diff -r d36be1e356c0 -r fff2e1b40dfc thys/Spec.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/Spec.thy	Wed Jul 19 14:55:46 2017 +0100
@@ -0,0 +1,746 @@
+   
+theory Spec
+  imports Main 
+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)"
+apply(induct s arbitrary: r)
+apply(simp_all add: Ders_def der_correctness Der_def)
+done
+
+
+section {* Lemmas about ders *}
+
+lemma ders_ZERO:
+  shows "ders s (ZERO) = ZERO"
+apply(induct s)
+apply(simp_all)
+done
+
+lemma ders_ONE:
+  shows "ders s (ONE) = (if s = [] then ONE else ZERO)"
+apply(induct s)
+apply(simp_all add: ders_ZERO)
+done
+
+lemma ders_CHAR:
+  shows "ders s (CHAR c) = 
+           (if s = [c] then ONE else 
+           (if s = [] then (CHAR c) else ZERO))"
+apply(induct s)
+apply(simp_all add: ders_ZERO ders_ONE)
+done
+
+lemma  ders_ALT:
+  shows "ders s (ALT r1 r2) = ALT (ders s r1) (ders s r2)"
+apply(induct s arbitrary: r1 r2)
+apply(simp_all)
+done
+
+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))" 
+
+lemma flat_Stars [simp]:
+ "flat (Stars vs) = concat (map flat vs)"
+by (induct vs) (auto)
+
+
+section {* Relation between values and regular expressions *}
+
+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"
+| "\<forall>v \<in> set vs. \<turnstile> v : r \<Longrightarrow> \<turnstile> Stars vs : STAR r"
+
+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"
+
+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_string:
+  assumes "s \<in> A\<star>"
+  shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A)"
+using assms
+apply(induct rule: Star.induct)
+apply(auto)
+apply(rule_tac x="[]" in exI)
+apply(simp)
+apply(rule_tac x="s1#ss" in exI)
+apply(simp)
+done
+
+lemma Star_val:
+  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
+  shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
+using assms
+apply(induct ss)
+apply(auto)
+apply(rule_tac x="[]" in exI)
+apply(simp)
+apply(rule_tac x="v#vs" in exI)
+apply(simp)
+done
+
+lemma Prf_Stars_append:
+  assumes "\<turnstile> Stars vs1 : STAR r" "\<turnstile> Stars vs2 : STAR r"
+  shows "\<turnstile> Stars (vs1 @ vs2) : STAR r"
+using assms
+by (auto intro!: Prf.intros elim!: Prf_elims)
+
+lemma Prf_flat_L:
+  assumes "\<turnstile> v : r" 
+  shows "flat v \<in> L r"
+using assms
+by (induct v r rule: Prf.induct)
+   (auto simp add: Sequ_def Star_concat)
+
+
+lemma L_flat_Prf1:
+  assumes "\<turnstile> v : r" 
+  shows "flat v \<in> L r"
+using assms
+by (induct) (auto simp add: Sequ_def Star_concat)
+
+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"
+  using Star_string by auto
+  then obtain vs where "concat (map flat vs) = s" "\<forall>v\<in>set vs. \<turnstile> v : r"
+  using IH Star_val by blast
+  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)
+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 {* CPT and CPTpre *}
+
+
+inductive 
+  CPrf :: "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"
+| "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> [] \<Longrightarrow> \<Turnstile> Stars vs : STAR r"
+
+lemma Prf_CPrf:
+  assumes "\<Turnstile> v : r"
+  shows "\<turnstile> v : r"
+using assms
+by (induct)(auto intro: Prf.intros)
+
+lemma CPrf_stars:
+  assumes "\<Turnstile> Stars vs : STAR r"
+  shows "\<forall>v \<in> set vs. flat v \<noteq> [] \<and> \<Turnstile> v : r"
+using assms
+apply(erule_tac CPrf.cases)
+apply(simp_all)
+done
+
+lemma CPrf_Stars_appendE:
+  assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
+  shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r" 
+using assms
+apply(erule_tac CPrf.cases)
+apply(auto intro: CPrf.intros elim: Prf.cases)
+done
+
+definition PT :: "rexp \<Rightarrow> string \<Rightarrow> val set"
+where "PT r s \<equiv> {v. flat v = s \<and> \<turnstile> v : r}"
+
+definition
+  "CPT r s = {v. flat v = s \<and> \<Turnstile> v : r}"
+
+definition
+  "CPTpre r s = {v. \<exists>s'. flat v @ s' = s \<and> \<Turnstile> v : r}"
+
+lemma CPT_CPTpre_subset:
+  shows "CPT r s \<subseteq> CPTpre r s"
+by(auto simp add: CPT_def CPTpre_def)
+
+lemma CPT_simps:
+  shows "CPT ZERO s = {}"
+  and   "CPT ONE s = (if s = [] then {Void} else {})"
+  and   "CPT (CHAR c) s = (if s = [c] then {Char c} else {})"
+  and   "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
+  and   "CPT (SEQ r1 r2) s = 
+          {Seq v1 v2 | v1 v2. flat v1 @ flat v2 = s \<and> v1 \<in> CPT r1 (flat v1) \<and> v2 \<in> CPT r2 (flat v2)}"
+  and   "CPT (STAR r) s = 
+          Stars ` {vs. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. v \<in> CPT r (flat v) \<and> flat v \<noteq> [])}"
+apply -
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+(* STAR case *)
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[6]
+done
+
+
+
+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)
+
+lemma Posix_Prf:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "\<turnstile> v : r"
+using assms
+apply(induct s r v rule: Posix.induct)
+apply(auto intro!: Prf.intros elim!: Prf_elims)
+done
+
+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 canonical value.
+*}
+
+lemma Posix_CPT:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "v \<in> CPT r s"
+using assms
+apply(induct rule: Posix.induct)
+apply(auto simp add: CPT_def intro: CPrf.intros elim: CPrf.cases)
+apply(rotate_tac 5)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule CPrf.intros)
+apply(simp_all)
+done
+
+
+
+(*
+lemma CPTpre_STAR_finite:
+  assumes "\<And>s. finite (CPT r s)"
+  shows "finite (CPT (STAR r) s)"
+apply(induct s rule: length_induct)
+apply(case_tac xs)
+apply(simp)
+apply(simp add: CPT_simps)
+apply(auto)
+apply(rule finite_imageI)
+using assms
+thm finite_Un
+prefer 2
+apply(simp add: CPT_simps)
+apply(rule finite_imageI)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
+apply(auto)[1]
+apply(rule finite_imageI)
+apply(simp add: Collect_case_prod_Sigma)
+apply(rule finite_SigmaI)
+apply(rule assms)
+apply(case_tac "flat v = []")
+apply(simp)
+apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
+apply(simp)
+apply(auto)[1]
+apply(rule test)
+apply(simp)
+done
+
+lemma CPTpre_subsets:
+  "CPTpre ZERO s = {}"
+  "CPTpre ONE s \<subseteq> {Void}"
+  "CPTpre (CHAR c) s \<subseteq> {Char c}"
+  "CPTpre (ALT r1 r2) s \<subseteq> Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
+  "CPTpre (SEQ r1 r2) s \<subseteq> {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
+  "CPTpre (STAR r) s \<subseteq> {Stars []} \<union>
+    {Stars (v#vs) | v vs. v \<in> CPTpre r s \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) s)}"
+  "CPTpre (STAR r) [] = {Stars []}"
+apply(auto simp add: CPTpre_def)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule CPrf.intros)
+done
+
+
+lemma CPTpre_simps:
+  shows "CPTpre ONE s = {Void}"
+  and   "CPTpre (CHAR c) (d#s) = (if c = d then {Char c} else {})"
+  and   "CPTpre (ALT r1 r2) s = Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
+  and   "CPTpre (SEQ r1 r2) s = 
+          {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
+apply -
+apply(rule subset_antisym)
+apply(rule CPTpre_subsets)
+apply(auto simp add: CPTpre_def intro: "CPrf.intros")[1]
+apply(case_tac "c = d")
+apply(simp)
+apply(rule subset_antisym)
+apply(rule CPTpre_subsets)
+apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
+apply(simp)
+apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)
+apply(rule subset_antisym)
+apply(rule CPTpre_subsets)
+apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
+apply(rule subset_antisym)
+apply(rule CPTpre_subsets)
+apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
+done
+
+lemma CPT_simps:
+  shows "CPT ONE s = (if s = [] then {Void} else {})"
+  and   "CPT (CHAR c) [d] = (if c = d then {Char c} else {})"
+  and   "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
+  and   "CPT (SEQ r1 r2) s = 
+          {Seq v1 v2 | v1 v2 s1 s2. s1 @ s2 = s \<and> v1 \<in> CPT r1 s1 \<and> v2 \<in> CPT r2 s2}"
+apply -
+apply(rule subset_antisym)
+apply(auto simp add: CPT_def)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+apply(clarify)
+apply blast
+apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
+apply(erule CPrf.cases)
+apply(simp_all)[7]
+done
+
+lemma test: 
+  assumes "finite A"
+  shows "finite {vs. Stars vs \<in> A}"
+using assms
+apply(induct A)
+apply(simp)
+apply(auto)
+apply(case_tac x)
+apply(simp_all)
+done
+
+lemma CPTpre_STAR_finite:
+  assumes "\<And>s. finite (CPTpre r s)"
+  shows "finite (CPTpre (STAR r) s)"
+apply(induct s rule: length_induct)
+apply(case_tac xs)
+apply(simp)
+apply(simp add: CPTpre_subsets)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
+apply(auto)[1]
+apply(rule finite_imageI)
+apply(simp add: Collect_case_prod_Sigma)
+apply(rule finite_SigmaI)
+apply(rule assms)
+apply(case_tac "flat v = []")
+apply(simp)
+apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
+apply(simp)
+apply(auto)[1]
+apply(rule test)
+apply(simp)
+done
+
+lemma CPTpre_finite:
+  shows "finite (CPTpre r s)"
+apply(induct r arbitrary: s)
+apply(simp add: CPTpre_subsets)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(rule_tac B="(\<lambda>(v1, v2). Seq v1 v2) ` {(v1, v2).  v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}" in finite_subset)
+apply(auto)[1]
+apply(rule finite_imageI)
+apply(simp add: Collect_case_prod_Sigma)
+apply(rule finite_subset)
+apply(rule CPTpre_subsets)
+apply(simp)
+by (simp add: CPTpre_STAR_finite)
+
+
+lemma CPT_finite:
+  shows "finite (CPT r s)"
+apply(rule finite_subset)
+apply(rule CPT_CPTpre_subset)
+apply(rule CPTpre_finite)
+done
+*)
+
+lemma test2: 
+  assumes "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
+  shows "(Stars vs) \<in> CPT (STAR r) (flat (Stars vs))" 
+using assms
+apply(induct vs)
+apply(auto simp add: CPT_def)
+apply(rule CPrf.intros)
+apply(simp)
+apply(rule CPrf.intros)
+apply(simp_all)
+by (metis (no_types, lifting) CPT_def Posix_CPT mem_Collect_eq)
+
+
+end
\ No newline at end of file