--- a/thys3/LexerSimp.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,246 +0,0 @@
-theory LexerSimp
- imports "Lexer"
-begin
-
-section {* Lexer including some simplifications *}
-
-
-fun F_RIGHT where
- "F_RIGHT f v = Right (f v)"
-
-fun F_LEFT where
- "F_LEFT f v = Left (f v)"
-
-fun F_ALT where
- "F_ALT f\<^sub>1 f\<^sub>2 (Right v) = Right (f\<^sub>2 v)"
-| "F_ALT f\<^sub>1 f\<^sub>2 (Left v) = Left (f\<^sub>1 v)"
-| "F_ALT f1 f2 v = v"
-
-
-fun F_SEQ1 where
- "F_SEQ1 f\<^sub>1 f\<^sub>2 v = Seq (f\<^sub>1 Void) (f\<^sub>2 v)"
-
-fun F_SEQ2 where
- "F_SEQ2 f\<^sub>1 f\<^sub>2 v = Seq (f\<^sub>1 v) (f\<^sub>2 Void)"
-
-fun F_SEQ where
- "F_SEQ f\<^sub>1 f\<^sub>2 (Seq v\<^sub>1 v\<^sub>2) = Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)"
-| "F_SEQ f1 f2 v = v"
-
-fun simp_ALT where
- "simp_ALT (ZERO, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (r\<^sub>2, F_RIGHT f\<^sub>2)"
-| "simp_ALT (r\<^sub>1, f\<^sub>1) (ZERO, f\<^sub>2) = (r\<^sub>1, F_LEFT f\<^sub>1)"
-| "simp_ALT (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (ALT r\<^sub>1 r\<^sub>2, F_ALT f\<^sub>1 f\<^sub>2)"
-
-
-fun simp_SEQ where
- "simp_SEQ (ONE, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (r\<^sub>2, F_SEQ1 f\<^sub>1 f\<^sub>2)"
-| "simp_SEQ (r\<^sub>1, f\<^sub>1) (ONE, f\<^sub>2) = (r\<^sub>1, F_SEQ2 f\<^sub>1 f\<^sub>2)"
-| "simp_SEQ (ZERO, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (ZERO, undefined)"
-| "simp_SEQ (r\<^sub>1, f\<^sub>1) (ZERO, f\<^sub>2) = (ZERO, undefined)"
-| "simp_SEQ (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (SEQ r\<^sub>1 r\<^sub>2, F_SEQ f\<^sub>1 f\<^sub>2)"
-
-lemma simp_SEQ_simps[simp]:
- "simp_SEQ p1 p2 = (if (fst p1 = ONE) then (fst p2, F_SEQ1 (snd p1) (snd p2))
- else (if (fst p2 = ONE) then (fst p1, F_SEQ2 (snd p1) (snd p2))
- else (if (fst p1 = ZERO) then (ZERO, undefined)
- else (if (fst p2 = ZERO) then (ZERO, undefined)
- else (SEQ (fst p1) (fst p2), F_SEQ (snd p1) (snd p2))))))"
-by (induct p1 p2 rule: simp_SEQ.induct) (auto)
-
-lemma simp_ALT_simps[simp]:
- "simp_ALT p1 p2 = (if (fst p1 = ZERO) then (fst p2, F_RIGHT (snd p2))
- else (if (fst p2 = ZERO) then (fst p1, F_LEFT (snd p1))
- else (ALT (fst p1) (fst p2), F_ALT (snd p1) (snd p2))))"
-by (induct p1 p2 rule: simp_ALT.induct) (auto)
-
-fun
- simp :: "rexp \<Rightarrow> rexp * (val \<Rightarrow> val)"
-where
- "simp (ALT r1 r2) = simp_ALT (simp r1) (simp r2)"
-| "simp (SEQ r1 r2) = simp_SEQ (simp r1) (simp r2)"
-| "simp r = (r, id)"
-
-fun
- slexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"
-where
- "slexer r [] = (if nullable r then Some(mkeps r) else None)"
-| "slexer r (c#s) = (let (rs, fr) = simp (der c r) in
- (case (slexer rs s) of
- None \<Rightarrow> None
- | Some(v) \<Rightarrow> Some(injval r c (fr v))))"
-
-
-lemma slexer_better_simp:
- "slexer r (c#s) = (case (slexer (fst (simp (der c r))) s) of
- None \<Rightarrow> None
- | Some(v) \<Rightarrow> Some(injval r c ((snd (simp (der c r))) v)))"
-by (auto split: prod.split option.split)
-
-
-lemma L_fst_simp:
- shows "L(r) = L(fst (simp r))"
-by (induct r) (auto)
-
-lemma Posix_simp:
- assumes "s \<in> (fst (simp r)) \<rightarrow> v"
- shows "s \<in> r \<rightarrow> ((snd (simp r)) v)"
-using assms
-proof(induct r arbitrary: s v rule: rexp.induct)
- case (ALT r1 r2 s v)
- have IH1: "\<And>s v. s \<in> fst (simp r1) \<rightarrow> v \<Longrightarrow> s \<in> r1 \<rightarrow> snd (simp r1) v" by fact
- have IH2: "\<And>s v. s \<in> fst (simp r2) \<rightarrow> v \<Longrightarrow> s \<in> r2 \<rightarrow> snd (simp r2) v" by fact
- have as: "s \<in> fst (simp (ALT r1 r2)) \<rightarrow> v" by fact
- consider (ZERO_ZERO) "fst (simp r1) = ZERO" "fst (simp r2) = ZERO"
- | (ZERO_NZERO) "fst (simp r1) = ZERO" "fst (simp r2) \<noteq> ZERO"
- | (NZERO_ZERO) "fst (simp r1) \<noteq> ZERO" "fst (simp r2) = ZERO"
- | (NZERO_NZERO) "fst (simp r1) \<noteq> ZERO" "fst (simp r2) \<noteq> ZERO" by auto
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v"
- proof(cases)
- case (ZERO_ZERO)
- with as have "s \<in> ZERO \<rightarrow> v" by simp
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v" by (rule Posix_elims(1))
- next
- case (ZERO_NZERO)
- with as have "s \<in> fst (simp r2) \<rightarrow> v" by simp
- with IH2 have "s \<in> r2 \<rightarrow> snd (simp r2) v" by simp
- moreover
- from ZERO_NZERO have "fst (simp r1) = ZERO" by simp
- then have "L (fst (simp r1)) = {}" by simp
- then have "L r1 = {}" using L_fst_simp by simp
- then have "s \<notin> L r1" by simp
- ultimately have "s \<in> ALT r1 r2 \<rightarrow> Right (snd (simp r2) v)" by (rule Posix_ALT2)
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v"
- using ZERO_NZERO by simp
- next
- case (NZERO_ZERO)
- with as have "s \<in> fst (simp r1) \<rightarrow> v" by simp
- with IH1 have "s \<in> r1 \<rightarrow> snd (simp r1) v" by simp
- then have "s \<in> ALT r1 r2 \<rightarrow> Left (snd (simp r1) v)" by (rule Posix_ALT1)
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v" using NZERO_ZERO by simp
- next
- case (NZERO_NZERO)
- with as have "s \<in> ALT (fst (simp r1)) (fst (simp r2)) \<rightarrow> v" by simp
- then consider (Left) v1 where "v = Left v1" "s \<in> (fst (simp r1)) \<rightarrow> v1"
- | (Right) v2 where "v = Right v2" "s \<in> (fst (simp r2)) \<rightarrow> v2" "s \<notin> L (fst (simp r1))"
- by (erule_tac Posix_elims(4))
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v"
- proof(cases)
- case (Left)
- then have "v = Left v1" "s \<in> r1 \<rightarrow> (snd (simp r1) v1)" using IH1 by simp_all
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v" using NZERO_NZERO
- by (simp_all add: Posix_ALT1)
- next
- case (Right)
- then have "v = Right v2" "s \<in> r2 \<rightarrow> (snd (simp r2) v2)" "s \<notin> L r1" using IH2 L_fst_simp by simp_all
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v" using NZERO_NZERO
- by (simp_all add: Posix_ALT2)
- qed
- qed
-next
- case (SEQ r1 r2 s v)
- have IH1: "\<And>s v. s \<in> fst (simp r1) \<rightarrow> v \<Longrightarrow> s \<in> r1 \<rightarrow> snd (simp r1) v" by fact
- have IH2: "\<And>s v. s \<in> fst (simp r2) \<rightarrow> v \<Longrightarrow> s \<in> r2 \<rightarrow> snd (simp r2) v" by fact
- have as: "s \<in> fst (simp (SEQ r1 r2)) \<rightarrow> v" by fact
- consider (ONE_ONE) "fst (simp r1) = ONE" "fst (simp r2) = ONE"
- | (ONE_NONE) "fst (simp r1) = ONE" "fst (simp r2) \<noteq> ONE"
- | (NONE_ONE) "fst (simp r1) \<noteq> ONE" "fst (simp r2) = ONE"
- | (NONE_NONE) "fst (simp r1) \<noteq> ONE" "fst (simp r2) \<noteq> ONE"
- by auto
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v"
- proof(cases)
- case (ONE_ONE)
- with as have b: "s \<in> ONE \<rightarrow> v" by simp
- from b have "s \<in> r1 \<rightarrow> snd (simp r1) v" using IH1 ONE_ONE by simp
- moreover
- from b have c: "s = []" "v = Void" using Posix_elims(2) by auto
- moreover
- have "[] \<in> ONE \<rightarrow> Void" by (simp add: Posix_ONE)
- then have "[] \<in> fst (simp r2) \<rightarrow> Void" using ONE_ONE by simp
- then have "[] \<in> r2 \<rightarrow> snd (simp r2) Void" using IH2 by simp
- ultimately have "([] @ []) \<in> SEQ r1 r2 \<rightarrow> Seq (snd (simp r1) Void) (snd (simp r2) Void)"
- using Posix_SEQ by blast
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v" using c ONE_ONE by simp
- next
- case (ONE_NONE)
- with as have b: "s \<in> fst (simp r2) \<rightarrow> v" by simp
- from b have "s \<in> r2 \<rightarrow> snd (simp r2) v" using IH2 ONE_NONE by simp
- moreover
- have "[] \<in> ONE \<rightarrow> Void" by (simp add: Posix_ONE)
- then have "[] \<in> fst (simp r1) \<rightarrow> Void" using ONE_NONE by simp
- then have "[] \<in> r1 \<rightarrow> snd (simp r1) Void" using IH1 by simp
- moreover
- from ONE_NONE(1) have "L (fst (simp r1)) = {[]}" by simp
- then have "L r1 = {[]}" by (simp add: L_fst_simp[symmetric])
- ultimately have "([] @ s) \<in> SEQ r1 r2 \<rightarrow> Seq (snd (simp r1) Void) (snd (simp r2) v)"
- by(rule_tac Posix_SEQ) auto
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v" using ONE_NONE by simp
- next
- case (NONE_ONE)
- with as have "s \<in> fst (simp r1) \<rightarrow> v" by simp
- with IH1 have "s \<in> r1 \<rightarrow> snd (simp r1) v" by simp
- moreover
- have "[] \<in> ONE \<rightarrow> Void" by (simp add: Posix_ONE)
- then have "[] \<in> fst (simp r2) \<rightarrow> Void" using NONE_ONE by simp
- then have "[] \<in> r2 \<rightarrow> snd (simp r2) Void" using IH2 by simp
- ultimately have "(s @ []) \<in> SEQ r1 r2 \<rightarrow> Seq (snd (simp r1) v) (snd (simp r2) Void)"
- by(rule_tac Posix_SEQ) auto
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v" using NONE_ONE by simp
- next
- case (NONE_NONE)
- from as have 00: "fst (simp r1) \<noteq> ZERO" "fst (simp r2) \<noteq> ZERO"
- apply(auto)
- apply(smt Posix_elims(1) fst_conv)
- by (smt NONE_NONE(2) Posix_elims(1) fstI)
- with NONE_NONE as have "s \<in> SEQ (fst (simp r1)) (fst (simp r2)) \<rightarrow> v" by simp
- then obtain s1 s2 v1 v2 where eqs: "s = s1 @ s2" "v = Seq v1 v2"
- "s1 \<in> (fst (simp r1)) \<rightarrow> v1" "s2 \<in> (fst (simp 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 (erule_tac Posix_elims(5)) (auto simp add: L_fst_simp[symmetric])
- then have "s1 \<in> r1 \<rightarrow> (snd (simp r1) v1)" "s2 \<in> r2 \<rightarrow> (snd (simp r2) v2)"
- using IH1 IH2 by auto
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v" using eqs NONE_NONE 00
- by(auto intro: Posix_SEQ)
- qed
-qed (simp_all)
-
-
-lemma slexer_correctness:
- shows "slexer r s = lexer r s"
-proof(induct s arbitrary: r)
- case Nil
- show "slexer r [] = lexer r []" by simp
-next
- case (Cons c s r)
- have IH: "\<And>r. slexer r s = lexer r s" by fact
- show "slexer r (c # s) = lexer r (c # s)"
- proof (cases "s \<in> L (der c r)")
- case True
- assume a1: "s \<in> L (der c r)"
- then obtain v1 where a2: "lexer (der c r) s = Some v1" "s \<in> der c r \<rightarrow> v1"
- using lexer_correct_Some by auto
- from a1 have "s \<in> L (fst (simp (der c r)))" using L_fst_simp[symmetric] by simp
- then obtain v2 where a3: "lexer (fst (simp (der c r))) s = Some v2" "s \<in> (fst (simp (der c r))) \<rightarrow> v2"
- using lexer_correct_Some by auto
- then have a4: "slexer (fst (simp (der c r))) s = Some v2" using IH by simp
- from a3(2) have "s \<in> der c r \<rightarrow> (snd (simp (der c r))) v2" using Posix_simp by simp
- with a2(2) have "v1 = (snd (simp (der c r))) v2" using Posix_determ by simp
- with a2(1) a4 show "slexer r (c # s) = lexer r (c # s)" by (auto split: prod.split)
- next
- case False
- assume b1: "s \<notin> L (der c r)"
- then have "lexer (der c r) s = None" using lexer_correct_None by simp
- moreover
- from b1 have "s \<notin> L (fst (simp (der c r)))" using L_fst_simp[symmetric] by simp
- then have "lexer (fst (simp (der c r))) s = None" using lexer_correct_None by simp
- then have "slexer (fst (simp (der c r))) s = None" using IH by simp
- ultimately show "slexer r (c # s) = lexer r (c # s)"
- by (simp del: slexer.simps add: slexer_better_simp)
- qed
- qed
-
-
-unused_thms
-
-
-end
\ No newline at end of file