+
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theory ReStar+
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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 seq_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 seq_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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+
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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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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 {* Kleene Star for Languages *}+
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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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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 a: "c # x \<in> A\<star>" +
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  shows "\<exists>a b. x = a @ b \<and> c # a \<in> A \<and> b \<in> A\<star>"+
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using a+
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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 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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    unfolding Sequ_def Der_def+
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    by (auto dest: star_decomp)+
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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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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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+
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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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+
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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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+
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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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+
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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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+
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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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+
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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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datatype_compat val+
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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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+
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section {* Relation between values and regular expressions *}+
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+
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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"+
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| "\<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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| "\<turnstile> Stars [] : STAR r"+
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| "\<lbrakk>\<turnstile> v : r; \<turnstile> Stars vs : STAR r\<rbrakk> \<Longrightarrow> \<turnstile> Stars (v # vs) : STAR r"+
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+
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inductive_cases Prf_elims:+
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  "\<turnstile> v : ZERO"+
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  "\<turnstile> v : SEQ r1 r2"+
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  "\<turnstile> v : ALT r1 r2"+
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  "\<turnstile> v : ONE"+
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  "\<turnstile> v : CHAR c"+
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(*  "\<turnstile> vs : STAR r"*)+
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+
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lemma Prf_flat_L:+
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  assumes "\<turnstile> v : r" shows "flat v \<in> L r"+
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using assms+
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apply(induct v r rule: Prf.induct)+
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apply(auto simp add: Sequ_def)+
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done+
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+
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lemma Prf_Stars:+
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  assumes "\<forall>v \<in> set vs. \<turnstile> v : r"+
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  shows "\<turnstile> Stars vs : STAR r"+
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using assms+
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apply(induct vs)+
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apply (metis Prf.intros(6))+
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by (metis Prf.intros(7) insert_iff set_simps(2))+
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+
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lemma Star_string:+
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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)"+
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using assms+
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apply(induct rule: Star.induct)+
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apply(auto)+
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apply(rule_tac x="[]" in exI)+
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apply(simp)+
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apply(rule_tac x="s1#ss" in exI)+
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apply(simp)+
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done+
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+
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+
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lemma Star_val:+
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  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"+
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  shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"+
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using assms+
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apply(induct ss)+
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apply(auto)+
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apply (metis empty_iff list.set(1))+
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by (metis concat.simps(2) list.simps(9) set_ConsD)+
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+
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lemma L_flat_Prf:+
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  "L(r) = {flat v | v. \<turnstile> v : r}"+
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apply(induct r)+
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apply(auto dest: Prf_flat_L simp add: Sequ_def)+
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apply (metis Prf.intros(4) flat.simps(1))+
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apply (metis Prf.intros(5) flat.simps(2))+
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apply (metis Prf.intros(1) flat.simps(5))+
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apply (metis Prf.intros(2) flat.simps(3))+
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apply (metis Prf.intros(3) flat.simps(4))+
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apply(auto elim!: Prf_elims)+
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apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = x \<and> (\<forall>v \<in> set vs. \<turnstile> v : r)")+
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apply(auto)[1]+
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apply(rule_tac x="Stars vs" in exI)+
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apply(simp)+
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apply(rule Prf_Stars)+
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apply(simp)+
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apply(drule Star_string)+
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apply(auto)+
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apply(rule Star_val)+
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apply(simp)+
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done+
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+
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+
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section {* Sulzmann and Lu functions *}+
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+
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fun +
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  mkeps :: "rexp \<Rightarrow> val"+
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where+
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  "mkeps(ONE) = Void"+
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| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"+
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| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"+
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| "mkeps(STAR r) = Stars []"+
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+
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fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"+
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where+
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  "injval (CHAR d) c Void = Char d"+
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| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"+
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| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"+
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| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"+
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| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"+
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| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"+
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| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" +
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+
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+
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section {* Mkeps, injval *}+
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+
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lemma mkeps_nullable:+
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  assumes "nullable(r)" +
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  shows "\<turnstile> mkeps r : r"+
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using assms+
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by (induct rule: nullable.induct) +
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   (auto intro: Prf.intros)+
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+
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lemma mkeps_flat:+
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  assumes "nullable(r)" +
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  shows "flat (mkeps r) = []"+
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using assms+
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by (induct rule: nullable.induct) (auto)+
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+
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+
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lemma Prf_injval:+
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  assumes "\<turnstile> v : der c r" +
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  shows "\<turnstile> (injval r c v) : r"+
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using assms+
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apply(induct r arbitrary: c v rule: rexp.induct)+
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apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)+
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(* STAR *)+
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apply(rotate_tac 2)+
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apply(erule Prf.cases)+
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apply(simp_all)[7]+
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apply(auto)+
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apply (metis Prf.intros(6) Prf.intros(7))+
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by (metis Prf.intros(7))+
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+
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lemma Prf_injval_flat:+
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  assumes "\<turnstile> v : der c r" +
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  shows "flat (injval r c v) = c # (flat v)"+
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using assms+
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apply(induct arbitrary: v rule: der.induct)+
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apply(auto elim!: Prf_elims split: if_splits)+
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apply(metis mkeps_flat)+
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apply(rotate_tac 2)+
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apply(erule Prf.cases)+
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apply(simp_all)[7]+
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done+
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+
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+
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+
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section {* Our Alternative Posix definition *}+
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+
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inductive +
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  Posix :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)+
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where+
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  Posix_ONE: "[] \<in> ONE \<rightarrow> Void"+
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| Posix_CHAR: "[c] \<in> (CHAR c) \<rightarrow> (Char c)"+
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| Posix_ALT1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"+
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| Posix_ALT2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"+
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| Posix_SEQ: "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;+
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    \<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> +
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    (s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"+
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| Posix_STAR1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];+
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    \<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>+
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    \<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"+
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| Posix_STAR2: "[] \<in> STAR r \<rightarrow> Stars []"+
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+
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inductive_cases Posix_elims:+
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  "s \<in> ZERO \<rightarrow> v"+
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  "s \<in> ONE \<rightarrow> v"+
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  "s \<in> CHAR c \<rightarrow> v"+
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  "s \<in> ALT r1 r2 \<rightarrow> v"+
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  "s \<in> SEQ r1 r2 \<rightarrow> v"+
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  "s \<in> STAR r \<rightarrow> v"+
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+
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lemma Posix1:+
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  assumes "s \<in> r \<rightarrow> v"+
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  shows "s \<in> L r" "flat v = s"+
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using assms+
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by (induct s r v rule: Posix.induct)+
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   (auto simp add: Sequ_def)+
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+
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+
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lemma Posix1a:+
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  assumes "s \<in> r \<rightarrow> v"+
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  shows "\<turnstile> v : r"+
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using assms+
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apply(induct s r v rule: Posix.induct)+
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apply(auto intro: Prf.intros)+
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done+
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+
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lemma Posix_mkeps:+
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  assumes "nullable r"+
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  shows "[] \<in> r \<rightarrow> mkeps r"+
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using assms+
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apply(induct r)+
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apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)+
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apply(subst append.simps(1)[symmetric])+
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apply(rule Posix.intros)+
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apply(auto)+
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done+
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+
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+
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lemma Posix_determ:+
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  assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"+
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  shows "v1 = v2"+
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using assms+
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proof (induct s r v1 arbitrary: v2 rule: Posix.induct)+
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  case (Posix_ONE v2)+
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  have "[] \<in> ONE \<rightarrow> v2" by fact+
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  then show "Void = v2" by cases auto+
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next +
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  case (Posix_CHAR c v2)+
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  have "[c] \<in> CHAR c \<rightarrow> v2" by fact+
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  then show "Char c = v2" by cases auto+
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next +
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  case (Posix_ALT1 s r1 v r2 v2)+
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  have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact+
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  moreover+
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  have "s \<in> r1 \<rightarrow> v" by fact+
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  then have "s \<in> L r1" by (simp add: Posix1)+
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  ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto +
−
  moreover+
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  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)+
−
qed+
−
+
−
+
−
(* a proof that does not need proj *)+
−
lemma Posix2_roy_version:+
−
  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+
−
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"+
−
using assms+
−
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)"+
−
using assms+
−
apply(induct s arbitrary: r)+
−
apply(auto simp add: Posix_mkeps nullable_correctness)[1]+
−
apply(drule_tac x="der a r" in meta_spec)+
−
apply(simp add: der_correctness Der_def)+
−
apply(rule iffI)+
−
apply(auto intro: Posix2_roy_version simp add: Posix1(1))+
−
done +
−
+
−
+
−
end+
−