
diff -r f9cdc295ccf7 -r f493a20feeb3 thys3/src/BasicIdentities.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys3/src/BasicIdentities.thy	Sat Apr 30 00:50:08 2022 +0100
@@ -0,0 +1,1175 @@
+theory BasicIdentities 
+  imports "Lexer" 
+begin
+
+datatype rrexp = 
+  RZERO
+| RONE 
+| RCHAR char
+| RSEQ rrexp rrexp
+| RALTS "rrexp list"
+| RSTAR rrexp
+
+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"
+
+
+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)"
+
+
+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)"
+
+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
+  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
+  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"
+
+
+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)
+  done  
+
+
+
+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)
+  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))
+  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
+  using idiot2 nonnested.simps(11) apply presburger
+  by (metis (mono_tags, lifting) Diff_empty image_iff list.set_map nn1bb nn1c rdistinct_set_equality1)
+
+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)
+
+  
+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 by blast
+
+
+
+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>"
+
+
+lemma RL_rnullable:
+  shows "rnullable r = ([] \<in> RL r)"
+  apply(induct r)
+  apply(auto simp add: Sequ_def)
+  done
+
+lemma RL_rder:
+  shows "RL (rder c r) = Der c (RL r)"
+  apply(induct r)
+  apply(auto simp add: Sequ_def Der_def)
+        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 (metis Star.step append_Cons)
+  using Star_decomp by auto
+
+
+
+
+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)
+  by simp
+
+lemma identity_wwo0:
+  shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
+  by (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))
+
+
+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(case_tac "listb")
+    apply simp+
+  apply (metis Cons_eq_appendI good1_flatten rflts.simps(3) rsimp.simps(2) rsimp_ALTs.simps(3))
+  by (metis (mono_tags, lifting) flts3 good1 image_iff list.set_map)
+
+
+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
+  by fastforce
+
+
+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) rflts_def_idiot2 rrexp.simps(20) rrexp.simps(6))
+      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_keeps_others rrexp.distinct(21) rrexp.distinct(3))
+  apply (metis distinct_removes_last(1) rflts_def_idiot2 rrexp.distinct(21) rrexp.distinct(3))
+
+    apply (metis distinct_removes_last(1) flts_keeps_others rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
+   prefer 2
+   apply (metis distinct_removes_last(1) flts_keeps_others flts_removes0 rflts_def_idiot2 rrexp.distinct(29))
+  apply(subgoal_tac "rflts (map rsimp rsa @ [rsimp a]) = rflts (map rsimp rsa) @ x")
+  prefer 2
+  apply (simp add: rflts_spills_last)
+  apply(subgoal_tac "\<forall> r \<in> set x. r \<in> set (rflts (map rsimp rsa))")
+    prefer 2
+  apply (metis (mono_tags, lifting) image_iff image_set spilled_alts_contained)
+  apply (metis rflts_spills_last)
+  by (metis distinct_removes_list spilled_alts_contained)
+
+
+
+(*some basic facts about rsimp*)
+
+unused_thms
+
+
+end
\ No newline at end of file




