--- a/thys3/ClosedForms.thy Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1682 +0,0 @@
-theory ClosedForms
- imports "BasicIdentities"
-begin
-
-lemma flts_middle0:
- shows "rflts (rsa @ RZERO # rsb) = rflts (rsa @ rsb)"
- apply(induct rsa)
- apply simp
- by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
-
-
-
-lemma simp_flatten_aux0:
- shows "rsimp (RALTS rs) = rsimp (RALTS (map rsimp rs))"
- 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)
-
-
-inductive
- hrewrite:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ h\<leadsto> _" [99, 99] 99)
-where
- "RSEQ RZERO r2 h\<leadsto> RZERO"
-| "RSEQ r1 RZERO h\<leadsto> RZERO"
-| "RSEQ RONE r h\<leadsto> r"
-| "r1 h\<leadsto> r2 \<Longrightarrow> RSEQ r1 r3 h\<leadsto> RSEQ r2 r3"
-| "r3 h\<leadsto> r4 \<Longrightarrow> RSEQ r1 r3 h\<leadsto> RSEQ r1 r4"
-| "r h\<leadsto> r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) h\<leadsto> (RALTS (rs1 @ [r'] @ rs2))"
-(*context rule for eliminating 0, alts--corresponds to the recursive call flts r::rs = r::(flts rs)*)
-| "RALTS (rsa @ [RZERO] @ rsb) h\<leadsto> RALTS (rsa @ rsb)"
-| "RALTS (rsa @ [RALTS rs1] @ rsb) h\<leadsto> RALTS (rsa @ rs1 @ rsb)"
-| "RALTS [] h\<leadsto> RZERO"
-| "RALTS [r] h\<leadsto> r"
-| "a1 = a2 \<Longrightarrow> RALTS (rsa@[a1]@rsb@[a2]@rsc) h\<leadsto> RALTS (rsa @ [a1] @ rsb @ rsc)"
-
-inductive
- hrewrites:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ h\<leadsto>* _" [100, 100] 100)
-where
- rs1[intro, simp]:"r h\<leadsto>* r"
-| rs2[intro]: "\<lbrakk>r1 h\<leadsto>* r2; r2 h\<leadsto> r3\<rbrakk> \<Longrightarrow> r1 h\<leadsto>* r3"
-
-
-lemma hr_in_rstar : "r1 h\<leadsto> r2 \<Longrightarrow> r1 h\<leadsto>* r2"
- using hrewrites.intros(1) hrewrites.intros(2) by blast
-
-lemma hreal_trans[trans]:
- assumes a1: "r1 h\<leadsto>* r2" and a2: "r2 h\<leadsto>* r3"
- shows "r1 h\<leadsto>* r3"
- using a2 a1
- apply(induct r2 r3 arbitrary: r1 rule: hrewrites.induct)
- apply(auto)
- done
-
-lemma hrewrites_seq_context:
- shows "r1 h\<leadsto>* r2 \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RSEQ r2 r3"
- apply(induct r1 r2 rule: hrewrites.induct)
- apply simp
- using hrewrite.intros(4) by blast
-
-lemma hrewrites_seq_context2:
- shows "r1 h\<leadsto>* r2 \<Longrightarrow> RSEQ r0 r1 h\<leadsto>* RSEQ r0 r2"
- apply(induct r1 r2 rule: hrewrites.induct)
- apply simp
- using hrewrite.intros(5) by blast
-
-
-lemma hrewrites_seq_contexts:
- shows "\<lbrakk>r1 h\<leadsto>* r2; r3 h\<leadsto>* r4\<rbrakk> \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RSEQ r2 r4"
- by (meson hreal_trans hrewrites_seq_context hrewrites_seq_context2)
-
-
-lemma simp_removes_duplicate1:
- shows " a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a])) = rsimp (RALTS (rsa))"
-and " rsimp (RALTS (a1 # rsa @ [a1])) = rsimp (RALTS (a1 # rsa))"
- apply(induct rsa arbitrary: a1)
- apply simp
- apply simp
- prefer 2
- apply(case_tac "a = aa")
- apply simp
- apply simp
- apply (metis Cons_eq_appendI Cons_eq_map_conv distinct_removes_duplicate_flts list.set_intros(2))
- apply (metis append_Cons append_Nil distinct_removes_duplicate_flts list.set_intros(1) list.simps(8) list.simps(9))
- 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))
-
-lemma simp_removes_duplicate2:
- shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a] @ rsb)) = rsimp (RALTS (rsa @ rsb))"
- apply(induct rsb arbitrary: rsa)
- apply simp
- using distinct_removes_duplicate_flts apply auto[1]
- by (metis append.assoc head_one_more_simp rsimp.simps(2) simp_flatten simp_removes_duplicate1(1))
-
-lemma simp_removes_duplicate3:
- shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ a # rsb)) = rsimp (RALTS (rsa @ rsb))"
- using simp_removes_duplicate2 by auto
-
-(*
-lemma distinct_removes_middle4:
- shows "a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ [a] @ rsb) rset = rdistinct (rsa @ rsb) rset"
- using distinct_removes_middle(1) by fastforce
-*)
-
-(*
-lemma distinct_removes_middle_list:
- shows "\<forall>a \<in> set x. a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ x @ rsb) rset = rdistinct (rsa @ rsb) rset"
- apply(induct x)
- apply simp
- by (simp add: distinct_removes_middle3)
-*)
-
-inductive frewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f _" [10, 10] 10)
- where
- "(RZERO # rs) \<leadsto>f rs"
-| "((RALTS rs) # rsa) \<leadsto>f (rs @ rsa)"
-| "rs1 \<leadsto>f rs2 \<Longrightarrow> (r # rs1) \<leadsto>f (r # rs2)"
-
-
-inductive
- frewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f* _" [10, 10] 10)
-where
- [intro, simp]:"rs \<leadsto>f* rs"
-| [intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>f* rs3"
-
-inductive grewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g _" [10, 10] 10)
- where
- "(RZERO # rs) \<leadsto>g rs"
-| "((RALTS rs) # rsa) \<leadsto>g (rs @ rsa)"
-| "rs1 \<leadsto>g rs2 \<Longrightarrow> (r # rs1) \<leadsto>g (r # rs2)"
-| "rsa @ [a] @ rsb @ [a] @ rsc \<leadsto>g rsa @ [a] @ rsb @ rsc"
-
-lemma grewrite_variant1:
- shows "a \<in> set rs1 \<Longrightarrow> rs1 @ a # rs \<leadsto>g rs1 @ rs"
- apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
- done
-
-
-inductive
- grewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g* _" [10, 10] 10)
-where
- [intro, simp]:"rs \<leadsto>g* rs"
-| [intro]: "\<lbrakk>rs1 \<leadsto>g* rs2; rs2 \<leadsto>g rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>g* rs3"
-
-
-
-(*
-inductive
- frewrites2:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ <\<leadsto>f* _" [10, 10] 10)
-where
- [intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f* rs1\<rbrakk> \<Longrightarrow> rs1 <\<leadsto>f* rs2"
-*)
-
-lemma fr_in_rstar : "r1 \<leadsto>f r2 \<Longrightarrow> r1 \<leadsto>f* r2"
- using frewrites.intros(1) frewrites.intros(2) by blast
-
-lemma freal_trans[trans]:
- assumes a1: "r1 \<leadsto>f* r2" and a2: "r2 \<leadsto>f* r3"
- shows "r1 \<leadsto>f* r3"
- using a2 a1
- apply(induct r2 r3 arbitrary: r1 rule: frewrites.induct)
- apply(auto)
- done
-
-
-lemma many_steps_later: "\<lbrakk>r1 \<leadsto>f r2; r2 \<leadsto>f* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>f* r3"
- by (meson fr_in_rstar freal_trans)
-
-
-lemma gr_in_rstar : "r1 \<leadsto>g r2 \<Longrightarrow> r1 \<leadsto>g* r2"
- using grewrites.intros(1) grewrites.intros(2) by blast
-
-lemma greal_trans[trans]:
- assumes a1: "r1 \<leadsto>g* r2" and a2: "r2 \<leadsto>g* r3"
- shows "r1 \<leadsto>g* r3"
- using a2 a1
- apply(induct r2 r3 arbitrary: r1 rule: grewrites.induct)
- apply(auto)
- done
-
-
-lemma gmany_steps_later: "\<lbrakk>r1 \<leadsto>g r2; r2 \<leadsto>g* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>g* r3"
- by (meson gr_in_rstar greal_trans)
-
-lemma gstar_rdistinct_general:
- shows "rs1 @ rs \<leadsto>g* rs1 @ (rdistinct rs (set rs1))"
- apply(induct rs arbitrary: rs1)
- apply simp
- apply(case_tac " a \<in> set rs1")
- apply simp
- apply(subgoal_tac "rs1 @ a # rs \<leadsto>g rs1 @ rs")
- using gmany_steps_later apply auto[1]
- apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
- apply simp
- apply(drule_tac x = "rs1 @ [a]" in meta_spec)
- by simp
-
-
-lemma gstar_rdistinct:
- shows "rs \<leadsto>g* rdistinct rs {}"
- apply(induct rs)
- apply simp
- by (metis append.left_neutral empty_set gstar_rdistinct_general)
-
-
-lemma grewrite_append:
- shows "\<lbrakk> rsa \<leadsto>g rsb \<rbrakk> \<Longrightarrow> rs @ rsa \<leadsto>g rs @ rsb"
- apply(induct rs)
- apply simp+
- using grewrite.intros(3) by blast
-
-
-
-lemma frewrites_cons:
- shows "\<lbrakk> rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>f* r # rsb"
- apply(induct rsa rsb rule: frewrites.induct)
- apply simp
- using frewrite.intros(3) by blast
-
-
-lemma grewrites_cons:
- shows "\<lbrakk> rsa \<leadsto>g* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>g* r # rsb"
- apply(induct rsa rsb rule: grewrites.induct)
- apply simp
- using grewrite.intros(3) by blast
-
-
-lemma frewrites_append:
- shows " \<lbrakk>rsa \<leadsto>f* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>f* (rs @ rsb)"
- apply(induct rs)
- apply simp
- by (simp add: frewrites_cons)
-
-lemma grewrites_append:
- shows " \<lbrakk>rsa \<leadsto>g* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>g* (rs @ rsb)"
- apply(induct rs)
- apply simp
- by (simp add: grewrites_cons)
-
-
-lemma grewrites_concat:
- shows "\<lbrakk>rs1 \<leadsto>g rs2; rsa \<leadsto>g* rsb \<rbrakk> \<Longrightarrow> (rs1 @ rsa) \<leadsto>g* (rs2 @ rsb)"
- apply(induct rs1 rs2 rule: grewrite.induct)
- apply(simp)
- apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>g (rs @ rsa)")
- prefer 2
- using grewrite.intros(1) apply blast
- apply(subgoal_tac "(rs @ rsa) \<leadsto>g* (rs @ rsb)")
- using gmany_steps_later apply blast
- apply (simp add: grewrites_append)
- apply (metis append.assoc append_Cons grewrite.intros(2) grewrites_append gmany_steps_later)
- using grewrites_cons apply auto
- apply(subgoal_tac "rsaa @ a # rsba @ a # rsc @ rsa \<leadsto>g* rsaa @ a # rsba @ a # rsc @ rsb")
- using grewrite.intros(4) grewrites.intros(2) apply force
- using grewrites_append by auto
-
-
-lemma grewritess_concat:
- shows "\<lbrakk>rsa \<leadsto>g* rsb; rsc \<leadsto>g* rsd \<rbrakk> \<Longrightarrow> (rsa @ rsc) \<leadsto>g* (rsb @ rsd)"
- apply(induct rsa rsb rule: grewrites.induct)
- apply(case_tac rs)
- apply simp
- using grewrites_append apply blast
- by (meson greal_trans grewrites.simps grewrites_concat)
-
-fun alt_set:: "rrexp \<Rightarrow> rrexp set"
- where
- "alt_set (RALTS rs) = set rs \<union> \<Union> (alt_set ` (set rs))"
-| "alt_set r = {r}"
-
-
-lemma grewrite_cases_middle:
- shows "rs1 \<leadsto>g rs2 \<Longrightarrow>
-(\<exists>rsa rsb rsc. rs1 = (rsa @ [RALTS rsb] @ rsc) \<and> rs2 = (rsa @ rsb @ rsc)) \<or>
-(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc) \<or>
-(\<exists>rsa rsb rsc a. rs1 = rsa @ [a] @ rsb @ [a] @ rsc \<and> rs2 = rsa @ [a] @ rsb @ rsc)"
- apply( induct rs1 rs2 rule: grewrite.induct)
- apply simp
- apply blast
- apply (metis append_Cons append_Nil)
- apply (metis append_Cons)
- by blast
-
-
-lemma good_singleton:
- shows "good a \<and> nonalt a \<Longrightarrow> rflts [a] = [a]"
- using good.simps(1) k0b by blast
-
-
-
-
-
-
-
-lemma all_that_same_elem:
- shows "\<lbrakk> a \<in> rset; rdistinct rs {a} = []\<rbrakk>
- \<Longrightarrow> rdistinct (rs @ rsb) rset = rdistinct rsb rset"
- apply(induct rs)
- apply simp
- apply(subgoal_tac "aa = a")
- apply simp
- by (metis empty_iff insert_iff list.discI rdistinct.simps(2))
-
-lemma distinct_early_app1:
- shows "rset1 \<subseteq> rset \<Longrightarrow> rdistinct rs rset = rdistinct (rdistinct rs rset1) rset"
- apply(induct rs arbitrary: rset rset1)
- apply simp
- apply simp
- apply(case_tac "a \<in> rset1")
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp+
-
- apply blast
- apply(case_tac "a \<in> rset1")
- apply simp+
- apply(case_tac "a \<in> rset")
- apply simp
- apply (metis insert_subsetI)
- apply simp
- by (meson insert_mono)
-
-
-lemma distinct_early_app:
- shows " rdistinct (rs @ rsb) rset = rdistinct (rdistinct rs {} @ rsb) rset"
- apply(induct rsb)
- apply simp
- using distinct_early_app1 apply blast
- by (metis distinct_early_app1 distinct_once_enough empty_subsetI)
-
-
-lemma distinct_eq_interesting1:
- shows "a \<in> rset \<Longrightarrow> rdistinct (rs @ rsb) rset = rdistinct (rdistinct (a # rs) {} @ rsb) rset"
- apply(subgoal_tac "rdistinct (rdistinct (a # rs) {} @ rsb) rset = rdistinct (rdistinct rs {} @ rsb) rset")
- apply(simp only:)
- using distinct_early_app apply blast
- by (metis append_Cons distinct_early_app rdistinct.simps(2))
-
-
-
-lemma good_flatten_aux_aux1:
- shows "\<lbrakk> size rs \<ge>2;
-\<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>
- \<Longrightarrow> rdistinct (rs @ rsb) rset =
- rdistinct (rflts [rsimp_ALTs (rdistinct rs {})] @ rsb) rset"
- apply(induct rs arbitrary: rset)
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp
- apply(case_tac "rdistinct rs {a}")
- apply simp
- apply(subst good_singleton)
- apply force
- apply simp
- apply (meson all_that_same_elem)
- apply(subgoal_tac "rflts [rsimp_ALTs (a # rdistinct rs {a})] = a # rdistinct rs {a} ")
- prefer 2
- using k0a rsimp_ALTs.simps(3) apply presburger
- apply(simp only:)
- apply(subgoal_tac "rdistinct (rs @ rsb) rset = rdistinct ((rdistinct (a # rs) {}) @ rsb) rset ")
- apply (metis insert_absorb insert_is_Un insert_not_empty rdistinct.simps(2))
- apply (meson distinct_eq_interesting1)
- apply simp
- apply(case_tac "rdistinct rs {a}")
- prefer 2
- apply(subgoal_tac "rsimp_ALTs (a # rdistinct rs {a}) = RALTS (a # rdistinct rs {a})")
- apply(simp only:)
- apply(subgoal_tac "a # rdistinct (rs @ rsb) (insert a rset) =
- rdistinct (rflts [RALTS (a # rdistinct rs {a})] @ rsb) rset")
- apply simp
- apply (metis append_Cons distinct_early_app empty_iff insert_is_Un k0a rdistinct.simps(2))
- using rsimp_ALTs.simps(3) apply presburger
- by (metis Un_insert_left append_Cons distinct_early_app empty_iff good_singleton rdistinct.simps(2) rsimp_ALTs.simps(2) sup_bot_left)
-
-
-
-
-
-lemma good_flatten_aux_aux:
- shows "\<lbrakk>\<exists>a aa lista list. rs = a # list \<and> list = aa # lista;
-\<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>
- \<Longrightarrow> rdistinct (rs @ rsb) rset =
- rdistinct (rflts [rsimp_ALTs (rdistinct rs {})] @ rsb) rset"
- apply(erule exE)+
- apply(subgoal_tac "size rs \<ge> 2")
- apply (metis good_flatten_aux_aux1)
- by (simp add: Suc_leI length_Cons less_add_Suc1)
-
-
-
-lemma good_flatten_aux:
- 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;
- rsimp (RALTS (rsa @ rs @ rsb)) = rsimp_ALTs (rdistinct (rflts (rsa @ rs @ rsb)) {});
- rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) =
- rsimp_ALTs (rdistinct (rflts (rsa @ [rsimp (RALTS rs)] @ rsb)) {});
- map rsimp rsa = rsa; map rsimp rsb = rsb; map rsimp rs = rs;
- rdistinct (rflts rsa @ rflts rs @ rflts rsb) {} =
- rdistinct (rflts rsa) {} @ rdistinct (rflts rs @ rflts rsb) (set (rflts rsa));
- rdistinct (rflts rsa @ rflts [rsimp (RALTS rs)] @ rflts rsb) {} =
- rdistinct (rflts rsa) {} @ rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))\<rbrakk>
- \<Longrightarrow> rdistinct (rflts rs @ rflts rsb) rset =
- rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) rset"
- apply simp
- apply(case_tac "rflts rs ")
- apply simp
- apply(case_tac "list")
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp
- apply (metis append.left_neutral append_Cons equals0D k0b list.set_intros(1) nonalt_flts_rd qqq1 rdistinct.simps(2))
- apply simp
- 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)
- apply(subgoal_tac "\<forall>r \<in> set (rflts rs). good r \<and> r \<noteq> RZERO \<and> nonalt r")
- prefer 2
- apply (metis Diff_empty flts3 nonalt_flts_rd qqq1 rdistinct_set_equality1)
- apply(subgoal_tac "\<forall>r \<in> set (rflts rsb). good r \<and> r \<noteq> RZERO \<and> nonalt r")
- prefer 2
- apply (metis Diff_empty flts3 good.simps(1) nonalt_flts_rd rdistinct_set_equality1)
- by (smt (verit, ccfv_threshold) good_flatten_aux_aux)
-
-
-
-
-lemma good_flatten_middle:
- 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>
-rsimp (RALTS (rsa @ rs @ rsb)) = rsimp (RALTS (rsa @ [RALTS rs] @ rsb))"
- apply(subgoal_tac "rsimp (RALTS (rsa @ rs @ rsb)) = rsimp_ALTs (rdistinct (rflts (map rsimp rsa @
-map rsimp rs @ map rsimp rsb)) {})")
- prefer 2
- apply simp
- apply(simp only:)
- apply(subgoal_tac "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) = rsimp_ALTs (rdistinct (rflts (map rsimp rsa @
-[rsimp (RALTS rs)] @ map rsimp rsb)) {})")
- prefer 2
- apply simp
- apply(simp only:)
- apply(subgoal_tac "map rsimp rsa = rsa")
- prefer 2
- apply (metis map_idI rsimp.simps(3) test)
- apply(simp only:)
- apply(subgoal_tac "map rsimp rsb = rsb")
- prefer 2
- apply (metis map_idI rsimp.simps(3) test)
- apply(simp only:)
- apply(subst k00)+
- apply(subgoal_tac "map rsimp rs = rs")
- apply(simp only:)
- prefer 2
- apply (metis map_idI rsimp.simps(3) test)
- apply(subgoal_tac "rdistinct (rflts rsa @ rflts rs @ rflts rsb) {} =
-rdistinct (rflts rsa) {} @ rdistinct (rflts rs @ rflts rsb) (set (rflts rsa))")
- apply(simp only:)
- prefer 2
- using rdistinct_concat_general apply blast
- apply(subgoal_tac "rdistinct (rflts rsa @ rflts [rsimp (RALTS rs)] @ rflts rsb) {} =
-rdistinct (rflts rsa) {} @ rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))")
- apply(simp only:)
- prefer 2
- using rdistinct_concat_general apply blast
- apply(subgoal_tac "rdistinct (rflts rs @ rflts rsb) (set (rflts rsa)) =
- rdistinct (rflts [rsimp (RALTS rs)] @ rflts rsb) (set (rflts rsa))")
- apply presburger
- using good_flatten_aux by blast
-
-
-lemma simp_flatten3:
- shows "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) = rsimp (RALTS (rsa @ rs @ rsb))"
- apply(subgoal_tac "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) =
- rsimp (RALTS (map rsimp rsa @ [rsimp (RALTS rs)] @ map rsimp rsb)) ")
- prefer 2
- apply (metis append.left_neutral append_Cons list.simps(9) map_append simp_flatten_aux0)
- apply (simp only:)
- apply(subgoal_tac "rsimp (RALTS (rsa @ rs @ rsb)) =
-rsimp (RALTS (map rsimp rsa @ map rsimp rs @ map rsimp rsb))")
- prefer 2
- apply (metis map_append simp_flatten_aux0)
- apply(simp only:)
- apply(subgoal_tac "rsimp (RALTS (map rsimp rsa @ map rsimp rs @ map rsimp rsb)) =
- rsimp (RALTS (map rsimp rsa @ [RALTS (map rsimp rs)] @ map rsimp rsb))")
-
- apply (metis (no_types, lifting) head_one_more_simp map_append simp_flatten_aux0)
- apply(subgoal_tac "\<forall>r \<in> set (map rsimp rsa). good r \<or> r = RZERO")
- apply(subgoal_tac "\<forall>r \<in> set (map rsimp rs). good r \<or> r = RZERO")
- apply(subgoal_tac "\<forall>r \<in> set (map rsimp rsb). good r \<or> r = RZERO")
-
- using good_flatten_middle apply presburger
-
- apply (simp add: good1)
- apply (simp add: good1)
- apply (simp add: good1)
-
- done
-
-
-
-
-
-lemma grewrite_equal_rsimp:
- shows "rs1 \<leadsto>g rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
- apply(frule grewrite_cases_middle)
- apply(case_tac "(\<exists>rsa rsb rsc. rs1 = rsa @ [RALTS rsb] @ rsc \<and> rs2 = rsa @ rsb @ rsc)")
- using simp_flatten3 apply auto[1]
- apply(case_tac "(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc)")
- 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)
- by (smt (verit) append.assoc append_Cons append_Nil in_set_conv_decomp simp_removes_duplicate3)
-
-
-lemma grewrites_equal_rsimp:
- shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
- apply (induct rs1 rs2 rule: grewrites.induct)
- apply simp
- using grewrite_equal_rsimp by presburger
-
-
-
-lemma grewrites_last:
- shows "r # [RALTS rs] \<leadsto>g* r # rs"
- by (metis gr_in_rstar grewrite.intros(2) grewrite.intros(3) self_append_conv)
-
-lemma simp_flatten2:
- shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
- using grewrites_equal_rsimp grewrites_last by blast
-
-
-lemma frewrites_alt:
- shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> (RALT r1 r2) # rs1 \<leadsto>f* r1 # r2 # rs2"
- by (metis Cons_eq_appendI append_self_conv2 frewrite.intros(2) frewrites_cons many_steps_later)
-
-lemma early_late_der_frewrites:
- shows "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)"
- apply(induct rs)
- apply simp
- apply(case_tac a)
- apply simp+
- using frewrite.intros(1) many_steps_later apply blast
- apply(case_tac "x = x3")
- apply simp
- using frewrites_cons apply presburger
- using frewrite.intros(1) many_steps_later apply fastforce
- apply(case_tac "rnullable x41")
- apply simp+
- apply (simp add: frewrites_alt)
- apply (simp add: frewrites_cons)
- apply (simp add: frewrites_append)
- by (simp add: frewrites_cons)
-
-
-lemma gstar0:
- shows "rsa @ (rdistinct rs (set rsa)) \<leadsto>g* rsa @ (rdistinct rs (insert RZERO (set rsa)))"
- apply(induct rs arbitrary: rsa)
- apply simp
- apply(case_tac "a = RZERO")
- apply simp
-
- using gr_in_rstar grewrite.intros(1) grewrites_append apply presburger
- apply(case_tac "a \<in> set rsa")
- apply simp+
- apply(drule_tac x = "rsa @ [a]" in meta_spec)
- by simp
-
-lemma grewrite_rdistinct_aux:
- shows "rs @ rdistinct rsa rset \<leadsto>g* rs @ rdistinct rsa (rset \<union> set rs)"
- apply(induct rsa arbitrary: rs rset)
- apply simp
- apply(case_tac " a \<in> rset")
- apply simp
- apply(case_tac "a \<in> set rs")
- apply simp
- apply (metis Un_insert_left Un_insert_right gmany_steps_later grewrite_variant1 insert_absorb)
- apply simp
- apply(drule_tac x = "rs @ [a]" in meta_spec)
- by (metis Un_insert_left Un_insert_right append.assoc append.right_neutral append_Cons append_Nil insert_absorb2 list.simps(15) set_append)
-
-
-lemma flts_gstar:
- shows "rs \<leadsto>g* rflts rs"
- apply(induct rs)
- apply simp
- apply(case_tac "a = RZERO")
- apply simp
- using gmany_steps_later grewrite.intros(1) apply blast
- apply(case_tac "\<exists>rsa. a = RALTS rsa")
- apply(erule exE)
- apply simp
- apply (meson grewrite.intros(2) grewrites.simps grewrites_cons)
- by (simp add: grewrites_cons rflts_def_idiot)
-
-lemma more_distinct1:
- shows " \<lbrakk>\<And>rsb rset rset2.
- rset2 \<subseteq> set rsb \<Longrightarrow> rsb @ rdistinct rs rset \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2);
- rset2 \<subseteq> set rsb; a \<notin> rset; a \<in> rset2\<rbrakk>
- \<Longrightarrow> rsb @ a # rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)"
- apply(subgoal_tac "rsb @ a # rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (insert a rset)")
- apply(subgoal_tac "rsb @ rdistinct rs (insert a rset) \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)")
- apply (meson greal_trans)
- apply (metis Un_iff Un_insert_left insert_absorb)
- by (simp add: gr_in_rstar grewrite_variant1 in_mono)
-
-
-
-
-
-lemma frewrite_rd_grewrites_aux:
- shows " RALTS rs \<notin> set rsb \<Longrightarrow>
- rsb @
- RALTS rs #
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) \<leadsto>g* rflts rsb @
- rdistinct rs (set rsb) @ rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
-
- apply simp
- apply(subgoal_tac "rsb @
- RALTS rs #
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) \<leadsto>g* rsb @
- rs @
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) ")
- apply(subgoal_tac " rsb @
- rs @
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) \<leadsto>g*
- rsb @
- rdistinct rs (set rsb) @
- rdistinct rsa
- (insert (RALTS rs)
- (set rsb)) ")
- 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)
- apply (metis append_assoc grewrites.intros(1) grewritess_concat gstar_rdistinct_general)
- by (simp add: gr_in_rstar grewrite.intros(2) grewrites_append)
-
-
-
-
-lemma list_dlist_union:
- shows "set rs \<subseteq> set rsb \<union> set (rdistinct rs (set rsb))"
- by (metis rdistinct_concat_general rdistinct_set_equality set_append sup_ge2)
-
-lemma r_finite1:
- shows "r = RALTS (r # rs) = False"
- apply(induct r)
- apply simp+
- apply (metis list.set_intros(1))
- by blast
-
-
-
-lemma grewrite_singleton:
- shows "[r] \<leadsto>g r # rs \<Longrightarrow> rs = []"
- apply (induct "[r]" "r # rs" rule: grewrite.induct)
- apply simp
- apply (metis r_finite1)
- using grewrite.simps apply blast
- by simp
-
-
-
-lemma concat_rdistinct_equality1:
- shows "rdistinct (rs @ rsa) rset = rdistinct rs rset @ rdistinct rsa (rset \<union> (set rs))"
- apply(induct rs arbitrary: rsa rset)
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp
- apply (simp add: insert_absorb)
- by auto
-
-
-lemma grewrites_rev_append:
- shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rs1 @ [x] \<leadsto>g* rs2 @ [x]"
- using grewritess_concat by auto
-
-lemma grewrites_inclusion:
- shows "set rs \<subseteq> set rs1 \<Longrightarrow> rs1 @ rs \<leadsto>g* rs1"
- apply(induct rs arbitrary: rs1)
- apply simp
- by (meson gmany_steps_later grewrite_variant1 list.set_intros(1) set_subset_Cons subset_code(1))
-
-lemma distinct_keeps_last:
- shows "\<lbrakk>x \<notin> rset; x \<notin> set xs \<rbrakk> \<Longrightarrow> rdistinct (xs @ [x]) rset = rdistinct xs rset @ [x]"
- by (simp add: concat_rdistinct_equality1)
-
-lemma grewrites_shape2_aux:
- shows " RALTS rs \<notin> set rsb \<Longrightarrow>
- rsb @
- rdistinct (rs @ rsa)
- (set rsb) \<leadsto>g* rsb @
- rdistinct rs (set rsb) @
- rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
- apply(subgoal_tac " rdistinct (rs @ rsa) (set rsb) = rdistinct rs (set rsb) @ rdistinct rsa (set rs \<union> set rsb)")
- apply (simp only:)
- prefer 2
- apply (simp add: Un_commute concat_rdistinct_equality1)
- apply(induct rsa arbitrary: rs rsb rule: rev_induct)
- apply simp
- apply(case_tac "x \<in> set rs")
- apply (simp add: distinct_removes_middle3)
- apply(case_tac "x = RALTS rs")
- apply simp
- apply(case_tac "x \<in> set rsb")
- apply simp
- apply (simp add: concat_rdistinct_equality1)
- apply (simp add: concat_rdistinct_equality1)
- apply simp
- apply(drule_tac x = "rs " in meta_spec)
- apply(drule_tac x = rsb in meta_spec)
- apply simp
- 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))")
- prefer 2
- apply (simp add: concat_rdistinct_equality1)
- apply(case_tac "x \<in> set xs")
- apply simp
- apply (simp add: distinct_removes_last)
- apply(case_tac "x \<in> set rsb")
- apply (smt (verit, ccfv_threshold) Un_iff append.right_neutral concat_rdistinct_equality1 insert_is_Un rdistinct.simps(2))
- 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]")
- apply(simp only:)
- apply(case_tac "x = RALTS rs")
- 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")
- 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) ")
- apply (smt (verit, ccfv_SIG) Un_insert_left append.right_neutral concat_rdistinct_equality1 greal_trans insert_iff rdistinct.simps(2))
- apply(subgoal_tac "set rs \<subseteq> set ( rsb @ rdistinct rs (set rsb) @ rdistinct xs (set rs \<union> set rsb))")
- apply (metis append.assoc grewrites_inclusion)
- apply (metis Un_upper1 append.assoc dual_order.trans list_dlist_union set_append)
- apply (metis append_Nil2 gr_in_rstar grewrite.intros(2) grewrite_append)
- 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]")
- apply(simp only:)
- apply (metis append.assoc grewrites_rev_append)
- apply (simp add: insert_absorb)
- apply (simp add: distinct_keeps_last)+
- done
-
-lemma grewrites_shape2:
- shows " RALTS rs \<notin> set rsb \<Longrightarrow>
- rsb @
- rdistinct (rs @ rsa)
- (set rsb) \<leadsto>g* rflts rsb @
- rdistinct rs (set rsb) @
- rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})"
- apply (meson flts_gstar greal_trans grewrites.simps grewrites_shape2_aux grewritess_concat)
- done
-
-lemma rdistinct_add_acc:
- shows "rset2 \<subseteq> set rsb \<Longrightarrow> rsb @ rdistinct rs rset \<leadsto>g* rsb @ rdistinct rs (rset \<union> rset2)"
- apply(induct rs arbitrary: rsb rset rset2)
- apply simp
- apply (case_tac "a \<in> rset")
- apply simp
- apply(case_tac "a \<in> rset2")
- apply simp
- apply (simp add: more_distinct1)
- apply simp
- apply(drule_tac x = "rsb @ [a]" in meta_spec)
- by (metis Un_insert_left append.assoc append_Cons append_Nil set_append sup.coboundedI1)
-
-
-lemma frewrite_fun1:
- shows " RALTS rs \<in> set rsb \<Longrightarrow>
- rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb \<union> set rs)"
- apply(subgoal_tac "rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb)")
- apply(subgoal_tac " set rs \<subseteq> set (rflts rsb)")
- prefer 2
- using spilled_alts_contained apply blast
- apply(subgoal_tac "rflts rsb @ rdistinct rsa (set rsb) \<leadsto>g* rflts rsb @ rdistinct rsa (set rsb \<union> set rs)")
- using greal_trans apply blast
- using rdistinct_add_acc apply presburger
- using flts_gstar grewritess_concat by auto
-
-lemma frewrite_rd_grewrites:
- shows "rs1 \<leadsto>f rs2 \<Longrightarrow>
-\<exists>rs3. (rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3) "
- apply(induct rs1 rs2 arbitrary: rs rule: frewrite.induct)
- apply(rule_tac x = "rsa @ (rdistinct rs ({RZERO} \<union> set rsa))" in exI)
- apply(rule conjI)
- apply(case_tac "RZERO \<in> set rsa")
- apply simp+
- using gstar0 apply fastforce
- apply (simp add: gr_in_rstar grewrite.intros(1) grewrites_append)
- apply (simp add: gstar0)
- prefer 2
- apply(case_tac "r \<in> set rs")
- apply simp
- apply(drule_tac x = "rs @ [r]" in meta_spec)
- apply(erule exE)
- apply(rule_tac x = "rs3" in exI)
- apply simp
- apply(case_tac "RALTS rs \<in> set rsb")
- apply simp
- apply(rule_tac x = "rflts rsb @ rdistinct rsa (set rsb \<union> set rs)" in exI)
- apply(rule conjI)
- using frewrite_fun1 apply force
- apply (metis frewrite_fun1 rdistinct_concat sup_ge2)
- apply(simp)
- apply(rule_tac x =
- "rflts rsb @
- rdistinct rs (set rsb) @
- rdistinct rsa (set rs \<union> set rsb \<union> {RALTS rs})" in exI)
- apply(rule conjI)
- prefer 2
- using grewrites_shape2 apply force
- using frewrite_rd_grewrites_aux by blast
-
-
-lemma frewrite_simpeq2:
- shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
- apply(subgoal_tac "\<exists> rs3. (rdistinct rs1 {} \<leadsto>g* rs3) \<and> (rdistinct rs2 {} \<leadsto>g* rs3)")
- using grewrites_equal_rsimp apply fastforce
- by (metis append_self_conv2 frewrite_rd_grewrites list.set(1))
-
-
-
-
-(*a more refined notion of h\<leadsto>* is needed,
-this lemma fails when rs1 contains some RALTS rs where elements
-of rs appear in later parts of rs1, which will be picked up by rs2
-and deduplicated*)
-lemma frewrites_simpeq:
- shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
- rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS ( rdistinct rs2 {})) "
- apply(induct rs1 rs2 rule: frewrites.induct)
- apply simp
- using frewrite_simpeq2 by presburger
-
-
-lemma frewrite_single_step:
- shows " rs2 \<leadsto>f rs3 \<Longrightarrow> rsimp (RALTS rs2) = rsimp (RALTS rs3)"
- apply(induct rs2 rs3 rule: frewrite.induct)
- apply simp
- using simp_flatten apply blast
- by (metis (no_types, opaque_lifting) list.simps(9) rsimp.simps(2) simp_flatten2)
-
-lemma grewrite_simpalts:
- shows " rs2 \<leadsto>g rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
- apply(induct rs2 rs3 rule : grewrite.induct)
- using identity_wwo0 apply presburger
- apply (metis frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_flatten)
- apply (smt (verit, ccfv_SIG) gmany_steps_later grewrites.intros(1) grewrites_cons grewrites_equal_rsimp identity_wwo0 rsimp_ALTs.simps(3))
- apply simp
- apply(subst rsimp_alts_equal)
- apply(case_tac "rsa = [] \<and> rsb = [] \<and> rsc = []")
- apply(subgoal_tac "rsa @ a # rsb @ rsc = [a]")
- apply (simp only:)
- apply (metis append_Nil frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_removes_duplicate1(2))
- apply simp
- 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)
-
-
-lemma grewrites_simpalts:
- shows " rs2 \<leadsto>g* rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
- apply(induct rs2 rs3 rule: grewrites.induct)
- apply simp
- using grewrite_simpalts by presburger
-
-
-lemma simp_der_flts:
- shows "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {})) =
- rsimp (RALTS (rdistinct (rflts (map (rder x) rs)) {}))"
- apply(subgoal_tac "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)")
- using frewrites_simpeq apply presburger
- using early_late_der_frewrites by auto
-
-
-lemma simp_der_pierce_flts_prelim:
- shows "rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts rs)) {}))
- = rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {}))"
- by (metis append.right_neutral grewrite.intros(2) grewrite_simpalts rsimp_ALTs.simps(2) simp_der_flts)
-
-
-lemma basic_regex_property1:
- shows "rnullable r \<Longrightarrow> rsimp r \<noteq> RZERO"
- apply(induct r rule: rsimp.induct)
- apply(auto)
- apply (metis idiot idiot2 rrexp.distinct(5))
- by (metis der_simp_nullability rnullable.simps(1) rnullable.simps(4) rsimp.simps(2))
-
-
-lemma inside_simp_seq_nullable:
- shows
-"\<And>r1 r2.
- \<lbrakk>rsimp (rder x (rsimp r1)) = rsimp (rder x r1); rsimp (rder x (rsimp r2)) = rsimp (rder x r2);
- rnullable r1\<rbrakk>
- \<Longrightarrow> rsimp (rder x (rsimp_SEQ (rsimp r1) (rsimp r2))) =
- rsimp_ALTs (rdistinct (rflts [rsimp_SEQ (rsimp (rder x r1)) (rsimp r2), rsimp (rder x r2)]) {})"
- apply(case_tac "rsimp r1 = RONE")
- apply(simp)
- apply(subst basic_rsimp_SEQ_property1)
- apply (simp add: idem_after_simp1)
- apply(case_tac "rsimp r1 = RZERO")
-
- using basic_regex_property1 apply blast
- apply(case_tac "rsimp r2 = RZERO")
-
- apply (simp add: basic_rsimp_SEQ_property3)
- apply(subst idiot2)
- apply simp+
- apply(subgoal_tac "rnullable (rsimp r1)")
- apply simp
- using rsimp_idem apply presburger
- using der_simp_nullability by presburger
-
-
-
-lemma grewrite_ralts:
- shows "rs \<leadsto>g rs' \<Longrightarrow> RALTS rs h\<leadsto>* RALTS rs'"
- by (smt (verit) grewrite_cases_middle hr_in_rstar hrewrite.intros(11) hrewrite.intros(7) hrewrite.intros(8))
-
-lemma grewrites_ralts:
- shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs h\<leadsto>* RALTS rs'"
- apply(induct rule: grewrites.induct)
- apply simp
- using grewrite_ralts hreal_trans by blast
-
-
-lemma distinct_grewrites_subgoal1:
- shows "
- \<lbrakk>rs1 \<leadsto>g* [a]; RALTS rs1 h\<leadsto>* a; [a] \<leadsto>g rs3\<rbrakk> \<Longrightarrow> RALTS rs1 h\<leadsto>* rsimp_ALTs rs3"
- apply(subgoal_tac "RALTS rs1 h\<leadsto>* RALTS rs3")
- apply (metis hrewrite.intros(10) hrewrite.intros(9) rs2 rsimp_ALTs.cases rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
- apply(subgoal_tac "rs1 \<leadsto>g* rs3")
- using grewrites_ralts apply blast
- using grewrites.intros(2) by presburger
-
-lemma grewrites_ralts_rsimpalts:
- shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs h\<leadsto>* rsimp_ALTs rs' "
- apply(induct rs rs' rule: grewrites.induct)
- apply(case_tac rs)
- using hrewrite.intros(9) apply force
- apply(case_tac list)
- apply simp
- using hr_in_rstar hrewrite.intros(10) rsimp_ALTs.simps(2) apply presburger
- apply simp
- apply(case_tac rs2)
- apply simp
- apply (metis grewrite.intros(3) grewrite_singleton rsimp_ALTs.simps(1))
- apply(case_tac list)
- apply(simp)
- using distinct_grewrites_subgoal1 apply blast
- apply simp
- apply(case_tac rs3)
- apply simp
- using grewrites_ralts hrewrite.intros(9) apply blast
- 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))
-
-lemma hrewrites_alts:
- shows " r h\<leadsto>* r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) h\<leadsto>* (RALTS (rs1 @ [r'] @ rs2))"
- apply(induct r r' rule: hrewrites.induct)
- apply simp
- using hrewrite.intros(6) by blast
-
-inductive
- srewritescf:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" (" _ scf\<leadsto>* _" [100, 100] 100)
-where
- ss1: "[] scf\<leadsto>* []"
-| ss2: "\<lbrakk>r h\<leadsto>* r'; rs scf\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (r#rs) scf\<leadsto>* (r'#rs')"
-
-
-lemma srewritescf_alt: "rs1 scf\<leadsto>* rs2 \<Longrightarrow> (RALTS (rs@rs1)) h\<leadsto>* (RALTS (rs@rs2))"
-
- apply(induct rs1 rs2 arbitrary: rs rule: srewritescf.induct)
- apply(rule rs1)
- apply(drule_tac x = "rsa@[r']" in meta_spec)
- apply simp
- apply(rule hreal_trans)
- prefer 2
- apply(assumption)
- apply(drule hrewrites_alts)
- by auto
-
-
-corollary srewritescf_alt1:
- assumes "rs1 scf\<leadsto>* rs2"
- shows "RALTS rs1 h\<leadsto>* RALTS rs2"
- using assms
- by (metis append_Nil srewritescf_alt)
-
-
-
-
-lemma trivialrsimp_srewrites:
- "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x h\<leadsto>* f x \<rbrakk> \<Longrightarrow> rs scf\<leadsto>* (map f rs)"
-
- apply(induction rs)
- apply simp
- apply(rule ss1)
- by (metis insert_iff list.simps(15) list.simps(9) srewritescf.simps)
-
-lemma hrewrites_list:
- shows
-" (\<And>xa. xa \<in> set x \<Longrightarrow> xa h\<leadsto>* rsimp xa) \<Longrightarrow> RALTS x h\<leadsto>* RALTS (map rsimp x)"
- apply(induct x)
- apply(simp)+
- by (simp add: srewritescf_alt1 ss2 trivialrsimp_srewrites)
-(* apply(subgoal_tac "RALTS x h\<leadsto>* RALTS (map rsimp x)")*)
-
-
-lemma hrewrite_simpeq:
- shows "r1 h\<leadsto> r2 \<Longrightarrow> rsimp r1 = rsimp r2"
- apply(induct rule: hrewrite.induct)
- apply simp+
- apply (simp add: basic_rsimp_SEQ_property3)
- apply (simp add: basic_rsimp_SEQ_property1)
- using rsimp.simps(1) apply presburger
- apply simp+
- using flts_middle0 apply force
-
-
- using simp_flatten3 apply presburger
-
- apply simp+
- apply (simp add: idem_after_simp1)
- using grewrite.intros(4) grewrite_equal_rsimp by presburger
-
-lemma hrewrites_simpeq:
- shows "r1 h\<leadsto>* r2 \<Longrightarrow> rsimp r1 = rsimp r2"
- apply(induct rule: hrewrites.induct)
- apply simp
- apply(subgoal_tac "rsimp r2 = rsimp r3")
- apply auto[1]
- using hrewrite_simpeq by presburger
-
-
-
-lemma simp_hrewrites:
- shows "r1 h\<leadsto>* rsimp r1"
- apply(induct r1)
- apply simp+
- apply(case_tac "rsimp r11 = RONE")
- apply simp
- apply(subst basic_rsimp_SEQ_property1)
- apply(subgoal_tac "RSEQ r11 r12 h\<leadsto>* RSEQ RONE r12")
- using hreal_trans hrewrite.intros(3) apply blast
- using hrewrites_seq_context apply presburger
- apply(case_tac "rsimp r11 = RZERO")
- apply simp
- using hrewrite.intros(1) hrewrites_seq_context apply blast
- apply(case_tac "rsimp r12 = RZERO")
- apply simp
- apply(subst basic_rsimp_SEQ_property3)
- apply (meson hrewrite.intros(2) hrewrites.simps hrewrites_seq_context2)
- apply(subst idiot2)
- apply simp+
- using hrewrites_seq_contexts apply presburger
- apply simp
- apply(subgoal_tac "RALTS x h\<leadsto>* RALTS (map rsimp x)")
- apply(subgoal_tac "RALTS (map rsimp x) h\<leadsto>* rsimp_ALTs (rdistinct (rflts (map rsimp x)) {}) ")
- using hreal_trans apply blast
- apply (meson flts_gstar greal_trans grewrites_ralts_rsimpalts gstar_rdistinct)
-
- apply (simp add: grewrites_ralts hrewrites_list)
- by simp
-
-lemma interleave_aux1:
- shows " RALT (RSEQ RZERO r1) r h\<leadsto>* r"
- apply(subgoal_tac "RSEQ RZERO r1 h\<leadsto>* RZERO")
- apply(subgoal_tac "RALT (RSEQ RZERO r1) r h\<leadsto>* RALT RZERO r")
- apply (meson grewrite.intros(1) grewrite_ralts hreal_trans hrewrite.intros(10) hrewrites.simps)
- using rs1 srewritescf_alt1 ss1 ss2 apply presburger
- by (simp add: hr_in_rstar hrewrite.intros(1))
-
-
-
-lemma rnullable_hrewrite:
- shows "r1 h\<leadsto> r2 \<Longrightarrow> rnullable r1 = rnullable r2"
- apply(induct rule: hrewrite.induct)
- apply simp+
- apply blast
- apply simp+
- done
-
-
-lemma interleave1:
- shows "r h\<leadsto> r' \<Longrightarrow> rder c r h\<leadsto>* rder c r'"
- apply(induct r r' rule: hrewrite.induct)
- apply (simp add: hr_in_rstar hrewrite.intros(1))
- 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)
- apply simp
- apply(subst interleave_aux1)
- apply simp
- apply(case_tac "rnullable r1")
- apply simp
-
- apply (simp add: hrewrites_seq_context rnullable_hrewrite srewritescf_alt1 ss1 ss2)
-
- apply (simp add: hrewrites_seq_context rnullable_hrewrite)
- apply(case_tac "rnullable r1")
- apply simp
-
- using hr_in_rstar hrewrites_seq_context2 srewritescf_alt1 ss1 ss2 apply presburger
- apply simp
- using hr_in_rstar hrewrites_seq_context2 apply blast
- apply simp
-
- using hrewrites_alts apply auto[1]
- apply simp
- using grewrite.intros(1) grewrite_append grewrite_ralts apply auto[1]
- apply simp
- apply (simp add: grewrite.intros(2) grewrite_append grewrite_ralts)
- apply (simp add: hr_in_rstar hrewrite.intros(9))
- apply (simp add: hr_in_rstar hrewrite.intros(10))
- apply simp
- using hrewrite.intros(11) by auto
-
-lemma interleave_star1:
- shows "r h\<leadsto>* r' \<Longrightarrow> rder c r h\<leadsto>* rder c r'"
- apply(induct rule : hrewrites.induct)
- apply simp
- by (meson hreal_trans interleave1)
-
-
-
-lemma inside_simp_removal:
- shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
- apply(induct r)
- apply simp+
- apply(case_tac "rnullable r1")
- apply simp
-
- using inside_simp_seq_nullable apply blast
- apply simp
- 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)
- apply(subgoal_tac "rder x (RALTS xa) h\<leadsto>* rder x (rsimp (RALTS xa))")
- using hrewrites_simpeq apply presburger
- using interleave_star1 simp_hrewrites apply presburger
- by simp
-
-
-
-
-lemma rders_simp_same_simpders:
- shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
- apply(induct s rule: rev_induct)
- apply simp
- apply(case_tac "xs = []")
- apply simp
- apply(simp add: rders_append rders_simp_append)
- using inside_simp_removal by blast
-
-
-
-
-lemma distinct_der:
- shows "rsimp (rsimp_ALTs (map (rder x) (rdistinct rs {}))) =
- rsimp (rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
- by (metis grewrites_simpalts gstar_rdistinct inside_simp_removal rder_rsimp_ALTs_commute)
-
-
-
-
-
-lemma rders_simp_lambda:
- shows " rsimp \<circ> rder x \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r (xs @ [x]))"
- using rders_simp_append by auto
-
-lemma rders_simp_nonempty_simped:
- shows "xs \<noteq> [] \<Longrightarrow> rsimp \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r xs)"
- using rders_simp_same_simpders rsimp_idem by auto
-
-lemma repeated_altssimp:
- shows "\<forall>r \<in> set rs. rsimp r = r \<Longrightarrow> rsimp (rsimp_ALTs (rdistinct (rflts rs) {})) =
- rsimp_ALTs (rdistinct (rflts rs) {})"
- by (metis map_idI rsimp.simps(2) rsimp_idem)
-
-
-
-lemma alts_closed_form:
- shows "rsimp (rders_simp (RALTS rs) s) = rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
- apply(induct s rule: rev_induct)
- apply simp
- apply simp
- apply(subst rders_simp_append)
- apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) =
- rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
- prefer 2
- apply (metis inside_simp_removal rders_simp_one_char)
- apply(simp only: )
- apply(subst rders_simp_one_char)
- apply(subst rsimp_idem)
- apply(subgoal_tac "rsimp (rder x (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {}))) =
- rsimp ((rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))) ")
- prefer 2
- using rder_rsimp_ALTs_commute apply presburger
- apply(simp only:)
- apply(subgoal_tac "rsimp (rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))
-= rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
- prefer 2
-
- using distinct_der apply presburger
- apply(simp only:)
- apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
- rsimp (rsimp_ALTs (rdistinct ( (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)))) {}))")
- apply(simp only:)
- apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
- rsimp (rsimp_ALTs (rdistinct (rflts ( (map (rsimp \<circ> (rder x) \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
- apply(simp only:)
- apply(subst rders_simp_lambda)
- apply(subst rders_simp_nonempty_simped)
- apply simp
- apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (xs @ [x])) rs). rsimp r = r")
- prefer 2
- apply (simp add: rders_simp_same_simpders rsimp_idem)
- apply(subst repeated_altssimp)
- apply simp
- apply fastforce
- apply (metis inside_simp_removal list.map_comp rder.simps(4) rsimp.simps(2) rsimp_idem)
- using simp_der_pierce_flts_prelim by blast
-
-
-lemma alts_closed_form_variant:
- shows "s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s = rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
- by (metis alts_closed_form comp_apply rders_simp_nonempty_simped)
-
-
-lemma rsimp_seq_equal1:
- shows "rsimp_SEQ (rsimp r1) (rsimp r2) = rsimp_ALTs (rdistinct (rflts [rsimp_SEQ (rsimp r1) (rsimp r2)]) {})"
- by (metis idem_after_simp1 rsimp.simps(1))
-
-
-fun sflat_aux :: "rrexp \<Rightarrow> rrexp list " where
- "sflat_aux (RALTS (r # rs)) = sflat_aux r @ rs"
-| "sflat_aux (RALTS []) = []"
-| "sflat_aux r = [r]"
-
-
-fun sflat :: "rrexp \<Rightarrow> rrexp" where
- "sflat (RALTS (r # [])) = r"
-| "sflat (RALTS (r # rs)) = RALTS (sflat_aux r @ rs)"
-| "sflat r = r"
-
-inductive created_by_seq:: "rrexp \<Rightarrow> bool" where
- "created_by_seq (RSEQ r1 r2) "
-| "created_by_seq r1 \<Longrightarrow> created_by_seq (RALT r1 r2)"
-
-lemma seq_ders_shape1:
- shows "\<forall>r1 r2. \<exists>r3 r4. (rders (RSEQ r1 r2) s) = RSEQ r3 r4 \<or> (rders (RSEQ r1 r2) s) = RALT r3 r4"
- apply(induct s rule: rev_induct)
- apply auto[1]
- apply(rule allI)+
- apply(subst rders_append)+
- apply(subgoal_tac " \<exists>r3 r4. rders (RSEQ r1 r2) xs = RSEQ r3 r4 \<or> rders (RSEQ r1 r2) xs = RALT r3 r4 ")
- apply(erule exE)+
- apply(erule disjE)
- apply simp+
- done
-
-lemma created_by_seq_der:
- shows "created_by_seq r \<Longrightarrow> created_by_seq (rder c r)"
- apply(induct r)
- apply simp+
-
- using created_by_seq.cases apply blast
-
- apply (meson created_by_seq.cases rrexp.distinct(19) rrexp.distinct(21))
- apply (metis created_by_seq.simps rder.simps(5))
- 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))
- using created_by_seq.intros(1) by force
-
-lemma createdbyseq_left_creatable:
- shows "created_by_seq (RALT r1 r2) \<Longrightarrow> created_by_seq r1"
- using created_by_seq.cases by blast
-
-
-
-lemma recursively_derseq:
- shows " created_by_seq (rders (RSEQ r1 r2) s)"
- apply(induct s rule: rev_induct)
- apply simp
- using created_by_seq.intros(1) apply force
- apply(subgoal_tac "created_by_seq (rders (RSEQ r1 r2) (xs @ [x]))")
- apply blast
- apply(subst rders_append)
- apply(subgoal_tac "\<exists>r3 r4. rders (RSEQ r1 r2) xs = RSEQ r3 r4 \<or>
- rders (RSEQ r1 r2) xs = RALT r3 r4")
- prefer 2
- using seq_ders_shape1 apply presburger
- apply(erule exE)+
- apply(erule disjE)
- apply(subgoal_tac "created_by_seq (rders (RSEQ r3 r4) [x])")
- apply presburger
- apply simp
- using created_by_seq.intros(1) created_by_seq.intros(2) apply presburger
- apply simp
- apply(subgoal_tac "created_by_seq r3")
- prefer 2
- using createdbyseq_left_creatable apply blast
- using created_by_seq.intros(2) created_by_seq_der by blast
-
-
-lemma recursively_derseq1:
- shows "r = rders (RSEQ r1 r2) s \<Longrightarrow> created_by_seq r"
- using recursively_derseq by blast
-
-
-lemma sfau_head:
- shows " created_by_seq r \<Longrightarrow> \<exists>ra rb rs. sflat_aux r = RSEQ ra rb # rs"
- apply(induction r rule: created_by_seq.induct)
- apply simp
- by fastforce
-
-
-lemma vsuf_prop1:
- shows "vsuf (xs @ [x]) r = (if (rnullable (rders r xs))
- then [x] # (map (\<lambda>s. s @ [x]) (vsuf xs r) )
- else (map (\<lambda>s. s @ [x]) (vsuf xs r)) )
- "
- apply(induct xs arbitrary: r)
- apply simp
- apply(case_tac "rnullable r")
- apply simp
- apply simp
- done
-
-fun breakHead :: "rrexp list \<Rightarrow> rrexp list" where
- "breakHead [] = [] "
-| "breakHead (RALT r1 r2 # rs) = r1 # r2 # rs"
-| "breakHead (r # rs) = r # rs"
-
-
-lemma sfau_idem_der:
- shows "created_by_seq r \<Longrightarrow> sflat_aux (rder c r) = breakHead (map (rder c) (sflat_aux r))"
- apply(induct rule: created_by_seq.induct)
- apply simp+
- using sfau_head by fastforce
-
-lemma vsuf_compose1:
- shows " \<not> rnullable (rders r1 xs)
- \<Longrightarrow> map (rder x \<circ> rders r2) (vsuf xs r1) = map (rders r2) (vsuf (xs @ [x]) r1)"
- apply(subst vsuf_prop1)
- apply simp
- by (simp add: rders_append)
-
-
-
-
-lemma seq_sfau0:
- shows "sflat_aux (rders (RSEQ r1 r2) s) = (RSEQ (rders r1 s) r2) #
- (map (rders r2) (vsuf s r1)) "
- apply(induct s rule: rev_induct)
- apply simp
- apply(subst rders_append)+
- apply(subgoal_tac "created_by_seq (rders (RSEQ r1 r2) xs)")
- prefer 2
- using recursively_derseq1 apply blast
- apply simp
- apply(subst sfau_idem_der)
-
- apply blast
- apply(case_tac "rnullable (rders r1 xs)")
- apply simp
- apply(subst vsuf_prop1)
- apply simp
- apply (simp add: rders_append)
- apply simp
- using vsuf_compose1 by blast
-
-
-
-
-
-
-
-
-
-thm sflat.elims
-
-
-
-
-
-lemma sflat_rsimpeq:
- shows "created_by_seq r1 \<Longrightarrow> sflat_aux r1 = rs \<Longrightarrow> rsimp r1 = rsimp (RALTS rs)"
- apply(induct r1 arbitrary: rs rule: created_by_seq.induct)
- apply simp
- using rsimp_seq_equal1 apply force
- by (metis head_one_more_simp rsimp.simps(2) sflat_aux.simps(1) simp_flatten)
-
-
-
-lemma seq_closed_form_general:
- shows "rsimp (rders (RSEQ r1 r2) s) =
-rsimp ( (RALTS ( (RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1))))))"
- apply(case_tac "s \<noteq> []")
- apply(subgoal_tac "created_by_seq (rders (RSEQ r1 r2) s)")
- apply(subgoal_tac "sflat_aux (rders (RSEQ r1 r2) s) = RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1))")
- using sflat_rsimpeq apply blast
- apply (simp add: seq_sfau0)
- using recursively_derseq1 apply blast
- apply simp
- by (metis idem_after_simp1 rsimp.simps(1))
-
-lemma seq_closed_form_aux1a:
- shows "rsimp (RALTS (RSEQ (rders r1 s) r2 # rs)) =
- rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # rs))"
- 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)
-
-
-lemma seq_closed_form_aux1:
- shows "rsimp (RALTS (RSEQ (rders r1 s) r2 # (map (rders r2) (vsuf s r1)))) =
- rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # (map (rders r2) (vsuf s r1))))"
- 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)
-
-lemma add_simp_to_rest:
- shows "rsimp (RALTS (r # rs)) = rsimp (RALTS (r # map rsimp rs))"
- by (metis append_Nil2 grewrite.intros(2) grewrite_simpalts head_one_more_simp list.simps(9) rsimp_ALTs.simps(2) spawn_simp_rsimpalts)
-
-lemma rsimp_compose_der2:
- shows "\<forall>s \<in> set ss. s \<noteq> [] \<Longrightarrow> map rsimp (map (rders r) ss) = map (\<lambda>s. (rders_simp r s)) ss"
- by (simp add: rders_simp_same_simpders)
-
-lemma vsuf_nonempty:
- shows "\<forall>s \<in> set ( vsuf s1 r). s \<noteq> []"
- apply(induct s1 arbitrary: r)
- apply simp
- apply simp
- done
-
-
-
-lemma seq_closed_form_aux2:
- shows "s \<noteq> [] \<Longrightarrow> rsimp ( (RALTS ( (RSEQ (rders_simp r1 s) r2 # (map (rders r2) (vsuf s r1)))))) =
- rsimp ( (RALTS ( (RSEQ (rders_simp r1 s) r2 # (map (rders_simp r2) (vsuf s r1))))))"
-
- by (metis add_simp_to_rest rsimp_compose_der2 vsuf_nonempty)
-
-
-lemma seq_closed_form:
- shows "rsimp (rders_simp (RSEQ r1 r2) s) =
- rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
-proof (cases s)
- case Nil
- then show ?thesis
- by (simp add: rsimp_seq_equal1[symmetric])
-next
- case (Cons a list)
- have "rsimp (rders_simp (RSEQ r1 r2) s) = rsimp (rsimp (rders (RSEQ r1 r2) s))"
- using local.Cons by (subst rders_simp_same_simpders)(simp_all)
- also have "... = rsimp (rders (RSEQ r1 r2) s)"
- by (simp add: rsimp_idem)
- also have "... = rsimp (RALTS (RSEQ (rders r1 s) r2 # map (rders r2) (vsuf s r1)))"
- using seq_closed_form_general by blast
- also have "... = rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders r2) (vsuf s r1)))"
- by (simp only: seq_closed_form_aux1)
- also have "... = rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)))"
- using local.Cons by (subst seq_closed_form_aux2)(simp_all)
- finally show ?thesis .
-qed
-
-lemma q: "s \<noteq> [] \<Longrightarrow> rders_simp (RSEQ r1 r2) s = rsimp (rders_simp (RSEQ r1 r2) s)"
- using rders_simp_same_simpders rsimp_idem by presburger
-
-
-lemma seq_closed_form_variant:
- assumes "s \<noteq> []"
- shows "rders_simp (RSEQ r1 r2) s =
- rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # (map (rders_simp r2) (vsuf s r1))))"
- using assms q seq_closed_form by force
-
-
-fun hflat_aux :: "rrexp \<Rightarrow> rrexp list" where
- "hflat_aux (RALT r1 r2) = hflat_aux r1 @ hflat_aux r2"
-| "hflat_aux r = [r]"
-
-
-fun hflat :: "rrexp \<Rightarrow> rrexp" where
- "hflat (RALT r1 r2) = RALTS ((hflat_aux r1) @ (hflat_aux r2))"
-| "hflat r = r"
-
-inductive created_by_star :: "rrexp \<Rightarrow> bool" where
- "created_by_star (RSEQ ra (RSTAR rb))"
-| "\<lbrakk>created_by_star r1; created_by_star r2\<rbrakk> \<Longrightarrow> created_by_star (RALT r1 r2)"
-
-fun hElem :: "rrexp \<Rightarrow> rrexp list" where
- "hElem (RALT r1 r2) = (hElem r1 ) @ (hElem r2)"
-| "hElem r = [r]"
-
-
-
-
-lemma cbs_ders_cbs:
- shows "created_by_star r \<Longrightarrow> created_by_star (rder c r)"
- apply(induct r rule: created_by_star.induct)
- apply simp
- using created_by_star.intros(1) created_by_star.intros(2) apply auto[1]
- by (metis (mono_tags, lifting) created_by_star.simps list.simps(8) list.simps(9) rder.simps(4))
-
-lemma star_ders_cbs:
- shows "created_by_star (rders (RSEQ r1 (RSTAR r2)) s)"
- apply(induct s rule: rev_induct)
- apply simp
- apply (simp add: created_by_star.intros(1))
- apply(subst rders_append)
- apply simp
- using cbs_ders_cbs by auto
-
-(*
-lemma created_by_star_cases:
- 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 "
- by (meson created_by_star.cases)
-*)
-
-
-lemma hfau_pushin:
- shows "created_by_star r \<Longrightarrow> hflat_aux (rder c r) = concat (map hElem (map (rder c) (hflat_aux r)))"
- apply(induct r rule: created_by_star.induct)
- apply simp
- apply(subgoal_tac "created_by_star (rder c r1)")
- prefer 2
- apply(subgoal_tac "created_by_star (rder c r2)")
- using cbs_ders_cbs apply blast
- using cbs_ders_cbs apply auto[1]
- apply simp
- done
-
-lemma stupdate_induct1:
- shows " concat (map (hElem \<circ> (rder x \<circ> (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)))) Ss) =
- map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_update x r0 Ss)"
- apply(induct Ss)
- apply simp+
- by (simp add: rders_append)
-
-
-
-lemma stupdates_join_general:
- shows "concat (map hElem (map (rder x) (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates xs r0 Ss)))) =
- map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates (xs @ [x]) r0 Ss)"
- apply(induct xs arbitrary: Ss)
- apply (simp)
- prefer 2
- apply auto[1]
- using stupdate_induct1 by blast
-
-lemma star_hfau_induct:
- shows "hflat_aux (rders (RSEQ (rder c r0) (RSTAR r0)) s) =
- map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates s r0 [[c]])"
- apply(induct s rule: rev_induct)
- apply simp
- apply(subst rders_append)+
- apply simp
- apply(subst stupdates_append)
- apply(subgoal_tac "created_by_star (rders (RSEQ (rder c r0) (RSTAR r0)) xs)")
- prefer 2
- apply (simp add: star_ders_cbs)
- apply(subst hfau_pushin)
- apply simp
- apply(subgoal_tac "concat (map hElem (map (rder x) (hflat_aux (rders (RSEQ (rder c r0) (RSTAR r0)) xs)))) =
- concat (map hElem (map (rder x) ( map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0)) (star_updates xs r0 [[c]])))) ")
- apply(simp only:)
- prefer 2
- apply presburger
- apply(subst stupdates_append[symmetric])
- using stupdates_join_general by blast
-
-lemma starders_hfau_also1:
- shows "hflat_aux (rders (RSTAR r) (c # xs)) = map (\<lambda>s1. RSEQ (rders r s1) (RSTAR r)) (star_updates xs r [[c]])"
- using star_hfau_induct by force
-
-lemma hflat_aux_grewrites:
- shows "a # rs \<leadsto>g* hflat_aux a @ rs"
- apply(induct a arbitrary: rs)
- apply simp+
- apply(case_tac x)
- apply simp
- apply(case_tac list)
-
- apply (metis append.right_neutral append_Cons append_eq_append_conv2 grewrites.simps hflat_aux.simps(7) same_append_eq)
- apply(case_tac lista)
- apply simp
- apply (metis (no_types, lifting) append_Cons append_eq_append_conv2 gmany_steps_later greal_trans grewrite.intros(2) grewrites_append self_append_conv)
- apply simp
- by simp
-
-
-
-
-lemma cbs_hfau_rsimpeq1:
- shows "rsimp (RALT a b) = rsimp (RALTS ((hflat_aux a) @ (hflat_aux b)))"
- apply(subgoal_tac "[a, b] \<leadsto>g* hflat_aux a @ hflat_aux b")
- using grewrites_equal_rsimp apply presburger
- by (metis append.right_neutral greal_trans grewrites_cons hflat_aux_grewrites)
-
-
-lemma hfau_rsimpeq2:
- shows "created_by_star r \<Longrightarrow> rsimp r = rsimp ( (RALTS (hflat_aux r)))"
- apply(induct r)
- apply simp+
-
- apply (metis rsimp_seq_equal1)
- prefer 2
- apply simp
- apply(case_tac x)
- apply simp
- apply(case_tac "list")
- apply simp
-
- apply (metis idem_after_simp1)
- apply(case_tac "lista")
- prefer 2
- apply (metis hflat_aux.simps(8) idem_after_simp1 list.simps(8) list.simps(9) rsimp.simps(2))
- apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux (RALT a aa)))")
- apply simp
- apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux a @ hflat_aux aa))")
- using hflat_aux.simps(1) apply presburger
- apply simp
- using cbs_hfau_rsimpeq1 by fastforce
-
-lemma star_closed_form1:
- shows "rsimp (rders (RSTAR r0) (c#s)) =
-rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
- using hfau_rsimpeq2 rder.simps(6) rders.simps(2) star_ders_cbs starders_hfau_also1 by presburger
-
-lemma star_closed_form2:
- shows "rsimp (rders_simp (RSTAR r0) (c#s)) =
-rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
- by (metis list.distinct(1) rders_simp_same_simpders rsimp_idem star_closed_form1)
-
-lemma star_closed_form3:
- shows "rsimp (rders_simp (RSTAR r0) (c#s)) = (rders_simp (RSTAR r0) (c#s))"
- by (metis list.distinct(1) rders_simp_same_simpders star_closed_form1 star_closed_form2)
-
-lemma star_closed_form4:
- shows " (rders_simp (RSTAR r0) (c#s)) =
-rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
- using star_closed_form2 star_closed_form3 by presburger
-
-lemma star_closed_form5:
- shows " rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rders r0 s1) (RSTAR r0) ) Ss )))) =
- rsimp ( ( RALTS ( (map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss ))))"
- by (metis (mono_tags, lifting) list.map_comp map_eq_conv o_apply rsimp.simps(2) rsimp_idem)
-
-lemma star_closed_form6_hrewrites:
- shows "
- (map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss )
- scf\<leadsto>*
-(map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss )"
- apply(induct Ss)
- apply simp
- apply (simp add: ss1)
- by (metis (no_types, lifting) list.simps(9) rsimp.simps(1) rsimp_idem simp_hrewrites ss2)
-
-lemma star_closed_form6:
- shows " rsimp ( ( RALTS ( (map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss )))) =
- rsimp ( ( RALTS ( (map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss ))))"
- apply(subgoal_tac " map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RSTAR r0)) ) Ss scf\<leadsto>*
- map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RSTAR r0)) ) Ss ")
- using hrewrites_simpeq srewritescf_alt1 apply fastforce
- using star_closed_form6_hrewrites by blast
-
-lemma stupdate_nonempty:
- shows "\<forall>s \<in> set Ss. s \<noteq> [] \<Longrightarrow> \<forall>s \<in> set (star_update c r Ss). s \<noteq> []"
- apply(induct Ss)
- apply simp
- apply(case_tac "rnullable (rders r a)")
- apply simp+
- done
-
-
-lemma stupdates_nonempty:
- shows "\<forall>s \<in> set Ss. s\<noteq> [] \<Longrightarrow> \<forall>s \<in> set (star_updates s r Ss). s \<noteq> []"
- apply(induct s arbitrary: Ss)
- apply simp
- apply simp
- using stupdate_nonempty by presburger
-
-
-lemma star_closed_form8:
- shows
-"rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ (rsimp (rders r0 s1)) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))) =
- rsimp ( ( RALTS ( (map (\<lambda>s1. RSEQ ( (rders_simp r0 s1)) (RSTAR r0) ) (star_updates s r0 [[c]]) ) )))"
- by (smt (verit, ccfv_SIG) list.simps(8) map_eq_conv rders__onechar rders_simp_same_simpders set_ConsD stupdates_nonempty)
-
-
-lemma star_closed_form:
- shows "rders_simp (RSTAR r0) (c#s) =
-rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
- apply(induct s)
- apply simp
- apply (metis idem_after_simp1 rsimp.simps(1) rsimp.simps(6) rsimp_idem)
- using star_closed_form4 star_closed_form5 star_closed_form6 star_closed_form8 by presburger
-
-
-unused_thms
-
-end
\ No newline at end of file