theory RegLangs+
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  imports Main "HOL-Library.Sublist"+
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begin+
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+
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section \<open>Sequential Composition of Languages\<close>+
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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 \<open>Two Simple Properties about Sequential Composition\<close>+
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+
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lemma Sequ_empty_string [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_empty [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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+
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section \<open>Semantic Derivative (Left Quotient) of Languages\<close>+
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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_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_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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+
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section \<open>Kleene Star for Languages\<close>+
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+
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inductive_set+
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  Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)+
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  for A :: "string set"+
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where+
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  start[intro]: "[] \<in> A\<star>"+
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| step[intro]:  "\<lbrakk>s1 \<in> A; s2 \<in> A\<star>\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> A\<star>"+
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+
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(* Arden's lemma *)+
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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 Sequ_def+
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by (auto) (metis Star.simps)+
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+
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lemma Star_decomp: +
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  assumes "c # x \<in> A\<star>" +
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  shows "\<exists>s1 s2. x = s1 @ s2 \<and> c # s1 \<in> A \<and> s2 \<in> A\<star>"+
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using assms+
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by (induct x\<equiv>"c # x" rule: Star.induct) +
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   (auto simp add: append_eq_Cons_conv)+
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+
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lemma Star_Der_Sequ: +
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  shows "Der c (A\<star>) \<subseteq> (Der c A) ;; A\<star>"+
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unfolding Der_def Sequ_def+
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by(auto simp add: Star_decomp)+
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+
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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> \<union> (if [] \<in> A then Der c (A\<star>) else {})"+
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    by simp+
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  also have "... =  (Der c A) ;; A\<star>"+
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    using Star_Der_Sequ by auto+
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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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lemma Star_concat:+
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  assumes "\<forall>s \<in> set ss. s \<in> A"  +
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  shows "concat ss \<in> A\<star>"+
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using assms by (induct ss) (auto)+
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+
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lemma Star_split:+
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  assumes "s \<in> A\<star>"+
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  shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A \<and> s \<noteq> [])"+
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using assms+
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  apply(induct rule: Star.induct)+
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  using concat.simps(1) apply fastforce+
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  apply(clarify)+
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  by (metis append_Nil concat.simps(2) set_ConsD)+
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+
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+
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+
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section \<open>Regular Expressions\<close>+
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+
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datatype rexp =+
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  ZERO+
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| ONE+
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| CH 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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+
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section \<open>Semantics of Regular Expressions\<close>+
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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 (CH 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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+
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+
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section \<open>Nullable, Derivatives\<close>+
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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 (CH 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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+
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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 (CH 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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+
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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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by (induct r) (auto simp add: Sequ_def) +
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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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by (induct r) (simp_all add: nullable_correctness)+
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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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  by (induct s arbitrary: r)+
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     (simp_all add: Ders_def der_correctness Der_def)+
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+
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lemma ders_append:+
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  shows "ders (s1 @ s2) r = ders s2 (ders s1 r)"+
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  by (induct s1 arbitrary: s2 r) (auto)+
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+
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lemma ders_snoc:+
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  shows "ders (s @ [c]) r = der c (ders s r)"+
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  by (simp add: ders_append)+
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+
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+
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(*+
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datatype ctxt = +
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    SeqC rexp bool+
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  | AltCL rexp+
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  | AltCH rexp +
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  | StarC rexp +
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+
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function+
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     down :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"+
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and  up :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"+
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where+
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  "down c (SEQ r1 r2) ctxts =+
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     (if (nullable r1) then down c r1 (SeqC r2 True # ctxts) +
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      else down c r1 (SeqC r2 False # ctxts))"+
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| "down c (CH d) ctxts = +
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     (if c = d then up c ONE ctxts else up c ZERO ctxts)"+
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| "down c ONE ctxts = up c ZERO ctxts"+
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| "down c ZERO ctxts = up c ZERO ctxts"+
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| "down c (ALT r1 r2) ctxts = down c r1 (AltCH r2 # ctxts)"+
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| "down c (STAR r1) ctxts = down c r1 (StarC r1 # ctxts)"+
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| "up c r [] = (r, [])"+
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| "up c r (SeqC r2 False # ctxts) = up c (SEQ r r2) ctxts"+
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| "up c r (SeqC r2 True # ctxts) = down c r2 (AltCL (SEQ r r2) # ctxts)"+
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| "up c r (AltCL r1 # ctxts) = up c (ALT r1 r) ctxts"+
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| "up c r (AltCH r2 # ctxts) = down c r2 (AltCL r # ctxts)"+
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| "up c r (StarC r1 # ctxts) = up c (SEQ r (STAR r1)) ctxts"+
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  apply(pat_completeness)+
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  apply(auto)+
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  done+
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+
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termination+
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  sorry+
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+
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*)+
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+
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+
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end+
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