--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys3/BlexerSimp.thy Thu Jun 23 16:10:04 2022 +0100
@@ -0,0 +1,702 @@
+theory BlexerSimp
+ imports Blexer
+begin
+
+fun distinctWith :: "'a list \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a list"
+ where
+ "distinctWith [] eq acc = []"
+| "distinctWith (x # xs) eq acc =
+ (if (\<exists> y \<in> acc. eq x y) then distinctWith xs eq acc
+ else x # (distinctWith xs eq ({x} \<union> acc)))"
+
+
+fun eq1 ("_ ~1 _" [80, 80] 80) where
+ "AZERO ~1 AZERO = True"
+| "(AONE bs1) ~1 (AONE bs2) = True"
+| "(ACHAR bs1 c) ~1 (ACHAR bs2 d) = (if c = d then True else False)"
+| "(ASEQ bs1 ra1 ra2) ~1 (ASEQ bs2 rb1 rb2) = (ra1 ~1 rb1 \<and> ra2 ~1 rb2)"
+| "(AALTs bs1 []) ~1 (AALTs bs2 []) = True"
+| "(AALTs bs1 (r1 # rs1)) ~1 (AALTs bs2 (r2 # rs2)) = (r1 ~1 r2 \<and> (AALTs bs1 rs1) ~1 (AALTs bs2 rs2))"
+| "(ASTAR bs1 r1) ~1 (ASTAR bs2 r2) = r1 ~1 r2"
+| "_ ~1 _ = False"
+
+
+
+lemma eq1_L:
+ assumes "x ~1 y"
+ shows "L (erase x) = L (erase y)"
+ using assms
+ apply(induct rule: eq1.induct)
+ apply(auto elim: eq1.elims)
+ apply presburger
+ done
+
+fun flts :: "arexp list \<Rightarrow> arexp list"
+ where
+ "flts [] = []"
+| "flts (AZERO # rs) = flts rs"
+| "flts ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
+| "flts (r1 # rs) = r1 # flts rs"
+
+
+
+fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
+ where
+ "bsimp_ASEQ _ AZERO _ = AZERO"
+| "bsimp_ASEQ _ _ AZERO = AZERO"
+| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+| "bsimp_ASEQ bs1 r1 r2 = ASEQ bs1 r1 r2"
+
+lemma bsimp_ASEQ0[simp]:
+ shows "bsimp_ASEQ bs r1 AZERO = AZERO"
+ by (case_tac r1)(simp_all)
+
+lemma bsimp_ASEQ1:
+ assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<nexists>bs. r1 = AONE bs"
+ shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
+ using assms
+ apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+ apply(auto)
+ done
+
+lemma bsimp_ASEQ2[simp]:
+ shows "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+ by (case_tac r2) (simp_all)
+
+
+fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+ where
+ "bsimp_AALTs _ [] = AZERO"
+| "bsimp_AALTs bs1 [r] = fuse bs1 r"
+| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
+
+
+
+
+fun bsimp :: "arexp \<Rightarrow> arexp"
+ where
+ "bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
+| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {}) "
+| "bsimp r = r"
+
+
+fun
+ bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+ "bders_simp r [] = r"
+| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
+
+definition blexer_simp where
+ "blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
+ decode (bmkeps (bders_simp (intern r) s)) r else None"
+
+
+
+lemma bders_simp_append:
+ shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
+ apply(induct s1 arbitrary: r s2)
+ apply(simp_all)
+ done
+
+lemma bmkeps_fuse:
+ assumes "bnullable r"
+ shows "bmkeps (fuse bs r) = bs @ bmkeps r"
+ using assms
+ by (induct r rule: bnullable.induct) (auto)
+
+lemma bmkepss_fuse:
+ assumes "bnullables rs"
+ shows "bmkepss (map (fuse bs) rs) = bs @ bmkepss rs"
+ using assms
+ apply(induct rs arbitrary: bs)
+ apply(auto simp add: bmkeps_fuse bnullable_fuse)
+ done
+
+lemma bder_fuse:
+ shows "bder c (fuse bs a) = fuse bs (bder c a)"
+ apply(induct a arbitrary: bs c)
+ apply(simp_all)
+ done
+
+
+
+
+inductive
+ rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)
+and
+ srewrite:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto> _" [100, 100] 100)
+where
+ bs1: "ASEQ bs AZERO r2 \<leadsto> AZERO"
+| bs2: "ASEQ bs r1 AZERO \<leadsto> AZERO"
+| bs3: "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"
+| bs4: "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
+| bs5: "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
+| bs6: "AALTs bs [] \<leadsto> AZERO"
+| bs7: "AALTs bs [r] \<leadsto> fuse bs r"
+| bs10: "rs1 s\<leadsto> rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto> AALTs bs rs2"
+| ss1: "[] s\<leadsto> []"
+| ss2: "rs1 s\<leadsto> rs2 \<Longrightarrow> (r # rs1) s\<leadsto> (r # rs2)"
+| ss3: "r1 \<leadsto> r2 \<Longrightarrow> (r1 # rs) s\<leadsto> (r2 # rs)"
+| ss4: "(AZERO # rs) s\<leadsto> rs"
+| ss5: "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"
+| ss6: "L (erase a2) \<subseteq> L (erase a1) \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"
+
+
+inductive
+ rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
+where
+ rs1[intro, simp]:"r \<leadsto>* r"
+| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
+
+inductive
+ srewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" ("_ s\<leadsto>* _" [100, 100] 100)
+where
+ sss1[intro, simp]:"rs s\<leadsto>* rs"
+| sss2[intro]: "\<lbrakk>rs1 s\<leadsto> rs2; rs2 s\<leadsto>* rs3\<rbrakk> \<Longrightarrow> rs1 s\<leadsto>* rs3"
+
+
+lemma r_in_rstar : "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
+ using rrewrites.intros(1) rrewrites.intros(2) by blast
+
+lemma rs_in_rstar:
+ shows "r1 s\<leadsto> r2 \<Longrightarrow> r1 s\<leadsto>* r2"
+ using rrewrites.intros(1) rrewrites.intros(2) by blast
+
+
+lemma rrewrites_trans[trans]:
+ assumes a1: "r1 \<leadsto>* r2" and a2: "r2 \<leadsto>* r3"
+ shows "r1 \<leadsto>* r3"
+ using a2 a1
+ apply(induct r2 r3 arbitrary: r1 rule: rrewrites.induct)
+ apply(auto)
+ done
+
+lemma srewrites_trans[trans]:
+ assumes a1: "r1 s\<leadsto>* r2" and a2: "r2 s\<leadsto>* r3"
+ shows "r1 s\<leadsto>* r3"
+ using a1 a2
+ apply(induct r1 r2 arbitrary: r3 rule: srewrites.induct)
+ apply(auto)
+ done
+
+
+
+lemma contextrewrites0:
+ "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
+ apply(induct rs1 rs2 rule: srewrites.inducts)
+ apply simp
+ using bs10 r_in_rstar rrewrites_trans by blast
+
+lemma contextrewrites1:
+ "r \<leadsto>* r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto>* AALTs bs (r' # rs)"
+ apply(induct r r' rule: rrewrites.induct)
+ apply simp
+ using bs10 ss3 by blast
+
+lemma srewrite1:
+ shows "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto> (rs @ rs2)"
+ apply(induct rs)
+ apply(auto)
+ using ss2 by auto
+
+lemma srewrites1:
+ shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto>* (rs @ rs2)"
+ apply(induct rs1 rs2 rule: srewrites.induct)
+ apply(auto)
+ using srewrite1 by blast
+
+lemma srewrite2:
+ shows "r1 \<leadsto> r2 \<Longrightarrow> True"
+ and "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
+ apply(induct rule: rrewrite_srewrite.inducts)
+ apply(auto)
+ apply (metis append_Cons append_Nil srewrites1)
+ apply(meson srewrites.simps ss3)
+ apply (meson srewrites.simps ss4)
+ apply (meson srewrites.simps ss5)
+ by (metis append_Cons append_Nil srewrites.simps ss6)
+
+
+lemma srewrites3:
+ shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
+ apply(induct rs1 rs2 arbitrary: rs rule: srewrites.induct)
+ apply(auto)
+ by (meson srewrite2(2) srewrites_trans)
+
+(*
+lemma srewrites4:
+ assumes "rs3 s\<leadsto>* rs4" "rs1 s\<leadsto>* rs2"
+ shows "(rs1 @ rs3) s\<leadsto>* (rs2 @ rs4)"
+ using assms
+ apply(induct rs3 rs4 arbitrary: rs1 rs2 rule: srewrites.induct)
+ apply (simp add: srewrites3)
+ using srewrite1 by blast
+*)
+
+lemma srewrites6:
+ assumes "r1 \<leadsto>* r2"
+ shows "[r1] s\<leadsto>* [r2]"
+ using assms
+ apply(induct r1 r2 rule: rrewrites.induct)
+ apply(auto)
+ by (meson srewrites.simps srewrites_trans ss3)
+
+lemma srewrites7:
+ assumes "rs3 s\<leadsto>* rs4" "r1 \<leadsto>* r2"
+ shows "(r1 # rs3) s\<leadsto>* (r2 # rs4)"
+ using assms
+ by (smt (verit, best) append_Cons append_Nil srewrites1 srewrites3 srewrites6 srewrites_trans)
+
+lemma ss6_stronger_aux:
+ shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctWith rs2 eq1 (set rs1))"
+ apply(induct rs2 arbitrary: rs1)
+ apply(auto)
+ prefer 2
+ apply(drule_tac x="rs1 @ [a]" in meta_spec)
+ apply(simp)
+ apply(drule_tac x="rs1" in meta_spec)
+ apply(subgoal_tac "(rs1 @ a # rs2) s\<leadsto>* (rs1 @ rs2)")
+ using srewrites_trans apply blast
+ apply(subgoal_tac "\<exists>rs1a rs1b. rs1 = rs1a @ [x] @ rs1b")
+ prefer 2
+ apply (simp add: split_list)
+ apply(erule exE)+
+ apply(simp)
+ using eq1_L rs_in_rstar ss6 by force
+
+lemma ss6_stronger:
+ shows "rs1 s\<leadsto>* distinctWith rs1 eq1 {}"
+ by (metis append_Nil list.set(1) ss6_stronger_aux)
+
+
+lemma rewrite_preserves_fuse:
+ shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"
+ and "rs2 s\<leadsto> rs3 \<Longrightarrow> map (fuse bs) rs2 s\<leadsto>* map (fuse bs) rs3"
+proof(induct rule: rrewrite_srewrite.inducts)
+ case (bs3 bs1 bs2 r)
+ then show ?case
+ by (metis fuse.simps(5) fuse_append rrewrite_srewrite.bs3)
+next
+ case (bs7 bs r)
+ then show ?case
+ by (metis fuse.simps(4) fuse_append rrewrite_srewrite.bs7)
+next
+ case (ss2 rs1 rs2 r)
+ then show ?case
+ using srewrites7 by force
+next
+ case (ss3 r1 r2 rs)
+ then show ?case by (simp add: r_in_rstar srewrites7)
+next
+ case (ss5 bs1 rs1 rsb)
+ then show ?case
+ apply(simp)
+ by (metis (mono_tags, lifting) comp_def fuse_append map_eq_conv rrewrite_srewrite.ss5 srewrites.simps)
+next
+ case (ss6 a1 a2 rsa rsb rsc)
+ then show ?case
+ apply(simp only: map_append)
+ by (smt (verit, best) erase_fuse list.simps(8) list.simps(9) rrewrite_srewrite.ss6 srewrites.simps)
+qed (auto intro: rrewrite_srewrite.intros)
+
+
+lemma rewrites_fuse:
+ assumes "r1 \<leadsto>* r2"
+ shows "fuse bs r1 \<leadsto>* fuse bs r2"
+using assms
+apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)
+apply(auto intro: rewrite_preserves_fuse rrewrites_trans)
+done
+
+
+lemma star_seq:
+ assumes "r1 \<leadsto>* r2"
+ shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"
+using assms
+apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)
+apply(auto intro: rrewrite_srewrite.intros)
+done
+
+lemma star_seq2:
+ assumes "r3 \<leadsto>* r4"
+ shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"
+ using assms
+apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)
+apply(auto intro: rrewrite_srewrite.intros)
+done
+
+lemma continuous_rewrite:
+ assumes "r1 \<leadsto>* AZERO"
+ shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+using assms bs1 star_seq by blast
+
+(*
+lemma continuous_rewrite2:
+ assumes "r1 \<leadsto>* AONE bs"
+ shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"
+ using assms by (meson bs3 rrewrites.simps star_seq)
+*)
+
+lemma bsimp_aalts_simpcases:
+ shows "AONE bs \<leadsto>* bsimp (AONE bs)"
+ and "AZERO \<leadsto>* bsimp AZERO"
+ and "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"
+ by (simp_all)
+
+lemma bsimp_AALTs_rewrites:
+ shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
+ by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)
+
+lemma trivialbsimp_srewrites:
+ "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x \<rbrakk> \<Longrightarrow> rs s\<leadsto>* (map f rs)"
+ apply(induction rs)
+ apply simp
+ apply(simp)
+ using srewrites7 by auto
+
+
+
+lemma fltsfrewrites: "rs s\<leadsto>* flts rs"
+ apply(induction rs rule: flts.induct)
+ apply(auto intro: rrewrite_srewrite.intros)
+ apply (meson srewrites.simps srewrites1 ss5)
+ using rs1 srewrites7 apply presburger
+ using srewrites7 apply force
+ apply (simp add: srewrites7)
+ by (simp add: srewrites7)
+
+lemma bnullable0:
+shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2"
+ and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 = bnullables rs2"
+ apply(induct rule: rrewrite_srewrite.inducts)
+ apply(auto simp add: bnullable_fuse)
+ apply (meson UnCI bnullable_fuse imageI)
+ using bnullable_correctness nullable_correctness by blast
+
+
+lemma rewritesnullable:
+ assumes "r1 \<leadsto>* r2"
+ shows "bnullable r1 = bnullable r2"
+using assms
+ apply(induction r1 r2 rule: rrewrites.induct)
+ apply simp
+ using bnullable0(1) by auto
+
+lemma rewrite_bmkeps_aux:
+ shows "r1 \<leadsto> r2 \<Longrightarrow> (bnullable r1 \<and> bnullable r2 \<Longrightarrow> bmkeps r1 = bmkeps r2)"
+ and "rs1 s\<leadsto> rs2 \<Longrightarrow> (bnullables rs1 \<and> bnullables rs2 \<Longrightarrow> bmkepss rs1 = bmkepss rs2)"
+proof (induct rule: rrewrite_srewrite.inducts)
+ case (bs3 bs1 bs2 r)
+ then show ?case by (simp add: bmkeps_fuse)
+next
+ case (bs7 bs r)
+ then show ?case by (simp add: bmkeps_fuse)
+next
+ case (ss3 r1 r2 rs)
+ then show ?case
+ using bmkepss.simps bnullable0(1) by presburger
+next
+ case (ss5 bs1 rs1 rsb)
+ then show ?case
+ by (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
+next
+ case (ss6 a1 a2 rsa rsb rsc)
+ then show ?case
+ by (smt (verit, best) Nil_is_append_conv bmkepss1 bmkepss2 bnullable_correctness in_set_conv_decomp list.distinct(1) list.set_intros(1) nullable_correctness set_ConsD subsetD)
+qed (auto)
+
+lemma rewrites_bmkeps:
+ assumes "r1 \<leadsto>* r2" "bnullable r1"
+ shows "bmkeps r1 = bmkeps r2"
+ using assms
+proof(induction r1 r2 rule: rrewrites.induct)
+ case (rs1 r)
+ then show "bmkeps r = bmkeps r" by simp
+next
+ case (rs2 r1 r2 r3)
+ then have IH: "bmkeps r1 = bmkeps r2" by simp
+ have a1: "bnullable r1" by fact
+ have a2: "r1 \<leadsto>* r2" by fact
+ have a3: "r2 \<leadsto> r3" by fact
+ have a4: "bnullable r2" using a1 a2 by (simp add: rewritesnullable)
+ then have "bmkeps r2 = bmkeps r3"
+ using a3 bnullable0(1) rewrite_bmkeps_aux(1) by blast
+ then show "bmkeps r1 = bmkeps r3" using IH by simp
+qed
+
+
+lemma rewrites_to_bsimp:
+ shows "r \<leadsto>* bsimp r"
+proof (induction r rule: bsimp.induct)
+ case (1 bs1 r1 r2)
+ have IH1: "r1 \<leadsto>* bsimp r1" by fact
+ have IH2: "r2 \<leadsto>* bsimp r2" by fact
+ { assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"
+ with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
+ then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+ by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto
+ }
+ moreover
+ { assume "\<exists>bs. bsimp r1 = AONE bs"
+ then obtain bs where as: "bsimp r1 = AONE bs" by blast
+ with IH1 have "r1 \<leadsto>* AONE bs" by simp
+ then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
+ with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"
+ using rewrites_fuse by (meson rrewrites_trans)
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as)
+ }
+ moreover
+ { assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)"
+ then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)"
+ by (simp add: bsimp_ASEQ1)
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2
+ by (metis rrewrites_trans star_seq star_seq2)
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp
+ }
+ ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast
+next
+ case (2 bs1 rs)
+ have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact
+ then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)
+ also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites)
+ also have "... s\<leadsto>* distinctWith (flts (map bsimp rs)) eq1 {}" by (simp add: ss6_stronger)
+ finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {})"
+ using contextrewrites0 by auto
+ also have "... \<leadsto>* bsimp_AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {})"
+ by (simp add: bsimp_AALTs_rewrites)
+ finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp
+qed (simp_all)
+
+
+lemma to_zero_in_alt:
+ shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
+ by (simp add: bs1 bs10 ss3)
+
+
+
+lemma bder_fuse_list:
+ shows "map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
+ apply(induction rs1)
+ apply(simp_all add: bder_fuse)
+ done
+
+lemma map1:
+ shows "(map f [a]) = [f a]"
+ by (simp)
+
+lemma rewrite_preserves_bder:
+ shows "r1 \<leadsto> r2 \<Longrightarrow> (bder c r1) \<leadsto>* (bder c r2)"
+ and "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"
+proof(induction rule: rrewrite_srewrite.inducts)
+ case (bs1 bs r2)
+ then show ?case
+ by (simp add: continuous_rewrite)
+next
+ case (bs2 bs r1)
+ then show ?case
+ apply(auto)
+ apply (meson bs6 contextrewrites0 rrewrite_srewrite.bs2 rs2 ss3 ss4 sss1 sss2)
+ by (simp add: r_in_rstar rrewrite_srewrite.bs2)
+next
+ case (bs3 bs1 bs2 r)
+ then show ?case
+ apply(simp)
+
+ by (metis (no_types, lifting) bder_fuse bs10 bs7 fuse_append rrewrites.simps ss4 to_zero_in_alt)
+next
+ case (bs4 r1 r2 bs r3)
+ have as: "r1 \<leadsto> r2" by fact
+ have IH: "bder c r1 \<leadsto>* bder c r2" by fact
+ from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r2 r3)"
+ by (metis bder.simps(5) bnullable0(1) contextrewrites1 rewrite_bmkeps_aux(1) star_seq)
+next
+ case (bs5 r3 r4 bs r1)
+ have as: "r3 \<leadsto> r4" by fact
+ have IH: "bder c r3 \<leadsto>* bder c r4" by fact
+ from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r1 r4)"
+ apply(simp)
+ apply(auto)
+ using contextrewrites0 r_in_rstar rewrites_fuse srewrites6 srewrites7 star_seq2 apply presburger
+ using star_seq2 by blast
+next
+ case (bs6 bs)
+ then show ?case
+ using rrewrite_srewrite.bs6 by force
+next
+ case (bs7 bs r)
+ then show ?case
+ by (simp add: bder_fuse r_in_rstar rrewrite_srewrite.bs7)
+next
+ case (bs10 rs1 rs2 bs)
+ then show ?case
+ using contextrewrites0 by force
+next
+ case ss1
+ then show ?case by simp
+next
+ case (ss2 rs1 rs2 r)
+ then show ?case
+ by (simp add: srewrites7)
+next
+ case (ss3 r1 r2 rs)
+ then show ?case
+ by (simp add: srewrites7)
+next
+ case (ss4 rs)
+ then show ?case
+ using rrewrite_srewrite.ss4 by fastforce
+next
+ case (ss5 bs1 rs1 rsb)
+ then show ?case
+ apply(simp)
+ using bder_fuse_list map_map rrewrite_srewrite.ss5 srewrites.simps by blast
+next
+ case (ss6 a1 a2 bs rsa rsb)
+ then show ?case
+ apply(simp only: map_append map1)
+ apply(rule srewrites_trans)
+ apply(rule rs_in_rstar)
+ apply(rule_tac rrewrite_srewrite.ss6)
+ using Der_def der_correctness apply auto[1]
+ by blast
+qed
+
+lemma rewrites_preserves_bder:
+ assumes "r1 \<leadsto>* r2"
+ shows "bder c r1 \<leadsto>* bder c r2"
+using assms
+apply(induction r1 r2 rule: rrewrites.induct)
+apply(simp_all add: rewrite_preserves_bder rrewrites_trans)
+done
+
+
+lemma central:
+ shows "bders r s \<leadsto>* bders_simp r s"
+proof(induct s arbitrary: r rule: rev_induct)
+ case Nil
+ then show "bders r [] \<leadsto>* bders_simp r []" by simp
+next
+ case (snoc x xs)
+ have IH: "\<And>r. bders r xs \<leadsto>* bders_simp r xs" by fact
+ have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
+ also have "... \<leadsto>* bders (bders_simp r xs) [x]" using IH
+ by (simp add: rewrites_preserves_bder)
+ also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
+ by (simp add: rewrites_to_bsimp)
+ finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])"
+ by (simp add: bders_simp_append)
+qed
+
+lemma main_aux:
+ assumes "bnullable (bders r s)"
+ shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
+proof -
+ have "bders r s \<leadsto>* bders_simp r s" by (rule central)
+ then
+ show "bmkeps (bders r s) = bmkeps (bders_simp r s)" using assms
+ by (rule rewrites_bmkeps)
+qed
+
+
+
+
+theorem main_blexer_simp:
+ shows "blexer r s = blexer_simp r s"
+ unfolding blexer_def blexer_simp_def
+ by (metis central main_aux rewritesnullable)
+
+theorem blexersimp_correctness:
+ shows "lexer r s = blexer_simp r s"
+ using blexer_correctness main_blexer_simp by simp
+
+
+
+
+fun bsimpStrong6 :: "arexp \<Rightarrow> arexp"
+ where
+ "bsimpStrong6 (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimpStrong6 r1) (bsimpStrong6 r2)"
+| "bsimpStrong6 (AALTs bs1 rs) = bsimp_AALTs bs1 (distinctWith (flts (map bsimpStrong6 rs)) eq1 {}) "
+| "bsimpStrong6 r = r"
+
+
+fun
+ bdersStrong6 :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+ "bdersStrong6 r [] = r"
+| "bdersStrong6 r (c # s) = bdersStrong6 (bsimpStrong6 (bder c r)) s"
+
+definition blexerStrong where
+ "blexerStrong r s \<equiv> if bnullable (bdersStrong6 (intern r) s) then
+ decode (bmkeps (bdersStrong6 (intern r) s)) r else None"
+
+fun ABIncludedByC :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> bool" where
+ "ABIncludedByC a b c f subseteqPred = subseteqPred (f a b) c"
+
+fun furtherSEQ :: "rexp \<Rightarrow> rexp \<Rightarrow> rexp list" and
+ turnIntoTerms :: "rexp \<Rightarrow> rexp list "
+ where
+ "furtherSEQ ONE r2 = turnIntoTerms r2 "
+| "furtherSEQ r11 r2 = [SEQ r11 r2]"
+| "turnIntoTerms (SEQ ONE r2) = turnIntoTerms r2"
+| "turnIntoTerms (SEQ r1 r2) = concat (map (\<lambda>r11. furtherSEQ r11 r2) (turnIntoTerms r1))"
+| "turnIntoTerms r = [r]"
+
+fun regConcat :: "rexp \<Rightarrow> rexp list \<Rightarrow> rexp" where
+ "regConcat acc [] = acc"
+| "regConcat ONE (r # rs1) = regConcat r rs1"
+| "regConcat acc (r # rs1) = regConcat (SEQ acc r) rs1"
+
+fun attachCtx :: "arexp \<Rightarrow> rexp list \<Rightarrow> rexp set" where
+ "attachCtx r ctx = set (turnIntoTerms (regConcat (erase r) ctx))"
+
+(*
+def regConcat(acc: Rexp, rs: List[Rexp]) : Rexp = rs match {
+ case Nil => acc
+ case r :: rs1 =>
+ // if(acc == ONE)
+ // regConcat(r, rs1)
+ // else
+ regConcat(SEQ(acc, r), rs1)
+}
+
+def attachCtx(r: ARexp, ctx: List[Rexp]) : Set[Rexp] = {
+ turnIntoTerms((regConcat(erase(r), ctx)))
+ .toSet
+}
+
+def attachCtxcc(r: Rexp, ctx: List[Rexp]) : Set[Rexp] =
+ turnIntoTerms(regConcat(r, ctx)).toSet
+
+def ABIncludedByC[A, B, C](a: A, b: B, c: C, f: (A, B) => C,
+subseteqPred: (C, C) => Boolean) : Boolean = {
+ subseteqPred(f(a, b), c)
+}
+def rs1_subseteq_rs2(rs1: Set[Rexp], rs2: Set[Rexp]) : Boolean =
+ //rs1 \subseteq rs2
+ rs1.forall(rs2.contains)
+
+
+}
+*)
+
+fun rs1_subseteq_rs2 :: "rexp set \<Rightarrow> rexp set \<Rightarrow> bool" where
+ "rs1_subseteq_rs2 rs1 rs2 = (rs1 \<subseteq> rs2)"
+
+fun prune6 :: "rexp set \<Rightarrow> arexp \<Rightarrow> rexp list \<Rightarrow> arexp" where
+ "prune6 acc a ctx = (if (ABIncludedByC a ctx acc attachCtx rs1_subseteq_rs2) then AZERO else
+ (case a of (ASEQ bs r1 r2) \<Rightarrow> bsimp_ASEQ bs (prune6 acc r1 (erase r2 # ctx)) r2
+ | AALTs bs rs0 \<Rightarrow> bsimp_AALTs bs (map (\<lambda>r. filter notZero (prune6 acc r ctx)) rs0)) )"
+
+
+fun dB6 :: "arexp list \<Rightarrow> rexp set \<Rightarrow> arexp list" where
+ "dB6 [] acc = []"
+| "dB6 (a # as) acc = (if (erase a \<in> acc) then (dB6 as acc)
+ else (let pruned = prune6 acc x in
+ (case pruned of
+ AZERO \<Rightarrow> dB6 as acc
+ |xPrime \<Rightarrow> xPrime # (dB6 xs ((turnIntoTerms (erase pruned)) \<union> acc) ) ) )) "
+
+
+end