--- a/thys2/ClosedForms.thy Sat Mar 19 09:53:48 2022 +0000
+++ b/thys2/ClosedForms.thy Sat Mar 19 10:36:52 2022 +0000
@@ -2,6 +2,32 @@
"BasicIdentities"
begin
+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
+ sorry
+
+(* apply (smt (z3) append.right_neutral empty_iff list.distinct(1) list.inject no_alt_short_list_after_simp no_further_dB_after_simp rdistinct.elims rflts.elims rflts.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2)))*)
+
+
+
lemma simp_rdistinct_f: shows
@@ -18,16 +44,365 @@
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 alts_preimage_cases:
+ shows "rder x r = RALT (RSEQ r1 r2) r3 \<Longrightarrow> (\<exists>ra rb. r = RSEQ ra rb) \<or> (\<exists>rc rd re. r = RALT (RSEQ rc rd) re)"
+ apply(case_tac r)
+ apply simp+
+
+ apply (metis rrexp.simps(12) rrexp.simps(20))
+ prefer 3
+ apply simp
+ apply blast
+ apply(frule alts_preimage_case2_2)
+ apply(case_tac "(\<exists>ra rb. r = RSEQ ra rb)")
+ apply blast
+ apply(subgoal_tac " (\<exists> rc rd. r = RALT rc rd )")
+ prefer 2
+ apply blast
+ apply(erule exE)+
+ apply(subgoal_tac "rder x r = RALT (rder x rc) (rder x rd)")
+ prefer 2
+ apply force
+ apply(subgoal_tac "rder x rc = RSEQ r1 r2")
+ oops
+
+
+lemma der_seq_eq_case:
+ shows "\<lbrakk>r1 \<noteq> r2 ; r1 = RSEQ ra rb; rder x r1 = rder x r2\<rbrakk> \<Longrightarrow> rsimp (rder x r1) = RZERO \<and> rsimp (rder x r2) = RZERO"
+ apply(case_tac "rnullable ra")
+ apply simp
+ oops
+
+
+
+
+lemma der_map_unequal_to_equal_zero_only:
+ shows "\<lbrakk>r1 \<noteq> r2 ; rder x r1 = rder x r2 \<rbrakk> \<Longrightarrow> rsimp (rder x r1) = RZERO"
+ apply(induct r1)
+ apply simp
+ apply simp
+ apply simp
+ apply(case_tac "x = xa")
+ apply simp
+ apply(subgoal_tac "r2 = RCHAR xa")
+ prefer 2
+ using inv_one_derx apply blast
+ apply simp
+ using rsimp.simps(3) apply presburger
+ apply(case_tac "rder x (RSEQ r11 r12)")
+ apply simp
+ apply (metis inv_one_derx)
+ apply (metis rrexp.distinct(21) rrexp.simps(24) shape_of_derseq)
+ apply(subgoal_tac "rder x r2 = RSEQ x41 x42")
+ prefer 2
+ apply presburger
+ apply(subgoal_tac "x41 = rder x r11")
+ prefer 2
+ apply (meson shape_of_derseq2)
+ apply(case_tac r2)
+ apply simp+
+ apply (metis rrexp.distinct(13) rrexp.simps(10))
+ apply(subgoal_tac "x42a = x42")
+ prefer 2
+ apply (metis rrexp.inject(2) rrexp.simps(30) shape_of_derseq)
+ apply(subgoal_tac "rder x x41a = x41")
+ prefer 2
+ apply (metis shape_of_derseq2)
+ apply(simp)
+ apply(subgoal_tac "\<not> rnullable r11")
+ prefer 2
+ apply force
+ apply simp
+ apply(subgoal_tac "\<not> rnullable x41a")
+ prefer 2
+ apply force
+ apply simp
+
+ oops
+
+
+
+lemma der_maps_1to1_except0:
+ shows "\<lbrakk>rder x ` rset = dset; a \<notin> rset; rder x a \<in> dset\<rbrakk> \<Longrightarrow> rsimp (rder x a) = RZERO"
+
+
+ sorry
+
+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_distinct:
+ shows "rsimp (rsimp_ALTs (rsa @ rs)) = rsimp (rsimp_ALTs (rsa @ (rdistinct rs (set rsa)))) \<and>
+ rsimp (rsimp_ALTs (rsb @ (rdistinct rs (set rsb)))) =
+ rsimp (rsimp_ALTs (rsb @ (rdistinct (rflts rs) (set rsb))))"
+ apply(induct rs arbitrary: rsa rsb)
+ apply simp
+
+ sorry
+
+lemma non_empty_list:
+ shows "a \<in> set as \<Longrightarrow> as \<noteq> []"
+
+ by (metis empty_iff empty_set)
+
+
+lemma distinct_removes_last:
+ shows "\<lbrakk>a \<in> set as; rsimp a \<in> set (map rsimp as)\<rbrakk>
+ \<Longrightarrow> rsimp_ALTs (rdistinct (rflts (map rsimp as @ [rsimp a])) {}) =
+ rsimp_ALTs (rdistinct (rflts (map rsimp as)) {})"
+ apply(induct "rsimp a" arbitrary: as)
+ apply(simp)
+ apply (metis append.right_neutral append_self_conv2 empty_set list.simps(9) map_append rflts.simps(2) rsimp.simps(2) rsimp_idem simp_more_distinct spawn_simp_rsimpalts)
+ apply simp
+ sorry
+
+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 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
+
+ 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)
- sorry
+
+
+
+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) =
@@ -54,6 +429,25 @@
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)
+
+(* apply (metis head_one_more_simp list.inject list.map_comp list.simps(9) rders_simp_lambda rsimp.simps(2))
+*)
sorry