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)
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 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)
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_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 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