theory BasicIdentities
imports "RfltsRdistinctProps"
begin
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
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 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
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 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)) {})\<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
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 der_simp_nullability:
shows "rnullable r = rnullable (rsimp r)"
using RL_rnullable RL_rsimp by auto
lemma qqq1:
shows "RZERO \<notin> set (rflts (map rsimp rs))"
by (metis ex_map_conv flts3 good.simps(1) good1)
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))
corollary head_one_more_simp:
shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
by (simp add: rsimp_idem)
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 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
(*equalities with rsimp *)
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))
(*some basic facts about rsimp*)
unused_thms
end