theory BasicIdentities imports
"Lexer" "PDerivs"
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 rdistinct_does_the_job:
shows "distinct (rdistinct rs s)"
apply(induct rs arbitrary: s)
apply simp
apply simp
sorry
lemma rdistinct_concat:
shows "set rs \<subseteq> rset \<Longrightarrow> rdistinct (rs @ rsa) rset = rdistinct rsa rset"
apply(induct rs)
apply simp+
done
lemma rdistinct_concat2:
shows "\<forall>r \<in> set rs. r \<in> rset \<Longrightarrow> rdistinct (rs @ rsa) rset = rdistinct rsa rset"
by (simp add: rdistinct_concat subsetI)
lemma distinct_not_exist:
shows "a \<notin> set rs \<Longrightarrow> rdistinct rs rset = rdistinct rs (insert a rset)"
apply(induct rs arbitrary: rset)
apply simp
apply(case_tac "aa \<in> rset")
apply simp
apply(subgoal_tac "a \<noteq> aa")
prefer 2
apply simp
apply simp
done
lemma rdistinct_on_distinct:
shows "distinct rs \<Longrightarrow> rdistinct rs {} = rs"
apply(induct rs)
apply simp
apply(subgoal_tac "rdistinct rs {} = rs")
prefer 2
apply simp
using distinct_not_exist by fastforce
lemma distinct_rdistinct_append:
shows "distinct rs1 \<Longrightarrow> rdistinct (rs1 @ rsa) {} = rs1 @ (rdistinct rsa (set rs1))"
sorry
lemma rdistinct_concat_general:
shows "rdistinct (rs1 @ rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
sorry
lemma rdistinct_set_equality:
shows "set (rdistinct rs {}) = set rs"
sorry
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_equality)
by (simp add: rdistinct_does_the_job)
fun rflts :: "rrexp list \<Rightarrow> rrexp list"
where
"rflts [] = []"
| "rflts (RZERO # rs) = rflts rs"
| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
| "rflts (r1 # rs) = r1 # rflts rs"
lemma rflts_def_idiot:
shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk>
\<Longrightarrow> rflts (a # rs) = a # rflts rs"
apply(case_tac a)
apply simp_all
done
lemma rflts_def_idiot2:
shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow> a \<in> set (rflts rs)"
apply(induct rs)
apply simp
by (metis append.assoc in_set_conv_decomp insert_iff list.simps(15) rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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_gte2elems:
shows "size rlist \<ge> 2 \<Longrightarrow> rsimp_ALTs rlist = RALTS rlist"
apply(induct rlist)
apply simp
apply(induct rlist)
apply simp
apply (metis Suc_le_length_iff rsimp_ALTs.simps(3))
by blast
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)"
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 ( sum_list (map rsize rs)) "
by simp
lemma rflts_mono:
shows "sum_list (map rsize (rflts rs))\<le> sum_list (map rsize 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 "sum_list (map rsize (rdistinct rs ss)) \<le>
sum_list (map rsize rs )"
apply (induct rs arbitrary: ss)
apply simp
by (simp add: trans_le_add2)
lemma rdistinct_phi_smaller: "sum_list (map rsize (rdistinct rs {})) \<le> sum_list (map rsize rs)"
by (simp add: rdistinct_smaller)
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 (sum_list (map rsize 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( sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})))")
prefer 2
using ralts_cap_mono apply blast
apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})) \<le>
sum_list (map rsize ( (rflts (map rsimp x))))")
prefer 2
using rdistinct_smaller apply presburger
apply(subgoal_tac "sum_list (map rsize (rflts (map rsimp x))) \<le>
sum_list (map rsize (map rsimp x))")
prefer 2
using rflts_mono apply blast
apply(subgoal_tac "sum_list (map rsize (map rsimp x)) \<le> sum_list (map rsize 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 no_alt_short_list_after_simp:
shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
sorry
lemma no_further_dB_after_simp:
shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
sorry
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 set_related_list:
shows "distinct rs \<Longrightarrow> length rs = card (set rs)"
by (simp add: distinct_card)
(*this section deals with the property of distinctBy: creates a list without duplicates*)
lemma rdistinct_never_added_twice:
shows "rdistinct (a # rs) {a} = rdistinct rs {a}"
by force
lemma rders_simp_one_char:
shows "rders_simp r [c] = rsimp (rder c r)"
apply auto
done
lemma rsimp_idem:
shows "rsimp (rsimp r) = rsimp r"
sorry
corollary rsimp_inner_idem1:
shows "rsimp r = RSEQ r1 r2 \<Longrightarrow> rsimp r1 = r1 \<and> rsimp r2 = r2"
sorry
corollary rsimp_inner_idem2:
shows "rsimp r = RALTS rs \<Longrightarrow> \<forall>r' \<in> (set rs). rsimp r' = r'"
sorry
corollary rsimp_inner_idem3:
shows "rsimp r = RALTS rs \<Longrightarrow> map rsimp rs = rs"
by (meson map_idI rsimp_inner_idem2)
corollary rsimp_inner_idem4:
shows "rsimp r = RALTS rs \<Longrightarrow> rflts rs = rs"
sorry
lemma head_one_more_simp:
shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
by (simp add: rsimp_idem)
lemma head_one_more_dersimp:
shows "map rsimp ((rder x (rders_simp r s) # rs)) = map rsimp ((rders_simp r (s@[x]) ) # rs)"
using head_one_more_simp rders_simp_append rders_simp_one_char by presburger
lemma ders_simp_nullability:
shows "rnullable (rders r s) = rnullable (rders_simp r s)"
sorry
lemma der_simp_nullability:
shows "rnullable r = rnullable (rsimp r)"
sorry
lemma first_elem_seqder:
shows "\<not>rnullable r1p \<Longrightarrow> map rsimp (rder x (RSEQ r1p r2)
# rs) = map rsimp ((RSEQ (rder x r1p) r2) # rs) "
by auto
lemma first_elem_seqder1:
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
map rsimp ( (RSEQ (rsimp (rder x (rders_simp r xs))) r2) # rs)"
by (simp add: rsimp_idem)
lemma first_elem_seqdersimps:
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
map rsimp ( (RSEQ (rders_simp r (xs @ [x])) r2) # rs)"
using first_elem_seqder1 rders_simp_append by auto
lemma seq_ders_closed_form1:
shows "\<exists>Ss. rders_simp (RSEQ r1 r2) [c] = rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) #
(map ( rders_simp r2 ) Ss)))"
apply(case_tac "rnullable r1")
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [[c]])))")
prefer 2
apply (simp add: rsimp_idem)
apply(rule_tac x = "[[c]]" in exI)
apply simp
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [])))")
apply blast
apply(simp add: rsimp_idem)
sorry
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_last2:
shows "\<lbrakk>a \<in> set as\<rbrakk>
\<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
by (simp add: distinct_removes_last(1))
lemma distinct_removes_middle2:
shows "a \<in> set as \<Longrightarrow> rdistinct (as @ [a] @ rs) {} = rdistinct (as @ rs) {}"
by (metis distinct_removes_middle(1))
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)
inductive good1 :: "rrexp \<Rightarrow> bool"
where
"\<lbrakk>RZERO \<notin> set rs; \<nexists>rs1. RALTS rs1 \<in> set rs\<rbrakk> \<Longrightarrow> good1 (RALTS rs)"
|"good1 RZERO"
|"good1 RONE"
|"good1 (RCHAR c)"
|"good1 (RSEQ r1 r2)"
|"good1 (RSTAR r0)"
inductive goods :: "rrexp list \<Rightarrow> bool"
where
"goods []"
|"\<lbrakk>goods rs; r \<noteq> RZERO; \<nexists>rs1. RALTS rs1 = r\<rbrakk> \<Longrightarrow> goods (r # rs)"
lemma goods_good1:
shows "goods rs = good1 (RALTS rs)"
apply(induct rs)
apply (simp add: good1.intros(1) goods.intros(1))
apply(case_tac "goods rs")
apply(case_tac a)
apply simp
using good1.simps goods.cases apply auto[1]
apply (simp add: good1.simps goods.intros(2))
apply (simp add: good1.simps goods.intros(2))
apply (simp add: good1.simps goods.intros(2))
using good1.simps goods.cases apply auto[1]
apply (metis good1.cases good1.intros(1) goods.intros(2) rrexp.distinct(15) rrexp.distinct(21) rrexp.distinct(25) rrexp.distinct(29) rrexp.distinct(7) rrexp.distinct(9) rrexp.inject(3) set_ConsD)
apply simp
by (metis good1.cases good1.intros(1) goods.cases list.distinct(1) list.inject list.set_intros(2) rrexp.distinct(15) rrexp.distinct(29) rrexp.distinct(7) rrexp.inject(3) rrexp.simps(26) rrexp.simps(30))
lemma rsimp_good1:
shows "rsimp r = r1 \<Longrightarrow> good1 r1"
sorry
lemma rsimp_no_dup:
shows "rsimp r = RALTS rs \<Longrightarrow> distinct rs"
sorry
lemma rsimp_good1_2:
shows "map rsimp rsa = [RALTS rs] \<Longrightarrow> good1 (RALTS rs)"
by (metis Cons_eq_map_D rsimp_good1)
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_flts:
shows "good1 (RALTS rs1) \<Longrightarrow> rflts rs1 = rs1"
apply(induct rs1)
apply simp
by (metis good1.cases good1.intros(1) list.set_intros(1) rflts_def_idiot rrexp.distinct(21) rrexp.distinct(25) rrexp.distinct(29) rrexp.inject(3) rsimp.simps(3) rsimp.simps(4) rsimp_inner_idem4 set_subset_Cons subsetD)
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
apply(subgoal_tac "good1 (RALTS rs1)")
prefer 2
using rsimp_good1 apply blast
using flts_append good1_flts 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(subgoal_tac "good1 (RALTS rs1)")
apply(subgoal_tac "distinct rs1")
apply(subst rdistinct_on_distinct)
apply blast
apply(subst rdistinct_on_distinct)
apply blast
using good1_flatten apply blast
using rsimp_no_dup apply force
using rsimp_good1_2 apply presburger
apply simp+
apply(case_tac "rflts (a # aa # lista)")
apply simp
by (smt (verit) append_Cons append_Nil empty_iff good1_flatten list.distinct(1) rdistinct.simps(2) rsimp.simps(2) rsimp_ALTs.elims rsimp_good1)
lemma flts_good_good:
shows "\<forall>r \<in> set rs. good1 r \<Longrightarrow> good1 (RALTS (rflts rs ))"
apply(induct rs)
apply simp
using goods.intros(1) goods_good1 apply auto[1]
apply(case_tac "a")
apply simp
apply (metis goods.intros(2) goods_good1 list.set_intros(2) rflts.simps(4) rrexp.distinct(1) rrexp.distinct(15))
apply simp
using goods.intros(2) goods_good1 apply blast
using goods.intros(2) goods_good1 apply auto[1]
apply(subgoal_tac "good1 a")
apply (metis Un_iff good1.cases good1.intros(1) list.set_intros(2) rflts.simps(3) rrexp.distinct(15) rrexp.distinct(21) rrexp.distinct(25) rrexp.distinct(29) rrexp.distinct(7) rrexp.inject(3) set_append)
apply simp
by (metis goods.intros(2) goods_good1 list.set_intros(2) rflts.simps(7) rrexp.distinct(29) rrexp.distinct(9))
lemma simp_flatten_aux1:
shows "\<forall>r \<in> set (map rsimp rsa). good1 r"
apply(induct rsa)
apply(simp add: goods.intros)
using rsimp_good1 by auto
lemma simp_flatten_aux:
shows "\<forall>r \<in> set rs. good1 r \<Longrightarrow> rflts (rdistinct (rflts rs) {}) = (rdistinct (rflts rs) {})"
sorry
lemma simp_flatten:
shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
apply simp
apply(subst flatten_rsimpalts)
apply(simp add: flts_append)
apply(subgoal_tac "\<forall>r \<in> set (map rsimp rsa). good1 r")
apply (metis distinct_once_enough simp_flatten_aux)
using simp_flatten_aux1 by blast
lemma simp_flatten3:
shows "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) = rsimp (RALTS (rsa @ rs @ rsb))"
sorry
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_simp 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 distinct_flts_no0:
shows " rflts (map rsimp (rdistinct rs (insert RZERO rset))) =
rflts (map rsimp (rdistinct rs rset)) "
apply(induct rs arbitrary: rset)
apply simp
apply(case_tac a)
apply simp+
apply (smt (verit, ccfv_SIG) rflts.simps(2) rflts.simps(3) rflts_def_idiot)
prefer 2
apply simp
by (smt (verit, ccfv_threshold) Un_insert_right insert_iff list.simps(9) rdistinct.simps(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot rrexp.distinct(7))
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 "a = RALTS rs \<Longrightarrow> rflts (rs1 @ [a]) = rflts rs1 @ rs"
apply (induct rs1)
apply simp
by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
lemma flts_keeps1:
shows " rflts (rs @ [RONE]) =
rflts rs @ [RONE] "
apply (induct rs)
apply simp
by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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)
apply simp
apply (simp add: rflts_def_idiot)
apply(case_tac a)
apply simp
using flts_keeps1 apply blast
apply (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
apply (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
apply blast
by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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_last2 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_last2 rflts_def_idiot2 rrexp.distinct(21) rrexp.distinct(3))
apply (metis distinct_removes_last2 flts_keeps_others rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
prefer 2
apply (metis distinct_removes_last2 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(simp only:)
apply(subgoal_tac "\<forall> r \<in> set x. r \<in> set (rflts (map rsimp rsa))")
prefer 2
using spilled_alts_contained apply presburger
using distinct_removes_list by blast
(*some basic facts about rsimp*)
end