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theory BasicIdentities imports
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"Lexer" "PDerivs"
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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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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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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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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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478
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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 arbitrary: s)
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apply simp
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apply simp
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sorry
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475
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lemma rdistinct_concat:
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shows "set rs \<subseteq> rset \<Longrightarrow> rdistinct (rs @ rsa) rset = rdistinct rsa rset"
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apply(induct rs)
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apply simp+
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done
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lemma rdistinct_concat2:
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shows "\<forall>r \<in> set rs. r \<in> rset \<Longrightarrow> rdistinct (rs @ rsa) rset = rdistinct rsa rset"
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by (simp add: rdistinct_concat subsetI)
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444
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467
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lemma distinct_not_exist:
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shows "a \<notin> set rs \<Longrightarrow> rdistinct rs rset = rdistinct rs (insert a rset)"
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apply(induct rs arbitrary: rset)
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apply simp
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apply(case_tac "aa \<in> rset")
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apply simp
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apply(subgoal_tac "a \<noteq> aa")
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prefer 2
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apply simp
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apply simp
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done
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478
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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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apply(subgoal_tac "rdistinct rs {} = rs")
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prefer 2
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apply simp
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using distinct_not_exist by fastforce
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481
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478
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lemma distinct_rdistinct_append:
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481
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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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478
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lemma rdistinct_concat_general:
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shows "rdistinct (rs1 @ rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
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sorry
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lemma rdistinct_set_equality:
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shows "set (rdistinct rs {}) = set rs"
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sorry
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lemma distinct_once_enough:
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shows "rdistinct (rs @ rsa) {} = rdistinct (rdistinct rs {} @ rsa) {}"
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apply(subgoal_tac "distinct (rdistinct rs {})")
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apply(subgoal_tac
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" rdistinct (rdistinct rs {} @ rsa) {} = rdistinct rs {} @ (rdistinct rsa (set rs))")
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apply(simp only:)
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using rdistinct_concat_general apply blast
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apply (simp add: distinct_rdistinct_append rdistinct_set_equality)
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by (simp add: rdistinct_does_the_job)
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444
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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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478
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lemma rflts_def_idiot:
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shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk>
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\<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)
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apply simp
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by (metis append.assoc in_set_conv_decomp insert_iff list.simps(15) rflts.simps(2) rflts.simps(3) rflts_def_idiot)
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lemma 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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465
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lemma rsimpalts_gte2elems:
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shows "size rlist \<ge> 2 \<Longrightarrow> rsimp_ALTs rlist = RALTS rlist"
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apply(induct rlist)
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apply simp
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apply(induct rlist)
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apply simp
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apply (metis Suc_le_length_iff rsimp_ALTs.simps(3))
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by blast
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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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476
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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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465
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444
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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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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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done
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lemma ralts_cap_mono:
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shows "rsize (RALTS rs) \<le> Suc ( sum_list (map rsize rs)) "
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by simp
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478
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444
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lemma rflts_mono:
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shows "sum_list (map rsize (rflts rs))\<le> sum_list (map rsize 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: shows "sum_list (map rsize (rdistinct rs ss)) \<le>
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sum_list (map rsize 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 rdistinct_phi_smaller: "sum_list (map rsize (rdistinct rs {})) \<le> sum_list (map rsize rs)"
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by (simp add: rdistinct_smaller)
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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 (sum_list (map rsize 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( sum_list (map rsize (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 "sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})) \<le>
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sum_list (map rsize ( (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 "sum_list (map rsize (rflts (map rsimp x))) \<le>
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sum_list (map rsize (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 "sum_list (map rsize (map rsimp x)) \<le> sum_list (map rsize 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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330 |
using rsimp_alts_mono by auto
|
|
|
331 |
|
|
|
332 |
lemma idiot:
|
|
|
333 |
shows "rsimp_SEQ RONE r = r"
|
|
|
334 |
apply(case_tac r)
|
|
|
335 |
apply simp_all
|
|
|
336 |
done
|
|
|
337 |
|
|
|
338 |
lemma no_alt_short_list_after_simp:
|
|
|
339 |
shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
|
|
|
340 |
sorry
|
|
|
341 |
|
|
|
342 |
lemma no_further_dB_after_simp:
|
|
|
343 |
shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
|
|
|
344 |
sorry
|
|
|
345 |
|
|
|
346 |
|
|
|
347 |
lemma idiot2:
|
|
|
348 |
shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
|
|
|
349 |
\<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
|
|
|
350 |
apply(case_tac r1)
|
|
|
351 |
apply(case_tac r2)
|
|
|
352 |
apply simp_all
|
|
|
353 |
apply(case_tac r2)
|
|
|
354 |
apply simp_all
|
|
|
355 |
apply(case_tac r2)
|
|
|
356 |
apply simp_all
|
|
|
357 |
apply(case_tac r2)
|
|
|
358 |
apply simp_all
|
|
|
359 |
apply(case_tac r2)
|
|
|
360 |
apply simp_all
|
|
|
361 |
done
|
|
|
362 |
|
|
|
363 |
lemma rders__onechar:
|
|
|
364 |
shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
|
|
|
365 |
by simp
|
|
|
366 |
|
|
|
367 |
lemma rders_append:
|
|
|
368 |
"rders c (s1 @ s2) = rders (rders c s1) s2"
|
|
|
369 |
apply(induct s1 arbitrary: c s2)
|
|
|
370 |
apply(simp_all)
|
|
|
371 |
done
|
|
|
372 |
|
|
|
373 |
lemma rders_simp_append:
|
|
|
374 |
"rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
|
|
|
375 |
apply(induct s1 arbitrary: c s2)
|
|
465
|
376 |
apply(simp_all)
|
|
444
|
377 |
done
|
|
|
378 |
|
|
480
|
379 |
|
|
444
|
380 |
|
|
|
381 |
lemma set_related_list:
|
|
|
382 |
shows "distinct rs \<Longrightarrow> length rs = card (set rs)"
|
|
|
383 |
by (simp add: distinct_card)
|
|
|
384 |
(*this section deals with the property of distinctBy: creates a list without duplicates*)
|
|
|
385 |
lemma rdistinct_never_added_twice:
|
|
|
386 |
shows "rdistinct (a # rs) {a} = rdistinct rs {a}"
|
|
|
387 |
by force
|
|
|
388 |
|
|
|
389 |
|
|
478
|
390 |
|
|
444
|
391 |
|
|
476
|
392 |
|
|
444
|
393 |
|
|
|
394 |
|
|
|
395 |
lemma rders_simp_one_char:
|
|
|
396 |
shows "rders_simp r [c] = rsimp (rder c r)"
|
|
|
397 |
apply auto
|
|
|
398 |
done
|
|
|
399 |
|
|
|
400 |
lemma rsimp_idem:
|
|
|
401 |
shows "rsimp (rsimp r) = rsimp r"
|
|
|
402 |
sorry
|
|
|
403 |
|
|
|
404 |
corollary rsimp_inner_idem1:
|
|
|
405 |
shows "rsimp r = RSEQ r1 r2 \<Longrightarrow> rsimp r1 = r1 \<and> rsimp r2 = r2"
|
|
|
406 |
|
|
|
407 |
sorry
|
|
|
408 |
|
|
|
409 |
corollary rsimp_inner_idem2:
|
|
|
410 |
shows "rsimp r = RALTS rs \<Longrightarrow> \<forall>r' \<in> (set rs). rsimp r' = r'"
|
|
|
411 |
sorry
|
|
|
412 |
|
|
|
413 |
corollary rsimp_inner_idem3:
|
|
|
414 |
shows "rsimp r = RALTS rs \<Longrightarrow> map rsimp rs = rs"
|
|
|
415 |
by (meson map_idI rsimp_inner_idem2)
|
|
|
416 |
|
|
|
417 |
corollary rsimp_inner_idem4:
|
|
465
|
418 |
shows "rsimp r = RALTS rs \<Longrightarrow> rflts rs = rs"
|
|
444
|
419 |
sorry
|
|
|
420 |
|
|
|
421 |
|
|
|
422 |
lemma head_one_more_simp:
|
|
|
423 |
shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
|
|
|
424 |
by (simp add: rsimp_idem)
|
|
|
425 |
|
|
|
426 |
lemma head_one_more_dersimp:
|
|
|
427 |
shows "map rsimp ((rder x (rders_simp r s) # rs)) = map rsimp ((rders_simp r (s@[x]) ) # rs)"
|
|
|
428 |
using head_one_more_simp rders_simp_append rders_simp_one_char by presburger
|
|
|
429 |
|
|
|
430 |
|
|
|
431 |
|
|
|
432 |
|
|
|
433 |
lemma ders_simp_nullability:
|
|
|
434 |
shows "rnullable (rders r s) = rnullable (rders_simp r s)"
|
|
|
435 |
sorry
|
|
|
436 |
|
|
480
|
437 |
lemma der_simp_nullability:
|
|
|
438 |
shows "rnullable r = rnullable (rsimp r)"
|
|
|
439 |
sorry
|
|
|
440 |
|
|
|
441 |
|
|
444
|
442 |
lemma first_elem_seqder:
|
|
|
443 |
shows "\<not>rnullable r1p \<Longrightarrow> map rsimp (rder x (RSEQ r1p r2)
|
|
|
444 |
# rs) = map rsimp ((RSEQ (rder x r1p) r2) # rs) "
|
|
|
445 |
by auto
|
|
|
446 |
|
|
|
447 |
lemma first_elem_seqder1:
|
|
|
448 |
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
|
|
|
449 |
map rsimp ( (RSEQ (rsimp (rder x (rders_simp r xs))) r2) # rs)"
|
|
|
450 |
by (simp add: rsimp_idem)
|
|
|
451 |
|
|
|
452 |
lemma first_elem_seqdersimps:
|
|
|
453 |
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
|
|
|
454 |
map rsimp ( (RSEQ (rders_simp r (xs @ [x])) r2) # rs)"
|
|
|
455 |
using first_elem_seqder1 rders_simp_append by auto
|
|
|
456 |
|
|
|
457 |
|
|
|
458 |
|
|
|
459 |
|
|
|
460 |
|
|
|
461 |
|
|
|
462 |
lemma seq_ders_closed_form1:
|
|
|
463 |
shows "\<exists>Ss. rders_simp (RSEQ r1 r2) [c] = rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) #
|
|
|
464 |
(map ( rders_simp r2 ) Ss)))"
|
|
|
465 |
apply(case_tac "rnullable r1")
|
|
|
466 |
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
|
|
|
467 |
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [[c]])))")
|
|
|
468 |
prefer 2
|
|
|
469 |
apply (simp add: rsimp_idem)
|
|
|
470 |
apply(rule_tac x = "[[c]]" in exI)
|
|
|
471 |
apply simp
|
|
|
472 |
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
|
|
|
473 |
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [])))")
|
|
|
474 |
apply blast
|
|
|
475 |
apply(simp add: rsimp_idem)
|
|
|
476 |
sorry
|
|
|
477 |
|
|
|
478 |
|
|
|
479 |
|
|
478
|
480 |
lemma idem_after_simp1:
|
|
|
481 |
shows "rsimp_ALTs (rdistinct (rflts [rsimp aa]) {}) = rsimp aa"
|
|
|
482 |
apply(case_tac "rsimp aa")
|
|
|
483 |
apply simp+
|
|
|
484 |
apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
|
|
|
485 |
by simp
|
|
|
486 |
|
|
|
487 |
lemma identity_wwo0:
|
|
|
488 |
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
|
|
|
489 |
by (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))
|
|
|
490 |
|
|
|
491 |
|
|
|
492 |
lemma distinct_removes_last:
|
|
|
493 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
|
494 |
\<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
|
|
|
495 |
and "rdistinct (ab # as @ [ab]) rset1 = rdistinct (ab # as) rset1"
|
|
|
496 |
apply(induct as arbitrary: rset ab rset1 a)
|
|
|
497 |
apply simp
|
|
|
498 |
apply simp
|
|
|
499 |
apply(case_tac "aa \<in> rset")
|
|
|
500 |
apply(case_tac "a = aa")
|
|
|
501 |
apply (metis append_Cons)
|
|
|
502 |
apply simp
|
|
|
503 |
apply(case_tac "a \<in> set as")
|
|
|
504 |
apply (metis append_Cons rdistinct.simps(2) set_ConsD)
|
|
|
505 |
apply(case_tac "a = aa")
|
|
|
506 |
prefer 2
|
|
|
507 |
apply simp
|
|
|
508 |
apply (metis append_Cons)
|
|
|
509 |
apply(case_tac "ab \<in> rset1")
|
|
|
510 |
prefer 2
|
|
|
511 |
apply(subgoal_tac "rdistinct (ab # (a # as) @ [ab]) rset1 =
|
|
|
512 |
ab # (rdistinct ((a # as) @ [ab]) (insert ab rset1))")
|
|
|
513 |
prefer 2
|
|
|
514 |
apply force
|
|
|
515 |
apply(simp only:)
|
|
|
516 |
apply(subgoal_tac "rdistinct (ab # a # as) rset1 = ab # (rdistinct (a # as) (insert ab rset1))")
|
|
|
517 |
apply(simp only:)
|
|
|
518 |
apply(subgoal_tac "rdistinct ((a # as) @ [ab]) (insert ab rset1) = rdistinct (a # as) (insert ab rset1)")
|
|
|
519 |
apply blast
|
|
|
520 |
apply(case_tac "a \<in> insert ab rset1")
|
|
|
521 |
apply simp
|
|
|
522 |
apply (metis insertI1)
|
|
|
523 |
apply simp
|
|
|
524 |
apply (meson insertI1)
|
|
|
525 |
apply simp
|
|
|
526 |
apply(subgoal_tac "rdistinct ((a # as) @ [ab]) rset1 = rdistinct (a # as) rset1")
|
|
|
527 |
apply simp
|
|
|
528 |
by (metis append_Cons insert_iff insert_is_Un rdistinct.simps(2))
|
|
|
529 |
|
|
|
530 |
|
|
|
531 |
lemma distinct_removes_middle:
|
|
|
532 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
|
533 |
\<Longrightarrow> rdistinct (as @ as2) rset = rdistinct (as @ [a] @ as2) rset"
|
|
|
534 |
and "rdistinct (ab # as @ [ab] @ as3) rset1 = rdistinct (ab # as @ as3) rset1"
|
|
|
535 |
apply(induct as arbitrary: rset rset1 ab as2 as3 a)
|
|
|
536 |
apply simp
|
|
|
537 |
apply simp
|
|
|
538 |
apply(case_tac "a \<in> rset")
|
|
|
539 |
apply simp
|
|
|
540 |
apply metis
|
|
|
541 |
apply simp
|
|
|
542 |
apply (metis insertI1)
|
|
|
543 |
apply(case_tac "a = ab")
|
|
|
544 |
apply simp
|
|
|
545 |
apply(case_tac "ab \<in> rset")
|
|
|
546 |
apply simp
|
|
|
547 |
apply presburger
|
|
|
548 |
apply (meson insertI1)
|
|
|
549 |
apply(case_tac "a \<in> rset")
|
|
|
550 |
apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
|
|
|
551 |
apply(case_tac "ab \<in> rset")
|
|
|
552 |
apply simp
|
|
|
553 |
apply (meson insert_iff)
|
|
|
554 |
apply simp
|
|
|
555 |
by (metis insertI1)
|
|
|
556 |
|
|
|
557 |
|
|
|
558 |
lemma distinct_removes_middle3:
|
|
|
559 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
|
560 |
\<Longrightarrow> rdistinct (as @ a #as2) rset = rdistinct (as @ as2) rset"
|
|
|
561 |
using distinct_removes_middle(1) by fastforce
|
|
|
562 |
|
|
|
563 |
lemma distinct_removes_last2:
|
|
|
564 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
|
565 |
\<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
|
|
|
566 |
by (simp add: distinct_removes_last(1))
|
|
|
567 |
|
|
|
568 |
lemma distinct_removes_middle2:
|
|
|
569 |
shows "a \<in> set as \<Longrightarrow> rdistinct (as @ [a] @ rs) {} = rdistinct (as @ rs) {}"
|
|
|
570 |
by (metis distinct_removes_middle(1))
|
|
|
571 |
|
|
|
572 |
lemma distinct_removes_list:
|
|
|
573 |
shows "\<lbrakk> \<forall>r \<in> set rs. r \<in> set as\<rbrakk> \<Longrightarrow> rdistinct (as @ rs) {} = rdistinct as {}"
|
|
|
574 |
apply(induct rs)
|
|
|
575 |
apply simp+
|
|
|
576 |
apply(subgoal_tac "rdistinct (as @ a # rs) {} = rdistinct (as @ rs) {}")
|
|
|
577 |
prefer 2
|
|
|
578 |
apply (metis append_Cons append_Nil distinct_removes_middle(1))
|
|
|
579 |
by presburger
|
|
|
580 |
|
|
|
581 |
|
|
|
582 |
lemma spawn_simp_rsimpalts:
|
|
|
583 |
shows "rsimp (rsimp_ALTs rs) = rsimp (rsimp_ALTs (map rsimp rs))"
|
|
|
584 |
apply(cases rs)
|
|
|
585 |
apply simp
|
|
|
586 |
apply(case_tac list)
|
|
|
587 |
apply simp
|
|
|
588 |
apply(subst rsimp_idem[symmetric])
|
|
|
589 |
apply simp
|
|
|
590 |
apply(subgoal_tac "rsimp_ALTs rs = RALTS rs")
|
|
|
591 |
apply(simp only:)
|
|
|
592 |
apply(subgoal_tac "rsimp_ALTs (map rsimp rs) = RALTS (map rsimp rs)")
|
|
|
593 |
apply(simp only:)
|
|
|
594 |
prefer 2
|
|
|
595 |
apply simp
|
|
|
596 |
prefer 2
|
|
|
597 |
using rsimp_ALTs.simps(3) apply presburger
|
|
|
598 |
apply auto
|
|
|
599 |
apply(subst rsimp_idem)+
|
|
|
600 |
by (metis comp_apply rsimp_idem)
|
|
444
|
601 |
|
|
|
602 |
|
|
|
603 |
|
|
|
604 |
|
|
478
|
605 |
inductive good1 :: "rrexp \<Rightarrow> bool"
|
|
|
606 |
where
|
|
|
607 |
"\<lbrakk>RZERO \<notin> set rs; \<nexists>rs1. RALTS rs1 \<in> set rs\<rbrakk> \<Longrightarrow> good1 (RALTS rs)"
|
|
|
608 |
|"good1 RZERO"
|
|
|
609 |
|"good1 RONE"
|
|
|
610 |
|"good1 (RCHAR c)"
|
|
|
611 |
|"good1 (RSEQ r1 r2)"
|
|
|
612 |
|"good1 (RSTAR r0)"
|
|
444
|
613 |
|
|
478
|
614 |
inductive goods :: "rrexp list \<Rightarrow> bool"
|
|
|
615 |
where
|
|
|
616 |
"goods []"
|
|
|
617 |
|"\<lbrakk>goods rs; r \<noteq> RZERO; \<nexists>rs1. RALTS rs1 = r\<rbrakk> \<Longrightarrow> goods (r # rs)"
|
|
|
618 |
|
|
|
619 |
lemma goods_good1:
|
|
|
620 |
shows "goods rs = good1 (RALTS rs)"
|
|
|
621 |
apply(induct rs)
|
|
|
622 |
apply (simp add: good1.intros(1) goods.intros(1))
|
|
|
623 |
apply(case_tac "goods rs")
|
|
|
624 |
apply(case_tac a)
|
|
|
625 |
apply simp
|
|
|
626 |
using good1.simps goods.cases apply auto[1]
|
|
|
627 |
apply (simp add: good1.simps goods.intros(2))
|
|
|
628 |
apply (simp add: good1.simps goods.intros(2))
|
|
|
629 |
apply (simp add: good1.simps goods.intros(2))
|
|
|
630 |
using good1.simps goods.cases apply auto[1]
|
|
|
631 |
apply (metis good1.cases good1.intros(1) goods.intros(2) rrexp.distinct(15) rrexp.distinct(21) rrexp.distinct(25) rrexp.distinct(29) rrexp.distinct(7) rrexp.distinct(9) rrexp.inject(3) set_ConsD)
|
|
|
632 |
apply simp
|
|
|
633 |
by (metis good1.cases good1.intros(1) goods.cases list.distinct(1) list.inject list.set_intros(2) rrexp.distinct(15) rrexp.distinct(29) rrexp.distinct(7) rrexp.inject(3) rrexp.simps(26) rrexp.simps(30))
|
|
|
634 |
|
|
|
635 |
lemma rsimp_good1:
|
|
|
636 |
shows "rsimp r = r1 \<Longrightarrow> good1 r1"
|
|
|
637 |
|
|
|
638 |
sorry
|
|
|
639 |
|
|
|
640 |
lemma rsimp_no_dup:
|
|
|
641 |
shows "rsimp r = RALTS rs \<Longrightarrow> distinct rs"
|
|
|
642 |
sorry
|
|
|
643 |
|
|
|
644 |
|
|
|
645 |
lemma rsimp_good1_2:
|
|
|
646 |
shows "map rsimp rsa = [RALTS rs] \<Longrightarrow> good1 (RALTS rs)"
|
|
|
647 |
by (metis Cons_eq_map_D rsimp_good1)
|
|
|
648 |
|
|
|
649 |
|
|
|
650 |
|
|
|
651 |
lemma simp_singlealt_flatten:
|
|
|
652 |
shows "rsimp (RALTS [RALTS rsa]) = rsimp (RALTS (rsa @ []))"
|
|
|
653 |
apply(induct rsa)
|
|
|
654 |
apply simp
|
|
|
655 |
apply simp
|
|
|
656 |
by (metis idem_after_simp1 list.simps(9) rsimp.simps(2))
|
|
|
657 |
|
|
|
658 |
|
|
|
659 |
lemma good1_rsimpalts:
|
|
|
660 |
shows "rsimp r = RALTS rs \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
|
|
|
661 |
by (metis no_alt_short_list_after_simp)
|
|
|
662 |
|
|
|
663 |
|
|
|
664 |
lemma good1_flts:
|
|
|
665 |
shows "good1 (RALTS rs1) \<Longrightarrow> rflts rs1 = rs1"
|
|
|
666 |
apply(induct rs1)
|
|
|
667 |
apply simp
|
|
|
668 |
by (metis good1.cases good1.intros(1) list.set_intros(1) rflts_def_idiot rrexp.distinct(21) rrexp.distinct(25) rrexp.distinct(29) rrexp.inject(3) rsimp.simps(3) rsimp.simps(4) rsimp_inner_idem4 set_subset_Cons subsetD)
|
|
|
669 |
|
|
|
670 |
|
|
|
671 |
|
|
|
672 |
lemma good1_flatten:
|
|
|
673 |
shows "\<lbrakk> rsimp r = (RALTS rs1)\<rbrakk>
|
|
|
674 |
\<Longrightarrow> rflts (rsimp_ALTs rs1 # map rsimp rsb) = rflts (rs1 @ map rsimp rsb)"
|
|
|
675 |
apply(subst good1_rsimpalts)
|
|
|
676 |
apply simp+
|
|
|
677 |
apply(subgoal_tac "rflts (rs1 @ map rsimp rsb) = rs1 @ rflts (map rsimp rsb)")
|
|
|
678 |
apply simp
|
|
|
679 |
apply(subgoal_tac "good1 (RALTS rs1)")
|
|
|
680 |
prefer 2
|
|
|
681 |
using rsimp_good1 apply blast
|
|
|
682 |
using flts_append good1_flts by presburger
|
|
|
683 |
|
|
|
684 |
lemma flatten_rsimpalts:
|
|
|
685 |
shows "rflts (rsimp_ALTs (rdistinct (rflts (map rsimp rsa)) {}) # map rsimp rsb) =
|
|
|
686 |
rflts ( (rdistinct (rflts (map rsimp rsa)) {}) @ map rsimp rsb)"
|
|
|
687 |
apply(case_tac "map rsimp rsa")
|
|
|
688 |
apply simp
|
|
|
689 |
apply(case_tac "list")
|
|
|
690 |
apply simp
|
|
|
691 |
apply(case_tac a)
|
|
|
692 |
apply simp+
|
|
|
693 |
apply(rename_tac rs1)
|
|
|
694 |
apply(subgoal_tac "good1 (RALTS rs1)")
|
|
|
695 |
apply(subgoal_tac "distinct rs1")
|
|
|
696 |
apply(subst rdistinct_on_distinct)
|
|
|
697 |
apply blast
|
|
|
698 |
apply(subst rdistinct_on_distinct)
|
|
|
699 |
apply blast
|
|
|
700 |
using good1_flatten apply blast
|
|
|
701 |
|
|
|
702 |
using rsimp_no_dup apply force
|
|
|
703 |
|
|
|
704 |
using rsimp_good1_2 apply presburger
|
|
|
705 |
|
|
|
706 |
apply simp+
|
|
|
707 |
apply(case_tac "rflts (a # aa # lista)")
|
|
|
708 |
apply simp
|
|
|
709 |
by (smt (verit) append_Cons append_Nil empty_iff good1_flatten list.distinct(1) rdistinct.simps(2) rsimp.simps(2) rsimp_ALTs.elims rsimp_good1)
|
|
|
710 |
|
|
|
711 |
|
|
|
712 |
lemma flts_good_good:
|
|
|
713 |
shows "\<forall>r \<in> set rs. good1 r \<Longrightarrow> good1 (RALTS (rflts rs ))"
|
|
|
714 |
apply(induct rs)
|
|
|
715 |
apply simp
|
|
|
716 |
using goods.intros(1) goods_good1 apply auto[1]
|
|
|
717 |
apply(case_tac "a")
|
|
|
718 |
apply simp
|
|
|
719 |
apply (metis goods.intros(2) goods_good1 list.set_intros(2) rflts.simps(4) rrexp.distinct(1) rrexp.distinct(15))
|
|
|
720 |
apply simp
|
|
|
721 |
using goods.intros(2) goods_good1 apply blast
|
|
|
722 |
using goods.intros(2) goods_good1 apply auto[1]
|
|
|
723 |
apply(subgoal_tac "good1 a")
|
|
|
724 |
apply (metis Un_iff good1.cases good1.intros(1) list.set_intros(2) rflts.simps(3) rrexp.distinct(15) rrexp.distinct(21) rrexp.distinct(25) rrexp.distinct(29) rrexp.distinct(7) rrexp.inject(3) set_append)
|
|
|
725 |
apply simp
|
|
|
726 |
by (metis goods.intros(2) goods_good1 list.set_intros(2) rflts.simps(7) rrexp.distinct(29) rrexp.distinct(9))
|
|
|
727 |
|
|
|
728 |
|
|
|
729 |
lemma simp_flatten_aux1:
|
|
|
730 |
shows "\<forall>r \<in> set (map rsimp rsa). good1 r"
|
|
|
731 |
apply(induct rsa)
|
|
|
732 |
apply(simp add: goods.intros)
|
|
|
733 |
using rsimp_good1 by auto
|
|
|
734 |
|
|
|
735 |
|
|
|
736 |
|
|
|
737 |
lemma simp_flatten_aux:
|
|
|
738 |
shows "\<forall>r \<in> set rs. good1 r \<Longrightarrow> rflts (rdistinct (rflts rs) {}) = (rdistinct (rflts rs) {})"
|
|
|
739 |
sorry
|
|
476
|
740 |
|
|
444
|
741 |
|
|
|
742 |
|
|
|
743 |
lemma simp_flatten:
|
|
|
744 |
shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
|
|
478
|
745 |
apply simp
|
|
|
746 |
apply(subst flatten_rsimpalts)
|
|
|
747 |
apply(simp add: flts_append)
|
|
|
748 |
apply(subgoal_tac "\<forall>r \<in> set (map rsimp rsa). good1 r")
|
|
|
749 |
apply (metis distinct_once_enough simp_flatten_aux)
|
|
|
750 |
using simp_flatten_aux1 by blast
|
|
444
|
751 |
|
|
478
|
752 |
lemma simp_flatten3:
|
|
|
753 |
shows "rsimp (RALTS (rsa @ [RALTS rs] @ rsb)) = rsimp (RALTS (rsa @ rs @ rsb))"
|
|
444
|
754 |
sorry
|
|
|
755 |
|
|
|
756 |
|
|
|
757 |
|
|
|
758 |
fun vsuf :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list" where
|
|
|
759 |
"vsuf [] _ = []"
|
|
|
760 |
|"vsuf (c#cs) r1 = (if (rnullable r1) then (vsuf cs (rder c r1)) @ [c # cs]
|
|
|
761 |
else (vsuf cs (rder c r1))
|
|
|
762 |
) "
|
|
|
763 |
|
|
|
764 |
|
|
|
765 |
|
|
|
766 |
|
|
|
767 |
|
|
|
768 |
|
|
|
769 |
fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list" where
|
|
|
770 |
"star_update c r [] = []"
|
|
486
|
771 |
|"star_update c r (s # Ss) = (if (rnullable (rders r s))
|
|
444
|
772 |
then (s@[c]) # [c] # (star_update c r Ss)
|
|
|
773 |
else (s@[c]) # (star_update c r Ss) )"
|
|
|
774 |
|
|
486
|
775 |
|
|
444
|
776 |
fun star_updates :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"
|
|
|
777 |
where
|
|
|
778 |
"star_updates [] r Ss = Ss"
|
|
|
779 |
| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"
|
|
|
780 |
|
|
486
|
781 |
lemma stupdates_append: shows
|
|
|
782 |
"star_updates (s @ [c]) r Ss = star_update c r (star_updates s r Ss)"
|
|
|
783 |
apply(induct s arbitrary: Ss)
|
|
|
784 |
apply simp
|
|
|
785 |
apply simp
|
|
|
786 |
done
|
|
|
787 |
|
|
444
|
788 |
|
|
478
|
789 |
lemma distinct_flts_no0:
|
|
|
790 |
shows " rflts (map rsimp (rdistinct rs (insert RZERO rset))) =
|
|
|
791 |
rflts (map rsimp (rdistinct rs rset)) "
|
|
|
792 |
|
|
|
793 |
apply(induct rs arbitrary: rset)
|
|
|
794 |
apply simp
|
|
|
795 |
apply(case_tac a)
|
|
|
796 |
apply simp+
|
|
|
797 |
apply (smt (verit, ccfv_SIG) rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
798 |
prefer 2
|
|
|
799 |
apply simp
|
|
|
800 |
by (smt (verit, ccfv_threshold) Un_insert_right insert_iff list.simps(9) rdistinct.simps(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot rrexp.distinct(7))
|
|
444
|
801 |
|
|
478
|
802 |
lemma flts_removes0:
|
|
|
803 |
shows " rflts (rs @ [RZERO]) =
|
|
|
804 |
rflts rs"
|
|
|
805 |
apply(induct rs)
|
|
|
806 |
apply simp
|
|
|
807 |
by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
808 |
|
|
|
809 |
|
|
|
810 |
lemma rflts_spills_last:
|
|
|
811 |
shows "a = RALTS rs \<Longrightarrow> rflts (rs1 @ [a]) = rflts rs1 @ rs"
|
|
|
812 |
apply (induct rs1)
|
|
|
813 |
apply simp
|
|
|
814 |
by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
815 |
|
|
|
816 |
lemma flts_keeps1:
|
|
|
817 |
shows " rflts (rs @ [RONE]) =
|
|
|
818 |
rflts rs @ [RONE] "
|
|
|
819 |
apply (induct rs)
|
|
|
820 |
apply simp
|
|
|
821 |
by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
822 |
|
|
|
823 |
lemma flts_keeps_others:
|
|
|
824 |
shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow>rflts (rs @ [a]) = rflts rs @ [a]"
|
|
|
825 |
apply(induct rs)
|
|
|
826 |
apply simp
|
|
|
827 |
apply (simp add: rflts_def_idiot)
|
|
|
828 |
apply(case_tac a)
|
|
|
829 |
apply simp
|
|
|
830 |
using flts_keeps1 apply blast
|
|
|
831 |
apply (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
832 |
apply (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
833 |
apply blast
|
|
|
834 |
by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
835 |
|
|
|
836 |
lemma spilled_alts_contained:
|
|
|
837 |
shows "\<lbrakk>a = RALTS rs ; a \<in> set rs1\<rbrakk> \<Longrightarrow> \<forall>r \<in> set rs. r \<in> set (rflts rs1)"
|
|
|
838 |
apply(induct rs1)
|
|
|
839 |
apply simp
|
|
|
840 |
apply(case_tac "a = aa")
|
|
|
841 |
apply simp
|
|
|
842 |
apply(subgoal_tac " a \<in> set rs1")
|
|
|
843 |
prefer 2
|
|
|
844 |
apply (meson set_ConsD)
|
|
|
845 |
apply(case_tac aa)
|
|
|
846 |
using rflts.simps(2) apply presburger
|
|
|
847 |
apply fastforce
|
|
|
848 |
apply fastforce
|
|
|
849 |
apply fastforce
|
|
|
850 |
apply fastforce
|
|
|
851 |
by fastforce
|
|
|
852 |
|
|
|
853 |
|
|
|
854 |
lemma distinct_removes_duplicate_flts:
|
|
|
855 |
shows " a \<in> set rsa
|
|
|
856 |
\<Longrightarrow> rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
|
|
857 |
rdistinct (rflts (map rsimp rsa)) {}"
|
|
|
858 |
apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
|
|
|
859 |
prefer 2
|
|
|
860 |
apply simp
|
|
|
861 |
apply(induct "rsimp a")
|
|
|
862 |
apply simp
|
|
|
863 |
using flts_removes0 apply presburger
|
|
|
864 |
apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
|
|
865 |
rdistinct (rflts (map rsimp rsa @ [RONE])) {}")
|
|
|
866 |
apply (simp only:)
|
|
|
867 |
apply(subst flts_keeps1)
|
|
|
868 |
apply (metis distinct_removes_last2 rflts_def_idiot2 rrexp.simps(20) rrexp.simps(6))
|
|
|
869 |
apply presburger
|
|
|
870 |
apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
|
|
|
871 |
rdistinct ((rflts (map rsimp rsa)) @ [RCHAR x]) {}")
|
|
|
872 |
apply (simp only:)
|
|
|
873 |
prefer 2
|
|
|
874 |
apply (metis flts_keeps_others rrexp.distinct(21) rrexp.distinct(3))
|
|
|
875 |
apply (metis distinct_removes_last2 rflts_def_idiot2 rrexp.distinct(21) rrexp.distinct(3))
|
|
|
876 |
|
|
|
877 |
apply (metis distinct_removes_last2 flts_keeps_others rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
|
|
|
878 |
prefer 2
|
|
|
879 |
apply (metis distinct_removes_last2 flts_keeps_others flts_removes0 rflts_def_idiot2 rrexp.distinct(29))
|
|
|
880 |
apply(subgoal_tac "rflts (map rsimp rsa @ [rsimp a]) = rflts (map rsimp rsa) @ x")
|
|
|
881 |
prefer 2
|
|
|
882 |
apply (simp add: rflts_spills_last)
|
|
|
883 |
apply(simp only:)
|
|
|
884 |
apply(subgoal_tac "\<forall> r \<in> set x. r \<in> set (rflts (map rsimp rsa))")
|
|
|
885 |
prefer 2
|
|
|
886 |
using spilled_alts_contained apply presburger
|
|
|
887 |
using distinct_removes_list by blast
|
|
444
|
888 |
|
|
|
889 |
|
|
|
890 |
|
|
|
891 |
(*some basic facts about rsimp*)
|
|
|
892 |
|
|
|
893 |
|
|
|
894 |
|
|
|
895 |
|
|
|
896 |
end |