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505
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theory ClosedForms
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imports "BasicIdentities"
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begin
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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 simp_flatten_aux0:
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shows "rsimp (RALTS rs) = rsimp (RALTS (map rsimp rs))"
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by (metis append_Nil head_one_more_simp identity_wwo0 list.simps(8) rdistinct.simps(1) rflts.simps(1) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(3) simp_flatten spawn_simp_rsimpalts)
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inductive
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hrewrite:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ h\<leadsto> _" [99, 99] 99)
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where
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"RSEQ RZERO r2 h\<leadsto> RZERO"
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| "RSEQ r1 RZERO h\<leadsto> RZERO"
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| "RSEQ RONE r h\<leadsto> r"
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| "r1 h\<leadsto> r2 \<Longrightarrow> RSEQ r1 r3 h\<leadsto> RSEQ r2 r3"
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| "r3 h\<leadsto> r4 \<Longrightarrow> RSEQ r1 r3 h\<leadsto> RSEQ r1 r4"
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| "r h\<leadsto> r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) h\<leadsto> (RALTS (rs1 @ [r'] @ rs2))"
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(*context rule for eliminating 0, alts--corresponds to the recursive call flts r::rs = r::(flts rs)*)
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| "RALTS (rsa @ [RZERO] @ rsb) h\<leadsto> RALTS (rsa @ rsb)"
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| "RALTS (rsa @ [RALTS rs1] @ rsb) h\<leadsto> RALTS (rsa @ rs1 @ rsb)"
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| "RALTS [] h\<leadsto> RZERO"
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| "RALTS [r] h\<leadsto> r"
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| "a1 = a2 \<Longrightarrow> RALTS (rsa@[a1]@rsb@[a2]@rsc) h\<leadsto> RALTS (rsa @ [a1] @ rsb @ rsc)"
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inductive
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hrewrites:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ h\<leadsto>* _" [100, 100] 100)
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where
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rs1[intro, simp]:"r h\<leadsto>* r"
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| rs2[intro]: "\<lbrakk>r1 h\<leadsto>* r2; r2 h\<leadsto> r3\<rbrakk> \<Longrightarrow> r1 h\<leadsto>* r3"
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lemma hr_in_rstar : "r1 h\<leadsto> r2 \<Longrightarrow> r1 h\<leadsto>* r2"
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using hrewrites.intros(1) hrewrites.intros(2) by blast
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lemma hreal_trans[trans]:
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assumes a1: "r1 h\<leadsto>* r2" and a2: "r2 h\<leadsto>* r3"
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shows "r1 h\<leadsto>* r3"
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using a2 a1
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apply(induct r2 r3 arbitrary: r1 rule: hrewrites.induct)
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apply(auto)
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done
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lemma hrewrites_seq_context:
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shows "r1 h\<leadsto>* r2 \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RSEQ r2 r3"
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apply(induct r1 r2 rule: hrewrites.induct)
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apply simp
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using hrewrite.intros(4) by blast
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lemma hrewrites_seq_context2:
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shows "r1 h\<leadsto>* r2 \<Longrightarrow> RSEQ r0 r1 h\<leadsto>* RSEQ r0 r2"
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apply(induct r1 r2 rule: hrewrites.induct)
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apply simp
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using hrewrite.intros(5) by blast
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lemma hrewrites_seq_contexts:
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shows "\<lbrakk>r1 h\<leadsto>* r2; r3 h\<leadsto>* r4\<rbrakk> \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RSEQ r2 r4"
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by (meson hreal_trans hrewrites_seq_context hrewrites_seq_context2)
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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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(*
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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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*)
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(*
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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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*)
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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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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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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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(*
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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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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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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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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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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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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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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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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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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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apply(induct rs)
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apply simp
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by (simp add: frewrites_cons)
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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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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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apply(induct rs1 rs2 rule: grewrite.induct)
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apply(simp)
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apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>g (rs @ rsa)")
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prefer 2
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using grewrite.intros(1) apply blast
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apply(subgoal_tac "(rs @ rsa) \<leadsto>g* (rs @ rsb)")
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using gmany_steps_later apply blast
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apply (simp add: grewrites_append)
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apply (metis append.assoc append_Cons grewrite.intros(2) grewrites_append gmany_steps_later)
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using grewrites_cons apply auto
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apply(subgoal_tac "rsaa @ a # rsba @ a # rsc @ rsa \<leadsto>g* rsaa @ a # rsba @ a # rsc @ rsb")
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using grewrite.intros(4) grewrites.intros(2) apply force
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using grewrites_append by auto
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lemma grewritess_concat:
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shows "\<lbrakk>rsa \<leadsto>g* rsb; rsc \<leadsto>g* rsd \<rbrakk> \<Longrightarrow> (rsa @ rsc) \<leadsto>g* (rsb @ rsd)"
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apply(induct rsa rsb rule: grewrites.induct)
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apply(case_tac rs)
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apply simp
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using grewrites_append apply blast
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by (meson greal_trans grewrites.simps grewrites_concat)
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fun alt_set:: "rrexp \<Rightarrow> rrexp set"
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where
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"alt_set (RALTS rs) = set rs \<union> \<Union> (alt_set ` (set rs))"
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| "alt_set r = {r}"
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lemma grewrite_cases_middle:
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shows "rs1 \<leadsto>g rs2 \<Longrightarrow>
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(\<exists>rsa rsb rsc. rs1 = (rsa @ [RALTS rsb] @ rsc) \<and> rs2 = (rsa @ rsb @ rsc)) \<or>
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(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc) \<or>
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(\<exists>rsa rsb rsc a. rs1 = rsa @ [a] @ rsb @ [a] @ rsc \<and> rs2 = rsa @ [a] @ rsb @ rsc)"
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apply( induct rs1 rs2 rule: grewrite.induct)
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apply simp
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apply blast
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apply (metis append_Cons append_Nil)
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apply (metis append_Cons)
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by blast
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lemma good_singleton:
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shows "good a \<and> nonalt a \<Longrightarrow> rflts [a] = [a]"
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using good.simps(1) k0b by blast
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284 |
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285 |
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286 |
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287 |
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288 |
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289 |
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290 |
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291 |
lemma all_that_same_elem:
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292 |
shows "\<lbrakk> a \<in> rset; rdistinct rs {a} = []\<rbrakk>
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293 |
\<Longrightarrow> rdistinct (rs @ rsb) rset = rdistinct rsb rset"
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294 |
apply(induct rs)
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295 |
apply simp
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296 |
apply(subgoal_tac "aa = a")
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297 |
apply simp
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298 |
by (metis empty_iff insert_iff list.discI rdistinct.simps(2))
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299 |
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300 |
lemma distinct_early_app1:
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301 |
shows "rset1 \<subseteq> rset \<Longrightarrow> rdistinct rs rset = rdistinct (rdistinct rs rset1) rset"
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302 |
apply(induct rs arbitrary: rset rset1)
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303 |
apply simp
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304 |
apply simp
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305 |
apply(case_tac "a \<in> rset1")
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306 |
apply simp
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307 |
apply(case_tac "a \<in> rset")
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308 |
apply simp+
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309 |
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310 |
apply blast
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311 |
apply(case_tac "a \<in> rset1")
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312 |
apply simp+
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313 |
apply(case_tac "a \<in> rset")
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314 |
apply simp
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315 |
apply (metis insert_subsetI)
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316 |
apply simp
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317 |
by (meson insert_mono)
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318 |
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319 |
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320 |
lemma distinct_early_app:
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321 |
shows " rdistinct (rs @ rsb) rset = rdistinct (rdistinct rs {} @ rsb) rset"
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322 |
apply(induct rsb)
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323 |
apply simp
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324 |
using distinct_early_app1 apply blast
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325 |
by (metis distinct_early_app1 distinct_once_enough empty_subsetI)
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326 |
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327 |
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328 |
lemma distinct_eq_interesting1:
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329 |
shows "a \<in> rset \<Longrightarrow> rdistinct (rs @ rsb) rset = rdistinct (rdistinct (a # rs) {} @ rsb) rset"
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330 |
apply(subgoal_tac "rdistinct (rdistinct (a # rs) {} @ rsb) rset = rdistinct (rdistinct rs {} @ rsb) rset")
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331 |
apply(simp only:)
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332 |
using distinct_early_app apply blast
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333 |
by (metis append_Cons distinct_early_app rdistinct.simps(2))
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334 |
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335 |
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336 |
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337 |
lemma good_flatten_aux_aux1:
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338 |
shows "\<lbrakk> size rs \<ge>2;
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339 |
\<forall>r \<in> set rs. good r \<and> r \<noteq> RZERO \<and> nonalt r; \<forall>r \<in> set rsb. good r \<and> r \<noteq> RZERO \<and> nonalt r \<rbrakk>
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340 |
\<Longrightarrow> rdistinct (rs @ rsb) rset =
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341 |
rdistinct (rflts [rsimp_ALTs (rdistinct rs {})] @ rsb) rset"
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342 |
apply(induct rs arbitrary: rset)
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343 |
apply simp
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344 |
apply(case_tac "a \<in> rset")
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345 |
apply simp
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346 |
apply(case_tac "rdistinct rs {a}")
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347 |
apply simp
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348 |
apply(subst good_singleton)
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349 |
apply force
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350 |
apply simp
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|
351 |
apply (meson all_that_same_elem)
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352 |
apply(subgoal_tac "rflts [rsimp_ALTs (a # rdistinct rs {a})] = a # rdistinct rs {a} ")
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353 |
prefer 2
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|
354 |
using k0a rsimp_ALTs.simps(3) apply presburger
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355 |
apply(simp only:)
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|
356 |
apply(subgoal_tac "rdistinct (rs @ rsb) rset = rdistinct ((rdistinct (a # rs) {}) @ rsb) rset ")
|
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|
357 |
apply (metis insert_absorb insert_is_Un insert_not_empty rdistinct.simps(2))
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|
358 |
apply (meson distinct_eq_interesting1)
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359 |
apply simp
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|
360 |
apply(case_tac "rdistinct rs {a}")
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|
361 |
prefer 2
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|
362 |
apply(subgoal_tac "rsimp_ALTs (a # rdistinct rs {a}) = RALTS (a # rdistinct rs {a})")
|
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|
363 |
apply(simp only:)
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|
364 |
apply(subgoal_tac "a # rdistinct (rs @ rsb) (insert a rset) =
|
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|
365 |
rdistinct (rflts [RALTS (a # rdistinct rs {a})] @ rsb) rset")
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|
366 |
apply simp
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|
367 |
apply (metis append_Cons distinct_early_app empty_iff insert_is_Un k0a rdistinct.simps(2))
|
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|
368 |
using rsimp_ALTs.simps(3) apply presburger
|
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|
369 |
by (metis Un_insert_left append_Cons distinct_early_app empty_iff good_singleton rdistinct.simps(2) rsimp_ALTs.simps(2) sup_bot_left)
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370 |
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371 |
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|
372 |
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373 |
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374 |
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|
375 |
lemma good_flatten_aux_aux:
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|
376 |
shows "\<lbrakk>\<exists>a aa lista list. rs = a # list \<and> list = aa # lista;
|
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|
377 |
\<forall>r \<in> set rs. good r \<and> r \<noteq> RZERO \<and> nonalt r; \<forall>r \<in> set rsb. good r \<and> r \<noteq> RZERO \<and> nonalt r \<rbrakk>
|
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|
378 |
\<Longrightarrow> rdistinct (rs @ rsb) rset =
|
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|
379 |
rdistinct (rflts [rsimp_ALTs (rdistinct rs {})] @ rsb) rset"
|
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|
380 |
apply(erule exE)+
|
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|
381 |
apply(subgoal_tac "size rs \<ge> 2")
|
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|
382 |
apply (metis good_flatten_aux_aux1)
|
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|
383 |
by (simp add: Suc_leI length_Cons less_add_Suc1)
|
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|
384 |
|
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|
385 |
|
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|
386 |
|
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|
387 |
lemma good_flatten_aux:
|
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|
388 |
shows " \<lbrakk>\<forall>r\<in>set rs. good r \<or> r = RZERO; \<forall>r\<in>set rsa . good r \<or> r = RZERO;
|
|
|
389 |
\<forall>r\<in>set rsb. good r \<or> r = RZERO;
|
|
|
390 |
rsimp (RALTS (rsa @ rs @ rsb)) = rsimp_ALTs (rdistinct (rflts (rsa @ rs @ rsb)) {});
|
|
|
391 |
rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) =
|
|
|
392 |
rsimp_ALTs (rdistinct (rflts (rsa @ [rsimp (RALTS rs)] @ rsb)) {});
|
|
|
393 |
map rsimp rsa = rsa; map rsimp rsb = rsb; map rsimp rs = rs;
|
|
|
394 |
rdistinct (rflts rsa @ rflts rs @ rflts rsb) {} =
|
|
|
395 |
rdistinct (rflts rsa) {} @ rdistinct (rflts rs @ rflts rsb) (set (rflts rsa));
|
|
|
396 |
rdistinct (rflts rsa @ rflts [rsimp (RALTS rs)] @ rflts rsb) {} =
|
|
|
397 |
rdistinct (rflts rsa) {} @ rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))\<rbrakk>
|
|
|
398 |
\<Longrightarrow> rdistinct (rflts rs @ rflts rsb) rset =
|
|
|
399 |
rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) rset"
|
|
|
400 |
apply simp
|
|
|
401 |
apply(case_tac "rflts rs ")
|
|
|
402 |
apply simp
|
|
|
403 |
apply(case_tac "list")
|
|
|
404 |
apply simp
|
|
|
405 |
apply(case_tac "a \<in> rset")
|
|
|
406 |
apply simp
|
|
|
407 |
apply (metis append.left_neutral append_Cons equals0D k0b list.set_intros(1) nonalt_flts_rd qqq1 rdistinct.simps(2))
|
|
|
408 |
apply simp
|
|
|
409 |
apply (metis Un_insert_left append_Cons append_Nil ex_in_conv flts_single1 insertI1 list.simps(15) nonalt_flts_rd nonazero.elims(3) qqq1 rdistinct.simps(2) sup_bot_left)
|
|
|
410 |
apply(subgoal_tac "\<forall>r \<in> set (rflts rs). good r \<and> r \<noteq> RZERO \<and> nonalt r")
|
|
|
411 |
prefer 2
|
|
|
412 |
apply (metis Diff_empty flts3 nonalt_flts_rd qqq1 rdistinct_set_equality1)
|
|
|
413 |
apply(subgoal_tac "\<forall>r \<in> set (rflts rsb). good r \<and> r \<noteq> RZERO \<and> nonalt r")
|
|
|
414 |
prefer 2
|
|
|
415 |
apply (metis Diff_empty flts3 good.simps(1) nonalt_flts_rd rdistinct_set_equality1)
|
|
|
416 |
by (smt (verit, ccfv_threshold) good_flatten_aux_aux)
|
|
|
417 |
|
|
|
418 |
|
|
|
419 |
|
|
|
420 |
|
|
|
421 |
lemma good_flatten_middle:
|
|
|
422 |
shows "\<lbrakk>\<forall>r \<in> set rs. good r \<or> r = RZERO; \<forall>r \<in> set rsa. good r \<or> r = RZERO; \<forall>r \<in> set rsb. good r \<or> r = RZERO\<rbrakk> \<Longrightarrow>
|
|
|
423 |
rsimp (RALTS (rsa @ rs @ rsb)) = rsimp (RALTS (rsa @ [RALTS rs] @ rsb))"
|
|
|
424 |
apply(subgoal_tac "rsimp (RALTS (rsa @ rs @ rsb)) = rsimp_ALTs (rdistinct (rflts (map rsimp rsa @
|
|
|
425 |
map rsimp rs @ map rsimp rsb)) {})")
|
|
|
426 |
prefer 2
|
|
|
427 |
apply simp
|
|
|
428 |
apply(simp only:)
|
|
|
429 |
apply(subgoal_tac "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) = rsimp_ALTs (rdistinct (rflts (map rsimp rsa @
|
|
|
430 |
[rsimp (RALTS rs)] @ map rsimp rsb)) {})")
|
|
|
431 |
prefer 2
|
|
|
432 |
apply simp
|
|
|
433 |
apply(simp only:)
|
|
|
434 |
apply(subgoal_tac "map rsimp rsa = rsa")
|
|
|
435 |
prefer 2
|
|
|
436 |
apply (metis map_idI rsimp.simps(3) test)
|
|
|
437 |
apply(simp only:)
|
|
|
438 |
apply(subgoal_tac "map rsimp rsb = rsb")
|
|
|
439 |
prefer 2
|
|
|
440 |
apply (metis map_idI rsimp.simps(3) test)
|
|
|
441 |
apply(simp only:)
|
|
|
442 |
apply(subst k00)+
|
|
|
443 |
apply(subgoal_tac "map rsimp rs = rs")
|
|
|
444 |
apply(simp only:)
|
|
|
445 |
prefer 2
|
|
|
446 |
apply (metis map_idI rsimp.simps(3) test)
|
|
|
447 |
apply(subgoal_tac "rdistinct (rflts rsa @ rflts rs @ rflts rsb) {} =
|
|
|
448 |
rdistinct (rflts rsa) {} @ rdistinct (rflts rs @ rflts rsb) (set (rflts rsa))")
|
|
|
449 |
apply(simp only:)
|
|
|
450 |
prefer 2
|
|
|
451 |
using rdistinct_concat_general apply blast
|
|
|
452 |
apply(subgoal_tac "rdistinct (rflts rsa @ rflts [rsimp (RALTS rs)] @ rflts rsb) {} =
|
|
|
453 |
rdistinct (rflts rsa) {} @ rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))")
|
|
|
454 |
apply(simp only:)
|
|
|
455 |
prefer 2
|
|
|
456 |
using rdistinct_concat_general apply blast
|
|
|
457 |
apply(subgoal_tac "rdistinct (rflts rs @ rflts rsb) (set (rflts rsa)) =
|
|
|
458 |
rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))")
|
|
|
459 |
apply presburger
|
|
|
460 |
using good_flatten_aux by blast
|
|
|
461 |
|
|
|
462 |
|
|
|
463 |
lemma simp_flatten3:
|
|
|
464 |
shows "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) = rsimp (RALTS (rsa @ rs @ rsb))"
|
|
511
|
465 |
proof -
|
|
|
466 |
have "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) =
|
|
|
467 |
rsimp (RALTS (map rsimp rsa @ [rsimp (RALTS rs)] @ map rsimp rsb)) "
|
|
|
468 |
by (metis append_Cons append_Nil list.simps(9) map_append simp_flatten_aux0)
|
|
|
469 |
apply(simp only:)
|
|
|
470 |
|
|
|
471 |
oops
|
|
|
472 |
|
|
|
473 |
lemma simp_flatten3:
|
|
|
474 |
shows "rsimp (RALTS (rsa @ [RALTS rs ] @ rsb)) = rsimp (RALTS (rsa @ rs @ rsb))"
|
|
505
|
475 |
apply(subgoal_tac "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) =
|
|
|
476 |
rsimp (RALTS (map rsimp rsa @ [rsimp (RALTS rs)] @ map rsimp rsb)) ")
|
|
|
477 |
prefer 2
|
|
|
478 |
apply (metis append.left_neutral append_Cons list.simps(9) map_append simp_flatten_aux0)
|
|
|
479 |
apply (simp only:)
|
|
|
480 |
apply(subgoal_tac "rsimp (RALTS (rsa @ rs @ rsb)) =
|
|
|
481 |
rsimp (RALTS (map rsimp rsa @ map rsimp rs @ map rsimp rsb))")
|
|
|
482 |
prefer 2
|
|
|
483 |
apply (metis map_append simp_flatten_aux0)
|
|
|
484 |
apply(simp only:)
|
|
|
485 |
apply(subgoal_tac "rsimp (RALTS (map rsimp rsa @ map rsimp rs @ map rsimp rsb)) =
|
|
|
486 |
rsimp (RALTS (map rsimp rsa @ [RALTS (map rsimp rs)] @ map rsimp rsb))")
|
|
|
487 |
|
|
|
488 |
apply (metis (no_types, lifting) head_one_more_simp map_append simp_flatten_aux0)
|
|
|
489 |
apply(subgoal_tac "\<forall>r \<in> set (map rsimp rsa). good r \<or> r = RZERO")
|
|
|
490 |
apply(subgoal_tac "\<forall>r \<in> set (map rsimp rs). good r \<or> r = RZERO")
|
|
|
491 |
apply(subgoal_tac "\<forall>r \<in> set (map rsimp rsb). good r \<or> r = RZERO")
|
|
|
492 |
|
|
|
493 |
using good_flatten_middle apply presburger
|
|
|
494 |
|
|
|
495 |
apply (simp add: good1)
|
|
|
496 |
apply (simp add: good1)
|
|
|
497 |
apply (simp add: good1)
|
|
|
498 |
|
|
|
499 |
done
|
|
|
500 |
|
|
|
501 |
|
|
|
502 |
|
|
|
503 |
|
|
|
504 |
|
|
|
505 |
lemma grewrite_equal_rsimp:
|
|
|
506 |
shows "rs1 \<leadsto>g rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
|
507 |
apply(frule grewrite_cases_middle)
|
|
|
508 |
apply(case_tac "(\<exists>rsa rsb rsc. rs1 = rsa @ [RALTS rsb] @ rsc \<and> rs2 = rsa @ rsb @ rsc)")
|
|
|
509 |
using simp_flatten3 apply auto[1]
|
|
|
510 |
apply(case_tac "(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc)")
|
|
|
511 |
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)
|
|
|
512 |
by (smt (verit) append.assoc append_Cons append_Nil in_set_conv_decomp simp_removes_duplicate3)
|
|
|
513 |
|
|
|
514 |
|
|
|
515 |
lemma grewrites_equal_rsimp:
|
|
|
516 |
shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
|
517 |
apply (induct rs1 rs2 rule: grewrites.induct)
|
|
|
518 |
apply simp
|
|
|
519 |
using grewrite_equal_rsimp by presburger
|
|
|
520 |
|
|
|
521 |
|
|
|
522 |
|
|
|
523 |
lemma grewrites_last:
|
|
|
524 |
shows "r # [RALTS rs] \<leadsto>g* r # rs"
|
|
|
525 |
by (metis gr_in_rstar grewrite.intros(2) grewrite.intros(3) self_append_conv)
|
|
|
526 |
|
|
|
527 |
lemma simp_flatten2:
|
|
|
528 |
shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
|
|
|
529 |
using grewrites_equal_rsimp grewrites_last by blast
|
|
|
530 |
|
|
|
531 |
|
|
|
532 |
lemma frewrites_alt:
|
|
|
533 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> (RALT r1 r2) # rs1 \<leadsto>f* r1 # r2 # rs2"
|
|
|
534 |
by (metis Cons_eq_appendI append_self_conv2 frewrite.intros(2) frewrites_cons many_steps_later)
|
|
|
535 |
|
|
|
536 |
lemma early_late_der_frewrites:
|
|
|
537 |
shows "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)"
|
|
|
538 |
apply(induct rs)
|
|
|
539 |
apply simp
|
|
|
540 |
apply(case_tac a)
|
|
|
541 |
apply simp+
|
|
|
542 |
using frewrite.intros(1) many_steps_later apply blast
|
|
|
543 |
apply(case_tac "x = x3")
|
|
|
544 |
apply simp
|
|
|
545 |
using frewrites_cons apply presburger
|
|
|
546 |
using frewrite.intros(1) many_steps_later apply fastforce
|
|
|
547 |
apply(case_tac "rnullable x41")
|
|
|
548 |
apply simp+
|
|
|
549 |
apply (simp add: frewrites_alt)
|
|
|
550 |
apply (simp add: frewrites_cons)
|
|
|
551 |
apply (simp add: frewrites_append)
|
|
|
552 |
by (simp add: frewrites_cons)
|
|
|
553 |
|
|
|
554 |
|
|
|
555 |
lemma gstar0:
|
|
|
556 |
shows "rsa @ (rdistinct rs (set rsa)) \<leadsto>g* rsa @ (rdistinct rs (insert RZERO (set rsa)))"
|
|
|
557 |
apply(induct rs arbitrary: rsa)
|
|
|
558 |
apply simp
|
|
|
559 |
apply(case_tac "a = RZERO")
|
|
|
560 |
apply simp
|
|
|
561 |
|
|
|
562 |
using gr_in_rstar grewrite.intros(1) grewrites_append apply presburger
|
|
|
563 |
apply(case_tac "a \<in> set rsa")
|
|
|
564 |
apply simp+
|
|
|
565 |
apply(drule_tac x = "rsa @ [a]" in meta_spec)
|
|
|
566 |
by simp
|
|
|
567 |
|
|
|
568 |
lemma grewrite_rdistinct_aux:
|
|
|
569 |
shows "rs @ rdistinct rsa rset \<leadsto>g* rs @ rdistinct rsa (rset \<union> set rs)"
|
|
|
570 |
apply(induct rsa arbitrary: rs rset)
|
|
|
571 |
apply simp
|
|
|
572 |
apply(case_tac " a \<in> rset")
|
|
|
573 |
apply simp
|
|
|
574 |
apply(case_tac "a \<in> set rs")
|
|
|
575 |
apply simp
|
|
|
576 |
apply (metis Un_insert_left Un_insert_right gmany_steps_later grewrite_variant1 insert_absorb)
|
|
|
577 |
apply simp
|
|
|
578 |
apply(drule_tac x = "rs @ [a]" in meta_spec)
|
|
|
579 |
by (metis Un_insert_left Un_insert_right append.assoc append.right_neutral append_Cons append_Nil insert_absorb2 list.simps(15) set_append)
|
|
|
580 |
|
|
|
581 |
|
|
|
582 |
lemma flts_gstar:
|
|
|
583 |
shows "rs \<leadsto>g* rflts rs"
|
|
|
584 |
apply(induct rs)
|
|
|
585 |
apply simp
|
|
|
586 |
apply(case_tac "a = RZERO")
|
|
|
587 |
apply simp
|
|
|
588 |
using gmany_steps_later grewrite.intros(1) apply blast
|
|
|
589 |
apply(case_tac "\<exists>rsa. a = RALTS rsa")
|
|
|
590 |
apply(erule exE)
|
|
|
591 |
apply simp
|
|
|
592 |
apply (meson grewrite.intros(2) grewrites.simps grewrites_cons)
|
|
|
593 |
by (simp add: grewrites_cons rflts_def_idiot)
|
|
|
594 |
|
|
|
595 |
lemma more_distinct1:
|
|
|
596 |
shows " \<lbrakk>\<And>rsb rset rset2.
|
|
|
597 |
rset2 \<subseteq> set rsb \<Longrightarrow> rsb @ rdistinct rs rset \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2);
|
|
|
598 |
rset2 \<subseteq> set rsb; a \<notin> rset; a \<in> rset2\<rbrakk>
|
|
|
599 |
\<Longrightarrow> rsb @ a # rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)"
|
|
|
600 |
apply(subgoal_tac "rsb @ a # rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (insert a rset)")
|
|
|
601 |
apply(subgoal_tac "rsb @ rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)")
|
|
|
602 |
apply (meson greal_trans)
|
|
|
603 |
apply (metis Un_iff Un_insert_left insert_absorb)
|
|
|
604 |
by (simp add: gr_in_rstar grewrite_variant1 in_mono)
|
|
|
605 |
|
|
|
606 |
|
|
|
607 |
|
|
|
608 |
|
|
|
609 |
|
|
|
610 |
lemma frewrite_rd_grewrites_aux:
|
|
|
611 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
612 |
rsb @
|
|
|
613 |
RALTS rs #
|
|
|
614 |
rdistinct rsa
|
|
|
615 |
(insert (RALTS rs)
|
|
|
616 |
(set rsb)) \<leadsto>g* rflts rsb @
|
|
|
617 |
rdistinct rs (set rsb) @ rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
|
|
|
618 |
|
|
|
619 |
apply simp
|
|
|
620 |
apply(subgoal_tac "rsb @
|
|
|
621 |
RALTS rs #
|
|
|
622 |
rdistinct rsa
|
|
|
623 |
(insert (RALTS rs)
|
|
|
624 |
(set rsb)) \<leadsto>g* rsb @
|
|
|
625 |
rs @
|
|
|
626 |
rdistinct rsa
|
|
|
627 |
(insert (RALTS rs)
|
|
|
628 |
(set rsb)) ")
|
|
|
629 |
apply(subgoal_tac " rsb @
|
|
|
630 |
rs @
|
|
|
631 |
rdistinct rsa
|
|
|
632 |
(insert (RALTS rs)
|
|
|
633 |
(set rsb)) \<leadsto>g*
|
|
|
634 |
rsb @
|
|
|
635 |
rdistinct rs (set rsb) @
|
|
|
636 |
rdistinct rsa
|
|
|
637 |
(insert (RALTS rs)
|
|
|
638 |
(set rsb)) ")
|
|
|
639 |
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)
|
|
|
640 |
apply (metis append_assoc grewrites.intros(1) grewritess_concat gstar_rdistinct_general)
|
|
|
641 |
by (simp add: gr_in_rstar grewrite.intros(2) grewrites_append)
|
|
|
642 |
|
|
|
643 |
|
|
|
644 |
|
|
|
645 |
|
|
|
646 |
lemma list_dlist_union:
|
|
|
647 |
shows "set rs \<subseteq> set rsb \<union> set (rdistinct rs (set rsb))"
|
|
|
648 |
by (metis rdistinct_concat_general rdistinct_set_equality set_append sup_ge2)
|
|
|
649 |
|
|
|
650 |
lemma r_finite1:
|
|
|
651 |
shows "r = RALTS (r # rs) = False"
|
|
|
652 |
apply(induct r)
|
|
|
653 |
apply simp+
|
|
|
654 |
apply (metis list.set_intros(1))
|
|
|
655 |
by blast
|
|
|
656 |
|
|
|
657 |
|
|
|
658 |
|
|
|
659 |
lemma grewrite_singleton:
|
|
|
660 |
shows "[r] \<leadsto>g r # rs \<Longrightarrow> rs = []"
|
|
|
661 |
apply (induct "[r]" "r # rs" rule: grewrite.induct)
|
|
|
662 |
apply simp
|
|
|
663 |
apply (metis r_finite1)
|
|
|
664 |
using grewrite.simps apply blast
|
|
|
665 |
by simp
|
|
|
666 |
|
|
|
667 |
|
|
|
668 |
|
|
|
669 |
lemma concat_rdistinct_equality1:
|
|
|
670 |
shows "rdistinct (rs @ rsa) rset = rdistinct rs rset @ rdistinct rsa (rset \<union> (set rs))"
|
|
|
671 |
apply(induct rs arbitrary: rsa rset)
|
|
|
672 |
apply simp
|
|
|
673 |
apply(case_tac "a \<in> rset")
|
|
|
674 |
apply simp
|
|
|
675 |
apply (simp add: insert_absorb)
|
|
|
676 |
by auto
|
|
|
677 |
|
|
|
678 |
|
|
|
679 |
lemma grewrites_rev_append:
|
|
|
680 |
shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rs1 @ [x] \<leadsto>g* rs2 @ [x]"
|
|
|
681 |
using grewritess_concat by auto
|
|
|
682 |
|
|
|
683 |
lemma grewrites_inclusion:
|
|
|
684 |
shows "set rs \<subseteq> set rs1 \<Longrightarrow> rs1 @ rs \<leadsto>g* rs1"
|
|
|
685 |
apply(induct rs arbitrary: rs1)
|
|
|
686 |
apply simp
|
|
|
687 |
by (meson gmany_steps_later grewrite_variant1 list.set_intros(1) set_subset_Cons subset_code(1))
|
|
|
688 |
|
|
|
689 |
lemma distinct_keeps_last:
|
|
|
690 |
shows "\<lbrakk>x \<notin> rset; x \<notin> set xs \<rbrakk> \<Longrightarrow> rdistinct (xs @ [x]) rset = rdistinct xs rset @ [x]"
|
|
|
691 |
by (simp add: concat_rdistinct_equality1)
|
|
|
692 |
|
|
|
693 |
lemma grewrites_shape2_aux:
|
|
|
694 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
695 |
rsb @
|
|
|
696 |
rdistinct (rs @ rsa)
|
|
|
697 |
(set rsb) \<leadsto>g* rsb @
|
|
|
698 |
rdistinct rs (set rsb) @
|
|
|
699 |
rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
|
|
|
700 |
apply(subgoal_tac " rdistinct (rs @ rsa) (set rsb) = rdistinct rs (set rsb) @ rdistinct rsa (set rs \<union> set rsb)")
|
|
|
701 |
apply (simp only:)
|
|
|
702 |
prefer 2
|
|
|
703 |
apply (simp add: Un_commute concat_rdistinct_equality1)
|
|
|
704 |
apply(induct rsa arbitrary: rs rsb rule: rev_induct)
|
|
|
705 |
apply simp
|
|
|
706 |
apply(case_tac "x \<in> set rs")
|
|
|
707 |
apply (simp add: distinct_removes_middle3)
|
|
|
708 |
apply(case_tac "x = RALTS rs")
|
|
|
709 |
apply simp
|
|
|
710 |
apply(case_tac "x \<in> set rsb")
|
|
|
711 |
apply simp
|
|
|
712 |
apply (simp add: concat_rdistinct_equality1)
|
|
|
713 |
apply (simp add: concat_rdistinct_equality1)
|
|
|
714 |
apply simp
|
|
|
715 |
apply(drule_tac x = "rs " in meta_spec)
|
|
|
716 |
apply(drule_tac x = rsb in meta_spec)
|
|
|
717 |
apply simp
|
|
|
718 |
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))")
|
|
|
719 |
prefer 2
|
|
|
720 |
apply (simp add: concat_rdistinct_equality1)
|
|
|
721 |
apply(case_tac "x \<in> set xs")
|
|
|
722 |
apply simp
|
|
|
723 |
apply (simp add: distinct_removes_last)
|
|
|
724 |
apply(case_tac "x \<in> set rsb")
|
|
|
725 |
apply (smt (verit, ccfv_threshold) Un_iff append.right_neutral concat_rdistinct_equality1 insert_is_Un rdistinct.simps(2))
|
|
|
726 |
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]")
|
|
|
727 |
apply(simp only:)
|
|
|
728 |
apply(case_tac "x = RALTS rs")
|
|
|
729 |
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")
|
|
|
730 |
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) ")
|
|
|
731 |
apply (smt (verit, ccfv_SIG) Un_insert_left append.right_neutral concat_rdistinct_equality1 greal_trans insert_iff rdistinct.simps(2))
|
|
|
732 |
apply(subgoal_tac "set rs \<subseteq> set ( rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb))")
|
|
|
733 |
apply (metis append.assoc grewrites_inclusion)
|
|
|
734 |
apply (metis Un_upper1 append.assoc dual_order.trans list_dlist_union set_append)
|
|
|
735 |
apply (metis append_Nil2 gr_in_rstar grewrite.intros(2) grewrite_append)
|
|
|
736 |
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]")
|
|
|
737 |
apply(simp only:)
|
|
|
738 |
apply (metis append.assoc grewrites_rev_append)
|
|
|
739 |
apply (simp add: insert_absorb)
|
|
|
740 |
apply (simp add: distinct_keeps_last)+
|
|
|
741 |
done
|
|
|
742 |
|
|
|
743 |
lemma grewrites_shape2:
|
|
|
744 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
745 |
rsb @
|
|
|
746 |
rdistinct (rs @ rsa)
|
|
|
747 |
(set rsb) \<leadsto>g* rflts rsb @
|
|
|
748 |
rdistinct rs (set rsb) @
|
|
|
749 |
rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
|
|
|
750 |
apply (meson flts_gstar greal_trans grewrites.simps grewrites_shape2_aux grewritess_concat)
|
|
|
751 |
done
|
|
|
752 |
|
|
|
753 |
lemma rdistinct_add_acc:
|
|
|
754 |
shows "rset2 \<subseteq> set rsb \<Longrightarrow> rsb @ rdistinct rs rset \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)"
|
|
|
755 |
apply(induct rs arbitrary: rsb rset rset2)
|
|
|
756 |
apply simp
|
|
|
757 |
apply (case_tac "a \<in> rset")
|
|
|
758 |
apply simp
|
|
|
759 |
apply(case_tac "a \<in> rset2")
|
|
|
760 |
apply simp
|
|
|
761 |
apply (simp add: more_distinct1)
|
|
|
762 |
apply simp
|
|
|
763 |
apply(drule_tac x = "rsb @ [a]" in meta_spec)
|
|
|
764 |
by (metis Un_insert_left append.assoc append_Cons append_Nil set_append sup.coboundedI1)
|
|
|
765 |
|
|
|
766 |
|
|
|
767 |
lemma frewrite_fun1:
|
|
|
768 |
shows " RALTS rs \<in> set rsb \<Longrightarrow>
|
|
|
769 |
rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb \<union> set rs)"
|
|
|
770 |
apply(subgoal_tac "rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb)")
|
|
|
771 |
apply(subgoal_tac " set rs \<subseteq> set (rflts rsb)")
|
|
|
772 |
prefer 2
|
|
|
773 |
using spilled_alts_contained apply blast
|
|
|
774 |
apply(subgoal_tac "rflts rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb \<union> set rs)")
|
|
|
775 |
using greal_trans apply blast
|
|
|
776 |
using rdistinct_add_acc apply presburger
|
|
|
777 |
using flts_gstar grewritess_concat by auto
|
|
|
778 |
|
|
|
779 |
lemma frewrite_rd_grewrites:
|
|
|
780 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow>
|
|
|
781 |
\<exists>rs3. (rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3) "
|
|
|
782 |
apply(induct rs1 rs2 arbitrary: rs rule: frewrite.induct)
|
|
|
783 |
apply(rule_tac x = "rsa @ (rdistinct rs ({RZERO} \<union> set rsa))" in exI)
|
|
|
784 |
apply(rule conjI)
|
|
|
785 |
apply(case_tac "RZERO \<in> set rsa")
|
|
|
786 |
apply simp+
|
|
|
787 |
using gstar0 apply fastforce
|
|
|
788 |
apply (simp add: gr_in_rstar grewrite.intros(1) grewrites_append)
|
|
|
789 |
apply (simp add: gstar0)
|
|
|
790 |
prefer 2
|
|
|
791 |
apply(case_tac "r \<in> set rs")
|
|
|
792 |
apply simp
|
|
|
793 |
apply(drule_tac x = "rs @ [r]" in meta_spec)
|
|
|
794 |
apply(erule exE)
|
|
|
795 |
apply(rule_tac x = "rs3" in exI)
|
|
|
796 |
apply simp
|
|
|
797 |
apply(case_tac "RALTS rs \<in> set rsb")
|
|
|
798 |
apply simp
|
|
|
799 |
apply(rule_tac x = "rflts rsb @ rdistinct rsa (set rsb \<union> set rs)" in exI)
|
|
|
800 |
apply(rule conjI)
|
|
|
801 |
using frewrite_fun1 apply force
|
|
|
802 |
apply (metis frewrite_fun1 rdistinct_concat sup_ge2)
|
|
|
803 |
apply(simp)
|
|
|
804 |
apply(rule_tac x =
|
|
|
805 |
"rflts rsb @
|
|
|
806 |
rdistinct rs (set rsb) @
|
|
|
807 |
rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})" in exI)
|
|
|
808 |
apply(rule conjI)
|
|
|
809 |
prefer 2
|
|
|
810 |
using grewrites_shape2 apply force
|
|
|
811 |
using frewrite_rd_grewrites_aux by blast
|
|
|
812 |
|
|
|
813 |
|
|
|
814 |
lemma frewrite_simpeq2:
|
|
|
815 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
|
|
|
816 |
apply(subgoal_tac "\<exists> rs3. (rdistinct rs1 {} \<leadsto>g* rs3) \<and> (rdistinct rs2 {} \<leadsto>g* rs3)")
|
|
|
817 |
using grewrites_equal_rsimp apply fastforce
|
|
|
818 |
by (metis append_self_conv2 frewrite_rd_grewrites list.set(1))
|
|
|
819 |
|
|
|
820 |
|
|
|
821 |
|
|
|
822 |
|
|
|
823 |
(*a more refined notion of h\<leadsto>* is needed,
|
|
|
824 |
this lemma fails when rs1 contains some RALTS rs where elements
|
|
|
825 |
of rs appear in later parts of rs1, which will be picked up by rs2
|
|
|
826 |
and deduplicated*)
|
|
|
827 |
lemma frewrites_simpeq:
|
|
|
828 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
|
|
|
829 |
rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS ( rdistinct rs2 {})) "
|
|
|
830 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
831 |
apply simp
|
|
|
832 |
using frewrite_simpeq2 by presburger
|
|
|
833 |
|
|
|
834 |
|
|
|
835 |
lemma frewrite_single_step:
|
|
|
836 |
shows " rs2 \<leadsto>f rs3 \<Longrightarrow> rsimp (RALTS rs2) = rsimp (RALTS rs3)"
|
|
|
837 |
apply(induct rs2 rs3 rule: frewrite.induct)
|
|
|
838 |
apply simp
|
|
|
839 |
using simp_flatten apply blast
|
|
|
840 |
by (metis (no_types, opaque_lifting) list.simps(9) rsimp.simps(2) simp_flatten2)
|
|
|
841 |
|
|
|
842 |
lemma grewrite_simpalts:
|
|
|
843 |
shows " rs2 \<leadsto>g rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
|
|
|
844 |
apply(induct rs2 rs3 rule : grewrite.induct)
|
|
|
845 |
using identity_wwo0 apply presburger
|
|
|
846 |
apply (metis frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_flatten)
|
|
|
847 |
apply (smt (verit, ccfv_SIG) gmany_steps_later grewrites.intros(1) grewrites_cons grewrites_equal_rsimp identity_wwo0 rsimp_ALTs.simps(3))
|
|
|
848 |
apply simp
|
|
|
849 |
apply(subst rsimp_alts_equal)
|
|
|
850 |
apply(case_tac "rsa = [] \<and> rsb = [] \<and> rsc = []")
|
|
|
851 |
apply(subgoal_tac "rsa @ a # rsb @ rsc = [a]")
|
|
|
852 |
apply (simp only:)
|
|
|
853 |
apply (metis append_Nil frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_removes_duplicate1(2))
|
|
|
854 |
apply simp
|
|
|
855 |
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)
|
|
|
856 |
|
|
|
857 |
|
|
|
858 |
lemma grewrites_simpalts:
|
|
|
859 |
shows " rs2 \<leadsto>g* rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
|
|
|
860 |
apply(induct rs2 rs3 rule: grewrites.induct)
|
|
|
861 |
apply simp
|
|
|
862 |
using grewrite_simpalts by presburger
|
|
|
863 |
|
|
|
864 |
|
|
|
865 |
lemma simp_der_flts:
|
|
|
866 |
shows "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {})) =
|
|
|
867 |
rsimp (RALTS (rdistinct (rflts (map (rder x) rs)) {}))"
|
|
|
868 |
apply(subgoal_tac "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)")
|
|
|
869 |
using frewrites_simpeq apply presburger
|
|
|
870 |
using early_late_der_frewrites by auto
|
|
|
871 |
|
|
|
872 |
|
|
|
873 |
lemma simp_der_pierce_flts_prelim:
|
|
|
874 |
shows "rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts rs)) {}))
|
|
|
875 |
= rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {}))"
|
|
|
876 |
by (metis append.right_neutral grewrite.intros(2) grewrite_simpalts rsimp_ALTs.simps(2) simp_der_flts)
|
|
|
877 |
|
|
|
878 |
|
|
|
879 |
lemma basic_regex_property1:
|
|
|
880 |
shows "rnullable r \<Longrightarrow> rsimp r \<noteq> RZERO"
|
|
|
881 |
apply(induct r rule: rsimp.induct)
|
|
|
882 |
apply(auto)
|
|
|
883 |
apply (metis idiot idiot2 rrexp.distinct(5))
|
|
|
884 |
by (metis der_simp_nullability rnullable.simps(1) rnullable.simps(4) rsimp.simps(2))
|
|
|
885 |
|
|
|
886 |
|
|
|
887 |
lemma inside_simp_seq_nullable:
|
|
|
888 |
shows
|
|
|
889 |
"\<And>r1 r2.
|
|
|
890 |
\<lbrakk>rsimp (rder x (rsimp r1)) = rsimp (rder x r1); rsimp (rder x (rsimp r2)) = rsimp (rder x r2);
|
|
|
891 |
rnullable r1\<rbrakk>
|
|
|
892 |
\<Longrightarrow> rsimp (rder x (rsimp_SEQ (rsimp r1) (rsimp r2))) =
|
|
|
893 |
rsimp_ALTs (rdistinct (rflts [rsimp_SEQ (rsimp (rder x r1)) (rsimp r2), rsimp (rder x r2)]) {})"
|
|
|
894 |
apply(case_tac "rsimp r1 = RONE")
|
|
|
895 |
apply(simp)
|
|
|
896 |
apply(subst basic_rsimp_SEQ_property1)
|
|
|
897 |
apply (simp add: idem_after_simp1)
|
|
|
898 |
apply(case_tac "rsimp r1 = RZERO")
|
|
|
899 |
|
|
|
900 |
using basic_regex_property1 apply blast
|
|
|
901 |
apply(case_tac "rsimp r2 = RZERO")
|
|
|
902 |
|
|
|
903 |
apply (simp add: basic_rsimp_SEQ_property3)
|
|
|
904 |
apply(subst idiot2)
|
|
|
905 |
apply simp+
|
|
|
906 |
apply(subgoal_tac "rnullable (rsimp r1)")
|
|
|
907 |
apply simp
|
|
|
908 |
using rsimp_idem apply presburger
|
|
|
909 |
using der_simp_nullability by presburger
|
|
|
910 |
|
|
|
911 |
|
|
|
912 |
|
|
|
913 |
lemma grewrite_ralts:
|
|
|
914 |
shows "rs \<leadsto>g rs' \<Longrightarrow> RALTS rs h\<leadsto>* RALTS rs'"
|
|
|
915 |
by (smt (verit) grewrite_cases_middle hr_in_rstar hrewrite.intros(11) hrewrite.intros(7) hrewrite.intros(8))
|
|
|
916 |
|
|
|
917 |
lemma grewrites_ralts:
|
|
|
918 |
shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs h\<leadsto>* RALTS rs'"
|
|
|
919 |
apply(induct rule: grewrites.induct)
|
|
|
920 |
apply simp
|
|
|
921 |
using grewrite_ralts hreal_trans by blast
|
|
|
922 |
|
|
|
923 |
|
|
|
924 |
lemma distinct_grewrites_subgoal1:
|
|
|
925 |
shows "
|
|
|
926 |
\<lbrakk>rs1 \<leadsto>g* [a]; RALTS rs1 h\<leadsto>* a; [a] \<leadsto>g rs3\<rbrakk> \<Longrightarrow> RALTS rs1 h\<leadsto>* rsimp_ALTs rs3"
|
|
|
927 |
apply(subgoal_tac "RALTS rs1 h\<leadsto>* RALTS rs3")
|
|
|
928 |
apply (metis hrewrite.intros(10) hrewrite.intros(9) rs2 rsimp_ALTs.cases rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
|
|
|
929 |
apply(subgoal_tac "rs1 \<leadsto>g* rs3")
|
|
|
930 |
using grewrites_ralts apply blast
|
|
|
931 |
using grewrites.intros(2) by presburger
|
|
|
932 |
|
|
|
933 |
lemma grewrites_ralts_rsimpalts:
|
|
|
934 |
shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs h\<leadsto>* rsimp_ALTs rs' "
|
|
|
935 |
apply(induct rs rs' rule: grewrites.induct)
|
|
|
936 |
apply(case_tac rs)
|
|
|
937 |
using hrewrite.intros(9) apply force
|
|
|
938 |
apply(case_tac list)
|
|
|
939 |
apply simp
|
|
|
940 |
using hr_in_rstar hrewrite.intros(10) rsimp_ALTs.simps(2) apply presburger
|
|
|
941 |
apply simp
|
|
|
942 |
apply(case_tac rs2)
|
|
|
943 |
apply simp
|
|
|
944 |
apply (metis grewrite.intros(3) grewrite_singleton rsimp_ALTs.simps(1))
|
|
|
945 |
apply(case_tac list)
|
|
|
946 |
apply(simp)
|
|
|
947 |
using distinct_grewrites_subgoal1 apply blast
|
|
|
948 |
apply simp
|
|
|
949 |
apply(case_tac rs3)
|
|
|
950 |
apply simp
|
|
|
951 |
using grewrites_ralts hrewrite.intros(9) apply blast
|
|
|
952 |
by (metis (no_types, opaque_lifting) grewrite_ralts hr_in_rstar hreal_trans hrewrite.intros(10) neq_Nil_conv rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
|
|
|
953 |
|
|
|
954 |
lemma hrewrites_alts:
|
|
|
955 |
shows " r h\<leadsto>* r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) h\<leadsto>* (RALTS (rs1 @ [r'] @ rs2))"
|
|
|
956 |
apply(induct r r' rule: hrewrites.induct)
|
|
|
957 |
apply simp
|
|
|
958 |
using hrewrite.intros(6) by blast
|
|
|
959 |
|
|
|
960 |
inductive
|
|
|
961 |
srewritescf:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" (" _ scf\<leadsto>* _" [100, 100] 100)
|
|
|
962 |
where
|
|
|
963 |
ss1: "[] scf\<leadsto>* []"
|
|
|
964 |
| ss2: "\<lbrakk>r h\<leadsto>* r'; rs scf\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (r#rs) scf\<leadsto>* (r'#rs')"
|
|
|
965 |
|
|
|
966 |
|
|
|
967 |
lemma srewritescf_alt: "rs1 scf\<leadsto>* rs2 \<Longrightarrow> (RALTS (rs@rs1)) h\<leadsto>* (RALTS (rs@rs2))"
|
|
|
968 |
|
|
|
969 |
apply(induct rs1 rs2 arbitrary: rs rule: srewritescf.induct)
|
|
|
970 |
apply(rule rs1)
|
|
|
971 |
apply(drule_tac x = "rsa@[r']" in meta_spec)
|
|
|
972 |
apply simp
|
|
|
973 |
apply(rule hreal_trans)
|
|
|
974 |
prefer 2
|
|
|
975 |
apply(assumption)
|
|
|
976 |
apply(drule hrewrites_alts)
|
|
|
977 |
by auto
|
|
|
978 |
|
|
|
979 |
|
|
|
980 |
corollary srewritescf_alt1:
|
|
|
981 |
assumes "rs1 scf\<leadsto>* rs2"
|
|
|
982 |
shows "RALTS rs1 h\<leadsto>* RALTS rs2"
|
|
|
983 |
using assms
|
|
|
984 |
by (metis append_Nil srewritescf_alt)
|
|
|
985 |
|
|
|
986 |
|
|
|
987 |
|
|
|
988 |
|
|
|
989 |
lemma trivialrsimp_srewrites:
|
|
|
990 |
"\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x h\<leadsto>* f x \<rbrakk> \<Longrightarrow> rs scf\<leadsto>* (map f rs)"
|
|
|
991 |
|
|
|
992 |
apply(induction rs)
|
|
|
993 |
apply simp
|
|
|
994 |
apply(rule ss1)
|
|
|
995 |
by (metis insert_iff list.simps(15) list.simps(9) srewritescf.simps)
|
|
|
996 |
|
|
|
997 |
lemma hrewrites_list:
|
|
|
998 |
shows
|
|
|
999 |
" (\<And>xa. xa \<in> set x \<Longrightarrow> xa h\<leadsto>* rsimp xa) \<Longrightarrow> RALTS x h\<leadsto>* RALTS (map rsimp x)"
|
|
|
1000 |
apply(induct x)
|
|
|
1001 |
apply(simp)+
|
|
|
1002 |
by (simp add: srewritescf_alt1 ss2 trivialrsimp_srewrites)
|
|
|
1003 |
(* apply(subgoal_tac "RALTS x h\<leadsto>* RALTS (map rsimp x)")*)
|
|
|
1004 |
|
|
|
1005 |
|
|
|
1006 |
lemma hrewrite_simpeq:
|
|
|
1007 |
shows "r1 h\<leadsto> r2 \<Longrightarrow> rsimp r1 = rsimp r2"
|
|
|
1008 |
apply(induct rule: hrewrite.induct)
|
|
|
1009 |
apply simp+
|
|
|
1010 |
apply (simp add: basic_rsimp_SEQ_property3)
|
|
|
1011 |
apply (simp add: basic_rsimp_SEQ_property1)
|
|
|
1012 |
using rsimp.simps(1) apply presburger
|
|
|
1013 |
apply simp+
|
|
|
1014 |
using flts_middle0 apply force
|
|
|
1015 |
|
|
|
1016 |
|
|
|
1017 |
using simp_flatten3 apply presburger
|
|
|
1018 |
|
|
|
1019 |
apply simp+
|
|
|
1020 |
apply (simp add: idem_after_simp1)
|
|
|
1021 |
using grewrite.intros(4) grewrite_equal_rsimp by presburger
|
|
|
1022 |
|
|
|
1023 |
lemma hrewrites_simpeq:
|
|
|
1024 |
shows "r1 h\<leadsto>* r2 \<Longrightarrow> rsimp r1 = rsimp r2"
|
|
|
1025 |
apply(induct rule: hrewrites.induct)
|
|
|
1026 |
apply simp
|
|
|
1027 |
apply(subgoal_tac "rsimp r2 = rsimp r3")
|
|
|
1028 |
apply auto[1]
|
|
|
1029 |
using hrewrite_simpeq by presburger
|
|
|
1030 |
|
|
|
1031 |
|
|
|
1032 |
|
|
|
1033 |
lemma simp_hrewrites:
|
|
|
1034 |
shows "r1 h\<leadsto>* rsimp r1"
|
|
|
1035 |
apply(induct r1)
|
|
|
1036 |
apply simp+
|
|
|
1037 |
apply(case_tac "rsimp r11 = RONE")
|
|
|
1038 |
apply simp
|
|
|
1039 |
apply(subst basic_rsimp_SEQ_property1)
|
|
|
1040 |
apply(subgoal_tac "RSEQ r11 r12 h\<leadsto>* RSEQ RONE r12")
|
|
|
1041 |
using hreal_trans hrewrite.intros(3) apply blast
|
|
|
1042 |
using hrewrites_seq_context apply presburger
|
|
|
1043 |
apply(case_tac "rsimp r11 = RZERO")
|
|
|
1044 |
apply simp
|
|
|
1045 |
using hrewrite.intros(1) hrewrites_seq_context apply blast
|
|
|
1046 |
apply(case_tac "rsimp r12 = RZERO")
|
|
|
1047 |
apply simp
|
|
|
1048 |
apply(subst basic_rsimp_SEQ_property3)
|
|
|
1049 |
apply (meson hrewrite.intros(2) hrewrites.simps hrewrites_seq_context2)
|
|
|
1050 |
apply(subst idiot2)
|
|
|
1051 |
apply simp+
|
|
|
1052 |
using hrewrites_seq_contexts apply presburger
|
|
|
1053 |
apply simp
|
|
|
1054 |
apply(subgoal_tac "RALTS x h\<leadsto>* RALTS (map rsimp x)")
|
|
|
1055 |
apply(subgoal_tac "RALTS (map rsimp x) h\<leadsto>* rsimp_ALTs (rdistinct (rflts (map rsimp x)) {}) ")
|
|
|
1056 |
using hreal_trans apply blast
|
|
|
1057 |
apply (meson flts_gstar greal_trans grewrites_ralts_rsimpalts gstar_rdistinct)
|
|
|
1058 |
|
|
|
1059 |
apply (simp add: grewrites_ralts hrewrites_list)
|
|
|
1060 |
by simp
|
|
|
1061 |
|
|
|
1062 |
lemma interleave_aux1:
|
|
|
1063 |
shows " RALT (RSEQ RZERO r1) r h\<leadsto>* r"
|
|
|
1064 |
apply(subgoal_tac "RSEQ RZERO r1 h\<leadsto>* RZERO")
|
|
|
1065 |
apply(subgoal_tac "RALT (RSEQ RZERO r1) r h\<leadsto>* RALT RZERO r")
|
|
|
1066 |
apply (meson grewrite.intros(1) grewrite_ralts hreal_trans hrewrite.intros(10) hrewrites.simps)
|
|
|
1067 |
using rs1 srewritescf_alt1 ss1 ss2 apply presburger
|
|
|
1068 |
by (simp add: hr_in_rstar hrewrite.intros(1))
|
|
|
1069 |
|
|
|
1070 |
|
|
|
1071 |
|
|
|
1072 |
lemma rnullable_hrewrite:
|
|
|
1073 |
shows "r1 h\<leadsto> r2 \<Longrightarrow> rnullable r1 = rnullable r2"
|
|
|
1074 |
apply(induct rule: hrewrite.induct)
|
|
|
1075 |
apply simp+
|
|
|
1076 |
apply blast
|
|
|
1077 |
apply simp+
|
|
|
1078 |
done
|
|
|
1079 |
|
|
|
1080 |
|
|
|
1081 |
lemma interleave1:
|
|
|
1082 |
shows "r h\<leadsto> r' \<Longrightarrow> rder c r h\<leadsto>* rder c r'"
|
|
|
1083 |
apply(induct r r' rule: hrewrite.induct)
|
|
|
1084 |
apply (simp add: hr_in_rstar hrewrite.intros(1))
|
|
|
1085 |
apply (metis (no_types, lifting) basic_rsimp_SEQ_property3 list.simps(8) list.simps(9) rder.simps(1) rder.simps(5) rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(1) rsimp.simps(2) rsimp.simps(3) rsimp_ALTs.simps(1) simp_hrewrites)
|
|
|
1086 |
apply simp
|
|
|
1087 |
apply(subst interleave_aux1)
|
|
|
1088 |
apply simp
|
|
|
1089 |
apply(case_tac "rnullable r1")
|
|
|
1090 |
apply simp
|
|
|
1091 |
|
|
|
1092 |
apply (simp add: hrewrites_seq_context rnullable_hrewrite srewritescf_alt1 ss1 ss2)
|
|
|
1093 |
|
|
|
1094 |
apply (simp add: hrewrites_seq_context rnullable_hrewrite)
|
|
|
1095 |
apply(case_tac "rnullable r1")
|
|
|
1096 |
apply simp
|
|
|
1097 |
|
|
|
1098 |
using hr_in_rstar hrewrites_seq_context2 srewritescf_alt1 ss1 ss2 apply presburger
|
|
|
1099 |
apply simp
|
|
|
1100 |
using hr_in_rstar hrewrites_seq_context2 apply blast
|
|
|
1101 |
apply simp
|
|
|
1102 |
|
|
|
1103 |
using hrewrites_alts apply auto[1]
|
|
|
1104 |
apply simp
|
|
|
1105 |
using grewrite.intros(1) grewrite_append grewrite_ralts apply auto[1]
|
|
|
1106 |
apply simp
|
|
|
1107 |
apply (simp add: grewrite.intros(2) grewrite_append grewrite_ralts)
|
|
|
1108 |
apply (simp add: hr_in_rstar hrewrite.intros(9))
|
|
|
1109 |
apply (simp add: hr_in_rstar hrewrite.intros(10))
|
|
|
1110 |
apply simp
|
|
|
1111 |
using hrewrite.intros(11) by auto
|
|
|
1112 |
|
|
|
1113 |
lemma interleave_star1:
|
|
|
1114 |
shows "r h\<leadsto>* r' \<Longrightarrow> rder c r h\<leadsto>* rder c r'"
|
|
|
1115 |
apply(induct rule : hrewrites.induct)
|
|
|
1116 |
apply simp
|
|
|
1117 |
by (meson hreal_trans interleave1)
|
|
|
1118 |
|
|
|
1119 |
|
|
|
1120 |
|
|
|
1121 |
lemma inside_simp_removal:
|
|
|
1122 |
shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
|
|
|
1123 |
apply(induct r)
|
|
|
1124 |
apply simp+
|
|
|
1125 |
apply(case_tac "rnullable r1")
|
|
|
1126 |
apply simp
|
|
|
1127 |
|
|
|
1128 |
using inside_simp_seq_nullable apply blast
|
|
|
1129 |
apply simp
|
|
|
1130 |
apply (smt (verit, del_insts) idiot2 basic_rsimp_SEQ_property3 der_simp_nullability rder.simps(1) rder.simps(5) rnullable.simps(2) rsimp.simps(1) rsimp_SEQ.simps(1) rsimp_idem)
|
|
|
1131 |
apply(subgoal_tac "rder x (RALTS xa) h\<leadsto>* rder x (rsimp (RALTS xa))")
|
|
|
1132 |
using hrewrites_simpeq apply presburger
|
|
|
1133 |
using interleave_star1 simp_hrewrites apply presburger
|
|
|
1134 |
by simp
|
|
|
1135 |
|
|
|
1136 |
|
|
|
1137 |
|
|
|
1138 |
|
|
|
1139 |
lemma rders_simp_same_simpders:
|
|
|
1140 |
shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
|
|
|
1141 |
apply(induct s rule: rev_induct)
|
|
|
1142 |
apply simp
|
|
|
1143 |
apply(case_tac "xs = []")
|
|
|
1144 |
apply simp
|
|
|
1145 |
apply(simp add: rders_append rders_simp_append)
|
|
|
1146 |
using inside_simp_removal by blast
|
|
|
1147 |
|
|
|
1148 |
|
|
|
1149 |
|
|
|
1150 |
|
|
|
1151 |
lemma distinct_der:
|
|
|
1152 |
shows "rsimp (rsimp_ALTs (map (rder x) (rdistinct rs {}))) =
|
|
|
1153 |
rsimp (rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
|
|
|
1154 |
by (metis grewrites_simpalts gstar_rdistinct inside_simp_removal rder_rsimp_ALTs_commute)
|
|
|
1155 |
|
|
|
1156 |
|
|
|
1157 |
|
|
|
1158 |
|
|
|
1159 |
|
|
|
1160 |
lemma rders_simp_lambda:
|
|
|
1161 |
shows " rsimp \<circ> rder x \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r (xs @ [x]))"
|
|
|
1162 |
using rders_simp_append by auto
|
|
|
1163 |
|
|
|
1164 |
lemma rders_simp_nonempty_simped:
|
|
|
1165 |
shows "xs \<noteq> [] \<Longrightarrow> rsimp \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r xs)"
|
|
|
1166 |
using rders_simp_same_simpders rsimp_idem by auto
|
|
|
1167 |
|
|
|
1168 |
lemma repeated_altssimp:
|
|
|
1169 |
shows "\<forall>r \<in> set rs. rsimp r = r \<Longrightarrow> rsimp (rsimp_ALTs (rdistinct (rflts rs) {})) =
|
|
|
1170 |
rsimp_ALTs (rdistinct (rflts rs) {})"
|
|
|
1171 |
by (metis map_idI rsimp.simps(2) rsimp_idem)
|
|
|
1172 |
|
|
|
1173 |
|
|
|
1174 |
|
|
|
1175 |
lemma alts_closed_form:
|
|
|
1176 |
shows "rsimp (rders_simp (RALTS rs) s) = rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
|
1177 |
apply(induct s rule: rev_induct)
|
|
|
1178 |
apply simp
|
|
|
1179 |
apply simp
|
|
|
1180 |
apply(subst rders_simp_append)
|
|
|
1181 |
apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) =
|
|
|
1182 |
rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
|
|
|
1183 |
prefer 2
|
|
|
1184 |
apply (metis inside_simp_removal rders_simp_one_char)
|
|
|
1185 |
apply(simp only: )
|
|
|
1186 |
apply(subst rders_simp_one_char)
|
|
|
1187 |
apply(subst rsimp_idem)
|
|
|
1188 |
apply(subgoal_tac "rsimp (rder x (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {}))) =
|
|
|
1189 |
rsimp ((rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))) ")
|
|
|
1190 |
prefer 2
|
|
|
1191 |
using rder_rsimp_ALTs_commute apply presburger
|
|
|
1192 |
apply(simp only:)
|
|
|
1193 |
apply(subgoal_tac "rsimp (rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))
|
|
|
1194 |
= rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
1195 |
prefer 2
|
|
|
1196 |
|
|
|
1197 |
using distinct_der apply presburger
|
|
|
1198 |
apply(simp only:)
|
|
|
1199 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
1200 |
rsimp (rsimp_ALTs (rdistinct ( (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)))) {}))")
|
|
|
1201 |
apply(simp only:)
|
|
|
1202 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
1203 |
rsimp (rsimp_ALTs (rdistinct (rflts ( (map (rsimp \<circ> (rder x) \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
1204 |
apply(simp only:)
|
|
|
1205 |
apply(subst rders_simp_lambda)
|
|
|
1206 |
apply(subst rders_simp_nonempty_simped)
|
|
|
1207 |
apply simp
|
|
|
1208 |
apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (xs @ [x])) rs). rsimp r = r")
|
|
|
1209 |
prefer 2
|
|
|
1210 |
apply (simp add: rders_simp_same_simpders rsimp_idem)
|
|
|
1211 |
apply(subst repeated_altssimp)
|
|
|
1212 |
apply simp
|
|
|
1213 |
apply fastforce
|
|
|
1214 |
apply (metis inside_simp_removal list.map_comp rder.simps(4) rsimp.simps(2) rsimp_idem)
|
|
|
1215 |
using simp_der_pierce_flts_prelim by blast
|
|
|
1216 |
|
|
|
1217 |
|
|
|
1218 |
lemma alts_closed_form_variant:
|
|
|
1219 |
shows "s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s = rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
|
1220 |
by (metis alts_closed_form comp_apply rders_simp_nonempty_simped)
|
|
|
1221 |
|
|
|
1222 |
|
|
|
1223 |
lemma rsimp_seq_equal1:
|
|
|
1224 |
shows "rsimp_SEQ (rsimp r1) (rsimp r2) = rsimp_ALTs (rdistinct (rflts [rsimp_SEQ (rsimp r1) (rsimp r2)]) {})"
|
|
|
1225 |
by (metis idem_after_simp1 rsimp.simps(1))
|
|
|
1226 |
|
|
|
1227 |
|
|
|
1228 |
fun sflat_aux :: "rrexp \<Rightarrow> rrexp list " where
|
|
|
1229 |
"sflat_aux (RALTS (r # rs)) = sflat_aux r @ rs"
|
|
|
1230 |
| "sflat_aux (RALTS []) = []"
|
|
|
1231 |
| "sflat_aux r = [r]"
|
|
|
1232 |
|
|
|
1233 |
|
|
|
1234 |
fun sflat :: "rrexp \<Rightarrow> rrexp" where
|
|
|
1235 |
"sflat (RALTS (r # [])) = r"
|
|
|
1236 |
| "sflat (RALTS (r # rs)) = RALTS (sflat_aux r @ rs)"
|
|
|
1237 |
| "sflat r = r"
|
|
|
1238 |
|
|
|
1239 |
inductive created_by_seq:: "rrexp \<Rightarrow> bool" where
|
|
|
1240 |
"created_by_seq (RSEQ r1 r2) "
|
|
|
1241 |
| "created_by_seq r1 \<Longrightarrow> created_by_seq (RALT r1 r2)"
|
|
|
1242 |
|
|
|
1243 |
lemma seq_ders_shape1:
|
|
|
1244 |
shows "\<forall>r1 r2. \<exists>r3 r4. (rders (RSEQ r1 r2) s) = RSEQ r3 r4 \<or> (rders (RSEQ r1 r2) s) = RALT r3 r4"
|
|
|
1245 |
apply(induct s rule: rev_induct)
|
|
|
1246 |
apply auto[1]
|
|
|
1247 |
apply(rule allI)+
|
|
|
1248 |
apply(subst rders_append)+
|
|
|
1249 |
apply(subgoal_tac " \<exists>r3 r4. rders (RSEQ r1 r2) xs = RSEQ r3 r4 \<or> rders (RSEQ r1 r2) xs = RALT r3 r4 ")
|
|
|
1250 |
apply(erule exE)+
|
|
|
1251 |
apply(erule disjE)
|
|
|
1252 |
apply simp+
|
|
|
1253 |
done
|
|
|
1254 |
|
|
|
1255 |
lemma created_by_seq_der:
|
|
|
1256 |
shows "created_by_seq r \<Longrightarrow> created_by_seq (rder c r)"
|
|
|
1257 |
apply(induct r)
|
|
|
1258 |
apply simp+
|
|
|
1259 |
|
|
|
1260 |
using created_by_seq.cases apply blast
|
|
|
1261 |
|
|
|
1262 |
apply (meson created_by_seq.cases rrexp.distinct(19) rrexp.distinct(21))
|
|
|
1263 |
apply (metis created_by_seq.simps rder.simps(5))
|
|
|
1264 |
apply (smt (verit, ccfv_threshold) created_by_seq.simps list.set_intros(1) list.simps(8) list.simps(9) rder.simps(4) rrexp.distinct(25) rrexp.inject(3))
|
|
|
1265 |
using created_by_seq.intros(1) by force
|
|
|
1266 |
|
|
|
1267 |
lemma createdbyseq_left_creatable:
|
|
|
1268 |
shows "created_by_seq (RALT r1 r2) \<Longrightarrow> created_by_seq r1"
|
|
|
1269 |
using created_by_seq.cases by blast
|
|
|
1270 |
|
|
|
1271 |
|
|
|
1272 |
|
|
|
1273 |
lemma recursively_derseq:
|
|
|
1274 |
shows " created_by_seq (rders (RSEQ r1 r2) s)"
|
|
|
1275 |
apply(induct s rule: rev_induct)
|
|
|
1276 |
apply simp
|
|
|
1277 |
using created_by_seq.intros(1) apply force
|
|
|
1278 |
apply(subgoal_tac "created_by_seq (rders (RSEQ r1 r2) (xs @ [x]))")
|
|
|
1279 |
apply blast
|
|
|
1280 |
apply(subst rders_append)
|
|
|
1281 |
apply(subgoal_tac "\<exists>r3 r4. rders (RSEQ r1 r2) xs = RSEQ r3 r4 \<or>
|
|
|
1282 |
rders (RSEQ r1 r2) xs = RALT r3 r4")
|
|
|
1283 |
prefer 2
|
|
|
1284 |
using seq_ders_shape1 apply presburger
|
|
|
1285 |
apply(erule exE)+
|
|
|
1286 |
apply(erule disjE)
|
|
|
1287 |
apply(subgoal_tac "created_by_seq (rders (RSEQ r3 r4) [x])")
|
|
|
1288 |
apply presburger
|
|
|
1289 |
apply simp
|
|
|
1290 |
using created_by_seq.intros(1) created_by_seq.intros(2) apply presburger
|
|
|
1291 |
apply simp
|
|
|
1292 |
apply(subgoal_tac "created_by_seq r3")
|
|
|
1293 |
prefer 2
|
|
|
1294 |
using createdbyseq_left_creatable apply blast
|
|
|
1295 |
using created_by_seq.intros(2) created_by_seq_der by blast
|
|
|
1296 |
|
|
|
1297 |
|
|
|
1298 |
lemma recursively_derseq1:
|
|
|
1299 |
shows "r = rders (RSEQ r1 r2) s \<Longrightarrow> created_by_seq r"
|
|
|
1300 |
using recursively_derseq by blast
|
|
|
1301 |
|
|
|
1302 |
|
|
|
1303 |
lemma sfau_head:
|
|
|
1304 |
shows " created_by_seq r \<Longrightarrow> \<exists>ra rb rs. sflat_aux r = RSEQ ra rb # rs"
|
|
|
1305 |
apply(induction r rule: created_by_seq.induct)
|
|
|
1306 |
apply simp
|
|
|
1307 |
by fastforce
|
|
|
1308 |
|
|
|
1309 |
|
|
|
1310 |
lemma vsuf_prop1:
|
|
|
1311 |
shows "vsuf (xs @ [x]) r = (if (rnullable (rders r xs))
|
|
|
1312 |
then [x] # (map (\<lambda>s. s @ [x]) (vsuf xs r) )
|
|
|
1313 |
else (map (\<lambda>s. s @ [x]) (vsuf xs r)) )
|
|
|
1314 |
"
|
|
|
1315 |
apply(induct xs arbitrary: r)
|
|
|
1316 |
apply simp
|
|
|
1317 |
apply(case_tac "rnullable r")
|
|
|
1318 |
apply simp
|
|
|
1319 |
apply simp
|
|
|
1320 |
done
|
|
|
1321 |
|
|
|
1322 |
fun breakHead :: "rrexp list \<Rightarrow> rrexp list" where
|
|
|
1323 |
"breakHead [] = [] "
|
|
|
1324 |
| "breakHead (RALT r1 r2 # rs) = r1 # r2 # rs"
|
|
|
1325 |
| "breakHead (r # rs) = r # rs"
|
|
|
1326 |
|
|
|
1327 |
|
|
|
1328 |
lemma sfau_idem_der:
|
|
|
1329 |
shows "created_by_seq r \<Longrightarrow> sflat_aux (rder c r) = breakHead (map (rder c) (sflat_aux r))"
|
|
|
1330 |
apply(induct rule: created_by_seq.induct)
|
|
|
1331 |
apply simp+
|
|
|
1332 |
using sfau_head by fastforce
|
|
|
1333 |
|
|
|
1334 |
lemma vsuf_compose1:
|
|
|
1335 |
shows " \<not> rnullable (rders r1 xs)
|
|
|
1336 |
\<Longrightarrow> map (rder x \<circ> rders r2) (vsuf xs r1) = map (rders r2) (vsuf (xs @ [x]) r1)"
|
|
|
1337 |
apply(subst vsuf_prop1)
|
|
|
1338 |
apply simp
|
|
|
1339 |
by (simp add: rders_append)
|
|
|
1340 |
|
|
|
1341 |
thm sfau_idem_der
|
|
|
1342 |
|
|
|
1343 |
lemma oneCharvsuf:
|
|
|
1344 |
shows "breakHead [rder x (RSEQ r1 r2)] = RSEQ (rder x r1) r2 # map (rders r2) (vsuf [x] r1)"
|
|
|
1345 |
by simp
|
|
|
1346 |
|
|
|
1347 |
|
|
|
1348 |
lemma vsuf_compose2:
|
|
|
1349 |
shows "(map (rders r2) (vsuf [x] (rders r1 xs))) @ map (rder x) (map (rders r2) (vsuf xs r1)) = map (rders r2) (vsuf (xs @ [x]) r1)"
|
|
|
1350 |
proof(induct xs arbitrary: r1)
|
|
|
1351 |
case Nil
|
|
|
1352 |
then show ?case
|
|
|
1353 |
by simp
|
|
|
1354 |
next
|
|
|
1355 |
case (Cons a xs)
|
|
|
1356 |
have "rnullable (rders r1 xs) \<longrightarrow> map (rders r2) (vsuf [x] (rders r1 (a # xs))) @ map (rder x) (map (rders r2) (vsuf (a # xs) r1)) =
|
|
|
1357 |
map (rders r2) (vsuf ((a # xs) @ [x]) r1)"
|
|
|
1358 |
proof
|
|
|
1359 |
assume nullableCond: "rnullable (rders r1 xs)"
|
|
|
1360 |
have "rnullable r1 \<longrightarrow> rder x (rders (rder a r2) xs) = rders (rder a r2) (xs @ [x])"
|
|
|
1361 |
by (simp add: rders_append)
|
|
|
1362 |
show " map (rders r2) (vsuf [x] (rders r1 (a # xs))) @ map (rder x) (map (rders r2) (vsuf (a # xs) r1)) =
|
|
|
1363 |
map (rders r2) (vsuf ((a # xs) @ [x]) r1)"
|
|
|
1364 |
using \<open>rnullable r1 \<longrightarrow> rder x (rders (rder a r2) xs) = rders (rder a r2) (xs @ [x])\<close> local.Cons by auto
|
|
|
1365 |
qed
|
|
|
1366 |
then have "\<not> rnullable (rders r1 xs) \<longrightarrow> map (rders r2) (vsuf [x] (rders r1 (a # xs))) @ map (rder x) (map (rders r2) (vsuf (a # xs) r1)) =
|
|
|
1367 |
map (rders r2) (vsuf ((a # xs) @ [x]) r1)"
|
|
|
1368 |
apply simp
|
|
|
1369 |
by (smt (verit, ccfv_threshold) append_Cons append_Nil list.map_comp list.simps(8) list.simps(9) local.Cons rders.simps(1) rders.simps(2) rders_append vsuf.simps(1) vsuf.simps(2))
|
|
|
1370 |
then show ?case
|
|
|
1371 |
using \<open>rnullable (rders r1 xs) \<longrightarrow> map (rders r2) (vsuf [x] (rders r1 (a # xs))) @ map (rder x) (map (rders r2) (vsuf (a # xs) r1)) = map (rders r2) (vsuf ((a # xs) @ [x]) r1)\<close> by blast
|
|
|
1372 |
qed
|
|
|
1373 |
|
|
|
1374 |
|
|
|
1375 |
lemma seq_sfau0:
|
|
|
1376 |
shows "sflat_aux (rders (RSEQ r1 r2) s) = (RSEQ (rders r1 s) r2) #
|
|
|
1377 |
(map (rders r2) (vsuf s r1)) "
|
|
|
1378 |
proof(induct s rule: rev_induct)
|
|
|
1379 |
case Nil
|
|
|
1380 |
then show ?case
|
|
|
1381 |
by simp
|
|
|
1382 |
next
|
|
|
1383 |
case (snoc x xs)
|
|
|
1384 |
then have LHS1:"sflat_aux (rders (RSEQ r1 r2) (xs @ [x])) = sflat_aux (rder x (rders (RSEQ r1 r2) xs)) "
|
|
|
1385 |
by (simp add: rders_append)
|
|
|
1386 |
then have LHS1A: "... = breakHead (map (rder x) (sflat_aux (rders (RSEQ r1 r2) xs)))"
|
|
|
1387 |
using recursively_derseq sfau_idem_der by auto
|
|
|
1388 |
then have LHS1B: "... = breakHead (map (rder x) (RSEQ (rders r1 xs) r2 # map (rders r2) (vsuf xs r1)))"
|
|
|
1389 |
using snoc by auto
|
|
|
1390 |
then have LHS1C: "... = breakHead (rder x (RSEQ (rders r1 xs) r2) # map (rder x) (map (rders r2) (vsuf xs r1)))"
|
|
|
1391 |
by simp
|
|
|
1392 |
then have LHS1D: "... = breakHead [rder x (RSEQ (rders r1 xs) r2)] @ map (rder x) (map (rders r2) (vsuf xs r1))"
|
|
|
1393 |
by simp
|
|
|
1394 |
then have LHS1E: "... = RSEQ (rder x (rders r1 xs)) r2 # (map (rders r2) (vsuf [x] (rders r1 xs))) @ map (rder x) (map (rders r2) (vsuf xs r1))"
|
|
|
1395 |
by force
|
|
|
1396 |
then have LHS1F: "... = RSEQ (rder x (rders r1 xs)) r2 # (map (rders r2) (vsuf (xs @ [x]) r1))"
|
|
|
1397 |
using vsuf_compose2 by blast
|
|
|
1398 |
then have LHS1G: "... = RSEQ (rders r1 (xs @ [x])) r2 # (map (rders r2) (vsuf (xs @ [x]) r1))"
|
|
|
1399 |
using rders.simps(1) rders.simps(2) rders_append by presburger
|
|
|
1400 |
then show ?case
|
|
|
1401 |
using LHS1 LHS1A LHS1C LHS1D LHS1E LHS1F snoc by presburger
|
|
|
1402 |
qed
|
|
|
1403 |
|
|
|
1404 |
|
|
|
1405 |
|
|
|
1406 |
|
|
|
1407 |
|
|
|
1408 |
|
|
|
1409 |
lemma sflat_rsimpeq:
|
|
|
1410 |
shows "created_by_seq r1 \<Longrightarrow> sflat_aux r1 = rs \<Longrightarrow> rsimp r1 = rsimp (RALTS rs)"
|
|
|
1411 |
apply(induct r1 arbitrary: rs rule: created_by_seq.induct)
|
|
|
1412 |
apply simp
|
|
|
1413 |
using rsimp_seq_equal1 apply force
|
|
|
1414 |
by (metis head_one_more_simp rsimp.simps(2) sflat_aux.simps(1) simp_flatten)
|
|
|
1415 |
|
|
|
1416 |
|
|
|
1417 |
|
|
|
1418 |
lemma seq_closed_form_general:
|
|
|
1419 |
shows "rsimp (rders (RSEQ r1 r2) s) =
|
|
|
1420 |
rsimp ( (RALTS ( (RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1))))))"
|
|
|
1421 |
apply(case_tac "s \<noteq> []")
|
|
|
1422 |
apply(subgoal_tac "created_by_seq (rders (RSEQ r1 r2) s)")
|
|
|
1423 |
apply(subgoal_tac "sflat_aux (rders (RSEQ r1 r2) s) = RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1))")
|
|
|
1424 |
using sflat_rsimpeq apply blast
|
|
|
1425 |
apply (simp add: seq_sfau0)
|
|
|
1426 |
using recursively_derseq1 apply blast
|
|
|
1427 |
apply simp
|
|
|
1428 |
by (metis idem_after_simp1 rsimp.simps(1))
|
|
|
1429 |
|
|
|
1430 |
lemma seq_closed_form_aux1a:
|
|
|
1431 |
shows "rsimp (RALTS (RSEQ (rders r1 s) r2 # rs)) =
|
|
|
1432 |
rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # rs))"
|
|
|
1433 |
by (metis head_one_more_simp rders.simps(1) rders_simp.simps(1) rders_simp_same_simpders rsimp.simps(1) rsimp_idem simp_flatten_aux0)
|
|
|
1434 |
|
|
|
1435 |
|
|
|
1436 |
lemma seq_closed_form_aux1:
|
|
|
1437 |
shows "rsimp (RALTS (RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1)))) =
|
|
|
1438 |
rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # (map (rders r2) (vsuf s r1))))"
|
|
|
1439 |
by (smt (verit, best) list.simps(9) rders.simps(1) rders_simp.simps(1) rders_simp_same_simpders rsimp.simps(1) rsimp.simps(2) rsimp_idem)
|
|
|
1440 |
|
|
|
1441 |
lemma add_simp_to_rest:
|
|
|
1442 |
shows "rsimp (RALTS (r # rs)) = rsimp (RALTS (r # map rsimp rs))"
|
|
|
1443 |
by (metis append_Nil2 grewrite.intros(2) grewrite_simpalts head_one_more_simp list.simps(9) rsimp_ALTs.simps(2) spawn_simp_rsimpalts)
|
|
|
1444 |
|
|
|
1445 |
lemma rsimp_compose_der2:
|
|
|
1446 |
shows "\<forall>s \<in> set ss. s \<noteq> [] \<Longrightarrow> map rsimp (map (rders r) ss) = map (\<lambda>s. (rders_simp r s)) ss"
|
|
|
1447 |
by (simp add: rders_simp_same_simpders)
|
|
|
1448 |
|
|
|
1449 |
lemma vsuf_nonempty:
|
|
|
1450 |
shows "\<forall>s \<in> set ( vsuf s1 r). s \<noteq> []"
|
|
|
1451 |
apply(induct s1 arbitrary: r)
|
|
|
1452 |
apply simp
|
|
|
1453 |
apply simp
|
|
|
1454 |
done
|
|
|
1455 |
|
|
|
1456 |
|
|
|
1457 |
|
|
|
1458 |
lemma seq_closed_form_aux2:
|
|
|
1459 |
shows "s \<noteq> [] \<Longrightarrow> rsimp ( (RALTS ( (RSEQ (rders_simp r1 s) r2 # (map (rders r2) (vsuf s r1)))))) =
|
|
|
1460 |
rsimp ( (RALTS ( (RSEQ (rders_simp r1 s) r2 # (map (rders_simp r2) (vsuf s r1))))))"
|
|
|
1461 |
|
|
|
1462 |
by (metis add_simp_to_rest rsimp_compose_der2 vsuf_nonempty)
|
|
|
1463 |
|
|
|
1464 |
|
|
|
1465 |
lemma seq_closed_form:
|
|
|
1466 |
shows "rsimp (rders_simp (RSEQ r1 r2) s) =
|
|
|
1467 |
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
|
|
|
1468 |
proof (cases s)
|
|
|
1469 |
case Nil
|
|
|
1470 |
then show ?thesis
|
|
|
1471 |
by (simp add: rsimp_seq_equal1[symmetric])
|
|
|
1472 |
next
|
|
|
1473 |
case (Cons a list)
|
|
|
1474 |
have "rsimp (rders_simp (RSEQ r1 r2) s) = rsimp (rsimp (rders (RSEQ r1 r2) s))"
|
|
|
1475 |
using local.Cons by (subst rders_simp_same_simpders)(simp_all)
|
|
|
1476 |
also have "... = rsimp (rders (RSEQ r1 r2) s)"
|
|
|
1477 |
by (simp add: rsimp_idem)
|
|
|
1478 |
also have "... = rsimp (RALTS (RSEQ (rders r1 s) r2 # map (rders r2) (vsuf s r1)))"
|
|
|
1479 |
using seq_closed_form_general by blast
|
|
|
1480 |
also have "... = rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders r2) (vsuf s r1)))"
|
|
|
1481 |
by (simp only: seq_closed_form_aux1)
|
|
|
1482 |
also have "... = rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)))"
|
|
|
1483 |
using local.Cons by (subst seq_closed_form_aux2)(simp_all)
|
|
|
1484 |
finally show ?thesis .
|
|
|
1485 |
qed
|
|
|
1486 |
|
|
|
1487 |
lemma q: "s \<noteq> [] \<Longrightarrow> rders_simp (RSEQ r1 r2) s = rsimp (rders_simp (RSEQ r1 r2) s)"
|
|
|
1488 |
using rders_simp_same_simpders rsimp_idem by presburger
|
|
|
1489 |
|
|
|
1490 |
|
|
|
1491 |
lemma seq_closed_form_variant:
|
|
|
1492 |
assumes "s \<noteq> []"
|
|
|
1493 |
shows "rders_simp (RSEQ r1 r2) s =
|
|
543
|
1494 |
rsimp (RALTS (RSEQ (rders_simp r1 s) r2 #
|
|
|
1495 |
(map (rders_simp r2) (vsuf s r1))))"
|
|
505
|
1496 |
using assms q seq_closed_form by force
|
|
|
1497 |
|
|
|
1498 |
|
|
|
1499 |
fun hflat_aux :: "rrexp \<Rightarrow> rrexp list" where
|
|
|
1500 |
"hflat_aux (RALT r1 r2) = hflat_aux r1 @ hflat_aux r2"
|
|
|
1501 |
| "hflat_aux r = [r]"
|
|
|
1502 |
|
|
|
1503 |
|
|
|
1504 |
fun hflat :: "rrexp \<Rightarrow> rrexp" where
|
|
|
1505 |
"hflat (RALT r1 r2) = RALTS ((hflat_aux r1) @ (hflat_aux r2))"
|
|
|
1506 |
| "hflat r = r"
|
|
|
1507 |
|
|
|
1508 |
inductive created_by_star :: "rrexp \<Rightarrow> bool" where
|
|
|
1509 |
"created_by_star (RSEQ ra (RSTAR rb))"
|
|
|
1510 |
| "\<lbrakk>created_by_star r1; created_by_star r2\<rbrakk> \<Longrightarrow> created_by_star (RALT r1 r2)"
|
|
|
1511 |
|
|
|
1512 |
|
|
|
1513 |
|
|
|
1514 |
|
|
|
1515 |
lemma cbs_ders_cbs:
|
|
|
1516 |
shows "created_by_star r \<Longrightarrow> created_by_star (rder c r)"
|
|
|
1517 |
apply(induct r rule: created_by_star.induct)
|
|
|
1518 |
apply simp
|
|
|
1519 |
using created_by_star.intros(1) created_by_star.intros(2) apply auto[1]
|
|
|
1520 |
by (metis (mono_tags, lifting) created_by_star.simps list.simps(8) list.simps(9) rder.simps(4))
|
|
|
1521 |
|
|
|
1522 |
lemma star_ders_cbs:
|
|
|
1523 |
shows "created_by_star (rders (RSEQ r1 (RSTAR r2)) s)"
|
|
|
1524 |
apply(induct s rule: rev_induct)
|
|
|
1525 |
apply simp
|
|
|
1526 |
apply (simp add: created_by_star.intros(1))
|
|
|
1527 |
apply(subst rders_append)
|
|
|
1528 |
apply simp
|
|
|
1529 |
using cbs_ders_cbs by auto
|
|
|
1530 |
|
|
|
1531 |
(*
|
|
|
1532 |
lemma created_by_star_cases:
|
|
|
1533 |
shows "created_by_star r \<Longrightarrow> \<exists>ra rb. (r = RALT ra rb \<and> created_by_star ra \<and> created_by_star rb) \<or> r = RSEQ ra rb "
|
|
|
1534 |
by (meson created_by_star.cases)
|
|
|
1535 |
*)
|
|
|
1536 |
|
|
|
1537 |
|
|
|
1538 |
lemma hfau_pushin:
|
|
|
1539 |
shows "created_by_star r \<Longrightarrow> hflat_aux (rder c r) = concat (map hflat_aux (map (rder c) (hflat_aux r)))"
|
|
506
|
1540 |
|
|
|
1541 |
proof(induct r rule: created_by_star.induct)
|
|
|
1542 |
case (1 ra rb)
|
|
|
1543 |
then show ?case by simp
|
|
|
1544 |
next
|
|
|
1545 |
case (2 r1 r2)
|
|
|
1546 |
then have "created_by_star (rder c r1)"
|
|
|
1547 |
using cbs_ders_cbs by blast
|
|
|
1548 |
then have "created_by_star (rder c r2)"
|
|
|
1549 |
using "2.hyps"(3) cbs_ders_cbs by auto
|
|
|
1550 |
then show ?case
|
|
|
1551 |
by (simp add: "2.hyps"(2) "2.hyps"(4))
|
|
|
1552 |
qed
|
|
|
1553 |
|
|
513
|
1554 |
(*AALTS [a\x . b.c, b\x .c, c \x]*)
|
|
|
1555 |
(*AALTS [a\x . b.c, AALTS [b\x .c, c\x]]*)
|
|
505
|
1556 |
|
|
|
1557 |
lemma stupdate_induct1:
|
|
|
1558 |
shows " concat (map (hflat_aux \<circ> (rder x \<circ> (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)))) Ss) =
|
|
|
1559 |
map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_update x r0 Ss)"
|
|
|
1560 |
apply(induct Ss)
|
|
|
1561 |
apply simp+
|
|
|
1562 |
by (simp add: rders_append)
|
|
|
1563 |
|
|
|
1564 |
|
|
|
1565 |
|
|
|
1566 |
lemma stupdates_join_general:
|
|
|
1567 |
shows "concat (map hflat_aux (map (rder x) (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates xs r0 Ss)))) =
|
|
|
1568 |
map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates (xs @ [x]) r0 Ss)"
|
|
|
1569 |
apply(induct xs arbitrary: Ss)
|
|
|
1570 |
apply (simp)
|
|
|
1571 |
prefer 2
|
|
|
1572 |
apply auto[1]
|
|
|
1573 |
using stupdate_induct1 by blast
|
|
|
1574 |
|
|
|
1575 |
lemma star_hfau_induct:
|
|
|
1576 |
shows "hflat_aux (rders (RSEQ (rder c r0) (RSTAR r0)) s) =
|
|
|
1577 |
map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates s r0 [[c]])"
|
|
|
1578 |
apply(induct s rule: rev_induct)
|
|
|
1579 |
apply simp
|
|
|
1580 |
apply(subst rders_append)+
|
|
|
1581 |
apply simp
|
|
|
1582 |
apply(subst stupdates_append)
|
|
|
1583 |
apply(subgoal_tac "created_by_star (rders (RSEQ (rder c r0) (RSTAR r0)) xs)")
|
|
|
1584 |
prefer 2
|
|
|
1585 |
apply (simp add: star_ders_cbs)
|
|
|
1586 |
apply(subst hfau_pushin)
|
|
|
1587 |
apply simp
|
|
|
1588 |
apply(subgoal_tac "concat (map hflat_aux (map (rder x) (hflat_aux (rders (RSEQ (rder c r0) (RSTAR r0)) xs)))) =
|
|
|
1589 |
concat (map hflat_aux (map (rder x) ( map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates xs r0 [[c]])))) ")
|
|
|
1590 |
apply(simp only:)
|
|
|
1591 |
prefer 2
|
|
|
1592 |
apply presburger
|
|
|
1593 |
apply(subst stupdates_append[symmetric])
|
|
|
1594 |
using stupdates_join_general by blast
|
|
|
1595 |
|
|
|
1596 |
lemma starders_hfau_also1:
|
|
|
1597 |
shows "hflat_aux (rders (RSTAR r) (c # xs)) = map (\<lambda>s1. RSEQ (rders r s1) (RSTAR r)) (star_updates xs r [[c]])"
|
|
|
1598 |
using star_hfau_induct by force
|
|
|
1599 |
|
|
|
1600 |
lemma hflat_aux_grewrites:
|
|
|
1601 |
shows "a # rs \<leadsto>g* hflat_aux a @ rs"
|
|
|
1602 |
apply(induct a arbitrary: rs)
|
|
|
1603 |
apply simp+
|
|
|
1604 |
apply(case_tac x)
|
|
|
1605 |
apply simp
|
|
|
1606 |
apply(case_tac list)
|
|
|
1607 |
|
|
|
1608 |
apply (metis append.right_neutral append_Cons append_eq_append_conv2 grewrites.simps hflat_aux.simps(7) same_append_eq)
|
|
|
1609 |
apply(case_tac lista)
|
|
|
1610 |
apply simp
|
|
|
1611 |
apply (metis (no_types, lifting) append_Cons append_eq_append_conv2 gmany_steps_later greal_trans grewrite.intros(2) grewrites_append self_append_conv)
|
|
|
1612 |
apply simp
|
|
|
1613 |
by simp
|
|
|
1614 |
|
|
|
1615 |
|
|
|
1616 |
|
|
|
1617 |
|
|
|
1618 |
lemma cbs_hfau_rsimpeq1:
|
|
|
1619 |
shows "rsimp (RALT a b) = rsimp (RALTS ((hflat_aux a) @ (hflat_aux b)))"
|
|
|
1620 |
apply(subgoal_tac "[a, b] \<leadsto>g* hflat_aux a @ hflat_aux b")
|
|
|
1621 |
using grewrites_equal_rsimp apply presburger
|
|
|
1622 |
by (metis append.right_neutral greal_trans grewrites_cons hflat_aux_grewrites)
|
|
|
1623 |
|
|
|
1624 |
|
|
|
1625 |
lemma hfau_rsimpeq2:
|
|
|
1626 |
shows "created_by_star r \<Longrightarrow> rsimp r = rsimp ( (RALTS (hflat_aux r)))"
|
|
506
|
1627 |
apply(induct rule: created_by_star.induct)
|
|
|
1628 |
apply simp
|
|
|
1629 |
apply (metis rsimp.simps(6) rsimp_seq_equal1)
|
|
|
1630 |
using cbs_hfau_rsimpeq1 hflat_aux.simps(1) by presburger
|
|
|
1631 |
|
|
|
1632 |
|
|
|
1633 |
(*
|
|
|
1634 |
lemma hfau_rsimpeq2_oldproof: shows "created_by_star r \<Longrightarrow> rsimp r = rsimp ( (RALTS (hflat_aux r)))"
|
|
505
|
1635 |
apply(induct r)
|
|
|
1636 |
apply simp+
|
|
|
1637 |
apply (metis rsimp_seq_equal1)
|
|
|
1638 |
prefer 2
|
|
|
1639 |
apply simp
|
|
|
1640 |
apply(case_tac x)
|
|
|
1641 |
apply simp
|
|
|
1642 |
apply(case_tac "list")
|
|
|
1643 |
apply simp
|
|
|
1644 |
apply (metis idem_after_simp1)
|
|
|
1645 |
apply(case_tac "lista")
|
|
|
1646 |
prefer 2
|
|
|
1647 |
apply (metis hflat_aux.simps(8) idem_after_simp1 list.simps(8) list.simps(9) rsimp.simps(2))
|
|
|
1648 |
apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux (RALT a aa)))")
|
|
|
1649 |
apply simp
|
|
|
1650 |
apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux a @ hflat_aux aa))")
|
|
|
1651 |
using hflat_aux.simps(1) apply presburger
|
|
|
1652 |
apply simp
|
|
|
1653 |
using cbs_hfau_rsimpeq1 by fastforce
|
|
506
|
1654 |
*)
|
|
505
|
1655 |
|
|
|
1656 |
lemma star_closed_form1:
|
|
|
1657 |
shows "rsimp (rders (RSTAR r0) (c#s)) =
|
|
|
1658 |
rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
|
|
|
1659 |
using hfau_rsimpeq2 rder.simps(6) rders.simps(2) star_ders_cbs starders_hfau_also1 by presburger
|
|
|
1660 |
|
|
|
1661 |
lemma star_closed_form2:
|
|
|
1662 |
shows "rsimp (rders_simp (RSTAR r0) (c#s)) =
|
|
|
1663 |
rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
|
|
|
1664 |
by (metis list.distinct(1) rders_simp_same_simpders rsimp_idem star_closed_form1)
|
|
|
1665 |
|
|
|
1666 |
lemma star_closed_form3:
|
|
|
1667 |
shows "rsimp (rders_simp (RSTAR r0) (c#s)) = (rders_simp (RSTAR r0) (c#s))"
|
|
|
1668 |
by (metis list.distinct(1) rders_simp_same_simpders star_closed_form1 star_closed_form2)
|
|
|
1669 |
|
|
|
1670 |
lemma star_closed_form4:
|
|
|
1671 |
shows " (rders_simp (RSTAR r0) (c#s)) =
|
|
|
1672 |
rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
|
|
|
1673 |
using star_closed_form2 star_closed_form3 by presburger
|
|
|
1674 |
|
|
|
1675 |
lemma star_closed_form5:
|
|
|
1676 |
shows " rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) Ss )))) =
|
|
|
1677 |
rsimp ( ( RALTS ( (map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss ))))"
|
|
|
1678 |
by (metis (mono_tags, lifting) list.map_comp map_eq_conv o_apply rsimp.simps(2) rsimp_idem)
|
|
|
1679 |
|
|
|
1680 |
lemma star_closed_form6_hrewrites:
|
|
|
1681 |
shows "
|
|
|
1682 |
(map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss )
|
|
|
1683 |
scf\<leadsto>*
|
|
|
1684 |
(map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss )"
|
|
|
1685 |
apply(induct Ss)
|
|
|
1686 |
apply simp
|
|
|
1687 |
apply (simp add: ss1)
|
|
|
1688 |
by (metis (no_types, lifting) list.simps(9) rsimp.simps(1) rsimp_idem simp_hrewrites ss2)
|
|
|
1689 |
|
|
|
1690 |
lemma star_closed_form6:
|
|
|
1691 |
shows " rsimp ( ( RALTS ( (map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss )))) =
|
|
|
1692 |
rsimp ( ( RALTS ( (map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss ))))"
|
|
|
1693 |
apply(subgoal_tac " map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss scf\<leadsto>*
|
|
|
1694 |
map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss ")
|
|
|
1695 |
using hrewrites_simpeq srewritescf_alt1 apply fastforce
|
|
|
1696 |
using star_closed_form6_hrewrites by blast
|
|
|
1697 |
|
|
|
1698 |
lemma stupdate_nonempty:
|
|
|
1699 |
shows "\<forall>s \<in> set Ss. s \<noteq> [] \<Longrightarrow> \<forall>s \<in> set (star_update c r Ss). s \<noteq> []"
|
|
|
1700 |
apply(induct Ss)
|
|
|
1701 |
apply simp
|
|
|
1702 |
apply(case_tac "rnullable (rders r a)")
|
|
|
1703 |
apply simp+
|
|
|
1704 |
done
|
|
|
1705 |
|
|
|
1706 |
|
|
|
1707 |
lemma stupdates_nonempty:
|
|
|
1708 |
shows "\<forall>s \<in> set Ss. s\<noteq> [] \<Longrightarrow> \<forall>s \<in> set (star_updates s r Ss). s \<noteq> []"
|
|
|
1709 |
apply(induct s arbitrary: Ss)
|
|
|
1710 |
apply simp
|
|
|
1711 |
apply simp
|
|
|
1712 |
using stupdate_nonempty by presburger
|
|
|
1713 |
|
|
|
1714 |
|
|
|
1715 |
lemma star_closed_form8:
|
|
|
1716 |
shows
|
|
|
1717 |
"rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rsimp (rders r0 s1)) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))) =
|
|
|
1718 |
rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ ( (rders_simp r0 s1)) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
|
|
|
1719 |
by (smt (verit, ccfv_SIG) list.simps(8) map_eq_conv rders__onechar rders_simp_same_simpders set_ConsD stupdates_nonempty)
|
|
|
1720 |
|
|
|
1721 |
|
|
|
1722 |
lemma star_closed_form:
|
|
|
1723 |
shows "rders_simp (RSTAR r0) (c#s) =
|
|
|
1724 |
rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
|
|
|
1725 |
apply(induct s)
|
|
|
1726 |
apply simp
|
|
|
1727 |
apply (metis idem_after_simp1 rsimp.simps(1) rsimp.simps(6) rsimp_idem)
|
|
|
1728 |
using star_closed_form4 star_closed_form5 star_closed_form6 star_closed_form8 by presburger
|
|
|
1729 |
|
|
|
1730 |
|
|
|
1731 |
unused_thms
|
|
|
1732 |
|
|
|
1733 |
end |