+ −
theory Lexer+ −
imports PosixSpec + −
begin+ −
+ − section {* The Lexer Functions by Sulzmann and Lu (without simplification) *}+ −
+ − 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 []"+ −
+ − fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"+ −
where+ −
"injval (CH 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)" + −
+ − 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_nullable:+ −
assumes "nullable(r)" + −
shows "\<Turnstile> mkeps r : r"+ −
using assms+ −
by (induct rule: nullable.induct) + −
(auto intro: Prf.intros)+ −
+ − lemma mkeps_flat:+ −
assumes "nullable(r)" + −
shows "flat (mkeps r) = []"+ −
using assms+ −
by (induct rule: nullable.induct) (auto)+ −
+ − lemma Prf_injval_flat:+ −
assumes "\<Turnstile> v : der c r" + −
shows "flat (injval r c v) = c # (flat v)"+ −
using assms+ −
apply(induct c r 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)+ −
apply(simp add: Prf_injval_flat)+ −
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 (CH d)+ −
consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast+ −
then show "(c # s) \<in> (CH d) \<rightarrow> (injval (CH d) c v)"+ −
proof (cases)+ −
case eq+ −
have "s \<in> der c (CH 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> CH d \<rightarrow> injval (CH d) c v" using eq eqs + −
by (auto intro: Posix.intros)+ −
next+ −
case ineq+ −
have "s \<in> der c (CH 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> CH d \<rightarrow> injval (CH 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 (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+ −
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)+ −
apply(simp add: nullable_correctness)+ −
apply(simp)+ −
apply(drule_tac x="der a r" in meta_spec) + −
apply(auto)+ −
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(simp only: lexer.simps)+ −
apply(simp)+ −
apply(simp add: nullable_correctness Posix_mkeps)+ −
apply(drule_tac x="der a r" in meta_spec)+ −
apply(simp (no_asm_use) add: der_correctness Der_def del: lexer.simps) + −
apply(simp del: lexer.simps)+ −
apply(simp only: lexer.simps)+ −
apply(case_tac "lexer (der a r) s = None")+ −
apply(auto)[1]+ −
apply(simp)+ −
apply(erule exE)+ −
apply(simp)+ −
apply(rule iffI)+ −
apply(simp add: Posix_injval)+ −
apply(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+ −
+ − + − subsection {* A slight reformulation of the lexer algorithm using stacked functions*}+ −
+ − fun flex :: "rexp \<Rightarrow> (val \<Rightarrow> val) => string \<Rightarrow> (val \<Rightarrow> val)"+ −
where+ −
"flex r f [] = f"+ −
| "flex r f (c#s) = flex (der c r) (\<lambda>v. f (injval r c v)) s" + −
+ − lemma flex_fun_apply:+ −
shows "g (flex r f s v) = flex r (g o f) s v"+ −
apply(induct s arbitrary: g f r v)+ −
apply(simp_all add: comp_def)+ −
by meson+ −
+ − lemma flex_fun_apply2:+ −
shows "g (flex r id s v) = flex r g s v"+ −
by (simp add: flex_fun_apply)+ −
+ − + − lemma flex_append:+ −
shows "flex r f (s1 @ s2) = flex (ders s1 r) (flex r f s1) s2"+ −
apply(induct s1 arbitrary: s2 r f)+ −
apply(simp_all)+ −
done + −
+ − lemma lexer_flex:+ −
shows "lexer r s = (if nullable (ders s r) + −
then Some(flex r id s (mkeps (ders s r))) else None)"+ −
apply(induct s arbitrary: r)+ −
apply(simp_all add: flex_fun_apply)+ −
done + −
+ − lemma Posix_flex:+ −
assumes "s2 \<in> (ders s1 r) \<rightarrow> v"+ −
shows "(s1 @ s2) \<in> r \<rightarrow> flex r id s1 v"+ −
using assms+ −
apply(induct s1 arbitrary: r v s2)+ −
apply(simp)+ −
apply(simp) + −
apply(drule_tac x="der a r" in meta_spec)+ −
apply(drule_tac x="v" in meta_spec)+ −
apply(drule_tac x="s2" in meta_spec)+ −
apply(simp)+ −
using Posix_injval+ −
apply(drule_tac Posix_injval)+ −
apply(subst (asm) (5) flex_fun_apply)+ −
apply(simp)+ −
done+ −
+ − lemma injval_inj:+ −
assumes "\<Turnstile> a : (der c r)" "\<Turnstile> v : (der c r)" "injval r c a = injval r c v" + −
shows "a = v"+ −
using assms+ −
apply(induct r arbitrary: a c v)+ −
apply(auto)+ −
using Prf_elims(1) apply blast+ −
using Prf_elims(1) apply blast+ −
apply(case_tac "c = x")+ −
apply(auto)+ −
using Prf_elims(4) apply auto[1]+ −
using Prf_elims(1) apply blast+ −
prefer 2+ −
apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) val.distinct(25) val.inject(3) val.inject(4))+ −
apply(case_tac "nullable r1")+ −
apply(auto)+ −
apply(erule Prf_elims)+ −
apply(erule Prf_elims)+ −
apply(erule Prf_elims)+ −
apply(erule Prf_elims)+ −
apply(auto)+ −
apply (metis Prf_injval_flat list.distinct(1) mkeps_flat)+ −
apply(erule Prf_elims)+ −
apply(erule Prf_elims)+ −
apply(auto)+ −
using Prf_injval_flat mkeps_flat apply fastforce+ −
apply(erule Prf_elims)+ −
apply(erule Prf_elims)+ −
apply(auto)+ −
apply(erule Prf_elims)+ −
apply(erule Prf_elims)+ −
apply(auto)+ −
apply (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5))+ −
by (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5))+ −
+ −
+ −
+ − lemma uu:+ −
assumes "(c # s) \<in> r \<rightarrow> injval r c v" "\<Turnstile> v : (der c r)"+ −
shows "s \<in> der c r \<rightarrow> v"+ −
using assms+ −
apply -+ −
apply(subgoal_tac "lexer r (c # s) = Some (injval r c v)")+ −
prefer 2+ −
using lexer_correctness(1) apply blast+ −
apply(simp add: )+ −
apply(case_tac "lexer (der c r) s")+ −
apply(simp)+ −
apply(simp)+ −
apply(case_tac "s \<in> der c r \<rightarrow> a")+ −
prefer 2+ −
apply (simp add: lexer_correctness(1))+ −
apply(subgoal_tac "\<Turnstile> a : (der c r)")+ −
prefer 2+ −
using Posix_Prf apply blast+ −
using injval_inj by blast+ −
+ −
+ − lemma Posix_flex2:+ −
assumes "(s1 @ s2) \<in> r \<rightarrow> flex r id s1 v" "\<Turnstile> v : ders s1 r"+ −
shows "s2 \<in> (ders s1 r) \<rightarrow> v"+ −
using assms+ −
apply(induct s1 arbitrary: r v s2 rule: rev_induct)+ −
apply(simp)+ −
apply(simp) + −
apply(drule_tac x="r" in meta_spec)+ −
apply(drule_tac x="injval (ders xs r) x v" in meta_spec)+ −
apply(drule_tac x="x#s2" in meta_spec)+ −
apply(simp add: flex_append ders_append)+ −
using Prf_injval uu by blast+ −
+ − lemma Posix_flex3:+ −
assumes "s1 \<in> r \<rightarrow> flex r id s1 v" "\<Turnstile> v : ders s1 r"+ −
shows "[] \<in> (ders s1 r) \<rightarrow> v"+ −
using assms+ −
by (simp add: Posix_flex2)+ −
+ − lemma flex_injval:+ −
shows "flex (der a r) (injval r a) s v = injval r a (flex (der a r) id s v)"+ −
by (simp add: flex_fun_apply)+ −
+ −
lemma Prf_flex:+ −
assumes "\<Turnstile> v : ders s r"+ −
shows "\<Turnstile> flex r id s v : r"+ −
using assms+ −
apply(induct s arbitrary: v r)+ −
apply(simp)+ −
apply(simp)+ −
by (simp add: Prf_injval flex_injval)+ −
+ − + − unused_thms+ −
+ − end+ −