst append.simps(1)[symmetric])+
−
apply(rule Posix.intros)+
−
apply(auto)+
−
apply(case_tac n)+
−
apply(simp)+
−
apply (simp add: Posix_NTIMES2)+
−
apply(simp)+
−
apply(subst append.simps(1)[symmetric])+
−
apply(rule Posix.intros)+
−
apply(auto)+
−
apply(rule Posix_NTIMES_mkeps)+
−
apply(simp)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(case_tac n)+
−
apply(simp)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(simp)+
−
apply(subst append.simps(1)[symmetric])+
−
apply(rule Posix.intros)+
−
apply(auto)+
−
apply(rule Posix_FROMNTIMES_mkeps)+
−
apply(simp)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(case_tac n)+
−
apply(simp)+
−
apply(rule Posix.intros)+
−
apply(rule Posix.intros)+
−
apply(simp)+
−
apply(case_tac m)+
−
apply(simp)+
−
apply(simp)+
−
apply(subst append_Nil[symmetric])+
−
apply(rule Posix.intros)+
−
apply(auto)+
−
apply(rule Posix_NMTIMES_mkeps)+
−
apply(auto)+
−
apply(subst append_Nil[symmetric])+
−
apply(rule Posix.intros)+
−
apply(auto)+
−
apply(rule Posix.intros)+
−
done+
−
+
−
+
−
lemma Posix_determ:+
−
  assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"+
−
  shows "v1 = v2"+
−
using assms+
−
proof (induct s r v1 arbitrary: v2 rule: Posix.induct)+
−
  case (Posix_ONE v2)+
−
  have "[] \<in> ONE \<rightarrow> v2" by fact+
−
  then show "Void = v2" by cases auto+
−
next +
−
  case (Posix_CHAR c v2)+
−
  have "[c] \<in> CHAR c \<rightarrow> v2" by fact+
−
  then show "Char c = v2" by cases auto+
−
next +
−
  case (Posix_ALT1 s r1 v r2 v2)+
−
  have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact+
−
  moreover+
−
  have "s \<in> r1 \<rightarrow> v" by fact+
−
  then have "s \<in> L r1" by (simp add: Posix1)+
−
  ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto +
−
  moreover+
−
  have IH: "\<And>v2. s \<in> r1 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact+
−
  ultimately have "v = v'" by simp+
−
  then show "Left v = v2" using eq by simp+
−
next +
−
  case (Posix_ALT2 s r2 v r1 v2)+
−
  have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact+
−
  moreover+
−
  have "s \<notin> L r1" by fact+
−
  ultimately obtain v' where eq: "v2 = Right v'" "s \<in> r2 \<rightarrow> v'" +
−
    by cases (auto simp add: Posix1) +
−
  moreover+
−
  have IH: "\<And>v2. s \<in> r2 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact+
−
  ultimately have "v = v'" by simp+
−
  then show "Right v = v2" using eq by simp+
−
next+
−
  case (Posix_SEQ s1 r1 v1 s2 r2 v2 v')+
−
  have "(s1 @ s2) \<in> SEQ r1 r2 \<rightarrow> v'" +
−
       "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2"+
−
       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact++
−
  then obtain v1' v2' where "v' = Seq v1' v2'" "s1 \<in> r1 \<rightarrow> v1'" "s2 \<in> r2 \<rightarrow> v2'"+
−
  apply(cases) apply (auto simp add: append_eq_append_conv2)+
−
  using Posix1(1) by fastforce++
−
  moreover+
−
  have IHs: "\<And>v1'. s1 \<in> r1 \<rightarrow> v1' \<Longrightarrow> v1 = v1'"+
−
            "\<And>v2'. s2 \<in> r2 \<rightarrow> v2' \<Longrightarrow> v2 = v2'" by fact++
−
  ultimately show "Seq v1 v2 = v'" by simp+
−
next+
−
  case (Posix_STAR1 s1 r v s2 vs v2)+
−
  have "(s1 @ s2) \<in> STAR r \<rightarrow> v2" +
−
       "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" "flat v \<noteq> []"+
−
       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact++
−
  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (STAR r) \<rightarrow> (Stars vs')"+
−
  apply(cases) apply (auto simp add: append_eq_append_conv2)+
−
  using Posix1(1) apply fastforce+
−
  apply (metis Posix1(1) Posix_STAR1.hyps(6) append_Nil append_Nil2)+
−
  using Posix1(2) by blast+
−
  moreover+
−
  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"+
−
            "\<And>v2. s2 \<in> STAR r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact++
−
  ultimately show "Stars (v # vs) = v2" by auto+
−
next+
−
  case (Posix_STAR2 r v2)+
−
  have "[] \<in> STAR r \<rightarrow> v2" by fact+
−
  then show "Stars [] = v2" by cases (auto simp add: Posix1)+
−
next+
−
  case (Posix_UPNTIMES1 s1 r v s2 n vs v2)+
−
  have "(s1 @ s2) \<in> UPNTIMES r (Suc n) \<rightarrow> v2" +
−
       "s1 \<in> r \<rightarrow> v" "s2 \<in> (UPNTIMES r n) \<rightarrow> Stars vs" "flat v \<noteq> []"+
−
       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r n))" by fact++
−
  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (UPNTIMES r n) \<rightarrow> (Stars vs')"+
−
  apply(cases) apply (auto simp add: append_eq_append_conv2)+
−
  using Posix1(1) apply fastforce+
−
  apply (metis Posix1(1) Posix_UPNTIMES1.hyps(6) append_Nil append_Nil2)+
−
  using Posix1(2) by blast+
−
  moreover+
−
  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"+
−
            "\<And>v2. s2 \<in> UPNTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact++
−
  ultimately show "Stars (v # vs) = v2" by auto+
−
next+
−
  case (Posix_UPNTIMES2 r n v2)+
−
  have "[] \<in> UPNTIMES r n \<rightarrow> v2" by fact+
−
  then show "Stars [] = v2" by cases (auto simp add: Posix1)+
−
next+
−
  case (Posix_NTIMES2 r v2)+
−
  have "[] \<in> NTIMES r 0 \<rightarrow> v2" by fact+
−
  then show "Stars [] = v2" by cases (auto simp add: Posix1)+
−
next+
−
  case (Posix_NTIMES1 s1 r v s2 n vs v2)+
−
  have "(s1 @ s2) \<in> NTIMES r (Suc n) \<rightarrow> v2"  "flat v = [] \<Longrightarrow> flat (Stars vs) = []"+
−
       "s1 \<in> r \<rightarrow> v" "s2 \<in> (NTIMES r n) \<rightarrow> Stars vs"+
−
       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r n))" by fact++
−
  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (NTIMES r n) \<rightarrow> (Stars vs')"+
−
  apply(cases) apply (auto simp add: append_eq_append_conv2)+
−
  using Posix1(1) apply fastforce+
−
  by (metis Posix1(1) Posix_NTIMES1.hyps(6) append_Nil append_Nil2)+
−
  moreover+
−
  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"+
−
            "\<And>v2. s2 \<in> NTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact++
−
  ultimately show "Stars (v # vs) = v2" by auto+
−
next+
−
  case (Posix_FROMNTIMES2 s r v1 v2)+
−
  have "s \<in> FROMNTIMES r 0 \<rightarrow> v2" "s \<in> STAR r \<rightarrow> Stars v1" +
−
       "\<And>v3. s \<in> STAR r \<rightarrow> v3 \<Longrightarrow> Stars v1 = v3" by fact++
−
  then show ?case +
−
  apply(cases)+
−
  apply(auto)+
−
  done+
−
next+
−
  case (Posix_FROMNTIMES1 s1 r v s2 n vs v2)+
−
  have "(s1 @ s2) \<in> FROMNTIMES r (Suc n) \<rightarrow> v2" +
−
       "s1 \<in> r \<rightarrow> v" "s2 \<in> (FROMNTIMES r n) \<rightarrow> Stars vs" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"+
−
       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r n))" by fact++
−
  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (FROMNTIMES r n) \<rightarrow> (Stars vs')"+
−
  apply(cases) apply (auto simp add: append_eq_append_conv2)+
−
  using Posix1(1) apply fastforce+
−
  by (metis Posix1(1) Posix_FROMNTIMES1.hyps(6) append_Nil append_Nil2)+
−
  moreover+
−
  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"+
−
            "\<And>v2. s2 \<in> FROMNTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact++
−
  ultimately show "Stars (v # vs) = v2" by auto+
−
next+
−
  case (Posix_NMTIMES2 s r m v1 v2)+
−
  have "s \<in> NMTIMES r 0 m \<rightarrow> v2" "s \<in> UPNTIMES r m \<rightarrow> Stars v1" +
−
       "\<And>v3. s \<in> UPNTIMES r m \<rightarrow> v3 \<Longrightarrow> Stars v1 = v3" by fact++
−
  then show ?case +
−
  apply(cases)+
−
  apply(auto)+
−
  done+
−
next+
−
  case (Posix_NMTIMES1 s1 r v s2 n m vs v2)+
−
  have "(s1 @ s2) \<in> NMTIMES r (Suc n) (Suc m) \<rightarrow> v2" +
−
       "s1 \<in> r \<rightarrow> v" "s2 \<in> (NMTIMES r n m) \<rightarrow> Stars vs" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"+
−
       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NMTIMES r n m))" by fact++
−
  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (NMTIMES r n m) \<rightarrow> (Stars vs')"+
−
  apply(cases) apply (auto simp add: append_eq_append_conv2)+
−
  using Posix1(1) apply fastforce+
−
  by (metis Posix1(1) Posix_NMTIMES1.hyps(6) self_append_conv self_append_conv2)+
−
  moreover+
−
  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"+
−
            "\<And>v2. s2 \<in> NMTIMES r n m \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact++
−
  ultimately show "Stars (v # vs) = v2" by auto+
−
next+
−
  case (Posix_PLUS1 s1 r v s2 vs v2)+
−
  have "(s1 @ s2) \<in> PLUS r \<rightarrow> v2" +
−
       "s1 \<in> r \<rightarrow> v" "s2 \<in> (STAR r) \<rightarrow> Stars vs" (*"flat v = [] \<Longrightarrow> flat (Stars vs) = []"*)+
−
       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact++
−
  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (STAR r) \<rightarrow> (Stars vs')"+
−
  apply(cases) apply (auto simp add: append_eq_append_conv2)+
−
  using Posix1(1) apply fastforce+
−
  by (metis Posix1(1) Posix_PLUS1.hyps(6) append_self_conv append_self_conv2)+
−
  moreover+
−
  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"+
−
            "\<And>v2. s2 \<in> STAR r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact++
−
  ultimately show "Stars (v # vs) = v2" by auto+
−
qed+
−
+
−
lemma NTIMES_Stars:+
−
 assumes "\<turnstile> v : NTIMES r n"+
−
 shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> length vs = n"+
−
using assms+
−
apply(induct n arbitrary: v)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
done+
−
+
−
lemma UPNTIMES_Stars:+
−
 assumes "\<turnstile> v : UPNTIMES r n"+
−
 shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> length vs \<le> n"+
−
using assms+
−
apply(induct n arbitrary: v)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
done+
−
+
−
lemma FROMNTIMES_Stars:+
−
 assumes "\<turnstile> v : FROMNTIMES r n"+
−
 shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> n \<le> length vs"+
−
using assms+
−
apply(induct n arbitrary: v)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
done+
−
+
−
lemma PLUS_Stars:+
−
 assumes "\<turnstile> v : PLUS r"+
−
 shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> 1 \<le> length vs"+
−
using assms+
−
apply(erule_tac Prf.cases)+
−
apply(simp_all)+
−
done+
−
+
−
lemma NMTIMES_Stars:+
−
 assumes "\<turnstile> v : NMTIMES r n m"+
−
 shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> n \<le> length vs \<and> length vs \<le> m"+
−
using assms+
−
apply(induct n arbitrary: m v)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
apply(erule Prf.cases)+
−
apply(simp_all)+
−
done+
−
+
−
+
−
lemma Posix_injval:+
−
  assumes "s \<in> (der c r) \<rightarrow> v"+
−
  shows "(c # s) \<in> r \<rightarrow> (injval r c v)"+
−
using assms+
−
proof(induct r arbitrary: s v rule: rexp.induct)+
−
  case ZERO+
−
  have "s \<in> der c ZERO \<rightarrow> v" by fact+
−
  then have "s \<in> ZERO \<rightarrow> v" by simp+
−
  then have "False" by cases+
−
  then show "(c # s) \<in> ZERO \<rightarrow> (injval ZERO c v)" by simp+
−
next+
−
  case ONE+
−
  have "s \<in> der c ONE \<rightarrow> v" by fact+
−
  then have "s \<in> ZERO \<rightarrow> v" by simp+
−
  then have "False" by cases+
−
  then show "(c # s) \<in> ONE \<rightarrow> (injval ONE c v)" by simp+
−
next +
−
  case (CHAR d)+
−
  consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast+
−
  then show "(c # s) \<in> (CHAR d) \<rightarrow> (injval (CHAR d) c v)"+
−
  proof (cases)+
−
    case eq+
−
    have "s \<in> der c (CHAR d) \<rightarrow> v" by fact+
−
    then have "s \<in> ONE \<rightarrow> v" using eq by simp+
−
    then have eqs: "s = [] \<and> v = Void" by cases simp+
−
    show "(c # s) \<in> CHAR d \<rightarrow> injval (CHAR d) c v" using eq eqs +
−
    by (auto intro: Posix.intros)+
−
  next+
−
    case ineq+
−
    have "s \<in> der c (CHAR d) \<rightarrow> v" by fact+
−
    then have "s \<in> ZERO \<rightarrow> v" using ineq by simp+
−
    then have "False" by cases+
−
    then show "(c # s) \<in> CHAR d \<rightarrow> injval (CHAR d) c v" by simp+
−
  qed+
−
next+
−
  case (ALT r1 r2)+
−
  have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact+
−
  have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact+
−
  have "s \<in> der c (ALT r1 r2) \<rightarrow> v" by fact+
−
  then have "s \<in> ALT (der c r1) (der c r2) \<rightarrow> v" by simp+
−
  then consider (left) v' where "v = Left v'" "s \<in> der c r1 \<rightarrow> v'" +
−
              | (right) v' where "v = Right v'" "s \<notin> L (der c r1)" "s \<in> der c r2 \<rightarrow> v'" +
−
              by cases auto+
−
  then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v"+
−
  proof (cases)+
−
    case left+
−
    have "s \<in> der c r1 \<rightarrow> v'" by fact+
−
    then have "(c # s) \<in> r1 \<rightarrow> injval r1 c v'" using IH1 by simp+
−
    then have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Left v')" by (auto intro: Posix.intros)+
−
    then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using left by simp+
−
  next +
−
    case right+
−
    have "s \<notin> L (der c r1)" by fact+
−
    then have "c # s \<notin> L r1" by (simp add: der_correctness Der_def)+
−
    moreover +
−
    have "s \<in> der c r2 \<rightarrow> v'" by fact+
−
    then have "(c # s) \<in> r2 \<rightarrow> injval r2 c v'" using IH2 by simp+
−
    ultimately have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Right v')" +
−
      by (auto intro: Posix.intros)+
−
    then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using right by simp+
−
  qed+
−
next+
−
  case (SEQ r1 r2)+
−
  have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact+
−
  have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact+
−
  have "s \<in> der c (SEQ r1 r2) \<rightarrow> v" by fact+
−
  then consider +
−
        (left_nullable) v1 v2 s1 s2 where +
−
        "v = Left (Seq v1 v2)"  "s = s1 @ s2" +
−
        "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "nullable r1" +
−
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)"+
−
      | (right_nullable) v1 s1 s2 where +
−
        "v = Right v1" "s = s1 @ s2"  +
−
        "s \<in> der c r2 \<rightarrow> v1" "nullable r1" "s1 @ s2 \<notin> L (SEQ (der c r1) r2)"+
−
      | (not_nullable) v1 v2 s1 s2 where+
−
        "v = Seq v1 v2" "s = s1 @ s2" +
−
        "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "\<not>nullable r1" +
−
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)"+
−
        by (force split: if_splits elim!: Posix_elims simp add: Sequ_def der_correctness Der_def)   +
−
  then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" +
−
    proof (cases)+
−
      case left_nullable+
−
      have "s1 \<in> der c r1 \<rightarrow> v1" by fact+
−
      then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp+
−
      moreover+
−
      have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact+
−
      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def)+
−
      ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using left_nullable by (rule_tac Posix.intros)+
−
      then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using left_nullable by simp+
−
    next+
−
      case right_nullable+
−
      have "nullable r1" by fact+
−
      then have "[] \<in> r1 \<rightarrow> (mkeps r1)" by (rule Posix_mkeps)+
−
      moreover+
−
      have "s \<in> der c r2 \<rightarrow> v1" by fact+
−
      then have "(c # s) \<in> r2 \<rightarrow> (injval r2 c v1)" using IH2 by simp+
−
      moreover+
−
      have "s1 @ s2 \<notin> L (SEQ (der c r1) r2)" by fact+
−
      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> [] @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" using right_nullable+
−
        by(auto simp add: der_correctness Der_def append_eq_Cons_conv Sequ_def)+
−
      ultimately have "([] @ (c # s)) \<in> SEQ r1 r2 \<rightarrow> Seq (mkeps r1) (injval r2 c v1)"+
−
      by(rule Posix.intros)+
−
      then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using right_nullable by simp+
−
    next+
−
      case not_nullable+
−
      have "s1 \<in> der c r1 \<rightarrow> v1" by fact+
−
      then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp+
−
      moreover+
−
      have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact+
−
      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def)+
−
      ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using not_nullable +
−
        by (rule_tac Posix.intros) (simp_all) +
−
      then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using not_nullable by simp+
−
    qed+
−
next+
−
  case (STAR r)+
−
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact+
−
  have "s \<in> der c (STAR r) \<rightarrow> v" by fact+
−
  then consider+
−
      (cons) v1 vs s1 s2 where +
−
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" +
−
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (STAR r) \<rightarrow> (Stars vs)"+
−
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" +
−
        apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)+
−
        apply(rotate_tac 3)+
−
        apply(erule_tac Posix_elims(6))+
−
        apply (simp add: Posix.intros(6))+
−
        using Posix.intros(7) by blast+
−
    then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" +
−
    proof (cases)+
−
      case cons+
−
          have "s1 \<in> der c r \<rightarrow> v1" by fact+
−
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp+
−
        moreover+
−
          have "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
−
        moreover +
−
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact +
−
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)+
−
          then have "flat (injval r c v1) \<noteq> []" by simp+
−
        moreover +
−
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" by fact+
−
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" +
−
            by (simp add: der_correctness Der_def)+
−
        ultimately +
−
        have "((c # s1) @ s2) \<in> STAR r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros)+
−
        then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp)+
−
    qed+
−
next +
−
  case (UPNTIMES r n)+
−
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact+
−
  have "s \<in> der c (UPNTIMES r n) \<rightarrow> v" by fact+
−
  then consider+
−
      (cons) m v1 vs s1 s2 where +
−
        "n = Suc m"+
−
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" +
−
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (UPNTIMES r m) \<rightarrow> (Stars vs)"+
−
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r m))" +
−
        apply(case_tac n)+
−
        apply(simp)+
−
        using Posix_elims(1) apply blast+
−
        apply(simp)+
−
        apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)        +
−
        by (metis Posix1a UPNTIMES_Stars)+
−
    then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" +
−
    proof (cases)+
−
      case cons+
−
        have "n = Suc m" by fact+
−
        moreover+
−
          have "s1 \<in> der c r \<rightarrow> v1" by fact+
−
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp+
−
        moreover+
−
          have "s2 \<in> UPNTIMES r m \<rightarrow> Stars vs" by fact+
−
        moreover +
−
          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact +
−
          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)+
−
          then have "flat (injval r c v1) \<noteq> []" by simp+
−
        moreover +
−
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r m))" by fact+
−
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r m))" +
−
            by (simp add: der_correctness Der_def)+
−
        ultimately +
−
        have "((c # s1) @ s2) \<in> UPNTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)" +
−
          apply(rule_tac Posix.intros(8))+
−
          apply(simp_all)+
−
          done+
−
        then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" using cons by(simp)+
−
    qed+
−
next +
−
  case (NTIMES r n)+
−
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact+
−
  have "s \<in> der c (NTIMES r n) \<rightarrow> v" by fact+
−
  then consider+
−
      (cons) m v1 vs s1 s2 where +
−
        "n = Suc m"+
−
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"+
−
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NTIMES r m) \<rightarrow> (Stars vs)"+
−
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r m))" +
−
        apply(case_tac n)+
−
        apply(simp)+
−
        using Posix_elims(1) apply blast+
−
        apply(simp)+
−
        apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)+
−
        by (metis NTIMES_Stars Posix1a)+
−
    then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" +
−
    proof (cases)+
−
      case cons+
−
        have "n = Suc m" by fact+
−
        moreover+
−
          have "s1 \<in> der c r \<rightarrow> v1" by fact+
−
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp+
−
        moreover+
−
          have "flat v = [] \<Longrightarrow> flat (Stars vs) = []" by fact+
−
        moreover+
−
          have "s2 \<in> NTIMES r m \<rightarrow> Stars vs" by fact+
−
        moreover +
−
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r m))" by fact+
−
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r m))" +
−
            by (simp add: der_correctness Der_def)+
−
        ultimately +
−
        have "((c # s1) @ s2) \<in> NTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)" +
−
          apply(rule_tac Posix.intros(10))+
−
          apply(simp_all)+
−
          by (simp add: Posix1(2))+
−
        then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" using cons by(simp)+
−
    qed+
−
next +
−
  case (FROMNTIMES r n)+
−
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact+
−
  have "s \<in> der c (FROMNTIMES r n) \<rightarrow> v" by fact+
−
  then consider+
−
        (null) v1 vs s1 s2 where +
−
        "n = 0"+
−
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" +
−
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r 0) \<rightarrow> (Stars vs)"+
−
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r 0))" +
−
      | (cons) m v1 vs s1 s2 where +
−
        "n = Suc m"+
−
        "v = Seq v1 (Stars vs)" "s = s1 @ s2" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"+
−
        "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r m) \<rightarrow> (Stars vs)"+
−
        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r m))" +
−
        apply(case_tac n)+
−
        apply(simp)+
−
        apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)+
−
        defer+
−
        apply (metis FROMNTIMES_Stars Posix1a)+
−
        apply(rotate_tac 5)+
−
        apply(erule Posix.cases)+
−
        apply(simp_all)+
−
        apply(clarify)+
−
        by (simp add: Posix_FROMNTIMES2)+
−
    then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" +
−
    proof (cases)+
−
      case cons+
−
        have "n = Suc m" by fact+
−
        moreover+
−
          have "s1 \<in> der c r \<rightarrow> v1" by fact+
−
          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp+
−
        moreover+
−
          have "s2 \<in> FROMNTIMES r m \<rightarrow> Stars vs" by fact+
−
        moreover+
−
          have "flat v = [] \<Longrightarrow> flat (Stars vs) = []" by fact+
−
        moreover +
−
          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r m))" by fact+
−
          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r m))" +
−
            by (simp add: der_correctness Der_def)+
−
        ultimately +
−
        have "((c # s1) @ s2) \<in> FROMNTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)" +
−
          apply(rule_tac Posix.intros(12))+
−
          apply(simp_all) +
−
          by (simp add: Posix1(2))+
−
        then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using cons by(simp)+
−
      next+
−
      case null+
−
      then have "((c # s1) @ s2) \<in> FROMNTIMES r 0 \<rightarrow> Stars (injval r c v1 # vs)"+
−
      apply(rule_tac Posix.intros)+
−
      apply(rule_tac Posix.intros)+
−
      apply(rule IH)+
−
      apply(simp)+
−
      apply(rotate_tac 4)+
−
      apply(erule Posix.cases)+
−
      apply(simp_all)+
−
      apply (simp add: Posix1a Prf_injval_flat)+
−
      apply(simp only: Star_def)+
−
      apply(simp only: der_correctness Der_def)+
−
      apply(simp)+
−
      done+
−
      then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using null by simp+
−
    qed+
−
next +
−
  case (NMTIMES r n m)+
−
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact+
−
  have "s \<in> der c (NMTIMES r n m) \<rightarrow> v" by fact+
−
  then show "(c # s) \<in> (NMTIMES r n m) \<rightarrow> injval (NMTIMES r n m) c v"+
−
        apply(case_tac "m < n")+
−
        apply(simp)+
−
        using Posix_elims(1) apply blast+
−
        apply(case_tac n)+
−
        apply(simp_all)+
−
        apply(case_tac m)+
−
        apply(simp)+
−
        apply(erule Posix_elims(1))+
−
        apply(simp)+
−
        apply(erule Posix.cases)+
−
        apply(simp_all)+
−
        apply(clarify)+
−
        apply(rotate_tac 5)+
−
        apply(frule Posix1a)+
−
        apply(drule UPNTIMES_Stars)+
−
        apply(clarify)+
−
        apply(simp)+
−
        apply(subst append_Cons[symmetric])+
−
        apply(rule Posix.intros)+
−
        apply(rule Posix.intros)+
−
        apply(auto)+
−
        apply(rule IH)+
−
        apply blast+
−
        using Posix1a Prf_injval_flat apply auto[1]+
−
        apply(simp add: der_correctness Der_def)+
−
        apply blast+
−
        apply(case_tac m)+
−
        apply(simp)+
−
        apply(simp)+
−
        apply(erule Posix.cases)+
−
        apply(simp_all)+
−
        apply(clarify)+
−
        apply(rotate_tac 6)+
−
        apply(frule Posix1a)+
−
        apply(drule NMTIMES_Stars)+
−
        apply(clarify)+
−
        apply(simp)+
−
        apply(subst append_Cons[symmetric])+
−
        apply(rule Posix.intros)+
−
        apply(rule IH)+
−
        apply(blast)+
−
        apply(simp)+
−
        apply(simp add: der_correctness Der_def)+
−
        using Posix1a Prf_injval_flat list.distinct(1) apply auto[1]+
−
        apply(simp add: der_correctness Der_def)+
−
        done+
−
next +
−
  case (PLUS r)+
−
  have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact+
−
  have "s \<in> der c (PLUS r) \<rightarrow> v" by fact+
−
  then show "(c # s) \<in> PLUS r \<rightarrow> injval (PLUS r) c v"+
−
  apply -+
−
  apply(erule Posix.cases)+
−
  apply(simp_all)+
−
  apply(auto)+
−
  apply(rotate_tac 3)+
−
  apply(erule Posix.cases)+
−
  apply(simp_all)+
−
  apply(subst append_Cons[symmetric])+
−
  apply(rule Posix.intros)+
−
  using PLUS.hyps apply auto[1]+
−
  apply(rule Posix.intros)+
−
  apply(simp)+
−
  apply(simp)+
−
  apply(simp)+
−
  apply(simp)+
−
  apply(simp)+
−
  using Posix1a Prf_injval_flat apply auto[1]+
−
  apply(simp add: der_correctness Der_def)+
−
  apply(subst append_Nil2[symmetric])+
−
  apply(rule Posix.intros)+
−
  using PLUS.hyps apply auto[1]+
−
  apply(rule Posix.intros)+
−
  apply(simp)+
−
  apply(simp)+
−
  done  +
−
qed+
−
+
−
+
−
+
−
section {* The Lexer by Sulzmann and Lu  *}+
−
+
−
fun +
−
  lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"+
−
where+
−
  "lexer r [] = (if nullable r then Some(mkeps r) else None)"+
−
| "lexer r (c#s) = (case (lexer (der c r) s) of  +
−
                    None \<Rightarrow> None+
−
                  | Some(v) \<Rightarrow> Some(injval r c v))"+
−
+
−
+
−
lemma lexer_correct_None:+
−
  shows "s \<notin> L r \<longleftrightarrow> lexer r s = None"+
−
apply(induct s arbitrary: r)+
−
apply(simp add: nullable_correctness)+
−
apply(drule_tac x="der a r" in meta_spec)+
−
apply(auto simp add: der_correctness Der_def)+
−
done+
−
+
−
lemma lexer_correct_Some:+
−
  shows "s \<in> L r \<longleftrightarrow> (\<exists>v. lexer r s = Some(v) \<and> s \<in> r \<rightarrow> v)"+
−
apply(induct s arbitrary: r)+
−
apply(auto simp add: Posix_mkeps nullable_correctness)[1]+
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apply(drule_tac x="der a r" in meta_spec)+
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apply(simp add: der_correctness Der_def)+
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apply(rule iffI)+
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apply(auto intro: Posix_injval simp add: Posix1(1))+
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done +
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+
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lemma lexer_correctness:+
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  shows "(lexer r s = Some v) \<longleftrightarrow> s \<in> r \<rightarrow> v"+
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  and   "(lexer r s = None) \<longleftrightarrow> \<not>(\<exists>v. s \<in> r \<rightarrow> v)"+
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using Posix1(1) Posix_determ lexer_correct_None lexer_correct_Some apply fastforce+
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using Posix1(1) lexer_correct_None lexer_correct_Some by blast+
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+
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value "lexer (PLUS (ALT ONE (SEQ (CHAR (CHR ''a'')) (CHAR (CHR ''b''))))) [CHR ''a'', CHR ''b'']"+
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+
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+
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unused_thms+
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end+
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