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theory BasicIdentities
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imports "Lexer"
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begin
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datatype rrexp =
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RZERO
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| RONE
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| RCHAR char
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| RSEQ rrexp rrexp
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| RALTS "rrexp list"
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| RSTAR rrexp
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| RNTIMES rrexp nat
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abbreviation
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"RALT r1 r2 \<equiv> RALTS [r1, r2]"
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fun
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rnullable :: "rrexp \<Rightarrow> bool"
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where
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"rnullable (RZERO) = False"
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| "rnullable (RONE) = True"
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| "rnullable (RCHAR c) = False"
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| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"
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| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"
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| "rnullable (RSTAR r) = True"
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| "rnullable (RNTIMES r n) = (if n = 0 then True else rnullable r)"
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fun
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rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
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where
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"rder c (RZERO) = RZERO"
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| "rder c (RONE) = RZERO"
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| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
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| "rder c (RALTS rs) = RALTS (map (rder c) rs)"
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| "rder c (RSEQ r1 r2) =
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(if rnullable r1
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then RALT (RSEQ (rder c r1) r2) (rder c r2)
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else RSEQ (rder c r1) r2)"
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| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
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| "rder c (RNTIMES r n) = (if n = 0 then RZERO else RSEQ (rder c r) (RNTIMES r (n - 1)))"
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fun
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rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
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where
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"rders r [] = r"
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| "rders r (c#s) = rders (rder c r) s"
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fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
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where
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"rdistinct [] acc = []"
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| "rdistinct (x#xs) acc =
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(if x \<in> acc then rdistinct xs acc
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else x # (rdistinct xs ({x} \<union> acc)))"
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lemma rdistinct1:
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assumes "a \<in> acc"
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shows "a \<notin> set (rdistinct rs acc)"
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using assms
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apply(induct rs arbitrary: acc a)
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apply(auto)
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done
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lemma rdistinct_does_the_job:
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shows "distinct (rdistinct rs s)"
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apply(induct rs s rule: rdistinct.induct)
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apply(auto simp add: rdistinct1)
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done
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lemma rdistinct_concat:
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assumes "set rs \<subseteq> rset"
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shows "rdistinct (rs @ rsa) rset = rdistinct rsa rset"
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using assms
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apply(induct rs)
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apply simp+
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done
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lemma distinct_not_exist:
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assumes "a \<notin> set rs"
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shows "rdistinct rs rset = rdistinct rs (insert a rset)"
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using assms
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apply(induct rs arbitrary: rset)
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apply(auto)
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done
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lemma rdistinct_on_distinct:
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shows "distinct rs \<Longrightarrow> rdistinct rs {} = rs"
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apply(induct rs)
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apply simp
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using distinct_not_exist by fastforce
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lemma distinct_rdistinct_append:
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assumes "distinct rs1" "\<forall>r \<in> set rs1. r \<notin> acc"
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shows "rdistinct (rs1 @ rsa) acc = rs1 @ (rdistinct rsa (acc \<union> set rs1))"
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using assms
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apply(induct rs1 arbitrary: rsa acc)
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apply(auto)[1]
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apply(auto)[1]
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apply(drule_tac x="rsa" in meta_spec)
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apply(drule_tac x="{a} \<union> acc" in meta_spec)
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apply(simp)
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apply(drule meta_mp)
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apply(auto)[1]
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apply(simp)
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done
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lemma rdistinct_set_equality1:
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shows "set (rdistinct rs acc) = set rs - acc"
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apply(induct rs acc rule: rdistinct.induct)
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apply(auto)
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done
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lemma rdistinct_set_equality:
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shows "set (rdistinct rs {}) = set rs"
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by (simp add: rdistinct_set_equality1)
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fun rflts :: "rrexp list \<Rightarrow> rrexp list"
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where
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"rflts [] = []"
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| "rflts (RZERO # rs) = rflts rs"
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| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
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| "rflts (r1 # rs) = r1 # rflts rs"
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lemma rflts_def_idiot:
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shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow> rflts (a # rs) = a # rflts rs"
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apply(case_tac a)
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apply simp_all
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done
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lemma rflts_def_idiot2:
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shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow> a \<in> set (rflts rs)"
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apply(induct rs rule: rflts.induct)
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apply(auto)
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done
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lemma flts_append:
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shows "rflts (rs1 @ rs2) = rflts rs1 @ rflts rs2"
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apply(induct rs1)
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apply simp
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apply(case_tac a)
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apply simp+
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done
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fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
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where
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"rsimp_ALTs [] = RZERO"
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| "rsimp_ALTs [r] = r"
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| "rsimp_ALTs rs = RALTS rs"
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lemma rsimpalts_conscons:
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shows "rsimp_ALTs (r1 # rsa @ r2 # rsb) = RALTS (r1 # rsa @ r2 # rsb)"
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by (metis Nil_is_append_conv list.exhaust rsimp_ALTs.simps(3))
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lemma rsimp_alts_equal:
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shows "rsimp_ALTs (rsa @ a # rsb @ a # rsc) = RALTS (rsa @ a # rsb @ a # rsc) "
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by (metis append_Cons append_Nil neq_Nil_conv rsimpalts_conscons)
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fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
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where
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"rsimp_SEQ RZERO _ = RZERO"
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| "rsimp_SEQ _ RZERO = RZERO"
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| "rsimp_SEQ RONE r2 = r2"
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| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
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fun rsimp :: "rrexp \<Rightarrow> rrexp"
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where
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"rsimp (RSEQ r1 r2) = rsimp_SEQ (rsimp r1) (rsimp r2)"
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| "rsimp (RALTS rs) = rsimp_ALTs (rdistinct (rflts (map rsimp rs)) {}) "
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| "rsimp r = r"
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fun
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rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
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where
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"rders_simp r [] = r"
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| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
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fun rsize :: "rrexp \<Rightarrow> nat" where
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"rsize RZERO = 1"
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| "rsize (RONE) = 1"
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| "rsize (RCHAR c) = 1"
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| "rsize (RALTS rs) = Suc (sum_list (map rsize rs))"
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| "rsize (RSEQ r1 r2) = Suc (rsize r1 + rsize r2)"
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| "rsize (RSTAR r) = Suc (rsize r)"
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| "rsize (RNTIMES r n) = Suc (rsize r) + n"
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abbreviation rsizes where
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"rsizes rs \<equiv> sum_list (map rsize rs)"
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lemma rder_rsimp_ALTs_commute:
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shows " (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"
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apply(induct rs)
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apply simp
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apply(case_tac rs)
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apply simp
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apply auto
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done
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lemma rsimp_aalts_smaller:
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shows "rsize (rsimp_ALTs rs) \<le> rsize (RALTS rs)"
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apply(induct rs)
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apply simp
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apply simp
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apply(case_tac "rs = []")
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apply simp
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apply(subgoal_tac "\<exists>rsp ap. rs = ap # rsp")
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apply(erule exE)+
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apply simp
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apply simp
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by(meson neq_Nil_conv)
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lemma rSEQ_mono:
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shows "rsize (rsimp_SEQ r1 r2) \<le>rsize (RSEQ r1 r2)"
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apply auto
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apply(induct r1)
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apply auto
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apply(case_tac "r2")
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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done
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lemma ralts_cap_mono:
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shows "rsize (RALTS rs) \<le> Suc (rsizes rs)"
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by simp
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lemma rflts_mono:
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shows "rsizes (rflts rs) \<le> rsizes rs"
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apply(induct rs)
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apply simp
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apply(case_tac "a = RZERO")
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apply simp
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apply(case_tac "\<exists>rs1. a = RALTS rs1")
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apply(erule exE)
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apply simp
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apply(subgoal_tac "rflts (a # rs) = a # (rflts rs)")
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prefer 2
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using rflts_def_idiot apply blast
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apply simp
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done
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lemma rdistinct_smaller:
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shows "rsizes (rdistinct rs ss) \<le> rsizes rs"
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apply (induct rs arbitrary: ss)
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apply simp
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by (simp add: trans_le_add2)
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lemma rsimp_alts_mono :
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shows "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> rsize (rsimp xa) \<le> rsize xa) \<Longrightarrow>
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rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {})) \<le> Suc (rsizes x)"
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apply(subgoal_tac "rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {} ))
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\<le> rsize (RALTS (rdistinct (rflts (map rsimp x)) {} ))")
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prefer 2
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using rsimp_aalts_smaller apply auto[1]
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apply(subgoal_tac "rsize (RALTS (rdistinct (rflts (map rsimp x)) {})) \<le>Suc (rsizes (rdistinct (rflts (map rsimp x)) {}))")
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prefer 2
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using ralts_cap_mono apply blast
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apply(subgoal_tac "rsizes (rdistinct (rflts (map rsimp x)) {}) \<le> rsizes (rflts (map rsimp x))")
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prefer 2
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using rdistinct_smaller apply presburger
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apply(subgoal_tac "rsizes (rflts (map rsimp x)) \<le> rsizes (map rsimp x)")
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prefer 2
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using rflts_mono apply blast
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apply(subgoal_tac "rsizes (map rsimp x) \<le> rsizes x")
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prefer 2
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apply (simp add: sum_list_mono)
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by linarith
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lemma rsimp_mono:
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shows "rsize (rsimp r) \<le> rsize r"
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apply(induct r)
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apply simp_all
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apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")
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apply force
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using rSEQ_mono
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apply presburger
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using rsimp_alts_mono by auto
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lemma idiot:
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shows "rsimp_SEQ RONE r = r"
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apply(case_tac r)
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apply simp_all
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done
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lemma idiot2:
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shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
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\<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
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apply(case_tac r1)
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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done
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lemma rders__onechar:
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|
342 |
shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
|
|
|
343 |
by simp
|
|
|
344 |
|
|
|
345 |
lemma rders_append:
|
|
|
346 |
"rders c (s1 @ s2) = rders (rders c s1) s2"
|
|
|
347 |
apply(induct s1 arbitrary: c s2)
|
|
|
348 |
apply(simp_all)
|
|
|
349 |
done
|
|
|
350 |
|
|
|
351 |
lemma rders_simp_append:
|
|
|
352 |
"rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
|
|
|
353 |
apply(induct s1 arbitrary: c s2)
|
|
|
354 |
apply(simp_all)
|
|
|
355 |
done
|
|
|
356 |
|
|
|
357 |
|
|
|
358 |
lemma rders_simp_one_char:
|
|
|
359 |
shows "rders_simp r [c] = rsimp (rder c r)"
|
|
|
360 |
apply auto
|
|
|
361 |
done
|
|
|
362 |
|
|
|
363 |
|
|
|
364 |
|
|
|
365 |
fun nonalt :: "rrexp \<Rightarrow> bool"
|
|
|
366 |
where
|
|
|
367 |
"nonalt (RALTS rs) = False"
|
|
|
368 |
| "nonalt r = True"
|
|
|
369 |
|
|
|
370 |
|
|
|
371 |
fun good :: "rrexp \<Rightarrow> bool" where
|
|
|
372 |
"good RZERO = False"
|
|
|
373 |
| "good (RONE) = True"
|
|
|
374 |
| "good (RCHAR c) = True"
|
|
|
375 |
| "good (RALTS []) = False"
|
|
|
376 |
| "good (RALTS [r]) = False"
|
|
|
377 |
| "good (RALTS (r1 # r2 # rs)) = ((distinct ( (r1 # r2 # rs))) \<and>(\<forall>r' \<in> set (r1 # r2 # rs). good r' \<and> nonalt r'))"
|
|
|
378 |
| "good (RSEQ RZERO _) = False"
|
|
|
379 |
| "good (RSEQ RONE _) = False"
|
|
|
380 |
| "good (RSEQ _ RZERO) = False"
|
|
|
381 |
| "good (RSEQ r1 r2) = (good r1 \<and> good r2)"
|
|
|
382 |
| "good (RSTAR r) = True"
|
|
|
383 |
| "good (RNTIMES r n) = True"
|
|
|
384 |
|
|
|
385 |
lemma k0a:
|
|
|
386 |
shows "rflts [RALTS rs] = rs"
|
|
|
387 |
apply(simp)
|
|
|
388 |
done
|
|
|
389 |
|
|
|
390 |
lemma bbbbs:
|
|
|
391 |
assumes "good r" "r = RALTS rs"
|
|
|
392 |
shows "rsimp_ALTs (rflts [r]) = RALTS rs"
|
|
|
393 |
using assms
|
|
|
394 |
by (metis good.simps(4) good.simps(5) k0a rsimp_ALTs.elims)
|
|
|
395 |
|
|
|
396 |
lemma bbbbs1:
|
|
|
397 |
shows "nonalt r \<or> (\<exists> rs. r = RALTS rs)"
|
|
|
398 |
by (meson nonalt.elims(3))
|
|
|
399 |
|
|
|
400 |
|
|
|
401 |
|
|
|
402 |
lemma good0:
|
|
|
403 |
assumes "rs \<noteq> Nil" "\<forall>r \<in> set rs. nonalt r" "distinct rs"
|
|
|
404 |
shows "good (rsimp_ALTs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. good r)"
|
|
|
405 |
using assms
|
|
|
406 |
apply(induct rs rule: rsimp_ALTs.induct)
|
|
|
407 |
apply(auto)
|
|
|
408 |
done
|
|
|
409 |
|
|
|
410 |
lemma flts1:
|
|
|
411 |
assumes "good r"
|
|
|
412 |
shows "rflts [r] \<noteq> []"
|
|
|
413 |
using assms
|
|
|
414 |
apply(induct r)
|
|
|
415 |
apply(simp_all)
|
|
|
416 |
using good.simps(4) by blast
|
|
|
417 |
|
|
|
418 |
lemma flts2:
|
|
|
419 |
assumes "good r"
|
|
|
420 |
shows "\<forall>r' \<in> set (rflts [r]). good r' \<and> nonalt r'"
|
|
|
421 |
using assms
|
|
|
422 |
apply(induct r)
|
|
|
423 |
apply(simp)
|
|
|
424 |
apply(simp)
|
|
|
425 |
apply(simp)
|
|
|
426 |
prefer 2
|
|
|
427 |
apply(simp)
|
|
|
428 |
apply(auto)[1]
|
|
|
429 |
|
|
|
430 |
apply (metis flts1 good.simps(5) good.simps(6) k0a neq_Nil_conv)
|
|
|
431 |
apply (metis flts1 good.simps(5) good.simps(6) k0a neq_Nil_conv)
|
|
|
432 |
apply fastforce
|
|
|
433 |
apply(simp)
|
|
|
434 |
by simp
|
|
|
435 |
|
|
|
436 |
|
|
|
437 |
lemma flts3:
|
|
|
438 |
assumes "\<forall>r \<in> set rs. good r \<or> r = RZERO"
|
|
|
439 |
shows "\<forall>r \<in> set (rflts rs). good r"
|
|
|
440 |
using assms
|
|
|
441 |
apply(induct rs arbitrary: rule: rflts.induct)
|
|
|
442 |
apply(simp_all)
|
|
|
443 |
by (metis UnE flts2 k0a)
|
|
|
444 |
|
|
|
445 |
|
|
|
446 |
lemma k0:
|
|
|
447 |
shows "rflts (r # rs1) = rflts [r] @ rflts rs1"
|
|
|
448 |
apply(induct r arbitrary: rs1)
|
|
|
449 |
apply(auto)
|
|
|
450 |
done
|
|
|
451 |
|
|
|
452 |
|
|
|
453 |
lemma good_SEQ:
|
|
|
454 |
assumes "r1 \<noteq> RZERO" "r2 \<noteq> RZERO" " r1 \<noteq> RONE"
|
|
|
455 |
shows "good (RSEQ r1 r2) = (good r1 \<and> good r2)"
|
|
|
456 |
using assms
|
|
|
457 |
apply(case_tac r1)
|
|
|
458 |
apply(simp_all)
|
|
|
459 |
apply(case_tac r2)
|
|
|
460 |
apply(simp_all)
|
|
|
461 |
apply(case_tac r2)
|
|
|
462 |
apply(simp_all)
|
|
|
463 |
apply(case_tac r2)
|
|
|
464 |
apply(simp_all)
|
|
|
465 |
apply(case_tac r2)
|
|
|
466 |
apply(simp_all)
|
|
|
467 |
apply(case_tac r2)
|
|
|
468 |
apply(simp_all)
|
|
|
469 |
done
|
|
|
470 |
|
|
|
471 |
lemma rsize0:
|
|
|
472 |
shows "0 < rsize r"
|
|
|
473 |
apply(induct r)
|
|
|
474 |
apply(auto)
|
|
|
475 |
done
|
|
|
476 |
|
|
|
477 |
|
|
|
478 |
fun nonnested :: "rrexp \<Rightarrow> bool"
|
|
|
479 |
where
|
|
|
480 |
"nonnested (RALTS []) = True"
|
|
|
481 |
| "nonnested (RALTS ((RALTS rs1) # rs2)) = False"
|
|
|
482 |
| "nonnested (RALTS (r # rs2)) = nonnested (RALTS rs2)"
|
|
|
483 |
| "nonnested r = True"
|
|
|
484 |
|
|
|
485 |
|
|
|
486 |
|
|
|
487 |
lemma k00:
|
|
|
488 |
shows "rflts (rs1 @ rs2) = rflts rs1 @ rflts rs2"
|
|
|
489 |
apply(induct rs1 arbitrary: rs2)
|
|
|
490 |
apply(auto)
|
|
|
491 |
by (metis append.assoc k0)
|
|
|
492 |
|
|
|
493 |
|
|
|
494 |
|
|
|
495 |
|
|
|
496 |
lemma k0b:
|
|
|
497 |
assumes "nonalt r" "r \<noteq> RZERO"
|
|
|
498 |
shows "rflts [r] = [r]"
|
|
|
499 |
using assms
|
|
|
500 |
apply(case_tac r)
|
|
|
501 |
apply(simp_all)
|
|
|
502 |
done
|
|
|
503 |
|
|
|
504 |
lemma nn1qq:
|
|
|
505 |
assumes "nonnested (RALTS rs)"
|
|
|
506 |
shows "\<nexists> rs1. RALTS rs1 \<in> set rs"
|
|
|
507 |
using assms
|
|
|
508 |
apply(induct rs rule: rflts.induct)
|
|
|
509 |
apply(auto)
|
|
|
510 |
done
|
|
|
511 |
|
|
|
512 |
|
|
|
513 |
|
|
|
514 |
lemma n0:
|
|
|
515 |
shows "nonnested (RALTS rs) \<longleftrightarrow> (\<forall>r \<in> set rs. nonalt r)"
|
|
|
516 |
apply(induct rs )
|
|
|
517 |
apply(auto)
|
|
|
518 |
apply (metis list.set_intros(1) nn1qq nonalt.elims(3))
|
|
|
519 |
apply (metis nonalt.elims(2) nonnested.simps(3) nonnested.simps(4) nonnested.simps(5) nonnested.simps(6) nonnested.simps(7) nonnested.simps(8))
|
|
|
520 |
using bbbbs1 apply fastforce
|
|
|
521 |
by (metis bbbbs1 list.set_intros(2) nn1qq)
|
|
|
522 |
|
|
|
523 |
|
|
|
524 |
|
|
|
525 |
|
|
|
526 |
lemma nn1c:
|
|
|
527 |
assumes "\<forall>r \<in> set rs. nonnested r"
|
|
|
528 |
shows "\<forall>r \<in> set (rflts rs). nonalt r"
|
|
|
529 |
using assms
|
|
|
530 |
apply(induct rs rule: rflts.induct)
|
|
|
531 |
apply(auto)
|
|
|
532 |
using n0 by blast
|
|
|
533 |
|
|
|
534 |
lemma nn1bb:
|
|
|
535 |
assumes "\<forall>r \<in> set rs. nonalt r"
|
|
|
536 |
shows "nonnested (rsimp_ALTs rs)"
|
|
|
537 |
using assms
|
|
|
538 |
apply(induct rs rule: rsimp_ALTs.induct)
|
|
|
539 |
apply(auto)
|
|
|
540 |
using nonalt.simps(1) nonnested.elims(3) apply blast
|
|
|
541 |
using n0 by auto
|
|
|
542 |
|
|
|
543 |
lemma bsimp_ASEQ0:
|
|
|
544 |
shows "rsimp_SEQ r1 RZERO = RZERO"
|
|
|
545 |
apply(induct r1)
|
|
|
546 |
apply(auto)
|
|
|
547 |
done
|
|
|
548 |
|
|
|
549 |
lemma nn1b:
|
|
|
550 |
shows "nonnested (rsimp r)"
|
|
|
551 |
apply(induct r)
|
|
|
552 |
apply(simp_all)
|
|
|
553 |
apply(case_tac "rsimp r1 = RZERO")
|
|
|
554 |
apply(simp)
|
|
|
555 |
apply(case_tac "rsimp r2 = RZERO")
|
|
|
556 |
apply(simp)
|
|
|
557 |
apply(subst bsimp_ASEQ0)
|
|
|
558 |
apply(simp)
|
|
|
559 |
apply(case_tac "\<exists>bs. rsimp r1 = RONE")
|
|
|
560 |
apply(auto)[1]
|
|
|
561 |
using idiot apply fastforce
|
|
|
562 |
apply (simp add: idiot2)
|
|
|
563 |
by (metis (mono_tags, lifting) image_iff list.set_map nn1bb nn1c rdistinct_set_equality)
|
|
|
564 |
|
|
|
565 |
lemma nonalt_flts_rd:
|
|
|
566 |
shows "\<lbrakk>xa \<in> set (rdistinct (rflts (map rsimp rs)) {})\<rbrakk>
|
|
|
567 |
\<Longrightarrow> nonalt xa"
|
|
|
568 |
by (metis Diff_empty ex_map_conv nn1b nn1c rdistinct_set_equality1)
|
|
|
569 |
|
|
|
570 |
|
|
|
571 |
lemma rsimpalts_implies1:
|
|
|
572 |
shows " rsimp_ALTs (a # rdistinct rs {a}) = RZERO \<Longrightarrow> a = RZERO"
|
|
|
573 |
using rsimp_ALTs.elims by auto
|
|
|
574 |
|
|
|
575 |
|
|
|
576 |
lemma rsimpalts_implies2:
|
|
|
577 |
shows "rsimp_ALTs (a # rdistinct rs rset) = RZERO \<Longrightarrow> rdistinct rs rset = []"
|
|
|
578 |
by (metis append_butlast_last_id rrexp.distinct(7) rsimpalts_conscons)
|
|
|
579 |
|
|
|
580 |
lemma rsimpalts_implies21:
|
|
|
581 |
shows "rsimp_ALTs (a # rdistinct rs {a}) = RZERO \<Longrightarrow> rdistinct rs {a} = []"
|
|
|
582 |
using rsimpalts_implies2 by blast
|
|
|
583 |
|
|
|
584 |
|
|
|
585 |
lemma bsimp_ASEQ2:
|
|
|
586 |
shows "rsimp_SEQ RONE r2 = r2"
|
|
|
587 |
apply(induct r2)
|
|
|
588 |
apply(auto)
|
|
|
589 |
done
|
|
|
590 |
|
|
|
591 |
lemma elem_smaller_than_set:
|
|
|
592 |
shows "xa \<in> set list \<Longrightarrow> rsize xa < Suc (rsizes list)"
|
|
|
593 |
apply(induct list)
|
|
|
594 |
apply simp
|
|
|
595 |
by (metis image_eqI le_imp_less_Suc list.set_map member_le_sum_list)
|
|
|
596 |
|
|
|
597 |
lemma rsimp_list_mono:
|
|
|
598 |
shows "rsizes (map rsimp rs) \<le> rsizes rs"
|
|
|
599 |
apply(induct rs)
|
|
|
600 |
apply simp+
|
|
|
601 |
by (simp add: add_mono_thms_linordered_semiring(1) rsimp_mono)
|
|
|
602 |
|
|
|
603 |
|
|
|
604 |
(*says anything coming out of simp+flts+db will be good*)
|
|
|
605 |
lemma good2_obv_simplified:
|
|
|
606 |
shows " \<lbrakk>\<forall>y. rsize y < Suc (rsizes rs) \<longrightarrow> good (rsimp y) \<or> rsimp y = RZERO;
|
|
|
607 |
xa \<in> set (rdistinct (rflts (map rsimp rs)) {}); good (rsimp xa) \<or> rsimp xa = RZERO\<rbrakk> \<Longrightarrow> good xa"
|
|
|
608 |
apply(subgoal_tac " \<forall>xa' \<in> set (map rsimp rs). good xa' \<or> xa' = RZERO")
|
|
|
609 |
prefer 2
|
|
|
610 |
apply (simp add: elem_smaller_than_set)
|
|
|
611 |
by (metis Diff_empty flts3 rdistinct_set_equality1)
|
|
|
612 |
|
|
|
613 |
thm Diff_empty flts3 rdistinct_set_equality1
|
|
|
614 |
|
|
|
615 |
lemma good1:
|
|
|
616 |
shows "good (rsimp a) \<or> rsimp a = RZERO"
|
|
|
617 |
apply(induct a taking: rsize rule: measure_induct)
|
|
|
618 |
apply(case_tac x)
|
|
|
619 |
apply(simp)
|
|
|
620 |
apply(simp)
|
|
|
621 |
apply(simp)
|
|
|
622 |
prefer 3
|
|
|
623 |
apply(simp)
|
|
|
624 |
prefer 2
|
|
|
625 |
apply(simp only:)
|
|
|
626 |
apply simp
|
|
|
627 |
apply (smt (verit, ccfv_threshold) add_mono_thms_linordered_semiring(1) elem_smaller_than_set good0 good2_obv_simplified le_eq_less_or_eq nonalt_flts_rd order_less_trans plus_1_eq_Suc rdistinct_does_the_job rdistinct_smaller rflts_mono rsimp_ALTs.simps(1) rsimp_list_mono)
|
|
|
628 |
apply simp
|
|
|
629 |
apply(subgoal_tac "good (rsimp x41) \<or> rsimp x41 = RZERO")
|
|
|
630 |
apply(subgoal_tac "good (rsimp x42) \<or> rsimp x42 = RZERO")
|
|
|
631 |
apply(case_tac "rsimp x41 = RZERO")
|
|
|
632 |
apply simp
|
|
|
633 |
apply(case_tac "rsimp x42 = RZERO")
|
|
|
634 |
apply simp
|
|
|
635 |
using bsimp_ASEQ0 apply blast
|
|
|
636 |
apply(subgoal_tac "good (rsimp x41)")
|
|
|
637 |
apply(subgoal_tac "good (rsimp x42)")
|
|
|
638 |
apply simp
|
|
|
639 |
apply (metis bsimp_ASEQ2 good_SEQ idiot2)
|
|
|
640 |
apply blast
|
|
|
641 |
apply fastforce
|
|
|
642 |
using less_add_Suc2 apply blast
|
|
|
643 |
using less_iff_Suc_add apply blast
|
|
|
644 |
using good.simps(45) rsimp.simps(7) by presburger
|
|
|
645 |
|
|
|
646 |
|
|
|
647 |
|
|
|
648 |
fun
|
|
|
649 |
RL :: "rrexp \<Rightarrow> string set"
|
|
|
650 |
where
|
|
|
651 |
"RL (RZERO) = {}"
|
|
|
652 |
| "RL (RONE) = {[]}"
|
|
|
653 |
| "RL (RCHAR c) = {[c]}"
|
|
|
654 |
| "RL (RSEQ r1 r2) = (RL r1) ;; (RL r2)"
|
|
|
655 |
| "RL (RALTS rs) = (\<Union> (set (map RL rs)))"
|
|
|
656 |
| "RL (RSTAR r) = (RL r)\<star>"
|
|
|
657 |
| "RL (RNTIMES r n) = (RL r) ^^ n"
|
|
|
658 |
|
|
|
659 |
lemma pow_rempty_iff:
|
|
|
660 |
shows "[] \<in> (RL r) ^^ n \<longleftrightarrow> (if n = 0 then True else [] \<in> (RL r))"
|
|
|
661 |
by (induct n) (auto simp add: Sequ_def)
|
|
|
662 |
|
|
|
663 |
lemma RL_rnullable:
|
|
|
664 |
shows "rnullable r = ([] \<in> RL r)"
|
|
|
665 |
apply(induct r)
|
|
|
666 |
apply(auto simp add: Sequ_def pow_rempty_iff)
|
|
|
667 |
done
|
|
|
668 |
|
|
|
669 |
lemma concI_if_Nil1: "[] \<in> A \<Longrightarrow> xs : B \<Longrightarrow> xs \<in> A ;; B"
|
|
|
670 |
by (metis append_Nil concI)
|
|
|
671 |
|
|
|
672 |
|
|
|
673 |
lemma empty_pow_add:
|
|
|
674 |
fixes A::"string set"
|
|
|
675 |
assumes "[] \<in> A" "s \<in> A ^^ n"
|
|
|
676 |
shows "s \<in> A ^^ (n + m)"
|
|
|
677 |
using assms
|
|
|
678 |
apply(induct m arbitrary: n)
|
|
|
679 |
apply(auto simp add: Sequ_def)
|
|
|
680 |
done
|
|
|
681 |
|
|
|
682 |
(*
|
|
|
683 |
lemma der_pow:
|
|
|
684 |
shows "Der c (A ^^ n) = (if n = 0 then {} else (Der c A) ;; (A ^^ (n - 1)))"
|
|
|
685 |
apply(induct n arbitrary: A)
|
|
|
686 |
apply(auto)
|
|
|
687 |
by (smt (verit, best) Suc_pred concE concI concI_if_Nil2 conc_pow_comm lang_pow.simps(2))
|
|
|
688 |
*)
|
|
|
689 |
|
|
|
690 |
lemma RL_rder:
|
|
|
691 |
shows "RL (rder c r) = Der c (RL r)"
|
|
|
692 |
apply(induct r)
|
|
|
693 |
apply(auto simp add: Sequ_def Der_def)[5]
|
|
|
694 |
apply (metis append_Cons)
|
|
|
695 |
using RL_rnullable apply blast
|
|
|
696 |
apply (metis append_eq_Cons_conv)
|
|
|
697 |
apply (metis append_Cons)
|
|
|
698 |
apply (metis RL_rnullable append_eq_Cons_conv)
|
|
|
699 |
apply simp
|
|
|
700 |
apply(simp)
|
|
|
701 |
done
|
|
|
702 |
|
|
|
703 |
lemma RL_rsimp_RSEQ:
|
|
|
704 |
shows "RL (rsimp_SEQ r1 r2) = (RL r1 ;; RL r2)"
|
|
|
705 |
apply(induct r1 r2 rule: rsimp_SEQ.induct)
|
|
|
706 |
apply(simp_all)
|
|
|
707 |
done
|
|
|
708 |
|
|
|
709 |
lemma RL_rsimp_RALTS:
|
|
|
710 |
shows "RL (rsimp_ALTs rs) = (\<Union> (set (map RL rs)))"
|
|
|
711 |
apply(induct rs rule: rsimp_ALTs.induct)
|
|
|
712 |
apply(simp_all)
|
|
|
713 |
done
|
|
|
714 |
|
|
|
715 |
lemma RL_rsimp_rdistinct:
|
|
|
716 |
shows "(\<Union> (set (map RL (rdistinct rs {})))) = (\<Union> (set (map RL rs)))"
|
|
|
717 |
apply(auto)
|
|
|
718 |
apply (metis Diff_iff rdistinct_set_equality1)
|
|
|
719 |
by (metis Diff_empty rdistinct_set_equality1)
|
|
|
720 |
|
|
|
721 |
lemma RL_rsimp_rflts:
|
|
|
722 |
shows "(\<Union> (set (map RL (rflts rs)))) = (\<Union> (set (map RL rs)))"
|
|
|
723 |
apply(induct rs rule: rflts.induct)
|
|
|
724 |
apply(simp_all)
|
|
|
725 |
done
|
|
|
726 |
|
|
|
727 |
lemma RL_rsimp:
|
|
|
728 |
shows "RL r = RL (rsimp r)"
|
|
|
729 |
apply(induct r rule: rsimp.induct)
|
|
|
730 |
apply(auto simp add: Sequ_def RL_rsimp_RSEQ)
|
|
|
731 |
using RL_rsimp_RALTS RL_rsimp_rdistinct RL_rsimp_rflts apply auto[1]
|
|
|
732 |
by (smt (verit, del_insts) RL_rsimp_RALTS RL_rsimp_rdistinct RL_rsimp_rflts UN_E image_iff list.set_map)
|
|
|
733 |
|
|
|
734 |
|
|
|
735 |
lemma qqq1:
|
|
|
736 |
shows "RZERO \<notin> set (rflts (map rsimp rs))"
|
|
|
737 |
by (metis ex_map_conv flts3 good.simps(1) good1)
|
|
|
738 |
|
|
|
739 |
|
|
|
740 |
fun nonazero :: "rrexp \<Rightarrow> bool"
|
|
|
741 |
where
|
|
|
742 |
"nonazero RZERO = False"
|
|
|
743 |
| "nonazero r = True"
|
|
|
744 |
|
|
|
745 |
|
|
|
746 |
lemma flts_single1:
|
|
|
747 |
assumes "nonalt r" "nonazero r"
|
|
|
748 |
shows "rflts [r] = [r]"
|
|
|
749 |
using assms
|
|
|
750 |
apply(induct r)
|
|
|
751 |
apply(auto)
|
|
|
752 |
done
|
|
|
753 |
|
|
|
754 |
lemma nonalt0_flts_keeps:
|
|
|
755 |
shows "(a \<noteq> RZERO) \<and> (\<forall>rs. a \<noteq> RALTS rs) \<Longrightarrow> rflts (a # xs) = a # rflts xs"
|
|
|
756 |
apply(case_tac a)
|
|
|
757 |
apply simp+
|
|
|
758 |
done
|
|
|
759 |
|
|
|
760 |
|
|
|
761 |
lemma nonalt0_fltseq:
|
|
|
762 |
shows "\<forall>r \<in> set rs. r \<noteq> RZERO \<and> nonalt r \<Longrightarrow> rflts rs = rs"
|
|
|
763 |
apply(induct rs)
|
|
|
764 |
apply simp
|
|
|
765 |
apply(case_tac "a = RZERO")
|
|
|
766 |
apply fastforce
|
|
|
767 |
apply(case_tac "\<exists>rs1. a = RALTS rs1")
|
|
|
768 |
apply(erule exE)
|
|
|
769 |
apply simp+
|
|
|
770 |
using nonalt0_flts_keeps by presburger
|
|
|
771 |
|
|
|
772 |
|
|
|
773 |
|
|
|
774 |
|
|
|
775 |
lemma goodalts_nonalt:
|
|
|
776 |
shows "good (RALTS rs) \<Longrightarrow> rflts rs = rs"
|
|
|
777 |
apply(induct x == "RALTS rs" arbitrary: rs rule: good.induct)
|
|
|
778 |
apply simp
|
|
|
779 |
|
|
|
780 |
using good.simps(5) apply blast
|
|
|
781 |
apply simp
|
|
|
782 |
apply(case_tac "r1 = RZERO")
|
|
|
783 |
using good.simps(1) apply force
|
|
|
784 |
apply(case_tac "r2 = RZERO")
|
|
|
785 |
using good.simps(1) apply force
|
|
|
786 |
apply(subgoal_tac "rflts (r1 # r2 # rs) = r1 # r2 # rflts rs")
|
|
|
787 |
prefer 2
|
|
|
788 |
apply (metis nonalt.simps(1) rflts_def_idiot)
|
|
|
789 |
apply(subgoal_tac "\<forall>r \<in> set rs. r \<noteq> RZERO \<and> nonalt r")
|
|
|
790 |
apply(subgoal_tac "rflts rs = rs")
|
|
|
791 |
apply presburger
|
|
|
792 |
using nonalt0_fltseq apply presburger
|
|
|
793 |
using good.simps(1) by blast
|
|
|
794 |
|
|
|
795 |
|
|
|
796 |
|
|
|
797 |
|
|
|
798 |
|
|
|
799 |
lemma test:
|
|
|
800 |
assumes "good r"
|
|
|
801 |
shows "rsimp r = r"
|
|
|
802 |
|
|
|
803 |
using assms
|
|
|
804 |
apply(induct rule: good.induct)
|
|
|
805 |
apply simp
|
|
|
806 |
apply simp
|
|
|
807 |
apply simp
|
|
|
808 |
apply simp
|
|
|
809 |
apply simp
|
|
|
810 |
apply(subgoal_tac "distinct (r1 # r2 # rs)")
|
|
|
811 |
prefer 2
|
|
|
812 |
using good.simps(6) apply blast
|
|
|
813 |
apply(subgoal_tac "rflts (r1 # r2 # rs ) = r1 # r2 # rs")
|
|
|
814 |
prefer 2
|
|
|
815 |
using goodalts_nonalt apply blast
|
|
|
816 |
|
|
|
817 |
apply(subgoal_tac "r1 \<noteq> r2")
|
|
|
818 |
prefer 2
|
|
|
819 |
apply (meson distinct_length_2_or_more)
|
|
|
820 |
apply(subgoal_tac "r1 \<notin> set rs")
|
|
|
821 |
apply(subgoal_tac "r2 \<notin> set rs")
|
|
|
822 |
apply(subgoal_tac "\<forall>r \<in> set rs. rsimp r = r")
|
|
|
823 |
apply(subgoal_tac "map rsimp rs = rs")
|
|
|
824 |
apply simp
|
|
|
825 |
apply(subgoal_tac "\<forall>r \<in> {r1, r2}. r \<notin> set rs")
|
|
|
826 |
apply (metis distinct_not_exist rdistinct_on_distinct)
|
|
|
827 |
|
|
|
828 |
apply blast
|
|
|
829 |
apply (meson map_idI)
|
|
|
830 |
apply (metis good.simps(6) insert_iff list.simps(15))
|
|
|
831 |
|
|
|
832 |
apply (meson distinct.simps(2))
|
|
|
833 |
apply (simp add: distinct_length_2_or_more)
|
|
|
834 |
apply simp+
|
|
|
835 |
done
|
|
|
836 |
|
|
|
837 |
|
|
|
838 |
|
|
|
839 |
lemma rsimp_idem:
|
|
|
840 |
shows "rsimp (rsimp r) = rsimp r"
|
|
|
841 |
using test good1
|
|
|
842 |
by force
|
|
|
843 |
|
|
|
844 |
corollary rsimp_inner_idem4:
|
|
|
845 |
shows "rsimp r = RALTS rs \<Longrightarrow> rflts rs = rs"
|
|
|
846 |
by (metis good1 goodalts_nonalt rrexp.simps(12))
|
|
|
847 |
|
|
|
848 |
|
|
|
849 |
lemma head_one_more_simp:
|
|
|
850 |
shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
|
|
|
851 |
by (simp add: rsimp_idem)
|
|
|
852 |
|
|
|
853 |
|
|
|
854 |
lemma der_simp_nullability:
|
|
|
855 |
shows "rnullable r = rnullable (rsimp r)"
|
|
|
856 |
using RL_rnullable RL_rsimp by auto
|
|
|
857 |
|
|
|
858 |
|
|
|
859 |
lemma no_alt_short_list_after_simp:
|
|
|
860 |
shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
|
|
|
861 |
by (metis bbbbs good1 k0a rrexp.simps(12))
|
|
|
862 |
|
|
|
863 |
|
|
|
864 |
lemma no_further_dB_after_simp:
|
|
|
865 |
shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
|
|
|
866 |
apply(subgoal_tac "good (RALTS rs)")
|
|
|
867 |
apply(subgoal_tac "distinct rs")
|
|
|
868 |
using rdistinct_on_distinct apply blast
|
|
|
869 |
apply (metis distinct.simps(1) distinct.simps(2) empty_iff good.simps(6) list.exhaust set_empty2)
|
|
|
870 |
using good1 by fastforce
|
|
|
871 |
|
|
|
872 |
|
|
|
873 |
lemma idem_after_simp1:
|
|
|
874 |
shows "rsimp_ALTs (rdistinct (rflts [rsimp aa]) {}) = rsimp aa"
|
|
|
875 |
apply(case_tac "rsimp aa")
|
|
|
876 |
apply simp+
|
|
|
877 |
apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
|
|
|
878 |
apply(simp)
|
|
|
879 |
apply(simp)
|
|
|
880 |
done
|
|
|
881 |
|
|
|
882 |
lemma identity_wwo0:
|
|
|
883 |
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
|
|
|
884 |
apply (metis idem_after_simp1 list.exhaust list.simps(8) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
|
|
|
885 |
done
|
|
|
886 |
|
|
|
887 |
lemma distinct_removes_last:
|
|
|
888 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
|
889 |
\<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
|
|
|
890 |
and "rdistinct (ab # as @ [ab]) rset1 = rdistinct (ab # as) rset1"
|
|
|
891 |
apply(induct as arbitrary: rset ab rset1 a)
|
|
|
892 |
apply simp
|
|
|
893 |
apply simp
|
|
|
894 |
apply(case_tac "aa \<in> rset")
|
|
|
895 |
apply(case_tac "a = aa")
|
|
|
896 |
apply (metis append_Cons)
|
|
|
897 |
apply simp
|
|
|
898 |
apply(case_tac "a \<in> set as")
|
|
|
899 |
apply (metis append_Cons rdistinct.simps(2) set_ConsD)
|
|
|
900 |
apply(case_tac "a = aa")
|
|
|
901 |
prefer 2
|
|
|
902 |
apply simp
|
|
|
903 |
apply (metis append_Cons)
|
|
|
904 |
apply(case_tac "ab \<in> rset1")
|
|
|
905 |
prefer 2
|
|
|
906 |
apply(subgoal_tac "rdistinct (ab # (a # as) @ [ab]) rset1 =
|
|
|
907 |
ab # (rdistinct ((a # as) @ [ab]) (insert ab rset1))")
|
|
|
908 |
prefer 2
|
|
|
909 |
apply force
|
|
|
910 |
apply(simp only:)
|
|
|
911 |
apply(subgoal_tac "rdistinct (ab # a # as) rset1 = ab # (rdistinct (a # as) (insert ab rset1))")
|
|
|
912 |
apply(simp only:)
|
|
|
913 |
apply(subgoal_tac "rdistinct ((a # as) @ [ab]) (insert ab rset1) = rdistinct (a # as) (insert ab rset1)")
|
|
|
914 |
apply blast
|
|
|
915 |
apply(case_tac "a \<in> insert ab rset1")
|
|
|
916 |
apply simp
|
|
|
917 |
apply (metis insertI1)
|
|
|
918 |
apply simp
|
|
|
919 |
apply (meson insertI1)
|
|
|
920 |
apply simp
|
|
|
921 |
apply(subgoal_tac "rdistinct ((a # as) @ [ab]) rset1 = rdistinct (a # as) rset1")
|
|
|
922 |
apply simp
|
|
|
923 |
by (metis append_Cons insert_iff insert_is_Un rdistinct.simps(2))
|
|
|
924 |
|
|
|
925 |
|
|
|
926 |
lemma distinct_removes_middle:
|
|
|
927 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
|
928 |
\<Longrightarrow> rdistinct (as @ as2) rset = rdistinct (as @ [a] @ as2) rset"
|
|
|
929 |
and "rdistinct (ab # as @ [ab] @ as3) rset1 = rdistinct (ab # as @ as3) rset1"
|
|
|
930 |
apply(induct as arbitrary: rset rset1 ab as2 as3 a)
|
|
|
931 |
apply simp
|
|
|
932 |
apply simp
|
|
|
933 |
apply(case_tac "a \<in> rset")
|
|
|
934 |
apply simp
|
|
|
935 |
apply metis
|
|
|
936 |
apply simp
|
|
|
937 |
apply (metis insertI1)
|
|
|
938 |
apply(case_tac "a = ab")
|
|
|
939 |
apply simp
|
|
|
940 |
apply(case_tac "ab \<in> rset")
|
|
|
941 |
apply simp
|
|
|
942 |
apply presburger
|
|
|
943 |
apply (meson insertI1)
|
|
|
944 |
apply(case_tac "a \<in> rset")
|
|
|
945 |
apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
|
|
|
946 |
apply(case_tac "ab \<in> rset")
|
|
|
947 |
apply simp
|
|
|
948 |
apply (meson insert_iff)
|
|
|
949 |
apply simp
|
|
|
950 |
by (metis insertI1)
|
|
|
951 |
|
|
|
952 |
|
|
|
953 |
lemma distinct_removes_middle3:
|
|
|
954 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
|
955 |
\<Longrightarrow> rdistinct (as @ a #as2) rset = rdistinct (as @ as2) rset"
|
|
|
956 |
using distinct_removes_middle(1) by fastforce
|
|
|
957 |
|
|
|
958 |
|
|
|
959 |
lemma distinct_removes_list:
|
|
|
960 |
shows "\<lbrakk> \<forall>r \<in> set rs. r \<in> set as\<rbrakk> \<Longrightarrow> rdistinct (as @ rs) {} = rdistinct as {}"
|
|
|
961 |
apply(induct rs)
|
|
|
962 |
apply simp+
|
|
|
963 |
apply(subgoal_tac "rdistinct (as @ a # rs) {} = rdistinct (as @ rs) {}")
|
|
|
964 |
prefer 2
|
|
|
965 |
apply (metis append_Cons append_Nil distinct_removes_middle(1))
|
|
|
966 |
by presburger
|
|
|
967 |
|
|
|
968 |
|
|
|
969 |
lemma spawn_simp_rsimpalts:
|
|
|
970 |
shows "rsimp (rsimp_ALTs rs) = rsimp (rsimp_ALTs (map rsimp rs))"
|
|
|
971 |
apply(cases rs)
|
|
|
972 |
apply simp
|
|
|
973 |
apply(case_tac list)
|
|
|
974 |
apply simp
|
|
|
975 |
apply(subst rsimp_idem[symmetric])
|
|
|
976 |
apply simp
|
|
|
977 |
apply(subgoal_tac "rsimp_ALTs rs = RALTS rs")
|
|
|
978 |
apply(simp only:)
|
|
|
979 |
apply(subgoal_tac "rsimp_ALTs (map rsimp rs) = RALTS (map rsimp rs)")
|
|
|
980 |
apply(simp only:)
|
|
|
981 |
prefer 2
|
|
|
982 |
apply simp
|
|
|
983 |
prefer 2
|
|
|
984 |
using rsimp_ALTs.simps(3) apply presburger
|
|
|
985 |
apply auto
|
|
|
986 |
apply(subst rsimp_idem)+
|
|
|
987 |
by (metis comp_apply rsimp_idem)
|
|
|
988 |
|
|
|
989 |
|
|
|
990 |
lemma simp_singlealt_flatten:
|
|
|
991 |
shows "rsimp (RALTS [RALTS rsa]) = rsimp (RALTS (rsa @ []))"
|
|
|
992 |
apply(induct rsa)
|
|
|
993 |
apply simp
|
|
|
994 |
apply simp
|
|
|
995 |
by (metis idem_after_simp1 list.simps(9) rsimp.simps(2))
|
|
|
996 |
|
|
|
997 |
|
|
|
998 |
lemma good1_rsimpalts:
|
|
|
999 |
shows "rsimp r = RALTS rs \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
|
|
|
1000 |
by (metis no_alt_short_list_after_simp)
|
|
|
1001 |
|
|
|
1002 |
|
|
|
1003 |
|
|
|
1004 |
|
|
|
1005 |
lemma good1_flatten:
|
|
|
1006 |
shows "\<lbrakk> rsimp r = (RALTS rs1)\<rbrakk>
|
|
|
1007 |
\<Longrightarrow> rflts (rsimp_ALTs rs1 # map rsimp rsb) = rflts (rs1 @ map rsimp rsb)"
|
|
|
1008 |
apply(subst good1_rsimpalts)
|
|
|
1009 |
apply simp+
|
|
|
1010 |
apply(subgoal_tac "rflts (rs1 @ map rsimp rsb) = rs1 @ rflts (map rsimp rsb)")
|
|
|
1011 |
apply simp
|
|
|
1012 |
using flts_append rsimp_inner_idem4 by presburger
|
|
|
1013 |
|
|
|
1014 |
|
|
|
1015 |
lemma flatten_rsimpalts:
|
|
|
1016 |
shows "rflts (rsimp_ALTs (rdistinct (rflts (map rsimp rsa)) {}) # map rsimp rsb) =
|
|
|
1017 |
rflts ( (rdistinct (rflts (map rsimp rsa)) {}) @ map rsimp rsb)"
|
|
|
1018 |
apply(case_tac "map rsimp rsa")
|
|
|
1019 |
apply simp
|
|
|
1020 |
apply(case_tac "list")
|
|
|
1021 |
apply simp
|
|
|
1022 |
apply(case_tac a)
|
|
|
1023 |
apply simp+
|
|
|
1024 |
apply(rename_tac rs1)
|
|
|
1025 |
apply (metis good1_flatten map_eq_Cons_D no_further_dB_after_simp)
|
|
|
1026 |
|
|
|
1027 |
apply simp
|
|
|
1028 |
|
|
|
1029 |
apply(subgoal_tac "\<forall>r \<in> set( rflts (map rsimp rsa)). good r")
|
|
|
1030 |
apply(case_tac "rdistinct (rflts (map rsimp rsa)) {}")
|
|
|
1031 |
apply simp
|
|
|
1032 |
apply auto[1]
|
|
|
1033 |
apply simp
|
|
|
1034 |
apply(simp)
|
|
|
1035 |
apply(case_tac "lista")
|
|
|
1036 |
apply simp_all
|
|
|
1037 |
|
|
|
1038 |
apply (metis append_Cons append_Nil good1_flatten rflts.simps(2) rsimp.simps(2) rsimp_ALTs.elims)
|
|
|
1039 |
by (metis (no_types, opaque_lifting) append_Cons append_Nil good1_flatten rflts.simps(2) rsimp.simps(2) rsimp_ALTs.elims)
|
|
|
1040 |
|
|
|
1041 |
lemma last_elem_out:
|
|
|
1042 |
shows "\<lbrakk>x \<notin> set xs; x \<notin> rset \<rbrakk> \<Longrightarrow> rdistinct (xs @ [x]) rset = rdistinct xs rset @ [x]"
|
|
|
1043 |
apply(induct xs arbitrary: rset)
|
|
|
1044 |
apply simp+
|
|
|
1045 |
done
|
|
|
1046 |
|
|
|
1047 |
|
|
|
1048 |
|
|
|
1049 |
|
|
|
1050 |
lemma rdistinct_concat_general:
|
|
|
1051 |
shows "rdistinct (rs1 @ rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
|
|
|
1052 |
apply(induct rs1 arbitrary: rs2 rule: rev_induct)
|
|
|
1053 |
apply simp
|
|
|
1054 |
apply(drule_tac x = "x # rs2" in meta_spec)
|
|
|
1055 |
apply simp
|
|
|
1056 |
apply(case_tac "x \<in> set xs")
|
|
|
1057 |
apply simp
|
|
|
1058 |
|
|
|
1059 |
apply (simp add: distinct_removes_middle3 insert_absorb)
|
|
|
1060 |
apply simp
|
|
|
1061 |
by (simp add: last_elem_out)
|
|
|
1062 |
|
|
|
1063 |
|
|
|
1064 |
|
|
|
1065 |
|
|
|
1066 |
lemma distinct_once_enough:
|
|
|
1067 |
shows "rdistinct (rs @ rsa) {} = rdistinct (rdistinct rs {} @ rsa) {}"
|
|
|
1068 |
apply(subgoal_tac "distinct (rdistinct rs {})")
|
|
|
1069 |
apply(subgoal_tac
|
|
|
1070 |
" rdistinct (rdistinct rs {} @ rsa) {} = rdistinct rs {} @ (rdistinct rsa (set rs))")
|
|
|
1071 |
apply(simp only:)
|
|
|
1072 |
using rdistinct_concat_general apply blast
|
|
|
1073 |
apply (simp add: distinct_rdistinct_append rdistinct_set_equality1)
|
|
|
1074 |
by (simp add: rdistinct_does_the_job)
|
|
|
1075 |
|
|
|
1076 |
|
|
|
1077 |
lemma simp_flatten:
|
|
|
1078 |
shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
|
|
|
1079 |
apply simp
|
|
|
1080 |
apply(subst flatten_rsimpalts)
|
|
|
1081 |
apply(simp add: flts_append)
|
|
|
1082 |
by (metis Diff_empty distinct_once_enough flts_append nonalt0_fltseq nonalt_flts_rd qqq1 rdistinct_set_equality1)
|
|
|
1083 |
|
|
|
1084 |
lemma basic_rsimp_SEQ_property1:
|
|
|
1085 |
shows "rsimp_SEQ RONE r = r"
|
|
|
1086 |
by (simp add: idiot)
|
|
|
1087 |
|
|
|
1088 |
|
|
|
1089 |
|
|
|
1090 |
lemma basic_rsimp_SEQ_property3:
|
|
|
1091 |
shows "rsimp_SEQ r RZERO = RZERO"
|
|
|
1092 |
using rsimp_SEQ.elims by blast
|
|
|
1093 |
|
|
|
1094 |
|
|
|
1095 |
|
|
|
1096 |
fun vsuf :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list" where
|
|
|
1097 |
"vsuf [] _ = []"
|
|
|
1098 |
|"vsuf (c#cs) r1 = (if (rnullable r1) then (vsuf cs (rder c r1)) @ [c # cs]
|
|
|
1099 |
else (vsuf cs (rder c r1))
|
|
|
1100 |
) "
|
|
|
1101 |
|
|
|
1102 |
|
|
|
1103 |
|
|
|
1104 |
|
|
|
1105 |
|
|
|
1106 |
|
|
|
1107 |
fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list" where
|
|
|
1108 |
"star_update c r [] = []"
|
|
|
1109 |
|"star_update c r (s # Ss) = (if (rnullable (rders r s))
|
|
|
1110 |
then (s@[c]) # [c] # (star_update c r Ss)
|
|
|
1111 |
else (s@[c]) # (star_update c r Ss) )"
|
|
|
1112 |
|
|
|
1113 |
|
|
|
1114 |
fun star_updates :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"
|
|
|
1115 |
where
|
|
|
1116 |
"star_updates [] r Ss = Ss"
|
|
|
1117 |
| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"
|
|
|
1118 |
|
|
|
1119 |
lemma stupdates_append: shows
|
|
|
1120 |
"star_updates (s @ [c]) r Ss = star_update c r (star_updates s r Ss)"
|
|
|
1121 |
apply(induct s arbitrary: Ss)
|
|
|
1122 |
apply simp
|
|
|
1123 |
apply simp
|
|
|
1124 |
done
|
|
|
1125 |
|
|
|
1126 |
lemma flts_removes0:
|
|
|
1127 |
shows " rflts (rs @ [RZERO]) =
|
|
|
1128 |
rflts rs"
|
|
|
1129 |
apply(induct rs)
|
|
|
1130 |
apply simp
|
|
|
1131 |
by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
1132 |
|
|
|
1133 |
|
|
|
1134 |
lemma rflts_spills_last:
|
|
|
1135 |
shows "rflts (rs1 @ [RALTS rs]) = rflts rs1 @ rs"
|
|
|
1136 |
apply (induct rs1 rule: rflts.induct)
|
|
|
1137 |
apply(auto)
|
|
|
1138 |
done
|
|
|
1139 |
|
|
|
1140 |
lemma flts_keeps1:
|
|
|
1141 |
shows "rflts (rs @ [RONE]) = rflts rs @ [RONE]"
|
|
|
1142 |
apply (induct rs rule: rflts.induct)
|
|
|
1143 |
apply(auto)
|
|
|
1144 |
done
|
|
|
1145 |
|
|
|
1146 |
lemma flts_keeps_others:
|
|
|
1147 |
shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow>rflts (rs @ [a]) = rflts rs @ [a]"
|
|
|
1148 |
apply(induct rs rule: rflts.induct)
|
|
|
1149 |
apply(auto)
|
|
|
1150 |
by (meson k0b nonalt.elims(3))
|
|
|
1151 |
|
|
|
1152 |
lemma spilled_alts_contained:
|
|
|
1153 |
shows "\<lbrakk>a = RALTS rs ; a \<in> set rs1\<rbrakk> \<Longrightarrow> \<forall>r \<in> set rs. r \<in> set (rflts rs1)"
|
|
|
1154 |
apply(induct rs1)
|
|
|
1155 |
apply simp
|
|
|
1156 |
apply(case_tac "a = aa")
|
|
|
1157 |
apply simp
|
|
|
1158 |
apply(subgoal_tac " a \<in> set rs1")
|
|
|
1159 |
prefer 2
|
|
|
1160 |
apply (meson set_ConsD)
|
|
|
1161 |
apply(case_tac aa)
|
|
|
1162 |
using rflts.simps(2) apply presburger
|
|
|
1163 |
apply fastforce
|
|
|
1164 |
apply fastforce
|
|
|
1165 |
apply fastforce
|
|
|
1166 |
apply fastforce
|
|
|
1167 |
apply fastforce
|
|
|
1168 |
by simp
|
|
|
1169 |
|
|
|
1170 |
|
|
|
1171 |
lemma distinct_removes_duplicate_flts:
|
|
|
1172 |
shows " a \<in> set rsa
|
|
|
1173 |
\<Longrightarrow> rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
|
|
1174 |
rdistinct (rflts (map rsimp rsa)) {}"
|
|
|
1175 |
apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
|
|
|
1176 |
prefer 2
|
|
|
1177 |
apply simp
|
|
|
1178 |
apply(induct "rsimp a")
|
|
|
1179 |
apply simp
|
|
|
1180 |
using flts_removes0 apply presburger
|
|
|
1181 |
apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
|
|
1182 |
rdistinct (rflts (map rsimp rsa @ [RONE])) {}")
|
|
|
1183 |
apply (simp only:)
|
|
|
1184 |
apply(subst flts_keeps1)
|
|
|
1185 |
apply (metis distinct_removes_last(1) flts_append in_set_conv_decomp rflts.simps(4))
|
|
|
1186 |
apply presburger
|
|
|
1187 |
apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
|
|
1188 |
rdistinct ((rflts (map rsimp rsa)) @ [RCHAR x]) {}")
|
|
|
1189 |
apply (simp only:)
|
|
|
1190 |
prefer 2
|
|
|
1191 |
apply (metis flts_append rflts.simps(1) rflts.simps(5))
|
|
|
1192 |
apply (metis distinct_removes_last(1) rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(3))
|
|
|
1193 |
apply (metis distinct_removes_last(1) flts_append rflts.simps(1) rflts.simps(6) rflts_def_idiot2 rrexp.distinct(31) rrexp.distinct(5))
|
|
|
1194 |
apply (metis distinct_removes_list rflts_spills_last spilled_alts_contained)
|
|
|
1195 |
apply (metis distinct_removes_last(1) flts_append good.simps(1) good.simps(44) rflts.simps(1) rflts.simps(7) rflts_def_idiot2 rrexp.distinct(37))
|
|
|
1196 |
by (metis distinct_removes_last(1) flts_append rflts.simps(1) rflts.simps(8) rflts_def_idiot2 rrexp.distinct(11) rrexp.distinct(39))
|
|
|
1197 |
|
|
|
1198 |
(*some basic facts about rsimp*)
|
|
|
1199 |
|
|
|
1200 |
unused_thms
|
|
|
1201 |
|
|
|
1202 |
|
|
|
1203 |
end |