52 lines
1.3 KiB
Plaintext
52 lines
1.3 KiB
Plaintext
(* Author: Tobias Nipkow *)
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section "Extended Regular Expressions"
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theory Regular_Exp2
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imports Regular_Set
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begin
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datatype (atoms: 'a) rexp =
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is_Zero: Zero |
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is_One: One |
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Atom 'a |
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Plus "('a rexp)" "('a rexp)" |
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Times "('a rexp)" "('a rexp)" |
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Star "('a rexp)" |
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Not "('a rexp)" |
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Inter "('a rexp)" "('a rexp)"
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context
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fixes S :: "'a set"
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begin
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primrec lang :: "'a rexp => 'a lang" where
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"lang Zero = {}" |
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"lang One = {[]}" |
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"lang (Atom a) = {[a]}" |
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"lang (Plus r s) = (lang r) Un (lang s)" |
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"lang (Times r s) = conc (lang r) (lang s)" |
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"lang (Star r) = star(lang r)" |
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"lang (Not r) = lists S - lang r" |
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"lang (Inter r s) = (lang r Int lang s)"
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end
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lemma lang_subset_lists: "atoms r \<subseteq> S \<Longrightarrow> lang S r \<subseteq> lists S"
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by(induction r)(auto simp: conc_subset_lists star_subset_lists)
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primrec nullable :: "'a rexp \<Rightarrow> bool" where
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"nullable Zero = False" |
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"nullable One = True" |
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"nullable (Atom c) = False" |
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"nullable (Plus r1 r2) = (nullable r1 \<or> nullable r2)" |
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"nullable (Times r1 r2) = (nullable r1 \<and> nullable r2)" |
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"nullable (Star r) = True" |
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"nullable (Not r) = (\<not> (nullable r))" |
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"nullable (Inter r s) = (nullable r \<and> nullable s)"
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lemma nullable_iff: "nullable r \<longleftrightarrow> [] \<in> lang S r"
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by (induct r) (auto simp add: conc_def split: if_splits)
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end
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