theory Simplifying imports "ReStar" beginsection {* Lexer including 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 (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 (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))"using assmsby (induct r) (auto)lemma Posix_simp: assumes "s \<in> (fst (simp r)) \<rightarrow> v" shows "s \<in> r \<rightarrow> ((snd (simp r)) v)"using assmsproof(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 qednext 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) with 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 by(auto intro: Posix_SEQ) qedqed (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 simpnext 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) qedqed end