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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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465
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lemma idem_after_simp1:
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shows "rsimp_ALTs (rdistinct (rflts [rsimp aa]) {}) = rsimp aa"
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apply(case_tac "rsimp aa")
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apply simp+
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apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
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by simp
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456
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lemma distinct_removes_last:
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465
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shows "\<lbrakk>a \<in> set as\<rbrakk>
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456
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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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465
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lemma distinct_removes_middle:
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shows "\<lbrakk>a \<in> set as\<rbrakk>
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\<Longrightarrow> rdistinct (as @ as2) rset = rdistinct (as @ [a] @ as2) rset"
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and "rdistinct (ab # as @ [ab] @ as3) rset1 = rdistinct (ab # as @ as3) rset1"
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apply(induct as arbitrary: rset rset1 ab as2 as3 a)
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apply simp
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apply simp
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apply(case_tac "a \<in> rset")
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apply simp
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apply metis
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453
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apply simp
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465
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apply (metis insertI1)
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apply(case_tac "a = ab")
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453
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apply simp
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465
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apply(case_tac "ab \<in> rset")
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453
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apply simp
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465
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apply presburger
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apply (meson insertI1)
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apply(case_tac "a \<in> rset")
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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(case_tac "ab \<in> rset")
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apply simp
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apply (meson insert_iff)
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apply simp
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by (metis insertI1)
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453
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465
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lemma distinct_removes_middle3:
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shows "\<lbrakk>a \<in> set as\<rbrakk>
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\<Longrightarrow> rdistinct (as @ a #as2) rset = rdistinct (as @ as2) rset"
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using distinct_removes_middle(1) by fastforce
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lemma distinct_removes_last2:
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shows "\<lbrakk>a \<in> set as\<rbrakk>
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\<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
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by (simp add: distinct_removes_last(1))
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lemma distinct_removes_middle2:
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shows "a \<in> set as \<Longrightarrow> rdistinct (as @ [a] @ rs) {} = rdistinct (as @ rs) {}"
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by (metis distinct_removes_middle(1))
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lemma distinct_removes_list:
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shows "\<lbrakk>a \<in> set as; \<forall>r \<in> set rs. r \<in> set as\<rbrakk> \<Longrightarrow> rdistinct (as @ rs) {} = rdistinct as {}"
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apply(induct rs)
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apply simp+
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apply(subgoal_tac "rdistinct (as @ aa # rs) {} = rdistinct (as @ rs) {}")
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prefer 2
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apply (metis append_Cons append_Nil distinct_removes_middle(1))
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by presburger
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453
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451
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lemma simp_rdistinct_f: shows
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465
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"f ` rset = frset \<Longrightarrow> rsimp (rsimp_ALTs (map f (rdistinct rs rset))) =
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rsimp (rsimp_ALTs (rdistinct (map f rs) frset)) "
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451
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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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465
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lemma simp_more_distinct1:
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shows "rsimp (rsimp_ALTs rs) = rsimp (rsimp_ALTs (rdistinct rs {}))"
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apply(induct rs)
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apply simp
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apply simp
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oops
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(*
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\<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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*)
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lemma simp_removes_duplicate2:
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shows "a "
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oops
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lemma flts_removes0:
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shows " rflts (rs @ [RZERO]) =
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rflts rs"
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apply(induct rs)
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apply simp
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by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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lemma flts_keeps1:
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shows " rflts (rs @ [RONE]) =
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rflts rs @ [RONE] "
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apply (induct rs)
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apply simp
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by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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lemma flts_keeps_others:
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shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow>rflts (rs @ [a]) = rflts rs @ [a]"
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apply(induct rs)
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apply simp
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apply (simp add: rflts_def_idiot)
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apply(case_tac a)
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apply simp
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using flts_keeps1 apply blast
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apply (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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apply (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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apply blast
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by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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lemma rflts_def_idiot2:
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shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow> a \<in> set (rflts rs)"
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287 |
apply(induct rs)
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apply simp
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by (metis append.assoc in_set_conv_decomp insert_iff list.simps(15) rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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lemma rflts_spills_last:
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shows "a = RALTS rs \<Longrightarrow> rflts (rs1 @ [a]) = rflts rs1 @ rs"
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apply (induct rs1)
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apply simp
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by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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lemma spilled_alts_contained:
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shows "\<lbrakk>a = RALTS rs ; a \<in> set rs1\<rbrakk> \<Longrightarrow> \<forall>r \<in> set rs. r \<in> set (rflts rs1)"
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apply(induct rs1)
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apply simp
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302 |
apply(case_tac "a = aa")
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apply simp
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304 |
apply(subgoal_tac " a \<in> set rs1")
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305 |
prefer 2
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|
306 |
apply (meson set_ConsD)
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307 |
apply(case_tac aa)
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|
308 |
using rflts.simps(2) apply presburger
|
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|
309 |
apply fastforce
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|
310 |
apply fastforce
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|
311 |
apply fastforce
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|
312 |
apply fastforce
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|
313 |
by fastforce
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314 |
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315 |
lemma distinct_removes_duplicate_flts:
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316 |
shows " a \<in> set rsa
|
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317 |
\<Longrightarrow> rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
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|
318 |
rdistinct (rflts (map rsimp rsa)) {}"
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|
319 |
apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
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320 |
prefer 2
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321 |
apply simp
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|
322 |
apply(induct "rsimp a")
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|
323 |
apply simp
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|
324 |
using flts_removes0 apply presburger
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|
325 |
apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
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326 |
rdistinct (rflts (map rsimp rsa @ [RONE])) {}")
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327 |
apply (simp only:)
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328 |
apply(subst flts_keeps1)
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329 |
apply (metis distinct_removes_last2 rflts_def_idiot2 rrexp.simps(20) rrexp.simps(6))
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|
330 |
apply presburger
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|
331 |
apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
|
|
332 |
rdistinct ((rflts (map rsimp rsa)) @ [RCHAR x]) {}")
|
|
|
333 |
apply (simp only:)
|
|
|
334 |
prefer 2
|
|
|
335 |
apply (metis flts_keeps_others rrexp.distinct(21) rrexp.distinct(3))
|
|
|
336 |
apply (metis distinct_removes_last2 rflts_def_idiot2 rrexp.distinct(21) rrexp.distinct(3))
|
|
|
337 |
|
|
|
338 |
apply (metis distinct_removes_last2 flts_keeps_others rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
|
|
|
339 |
prefer 2
|
|
|
340 |
apply (metis distinct_removes_last2 flts_keeps_others flts_removes0 rflts_def_idiot2 rrexp.distinct(29))
|
|
|
341 |
apply(subgoal_tac "rflts (map rsimp rsa @ [rsimp a]) = rflts (map rsimp rsa) @ x")
|
|
|
342 |
prefer 2
|
|
|
343 |
apply (simp add: rflts_spills_last)
|
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|
344 |
apply(simp only:)
|
|
|
345 |
apply(subgoal_tac "\<forall> r \<in> set x. r \<in> set (rflts (map rsimp rsa))")
|
|
|
346 |
prefer 2
|
|
|
347 |
using spilled_alts_contained apply presburger
|
|
|
348 |
by (metis append_self_conv distinct_removes_list in_set_conv_decomp rev_exhaust)
|
|
|
349 |
|
|
|
350 |
lemma flts_middle0:
|
|
|
351 |
shows "rflts (rsa @ RZERO # rsb) = rflts (rsa @ rsb)"
|
|
|
352 |
apply(induct rsa)
|
|
|
353 |
apply simp
|
|
|
354 |
by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
355 |
|
|
|
356 |
lemma flts_middle01:
|
|
|
357 |
shows "rflts (rsa @ [RZERO] @ rsb) = rflts (rsa @ rsb)"
|
|
|
358 |
by (simp add: flts_middle0)
|
|
|
359 |
|
|
|
360 |
lemma flts_append1:
|
|
|
361 |
shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow>
|
|
|
362 |
rflts (rsa @ [a] @ rsb) = rflts rsa @ [a] @ (rflts rsb)"
|
|
|
363 |
apply(induct rsa arbitrary: rsb)
|
|
|
364 |
apply simp
|
|
|
365 |
using rflts_def_idiot apply presburger
|
|
|
366 |
apply(case_tac aa)
|
|
|
367 |
apply simp+
|
|
|
368 |
done
|
|
|
369 |
|
|
|
370 |
lemma flts_append:
|
|
|
371 |
shows "rflts (rs1 @ rs2) = rflts rs1 @ rflts rs2"
|
|
|
372 |
apply(induct rs1)
|
|
|
373 |
apply simp
|
|
|
374 |
apply(case_tac a)
|
|
|
375 |
apply simp+
|
|
|
376 |
done
|
|
|
377 |
|
|
|
378 |
lemma simp_removes_duplicate1:
|
|
|
379 |
shows " a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a])) = rsimp (RALTS (rsa))"
|
|
|
380 |
and " rsimp (RALTS (a1 # rsa @ [a1])) = rsimp (RALTS (a1 # rsa))"
|
|
|
381 |
apply(induct rsa arbitrary: a1)
|
|
|
382 |
apply simp
|
|
|
383 |
apply simp
|
|
|
384 |
prefer 2
|
|
|
385 |
apply(case_tac "a = aa")
|
|
|
386 |
apply simp
|
|
|
387 |
apply simp
|
|
|
388 |
apply (metis Cons_eq_appendI Cons_eq_map_conv distinct_removes_duplicate_flts list.set_intros(2))
|
|
|
389 |
apply (metis append_Cons append_Nil distinct_removes_duplicate_flts list.set_intros(1) list.simps(8) list.simps(9))
|
|
|
390 |
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))
|
|
|
391 |
|
|
|
392 |
lemma simp_removes_duplicate2:
|
|
|
393 |
shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a] @ rsb)) = rsimp (RALTS (rsa @ rsb))"
|
|
|
394 |
apply(induct rsb arbitrary: rsa)
|
|
|
395 |
apply simp
|
|
|
396 |
using distinct_removes_duplicate_flts apply auto[1]
|
|
|
397 |
by (metis append.assoc head_one_more_simp rsimp.simps(2) simp_flatten simp_removes_duplicate1(1))
|
|
|
398 |
|
|
|
399 |
lemma simp_removes_duplicate3:
|
|
|
400 |
shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ a # rsb)) = rsimp (RALTS (rsa @ rsb))"
|
|
|
401 |
using simp_removes_duplicate2 by auto
|
|
|
402 |
|
|
|
403 |
lemma distinct_removes_middle4:
|
|
|
404 |
shows "a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ [a] @ rsb) rset = rdistinct (rsa @ rsb) rset"
|
|
|
405 |
using distinct_removes_middle(1) by fastforce
|
|
|
406 |
|
|
|
407 |
lemma distinct_removes_middle_list:
|
|
|
408 |
shows "\<forall>a \<in> set x. a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ x @ rsb) rset = rdistinct (rsa @ rsb) rset"
|
|
|
409 |
apply(induct x)
|
|
|
410 |
apply simp
|
|
|
411 |
by (simp add: distinct_removes_middle3)
|
|
|
412 |
|
|
|
413 |
|
|
|
414 |
lemma distinct_removes_duplicate_flts2:
|
|
|
415 |
shows " a \<in> set rsa
|
|
|
416 |
\<Longrightarrow> rdistinct (rflts (rsa @ [a] @ rsb)) {} =
|
|
|
417 |
rdistinct (rflts (rsa @ rsb)) {}"
|
|
|
418 |
apply(induct a arbitrary: rsb)
|
|
|
419 |
using flts_middle01 apply presburger
|
|
|
420 |
apply(subgoal_tac "rflts (rsa @ [RONE] @ rsb) = rflts rsa @ [RONE] @ rflts rsb")
|
|
|
421 |
prefer 2
|
|
|
422 |
using flts_append1 apply blast
|
|
|
423 |
apply simp
|
|
|
424 |
apply(subgoal_tac "RONE \<in> set (rflts rsa)")
|
|
|
425 |
prefer 2
|
|
|
426 |
using rflts_def_idiot2 apply blast
|
|
|
427 |
apply(subst distinct_removes_middle3)
|
|
|
428 |
apply simp
|
|
|
429 |
using flts_append apply presburger
|
|
|
430 |
apply simp
|
|
|
431 |
apply (metis distinct_removes_middle3 flts_append in_set_conv_decomp rflts.simps(5))
|
|
|
432 |
apply (metis distinct_removes_middle(1) flts_append flts_append1 rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
|
|
|
433 |
apply(subgoal_tac "rflts (rsa @ [RALTS x] @ rsb) = rflts rsa @ x @ rflts rsb")
|
|
|
434 |
prefer 2
|
|
|
435 |
apply (simp add: flts_append)
|
|
|
436 |
apply (simp only:)
|
|
|
437 |
|
|
|
438 |
apply(subgoal_tac "\<forall>r1 \<in> set x. r1 \<in> set (rflts rsa)")
|
|
|
439 |
prefer 2
|
|
|
440 |
using spilled_alts_contained apply blast
|
|
|
441 |
apply(subst flts_append)
|
|
|
442 |
using distinct_removes_middle_list apply blast
|
|
|
443 |
using distinct_removes_middle2 flts_append rflts_def_idiot2 by fastforce
|
|
|
444 |
|
|
|
445 |
|
|
|
446 |
lemma simp_removes_duplicate:
|
|
|
447 |
shows "a \<in> set rsa \<Longrightarrow> rsimp (rsimp_ALTs (rsa @ a # rs)) = rsimp (rsimp_ALTs (rsa @ rs))"
|
|
|
448 |
apply(subgoal_tac "rsimp (rsimp_ALTs (rsa @ a # rs)) = rsimp (RALTS (rsa @ a # rs))")
|
|
|
449 |
prefer 2
|
|
|
450 |
apply (smt (verit, best) Cons_eq_append_conv append_is_Nil_conv empty_set equals0D list.distinct(1) rsimp_ALTs.elims)
|
|
|
451 |
apply(simp only:)
|
|
|
452 |
apply simp
|
|
|
453 |
apply(subgoal_tac "(rdistinct (rflts (map rsimp rsa @ rsimp a # map rsimp rs)) {}) = (rdistinct (rflts (map rsimp rsa @ map rsimp rs)) {})")
|
|
|
454 |
apply(simp only:)
|
|
|
455 |
prefer 2
|
|
|
456 |
apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
|
|
|
457 |
prefer 2
|
|
|
458 |
apply simp
|
|
|
459 |
using distinct_removes_duplicate_flts2 apply force
|
|
|
460 |
apply(case_tac rsa)
|
|
|
461 |
apply simp
|
|
|
462 |
apply(case_tac rs)
|
|
|
463 |
apply simp
|
|
|
464 |
apply(case_tac list)
|
|
|
465 |
apply simp
|
|
|
466 |
using idem_after_simp1 apply presburger
|
|
|
467 |
apply simp+
|
|
|
468 |
apply(subgoal_tac "rsimp_ALTs (aa # list @ aaa # lista) = RALTS (aa # list @ aaa # lista)")
|
|
|
469 |
apply simp
|
|
|
470 |
using rsimpalts_conscons by presburger
|
|
467
|
471 |
|
|
468
|
472 |
lemma no0_flts:
|
|
|
473 |
shows "RZERO \<notin> set (rflts rs)"
|
|
|
474 |
apply (induct rs)
|
|
|
475 |
apply simp
|
|
|
476 |
apply(case_tac a)
|
|
|
477 |
apply simp+
|
|
|
478 |
oops
|
|
467
|
479 |
|
|
|
480 |
|
|
|
481 |
|
|
|
482 |
lemma distinct_flts_no0:
|
|
468
|
483 |
shows " rflts (map rsimp (rdistinct rs (insert RZERO rset))) =
|
|
|
484 |
rflts (map rsimp (rdistinct rs rset)) "
|
|
|
485 |
|
|
|
486 |
apply(induct rs arbitrary: rset)
|
|
467
|
487 |
apply simp
|
|
|
488 |
apply(case_tac a)
|
|
468
|
489 |
apply simp+
|
|
|
490 |
apply (smt (verit, ccfv_SIG) rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
491 |
prefer 2
|
|
|
492 |
apply simp
|
|
|
493 |
by (smt (verit, ccfv_threshold) Un_insert_right insert_iff list.simps(9) rdistinct.simps(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot rrexp.distinct(7))
|
|
|
494 |
|
|
467
|
495 |
|
|
|
496 |
|
|
471
|
497 |
|
|
|
498 |
inductive frewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f _" [10, 10] 10)
|
|
|
499 |
where
|
|
|
500 |
"(RZERO # rs) \<leadsto>f rs"
|
|
|
501 |
| "((RALTS rs) # rsa) \<leadsto>f (rs @ rsa)"
|
|
|
502 |
| "rs1 \<leadsto>f rs2 \<Longrightarrow> (r # rs1) \<leadsto>f (r # rs2)"
|
|
|
503 |
|
|
|
504 |
|
|
|
505 |
inductive
|
|
|
506 |
frewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f* _" [10, 10] 10)
|
|
|
507 |
where
|
|
473
|
508 |
[intro, simp]:"rs \<leadsto>f* rs"
|
|
|
509 |
| [intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>f* rs3"
|
|
|
510 |
|
|
|
511 |
inductive grewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g _" [10, 10] 10)
|
|
|
512 |
where
|
|
|
513 |
"(RZERO # rs) \<leadsto>g rs"
|
|
|
514 |
| "((RALTS rs) # rsa) \<leadsto>g (rs @ rsa)"
|
|
|
515 |
| "rs1 \<leadsto>g rs2 \<Longrightarrow> (r # rs1) \<leadsto>g (r # rs2)"
|
|
|
516 |
| "rsa @ [a] @ rsb @ [a] @ rsc \<leadsto>g rsa @ [a] @ rsb @ rsc"
|
|
|
517 |
|
|
|
518 |
|
|
|
519 |
inductive
|
|
|
520 |
grewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g* _" [10, 10] 10)
|
|
|
521 |
where
|
|
|
522 |
[intro, simp]:"rs \<leadsto>g* rs"
|
|
|
523 |
| [intro]: "\<lbrakk>rs1 \<leadsto>g* rs2; rs2 \<leadsto>g rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>g* rs3"
|
|
|
524 |
(*
|
|
|
525 |
inductive
|
|
|
526 |
frewrites2:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ <\<leadsto>f* _" [10, 10] 10)
|
|
|
527 |
where
|
|
|
528 |
[intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f* rs1\<rbrakk> \<Longrightarrow> rs1 <\<leadsto>f* rs2"
|
|
|
529 |
*)
|
|
471
|
530 |
|
|
|
531 |
lemma fr_in_rstar : "r1 \<leadsto>f r2 \<Longrightarrow> r1 \<leadsto>f* r2"
|
|
|
532 |
using frewrites.intros(1) frewrites.intros(2) by blast
|
|
|
533 |
|
|
|
534 |
lemma freal_trans[trans]:
|
|
|
535 |
assumes a1: "r1 \<leadsto>f* r2" and a2: "r2 \<leadsto>f* r3"
|
|
|
536 |
shows "r1 \<leadsto>f* r3"
|
|
|
537 |
using a2 a1
|
|
|
538 |
apply(induct r2 r3 arbitrary: r1 rule: frewrites.induct)
|
|
|
539 |
apply(auto)
|
|
|
540 |
done
|
|
|
541 |
|
|
|
542 |
|
|
|
543 |
lemma many_steps_later: "\<lbrakk>r1 \<leadsto>f r2; r2 \<leadsto>f* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>f* r3"
|
|
|
544 |
by (meson fr_in_rstar freal_trans)
|
|
|
545 |
|
|
|
546 |
|
|
|
547 |
lemma frewrite_append:
|
|
|
548 |
shows "\<lbrakk> rsa \<leadsto>f rsb \<rbrakk> \<Longrightarrow> rs @ rsa \<leadsto>f rs @ rsb"
|
|
|
549 |
apply(induct rs)
|
|
472
|
550 |
apply simp+
|
|
|
551 |
using frewrite.intros(3) by blast
|
|
|
552 |
|
|
471
|
553 |
|
|
|
554 |
|
|
|
555 |
lemma frewrites_cons:
|
|
|
556 |
shows "\<lbrakk> rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>f* r # rsb"
|
|
472
|
557 |
apply(induct rsa rsb rule: frewrites.induct)
|
|
|
558 |
apply simp
|
|
|
559 |
using frewrite.intros(3) by blast
|
|
471
|
560 |
|
|
|
561 |
|
|
|
562 |
lemma frewrites_append:
|
|
|
563 |
shows " \<lbrakk>rsa \<leadsto>f* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>f* (rs @ rsb)"
|
|
472
|
564 |
apply(induct rs)
|
|
471
|
565 |
apply simp
|
|
472
|
566 |
by (simp add: frewrites_cons)
|
|
471
|
567 |
|
|
|
568 |
|
|
|
569 |
|
|
|
570 |
lemma frewrites_concat:
|
|
|
571 |
shows "\<lbrakk>rs1 \<leadsto>f rs2; rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> (rs1 @ rsa) \<leadsto>f* (rs2 @ rsb)"
|
|
|
572 |
apply(induct rs1 rs2 rule: frewrite.induct)
|
|
|
573 |
apply(simp)
|
|
|
574 |
apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>f (rs @ rsa)")
|
|
|
575 |
prefer 2
|
|
|
576 |
using frewrite.intros(1) apply blast
|
|
|
577 |
apply(subgoal_tac "(rs @ rsa) \<leadsto>f* (rs @ rsb)")
|
|
|
578 |
using many_steps_later apply blast
|
|
472
|
579 |
apply (simp add: frewrites_append)
|
|
|
580 |
apply (metis append.assoc append_Cons frewrite.intros(2) frewrites_append many_steps_later)
|
|
|
581 |
using frewrites_cons by auto
|
|
471
|
582 |
|
|
473
|
583 |
lemma frewrites_middle:
|
|
|
584 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> r # (RALTS rs # rs1) \<leadsto>f* r # (rs @ rs1)"
|
|
|
585 |
by (simp add: fr_in_rstar frewrite.intros(2) frewrite.intros(3))
|
|
471
|
586 |
|
|
473
|
587 |
lemma frewrites_alt:
|
|
|
588 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> (RALT r1 r2) # rs1 \<leadsto>f* r1 # r2 # rs2"
|
|
|
589 |
by (metis Cons_eq_appendI append_self_conv2 frewrite.intros(2) frewrites_cons many_steps_later)
|
|
471
|
590 |
|
|
|
591 |
lemma many_steps_later1:
|
|
|
592 |
shows " \<lbrakk>rs1 \<leadsto>f* rs2\<rbrakk>
|
|
|
593 |
\<Longrightarrow> (RONE # rs1) \<leadsto>f* (RONE # rs2)"
|
|
|
594 |
oops
|
|
|
595 |
|
|
|
596 |
lemma early_late_der_frewrites:
|
|
|
597 |
shows "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)"
|
|
|
598 |
apply(induct rs)
|
|
|
599 |
apply simp
|
|
|
600 |
apply(case_tac a)
|
|
|
601 |
apply simp+
|
|
|
602 |
using frewrite.intros(1) many_steps_later apply blast
|
|
|
603 |
apply(case_tac "x = x3")
|
|
472
|
604 |
apply simp
|
|
|
605 |
using frewrites_cons apply presburger
|
|
|
606 |
using frewrite.intros(1) many_steps_later apply fastforce
|
|
|
607 |
apply(case_tac "rnullable x41")
|
|
473
|
608 |
apply simp+
|
|
|
609 |
apply (simp add: frewrites_alt)
|
|
|
610 |
apply (simp add: frewrites_cons)
|
|
|
611 |
apply (simp add: frewrites_append)
|
|
|
612 |
by (simp add: frewrites_cons)
|
|
471
|
613 |
|
|
|
614 |
|
|
|
615 |
fun alt_set:: "rrexp \<Rightarrow> rrexp set"
|
|
|
616 |
where
|
|
|
617 |
"alt_set (RALTS rs) = set rs"
|
|
|
618 |
| "alt_set r = {r}"
|
|
|
619 |
|
|
|
620 |
|
|
|
621 |
|
|
|
622 |
lemma rd_flts_set:
|
|
|
623 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rdistinct rs1 (insert RZERO (rset \<union> (\<Union>(alt_set ` rset)))) \<leadsto>f*
|
|
|
624 |
rdistinct rs2 (rset \<union> (\<Union>(alt_set ` rset)))"
|
|
473
|
625 |
|
|
|
626 |
oops
|
|
|
627 |
|
|
|
628 |
|
|
471
|
629 |
|
|
|
630 |
lemma rd_flts_set2:
|
|
|
631 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rdistinct rs1 ((rset \<union> (\<Union>(alt_set ` rset)))) \<leadsto>f*
|
|
|
632 |
rdistinct rs2 (rset \<union> (\<Union>(alt_set ` rset)))"
|
|
473
|
633 |
oops
|
|
471
|
634 |
|
|
473
|
635 |
|
|
471
|
636 |
|
|
|
637 |
lemma flts_does_rewrite:
|
|
|
638 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rflts rs1 = rflts rs2"
|
|
|
639 |
oops
|
|
|
640 |
|
|
473
|
641 |
lemma with_wo0_distinct:
|
|
|
642 |
shows "rdistinct rs rset \<leadsto>f* rdistinct rs (insert RZERO rset)"
|
|
|
643 |
apply(induct rs arbitrary: rset)
|
|
|
644 |
apply simp
|
|
|
645 |
apply(case_tac a)
|
|
|
646 |
apply(case_tac "RZERO \<in> rset")
|
|
|
647 |
apply simp+
|
|
|
648 |
using fr_in_rstar frewrite.intros(1) apply presburger
|
|
|
649 |
apply (case_tac "RONE \<in> rset")
|
|
|
650 |
apply simp+
|
|
|
651 |
using frewrites_cons apply presburger
|
|
|
652 |
apply(case_tac "a \<in> rset")
|
|
|
653 |
apply simp
|
|
|
654 |
apply (simp add: frewrites_cons)
|
|
|
655 |
apply(case_tac "a \<in> rset")
|
|
|
656 |
apply simp
|
|
|
657 |
apply (simp add: frewrites_cons)
|
|
|
658 |
apply(case_tac "a \<in> rset")
|
|
|
659 |
apply simp
|
|
|
660 |
apply (simp add: frewrites_cons)
|
|
|
661 |
apply(case_tac "a \<in> rset")
|
|
|
662 |
apply simp
|
|
|
663 |
apply (simp add: frewrites_cons)
|
|
|
664 |
done
|
|
|
665 |
|
|
|
666 |
lemma rdistinct_concat:
|
|
|
667 |
shows "set rs \<subseteq> rset \<Longrightarrow> rdistinct (rs @ rsa) rset = rdistinct rsa rset"
|
|
|
668 |
apply(induct rs)
|
|
|
669 |
apply simp+
|
|
|
670 |
done
|
|
|
671 |
|
|
|
672 |
lemma rdistinct_concat2:
|
|
|
673 |
shows "\<forall>r \<in> set rs. r \<in> rset \<Longrightarrow> rdistinct (rs @ rsa) rset = rdistinct rsa rset"
|
|
|
674 |
by (simp add: rdistinct_concat subsetI)
|
|
|
675 |
|
|
|
676 |
lemma frewrite_with_distinct:
|
|
|
677 |
shows " \<lbrakk>rs2 \<leadsto>f rs3\<rbrakk>
|
|
|
678 |
\<Longrightarrow> rdistinct rs2
|
|
|
679 |
(insert RZERO (rset \<union> \<Union> (alt_set ` rset))) \<leadsto>f*
|
|
|
680 |
rdistinct rs3
|
|
|
681 |
(insert RZERO (rset \<union> \<Union> (alt_set ` rset)))"
|
|
|
682 |
apply(induct rs2 rs3 rule: frewrite.induct)
|
|
|
683 |
apply(case_tac "RZERO \<in> (rset \<union> \<Union> (alt_set ` rset))")
|
|
|
684 |
apply simp
|
|
|
685 |
apply simp
|
|
|
686 |
apply(case_tac "RALTS rs \<in> rset")
|
|
|
687 |
apply simp
|
|
|
688 |
apply(subgoal_tac "\<forall>r \<in> set rs. r \<in> \<Union> (alt_set ` rset)")
|
|
|
689 |
apply(subgoal_tac " rdistinct (rs @ rsa) (insert RZERO (rset \<union> \<Union> (alt_set ` rset))) =
|
|
|
690 |
rdistinct rsa (insert RZERO (rset \<union> \<Union> (alt_set ` rset)))")
|
|
|
691 |
using frewrites.intros(1) apply presburger
|
|
|
692 |
apply (simp add: rdistinct_concat2)
|
|
|
693 |
apply simp
|
|
|
694 |
using alt_set.simps(1) apply blast
|
|
|
695 |
apply(case_tac "RALTS rs \<in> rset \<union> \<Union>(alt_set ` rset)")
|
|
|
696 |
|
|
|
697 |
|
|
469
|
698 |
sorry
|
|
|
699 |
|
|
|
700 |
|
|
473
|
701 |
lemma frewrites_with_distinct:
|
|
|
702 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
|
|
|
703 |
rs1 @ (rdistinct rsa (insert RZERO (set rs1 \<union> \<Union>(alt_set ` (set rs1) )))) \<leadsto>f*
|
|
|
704 |
rs2 @ (rdistinct rsb (insert RZERO (set rs2 \<union> \<Union>(alt_set ` (set rs2) ))))"
|
|
|
705 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
706 |
apply simp
|
|
|
707 |
|
|
|
708 |
|
|
|
709 |
sorry
|
|
|
710 |
|
|
|
711 |
(*a more refined notion of \<leadsto>* is needed,
|
|
|
712 |
this lemma fails when rs1 contains some RALTS rs where elements
|
|
|
713 |
of rs appear in later parts of rs1, which will be picked up by rs2
|
|
|
714 |
and deduplicated*)
|
|
|
715 |
lemma wrong_frewrites_with_distinct2:
|
|
|
716 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
|
|
|
717 |
(rdistinct rs1 {RZERO}) \<leadsto>f* rdistinct rs2 {RZERO}"
|
|
|
718 |
oops
|
|
|
719 |
|
|
|
720 |
lemma frewrite_single_step:
|
|
|
721 |
shows " rs2 \<leadsto>f rs3 \<Longrightarrow> rsimp (RALTS rs2) = rsimp (RALTS rs3)"
|
|
|
722 |
apply(induct rs2 rs3 rule: frewrite.induct)
|
|
|
723 |
apply simp
|
|
|
724 |
using simp_flatten apply blast
|
|
|
725 |
by (metis (no_types, opaque_lifting) list.simps(9) rsimp.simps(2) simp_flatten2)
|
|
|
726 |
|
|
|
727 |
lemma frewrites_equivalent_simp:
|
|
|
728 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
|
729 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
730 |
apply simp
|
|
|
731 |
using frewrite_single_step by presburger
|
|
|
732 |
|
|
|
733 |
lemma frewrites_dB_wwo0_simp:
|
|
|
734 |
shows "rdistinct rs1 {RZERO} \<leadsto>f* rdistinct rs2 {RZERO}
|
|
|
735 |
\<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
|
|
|
736 |
|
|
|
737 |
sorry
|
|
|
738 |
|
|
|
739 |
|
|
|
740 |
|
|
467
|
741 |
lemma simp_der_flts:
|
|
471
|
742 |
shows "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {})) =
|
|
|
743 |
rsimp (RALTS (rdistinct (rflts (map (rder x) rs)) {}))"
|
|
473
|
744 |
apply(subgoal_tac "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)")
|
|
|
745 |
apply(subgoal_tac "rdistinct (map (rder x) (rflts rs)) {RZERO}
|
|
|
746 |
\<leadsto>f* rdistinct ( rflts (map (rder x) rs)) {RZERO}")
|
|
|
747 |
apply(subgoal_tac "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {}))
|
|
|
748 |
= rsimp (RALTS ( rdistinct ( rflts (map (rder x) rs)) {}))")
|
|
|
749 |
apply meson
|
|
|
750 |
using frewrites_dB_wwo0_simp apply blast
|
|
|
751 |
using frewrites_with_distinct2 apply blast
|
|
|
752 |
using early_late_der_frewrites by blast
|
|
467
|
753 |
|
|
|
754 |
|
|
465
|
755 |
lemma simp_der_pierce_flts:
|
|
471
|
756 |
shows " rsimp (
|
|
|
757 |
rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})
|
|
|
758 |
) =
|
|
|
759 |
rsimp (
|
|
|
760 |
rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})
|
|
|
761 |
)"
|
|
|
762 |
|
|
467
|
763 |
sorry
|
|
|
764 |
|
|
465
|
765 |
|
|
453
|
766 |
|
|
|
767 |
|
|
|
768 |
lemma simp_more_distinct:
|
|
465
|
769 |
shows "rsimp (rsimp_ALTs (rsa @ rs)) = rsimp (rsimp_ALTs (rsa @ (rdistinct rs (set rsa)))) "
|
|
467
|
770 |
|
|
|
771 |
|
|
465
|
772 |
|
|
453
|
773 |
|
|
|
774 |
sorry
|
|
|
775 |
|
|
|
776 |
lemma non_empty_list:
|
|
|
777 |
shows "a \<in> set as \<Longrightarrow> as \<noteq> []"
|
|
|
778 |
by (metis empty_iff empty_set)
|
|
|
779 |
|
|
456
|
780 |
lemma distinct_comp:
|
|
|
781 |
shows "rdistinct (rs1@rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
|
|
|
782 |
apply(induct rs2 arbitrary: rs1)
|
|
|
783 |
apply simp
|
|
|
784 |
apply(subgoal_tac "rs1 @ a # rs2 = (rs1 @ [a]) @ rs2")
|
|
|
785 |
apply(simp only:)
|
|
|
786 |
apply(case_tac "a \<in> set rs1")
|
|
|
787 |
apply simp
|
|
|
788 |
oops
|
|
453
|
789 |
|
|
456
|
790 |
lemma instantiate1:
|
|
|
791 |
shows "\<lbrakk>\<And>ab rset1. rdistinct (ab # as) rset1 = rdistinct (ab # as @ [ab]) rset1\<rbrakk> \<Longrightarrow>
|
|
|
792 |
rdistinct (aa # as) rset = rdistinct (aa # as @ [aa]) rset"
|
|
|
793 |
apply(drule_tac x = "aa" in meta_spec)
|
|
|
794 |
apply(drule_tac x = "rset" in meta_spec)
|
|
453
|
795 |
apply simp
|
|
456
|
796 |
done
|
|
|
797 |
|
|
|
798 |
|
|
|
799 |
lemma not_head_elem:
|
|
|
800 |
shows " \<lbrakk>aa \<in> set (a # as); aa \<notin> (set as)\<rbrakk> \<Longrightarrow> a = aa"
|
|
|
801 |
|
|
|
802 |
by fastforce
|
|
|
803 |
|
|
|
804 |
(*
|
|
|
805 |
apply simp
|
|
|
806 |
apply (metis append_Cons)
|
|
|
807 |
apply(case_tac "ab \<in> rset1")
|
|
|
808 |
apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
|
|
|
809 |
apply(subgoal_tac "rdistinct (ab # (aa # as) @ [ab]) rset1 =
|
|
|
810 |
ab # (rdistinct ((aa # as) @ [ab]) (insert ab rset1))")
|
|
|
811 |
apply(simp only:)
|
|
|
812 |
apply(subgoal_tac "rdistinct (ab # aa # as) rset1 = ab # (rdistinct (aa # as) (insert ab rset1))")
|
|
|
813 |
apply(simp only:)
|
|
|
814 |
apply(subgoal_tac "rdistinct ((aa # as) @ [ab]) (insert ab rset1) = rdistinct (aa # as) (insert ab rset1)")
|
|
|
815 |
apply blast
|
|
|
816 |
*)
|
|
|
817 |
|
|
453
|
818 |
|
|
|
819 |
lemma flts_identity1:
|
|
|
820 |
shows "rflts (rs @ [RONE]) = rflts rs @ [RONE] "
|
|
|
821 |
apply(induct rs)
|
|
|
822 |
apply simp+
|
|
|
823 |
apply(case_tac a)
|
|
|
824 |
apply simp
|
|
|
825 |
apply simp+
|
|
|
826 |
done
|
|
|
827 |
|
|
|
828 |
lemma flts_identity10:
|
|
|
829 |
shows " rflts (rs @ [RCHAR c]) = rflts rs @ [RCHAR c]"
|
|
|
830 |
apply(induct rs)
|
|
|
831 |
apply simp+
|
|
|
832 |
apply(case_tac a)
|
|
|
833 |
apply simp+
|
|
|
834 |
done
|
|
|
835 |
|
|
|
836 |
lemma flts_identity11:
|
|
|
837 |
shows " rflts (rs @ [RSEQ r1 r2]) = rflts rs @ [RSEQ r1 r2]"
|
|
|
838 |
apply(induct rs)
|
|
|
839 |
apply simp+
|
|
|
840 |
apply(case_tac a)
|
|
|
841 |
apply simp+
|
|
|
842 |
done
|
|
|
843 |
|
|
|
844 |
lemma flts_identity12:
|
|
|
845 |
shows " rflts (rs @ [RSTAR r0]) = rflts rs @ [RSTAR r0]"
|
|
|
846 |
apply(induct rs)
|
|
|
847 |
apply simp+
|
|
|
848 |
apply(case_tac a)
|
|
|
849 |
apply simp+
|
|
|
850 |
done
|
|
|
851 |
|
|
|
852 |
lemma flts_identity2:
|
|
|
853 |
shows "a \<noteq> RZERO \<and> (\<forall>rs. a \<noteq> RALTS rs) \<Longrightarrow> rflts (rs @ [a]) = rflts rs @ [a]"
|
|
|
854 |
apply(case_tac a)
|
|
|
855 |
apply simp
|
|
|
856 |
using flts_identity1 apply auto[1]
|
|
|
857 |
using flts_identity10 apply blast
|
|
|
858 |
using flts_identity11 apply auto[1]
|
|
|
859 |
apply blast
|
|
|
860 |
using flts_identity12 by presburger
|
|
456
|
861 |
|
|
|
862 |
lemma flts_identity3:
|
|
|
863 |
shows "a = RZERO \<Longrightarrow> rflts (rs @ [a]) = rflts rs"
|
|
|
864 |
apply simp
|
|
|
865 |
apply(induct rs)
|
|
|
866 |
apply simp+
|
|
|
867 |
apply(case_tac aa)
|
|
|
868 |
apply simp+
|
|
|
869 |
done
|
|
|
870 |
|
|
|
871 |
lemma distinct_removes_last3:
|
|
465
|
872 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
456
|
873 |
\<Longrightarrow> rdistinct as {} = rdistinct (as @ [a]) {}"
|
|
465
|
874 |
by (simp add: distinct_removes_last2)
|
|
456
|
875 |
|
|
|
876 |
lemma set_inclusion_with_flts1:
|
|
|
877 |
shows " \<lbrakk>RONE \<in> set rs\<rbrakk> \<Longrightarrow> RONE \<in> set (rflts rs)"
|
|
|
878 |
apply(induct rs)
|
|
|
879 |
apply simp
|
|
|
880 |
apply(case_tac " RONE \<in> set rs")
|
|
|
881 |
apply simp
|
|
|
882 |
apply (metis Un_upper2 insert_absorb insert_subset list.set_intros(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append)
|
|
|
883 |
apply(case_tac "RONE = a")
|
|
|
884 |
apply simp
|
|
|
885 |
apply simp
|
|
|
886 |
done
|
|
|
887 |
|
|
|
888 |
lemma set_inclusion_with_flts10:
|
|
|
889 |
shows " \<lbrakk>RCHAR x \<in> set rs\<rbrakk> \<Longrightarrow> RCHAR x \<in> set (rflts rs)"
|
|
|
890 |
apply(induct rs)
|
|
|
891 |
apply simp
|
|
|
892 |
apply(case_tac " RCHAR x \<in> set rs")
|
|
|
893 |
apply simp
|
|
|
894 |
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
|
|
|
895 |
apply(case_tac "RCHAR x = a")
|
|
|
896 |
apply simp
|
|
|
897 |
apply fastforce
|
|
|
898 |
apply simp
|
|
|
899 |
done
|
|
|
900 |
|
|
|
901 |
lemma set_inclusion_with_flts11:
|
|
|
902 |
shows " \<lbrakk>RSEQ r1 r2 \<in> set rs\<rbrakk> \<Longrightarrow> RSEQ r1 r2 \<in> set (rflts rs)"
|
|
|
903 |
apply(induct rs)
|
|
|
904 |
apply simp
|
|
|
905 |
apply(case_tac " RSEQ r1 r2 \<in> set rs")
|
|
|
906 |
apply simp
|
|
|
907 |
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
|
|
|
908 |
apply(case_tac "RSEQ r1 r2 = a")
|
|
|
909 |
apply simp
|
|
|
910 |
apply fastforce
|
|
|
911 |
apply simp
|
|
|
912 |
done
|
|
|
913 |
|
|
|
914 |
|
|
|
915 |
lemma set_inclusion_with_flts:
|
|
|
916 |
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))"
|
|
|
917 |
by (simp add: set_inclusion_with_flts1)
|
|
453
|
918 |
|
|
456
|
919 |
lemma "\<And>x5. \<lbrakk>a \<in> set as; rsimp a \<in> set (map rsimp as); rsimp a = RALTS x5\<rbrakk>
|
|
|
920 |
\<Longrightarrow> rsimp_ALTs (rdistinct (rflts (map rsimp as @ [rsimp a])) {}) =
|
|
|
921 |
rsimp_ALTs (rdistinct (rflts (map rsimp as @ x5)) {})"
|
|
|
922 |
|
|
465
|
923 |
sorry
|
|
|
924 |
|
|
453
|
925 |
|
|
|
926 |
lemma last_elem_dup1:
|
|
|
927 |
shows " a \<in> set as \<Longrightarrow> rsimp (RALTS (as @ [a] )) = rsimp (RALTS (as ))"
|
|
|
928 |
apply simp
|
|
|
929 |
apply(subgoal_tac "rsimp a \<in> set (map rsimp as)")
|
|
|
930 |
prefer 2
|
|
|
931 |
apply simp
|
|
456
|
932 |
apply(case_tac "rsimp a")
|
|
|
933 |
apply simp
|
|
|
934 |
|
|
|
935 |
using flts_identity3 apply presburger
|
|
|
936 |
apply(subst flts_identity2)
|
|
|
937 |
using rrexp.distinct(1) rrexp.distinct(15) apply presburger
|
|
|
938 |
apply(subst distinct_removes_last3[symmetric])
|
|
|
939 |
using set_inclusion_with_flts apply blast
|
|
|
940 |
apply simp
|
|
|
941 |
apply (metis distinct_removes_last3 flts_identity10 set_inclusion_with_flts10)
|
|
|
942 |
apply (metis distinct_removes_last3 flts_identity11 set_inclusion_with_flts11)
|
|
453
|
943 |
sorry
|
|
|
944 |
|
|
|
945 |
lemma last_elem_dup:
|
|
|
946 |
shows " a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs (as @ [a] )) = rsimp (rsimp_ALTs (as ))"
|
|
|
947 |
apply(induct as rule: rev_induct)
|
|
|
948 |
apply simp
|
|
|
949 |
apply simp
|
|
|
950 |
apply(subgoal_tac "xs \<noteq> []")
|
|
|
951 |
prefer 2
|
|
|
952 |
|
|
|
953 |
|
|
|
954 |
|
|
|
955 |
|
|
|
956 |
sorry
|
|
|
957 |
|
|
|
958 |
lemma appeared_before_remove_later:
|
|
|
959 |
shows "a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs ( as @ a # rs)) = rsimp (rsimp_ALTs (as @ rs))"
|
|
|
960 |
and "a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs as ) = rsimp (rsimp_ALTs (as @ [a]))"
|
|
|
961 |
apply(induct rs arbitrary: as)
|
|
|
962 |
apply simp
|
|
|
963 |
|
|
|
964 |
|
|
|
965 |
sorry
|
|
|
966 |
|
|
|
967 |
lemma distinct_remove_later:
|
|
|
968 |
shows "\<lbrakk>rder x a \<in> rder x ` set rsa\<rbrakk>
|
|
|
969 |
\<Longrightarrow> rsimp (rsimp_ALTs (map (rder x) rsa @ rder x a # map (rder x) (rdistinct rs (insert a (set rsa))))) =
|
|
|
970 |
rsimp (rsimp_ALTs (map (rder x) rsa @ map (rder x) (rdistinct rs (set rsa))))"
|
|
451
|
971 |
|
|
|
972 |
sorry
|
|
|
973 |
|
|
|
974 |
|
|
453
|
975 |
lemma distinct_der_general:
|
|
|
976 |
shows "rsimp (rsimp_ALTs (map (rder x) (rsa @ (rdistinct rs (set rsa))))) =
|
|
|
977 |
rsimp ( rsimp_ALTs ((map (rder x) rsa)@(rdistinct (map (rder x) rs) (set (map (rder x) rsa)))) )"
|
|
|
978 |
apply(induct rs arbitrary: rsa)
|
|
|
979 |
apply simp
|
|
|
980 |
apply(case_tac "a \<in> set rsa")
|
|
|
981 |
apply(subgoal_tac "rder x a \<in> set (map (rder x) rsa)")
|
|
|
982 |
apply simp
|
|
|
983 |
apply simp
|
|
|
984 |
apply(case_tac "rder x a \<notin> set (map (rder x) rsa)")
|
|
|
985 |
apply(simp)
|
|
|
986 |
apply(subst map_concat_cons)+
|
|
|
987 |
apply(drule_tac x = "rsa @ [a]" in meta_spec)
|
|
|
988 |
apply simp
|
|
|
989 |
apply(drule neg_removal_element_of)
|
|
|
990 |
apply simp
|
|
|
991 |
apply(subst distinct_remove_later)
|
|
|
992 |
apply simp
|
|
|
993 |
apply(drule_tac x = "rsa" in meta_spec)
|
|
|
994 |
by blast
|
|
|
995 |
|
|
|
996 |
|
|
|
997 |
|
|
|
998 |
|
|
451
|
999 |
lemma distinct_der:
|
|
|
1000 |
shows "rsimp (rsimp_ALTs (map (rder x) (rdistinct rs {}))) = rsimp ( rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
|
|
453
|
1001 |
by (metis distinct_der_general list.simps(8) self_append_conv2 set_empty)
|
|
451
|
1002 |
|
|
453
|
1003 |
|
|
|
1004 |
|
|
|
1005 |
|
|
|
1006 |
lemma rders_simp_lambda:
|
|
|
1007 |
shows " rsimp \<circ> rder x \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r (xs @ [x]))"
|
|
|
1008 |
using rders_simp_append by auto
|
|
451
|
1009 |
|
|
453
|
1010 |
lemma rders_simp_nonempty_simped:
|
|
|
1011 |
shows "xs \<noteq> [] \<Longrightarrow> rsimp \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r xs)"
|
|
|
1012 |
using rders_simp_same_simpders rsimp_idem by auto
|
|
|
1013 |
|
|
|
1014 |
lemma repeated_altssimp:
|
|
|
1015 |
shows "\<forall>r \<in> set rs. rsimp r = r \<Longrightarrow> rsimp (rsimp_ALTs (rdistinct (rflts rs) {})) =
|
|
|
1016 |
rsimp_ALTs (rdistinct (rflts rs) {})"
|
|
|
1017 |
by (metis map_idI rsimp.simps(2) rsimp_idem)
|
|
451
|
1018 |
|
|
465
|
1019 |
|
|
|
1020 |
lemma add0_isomorphic:
|
|
|
1021 |
shows "rsimp_ALTs (rdistinct (rflts [rsimp r, RZERO]) {}) = rsimp r"
|
|
|
1022 |
sorry
|
|
|
1023 |
|
|
|
1024 |
|
|
|
1025 |
lemma distinct_append_simp:
|
|
|
1026 |
shows " rsimp (rsimp_ALTs rs1) = rsimp (rsimp_ALTs rs2) \<Longrightarrow>
|
|
|
1027 |
rsimp (rsimp_ALTs (f a # rs1)) =
|
|
|
1028 |
rsimp (rsimp_ALTs (f a # rs2))"
|
|
|
1029 |
apply(case_tac rs1)
|
|
|
1030 |
apply simp
|
|
|
1031 |
apply(case_tac rs2)
|
|
|
1032 |
apply simp
|
|
|
1033 |
apply simp
|
|
|
1034 |
prefer 2
|
|
|
1035 |
apply(case_tac list)
|
|
|
1036 |
apply(case_tac rs2)
|
|
|
1037 |
apply simp
|
|
|
1038 |
using add0_isomorphic apply blast
|
|
|
1039 |
apply simp
|
|
467
|
1040 |
oops
|
|
465
|
1041 |
|
|
444
|
1042 |
lemma alts_closed_form: shows
|
|
|
1043 |
"rsimp (rders_simp (RALTS rs) s) =
|
|
|
1044 |
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
|
1045 |
apply(induct s rule: rev_induct)
|
|
|
1046 |
apply simp
|
|
|
1047 |
apply simp
|
|
|
1048 |
apply(subst rders_simp_append)
|
|
|
1049 |
apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) =
|
|
|
1050 |
rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
|
|
|
1051 |
prefer 2
|
|
|
1052 |
apply (metis inside_simp_removal rders_simp_one_char)
|
|
|
1053 |
apply(simp only: )
|
|
451
|
1054 |
apply(subst rders_simp_one_char)
|
|
|
1055 |
apply(subst rsimp_idem)
|
|
|
1056 |
apply(subgoal_tac "rsimp (rder x (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {}))) =
|
|
|
1057 |
rsimp ((rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))) ")
|
|
|
1058 |
prefer 2
|
|
|
1059 |
using rder_rsimp_ALTs_commute apply presburger
|
|
|
1060 |
apply(simp only:)
|
|
|
1061 |
apply(subgoal_tac "rsimp (rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))
|
|
|
1062 |
= rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
1063 |
prefer 2
|
|
|
1064 |
|
|
|
1065 |
using distinct_der apply presburger
|
|
|
1066 |
apply(simp only:)
|
|
453
|
1067 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
1068 |
rsimp (rsimp_ALTs (rdistinct ( (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)))) {}))")
|
|
|
1069 |
apply(simp only:)
|
|
|
1070 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
1071 |
rsimp (rsimp_ALTs (rdistinct (rflts ( (map (rsimp \<circ> (rder x) \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
1072 |
apply(simp only:)
|
|
|
1073 |
apply(subst rders_simp_lambda)
|
|
|
1074 |
apply(subst rders_simp_nonempty_simped)
|
|
|
1075 |
apply simp
|
|
|
1076 |
apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (xs @ [x])) rs). rsimp r = r")
|
|
|
1077 |
prefer 2
|
|
|
1078 |
apply (simp add: rders_simp_same_simpders rsimp_idem)
|
|
|
1079 |
apply(subst repeated_altssimp)
|
|
|
1080 |
apply simp
|
|
|
1081 |
apply fastforce
|
|
465
|
1082 |
apply (metis inside_simp_removal list.map_comp rder.simps(4) rsimp.simps(2) rsimp_idem)
|
|
|
1083 |
sledgehammer
|
|
|
1084 |
(* by (metis inside_simp_removal rder_rsimp_ALTs_commute self_append_conv2 set_empty simp_more_distinct)
|
|
451
|
1085 |
|
|
465
|
1086 |
*)
|
|
443
|
1087 |
|
|
444
|
1088 |
lemma alts_closed_form_variant: shows
|
|
|
1089 |
"s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s =
|
|
|
1090 |
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
|
1091 |
sorry
|
|
443
|
1092 |
|
|
|
1093 |
|
|
|
1094 |
|
|
444
|
1095 |
lemma star_closed_form:
|
|
|
1096 |
shows "rders_simp (RSTAR r0) (c#s) =
|
|
|
1097 |
rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
|
|
|
1098 |
apply(induct s)
|
|
|
1099 |
apply simp
|
|
|
1100 |
sorry
|
|
443
|
1101 |
|
|
|
1102 |
|
|
|
1103 |
|
|
|
1104 |
lemma seq_closed_form: shows
|
|
|
1105 |
"rsimp (rders_simp (RSEQ r1 r2) s) =
|
|
|
1106 |
rsimp ( RALTS ( (RSEQ (rders_simp r1 s) r2) #
|
|
445
|
1107 |
(map (rders_simp r2) (vsuf s r1))
|
|
443
|
1108 |
)
|
|
|
1109 |
)"
|
|
|
1110 |
apply(induct s)
|
|
|
1111 |
apply simp
|
|
|
1112 |
sorry
|
|
|
1113 |
|
|
|
1114 |
|
|
444
|
1115 |
lemma seq_closed_form_variant: shows
|
|
|
1116 |
"s \<noteq> [] \<Longrightarrow> (rders_simp (RSEQ r1 r2) s) =
|
|
|
1117 |
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
|
|
445
|
1118 |
apply(induct s rule: rev_induct)
|
|
|
1119 |
apply simp
|
|
|
1120 |
apply(subst rders_simp_append)
|
|
|
1121 |
apply(subst rders_simp_one_char)
|
|
|
1122 |
apply(subst rsimp_idem[symmetric])
|
|
|
1123 |
apply(subst rders_simp_one_char[symmetric])
|
|
|
1124 |
apply(subst rders_simp_append[symmetric])
|
|
|
1125 |
apply(insert seq_closed_form)
|
|
|
1126 |
apply(subgoal_tac "rsimp (rders_simp (RSEQ r1 r2) (xs @ [x]))
|
|
|
1127 |
= rsimp (RALTS (RSEQ (rders_simp r1 (xs @ [x])) r2 # map (rders_simp r2) (vsuf (xs @ [x]) r1)))")
|
|
|
1128 |
apply force
|
|
|
1129 |
by presburger
|
|
443
|
1130 |
|
|
444
|
1131 |
end |