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