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444
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theory ClosedForms imports
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"BasicIdentities"
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443
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begin
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453
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lemma add0_isomorphic:
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shows "rsimp_ALTs (rdistinct (rflts [rsimp r, RZERO]) {}) = rsimp r"
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sorry
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456
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lemma distinct_removes_last:
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shows "\<lbrakk>(a::rrexp) \<in> set as\<rbrakk>
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\<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
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and "rdistinct (ab # as @ [ab]) rset1 = rdistinct (ab # as) rset1"
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apply(induct as arbitrary: rset ab rset1 a)
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apply simp
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apply simp
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apply(case_tac "aa \<in> rset")
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apply(case_tac "a = aa")
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apply (metis append_Cons)
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apply simp
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apply(case_tac "a \<in> set as")
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apply (metis append_Cons rdistinct.simps(2) set_ConsD)
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apply(case_tac "a = aa")
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prefer 2
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apply simp
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apply (metis append_Cons)
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apply(case_tac "ab \<in> rset1")
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prefer 2
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apply(subgoal_tac "rdistinct (ab # (a # as) @ [ab]) rset1 =
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ab # (rdistinct ((a # as) @ [ab]) (insert ab rset1))")
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prefer 2
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apply force
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apply(simp only:)
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apply(subgoal_tac "rdistinct (ab # a # as) rset1 = ab # (rdistinct (a # as) (insert ab rset1))")
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apply(simp only:)
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apply(subgoal_tac "rdistinct ((a # as) @ [ab]) (insert ab rset1) = rdistinct (a # as) (insert ab rset1)")
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apply blast
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apply(case_tac "a \<in> insert ab rset1")
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apply simp
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apply (metis insertI1)
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apply simp
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apply (meson insertI1)
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apply simp
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apply(subgoal_tac "rdistinct ((a # as) @ [ab]) rset1 = rdistinct (a # as) rset1")
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apply simp
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by (metis append_Cons insert_iff insert_is_Un rdistinct.simps(2))
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lemma distinct_removes_last2:
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shows "\<lbrakk>(a::rrexp) \<in> set as\<rbrakk>
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\<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
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using distinct_removes_last(1) by presburger
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453
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lemma distinct_append_simp:
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shows " rsimp (rsimp_ALTs rs1) = rsimp (rsimp_ALTs rs2) \<Longrightarrow>
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rsimp (rsimp_ALTs (f a # rs1)) =
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rsimp (rsimp_ALTs (f a # rs2))"
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apply(case_tac rs1)
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apply simp
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apply(case_tac rs2)
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apply simp
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apply simp
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prefer 2
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apply(case_tac list)
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apply(case_tac rs2)
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apply simp
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using add0_isomorphic apply blast
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apply simp
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sorry
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(* 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)))*)
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451
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lemma simp_rdistinct_f: shows
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"f ` rset = frset \<Longrightarrow> rsimp (rsimp_ALTs (map f (rdistinct rs rset))) = rsimp (rsimp_ALTs (rdistinct (map f rs) frset)) "
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apply(induct rs arbitrary: rset)
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apply simp
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apply(case_tac "a \<in> rset")
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apply(case_tac " f a \<in> frset")
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apply simp
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apply blast
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apply(subgoal_tac "f a \<notin> frset")
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apply(simp)
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apply(subgoal_tac "f ` (insert a rset) = insert (f a) frset")
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prefer 2
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apply (meson image_insert)
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453
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oops
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451
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453
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lemma spawn_simp_rsimpalts:
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shows "rsimp (rsimp_ALTs rs) = rsimp (rsimp_ALTs (map rsimp rs))"
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apply(cases rs)
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apply simp
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apply(case_tac list)
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apply simp
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apply(subst rsimp_idem[symmetric])
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apply simp
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apply(subgoal_tac "rsimp_ALTs rs = RALTS rs")
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apply(simp only:)
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apply(subgoal_tac "rsimp_ALTs (map rsimp rs) = RALTS (map rsimp rs)")
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apply(simp only:)
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prefer 2
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apply simp
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prefer 2
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using rsimp_ALTs.simps(3) apply presburger
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apply auto
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apply(subst rsimp_idem)+
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by (metis comp_apply rsimp_idem)
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lemma spawn_simp_distinct:
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shows "rsimp (rsimp_ALTs (rsa @ (rdistinct rs (set rsa)))) = rsimp (rsimp_ALTs (rsa @ rs))
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\<and> (a1 \<in> set rsa1 \<longrightarrow> rsimp (rsimp_ALTs (rsa1 @ rs)) = rsimp (rsimp_ALTs (rsa1 @ a1 # rs)))
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\<and> rsimp (rsimp_ALTs (rsc @ rs)) = rsimp (rsimp_ALTs (rsc @ (rdistinct rs (set rsc))))"
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apply(induct rs arbitrary: rsa rsa1 a1 rsc)
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apply simp
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apply(subgoal_tac "rsimp (rsimp_ALTs (rsa1 @ [a1])) = rsimp (rsimp_ALTs (rsa1 @ (rdistinct [a1] (set rsa1))))")
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prefer 2
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oops
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lemma inv_one_derx:
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shows " RONE = rder xa r2 \<Longrightarrow> r2 = RCHAR xa"
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apply(case_tac r2)
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apply simp+
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using rrexp.distinct(1) apply presburger
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apply (metis rder.simps(5) rrexp.distinct(13) rrexp.simps(20))
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apply simp+
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done
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lemma shape_of_derseq:
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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))"
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using rder.simps(5) by presburger
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lemma shape_of_derseq2:
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shows "rder x (RSEQ r11 r12) = RSEQ x41 x42 \<Longrightarrow> x41 = rder x r11"
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by (metis rrexp.distinct(25) rrexp.inject(2) shape_of_derseq)
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lemma alts_preimage_case1:
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shows "rder x r = RALTS [r] \<Longrightarrow> \<exists>ra. r = RALTS [ra]"
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apply(case_tac r)
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apply simp+
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apply (metis rrexp.simps(12) rrexp.simps(20))
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apply (metis rrexp.inject(3) rrexp.simps(30) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) shape_of_derseq)
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apply auto[1]
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by auto
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lemma alts_preimage_case2:
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shows "rder x r = RALT r1 r2 \<Longrightarrow> \<exists>ra rb. (r = RSEQ ra rb \<or> r = RALT ra rb)"
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apply(case_tac r)
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apply simp+
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apply (metis rrexp.distinct(15) rrexp.distinct(7))
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apply simp
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apply auto[1]
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by auto
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lemma alts_preimage_case2_2:
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shows "rder x r = RALT r1 r2 \<Longrightarrow> (\<exists>ra rb. r = RSEQ ra rb) \<or> (\<exists>rc rd. r = RALT rc rd)"
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using alts_preimage_case2 by blast
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lemma alts_preimage_case3:
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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])"
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using alts_preimage_case2 by blast
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lemma star_seq:
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shows "rder x (RSEQ (RSTAR a) b) = RALT (RSEQ (RSEQ (rder x a) (RSTAR a)) b) (rder x b)"
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using rder.simps(5) rder.simps(6) rnullable.simps(6) by presburger
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lemma language_equality_id1:
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shows "\<not>rnullable a \<Longrightarrow> rder x (RSEQ (RSTAR a) b) = rder x (RALT (RSEQ (RSEQ a (RSTAR a)) b) b)"
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apply (subst star_seq)
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apply simp
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done
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lemma distinct_der_set:
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shows "(rder x) ` rset = dset \<Longrightarrow>
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rsimp (rsimp_ALTs (map (rder x) (rdistinct rs rset))) = rsimp ( rsimp_ALTs (rdistinct (map (rder x) rs) dset))"
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apply(induct rs arbitrary: rset dset)
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apply simp
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apply(case_tac "a \<in> rset")
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apply(subgoal_tac "rder x a \<in> dset")
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prefer 2
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apply blast
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apply simp
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apply(case_tac "rder x a \<notin> dset")
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prefer 2
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apply simp
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oops
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lemma map_concat_cons:
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shows "map f rsa @ f a # rs = map f (rsa @ [a]) @ rs"
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by simp
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lemma neg_removal_element_of:
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shows " \<not> a \<notin> aset \<Longrightarrow> a \<in> aset"
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by simp
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lemma simp_more_flts:
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shows "rsimp (rsimp_ALTs (rdistinct rs {})) = rsimp (rsimp_ALTs (rdistinct (rflts rs) {}))"
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oops
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lemma simp_more_distinct:
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shows "rsimp (rsimp_ALTs (rsa @ rs)) = rsimp (rsimp_ALTs (rsa @ (rdistinct rs (set rsa)))) \<and>
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rsimp (rsimp_ALTs (rsb @ (rdistinct rs (set rsb)))) =
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rsimp (rsimp_ALTs (rsb @ (rdistinct (rflts rs) (set rsb))))"
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apply(induct rs arbitrary: rsa rsb)
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apply simp
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sorry
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lemma non_empty_list:
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shows "a \<in> set as \<Longrightarrow> as \<noteq> []"
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by (metis empty_iff empty_set)
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456
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lemma distinct_comp:
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shows "rdistinct (rs1@rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
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apply(induct rs2 arbitrary: rs1)
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apply simp
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apply(subgoal_tac "rs1 @ a # rs2 = (rs1 @ [a]) @ rs2")
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apply(simp only:)
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apply(case_tac "a \<in> set rs1")
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apply simp
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oops
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453
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456
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lemma instantiate1:
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shows "\<lbrakk>\<And>ab rset1. rdistinct (ab # as) rset1 = rdistinct (ab # as @ [ab]) rset1\<rbrakk> \<Longrightarrow>
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rdistinct (aa # as) rset = rdistinct (aa # as @ [aa]) rset"
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apply(drule_tac x = "aa" in meta_spec)
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apply(drule_tac x = "rset" in meta_spec)
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453
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apply simp
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456
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done
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lemma not_head_elem:
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shows " \<lbrakk>aa \<in> set (a # as); aa \<notin> (set as)\<rbrakk> \<Longrightarrow> a = aa"
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by fastforce
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(*
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apply simp
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apply (metis append_Cons)
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apply(case_tac "ab \<in> rset1")
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apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
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apply(subgoal_tac "rdistinct (ab # (aa # as) @ [ab]) rset1 =
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ab # (rdistinct ((aa # as) @ [ab]) (insert ab rset1))")
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apply(simp only:)
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apply(subgoal_tac "rdistinct (ab # aa # as) rset1 = ab # (rdistinct (aa # as) (insert ab rset1))")
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apply(simp only:)
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apply(subgoal_tac "rdistinct ((aa # as) @ [ab]) (insert ab rset1) = rdistinct (aa # as) (insert ab rset1)")
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apply blast
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*)
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453
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lemma flts_identity1:
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shows "rflts (rs @ [RONE]) = rflts rs @ [RONE] "
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apply(induct rs)
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apply simp+
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apply(case_tac a)
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apply simp
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apply simp+
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done
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lemma flts_identity10:
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shows " rflts (rs @ [RCHAR c]) = rflts rs @ [RCHAR c]"
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apply(induct rs)
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apply simp+
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apply(case_tac a)
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apply simp+
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done
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lemma flts_identity11:
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shows " rflts (rs @ [RSEQ r1 r2]) = rflts rs @ [RSEQ r1 r2]"
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apply(induct rs)
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apply simp+
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apply(case_tac a)
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apply simp+
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done
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lemma flts_identity12:
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shows " rflts (rs @ [RSTAR r0]) = rflts rs @ [RSTAR r0]"
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apply(induct rs)
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apply simp+
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apply(case_tac a)
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apply simp+
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done
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lemma flts_identity2:
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shows "a \<noteq> RZERO \<and> (\<forall>rs. a \<noteq> RALTS rs) \<Longrightarrow> rflts (rs @ [a]) = rflts rs @ [a]"
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apply(case_tac a)
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apply simp
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using flts_identity1 apply auto[1]
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using flts_identity10 apply blast
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using flts_identity11 apply auto[1]
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apply blast
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using flts_identity12 by presburger
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456
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lemma flts_identity3:
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shows "a = RZERO \<Longrightarrow> rflts (rs @ [a]) = rflts rs"
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apply simp
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apply(induct rs)
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apply simp+
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apply(case_tac aa)
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315 |
apply simp+
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316 |
done
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317 |
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318 |
lemma distinct_removes_last3:
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319 |
shows "\<lbrakk>(a::rrexp) \<in> set as\<rbrakk>
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320 |
\<Longrightarrow> rdistinct as {} = rdistinct (as @ [a]) {}"
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321 |
using distinct_removes_last2 by blast
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322 |
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323 |
lemma set_inclusion_with_flts1:
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324 |
shows " \<lbrakk>RONE \<in> set rs\<rbrakk> \<Longrightarrow> RONE \<in> set (rflts rs)"
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325 |
apply(induct rs)
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326 |
apply simp
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327 |
apply(case_tac " RONE \<in> set rs")
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328 |
apply simp
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329 |
apply (metis Un_upper2 insert_absorb insert_subset list.set_intros(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append)
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330 |
apply(case_tac "RONE = a")
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331 |
apply simp
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332 |
apply simp
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333 |
done
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334 |
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335 |
lemma set_inclusion_with_flts10:
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336 |
shows " \<lbrakk>RCHAR x \<in> set rs\<rbrakk> \<Longrightarrow> RCHAR x \<in> set (rflts rs)"
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337 |
apply(induct rs)
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338 |
apply simp
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339 |
apply(case_tac " RCHAR x \<in> set rs")
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340 |
apply simp
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341 |
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
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342 |
apply(case_tac "RCHAR x = a")
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343 |
apply simp
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344 |
apply fastforce
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345 |
apply simp
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346 |
done
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347 |
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348 |
lemma set_inclusion_with_flts11:
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349 |
shows " \<lbrakk>RSEQ r1 r2 \<in> set rs\<rbrakk> \<Longrightarrow> RSEQ r1 r2 \<in> set (rflts rs)"
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350 |
apply(induct rs)
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351 |
apply simp
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352 |
apply(case_tac " RSEQ r1 r2 \<in> set rs")
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353 |
apply simp
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354 |
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
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355 |
apply(case_tac "RSEQ r1 r2 = a")
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356 |
apply simp
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357 |
apply fastforce
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358 |
apply simp
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359 |
done
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360 |
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361 |
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362 |
lemma set_inclusion_with_flts:
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363 |
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))"
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364 |
by (simp add: set_inclusion_with_flts1)
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453
|
365 |
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456
|
366 |
lemma "\<And>x5. \<lbrakk>a \<in> set as; rsimp a \<in> set (map rsimp as); rsimp a = RALTS x5\<rbrakk>
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367 |
\<Longrightarrow> rsimp_ALTs (rdistinct (rflts (map rsimp as @ [rsimp a])) {}) =
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368 |
rsimp_ALTs (rdistinct (rflts (map rsimp as @ x5)) {})"
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369 |
|
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453
|
370 |
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371 |
lemma last_elem_dup1:
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372 |
shows " a \<in> set as \<Longrightarrow> rsimp (RALTS (as @ [a] )) = rsimp (RALTS (as ))"
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373 |
apply simp
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374 |
apply(subgoal_tac "rsimp a \<in> set (map rsimp as)")
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|
375 |
prefer 2
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|
376 |
apply simp
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456
|
377 |
apply(case_tac "rsimp a")
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|
378 |
apply simp
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379 |
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|
380 |
using flts_identity3 apply presburger
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|
381 |
apply(subst flts_identity2)
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|
382 |
using rrexp.distinct(1) rrexp.distinct(15) apply presburger
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|
383 |
apply(subst distinct_removes_last3[symmetric])
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|
384 |
using set_inclusion_with_flts apply blast
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|
385 |
apply simp
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|
386 |
apply (metis distinct_removes_last3 flts_identity10 set_inclusion_with_flts10)
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|
387 |
apply (metis distinct_removes_last3 flts_identity11 set_inclusion_with_flts11)
|
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453
|
388 |
sorry
|
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|
389 |
|
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|
390 |
lemma last_elem_dup:
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|
391 |
shows " a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs (as @ [a] )) = rsimp (rsimp_ALTs (as ))"
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|
392 |
apply(induct as rule: rev_induct)
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|
393 |
apply simp
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|
394 |
apply simp
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|
395 |
apply(subgoal_tac "xs \<noteq> []")
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|
396 |
prefer 2
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|
397 |
|
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|
398 |
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|
399 |
|
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|
400 |
|
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|
401 |
sorry
|
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|
402 |
|
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|
403 |
lemma appeared_before_remove_later:
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|
404 |
shows "a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs ( as @ a # rs)) = rsimp (rsimp_ALTs (as @ rs))"
|
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|
405 |
and "a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs as ) = rsimp (rsimp_ALTs (as @ [a]))"
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|
406 |
apply(induct rs arbitrary: as)
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|
407 |
apply simp
|
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|
408 |
|
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|
409 |
|
|
|
410 |
sorry
|
|
|
411 |
|
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|
412 |
lemma distinct_remove_later:
|
|
|
413 |
shows "\<lbrakk>rder x a \<in> rder x ` set rsa\<rbrakk>
|
|
|
414 |
\<Longrightarrow> rsimp (rsimp_ALTs (map (rder x) rsa @ rder x a # map (rder x) (rdistinct rs (insert a (set rsa))))) =
|
|
|
415 |
rsimp (rsimp_ALTs (map (rder x) rsa @ map (rder x) (rdistinct rs (set rsa))))"
|
|
451
|
416 |
|
|
|
417 |
sorry
|
|
|
418 |
|
|
|
419 |
|
|
453
|
420 |
lemma distinct_der_general:
|
|
|
421 |
shows "rsimp (rsimp_ALTs (map (rder x) (rsa @ (rdistinct rs (set rsa))))) =
|
|
|
422 |
rsimp ( rsimp_ALTs ((map (rder x) rsa)@(rdistinct (map (rder x) rs) (set (map (rder x) rsa)))) )"
|
|
|
423 |
apply(induct rs arbitrary: rsa)
|
|
|
424 |
apply simp
|
|
|
425 |
apply(case_tac "a \<in> set rsa")
|
|
|
426 |
apply(subgoal_tac "rder x a \<in> set (map (rder x) rsa)")
|
|
|
427 |
apply simp
|
|
|
428 |
apply simp
|
|
|
429 |
apply(case_tac "rder x a \<notin> set (map (rder x) rsa)")
|
|
|
430 |
apply(simp)
|
|
|
431 |
apply(subst map_concat_cons)+
|
|
|
432 |
apply(drule_tac x = "rsa @ [a]" in meta_spec)
|
|
|
433 |
apply simp
|
|
|
434 |
apply(drule neg_removal_element_of)
|
|
|
435 |
apply simp
|
|
|
436 |
apply(subst distinct_remove_later)
|
|
|
437 |
apply simp
|
|
|
438 |
apply(drule_tac x = "rsa" in meta_spec)
|
|
|
439 |
by blast
|
|
|
440 |
|
|
|
441 |
|
|
|
442 |
|
|
|
443 |
|
|
451
|
444 |
lemma distinct_der:
|
|
|
445 |
shows "rsimp (rsimp_ALTs (map (rder x) (rdistinct rs {}))) = rsimp ( rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
|
|
453
|
446 |
by (metis distinct_der_general list.simps(8) self_append_conv2 set_empty)
|
|
451
|
447 |
|
|
453
|
448 |
|
|
|
449 |
|
|
|
450 |
|
|
|
451 |
lemma rders_simp_lambda:
|
|
|
452 |
shows " rsimp \<circ> rder x \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r (xs @ [x]))"
|
|
|
453 |
using rders_simp_append by auto
|
|
451
|
454 |
|
|
453
|
455 |
lemma rders_simp_nonempty_simped:
|
|
|
456 |
shows "xs \<noteq> [] \<Longrightarrow> rsimp \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r xs)"
|
|
|
457 |
using rders_simp_same_simpders rsimp_idem by auto
|
|
|
458 |
|
|
|
459 |
lemma repeated_altssimp:
|
|
|
460 |
shows "\<forall>r \<in> set rs. rsimp r = r \<Longrightarrow> rsimp (rsimp_ALTs (rdistinct (rflts rs) {})) =
|
|
|
461 |
rsimp_ALTs (rdistinct (rflts rs) {})"
|
|
|
462 |
by (metis map_idI rsimp.simps(2) rsimp_idem)
|
|
451
|
463 |
|
|
444
|
464 |
lemma alts_closed_form: shows
|
|
|
465 |
"rsimp (rders_simp (RALTS rs) s) =
|
|
|
466 |
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
|
467 |
apply(induct s rule: rev_induct)
|
|
|
468 |
apply simp
|
|
|
469 |
apply simp
|
|
|
470 |
apply(subst rders_simp_append)
|
|
|
471 |
apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) =
|
|
|
472 |
rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
|
|
|
473 |
prefer 2
|
|
|
474 |
apply (metis inside_simp_removal rders_simp_one_char)
|
|
|
475 |
apply(simp only: )
|
|
451
|
476 |
apply(subst rders_simp_one_char)
|
|
|
477 |
apply(subst rsimp_idem)
|
|
|
478 |
apply(subgoal_tac "rsimp (rder x (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {}))) =
|
|
|
479 |
rsimp ((rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))) ")
|
|
|
480 |
prefer 2
|
|
|
481 |
using rder_rsimp_ALTs_commute apply presburger
|
|
|
482 |
apply(simp only:)
|
|
|
483 |
apply(subgoal_tac "rsimp (rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))
|
|
|
484 |
= rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
485 |
prefer 2
|
|
|
486 |
|
|
|
487 |
using distinct_der apply presburger
|
|
|
488 |
apply(simp only:)
|
|
453
|
489 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
490 |
rsimp (rsimp_ALTs (rdistinct ( (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)))) {}))")
|
|
|
491 |
apply(simp only:)
|
|
|
492 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
493 |
rsimp (rsimp_ALTs (rdistinct (rflts ( (map (rsimp \<circ> (rder x) \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
494 |
apply(simp only:)
|
|
|
495 |
apply(subst rders_simp_lambda)
|
|
|
496 |
apply(subst rders_simp_nonempty_simped)
|
|
|
497 |
apply simp
|
|
|
498 |
apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (xs @ [x])) rs). rsimp r = r")
|
|
|
499 |
prefer 2
|
|
|
500 |
apply (simp add: rders_simp_same_simpders rsimp_idem)
|
|
|
501 |
apply(subst repeated_altssimp)
|
|
|
502 |
apply simp
|
|
|
503 |
apply fastforce
|
|
|
504 |
apply (metis inside_simp_removal list.map_comp rder.simps(4) rsimp.simps(2) rsimp_idem)
|
|
|
505 |
|
|
|
506 |
(* apply (metis head_one_more_simp list.inject list.map_comp list.simps(9) rders_simp_lambda rsimp.simps(2))
|
|
|
507 |
*)
|
|
451
|
508 |
|
|
444
|
509 |
sorry
|
|
443
|
510 |
|
|
444
|
511 |
lemma alts_closed_form_variant: shows
|
|
|
512 |
"s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s =
|
|
|
513 |
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
|
514 |
sorry
|
|
443
|
515 |
|
|
|
516 |
|
|
|
517 |
|
|
444
|
518 |
lemma star_closed_form:
|
|
|
519 |
shows "rders_simp (RSTAR r0) (c#s) =
|
|
|
520 |
rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
|
|
|
521 |
apply(induct s)
|
|
|
522 |
apply simp
|
|
|
523 |
sorry
|
|
443
|
524 |
|
|
|
525 |
|
|
|
526 |
|
|
|
527 |
lemma seq_closed_form: shows
|
|
|
528 |
"rsimp (rders_simp (RSEQ r1 r2) s) =
|
|
|
529 |
rsimp ( RALTS ( (RSEQ (rders_simp r1 s) r2) #
|
|
445
|
530 |
(map (rders_simp r2) (vsuf s r1))
|
|
443
|
531 |
)
|
|
|
532 |
)"
|
|
|
533 |
apply(induct s)
|
|
|
534 |
apply simp
|
|
|
535 |
sorry
|
|
|
536 |
|
|
|
537 |
|
|
444
|
538 |
lemma seq_closed_form_variant: shows
|
|
|
539 |
"s \<noteq> [] \<Longrightarrow> (rders_simp (RSEQ r1 r2) s) =
|
|
|
540 |
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
|
|
445
|
541 |
apply(induct s rule: rev_induct)
|
|
|
542 |
apply simp
|
|
|
543 |
apply(subst rders_simp_append)
|
|
|
544 |
apply(subst rders_simp_one_char)
|
|
|
545 |
apply(subst rsimp_idem[symmetric])
|
|
|
546 |
apply(subst rders_simp_one_char[symmetric])
|
|
|
547 |
apply(subst rders_simp_append[symmetric])
|
|
|
548 |
apply(insert seq_closed_form)
|
|
|
549 |
apply(subgoal_tac "rsimp (rders_simp (RSEQ r1 r2) (xs @ [x]))
|
|
|
550 |
= rsimp (RALTS (RSEQ (rders_simp r1 (xs @ [x])) r2 # map (rders_simp r2) (vsuf (xs @ [x]) r1)))")
|
|
|
551 |
apply force
|
|
|
552 |
by presburger
|
|
443
|
553 |
|
|
444
|
554 |
end |