--- a/thys3/BasicIdentities.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1175 +0,0 @@
-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