--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/LexerExt.thy Sun Oct 10 18:35:21 2021 +0100
@@ -0,0 +1,649 @@
+
+theory LexerExt
+ imports SpecExt
+begin
+
+
+section {* The Lexer Functions by Sulzmann and Lu *}
+
+fun
+ mkeps :: "rexp \<Rightarrow> val"
+where
+ "mkeps(ONE) = Void"
+| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"
+| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
+| "mkeps(STAR r) = Stars []"
+| "mkeps(UPNTIMES r n) = Stars []"
+| "mkeps(NTIMES r n) = Stars (replicate n (mkeps r))"
+| "mkeps(FROMNTIMES r n) = Stars (replicate n (mkeps r))"
+| "mkeps(NMTIMES r n m) = Stars (replicate n (mkeps r))"
+
+fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
+where
+ "injval (CHAR d) c Void = Char d"
+| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"
+| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"
+| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
+| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
+| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
+| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+| "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+| "injval (FROMNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+| "injval (UPNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+| "injval (NMTIMES r n m) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+
+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))"
+
+
+
+section {* Mkeps, Injval Properties *}
+
+lemma mkeps_flat:
+ assumes "nullable(r)"
+ shows "flat (mkeps r) = []"
+using assms
+ apply(induct rule: nullable.induct)
+ apply(auto)
+ by presburger
+
+
+lemma mkeps_nullable:
+ assumes "nullable(r)"
+ shows "\<Turnstile> mkeps r : r"
+using assms
+apply(induct rule: nullable.induct)
+ apply(auto intro: Prf.intros split: if_splits)
+ using Prf.intros(8) apply force
+ apply(subst append.simps(1)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp)
+ apply(simp)
+ apply (simp add: mkeps_flat)
+ apply(simp)
+ using Prf.intros(9) apply force
+ apply(subst append.simps(1)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp)
+ apply(simp)
+ apply (simp add: mkeps_flat)
+ apply(simp)
+ using Prf.intros(11) apply force
+ apply(subst append.simps(1)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp)
+ apply(simp)
+ apply (simp add: mkeps_flat)
+ apply(simp)
+ apply(simp)
+done
+
+
+lemma Prf_injval_flat:
+ assumes "\<Turnstile> v : der c r"
+ shows "flat (injval r c v) = c # (flat v)"
+using assms
+apply(induct arbitrary: v rule: der.induct)
+apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits)
+done
+
+lemma Prf_injval:
+ assumes "\<Turnstile> v : der c r"
+ shows "\<Turnstile> (injval r c v) : r"
+using assms
+apply(induct r arbitrary: c v rule: rexp.induct)
+apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)[6]
+ apply(simp add: Prf_injval_flat)
+ apply(simp)
+ apply(case_tac x2)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ using Prf.intros(7) Prf_injval_flat apply auto[1]
+ apply(simp)
+ apply(case_tac x2)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(subst append.simps(2)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp add: Prf_injval_flat)
+ apply(simp)
+ apply(simp)
+ prefer 2
+ apply(simp)
+ apply(case_tac "x3a < x2")
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(case_tac x2)
+ apply(simp)
+ apply(case_tac x3a)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ using Prf.intros(12) Prf_injval_flat apply auto[1]
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(subst append.simps(2)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp add: Prf_injval_flat)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ using Prf.intros(12) Prf_injval_flat apply auto[1]
+ apply(case_tac x2)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp_all)
+ apply (simp add: Prf.intros(10) Prf_injval_flat)
+ using Prf.intros(10) Prf_injval_flat apply auto[1]
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp_all)
+ apply(subst append.simps(2)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp add: Prf_injval_flat)
+ apply(simp)
+ apply(simp)
+done
+
+
+
+text {*
+ Mkeps and injval produce, or preserve, Posix values.
+*}
+
+lemma Posix_mkeps:
+ assumes "nullable r"
+ shows "[] \<in> r \<rightarrow> mkeps r"
+using assms
+apply(induct r rule: nullable.induct)
+apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)
+apply(subst append.simps(1)[symmetric])
+apply(rule Posix.intros)
+ apply(auto)
+ done
+
+
+lemma Posix_injval:
+ 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 (UPNTIMES r n s v)
+ 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 (UPNTIMES r n) \<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> (UPNTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+ "\<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 (UPNTIMES r (n - 1)))"
+ (* here *)
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ apply(erule Posix_elims)
+ apply(simp)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply(drule_tac x="v1" in meta_spec)
+ apply(drule_tac x="vss" in meta_spec)
+ apply(drule_tac x="s1" in meta_spec)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp add: der_correctness Der_def)
+ apply(erule Posix_elims)
+ apply(auto)
+ done
+ then show "(c # s) \<in> (UPNTIMES r n) \<rightarrow> injval (UPNTIMES r n) 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> (UPNTIMES r (n - 1)) \<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 (UPNTIMES r (n - 1)))" 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 (UPNTIMES r (n - 1)))"
+ by (simp add: der_correctness Der_def)
+ ultimately
+ have "((c # s1) @ s2) \<in> UPNTIMES r n \<rightarrow> Stars (injval r c v1 # vs)"
+ thm Posix.intros
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ apply(case_tac n)
+ apply(simp)
+ using Posix_elims(1) UPNTIMES.prems apply auto[1]
+ apply(simp)
+ done
+ then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" using cons 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
+ next
+ case (NTIMES r n s v)
+ 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 (NTIMES r n) \<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> (NTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+ "\<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 (NTIMES r (n - 1)))"
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ apply(erule Posix_elims)
+ apply(simp)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply(drule_tac x="v1" in meta_spec)
+ apply(drule_tac x="vss" in meta_spec)
+ apply(drule_tac x="s1" in meta_spec)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp add: der_correctness Der_def)
+ apply(erule Posix_elims)
+ apply(auto)
+ done
+ then show "(c # s) \<in> (NTIMES r n) \<rightarrow> injval (NTIMES r n) 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> (NTIMES r (n - 1)) \<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 (NTIMES r (n - 1)))" 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 (NTIMES r (n - 1)))"
+ by (simp add: der_correctness Der_def)
+ ultimately
+ have "((c # s1) @ s2) \<in> NTIMES r n \<rightarrow> Stars (injval r c v1 # vs)"
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ apply(case_tac n)
+ apply(simp)
+ using Posix_elims(1) NTIMES.prems apply auto[1]
+ apply(simp)
+ done
+ then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" using cons by(simp)
+ qed
+ next
+ case (FROMNTIMES r n s v)
+ 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 (FROMNTIMES r n) \<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> (FROMNTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+ "\<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 (FROMNTIMES r (n - 1)))"
+ | (null) v1 vs s1 s2 where
+ "v = Seq v1 (Stars vs)" "s = s1 @ s2" "s2 \<in> (STAR r) \<rightarrow> (Stars vs)"
+ "s1 \<in> der c r \<rightarrow> v1" "n = 0"
+ "\<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 simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ prefer 2
+ apply(erule Posix_elims)
+ apply(simp)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply(drule_tac x="v1" in meta_spec)
+ apply(drule_tac x="vss" in meta_spec)
+ apply(drule_tac x="s1" in meta_spec)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp add: der_correctness Der_def)
+ apply(rotate_tac 5)
+ apply(erule Posix_elims)
+ apply(auto)[2]
+ apply(erule Posix_elims)
+ apply(simp)
+ apply blast
+ apply(erule Posix_elims)
+ apply(auto)
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply simp
+ apply(rotate_tac 6)
+ apply(erule Posix_elims)
+ apply(auto)[2]
+ done
+ then show "(c # s) \<in> (FROMNTIMES r n) \<rightarrow> injval (FROMNTIMES r n) 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> (FROMNTIMES r (n - 1)) \<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 (FROMNTIMES r (n - 1)))" 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 (FROMNTIMES r (n - 1)))"
+ by (simp add: der_correctness Der_def)
+ ultimately
+ have "((c # s1) @ s2) \<in> FROMNTIMES r n \<rightarrow> Stars (injval r c v1 # vs)"
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ apply(case_tac n)
+ apply(simp)
+ using Posix_elims(1) FROMNTIMES.prems apply auto[1]
+ using cons(5) apply blast
+ apply(simp)
+ done
+ then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using cons by(simp)
+ next
+ case null
+ 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
+ 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> FROMNTIMES r 0 \<rightarrow> Stars (injval r c v1 # vs)"
+ apply (rule_tac Posix.intros) back
+ apply(simp_all)
+ done
+ then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using null
+ apply(simp)
+ done
+ qed
+ next
+ case (NMTIMES r n m s v)
+ 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 (NMTIMES r n m) \<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> (NMTIMES r (n - 1) (m - 1)) \<rightarrow> (Stars vs)" "0 < n" "n \<le> m"
+ "\<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 (NMTIMES r (n - 1) (m - 1)))"
+ | (null) v1 vs s1 s2 where
+ "v = Seq v1 (Stars vs)" "s = s1 @ s2" "s2 \<in> (UPNTIMES r (m - 1)) \<rightarrow> (Stars vs)"
+ "s1 \<in> der c r \<rightarrow> v1" "n = 0" "0 < m"
+ "\<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 (UPNTIMES r (m - 1)))"
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ prefer 2
+ apply(erule Posix_elims)
+ apply(simp)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply(drule_tac x="v1" in meta_spec)
+ apply(drule_tac x="vss" in meta_spec)
+ apply(drule_tac x="s1" in meta_spec)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp add: der_correctness Der_def)
+ apply(rotate_tac 5)
+ apply(erule Posix_elims)
+ apply(auto)[2]
+ apply(erule Posix_elims)
+ apply(simp)
+ apply blast
+
+ apply(erule Posix_elims)
+ apply(auto)
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply simp
+ apply(rotate_tac 6)
+ apply(erule Posix_elims)
+ apply(auto)[2]
+ done
+ then show "(c # s) \<in> (NMTIMES r n m) \<rightarrow> injval (NMTIMES r n m) 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> (NMTIMES r (n - 1) (m - 1)) \<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 (NMTIMES r (n - 1) (m - 1)))" 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 (NMTIMES r (n - 1) (m - 1)))"
+ by (simp add: der_correctness Der_def)
+ ultimately
+ have "((c # s1) @ s2) \<in> NMTIMES r n m \<rightarrow> Stars (injval r c v1 # vs)"
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ apply(case_tac n)
+ apply(simp)
+ using Posix_elims(1) NMTIMES.prems apply auto[1]
+ using cons(5) apply blast
+ apply(simp)
+ apply(rule cons)
+ done
+ then show "(c # s) \<in> NMTIMES r n m \<rightarrow> injval (NMTIMES r n m) c v" using cons by(simp)
+ next
+ case null
+ 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> UPNTIMES r (m - 1) \<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
+ 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 (UPNTIMES r (m - 1)))" 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 (UPNTIMES r (m - 1)))"
+ by (simp add: der_correctness Der_def)
+ ultimately
+ have "((c # s1) @ s2) \<in> NMTIMES r 0 m \<rightarrow> Stars (injval r c v1 # vs)"
+ apply (rule_tac Posix.intros) back
+ apply(simp_all)
+ apply(rule null)
+ done
+ then show "(c # s) \<in> NMTIMES r n m \<rightarrow> injval (NMTIMES r n m) c v" using null
+ apply(simp)
+ done
+ qed
+qed
+
+section {* Lexer Correctness *}
+
+lemma lexer_correct_None:
+ shows "s \<notin> L r \<longleftrightarrow> lexer r s = None"
+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)"
+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: Posix_injval simp add: Posix1(1))
+done
+
+lemma lexer_correctness:
+ shows "(lexer r s = Some v) \<longleftrightarrow> s \<in> r \<rightarrow> v"
+ and "(lexer r s = None) \<longleftrightarrow> \<not>(\<exists>v. s \<in> r \<rightarrow> v)"
+using Posix1(1) Posix_determ lexer_correct_None lexer_correct_Some apply fastforce
+using Posix1(1) lexer_correct_None lexer_correct_Some by blast
+
+end
\ No newline at end of file