
diff -r f16019b11179 -r e85099ac4c6c thys/SpecExt.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/SpecExt.thy	Wed Sep 06 00:52:08 2017 +0100
@@ -0,0 +1,978 @@
+   
+theory SpecExt
+  imports Main "~~/src/HOL/Library/Sublist"
+begin
+
+section {* Sequential Composition of Languages *}
+
+definition
+  Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
+where 
+  "A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"
+
+text {* Two Simple Properties about Sequential Composition *}
+
+lemma Sequ_empty_string [simp]:
+  shows "A ;; {[]} = A"
+  and   "{[]} ;; A = A"
+by (simp_all add: Sequ_def)
+
+lemma Sequ_empty [simp]:
+  shows "A ;; {} = {}"
+  and   "{} ;; A = {}"
+by (simp_all add: Sequ_def)
+
+lemma Sequ_assoc:
+  shows "(A ;; B) ;; C = A ;; (B ;; C)"
+apply(auto simp add: Sequ_def)
+apply blast
+by (metis append_assoc)
+
+lemma Sequ_Union_in:
+  shows "(A ;; (\<Union>x\<in> B. C x)) = (\<Union>x\<in> B. A ;; C x)" 
+by (auto simp add: Sequ_def)
+
+section {* Semantic Derivative (Left Quotient) of Languages *}
+
+definition
+  Der :: "char \<Rightarrow> string set \<Rightarrow> string set"
+where
+  "Der c A \<equiv> {s. c # s \<in> A}"
+
+definition
+  Ders :: "string \<Rightarrow> string set \<Rightarrow> string set"
+where
+  "Ders s A \<equiv> {s'. s @ s' \<in> A}"
+
+lemma Der_null [simp]:
+  shows "Der c {} = {}"
+unfolding Der_def
+by auto
+
+lemma Der_empty [simp]:
+  shows "Der c {[]} = {}"
+unfolding Der_def
+by auto
+
+lemma Der_char [simp]:
+  shows "Der c {[d]} = (if c = d then {[]} else {})"
+unfolding Der_def
+by auto
+
+lemma Der_union [simp]:
+  shows "Der c (A \<union> B) = Der c A \<union> Der c B"
+unfolding Der_def
+by auto
+
+lemma Der_UNION [simp]: 
+  shows "Der c (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. Der c (B x))"
+by (auto simp add: Der_def)
+
+lemma Der_Sequ [simp]:
+  shows "Der c (A ;; B) = (Der c A) ;; B \<union> (if [] \<in> A then Der c B else {})"
+unfolding Der_def Sequ_def
+by (auto simp add: Cons_eq_append_conv)
+
+
+section {* Kleene Star for Languages *}
+
+inductive_set
+  Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
+  for A :: "string set"
+where
+  start[intro]: "[] \<in> A\<star>"
+| step[intro]:  "\<lbrakk>s1 \<in> A; s2 \<in> A\<star>\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> A\<star>"
+
+(* Arden's lemma *)
+
+lemma Star_cases:
+  shows "A\<star> = {[]} \<union> A ;; A\<star>"
+unfolding Sequ_def
+by (auto) (metis Star.simps)
+
+lemma Star_decomp: 
+  assumes "c # x \<in> A\<star>" 
+  shows "\<exists>s1 s2. x = s1 @ s2 \<and> c # s1 \<in> A \<and> s2 \<in> A\<star>"
+using assms
+by (induct x\<equiv>"c # x" rule: Star.induct) 
+   (auto simp add: append_eq_Cons_conv)
+
+lemma Star_Der_Sequ: 
+  shows "Der c (A\<star>) \<subseteq> (Der c A) ;; A\<star>"
+unfolding Der_def Sequ_def
+by(auto simp add: Star_decomp)
+
+
+lemma Der_star [simp]:
+  shows "Der c (A\<star>) = (Der c A) ;; A\<star>"
+proof -    
+  have "Der c (A\<star>) = Der c ({[]} \<union> A ;; A\<star>)"  
+    by (simp only: Star_cases[symmetric])
+  also have "... = Der c (A ;; A\<star>)"
+    by (simp only: Der_union Der_empty) (simp)
+  also have "... = (Der c A) ;; A\<star> \<union> (if [] \<in> A then Der c (A\<star>) else {})"
+    by simp
+  also have "... =  (Der c A) ;; A\<star>"
+    using Star_Der_Sequ by auto
+  finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .
+qed
+
+section {* Power operation for Sets *}
+
+fun 
+  Pow :: "string set \<Rightarrow> nat \<Rightarrow> string set" ("_ \<up> _" [101, 102] 101)
+where
+   "A \<up> 0 = {[]}"
+|  "A \<up> (Suc n) = A ;; (A \<up> n)"
+
+lemma Pow_empty [simp]:
+  shows "[] \<in> A \<up> n \<longleftrightarrow> (n = 0 \<or> [] \<in> A)"
+by(induct n) (auto simp add: Sequ_def)
+
+lemma Pow_Suc_rev:
+  "A \<up> (Suc n) =  (A \<up> n) ;; A"
+apply(induct n arbitrary: A)
+apply(simp_all)
+by (metis Sequ_assoc)
+
+
+lemma Pow_decomp: 
+  assumes "c # x \<in> A \<up> n" 
+  shows "\<exists>s1 s2. x = s1 @ s2 \<and> c # s1 \<in> A \<and> s2 \<in> A \<up> (n - 1)"
+using assms
+apply(induct n) 
+apply(auto simp add: Cons_eq_append_conv Sequ_def)
+apply(case_tac n)
+apply(auto simp add: Sequ_def)
+apply(blast)
+done
+
+lemma Star_Pow:
+  assumes "s \<in> A\<star>"
+  shows "\<exists>n. s \<in> A \<up> n"
+using assms
+apply(induct)
+apply(auto)
+apply(rule_tac x="Suc n" in exI)
+apply(auto simp add: Sequ_def)
+done
+
+lemma Pow_Star:
+  assumes "s \<in> A \<up> n"
+  shows "s \<in> A\<star>"
+using assms
+apply(induct n arbitrary: s)
+apply(auto simp add: Sequ_def)
+done
+
+lemma Der_Pow_0:
+  shows "Der c (A \<up> 0) = {}"
+by(simp add: Der_def)
+
+lemma Der_Pow_Suc:
+  shows "Der c (A \<up> (Suc n)) = (Der c A) ;; (A \<up> n)"
+unfolding Der_def Sequ_def 
+apply(auto simp add: Cons_eq_append_conv Sequ_def dest!: Pow_decomp)
+apply(case_tac n)
+apply(force simp add: Sequ_def)+
+done
+
+lemma Der_Pow [simp]:
+  shows "Der c (A \<up> n) = (if n = 0 then {} else (Der c A) ;; (A \<up> (n - 1)))"
+apply(case_tac n)
+apply(simp_all del: Pow.simps add: Der_Pow_0 Der_Pow_Suc)
+done
+
+lemma Der_Pow_Sequ [simp]:
+  shows "Der c (A ;; A \<up> n) = (Der c A) ;; (A \<up> n)"
+by (simp only: Pow.simps[symmetric] Der_Pow) (simp)
+
+
+lemma Pow_Sequ_Un:
+  assumes "0 < x"
+  shows "(\<Union>n \<in> {..x}. (A \<up> n)) = ({[]} \<union> (\<Union>n \<in> {..x - Suc 0}. A ;; (A \<up> n)))"
+using assms
+apply(auto simp add: Sequ_def)
+apply(smt Pow.elims Sequ_def Suc_le_mono Suc_pred atMost_iff empty_iff insert_iff mem_Collect_eq)
+apply(rule_tac x="Suc xa" in bexI)
+apply(auto simp add: Sequ_def)
+done
+
+lemma Pow_Sequ_Un2:
+  assumes "0 < x"
+  shows "(\<Union>n \<in> {x..}. (A \<up> n)) = (\<Union>n \<in> {x - Suc 0..}. A ;; (A \<up> n))"
+using assms
+apply(auto simp add: Sequ_def)
+apply(case_tac n)
+apply(auto simp add: Sequ_def)
+apply fastforce
+apply(case_tac x)
+apply(auto)
+apply(rule_tac x="Suc xa" in bexI)
+apply(auto simp add: Sequ_def)
+done
+
+section {* Regular Expressions *}
+
+datatype rexp =
+  ZERO
+| ONE
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+| UPNTIMES rexp nat
+| NTIMES rexp nat
+| FROMNTIMES rexp nat
+| NMTIMES rexp nat nat
+
+section {* Semantics of Regular Expressions *}
+ 
+fun
+  L :: "rexp \<Rightarrow> string set"
+where
+  "L (ZERO) = {}"
+| "L (ONE) = {[]}"
+| "L (CHAR c) = {[c]}"
+| "L (SEQ r1 r2) = (L r1) ;; (L r2)"
+| "L (ALT r1 r2) = (L r1) \<union> (L r2)"
+| "L (STAR r) = (L r)\<star>"
+| "L (UPNTIMES r n) = (\<Union>i\<in> {..n} . (L r) \<up> i)"
+| "L (NTIMES r n) = (L r) \<up> n"
+| "L (FROMNTIMES r n) = (\<Union>i\<in> {n..} . (L r) \<up> i)"
+| "L (NMTIMES r n m) = (\<Union>i\<in>{n..m} . (L r) \<up> i)" 
+
+section {* Nullable, Derivatives *}
+
+fun
+ nullable :: "rexp \<Rightarrow> bool"
+where
+  "nullable (ZERO) = False"
+| "nullable (ONE) = True"
+| "nullable (CHAR c) = False"
+| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
+| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
+| "nullable (STAR r) = True"
+| "nullable (UPNTIMES r n) = True"
+| "nullable (NTIMES r n) = (if n = 0 then True else nullable r)"
+| "nullable (FROMNTIMES r n) = (if n = 0 then True else nullable r)"
+| "nullable (NMTIMES r n m) = (if m < n then False else (if n = 0 then True else nullable r))"
+
+fun
+ der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+  "der c (ZERO) = ZERO"
+| "der c (ONE) = ZERO"
+| "der c (CHAR d) = (if c = d then ONE else ZERO)"
+| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
+| "der c (SEQ r1 r2) = 
+     (if nullable r1
+      then ALT (SEQ (der c r1) r2) (der c r2)
+      else SEQ (der c r1) r2)"
+| "der c (STAR r) = SEQ (der c r) (STAR r)"
+| "der c (UPNTIMES r n) = (if n = 0 then ZERO else SEQ (der c r) (UPNTIMES r (n - 1)))"
+| "der c (NTIMES r n) = (if n = 0 then ZERO else SEQ (der c r) (NTIMES r (n - 1)))"
+| "der c (FROMNTIMES r n) = SEQ (der c r) (FROMNTIMES r (n - 1))"
+| "der c (NMTIMES r n m) = 
+     (if m < n then ZERO 
+      else (if n = 0 then (if m = 0 then ZERO else 
+                           SEQ (der c r) (UPNTIMES r (m - 1))) else 
+                           SEQ (der c r) (NMTIMES r (n - 1) (m - 1))))" 
+
+fun 
+ ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
+where
+  "ders [] r = r"
+| "ders (c # s) r = ders s (der c r)"
+
+
+lemma nullable_correctness:
+  shows "nullable r  \<longleftrightarrow> [] \<in> (L r)"
+by(induct r) (auto simp add: Sequ_def) 
+
+
+lemma der_correctness:
+  shows "L (der c r) = Der c (L r)"
+apply(induct r) 
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(simp add: nullable_correctness del: Der_UNION)
+prefer 2
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(rule impI)
+apply(subst Sequ_Union_in)
+apply(subst Der_Pow_Sequ[symmetric])
+apply(subst Pow.simps[symmetric])
+apply(subst Der_UNION[symmetric])
+apply(subst Pow_Sequ_Un)
+apply(simp)
+apply(simp only: Der_union Der_empty)
+apply(simp)
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(subst Sequ_Union_in)
+apply(subst Der_Pow_Sequ[symmetric])
+apply(subst Pow.simps[symmetric])
+apply(case_tac x2)
+prefer 2
+apply(subst Pow_Sequ_Un2)
+apply(simp)
+apply(simp)
+apply(auto simp add: Sequ_def Der_def)[1]
+apply(rule_tac x="Suc xa" in exI)
+apply(auto simp add: Sequ_def)[1]
+apply(drule Pow_decomp)
+apply(auto)[1]
+apply (metis append_Cons)
+apply(simp add: nullable_correctness del: Der_UNION)
+apply(rule impI)
+apply(rule conjI)
+apply(rule impI)
+apply(subst Sequ_Union_in)
+apply(subst Der_Pow_Sequ[symmetric])
+apply(subst Pow.simps[symmetric])
+apply(subst Der_UNION[symmetric])
+apply(case_tac x3a)
+apply(simp)
+apply(clarify)
+apply(auto simp add: Sequ_def Der_def Cons_eq_append_conv)[1]
+apply(rule_tac x="Suc xa" in bexI)
+apply(auto simp add: Sequ_def)[2]
+apply (metis append_Cons)
+apply (metis (no_types, hide_lams) Pow_decomp atMost_iff diff_Suc_eq_diff_pred diff_is_0_eq)
+apply(rule impI)+
+apply(subst Sequ_Union_in)
+apply(subst Der_Pow_Sequ[symmetric])
+apply(subst Pow.simps[symmetric])
+apply(subst Der_UNION[symmetric])
+apply(case_tac x2)
+apply(simp)
+apply(simp del: Pow.simps)
+apply(auto simp add: Sequ_def Der_def)
+apply (metis One_nat_def Suc_le_D Suc_le_mono atLeastAtMost_iff diff_Suc_1 not_le)
+by fastforce
+
+
+
+lemma ders_correctness:
+  shows "L (ders s r) = Ders s (L r)"
+by (induct s arbitrary: r)
+   (simp_all add: Ders_def der_correctness Der_def)
+
+
+
+section {* Values *}
+
+datatype val = 
+  Void
+| Char char
+| Seq val val
+| Right val
+| Left val
+| Stars "val list"
+
+
+section {* The string behind a value *}
+
+fun 
+  flat :: "val \<Rightarrow> string"
+where
+  "flat (Void) = []"
+| "flat (Char c) = [c]"
+| "flat (Left v) = flat v"
+| "flat (Right v) = flat v"
+| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"
+| "flat (Stars []) = []"
+| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))" 
+
+abbreviation
+  "flats vs \<equiv> concat (map flat vs)"
+
+lemma flat_Stars [simp]:
+ "flat (Stars vs) = flats vs"
+by (induct vs) (auto)
+
+lemma Star_concat:
+  assumes "\<forall>s \<in> set ss. s \<in> A"  
+  shows "concat ss \<in> A\<star>"
+using assms by (induct ss) (auto)
+
+lemma Star_cstring:
+  assumes "s \<in> A\<star>"
+  shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A \<and> s \<noteq> [])"
+using assms
+apply(induct rule: Star.induct)
+apply(auto)[1]
+apply(rule_tac x="[]" in exI)
+apply(simp)
+apply(erule exE)
+apply(clarify)
+apply(case_tac "s1 = []")
+apply(rule_tac x="ss" in exI)
+apply(simp)
+apply(rule_tac x="s1#ss" in exI)
+apply(simp)
+done
+
+lemma Aux:
+  assumes "\<forall>s\<in>set ss. s = []"
+  shows "concat ss = []"
+using assms
+by (induct ss) (auto)
+
+lemma Pow_cstring_nonempty:
+  assumes "s \<in> A \<up> n"
+  shows "\<exists>ss. concat ss = s \<and> length ss \<le> n \<and> (\<forall>s \<in> set ss. s \<in> A \<and> s \<noteq> [])"
+using assms
+apply(induct n arbitrary: s)
+apply(auto)
+apply(simp add: Sequ_def)
+apply(erule exE)+
+apply(clarify)
+apply(drule_tac x="s2" in meta_spec)
+apply(simp)
+apply(clarify)
+apply(case_tac "s1 = []")
+apply(simp)
+apply(rule_tac x="ss" in exI)
+apply(simp)
+apply(rule_tac x="s1 # ss" in exI)
+apply(simp)
+done
+
+lemma Pow_cstring:
+  assumes "s \<in> A \<up> n"
+  shows "\<exists>ss1 ss2. concat (ss1 @ ss2) = s \<and> length (ss1 @ ss2) = n \<and> 
+         (\<forall>s \<in> set ss1. s \<in> A \<and> s \<noteq> []) \<and> (\<forall>s \<in> set ss2. s \<in> A \<and> s = [])"
+using assms
+apply(induct n arbitrary: s)
+apply(auto)[1]
+apply(simp only: Pow_Suc_rev)
+apply(simp add: Sequ_def)
+apply(erule exE)+
+apply(clarify)
+apply(drule_tac x="s1" in meta_spec)
+apply(simp)
+apply(erule exE)+
+apply(clarify)
+apply(case_tac "s2 = []")
+apply(simp)
+apply(rule_tac x="ss1" in exI)
+apply(rule_tac x="s2#ss2" in exI)
+apply(simp)
+apply(rule_tac x="ss1 @ [s2]" in exI)
+apply(rule_tac x="ss2" in exI)
+apply(simp)
+apply(subst Aux)
+apply(auto)[1]
+apply(subst Aux)
+apply(auto)[1]
+apply(simp)
+done
+
+
+section {* Lexical Values *}
+
+
+
+inductive 
+  Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<Turnstile> _ : _" [100, 100] 100)
+where
+ "\<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"
+| "\<Turnstile> v2 : r2 \<Longrightarrow> \<Turnstile> Right v2 : ALT r1 r2"
+| "\<Turnstile> Void : ONE"
+| "\<Turnstile> Char c : CHAR c"
+| "\<lbrakk>\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> []\<rbrakk> \<Longrightarrow> \<Turnstile> Stars vs : STAR r"
+| "\<lbrakk>\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> []; length vs \<le> n\<rbrakk> \<Longrightarrow> \<Turnstile> Stars vs : UPNTIMES r n"
+| "\<lbrakk>\<forall>v \<in> set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []; 
+    \<forall>v \<in> set vs2. \<Turnstile> v : r \<and> flat v = []; 
+    length (vs1 @ vs2) = n\<rbrakk> \<Longrightarrow> \<Turnstile> Stars (vs1 @ vs2) : NTIMES r n"
+| "\<lbrakk>\<forall>v \<in> set vs1. \<Turnstile> v : r  \<and> flat v \<noteq> []; 
+    \<forall>v \<in> set vs2. \<Turnstile> v : r \<and> flat v = []; 
+    length (vs1 @ vs2) \<ge> n\<rbrakk> \<Longrightarrow> \<Turnstile> Stars (vs1 @ vs2) : FROMNTIMES r n"
+| "\<lbrakk>\<forall>v \<in> set vs1. \<Turnstile> v : r \<and> flat v \<noteq> [];
+    \<forall>v \<in> set vs2. \<Turnstile> v : r \<and> flat v = []; 
+    length (vs1 @ vs2) \<ge> n; length (vs1 @ vs2) \<le> m\<rbrakk> \<Longrightarrow> \<Turnstile> Stars (vs1 @ vs2) : NMTIMES r n m"
+
+inductive_cases Prf_elims:
+  "\<Turnstile> v : ZERO"
+  "\<Turnstile> v : SEQ r1 r2"
+  "\<Turnstile> v : ALT r1 r2"
+  "\<Turnstile> v : ONE"
+  "\<Turnstile> v : CHAR c"
+  "\<Turnstile> vs : STAR r"
+  "\<Turnstile> vs : UPNTIMES r n"
+  "\<Turnstile> vs : NTIMES r n"
+  "\<Turnstile> vs : FROMNTIMES r n"
+  "\<Turnstile> vs : NMTIMES r n m"
+
+lemma Prf_Stars_appendE:
+  assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
+  shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r" 
+using assms
+by (auto intro: Prf.intros elim!: Prf_elims)
+
+lemma flats_empty:
+  assumes "(\<forall>v\<in>set vs. flat v = [])"
+  shows "flats vs = []"
+using assms
+by(induct vs) (simp_all)
+
+lemma Star_cval:
+  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"
+  shows "\<exists>vs. flats vs = concat ss \<and> (\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> [])"
+using assms
+apply(induct ss)
+apply(auto)
+apply(rule_tac x="[]" in exI)
+apply(simp)
+apply(case_tac "flat v = []")
+apply(rule_tac x="vs" in exI)
+apply(simp)
+apply(rule_tac x="v#vs" in exI)
+apply(simp)
+done
+
+
+lemma flats_cval:
+  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"
+  shows "\<exists>vs1 vs2. flats (vs1 @ vs2) = concat ss \<and> length (vs1 @ vs2) = length ss \<and> 
+          (\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []) \<and>
+          (\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = [])"
+using assms
+apply(induct ss rule: rev_induct)
+apply(rule_tac x="[]" in exI)+
+apply(simp)
+apply(simp)
+apply(clarify)
+apply(case_tac "flat v = []")
+apply(rule_tac x="vs1" in exI)
+apply(rule_tac x="v#vs2" in exI)
+apply(simp)
+apply(rule_tac x="vs1 @ [v]" in exI)
+apply(rule_tac x="vs2" in exI)
+apply(simp)
+apply(subst (asm) (2) flats_empty)
+apply(simp)
+apply(simp)
+done
+
+lemma flats_cval_nonempty:
+  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"
+  shows "\<exists>vs. flats vs = concat ss \<and> length vs \<le> length ss \<and> 
+          (\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> [])" 
+using assms
+apply(induct ss)
+apply(rule_tac x="[]" in exI)
+apply(simp)
+apply(simp)
+apply(clarify)
+apply(case_tac "flat v = []")
+apply(rule_tac x="vs" in exI)
+apply(simp)
+apply(rule_tac x="v # vs" in exI)
+apply(simp)
+done
+
+lemma Pow_flats:
+  assumes "\<forall>v \<in> set vs. flat v \<in> A"
+  shows "flats vs \<in> A \<up> length vs"
+using assms
+by(induct vs)(auto simp add: Sequ_def)
+
+lemma Pow_flats_appends:
+  assumes "\<forall>v \<in> set vs1. flat v \<in> A" "\<forall>v \<in> set vs2. flat v \<in> A"
+  shows "flats vs1 @ flats vs2 \<in> A \<up> (length vs1 + length vs2)"
+using assms
+apply(induct vs1)
+apply(auto simp add: Sequ_def Pow_flats)
+done
+
+lemma L_flat_Prf1:
+  assumes "\<Turnstile> v : r" 
+  shows "flat v \<in> L r"
+using assms
+apply(induct) 
+apply(auto simp add: Sequ_def Star_concat Pow_flats)
+apply(meson Pow_flats atMost_iff)
+using Pow_flats_appends apply blast
+apply(meson Pow_flats_appends atLeast_iff)
+apply(meson Pow_flats_appends atLeastAtMost_iff)
+done
+
+lemma L_flat_Prf2:
+  assumes "s \<in> L r" 
+  shows "\<exists>v. \<Turnstile> v : r \<and> flat v = s"
+using assms
+proof(induct r arbitrary: s)
+  case (STAR r s)
+  have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
+  have "s \<in> L (STAR r)" by fact
+  then obtain ss where "concat ss = s" "\<forall>s \<in> set ss. s \<in> L r \<and> s \<noteq> []"
+  using Star_cstring by auto  
+  then obtain vs where "flats vs = s" "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> []"
+  using IH Star_cval by metis 
+  then show "\<exists>v. \<Turnstile> v : STAR r \<and> flat v = s"
+  using Prf.intros(6) flat_Stars by blast
+next 
+  case (SEQ r1 r2 s)
+  then show "\<exists>v. \<Turnstile> v : SEQ r1 r2 \<and> flat v = s"
+  unfolding Sequ_def L.simps by (fastforce intro: Prf.intros)
+next
+  case (ALT r1 r2 s)
+  then show "\<exists>v. \<Turnstile> v : ALT r1 r2 \<and> flat v = s"
+  unfolding L.simps by (fastforce intro: Prf.intros)
+next
+  case (NTIMES r n)
+  have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
+  have "s \<in> L (NTIMES r n)" by fact
+  then obtain ss1 ss2 where "concat (ss1 @ ss2) = s" "length (ss1 @ ss2) = n" 
+    "\<forall>s \<in> set ss1. s \<in> L r \<and> s \<noteq> []" "\<forall>s \<in> set ss2. s \<in> L r \<and> s = []"
+  using Pow_cstring by force
+  then obtain vs1 vs2 where "flats (vs1 @ vs2) = s" "length (vs1 @ vs2) = n" 
+      "\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []" "\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = []"
+  using IH flats_cval 
+  apply -
+  apply(drule_tac x="ss1 @ ss2" in meta_spec)
+  apply(drule_tac x="r" in meta_spec)
+  apply(drule meta_mp)
+  apply(simp)
+  apply (metis Un_iff)
+  apply(clarify)
+  apply(drule_tac x="vs1" in meta_spec)
+  apply(drule_tac x="vs2" in meta_spec)
+  apply(simp)
+  done
+  then show "\<exists>v. \<Turnstile> v : NTIMES r n \<and> flat v = s"
+  using Prf.intros(8) flat_Stars by blast
+next 
+  case (FROMNTIMES r n)
+  have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
+  have "s \<in> L (FROMNTIMES r n)" by fact 
+  then obtain ss1 ss2 m where "concat (ss1 @ ss2) = s" "length (ss1 @ ss2) = m" "n \<le> m" 
+    "\<forall>s \<in> set ss1. s \<in> L r \<and> s \<noteq> []" "\<forall>s \<in> set ss2. s \<in> L r \<and> s = []"
+  using Pow_cstring by auto blast
+  then obtain vs1 vs2 where "flats (vs1 @ vs2) = s" "length (vs1 @ vs2) = m" "n \<le> m"
+      "\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []" "\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = []"
+  using IH flats_cval 
+  apply -
+  apply(drule_tac x="ss1 @ ss2" in meta_spec)
+  apply(drule_tac x="r" in meta_spec)
+  apply(drule meta_mp)
+  apply(simp)
+  apply (metis Un_iff)
+  apply(clarify)
+  apply(drule_tac x="vs1" in meta_spec)
+  apply(drule_tac x="vs2" in meta_spec)
+  apply(simp)
+  done
+  then show "\<exists>v. \<Turnstile> v : FROMNTIMES r n \<and> flat v = s"
+  using Prf.intros(9) flat_Stars by blast
+next 
+  case (NMTIMES r n m)
+  have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
+  have "s \<in> L (NMTIMES r n m)" by fact 
+  then obtain ss1 ss2 k where "concat (ss1 @ ss2) = s" "length (ss1 @ ss2) = k" "n \<le> k" "k \<le> m" 
+    "\<forall>s \<in> set ss1. s \<in> L r \<and> s \<noteq> []" "\<forall>s \<in> set ss2. s \<in> L r \<and> s = []"
+  using Pow_cstring by (auto, blast)
+  then obtain vs1 vs2 where "flats (vs1 @ vs2) = s" "length (vs1 @ vs2) = k" "n \<le> k" "k \<le> m"
+      "\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []" "\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = []"
+  using IH flats_cval 
+  apply -
+  apply(drule_tac x="ss1 @ ss2" in meta_spec)
+  apply(drule_tac x="r" in meta_spec)
+  apply(drule meta_mp)
+  apply(simp)
+  apply (metis Un_iff)
+  apply(clarify)
+  apply(drule_tac x="vs1" in meta_spec)
+  apply(drule_tac x="vs2" in meta_spec)
+  apply(simp)
+  done
+  then show "\<exists>v. \<Turnstile> v : NMTIMES r n m \<and> flat v = s"
+  apply(rule_tac x="Stars (vs1 @ vs2)" in exI)
+  apply(simp)
+  apply(rule Prf.intros)
+  apply(auto) 
+  done
+next 
+  case (UPNTIMES r n s)
+  have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
+  have "s \<in> L (UPNTIMES r n)" by fact
+  then obtain ss where "concat ss = s" "\<forall>s \<in> set ss. s \<in> L r \<and> s \<noteq> []" "length ss \<le> n"
+  using Pow_cstring_nonempty by force
+  then obtain vs where "flats vs = s" "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> []" "length vs \<le> n"
+  using IH flats_cval_nonempty by (smt order.trans) 
+  then show "\<exists>v. \<Turnstile> v : UPNTIMES r n \<and> flat v = s"
+  using Prf.intros(7) flat_Stars by blast
+qed (auto intro: Prf.intros)
+
+
+lemma L_flat_Prf:
+  shows "L(r) = {flat v | v. \<Turnstile> v : r}"
+using L_flat_Prf1 L_flat_Prf2 by blast
+
+
+
+section {* Sets of Lexical Values *}
+
+text {*
+  Shows that lexical values are finite for a given regex and string.
+*}
+
+definition
+  LV :: "rexp \<Rightarrow> string \<Rightarrow> val set"
+where  "LV r s \<equiv> {v. \<Turnstile> v : r \<and> flat v = s}"
+
+lemma LV_simps:
+  shows "LV ZERO s = {}"
+  and   "LV ONE s = (if s = [] then {Void} else {})"
+  and   "LV (CHAR c) s = (if s = [c] then {Char c} else {})"
+  and   "LV (ALT r1 r2) s = Left ` LV r1 s \<union> Right ` LV r2 s"
+unfolding LV_def
+by (auto intro: Prf.intros elim: Prf.cases)
+
+
+abbreviation
+  "Prefixes s \<equiv> {s'. prefixeq s' s}"
+
+abbreviation
+  "Suffixes s \<equiv> {s'. suffixeq s' s}"
+
+abbreviation
+  "SSuffixes s \<equiv> {s'. suffix s' s}"
+
+lemma Suffixes_cons [simp]:
+  shows "Suffixes (c # s) = Suffixes s \<union> {c # s}"
+by (auto simp add: suffixeq_def Cons_eq_append_conv)
+
+
+lemma finite_Suffixes: 
+  shows "finite (Suffixes s)"
+by (induct s) (simp_all)
+
+lemma finite_SSuffixes: 
+  shows "finite (SSuffixes s)"
+proof -
+  have "SSuffixes s \<subseteq> Suffixes s"
+   unfolding suffix_def suffixeq_def by auto
+  then show "finite (SSuffixes s)"
+   using finite_Suffixes finite_subset by blast
+qed
+
+lemma finite_Prefixes: 
+  shows "finite (Prefixes s)"
+proof -
+  have "finite (Suffixes (rev s))" 
+    by (rule finite_Suffixes)
+  then have "finite (rev ` Suffixes (rev s))" by simp
+  moreover
+  have "rev ` (Suffixes (rev s)) = Prefixes s"
+  unfolding suffixeq_def prefixeq_def image_def
+   by (auto)(metis rev_append rev_rev_ident)+
+  ultimately show "finite (Prefixes s)" by simp
+qed
+
+lemma LV_STAR_finite:
+  assumes "\<forall>s. finite (LV r s)"
+  shows "finite (LV (STAR r) s)"
+proof(induct s rule: length_induct)
+  fix s::"char list"
+  assume "\<forall>s'. length s' < length s \<longrightarrow> finite (LV (STAR r) s')"
+  then have IH: "\<forall>s' \<in> SSuffixes s. finite (LV (STAR r) s')"
+    by (auto simp add: suffix_def) 
+  def f \<equiv> "\<lambda>(v, vs). Stars (v # vs)"
+  def S1 \<equiv> "\<Union>s' \<in> Prefixes s. LV r s'"
+  def S2 \<equiv> "\<Union>s2 \<in> SSuffixes s. Stars -` (LV (STAR r) s2)"
+  have "finite S1" using assms
+    unfolding S1_def by (simp_all add: finite_Prefixes)
+  moreover 
+  with IH have "finite S2" unfolding S2_def
+    by (auto simp add: finite_SSuffixes inj_on_def finite_vimageI)
+  ultimately 
+  have "finite ({Stars []} \<union> f ` (S1 \<times> S2))" by simp
+  moreover 
+  have "LV (STAR r) s \<subseteq> {Stars []} \<union> f ` (S1 \<times> S2)" 
+  unfolding S1_def S2_def f_def
+  unfolding LV_def image_def prefixeq_def suffix_def
+  apply(auto elim: Prf_elims)
+  apply(erule Prf_elims)
+  apply(auto)
+  apply(case_tac vs)
+  apply(auto intro: Prf.intros)
+  done  
+  ultimately
+  show "finite (LV (STAR r) s)" by (simp add: finite_subset)
+qed  
+    
+lemma LV_UPNTIMES_STAR:
+  "LV (UPNTIMES r n) s \<subseteq> LV (STAR r) s"
+by(auto simp add: LV_def intro: Prf.intros elim: Prf_elims)
+
+lemma LV_FROMNTIMES_STAR:
+  "LV (FROMNTIMES r n) s \<subseteq> LV (STAR r) s"
+apply(auto simp add: LV_def intro: Prf.intros elim!: Prf_elims)
+oops
+
+lemma LV_finite:
+  shows "finite (LV r s)"
+proof(induct r arbitrary: s)
+  case (ZERO s) 
+  show "finite (LV ZERO s)" by (simp add: LV_simps)
+next
+  case (ONE s)
+  show "finite (LV ONE s)" by (simp add: LV_simps)
+next
+  case (CHAR c s)
+  show "finite (LV (CHAR c) s)" by (simp add: LV_simps)
+next 
+  case (ALT r1 r2 s)
+  then show "finite (LV (ALT r1 r2) s)" by (simp add: LV_simps)
+next 
+  case (SEQ r1 r2 s)
+  def f \<equiv> "\<lambda>(v1, v2). Seq v1 v2"
+  def S1 \<equiv> "\<Union>s' \<in> Prefixes s. LV r1 s'"
+  def S2 \<equiv> "\<Union>s' \<in> Suffixes s. LV r2 s'"
+  have IHs: "\<And>s. finite (LV r1 s)" "\<And>s. finite (LV r2 s)" by fact+
+  then have "finite S1" "finite S2" unfolding S1_def S2_def
+    by (simp_all add: finite_Prefixes finite_Suffixes)
+  moreover
+  have "LV (SEQ r1 r2) s \<subseteq> f ` (S1 \<times> S2)"
+    unfolding f_def S1_def S2_def 
+    unfolding LV_def image_def prefixeq_def suffixeq_def
+    by (auto elim: Prf.cases)
+  ultimately 
+  show "finite (LV (SEQ r1 r2) s)"
+    by (simp add: finite_subset)
+next
+  case (STAR r s)
+  then show "finite (LV (STAR r) s)" by (simp add: LV_STAR_finite)
+next 
+  case (NTIMES r n s)
+  have "\<And>s. finite (LV r s)" by fact
+  then show "finite (LV (NTIMES r n) s)"
+  apply(simp add: LV_def)
+qed
+
+
+
+section {* Our 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 []"
+
+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
+by (induct s r v rule: Posix.induct)
+   (auto simp add: Sequ_def)
+
+text {*
+  Our Posix definition determines a unique value.
+*}
+
+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)
+qed
+
+
+text {*
+  Our POSIX value is a lexical value.
+*}
+
+lemma Posix_LV:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "v \<in> LV r s"
+using assms unfolding LV_def
+apply(induct rule: Posix.induct)
+apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)
+done
+
+end
\ No newline at end of file




