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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 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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465
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478
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465
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478
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465
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lemma flts_middle0:
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shows "rflts (rsa @ RZERO # rsb) = rflts (rsa @ rsb)"
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apply(induct rsa)
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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_middle01:
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shows "rflts (rsa @ [RZERO] @ rsb) = rflts (rsa @ rsb)"
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by (simp add: flts_middle0)
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lemma flts_append1:
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shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow>
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rflts (rsa @ [a] @ rsb) = rflts rsa @ [a] @ (rflts rsb)"
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apply(induct rsa arbitrary: rsb)
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apply simp
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using rflts_def_idiot apply presburger
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apply(case_tac aa)
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apply simp+
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done
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478
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465
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lemma simp_removes_duplicate1:
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shows " a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a])) = rsimp (RALTS (rsa))"
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and " rsimp (RALTS (a1 # rsa @ [a1])) = rsimp (RALTS (a1 # rsa))"
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apply(induct rsa arbitrary: a1)
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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 "a = aa")
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apply simp
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apply simp
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apply (metis Cons_eq_appendI Cons_eq_map_conv distinct_removes_duplicate_flts list.set_intros(2))
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apply (metis append_Cons append_Nil distinct_removes_duplicate_flts list.set_intros(1) list.simps(8) list.simps(9))
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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))
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lemma simp_removes_duplicate2:
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shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a] @ rsb)) = rsimp (RALTS (rsa @ rsb))"
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apply(induct rsb arbitrary: rsa)
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apply simp
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using distinct_removes_duplicate_flts apply auto[1]
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by (metis append.assoc head_one_more_simp rsimp.simps(2) simp_flatten simp_removes_duplicate1(1))
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lemma simp_removes_duplicate3:
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shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ a # rsb)) = rsimp (RALTS (rsa @ rsb))"
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using simp_removes_duplicate2 by auto
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lemma distinct_removes_middle4:
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shows "a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ [a] @ rsb) rset = rdistinct (rsa @ rsb) rset"
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using distinct_removes_middle(1) by fastforce
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lemma distinct_removes_middle_list:
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shows "\<forall>a \<in> set x. a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ x @ rsb) rset = rdistinct (rsa @ rsb) rset"
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apply(induct x)
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apply simp
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by (simp add: distinct_removes_middle3)
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lemma distinct_removes_duplicate_flts2:
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shows " a \<in> set rsa
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\<Longrightarrow> rdistinct (rflts (rsa @ [a] @ rsb)) {} =
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rdistinct (rflts (rsa @ rsb)) {}"
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apply(induct a arbitrary: rsb)
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using flts_middle01 apply presburger
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apply(subgoal_tac "rflts (rsa @ [RONE] @ rsb) = rflts rsa @ [RONE] @ rflts rsb")
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prefer 2
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using flts_append1 apply blast
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apply simp
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apply(subgoal_tac "RONE \<in> set (rflts rsa)")
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prefer 2
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using rflts_def_idiot2 apply blast
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apply(subst distinct_removes_middle3)
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apply simp
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using flts_append apply presburger
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apply simp
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apply (metis distinct_removes_middle3 flts_append in_set_conv_decomp rflts.simps(5))
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apply (metis distinct_removes_middle(1) flts_append flts_append1 rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
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apply(subgoal_tac "rflts (rsa @ [RALTS x] @ rsb) = rflts rsa @ x @ rflts rsb")
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prefer 2
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apply (simp add: flts_append)
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apply (simp only:)
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apply(subgoal_tac "\<forall>r1 \<in> set x. r1 \<in> set (rflts rsa)")
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prefer 2
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using spilled_alts_contained apply blast
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apply(subst flts_append)
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using distinct_removes_middle_list apply blast
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using distinct_removes_middle2 flts_append rflts_def_idiot2 by fastforce
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lemma simp_removes_duplicate:
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shows "a \<in> set rsa \<Longrightarrow> rsimp (rsimp_ALTs (rsa @ a # rs)) = rsimp (rsimp_ALTs (rsa @ rs))"
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apply(subgoal_tac "rsimp (rsimp_ALTs (rsa @ a # rs)) = rsimp (RALTS (rsa @ a # rs))")
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prefer 2
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apply (smt (verit, best) Cons_eq_append_conv append_is_Nil_conv empty_set equals0D list.distinct(1) rsimp_ALTs.elims)
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apply(simp only:)
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apply simp
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apply(subgoal_tac "(rdistinct (rflts (map rsimp rsa @ rsimp a # map rsimp rs)) {}) = (rdistinct (rflts (map rsimp rsa @ map rsimp rs)) {})")
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apply(simp only:)
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prefer 2
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apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
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prefer 2
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apply simp
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using distinct_removes_duplicate_flts2 apply force
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apply(case_tac rsa)
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apply simp
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apply(case_tac 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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using idem_after_simp1 apply presburger
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apply simp+
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apply(subgoal_tac "rsimp_ALTs (aa # list @ aaa # lista) = RALTS (aa # list @ aaa # lista)")
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apply simp
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using rsimpalts_conscons by presburger
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467
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468
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467
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471
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inductive frewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f _" [10, 10] 10)
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where
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"(RZERO # rs) \<leadsto>f rs"
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| "((RALTS rs) # rsa) \<leadsto>f (rs @ rsa)"
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| "rs1 \<leadsto>f rs2 \<Longrightarrow> (r # rs1) \<leadsto>f (r # rs2)"
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inductive
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frewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f* _" [10, 10] 10)
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where
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[intro, simp]:"rs \<leadsto>f* rs"
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| [intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>f* rs3"
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inductive grewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g _" [10, 10] 10)
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where
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"(RZERO # rs) \<leadsto>g rs"
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| "((RALTS rs) # rsa) \<leadsto>g (rs @ rsa)"
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| "rs1 \<leadsto>g rs2 \<Longrightarrow> (r # rs1) \<leadsto>g (r # rs2)"
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| "rsa @ [a] @ rsb @ [a] @ rsc \<leadsto>g rsa @ [a] @ rsb @ rsc"
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478
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lemma grewrite_variant1:
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shows "a \<in> set rs1 \<Longrightarrow> rs1 @ a # rs \<leadsto>g rs1 @ rs"
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apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
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done
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473
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inductive
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grewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g* _" [10, 10] 10)
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where
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[intro, simp]:"rs \<leadsto>g* rs"
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| [intro]: "\<lbrakk>rs1 \<leadsto>g* rs2; rs2 \<leadsto>g rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>g* rs3"
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478
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473
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(*
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inductive
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frewrites2:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ <\<leadsto>f* _" [10, 10] 10)
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where
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[intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f* rs1\<rbrakk> \<Longrightarrow> rs1 <\<leadsto>f* rs2"
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*)
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471
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lemma fr_in_rstar : "r1 \<leadsto>f r2 \<Longrightarrow> r1 \<leadsto>f* r2"
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using frewrites.intros(1) frewrites.intros(2) by blast
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lemma freal_trans[trans]:
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assumes a1: "r1 \<leadsto>f* r2" and a2: "r2 \<leadsto>f* r3"
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shows "r1 \<leadsto>f* r3"
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using a2 a1
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apply(induct r2 r3 arbitrary: r1 rule: frewrites.induct)
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apply(auto)
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done
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lemma many_steps_later: "\<lbrakk>r1 \<leadsto>f r2; r2 \<leadsto>f* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>f* r3"
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by (meson fr_in_rstar freal_trans)
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475
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lemma gr_in_rstar : "r1 \<leadsto>g r2 \<Longrightarrow> r1 \<leadsto>g* r2"
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using grewrites.intros(1) grewrites.intros(2) by blast
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lemma greal_trans[trans]:
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assumes a1: "r1 \<leadsto>g* r2" and a2: "r2 \<leadsto>g* r3"
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shows "r1 \<leadsto>g* r3"
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using a2 a1
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apply(induct r2 r3 arbitrary: r1 rule: grewrites.induct)
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apply(auto)
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done
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lemma gmany_steps_later: "\<lbrakk>r1 \<leadsto>g r2; r2 \<leadsto>g* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>g* r3"
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by (meson gr_in_rstar greal_trans)
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478
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lemma gstar_rdistinct_general:
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shows "rs1 @ rs \<leadsto>g* rs1 @ (rdistinct rs (set rs1))"
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apply(induct rs arbitrary: rs1)
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apply simp
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apply(case_tac " a \<in> set rs1")
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apply simp
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apply(subgoal_tac "rs1 @ a # rs \<leadsto>g rs1 @ rs")
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using gmany_steps_later apply auto[1]
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apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
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apply simp
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apply(drule_tac x = "rs1 @ [a]" in meta_spec)
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by simp
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lemma gstar_rdistinct:
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shows "rs \<leadsto>g* rdistinct rs {}"
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apply(induct rs)
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apply simp
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by (metis append.left_neutral empty_set gstar_rdistinct_general)
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475
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471
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lemma frewrite_append:
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shows "\<lbrakk> rsa \<leadsto>f rsb \<rbrakk> \<Longrightarrow> rs @ rsa \<leadsto>f rs @ rsb"
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apply(induct rs)
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472
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apply simp+
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using frewrite.intros(3) by blast
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475
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lemma grewrite_append:
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shows "\<lbrakk> rsa \<leadsto>g rsb \<rbrakk> \<Longrightarrow> rs @ rsa \<leadsto>g rs @ rsb"
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apply(induct rs)
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apply simp+
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using grewrite.intros(3) by blast
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472
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471
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lemma frewrites_cons:
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shows "\<lbrakk> rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>f* r # rsb"
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472
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apply(induct rsa rsb rule: frewrites.induct)
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apply simp
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using frewrite.intros(3) by blast
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471
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475
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lemma grewrites_cons:
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shows "\<lbrakk> rsa \<leadsto>g* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>g* r # rsb"
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apply(induct rsa rsb rule: grewrites.induct)
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apply simp
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using grewrite.intros(3) by blast
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265 |
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471
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lemma frewrites_append:
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shows " \<lbrakk>rsa \<leadsto>f* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>f* (rs @ rsb)"
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472
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apply(induct rs)
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471
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apply simp
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472
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by (simp add: frewrites_cons)
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471
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475
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lemma grewrites_append:
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shows " \<lbrakk>rsa \<leadsto>g* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>g* (rs @ rsb)"
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apply(induct rs)
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apply simp
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by (simp add: grewrites_cons)
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471
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lemma frewrites_concat:
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shows "\<lbrakk>rs1 \<leadsto>f rs2; rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> (rs1 @ rsa) \<leadsto>f* (rs2 @ rsb)"
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apply(induct rs1 rs2 rule: frewrite.induct)
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apply(simp)
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apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>f (rs @ rsa)")
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prefer 2
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using frewrite.intros(1) apply blast
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apply(subgoal_tac "(rs @ rsa) \<leadsto>f* (rs @ rsb)")
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using many_steps_later apply blast
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472
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apply (simp add: frewrites_append)
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apply (metis append.assoc append_Cons frewrite.intros(2) frewrites_append many_steps_later)
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using frewrites_cons by auto
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471
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475
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lemma grewrites_concat:
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shows "\<lbrakk>rs1 \<leadsto>g rs2; rsa \<leadsto>g* rsb \<rbrakk> \<Longrightarrow> (rs1 @ rsa) \<leadsto>g* (rs2 @ rsb)"
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296 |
apply(induct rs1 rs2 rule: grewrite.induct)
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297 |
apply(simp)
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|
298 |
apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>g (rs @ rsa)")
|
|
|
299 |
prefer 2
|
|
|
300 |
using grewrite.intros(1) apply blast
|
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|
301 |
apply(subgoal_tac "(rs @ rsa) \<leadsto>g* (rs @ rsb)")
|
|
|
302 |
using gmany_steps_later apply blast
|
|
|
303 |
apply (simp add: grewrites_append)
|
|
|
304 |
apply (metis append.assoc append_Cons grewrite.intros(2) grewrites_append gmany_steps_later)
|
|
|
305 |
using grewrites_cons apply auto
|
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|
306 |
apply(subgoal_tac "rsaa @ a # rsba @ a # rsc @ rsa \<leadsto>g* rsaa @ a # rsba @ a # rsc @ rsb")
|
|
|
307 |
using grewrite.intros(4) grewrites.intros(2) apply force
|
|
|
308 |
using grewrites_append by auto
|
|
|
309 |
|
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|
310 |
|
|
|
311 |
lemma grewritess_concat:
|
|
|
312 |
shows "\<lbrakk>rsa \<leadsto>g* rsb; rsc \<leadsto>g* rsd \<rbrakk> \<Longrightarrow> (rsa @ rsc) \<leadsto>g* (rsb @ rsd)"
|
|
|
313 |
apply(induct rsa rsb rule: grewrites.induct)
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|
314 |
apply(case_tac rs)
|
|
|
315 |
apply simp
|
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|
316 |
using grewrites_append apply blast
|
|
|
317 |
by (meson greal_trans grewrites.simps grewrites_concat)
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|
318 |
|
|
476
|
319 |
fun alt_set:: "rrexp \<Rightarrow> rrexp set"
|
|
|
320 |
where
|
|
|
321 |
"alt_set (RALTS rs) = set rs \<union> \<Union> (alt_set ` (set rs))"
|
|
|
322 |
| "alt_set r = {r}"
|
|
|
323 |
|
|
|
324 |
lemma alt_set_has_all:
|
|
|
325 |
shows "RALTS rs \<in> alt_set rx \<Longrightarrow> set rs \<subseteq> alt_set rx"
|
|
|
326 |
apply(induct rx arbitrary: rs)
|
|
|
327 |
apply simp_all
|
|
|
328 |
apply(rename_tac rSS rss)
|
|
|
329 |
using in_mono by fastforce
|
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|
330 |
|
|
|
331 |
|
|
|
332 |
|
|
|
333 |
|
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|
334 |
lemma grewrite_equal_rsimp:
|
|
|
335 |
shows "\<lbrakk>rs1 \<leadsto>g rs2; rsimp_ALTs (rdistinct (rflts (map rsimp rs1)) (rset \<union> \<Union>(alt_set ` rset))) =
|
|
|
336 |
rsimp_ALTs (rdistinct (rflts (map rsimp rs2)) (rset \<union> \<Union>(alt_set ` rset)))\<rbrakk>
|
|
|
337 |
\<Longrightarrow> rsimp_ALTs (rdistinct (rflts (rsimp r # map rsimp rs1)) (rset \<union> \<Union>(alt_set ` rset))) =
|
|
|
338 |
rsimp_ALTs (rdistinct (rflts (rsimp r # map rsimp rs2)) (rset \<union> \<Union>(alt_set ` rset)))"
|
|
|
339 |
apply(induct rs1 rs2 arbitrary:rset rule: grewrite.induct)
|
|
|
340 |
apply simp
|
|
|
341 |
apply (metis append_Cons append_Nil flts_middle0)
|
|
|
342 |
apply(case_tac "rsimp r \<in> rset")
|
|
|
343 |
apply simp
|
|
|
344 |
oops
|
|
|
345 |
|
|
478
|
346 |
lemma grewrite_cases_middle:
|
|
|
347 |
shows "rs1 \<leadsto>g rs2 \<Longrightarrow>
|
|
|
348 |
(\<exists>rsa rsb rsc. rs1 = (rsa @ [RALTS rsb] @ rsc) \<and> rs2 = (rsa @ rsb @ rsc)) \<or>
|
|
|
349 |
(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc) \<or>
|
|
|
350 |
(\<exists>rsa rsb rsc a. rs1 = rsa @ [a] @ rsb @ [a] @ rsc \<and> rs2 = rsa @ [a] @ rsb @ rsc)"
|
|
|
351 |
apply( induct rs1 rs2 rule: grewrite.induct)
|
|
|
352 |
apply simp
|
|
|
353 |
apply blast
|
|
|
354 |
apply (metis append_Cons append_Nil)
|
|
|
355 |
apply (metis append_Cons)
|
|
|
356 |
by blast
|
|
476
|
357 |
|
|
|
358 |
|
|
|
359 |
lemma grewrite_equal_rsimp:
|
|
|
360 |
shows "rs1 \<leadsto>g rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
478
|
361 |
apply(frule grewrite_cases_middle)
|
|
|
362 |
apply(case_tac "(\<exists>rsa rsb rsc. rs1 = rsa @ [RALTS rsb] @ rsc \<and> rs2 = rsa @ rsb @ rsc)")
|
|
|
363 |
using simp_flatten3 apply auto[1]
|
|
|
364 |
apply(case_tac "(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc)")
|
|
|
365 |
apply (metis (mono_tags, opaque_lifting) append_Cons append_Nil list.set_intros(1) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) simp_removes_duplicate3)
|
|
|
366 |
by (smt (verit) append.assoc append_Cons append_Nil in_set_conv_decomp simp_removes_duplicate3)
|
|
476
|
367 |
|
|
|
368 |
|
|
475
|
369 |
lemma grewrites_equal_rsimp:
|
|
|
370 |
shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
476
|
371 |
apply (induct rs1 rs2 rule: grewrites.induct)
|
|
475
|
372 |
apply simp
|
|
476
|
373 |
using grewrite_equal_rsimp by presburger
|
|
|
374 |
|
|
|
375 |
|
|
|
376 |
|
|
|
377 |
|
|
|
378 |
lemma grewrites_equal_simp_2:
|
|
|
379 |
shows "rsimp (RALTS rs1) = rsimp (RALTS rs2) \<Longrightarrow> rs1 \<leadsto>g* rs2"
|
|
478
|
380 |
oops
|
|
|
381 |
|
|
|
382 |
|
|
|
383 |
|
|
475
|
384 |
|
|
476
|
385 |
lemma grewrites_last:
|
|
|
386 |
shows "r # [RALTS rs] \<leadsto>g* r # rs"
|
|
|
387 |
by (metis gr_in_rstar grewrite.intros(2) grewrite.intros(3) self_append_conv)
|
|
|
388 |
|
|
|
389 |
lemma simp_flatten2:
|
|
|
390 |
shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
|
|
|
391 |
using grewrites_equal_rsimp grewrites_last by blast
|
|
475
|
392 |
|
|
473
|
393 |
lemma frewrites_middle:
|
|
|
394 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> r # (RALTS rs # rs1) \<leadsto>f* r # (rs @ rs1)"
|
|
|
395 |
by (simp add: fr_in_rstar frewrite.intros(2) frewrite.intros(3))
|
|
471
|
396 |
|
|
473
|
397 |
lemma frewrites_alt:
|
|
|
398 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> (RALT r1 r2) # rs1 \<leadsto>f* r1 # r2 # rs2"
|
|
|
399 |
by (metis Cons_eq_appendI append_self_conv2 frewrite.intros(2) frewrites_cons many_steps_later)
|
|
471
|
400 |
|
|
|
401 |
lemma early_late_der_frewrites:
|
|
|
402 |
shows "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)"
|
|
|
403 |
apply(induct rs)
|
|
|
404 |
apply simp
|
|
|
405 |
apply(case_tac a)
|
|
|
406 |
apply simp+
|
|
|
407 |
using frewrite.intros(1) many_steps_later apply blast
|
|
|
408 |
apply(case_tac "x = x3")
|
|
472
|
409 |
apply simp
|
|
|
410 |
using frewrites_cons apply presburger
|
|
|
411 |
using frewrite.intros(1) many_steps_later apply fastforce
|
|
|
412 |
apply(case_tac "rnullable x41")
|
|
473
|
413 |
apply simp+
|
|
|
414 |
apply (simp add: frewrites_alt)
|
|
|
415 |
apply (simp add: frewrites_cons)
|
|
|
416 |
apply (simp add: frewrites_append)
|
|
|
417 |
by (simp add: frewrites_cons)
|
|
471
|
418 |
|
|
|
419 |
|
|
|
420 |
|
|
|
421 |
|
|
473
|
422 |
lemma with_wo0_distinct:
|
|
|
423 |
shows "rdistinct rs rset \<leadsto>f* rdistinct rs (insert RZERO rset)"
|
|
|
424 |
apply(induct rs arbitrary: rset)
|
|
|
425 |
apply simp
|
|
|
426 |
apply(case_tac a)
|
|
|
427 |
apply(case_tac "RZERO \<in> rset")
|
|
|
428 |
apply simp+
|
|
|
429 |
using fr_in_rstar frewrite.intros(1) apply presburger
|
|
|
430 |
apply (case_tac "RONE \<in> rset")
|
|
|
431 |
apply simp+
|
|
|
432 |
using frewrites_cons apply presburger
|
|
|
433 |
apply(case_tac "a \<in> rset")
|
|
|
434 |
apply simp
|
|
|
435 |
apply (simp add: frewrites_cons)
|
|
|
436 |
apply(case_tac "a \<in> rset")
|
|
|
437 |
apply simp
|
|
|
438 |
apply (simp add: frewrites_cons)
|
|
|
439 |
apply(case_tac "a \<in> rset")
|
|
|
440 |
apply simp
|
|
|
441 |
apply (simp add: frewrites_cons)
|
|
|
442 |
apply(case_tac "a \<in> rset")
|
|
|
443 |
apply simp
|
|
|
444 |
apply (simp add: frewrites_cons)
|
|
|
445 |
done
|
|
|
446 |
|
|
476
|
447 |
(*Interesting lemma: not obvious but easily proven by sledgehammer*)
|
|
473
|
448 |
|
|
476
|
449 |
|
|
|
450 |
|
|
|
451 |
|
|
|
452 |
(*lemma induction last rule not go through
|
|
|
453 |
example:
|
|
|
454 |
r #
|
|
|
455 |
rdistinct rs1
|
|
|
456 |
(insert RZERO
|
|
|
457 |
(insert r
|
|
|
458 |
(rset \<union>
|
|
|
459 |
\<Union> (alt_set `
|
|
|
460 |
rset)))) \<leadsto>g* r #
|
|
|
461 |
rdistinct rs2
|
|
|
462 |
(insert RZERO (insert r (rset \<union> \<Union> (alt_set ` rset))))
|
|
|
463 |
rs2 = [+rs] rs3 = rs,
|
|
|
464 |
r = +rs
|
|
|
465 |
[] \<leadsto>g* rs which is wrong
|
|
|
466 |
*)
|
|
|
467 |
|
|
469
|
468 |
|
|
|
469 |
|
|
476
|
470 |
|
|
|
471 |
|
|
|
472 |
|
|
|
473 |
|
|
|
474 |
lemma frewrite_simpeq:
|
|
|
475 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
|
476 |
apply(induct rs1 rs2 rule: frewrite.induct)
|
|
|
477 |
apply simp
|
|
|
478 |
using simp_flatten apply presburger
|
|
478
|
479 |
by (metis (no_types, opaque_lifting) grewrites_equal_rsimp grewrites_last list.simps(9) rsimp.simps(2))
|
|
|
480 |
|
|
|
481 |
lemma gstar0:
|
|
|
482 |
shows "rsa @ (rdistinct rs (set rsa)) \<leadsto>g* rsa @ (rdistinct rs (insert RZERO (set rsa)))"
|
|
|
483 |
apply(induct rs arbitrary: rsa)
|
|
|
484 |
apply simp
|
|
|
485 |
apply(case_tac "a = RZERO")
|
|
|
486 |
apply simp
|
|
|
487 |
|
|
|
488 |
using gr_in_rstar grewrite.intros(1) grewrites_append apply presburger
|
|
|
489 |
apply(case_tac "a \<in> set rsa")
|
|
|
490 |
apply simp+
|
|
|
491 |
apply(drule_tac x = "rsa @ [a]" in meta_spec)
|
|
|
492 |
by simp
|
|
|
493 |
|
|
|
494 |
lemma gstar01:
|
|
|
495 |
shows "rdistinct rs {} \<leadsto>g* rdistinct rs {RZERO}"
|
|
|
496 |
by (metis empty_set gstar0 self_append_conv2)
|
|
|
497 |
|
|
|
498 |
|
|
|
499 |
lemma grewrite_rdistinct_aux:
|
|
|
500 |
shows "rs @ rdistinct rsa rset \<leadsto>g* rs @ rdistinct rsa (rset \<union> set rs)"
|
|
480
|
501 |
apply(induct rsa arbitrary: rs rset)
|
|
|
502 |
apply simp
|
|
|
503 |
apply(case_tac " a \<in> rset")
|
|
|
504 |
apply simp
|
|
|
505 |
apply(case_tac "a \<in> set rs")
|
|
|
506 |
apply simp
|
|
|
507 |
apply (metis Un_insert_left Un_insert_right gmany_steps_later grewrite_variant1 insert_absorb)
|
|
|
508 |
apply simp
|
|
|
509 |
apply(drule_tac x = "rs @ [a]" in meta_spec)
|
|
|
510 |
by (metis Un_insert_left Un_insert_right append.assoc append.right_neutral append_Cons append_Nil insert_absorb2 list.simps(15) set_append)
|
|
|
511 |
|
|
478
|
512 |
|
|
|
513 |
lemma grewrite_rdistinct_worth1:
|
|
|
514 |
shows "(rsb @ [a]) @ rdistinct rs set1 \<leadsto>g* (rsb @ [a]) @ rdistinct rs (insert a set1)"
|
|
|
515 |
by (metis append.assoc empty_set grewrite_rdistinct_aux grewrites_append inf_sup_aci(5) insert_is_Un list.simps(15))
|
|
|
516 |
|
|
|
517 |
lemma grewrite_rdisitinct:
|
|
|
518 |
shows "rs @ rdistinct rsa {RALTS rs} \<leadsto>g* rs @ rdistinct rsa (insert (RALTS rs) (set rs))"
|
|
|
519 |
apply(induct rsa arbitrary: rs)
|
|
|
520 |
apply simp
|
|
|
521 |
apply(case_tac "a = RALTS rs")
|
|
|
522 |
apply simp
|
|
|
523 |
apply(case_tac "a \<in> set rs")
|
|
|
524 |
apply simp
|
|
|
525 |
apply(subgoal_tac "rs @
|
|
|
526 |
a # rdistinct rsa {RALTS rs, a} \<leadsto>g rs @ rdistinct rsa {RALTS rs, a}")
|
|
|
527 |
apply(subgoal_tac
|
|
|
528 |
"rs @ rdistinct rsa {RALTS rs, a} \<leadsto>g* rs @ rdistinct rsa (insert (RALTS rs) (set rs))")
|
|
|
529 |
using gmany_steps_later apply blast
|
|
|
530 |
apply(subgoal_tac
|
|
|
531 |
" rs @ rdistinct rsa {RALTS rs, a} \<leadsto>g* rs @ rdistinct rsa ({RALTS rs, a} \<union> set rs)")
|
|
|
532 |
apply (simp add: insert_absorb)
|
|
|
533 |
using grewrite_rdistinct_aux apply blast
|
|
|
534 |
using grewrite_variant1 apply blast
|
|
|
535 |
by (metis grewrite_rdistinct_aux insert_is_Un)
|
|
|
536 |
|
|
|
537 |
|
|
|
538 |
lemma frewrite_rd_grewrites_general:
|
|
|
539 |
shows "\<lbrakk>rs1 \<leadsto>f rs2; \<And>rs. \<exists>rs3.
|
|
|
540 |
(rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3)\<rbrakk>
|
|
|
541 |
\<Longrightarrow>
|
|
|
542 |
\<exists>rs3. (rs @ (r # rdistinct rs1 (set rs \<union> {r})) \<leadsto>g* rs3) \<and> (rs @ (r # rdistinct rs2 (set rs \<union> {r})) \<leadsto>g* rs3)"
|
|
|
543 |
apply(drule_tac x = "rs @ [r]" in meta_spec )
|
|
|
544 |
by simp
|
|
|
545 |
|
|
|
546 |
|
|
479
|
547 |
lemma alts_g_can_flts:
|
|
|
548 |
shows "RALTS rs \<in> set rsb \<Longrightarrow> \<exists>rs1 rs2.( rflts rsb = rs1 @ rs @ rs2)"
|
|
|
549 |
by (metis flts_append rflts.simps(3) split_list_last)
|
|
|
550 |
|
|
|
551 |
lemma flts_gstar:
|
|
|
552 |
shows "rs \<leadsto>g* rflts rs"
|
|
480
|
553 |
apply(induct rs)
|
|
|
554 |
apply simp
|
|
|
555 |
apply(case_tac "a = RZERO")
|
|
|
556 |
apply simp
|
|
|
557 |
using gmany_steps_later grewrite.intros(1) apply blast
|
|
|
558 |
apply(case_tac "\<exists>rsa. a = RALTS rsa")
|
|
|
559 |
apply(erule exE)
|
|
|
560 |
apply simp
|
|
|
561 |
apply (meson grewrite.intros(2) grewrites.simps grewrites_cons)
|
|
|
562 |
by (simp add: grewrites_cons rflts_def_idiot)
|
|
|
563 |
|
|
479
|
564 |
|
|
480
|
565 |
lemma wrong1:
|
|
479
|
566 |
shows "a \<in> set rs1 \<Longrightarrow> rs1 @ (rdistinct rs (insert a rset)) \<leadsto>g* rs1 @ (rdistinct rs (rset))"
|
|
480
|
567 |
|
|
|
568 |
oops
|
|
|
569 |
|
|
|
570 |
lemma more_distinct1:
|
|
|
571 |
shows " \<lbrakk>\<And>rsb rset rset2.
|
|
|
572 |
rset2 \<subseteq> set rsb \<Longrightarrow> rsb @ rdistinct rs rset \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2);
|
|
|
573 |
rset2 \<subseteq> set rsb; a \<notin> rset; a \<in> rset2\<rbrakk>
|
|
|
574 |
\<Longrightarrow> rsb @ a # rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)"
|
|
|
575 |
apply(subgoal_tac "rsb @ a # rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (insert a rset)")
|
|
|
576 |
apply(subgoal_tac "rsb @ rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)")
|
|
|
577 |
apply (meson greal_trans)
|
|
|
578 |
apply (metis Un_iff Un_insert_left insert_absorb)
|
|
|
579 |
by (simp add: gr_in_rstar grewrite_variant1 in_mono)
|
|
|
580 |
|
|
|
581 |
|
|
|
582 |
|
|
|
583 |
|
|
478
|
584 |
|
|
479
|
585 |
lemma grewrites_in_distinct0:
|
|
|
586 |
shows "a \<in> set rs1 \<Longrightarrow> rs1 @ (rdistinct (a # rs) rset) \<leadsto>g* rs1 @ (rdistinct rs rset)"
|
|
|
587 |
apply(case_tac "a \<in> rset")
|
|
|
588 |
apply simp
|
|
|
589 |
apply simp
|
|
|
590 |
oops
|
|
|
591 |
|
|
|
592 |
|
|
|
593 |
|
|
|
594 |
|
|
|
595 |
|
|
|
596 |
|
|
|
597 |
|
|
478
|
598 |
|
|
|
599 |
|
|
|
600 |
lemma frewrite_rd_grewrites_aux:
|
|
480
|
601 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
602 |
rsb @
|
|
|
603 |
RALTS rs #
|
|
|
604 |
rdistinct rsa
|
|
|
605 |
(insert (RALTS rs)
|
|
|
606 |
(set rsb)) \<leadsto>g* rflts rsb @
|
|
|
607 |
rdistinct rs (set rsb) @ rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
|
|
478
|
608 |
|
|
480
|
609 |
apply simp
|
|
|
610 |
apply(subgoal_tac "rsb @
|
|
|
611 |
RALTS rs #
|
|
|
612 |
rdistinct rsa
|
|
|
613 |
(insert (RALTS rs)
|
|
|
614 |
(set rsb)) \<leadsto>g* rsb @
|
|
|
615 |
rs @
|
|
|
616 |
rdistinct rsa
|
|
|
617 |
(insert (RALTS rs)
|
|
|
618 |
(set rsb)) ")
|
|
|
619 |
apply(subgoal_tac " rsb @
|
|
|
620 |
rs @
|
|
|
621 |
rdistinct rsa
|
|
|
622 |
(insert (RALTS rs)
|
|
|
623 |
(set rsb)) \<leadsto>g*
|
|
|
624 |
rsb @
|
|
|
625 |
rdistinct rs (set rsb) @
|
|
|
626 |
rdistinct rsa
|
|
|
627 |
(insert (RALTS rs)
|
|
|
628 |
(set rsb)) ")
|
|
|
629 |
apply (smt (verit, ccfv_SIG) Un_insert_left flts_gstar greal_trans grewrite_rdistinct_aux grewritess_concat inf_sup_aci(5) rdistinct_concat_general rdistinct_set_equality set_append)
|
|
|
630 |
apply (metis append_assoc grewrites.intros(1) grewritess_concat gstar_rdistinct_general)
|
|
|
631 |
by (simp add: gr_in_rstar grewrite.intros(2) grewrites_append)
|
|
478
|
632 |
|
|
|
633 |
|
|
479
|
634 |
|
|
478
|
635 |
|
|
|
636 |
lemma list_dlist_union:
|
|
|
637 |
shows "set rs \<subseteq> set rsb \<union> set (rdistinct rs (set rsb))"
|
|
|
638 |
by (metis rdistinct_concat_general rdistinct_set_equality set_append sup_ge2)
|
|
|
639 |
|
|
|
640 |
lemma subset_distinct_rewrite1:
|
|
|
641 |
shows "set1 \<subseteq> set rsb \<Longrightarrow> rsb @ rs \<leadsto>g* rsb @ (rdistinct rs set1)"
|
|
|
642 |
apply(induct rs arbitrary: rsb)
|
|
|
643 |
apply simp
|
|
|
644 |
apply(case_tac "a \<in> set1")
|
|
|
645 |
apply simp
|
|
|
646 |
|
|
|
647 |
using gmany_steps_later grewrite_variant1 apply blast
|
|
|
648 |
apply simp
|
|
|
649 |
apply(drule_tac x = "rsb @ [a]" in meta_spec)
|
|
|
650 |
apply(subgoal_tac "set1 \<subseteq> set (rsb @ [a])")
|
|
|
651 |
apply (simp only:)
|
|
|
652 |
apply(subgoal_tac "(rsb @ [a]) @ rdistinct rs set1 \<leadsto>g* (rsb @ [a]) @ rdistinct rs (insert a set1)")
|
|
|
653 |
apply (metis (no_types, opaque_lifting) append.assoc append_Cons append_Nil greal_trans)
|
|
|
654 |
apply (metis append.assoc empty_set grewrite_rdistinct_aux grewrites_append inf_sup_aci(5) insert_is_Un list.simps(15))
|
|
|
655 |
by auto
|
|
|
656 |
|
|
|
657 |
|
|
|
658 |
lemma subset_distinct_rewrite:
|
|
|
659 |
shows "set rsb' \<subseteq> set rsb \<Longrightarrow> rsb @ rs \<leadsto>g* rsb @ (rdistinct rs (set rsb'))"
|
|
|
660 |
by (simp add: subset_distinct_rewrite1)
|
|
|
661 |
|
|
|
662 |
|
|
|
663 |
|
|
|
664 |
lemma distinct_3list:
|
|
|
665 |
shows "rsb @ (rdistinct rs (set rsb)) @ rsa \<leadsto>g*
|
|
|
666 |
rsb @ (rdistinct rs (set rsb)) @ (rdistinct rsa (set rs))"
|
|
|
667 |
by (metis append.assoc list_dlist_union set_append subset_distinct_rewrite)
|
|
|
668 |
|
|
|
669 |
|
|
|
670 |
|
|
|
671 |
|
|
|
672 |
lemma grewrites_shape1:
|
|
|
673 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
674 |
rsb @
|
|
|
675 |
RALTS rs #
|
|
|
676 |
rdistinct rsa
|
|
|
677 |
(
|
|
|
678 |
(set rsb)) \<leadsto>g* rsb @
|
|
|
679 |
rdistinct rs (set rsb) @
|
|
|
680 |
rdistinct (rflts (rdistinct rsa ( (set rsb \<union> set rs)))) (set rs)"
|
|
|
681 |
|
|
|
682 |
|
|
|
683 |
apply (subgoal_tac " rsb @
|
|
|
684 |
RALTS rs #
|
|
|
685 |
rdistinct rsa
|
|
|
686 |
(
|
|
|
687 |
(set rsb)) \<leadsto>g* rsb @
|
|
|
688 |
rs @
|
|
|
689 |
rdistinct rsa
|
|
|
690 |
(
|
|
|
691 |
(set rsb)) ")
|
|
|
692 |
prefer 2
|
|
|
693 |
using gr_in_rstar grewrite.intros(2) grewrites_append apply presburger
|
|
|
694 |
apply(subgoal_tac "rsb @ rs @ rdistinct rsa ( (set rsb)) \<leadsto>g* rsb @
|
|
|
695 |
(rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb)))")
|
|
|
696 |
prefer 2
|
|
|
697 |
apply (metis append_assoc grewrites.intros(1) grewritess_concat gstar_rdistinct_general)
|
|
|
698 |
apply(subgoal_tac " rsb @ rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb))
|
|
|
699 |
\<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb) \<union> (set rs))")
|
|
|
700 |
prefer 2
|
|
|
701 |
apply (smt (verit, best) append.assoc append_assoc boolean_algebra_cancel.sup2 grewrite_rdistinct_aux inf_sup_aci(5) insert_is_Un rdistinct_concat_general rdistinct_set_equality set_append sup.commute sup.right_idem sup_commute)
|
|
|
702 |
apply(subgoal_tac "rdistinct rsa ( (set rsb) \<union> set rs) \<leadsto>g*
|
|
|
703 |
rflts (rdistinct rsa ( (set rsb) \<union> set rs))")
|
|
|
704 |
apply(subgoal_tac "rsb @ (rdistinct rs (set rsb)) @ rflts (rdistinct rsa ( (set rsb) \<union> set rs)) \<leadsto>g*
|
|
|
705 |
rsb @ (rdistinct rs (set rsb)) @ (rdistinct (rflts (rdistinct rsa ( (set rsb) \<union> set rs))) (set rs))")
|
|
|
706 |
apply (smt (verit, ccfv_SIG) Un_insert_left greal_trans grewrites_append)
|
|
|
707 |
using distinct_3list apply presburger
|
|
|
708 |
using flts_gstar apply blast
|
|
|
709 |
done
|
|
|
710 |
|
|
|
711 |
lemma r_finite1:
|
|
|
712 |
shows "r = RALTS (r # rs) = False"
|
|
|
713 |
apply(induct r)
|
|
|
714 |
apply simp+
|
|
|
715 |
apply (metis list.set_intros(1))
|
|
|
716 |
by blast
|
|
|
717 |
|
|
|
718 |
|
|
|
719 |
|
|
|
720 |
lemma grewrite_singleton:
|
|
|
721 |
shows "[r] \<leadsto>g r # rs \<Longrightarrow> rs = []"
|
|
|
722 |
apply (induct "[r]" "r # rs" rule: grewrite.induct)
|
|
|
723 |
apply simp
|
|
|
724 |
apply (metis r_finite1)
|
|
|
725 |
using grewrite.simps apply blast
|
|
|
726 |
by simp
|
|
|
727 |
|
|
|
728 |
lemma impossible_grewrite1:
|
|
|
729 |
shows "\<not>( [RONE] \<leadsto>g [])"
|
|
|
730 |
using grewrite.cases by fastforce
|
|
|
731 |
|
|
|
732 |
|
|
|
733 |
lemma impossible_grewrite2:
|
|
|
734 |
shows "\<not> ([RALTS rs] \<leadsto>g (RALTS rs) # a # rs)"
|
|
|
735 |
using grewrite_singleton by blast
|
|
|
736 |
|
|
|
737 |
|
|
|
738 |
|
|
|
739 |
|
|
479
|
740 |
lemma wront_sublist_grewrites:
|
|
478
|
741 |
shows "rs1 @ rs2 \<leadsto>g* rs1 @ rs3 \<Longrightarrow> rs2 \<leadsto>g* rs3"
|
|
|
742 |
apply(induct rs1 arbitrary: rs2 rs3 rule: rev_induct)
|
|
|
743 |
apply simp
|
|
|
744 |
apply(drule_tac x = "[x] @ rs2" in meta_spec)
|
|
|
745 |
apply(drule_tac x = "[x] @ rs3" in meta_spec)
|
|
|
746 |
apply(simp)
|
|
|
747 |
|
|
|
748 |
oops
|
|
|
749 |
|
|
|
750 |
|
|
|
751 |
|
|
|
752 |
|
|
479
|
753 |
lemma concat_rdistinct_equality1:
|
|
|
754 |
shows "rdistinct (rs @ rsa) rset = rdistinct rs rset @ rdistinct rsa (rset \<union> (set rs))"
|
|
|
755 |
apply(induct rs arbitrary: rsa rset)
|
|
478
|
756 |
apply simp
|
|
479
|
757 |
apply(case_tac "a \<in> rset")
|
|
|
758 |
apply simp
|
|
|
759 |
apply (simp add: insert_absorb)
|
|
|
760 |
by auto
|
|
478
|
761 |
|
|
480
|
762 |
lemma middle_grewrites:
|
|
|
763 |
"rs1 \<leadsto>g* rs2 \<Longrightarrow> rsa @ rs1 @ rsb \<leadsto>g* rsa @ rs2 @ rsb "
|
|
|
764 |
by (simp add: grewritess_concat)
|
|
|
765 |
|
|
|
766 |
lemma rdistinct_removes_all:
|
|
|
767 |
shows "set rs \<subseteq> rset \<Longrightarrow> rdistinct rs rset = []"
|
|
|
768 |
|
|
|
769 |
by (metis append.right_neutral rdistinct.simps(1) rdistinct_concat)
|
|
478
|
770 |
|
|
479
|
771 |
lemma ends_removal:
|
|
|
772 |
shows " rsb @ rdistinct rs (set rsb) @ RALTS rs # rsc \<leadsto>g* rsb @ rdistinct rs (set rsb) @ rsc"
|
|
480
|
773 |
apply(subgoal_tac "rsb @ rdistinct rs (set rsb) @ RALTS rs # rsc \<leadsto>g*
|
|
|
774 |
rsb @ rdistinct rs (set rsb) @ rs @ rsc")
|
|
|
775 |
apply(subgoal_tac "rsb @ rdistinct rs (set rsb) @ rs @ rsc \<leadsto>g*
|
|
|
776 |
rsb @ rdistinct rs (set rsb) @ rdistinct rs (set (rsb @ rdistinct rs (set rsb))) @ rsc")
|
|
|
777 |
apply (metis (full_types) append_Nil2 append_eq_appendI greal_trans list_dlist_union rdistinct_removes_all set_append)
|
|
|
778 |
apply (metis append.assoc append_Nil gstar_rdistinct_general middle_grewrites)
|
|
|
779 |
using gr_in_rstar grewrite.intros(2) grewrites_append by presburger
|
|
478
|
780 |
|
|
479
|
781 |
lemma grewrites_rev_append:
|
|
|
782 |
shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rs1 @ [x] \<leadsto>g* rs2 @ [x]"
|
|
|
783 |
using grewritess_concat by auto
|
|
478
|
784 |
|
|
479
|
785 |
lemma grewrites_inclusion:
|
|
|
786 |
shows "set rs \<subseteq> set rs1 \<Longrightarrow> rs1 @ rs \<leadsto>g* rs1"
|
|
|
787 |
apply(induct rs arbitrary: rs1)
|
|
|
788 |
apply simp
|
|
|
789 |
by (meson gmany_steps_later grewrite_variant1 list.set_intros(1) set_subset_Cons subset_code(1))
|
|
478
|
790 |
|
|
479
|
791 |
lemma distinct_keeps_last:
|
|
|
792 |
shows "\<lbrakk>x \<notin> rset; x \<notin> set xs \<rbrakk> \<Longrightarrow> rdistinct (xs @ [x]) rset = rdistinct xs rset @ [x]"
|
|
|
793 |
by (simp add: concat_rdistinct_equality1)
|
|
478
|
794 |
|
|
480
|
795 |
lemma grewrites_shape2_aux:
|
|
478
|
796 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
797 |
rsb @
|
|
|
798 |
rdistinct (rs @ rsa)
|
|
|
799 |
(set rsb) \<leadsto>g* rsb @
|
|
|
800 |
rdistinct rs (set rsb) @
|
|
479
|
801 |
rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
|
|
|
802 |
apply(subgoal_tac " rdistinct (rs @ rsa) (set rsb) = rdistinct rs (set rsb) @ rdistinct rsa (set rs \<union> set rsb)")
|
|
|
803 |
apply (simp only:)
|
|
|
804 |
prefer 2
|
|
|
805 |
apply (simp add: Un_commute concat_rdistinct_equality1)
|
|
|
806 |
apply(induct rsa arbitrary: rs rsb rule: rev_induct)
|
|
|
807 |
apply simp
|
|
|
808 |
apply(case_tac "x \<in> set rs")
|
|
|
809 |
apply (simp add: distinct_removes_middle3)
|
|
|
810 |
apply(case_tac "x = RALTS rs")
|
|
|
811 |
apply simp
|
|
|
812 |
apply(case_tac "x \<in> set rsb")
|
|
|
813 |
apply simp
|
|
|
814 |
apply (simp add: concat_rdistinct_equality1)
|
|
|
815 |
apply (simp add: concat_rdistinct_equality1)
|
|
|
816 |
apply simp
|
|
|
817 |
apply(drule_tac x = "rs " in meta_spec)
|
|
|
818 |
apply(drule_tac x = rsb in meta_spec)
|
|
|
819 |
apply simp
|
|
|
820 |
apply(subgoal_tac " rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) \<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct xs (insert (RALTS rs) (set rs \<union> set rsb))")
|
|
|
821 |
prefer 2
|
|
|
822 |
apply (simp add: concat_rdistinct_equality1)
|
|
|
823 |
apply(case_tac "x \<in> set xs")
|
|
|
824 |
apply simp
|
|
|
825 |
apply (simp add: distinct_removes_last2)
|
|
|
826 |
apply(case_tac "x \<in> set rsb")
|
|
|
827 |
apply (smt (verit, ccfv_threshold) Un_iff append.right_neutral concat_rdistinct_equality1 insert_is_Un rdistinct.simps(2))
|
|
|
828 |
apply(subgoal_tac "rsb @ rdistinct rs (set rsb) @ rdistinct (xs @ [x]) (set rs \<union> set rsb) = rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) @ [x]")
|
|
|
829 |
apply(simp only:)
|
|
|
830 |
apply(case_tac "x = RALTS rs")
|
|
|
831 |
apply(subgoal_tac "rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) @ [x] \<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) @ rs")
|
|
|
832 |
apply(subgoal_tac "rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) @ rs \<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb) ")
|
|
|
833 |
apply (smt (verit, ccfv_SIG) Un_insert_left append.right_neutral concat_rdistinct_equality1 greal_trans insert_iff rdistinct.simps(2))
|
|
|
834 |
apply(subgoal_tac "set rs \<subseteq> set ( rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb))")
|
|
|
835 |
apply (metis append.assoc grewrites_inclusion)
|
|
|
836 |
apply (metis Un_upper1 append.assoc dual_order.trans list_dlist_union set_append)
|
|
|
837 |
apply (metis append_Nil2 gr_in_rstar grewrite.intros(2) grewrite_append)
|
|
|
838 |
apply(subgoal_tac " rsb @ rdistinct rs (set rsb) @ rdistinct (xs @ [x]) (insert (RALTS rs) (set rs \<union> set rsb)) = rsb @ rdistinct rs (set rsb) @ rdistinct (xs) (insert (RALTS rs) (set rs \<union> set rsb)) @ [x]")
|
|
|
839 |
apply(simp only:)
|
|
|
840 |
apply (metis append.assoc grewrites_rev_append)
|
|
|
841 |
apply (simp add: insert_absorb)
|
|
|
842 |
apply (simp add: distinct_keeps_last)+
|
|
|
843 |
done
|
|
478
|
844 |
|
|
480
|
845 |
lemma grewrites_shape2:
|
|
|
846 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
847 |
rsb @
|
|
|
848 |
rdistinct (rs @ rsa)
|
|
|
849 |
(set rsb) \<leadsto>g* rflts rsb @
|
|
|
850 |
rdistinct rs (set rsb) @
|
|
|
851 |
rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
|
|
|
852 |
apply (meson flts_gstar greal_trans grewrites.simps grewrites_shape2_aux grewritess_concat)
|
|
|
853 |
done
|
|
|
854 |
|
|
479
|
855 |
lemma rdistinct_add_acc:
|
|
|
856 |
shows "rset2 \<subseteq> set rsb \<Longrightarrow> rsb @ rdistinct rs rset \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)"
|
|
|
857 |
apply(induct rs arbitrary: rsb rset rset2)
|
|
|
858 |
apply simp
|
|
|
859 |
apply (case_tac "a \<in> rset")
|
|
|
860 |
apply simp
|
|
|
861 |
apply(case_tac "a \<in> rset2")
|
|
480
|
862 |
apply simp
|
|
|
863 |
apply (simp add: more_distinct1)
|
|
479
|
864 |
apply simp
|
|
|
865 |
apply(drule_tac x = "rsb @ [a]" in meta_spec)
|
|
|
866 |
by (metis Un_insert_left append.assoc append_Cons append_Nil set_append sup.coboundedI1)
|
|
|
867 |
|
|
478
|
868 |
|
|
479
|
869 |
lemma frewrite_fun1:
|
|
|
870 |
shows " RALTS rs \<in> set rsb \<Longrightarrow>
|
|
|
871 |
rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb \<union> set rs)"
|
|
|
872 |
apply(subgoal_tac "rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb)")
|
|
|
873 |
apply(subgoal_tac " set rs \<subseteq> set (rflts rsb)")
|
|
|
874 |
prefer 2
|
|
|
875 |
using spilled_alts_contained apply blast
|
|
|
876 |
apply(subgoal_tac "rflts rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb \<union> set rs)")
|
|
|
877 |
using greal_trans apply blast
|
|
|
878 |
using rdistinct_add_acc apply presburger
|
|
|
879 |
using flts_gstar grewritess_concat by auto
|
|
|
880 |
|
|
478
|
881 |
|
|
|
882 |
|
|
|
883 |
|
|
476
|
884 |
|
|
|
885 |
lemma frewrite_rd_grewrites:
|
|
|
886 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow>
|
|
478
|
887 |
\<exists>rs3. (rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3) "
|
|
|
888 |
apply(induct rs1 rs2 arbitrary: rs rule: frewrite.induct)
|
|
|
889 |
apply(rule_tac x = "rsa @ (rdistinct rs ({RZERO} \<union> set rsa))" in exI)
|
|
476
|
890 |
apply(rule conjI)
|
|
478
|
891 |
apply(case_tac "RZERO \<in> set rsa")
|
|
|
892 |
apply simp+
|
|
|
893 |
using gstar0 apply fastforce
|
|
|
894 |
apply (simp add: gr_in_rstar grewrite.intros(1) grewrites_append)
|
|
|
895 |
apply (simp add: gstar0)
|
|
|
896 |
prefer 2
|
|
|
897 |
apply(case_tac "r \<in> set rs")
|
|
|
898 |
apply simp
|
|
|
899 |
apply(drule_tac x = "rs @ [r]" in meta_spec)
|
|
|
900 |
apply(erule exE)
|
|
|
901 |
apply(rule_tac x = "rs3" in exI)
|
|
|
902 |
apply simp
|
|
|
903 |
apply(case_tac "RALTS rs \<in> set rsb")
|
|
|
904 |
apply simp
|
|
479
|
905 |
apply(rule_tac x = "rflts rsb @ rdistinct rsa (set rsb \<union> set rs)" in exI)
|
|
478
|
906 |
apply(rule conjI)
|
|
479
|
907 |
using frewrite_fun1 apply force
|
|
|
908 |
apply (metis frewrite_fun1 rdistinct_concat sup_ge2)
|
|
478
|
909 |
apply(simp)
|
|
|
910 |
apply(rule_tac x =
|
|
480
|
911 |
"rflts rsb @
|
|
479
|
912 |
rdistinct rs (set rsb) @
|
|
|
913 |
rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})" in exI)
|
|
478
|
914 |
apply(rule conjI)
|
|
|
915 |
prefer 2
|
|
|
916 |
using grewrites_shape2 apply force
|
|
480
|
917 |
using frewrite_rd_grewrites_aux by blast
|
|
478
|
918 |
|
|
473
|
919 |
|
|
476
|
920 |
|
|
|
921 |
lemma frewrites_rd_grewrites:
|
|
|
922 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
|
|
478
|
923 |
rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
476
|
924 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
925 |
apply simp
|
|
479
|
926 |
using frewrite_simpeq by presburger
|
|
476
|
927 |
|
|
|
928 |
lemma frewrite_simpeq2:
|
|
|
929 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
|
|
478
|
930 |
apply(subgoal_tac "\<exists> rs3. (rdistinct rs1 {} \<leadsto>g* rs3) \<and> (rdistinct rs2 {} \<leadsto>g* rs3)")
|
|
|
931 |
using grewrites_equal_rsimp apply fastforce
|
|
479
|
932 |
by (metis append_self_conv2 frewrite_rd_grewrites list.set(1))
|
|
|
933 |
|
|
|
934 |
|
|
|
935 |
|
|
478
|
936 |
|
|
473
|
937 |
(*a more refined notion of \<leadsto>* is needed,
|
|
|
938 |
this lemma fails when rs1 contains some RALTS rs where elements
|
|
|
939 |
of rs appear in later parts of rs1, which will be picked up by rs2
|
|
|
940 |
and deduplicated*)
|
|
476
|
941 |
lemma frewrites_simpeq:
|
|
473
|
942 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
|
|
476
|
943 |
rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS ( rdistinct rs2 {})) "
|
|
|
944 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
945 |
apply simp
|
|
478
|
946 |
using frewrite_simpeq2 by presburger
|
|
476
|
947 |
|
|
473
|
948 |
|
|
|
949 |
lemma frewrite_single_step:
|
|
|
950 |
shows " rs2 \<leadsto>f rs3 \<Longrightarrow> rsimp (RALTS rs2) = rsimp (RALTS rs3)"
|
|
|
951 |
apply(induct rs2 rs3 rule: frewrite.induct)
|
|
|
952 |
apply simp
|
|
|
953 |
using simp_flatten apply blast
|
|
|
954 |
by (metis (no_types, opaque_lifting) list.simps(9) rsimp.simps(2) simp_flatten2)
|
|
|
955 |
|
|
|
956 |
lemma frewrites_equivalent_simp:
|
|
|
957 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
|
958 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
959 |
apply simp
|
|
|
960 |
using frewrite_single_step by presburger
|
|
|
961 |
|
|
476
|
962 |
lemma grewrite_simpalts:
|
|
|
963 |
shows " rs2 \<leadsto>g rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
|
|
|
964 |
apply(induct rs2 rs3 rule : grewrite.induct)
|
|
|
965 |
using identity_wwo0 apply presburger
|
|
|
966 |
apply (metis frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_flatten)
|
|
|
967 |
apply (smt (verit, ccfv_SIG) gmany_steps_later grewrites.intros(1) grewrites_cons grewrites_equal_rsimp identity_wwo0 rsimp_ALTs.simps(3))
|
|
|
968 |
apply simp
|
|
|
969 |
apply(subst rsimp_alts_equal)
|
|
|
970 |
apply(case_tac "rsa = [] \<and> rsb = [] \<and> rsc = []")
|
|
|
971 |
apply(subgoal_tac "rsa @ a # rsb @ rsc = [a]")
|
|
|
972 |
apply (simp only:)
|
|
|
973 |
apply (metis append_Nil frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_removes_duplicate1(2))
|
|
|
974 |
apply simp
|
|
|
975 |
by (smt (verit, best) append.assoc append_Cons frewrite.intros(1) frewrite_single_step identity_wwo0 in_set_conv_decomp rsimp_ALTs.simps(3) simp_removes_duplicate3)
|
|
|
976 |
|
|
|
977 |
|
|
|
978 |
lemma grewrites_simpalts:
|
|
|
979 |
shows " rs2 \<leadsto>g* rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
|
|
|
980 |
apply(induct rs2 rs3 rule: grewrites.induct)
|
|
|
981 |
apply simp
|
|
|
982 |
using grewrite_simpalts by presburger
|
|
473
|
983 |
|
|
476
|
984 |
|
|
473
|
985 |
|
|
|
986 |
|
|
|
987 |
|
|
467
|
988 |
lemma simp_der_flts:
|
|
471
|
989 |
shows "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {})) =
|
|
|
990 |
rsimp (RALTS (rdistinct (rflts (map (rder x) rs)) {}))"
|
|
473
|
991 |
apply(subgoal_tac "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)")
|
|
478
|
992 |
using frewrites_simpeq apply presburger
|
|
|
993 |
using early_late_der_frewrites by auto
|
|
|
994 |
|
|
|
995 |
|
|
476
|
996 |
|
|
|
997 |
|
|
|
998 |
|
|
|
999 |
|
|
|
1000 |
|
|
|
1001 |
|
|
|
1002 |
|
|
|
1003 |
|
|
|
1004 |
lemma simp_der_pierce_flts_prelim:
|
|
|
1005 |
shows "rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts rs)) {}))
|
|
|
1006 |
= rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {}))"
|
|
478
|
1007 |
by (metis append.right_neutral grewrite.intros(2) grewrite_simpalts rsimp_ALTs.simps(2) simp_der_flts)
|
|
467
|
1008 |
|
|
|
1009 |
|
|
465
|
1010 |
lemma simp_der_pierce_flts:
|
|
471
|
1011 |
shows " rsimp (
|
|
|
1012 |
rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})
|
|
|
1013 |
) =
|
|
|
1014 |
rsimp (
|
|
|
1015 |
rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})
|
|
|
1016 |
)"
|
|
476
|
1017 |
using simp_der_pierce_flts_prelim by presburger
|
|
465
|
1018 |
|
|
453
|
1019 |
|
|
|
1020 |
|
|
|
1021 |
lemma simp_more_distinct:
|
|
465
|
1022 |
shows "rsimp (rsimp_ALTs (rsa @ rs)) = rsimp (rsimp_ALTs (rsa @ (rdistinct rs (set rsa)))) "
|
|
480
|
1023 |
oops
|
|
465
|
1024 |
|
|
453
|
1025 |
|
|
|
1026 |
lemma non_empty_list:
|
|
|
1027 |
shows "a \<in> set as \<Longrightarrow> as \<noteq> []"
|
|
|
1028 |
by (metis empty_iff empty_set)
|
|
|
1029 |
|
|
456
|
1030 |
lemma distinct_comp:
|
|
|
1031 |
shows "rdistinct (rs1@rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
|
|
|
1032 |
apply(induct rs2 arbitrary: rs1)
|
|
|
1033 |
apply simp
|
|
|
1034 |
apply(subgoal_tac "rs1 @ a # rs2 = (rs1 @ [a]) @ rs2")
|
|
|
1035 |
apply(simp only:)
|
|
|
1036 |
apply(case_tac "a \<in> set rs1")
|
|
|
1037 |
apply simp
|
|
|
1038 |
oops
|
|
453
|
1039 |
|
|
456
|
1040 |
lemma instantiate1:
|
|
|
1041 |
shows "\<lbrakk>\<And>ab rset1. rdistinct (ab # as) rset1 = rdistinct (ab # as @ [ab]) rset1\<rbrakk> \<Longrightarrow>
|
|
|
1042 |
rdistinct (aa # as) rset = rdistinct (aa # as @ [aa]) rset"
|
|
|
1043 |
apply(drule_tac x = "aa" in meta_spec)
|
|
|
1044 |
apply(drule_tac x = "rset" in meta_spec)
|
|
453
|
1045 |
apply simp
|
|
456
|
1046 |
done
|
|
|
1047 |
|
|
|
1048 |
|
|
|
1049 |
lemma not_head_elem:
|
|
|
1050 |
shows " \<lbrakk>aa \<in> set (a # as); aa \<notin> (set as)\<rbrakk> \<Longrightarrow> a = aa"
|
|
|
1051 |
|
|
|
1052 |
by fastforce
|
|
|
1053 |
|
|
|
1054 |
(*
|
|
|
1055 |
apply simp
|
|
|
1056 |
apply (metis append_Cons)
|
|
|
1057 |
apply(case_tac "ab \<in> rset1")
|
|
|
1058 |
apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
|
|
|
1059 |
apply(subgoal_tac "rdistinct (ab # (aa # as) @ [ab]) rset1 =
|
|
|
1060 |
ab # (rdistinct ((aa # as) @ [ab]) (insert ab rset1))")
|
|
|
1061 |
apply(simp only:)
|
|
|
1062 |
apply(subgoal_tac "rdistinct (ab # aa # as) rset1 = ab # (rdistinct (aa # as) (insert ab rset1))")
|
|
|
1063 |
apply(simp only:)
|
|
|
1064 |
apply(subgoal_tac "rdistinct ((aa # as) @ [ab]) (insert ab rset1) = rdistinct (aa # as) (insert ab rset1)")
|
|
|
1065 |
apply blast
|
|
|
1066 |
*)
|
|
|
1067 |
|
|
453
|
1068 |
|
|
|
1069 |
lemma flts_identity1:
|
|
|
1070 |
shows "rflts (rs @ [RONE]) = rflts rs @ [RONE] "
|
|
|
1071 |
apply(induct rs)
|
|
|
1072 |
apply simp+
|
|
|
1073 |
apply(case_tac a)
|
|
|
1074 |
apply simp
|
|
|
1075 |
apply simp+
|
|
|
1076 |
done
|
|
|
1077 |
|
|
|
1078 |
lemma flts_identity10:
|
|
|
1079 |
shows " rflts (rs @ [RCHAR c]) = rflts rs @ [RCHAR c]"
|
|
|
1080 |
apply(induct rs)
|
|
|
1081 |
apply simp+
|
|
|
1082 |
apply(case_tac a)
|
|
|
1083 |
apply simp+
|
|
|
1084 |
done
|
|
|
1085 |
|
|
|
1086 |
lemma flts_identity11:
|
|
|
1087 |
shows " rflts (rs @ [RSEQ r1 r2]) = rflts rs @ [RSEQ r1 r2]"
|
|
|
1088 |
apply(induct rs)
|
|
|
1089 |
apply simp+
|
|
|
1090 |
apply(case_tac a)
|
|
|
1091 |
apply simp+
|
|
|
1092 |
done
|
|
|
1093 |
|
|
|
1094 |
lemma flts_identity12:
|
|
|
1095 |
shows " rflts (rs @ [RSTAR r0]) = rflts rs @ [RSTAR r0]"
|
|
|
1096 |
apply(induct rs)
|
|
|
1097 |
apply simp+
|
|
|
1098 |
apply(case_tac a)
|
|
|
1099 |
apply simp+
|
|
|
1100 |
done
|
|
|
1101 |
|
|
|
1102 |
lemma flts_identity2:
|
|
|
1103 |
shows "a \<noteq> RZERO \<and> (\<forall>rs. a \<noteq> RALTS rs) \<Longrightarrow> rflts (rs @ [a]) = rflts rs @ [a]"
|
|
|
1104 |
apply(case_tac a)
|
|
|
1105 |
apply simp
|
|
|
1106 |
using flts_identity1 apply auto[1]
|
|
|
1107 |
using flts_identity10 apply blast
|
|
|
1108 |
using flts_identity11 apply auto[1]
|
|
|
1109 |
apply blast
|
|
|
1110 |
using flts_identity12 by presburger
|
|
456
|
1111 |
|
|
|
1112 |
lemma flts_identity3:
|
|
|
1113 |
shows "a = RZERO \<Longrightarrow> rflts (rs @ [a]) = rflts rs"
|
|
|
1114 |
apply simp
|
|
|
1115 |
apply(induct rs)
|
|
|
1116 |
apply simp+
|
|
|
1117 |
apply(case_tac aa)
|
|
|
1118 |
apply simp+
|
|
|
1119 |
done
|
|
|
1120 |
|
|
|
1121 |
lemma distinct_removes_last3:
|
|
465
|
1122 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
456
|
1123 |
\<Longrightarrow> rdistinct as {} = rdistinct (as @ [a]) {}"
|
|
465
|
1124 |
by (simp add: distinct_removes_last2)
|
|
456
|
1125 |
|
|
|
1126 |
lemma set_inclusion_with_flts1:
|
|
|
1127 |
shows " \<lbrakk>RONE \<in> set rs\<rbrakk> \<Longrightarrow> RONE \<in> set (rflts rs)"
|
|
|
1128 |
apply(induct rs)
|
|
|
1129 |
apply simp
|
|
|
1130 |
apply(case_tac " RONE \<in> set rs")
|
|
|
1131 |
apply simp
|
|
|
1132 |
apply (metis Un_upper2 insert_absorb insert_subset list.set_intros(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append)
|
|
|
1133 |
apply(case_tac "RONE = a")
|
|
|
1134 |
apply simp
|
|
|
1135 |
apply simp
|
|
|
1136 |
done
|
|
|
1137 |
|
|
|
1138 |
lemma set_inclusion_with_flts10:
|
|
|
1139 |
shows " \<lbrakk>RCHAR x \<in> set rs\<rbrakk> \<Longrightarrow> RCHAR x \<in> set (rflts rs)"
|
|
|
1140 |
apply(induct rs)
|
|
|
1141 |
apply simp
|
|
|
1142 |
apply(case_tac " RCHAR x \<in> set rs")
|
|
|
1143 |
apply simp
|
|
|
1144 |
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
|
|
|
1145 |
apply(case_tac "RCHAR x = a")
|
|
|
1146 |
apply simp
|
|
|
1147 |
apply fastforce
|
|
|
1148 |
apply simp
|
|
|
1149 |
done
|
|
|
1150 |
|
|
|
1151 |
lemma set_inclusion_with_flts11:
|
|
|
1152 |
shows " \<lbrakk>RSEQ r1 r2 \<in> set rs\<rbrakk> \<Longrightarrow> RSEQ r1 r2 \<in> set (rflts rs)"
|
|
|
1153 |
apply(induct rs)
|
|
|
1154 |
apply simp
|
|
|
1155 |
apply(case_tac " RSEQ r1 r2 \<in> set rs")
|
|
|
1156 |
apply simp
|
|
|
1157 |
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
|
|
|
1158 |
apply(case_tac "RSEQ r1 r2 = a")
|
|
|
1159 |
apply simp
|
|
|
1160 |
apply fastforce
|
|
|
1161 |
apply simp
|
|
|
1162 |
done
|
|
|
1163 |
|
|
|
1164 |
|
|
|
1165 |
lemma set_inclusion_with_flts:
|
|
|
1166 |
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))"
|
|
|
1167 |
by (simp add: set_inclusion_with_flts1)
|
|
453
|
1168 |
|
|
480
|
1169 |
lemma can_spill_lst:"\<And>x5. \<lbrakk>a \<in> set as; rsimp a \<in> set (map rsimp as); rsimp a = RALTS x5\<rbrakk>
|
|
456
|
1170 |
\<Longrightarrow> rsimp_ALTs (rdistinct (rflts (map rsimp as @ [rsimp a])) {}) =
|
|
|
1171 |
rsimp_ALTs (rdistinct (rflts (map rsimp as @ x5)) {})"
|
|
480
|
1172 |
|
|
|
1173 |
using flts_append rflts_spills_last rsimp_inner_idem4 by presburger
|
|
|
1174 |
|
|
|
1175 |
|
|
|
1176 |
|
|
|
1177 |
|
|
|
1178 |
|
|
|
1179 |
lemma common_rewrites_equal:
|
|
|
1180 |
shows "(rs1 \<leadsto>g* rs3) \<and> (rs2 \<leadsto>g* rs3) \<Longrightarrow> rsimp (rsimp_ALTs rs1 ) = rsimp (rsimp_ALTs rs2)"
|
|
|
1181 |
using grewrites_simpalts by force
|
|
|
1182 |
|
|
|
1183 |
|
|
|
1184 |
lemma basic_regex_property1:
|
|
|
1185 |
shows "rnullable r \<Longrightarrow> rsimp r \<noteq> RZERO"
|
|
|
1186 |
sorry
|
|
|
1187 |
|
|
|
1188 |
lemma basic_rsimp_SEQ_property1:
|
|
|
1189 |
shows "rsimp_SEQ RONE r = r"
|
|
|
1190 |
by (simp add: idiot)
|
|
|
1191 |
|
|
|
1192 |
lemma basic_rsimp_SEQ_property3:
|
|
|
1193 |
shows "rsimp_SEQ r RZERO = RZERO"
|
|
|
1194 |
using rsimp_SEQ.elims by blast
|
|
|
1195 |
|
|
|
1196 |
thm rsimp_SEQ.elims
|
|
|
1197 |
|
|
|
1198 |
|
|
|
1199 |
lemma basic_rsimp_SEQ_property2:
|
|
|
1200 |
shows "\<lbrakk>r1 \<noteq> RZERO ; r1 \<noteq> RONE; r2 \<noteq> RZERO\<rbrakk> \<Longrightarrow>rsimp_SEQ r1 r2 = RSEQ r1 r2"
|
|
|
1201 |
apply(case_tac r1)
|
|
|
1202 |
apply simp+
|
|
|
1203 |
apply (simp add: idiot2)
|
|
|
1204 |
using idiot2 apply blast
|
|
|
1205 |
using idiot2 apply auto[1]
|
|
|
1206 |
using idiot2 by blast
|
|
|
1207 |
|
|
|
1208 |
|
|
|
1209 |
(*
|
|
|
1210 |
lemma rderssimp_same_rewrites_rder_induct1:
|
|
|
1211 |
shows "\<lbrakk> ([rder x (rsimp r1)] \<leadsto>g* rs1) \<and> ([rder x r1] \<leadsto>g* rs1) ;
|
|
|
1212 |
([rder x (rsimp r2)] \<leadsto>g* rs2) \<and> ([rder x r2] \<leadsto>g* rs2) \<rbrakk> \<Longrightarrow>
|
|
|
1213 |
\<exists>rs3. ([rder x (rsimp (RSEQ r1 r2))] \<leadsto>g* rs3) \<and> ([rder x (RSEQ r1 r2)] \<leadsto>g* rs3) "
|
|
|
1214 |
sorry
|
|
|
1215 |
|
|
|
1216 |
lemma rderssimp_same_rewrites_rder_induct2:
|
|
|
1217 |
shows "\<lbrakk> ([rder x (rsimp r1)] \<leadsto>g* rs1) \<and> ([rder x r1] \<leadsto>g* rs1) \<rbrakk> \<Longrightarrow>
|
|
|
1218 |
\<exists>rs3. ([rder x (rsimp (RSTAR r1))] \<leadsto>g* rs3) \<and> ([rder x (RSTAR r1)] \<leadsto>g* rs3) "
|
|
|
1219 |
sorry
|
|
|
1220 |
|
|
|
1221 |
lemma rderssimp_same_rewrites_rder_induct3:
|
|
|
1222 |
shows "\<lbrakk> ([rder x (rsimp r1)] \<leadsto>g* rs1) \<and> ([rder x r1] \<leadsto>g* rs1) ;
|
|
|
1223 |
([rder x (rsimp r2)] \<leadsto>g* rs2) \<and> ([rder x r2] \<leadsto>g* rs2) \<rbrakk> \<Longrightarrow>
|
|
|
1224 |
\<exists>rs3. ([rder x (rsimp (RALT r1 r2))] \<leadsto>g* rs3) \<and> ([rder x (RALT r1 r2)] \<leadsto>g* rs3) "
|
|
|
1225 |
sorry
|
|
|
1226 |
|
|
|
1227 |
lemma rderssimp_same_rewrites_rder_induct4:
|
|
|
1228 |
shows "\<lbrakk>\<forall>r \<in> set rs. \<exists> rsa. ([rder x (rsimp r)] \<leadsto>g* rsa ) \<and> ([rder x r] \<leadsto>g* rsa) \<rbrakk> \<Longrightarrow>
|
|
|
1229 |
\<exists>rsb. ([rder x (rsimp (RALTS rs))] \<leadsto>g* rsb) \<and> ([rder x (RALTS rs)] \<leadsto>g* rsb) "
|
|
|
1230 |
sorry
|
|
|
1231 |
|
|
|
1232 |
lemma rderssimp_same_rewrites_rder_base1:
|
|
|
1233 |
shows "([rder x (rsimp RONE)] \<leadsto>g* [] ) \<and> ([rder x RONE] \<leadsto>g* [])"
|
|
|
1234 |
by (simp add: gr_in_rstar grewrite.intros(1))
|
|
|
1235 |
|
|
|
1236 |
lemma rderssimp_same_rewrites_rder_base2:
|
|
|
1237 |
shows " ([rder x (rsimp RZERO)] \<leadsto>g* [] ) \<and> ([rder x RZERO] \<leadsto>g* [])"
|
|
|
1238 |
using rderssimp_same_rewrites_rder_base1 by auto
|
|
|
1239 |
|
|
|
1240 |
lemma rderssimp_same_rewrites_rder_base3:
|
|
|
1241 |
shows " ([rder x (rsimp (RCHAR c))] \<leadsto>g* [] ) \<and> ([rder x (RCHAR c)] \<leadsto>g* [])"
|
|
|
1242 |
sorry
|
|
|
1243 |
*)
|
|
|
1244 |
|
|
|
1245 |
lemma inside_simp_seq_nullable:
|
|
|
1246 |
shows
|
|
|
1247 |
"\<And>r1 r2.
|
|
|
1248 |
\<lbrakk>rsimp (rder x (rsimp r1)) = rsimp (rder x r1); rsimp (rder x (rsimp r2)) = rsimp (rder x r2);
|
|
|
1249 |
rnullable r1\<rbrakk>
|
|
|
1250 |
\<Longrightarrow> rsimp (rder x (rsimp_SEQ (rsimp r1) (rsimp r2))) =
|
|
|
1251 |
rsimp_ALTs (rdistinct (rflts [rsimp_SEQ (rsimp (rder x r1)) (rsimp r2), rsimp (rder x r2)]) {})"
|
|
|
1252 |
apply(case_tac "rsimp r1 = RONE")
|
|
|
1253 |
apply(simp)
|
|
|
1254 |
apply(subst basic_rsimp_SEQ_property1)
|
|
|
1255 |
apply (simp add: idem_after_simp1)
|
|
|
1256 |
apply(case_tac "rsimp r1 = RZERO")
|
|
|
1257 |
|
|
|
1258 |
using basic_regex_property1 apply blast
|
|
|
1259 |
apply(case_tac "rsimp r2 = RZERO")
|
|
|
1260 |
|
|
|
1261 |
apply (simp add: basic_rsimp_SEQ_property3)
|
|
|
1262 |
apply(subst basic_rsimp_SEQ_property2)
|
|
|
1263 |
apply simp+
|
|
|
1264 |
apply(subgoal_tac "rnullable (rsimp r1)")
|
|
|
1265 |
apply simp
|
|
|
1266 |
using rsimp_idem apply presburger
|
|
456
|
1267 |
|
|
465
|
1268 |
sorry
|
|
|
1269 |
|
|
453
|
1270 |
|
|
480
|
1271 |
lemma inside_simp_removal:
|
|
|
1272 |
shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
|
|
|
1273 |
apply(induct r)
|
|
|
1274 |
apply simp+
|
|
|
1275 |
apply(case_tac "rnullable r1")
|
|
|
1276 |
apply simp
|
|
456
|
1277 |
|
|
480
|
1278 |
using inside_simp_seq_nullable apply blast
|
|
451
|
1279 |
sorry
|
|
|
1280 |
|
|
|
1281 |
|
|
480
|
1282 |
lemma rders_simp_same_simpders:
|
|
|
1283 |
shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
|
|
|
1284 |
apply(induct s rule: rev_induct)
|
|
453
|
1285 |
apply simp
|
|
480
|
1286 |
apply(case_tac "xs = []")
|
|
453
|
1287 |
apply simp
|
|
480
|
1288 |
apply(simp add: rders_append rders_simp_append)
|
|
|
1289 |
using inside_simp_removal by blast
|
|
453
|
1290 |
|
|
480
|
1291 |
|
|
453
|
1292 |
|
|
|
1293 |
|
|
451
|
1294 |
lemma distinct_der:
|
|
480
|
1295 |
shows "rsimp (rsimp_ALTs (map (rder x) (rdistinct rs {}))) =
|
|
|
1296 |
rsimp (rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
|
|
|
1297 |
by (metis grewrites_simpalts gstar_rdistinct inside_simp_removal rder_rsimp_ALTs_commute)
|
|
|
1298 |
|
|
451
|
1299 |
|
|
453
|
1300 |
|
|
|
1301 |
|
|
|
1302 |
|
|
|
1303 |
lemma rders_simp_lambda:
|
|
|
1304 |
shows " rsimp \<circ> rder x \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r (xs @ [x]))"
|
|
|
1305 |
using rders_simp_append by auto
|
|
451
|
1306 |
|
|
453
|
1307 |
lemma rders_simp_nonempty_simped:
|
|
|
1308 |
shows "xs \<noteq> [] \<Longrightarrow> rsimp \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r xs)"
|
|
|
1309 |
using rders_simp_same_simpders rsimp_idem by auto
|
|
|
1310 |
|
|
|
1311 |
lemma repeated_altssimp:
|
|
|
1312 |
shows "\<forall>r \<in> set rs. rsimp r = r \<Longrightarrow> rsimp (rsimp_ALTs (rdistinct (rflts rs) {})) =
|
|
|
1313 |
rsimp_ALTs (rdistinct (rflts rs) {})"
|
|
|
1314 |
by (metis map_idI rsimp.simps(2) rsimp_idem)
|
|
451
|
1315 |
|
|
465
|
1316 |
|
|
|
1317 |
lemma add0_isomorphic:
|
|
|
1318 |
shows "rsimp_ALTs (rdistinct (rflts [rsimp r, RZERO]) {}) = rsimp r"
|
|
|
1319 |
sorry
|
|
|
1320 |
|
|
|
1321 |
|
|
|
1322 |
lemma distinct_append_simp:
|
|
|
1323 |
shows " rsimp (rsimp_ALTs rs1) = rsimp (rsimp_ALTs rs2) \<Longrightarrow>
|
|
|
1324 |
rsimp (rsimp_ALTs (f a # rs1)) =
|
|
|
1325 |
rsimp (rsimp_ALTs (f a # rs2))"
|
|
|
1326 |
apply(case_tac rs1)
|
|
|
1327 |
apply simp
|
|
|
1328 |
apply(case_tac rs2)
|
|
|
1329 |
apply simp
|
|
|
1330 |
apply simp
|
|
|
1331 |
prefer 2
|
|
|
1332 |
apply(case_tac list)
|
|
|
1333 |
apply(case_tac rs2)
|
|
|
1334 |
apply simp
|
|
|
1335 |
using add0_isomorphic apply blast
|
|
|
1336 |
apply simp
|
|
467
|
1337 |
oops
|
|
465
|
1338 |
|
|
444
|
1339 |
lemma alts_closed_form: shows
|
|
|
1340 |
"rsimp (rders_simp (RALTS rs) s) =
|
|
|
1341 |
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
|
1342 |
apply(induct s rule: rev_induct)
|
|
|
1343 |
apply simp
|
|
|
1344 |
apply simp
|
|
|
1345 |
apply(subst rders_simp_append)
|
|
|
1346 |
apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) =
|
|
|
1347 |
rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
|
|
|
1348 |
prefer 2
|
|
|
1349 |
apply (metis inside_simp_removal rders_simp_one_char)
|
|
|
1350 |
apply(simp only: )
|
|
451
|
1351 |
apply(subst rders_simp_one_char)
|
|
|
1352 |
apply(subst rsimp_idem)
|
|
|
1353 |
apply(subgoal_tac "rsimp (rder x (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {}))) =
|
|
|
1354 |
rsimp ((rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))) ")
|
|
|
1355 |
prefer 2
|
|
|
1356 |
using rder_rsimp_ALTs_commute apply presburger
|
|
|
1357 |
apply(simp only:)
|
|
|
1358 |
apply(subgoal_tac "rsimp (rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))
|
|
|
1359 |
= rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
1360 |
prefer 2
|
|
|
1361 |
|
|
|
1362 |
using distinct_der apply presburger
|
|
|
1363 |
apply(simp only:)
|
|
453
|
1364 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
1365 |
rsimp (rsimp_ALTs (rdistinct ( (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)))) {}))")
|
|
|
1366 |
apply(simp only:)
|
|
|
1367 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
1368 |
rsimp (rsimp_ALTs (rdistinct (rflts ( (map (rsimp \<circ> (rder x) \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
1369 |
apply(simp only:)
|
|
|
1370 |
apply(subst rders_simp_lambda)
|
|
|
1371 |
apply(subst rders_simp_nonempty_simped)
|
|
|
1372 |
apply simp
|
|
|
1373 |
apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (xs @ [x])) rs). rsimp r = r")
|
|
|
1374 |
prefer 2
|
|
|
1375 |
apply (simp add: rders_simp_same_simpders rsimp_idem)
|
|
|
1376 |
apply(subst repeated_altssimp)
|
|
|
1377 |
apply simp
|
|
|
1378 |
apply fastforce
|
|
465
|
1379 |
apply (metis inside_simp_removal list.map_comp rder.simps(4) rsimp.simps(2) rsimp_idem)
|
|
476
|
1380 |
using simp_der_pierce_flts by blast
|
|
443
|
1381 |
|
|
444
|
1382 |
lemma alts_closed_form_variant: shows
|
|
|
1383 |
"s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s =
|
|
|
1384 |
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
478
|
1385 |
by (metis alts_closed_form comp_apply rders_simp_nonempty_simped)
|
|
|
1386 |
|
|
443
|
1387 |
|
|
|
1388 |
|
|
|
1389 |
|
|
444
|
1390 |
lemma star_closed_form:
|
|
|
1391 |
shows "rders_simp (RSTAR r0) (c#s) =
|
|
|
1392 |
rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
|
|
|
1393 |
apply(induct s)
|
|
|
1394 |
apply simp
|
|
|
1395 |
sorry
|
|
443
|
1396 |
|
|
|
1397 |
|
|
|
1398 |
|
|
|
1399 |
lemma seq_closed_form: shows
|
|
|
1400 |
"rsimp (rders_simp (RSEQ r1 r2) s) =
|
|
|
1401 |
rsimp ( RALTS ( (RSEQ (rders_simp r1 s) r2) #
|
|
445
|
1402 |
(map (rders_simp r2) (vsuf s r1))
|
|
443
|
1403 |
)
|
|
|
1404 |
)"
|
|
|
1405 |
apply(induct s)
|
|
|
1406 |
apply simp
|
|
|
1407 |
sorry
|
|
|
1408 |
|
|
|
1409 |
|
|
444
|
1410 |
lemma seq_closed_form_variant: shows
|
|
|
1411 |
"s \<noteq> [] \<Longrightarrow> (rders_simp (RSEQ r1 r2) s) =
|
|
|
1412 |
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
|
|
445
|
1413 |
apply(induct s rule: rev_induct)
|
|
|
1414 |
apply simp
|
|
|
1415 |
apply(subst rders_simp_append)
|
|
|
1416 |
apply(subst rders_simp_one_char)
|
|
|
1417 |
apply(subst rsimp_idem[symmetric])
|
|
|
1418 |
apply(subst rders_simp_one_char[symmetric])
|
|
|
1419 |
apply(subst rders_simp_append[symmetric])
|
|
|
1420 |
apply(insert seq_closed_form)
|
|
|
1421 |
apply(subgoal_tac "rsimp (rders_simp (RSEQ r1 r2) (xs @ [x]))
|
|
|
1422 |
= rsimp (RALTS (RSEQ (rders_simp r1 (xs @ [x])) r2 # map (rders_simp r2) (vsuf (xs @ [x]) r1)))")
|
|
|
1423 |
apply force
|
|
|
1424 |
by presburger
|
|
443
|
1425 |
|
|
476
|
1426 |
lemma simp_helps_der_pierce:
|
|
|
1427 |
shows " rsimp
|
|
|
1428 |
(rder x
|
|
|
1429 |
(rsimp_ALTs rs)) =
|
|
|
1430 |
rsimp
|
|
|
1431 |
(rsimp_ALTs
|
|
|
1432 |
(map (rder x )
|
|
|
1433 |
rs
|
|
|
1434 |
)
|
|
|
1435 |
)"
|
|
|
1436 |
sorry
|
|
|
1437 |
|
|
444
|
1438 |
end |