theory Lexer imports PosixSpec beginsection \<open>The Lexer Functions by Sulzmann and Lu (without simplification)\<close>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(NTIMES r n) = Stars (replicate n (mkeps r))" 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)" | "injval (NTIMES r n) 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 \<open>Mkeps, Injval Properties\<close>lemma mkeps_flat: assumes "nullable(r)" shows "flat (mkeps r) = []"using assms by (induct rule: mkeps.induct) (auto)lemma Prf_NTimes_empty: assumes "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v = []" and "length vs = n" shows "\<Turnstile> Stars vs : NTIMES r n" using assms by (metis Prf.intros(7) empty_iff eq_Nil_appendI list.set(1))lemma mkeps_nullable: assumes "nullable(r)" shows "\<Turnstile> mkeps r : r"using assms apply (induct rule: mkeps.induct) apply(auto intro: Prf.intros split: if_splits) apply (metis Prf.intros(7) append_is_Nil_conv empty_iff list.set(1) list.size(3)) apply(rule Prf_NTimes_empty) apply(auto simp add: mkeps_flat) donelemma Prf_injval_flat: assumes "\<Turnstile> v : der c r" shows "flat (injval r c v) = c # (flat v)"using assmsapply(induct c r arbitrary: v rule: der.induct)apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits)donelemma Prf_injval: assumes "\<Turnstile> v : der c r" shows "\<Turnstile> (injval r c v) : r"using assmsapply(induct r arbitrary: c v rule: rexp.induct)apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)(* Star *)apply(simp add: Prf_injval_flat)(* NTimes *) apply(case_tac x2) apply(simp) apply(simp) apply(subst append.simps(2)[symmetric]) apply(rule Prf.intros) apply(auto simp add: Prf_injval_flat) donetext \<open>Mkeps and injval produce, or preserve, Posix values.\<close>lemma mkepsPosixSeq_pf2: shows " \<And>x1 x2 s v. \<lbrakk>\<And>s v. s \<in> der c x1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> x1 \<rightarrow> injval x1 c v; \<And>s v. s \<in> der c x2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> x2 \<rightarrow> injval x2 c v; s \<in> der c (SEQ x1 x2) \<rightarrow> v\<rbrakk> \<Longrightarrow> (c # s) \<in> (SEQ x1 x2) \<rightarrow> (injval (SEQ x1 x2) c v) " apply(case_tac "v ") apply (metis Posix1a Posix_elims(4) Prf_elims(2) der.simps(5) val.distinct(3) val.distinct(5) val.distinct(7)) apply (metis Posix1a Posix_elims(4) Prf_elims(2) der.simps(5) val.distinct(11) val.distinct(13) val.distinct(15)) sorrylemma Posix_mkeps: assumes "nullable r" shows "[] \<in> r \<rightarrow> mkeps r"using assmsapply(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)by (simp add: Posix_NTIMES2 pow_empty_iff)lemma Posix_injval: assumes "s \<in> (der c r) \<rightarrow> v" shows "(c # s) \<in> r \<rightarrow> (injval r c v)"using assmsproof(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 simpnext 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 simpnext 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 qednext 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 qednext 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 qednext 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) qednext case (NTIMES r n) 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="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 qedsection \<open>Lexer Correctness\<close>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)donelemma 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 fastforceusing Posix1(1) lexer_correct_None lexer_correct_Some by blastsubsection {* 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 mesonlemma 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) donelemma 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)) apply (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5)) by (smt (verit, best) Prf_elims(1) Prf_elims(2) Prf_elims(7) injval.simps(8) list.inject val.simps(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 Posix1a apply blast using injval_inj by blastlemma 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 blastlemma 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_thmsend