--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys3/src/ClosedForms.thy Sat Apr 30 00:50:08 2022 +0100
@@ -0,0 +1,1682 @@
+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