diff -r c090baa7059d -r 8b8db9558ecf thys/Spec.thy
--- a/thys/Spec.thy	Mon Feb 11 23:18:05 2019 +0000
+++ b/thys/Spec.thy	Sun Feb 17 22:15:06 2019 +0000
@@ -1,187 +1,10 @@
    
 theory Spec
-  imports Main "~~/src/HOL/Library/Sublist"
+  imports RegLangs
 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)
-
-
-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_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 {* Regular Expressions *}
-
-datatype rexp =
-  ZERO
-| ONE
-| CHAR char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-
-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>"
-
-
-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"
-
-
-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)"
-
-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)"
-by (induct r) (simp_all add: nullable_correctness)
-
-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)
-
-lemma ders_append:
-  shows "ders (s1 @ s2) r = ders s2 (ders s1 r)"
-  apply(induct s1 arbitrary: s2 r)
-  apply(auto)
-  done
-
-
-section {* Values *}
+section {* "Plain" Values *}
 
 datatype val = 
   Void
@@ -212,28 +35,6 @@
  "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
-
 
 section {* Lexical Values *}
 
@@ -262,7 +63,7 @@
 by (auto intro: Prf.intros elim!: Prf_elims)
 
 
-lemma Star_cval:
+lemma flats_Prf_value:
   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
@@ -293,9 +94,9 @@
   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  
+  using Star_split 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 
+  using IH flats_Prf_value by metis 
   then show "\<exists>v. \<Turnstile> v : STAR r \<and> flat v = s"
   using Prf.intros(6) flat_Stars by blast
 next 
@@ -455,7 +256,7 @@
 
 
 
-section {* Our POSIX Definition *}
+section {* Our inductive POSIX Definition *}
 
 inductive 
   Posix :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
@@ -488,7 +289,8 @@
    (auto simp add: Sequ_def)
 
 text {*
-  Our Posix definition determines a unique value.
+  For a give value and string, our Posix definition 
+  determines a unique value.
 *}
 
 lemma Posix_determ: