theory ClosedForms imports
"BasicIdentities"
begin
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 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>a \<in> set as; \<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 @ aa # rs) {} = rdistinct (as @ rs) {}")
prefer 2
apply (metis append_Cons append_Nil distinct_removes_middle(1))
by presburger
lemma simp_rdistinct_f: shows
"f ` rset = frset \<Longrightarrow> rsimp (rsimp_ALTs (map f (rdistinct rs rset))) =
rsimp (rsimp_ALTs (rdistinct (map f rs) frset)) "
apply(induct rs arbitrary: rset)
apply simp
apply(case_tac "a \<in> rset")
apply(case_tac " f a \<in> frset")
apply simp
apply blast
apply(subgoal_tac "f a \<notin> frset")
apply(simp)
apply(subgoal_tac "f ` (insert a rset) = insert (f a) frset")
prefer 2
apply (meson image_insert)
oops
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 spawn_simp_distinct:
shows "rsimp (rsimp_ALTs (rsa @ (rdistinct rs (set rsa)))) = rsimp (rsimp_ALTs (rsa @ rs))
\<and> (a1 \<in> set rsa1 \<longrightarrow> rsimp (rsimp_ALTs (rsa1 @ rs)) = rsimp (rsimp_ALTs (rsa1 @ a1 # rs)))
\<and> rsimp (rsimp_ALTs (rsc @ rs)) = rsimp (rsimp_ALTs (rsc @ (rdistinct rs (set rsc))))"
apply(induct rs arbitrary: rsa rsa1 a1 rsc)
apply simp
apply(subgoal_tac "rsimp (rsimp_ALTs (rsa1 @ [a1])) = rsimp (rsimp_ALTs (rsa1 @ (rdistinct [a1] (set rsa1))))")
prefer 2
oops
lemma inv_one_derx:
shows " RONE = rder xa r2 \<Longrightarrow> r2 = RCHAR xa"
apply(case_tac r2)
apply simp+
using rrexp.distinct(1) apply presburger
apply (metis rder.simps(5) rrexp.distinct(13) rrexp.simps(20))
apply simp+
done
lemma shape_of_derseq:
shows "rder x (RSEQ r1 r2) = RSEQ (rder x r1) r2 \<or> rder x (RSEQ r1 r2) = (RALT (RSEQ (rder x r1) r2) (rder x r2))"
using rder.simps(5) by presburger
lemma shape_of_derseq2:
shows "rder x (RSEQ r11 r12) = RSEQ x41 x42 \<Longrightarrow> x41 = rder x r11"
by (metis rrexp.distinct(25) rrexp.inject(2) shape_of_derseq)
lemma alts_preimage_case1:
shows "rder x r = RALTS [r] \<Longrightarrow> \<exists>ra. r = RALTS [ra]"
apply(case_tac r)
apply simp+
apply (metis rrexp.simps(12) rrexp.simps(20))
apply (metis rrexp.inject(3) rrexp.simps(30) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) shape_of_derseq)
apply auto[1]
by auto
lemma alts_preimage_case2:
shows "rder x r = RALT r1 r2 \<Longrightarrow> \<exists>ra rb. (r = RSEQ ra rb \<or> r = RALT ra rb)"
apply(case_tac r)
apply simp+
apply (metis rrexp.distinct(15) rrexp.distinct(7))
apply simp
apply auto[1]
by auto
lemma alts_preimage_case2_2:
shows "rder x r = RALT r1 r2 \<Longrightarrow> (\<exists>ra rb. r = RSEQ ra rb) \<or> (\<exists>rc rd. r = RALT rc rd)"
using alts_preimage_case2 by blast
lemma alts_preimage_case3:
shows "rder x r = RALT r1 r2 \<Longrightarrow> (\<exists>ra rb. r = RSEQ ra rb) \<or> (\<exists>rcs rc rd. r = RALTS rcs \<and> rcs = [rc, rd])"
using alts_preimage_case2 by blast
lemma star_seq:
shows "rder x (RSEQ (RSTAR a) b) = RALT (RSEQ (RSEQ (rder x a) (RSTAR a)) b) (rder x b)"
using rder.simps(5) rder.simps(6) rnullable.simps(6) by presburger
lemma language_equality_id1:
shows "\<not>rnullable a \<Longrightarrow> rder x (RSEQ (RSTAR a) b) = rder x (RALT (RSEQ (RSEQ a (RSTAR a)) b) b)"
apply (subst star_seq)
apply simp
done
lemma distinct_der_set:
shows "(rder x) ` rset = dset \<Longrightarrow>
rsimp (rsimp_ALTs (map (rder x) (rdistinct rs rset))) = rsimp ( rsimp_ALTs (rdistinct (map (rder x) rs) dset))"
apply(induct rs arbitrary: rset dset)
apply simp
apply(case_tac "a \<in> rset")
apply(subgoal_tac "rder x a \<in> dset")
prefer 2
apply blast
apply simp
apply(case_tac "rder x a \<notin> dset")
prefer 2
apply simp
oops
lemma map_concat_cons:
shows "map f rsa @ f a # rs = map f (rsa @ [a]) @ rs"
by simp
lemma neg_removal_element_of:
shows " \<not> a \<notin> aset \<Longrightarrow> a \<in> aset"
by simp
lemma simp_more_flts:
shows "rsimp (rsimp_ALTs (rdistinct rs {})) = rsimp (rsimp_ALTs (rdistinct (rflts rs) {}))"
oops
lemma simp_more_distinct1:
shows "rsimp (rsimp_ALTs rs) = rsimp (rsimp_ALTs (rdistinct rs {}))"
apply(induct rs)
apply simp
apply simp
oops
(*
\<and>
rsimp (rsimp_ALTs (rsb @ (rdistinct rs (set rsb)))) =
rsimp (rsimp_ALTs (rsb @ (rdistinct (rflts rs) (set rsb))))
*)
lemma simp_removes_duplicate2:
shows "a "
oops
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 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 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 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 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
by (metis append_self_conv distinct_removes_list in_set_conv_decomp rev_exhaust)
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 flts_middle01:
shows "rflts (rsa @ [RZERO] @ rsb) = rflts (rsa @ rsb)"
by (simp add: flts_middle0)
lemma flts_append1:
shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow>
rflts (rsa @ [a] @ rsb) = rflts rsa @ [a] @ (rflts rsb)"
apply(induct rsa arbitrary: rsb)
apply simp
using rflts_def_idiot apply presburger
apply(case_tac aa)
apply simp+
done
lemma flts_append:
shows "rflts (rs1 @ rs2) = rflts rs1 @ rflts rs2"
apply(induct rs1)
apply simp
apply(case_tac a)
apply simp+
done
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)
lemma distinct_removes_duplicate_flts2:
shows " a \<in> set rsa
\<Longrightarrow> rdistinct (rflts (rsa @ [a] @ rsb)) {} =
rdistinct (rflts (rsa @ rsb)) {}"
apply(induct a arbitrary: rsb)
using flts_middle01 apply presburger
apply(subgoal_tac "rflts (rsa @ [RONE] @ rsb) = rflts rsa @ [RONE] @ rflts rsb")
prefer 2
using flts_append1 apply blast
apply simp
apply(subgoal_tac "RONE \<in> set (rflts rsa)")
prefer 2
using rflts_def_idiot2 apply blast
apply(subst distinct_removes_middle3)
apply simp
using flts_append apply presburger
apply simp
apply (metis distinct_removes_middle3 flts_append in_set_conv_decomp rflts.simps(5))
apply (metis distinct_removes_middle(1) flts_append flts_append1 rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
apply(subgoal_tac "rflts (rsa @ [RALTS x] @ rsb) = rflts rsa @ x @ rflts rsb")
prefer 2
apply (simp add: flts_append)
apply (simp only:)
apply(subgoal_tac "\<forall>r1 \<in> set x. r1 \<in> set (rflts rsa)")
prefer 2
using spilled_alts_contained apply blast
apply(subst flts_append)
using distinct_removes_middle_list apply blast
using distinct_removes_middle2 flts_append rflts_def_idiot2 by fastforce
lemma simp_removes_duplicate:
shows "a \<in> set rsa \<Longrightarrow> rsimp (rsimp_ALTs (rsa @ a # rs)) = rsimp (rsimp_ALTs (rsa @ rs))"
apply(subgoal_tac "rsimp (rsimp_ALTs (rsa @ a # rs)) = rsimp (RALTS (rsa @ a # rs))")
prefer 2
apply (smt (verit, best) Cons_eq_append_conv append_is_Nil_conv empty_set equals0D list.distinct(1) rsimp_ALTs.elims)
apply(simp only:)
apply simp
apply(subgoal_tac "(rdistinct (rflts (map rsimp rsa @ rsimp a # map rsimp rs)) {}) = (rdistinct (rflts (map rsimp rsa @ map rsimp rs)) {})")
apply(simp only:)
prefer 2
apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
prefer 2
apply simp
using distinct_removes_duplicate_flts2 apply force
apply(case_tac rsa)
apply simp
apply(case_tac rs)
apply simp
apply(case_tac list)
apply simp
using idem_after_simp1 apply presburger
apply simp+
apply(subgoal_tac "rsimp_ALTs (aa # list @ aaa # lista) = RALTS (aa # list @ aaa # lista)")
apply simp
using rsimpalts_conscons by presburger
lemma no0_flts:
shows "RZERO \<notin> set (rflts rs)"
apply (induct rs)
apply simp
apply(case_tac a)
apply simp+
oops
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))
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"
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 frewrite_append:
shows "\<lbrakk> rsa \<leadsto>f rsb \<rbrakk> \<Longrightarrow> rs @ rsa \<leadsto>f rs @ rsb"
apply(induct rs)
apply simp+
using frewrite.intros(3) by blast
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 frewrites_concat:
shows "\<lbrakk>rs1 \<leadsto>f rs2; rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> (rs1 @ rsa) \<leadsto>f* (rs2 @ rsb)"
apply(induct rs1 rs2 rule: frewrite.induct)
apply(simp)
apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>f (rs @ rsa)")
prefer 2
using frewrite.intros(1) apply blast
apply(subgoal_tac "(rs @ rsa) \<leadsto>f* (rs @ rsb)")
using many_steps_later apply blast
apply (simp add: frewrites_append)
apply (metis append.assoc append_Cons frewrite.intros(2) frewrites_append many_steps_later)
using frewrites_cons by auto
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)
lemma grewrites_equal_rsimp:
shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
apply simp
sorry
lemma frewrites_middle:
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> r # (RALTS rs # rs1) \<leadsto>f* r # (rs @ rs1)"
by (simp add: fr_in_rstar frewrite.intros(2) frewrite.intros(3))
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)
fun alt_set:: "rrexp \<Rightarrow> rrexp set"
where
"alt_set (RALTS rs) = set rs"
| "alt_set r = {r}"
lemma rd_flts_set:
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rdistinct rs1 ({RZERO} \<union> (rset \<union> \<Union>(alt_set ` rset))) \<leadsto>g*
rdistinct rs2 ({RZERO} \<union> (rset \<union> \<Union>(alt_set ` rset)))"
sorry
lemma with_wo0_distinct:
shows "rdistinct rs rset \<leadsto>f* rdistinct rs (insert RZERO rset)"
apply(induct rs arbitrary: rset)
apply simp
apply(case_tac a)
apply(case_tac "RZERO \<in> rset")
apply simp+
using fr_in_rstar frewrite.intros(1) apply presburger
apply (case_tac "RONE \<in> rset")
apply simp+
using frewrites_cons apply presburger
apply(case_tac "a \<in> rset")
apply simp
apply (simp add: frewrites_cons)
apply(case_tac "a \<in> rset")
apply simp
apply (simp add: frewrites_cons)
apply(case_tac "a \<in> rset")
apply simp
apply (simp add: frewrites_cons)
apply(case_tac "a \<in> rset")
apply simp
apply (simp add: frewrites_cons)
done
lemma frewrite_with_distinct:
shows " \<lbrakk>rs2 \<leadsto>f rs3\<rbrakk>
\<Longrightarrow> rdistinct rs2
(insert RZERO (rset \<union> \<Union> (alt_set ` rset))) \<leadsto>f*
rdistinct rs3
(insert RZERO (rset \<union> \<Union> (alt_set ` rset)))"
apply(induct rs2 rs3 rule: frewrite.induct)
apply(case_tac "RZERO \<in> (rset \<union> \<Union> (alt_set ` rset))")
apply simp
apply simp
apply(case_tac "RALTS rs \<in> rset")
apply simp
apply(subgoal_tac "\<forall>r \<in> set rs. r \<in> \<Union> (alt_set ` rset)")
apply(subgoal_tac " rdistinct (rs @ rsa) (insert RZERO (rset \<union> \<Union> (alt_set ` rset))) =
rdistinct rsa (insert RZERO (rset \<union> \<Union> (alt_set ` rset)))")
using frewrites.intros(1) apply presburger
apply (simp add: rdistinct_concat2)
apply simp
using alt_set.simps(1) apply blast
apply(case_tac "RALTS rs \<in> rset \<union> \<Union>(alt_set ` rset)")
sorry
lemma frewrites_with_distinct:
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
rs1 @ (rdistinct rsa (insert RZERO (set rs1 \<union> \<Union>(alt_set ` (set rs1) )))) \<leadsto>f*
rs2 @ (rdistinct rsb (insert RZERO (set rs2 \<union> \<Union>(alt_set ` (set rs2) ))))"
apply(induct rs1 rs2 rule: frewrites.induct)
apply simp
sorry
(*a more refined notion of \<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 wrong_frewrites_with_distinct2:
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
(rdistinct rs1 {RZERO}) \<leadsto>f* rdistinct rs2 {RZERO}"
oops
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 frewrites_equivalent_simp:
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
apply(induct rs1 rs2 rule: frewrites.induct)
apply simp
using frewrite_single_step by presburger
lemma frewrites_dB_wwo0_simp:
shows "rdistinct rs1 {RZERO} \<leadsto>f* rdistinct rs2 {RZERO}
\<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
sorry
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)")
apply(subgoal_tac "rdistinct (map (rder x) (rflts rs)) {RZERO}
\<leadsto>f* rdistinct ( rflts (map (rder x) rs)) {RZERO}")
apply(subgoal_tac "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {}))
= rsimp (RALTS ( rdistinct ( rflts (map (rder x) rs)) {}))")
apply meson
using frewrites_dB_wwo0_simp apply blast
using frewrites_with_distinct2 apply blast
using early_late_der_frewrites by blast
lemma simp_der_pierce_flts:
shows " 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))) {})
)"
sorry
lemma simp_more_distinct:
shows "rsimp (rsimp_ALTs (rsa @ rs)) = rsimp (rsimp_ALTs (rsa @ (rdistinct rs (set rsa)))) "
sorry
lemma non_empty_list:
shows "a \<in> set as \<Longrightarrow> as \<noteq> []"
by (metis empty_iff empty_set)
lemma distinct_comp:
shows "rdistinct (rs1@rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
apply(induct rs2 arbitrary: rs1)
apply simp
apply(subgoal_tac "rs1 @ a # rs2 = (rs1 @ [a]) @ rs2")
apply(simp only:)
apply(case_tac "a \<in> set rs1")
apply simp
oops
lemma instantiate1:
shows "\<lbrakk>\<And>ab rset1. rdistinct (ab # as) rset1 = rdistinct (ab # as @ [ab]) rset1\<rbrakk> \<Longrightarrow>
rdistinct (aa # as) rset = rdistinct (aa # as @ [aa]) rset"
apply(drule_tac x = "aa" in meta_spec)
apply(drule_tac x = "rset" in meta_spec)
apply simp
done
lemma not_head_elem:
shows " \<lbrakk>aa \<in> set (a # as); aa \<notin> (set as)\<rbrakk> \<Longrightarrow> a = aa"
by fastforce
(*
apply simp
apply (metis append_Cons)
apply(case_tac "ab \<in> rset1")
apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
apply(subgoal_tac "rdistinct (ab # (aa # as) @ [ab]) rset1 =
ab # (rdistinct ((aa # as) @ [ab]) (insert ab rset1))")
apply(simp only:)
apply(subgoal_tac "rdistinct (ab # aa # as) rset1 = ab # (rdistinct (aa # as) (insert ab rset1))")
apply(simp only:)
apply(subgoal_tac "rdistinct ((aa # as) @ [ab]) (insert ab rset1) = rdistinct (aa # as) (insert ab rset1)")
apply blast
*)
lemma flts_identity1:
shows "rflts (rs @ [RONE]) = rflts rs @ [RONE] "
apply(induct rs)
apply simp+
apply(case_tac a)
apply simp
apply simp+
done
lemma flts_identity10:
shows " rflts (rs @ [RCHAR c]) = rflts rs @ [RCHAR c]"
apply(induct rs)
apply simp+
apply(case_tac a)
apply simp+
done
lemma flts_identity11:
shows " rflts (rs @ [RSEQ r1 r2]) = rflts rs @ [RSEQ r1 r2]"
apply(induct rs)
apply simp+
apply(case_tac a)
apply simp+
done
lemma flts_identity12:
shows " rflts (rs @ [RSTAR r0]) = rflts rs @ [RSTAR r0]"
apply(induct rs)
apply simp+
apply(case_tac a)
apply simp+
done
lemma flts_identity2:
shows "a \<noteq> RZERO \<and> (\<forall>rs. a \<noteq> RALTS rs) \<Longrightarrow> rflts (rs @ [a]) = rflts rs @ [a]"
apply(case_tac a)
apply simp
using flts_identity1 apply auto[1]
using flts_identity10 apply blast
using flts_identity11 apply auto[1]
apply blast
using flts_identity12 by presburger
lemma flts_identity3:
shows "a = RZERO \<Longrightarrow> rflts (rs @ [a]) = rflts rs"
apply simp
apply(induct rs)
apply simp+
apply(case_tac aa)
apply simp+
done
lemma distinct_removes_last3:
shows "\<lbrakk>a \<in> set as\<rbrakk>
\<Longrightarrow> rdistinct as {} = rdistinct (as @ [a]) {}"
by (simp add: distinct_removes_last2)
lemma set_inclusion_with_flts1:
shows " \<lbrakk>RONE \<in> set rs\<rbrakk> \<Longrightarrow> RONE \<in> set (rflts rs)"
apply(induct rs)
apply simp
apply(case_tac " RONE \<in> set rs")
apply simp
apply (metis Un_upper2 insert_absorb insert_subset list.set_intros(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append)
apply(case_tac "RONE = a")
apply simp
apply simp
done
lemma set_inclusion_with_flts10:
shows " \<lbrakk>RCHAR x \<in> set rs\<rbrakk> \<Longrightarrow> RCHAR x \<in> set (rflts rs)"
apply(induct rs)
apply simp
apply(case_tac " RCHAR x \<in> set rs")
apply simp
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
apply(case_tac "RCHAR x = a")
apply simp
apply fastforce
apply simp
done
lemma set_inclusion_with_flts11:
shows " \<lbrakk>RSEQ r1 r2 \<in> set rs\<rbrakk> \<Longrightarrow> RSEQ r1 r2 \<in> set (rflts rs)"
apply(induct rs)
apply simp
apply(case_tac " RSEQ r1 r2 \<in> set rs")
apply simp
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
apply(case_tac "RSEQ r1 r2 = a")
apply simp
apply fastforce
apply simp
done
lemma set_inclusion_with_flts:
shows " \<lbrakk>a \<in> set as; rsimp a \<in> set (map rsimp as); rsimp a = RONE\<rbrakk> \<Longrightarrow> rsimp a \<in> set (rflts (map rsimp as))"
by (simp add: set_inclusion_with_flts1)
lemma "\<And>x5. \<lbrakk>a \<in> set as; rsimp a \<in> set (map rsimp as); rsimp a = RALTS x5\<rbrakk>
\<Longrightarrow> rsimp_ALTs (rdistinct (rflts (map rsimp as @ [rsimp a])) {}) =
rsimp_ALTs (rdistinct (rflts (map rsimp as @ x5)) {})"
sorry
lemma last_elem_dup1:
shows " a \<in> set as \<Longrightarrow> rsimp (RALTS (as @ [a] )) = rsimp (RALTS (as ))"
apply simp
apply(subgoal_tac "rsimp a \<in> set (map rsimp as)")
prefer 2
apply simp
apply(case_tac "rsimp a")
apply simp
using flts_identity3 apply presburger
apply(subst flts_identity2)
using rrexp.distinct(1) rrexp.distinct(15) apply presburger
apply(subst distinct_removes_last3[symmetric])
using set_inclusion_with_flts apply blast
apply simp
apply (metis distinct_removes_last3 flts_identity10 set_inclusion_with_flts10)
apply (metis distinct_removes_last3 flts_identity11 set_inclusion_with_flts11)
sorry
lemma last_elem_dup:
shows " a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs (as @ [a] )) = rsimp (rsimp_ALTs (as ))"
apply(induct as rule: rev_induct)
apply simp
apply simp
apply(subgoal_tac "xs \<noteq> []")
prefer 2
sorry
lemma appeared_before_remove_later:
shows "a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs ( as @ a # rs)) = rsimp (rsimp_ALTs (as @ rs))"
and "a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs as ) = rsimp (rsimp_ALTs (as @ [a]))"
apply(induct rs arbitrary: as)
apply simp
sorry
lemma distinct_remove_later:
shows "\<lbrakk>rder x a \<in> rder x ` set rsa\<rbrakk>
\<Longrightarrow> rsimp (rsimp_ALTs (map (rder x) rsa @ rder x a # map (rder x) (rdistinct rs (insert a (set rsa))))) =
rsimp (rsimp_ALTs (map (rder x) rsa @ map (rder x) (rdistinct rs (set rsa))))"
sorry
lemma distinct_der_general:
shows "rsimp (rsimp_ALTs (map (rder x) (rsa @ (rdistinct rs (set rsa))))) =
rsimp ( rsimp_ALTs ((map (rder x) rsa)@(rdistinct (map (rder x) rs) (set (map (rder x) rsa)))) )"
apply(induct rs arbitrary: rsa)
apply simp
apply(case_tac "a \<in> set rsa")
apply(subgoal_tac "rder x a \<in> set (map (rder x) rsa)")
apply simp
apply simp
apply(case_tac "rder x a \<notin> set (map (rder x) rsa)")
apply(simp)
apply(subst map_concat_cons)+
apply(drule_tac x = "rsa @ [a]" in meta_spec)
apply simp
apply(drule neg_removal_element_of)
apply simp
apply(subst distinct_remove_later)
apply simp
apply(drule_tac x = "rsa" in meta_spec)
by blast
lemma distinct_der:
shows "rsimp (rsimp_ALTs (map (rder x) (rdistinct rs {}))) = rsimp ( rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
by (metis distinct_der_general list.simps(8) self_append_conv2 set_empty)
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 add0_isomorphic:
shows "rsimp_ALTs (rdistinct (rflts [rsimp r, RZERO]) {}) = rsimp r"
sorry
lemma distinct_append_simp:
shows " rsimp (rsimp_ALTs rs1) = rsimp (rsimp_ALTs rs2) \<Longrightarrow>
rsimp (rsimp_ALTs (f a # rs1)) =
rsimp (rsimp_ALTs (f a # rs2))"
apply(case_tac rs1)
apply simp
apply(case_tac rs2)
apply simp
apply simp
prefer 2
apply(case_tac list)
apply(case_tac rs2)
apply simp
using add0_isomorphic apply blast
apply simp
oops
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)
sledgehammer
(* by (metis inside_simp_removal rder_rsimp_ALTs_commute self_append_conv2 set_empty simp_more_distinct)
*)
lemma alts_closed_form_variant: shows
"s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s =
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
sorry
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
sorry
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))
)
)"
apply(induct s)
apply simp
sorry
lemma seq_closed_form_variant: shows
"s \<noteq> [] \<Longrightarrow> (rders_simp (RSEQ r1 r2) s) =
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
apply(induct s rule: rev_induct)
apply simp
apply(subst rders_simp_append)
apply(subst rders_simp_one_char)
apply(subst rsimp_idem[symmetric])
apply(subst rders_simp_one_char[symmetric])
apply(subst rders_simp_append[symmetric])
apply(insert seq_closed_form)
apply(subgoal_tac "rsimp (rders_simp (RSEQ r1 r2) (xs @ [x]))
= rsimp (RALTS (RSEQ (rders_simp r1 (xs @ [x])) r2 # map (rders_simp r2) (vsuf (xs @ [x]) r1)))")
apply force
by presburger
end