--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys4/posix/BasicIdentities.thy Mon Aug 29 23:16:28 2022 +0100
@@ -0,0 +1,1203 @@
+theory BasicIdentities
+ imports "Lexer"
+begin
+
+datatype rrexp =
+ RZERO
+| RONE
+| RCHAR char
+| RSEQ rrexp rrexp
+| RALTS "rrexp list"
+| RSTAR rrexp
+| RNTIMES rrexp nat
+
+abbreviation
+ "RALT r1 r2 \<equiv> RALTS [r1, r2]"
+
+
+fun
+ rnullable :: "rrexp \<Rightarrow> bool"
+where
+ "rnullable (RZERO) = False"
+| "rnullable (RONE) = True"
+| "rnullable (RCHAR c) = False"
+| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"
+| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"
+| "rnullable (RSTAR r) = True"
+| "rnullable (RNTIMES r n) = (if n = 0 then True else rnullable r)"
+
+fun
+ rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
+where
+ "rder c (RZERO) = RZERO"
+| "rder c (RONE) = RZERO"
+| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
+| "rder c (RALTS rs) = RALTS (map (rder c) rs)"
+| "rder c (RSEQ r1 r2) =
+ (if rnullable r1
+ then RALT (RSEQ (rder c r1) r2) (rder c r2)
+ else RSEQ (rder c r1) r2)"
+| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
+| "rder c (RNTIMES r n) = (if n = 0 then RZERO else RSEQ (rder c r) (RNTIMES r (n - 1)))"
+
+fun
+ rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
+where
+ "rders r [] = r"
+| "rders r (c#s) = rders (rder c r) s"
+
+fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
+ where
+ "rdistinct [] acc = []"
+| "rdistinct (x#xs) acc =
+ (if x \<in> acc then rdistinct xs acc
+ else x # (rdistinct xs ({x} \<union> acc)))"
+
+lemma rdistinct1:
+ assumes "a \<in> acc"
+ shows "a \<notin> set (rdistinct rs acc)"
+ using assms
+ apply(induct rs arbitrary: acc a)
+ apply(auto)
+ done
+
+
+lemma rdistinct_does_the_job:
+ shows "distinct (rdistinct rs s)"
+ apply(induct rs s rule: rdistinct.induct)
+ apply(auto simp add: rdistinct1)
+ done
+
+
+
+lemma rdistinct_concat:
+ assumes "set rs \<subseteq> rset"
+ shows "rdistinct (rs @ rsa) rset = rdistinct rsa rset"
+ using assms
+ apply(induct rs)
+ apply simp+
+ done
+
+lemma distinct_not_exist:
+ assumes "a \<notin> set rs"
+ shows "rdistinct rs rset = rdistinct rs (insert a rset)"
+ using assms
+ apply(induct rs arbitrary: rset)
+ apply(auto)
+ done
+
+lemma rdistinct_on_distinct:
+ shows "distinct rs \<Longrightarrow> rdistinct rs {} = rs"
+ apply(induct rs)
+ apply simp
+ using distinct_not_exist by fastforce
+
+lemma distinct_rdistinct_append:
+ assumes "distinct rs1" "\<forall>r \<in> set rs1. r \<notin> acc"
+ shows "rdistinct (rs1 @ rsa) acc = rs1 @ (rdistinct rsa (acc \<union> set rs1))"
+ using assms
+ apply(induct rs1 arbitrary: rsa acc)
+ apply(auto)[1]
+ apply(auto)[1]
+ apply(drule_tac x="rsa" in meta_spec)
+ apply(drule_tac x="{a} \<union> acc" in meta_spec)
+ apply(simp)
+ apply(drule meta_mp)
+ apply(auto)[1]
+ apply(simp)
+ done
+
+
+lemma rdistinct_set_equality1:
+ shows "set (rdistinct rs acc) = set rs - acc"
+ apply(induct rs acc rule: rdistinct.induct)
+ apply(auto)
+ done
+
+
+lemma rdistinct_set_equality:
+ shows "set (rdistinct rs {}) = set rs"
+ by (simp add: rdistinct_set_equality1)
+
+
+fun rflts :: "rrexp list \<Rightarrow> rrexp list"
+ where
+ "rflts [] = []"
+| "rflts (RZERO # rs) = rflts rs"
+| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
+| "rflts (r1 # rs) = r1 # rflts rs"
+
+
+lemma rflts_def_idiot:
+ shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow> rflts (a # rs) = a # rflts rs"
+ apply(case_tac a)
+ apply simp_all
+ done
+
+lemma rflts_def_idiot2:
+ shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow> a \<in> set (rflts rs)"
+ apply(induct rs rule: rflts.induct)
+ apply(auto)
+ done
+
+lemma flts_append:
+ shows "rflts (rs1 @ rs2) = rflts rs1 @ rflts rs2"
+ apply(induct rs1)
+ apply simp
+ apply(case_tac a)
+ apply simp+
+ done
+
+
+fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
+ where
+ "rsimp_ALTs [] = RZERO"
+| "rsimp_ALTs [r] = r"
+| "rsimp_ALTs rs = RALTS rs"
+
+lemma rsimpalts_conscons:
+ shows "rsimp_ALTs (r1 # rsa @ r2 # rsb) = RALTS (r1 # rsa @ r2 # rsb)"
+ by (metis Nil_is_append_conv list.exhaust rsimp_ALTs.simps(3))
+
+lemma rsimp_alts_equal:
+ shows "rsimp_ALTs (rsa @ a # rsb @ a # rsc) = RALTS (rsa @ a # rsb @ a # rsc) "
+ by (metis append_Cons append_Nil neq_Nil_conv rsimpalts_conscons)
+
+
+fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
+ where
+ "rsimp_SEQ RZERO _ = RZERO"
+| "rsimp_SEQ _ RZERO = RZERO"
+| "rsimp_SEQ RONE r2 = r2"
+| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
+
+
+fun rsimp :: "rrexp \<Rightarrow> rrexp"
+ where
+ "rsimp (RSEQ r1 r2) = rsimp_SEQ (rsimp r1) (rsimp r2)"
+| "rsimp (RALTS rs) = rsimp_ALTs (rdistinct (rflts (map rsimp rs)) {}) "
+| "rsimp r = r"
+
+
+fun
+ rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
+where
+ "rders_simp r [] = r"
+| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
+
+fun rsize :: "rrexp \<Rightarrow> nat" where
+ "rsize RZERO = 1"
+| "rsize (RONE) = 1"
+| "rsize (RCHAR c) = 1"
+| "rsize (RALTS rs) = Suc (sum_list (map rsize rs))"
+| "rsize (RSEQ r1 r2) = Suc (rsize r1 + rsize r2)"
+| "rsize (RSTAR r) = Suc (rsize r)"
+| "rsize (RNTIMES r n) = Suc (rsize r) + n"
+
+abbreviation rsizes where
+ "rsizes rs \<equiv> sum_list (map rsize rs)"
+
+
+lemma rder_rsimp_ALTs_commute:
+ shows " (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"
+ apply(induct rs)
+ apply simp
+ apply(case_tac rs)
+ apply simp
+ apply auto
+ done
+
+
+lemma rsimp_aalts_smaller:
+ shows "rsize (rsimp_ALTs rs) \<le> rsize (RALTS rs)"
+ apply(induct rs)
+ apply simp
+ apply simp
+ apply(case_tac "rs = []")
+ apply simp
+ apply(subgoal_tac "\<exists>rsp ap. rs = ap # rsp")
+ apply(erule exE)+
+ apply simp
+ apply simp
+ by(meson neq_Nil_conv)
+
+
+
+
+
+lemma rSEQ_mono:
+ shows "rsize (rsimp_SEQ r1 r2) \<le>rsize (RSEQ r1 r2)"
+ apply auto
+ apply(induct r1)
+ apply auto
+ apply(case_tac "r2")
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+ done
+
+lemma ralts_cap_mono:
+ shows "rsize (RALTS rs) \<le> Suc (rsizes rs)"
+ by simp
+
+
+
+
+lemma rflts_mono:
+ shows "rsizes (rflts rs) \<le> rsizes rs"
+ apply(induct rs)
+ apply simp
+ apply(case_tac "a = RZERO")
+ apply simp
+ apply(case_tac "\<exists>rs1. a = RALTS rs1")
+ apply(erule exE)
+ apply simp
+ apply(subgoal_tac "rflts (a # rs) = a # (rflts rs)")
+ prefer 2
+
+ using rflts_def_idiot apply blast
+ apply simp
+ done
+
+lemma rdistinct_smaller:
+ shows "rsizes (rdistinct rs ss) \<le> rsizes rs"
+ apply (induct rs arbitrary: ss)
+ apply simp
+ by (simp add: trans_le_add2)
+
+
+lemma rsimp_alts_mono :
+ shows "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> rsize (rsimp xa) \<le> rsize xa) \<Longrightarrow>
+ rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {})) \<le> Suc (rsizes x)"
+ apply(subgoal_tac "rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {} ))
+ \<le> rsize (RALTS (rdistinct (rflts (map rsimp x)) {} ))")
+ prefer 2
+ using rsimp_aalts_smaller apply auto[1]
+ apply(subgoal_tac "rsize (RALTS (rdistinct (rflts (map rsimp x)) {})) \<le>Suc (rsizes (rdistinct (rflts (map rsimp x)) {}))")
+ prefer 2
+ using ralts_cap_mono apply blast
+ apply(subgoal_tac "rsizes (rdistinct (rflts (map rsimp x)) {}) \<le> rsizes (rflts (map rsimp x))")
+ prefer 2
+ using rdistinct_smaller apply presburger
+ apply(subgoal_tac "rsizes (rflts (map rsimp x)) \<le> rsizes (map rsimp x)")
+ prefer 2
+ using rflts_mono apply blast
+ apply(subgoal_tac "rsizes (map rsimp x) \<le> rsizes x")
+ prefer 2
+
+ apply (simp add: sum_list_mono)
+ by linarith
+
+
+
+
+
+lemma rsimp_mono:
+ shows "rsize (rsimp r) \<le> rsize r"
+ apply(induct r)
+ apply simp_all
+ apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")
+ apply force
+ using rSEQ_mono
+ apply presburger
+ using rsimp_alts_mono by auto
+
+lemma idiot:
+ shows "rsimp_SEQ RONE r = r"
+ apply(case_tac r)
+ apply simp_all
+ done
+
+
+
+
+
+lemma idiot2:
+ shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
+ \<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
+ apply(case_tac r1)
+ apply(case_tac r2)
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
+apply(case_tac r2)
+ apply simp_all
+ done
+
+lemma rders__onechar:
+ shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
+ by simp
+
+lemma rders_append:
+ "rders c (s1 @ s2) = rders (rders c s1) s2"
+ apply(induct s1 arbitrary: c s2)
+ apply(simp_all)
+ done
+
+lemma rders_simp_append:
+ "rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
+ apply(induct s1 arbitrary: c s2)
+ apply(simp_all)
+ done
+
+
+lemma rders_simp_one_char:
+ shows "rders_simp r [c] = rsimp (rder c r)"
+ apply auto
+ done
+
+
+
+fun nonalt :: "rrexp \<Rightarrow> bool"
+ where
+ "nonalt (RALTS rs) = False"
+| "nonalt r = True"
+
+
+fun good :: "rrexp \<Rightarrow> bool" where
+ "good RZERO = False"
+| "good (RONE) = True"
+| "good (RCHAR c) = True"
+| "good (RALTS []) = False"
+| "good (RALTS [r]) = False"
+| "good (RALTS (r1 # r2 # rs)) = ((distinct ( (r1 # r2 # rs))) \<and>(\<forall>r' \<in> set (r1 # r2 # rs). good r' \<and> nonalt r'))"
+| "good (RSEQ RZERO _) = False"
+| "good (RSEQ RONE _) = False"
+| "good (RSEQ _ RZERO) = False"
+| "good (RSEQ r1 r2) = (good r1 \<and> good r2)"
+| "good (RSTAR r) = True"
+| "good (RNTIMES r n) = True"
+
+lemma k0a:
+ shows "rflts [RALTS rs] = rs"
+ apply(simp)
+ done
+
+lemma bbbbs:
+ assumes "good r" "r = RALTS rs"
+ shows "rsimp_ALTs (rflts [r]) = RALTS rs"
+ using assms
+ by (metis good.simps(4) good.simps(5) k0a rsimp_ALTs.elims)
+
+lemma bbbbs1:
+ shows "nonalt r \<or> (\<exists> rs. r = RALTS rs)"
+ by (meson nonalt.elims(3))
+
+
+
+lemma good0:
+ assumes "rs \<noteq> Nil" "\<forall>r \<in> set rs. nonalt r" "distinct rs"
+ shows "good (rsimp_ALTs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. good r)"
+ using assms
+ apply(induct rs rule: rsimp_ALTs.induct)
+ apply(auto)
+ done
+
+lemma flts1:
+ assumes "good r"
+ shows "rflts [r] \<noteq> []"
+ using assms
+ apply(induct r)
+ apply(simp_all)
+ using good.simps(4) by blast
+
+lemma flts2:
+ assumes "good r"
+ shows "\<forall>r' \<in> set (rflts [r]). good r' \<and> nonalt r'"
+ using assms
+ apply(induct r)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 2
+ apply(simp)
+ apply(auto)[1]
+
+ apply (metis flts1 good.simps(5) good.simps(6) k0a neq_Nil_conv)
+ apply (metis flts1 good.simps(5) good.simps(6) k0a neq_Nil_conv)
+ apply fastforce
+ apply(simp)
+ by simp
+
+
+lemma flts3:
+ assumes "\<forall>r \<in> set rs. good r \<or> r = RZERO"
+ shows "\<forall>r \<in> set (rflts rs). good r"
+ using assms
+ apply(induct rs arbitrary: rule: rflts.induct)
+ apply(simp_all)
+ by (metis UnE flts2 k0a)
+
+
+lemma k0:
+ shows "rflts (r # rs1) = rflts [r] @ rflts rs1"
+ apply(induct r arbitrary: rs1)
+ apply(auto)
+ done
+
+
+lemma good_SEQ:
+ assumes "r1 \<noteq> RZERO" "r2 \<noteq> RZERO" " r1 \<noteq> RONE"
+ shows "good (RSEQ r1 r2) = (good r1 \<and> good r2)"
+ using assms
+ apply(case_tac r1)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+apply(case_tac r2)
+ apply(simp_all)
+ done
+
+lemma rsize0:
+ shows "0 < rsize r"
+ apply(induct r)
+ apply(auto)
+ done
+
+
+fun nonnested :: "rrexp \<Rightarrow> bool"
+ where
+ "nonnested (RALTS []) = True"
+| "nonnested (RALTS ((RALTS rs1) # rs2)) = False"
+| "nonnested (RALTS (r # rs2)) = nonnested (RALTS rs2)"
+| "nonnested r = True"
+
+
+
+lemma k00:
+ shows "rflts (rs1 @ rs2) = rflts rs1 @ rflts rs2"
+ apply(induct rs1 arbitrary: rs2)
+ apply(auto)
+ by (metis append.assoc k0)
+
+
+
+
+lemma k0b:
+ assumes "nonalt r" "r \<noteq> RZERO"
+ shows "rflts [r] = [r]"
+ using assms
+ apply(case_tac r)
+ apply(simp_all)
+ done
+
+lemma nn1qq:
+ assumes "nonnested (RALTS rs)"
+ shows "\<nexists> rs1. RALTS rs1 \<in> set rs"
+ using assms
+ apply(induct rs rule: rflts.induct)
+ apply(auto)
+ done
+
+
+
+lemma n0:
+ shows "nonnested (RALTS rs) \<longleftrightarrow> (\<forall>r \<in> set rs. nonalt r)"
+ apply(induct rs )
+ apply(auto)
+ apply (metis list.set_intros(1) nn1qq nonalt.elims(3))
+ apply (metis nonalt.elims(2) nonnested.simps(3) nonnested.simps(4) nonnested.simps(5) nonnested.simps(6) nonnested.simps(7) nonnested.simps(8))
+ using bbbbs1 apply fastforce
+ by (metis bbbbs1 list.set_intros(2) nn1qq)
+
+
+
+
+lemma nn1c:
+ assumes "\<forall>r \<in> set rs. nonnested r"
+ shows "\<forall>r \<in> set (rflts rs). nonalt r"
+ using assms
+ apply(induct rs rule: rflts.induct)
+ apply(auto)
+ using n0 by blast
+
+lemma nn1bb:
+ assumes "\<forall>r \<in> set rs. nonalt r"
+ shows "nonnested (rsimp_ALTs rs)"
+ using assms
+ apply(induct rs rule: rsimp_ALTs.induct)
+ apply(auto)
+ using nonalt.simps(1) nonnested.elims(3) apply blast
+ using n0 by auto
+
+lemma bsimp_ASEQ0:
+ shows "rsimp_SEQ r1 RZERO = RZERO"
+ apply(induct r1)
+ apply(auto)
+ done
+
+lemma nn1b:
+ shows "nonnested (rsimp r)"
+ apply(induct r)
+ apply(simp_all)
+ apply(case_tac "rsimp r1 = RZERO")
+ apply(simp)
+ apply(case_tac "rsimp r2 = RZERO")
+ apply(simp)
+ apply(subst bsimp_ASEQ0)
+ apply(simp)
+ apply(case_tac "\<exists>bs. rsimp r1 = RONE")
+ apply(auto)[1]
+ using idiot apply fastforce
+ apply (simp add: idiot2)
+ by (metis (mono_tags, lifting) image_iff list.set_map nn1bb nn1c rdistinct_set_equality)
+
+lemma nonalt_flts_rd:
+ shows "\<lbrakk>xa \<in> set (rdistinct (rflts (map rsimp rs)) {})\<rbrakk>
+ \<Longrightarrow> nonalt xa"
+ by (metis Diff_empty ex_map_conv nn1b nn1c rdistinct_set_equality1)
+
+
+lemma rsimpalts_implies1:
+ shows " rsimp_ALTs (a # rdistinct rs {a}) = RZERO \<Longrightarrow> a = RZERO"
+ using rsimp_ALTs.elims by auto
+
+
+lemma rsimpalts_implies2:
+ shows "rsimp_ALTs (a # rdistinct rs rset) = RZERO \<Longrightarrow> rdistinct rs rset = []"
+ by (metis append_butlast_last_id rrexp.distinct(7) rsimpalts_conscons)
+
+lemma rsimpalts_implies21:
+ shows "rsimp_ALTs (a # rdistinct rs {a}) = RZERO \<Longrightarrow> rdistinct rs {a} = []"
+ using rsimpalts_implies2 by blast
+
+
+lemma bsimp_ASEQ2:
+ shows "rsimp_SEQ RONE r2 = r2"
+ apply(induct r2)
+ apply(auto)
+ done
+
+lemma elem_smaller_than_set:
+ shows "xa \<in> set list \<Longrightarrow> rsize xa < Suc (rsizes list)"
+ apply(induct list)
+ apply simp
+ by (metis image_eqI le_imp_less_Suc list.set_map member_le_sum_list)
+
+lemma rsimp_list_mono:
+ shows "rsizes (map rsimp rs) \<le> rsizes rs"
+ apply(induct rs)
+ apply simp+
+ by (simp add: add_mono_thms_linordered_semiring(1) rsimp_mono)
+
+
+(*says anything coming out of simp+flts+db will be good*)
+lemma good2_obv_simplified:
+ shows " \<lbrakk>\<forall>y. rsize y < Suc (rsizes rs) \<longrightarrow> good (rsimp y) \<or> rsimp y = RZERO;
+ xa \<in> set (rdistinct (rflts (map rsimp rs)) {}); good (rsimp xa) \<or> rsimp xa = RZERO\<rbrakk> \<Longrightarrow> good xa"
+ apply(subgoal_tac " \<forall>xa' \<in> set (map rsimp rs). good xa' \<or> xa' = RZERO")
+ prefer 2
+ apply (simp add: elem_smaller_than_set)
+ by (metis Diff_empty flts3 rdistinct_set_equality1)
+
+thm Diff_empty flts3 rdistinct_set_equality1
+
+lemma good1:
+ shows "good (rsimp a) \<or> rsimp a = RZERO"
+ apply(induct a taking: rsize rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ prefer 2
+ apply(simp only:)
+ apply simp
+ apply (smt (verit, ccfv_threshold) add_mono_thms_linordered_semiring(1) elem_smaller_than_set good0 good2_obv_simplified le_eq_less_or_eq nonalt_flts_rd order_less_trans plus_1_eq_Suc rdistinct_does_the_job rdistinct_smaller rflts_mono rsimp_ALTs.simps(1) rsimp_list_mono)
+ apply simp
+ apply(subgoal_tac "good (rsimp x41) \<or> rsimp x41 = RZERO")
+ apply(subgoal_tac "good (rsimp x42) \<or> rsimp x42 = RZERO")
+ apply(case_tac "rsimp x41 = RZERO")
+ apply simp
+ apply(case_tac "rsimp x42 = RZERO")
+ apply simp
+ using bsimp_ASEQ0 apply blast
+ apply(subgoal_tac "good (rsimp x41)")
+ apply(subgoal_tac "good (rsimp x42)")
+ apply simp
+ apply (metis bsimp_ASEQ2 good_SEQ idiot2)
+ apply blast
+ apply fastforce
+ using less_add_Suc2 apply blast
+ using less_iff_Suc_add apply blast
+ using good.simps(45) rsimp.simps(7) by presburger
+
+
+
+fun
+ RL :: "rrexp \<Rightarrow> string set"
+where
+ "RL (RZERO) = {}"
+| "RL (RONE) = {[]}"
+| "RL (RCHAR c) = {[c]}"
+| "RL (RSEQ r1 r2) = (RL r1) ;; (RL r2)"
+| "RL (RALTS rs) = (\<Union> (set (map RL rs)))"
+| "RL (RSTAR r) = (RL r)\<star>"
+| "RL (RNTIMES r n) = (RL r) ^^ n"
+
+lemma pow_rempty_iff:
+ shows "[] \<in> (RL r) ^^ n \<longleftrightarrow> (if n = 0 then True else [] \<in> (RL r))"
+ by (induct n) (auto simp add: Sequ_def)
+
+lemma RL_rnullable:
+ shows "rnullable r = ([] \<in> RL r)"
+ apply(induct r)
+ apply(auto simp add: Sequ_def pow_rempty_iff)
+ done
+
+lemma concI_if_Nil1: "[] \<in> A \<Longrightarrow> xs : B \<Longrightarrow> xs \<in> A ;; B"
+by (metis append_Nil concI)
+
+
+lemma empty_pow_add:
+ fixes A::"string set"
+ assumes "[] \<in> A" "s \<in> A ^^ n"
+ shows "s \<in> A ^^ (n + m)"
+ using assms
+ apply(induct m arbitrary: n)
+ apply(auto simp add: Sequ_def)
+ done
+
+(*
+lemma der_pow:
+ shows "Der c (A ^^ n) = (if n = 0 then {} else (Der c A) ;; (A ^^ (n - 1)))"
+ apply(induct n arbitrary: A)
+ apply(auto)
+ by (smt (verit, best) Suc_pred concE concI concI_if_Nil2 conc_pow_comm lang_pow.simps(2))
+*)
+
+lemma RL_rder:
+ shows "RL (rder c r) = Der c (RL r)"
+ apply(induct r)
+ apply(auto simp add: Sequ_def Der_def)[5]
+ apply (metis append_Cons)
+ using RL_rnullable apply blast
+ apply (metis append_eq_Cons_conv)
+ apply (metis append_Cons)
+ apply (metis RL_rnullable append_eq_Cons_conv)
+ apply simp
+ apply(simp)
+ done
+
+lemma RL_rsimp_RSEQ:
+ shows "RL (rsimp_SEQ r1 r2) = (RL r1 ;; RL r2)"
+ apply(induct r1 r2 rule: rsimp_SEQ.induct)
+ apply(simp_all)
+ done
+
+lemma RL_rsimp_RALTS:
+ shows "RL (rsimp_ALTs rs) = (\<Union> (set (map RL rs)))"
+ apply(induct rs rule: rsimp_ALTs.induct)
+ apply(simp_all)
+ done
+
+lemma RL_rsimp_rdistinct:
+ shows "(\<Union> (set (map RL (rdistinct rs {})))) = (\<Union> (set (map RL rs)))"
+ apply(auto)
+ apply (metis Diff_iff rdistinct_set_equality1)
+ by (metis Diff_empty rdistinct_set_equality1)
+
+lemma RL_rsimp_rflts:
+ shows "(\<Union> (set (map RL (rflts rs)))) = (\<Union> (set (map RL rs)))"
+ apply(induct rs rule: rflts.induct)
+ apply(simp_all)
+ done
+
+lemma RL_rsimp:
+ shows "RL r = RL (rsimp r)"
+ apply(induct r rule: rsimp.induct)
+ apply(auto simp add: Sequ_def RL_rsimp_RSEQ)
+ using RL_rsimp_RALTS RL_rsimp_rdistinct RL_rsimp_rflts apply auto[1]
+ by (smt (verit, del_insts) RL_rsimp_RALTS RL_rsimp_rdistinct RL_rsimp_rflts UN_E image_iff list.set_map)
+
+
+lemma qqq1:
+ shows "RZERO \<notin> set (rflts (map rsimp rs))"
+ by (metis ex_map_conv flts3 good.simps(1) good1)
+
+
+fun nonazero :: "rrexp \<Rightarrow> bool"
+ where
+ "nonazero RZERO = False"
+| "nonazero r = True"
+
+
+lemma flts_single1:
+ assumes "nonalt r" "nonazero r"
+ shows "rflts [r] = [r]"
+ using assms
+ apply(induct r)
+ apply(auto)
+ done
+
+lemma nonalt0_flts_keeps:
+ shows "(a \<noteq> RZERO) \<and> (\<forall>rs. a \<noteq> RALTS rs) \<Longrightarrow> rflts (a # xs) = a # rflts xs"
+ apply(case_tac a)
+ apply simp+
+ done
+
+
+lemma nonalt0_fltseq:
+ shows "\<forall>r \<in> set rs. r \<noteq> RZERO \<and> nonalt r \<Longrightarrow> rflts rs = rs"
+ apply(induct rs)
+ apply simp
+ apply(case_tac "a = RZERO")
+ apply fastforce
+ apply(case_tac "\<exists>rs1. a = RALTS rs1")
+ apply(erule exE)
+ apply simp+
+ using nonalt0_flts_keeps by presburger
+
+
+
+
+lemma goodalts_nonalt:
+ shows "good (RALTS rs) \<Longrightarrow> rflts rs = rs"
+ apply(induct x == "RALTS rs" arbitrary: rs rule: good.induct)
+ apply simp
+
+ using good.simps(5) apply blast
+ apply simp
+ apply(case_tac "r1 = RZERO")
+ using good.simps(1) apply force
+ apply(case_tac "r2 = RZERO")
+ using good.simps(1) apply force
+ apply(subgoal_tac "rflts (r1 # r2 # rs) = r1 # r2 # rflts rs")
+ prefer 2
+ apply (metis nonalt.simps(1) rflts_def_idiot)
+ apply(subgoal_tac "\<forall>r \<in> set rs. r \<noteq> RZERO \<and> nonalt r")
+ apply(subgoal_tac "rflts rs = rs")
+ apply presburger
+ using nonalt0_fltseq apply presburger
+ using good.simps(1) by blast
+
+
+
+
+
+lemma test:
+ assumes "good r"
+ shows "rsimp r = r"
+
+ using assms
+ apply(induct rule: good.induct)
+ apply simp
+ apply simp
+ apply simp
+ apply simp
+ apply simp
+ apply(subgoal_tac "distinct (r1 # r2 # rs)")
+ prefer 2
+ using good.simps(6) apply blast
+ apply(subgoal_tac "rflts (r1 # r2 # rs ) = r1 # r2 # rs")
+ prefer 2
+ using goodalts_nonalt apply blast
+
+ apply(subgoal_tac "r1 \<noteq> r2")
+ prefer 2
+ apply (meson distinct_length_2_or_more)
+ apply(subgoal_tac "r1 \<notin> set rs")
+ apply(subgoal_tac "r2 \<notin> set rs")
+ apply(subgoal_tac "\<forall>r \<in> set rs. rsimp r = r")
+ apply(subgoal_tac "map rsimp rs = rs")
+ apply simp
+ apply(subgoal_tac "\<forall>r \<in> {r1, r2}. r \<notin> set rs")
+ apply (metis distinct_not_exist rdistinct_on_distinct)
+
+ apply blast
+ apply (meson map_idI)
+ apply (metis good.simps(6) insert_iff list.simps(15))
+
+ apply (meson distinct.simps(2))
+ apply (simp add: distinct_length_2_or_more)
+ apply simp+
+ done
+
+
+
+lemma rsimp_idem:
+ shows "rsimp (rsimp r) = rsimp r"
+ using test good1
+ by force
+
+corollary rsimp_inner_idem4:
+ shows "rsimp r = RALTS rs \<Longrightarrow> rflts rs = rs"
+ by (metis good1 goodalts_nonalt rrexp.simps(12))
+
+
+lemma head_one_more_simp:
+ shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
+ by (simp add: rsimp_idem)
+
+
+lemma der_simp_nullability:
+ shows "rnullable r = rnullable (rsimp r)"
+ using RL_rnullable RL_rsimp by auto
+
+
+lemma no_alt_short_list_after_simp:
+ shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
+ by (metis bbbbs good1 k0a rrexp.simps(12))
+
+
+lemma no_further_dB_after_simp:
+ shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
+ apply(subgoal_tac "good (RALTS rs)")
+ apply(subgoal_tac "distinct rs")
+ using rdistinct_on_distinct apply blast
+ apply (metis distinct.simps(1) distinct.simps(2) empty_iff good.simps(6) list.exhaust set_empty2)
+ using good1 by fastforce
+
+
+lemma idem_after_simp1:
+ shows "rsimp_ALTs (rdistinct (rflts [rsimp aa]) {}) = rsimp aa"
+ apply(case_tac "rsimp aa")
+ apply simp+
+ apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
+ apply(simp)
+ apply(simp)
+ done
+
+lemma identity_wwo0:
+ shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
+ apply (metis idem_after_simp1 list.exhaust list.simps(8) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
+ done
+
+lemma distinct_removes_last:
+ shows "\<lbrakk>a \<in> set as\<rbrakk>
+ \<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
+and "rdistinct (ab # as @ [ab]) rset1 = rdistinct (ab # as) rset1"
+ apply(induct as arbitrary: rset ab rset1 a)
+ apply simp
+ apply simp
+ apply(case_tac "aa \<in> rset")
+ apply(case_tac "a = aa")
+ apply (metis append_Cons)
+ apply simp
+ apply(case_tac "a \<in> set as")
+ apply (metis append_Cons rdistinct.simps(2) set_ConsD)
+ apply(case_tac "a = aa")
+ prefer 2
+ apply simp
+ apply (metis append_Cons)
+ apply(case_tac "ab \<in> rset1")
+ prefer 2
+ apply(subgoal_tac "rdistinct (ab # (a # as) @ [ab]) rset1 =
+ ab # (rdistinct ((a # as) @ [ab]) (insert ab rset1))")
+ prefer 2
+ apply force
+ apply(simp only:)
+ apply(subgoal_tac "rdistinct (ab # a # as) rset1 = ab # (rdistinct (a # as) (insert ab rset1))")
+ apply(simp only:)
+ apply(subgoal_tac "rdistinct ((a # as) @ [ab]) (insert ab rset1) = rdistinct (a # as) (insert ab rset1)")
+ apply blast
+ apply(case_tac "a \<in> insert ab rset1")
+ apply simp
+ apply (metis insertI1)
+ apply simp
+ apply (meson insertI1)
+ apply simp
+ apply(subgoal_tac "rdistinct ((a # as) @ [ab]) rset1 = rdistinct (a # as) rset1")
+ apply simp
+ by (metis append_Cons insert_iff insert_is_Un rdistinct.simps(2))
+
+
+lemma distinct_removes_middle:
+ shows "\<lbrakk>a \<in> set as\<rbrakk>
+ \<Longrightarrow> rdistinct (as @ as2) rset = rdistinct (as @ [a] @ as2) rset"
+and "rdistinct (ab # as @ [ab] @ as3) rset1 = rdistinct (ab # as @ as3) rset1"
+ apply(induct as arbitrary: rset rset1 ab as2 as3 a)
+ apply simp
+ apply simp
+ apply(case_tac "a \<in> rset")
+ apply simp
+ apply metis
+ apply simp
+ apply (metis insertI1)
+ apply(case_tac "a = ab")
+ apply simp
+ apply(case_tac "ab \<in> rset")
+ apply simp
+ apply presburger
+ apply (meson insertI1)
+ apply(case_tac "a \<in> rset")
+ apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
+ apply(case_tac "ab \<in> rset")
+ apply simp
+ apply (meson insert_iff)
+ apply simp
+ by (metis insertI1)
+
+
+lemma distinct_removes_middle3:
+ shows "\<lbrakk>a \<in> set as\<rbrakk>
+ \<Longrightarrow> rdistinct (as @ a #as2) rset = rdistinct (as @ as2) rset"
+ using distinct_removes_middle(1) by fastforce
+
+
+lemma distinct_removes_list:
+ shows "\<lbrakk> \<forall>r \<in> set rs. r \<in> set as\<rbrakk> \<Longrightarrow> rdistinct (as @ rs) {} = rdistinct as {}"
+ apply(induct rs)
+ apply simp+
+ apply(subgoal_tac "rdistinct (as @ a # rs) {} = rdistinct (as @ rs) {}")
+ prefer 2
+ apply (metis append_Cons append_Nil distinct_removes_middle(1))
+ by presburger
+
+
+lemma spawn_simp_rsimpalts:
+ shows "rsimp (rsimp_ALTs rs) = rsimp (rsimp_ALTs (map rsimp rs))"
+ apply(cases rs)
+ apply simp
+ apply(case_tac list)
+ apply simp
+ apply(subst rsimp_idem[symmetric])
+ apply simp
+ apply(subgoal_tac "rsimp_ALTs rs = RALTS rs")
+ apply(simp only:)
+ apply(subgoal_tac "rsimp_ALTs (map rsimp rs) = RALTS (map rsimp rs)")
+ apply(simp only:)
+ prefer 2
+ apply simp
+ prefer 2
+ using rsimp_ALTs.simps(3) apply presburger
+ apply auto
+ apply(subst rsimp_idem)+
+ by (metis comp_apply rsimp_idem)
+
+
+lemma simp_singlealt_flatten:
+ shows "rsimp (RALTS [RALTS rsa]) = rsimp (RALTS (rsa @ []))"
+ apply(induct rsa)
+ apply simp
+ apply simp
+ by (metis idem_after_simp1 list.simps(9) rsimp.simps(2))
+
+
+lemma good1_rsimpalts:
+ shows "rsimp r = RALTS rs \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
+ by (metis no_alt_short_list_after_simp)
+
+
+
+
+lemma good1_flatten:
+ shows "\<lbrakk> rsimp r = (RALTS rs1)\<rbrakk>
+ \<Longrightarrow> rflts (rsimp_ALTs rs1 # map rsimp rsb) = rflts (rs1 @ map rsimp rsb)"
+ apply(subst good1_rsimpalts)
+ apply simp+
+ apply(subgoal_tac "rflts (rs1 @ map rsimp rsb) = rs1 @ rflts (map rsimp rsb)")
+ apply simp
+ using flts_append rsimp_inner_idem4 by presburger
+
+
+lemma flatten_rsimpalts:
+ shows "rflts (rsimp_ALTs (rdistinct (rflts (map rsimp rsa)) {}) # map rsimp rsb) =
+ rflts ( (rdistinct (rflts (map rsimp rsa)) {}) @ map rsimp rsb)"
+ apply(case_tac "map rsimp rsa")
+ apply simp
+ apply(case_tac "list")
+ apply simp
+ apply(case_tac a)
+ apply simp+
+ apply(rename_tac rs1)
+ apply (metis good1_flatten map_eq_Cons_D no_further_dB_after_simp)
+
+ apply simp
+
+ apply(subgoal_tac "\<forall>r \<in> set( rflts (map rsimp rsa)). good r")
+ apply(case_tac "rdistinct (rflts (map rsimp rsa)) {}")
+ apply simp
+ apply auto[1]
+ apply simp
+ apply(simp)
+ apply(case_tac "lista")
+ apply simp_all
+
+ apply (metis append_Cons append_Nil good1_flatten rflts.simps(2) rsimp.simps(2) rsimp_ALTs.elims)
+ by (metis (no_types, opaque_lifting) append_Cons append_Nil good1_flatten rflts.simps(2) rsimp.simps(2) rsimp_ALTs.elims)
+
+lemma last_elem_out:
+ shows "\<lbrakk>x \<notin> set xs; x \<notin> rset \<rbrakk> \<Longrightarrow> rdistinct (xs @ [x]) rset = rdistinct xs rset @ [x]"
+ apply(induct xs arbitrary: rset)
+ apply simp+
+ done
+
+
+
+
+lemma rdistinct_concat_general:
+ shows "rdistinct (rs1 @ rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
+ apply(induct rs1 arbitrary: rs2 rule: rev_induct)
+ apply simp
+ apply(drule_tac x = "x # rs2" in meta_spec)
+ apply simp
+ apply(case_tac "x \<in> set xs")
+ apply simp
+
+ apply (simp add: distinct_removes_middle3 insert_absorb)
+ apply simp
+ by (simp add: last_elem_out)
+
+
+
+
+lemma distinct_once_enough:
+ shows "rdistinct (rs @ rsa) {} = rdistinct (rdistinct rs {} @ rsa) {}"
+ apply(subgoal_tac "distinct (rdistinct rs {})")
+ apply(subgoal_tac
+" rdistinct (rdistinct rs {} @ rsa) {} = rdistinct rs {} @ (rdistinct rsa (set rs))")
+ apply(simp only:)
+ using rdistinct_concat_general apply blast
+ apply (simp add: distinct_rdistinct_append rdistinct_set_equality1)
+ by (simp add: rdistinct_does_the_job)
+
+
+lemma simp_flatten:
+ shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
+ apply simp
+ apply(subst flatten_rsimpalts)
+ apply(simp add: flts_append)
+ by (metis Diff_empty distinct_once_enough flts_append nonalt0_fltseq nonalt_flts_rd qqq1 rdistinct_set_equality1)
+
+lemma basic_rsimp_SEQ_property1:
+ shows "rsimp_SEQ RONE r = r"
+ by (simp add: idiot)
+
+
+
+lemma basic_rsimp_SEQ_property3:
+ shows "rsimp_SEQ r RZERO = RZERO"
+ using rsimp_SEQ.elims by blast
+
+
+
+fun vsuf :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list" where
+"vsuf [] _ = []"
+|"vsuf (c#cs) r1 = (if (rnullable r1) then (vsuf cs (rder c r1)) @ [c # cs]
+ else (vsuf cs (rder c r1))
+ ) "
+
+
+
+
+
+
+fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list" where
+"star_update c r [] = []"
+|"star_update c r (s # Ss) = (if (rnullable (rders r s))
+ then (s@[c]) # [c] # (star_update c r Ss)
+ else (s@[c]) # (star_update c r Ss) )"
+
+
+fun star_updates :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"
+ where
+"star_updates [] r Ss = Ss"
+| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"
+
+lemma stupdates_append: shows
+"star_updates (s @ [c]) r Ss = star_update c r (star_updates s r Ss)"
+ apply(induct s arbitrary: Ss)
+ apply simp
+ apply simp
+ done
+
+lemma flts_removes0:
+ shows " rflts (rs @ [RZERO]) =
+ rflts rs"
+ apply(induct rs)
+ apply simp
+ by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
+
+
+lemma rflts_spills_last:
+ shows "rflts (rs1 @ [RALTS rs]) = rflts rs1 @ rs"
+ apply (induct rs1 rule: rflts.induct)
+ apply(auto)
+ done
+
+lemma flts_keeps1:
+ shows "rflts (rs @ [RONE]) = rflts rs @ [RONE]"
+ apply (induct rs rule: rflts.induct)
+ apply(auto)
+ done
+
+lemma flts_keeps_others:
+ shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow>rflts (rs @ [a]) = rflts rs @ [a]"
+ apply(induct rs rule: rflts.induct)
+ apply(auto)
+ by (meson k0b nonalt.elims(3))
+
+lemma spilled_alts_contained:
+ shows "\<lbrakk>a = RALTS rs ; a \<in> set rs1\<rbrakk> \<Longrightarrow> \<forall>r \<in> set rs. r \<in> set (rflts rs1)"
+ apply(induct rs1)
+ apply simp
+ apply(case_tac "a = aa")
+ apply simp
+ apply(subgoal_tac " a \<in> set rs1")
+ prefer 2
+ apply (meson set_ConsD)
+ apply(case_tac aa)
+ using rflts.simps(2) apply presburger
+ apply fastforce
+ apply fastforce
+ apply fastforce
+ apply fastforce
+ apply fastforce
+ by simp
+
+
+lemma distinct_removes_duplicate_flts:
+ shows " a \<in> set rsa
+ \<Longrightarrow> rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
+ rdistinct (rflts (map rsimp rsa)) {}"
+ apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
+ prefer 2
+ apply simp
+ apply(induct "rsimp a")
+ apply simp
+ using flts_removes0 apply presburger
+ apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
+ rdistinct (rflts (map rsimp rsa @ [RONE])) {}")
+ apply (simp only:)
+ apply(subst flts_keeps1)
+ apply (metis distinct_removes_last(1) flts_append in_set_conv_decomp rflts.simps(4))
+ apply presburger
+ apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
+ rdistinct ((rflts (map rsimp rsa)) @ [RCHAR x]) {}")
+ apply (simp only:)
+ prefer 2
+ apply (metis flts_append rflts.simps(1) rflts.simps(5))
+ apply (metis distinct_removes_last(1) rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(3))
+ apply (metis distinct_removes_last(1) flts_append rflts.simps(1) rflts.simps(6) rflts_def_idiot2 rrexp.distinct(31) rrexp.distinct(5))
+ apply (metis distinct_removes_list rflts_spills_last spilled_alts_contained)
+ apply (metis distinct_removes_last(1) flts_append good.simps(1) good.simps(44) rflts.simps(1) rflts.simps(7) rflts_def_idiot2 rrexp.distinct(37))
+ by (metis distinct_removes_last(1) flts_append rflts.simps(1) rflts.simps(8) rflts_def_idiot2 rrexp.distinct(11) rrexp.distinct(39))
+
+(*some basic facts about rsimp*)
+
+unused_thms
+
+
+end
\ No newline at end of file