# HG changeset patch
# User Christian Urban <christian.urban@kcl.ac.uk>
# Date 1651276530 -3600
# Node ID 04b5e904a2206d65ab480c0671a217622f76f9e8
# Parent f493a20feeb365200b6ea2674db28b678494c998
cleaned up
diff -r f493a20feeb3 -r 04b5e904a220 thys3/BasicIdentities.thy
--- 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
diff -r f493a20feeb3 -r 04b5e904a220 thys3/Blexer.thy
--- a/thys3/Blexer.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,454 +0,0 @@
-
-theory Blexer
- imports "Lexer" "PDerivs"
-begin
-
-section \<open>Bit-Encodings\<close>
-
-datatype bit = Z | S
-
-fun code :: "val \<Rightarrow> bit list"
-where
- "code Void = []"
-| "code (Char c) = []"
-| "code (Left v) = Z # (code v)"
-| "code (Right v) = S # (code v)"
-| "code (Seq v1 v2) = (code v1) @ (code v2)"
-| "code (Stars []) = [S]"
-| "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)"
-
-
-fun
- Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"
-where
- "Stars_add v (Stars vs) = Stars (v # vs)"
-
-function
- decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)"
-where
- "decode' bs ZERO = (undefined, bs)"
-| "decode' bs ONE = (Void, bs)"
-| "decode' bs (CH d) = (Char d, bs)"
-| "decode' [] (ALT r1 r2) = (Void, [])"
-| "decode' (Z # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r1 in (Left v, bs'))"
-| "decode' (S # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r2 in (Right v, bs'))"
-| "decode' bs (SEQ r1 r2) = (let (v1, bs') = decode' bs r1 in
- let (v2, bs'') = decode' bs' r2 in (Seq v1 v2, bs''))"
-| "decode' [] (STAR r) = (Void, [])"
-| "decode' (S # bs) (STAR r) = (Stars [], bs)"
-| "decode' (Z # bs) (STAR r) = (let (v, bs') = decode' bs r in
- let (vs, bs'') = decode' bs' (STAR r)
- in (Stars_add v vs, bs''))"
-by pat_completeness auto
-
-lemma decode'_smaller:
- assumes "decode'_dom (bs, r)"
- shows "length (snd (decode' bs r)) \<le> length bs"
-using assms
-apply(induct bs r)
-apply(auto simp add: decode'.psimps split: prod.split)
-using dual_order.trans apply blast
-by (meson dual_order.trans le_SucI)
-
-termination "decode'"
-apply(relation "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))")
-apply(auto dest!: decode'_smaller)
-by (metis less_Suc_eq_le snd_conv)
-
-definition
- decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option"
-where
- "decode ds r \<equiv> (let (v, ds') = decode' ds r
- in (if ds' = [] then Some v else None))"
-
-lemma decode'_code_Stars:
- assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x)) \<and> flat v \<noteq> []"
- shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)"
- using assms
- apply(induct vs)
- apply(auto)
- done
-
-lemma decode'_code:
- assumes "\<Turnstile> v : r"
- shows "decode' ((code v) @ ds) r = (v, ds)"
-using assms
- apply(induct v r arbitrary: ds)
- apply(auto)
- using decode'_code_Stars by blast
-
-lemma decode_code:
- assumes "\<Turnstile> v : r"
- shows "decode (code v) r = Some v"
- using assms unfolding decode_def
- by (smt append_Nil2 decode'_code old.prod.case)
-
-
-section {* Annotated Regular Expressions *}
-
-datatype arexp =
- AZERO
-| AONE "bit list"
-| ACHAR "bit list" char
-| ASEQ "bit list" arexp arexp
-| AALTs "bit list" "arexp list"
-| ASTAR "bit list" arexp
-
-abbreviation
- "AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]"
-
-fun asize :: "arexp \<Rightarrow> nat" where
- "asize AZERO = 1"
-| "asize (AONE cs) = 1"
-| "asize (ACHAR cs c) = 1"
-| "asize (AALTs cs rs) = Suc (sum_list (map asize rs))"
-| "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)"
-| "asize (ASTAR cs r) = Suc (asize r)"
-
-fun
- erase :: "arexp \<Rightarrow> rexp"
-where
- "erase AZERO = ZERO"
-| "erase (AONE _) = ONE"
-| "erase (ACHAR _ c) = CH c"
-| "erase (AALTs _ []) = ZERO"
-| "erase (AALTs _ [r]) = (erase r)"
-| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"
-| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
-| "erase (ASTAR _ r) = STAR (erase r)"
-
-
-fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
- "fuse bs AZERO = AZERO"
-| "fuse bs (AONE cs) = AONE (bs @ cs)"
-| "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c"
-| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
-| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
-| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
-
-lemma fuse_append:
- shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"
- apply(induct r)
- apply(auto)
- done
-
-
-fun intern :: "rexp \<Rightarrow> arexp" where
- "intern ZERO = AZERO"
-| "intern ONE = AONE []"
-| "intern (CH c) = ACHAR [] c"
-| "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1))
- (fuse [S] (intern r2))"
-| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"
-| "intern (STAR r) = ASTAR [] (intern r)"
-
-
-fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where
- "retrieve (AONE bs) Void = bs"
-| "retrieve (ACHAR bs c) (Char d) = bs"
-| "retrieve (AALTs bs [r]) v = bs @ retrieve r v"
-| "retrieve (AALTs bs (r#rs)) (Left v) = bs @ retrieve r v"
-| "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v"
-| "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2"
-| "retrieve (ASTAR bs r) (Stars []) = bs @ [S]"
-| "retrieve (ASTAR bs r) (Stars (v#vs)) =
- bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
-
-
-
-fun
- bnullable :: "arexp \<Rightarrow> bool"
-where
- "bnullable (AZERO) = False"
-| "bnullable (AONE bs) = True"
-| "bnullable (ACHAR bs c) = False"
-| "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
-| "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)"
-| "bnullable (ASTAR bs r) = True"
-
-abbreviation
- bnullables :: "arexp list \<Rightarrow> bool"
-where
- "bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)"
-
-fun
- bmkeps :: "arexp \<Rightarrow> bit list" and
- bmkepss :: "arexp list \<Rightarrow> bit list"
-where
- "bmkeps(AONE bs) = bs"
-| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
-| "bmkeps(AALTs bs rs) = bs @ (bmkepss rs)"
-| "bmkeps(ASTAR bs r) = bs @ [S]"
-| "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))"
-
-lemma bmkepss1:
- assumes "\<not> bnullables rs1"
- shows "bmkepss (rs1 @ rs2) = bmkepss rs2"
- using assms
- by (induct rs1) (auto)
-
-lemma bmkepss2:
- assumes "bnullables rs1"
- shows "bmkepss (rs1 @ rs2) = bmkepss rs1"
- using assms
- by (induct rs1) (auto)
-
-
-fun
- bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
-where
- "bder c (AZERO) = AZERO"
-| "bder c (AONE bs) = AZERO"
-| "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"
-| "bder c (AALTs bs rs) = AALTs bs (map (bder c) rs)"
-| "bder c (ASEQ bs r1 r2) =
- (if bnullable r1
- then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2))
- else ASEQ bs (bder c r1) r2)"
-| "bder c (ASTAR bs r) = ASEQ (bs @ [Z]) (bder c r) (ASTAR [] r)"
-
-
-fun
- bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
-where
- "bders r [] = r"
-| "bders r (c#s) = bders (bder c r) s"
-
-lemma bders_append:
- "bders c (s1 @ s2) = bders (bders c s1) s2"
- apply(induct s1 arbitrary: c s2)
- apply(simp_all)
- done
-
-lemma bnullable_correctness:
- shows "nullable (erase r) = bnullable r"
- apply(induct r rule: erase.induct)
- apply(simp_all)
- done
-
-lemma erase_fuse:
- shows "erase (fuse bs r) = erase r"
- apply(induct r rule: erase.induct)
- apply(simp_all)
- done
-
-lemma erase_intern [simp]:
- shows "erase (intern r) = r"
- apply(induct r)
- apply(simp_all add: erase_fuse)
- done
-
-lemma erase_bder [simp]:
- shows "erase (bder a r) = der a (erase r)"
- apply(induct r rule: erase.induct)
- apply(simp_all add: erase_fuse bnullable_correctness)
- done
-
-lemma erase_bders [simp]:
- shows "erase (bders r s) = ders s (erase r)"
- apply(induct s arbitrary: r )
- apply(simp_all)
- done
-
-lemma bnullable_fuse:
- shows "bnullable (fuse bs r) = bnullable r"
- apply(induct r arbitrary: bs)
- apply(auto)
- done
-
-lemma retrieve_encode_STARS:
- assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v"
- shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)"
- using assms
- apply(induct vs)
- apply(simp_all)
- done
-
-lemma retrieve_fuse2:
- assumes "\<Turnstile> v : (erase r)"
- shows "retrieve (fuse bs r) v = bs @ retrieve r v"
- using assms
- apply(induct r arbitrary: v bs)
- apply(auto elim: Prf_elims)[4]
- apply(case_tac x2a)
- apply(simp)
- using Prf_elims(1) apply blast
- apply(case_tac x2a)
- apply(simp)
- apply(simp)
- apply(case_tac list)
- apply(simp)
- apply(simp)
- apply (smt (verit, best) Prf_elims(3) append_assoc retrieve.simps(4) retrieve.simps(5))
- apply(simp)
- using retrieve_encode_STARS
- apply(auto elim!: Prf_elims)[1]
- apply(case_tac vs)
- apply(simp)
- apply(simp)
- done
-
-lemma retrieve_fuse:
- assumes "\<Turnstile> v : r"
- shows "retrieve (fuse bs (intern r)) v = bs @ retrieve (intern r) v"
- using assms
- by (simp_all add: retrieve_fuse2)
-
-
-lemma retrieve_code:
- assumes "\<Turnstile> v : r"
- shows "code v = retrieve (intern r) v"
- using assms
- apply(induct v r )
- apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
- done
-
-
-lemma retrieve_AALTs_bnullable1:
- assumes "bnullable r"
- shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))
- = bs @ retrieve r (mkeps (erase r))"
- using assms
- apply(case_tac rs)
- apply(auto simp add: bnullable_correctness)
- done
-
-lemma retrieve_AALTs_bnullable2:
- assumes "\<not>bnullable r" "bnullables rs"
- shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))
- = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
- using assms
- apply(induct rs arbitrary: r bs)
- apply(auto)
- using bnullable_correctness apply blast
- apply(case_tac rs)
- apply(auto)
- using bnullable_correctness apply blast
- apply(case_tac rs)
- apply(auto)
- done
-
-lemma bmkeps_retrieve_AALTs:
- assumes "\<forall>r \<in> set rs. bnullable r \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))"
- "bnullables rs"
- shows "bs @ bmkepss rs = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
- using assms
- apply(induct rs arbitrary: bs)
- apply(auto)
- using retrieve_AALTs_bnullable1 apply presburger
- apply (metis retrieve_AALTs_bnullable2)
- apply (simp add: retrieve_AALTs_bnullable1)
- by (metis retrieve_AALTs_bnullable2)
-
-
-lemma bmkeps_retrieve:
- assumes "bnullable r"
- shows "bmkeps r = retrieve r (mkeps (erase r))"
- using assms
- apply(induct r)
- apply(auto)
- using bmkeps_retrieve_AALTs by auto
-
-lemma bder_retrieve:
- assumes "\<Turnstile> v : der c (erase r)"
- shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
- using assms
- apply(induct r arbitrary: v rule: erase.induct)
- using Prf_elims(1) apply auto[1]
- using Prf_elims(1) apply auto[1]
- apply(auto)[1]
- apply (metis Prf_elims(4) injval.simps(1) retrieve.simps(1) retrieve.simps(2))
- using Prf_elims(1) apply blast
- (* AALTs case *)
- apply(simp)
- apply(erule Prf_elims)
- apply(simp)
- apply(simp)
- apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)
- apply(erule Prf_elims)
- apply(simp)
- apply(simp)
- apply(case_tac rs)
- apply(simp)
- apply(simp)
- using Prf_elims(3) apply fastforce
- (* ASEQ case *)
- apply(simp)
- apply(case_tac "nullable (erase r1)")
- apply(simp)
- apply(erule Prf_elims)
- using Prf_elims(2) bnullable_correctness apply force
- apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
- apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
- using Prf_elims(2) apply force
- (* ASTAR case *)
- apply(rename_tac bs r v)
- apply(simp)
- apply(erule Prf_elims)
- apply(clarify)
- apply(erule Prf_elims)
- apply(clarify)
- by (simp add: retrieve_fuse2)
-
-
-lemma MAIN_decode:
- assumes "\<Turnstile> v : ders s r"
- shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
- using assms
-proof (induct s arbitrary: v rule: rev_induct)
- case Nil
- have "\<Turnstile> v : ders [] r" by fact
- then have "\<Turnstile> v : r" by simp
- then have "Some v = decode (retrieve (intern r) v) r"
- using decode_code retrieve_code by auto
- then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
- by simp
-next
- case (snoc c s v)
- have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow>
- Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
- have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
- then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r"
- by (simp add: Prf_injval ders_append)
- have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
- by (simp add: flex_append)
- also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
- using asm2 IH by simp
- also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
- using asm by (simp_all add: bder_retrieve ders_append)
- finally show "Some (flex r id (s @ [c]) v) =
- decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
-qed
-
-definition blexer where
- "blexer r s \<equiv> if bnullable (bders (intern r) s) then
- decode (bmkeps (bders (intern r) s)) r else None"
-
-lemma blexer_correctness:
- shows "blexer r s = lexer r s"
-proof -
- { define bds where "bds \<equiv> bders (intern r) s"
- define ds where "ds \<equiv> ders s r"
- assume asm: "nullable ds"
- have era: "erase bds = ds"
- unfolding ds_def bds_def by simp
- have mke: "\<Turnstile> mkeps ds : ds"
- using asm by (simp add: mkeps_nullable)
- have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
- using bmkeps_retrieve
- using asm era
- using bnullable_correctness by force
- also have "... = Some (flex r id s (mkeps ds))"
- using mke by (simp_all add: MAIN_decode ds_def bds_def)
- finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))"
- unfolding bds_def ds_def .
- }
- then show "blexer r s = lexer r s"
- unfolding blexer_def lexer_flex
- by (auto simp add: bnullable_correctness[symmetric])
-qed
-
-
-unused_thms
-
-end
diff -r f493a20feeb3 -r 04b5e904a220 thys3/BlexerSimp.thy
--- a/thys3/BlexerSimp.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,617 +0,0 @@
-theory BlexerSimp
- imports Blexer
-begin
-
-fun distinctWith :: "'a list \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a list"
- where
- "distinctWith [] eq acc = []"
-| "distinctWith (x # xs) eq acc =
- (if (\<exists> y \<in> acc. eq x y) then distinctWith xs eq acc
- else x # (distinctWith xs eq ({x} \<union> acc)))"
-
-
-fun eq1 ("_ ~1 _" [80, 80] 80) where
- "AZERO ~1 AZERO = True"
-| "(AONE bs1) ~1 (AONE bs2) = True"
-| "(ACHAR bs1 c) ~1 (ACHAR bs2 d) = (if c = d then True else False)"
-| "(ASEQ bs1 ra1 ra2) ~1 (ASEQ bs2 rb1 rb2) = (ra1 ~1 rb1 \<and> ra2 ~1 rb2)"
-| "(AALTs bs1 []) ~1 (AALTs bs2 []) = True"
-| "(AALTs bs1 (r1 # rs1)) ~1 (AALTs bs2 (r2 # rs2)) = (r1 ~1 r2 \<and> (AALTs bs1 rs1) ~1 (AALTs bs2 rs2))"
-| "(ASTAR bs1 r1) ~1 (ASTAR bs2 r2) = r1 ~1 r2"
-| "_ ~1 _ = False"
-
-
-
-lemma eq1_L:
- assumes "x ~1 y"
- shows "L (erase x) = L (erase y)"
- using assms
- apply(induct rule: eq1.induct)
- apply(auto elim: eq1.elims)
- apply presburger
- done
-
-fun flts :: "arexp list \<Rightarrow> arexp list"
- where
- "flts [] = []"
-| "flts (AZERO # rs) = flts rs"
-| "flts ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
-| "flts (r1 # rs) = r1 # flts rs"
-
-
-
-fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
- where
- "bsimp_ASEQ _ AZERO _ = AZERO"
-| "bsimp_ASEQ _ _ AZERO = AZERO"
-| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
-| "bsimp_ASEQ bs1 r1 r2 = ASEQ bs1 r1 r2"
-
-lemma bsimp_ASEQ0[simp]:
- shows "bsimp_ASEQ bs r1 AZERO = AZERO"
- by (case_tac r1)(simp_all)
-
-lemma bsimp_ASEQ1:
- assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<nexists>bs. r1 = AONE bs"
- shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
- using assms
- apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
- apply(auto)
- done
-
-lemma bsimp_ASEQ2[simp]:
- shows "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
- by (case_tac r2) (simp_all)
-
-
-fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
- where
- "bsimp_AALTs _ [] = AZERO"
-| "bsimp_AALTs bs1 [r] = fuse bs1 r"
-| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
-
-
-
-
-fun bsimp :: "arexp \<Rightarrow> arexp"
- where
- "bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
-| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {}) "
-| "bsimp r = r"
-
-
-fun
- bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
-where
- "bders_simp r [] = r"
-| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
-
-definition blexer_simp where
- "blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
- decode (bmkeps (bders_simp (intern r) s)) r else None"
-
-
-
-lemma bders_simp_append:
- shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
- apply(induct s1 arbitrary: r s2)
- apply(simp_all)
- done
-
-lemma bmkeps_fuse:
- assumes "bnullable r"
- shows "bmkeps (fuse bs r) = bs @ bmkeps r"
- using assms
- by (induct r rule: bnullable.induct) (auto)
-
-lemma bmkepss_fuse:
- assumes "bnullables rs"
- shows "bmkepss (map (fuse bs) rs) = bs @ bmkepss rs"
- using assms
- apply(induct rs arbitrary: bs)
- apply(auto simp add: bmkeps_fuse bnullable_fuse)
- done
-
-lemma bder_fuse:
- shows "bder c (fuse bs a) = fuse bs (bder c a)"
- apply(induct a arbitrary: bs c)
- apply(simp_all)
- done
-
-
-
-
-inductive
- rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)
-and
- srewrite:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto> _" [100, 100] 100)
-where
- bs1: "ASEQ bs AZERO r2 \<leadsto> AZERO"
-| bs2: "ASEQ bs r1 AZERO \<leadsto> AZERO"
-| bs3: "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"
-| bs4: "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
-| bs5: "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
-| bs6: "AALTs bs [] \<leadsto> AZERO"
-| bs7: "AALTs bs [r] \<leadsto> fuse bs r"
-| bs10: "rs1 s\<leadsto> rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto> AALTs bs rs2"
-| ss1: "[] s\<leadsto> []"
-| ss2: "rs1 s\<leadsto> rs2 \<Longrightarrow> (r # rs1) s\<leadsto> (r # rs2)"
-| ss3: "r1 \<leadsto> r2 \<Longrightarrow> (r1 # rs) s\<leadsto> (r2 # rs)"
-| ss4: "(AZERO # rs) s\<leadsto> rs"
-| ss5: "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"
-| ss6: "L (erase a2) \<subseteq> L (erase a1) \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"
-
-
-inductive
- rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
-where
- rs1[intro, simp]:"r \<leadsto>* r"
-| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
-
-inductive
- srewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" ("_ s\<leadsto>* _" [100, 100] 100)
-where
- sss1[intro, simp]:"rs s\<leadsto>* rs"
-| sss2[intro]: "\<lbrakk>rs1 s\<leadsto> rs2; rs2 s\<leadsto>* rs3\<rbrakk> \<Longrightarrow> rs1 s\<leadsto>* rs3"
-
-
-lemma r_in_rstar : "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
- using rrewrites.intros(1) rrewrites.intros(2) by blast
-
-lemma rs_in_rstar:
- shows "r1 s\<leadsto> r2 \<Longrightarrow> r1 s\<leadsto>* r2"
- using rrewrites.intros(1) rrewrites.intros(2) by blast
-
-
-lemma rrewrites_trans[trans]:
- assumes a1: "r1 \<leadsto>* r2" and a2: "r2 \<leadsto>* r3"
- shows "r1 \<leadsto>* r3"
- using a2 a1
- apply(induct r2 r3 arbitrary: r1 rule: rrewrites.induct)
- apply(auto)
- done
-
-lemma srewrites_trans[trans]:
- assumes a1: "r1 s\<leadsto>* r2" and a2: "r2 s\<leadsto>* r3"
- shows "r1 s\<leadsto>* r3"
- using a1 a2
- apply(induct r1 r2 arbitrary: r3 rule: srewrites.induct)
- apply(auto)
- done
-
-
-
-lemma contextrewrites0:
- "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
- apply(induct rs1 rs2 rule: srewrites.inducts)
- apply simp
- using bs10 r_in_rstar rrewrites_trans by blast
-
-lemma contextrewrites1:
- "r \<leadsto>* r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto>* AALTs bs (r' # rs)"
- apply(induct r r' rule: rrewrites.induct)
- apply simp
- using bs10 ss3 by blast
-
-lemma srewrite1:
- shows "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto> (rs @ rs2)"
- apply(induct rs)
- apply(auto)
- using ss2 by auto
-
-lemma srewrites1:
- shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto>* (rs @ rs2)"
- apply(induct rs1 rs2 rule: srewrites.induct)
- apply(auto)
- using srewrite1 by blast
-
-lemma srewrite2:
- shows "r1 \<leadsto> r2 \<Longrightarrow> True"
- and "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
- apply(induct rule: rrewrite_srewrite.inducts)
- apply(auto)
- apply (metis append_Cons append_Nil srewrites1)
- apply(meson srewrites.simps ss3)
- apply (meson srewrites.simps ss4)
- apply (meson srewrites.simps ss5)
- by (metis append_Cons append_Nil srewrites.simps ss6)
-
-
-lemma srewrites3:
- shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
- apply(induct rs1 rs2 arbitrary: rs rule: srewrites.induct)
- apply(auto)
- by (meson srewrite2(2) srewrites_trans)
-
-(*
-lemma srewrites4:
- assumes "rs3 s\<leadsto>* rs4" "rs1 s\<leadsto>* rs2"
- shows "(rs1 @ rs3) s\<leadsto>* (rs2 @ rs4)"
- using assms
- apply(induct rs3 rs4 arbitrary: rs1 rs2 rule: srewrites.induct)
- apply (simp add: srewrites3)
- using srewrite1 by blast
-*)
-
-lemma srewrites6:
- assumes "r1 \<leadsto>* r2"
- shows "[r1] s\<leadsto>* [r2]"
- using assms
- apply(induct r1 r2 rule: rrewrites.induct)
- apply(auto)
- by (meson srewrites.simps srewrites_trans ss3)
-
-lemma srewrites7:
- assumes "rs3 s\<leadsto>* rs4" "r1 \<leadsto>* r2"
- shows "(r1 # rs3) s\<leadsto>* (r2 # rs4)"
- using assms
- by (smt (verit, best) append_Cons append_Nil srewrites1 srewrites3 srewrites6 srewrites_trans)
-
-lemma ss6_stronger_aux:
- shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctWith rs2 eq1 (set rs1))"
- apply(induct rs2 arbitrary: rs1)
- apply(auto)
- prefer 2
- apply(drule_tac x="rs1 @ [a]" in meta_spec)
- apply(simp)
- apply(drule_tac x="rs1" in meta_spec)
- apply(subgoal_tac "(rs1 @ a # rs2) s\<leadsto>* (rs1 @ rs2)")
- using srewrites_trans apply blast
- apply(subgoal_tac "\<exists>rs1a rs1b. rs1 = rs1a @ [x] @ rs1b")
- prefer 2
- apply (simp add: split_list)
- apply(erule exE)+
- apply(simp)
- using eq1_L rs_in_rstar ss6 by force
-
-lemma ss6_stronger:
- shows "rs1 s\<leadsto>* distinctWith rs1 eq1 {}"
- by (metis append_Nil list.set(1) ss6_stronger_aux)
-
-
-lemma rewrite_preserves_fuse:
- shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"
- and "rs2 s\<leadsto> rs3 \<Longrightarrow> map (fuse bs) rs2 s\<leadsto>* map (fuse bs) rs3"
-proof(induct rule: rrewrite_srewrite.inducts)
- case (bs3 bs1 bs2 r)
- then show ?case
- by (metis fuse.simps(5) fuse_append rrewrite_srewrite.bs3)
-next
- case (bs7 bs r)
- then show ?case
- by (metis fuse.simps(4) fuse_append rrewrite_srewrite.bs7)
-next
- case (ss2 rs1 rs2 r)
- then show ?case
- using srewrites7 by force
-next
- case (ss3 r1 r2 rs)
- then show ?case by (simp add: r_in_rstar srewrites7)
-next
- case (ss5 bs1 rs1 rsb)
- then show ?case
- apply(simp)
- by (metis (mono_tags, lifting) comp_def fuse_append map_eq_conv rrewrite_srewrite.ss5 srewrites.simps)
-next
- case (ss6 a1 a2 rsa rsb rsc)
- then show ?case
- apply(simp only: map_append)
- by (smt (verit, best) erase_fuse list.simps(8) list.simps(9) rrewrite_srewrite.ss6 srewrites.simps)
-qed (auto intro: rrewrite_srewrite.intros)
-
-
-lemma rewrites_fuse:
- assumes "r1 \<leadsto>* r2"
- shows "fuse bs r1 \<leadsto>* fuse bs r2"
-using assms
-apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)
-apply(auto intro: rewrite_preserves_fuse rrewrites_trans)
-done
-
-
-lemma star_seq:
- assumes "r1 \<leadsto>* r2"
- shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"
-using assms
-apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)
-apply(auto intro: rrewrite_srewrite.intros)
-done
-
-lemma star_seq2:
- assumes "r3 \<leadsto>* r4"
- shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"
- using assms
-apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)
-apply(auto intro: rrewrite_srewrite.intros)
-done
-
-lemma continuous_rewrite:
- assumes "r1 \<leadsto>* AZERO"
- shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
-using assms bs1 star_seq by blast
-
-(*
-lemma continuous_rewrite2:
- assumes "r1 \<leadsto>* AONE bs"
- shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"
- using assms by (meson bs3 rrewrites.simps star_seq)
-*)
-
-lemma bsimp_aalts_simpcases:
- shows "AONE bs \<leadsto>* bsimp (AONE bs)"
- and "AZERO \<leadsto>* bsimp AZERO"
- and "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"
- by (simp_all)
-
-lemma bsimp_AALTs_rewrites:
- shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
- by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)
-
-lemma trivialbsimp_srewrites:
- "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x \<rbrakk> \<Longrightarrow> rs s\<leadsto>* (map f rs)"
- apply(induction rs)
- apply simp
- apply(simp)
- using srewrites7 by auto
-
-
-
-lemma fltsfrewrites: "rs s\<leadsto>* flts rs"
- apply(induction rs rule: flts.induct)
- apply(auto intro: rrewrite_srewrite.intros)
- apply (meson srewrites.simps srewrites1 ss5)
- using rs1 srewrites7 apply presburger
- using srewrites7 apply force
- apply (simp add: srewrites7)
- by (simp add: srewrites7)
-
-lemma bnullable0:
-shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2"
- and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 = bnullables rs2"
- apply(induct rule: rrewrite_srewrite.inducts)
- apply(auto simp add: bnullable_fuse)
- apply (meson UnCI bnullable_fuse imageI)
- using bnullable_correctness nullable_correctness by blast
-
-
-lemma rewritesnullable:
- assumes "r1 \<leadsto>* r2"
- shows "bnullable r1 = bnullable r2"
-using assms
- apply(induction r1 r2 rule: rrewrites.induct)
- apply simp
- using bnullable0(1) by auto
-
-lemma rewrite_bmkeps_aux:
- shows "r1 \<leadsto> r2 \<Longrightarrow> (bnullable r1 \<and> bnullable r2 \<Longrightarrow> bmkeps r1 = bmkeps r2)"
- and "rs1 s\<leadsto> rs2 \<Longrightarrow> (bnullables rs1 \<and> bnullables rs2 \<Longrightarrow> bmkepss rs1 = bmkepss rs2)"
-proof (induct rule: rrewrite_srewrite.inducts)
- case (bs3 bs1 bs2 r)
- then show ?case by (simp add: bmkeps_fuse)
-next
- case (bs7 bs r)
- then show ?case by (simp add: bmkeps_fuse)
-next
- case (ss3 r1 r2 rs)
- then show ?case
- using bmkepss.simps bnullable0(1) by presburger
-next
- case (ss5 bs1 rs1 rsb)
- then show ?case
- by (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
-next
- case (ss6 a1 a2 rsa rsb rsc)
- then show ?case
- by (smt (verit, best) Nil_is_append_conv bmkepss1 bmkepss2 bnullable_correctness in_set_conv_decomp list.distinct(1) list.set_intros(1) nullable_correctness set_ConsD subsetD)
-qed (auto)
-
-lemma rewrites_bmkeps:
- assumes "r1 \<leadsto>* r2" "bnullable r1"
- shows "bmkeps r1 = bmkeps r2"
- using assms
-proof(induction r1 r2 rule: rrewrites.induct)
- case (rs1 r)
- then show "bmkeps r = bmkeps r" by simp
-next
- case (rs2 r1 r2 r3)
- then have IH: "bmkeps r1 = bmkeps r2" by simp
- have a1: "bnullable r1" by fact
- have a2: "r1 \<leadsto>* r2" by fact
- have a3: "r2 \<leadsto> r3" by fact
- have a4: "bnullable r2" using a1 a2 by (simp add: rewritesnullable)
- then have "bmkeps r2 = bmkeps r3"
- using a3 bnullable0(1) rewrite_bmkeps_aux(1) by blast
- then show "bmkeps r1 = bmkeps r3" using IH by simp
-qed
-
-
-lemma rewrites_to_bsimp:
- shows "r \<leadsto>* bsimp r"
-proof (induction r rule: bsimp.induct)
- case (1 bs1 r1 r2)
- have IH1: "r1 \<leadsto>* bsimp r1" by fact
- have IH2: "r2 \<leadsto>* bsimp r2" by fact
- { assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"
- with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
- then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
- by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)
- then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto
- }
- moreover
- { assume "\<exists>bs. bsimp r1 = AONE bs"
- then obtain bs where as: "bsimp r1 = AONE bs" by blast
- with IH1 have "r1 \<leadsto>* AONE bs" by simp
- then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
- with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"
- using rewrites_fuse by (meson rrewrites_trans)
- then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp
- then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as)
- }
- moreover
- { assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)"
- then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)"
- by (simp add: bsimp_ASEQ1)
- then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2
- by (metis rrewrites_trans star_seq star_seq2)
- then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp
- }
- ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast
-next
- case (2 bs1 rs)
- have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact
- then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)
- also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites)
- also have "... s\<leadsto>* distinctWith (flts (map bsimp rs)) eq1 {}" by (simp add: ss6_stronger)
- finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {})"
- using contextrewrites0 by auto
- also have "... \<leadsto>* bsimp_AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {})"
- by (simp add: bsimp_AALTs_rewrites)
- finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp
-qed (simp_all)
-
-
-lemma to_zero_in_alt:
- shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
- by (simp add: bs1 bs10 ss3)
-
-
-
-lemma bder_fuse_list:
- shows "map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
- apply(induction rs1)
- apply(simp_all add: bder_fuse)
- done
-
-lemma map1:
- shows "(map f [a]) = [f a]"
- by (simp)
-
-lemma rewrite_preserves_bder:
- shows "r1 \<leadsto> r2 \<Longrightarrow> (bder c r1) \<leadsto>* (bder c r2)"
- and "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"
-proof(induction rule: rrewrite_srewrite.inducts)
- case (bs1 bs r2)
- then show ?case
- by (simp add: continuous_rewrite)
-next
- case (bs2 bs r1)
- then show ?case
- apply(auto)
- apply (meson bs6 contextrewrites0 rrewrite_srewrite.bs2 rs2 ss3 ss4 sss1 sss2)
- by (simp add: r_in_rstar rrewrite_srewrite.bs2)
-next
- case (bs3 bs1 bs2 r)
- then show ?case
- apply(simp)
-
- by (metis (no_types, lifting) bder_fuse bs10 bs7 fuse_append rrewrites.simps ss4 to_zero_in_alt)
-next
- case (bs4 r1 r2 bs r3)
- have as: "r1 \<leadsto> r2" by fact
- have IH: "bder c r1 \<leadsto>* bder c r2" by fact
- from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r2 r3)"
- by (metis bder.simps(5) bnullable0(1) contextrewrites1 rewrite_bmkeps_aux(1) star_seq)
-next
- case (bs5 r3 r4 bs r1)
- have as: "r3 \<leadsto> r4" by fact
- have IH: "bder c r3 \<leadsto>* bder c r4" by fact
- from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r1 r4)"
- apply(simp)
- apply(auto)
- using contextrewrites0 r_in_rstar rewrites_fuse srewrites6 srewrites7 star_seq2 apply presburger
- using star_seq2 by blast
-next
- case (bs6 bs)
- then show ?case
- using rrewrite_srewrite.bs6 by force
-next
- case (bs7 bs r)
- then show ?case
- by (simp add: bder_fuse r_in_rstar rrewrite_srewrite.bs7)
-next
- case (bs10 rs1 rs2 bs)
- then show ?case
- using contextrewrites0 by force
-next
- case ss1
- then show ?case by simp
-next
- case (ss2 rs1 rs2 r)
- then show ?case
- by (simp add: srewrites7)
-next
- case (ss3 r1 r2 rs)
- then show ?case
- by (simp add: srewrites7)
-next
- case (ss4 rs)
- then show ?case
- using rrewrite_srewrite.ss4 by fastforce
-next
- case (ss5 bs1 rs1 rsb)
- then show ?case
- apply(simp)
- using bder_fuse_list map_map rrewrite_srewrite.ss5 srewrites.simps by blast
-next
- case (ss6 a1 a2 bs rsa rsb)
- then show ?case
- apply(simp only: map_append map1)
- apply(rule srewrites_trans)
- apply(rule rs_in_rstar)
- apply(rule_tac rrewrite_srewrite.ss6)
- using Der_def der_correctness apply auto[1]
- by blast
-qed
-
-lemma rewrites_preserves_bder:
- assumes "r1 \<leadsto>* r2"
- shows "bder c r1 \<leadsto>* bder c r2"
-using assms
-apply(induction r1 r2 rule: rrewrites.induct)
-apply(simp_all add: rewrite_preserves_bder rrewrites_trans)
-done
-
-
-lemma central:
- shows "bders r s \<leadsto>* bders_simp r s"
-proof(induct s arbitrary: r rule: rev_induct)
- case Nil
- then show "bders r [] \<leadsto>* bders_simp r []" by simp
-next
- case (snoc x xs)
- have IH: "\<And>r. bders r xs \<leadsto>* bders_simp r xs" by fact
- have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
- also have "... \<leadsto>* bders (bders_simp r xs) [x]" using IH
- by (simp add: rewrites_preserves_bder)
- also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
- by (simp add: rewrites_to_bsimp)
- finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])"
- by (simp add: bders_simp_append)
-qed
-
-lemma main_aux:
- assumes "bnullable (bders r s)"
- shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
-proof -
- have "bders r s \<leadsto>* bders_simp r s" by (rule central)
- then
- show "bmkeps (bders r s) = bmkeps (bders_simp r s)" using assms
- by (rule rewrites_bmkeps)
-qed
-
-
-
-
-theorem main_blexer_simp:
- shows "blexer r s = blexer_simp r s"
- unfolding blexer_def blexer_simp_def
- by (metis central main_aux rewritesnullable)
-
-theorem blexersimp_correctness:
- shows "lexer r s = blexer_simp r s"
- using blexer_correctness main_blexer_simp by simp
-
-
-unused_thms
-
-end
diff -r f493a20feeb3 -r 04b5e904a220 thys3/ClosedForms.thy
--- a/thys3/ClosedForms.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1682 +0,0 @@
-theory ClosedForms
- imports "BasicIdentities"
-begin
-
-lemma flts_middle0:
- shows "rflts (rsa @ RZERO # rsb) = rflts (rsa @ rsb)"
- apply(induct rsa)
- apply simp
- by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
-
-
-
-lemma simp_flatten_aux0:
- shows "rsimp (RALTS rs) = rsimp (RALTS (map rsimp rs))"
- by (metis append_Nil head_one_more_simp identity_wwo0 list.simps(8) rdistinct.simps(1) rflts.simps(1) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(3) simp_flatten spawn_simp_rsimpalts)
-
-
-inductive
- hrewrite:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ h\<leadsto> _" [99, 99] 99)
-where
- "RSEQ RZERO r2 h\<leadsto> RZERO"
-| "RSEQ r1 RZERO h\<leadsto> RZERO"
-| "RSEQ RONE r h\<leadsto> r"
-| "r1 h\<leadsto> r2 \<Longrightarrow> RSEQ r1 r3 h\<leadsto> RSEQ r2 r3"
-| "r3 h\<leadsto> r4 \<Longrightarrow> RSEQ r1 r3 h\<leadsto> RSEQ r1 r4"
-| "r h\<leadsto> r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) h\<leadsto> (RALTS (rs1 @ [r'] @ rs2))"
-(*context rule for eliminating 0, alts--corresponds to the recursive call flts r::rs = r::(flts rs)*)
-| "RALTS (rsa @ [RZERO] @ rsb) h\<leadsto> RALTS (rsa @ rsb)"
-| "RALTS (rsa @ [RALTS rs1] @ rsb) h\<leadsto> RALTS (rsa @ rs1 @ rsb)"
-| "RALTS [] h\<leadsto> RZERO"
-| "RALTS [r] h\<leadsto> r"
-| "a1 = a2 \<Longrightarrow> RALTS (rsa@[a1]@rsb@[a2]@rsc) h\<leadsto> RALTS (rsa @ [a1] @ rsb @ rsc)"
-
-inductive
- hrewrites:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ h\<leadsto>* _" [100, 100] 100)
-where
- rs1[intro, simp]:"r h\<leadsto>* r"
-| rs2[intro]: "\<lbrakk>r1 h\<leadsto>* r2; r2 h\<leadsto> r3\<rbrakk> \<Longrightarrow> r1 h\<leadsto>* r3"
-
-
-lemma hr_in_rstar : "r1 h\<leadsto> r2 \<Longrightarrow> r1 h\<leadsto>* r2"
- using hrewrites.intros(1) hrewrites.intros(2) by blast
-
-lemma hreal_trans[trans]:
- assumes a1: "r1 h\<leadsto>* r2" and a2: "r2 h\<leadsto>* r3"
- shows "r1 h\<leadsto>* r3"
- using a2 a1
- apply(induct r2 r3 arbitrary: r1 rule: hrewrites.induct)
- apply(auto)
- done
-
-lemma hrewrites_seq_context:
- shows "r1 h\<leadsto>* r2 \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RSEQ r2 r3"
- apply(induct r1 r2 rule: hrewrites.induct)
- apply simp
- using hrewrite.intros(4) by blast
-
-lemma hrewrites_seq_context2:
- shows "r1 h\<leadsto>* r2 \<Longrightarrow> RSEQ r0 r1 h\<leadsto>* RSEQ r0 r2"
- apply(induct r1 r2 rule: hrewrites.induct)
- apply simp
- using hrewrite.intros(5) by blast
-
-
-lemma hrewrites_seq_contexts:
- shows "\<lbrakk>r1 h\<leadsto>* r2; r3 h\<leadsto>* r4\<rbrakk> \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RSEQ r2 r4"
- by (meson hreal_trans hrewrites_seq_context hrewrites_seq_context2)
-
-
-lemma simp_removes_duplicate1:
- shows " a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a])) = rsimp (RALTS (rsa))"
-and " rsimp (RALTS (a1 # rsa @ [a1])) = rsimp (RALTS (a1 # rsa))"
- apply(induct rsa arbitrary: a1)
- apply simp
- apply simp
- prefer 2
- apply(case_tac "a = aa")
- apply simp
- apply simp
- apply (metis Cons_eq_appendI Cons_eq_map_conv distinct_removes_duplicate_flts list.set_intros(2))
- apply (metis append_Cons append_Nil distinct_removes_duplicate_flts list.set_intros(1) list.simps(8) list.simps(9))
- by (metis (mono_tags, lifting) append_Cons distinct_removes_duplicate_flts list.set_intros(1) list.simps(8) list.simps(9) map_append rsimp.simps(2))
-
-lemma simp_removes_duplicate2:
- shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a] @ rsb)) = rsimp (RALTS (rsa @ rsb))"
- apply(induct rsb arbitrary: rsa)
- apply simp
- using distinct_removes_duplicate_flts apply auto[1]
- by (metis append.assoc head_one_more_simp rsimp.simps(2) simp_flatten simp_removes_duplicate1(1))
-
-lemma simp_removes_duplicate3:
- shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ a # rsb)) = rsimp (RALTS (rsa @ rsb))"
- using simp_removes_duplicate2 by auto
-
-(*
-lemma distinct_removes_middle4:
- shows "a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ [a] @ rsb) rset = rdistinct (rsa @ rsb) rset"
- using distinct_removes_middle(1) by fastforce
-*)
-
-(*
-lemma distinct_removes_middle_list:
- shows "\<forall>a \<in> set x. a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ x @ rsb) rset = rdistinct (rsa @ rsb) rset"
- apply(induct x)
- apply simp
- by (simp add: distinct_removes_middle3)
-*)
-
-inductive frewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f _" [10, 10] 10)
- where
- "(RZERO # rs) \<leadsto>f rs"
-| "((RALTS rs) # rsa) \<leadsto>f (rs @ rsa)"
-| "rs1 \<leadsto>f rs2 \<Longrightarrow> (r # rs1) \<leadsto>f (r # rs2)"
-
-
-inductive
- frewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f* _" [10, 10] 10)
-where
- [intro, simp]:"rs \<leadsto>f* rs"
-| [intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>f* rs3"
-
-inductive grewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g _" [10, 10] 10)
- where
- "(RZERO # rs) \<leadsto>g rs"
-| "((RALTS rs) # rsa) \<leadsto>g (rs @ rsa)"
-| "rs1 \<leadsto>g rs2 \<Longrightarrow> (r # rs1) \<leadsto>g (r # rs2)"
-| "rsa @ [a] @ rsb @ [a] @ rsc \<leadsto>g rsa @ [a] @ rsb @ rsc"
-
-lemma grewrite_variant1:
- shows "a \<in> set rs1 \<Longrightarrow> rs1 @ a # rs \<leadsto>g rs1 @ rs"
- apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
- done
-
-
-inductive
- grewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g* _" [10, 10] 10)
-where
- [intro, simp]:"rs \<leadsto>g* rs"
-| [intro]: "\<lbrakk>rs1 \<leadsto>g* rs2; rs2 \<leadsto>g rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>g* rs3"
-
-
-
-(*
-inductive
- frewrites2:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ <\<leadsto>f* _" [10, 10] 10)
-where
- [intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f* rs1\<rbrakk> \<Longrightarrow> rs1 <\<leadsto>f* rs2"
-*)
-
-lemma fr_in_rstar : "r1 \<leadsto>f r2 \<Longrightarrow> r1 \<leadsto>f* r2"
- using frewrites.intros(1) frewrites.intros(2) by blast
-
-lemma freal_trans[trans]:
- assumes a1: "r1 \<leadsto>f* r2" and a2: "r2 \<leadsto>f* r3"
- shows "r1 \<leadsto>f* r3"
- using a2 a1
- apply(induct r2 r3 arbitrary: r1 rule: frewrites.induct)
- apply(auto)
- done
-
-
-lemma many_steps_later: "\<lbrakk>r1 \<leadsto>f r2; r2 \<leadsto>f* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>f* r3"
- by (meson fr_in_rstar freal_trans)
-
-
-lemma gr_in_rstar : "r1 \<leadsto>g r2 \<Longrightarrow> r1 \<leadsto>g* r2"
- using grewrites.intros(1) grewrites.intros(2) by blast
-
-lemma greal_trans[trans]:
- assumes a1: "r1 \<leadsto>g* r2" and a2: "r2 \<leadsto>g* r3"
- shows "r1 \<leadsto>g* r3"
- using a2 a1
- apply(induct r2 r3 arbitrary: r1 rule: grewrites.induct)
- apply(auto)
- done
-
-
-lemma gmany_steps_later: "\<lbrakk>r1 \<leadsto>g r2; r2 \<leadsto>g* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>g* r3"
- by (meson gr_in_rstar greal_trans)
-
-lemma gstar_rdistinct_general:
- shows "rs1 @ rs \<leadsto>g* rs1 @ (rdistinct rs (set rs1))"
- apply(induct rs arbitrary: rs1)
- apply simp
- apply(case_tac " a \<in> set rs1")
- apply simp
- apply(subgoal_tac "rs1 @ a # rs \<leadsto>g rs1 @ rs")
- using gmany_steps_later apply auto[1]
- apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
- apply simp
- apply(drule_tac x = "rs1 @ [a]" in meta_spec)
- by simp
-
-
-lemma gstar_rdistinct:
- shows "rs \<leadsto>g* rdistinct rs {}"
- apply(induct rs)
- apply simp
- by (metis append.left_neutral empty_set gstar_rdistinct_general)
-
-
-lemma grewrite_append:
- shows "\<lbrakk> rsa \<leadsto>g rsb \<rbrakk> \<Longrightarrow> rs @ rsa \<leadsto>g rs @ rsb"
- apply(induct rs)
- apply simp+
- using grewrite.intros(3) by blast
-
-
-
-lemma frewrites_cons:
- shows "\<lbrakk> rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>f* r # rsb"
- apply(induct rsa rsb rule: frewrites.induct)
- apply simp
- using frewrite.intros(3) by blast
-
-
-lemma grewrites_cons:
- shows "\<lbrakk> rsa \<leadsto>g* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>g* r # rsb"
- apply(induct rsa rsb rule: grewrites.induct)
- apply simp
- using grewrite.intros(3) by blast
-
-
-lemma frewrites_append:
- shows " \<lbrakk>rsa \<leadsto>f* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>f* (rs @ rsb)"
- apply(induct rs)
- apply simp
- by (simp add: frewrites_cons)
-
-lemma grewrites_append:
- shows " \<lbrakk>rsa \<leadsto>g* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>g* (rs @ rsb)"
- apply(induct rs)
- apply simp
- by (simp add: grewrites_cons)
-
-
-lemma grewrites_concat:
- shows "\<lbrakk>rs1 \<leadsto>g rs2; rsa \<leadsto>g* rsb \<rbrakk> \<Longrightarrow> (rs1 @ rsa) \<leadsto>g* (rs2 @ rsb)"
- apply(induct rs1 rs2 rule: grewrite.induct)
- apply(simp)
- apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>g (rs @ rsa)")
- prefer 2
- using grewrite.intros(1) apply blast
- apply(subgoal_tac "(rs @ rsa) \<leadsto>g* (rs @ rsb)")
- using gmany_steps_later apply blast
- apply (simp add: grewrites_append)
- apply (metis append.assoc append_Cons grewrite.intros(2) grewrites_append gmany_steps_later)
- using grewrites_cons apply auto
- apply(subgoal_tac "rsaa @ a # rsba @ a # rsc @ rsa \<leadsto>g* rsaa @ a # rsba @ a # rsc @ rsb")
- using grewrite.intros(4) grewrites.intros(2) apply force
- using grewrites_append by auto
-
-
-lemma grewritess_concat:
- shows "\<lbrakk>rsa \<leadsto>g* rsb; rsc \<leadsto>g* rsd \<rbrakk> \<Longrightarrow> (rsa @ rsc) \<leadsto>g* (rsb @ rsd)"
- apply(induct rsa rsb rule: grewrites.induct)
- apply(case_tac rs)
- apply simp
- using grewrites_append apply blast
- by (meson greal_trans grewrites.simps grewrites_concat)
-
-fun alt_set:: "rrexp \<Rightarrow> rrexp set"
- where
- "alt_set (RALTS rs) = set rs \<union> \<Union> (alt_set ` (set rs))"
-| "alt_set r = {r}"
-
-
-lemma grewrite_cases_middle:
- shows "rs1 \<leadsto>g rs2 \<Longrightarrow>
-(\<exists>rsa rsb rsc. rs1 = (rsa @ [RALTS rsb] @ rsc) \<and> rs2 = (rsa @ rsb @ rsc)) \<or>
-(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc) \<or>
-(\<exists>rsa rsb rsc a. rs1 = rsa @ [a] @ rsb @ [a] @ rsc \<and> rs2 = rsa @ [a] @ rsb @ rsc)"
- apply( induct rs1 rs2 rule: grewrite.induct)
- apply simp
- apply blast
- apply (metis append_Cons append_Nil)
- apply (metis append_Cons)
- by blast
-
-
-lemma good_singleton:
- shows "good a \<and> nonalt a \<Longrightarrow> rflts [a] = [a]"
- using good.simps(1) k0b by blast
-
-
-
-
-
-
-
-lemma all_that_same_elem:
- shows "\<lbrakk> a \<in> rset; rdistinct rs {a} = []\<rbrakk>
- \<Longrightarrow> rdistinct (rs @ rsb) rset = rdistinct rsb rset"
- apply(induct rs)
- apply simp
- apply(subgoal_tac "aa = a")
- apply simp
- by (metis empty_iff insert_iff list.discI rdistinct.simps(2))
-
-lemma distinct_early_app1:
- shows "rset1 \<subseteq> rset \<Longrightarrow> rdistinct rs rset = rdistinct (rdistinct rs rset1) rset"
- apply(induct rs arbitrary: rset rset1)
- apply simp
- apply simp
- apply(case_tac "a \<in> rset1")
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp+
-
- apply blast
- apply(case_tac "a \<in> rset1")
- apply simp+
- apply(case_tac "a \<in> rset")
- apply simp
- apply (metis insert_subsetI)
- apply simp
- by (meson insert_mono)
-
-
-lemma distinct_early_app:
- shows " rdistinct (rs @ rsb) rset = rdistinct (rdistinct rs {} @ rsb) rset"
- apply(induct rsb)
- apply simp
- using distinct_early_app1 apply blast
- by (metis distinct_early_app1 distinct_once_enough empty_subsetI)
-
-
-lemma distinct_eq_interesting1:
- shows "a \<in> rset \<Longrightarrow> rdistinct (rs @ rsb) rset = rdistinct (rdistinct (a # rs) {} @ rsb) rset"
- apply(subgoal_tac "rdistinct (rdistinct (a # rs) {} @ rsb) rset = rdistinct (rdistinct rs {} @ rsb) rset")
- apply(simp only:)
- using distinct_early_app apply blast
- by (metis append_Cons distinct_early_app rdistinct.simps(2))
-
-
-
-lemma good_flatten_aux_aux1:
- shows "\<lbrakk> size rs \<ge>2;
-\<forall>r \<in> set rs. good r \<and> r \<noteq> RZERO \<and> nonalt r; \<forall>r \<in> set rsb. good r \<and> r \<noteq> RZERO \<and> nonalt r \<rbrakk>
- \<Longrightarrow> rdistinct (rs @ rsb) rset =
- rdistinct (rflts [rsimp_ALTs (rdistinct rs {})] @ rsb) rset"
- apply(induct rs arbitrary: rset)
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp
- apply(case_tac "rdistinct rs {a}")
- apply simp
- apply(subst good_singleton)
- apply force
- apply simp
- apply (meson all_that_same_elem)
- apply(subgoal_tac "rflts [rsimp_ALTs (a # rdistinct rs {a})] = a # rdistinct rs {a} ")
- prefer 2
- using k0a rsimp_ALTs.simps(3) apply presburger
- apply(simp only:)
- apply(subgoal_tac "rdistinct (rs @ rsb) rset = rdistinct ((rdistinct (a # rs) {}) @ rsb) rset ")
- apply (metis insert_absorb insert_is_Un insert_not_empty rdistinct.simps(2))
- apply (meson distinct_eq_interesting1)
- apply simp
- apply(case_tac "rdistinct rs {a}")
- prefer 2
- apply(subgoal_tac "rsimp_ALTs (a # rdistinct rs {a}) = RALTS (a # rdistinct rs {a})")
- apply(simp only:)
- apply(subgoal_tac "a # rdistinct (rs @ rsb) (insert a rset) =
- rdistinct (rflts [RALTS (a # rdistinct rs {a})] @ rsb) rset")
- apply simp
- apply (metis append_Cons distinct_early_app empty_iff insert_is_Un k0a rdistinct.simps(2))
- using rsimp_ALTs.simps(3) apply presburger
- by (metis Un_insert_left append_Cons distinct_early_app empty_iff good_singleton rdistinct.simps(2) rsimp_ALTs.simps(2) sup_bot_left)
-
-
-
-
-
-lemma good_flatten_aux_aux:
- shows "\<lbrakk>\<exists>a aa lista list. rs = a # list \<and> list = aa # lista;
-\<forall>r \<in> set rs. good r \<and> r \<noteq> RZERO \<and> nonalt r; \<forall>r \<in> set rsb. good r \<and> r \<noteq> RZERO \<and> nonalt r \<rbrakk>
- \<Longrightarrow> rdistinct (rs @ rsb) rset =
- rdistinct (rflts [rsimp_ALTs (rdistinct rs {})] @ rsb) rset"
- apply(erule exE)+
- apply(subgoal_tac "size rs \<ge> 2")
- apply (metis good_flatten_aux_aux1)
- by (simp add: Suc_leI length_Cons less_add_Suc1)
-
-
-
-lemma good_flatten_aux:
- shows " \<lbrakk>\<forall>r\<in>set rs. good r \<or> r = RZERO; \<forall>r\<in>set rsa . good r \<or> r = RZERO;
- \<forall>r\<in>set rsb. good r \<or> r = RZERO;
- rsimp (RALTS (rsa @ rs @ rsb)) = rsimp_ALTs (rdistinct (rflts (rsa @ rs @ rsb)) {});
- rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) =
- rsimp_ALTs (rdistinct (rflts (rsa @ [rsimp (RALTS rs)] @ rsb)) {});
- map rsimp rsa = rsa; map rsimp rsb = rsb; map rsimp rs = rs;
- rdistinct (rflts rsa @ rflts rs @ rflts rsb) {} =
- rdistinct (rflts rsa) {} @ rdistinct (rflts rs @ rflts rsb) (set (rflts rsa));
- rdistinct (rflts rsa @ rflts [rsimp (RALTS rs)] @ rflts rsb) {} =
- rdistinct (rflts rsa) {} @ rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))\<rbrakk>
- \<Longrightarrow> rdistinct (rflts rs @ rflts rsb) rset =
- rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) rset"
- apply simp
- apply(case_tac "rflts rs ")
- apply simp
- apply(case_tac "list")
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp
- apply (metis append.left_neutral append_Cons equals0D k0b list.set_intros(1) nonalt_flts_rd qqq1 rdistinct.simps(2))
- apply simp
- apply (metis Un_insert_left append_Cons append_Nil ex_in_conv flts_single1 insertI1 list.simps(15) nonalt_flts_rd nonazero.elims(3) qqq1 rdistinct.simps(2) sup_bot_left)
- apply(subgoal_tac "\<forall>r \<in> set (rflts rs). good r \<and> r \<noteq> RZERO \<and> nonalt r")
- prefer 2
- apply (metis Diff_empty flts3 nonalt_flts_rd qqq1 rdistinct_set_equality1)
- apply(subgoal_tac "\<forall>r \<in> set (rflts rsb). good r \<and> r \<noteq> RZERO \<and> nonalt r")
- prefer 2
- apply (metis Diff_empty flts3 good.simps(1) nonalt_flts_rd rdistinct_set_equality1)
- by (smt (verit, ccfv_threshold) good_flatten_aux_aux)
-
-
-
-
-lemma good_flatten_middle:
- shows "\<lbrakk>\<forall>r \<in> set rs. good r \<or> r = RZERO; \<forall>r \<in> set rsa. good r \<or> r = RZERO; \<forall>r \<in> set rsb. good r \<or> r = RZERO\<rbrakk> \<Longrightarrow>
-rsimp (RALTS (rsa @ rs @ rsb)) = rsimp (RALTS (rsa @ [RALTS rs] @ rsb))"
- apply(subgoal_tac "rsimp (RALTS (rsa @ rs @ rsb)) = rsimp_ALTs (rdistinct (rflts (map rsimp rsa @
-map rsimp rs @ map rsimp rsb)) {})")
- prefer 2
- apply simp
- apply(simp only:)
- apply(subgoal_tac "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) = rsimp_ALTs (rdistinct (rflts (map rsimp rsa @
-[rsimp (RALTS rs)] @ map rsimp rsb)) {})")
- prefer 2
- apply simp
- apply(simp only:)
- apply(subgoal_tac "map rsimp rsa = rsa")
- prefer 2
- apply (metis map_idI rsimp.simps(3) test)
- apply(simp only:)
- apply(subgoal_tac "map rsimp rsb = rsb")
- prefer 2
- apply (metis map_idI rsimp.simps(3) test)
- apply(simp only:)
- apply(subst k00)+
- apply(subgoal_tac "map rsimp rs = rs")
- apply(simp only:)
- prefer 2
- apply (metis map_idI rsimp.simps(3) test)
- apply(subgoal_tac "rdistinct (rflts rsa @ rflts rs @ rflts rsb) {} =
-rdistinct (rflts rsa) {} @ rdistinct (rflts rs @ rflts rsb) (set (rflts rsa))")
- apply(simp only:)
- prefer 2
- using rdistinct_concat_general apply blast
- apply(subgoal_tac "rdistinct (rflts rsa @ rflts [rsimp (RALTS rs)] @ rflts rsb) {} =
-rdistinct (rflts rsa) {} @ rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))")
- apply(simp only:)
- prefer 2
- using rdistinct_concat_general apply blast
- apply(subgoal_tac "rdistinct (rflts rs @ rflts rsb) (set (rflts rsa)) =
- rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))")
- apply presburger
- using good_flatten_aux by blast
-
-
-lemma simp_flatten3:
- shows "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) = rsimp (RALTS (rsa @ rs @ rsb))"
- apply(subgoal_tac "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) =
- rsimp (RALTS (map rsimp rsa @ [rsimp (RALTS rs)] @ map rsimp rsb)) ")
- prefer 2
- apply (metis append.left_neutral append_Cons list.simps(9) map_append simp_flatten_aux0)
- apply (simp only:)
- apply(subgoal_tac "rsimp (RALTS (rsa @ rs @ rsb)) =
-rsimp (RALTS (map rsimp rsa @ map rsimp rs @ map rsimp rsb))")
- prefer 2
- apply (metis map_append simp_flatten_aux0)
- apply(simp only:)
- apply(subgoal_tac "rsimp (RALTS (map rsimp rsa @ map rsimp rs @ map rsimp rsb)) =
- rsimp (RALTS (map rsimp rsa @ [RALTS (map rsimp rs)] @ map rsimp rsb))")
-
- apply (metis (no_types, lifting) head_one_more_simp map_append simp_flatten_aux0)
- apply(subgoal_tac "\<forall>r \<in> set (map rsimp rsa). good r \<or> r = RZERO")
- apply(subgoal_tac "\<forall>r \<in> set (map rsimp rs). good r \<or> r = RZERO")
- apply(subgoal_tac "\<forall>r \<in> set (map rsimp rsb). good r \<or> r = RZERO")
-
- using good_flatten_middle apply presburger
-
- apply (simp add: good1)
- apply (simp add: good1)
- apply (simp add: good1)
-
- done
-
-
-
-
-
-lemma grewrite_equal_rsimp:
- shows "rs1 \<leadsto>g rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
- apply(frule grewrite_cases_middle)
- apply(case_tac "(\<exists>rsa rsb rsc. rs1 = rsa @ [RALTS rsb] @ rsc \<and> rs2 = rsa @ rsb @ rsc)")
- using simp_flatten3 apply auto[1]
- apply(case_tac "(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc)")
- apply (metis (mono_tags, opaque_lifting) append_Cons append_Nil list.set_intros(1) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) simp_removes_duplicate3)
- by (smt (verit) append.assoc append_Cons append_Nil in_set_conv_decomp simp_removes_duplicate3)
-
-
-lemma grewrites_equal_rsimp:
- shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
- apply (induct rs1 rs2 rule: grewrites.induct)
- apply simp
- using grewrite_equal_rsimp by presburger
-
-
-
-lemma grewrites_last:
- shows "r # [RALTS rs] \<leadsto>g* r # rs"
- by (metis gr_in_rstar grewrite.intros(2) grewrite.intros(3) self_append_conv)
-
-lemma simp_flatten2:
- shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
- using grewrites_equal_rsimp grewrites_last by blast
-
-
-lemma frewrites_alt:
- shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> (RALT r1 r2) # rs1 \<leadsto>f* r1 # r2 # rs2"
- by (metis Cons_eq_appendI append_self_conv2 frewrite.intros(2) frewrites_cons many_steps_later)
-
-lemma early_late_der_frewrites:
- shows "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)"
- apply(induct rs)
- apply simp
- apply(case_tac a)
- apply simp+
- using frewrite.intros(1) many_steps_later apply blast
- apply(case_tac "x = x3")
- apply simp
- using frewrites_cons apply presburger
- using frewrite.intros(1) many_steps_later apply fastforce
- apply(case_tac "rnullable x41")
- apply simp+
- apply (simp add: frewrites_alt)
- apply (simp add: frewrites_cons)
- apply (simp add: frewrites_append)
- by (simp add: frewrites_cons)
-
-
-lemma gstar0:
- shows "rsa @ (rdistinct rs (set rsa)) \<leadsto>g* rsa @ (rdistinct rs (insert RZERO (set rsa)))"
- apply(induct rs arbitrary: rsa)
- apply simp
- apply(case_tac "a = RZERO")
- apply simp
-
- using gr_in_rstar grewrite.intros(1) grewrites_append apply presburger
- apply(case_tac "a \<in> set rsa")
- apply simp+
- apply(drule_tac x = "rsa @ [a]" in meta_spec)
- by simp
-
-lemma grewrite_rdistinct_aux:
- shows "rs @ rdistinct rsa rset \<leadsto>g* rs @ rdistinct rsa (rset \<union> set rs)"
- apply(induct rsa arbitrary: rs rset)
- apply simp
- apply(case_tac " a \<in> rset")
- apply simp
- apply(case_tac "a \<in> set rs")
- apply simp
- apply (metis Un_insert_left Un_insert_right gmany_steps_later grewrite_variant1 insert_absorb)
- apply simp
- apply(drule_tac x = "rs @ [a]" in meta_spec)
- by (metis Un_insert_left Un_insert_right append.assoc append.right_neutral append_Cons append_Nil insert_absorb2 list.simps(15) set_append)
-
-
-lemma flts_gstar:
- shows "rs \<leadsto>g* rflts rs"
- apply(induct rs)
- apply simp
- apply(case_tac "a = RZERO")
- apply simp
- using gmany_steps_later grewrite.intros(1) apply blast
- apply(case_tac "\<exists>rsa. a = RALTS rsa")
- apply(erule exE)
- apply simp
- apply (meson grewrite.intros(2) grewrites.simps grewrites_cons)
- by (simp add: grewrites_cons rflts_def_idiot)
-
-lemma more_distinct1:
- shows " \<lbrakk>\<And>rsb rset rset2.
- rset2 \<subseteq> set rsb \<Longrightarrow> rsb @ rdistinct rs rset \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2);
- rset2 \<subseteq> set rsb; a \<notin> rset; a \<in> rset2\<rbrakk>
- \<Longrightarrow> rsb @ a # rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)"
- apply(subgoal_tac "rsb @ a # rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (insert a rset)")
- apply(subgoal_tac "rsb @ rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)")
- apply (meson greal_trans)
- apply (metis Un_iff Un_insert_left insert_absorb)
- by (simp add: gr_in_rstar grewrite_variant1 in_mono)
-
-
-
-
-
-lemma frewrite_rd_grewrites_aux:
- shows " RALTS rs \<notin> set rsb \<Longrightarrow>
- rsb @
- RALTS rs #
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) \<leadsto>g* rflts rsb @
- rdistinct rs (set rsb) @ rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
-
- apply simp
- apply(subgoal_tac "rsb @
- RALTS rs #
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) \<leadsto>g* rsb @
- rs @
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) ")
- apply(subgoal_tac " rsb @
- rs @
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) \<leadsto>g*
- rsb @
- rdistinct rs (set rsb) @
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) ")
- apply (smt (verit, ccfv_SIG) Un_insert_left flts_gstar greal_trans grewrite_rdistinct_aux grewritess_concat inf_sup_aci(5) rdistinct_concat_general rdistinct_set_equality set_append)
- apply (metis append_assoc grewrites.intros(1) grewritess_concat gstar_rdistinct_general)
- by (simp add: gr_in_rstar grewrite.intros(2) grewrites_append)
-
-
-
-
-lemma list_dlist_union:
- shows "set rs \<subseteq> set rsb \<union> set (rdistinct rs (set rsb))"
- by (metis rdistinct_concat_general rdistinct_set_equality set_append sup_ge2)
-
-lemma r_finite1:
- shows "r = RALTS (r # rs) = False"
- apply(induct r)
- apply simp+
- apply (metis list.set_intros(1))
- by blast
-
-
-
-lemma grewrite_singleton:
- shows "[r] \<leadsto>g r # rs \<Longrightarrow> rs = []"
- apply (induct "[r]" "r # rs" rule: grewrite.induct)
- apply simp
- apply (metis r_finite1)
- using grewrite.simps apply blast
- by simp
-
-
-
-lemma concat_rdistinct_equality1:
- shows "rdistinct (rs @ rsa) rset = rdistinct rs rset @ rdistinct rsa (rset \<union> (set rs))"
- apply(induct rs arbitrary: rsa rset)
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp
- apply (simp add: insert_absorb)
- by auto
-
-
-lemma grewrites_rev_append:
- shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rs1 @ [x] \<leadsto>g* rs2 @ [x]"
- using grewritess_concat by auto
-
-lemma grewrites_inclusion:
- shows "set rs \<subseteq> set rs1 \<Longrightarrow> rs1 @ rs \<leadsto>g* rs1"
- apply(induct rs arbitrary: rs1)
- apply simp
- by (meson gmany_steps_later grewrite_variant1 list.set_intros(1) set_subset_Cons subset_code(1))
-
-lemma distinct_keeps_last:
- shows "\<lbrakk>x \<notin> rset; x \<notin> set xs \<rbrakk> \<Longrightarrow> rdistinct (xs @ [x]) rset = rdistinct xs rset @ [x]"
- by (simp add: concat_rdistinct_equality1)
-
-lemma grewrites_shape2_aux:
- shows " RALTS rs \<notin> set rsb \<Longrightarrow>
- rsb @
- rdistinct (rs @ rsa)
- (set rsb) \<leadsto>g* rsb @
- rdistinct rs (set rsb) @
- rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
- apply(subgoal_tac " rdistinct (rs @ rsa) (set rsb) = rdistinct rs (set rsb) @ rdistinct rsa (set rs \<union> set rsb)")
- apply (simp only:)
- prefer 2
- apply (simp add: Un_commute concat_rdistinct_equality1)
- apply(induct rsa arbitrary: rs rsb rule: rev_induct)
- apply simp
- apply(case_tac "x \<in> set rs")
- apply (simp add: distinct_removes_middle3)
- apply(case_tac "x = RALTS rs")
- apply simp
- apply(case_tac "x \<in> set rsb")
- apply simp
- apply (simp add: concat_rdistinct_equality1)
- apply (simp add: concat_rdistinct_equality1)
- apply simp
- apply(drule_tac x = "rs " in meta_spec)
- apply(drule_tac x = rsb in meta_spec)
- apply simp
- apply(subgoal_tac " rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) \<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct xs (insert (RALTS rs) (set rs \<union> set rsb))")
- prefer 2
- apply (simp add: concat_rdistinct_equality1)
- apply(case_tac "x \<in> set xs")
- apply simp
- apply (simp add: distinct_removes_last)
- apply(case_tac "x \<in> set rsb")
- apply (smt (verit, ccfv_threshold) Un_iff append.right_neutral concat_rdistinct_equality1 insert_is_Un rdistinct.simps(2))
- apply(subgoal_tac "rsb @ rdistinct rs (set rsb) @ rdistinct (xs @ [x]) (set rs \<union> set rsb) = rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) @ [x]")
- apply(simp only:)
- apply(case_tac "x = RALTS rs")
- apply(subgoal_tac "rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) @ [x] \<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) @ rs")
- apply(subgoal_tac "rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) @ rs \<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) ")
- apply (smt (verit, ccfv_SIG) Un_insert_left append.right_neutral concat_rdistinct_equality1 greal_trans insert_iff rdistinct.simps(2))
- apply(subgoal_tac "set rs \<subseteq> set ( rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb))")
- apply (metis append.assoc grewrites_inclusion)
- apply (metis Un_upper1 append.assoc dual_order.trans list_dlist_union set_append)
- apply (metis append_Nil2 gr_in_rstar grewrite.intros(2) grewrite_append)
- apply(subgoal_tac " rsb @ rdistinct rs (set rsb) @ rdistinct (xs @ [x]) (insert (RALTS rs) (set rs \<union> set rsb)) = rsb @ rdistinct rs (set rsb) @ rdistinct (xs) (insert (RALTS rs) (set rs \<union> set rsb)) @ [x]")
- apply(simp only:)
- apply (metis append.assoc grewrites_rev_append)
- apply (simp add: insert_absorb)
- apply (simp add: distinct_keeps_last)+
- done
-
-lemma grewrites_shape2:
- shows " RALTS rs \<notin> set rsb \<Longrightarrow>
- rsb @
- rdistinct (rs @ rsa)
- (set rsb) \<leadsto>g* rflts rsb @
- rdistinct rs (set rsb) @
- rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
- apply (meson flts_gstar greal_trans grewrites.simps grewrites_shape2_aux grewritess_concat)
- done
-
-lemma rdistinct_add_acc:
- shows "rset2 \<subseteq> set rsb \<Longrightarrow> rsb @ rdistinct rs rset \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)"
- apply(induct rs arbitrary: rsb rset rset2)
- apply simp
- apply (case_tac "a \<in> rset")
- apply simp
- apply(case_tac "a \<in> rset2")
- apply simp
- apply (simp add: more_distinct1)
- apply simp
- apply(drule_tac x = "rsb @ [a]" in meta_spec)
- by (metis Un_insert_left append.assoc append_Cons append_Nil set_append sup.coboundedI1)
-
-
-lemma frewrite_fun1:
- shows " RALTS rs \<in> set rsb \<Longrightarrow>
- rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb \<union> set rs)"
- apply(subgoal_tac "rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb)")
- apply(subgoal_tac " set rs \<subseteq> set (rflts rsb)")
- prefer 2
- using spilled_alts_contained apply blast
- apply(subgoal_tac "rflts rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb \<union> set rs)")
- using greal_trans apply blast
- using rdistinct_add_acc apply presburger
- using flts_gstar grewritess_concat by auto
-
-lemma frewrite_rd_grewrites:
- shows "rs1 \<leadsto>f rs2 \<Longrightarrow>
-\<exists>rs3. (rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3) "
- apply(induct rs1 rs2 arbitrary: rs rule: frewrite.induct)
- apply(rule_tac x = "rsa @ (rdistinct rs ({RZERO} \<union> set rsa))" in exI)
- apply(rule conjI)
- apply(case_tac "RZERO \<in> set rsa")
- apply simp+
- using gstar0 apply fastforce
- apply (simp add: gr_in_rstar grewrite.intros(1) grewrites_append)
- apply (simp add: gstar0)
- prefer 2
- apply(case_tac "r \<in> set rs")
- apply simp
- apply(drule_tac x = "rs @ [r]" in meta_spec)
- apply(erule exE)
- apply(rule_tac x = "rs3" in exI)
- apply simp
- apply(case_tac "RALTS rs \<in> set rsb")
- apply simp
- apply(rule_tac x = "rflts rsb @ rdistinct rsa (set rsb \<union> set rs)" in exI)
- apply(rule conjI)
- using frewrite_fun1 apply force
- apply (metis frewrite_fun1 rdistinct_concat sup_ge2)
- apply(simp)
- apply(rule_tac x =
- "rflts rsb @
- rdistinct rs (set rsb) @
- rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})" in exI)
- apply(rule conjI)
- prefer 2
- using grewrites_shape2 apply force
- using frewrite_rd_grewrites_aux by blast
-
-
-lemma frewrite_simpeq2:
- shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
- apply(subgoal_tac "\<exists> rs3. (rdistinct rs1 {} \<leadsto>g* rs3) \<and> (rdistinct rs2 {} \<leadsto>g* rs3)")
- using grewrites_equal_rsimp apply fastforce
- by (metis append_self_conv2 frewrite_rd_grewrites list.set(1))
-
-
-
-
-(*a more refined notion of h\<leadsto>* is needed,
-this lemma fails when rs1 contains some RALTS rs where elements
-of rs appear in later parts of rs1, which will be picked up by rs2
-and deduplicated*)
-lemma frewrites_simpeq:
- shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
- rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS ( rdistinct rs2 {})) "
- apply(induct rs1 rs2 rule: frewrites.induct)
- apply simp
- using frewrite_simpeq2 by presburger
-
-
-lemma frewrite_single_step:
- shows " rs2 \<leadsto>f rs3 \<Longrightarrow> rsimp (RALTS rs2) = rsimp (RALTS rs3)"
- apply(induct rs2 rs3 rule: frewrite.induct)
- apply simp
- using simp_flatten apply blast
- by (metis (no_types, opaque_lifting) list.simps(9) rsimp.simps(2) simp_flatten2)
-
-lemma grewrite_simpalts:
- shows " rs2 \<leadsto>g rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
- apply(induct rs2 rs3 rule : grewrite.induct)
- using identity_wwo0 apply presburger
- apply (metis frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_flatten)
- apply (smt (verit, ccfv_SIG) gmany_steps_later grewrites.intros(1) grewrites_cons grewrites_equal_rsimp identity_wwo0 rsimp_ALTs.simps(3))
- apply simp
- apply(subst rsimp_alts_equal)
- apply(case_tac "rsa = [] \<and> rsb = [] \<and> rsc = []")
- apply(subgoal_tac "rsa @ a # rsb @ rsc = [a]")
- apply (simp only:)
- apply (metis append_Nil frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_removes_duplicate1(2))
- apply simp
- by (smt (verit, best) append.assoc append_Cons frewrite.intros(1) frewrite_single_step identity_wwo0 in_set_conv_decomp rsimp_ALTs.simps(3) simp_removes_duplicate3)
-
-
-lemma grewrites_simpalts:
- shows " rs2 \<leadsto>g* rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
- apply(induct rs2 rs3 rule: grewrites.induct)
- apply simp
- using grewrite_simpalts by presburger
-
-
-lemma simp_der_flts:
- shows "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {})) =
- rsimp (RALTS (rdistinct (rflts (map (rder x) rs)) {}))"
- apply(subgoal_tac "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)")
- using frewrites_simpeq apply presburger
- using early_late_der_frewrites by auto
-
-
-lemma simp_der_pierce_flts_prelim:
- shows "rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts rs)) {}))
- = rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {}))"
- by (metis append.right_neutral grewrite.intros(2) grewrite_simpalts rsimp_ALTs.simps(2) simp_der_flts)
-
-
-lemma basic_regex_property1:
- shows "rnullable r \<Longrightarrow> rsimp r \<noteq> RZERO"
- apply(induct r rule: rsimp.induct)
- apply(auto)
- apply (metis idiot idiot2 rrexp.distinct(5))
- by (metis der_simp_nullability rnullable.simps(1) rnullable.simps(4) rsimp.simps(2))
-
-
-lemma inside_simp_seq_nullable:
- shows
-"\<And>r1 r2.
- \<lbrakk>rsimp (rder x (rsimp r1)) = rsimp (rder x r1); rsimp (rder x (rsimp r2)) = rsimp (rder x r2);
- rnullable r1\<rbrakk>
- \<Longrightarrow> rsimp (rder x (rsimp_SEQ (rsimp r1) (rsimp r2))) =
- rsimp_ALTs (rdistinct (rflts [rsimp_SEQ (rsimp (rder x r1)) (rsimp r2), rsimp (rder x r2)]) {})"
- apply(case_tac "rsimp r1 = RONE")
- apply(simp)
- apply(subst basic_rsimp_SEQ_property1)
- apply (simp add: idem_after_simp1)
- apply(case_tac "rsimp r1 = RZERO")
-
- using basic_regex_property1 apply blast
- apply(case_tac "rsimp r2 = RZERO")
-
- apply (simp add: basic_rsimp_SEQ_property3)
- apply(subst idiot2)
- apply simp+
- apply(subgoal_tac "rnullable (rsimp r1)")
- apply simp
- using rsimp_idem apply presburger
- using der_simp_nullability by presburger
-
-
-
-lemma grewrite_ralts:
- shows "rs \<leadsto>g rs' \<Longrightarrow> RALTS rs h\<leadsto>* RALTS rs'"
- by (smt (verit) grewrite_cases_middle hr_in_rstar hrewrite.intros(11) hrewrite.intros(7) hrewrite.intros(8))
-
-lemma grewrites_ralts:
- shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs h\<leadsto>* RALTS rs'"
- apply(induct rule: grewrites.induct)
- apply simp
- using grewrite_ralts hreal_trans by blast
-
-
-lemma distinct_grewrites_subgoal1:
- shows "
- \<lbrakk>rs1 \<leadsto>g* [a]; RALTS rs1 h\<leadsto>* a; [a] \<leadsto>g rs3\<rbrakk> \<Longrightarrow> RALTS rs1 h\<leadsto>* rsimp_ALTs rs3"
- apply(subgoal_tac "RALTS rs1 h\<leadsto>* RALTS rs3")
- apply (metis hrewrite.intros(10) hrewrite.intros(9) rs2 rsimp_ALTs.cases rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
- apply(subgoal_tac "rs1 \<leadsto>g* rs3")
- using grewrites_ralts apply blast
- using grewrites.intros(2) by presburger
-
-lemma grewrites_ralts_rsimpalts:
- shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs h\<leadsto>* rsimp_ALTs rs' "
- apply(induct rs rs' rule: grewrites.induct)
- apply(case_tac rs)
- using hrewrite.intros(9) apply force
- apply(case_tac list)
- apply simp
- using hr_in_rstar hrewrite.intros(10) rsimp_ALTs.simps(2) apply presburger
- apply simp
- apply(case_tac rs2)
- apply simp
- apply (metis grewrite.intros(3) grewrite_singleton rsimp_ALTs.simps(1))
- apply(case_tac list)
- apply(simp)
- using distinct_grewrites_subgoal1 apply blast
- apply simp
- apply(case_tac rs3)
- apply simp
- using grewrites_ralts hrewrite.intros(9) apply blast
- by (metis (no_types, opaque_lifting) grewrite_ralts hr_in_rstar hreal_trans hrewrite.intros(10) neq_Nil_conv rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
-
-lemma hrewrites_alts:
- shows " r h\<leadsto>* r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) h\<leadsto>* (RALTS (rs1 @ [r'] @ rs2))"
- apply(induct r r' rule: hrewrites.induct)
- apply simp
- using hrewrite.intros(6) by blast
-
-inductive
- srewritescf:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" (" _ scf\<leadsto>* _" [100, 100] 100)
-where
- ss1: "[] scf\<leadsto>* []"
-| ss2: "\<lbrakk>r h\<leadsto>* r'; rs scf\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (r#rs) scf\<leadsto>* (r'#rs')"
-
-
-lemma srewritescf_alt: "rs1 scf\<leadsto>* rs2 \<Longrightarrow> (RALTS (rs@rs1)) h\<leadsto>* (RALTS (rs@rs2))"
-
- apply(induct rs1 rs2 arbitrary: rs rule: srewritescf.induct)
- apply(rule rs1)
- apply(drule_tac x = "rsa@[r']" in meta_spec)
- apply simp
- apply(rule hreal_trans)
- prefer 2
- apply(assumption)
- apply(drule hrewrites_alts)
- by auto
-
-
-corollary srewritescf_alt1:
- assumes "rs1 scf\<leadsto>* rs2"
- shows "RALTS rs1 h\<leadsto>* RALTS rs2"
- using assms
- by (metis append_Nil srewritescf_alt)
-
-
-
-
-lemma trivialrsimp_srewrites:
- "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x h\<leadsto>* f x \<rbrakk> \<Longrightarrow> rs scf\<leadsto>* (map f rs)"
-
- apply(induction rs)
- apply simp
- apply(rule ss1)
- by (metis insert_iff list.simps(15) list.simps(9) srewritescf.simps)
-
-lemma hrewrites_list:
- shows
-" (\<And>xa. xa \<in> set x \<Longrightarrow> xa h\<leadsto>* rsimp xa) \<Longrightarrow> RALTS x h\<leadsto>* RALTS (map rsimp x)"
- apply(induct x)
- apply(simp)+
- by (simp add: srewritescf_alt1 ss2 trivialrsimp_srewrites)
-(* apply(subgoal_tac "RALTS x h\<leadsto>* RALTS (map rsimp x)")*)
-
-
-lemma hrewrite_simpeq:
- shows "r1 h\<leadsto> r2 \<Longrightarrow> rsimp r1 = rsimp r2"
- apply(induct rule: hrewrite.induct)
- apply simp+
- apply (simp add: basic_rsimp_SEQ_property3)
- apply (simp add: basic_rsimp_SEQ_property1)
- using rsimp.simps(1) apply presburger
- apply simp+
- using flts_middle0 apply force
-
-
- using simp_flatten3 apply presburger
-
- apply simp+
- apply (simp add: idem_after_simp1)
- using grewrite.intros(4) grewrite_equal_rsimp by presburger
-
-lemma hrewrites_simpeq:
- shows "r1 h\<leadsto>* r2 \<Longrightarrow> rsimp r1 = rsimp r2"
- apply(induct rule: hrewrites.induct)
- apply simp
- apply(subgoal_tac "rsimp r2 = rsimp r3")
- apply auto[1]
- using hrewrite_simpeq by presburger
-
-
-
-lemma simp_hrewrites:
- shows "r1 h\<leadsto>* rsimp r1"
- apply(induct r1)
- apply simp+
- apply(case_tac "rsimp r11 = RONE")
- apply simp
- apply(subst basic_rsimp_SEQ_property1)
- apply(subgoal_tac "RSEQ r11 r12 h\<leadsto>* RSEQ RONE r12")
- using hreal_trans hrewrite.intros(3) apply blast
- using hrewrites_seq_context apply presburger
- apply(case_tac "rsimp r11 = RZERO")
- apply simp
- using hrewrite.intros(1) hrewrites_seq_context apply blast
- apply(case_tac "rsimp r12 = RZERO")
- apply simp
- apply(subst basic_rsimp_SEQ_property3)
- apply (meson hrewrite.intros(2) hrewrites.simps hrewrites_seq_context2)
- apply(subst idiot2)
- apply simp+
- using hrewrites_seq_contexts apply presburger
- apply simp
- apply(subgoal_tac "RALTS x h\<leadsto>* RALTS (map rsimp x)")
- apply(subgoal_tac "RALTS (map rsimp x) h\<leadsto>* rsimp_ALTs (rdistinct (rflts (map rsimp x)) {}) ")
- using hreal_trans apply blast
- apply (meson flts_gstar greal_trans grewrites_ralts_rsimpalts gstar_rdistinct)
-
- apply (simp add: grewrites_ralts hrewrites_list)
- by simp
-
-lemma interleave_aux1:
- shows " RALT (RSEQ RZERO r1) r h\<leadsto>* r"
- apply(subgoal_tac "RSEQ RZERO r1 h\<leadsto>* RZERO")
- apply(subgoal_tac "RALT (RSEQ RZERO r1) r h\<leadsto>* RALT RZERO r")
- apply (meson grewrite.intros(1) grewrite_ralts hreal_trans hrewrite.intros(10) hrewrites.simps)
- using rs1 srewritescf_alt1 ss1 ss2 apply presburger
- by (simp add: hr_in_rstar hrewrite.intros(1))
-
-
-
-lemma rnullable_hrewrite:
- shows "r1 h\<leadsto> r2 \<Longrightarrow> rnullable r1 = rnullable r2"
- apply(induct rule: hrewrite.induct)
- apply simp+
- apply blast
- apply simp+
- done
-
-
-lemma interleave1:
- shows "r h\<leadsto> r' \<Longrightarrow> rder c r h\<leadsto>* rder c r'"
- apply(induct r r' rule: hrewrite.induct)
- apply (simp add: hr_in_rstar hrewrite.intros(1))
- apply (metis (no_types, lifting) basic_rsimp_SEQ_property3 list.simps(8) list.simps(9) rder.simps(1) rder.simps(5) rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(1) rsimp.simps(2) rsimp.simps(3) rsimp_ALTs.simps(1) simp_hrewrites)
- apply simp
- apply(subst interleave_aux1)
- apply simp
- apply(case_tac "rnullable r1")
- apply simp
-
- apply (simp add: hrewrites_seq_context rnullable_hrewrite srewritescf_alt1 ss1 ss2)
-
- apply (simp add: hrewrites_seq_context rnullable_hrewrite)
- apply(case_tac "rnullable r1")
- apply simp
-
- using hr_in_rstar hrewrites_seq_context2 srewritescf_alt1 ss1 ss2 apply presburger
- apply simp
- using hr_in_rstar hrewrites_seq_context2 apply blast
- apply simp
-
- using hrewrites_alts apply auto[1]
- apply simp
- using grewrite.intros(1) grewrite_append grewrite_ralts apply auto[1]
- apply simp
- apply (simp add: grewrite.intros(2) grewrite_append grewrite_ralts)
- apply (simp add: hr_in_rstar hrewrite.intros(9))
- apply (simp add: hr_in_rstar hrewrite.intros(10))
- apply simp
- using hrewrite.intros(11) by auto
-
-lemma interleave_star1:
- shows "r h\<leadsto>* r' \<Longrightarrow> rder c r h\<leadsto>* rder c r'"
- apply(induct rule : hrewrites.induct)
- apply simp
- by (meson hreal_trans interleave1)
-
-
-
-lemma inside_simp_removal:
- shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
- apply(induct r)
- apply simp+
- apply(case_tac "rnullable r1")
- apply simp
-
- using inside_simp_seq_nullable apply blast
- apply simp
- apply (smt (verit, del_insts) idiot2 basic_rsimp_SEQ_property3 der_simp_nullability rder.simps(1) rder.simps(5) rnullable.simps(2) rsimp.simps(1) rsimp_SEQ.simps(1) rsimp_idem)
- apply(subgoal_tac "rder x (RALTS xa) h\<leadsto>* rder x (rsimp (RALTS xa))")
- using hrewrites_simpeq apply presburger
- using interleave_star1 simp_hrewrites apply presburger
- by simp
-
-
-
-
-lemma rders_simp_same_simpders:
- shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
- apply(induct s rule: rev_induct)
- apply simp
- apply(case_tac "xs = []")
- apply simp
- apply(simp add: rders_append rders_simp_append)
- using inside_simp_removal by blast
-
-
-
-
-lemma distinct_der:
- shows "rsimp (rsimp_ALTs (map (rder x) (rdistinct rs {}))) =
- rsimp (rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
- by (metis grewrites_simpalts gstar_rdistinct inside_simp_removal rder_rsimp_ALTs_commute)
-
-
-
-
-
-lemma rders_simp_lambda:
- shows " rsimp \<circ> rder x \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r (xs @ [x]))"
- using rders_simp_append by auto
-
-lemma rders_simp_nonempty_simped:
- shows "xs \<noteq> [] \<Longrightarrow> rsimp \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r xs)"
- using rders_simp_same_simpders rsimp_idem by auto
-
-lemma repeated_altssimp:
- shows "\<forall>r \<in> set rs. rsimp r = r \<Longrightarrow> rsimp (rsimp_ALTs (rdistinct (rflts rs) {})) =
- rsimp_ALTs (rdistinct (rflts rs) {})"
- by (metis map_idI rsimp.simps(2) rsimp_idem)
-
-
-
-lemma alts_closed_form:
- shows "rsimp (rders_simp (RALTS rs) s) = rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
- apply(induct s rule: rev_induct)
- apply simp
- apply simp
- apply(subst rders_simp_append)
- apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) =
- rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
- prefer 2
- apply (metis inside_simp_removal rders_simp_one_char)
- apply(simp only: )
- apply(subst rders_simp_one_char)
- apply(subst rsimp_idem)
- apply(subgoal_tac "rsimp (rder x (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {}))) =
- rsimp ((rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))) ")
- prefer 2
- using rder_rsimp_ALTs_commute apply presburger
- apply(simp only:)
- apply(subgoal_tac "rsimp (rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))
-= rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
- prefer 2
-
- using distinct_der apply presburger
- apply(simp only:)
- apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
- rsimp (rsimp_ALTs (rdistinct ( (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)))) {}))")
- apply(simp only:)
- apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
- rsimp (rsimp_ALTs (rdistinct (rflts ( (map (rsimp \<circ> (rder x) \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
- apply(simp only:)
- apply(subst rders_simp_lambda)
- apply(subst rders_simp_nonempty_simped)
- apply simp
- apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (xs @ [x])) rs). rsimp r = r")
- prefer 2
- apply (simp add: rders_simp_same_simpders rsimp_idem)
- apply(subst repeated_altssimp)
- apply simp
- apply fastforce
- apply (metis inside_simp_removal list.map_comp rder.simps(4) rsimp.simps(2) rsimp_idem)
- using simp_der_pierce_flts_prelim by blast
-
-
-lemma alts_closed_form_variant:
- shows "s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s = rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
- by (metis alts_closed_form comp_apply rders_simp_nonempty_simped)
-
-
-lemma rsimp_seq_equal1:
- shows "rsimp_SEQ (rsimp r1) (rsimp r2) = rsimp_ALTs (rdistinct (rflts [rsimp_SEQ (rsimp r1) (rsimp r2)]) {})"
- by (metis idem_after_simp1 rsimp.simps(1))
-
-
-fun sflat_aux :: "rrexp \<Rightarrow> rrexp list " where
- "sflat_aux (RALTS (r # rs)) = sflat_aux r @ rs"
-| "sflat_aux (RALTS []) = []"
-| "sflat_aux r = [r]"
-
-
-fun sflat :: "rrexp \<Rightarrow> rrexp" where
- "sflat (RALTS (r # [])) = r"
-| "sflat (RALTS (r # rs)) = RALTS (sflat_aux r @ rs)"
-| "sflat r = r"
-
-inductive created_by_seq:: "rrexp \<Rightarrow> bool" where
- "created_by_seq (RSEQ r1 r2) "
-| "created_by_seq r1 \<Longrightarrow> created_by_seq (RALT r1 r2)"
-
-lemma seq_ders_shape1:
- shows "\<forall>r1 r2. \<exists>r3 r4. (rders (RSEQ r1 r2) s) = RSEQ r3 r4 \<or> (rders (RSEQ r1 r2) s) = RALT r3 r4"
- apply(induct s rule: rev_induct)
- apply auto[1]
- apply(rule allI)+
- apply(subst rders_append)+
- apply(subgoal_tac " \<exists>r3 r4. rders (RSEQ r1 r2) xs = RSEQ r3 r4 \<or> rders (RSEQ r1 r2) xs = RALT r3 r4 ")
- apply(erule exE)+
- apply(erule disjE)
- apply simp+
- done
-
-lemma created_by_seq_der:
- shows "created_by_seq r \<Longrightarrow> created_by_seq (rder c r)"
- apply(induct r)
- apply simp+
-
- using created_by_seq.cases apply blast
-
- apply (meson created_by_seq.cases rrexp.distinct(19) rrexp.distinct(21))
- apply (metis created_by_seq.simps rder.simps(5))
- apply (smt (verit, ccfv_threshold) created_by_seq.simps list.set_intros(1) list.simps(8) list.simps(9) rder.simps(4) rrexp.distinct(25) rrexp.inject(3))
- using created_by_seq.intros(1) by force
-
-lemma createdbyseq_left_creatable:
- shows "created_by_seq (RALT r1 r2) \<Longrightarrow> created_by_seq r1"
- using created_by_seq.cases by blast
-
-
-
-lemma recursively_derseq:
- shows " created_by_seq (rders (RSEQ r1 r2) s)"
- apply(induct s rule: rev_induct)
- apply simp
- using created_by_seq.intros(1) apply force
- apply(subgoal_tac "created_by_seq (rders (RSEQ r1 r2) (xs @ [x]))")
- apply blast
- apply(subst rders_append)
- apply(subgoal_tac "\<exists>r3 r4. rders (RSEQ r1 r2) xs = RSEQ r3 r4 \<or>
- rders (RSEQ r1 r2) xs = RALT r3 r4")
- prefer 2
- using seq_ders_shape1 apply presburger
- apply(erule exE)+
- apply(erule disjE)
- apply(subgoal_tac "created_by_seq (rders (RSEQ r3 r4) [x])")
- apply presburger
- apply simp
- using created_by_seq.intros(1) created_by_seq.intros(2) apply presburger
- apply simp
- apply(subgoal_tac "created_by_seq r3")
- prefer 2
- using createdbyseq_left_creatable apply blast
- using created_by_seq.intros(2) created_by_seq_der by blast
-
-
-lemma recursively_derseq1:
- shows "r = rders (RSEQ r1 r2) s \<Longrightarrow> created_by_seq r"
- using recursively_derseq by blast
-
-
-lemma sfau_head:
- shows " created_by_seq r \<Longrightarrow> \<exists>ra rb rs. sflat_aux r = RSEQ ra rb # rs"
- apply(induction r rule: created_by_seq.induct)
- apply simp
- by fastforce
-
-
-lemma vsuf_prop1:
- shows "vsuf (xs @ [x]) r = (if (rnullable (rders r xs))
- then [x] # (map (\<lambda>s. s @ [x]) (vsuf xs r) )
- else (map (\<lambda>s. s @ [x]) (vsuf xs r)) )
- "
- apply(induct xs arbitrary: r)
- apply simp
- apply(case_tac "rnullable r")
- apply simp
- apply simp
- done
-
-fun breakHead :: "rrexp list \<Rightarrow> rrexp list" where
- "breakHead [] = [] "
-| "breakHead (RALT r1 r2 # rs) = r1 # r2 # rs"
-| "breakHead (r # rs) = r # rs"
-
-
-lemma sfau_idem_der:
- shows "created_by_seq r \<Longrightarrow> sflat_aux (rder c r) = breakHead (map (rder c) (sflat_aux r))"
- apply(induct rule: created_by_seq.induct)
- apply simp+
- using sfau_head by fastforce
-
-lemma vsuf_compose1:
- shows " \<not> rnullable (rders r1 xs)
- \<Longrightarrow> map (rder x \<circ> rders r2) (vsuf xs r1) = map (rders r2) (vsuf (xs @ [x]) r1)"
- apply(subst vsuf_prop1)
- apply simp
- by (simp add: rders_append)
-
-
-
-
-lemma seq_sfau0:
- shows "sflat_aux (rders (RSEQ r1 r2) s) = (RSEQ (rders r1 s) r2) #
- (map (rders r2) (vsuf s r1)) "
- apply(induct s rule: rev_induct)
- apply simp
- apply(subst rders_append)+
- apply(subgoal_tac "created_by_seq (rders (RSEQ r1 r2) xs)")
- prefer 2
- using recursively_derseq1 apply blast
- apply simp
- apply(subst sfau_idem_der)
-
- apply blast
- apply(case_tac "rnullable (rders r1 xs)")
- apply simp
- apply(subst vsuf_prop1)
- apply simp
- apply (simp add: rders_append)
- apply simp
- using vsuf_compose1 by blast
-
-
-
-
-
-
-
-
-
-thm sflat.elims
-
-
-
-
-
-lemma sflat_rsimpeq:
- shows "created_by_seq r1 \<Longrightarrow> sflat_aux r1 = rs \<Longrightarrow> rsimp r1 = rsimp (RALTS rs)"
- apply(induct r1 arbitrary: rs rule: created_by_seq.induct)
- apply simp
- using rsimp_seq_equal1 apply force
- by (metis head_one_more_simp rsimp.simps(2) sflat_aux.simps(1) simp_flatten)
-
-
-
-lemma seq_closed_form_general:
- shows "rsimp (rders (RSEQ r1 r2) s) =
-rsimp ( (RALTS ( (RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1))))))"
- apply(case_tac "s \<noteq> []")
- apply(subgoal_tac "created_by_seq (rders (RSEQ r1 r2) s)")
- apply(subgoal_tac "sflat_aux (rders (RSEQ r1 r2) s) = RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1))")
- using sflat_rsimpeq apply blast
- apply (simp add: seq_sfau0)
- using recursively_derseq1 apply blast
- apply simp
- by (metis idem_after_simp1 rsimp.simps(1))
-
-lemma seq_closed_form_aux1a:
- shows "rsimp (RALTS (RSEQ (rders r1 s) r2 # rs)) =
- rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # rs))"
- by (metis head_one_more_simp rders.simps(1) rders_simp.simps(1) rders_simp_same_simpders rsimp.simps(1) rsimp_idem simp_flatten_aux0)
-
-
-lemma seq_closed_form_aux1:
- shows "rsimp (RALTS (RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1)))) =
- rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # (map (rders r2) (vsuf s r1))))"
- by (smt (verit, best) list.simps(9) rders.simps(1) rders_simp.simps(1) rders_simp_same_simpders rsimp.simps(1) rsimp.simps(2) rsimp_idem)
-
-lemma add_simp_to_rest:
- shows "rsimp (RALTS (r # rs)) = rsimp (RALTS (r # map rsimp rs))"
- by (metis append_Nil2 grewrite.intros(2) grewrite_simpalts head_one_more_simp list.simps(9) rsimp_ALTs.simps(2) spawn_simp_rsimpalts)
-
-lemma rsimp_compose_der2:
- shows "\<forall>s \<in> set ss. s \<noteq> [] \<Longrightarrow> map rsimp (map (rders r) ss) = map (\<lambda>s. (rders_simp r s)) ss"
- by (simp add: rders_simp_same_simpders)
-
-lemma vsuf_nonempty:
- shows "\<forall>s \<in> set ( vsuf s1 r). s \<noteq> []"
- apply(induct s1 arbitrary: r)
- apply simp
- apply simp
- done
-
-
-
-lemma seq_closed_form_aux2:
- shows "s \<noteq> [] \<Longrightarrow> rsimp ( (RALTS ( (RSEQ (rders_simp r1 s) r2 # (map (rders r2) (vsuf s r1)))))) =
- rsimp ( (RALTS ( (RSEQ (rders_simp r1 s) r2 # (map (rders_simp r2) (vsuf s r1))))))"
-
- by (metis add_simp_to_rest rsimp_compose_der2 vsuf_nonempty)
-
-
-lemma seq_closed_form:
- shows "rsimp (rders_simp (RSEQ r1 r2) s) =
- rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
-proof (cases s)
- case Nil
- then show ?thesis
- by (simp add: rsimp_seq_equal1[symmetric])
-next
- case (Cons a list)
- have "rsimp (rders_simp (RSEQ r1 r2) s) = rsimp (rsimp (rders (RSEQ r1 r2) s))"
- using local.Cons by (subst rders_simp_same_simpders)(simp_all)
- also have "... = rsimp (rders (RSEQ r1 r2) s)"
- by (simp add: rsimp_idem)
- also have "... = rsimp (RALTS (RSEQ (rders r1 s) r2 # map (rders r2) (vsuf s r1)))"
- using seq_closed_form_general by blast
- also have "... = rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders r2) (vsuf s r1)))"
- by (simp only: seq_closed_form_aux1)
- also have "... = rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)))"
- using local.Cons by (subst seq_closed_form_aux2)(simp_all)
- finally show ?thesis .
-qed
-
-lemma q: "s \<noteq> [] \<Longrightarrow> rders_simp (RSEQ r1 r2) s = rsimp (rders_simp (RSEQ r1 r2) s)"
- using rders_simp_same_simpders rsimp_idem by presburger
-
-
-lemma seq_closed_form_variant:
- assumes "s \<noteq> []"
- shows "rders_simp (RSEQ r1 r2) s =
- rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # (map (rders_simp r2) (vsuf s r1))))"
- using assms q seq_closed_form by force
-
-
-fun hflat_aux :: "rrexp \<Rightarrow> rrexp list" where
- "hflat_aux (RALT r1 r2) = hflat_aux r1 @ hflat_aux r2"
-| "hflat_aux r = [r]"
-
-
-fun hflat :: "rrexp \<Rightarrow> rrexp" where
- "hflat (RALT r1 r2) = RALTS ((hflat_aux r1) @ (hflat_aux r2))"
-| "hflat r = r"
-
-inductive created_by_star :: "rrexp \<Rightarrow> bool" where
- "created_by_star (RSEQ ra (RSTAR rb))"
-| "\<lbrakk>created_by_star r1; created_by_star r2\<rbrakk> \<Longrightarrow> created_by_star (RALT r1 r2)"
-
-fun hElem :: "rrexp \<Rightarrow> rrexp list" where
- "hElem (RALT r1 r2) = (hElem r1 ) @ (hElem r2)"
-| "hElem r = [r]"
-
-
-
-
-lemma cbs_ders_cbs:
- shows "created_by_star r \<Longrightarrow> created_by_star (rder c r)"
- apply(induct r rule: created_by_star.induct)
- apply simp
- using created_by_star.intros(1) created_by_star.intros(2) apply auto[1]
- by (metis (mono_tags, lifting) created_by_star.simps list.simps(8) list.simps(9) rder.simps(4))
-
-lemma star_ders_cbs:
- shows "created_by_star (rders (RSEQ r1 (RSTAR r2)) s)"
- apply(induct s rule: rev_induct)
- apply simp
- apply (simp add: created_by_star.intros(1))
- apply(subst rders_append)
- apply simp
- using cbs_ders_cbs by auto
-
-(*
-lemma created_by_star_cases:
- shows "created_by_star r \<Longrightarrow> \<exists>ra rb. (r = RALT ra rb \<and> created_by_star ra \<and> created_by_star rb) \<or> r = RSEQ ra rb "
- by (meson created_by_star.cases)
-*)
-
-
-lemma hfau_pushin:
- shows "created_by_star r \<Longrightarrow> hflat_aux (rder c r) = concat (map hElem (map (rder c) (hflat_aux r)))"
- apply(induct r rule: created_by_star.induct)
- apply simp
- apply(subgoal_tac "created_by_star (rder c r1)")
- prefer 2
- apply(subgoal_tac "created_by_star (rder c r2)")
- using cbs_ders_cbs apply blast
- using cbs_ders_cbs apply auto[1]
- apply simp
- done
-
-lemma stupdate_induct1:
- shows " concat (map (hElem \<circ> (rder x \<circ> (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)))) Ss) =
- map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_update x r0 Ss)"
- apply(induct Ss)
- apply simp+
- by (simp add: rders_append)
-
-
-
-lemma stupdates_join_general:
- shows "concat (map hElem (map (rder x) (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates xs r0 Ss)))) =
- map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates (xs @ [x]) r0 Ss)"
- apply(induct xs arbitrary: Ss)
- apply (simp)
- prefer 2
- apply auto[1]
- using stupdate_induct1 by blast
-
-lemma star_hfau_induct:
- shows "hflat_aux (rders (RSEQ (rder c r0) (RSTAR r0)) s) =
- map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates s r0 [[c]])"
- apply(induct s rule: rev_induct)
- apply simp
- apply(subst rders_append)+
- apply simp
- apply(subst stupdates_append)
- apply(subgoal_tac "created_by_star (rders (RSEQ (rder c r0) (RSTAR r0)) xs)")
- prefer 2
- apply (simp add: star_ders_cbs)
- apply(subst hfau_pushin)
- apply simp
- apply(subgoal_tac "concat (map hElem (map (rder x) (hflat_aux (rders (RSEQ (rder c r0) (RSTAR r0)) xs)))) =
- concat (map hElem (map (rder x) ( map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates xs r0 [[c]])))) ")
- apply(simp only:)
- prefer 2
- apply presburger
- apply(subst stupdates_append[symmetric])
- using stupdates_join_general by blast
-
-lemma starders_hfau_also1:
- shows "hflat_aux (rders (RSTAR r) (c # xs)) = map (\<lambda>s1. RSEQ (rders r s1) (RSTAR r)) (star_updates xs r [[c]])"
- using star_hfau_induct by force
-
-lemma hflat_aux_grewrites:
- shows "a # rs \<leadsto>g* hflat_aux a @ rs"
- apply(induct a arbitrary: rs)
- apply simp+
- apply(case_tac x)
- apply simp
- apply(case_tac list)
-
- apply (metis append.right_neutral append_Cons append_eq_append_conv2 grewrites.simps hflat_aux.simps(7) same_append_eq)
- apply(case_tac lista)
- apply simp
- apply (metis (no_types, lifting) append_Cons append_eq_append_conv2 gmany_steps_later greal_trans grewrite.intros(2) grewrites_append self_append_conv)
- apply simp
- by simp
-
-
-
-
-lemma cbs_hfau_rsimpeq1:
- shows "rsimp (RALT a b) = rsimp (RALTS ((hflat_aux a) @ (hflat_aux b)))"
- apply(subgoal_tac "[a, b] \<leadsto>g* hflat_aux a @ hflat_aux b")
- using grewrites_equal_rsimp apply presburger
- by (metis append.right_neutral greal_trans grewrites_cons hflat_aux_grewrites)
-
-
-lemma hfau_rsimpeq2:
- shows "created_by_star r \<Longrightarrow> rsimp r = rsimp ( (RALTS (hflat_aux r)))"
- apply(induct r)
- apply simp+
-
- apply (metis rsimp_seq_equal1)
- prefer 2
- apply simp
- apply(case_tac x)
- apply simp
- apply(case_tac "list")
- apply simp
-
- apply (metis idem_after_simp1)
- apply(case_tac "lista")
- prefer 2
- apply (metis hflat_aux.simps(8) idem_after_simp1 list.simps(8) list.simps(9) rsimp.simps(2))
- apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux (RALT a aa)))")
- apply simp
- apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux a @ hflat_aux aa))")
- using hflat_aux.simps(1) apply presburger
- apply simp
- using cbs_hfau_rsimpeq1 by fastforce
-
-lemma star_closed_form1:
- shows "rsimp (rders (RSTAR r0) (c#s)) =
-rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
- using hfau_rsimpeq2 rder.simps(6) rders.simps(2) star_ders_cbs starders_hfau_also1 by presburger
-
-lemma star_closed_form2:
- shows "rsimp (rders_simp (RSTAR r0) (c#s)) =
-rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
- by (metis list.distinct(1) rders_simp_same_simpders rsimp_idem star_closed_form1)
-
-lemma star_closed_form3:
- shows "rsimp (rders_simp (RSTAR r0) (c#s)) = (rders_simp (RSTAR r0) (c#s))"
- by (metis list.distinct(1) rders_simp_same_simpders star_closed_form1 star_closed_form2)
-
-lemma star_closed_form4:
- shows " (rders_simp (RSTAR r0) (c#s)) =
-rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
- using star_closed_form2 star_closed_form3 by presburger
-
-lemma star_closed_form5:
- shows " rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) Ss )))) =
- rsimp ( ( RALTS ( (map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss ))))"
- by (metis (mono_tags, lifting) list.map_comp map_eq_conv o_apply rsimp.simps(2) rsimp_idem)
-
-lemma star_closed_form6_hrewrites:
- shows "
- (map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss )
- scf\<leadsto>*
-(map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss )"
- apply(induct Ss)
- apply simp
- apply (simp add: ss1)
- by (metis (no_types, lifting) list.simps(9) rsimp.simps(1) rsimp_idem simp_hrewrites ss2)
-
-lemma star_closed_form6:
- shows " rsimp ( ( RALTS ( (map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss )))) =
- rsimp ( ( RALTS ( (map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss ))))"
- apply(subgoal_tac " map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss scf\<leadsto>*
- map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss ")
- using hrewrites_simpeq srewritescf_alt1 apply fastforce
- using star_closed_form6_hrewrites by blast
-
-lemma stupdate_nonempty:
- shows "\<forall>s \<in> set Ss. s \<noteq> [] \<Longrightarrow> \<forall>s \<in> set (star_update c r Ss). s \<noteq> []"
- apply(induct Ss)
- apply simp
- apply(case_tac "rnullable (rders r a)")
- apply simp+
- done
-
-
-lemma stupdates_nonempty:
- shows "\<forall>s \<in> set Ss. s\<noteq> [] \<Longrightarrow> \<forall>s \<in> set (star_updates s r Ss). s \<noteq> []"
- apply(induct s arbitrary: Ss)
- apply simp
- apply simp
- using stupdate_nonempty by presburger
-
-
-lemma star_closed_form8:
- shows
-"rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rsimp (rders r0 s1)) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))) =
- rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ ( (rders_simp r0 s1)) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
- by (smt (verit, ccfv_SIG) list.simps(8) map_eq_conv rders__onechar rders_simp_same_simpders set_ConsD stupdates_nonempty)
-
-
-lemma star_closed_form:
- shows "rders_simp (RSTAR r0) (c#s) =
-rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
- apply(induct s)
- apply simp
- apply (metis idem_after_simp1 rsimp.simps(1) rsimp.simps(6) rsimp_idem)
- using star_closed_form4 star_closed_form5 star_closed_form6 star_closed_form8 by presburger
-
-
-unused_thms
-
-end
\ No newline at end of file
diff -r f493a20feeb3 -r 04b5e904a220 thys3/ClosedFormsBounds.thy
--- a/thys3/ClosedFormsBounds.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,448 +0,0 @@
-
-theory ClosedFormsBounds
- imports "GeneralRegexBound" "ClosedForms"
-begin
-lemma alts_ders_lambda_shape_ders:
- shows "\<forall>r \<in> set (map (\<lambda>r. rders_simp r ( s)) rs ). \<exists>r1 \<in> set rs. r = rders_simp r1 s"
- by (simp add: image_iff)
-
-lemma rlist_bound:
- assumes "\<forall>r \<in> set rs. rsize r \<le> N"
- shows "rsizes rs \<le> N * (length rs)"
- using assms
- apply(induct rs)
- apply simp
- by simp
-
-lemma alts_closed_form_bounded:
- assumes "\<forall>r \<in> set rs. \<forall>s. rsize (rders_simp r s) \<le> N"
- shows "rsize (rders_simp (RALTS rs) s) \<le> max (Suc (N * (length rs))) (rsize (RALTS rs))"
-proof (cases s)
- case Nil
- then show "rsize (rders_simp (RALTS rs) s) \<le> max (Suc (N * length rs)) (rsize (RALTS rs))"
- by simp
-next
- case (Cons a s)
-
- from assms have "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (a # s)) rs ). rsize r \<le> N"
- by (metis alts_ders_lambda_shape_ders)
- then have a: "rsizes (map (\<lambda>r. rders_simp r (a # s)) rs ) \<le> N * (length rs)"
- by (metis length_map rlist_bound)
-
- have "rsize (rders_simp (RALTS rs) (a # s))
- = rsize (rsimp (RALTS (map (\<lambda>r. rders_simp r (a # s)) rs)))"
- by (metis alts_closed_form_variant list.distinct(1))
- also have "... \<le> rsize (RALTS (map (\<lambda>r. rders_simp r (a # s)) rs))"
- using rsimp_mono by blast
- also have "... = Suc (rsizes (map (\<lambda>r. rders_simp r (a # s)) rs))"
- by simp
- also have "... \<le> Suc (N * (length rs))"
- using a by blast
- finally have "rsize (rders_simp (RALTS rs) (a # s)) \<le> max (Suc (N * length rs)) (rsize (RALTS rs))"
- by auto
- then show ?thesis using local.Cons by simp
-qed
-
-lemma alts_simp_ineq_unfold:
- shows "rsize (rsimp (RALTS rs)) \<le> Suc (rsizes (rdistinct (rflts (map rsimp rs)) {}))"
- using rsimp_aalts_smaller by auto
-
-
-lemma rdistinct_mono_list:
- shows "rsizes (rdistinct (x5 @ rs) rset) \<le> rsizes x5 + rsizes (rdistinct rs ((set x5 ) \<union> rset))"
- apply(induct x5 arbitrary: rs rset)
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp
- apply (simp add: add.assoc insert_absorb trans_le_add2)
- apply simp
- by (metis Un_insert_right)
-
-
-lemma flts_size_reduction_alts:
- assumes a: "\<And>noalts_set alts_set corr_set.
- (\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs) \<and>
- (\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set) \<Longrightarrow>
- Suc (rsizes (rdistinct (rflts rs) (noalts_set \<union> corr_set)))
- \<le> Suc (rsizes (rdistinct rs (insert RZERO (noalts_set \<union> alts_set))))"
- and b: "\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs"
- and c: "\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set"
- and d: "a = RALTS x5"
- shows "rsizes (rdistinct (rflts (a # rs)) (noalts_set \<union> corr_set))
- \<le> rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))"
-
- apply(case_tac "a \<in> alts_set")
- using a b c d
- apply simp
- apply(subgoal_tac "set x5 \<subseteq> corr_set")
- apply(subst rdistinct_concat)
- apply auto[1]
- apply presburger
- apply fastforce
- using a b c d
- apply (subgoal_tac "a \<notin> noalts_set")
- prefer 2
- apply blast
- apply simp
- apply(subgoal_tac "rsizes (rdistinct (x5 @ rflts rs) (noalts_set \<union> corr_set))
- \<le> rsizes x5 + rsizes (rdistinct (rflts rs) ((set x5) \<union> (noalts_set \<union> corr_set)))")
- prefer 2
- using rdistinct_mono_list apply presburger
- apply(subgoal_tac "insert (RALTS x5) (noalts_set \<union> alts_set) = noalts_set \<union> (insert (RALTS x5) alts_set)")
- apply(simp only:)
- apply(subgoal_tac "rsizes x5 + rsizes (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> (set x5)))) \<le>
- rsizes x5 + rsizes (rdistinct rs (insert RZERO (noalts_set \<union> insert (RALTS x5) alts_set)))")
-
- apply (simp add: Un_left_commute inf_sup_aci(5))
- apply(subgoal_tac "rsizes (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> set x5))) \<le>
- rsizes (rdistinct rs (insert RZERO (noalts_set \<union> insert (RALTS x5) alts_set)))")
- apply linarith
- apply(subgoal_tac "\<forall>r \<in> insert (RALTS x5) alts_set. \<exists>xs1.( r = RALTS xs1 \<and> set xs1 \<subseteq> corr_set \<union> set x5)")
- apply presburger
- apply (meson insert_iff sup.cobounded2 sup.coboundedI1)
- by blast
-
-
-lemma flts_vs_nflts1:
- assumes "\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs"
- and "\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set)"
- shows "rsizes (rdistinct (rflts rs) (noalts_set \<union> corr_set))
- \<le> rsizes (rdistinct rs (insert RZERO (noalts_set \<union> alts_set)))"
- using assms
- apply(induct rs arbitrary: noalts_set alts_set corr_set)
- apply simp
- apply(case_tac a)
- apply(case_tac "RZERO \<in> noalts_set")
- apply simp
- apply(subgoal_tac "RZERO \<notin> alts_set")
- apply simp
- apply fastforce
- apply(case_tac "RONE \<in> noalts_set")
- apply simp
- apply(subgoal_tac "RONE \<notin> alts_set")
- prefer 2
- apply fastforce
- apply(case_tac "RONE \<in> corr_set")
- apply(subgoal_tac "rflts (a # rs) = RONE # rflts rs")
- apply(simp only:)
- apply(subgoal_tac "rdistinct (RONE # rflts rs) (noalts_set \<union> corr_set) =
- rdistinct (rflts rs) (noalts_set \<union> corr_set)")
- apply(simp only:)
- apply(subgoal_tac "rdistinct (RONE # rs) (insert RZERO (noalts_set \<union> alts_set)) =
- RONE # (rdistinct rs (insert RONE (insert RZERO (noalts_set \<union> alts_set)))) ")
- apply(simp only:)
- apply(subgoal_tac "rdistinct (rflts rs) (noalts_set \<union> corr_set) =
- rdistinct (rflts rs) (insert RONE (noalts_set \<union> corr_set))")
- apply (simp only:)
- apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
- apply(simp only:)
- apply(subgoal_tac "insert RONE (insert RZERO (noalts_set \<union> alts_set)) =
- insert RZERO ((insert RONE noalts_set) \<union> alts_set)")
- apply(simp only:)
- apply(subgoal_tac "rsizes (rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))
- \<le> rsizes (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))")
- apply (smt (verit, best) dual_order.trans insert_iff rrexp.distinct(15))
- apply (metis (no_types, opaque_lifting) le_add_same_cancel2 list.simps(9) sum_list.Cons zero_le)
- apply fastforce
- apply fastforce
- apply (metis Un_iff insert_absorb)
- apply (metis UnE insertE insert_is_Un rdistinct.simps(2) rrexp.distinct(1))
- apply (meson UnCI rdistinct.simps(2))
- using rflts.simps(4) apply presburger
- apply simp
- apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
- apply(simp only:)
- apply (metis Un_insert_left insertE rrexp.distinct(15))
- apply fastforce
- apply(case_tac "a \<in> noalts_set")
- apply simp
- apply(subgoal_tac "a \<notin> alts_set")
- prefer 2
- apply blast
- apply(case_tac "a \<in> corr_set")
- apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
- prefer 2
- apply fastforce
- apply(simp only:)
- apply(subgoal_tac "rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))) \<le>
- rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))")
-
- apply(subgoal_tac "rsizes (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)) \<le>
- rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set)))")
- apply fastforce
- apply simp
- apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
- apply(simp only:)
- apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
- apply(simp only:)
- apply (metis insertE rrexp.distinct(21))
- apply blast
-
- apply fastforce
- apply force
- apply simp
- apply (metis Un_insert_left insert_iff rrexp.distinct(21))
- apply(case_tac "a \<in> noalts_set")
- apply simp
- apply(subgoal_tac "a \<notin> alts_set")
- prefer 2
- apply blast
- apply(case_tac "a \<in> corr_set")
- apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
- prefer 2
- apply fastforce
- apply(simp only:)
- apply(subgoal_tac "rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))) \<le>
- rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))")
-
- apply(subgoal_tac "rsizes (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)) \<le>
- rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set)))")
- apply fastforce
- apply simp
- apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
- apply(simp only:)
- apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
- apply(simp only:)
-
-
- apply (metis insertE rrexp.distinct(25))
- apply blast
- apply fastforce
- apply force
- apply simp
-
- apply (metis Un_insert_left insertE rrexp.distinct(25))
-
- using Suc_le_mono flts_size_reduction_alts apply presburger
- apply(case_tac "a \<in> noalts_set")
- apply simp
- apply(subgoal_tac "a \<notin> alts_set")
- prefer 2
- apply blast
- apply(case_tac "a \<in> corr_set")
- apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
- prefer 2
- apply fastforce
- apply(simp only:)
- apply(subgoal_tac "rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))) \<le>
- rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))")
-
- apply(subgoal_tac "rsizes (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)) \<le>
- rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set)))")
- apply fastforce
- apply simp
- apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
- apply(simp only:)
- apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
- apply(simp only:)
- apply (metis insertE rrexp.distinct(29))
-
- apply blast
-
- apply fastforce
- apply force
- apply simp
- apply (metis Un_insert_left insert_iff rrexp.distinct(29))
- done
-
-
-lemma flts_vs_nflts:
- assumes "\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs"
- and "\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set)"
- shows "rsizes (rdistinct (rflts rs) (noalts_set \<union> corr_set))
- \<le> rsizes (rdistinct rs (insert RZERO (noalts_set \<union> alts_set)))"
- by (simp add: assms flts_vs_nflts1)
-
-lemma distinct_simp_ineq_general:
- assumes "rsimp ` no_simp = has_simp" "finite no_simp"
- shows "rsizes (rdistinct (map rsimp rs) has_simp) \<le> rsizes (rdistinct rs no_simp)"
- using assms
- apply(induct rs no_simp arbitrary: has_simp rule: rdistinct.induct)
- apply simp
- apply(auto)
- using add_le_mono rsimp_mono by presburger
-
-lemma larger_acc_smaller_distinct_res0:
- assumes "ss \<subseteq> SS"
- shows "rsizes (rdistinct rs SS) \<le> rsizes (rdistinct rs ss)"
- using assms
- apply(induct rs arbitrary: ss SS)
- apply simp
- by (metis distinct_early_app1 rdistinct_smaller)
-
-lemma without_flts_ineq:
- shows "rsizes (rdistinct (rflts rs) {}) \<le> rsizes (rdistinct rs {})"
-proof -
- have "rsizes (rdistinct (rflts rs) {}) \<le> rsizes (rdistinct rs (insert RZERO {}))"
- by (metis empty_iff flts_vs_nflts sup_bot_left)
- also have "... \<le> rsizes (rdistinct rs {})"
- by (simp add: larger_acc_smaller_distinct_res0)
- finally show ?thesis
- by blast
-qed
-
-
-lemma distinct_simp_ineq:
- shows "rsizes (rdistinct (map rsimp rs) {}) \<le> rsizes (rdistinct rs {})"
- using distinct_simp_ineq_general by blast
-
-
-lemma alts_simp_control:
- shows "rsize (rsimp (RALTS rs)) \<le> Suc (rsizes (rdistinct rs {}))"
-proof -
- have "rsize (rsimp (RALTS rs)) \<le> Suc (rsizes (rdistinct (rflts (map rsimp rs)) {}))"
- using alts_simp_ineq_unfold by auto
- moreover have "\<dots> \<le> Suc (rsizes (rdistinct (map rsimp rs) {}))"
- using without_flts_ineq by blast
- ultimately show "rsize (rsimp (RALTS rs)) \<le> Suc (rsizes (rdistinct rs {}))"
- by (meson Suc_le_mono distinct_simp_ineq le_trans)
-qed
-
-
-lemma larger_acc_smaller_distinct_res:
- shows "rsizes (rdistinct rs (insert a ss)) \<le> rsizes (rdistinct rs ss)"
- by (simp add: larger_acc_smaller_distinct_res0 subset_insertI)
-
-lemma triangle_inequality_distinct:
- shows "rsizes (rdistinct (a # rs) ss) \<le> rsize a + rsizes (rdistinct rs ss)"
- apply(case_tac "a \<in> ss")
- apply simp
- by (simp add: larger_acc_smaller_distinct_res)
-
-
-lemma distinct_list_size_len_bounded:
- assumes "\<forall>r \<in> set rs. rsize r \<le> N" "length rs \<le> lrs"
- shows "rsizes rs \<le> lrs * N "
- using assms
- by (metis rlist_bound dual_order.trans mult.commute mult_le_mono1)
-
-
-
-lemma rdistinct_same_set:
- shows "r \<in> set rs \<longleftrightarrow> r \<in> set (rdistinct rs {})"
- apply(induct rs)
- apply simp
- by (metis rdistinct_set_equality)
-
-(* distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size *)
-lemma distinct_list_rexp_upto:
- assumes "\<forall>r\<in> set rs. (rsize r) \<le> N"
- shows "rsizes (rdistinct rs {}) \<le> (card (sizeNregex N)) * N"
-
- apply(subgoal_tac "distinct (rdistinct rs {})")
- prefer 2
- using rdistinct_does_the_job apply blast
- apply(subgoal_tac "length (rdistinct rs {}) \<le> card (sizeNregex N)")
- apply(rule distinct_list_size_len_bounded)
- using assms
- apply (meson rdistinct_same_set)
- apply blast
- apply(subgoal_tac "\<forall>r \<in> set (rdistinct rs {}). rsize r \<le> N")
- prefer 2
- using assms
- apply (meson rdistinct_same_set)
- apply(subgoal_tac "length (rdistinct rs {}) = card (set (rdistinct rs {}))")
- prefer 2
- apply (simp add: distinct_card)
- apply(simp)
- by (metis card_mono finite_size_n mem_Collect_eq sizeNregex_def subsetI)
-
-
-lemma star_control_bounded:
- assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
- shows "rsizes (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates s r [[c]])) {})
- \<le> (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r)))"
- by (smt (verit) add_Suc_shift add_mono_thms_linordered_semiring(3) assms distinct_list_rexp_upto image_iff list.set_map plus_nat.simps(2) rsize.simps(5))
-
-
-lemma star_closed_form_bounded:
- assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
- shows "rsize (rders_simp (RSTAR r) s) \<le>
- max ((Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r))))) (rsize (RSTAR r))"
-proof(cases s)
- case Nil
- then show "rsize (rders_simp (RSTAR r) s)
- \<le> max (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * Suc (N + rsize (RSTAR r))) (rsize (RSTAR r))"
- by simp
-next
- case (Cons a list)
- then have "rsize (rders_simp (RSTAR r) s) =
- rsize (rsimp (RALTS ((map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates list r [[a]])))))"
- using star_closed_form by fastforce
- also have "... \<le> Suc (rsizes (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates list r [[a]])) {}))"
- using alts_simp_control by blast
- also have "... \<le> Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r)))"
- using star_control_bounded[OF assms] by (metis add_mono le_add1 mult_Suc plus_1_eq_Suc)
- also have "... \<le> max (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * Suc (N + rsize (RSTAR r))) (rsize (RSTAR r))"
- by simp
- finally show ?thesis by simp
-qed
-
-
-lemma seq_estimate_bounded:
- assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1"
- and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
- shows
- "rsizes (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})
- \<le> (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))"
-proof -
- have a: "rsizes (rdistinct (map (rders_simp r2) (vsuf s r1)) {}) \<le> N2 * card (sizeNregex N2)"
- by (metis assms(2) distinct_list_rexp_upto ex_map_conv mult.commute)
-
- have "rsizes (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}) \<le>
- rsize (RSEQ (rders_simp r1 s) r2) + rsizes (rdistinct (map (rders_simp r2) (vsuf s r1)) {})"
- using triangle_inequality_distinct by blast
- also have "... \<le> rsize (RSEQ (rders_simp r1 s) r2) + N2 * card (sizeNregex N2)"
- by (simp add: a)
- also have "... \<le> Suc (N1 + (rsize r2) + N2 * card (sizeNregex N2))"
- by (simp add: assms(1))
- finally show ?thesis
- by force
-qed
-
-
-lemma seq_closed_form_bounded2:
- assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1"
- and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
-shows "rsize (rders_simp (RSEQ r1 r2) s)
- \<le> max (2 + N1 + (rsize r2) + (N2 * card (sizeNregex N2))) (rsize (RSEQ r1 r2))"
-proof(cases s)
- case Nil
- then show "rsize (rders_simp (RSEQ r1 r2) s)
- \<le> max (2 + N1 + (rsize r2) + (N2 * card (sizeNregex N2))) (rsize (RSEQ r1 r2))"
- by simp
-next
- case (Cons a list)
- then have "rsize (rders_simp (RSEQ r1 r2) s) =
- rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1)))))"
- using seq_closed_form_variant by (metis list.distinct(1))
- also have "... \<le> Suc (rsizes (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))"
- using alts_simp_control by blast
- also have "... \<le> 2 + N1 + (rsize r2) + (N2 * card (sizeNregex N2))"
- using seq_estimate_bounded[OF assms] by auto
- ultimately show "rsize (rders_simp (RSEQ r1 r2) s)
- \<le> max (2 + N1 + (rsize r2) + N2 * card (sizeNregex N2)) (rsize (RSEQ r1 r2))"
- by auto
-qed
-
-
-lemma rders_simp_bounded:
- shows "\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
- apply(induct r)
- apply(rule_tac x = "Suc 0 " in exI)
- using three_easy_cases0 apply force
- using three_easy_cases1 apply blast
- using three_easy_casesC apply blast
- apply(erule exE)+
- apply(rule exI)
- apply(rule allI)
- apply(rule seq_closed_form_bounded2)
- apply(assumption)
- apply(assumption)
- apply (metis alts_closed_form_bounded size_list_estimation')
- using star_closed_form_bounded by blast
-
-
-unused_thms
-
-end
diff -r f493a20feeb3 -r 04b5e904a220 thys3/FBound.thy
--- a/thys3/FBound.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,180 +0,0 @@
-
-theory FBound
- imports "BlexerSimp" "ClosedFormsBounds"
-begin
-
-fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
- where
- "distinctBy [] f acc = []"
-| "distinctBy (x#xs) f acc =
- (if (f x) \<in> acc then distinctBy xs f acc
- else x # (distinctBy xs f ({f x} \<union> acc)))"
-
-fun rerase :: "arexp \<Rightarrow> rrexp"
-where
- "rerase AZERO = RZERO"
-| "rerase (AONE _) = RONE"
-| "rerase (ACHAR _ c) = RCHAR c"
-| "rerase (AALTs bs rs) = RALTS (map rerase rs)"
-| "rerase (ASEQ _ r1 r2) = RSEQ (rerase r1) (rerase r2)"
-| "rerase (ASTAR _ r) = RSTAR (rerase r)"
-
-lemma eq1_rerase:
- shows "x ~1 y \<longleftrightarrow> (rerase x) = (rerase y)"
- apply(induct x y rule: eq1.induct)
- apply(auto)
- done
-
-
-lemma distinctBy_distinctWith:
- shows "distinctBy xs f (f ` acc) = distinctWith xs (\<lambda>x y. f x = f y) acc"
- apply(induct xs arbitrary: acc)
- apply(auto)
- by (metis image_insert)
-
-lemma distinctBy_distinctWith2:
- shows "distinctBy xs rerase {} = distinctWith xs eq1 {}"
- apply(subst distinctBy_distinctWith[of _ _ "{}", simplified])
- using eq1_rerase by presburger
-
-lemma asize_rsize:
- shows "rsize (rerase r) = asize r"
- apply(induct r rule: rerase.induct)
- apply(auto)
- apply (metis (mono_tags, lifting) comp_apply map_eq_conv)
- done
-
-lemma rerase_fuse:
- shows "rerase (fuse bs r) = rerase r"
- apply(induct r)
- apply simp+
- done
-
-lemma rerase_bsimp_ASEQ:
- shows "rerase (bsimp_ASEQ x1 a1 a2) = rsimp_SEQ (rerase a1) (rerase a2)"
- apply(induct x1 a1 a2 rule: bsimp_ASEQ.induct)
- apply(auto)
- done
-
-lemma rerase_bsimp_AALTs:
- shows "rerase (bsimp_AALTs bs rs) = rsimp_ALTs (map rerase rs)"
- apply(induct bs rs rule: bsimp_AALTs.induct)
- apply(auto simp add: rerase_fuse)
- done
-
-fun anonalt :: "arexp \<Rightarrow> bool"
- where
- "anonalt (AALTs bs2 rs) = False"
-| "anonalt r = True"
-
-
-fun agood :: "arexp \<Rightarrow> bool" where
- "agood AZERO = False"
-| "agood (AONE cs) = True"
-| "agood (ACHAR cs c) = True"
-| "agood (AALTs cs []) = False"
-| "agood (AALTs cs [r]) = False"
-| "agood (AALTs cs (r1#r2#rs)) = (distinct (map rerase (r1 # r2 # rs)) \<and>(\<forall>r' \<in> set (r1#r2#rs). agood r' \<and> anonalt r'))"
-| "agood (ASEQ _ AZERO _) = False"
-| "agood (ASEQ _ (AONE _) _) = False"
-| "agood (ASEQ _ _ AZERO) = False"
-| "agood (ASEQ cs r1 r2) = (agood r1 \<and> agood r2)"
-| "agood (ASTAR cs r) = True"
-
-
-fun anonnested :: "arexp \<Rightarrow> bool"
- where
- "anonnested (AALTs bs2 []) = True"
-| "anonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False"
-| "anonnested (AALTs bs2 (r # rs2)) = anonnested (AALTs bs2 rs2)"
-| "anonnested r = True"
-
-
-lemma asize0:
- shows "0 < asize r"
- apply(induct r)
- apply(auto)
- done
-
-lemma rnullable:
- shows "rnullable (rerase r) = bnullable r"
- apply(induct r rule: rerase.induct)
- apply(auto)
- done
-
-lemma rder_bder_rerase:
- shows "rder c (rerase r ) = rerase (bder c r)"
- apply (induct r)
- apply (auto)
- using rerase_fuse apply presburger
- using rnullable apply blast
- using rnullable by blast
-
-lemma rerase_map_bsimp:
- assumes "\<And> r. r \<in> set rs \<Longrightarrow> rerase (bsimp r) = (rsimp \<circ> rerase) r"
- shows "map rerase (map bsimp rs) = map (rsimp \<circ> rerase) rs"
- using assms
- apply(induct rs)
- by simp_all
-
-
-lemma rerase_flts:
- shows "map rerase (flts rs) = rflts (map rerase rs)"
- apply(induct rs rule: flts.induct)
- apply(auto simp add: rerase_fuse)
- done
-
-lemma rerase_dB:
- shows "map rerase (distinctBy rs rerase acc) = rdistinct (map rerase rs) acc"
- apply(induct rs arbitrary: acc)
- apply simp+
- done
-
-lemma rerase_earlier_later_same:
- assumes " \<And>r. r \<in> set rs \<Longrightarrow> rerase (bsimp r) = rsimp (rerase r)"
- shows " (map rerase (distinctBy (flts (map bsimp rs)) rerase {})) =
- (rdistinct (rflts (map (rsimp \<circ> rerase) rs)) {})"
- apply(subst rerase_dB)
- apply(subst rerase_flts)
- apply(subst rerase_map_bsimp)
- apply auto
- using assms
- apply simp
- done
-
-lemma bsimp_rerase:
- shows "rerase (bsimp a) = rsimp (rerase a)"
- apply(induct a rule: bsimp.induct)
- apply(auto)
- using rerase_bsimp_ASEQ apply presburger
- using distinctBy_distinctWith2 rerase_bsimp_AALTs rerase_earlier_later_same by fastforce
-
-lemma rders_simp_size:
- shows "rders_simp (rerase r) s = rerase (bders_simp r s)"
- apply(induct s rule: rev_induct)
- apply simp
- by (simp add: bders_simp_append rder_bder_rerase rders_simp_append bsimp_rerase)
-
-
-corollary aders_simp_finiteness:
- assumes "\<exists>N. \<forall>s. rsize (rders_simp (rerase r) s) \<le> N"
- shows " \<exists>N. \<forall>s. asize (bders_simp r s) \<le> N"
-proof -
- from assms obtain N where "\<forall>s. rsize (rders_simp (rerase r) s) \<le> N"
- by blast
- then have "\<forall>s. rsize (rerase (bders_simp r s)) \<le> N"
- by (simp add: rders_simp_size)
- then have "\<forall>s. asize (bders_simp r s) \<le> N"
- by (simp add: asize_rsize)
- then show "\<exists>N. \<forall>s. asize (bders_simp r s) \<le> N" by blast
-qed
-
-theorem annotated_size_bound:
- shows "\<exists>N. \<forall>s. asize (bders_simp r s) \<le> N"
- apply(insert aders_simp_finiteness)
- by (simp add: rders_simp_bounded)
-
-
-unused_thms
-
-end
diff -r f493a20feeb3 -r 04b5e904a220 thys3/GeneralRegexBound.thy
--- a/thys3/GeneralRegexBound.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,212 +0,0 @@
-theory GeneralRegexBound
- imports "BasicIdentities"
-begin
-
-lemma size_geq1:
- shows "rsize r \<ge> 1"
- by (induct r) auto
-
-definition RSEQ_set where
- "RSEQ_set A n \<equiv> {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A \<and> rsize r1 + rsize r2 \<le> n}"
-
-definition RSEQ_set_cartesian where
- "RSEQ_set_cartesian A = {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A}"
-
-definition RALT_set where
- "RALT_set A n \<equiv> {RALTS rs | rs. set rs \<subseteq> A \<and> rsizes rs \<le> n}"
-
-definition RALTs_set where
- "RALTs_set A n \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A \<and> rsizes rs \<le> n}"
-
-definition
- "sizeNregex N \<equiv> {r. rsize r \<le> N}"
-
-
-lemma sizenregex_induct1:
- "sizeNregex (Suc n) = (({RZERO, RONE} \<union> {RCHAR c| c. True})
- \<union> (RSTAR ` sizeNregex n)
- \<union> (RSEQ_set (sizeNregex n) n)
- \<union> (RALTs_set (sizeNregex n) n))"
- apply(auto)
- apply(case_tac x)
- apply(auto simp add: RSEQ_set_def)
- using sizeNregex_def apply force
- using sizeNregex_def apply auto[1]
- apply (simp add: sizeNregex_def)
- apply (simp add: sizeNregex_def)
- apply (simp add: RALTs_set_def)
- apply (metis imageI list.set_map member_le_sum_list order_trans)
- apply (simp add: sizeNregex_def)
- apply (simp add: sizeNregex_def)
- apply (simp add: sizeNregex_def)
- using sizeNregex_def apply force
- apply (simp add: sizeNregex_def)
- apply (simp add: sizeNregex_def)
- apply (simp add: RALTs_set_def)
- apply(simp add: sizeNregex_def)
- apply(auto)
- using ex_in_conv by fastforce
-
-lemma s4:
- "RSEQ_set A n \<subseteq> RSEQ_set_cartesian A"
- using RSEQ_set_cartesian_def RSEQ_set_def by fastforce
-
-lemma s5:
- assumes "finite A"
- shows "finite (RSEQ_set_cartesian A)"
- using assms
- apply(subgoal_tac "RSEQ_set_cartesian A = (\<lambda>(x1, x2). RSEQ x1 x2) ` (A \<times> A)")
- apply simp
- unfolding RSEQ_set_cartesian_def
- apply(auto)
- done
-
-
-definition RALTs_set_length
- where
- "RALTs_set_length A n l \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A \<and> rsizes rs \<le> n \<and> length rs \<le> l}"
-
-
-definition RALTs_set_length2
- where
- "RALTs_set_length2 A l \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A \<and> length rs \<le> l}"
-
-definition set_length2
- where
- "set_length2 A l \<equiv> {rs. \<forall>r \<in> set rs. r \<in> A \<and> length rs \<le> l}"
-
-
-lemma r000:
- shows "RALTs_set_length A n l \<subseteq> RALTs_set_length2 A l"
- apply(auto simp add: RALTs_set_length2_def RALTs_set_length_def)
- done
-
-
-lemma r02:
- shows "set_length2 A 0 \<subseteq> {[]}"
- apply(auto simp add: set_length2_def)
- apply(case_tac x)
- apply(auto)
- done
-
-lemma r03:
- shows "set_length2 A (Suc n) \<subseteq>
- {[]} \<union> (\<lambda>(h, t). h # t) ` (A \<times> (set_length2 A n))"
- apply(auto simp add: set_length2_def)
- apply(case_tac x)
- apply(auto)
- done
-
-lemma r1:
- assumes "finite A"
- shows "finite (set_length2 A n)"
- using assms
- apply(induct n)
- apply(rule finite_subset)
- apply(rule r02)
- apply(simp)
- apply(rule finite_subset)
- apply(rule r03)
- apply(simp)
- done
-
-lemma size_sum_more_than_len:
- shows "rsizes rs \<ge> length rs"
- apply(induct rs)
- apply simp
- apply simp
- apply(subgoal_tac "rsize a \<ge> 1")
- apply linarith
- using size_geq1 by auto
-
-
-lemma sum_list_len:
- shows "rsizes rs \<le> n \<Longrightarrow> length rs \<le> n"
- by (meson order.trans size_sum_more_than_len)
-
-
-lemma t2:
- shows "RALTs_set A n \<subseteq> RALTs_set_length A n n"
- unfolding RALTs_set_length_def RALTs_set_def
- apply(auto)
- using sum_list_len by blast
-
-lemma s8_aux:
- assumes "finite A"
- shows "finite (RALTs_set_length A n n)"
-proof -
- have "finite A" by fact
- then have "finite (set_length2 A n)"
- by (simp add: r1)
- moreover have "(RALTS ` (set_length2 A n)) = RALTs_set_length2 A n"
- unfolding RALTs_set_length2_def set_length2_def
- by (auto)
- ultimately have "finite (RALTs_set_length2 A n)"
- by (metis finite_imageI)
- then show ?thesis
- by (metis infinite_super r000)
-qed
-
-lemma char_finite:
- shows "finite {RCHAR c |c. True}"
- apply simp
- apply(subgoal_tac "finite (RCHAR ` (UNIV::char set))")
- prefer 2
- apply simp
- by (simp add: full_SetCompr_eq)
-
-
-lemma finite_size_n:
- shows "finite (sizeNregex n)"
- apply(induct n)
- apply(simp add: sizeNregex_def)
- apply (metis (mono_tags, lifting) not_finite_existsD not_one_le_zero size_geq1)
- apply(subst sizenregex_induct1)
- apply(simp only: finite_Un)
- apply(rule conjI)+
- apply(simp)
-
- using char_finite apply blast
- apply(simp)
- apply(rule finite_subset)
- apply(rule s4)
- apply(rule s5)
- apply(simp)
- apply(rule finite_subset)
- apply(rule t2)
- apply(rule s8_aux)
- apply(simp)
- done
-
-lemma three_easy_cases0:
- shows "rsize (rders_simp RZERO s) \<le> Suc 0"
- apply(induct s)
- apply simp
- apply simp
- done
-
-
-lemma three_easy_cases1:
- shows "rsize (rders_simp RONE s) \<le> Suc 0"
- apply(induct s)
- apply simp
- apply simp
- using three_easy_cases0 by auto
-
-
-lemma three_easy_casesC:
- shows "rsize (rders_simp (RCHAR c) s) \<le> Suc 0"
- apply(induct s)
- apply simp
- apply simp
- apply(case_tac " a = c")
- using three_easy_cases1 apply blast
- apply simp
- using three_easy_cases0 by force
-
-
-unused_thms
-
-
-end
-
diff -r f493a20feeb3 -r 04b5e904a220 thys3/Lexer.thy
--- a/thys3/Lexer.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,417 +0,0 @@
-
-theory Lexer
- imports PosixSpec
-begin
-
-section {* The Lexer Functions by Sulzmann and Lu (without simplification) *}
-
-fun
- mkeps :: "rexp \<Rightarrow> val"
-where
- "mkeps(ONE) = Void"
-| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"
-| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
-| "mkeps(STAR r) = Stars []"
-
-fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
-where
- "injval (CH d) c Void = Char d"
-| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"
-| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"
-| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
-| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
-| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
-| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-
-fun
- lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"
-where
- "lexer r [] = (if nullable r then Some(mkeps r) else None)"
-| "lexer r (c#s) = (case (lexer (der c r) s) of
- None \<Rightarrow> None
- | Some(v) \<Rightarrow> Some(injval r c v))"
-
-
-
-section {* Mkeps, Injval Properties *}
-
-lemma mkeps_nullable:
- assumes "nullable(r)"
- shows "\<Turnstile> mkeps r : r"
-using assms
-by (induct rule: nullable.induct)
- (auto intro: Prf.intros)
-
-lemma mkeps_flat:
- assumes "nullable(r)"
- shows "flat (mkeps r) = []"
-using assms
-by (induct rule: nullable.induct) (auto)
-
-lemma Prf_injval_flat:
- assumes "\<Turnstile> v : der c r"
- shows "flat (injval r c v) = c # (flat v)"
-using assms
-apply(induct c r arbitrary: v rule: der.induct)
-apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits)
-done
-
-lemma Prf_injval:
- assumes "\<Turnstile> v : der c r"
- shows "\<Turnstile> (injval r c v) : r"
-using assms
-apply(induct r arbitrary: c v rule: rexp.induct)
-apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)
-apply(simp add: Prf_injval_flat)
-done
-
-
-
-text {*
- Mkeps and injval produce, or preserve, Posix values.
-*}
-
-lemma Posix_mkeps:
- assumes "nullable r"
- shows "[] \<in> r \<rightarrow> mkeps r"
-using assms
-apply(induct r rule: nullable.induct)
-apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)
-apply(subst append.simps(1)[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-done
-
-lemma Posix_injval:
- assumes "s \<in> (der c r) \<rightarrow> v"
- shows "(c # s) \<in> r \<rightarrow> (injval r c v)"
-using assms
-proof(induct r arbitrary: s v rule: rexp.induct)
- case ZERO
- have "s \<in> der c ZERO \<rightarrow> v" by fact
- then have "s \<in> ZERO \<rightarrow> v" by simp
- then have "False" by cases
- then show "(c # s) \<in> ZERO \<rightarrow> (injval ZERO c v)" by simp
-next
- case ONE
- have "s \<in> der c ONE \<rightarrow> v" by fact
- then have "s \<in> ZERO \<rightarrow> v" by simp
- then have "False" by cases
- then show "(c # s) \<in> ONE \<rightarrow> (injval ONE c v)" by simp
-next
- case (CH d)
- consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast
- then show "(c # s) \<in> (CH d) \<rightarrow> (injval (CH d) c v)"
- proof (cases)
- case eq
- have "s \<in> der c (CH d) \<rightarrow> v" by fact
- then have "s \<in> ONE \<rightarrow> v" using eq by simp
- then have eqs: "s = [] \<and> v = Void" by cases simp
- show "(c # s) \<in> CH d \<rightarrow> injval (CH d) c v" using eq eqs
- by (auto intro: Posix.intros)
- next
- case ineq
- have "s \<in> der c (CH d) \<rightarrow> v" by fact
- then have "s \<in> ZERO \<rightarrow> v" using ineq by simp
- then have "False" by cases
- then show "(c # s) \<in> CH d \<rightarrow> injval (CH d) c v" by simp
- qed
-next
- case (ALT r1 r2)
- have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
- have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
- have "s \<in> der c (ALT r1 r2) \<rightarrow> v" by fact
- then have "s \<in> ALT (der c r1) (der c r2) \<rightarrow> v" by simp
- then consider (left) v' where "v = Left v'" "s \<in> der c r1 \<rightarrow> v'"
- | (right) v' where "v = Right v'" "s \<notin> L (der c r1)" "s \<in> der c r2 \<rightarrow> v'"
- by cases auto
- then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v"
- proof (cases)
- case left
- have "s \<in> der c r1 \<rightarrow> v'" by fact
- then have "(c # s) \<in> r1 \<rightarrow> injval r1 c v'" using IH1 by simp
- then have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Left v')" by (auto intro: Posix.intros)
- then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using left by simp
- next
- case right
- have "s \<notin> L (der c r1)" by fact
- then have "c # s \<notin> L r1" by (simp add: der_correctness Der_def)
- moreover
- have "s \<in> der c r2 \<rightarrow> v'" by fact
- then have "(c # s) \<in> r2 \<rightarrow> injval r2 c v'" using IH2 by simp
- ultimately have "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c (Right v')"
- by (auto intro: Posix.intros)
- then show "(c # s) \<in> ALT r1 r2 \<rightarrow> injval (ALT r1 r2) c v" using right by simp
- qed
-next
- case (SEQ r1 r2)
- have IH1: "\<And>s v. s \<in> der c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
- have IH2: "\<And>s v. s \<in> der c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
- have "s \<in> der c (SEQ r1 r2) \<rightarrow> v" by fact
- then consider
- (left_nullable) v1 v2 s1 s2 where
- "v = Left (Seq v1 v2)" "s = s1 @ s2"
- "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "nullable r1"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)"
- | (right_nullable) v1 s1 s2 where
- "v = Right v1" "s = s1 @ s2"
- "s \<in> der c r2 \<rightarrow> v1" "nullable r1" "s1 @ s2 \<notin> L (SEQ (der c r1) r2)"
- | (not_nullable) v1 v2 s1 s2 where
- "v = Seq v1 v2" "s = s1 @ s2"
- "s1 \<in> der c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "\<not>nullable r1"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)"
- by (force split: if_splits elim!: Posix_elims simp add: Sequ_def der_correctness Der_def)
- then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v"
- proof (cases)
- case left_nullable
- have "s1 \<in> der c r1 \<rightarrow> v1" by fact
- then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
- moreover
- have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact
- then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def)
- ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using left_nullable by (rule_tac Posix.intros)
- then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using left_nullable by simp
- next
- case right_nullable
- have "nullable r1" by fact
- then have "[] \<in> r1 \<rightarrow> (mkeps r1)" by (rule Posix_mkeps)
- moreover
- have "s \<in> der c r2 \<rightarrow> v1" by fact
- then have "(c # s) \<in> r2 \<rightarrow> (injval r2 c v1)" using IH2 by simp
- moreover
- have "s1 @ s2 \<notin> L (SEQ (der c r1) r2)" by fact
- then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> [] @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" using right_nullable
- by(auto simp add: der_correctness Der_def append_eq_Cons_conv Sequ_def)
- ultimately have "([] @ (c # s)) \<in> SEQ r1 r2 \<rightarrow> Seq (mkeps r1) (injval r2 c v1)"
- by(rule Posix.intros)
- then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using right_nullable by simp
- next
- case not_nullable
- have "s1 \<in> der c r1 \<rightarrow> v1" by fact
- then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
- moreover
- have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r1) \<and> s\<^sub>4 \<in> L r2)" by fact
- then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by (simp add: der_correctness Der_def)
- ultimately have "((c # s1) @ s2) \<in> SEQ r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using not_nullable
- by (rule_tac Posix.intros) (simp_all)
- then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using not_nullable by simp
- qed
-next
- case (STAR r)
- have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
- have "s \<in> der c (STAR r) \<rightarrow> v" by fact
- then consider
- (cons) v1 vs s1 s2 where
- "v = Seq v1 (Stars vs)" "s = s1 @ s2"
- "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (STAR r) \<rightarrow> (Stars vs)"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))"
- apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)
- apply(rotate_tac 3)
- apply(erule_tac Posix_elims(6))
- apply (simp add: Posix.intros(6))
- using Posix.intros(7) by blast
- then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v"
- proof (cases)
- case cons
- have "s1 \<in> der c r \<rightarrow> v1" by fact
- then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
- moreover
- have "s2 \<in> STAR r \<rightarrow> Stars vs" by fact
- moreover
- have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact
- then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
- then have "flat (injval r c v1) \<noteq> []" by simp
- moreover
- have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (STAR r))" by fact
- then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))"
- by (simp add: der_correctness Der_def)
- ultimately
- have "((c # s1) @ s2) \<in> STAR r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros)
- then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp)
- qed
-qed
-
-
-section {* Lexer Correctness *}
-
-
-lemma lexer_correct_None:
- shows "s \<notin> L r \<longleftrightarrow> lexer r s = None"
- apply(induct s arbitrary: r)
- apply(simp)
- apply(simp add: nullable_correctness)
- apply(simp)
- apply(drule_tac x="der a r" in meta_spec)
- apply(auto)
- apply(auto simp add: der_correctness Der_def)
-done
-
-lemma lexer_correct_Some:
- shows "s \<in> L r \<longleftrightarrow> (\<exists>v. lexer r s = Some(v) \<and> s \<in> r \<rightarrow> v)"
- apply(induct s arbitrary : r)
- apply(simp only: lexer.simps)
- apply(simp)
- apply(simp add: nullable_correctness Posix_mkeps)
- apply(drule_tac x="der a r" in meta_spec)
- apply(simp (no_asm_use) add: der_correctness Der_def del: lexer.simps)
- apply(simp del: lexer.simps)
- apply(simp only: lexer.simps)
- apply(case_tac "lexer (der a r) s = None")
- apply(auto)[1]
- apply(simp)
- apply(erule exE)
- apply(simp)
- apply(rule iffI)
- apply(simp add: Posix_injval)
- apply(simp add: Posix1(1))
-done
-
-lemma lexer_correctness:
- shows "(lexer r s = Some v) \<longleftrightarrow> s \<in> r \<rightarrow> v"
- and "(lexer r s = None) \<longleftrightarrow> \<not>(\<exists>v. s \<in> r \<rightarrow> v)"
-using Posix1(1) Posix_determ lexer_correct_None lexer_correct_Some apply fastforce
-using Posix1(1) lexer_correct_None lexer_correct_Some by blast
-
-
-subsection {* A slight reformulation of the lexer algorithm using stacked functions*}
-
-fun flex :: "rexp \<Rightarrow> (val \<Rightarrow> val) => string \<Rightarrow> (val \<Rightarrow> val)"
- where
- "flex r f [] = f"
-| "flex r f (c#s) = flex (der c r) (\<lambda>v. f (injval r c v)) s"
-
-lemma flex_fun_apply:
- shows "g (flex r f s v) = flex r (g o f) s v"
- apply(induct s arbitrary: g f r v)
- apply(simp_all add: comp_def)
- by meson
-
-lemma flex_fun_apply2:
- shows "g (flex r id s v) = flex r g s v"
- by (simp add: flex_fun_apply)
-
-
-lemma flex_append:
- shows "flex r f (s1 @ s2) = flex (ders s1 r) (flex r f s1) s2"
- apply(induct s1 arbitrary: s2 r f)
- apply(simp_all)
- done
-
-lemma lexer_flex:
- shows "lexer r s = (if nullable (ders s r)
- then Some(flex r id s (mkeps (ders s r))) else None)"
- apply(induct s arbitrary: r)
- apply(simp_all add: flex_fun_apply)
- done
-
-lemma Posix_flex:
- assumes "s2 \<in> (ders s1 r) \<rightarrow> v"
- shows "(s1 @ s2) \<in> r \<rightarrow> flex r id s1 v"
- using assms
- apply(induct s1 arbitrary: r v s2)
- apply(simp)
- apply(simp)
- apply(drule_tac x="der a r" in meta_spec)
- apply(drule_tac x="v" in meta_spec)
- apply(drule_tac x="s2" in meta_spec)
- apply(simp)
- using Posix_injval
- apply(drule_tac Posix_injval)
- apply(subst (asm) (5) flex_fun_apply)
- apply(simp)
- done
-
-lemma injval_inj:
- assumes "\<Turnstile> a : (der c r)" "\<Turnstile> v : (der c r)" "injval r c a = injval r c v"
- shows "a = v"
- using assms
- apply(induct r arbitrary: a c v)
- apply(auto)
- using Prf_elims(1) apply blast
- using Prf_elims(1) apply blast
- apply(case_tac "c = x")
- apply(auto)
- using Prf_elims(4) apply auto[1]
- using Prf_elims(1) apply blast
- prefer 2
- apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) val.distinct(25) val.inject(3) val.inject(4))
- apply(case_tac "nullable r1")
- apply(auto)
- apply(erule Prf_elims)
- apply(erule Prf_elims)
- apply(erule Prf_elims)
- apply(erule Prf_elims)
- apply(auto)
- apply (metis Prf_injval_flat list.distinct(1) mkeps_flat)
- apply(erule Prf_elims)
- apply(erule Prf_elims)
- apply(auto)
- using Prf_injval_flat mkeps_flat apply fastforce
- apply(erule Prf_elims)
- apply(erule Prf_elims)
- apply(auto)
- apply(erule Prf_elims)
- apply(erule Prf_elims)
- apply(auto)
- apply (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5))
- by (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5))
-
-
-
-lemma uu:
- assumes "(c # s) \<in> r \<rightarrow> injval r c v" "\<Turnstile> v : (der c r)"
- shows "s \<in> der c r \<rightarrow> v"
- using assms
- apply -
- apply(subgoal_tac "lexer r (c # s) = Some (injval r c v)")
- prefer 2
- using lexer_correctness(1) apply blast
- apply(simp add: )
- apply(case_tac "lexer (der c r) s")
- apply(simp)
- apply(simp)
- apply(case_tac "s \<in> der c r \<rightarrow> a")
- prefer 2
- apply (simp add: lexer_correctness(1))
- apply(subgoal_tac "\<Turnstile> a : (der c r)")
- prefer 2
- using Posix_Prf apply blast
- using injval_inj by blast
-
-
-lemma Posix_flex2:
- assumes "(s1 @ s2) \<in> r \<rightarrow> flex r id s1 v" "\<Turnstile> v : ders s1 r"
- shows "s2 \<in> (ders s1 r) \<rightarrow> v"
- using assms
- apply(induct s1 arbitrary: r v s2 rule: rev_induct)
- apply(simp)
- apply(simp)
- apply(drule_tac x="r" in meta_spec)
- apply(drule_tac x="injval (ders xs r) x v" in meta_spec)
- apply(drule_tac x="x#s2" in meta_spec)
- apply(simp add: flex_append ders_append)
- using Prf_injval uu by blast
-
-lemma Posix_flex3:
- assumes "s1 \<in> r \<rightarrow> flex r id s1 v" "\<Turnstile> v : ders s1 r"
- shows "[] \<in> (ders s1 r) \<rightarrow> v"
- using assms
- by (simp add: Posix_flex2)
-
-lemma flex_injval:
- shows "flex (der a r) (injval r a) s v = injval r a (flex (der a r) id s v)"
- by (simp add: flex_fun_apply)
-
-lemma Prf_flex:
- assumes "\<Turnstile> v : ders s r"
- shows "\<Turnstile> flex r id s v : r"
- using assms
- apply(induct s arbitrary: v r)
- apply(simp)
- apply(simp)
- by (simp add: Prf_injval flex_injval)
-
-
-unused_thms
-
-end
\ No newline at end of file
diff -r f493a20feeb3 -r 04b5e904a220 thys3/LexerSimp.thy
--- a/thys3/LexerSimp.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,246 +0,0 @@
-theory LexerSimp
- imports "Lexer"
-begin
-
-section {* Lexer including some simplifications *}
-
-
-fun F_RIGHT where
- "F_RIGHT f v = Right (f v)"
-
-fun F_LEFT where
- "F_LEFT f v = Left (f v)"
-
-fun F_ALT where
- "F_ALT f\<^sub>1 f\<^sub>2 (Right v) = Right (f\<^sub>2 v)"
-| "F_ALT f\<^sub>1 f\<^sub>2 (Left v) = Left (f\<^sub>1 v)"
-| "F_ALT f1 f2 v = v"
-
-
-fun F_SEQ1 where
- "F_SEQ1 f\<^sub>1 f\<^sub>2 v = Seq (f\<^sub>1 Void) (f\<^sub>2 v)"
-
-fun F_SEQ2 where
- "F_SEQ2 f\<^sub>1 f\<^sub>2 v = Seq (f\<^sub>1 v) (f\<^sub>2 Void)"
-
-fun F_SEQ where
- "F_SEQ f\<^sub>1 f\<^sub>2 (Seq v\<^sub>1 v\<^sub>2) = Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)"
-| "F_SEQ f1 f2 v = v"
-
-fun simp_ALT where
- "simp_ALT (ZERO, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (r\<^sub>2, F_RIGHT f\<^sub>2)"
-| "simp_ALT (r\<^sub>1, f\<^sub>1) (ZERO, f\<^sub>2) = (r\<^sub>1, F_LEFT f\<^sub>1)"
-| "simp_ALT (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (ALT r\<^sub>1 r\<^sub>2, F_ALT f\<^sub>1 f\<^sub>2)"
-
-
-fun simp_SEQ where
- "simp_SEQ (ONE, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (r\<^sub>2, F_SEQ1 f\<^sub>1 f\<^sub>2)"
-| "simp_SEQ (r\<^sub>1, f\<^sub>1) (ONE, f\<^sub>2) = (r\<^sub>1, F_SEQ2 f\<^sub>1 f\<^sub>2)"
-| "simp_SEQ (ZERO, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (ZERO, undefined)"
-| "simp_SEQ (r\<^sub>1, f\<^sub>1) (ZERO, f\<^sub>2) = (ZERO, undefined)"
-| "simp_SEQ (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (SEQ r\<^sub>1 r\<^sub>2, F_SEQ f\<^sub>1 f\<^sub>2)"
-
-lemma simp_SEQ_simps[simp]:
- "simp_SEQ p1 p2 = (if (fst p1 = ONE) then (fst p2, F_SEQ1 (snd p1) (snd p2))
- else (if (fst p2 = ONE) then (fst p1, F_SEQ2 (snd p1) (snd p2))
- else (if (fst p1 = ZERO) then (ZERO, undefined)
- else (if (fst p2 = ZERO) then (ZERO, undefined)
- else (SEQ (fst p1) (fst p2), F_SEQ (snd p1) (snd p2))))))"
-by (induct p1 p2 rule: simp_SEQ.induct) (auto)
-
-lemma simp_ALT_simps[simp]:
- "simp_ALT p1 p2 = (if (fst p1 = ZERO) then (fst p2, F_RIGHT (snd p2))
- else (if (fst p2 = ZERO) then (fst p1, F_LEFT (snd p1))
- else (ALT (fst p1) (fst p2), F_ALT (snd p1) (snd p2))))"
-by (induct p1 p2 rule: simp_ALT.induct) (auto)
-
-fun
- simp :: "rexp \<Rightarrow> rexp * (val \<Rightarrow> val)"
-where
- "simp (ALT r1 r2) = simp_ALT (simp r1) (simp r2)"
-| "simp (SEQ r1 r2) = simp_SEQ (simp r1) (simp r2)"
-| "simp r = (r, id)"
-
-fun
- slexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"
-where
- "slexer r [] = (if nullable r then Some(mkeps r) else None)"
-| "slexer r (c#s) = (let (rs, fr) = simp (der c r) in
- (case (slexer rs s) of
- None \<Rightarrow> None
- | Some(v) \<Rightarrow> Some(injval r c (fr v))))"
-
-
-lemma slexer_better_simp:
- "slexer r (c#s) = (case (slexer (fst (simp (der c r))) s) of
- None \<Rightarrow> None
- | Some(v) \<Rightarrow> Some(injval r c ((snd (simp (der c r))) v)))"
-by (auto split: prod.split option.split)
-
-
-lemma L_fst_simp:
- shows "L(r) = L(fst (simp r))"
-by (induct r) (auto)
-
-lemma Posix_simp:
- assumes "s \<in> (fst (simp r)) \<rightarrow> v"
- shows "s \<in> r \<rightarrow> ((snd (simp r)) v)"
-using assms
-proof(induct r arbitrary: s v rule: rexp.induct)
- case (ALT r1 r2 s v)
- have IH1: "\<And>s v. s \<in> fst (simp r1) \<rightarrow> v \<Longrightarrow> s \<in> r1 \<rightarrow> snd (simp r1) v" by fact
- have IH2: "\<And>s v. s \<in> fst (simp r2) \<rightarrow> v \<Longrightarrow> s \<in> r2 \<rightarrow> snd (simp r2) v" by fact
- have as: "s \<in> fst (simp (ALT r1 r2)) \<rightarrow> v" by fact
- consider (ZERO_ZERO) "fst (simp r1) = ZERO" "fst (simp r2) = ZERO"
- | (ZERO_NZERO) "fst (simp r1) = ZERO" "fst (simp r2) \<noteq> ZERO"
- | (NZERO_ZERO) "fst (simp r1) \<noteq> ZERO" "fst (simp r2) = ZERO"
- | (NZERO_NZERO) "fst (simp r1) \<noteq> ZERO" "fst (simp r2) \<noteq> ZERO" by auto
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v"
- proof(cases)
- case (ZERO_ZERO)
- with as have "s \<in> ZERO \<rightarrow> v" by simp
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v" by (rule Posix_elims(1))
- next
- case (ZERO_NZERO)
- with as have "s \<in> fst (simp r2) \<rightarrow> v" by simp
- with IH2 have "s \<in> r2 \<rightarrow> snd (simp r2) v" by simp
- moreover
- from ZERO_NZERO have "fst (simp r1) = ZERO" by simp
- then have "L (fst (simp r1)) = {}" by simp
- then have "L r1 = {}" using L_fst_simp by simp
- then have "s \<notin> L r1" by simp
- ultimately have "s \<in> ALT r1 r2 \<rightarrow> Right (snd (simp r2) v)" by (rule Posix_ALT2)
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v"
- using ZERO_NZERO by simp
- next
- case (NZERO_ZERO)
- with as have "s \<in> fst (simp r1) \<rightarrow> v" by simp
- with IH1 have "s \<in> r1 \<rightarrow> snd (simp r1) v" by simp
- then have "s \<in> ALT r1 r2 \<rightarrow> Left (snd (simp r1) v)" by (rule Posix_ALT1)
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v" using NZERO_ZERO by simp
- next
- case (NZERO_NZERO)
- with as have "s \<in> ALT (fst (simp r1)) (fst (simp r2)) \<rightarrow> v" by simp
- then consider (Left) v1 where "v = Left v1" "s \<in> (fst (simp r1)) \<rightarrow> v1"
- | (Right) v2 where "v = Right v2" "s \<in> (fst (simp r2)) \<rightarrow> v2" "s \<notin> L (fst (simp r1))"
- by (erule_tac Posix_elims(4))
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v"
- proof(cases)
- case (Left)
- then have "v = Left v1" "s \<in> r1 \<rightarrow> (snd (simp r1) v1)" using IH1 by simp_all
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v" using NZERO_NZERO
- by (simp_all add: Posix_ALT1)
- next
- case (Right)
- then have "v = Right v2" "s \<in> r2 \<rightarrow> (snd (simp r2) v2)" "s \<notin> L r1" using IH2 L_fst_simp by simp_all
- then show "s \<in> ALT r1 r2 \<rightarrow> snd (simp (ALT r1 r2)) v" using NZERO_NZERO
- by (simp_all add: Posix_ALT2)
- qed
- qed
-next
- case (SEQ r1 r2 s v)
- have IH1: "\<And>s v. s \<in> fst (simp r1) \<rightarrow> v \<Longrightarrow> s \<in> r1 \<rightarrow> snd (simp r1) v" by fact
- have IH2: "\<And>s v. s \<in> fst (simp r2) \<rightarrow> v \<Longrightarrow> s \<in> r2 \<rightarrow> snd (simp r2) v" by fact
- have as: "s \<in> fst (simp (SEQ r1 r2)) \<rightarrow> v" by fact
- consider (ONE_ONE) "fst (simp r1) = ONE" "fst (simp r2) = ONE"
- | (ONE_NONE) "fst (simp r1) = ONE" "fst (simp r2) \<noteq> ONE"
- | (NONE_ONE) "fst (simp r1) \<noteq> ONE" "fst (simp r2) = ONE"
- | (NONE_NONE) "fst (simp r1) \<noteq> ONE" "fst (simp r2) \<noteq> ONE"
- by auto
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v"
- proof(cases)
- case (ONE_ONE)
- with as have b: "s \<in> ONE \<rightarrow> v" by simp
- from b have "s \<in> r1 \<rightarrow> snd (simp r1) v" using IH1 ONE_ONE by simp
- moreover
- from b have c: "s = []" "v = Void" using Posix_elims(2) by auto
- moreover
- have "[] \<in> ONE \<rightarrow> Void" by (simp add: Posix_ONE)
- then have "[] \<in> fst (simp r2) \<rightarrow> Void" using ONE_ONE by simp
- then have "[] \<in> r2 \<rightarrow> snd (simp r2) Void" using IH2 by simp
- ultimately have "([] @ []) \<in> SEQ r1 r2 \<rightarrow> Seq (snd (simp r1) Void) (snd (simp r2) Void)"
- using Posix_SEQ by blast
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v" using c ONE_ONE by simp
- next
- case (ONE_NONE)
- with as have b: "s \<in> fst (simp r2) \<rightarrow> v" by simp
- from b have "s \<in> r2 \<rightarrow> snd (simp r2) v" using IH2 ONE_NONE by simp
- moreover
- have "[] \<in> ONE \<rightarrow> Void" by (simp add: Posix_ONE)
- then have "[] \<in> fst (simp r1) \<rightarrow> Void" using ONE_NONE by simp
- then have "[] \<in> r1 \<rightarrow> snd (simp r1) Void" using IH1 by simp
- moreover
- from ONE_NONE(1) have "L (fst (simp r1)) = {[]}" by simp
- then have "L r1 = {[]}" by (simp add: L_fst_simp[symmetric])
- ultimately have "([] @ s) \<in> SEQ r1 r2 \<rightarrow> Seq (snd (simp r1) Void) (snd (simp r2) v)"
- by(rule_tac Posix_SEQ) auto
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v" using ONE_NONE by simp
- next
- case (NONE_ONE)
- with as have "s \<in> fst (simp r1) \<rightarrow> v" by simp
- with IH1 have "s \<in> r1 \<rightarrow> snd (simp r1) v" by simp
- moreover
- have "[] \<in> ONE \<rightarrow> Void" by (simp add: Posix_ONE)
- then have "[] \<in> fst (simp r2) \<rightarrow> Void" using NONE_ONE by simp
- then have "[] \<in> r2 \<rightarrow> snd (simp r2) Void" using IH2 by simp
- ultimately have "(s @ []) \<in> SEQ r1 r2 \<rightarrow> Seq (snd (simp r1) v) (snd (simp r2) Void)"
- by(rule_tac Posix_SEQ) auto
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v" using NONE_ONE by simp
- next
- case (NONE_NONE)
- from as have 00: "fst (simp r1) \<noteq> ZERO" "fst (simp r2) \<noteq> ZERO"
- apply(auto)
- apply(smt Posix_elims(1) fst_conv)
- by (smt NONE_NONE(2) Posix_elims(1) fstI)
- with NONE_NONE as have "s \<in> SEQ (fst (simp r1)) (fst (simp r2)) \<rightarrow> v" by simp
- then obtain s1 s2 v1 v2 where eqs: "s = s1 @ s2" "v = Seq v1 v2"
- "s1 \<in> (fst (simp r1)) \<rightarrow> v1" "s2 \<in> (fst (simp r2)) \<rightarrow> v2"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)"
- by (erule_tac Posix_elims(5)) (auto simp add: L_fst_simp[symmetric])
- then have "s1 \<in> r1 \<rightarrow> (snd (simp r1) v1)" "s2 \<in> r2 \<rightarrow> (snd (simp r2) v2)"
- using IH1 IH2 by auto
- then show "s \<in> SEQ r1 r2 \<rightarrow> snd (simp (SEQ r1 r2)) v" using eqs NONE_NONE 00
- by(auto intro: Posix_SEQ)
- qed
-qed (simp_all)
-
-
-lemma slexer_correctness:
- shows "slexer r s = lexer r s"
-proof(induct s arbitrary: r)
- case Nil
- show "slexer r [] = lexer r []" by simp
-next
- case (Cons c s r)
- have IH: "\<And>r. slexer r s = lexer r s" by fact
- show "slexer r (c # s) = lexer r (c # s)"
- proof (cases "s \<in> L (der c r)")
- case True
- assume a1: "s \<in> L (der c r)"
- then obtain v1 where a2: "lexer (der c r) s = Some v1" "s \<in> der c r \<rightarrow> v1"
- using lexer_correct_Some by auto
- from a1 have "s \<in> L (fst (simp (der c r)))" using L_fst_simp[symmetric] by simp
- then obtain v2 where a3: "lexer (fst (simp (der c r))) s = Some v2" "s \<in> (fst (simp (der c r))) \<rightarrow> v2"
- using lexer_correct_Some by auto
- then have a4: "slexer (fst (simp (der c r))) s = Some v2" using IH by simp
- from a3(2) have "s \<in> der c r \<rightarrow> (snd (simp (der c r))) v2" using Posix_simp by simp
- with a2(2) have "v1 = (snd (simp (der c r))) v2" using Posix_determ by simp
- with a2(1) a4 show "slexer r (c # s) = lexer r (c # s)" by (auto split: prod.split)
- next
- case False
- assume b1: "s \<notin> L (der c r)"
- then have "lexer (der c r) s = None" using lexer_correct_None by simp
- moreover
- from b1 have "s \<notin> L (fst (simp (der c r)))" using L_fst_simp[symmetric] by simp
- then have "lexer (fst (simp (der c r))) s = None" using lexer_correct_None by simp
- then have "slexer (fst (simp (der c r))) s = None" using IH by simp
- ultimately show "slexer r (c # s) = lexer r (c # s)"
- by (simp del: slexer.simps add: slexer_better_simp)
- qed
- qed
-
-
-unused_thms
-
-
-end
\ No newline at end of file
diff -r f493a20feeb3 -r 04b5e904a220 thys3/PDerivs.thy
--- a/thys3/PDerivs.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,603 +0,0 @@
-
-theory PDerivs
- imports PosixSpec
-begin
-
-
-
-abbreviation
- "SEQs rs r \<equiv> (\<Union>r' \<in> rs. {SEQ r' r})"
-
-lemma SEQs_eq_image:
- "SEQs rs r = (\<lambda>r'. SEQ r' r) ` rs"
- by auto
-
-fun
- pder :: "char \<Rightarrow> rexp \<Rightarrow> rexp set"
-where
- "pder c ZERO = {}"
-| "pder c ONE = {}"
-| "pder c (CH d) = (if c = d then {ONE} else {})"
-| "pder c (ALT r1 r2) = (pder c r1) \<union> (pder c r2)"
-| "pder c (SEQ r1 r2) =
- (if nullable r1 then SEQs (pder c r1) r2 \<union> pder c r2 else SEQs (pder c r1) r2)"
-| "pder c (STAR r) = SEQs (pder c r) (STAR r)"
-
-fun
- pders :: "char list \<Rightarrow> rexp \<Rightarrow> rexp set"
-where
- "pders [] r = {r}"
-| "pders (c # s) r = \<Union> (pders s ` pder c r)"
-
-abbreviation
- pder_set :: "char \<Rightarrow> rexp set \<Rightarrow> rexp set"
-where
- "pder_set c rs \<equiv> \<Union> (pder c ` rs)"
-
-abbreviation
- pders_set :: "char list \<Rightarrow> rexp set \<Rightarrow> rexp set"
-where
- "pders_set s rs \<equiv> \<Union> (pders s ` rs)"
-
-lemma pders_append:
- "pders (s1 @ s2) r = \<Union> (pders s2 ` pders s1 r)"
-by (induct s1 arbitrary: r) (simp_all)
-
-lemma pders_snoc:
- shows "pders (s @ [c]) r = pder_set c (pders s r)"
-by (simp add: pders_append)
-
-lemma pders_simps [simp]:
- shows "pders s ZERO = (if s = [] then {ZERO} else {})"
- and "pders s ONE = (if s = [] then {ONE} else {})"
- and "pders s (ALT r1 r2) = (if s = [] then {ALT r1 r2} else (pders s r1) \<union> (pders s r2))"
-by (induct s) (simp_all)
-
-lemma pders_CHAR:
- shows "pders s (CH c) \<subseteq> {CH c, ONE}"
-by (induct s) (simp_all)
-
-subsection \<open>Relating left-quotients and partial derivatives\<close>
-
-lemma Sequ_UNION_distrib:
-shows "A ;; \<Union>(M ` I) = \<Union>((\<lambda>i. A ;; M i) ` I)"
-and "\<Union>(M ` I) ;; A = \<Union>((\<lambda>i. M i ;; A) ` I)"
-by (auto simp add: Sequ_def)
-
-
-lemma Der_pder:
- shows "Der c (L r) = \<Union> (L ` pder c r)"
-by (induct r) (simp_all add: nullable_correctness Sequ_UNION_distrib)
-
-lemma Ders_pders:
- shows "Ders s (L r) = \<Union> (L ` pders s r)"
-proof (induct s arbitrary: r)
- case (Cons c s)
- have ih: "\<And>r. Ders s (L r) = \<Union> (L ` pders s r)" by fact
- have "Ders (c # s) (L r) = Ders s (Der c (L r))" by (simp add: Ders_def Der_def)
- also have "\<dots> = Ders s (\<Union> (L ` pder c r))" by (simp add: Der_pder)
- also have "\<dots> = (\<Union>A\<in>(L ` (pder c r)). (Ders s A))"
- by (auto simp add: Ders_def)
- also have "\<dots> = \<Union> (L ` (pders_set s (pder c r)))"
- using ih by auto
- also have "\<dots> = \<Union> (L ` (pders (c # s) r))" by simp
- finally show "Ders (c # s) (L r) = \<Union> (L ` pders (c # s) r)" .
-qed (simp add: Ders_def)
-
-subsection \<open>Relating derivatives and partial derivatives\<close>
-
-lemma der_pder:
- shows "\<Union> (L ` (pder c r)) = L (der c r)"
-unfolding der_correctness Der_pder by simp
-
-lemma ders_pders:
- shows "\<Union> (L ` (pders s r)) = L (ders s r)"
-unfolding der_correctness ders_correctness Ders_pders by simp
-
-
-subsection \<open>Finiteness property of partial derivatives\<close>
-
-definition
- pders_Set :: "string set \<Rightarrow> rexp \<Rightarrow> rexp set"
-where
- "pders_Set A r \<equiv> \<Union>x \<in> A. pders x r"
-
-lemma pders_Set_subsetI:
- assumes "\<And>s. s \<in> A \<Longrightarrow> pders s r \<subseteq> C"
- shows "pders_Set A r \<subseteq> C"
-using assms unfolding pders_Set_def by (rule UN_least)
-
-lemma pders_Set_union:
- shows "pders_Set (A \<union> B) r = (pders_Set A r \<union> pders_Set B r)"
-by (simp add: pders_Set_def)
-
-lemma pders_Set_subset:
- shows "A \<subseteq> B \<Longrightarrow> pders_Set A r \<subseteq> pders_Set B r"
-by (auto simp add: pders_Set_def)
-
-definition
- "UNIV1 \<equiv> UNIV - {[]}"
-
-lemma pders_Set_ZERO [simp]:
- shows "pders_Set UNIV1 ZERO = {}"
-unfolding UNIV1_def pders_Set_def by auto
-
-lemma pders_Set_ONE [simp]:
- shows "pders_Set UNIV1 ONE = {}"
-unfolding UNIV1_def pders_Set_def by (auto split: if_splits)
-
-lemma pders_Set_CHAR [simp]:
- shows "pders_Set UNIV1 (CH c) = {ONE}"
-unfolding UNIV1_def pders_Set_def
-apply(auto)
-apply(frule rev_subsetD)
-apply(rule pders_CHAR)
-apply(simp)
-apply(case_tac xa)
-apply(auto split: if_splits)
-done
-
-lemma pders_Set_ALT [simp]:
- shows "pders_Set UNIV1 (ALT r1 r2) = pders_Set UNIV1 r1 \<union> pders_Set UNIV1 r2"
-unfolding UNIV1_def pders_Set_def by auto
-
-
-text \<open>Non-empty suffixes of a string (needed for the cases of @{const SEQ} and @{const STAR} below)\<close>
-
-definition
- "PSuf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
-
-lemma PSuf_snoc:
- shows "PSuf (s @ [c]) = (PSuf s) ;; {[c]} \<union> {[c]}"
-unfolding PSuf_def Sequ_def
-by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)
-
-lemma PSuf_Union:
- shows "(\<Union>v \<in> PSuf s ;; {[c]}. f v) = (\<Union>v \<in> PSuf s. f (v @ [c]))"
-by (auto simp add: Sequ_def)
-
-lemma pders_Set_snoc:
- shows "pders_Set (PSuf s ;; {[c]}) r = (pder_set c (pders_Set (PSuf s) r))"
-unfolding pders_Set_def
-by (simp add: PSuf_Union pders_snoc)
-
-lemma pders_SEQ:
- shows "pders s (SEQ r1 r2) \<subseteq> SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2)"
-proof (induct s rule: rev_induct)
- case (snoc c s)
- have ih: "pders s (SEQ r1 r2) \<subseteq> SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2)"
- by fact
- have "pders (s @ [c]) (SEQ r1 r2) = pder_set c (pders s (SEQ r1 r2))"
- by (simp add: pders_snoc)
- also have "\<dots> \<subseteq> pder_set c (SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2))"
- using ih by fastforce
- also have "\<dots> = pder_set c (SEQs (pders s r1) r2) \<union> pder_set c (pders_Set (PSuf s) r2)"
- by (simp)
- also have "\<dots> = pder_set c (SEQs (pders s r1) r2) \<union> pders_Set (PSuf s ;; {[c]}) r2"
- by (simp add: pders_Set_snoc)
- also
- have "\<dots> \<subseteq> pder_set c (SEQs (pders s r1) r2) \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"
- by auto
- also
- have "\<dots> \<subseteq> SEQs (pder_set c (pders s r1)) r2 \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"
- by (auto simp add: if_splits)
- also have "\<dots> = SEQs (pders (s @ [c]) r1) r2 \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"
- by (simp add: pders_snoc)
- also have "\<dots> \<subseteq> SEQs (pders (s @ [c]) r1) r2 \<union> pders_Set (PSuf (s @ [c])) r2"
- unfolding pders_Set_def by (auto simp add: PSuf_snoc)
- finally show ?case .
-qed (simp)
-
-lemma pders_Set_SEQ_aux1:
- assumes a: "s \<in> UNIV1"
- shows "pders_Set (PSuf s) r \<subseteq> pders_Set UNIV1 r"
-using a unfolding UNIV1_def PSuf_def pders_Set_def by auto
-
-lemma pders_Set_SEQ_aux2:
- assumes a: "s \<in> UNIV1"
- shows "SEQs (pders s r1) r2 \<subseteq> SEQs (pders_Set UNIV1 r1) r2"
-using a unfolding pders_Set_def by auto
-
-lemma pders_Set_SEQ:
- shows "pders_Set UNIV1 (SEQ r1 r2) \<subseteq> SEQs (pders_Set UNIV1 r1) r2 \<union> pders_Set UNIV1 r2"
-apply(rule pders_Set_subsetI)
-apply(rule subset_trans)
-apply(rule pders_SEQ)
-using pders_Set_SEQ_aux1 pders_Set_SEQ_aux2
-apply auto
-apply blast
-done
-
-lemma pders_STAR:
- assumes a: "s \<noteq> []"
- shows "pders s (STAR r) \<subseteq> SEQs (pders_Set (PSuf s) r) (STAR r)"
-using a
-proof (induct s rule: rev_induct)
- case (snoc c s)
- have ih: "s \<noteq> [] \<Longrightarrow> pders s (STAR r) \<subseteq> SEQs (pders_Set (PSuf s) r) (STAR r)" by fact
- { assume asm: "s \<noteq> []"
- have "pders (s @ [c]) (STAR r) = pder_set c (pders s (STAR r))" by (simp add: pders_snoc)
- also have "\<dots> \<subseteq> pder_set c (SEQs (pders_Set (PSuf s) r) (STAR r))"
- using ih[OF asm] by fast
- also have "\<dots> \<subseteq> SEQs (pder_set c (pders_Set (PSuf s) r)) (STAR r) \<union> pder c (STAR r)"
- by (auto split: if_splits)
- also have "\<dots> \<subseteq> SEQs (pders_Set (PSuf (s @ [c])) r) (STAR r) \<union> (SEQs (pder c r) (STAR r))"
- by (simp only: PSuf_snoc pders_Set_snoc pders_Set_union)
- (auto simp add: pders_Set_def)
- also have "\<dots> = SEQs (pders_Set (PSuf (s @ [c])) r) (STAR r)"
- by (auto simp add: PSuf_snoc PSuf_Union pders_snoc pders_Set_def)
- finally have ?case .
- }
- moreover
- { assume asm: "s = []"
- then have ?case by (auto simp add: pders_Set_def pders_snoc PSuf_def)
- }
- ultimately show ?case by blast
-qed (simp)
-
-lemma pders_Set_STAR:
- shows "pders_Set UNIV1 (STAR r) \<subseteq> SEQs (pders_Set UNIV1 r) (STAR r)"
-apply(rule pders_Set_subsetI)
-apply(rule subset_trans)
-apply(rule pders_STAR)
-apply(simp add: UNIV1_def)
-apply(simp add: UNIV1_def PSuf_def)
-apply(auto simp add: pders_Set_def)
-done
-
-lemma finite_SEQs [simp]:
- assumes a: "finite A"
- shows "finite (SEQs A r)"
-using a by auto
-
-
-lemma finite_pders_Set_UNIV1:
- shows "finite (pders_Set UNIV1 r)"
-apply(induct r)
-apply(simp_all add:
- finite_subset[OF pders_Set_SEQ]
- finite_subset[OF pders_Set_STAR])
-done
-
-lemma pders_Set_UNIV:
- shows "pders_Set UNIV r = pders [] r \<union> pders_Set UNIV1 r"
-unfolding UNIV1_def pders_Set_def
-by blast
-
-lemma finite_pders_Set_UNIV:
- shows "finite (pders_Set UNIV r)"
-unfolding pders_Set_UNIV
-by (simp add: finite_pders_Set_UNIV1)
-
-lemma finite_pders_set:
- shows "finite (pders_Set A r)"
-by (metis finite_pders_Set_UNIV pders_Set_subset rev_finite_subset subset_UNIV)
-
-
-text \<open>The following relationship between the alphabetic width of regular expressions
-(called \<open>awidth\<close> below) and the number of partial derivatives was proved
-by Antimirov~\cite{Antimirov95} and formalized by Max Haslbeck.\<close>
-
-fun awidth :: "rexp \<Rightarrow> nat" where
-"awidth ZERO = 0" |
-"awidth ONE = 0" |
-"awidth (CH a) = 1" |
-"awidth (ALT r1 r2) = awidth r1 + awidth r2" |
-"awidth (SEQ r1 r2) = awidth r1 + awidth r2" |
-"awidth (STAR r1) = awidth r1"
-
-lemma card_SEQs_pders_Set_le:
- shows "card (SEQs (pders_Set A r) s) \<le> card (pders_Set A r)"
- using finite_pders_set
- unfolding SEQs_eq_image
- by (rule card_image_le)
-
-lemma card_pders_set_UNIV1_le_awidth:
- shows "card (pders_Set UNIV1 r) \<le> awidth r"
-proof (induction r)
- case (ALT r1 r2)
- have "card (pders_Set UNIV1 (ALT r1 r2)) = card (pders_Set UNIV1 r1 \<union> pders_Set UNIV1 r2)" by simp
- also have "\<dots> \<le> card (pders_Set UNIV1 r1) + card (pders_Set UNIV1 r2)"
- by(simp add: card_Un_le)
- also have "\<dots> \<le> awidth (ALT r1 r2)" using ALT.IH by simp
- finally show ?case .
-next
- case (SEQ r1 r2)
- have "card (pders_Set UNIV1 (SEQ r1 r2)) \<le> card (SEQs (pders_Set UNIV1 r1) r2 \<union> pders_Set UNIV1 r2)"
- by (simp add: card_mono finite_pders_set pders_Set_SEQ)
- also have "\<dots> \<le> card (SEQs (pders_Set UNIV1 r1) r2) + card (pders_Set UNIV1 r2)"
- by (simp add: card_Un_le)
- also have "\<dots> \<le> card (pders_Set UNIV1 r1) + card (pders_Set UNIV1 r2)"
- by (simp add: card_SEQs_pders_Set_le)
- also have "\<dots> \<le> awidth (SEQ r1 r2)" using SEQ.IH by simp
- finally show ?case .
-next
- case (STAR r)
- have "card (pders_Set UNIV1 (STAR r)) \<le> card (SEQs (pders_Set UNIV1 r) (STAR r))"
- by (simp add: card_mono finite_pders_set pders_Set_STAR)
- also have "\<dots> \<le> card (pders_Set UNIV1 r)" by (rule card_SEQs_pders_Set_le)
- also have "\<dots> \<le> awidth (STAR r)" by (simp add: STAR.IH)
- finally show ?case .
-qed (auto)
-
-text\<open>Antimirov's Theorem 3.4:\<close>
-
-theorem card_pders_set_UNIV_le_awidth:
- shows "card (pders_Set UNIV r) \<le> awidth r + 1"
-proof -
- have "card (insert r (pders_Set UNIV1 r)) \<le> Suc (card (pders_Set UNIV1 r))"
- by(auto simp: card_insert_if[OF finite_pders_Set_UNIV1])
- also have "\<dots> \<le> Suc (awidth r)" by(simp add: card_pders_set_UNIV1_le_awidth)
- finally show ?thesis by(simp add: pders_Set_UNIV)
-qed
-
-text\<open>Antimirov's Corollary 3.5:\<close>
-(*W stands for word set*)
-corollary card_pders_set_le_awidth:
- shows "card (pders_Set W r) \<le> awidth r + 1"
-proof -
- have "card (pders_Set W r) \<le> card (pders_Set UNIV r)"
- by (simp add: card_mono finite_pders_set pders_Set_subset)
- also have "... \<le> awidth r + 1"
- by (rule card_pders_set_UNIV_le_awidth)
- finally show "card (pders_Set W r) \<le> awidth r + 1" by simp
-qed
-
-(* other result by antimirov *)
-
-lemma card_pders_awidth:
- shows "card (pders s r) \<le> awidth r + 1"
-proof -
- have "pders s r \<subseteq> pders_Set UNIV r"
- using pders_Set_def by auto
- then have "card (pders s r) \<le> card (pders_Set UNIV r)"
- by (simp add: card_mono finite_pders_set)
- then show "card (pders s r) \<le> awidth r + 1"
- using card_pders_set_le_awidth order_trans by blast
-qed
-
-
-
-
-
-fun subs :: "rexp \<Rightarrow> rexp set" where
-"subs ZERO = {ZERO}" |
-"subs ONE = {ONE}" |
-"subs (CH a) = {CH a, ONE}" |
-"subs (ALT r1 r2) = (subs r1 \<union> subs r2 \<union> {ALT r1 r2})" |
-"subs (SEQ r1 r2) = (subs r1 \<union> subs r2 \<union> {SEQ r1 r2} \<union> SEQs (subs r1) r2)" |
-"subs (STAR r1) = (subs r1 \<union> {STAR r1} \<union> SEQs (subs r1) (STAR r1))"
-
-lemma subs_finite:
- shows "finite (subs r)"
- apply(induct r)
- apply(simp_all)
- done
-
-
-
-lemma pders_subs:
- shows "pders s r \<subseteq> subs r"
- apply(induct r arbitrary: s)
- apply(simp)
- apply(simp)
- apply(simp add: pders_CHAR)
-(* SEQ case *)
- apply(simp)
- apply(rule subset_trans)
- apply(rule pders_SEQ)
- defer
-(* ALT case *)
- apply(simp)
- apply(rule impI)
- apply(rule conjI)
- apply blast
- apply blast
-(* STAR case *)
- apply(case_tac s)
- apply(simp)
- apply(rule subset_trans)
- thm pders_STAR
- apply(rule pders_STAR)
- apply(simp)
- apply(auto simp add: pders_Set_def)[1]
- apply(simp)
- apply(rule conjI)
- apply blast
-apply(auto simp add: pders_Set_def)[1]
- done
-
-fun size2 :: "rexp \<Rightarrow> nat" where
- "size2 ZERO = 1" |
- "size2 ONE = 1" |
- "size2 (CH c) = 1" |
- "size2 (ALT r1 r2) = Suc (size2 r1 + size2 r2)" |
- "size2 (SEQ r1 r2) = Suc (size2 r1 + size2 r2)" |
- "size2 (STAR r1) = Suc (size2 r1)"
-
-
-lemma size_rexp:
- fixes r :: rexp
- shows "1 \<le> size2 r"
- apply(induct r)
- apply(simp)
- apply(simp_all)
- done
-
-lemma subs_size2:
- shows "\<forall>r1 \<in> subs r. size2 r1 \<le> Suc (size2 r * size2 r)"
- apply(induct r)
- apply(simp)
- apply(simp)
- apply(simp)
-(* SEQ case *)
- apply(simp)
- apply(auto)[1]
- apply (smt Suc_n_not_le_n add.commute distrib_left le_Suc_eq left_add_mult_distrib nat_le_linear trans_le_add1)
- apply (smt Suc_le_mono Suc_n_not_le_n le_trans nat_le_linear power2_eq_square power2_sum semiring_normalization_rules(23) trans_le_add2)
- apply (smt Groups.add_ac(3) Suc_n_not_le_n distrib_left le_Suc_eq left_add_mult_distrib nat_le_linear trans_le_add1)
-(* ALT case *)
- apply(simp)
- apply(auto)[1]
- apply (smt Groups.add_ac(2) Suc_le_mono Suc_n_not_le_n le_add2 linear order_trans power2_eq_square power2_sum)
- apply (smt Groups.add_ac(2) Suc_le_mono Suc_n_not_le_n left_add_mult_distrib linear mult.commute order.trans trans_le_add1)
-(* STAR case *)
- apply(auto)[1]
- apply(drule_tac x="r'" in bspec)
- apply(simp)
- apply(rule le_trans)
- apply(assumption)
- apply(simp)
- using size_rexp
- apply(simp)
- done
-
-lemma awidth_size:
- shows "awidth r \<le> size2 r"
- apply(induct r)
- apply(simp_all)
- done
-
-lemma Sum1:
- fixes A B :: "nat set"
- assumes "A \<subseteq> B" "finite A" "finite B"
- shows "\<Sum>A \<le> \<Sum>B"
- using assms
- by (simp add: sum_mono2)
-
-lemma Sum2:
- fixes A :: "rexp set"
- and f g :: "rexp \<Rightarrow> nat"
- assumes "finite A" "\<forall>x \<in> A. f x \<le> g x"
- shows "sum f A \<le> sum g A"
- using assms
- apply(induct A)
- apply(auto)
- done
-
-
-
-
-
-lemma pders_max_size:
- shows "(sum size2 (pders s r)) \<le> (Suc (size2 r)) ^ 3"
-proof -
- have "(sum size2 (pders s r)) \<le> sum (\<lambda>_. Suc (size2 r * size2 r)) (pders s r)"
- apply(rule_tac Sum2)
- apply (meson pders_subs rev_finite_subset subs_finite)
- using pders_subs subs_size2 by blast
- also have "... \<le> (Suc (size2 r * size2 r)) * (sum (\<lambda>_. 1) (pders s r))"
- by simp
- also have "... \<le> (Suc (size2 r * size2 r)) * card (pders s r)"
- by simp
- also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (awidth r))"
- using Suc_eq_plus1 card_pders_awidth mult_le_mono2 by presburger
- also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (size2 r))"
- using Suc_le_mono awidth_size mult_le_mono2 by presburger
- also have "... \<le> (Suc (size2 r)) ^ 3"
- by (smt One_nat_def Suc_1 Suc_mult_le_cancel1 Suc_n_not_le_n antisym_conv le_Suc_eq mult.commute nat_le_linear numeral_3_eq_3 power2_eq_square power2_le_imp_le power_Suc size_rexp)
- finally show ?thesis .
-qed
-
-lemma pders_Set_max_size:
- shows "(sum size2 (pders_Set A r)) \<le> (Suc (size2 r)) ^ 3"
-proof -
- have "(sum size2 (pders_Set A r)) \<le> sum (\<lambda>_. Suc (size2 r * size2 r)) (pders_Set A r)"
- apply(rule_tac Sum2)
- apply (simp add: finite_pders_set)
- by (meson pders_Set_subsetI pders_subs subs_size2 subsetD)
- also have "... \<le> (Suc (size2 r * size2 r)) * (sum (\<lambda>_. 1) (pders_Set A r))"
- by simp
- also have "... \<le> (Suc (size2 r * size2 r)) * card (pders_Set A r)"
- by simp
- also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (awidth r))"
- using Suc_eq_plus1 card_pders_set_le_awidth mult_le_mono2 by presburger
- also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (size2 r))"
- using Suc_le_mono awidth_size mult_le_mono2 by presburger
- also have "... \<le> (Suc (size2 r)) ^ 3"
- by (smt One_nat_def Suc_1 Suc_mult_le_cancel1 Suc_n_not_le_n antisym_conv le_Suc_eq mult.commute nat_le_linear numeral_3_eq_3 power2_eq_square power2_le_imp_le power_Suc size_rexp)
- finally show ?thesis .
-qed
-
-fun height :: "rexp \<Rightarrow> nat" where
- "height ZERO = 1" |
- "height ONE = 1" |
- "height (CH c) = 1" |
- "height (ALT r1 r2) = Suc (max (height r1) (height r2))" |
- "height (SEQ r1 r2) = Suc (max (height r1) (height r2))" |
- "height (STAR r1) = Suc (height r1)"
-
-lemma height_size2:
- shows "height r \<le> size2 r"
- apply(induct r)
- apply(simp_all)
- done
-
-lemma height_rexp:
- fixes r :: rexp
- shows "1 \<le> height r"
- apply(induct r)
- apply(simp_all)
- done
-
-lemma subs_height:
- shows "\<forall>r1 \<in> subs r. height r1 \<le> Suc (height r)"
- apply(induct r)
- apply(auto)+
- done
-
-fun lin_concat :: "(char \<times> rexp) \<Rightarrow> rexp \<Rightarrow> (char \<times> rexp)" (infixl "[.]" 91)
- where
-"(c, ZERO) [.] t = (c, ZERO)"
-| "(c, ONE) [.] t = (c, t)"
-| "(c, p) [.] t = (c, SEQ p t)"
-
-
-fun circle_concat :: "(char \<times> rexp ) set \<Rightarrow> rexp \<Rightarrow> (char \<times> rexp) set" ( infixl "\<circle>" 90)
- where
-"l \<circle> ZERO = {}"
-| "l \<circle> ONE = l"
-| "l \<circle> t = ( (\<lambda>p. p [.] t) ` l ) "
-
-
-
-fun linear_form :: "rexp \<Rightarrow>( char \<times> rexp ) set"
- where
- "linear_form ZERO = {}"
-| "linear_form ONE = {}"
-| "linear_form (CH c) = {(c, ONE)}"
-| "linear_form (ALT r1 r2) = (linear_form) r1 \<union> (linear_form r2)"
-| "linear_form (SEQ r1 r2) = (if (nullable r1) then (linear_form r1) \<circle> r2 \<union> linear_form r2 else (linear_form r1) \<circle> r2) "
-| "linear_form (STAR r ) = (linear_form r) \<circle> (STAR r)"
-
-
-value "linear_form (SEQ (STAR (CH x)) (STAR (ALT (SEQ (CH x) (CH x)) (CH y) )) )"
-
-
-value "linear_form (STAR (ALT (SEQ (CH x) (CH x)) (CH y) )) "
-
-fun pdero :: "char \<Rightarrow> rexp \<Rightarrow> rexp set"
- where
-"pdero c t = \<Union> ((\<lambda>(d, p). if d = c then {p} else {}) ` (linear_form t) )"
-
-fun pderso :: "char list \<Rightarrow> rexp \<Rightarrow> rexp set"
- where
- "pderso [] r = {r}"
-| "pderso (c # s) r = \<Union> ( pderso s ` (pdero c r) )"
-
-lemma pdero_result:
- shows "pdero c (STAR (ALT (CH c) (SEQ (CH c) (CH c)))) = {SEQ (CH c)(STAR (ALT (CH c) (SEQ (CH c) (CH c)))),(STAR (ALT (CH c) (SEQ (CH c) (CH c))))}"
- apply(simp)
- by auto
-
-fun concatLen :: "rexp \<Rightarrow> nat" where
-"concatLen ZERO = 0" |
-"concatLen ONE = 0" |
-"concatLen (CH c) = 0" |
-"concatLen (SEQ v1 v2) = Suc (max (concatLen v1) (concatLen v2))" |
-" concatLen (ALT v1 v2) = max (concatLen v1) (concatLen v2)" |
-" concatLen (STAR v) = Suc (concatLen v)"
-
-
-
-end
\ No newline at end of file
diff -r f493a20feeb3 -r 04b5e904a220 thys3/Positions.thy
--- a/thys3/Positions.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,773 +0,0 @@
-
-theory Positions
- imports PosixSpec Lexer
-begin
-
-chapter \<open>An alternative definition for POSIX values\<close>
-
-section \<open>Positions in Values\<close>
-
-fun
- at :: "val \<Rightarrow> nat list \<Rightarrow> val"
-where
- "at v [] = v"
-| "at (Left v) (0#ps)= at v ps"
-| "at (Right v) (Suc 0#ps)= at v ps"
-| "at (Seq v1 v2) (0#ps)= at v1 ps"
-| "at (Seq v1 v2) (Suc 0#ps)= at v2 ps"
-| "at (Stars vs) (n#ps)= at (nth vs n) ps"
-
-
-
-fun Pos :: "val \<Rightarrow> (nat list) set"
-where
- "Pos (Void) = {[]}"
-| "Pos (Char c) = {[]}"
-| "Pos (Left v) = {[]} \<union> {0#ps | ps. ps \<in> Pos v}"
-| "Pos (Right v) = {[]} \<union> {1#ps | ps. ps \<in> Pos v}"
-| "Pos (Seq v1 v2) = {[]} \<union> {0#ps | ps. ps \<in> Pos v1} \<union> {1#ps | ps. ps \<in> Pos v2}"
-| "Pos (Stars []) = {[]}"
-| "Pos (Stars (v#vs)) = {[]} \<union> {0#ps | ps. ps \<in> Pos v} \<union> {Suc n#ps | n ps. n#ps \<in> Pos (Stars vs)}"
-
-
-lemma Pos_stars:
- "Pos (Stars vs) = {[]} \<union> (\<Union>n < length vs. {n#ps | ps. ps \<in> Pos (vs ! n)})"
-apply(induct vs)
-apply(auto simp add: insert_ident less_Suc_eq_0_disj)
-done
-
-lemma Pos_empty:
- shows "[] \<in> Pos v"
-by (induct v rule: Pos.induct)(auto)
-
-
-abbreviation
- "intlen vs \<equiv> int (length vs)"
-
-
-definition pflat_len :: "val \<Rightarrow> nat list => int"
-where
- "pflat_len v p \<equiv> (if p \<in> Pos v then intlen (flat (at v p)) else -1)"
-
-lemma pflat_len_simps:
- shows "pflat_len (Seq v1 v2) (0#p) = pflat_len v1 p"
- and "pflat_len (Seq v1 v2) (Suc 0#p) = pflat_len v2 p"
- and "pflat_len (Left v) (0#p) = pflat_len v p"
- and "pflat_len (Left v) (Suc 0#p) = -1"
- and "pflat_len (Right v) (Suc 0#p) = pflat_len v p"
- and "pflat_len (Right v) (0#p) = -1"
- and "pflat_len (Stars (v#vs)) (Suc n#p) = pflat_len (Stars vs) (n#p)"
- and "pflat_len (Stars (v#vs)) (0#p) = pflat_len v p"
- and "pflat_len v [] = intlen (flat v)"
-by (auto simp add: pflat_len_def Pos_empty)
-
-lemma pflat_len_Stars_simps:
- assumes "n < length vs"
- shows "pflat_len (Stars vs) (n#p) = pflat_len (vs!n) p"
-using assms
-apply(induct vs arbitrary: n p)
-apply(auto simp add: less_Suc_eq_0_disj pflat_len_simps)
-done
-
-lemma pflat_len_outside:
- assumes "p \<notin> Pos v1"
- shows "pflat_len v1 p = -1 "
-using assms by (simp add: pflat_len_def)
-
-
-
-section \<open>Orderings\<close>
-
-
-definition prefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubseteq>pre _" [60,59] 60)
-where
- "ps1 \<sqsubseteq>pre ps2 \<equiv> \<exists>ps'. ps1 @ps' = ps2"
-
-definition sprefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubset>spre _" [60,59] 60)
-where
- "ps1 \<sqsubset>spre ps2 \<equiv> ps1 \<sqsubseteq>pre ps2 \<and> ps1 \<noteq> ps2"
-
-inductive lex_list :: "nat list \<Rightarrow> nat list \<Rightarrow> bool" ("_ \<sqsubset>lex _" [60,59] 60)
-where
- "[] \<sqsubset>lex (p#ps)"
-| "ps1 \<sqsubset>lex ps2 \<Longrightarrow> (p#ps1) \<sqsubset>lex (p#ps2)"
-| "p1 < p2 \<Longrightarrow> (p1#ps1) \<sqsubset>lex (p2#ps2)"
-
-lemma lex_irrfl:
- fixes ps1 ps2 :: "nat list"
- assumes "ps1 \<sqsubset>lex ps2"
- shows "ps1 \<noteq> ps2"
-using assms
-by(induct rule: lex_list.induct)(auto)
-
-lemma lex_simps [simp]:
- fixes xs ys :: "nat list"
- shows "[] \<sqsubset>lex ys \<longleftrightarrow> ys \<noteq> []"
- and "xs \<sqsubset>lex [] \<longleftrightarrow> False"
- and "(x # xs) \<sqsubset>lex (y # ys) \<longleftrightarrow> (x < y \<or> (x = y \<and> xs \<sqsubset>lex ys))"
-by (auto simp add: neq_Nil_conv elim: lex_list.cases intro: lex_list.intros)
-
-lemma lex_trans:
- fixes ps1 ps2 ps3 :: "nat list"
- assumes "ps1 \<sqsubset>lex ps2" "ps2 \<sqsubset>lex ps3"
- shows "ps1 \<sqsubset>lex ps3"
-using assms
-by (induct arbitrary: ps3 rule: lex_list.induct)
- (auto elim: lex_list.cases)
-
-
-lemma lex_trichotomous:
- fixes p q :: "nat list"
- shows "p = q \<or> p \<sqsubset>lex q \<or> q \<sqsubset>lex p"
-apply(induct p arbitrary: q)
-apply(auto elim: lex_list.cases)
-apply(case_tac q)
-apply(auto)
-done
-
-
-
-
-section \<open>POSIX Ordering of Values According to Okui \& Suzuki\<close>
-
-
-definition PosOrd:: "val \<Rightarrow> nat list \<Rightarrow> val \<Rightarrow> bool" ("_ \<sqsubset>val _ _" [60, 60, 59] 60)
-where
- "v1 \<sqsubset>val p v2 \<equiv> pflat_len v1 p > pflat_len v2 p \<and>
- (\<forall>q \<in> Pos v1 \<union> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
-
-lemma PosOrd_def2:
- shows "v1 \<sqsubset>val p v2 \<longleftrightarrow>
- pflat_len v1 p > pflat_len v2 p \<and>
- (\<forall>q \<in> Pos v1. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q) \<and>
- (\<forall>q \<in> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
-unfolding PosOrd_def
-apply(auto)
-done
-
-
-definition PosOrd_ex:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubset>val _" [60, 59] 60)
-where
- "v1 :\<sqsubset>val v2 \<equiv> \<exists>p. v1 \<sqsubset>val p v2"
-
-definition PosOrd_ex_eq:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubseteq>val _" [60, 59] 60)
-where
- "v1 :\<sqsubseteq>val v2 \<equiv> v1 :\<sqsubset>val v2 \<or> v1 = v2"
-
-
-lemma PosOrd_trans:
- assumes "v1 :\<sqsubset>val v2" "v2 :\<sqsubset>val v3"
- shows "v1 :\<sqsubset>val v3"
-proof -
- from assms obtain p p'
- where as: "v1 \<sqsubset>val p v2" "v2 \<sqsubset>val p' v3" unfolding PosOrd_ex_def by blast
- then have pos: "p \<in> Pos v1" "p' \<in> Pos v2" unfolding PosOrd_def pflat_len_def
- by (smt not_int_zless_negative)+
- have "p = p' \<or> p \<sqsubset>lex p' \<or> p' \<sqsubset>lex p"
- by (rule lex_trichotomous)
- moreover
- { assume "p = p'"
- with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def
- by (smt Un_iff)
- then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
- }
- moreover
- { assume "p \<sqsubset>lex p'"
- with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def
- by (smt Un_iff lex_trans)
- then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
- }
- moreover
- { assume "p' \<sqsubset>lex p"
- with as have "v1 \<sqsubset>val p' v3" unfolding PosOrd_def
- by (smt Un_iff lex_trans pflat_len_def)
- then have "v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
- }
- ultimately show "v1 :\<sqsubset>val v3" by blast
-qed
-
-lemma PosOrd_irrefl:
- assumes "v :\<sqsubset>val v"
- shows "False"
-using assms unfolding PosOrd_ex_def PosOrd_def
-by auto
-
-lemma PosOrd_assym:
- assumes "v1 :\<sqsubset>val v2"
- shows "\<not>(v2 :\<sqsubset>val v1)"
-using assms
-using PosOrd_irrefl PosOrd_trans by blast
-
-(*
- :\<sqsubseteq>val and :\<sqsubset>val are partial orders.
-*)
-
-lemma PosOrd_ordering:
- shows "ordering (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)"
-unfolding ordering_def PosOrd_ex_eq_def
-apply(auto)
-using PosOrd_trans partial_preordering_def apply blast
-using PosOrd_assym ordering_axioms_def by blast
-
-lemma PosOrd_order:
- shows "class.order (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)"
-using PosOrd_ordering
-apply(simp add: class.order_def class.preorder_def class.order_axioms_def)
- by (metis (full_types) PosOrd_ex_eq_def PosOrd_irrefl PosOrd_trans)
-
-
-lemma PosOrd_ex_eq2:
- shows "v1 :\<sqsubset>val v2 \<longleftrightarrow> (v1 :\<sqsubseteq>val v2 \<and> v1 \<noteq> v2)"
- using PosOrd_ordering
- using PosOrd_ex_eq_def PosOrd_irrefl by blast
-
-lemma PosOrdeq_trans:
- assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v3"
- shows "v1 :\<sqsubseteq>val v3"
-using assms PosOrd_ordering
- unfolding ordering_def
- by (metis partial_preordering.trans)
-
-lemma PosOrdeq_antisym:
- assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v1"
- shows "v1 = v2"
-using assms PosOrd_ordering
- unfolding ordering_def
- by (simp add: ordering_axioms_def)
-
-lemma PosOrdeq_refl:
- shows "v :\<sqsubseteq>val v"
-unfolding PosOrd_ex_eq_def
-by auto
-
-
-lemma PosOrd_shorterE:
- assumes "v1 :\<sqsubset>val v2"
- shows "length (flat v2) \<le> length (flat v1)"
-using assms unfolding PosOrd_ex_def PosOrd_def
-apply(auto)
-apply(case_tac p)
-apply(simp add: pflat_len_simps)
-apply(drule_tac x="[]" in bspec)
-apply(simp add: Pos_empty)
-apply(simp add: pflat_len_simps)
-done
-
-lemma PosOrd_shorterI:
- assumes "length (flat v2) < length (flat v1)"
- shows "v1 :\<sqsubset>val v2"
-unfolding PosOrd_ex_def PosOrd_def pflat_len_def
-using assms Pos_empty by force
-
-lemma PosOrd_spreI:
- assumes "flat v' \<sqsubset>spre flat v"
- shows "v :\<sqsubset>val v'"
-using assms
-apply(rule_tac PosOrd_shorterI)
-unfolding prefix_list_def sprefix_list_def
-by (metis append_Nil2 append_eq_conv_conj drop_all le_less_linear)
-
-lemma pflat_len_inside:
- assumes "pflat_len v2 p < pflat_len v1 p"
- shows "p \<in> Pos v1"
-using assms
-unfolding pflat_len_def
-by (auto split: if_splits)
-
-
-lemma PosOrd_Left_Right:
- assumes "flat v1 = flat v2"
- shows "Left v1 :\<sqsubset>val Right v2"
-unfolding PosOrd_ex_def
-apply(rule_tac x="[0]" in exI)
-apply(auto simp add: PosOrd_def pflat_len_simps assms)
-done
-
-lemma PosOrd_LeftE:
- assumes "Left v1 :\<sqsubset>val Left v2" "flat v1 = flat v2"
- shows "v1 :\<sqsubset>val v2"
-using assms
-unfolding PosOrd_ex_def PosOrd_def2
-apply(auto simp add: pflat_len_simps)
-apply(frule pflat_len_inside)
-apply(auto simp add: pflat_len_simps)
-by (metis lex_simps(3) pflat_len_simps(3))
-
-lemma PosOrd_LeftI:
- assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2"
- shows "Left v1 :\<sqsubset>val Left v2"
-using assms
-unfolding PosOrd_ex_def PosOrd_def2
-apply(auto simp add: pflat_len_simps)
-by (metis less_numeral_extra(3) lex_simps(3) pflat_len_simps(3))
-
-lemma PosOrd_Left_eq:
- assumes "flat v1 = flat v2"
- shows "Left v1 :\<sqsubset>val Left v2 \<longleftrightarrow> v1 :\<sqsubset>val v2"
-using assms PosOrd_LeftE PosOrd_LeftI
-by blast
-
-
-lemma PosOrd_RightE:
- assumes "Right v1 :\<sqsubset>val Right v2" "flat v1 = flat v2"
- shows "v1 :\<sqsubset>val v2"
-using assms
-unfolding PosOrd_ex_def PosOrd_def2
-apply(auto simp add: pflat_len_simps)
-apply(frule pflat_len_inside)
-apply(auto simp add: pflat_len_simps)
-by (metis lex_simps(3) pflat_len_simps(5))
-
-lemma PosOrd_RightI:
- assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2"
- shows "Right v1 :\<sqsubset>val Right v2"
-using assms
-unfolding PosOrd_ex_def PosOrd_def2
-apply(auto simp add: pflat_len_simps)
-by (metis lex_simps(3) nat_neq_iff pflat_len_simps(5))
-
-
-lemma PosOrd_Right_eq:
- assumes "flat v1 = flat v2"
- shows "Right v1 :\<sqsubset>val Right v2 \<longleftrightarrow> v1 :\<sqsubset>val v2"
-using assms PosOrd_RightE PosOrd_RightI
-by blast
-
-
-lemma PosOrd_SeqI1:
- assumes "v1 :\<sqsubset>val w1" "flat (Seq v1 v2) = flat (Seq w1 w2)"
- shows "Seq v1 v2 :\<sqsubset>val Seq w1 w2"
-using assms(1)
-apply(subst (asm) PosOrd_ex_def)
-apply(subst (asm) PosOrd_def)
-apply(clarify)
-apply(subst PosOrd_ex_def)
-apply(rule_tac x="0#p" in exI)
-apply(subst PosOrd_def)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-apply(rule ballI)
-apply(rule impI)
-apply(simp only: Pos.simps)
-apply(auto)[1]
-apply(simp add: pflat_len_simps)
-apply(auto simp add: pflat_len_simps)
-using assms(2)
-apply(simp)
-apply(metis length_append of_nat_add)
-done
-
-lemma PosOrd_SeqI2:
- assumes "v2 :\<sqsubset>val w2" "flat v2 = flat w2"
- shows "Seq v v2 :\<sqsubset>val Seq v w2"
-using assms(1)
-apply(subst (asm) PosOrd_ex_def)
-apply(subst (asm) PosOrd_def)
-apply(clarify)
-apply(subst PosOrd_ex_def)
-apply(rule_tac x="Suc 0#p" in exI)
-apply(subst PosOrd_def)
-apply(rule conjI)
-apply(simp add: pflat_len_simps)
-apply(rule ballI)
-apply(rule impI)
-apply(simp only: Pos.simps)
-apply(auto)[1]
-apply(simp add: pflat_len_simps)
-using assms(2)
-apply(simp)
-apply(auto simp add: pflat_len_simps)
-done
-
-lemma PosOrd_Seq_eq:
- assumes "flat v2 = flat w2"
- shows "(Seq v v2) :\<sqsubset>val (Seq v w2) \<longleftrightarrow> v2 :\<sqsubset>val w2"
-using assms
-apply(auto)
-prefer 2
-apply(simp add: PosOrd_SeqI2)
-apply(simp add: PosOrd_ex_def)
-apply(auto)
-apply(case_tac p)
-apply(simp add: PosOrd_def pflat_len_simps)
-apply(case_tac a)
-apply(simp add: PosOrd_def pflat_len_simps)
-apply(clarify)
-apply(case_tac nat)
-prefer 2
-apply(simp add: PosOrd_def pflat_len_simps pflat_len_outside)
-apply(rule_tac x="list" in exI)
-apply(auto simp add: PosOrd_def2 pflat_len_simps)
-apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2))
-apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2))
-done
-
-
-
-lemma PosOrd_StarsI:
- assumes "v1 :\<sqsubset>val v2" "flats (v1#vs1) = flats (v2#vs2)"
- shows "Stars (v1#vs1) :\<sqsubset>val Stars (v2#vs2)"
-using assms(1)
-apply(subst (asm) PosOrd_ex_def)
-apply(subst (asm) PosOrd_def)
-apply(clarify)
-apply(subst PosOrd_ex_def)
-apply(subst PosOrd_def)
-apply(rule_tac x="0#p" in exI)
-apply(simp add: pflat_len_Stars_simps pflat_len_simps)
-using assms(2)
-apply(simp add: pflat_len_simps)
-apply(auto simp add: pflat_len_Stars_simps pflat_len_simps)
-by (metis length_append of_nat_add)
-
-lemma PosOrd_StarsI2:
- assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flats vs1 = flats vs2"
- shows "Stars (v#vs1) :\<sqsubset>val Stars (v#vs2)"
-using assms(1)
-apply(subst (asm) PosOrd_ex_def)
-apply(subst (asm) PosOrd_def)
-apply(clarify)
-apply(subst PosOrd_ex_def)
-apply(subst PosOrd_def)
-apply(case_tac p)
-apply(simp add: pflat_len_simps)
-apply(rule_tac x="Suc a#list" in exI)
-apply(auto simp add: pflat_len_Stars_simps pflat_len_simps assms(2))
-done
-
-lemma PosOrd_Stars_appendI:
- assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flat (Stars vs1) = flat (Stars vs2)"
- shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
-using assms
-apply(induct vs)
-apply(simp)
-apply(simp add: PosOrd_StarsI2)
-done
-
-lemma PosOrd_StarsE2:
- assumes "Stars (v # vs1) :\<sqsubset>val Stars (v # vs2)"
- shows "Stars vs1 :\<sqsubset>val Stars vs2"
-using assms
-apply(subst (asm) PosOrd_ex_def)
-apply(erule exE)
-apply(case_tac p)
-apply(simp)
-apply(simp add: PosOrd_def pflat_len_simps)
-apply(subst PosOrd_ex_def)
-apply(rule_tac x="[]" in exI)
-apply(simp add: PosOrd_def pflat_len_simps Pos_empty)
-apply(simp)
-apply(case_tac a)
-apply(clarify)
-apply(auto simp add: pflat_len_simps PosOrd_def pflat_len_def split: if_splits)[1]
-apply(clarify)
-apply(simp add: PosOrd_ex_def)
-apply(rule_tac x="nat#list" in exI)
-apply(auto simp add: PosOrd_def pflat_len_simps)[1]
-apply(case_tac q)
-apply(simp add: PosOrd_def pflat_len_simps)
-apply(clarify)
-apply(drule_tac x="Suc a # lista" in bspec)
-apply(simp)
-apply(auto simp add: PosOrd_def pflat_len_simps)[1]
-apply(case_tac q)
-apply(simp add: PosOrd_def pflat_len_simps)
-apply(clarify)
-apply(drule_tac x="Suc a # lista" in bspec)
-apply(simp)
-apply(auto simp add: PosOrd_def pflat_len_simps)[1]
-done
-
-lemma PosOrd_Stars_appendE:
- assumes "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
- shows "Stars vs1 :\<sqsubset>val Stars vs2"
-using assms
-apply(induct vs)
-apply(simp)
-apply(simp add: PosOrd_StarsE2)
-done
-
-lemma PosOrd_Stars_append_eq:
- assumes "flats vs1 = flats vs2"
- shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2) \<longleftrightarrow> Stars vs1 :\<sqsubset>val Stars vs2"
-using assms
-apply(rule_tac iffI)
-apply(erule PosOrd_Stars_appendE)
-apply(rule PosOrd_Stars_appendI)
-apply(auto)
-done
-
-lemma PosOrd_almost_trichotomous:
- shows "v1 :\<sqsubset>val v2 \<or> v2 :\<sqsubset>val v1 \<or> (length (flat v1) = length (flat v2))"
-apply(auto simp add: PosOrd_ex_def)
-apply(auto simp add: PosOrd_def)
-apply(rule_tac x="[]" in exI)
-apply(auto simp add: Pos_empty pflat_len_simps)
-apply(drule_tac x="[]" in spec)
-apply(auto simp add: Pos_empty pflat_len_simps)
-done
-
-
-
-section \<open>The Posix Value is smaller than any other Value\<close>
-
-
-lemma Posix_PosOrd:
- assumes "s \<in> r \<rightarrow> v1" "v2 \<in> LV r s"
- shows "v1 :\<sqsubseteq>val v2"
-using assms
-proof (induct arbitrary: v2 rule: Posix.induct)
- case (Posix_ONE v)
- have "v \<in> LV ONE []" by fact
- then have "v = Void"
- by (simp add: LV_simps)
- then show "Void :\<sqsubseteq>val v"
- by (simp add: PosOrd_ex_eq_def)
-next
- case (Posix_CH c v)
- have "v \<in> LV (CH c) [c]" by fact
- then have "v = Char c"
- by (simp add: LV_simps)
- then show "Char c :\<sqsubseteq>val v"
- by (simp add: PosOrd_ex_eq_def)
-next
- case (Posix_ALT1 s r1 v r2 v2)
- have as1: "s \<in> r1 \<rightarrow> v" by fact
- have IH: "\<And>v2. v2 \<in> LV r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
- have "v2 \<in> LV (ALT r1 r2) s" by fact
- then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
- by(auto simp add: LV_def prefix_list_def)
- then consider
- (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s"
- | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
- by (auto elim: Prf.cases)
- then show "Left v :\<sqsubseteq>val v2"
- proof(cases)
- case (Left v3)
- have "v3 \<in> LV r1 s" using Left(2,3)
- by (auto simp add: LV_def prefix_list_def)
- with IH have "v :\<sqsubseteq>val v3" by simp
- moreover
- have "flat v3 = flat v" using as1 Left(3)
- by (simp add: Posix1(2))
- ultimately have "Left v :\<sqsubseteq>val Left v3"
- by (simp add: PosOrd_ex_eq_def PosOrd_Left_eq)
- then show "Left v :\<sqsubseteq>val v2" unfolding Left .
- next
- case (Right v3)
- have "flat v3 = flat v" using as1 Right(3)
- by (simp add: Posix1(2))
- then have "Left v :\<sqsubseteq>val Right v3"
- unfolding PosOrd_ex_eq_def
- by (simp add: PosOrd_Left_Right)
- then show "Left v :\<sqsubseteq>val v2" unfolding Right .
- qed
-next
- case (Posix_ALT2 s r2 v r1 v2)
- have as1: "s \<in> r2 \<rightarrow> v" by fact
- have as2: "s \<notin> L r1" by fact
- have IH: "\<And>v2. v2 \<in> LV r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
- have "v2 \<in> LV (ALT r1 r2) s" by fact
- then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
- by(auto simp add: LV_def prefix_list_def)
- then consider
- (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s"
- | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
- by (auto elim: Prf.cases)
- then show "Right v :\<sqsubseteq>val v2"
- proof (cases)
- case (Right v3)
- have "v3 \<in> LV r2 s" using Right(2,3)
- by (auto simp add: LV_def prefix_list_def)
- with IH have "v :\<sqsubseteq>val v3" by simp
- moreover
- have "flat v3 = flat v" using as1 Right(3)
- by (simp add: Posix1(2))
- ultimately have "Right v :\<sqsubseteq>val Right v3"
- by (auto simp add: PosOrd_ex_eq_def PosOrd_RightI)
- then show "Right v :\<sqsubseteq>val v2" unfolding Right .
- next
- case (Left v3)
- have "v3 \<in> LV r1 s" using Left(2,3) as2
- by (auto simp add: LV_def prefix_list_def)
- then have "flat v3 = flat v \<and> \<Turnstile> v3 : r1" using as1 Left(3)
- by (simp add: Posix1(2) LV_def)
- then have "False" using as1 as2 Left
- by (auto simp add: Posix1(2) L_flat_Prf1)
- then show "Right v :\<sqsubseteq>val v2" by simp
- qed
-next
- case (Posix_SEQ s1 r1 v1 s2 r2 v2 v3)
- have "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" by fact+
- then have as1: "s1 = flat v1" "s2 = flat v2" by (simp_all add: Posix1(2))
- have IH1: "\<And>v3. v3 \<in> LV r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
- have IH2: "\<And>v3. v3 \<in> LV r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
- have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact
- have "v3 \<in> LV (SEQ r1 r2) (s1 @ s2)" by fact
- then obtain v3a v3b where eqs:
- "v3 = Seq v3a v3b" "\<Turnstile> v3a : r1" "\<Turnstile> v3b : r2"
- "flat v3a @ flat v3b = s1 @ s2"
- by (force simp add: prefix_list_def LV_def elim: Prf.cases)
- with cond have "flat v3a \<sqsubseteq>pre s1" unfolding prefix_list_def
- by (smt L_flat_Prf1 append_eq_append_conv2 append_self_conv)
- then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat v3b = s2)" using eqs
- by (simp add: sprefix_list_def append_eq_conv_conj)
- then have q2: "v1 :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat v3b = s2)"
- using PosOrd_spreI as1(1) eqs by blast
- then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> LV r1 s1 \<and> v3b \<in> LV r2 s2)" using eqs(2,3)
- by (auto simp add: LV_def)
- then have "v1 :\<sqsubset>val v3a \<or> (v1 :\<sqsubseteq>val v3a \<and> v2 :\<sqsubseteq>val v3b)" using IH1 IH2 by blast
- then have "Seq v1 v2 :\<sqsubseteq>val Seq v3a v3b" using eqs q2 as1
- unfolding PosOrd_ex_eq_def by (auto simp add: PosOrd_SeqI1 PosOrd_Seq_eq)
- then show "Seq v1 v2 :\<sqsubseteq>val v3" unfolding eqs by blast
-next
- case (Posix_STAR1 s1 r v s2 vs v3)
- have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
- then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
- have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
- have IH2: "\<And>v3. v3 \<in> LV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
- have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact
- have cond2: "flat v \<noteq> []" by fact
- have "v3 \<in> LV (STAR r) (s1 @ s2)" by fact
- then consider
- (NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
- "\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r"
- "flat (Stars (v3a # vs3)) = s1 @ s2"
- | (Empty) "v3 = Stars []"
- unfolding LV_def
- apply(auto)
- apply(erule Prf.cases)
- apply(auto)
- apply(case_tac vs)
- apply(auto intro: Prf.intros)
- done
- then show "Stars (v # vs) :\<sqsubseteq>val v3"
- proof (cases)
- case (NonEmpty v3a vs3)
- have "flat (Stars (v3a # vs3)) = s1 @ s2" using NonEmpty(4) .
- with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
- unfolding prefix_list_def
- by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7))
- then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
- by (simp add: sprefix_list_def append_eq_conv_conj)
- then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
- using PosOrd_spreI as1(1) NonEmpty(4) by blast
- then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (STAR r) s2)"
- using NonEmpty(2,3) by (auto simp add: LV_def)
- then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
- then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
- unfolding PosOrd_ex_eq_def by auto
- then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
- unfolding PosOrd_ex_eq_def
- using PosOrd_StarsI PosOrd_StarsI2 by auto
- then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
- next
- case Empty
- have "v3 = Stars []" by fact
- then show "Stars (v # vs) :\<sqsubseteq>val v3"
- unfolding PosOrd_ex_eq_def using cond2
- by (simp add: PosOrd_shorterI)
- qed
-next
- case (Posix_STAR2 r v2)
- have "v2 \<in> LV (STAR r) []" by fact
- then have "v2 = Stars []"
- unfolding LV_def by (auto elim: Prf.cases)
- then show "Stars [] :\<sqsubseteq>val v2"
- by (simp add: PosOrd_ex_eq_def)
-qed
-
-
-lemma Posix_PosOrd_reverse:
- assumes "s \<in> r \<rightarrow> v1"
- shows "\<not>(\<exists>v2 \<in> LV r s. v2 :\<sqsubset>val v1)"
-using assms
-by (metis Posix_PosOrd less_irrefl PosOrd_def
- PosOrd_ex_eq_def PosOrd_ex_def PosOrd_trans)
-
-lemma PosOrd_Posix:
- assumes "v1 \<in> LV r s" "\<forall>v\<^sub>2 \<in> LV r s. \<not> v\<^sub>2 :\<sqsubset>val v1"
- shows "s \<in> r \<rightarrow> v1"
-proof -
- have "s \<in> L r" using assms(1) unfolding LV_def
- using L_flat_Prf1 by blast
- then obtain vposix where vp: "s \<in> r \<rightarrow> vposix"
- using lexer_correct_Some by blast
- with assms(1) have "vposix :\<sqsubseteq>val v1" by (simp add: Posix_PosOrd)
- then have "vposix = v1 \<or> vposix :\<sqsubset>val v1" unfolding PosOrd_ex_eq2 by auto
- moreover
- { assume "vposix :\<sqsubset>val v1"
- moreover
- have "vposix \<in> LV r s" using vp
- using Posix_LV by blast
- ultimately have "False" using assms(2) by blast
- }
- ultimately show "s \<in> r \<rightarrow> v1" using vp by blast
-qed
-
-lemma Least_existence:
- assumes "LV r s \<noteq> {}"
- shows " \<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
-proof -
- from assms
- obtain vposix where "s \<in> r \<rightarrow> vposix"
- unfolding LV_def
- using L_flat_Prf1 lexer_correct_Some by blast
- then have "\<forall>v \<in> LV r s. vposix :\<sqsubseteq>val v"
- by (simp add: Posix_PosOrd)
- then show "\<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
- using Posix_LV \<open>s \<in> r \<rightarrow> vposix\<close> by blast
-qed
-
-lemma Least_existence1:
- assumes "LV r s \<noteq> {}"
- shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
-using Least_existence[OF assms] assms
-using PosOrdeq_antisym by blast
-
-lemma Least_existence2:
- assumes "LV r s \<noteq> {}"
- shows " \<exists>!vmin \<in> LV r s. lexer r s = Some vmin \<and> (\<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v)"
-using Least_existence[OF assms] assms
-using PosOrdeq_antisym
- using PosOrd_Posix PosOrd_ex_eq2 lexer_correctness(1) by auto
-
-
-lemma Least_existence1_pre:
- assumes "LV r s \<noteq> {}"
- shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> (LV r s \<union> {v'. flat v' \<sqsubset>spre s}). vmin :\<sqsubseteq>val v"
-using Least_existence[OF assms] assms
-apply -
-apply(erule bexE)
-apply(rule_tac a="vmin" in ex1I)
-apply(auto)[1]
-apply (metis PosOrd_Posix PosOrd_ex_eq2 PosOrd_spreI PosOrdeq_antisym Posix1(2))
-apply(auto)[1]
-apply(simp add: PosOrdeq_antisym)
-done
-
-lemma
- shows "partial_order_on UNIV {(v1, v2). v1 :\<sqsubseteq>val v2}"
-apply(simp add: partial_order_on_def)
-apply(simp add: preorder_on_def refl_on_def)
-apply(simp add: PosOrdeq_refl)
-apply(auto)
-apply(rule transI)
-apply(auto intro: PosOrdeq_trans)[1]
-apply(rule antisymI)
-apply(simp add: PosOrdeq_antisym)
-done
-
-lemma
- "wf {(v1, v2). v1 :\<sqsubset>val v2 \<and> v1 \<in> LV r s \<and> v2 \<in> LV r s}"
-apply(rule finite_acyclic_wf)
-prefer 2
-apply(simp add: acyclic_def)
-apply(induct_tac rule: trancl.induct)
-apply(auto)[1]
-oops
-
-
-unused_thms
-
-end
\ No newline at end of file
diff -r f493a20feeb3 -r 04b5e904a220 thys3/PosixSpec.thy
--- a/thys3/PosixSpec.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,380 +0,0 @@
-
-theory PosixSpec
- imports RegLangs
-begin
-
-section \<open>"Plain" Values\<close>
-
-datatype val =
- Void
-| Char char
-| Seq val val
-| Right val
-| Left val
-| Stars "val list"
-
-
-section \<open>The string behind a value\<close>
-
-fun
- flat :: "val \<Rightarrow> string"
-where
- "flat (Void) = []"
-| "flat (Char c) = [c]"
-| "flat (Left v) = flat v"
-| "flat (Right v) = flat v"
-| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"
-| "flat (Stars []) = []"
-| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))"
-
-abbreviation
- "flats vs \<equiv> concat (map flat vs)"
-
-lemma flat_Stars [simp]:
- "flat (Stars vs) = flats vs"
-by (induct vs) (auto)
-
-
-section \<open>Lexical Values\<close>
-
-inductive
- Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<Turnstile> _ : _" [100, 100] 100)
-where
- "\<lbrakk>\<Turnstile> v1 : r1; \<Turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<Turnstile> Seq v1 v2 : SEQ r1 r2"
-| "\<Turnstile> v1 : r1 \<Longrightarrow> \<Turnstile> Left v1 : ALT r1 r2"
-| "\<Turnstile> v2 : r2 \<Longrightarrow> \<Turnstile> Right v2 : ALT r1 r2"
-| "\<Turnstile> Void : ONE"
-| "\<Turnstile> Char c : CH c"
-| "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> [] \<Longrightarrow> \<Turnstile> Stars vs : STAR r"
-
-inductive_cases Prf_elims:
- "\<Turnstile> v : ZERO"
- "\<Turnstile> v : SEQ r1 r2"
- "\<Turnstile> v : ALT r1 r2"
- "\<Turnstile> v : ONE"
- "\<Turnstile> v : CH c"
- "\<Turnstile> vs : STAR r"
-
-lemma Prf_Stars_appendE:
- assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
- shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r"
-using assms
-by (auto intro: Prf.intros elim!: Prf_elims)
-
-
-lemma flats_Prf_value:
- assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"
- shows "\<exists>vs. flats vs = concat ss \<and> (\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> [])"
-using assms
-apply(induct ss)
-apply(auto)
-apply(rule_tac x="[]" in exI)
-apply(simp)
-apply(case_tac "flat v = []")
-apply(rule_tac x="vs" in exI)
-apply(simp)
-apply(rule_tac x="v#vs" in exI)
-apply(simp)
-done
-
-
-lemma L_flat_Prf1:
- assumes "\<Turnstile> v : r"
- shows "flat v \<in> L r"
-using assms
-by (induct) (auto simp add: Sequ_def Star_concat)
-
-lemma L_flat_Prf2:
- assumes "s \<in> L r"
- shows "\<exists>v. \<Turnstile> v : r \<and> flat v = s"
-using assms
-proof(induct r arbitrary: s)
- case (STAR r s)
- have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
- have "s \<in> L (STAR r)" by fact
- then obtain ss where "concat ss = s" "\<forall>s \<in> set ss. s \<in> L r \<and> s \<noteq> []"
- using Star_split by auto
- then obtain vs where "flats vs = s" "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<noteq> []"
- using IH flats_Prf_value by metis
- then show "\<exists>v. \<Turnstile> v : STAR r \<and> flat v = s"
- using Prf.intros(6) flat_Stars by blast
-next
- case (SEQ r1 r2 s)
- then show "\<exists>v. \<Turnstile> v : SEQ r1 r2 \<and> flat v = s"
- unfolding Sequ_def L.simps by (fastforce intro: Prf.intros)
-next
- case (ALT r1 r2 s)
- then show "\<exists>v. \<Turnstile> v : ALT r1 r2 \<and> flat v = s"
- unfolding L.simps by (fastforce intro: Prf.intros)
-qed (auto intro: Prf.intros)
-
-
-lemma L_flat_Prf:
- shows "L(r) = {flat v | v. \<Turnstile> v : r}"
-using L_flat_Prf1 L_flat_Prf2 by blast
-
-
-
-section \<open>Sets of Lexical Values\<close>
-
-text \<open>
- Shows that lexical values are finite for a given regex and string.
-\<close>
-
-definition
- LV :: "rexp \<Rightarrow> string \<Rightarrow> val set"
-where "LV r s \<equiv> {v. \<Turnstile> v : r \<and> flat v = s}"
-
-lemma LV_simps:
- shows "LV ZERO s = {}"
- and "LV ONE s = (if s = [] then {Void} else {})"
- and "LV (CH c) s = (if s = [c] then {Char c} else {})"
- and "LV (ALT r1 r2) s = Left ` LV r1 s \<union> Right ` LV r2 s"
-unfolding LV_def
-by (auto intro: Prf.intros elim: Prf.cases)
-
-
-abbreviation
- "Prefixes s \<equiv> {s'. prefix s' s}"
-
-abbreviation
- "Suffixes s \<equiv> {s'. suffix s' s}"
-
-abbreviation
- "SSuffixes s \<equiv> {s'. strict_suffix s' s}"
-
-lemma Suffixes_cons [simp]:
- shows "Suffixes (c # s) = Suffixes s \<union> {c # s}"
-by (auto simp add: suffix_def Cons_eq_append_conv)
-
-
-lemma finite_Suffixes:
- shows "finite (Suffixes s)"
-by (induct s) (simp_all)
-
-lemma finite_SSuffixes:
- shows "finite (SSuffixes s)"
-proof -
- have "SSuffixes s \<subseteq> Suffixes s"
- unfolding strict_suffix_def suffix_def by auto
- then show "finite (SSuffixes s)"
- using finite_Suffixes finite_subset by blast
-qed
-
-lemma finite_Prefixes:
- shows "finite (Prefixes s)"
-proof -
- have "finite (Suffixes (rev s))"
- by (rule finite_Suffixes)
- then have "finite (rev ` Suffixes (rev s))" by simp
- moreover
- have "rev ` (Suffixes (rev s)) = Prefixes s"
- unfolding suffix_def prefix_def image_def
- by (auto)(metis rev_append rev_rev_ident)+
- ultimately show "finite (Prefixes s)" by simp
-qed
-
-lemma LV_STAR_finite:
- assumes "\<forall>s. finite (LV r s)"
- shows "finite (LV (STAR r) s)"
-proof(induct s rule: length_induct)
- fix s::"char list"
- assume "\<forall>s'. length s' < length s \<longrightarrow> finite (LV (STAR r) s')"
- then have IH: "\<forall>s' \<in> SSuffixes s. finite (LV (STAR r) s')"
- by (force simp add: strict_suffix_def suffix_def)
- define f where "f \<equiv> \<lambda>(v, vs). Stars (v # vs)"
- define S1 where "S1 \<equiv> \<Union>s' \<in> Prefixes s. LV r s'"
- define S2 where "S2 \<equiv> \<Union>s2 \<in> SSuffixes s. Stars -` (LV (STAR r) s2)"
- have "finite S1" using assms
- unfolding S1_def by (simp_all add: finite_Prefixes)
- moreover
- with IH have "finite S2" unfolding S2_def
- by (auto simp add: finite_SSuffixes inj_on_def finite_vimageI)
- ultimately
- have "finite ({Stars []} \<union> f ` (S1 \<times> S2))" by simp
- moreover
- have "LV (STAR r) s \<subseteq> {Stars []} \<union> f ` (S1 \<times> S2)"
- unfolding S1_def S2_def f_def
- unfolding LV_def image_def prefix_def strict_suffix_def
- apply(auto)
- apply(case_tac x)
- apply(auto elim: Prf_elims)
- apply(erule Prf_elims)
- apply(auto)
- apply(case_tac vs)
- apply(auto intro: Prf.intros)
- apply(rule exI)
- apply(rule conjI)
- apply(rule_tac x="flat a" in exI)
- apply(rule conjI)
- apply(rule_tac x="flats list" in exI)
- apply(simp)
- apply(blast)
- apply(simp add: suffix_def)
- using Prf.intros(6) by blast
- ultimately
- show "finite (LV (STAR r) s)" by (simp add: finite_subset)
-qed
-
-
-lemma LV_finite:
- shows "finite (LV r s)"
-proof(induct r arbitrary: s)
- case (ZERO s)
- show "finite (LV ZERO s)" by (simp add: LV_simps)
-next
- case (ONE s)
- show "finite (LV ONE s)" by (simp add: LV_simps)
-next
- case (CH c s)
- show "finite (LV (CH c) s)" by (simp add: LV_simps)
-next
- case (ALT r1 r2 s)
- then show "finite (LV (ALT r1 r2) s)" by (simp add: LV_simps)
-next
- case (SEQ r1 r2 s)
- define f where "f \<equiv> \<lambda>(v1, v2). Seq v1 v2"
- define S1 where "S1 \<equiv> \<Union>s' \<in> Prefixes s. LV r1 s'"
- define S2 where "S2 \<equiv> \<Union>s' \<in> Suffixes s. LV r2 s'"
- have IHs: "\<And>s. finite (LV r1 s)" "\<And>s. finite (LV r2 s)" by fact+
- then have "finite S1" "finite S2" unfolding S1_def S2_def
- by (simp_all add: finite_Prefixes finite_Suffixes)
- moreover
- have "LV (SEQ r1 r2) s \<subseteq> f ` (S1 \<times> S2)"
- unfolding f_def S1_def S2_def
- unfolding LV_def image_def prefix_def suffix_def
- apply (auto elim!: Prf_elims)
- by (metis (mono_tags, lifting) mem_Collect_eq)
- ultimately
- show "finite (LV (SEQ r1 r2) s)"
- by (simp add: finite_subset)
-next
- case (STAR r s)
- then show "finite (LV (STAR r) s)" by (simp add: LV_STAR_finite)
-qed
-
-
-
-section \<open>Our inductive POSIX Definition\<close>
-
-inductive
- Posix :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
-where
- Posix_ONE: "[] \<in> ONE \<rightarrow> Void"
-| Posix_CH: "[c] \<in> (CH c) \<rightarrow> (Char c)"
-| Posix_ALT1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"
-| Posix_ALT2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"
-| Posix_SEQ: "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)\<rbrakk> \<Longrightarrow>
- (s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"
-| Posix_STAR1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>
- \<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"
-| Posix_STAR2: "[] \<in> STAR r \<rightarrow> Stars []"
-
-inductive_cases Posix_elims:
- "s \<in> ZERO \<rightarrow> v"
- "s \<in> ONE \<rightarrow> v"
- "s \<in> CH c \<rightarrow> v"
- "s \<in> ALT r1 r2 \<rightarrow> v"
- "s \<in> SEQ r1 r2 \<rightarrow> v"
- "s \<in> STAR r \<rightarrow> v"
-
-lemma Posix1:
- assumes "s \<in> r \<rightarrow> v"
- shows "s \<in> L r" "flat v = s"
-using assms
- by(induct s r v rule: Posix.induct)
- (auto simp add: Sequ_def)
-
-text \<open>
- For a give value and string, our Posix definition
- determines a unique value.
-\<close>
-
-lemma Posix_determ:
- assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
- shows "v1 = v2"
-using assms
-proof (induct s r v1 arbitrary: v2 rule: Posix.induct)
- case (Posix_ONE v2)
- have "[] \<in> ONE \<rightarrow> v2" by fact
- then show "Void = v2" by cases auto
-next
- case (Posix_CH c v2)
- have "[c] \<in> CH c \<rightarrow> v2" by fact
- then show "Char c = v2" by cases auto
-next
- case (Posix_ALT1 s r1 v r2 v2)
- have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact
- moreover
- have "s \<in> r1 \<rightarrow> v" by fact
- then have "s \<in> L r1" by (simp add: Posix1)
- ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto
- moreover
- have IH: "\<And>v2. s \<in> r1 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
- ultimately have "v = v'" by simp
- then show "Left v = v2" using eq by simp
-next
- case (Posix_ALT2 s r2 v r1 v2)
- have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact
- moreover
- have "s \<notin> L r1" by fact
- ultimately obtain v' where eq: "v2 = Right v'" "s \<in> r2 \<rightarrow> v'"
- by cases (auto simp add: Posix1)
- moreover
- have IH: "\<And>v2. s \<in> r2 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
- ultimately have "v = v'" by simp
- then show "Right v = v2" using eq by simp
-next
- case (Posix_SEQ s1 r1 v1 s2 r2 v2 v')
- have "(s1 @ s2) \<in> SEQ r1 r2 \<rightarrow> v'"
- "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact+
- then obtain v1' v2' where "v' = Seq v1' v2'" "s1 \<in> r1 \<rightarrow> v1'" "s2 \<in> r2 \<rightarrow> v2'"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) by fastforce+
- moreover
- have IHs: "\<And>v1'. s1 \<in> r1 \<rightarrow> v1' \<Longrightarrow> v1 = v1'"
- "\<And>v2'. s2 \<in> r2 \<rightarrow> v2' \<Longrightarrow> v2 = v2'" by fact+
- ultimately show "Seq v1 v2 = v'" by simp
-next
- case (Posix_STAR1 s1 r v s2 vs v2)
- have "(s1 @ s2) \<in> STAR r \<rightarrow> v2"
- "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" "flat v \<noteq> []"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact+
- then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (STAR r) \<rightarrow> (Stars vs')"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) apply fastforce
- apply (metis Posix1(1) Posix_STAR1.hyps(6) append_Nil append_Nil2)
- using Posix1(2) by blast
- moreover
- have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
- "\<And>v2. s2 \<in> STAR r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
- ultimately show "Stars (v # vs) = v2" by auto
-next
- case (Posix_STAR2 r v2)
- have "[] \<in> STAR r \<rightarrow> v2" by fact
- then show "Stars [] = v2" by cases (auto simp add: Posix1)
-qed
-
-
-text \<open>
- Our POSIX values are lexical values.
-\<close>
-
-lemma Posix_LV:
- assumes "s \<in> r \<rightarrow> v"
- shows "v \<in> LV r s"
- using assms unfolding LV_def
- apply(induct rule: Posix.induct)
- apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)
- done
-
-lemma Posix_Prf:
- assumes "s \<in> r \<rightarrow> v"
- shows "\<Turnstile> v : r"
- using assms Posix_LV LV_def
- by simp
-
-end
diff -r f493a20feeb3 -r 04b5e904a220 thys3/RegLangs.thy
--- a/thys3/RegLangs.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,236 +0,0 @@
-theory RegLangs
- imports Main "HOL-Library.Sublist"
-begin
-
-section \<open>Sequential Composition of Languages\<close>
-
-definition
- Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
-where
- "A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"
-
-text \<open>Two Simple Properties about Sequential Composition\<close>
-
-lemma Sequ_empty_string [simp]:
- shows "A ;; {[]} = A"
- and "{[]} ;; A = A"
-by (simp_all add: Sequ_def)
-
-lemma Sequ_empty [simp]:
- shows "A ;; {} = {}"
- and "{} ;; A = {}"
- by (simp_all add: Sequ_def)
-
-
-section \<open>Semantic Derivative (Left Quotient) of Languages\<close>
-
-definition
- Der :: "char \<Rightarrow> string set \<Rightarrow> string set"
-where
- "Der c A \<equiv> {s. c # s \<in> A}"
-
-definition
- Ders :: "string \<Rightarrow> string set \<Rightarrow> string set"
-where
- "Ders s A \<equiv> {s'. s @ s' \<in> A}"
-
-lemma Der_null [simp]:
- shows "Der c {} = {}"
-unfolding Der_def
-by auto
-
-lemma Der_empty [simp]:
- shows "Der c {[]} = {}"
-unfolding Der_def
-by auto
-
-lemma Der_char [simp]:
- shows "Der c {[d]} = (if c = d then {[]} else {})"
-unfolding Der_def
-by auto
-
-lemma Der_union [simp]:
- shows "Der c (A \<union> B) = Der c A \<union> Der c B"
-unfolding Der_def
-by auto
-
-lemma Der_Sequ [simp]:
- shows "Der c (A ;; B) = (Der c A) ;; B \<union> (if [] \<in> A then Der c B else {})"
-unfolding Der_def Sequ_def
-by (auto simp add: Cons_eq_append_conv)
-
-
-section \<open>Kleene Star for Languages\<close>
-
-inductive_set
- Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
- for A :: "string set"
-where
- start[intro]: "[] \<in> A\<star>"
-| step[intro]: "\<lbrakk>s1 \<in> A; s2 \<in> A\<star>\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> A\<star>"
-
-(* Arden's lemma *)
-
-lemma Star_cases:
- shows "A\<star> = {[]} \<union> A ;; A\<star>"
-unfolding Sequ_def
-by (auto) (metis Star.simps)
-
-lemma Star_decomp:
- assumes "c # x \<in> A\<star>"
- shows "\<exists>s1 s2. x = s1 @ s2 \<and> c # s1 \<in> A \<and> s2 \<in> A\<star>"
-using assms
-by (induct x\<equiv>"c # x" rule: Star.induct)
- (auto simp add: append_eq_Cons_conv)
-
-lemma Star_Der_Sequ:
- shows "Der c (A\<star>) \<subseteq> (Der c A) ;; A\<star>"
-unfolding Der_def Sequ_def
-by(auto simp add: Star_decomp)
-
-
-lemma Der_star[simp]:
- shows "Der c (A\<star>) = (Der c A) ;; A\<star>"
-proof -
- have "Der c (A\<star>) = Der c ({[]} \<union> A ;; A\<star>)"
- by (simp only: Star_cases[symmetric])
- also have "... = Der c (A ;; A\<star>)"
- by (simp only: Der_union Der_empty) (simp)
- also have "... = (Der c A) ;; A\<star> \<union> (if [] \<in> A then Der c (A\<star>) else {})"
- by simp
- also have "... = (Der c A) ;; A\<star>"
- using Star_Der_Sequ by auto
- finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .
-qed
-
-lemma Star_concat:
- assumes "\<forall>s \<in> set ss. s \<in> A"
- shows "concat ss \<in> A\<star>"
-using assms by (induct ss) (auto)
-
-lemma Star_split:
- assumes "s \<in> A\<star>"
- shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A \<and> s \<noteq> [])"
-using assms
- apply(induct rule: Star.induct)
- using concat.simps(1) apply fastforce
- apply(clarify)
- by (metis append_Nil concat.simps(2) set_ConsD)
-
-
-
-section \<open>Regular Expressions\<close>
-
-datatype rexp =
- ZERO
-| ONE
-| CH char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-
-section \<open>Semantics of Regular Expressions\<close>
-
-fun
- L :: "rexp \<Rightarrow> string set"
-where
- "L (ZERO) = {}"
-| "L (ONE) = {[]}"
-| "L (CH c) = {[c]}"
-| "L (SEQ r1 r2) = (L r1) ;; (L r2)"
-| "L (ALT r1 r2) = (L r1) \<union> (L r2)"
-| "L (STAR r) = (L r)\<star>"
-
-
-section \<open>Nullable, Derivatives\<close>
-
-fun
- nullable :: "rexp \<Rightarrow> bool"
-where
- "nullable (ZERO) = False"
-| "nullable (ONE) = True"
-| "nullable (CH c) = False"
-| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
-| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
-| "nullable (STAR r) = True"
-
-
-fun
- der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
-where
- "der c (ZERO) = ZERO"
-| "der c (ONE) = ZERO"
-| "der c (CH d) = (if c = d then ONE else ZERO)"
-| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
-| "der c (SEQ r1 r2) =
- (if nullable r1
- then ALT (SEQ (der c r1) r2) (der c r2)
- else SEQ (der c r1) r2)"
-| "der c (STAR r) = SEQ (der c r) (STAR r)"
-
-fun
- ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
-where
- "ders [] r = r"
-| "ders (c # s) r = ders s (der c r)"
-
-
-lemma nullable_correctness:
- shows "nullable r \<longleftrightarrow> [] \<in> (L r)"
-by (induct r) (auto simp add: Sequ_def)
-
-lemma der_correctness:
- shows "L (der c r) = Der c (L r)"
-by (induct r) (simp_all add: nullable_correctness)
-
-lemma ders_correctness:
- shows "L (ders s r) = Ders s (L r)"
- by (induct s arbitrary: r)
- (simp_all add: Ders_def der_correctness Der_def)
-
-lemma ders_append:
- shows "ders (s1 @ s2) r = ders s2 (ders s1 r)"
- by (induct s1 arbitrary: s2 r) (auto)
-
-lemma ders_snoc:
- shows "ders (s @ [c]) r = der c (ders s r)"
- by (simp add: ders_append)
-
-
-(*
-datatype ctxt =
- SeqC rexp bool
- | AltCL rexp
- | AltCH rexp
- | StarC rexp
-
-function
- down :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"
-and up :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"
-where
- "down c (SEQ r1 r2) ctxts =
- (if (nullable r1) then down c r1 (SeqC r2 True # ctxts)
- else down c r1 (SeqC r2 False # ctxts))"
-| "down c (CH d) ctxts =
- (if c = d then up c ONE ctxts else up c ZERO ctxts)"
-| "down c ONE ctxts = up c ZERO ctxts"
-| "down c ZERO ctxts = up c ZERO ctxts"
-| "down c (ALT r1 r2) ctxts = down c r1 (AltCH r2 # ctxts)"
-| "down c (STAR r1) ctxts = down c r1 (StarC r1 # ctxts)"
-| "up c r [] = (r, [])"
-| "up c r (SeqC r2 False # ctxts) = up c (SEQ r r2) ctxts"
-| "up c r (SeqC r2 True # ctxts) = down c r2 (AltCL (SEQ r r2) # ctxts)"
-| "up c r (AltCL r1 # ctxts) = up c (ALT r1 r) ctxts"
-| "up c r (AltCH r2 # ctxts) = down c r2 (AltCL r # ctxts)"
-| "up c r (StarC r1 # ctxts) = up c (SEQ r (STAR r1)) ctxts"
- apply(pat_completeness)
- apply(auto)
- done
-
-termination
- sorry
-
-*)
-
-
-end
\ No newline at end of file