--- a/AFP-Submission/Regular_Exp.thy Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,160 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-section "Regular expressions"
-
-theory Regular_Exp
-imports Regular_Set
-begin
-
-datatype (atoms: 'a) rexp =
- is_Zero: Zero |
- is_One: One |
- Atom 'a |
- Plus "('a rexp)" "('a rexp)" |
- Times "('a rexp)" "('a rexp)" |
- Star "('a rexp)"
-
-primrec lang :: "'a rexp => 'a lang" where
-"lang Zero = {}" |
-"lang One = {[]}" |
-"lang (Atom a) = {[a]}" |
-"lang (Plus r s) = (lang r) Un (lang s)" |
-"lang (Times r s) = conc (lang r) (lang s)" |
-"lang (Star r) = star(lang r)"
-
-primrec nullable :: "'a rexp \<Rightarrow> bool" where
-"nullable Zero = False" |
-"nullable One = True" |
-"nullable (Atom c) = False" |
-"nullable (Plus r1 r2) = (nullable r1 \<or> nullable r2)" |
-"nullable (Times r1 r2) = (nullable r1 \<and> nullable r2)" |
-"nullable (Star r) = True"
-
-lemma nullable_iff: "nullable r \<longleftrightarrow> [] \<in> lang r"
-by (induct r) (auto simp add: conc_def split: if_splits)
-
-text{* Composition on rhs usually complicates matters: *}
-lemma map_map_rexp:
- "map_rexp f (map_rexp g r) = map_rexp (\<lambda>r. f (g r)) r"
- unfolding rexp.map_comp o_def ..
-
-lemma map_rexp_ident[simp]: "map_rexp (\<lambda>x. x) = (\<lambda>r. r)"
- unfolding id_def[symmetric] fun_eq_iff rexp.map_id id_apply by (intro allI refl)
-
-lemma atoms_lang: "w : lang r \<Longrightarrow> set w \<subseteq> atoms r"
-proof(induction r arbitrary: w)
- case Times thus ?case by fastforce
-next
- case Star thus ?case by (fastforce simp add: star_conv_concat)
-qed auto
-
-lemma lang_eq_ext: "(lang r = lang s) =
- (\<forall>w \<in> lists(atoms r \<union> atoms s). w \<in> lang r \<longleftrightarrow> w \<in> lang s)"
- by (auto simp: atoms_lang[unfolded subset_iff])
-
-lemma lang_eq_ext_Nil_fold_Deriv:
- fixes r s
- defines "\<BB> \<equiv> {(fold Deriv w (lang r), fold Deriv w (lang s))| w. w\<in>lists (atoms r \<union> atoms s)}"
- shows "lang r = lang s \<longleftrightarrow> (\<forall>(K, L) \<in> \<BB>. [] \<in> K \<longleftrightarrow> [] \<in> L)"
- unfolding lang_eq_ext \<BB>_def by (subst (1 2) in_fold_Deriv[of "[]", simplified, symmetric]) auto
-
-
-subsection {* Term ordering *}
-
-instantiation rexp :: (order) "{order}"
-begin
-
-fun le_rexp :: "('a::order) rexp \<Rightarrow> ('a::order) rexp \<Rightarrow> bool"
-where
- "le_rexp Zero _ = True"
-| "le_rexp _ Zero = False"
-| "le_rexp One _ = True"
-| "le_rexp _ One = False"
-| "le_rexp (Atom a) (Atom b) = (a <= b)"
-| "le_rexp (Atom _) _ = True"
-| "le_rexp _ (Atom _) = False"
-| "le_rexp (Star r) (Star s) = le_rexp r s"
-| "le_rexp (Star _) _ = True"
-| "le_rexp _ (Star _) = False"
-| "le_rexp (Plus r r') (Plus s s') =
- (if r = s then le_rexp r' s' else le_rexp r s)"
-| "le_rexp (Plus _ _) _ = True"
-| "le_rexp _ (Plus _ _) = False"
-| "le_rexp (Times r r') (Times s s') =
- (if r = s then le_rexp r' s' else le_rexp r s)"
-
-(* The class instance stuff is by Dmitriy Traytel *)
-
-definition less_eq_rexp where "r \<le> s \<equiv> le_rexp r s"
-definition less_rexp where "r < s \<equiv> le_rexp r s \<and> r \<noteq> s"
-
-lemma le_rexp_Zero: "le_rexp r Zero \<Longrightarrow> r = Zero"
-by (induction r) auto
-
-lemma le_rexp_refl: "le_rexp r r"
-by (induction r) auto
-
-lemma le_rexp_antisym: "\<lbrakk>le_rexp r s; le_rexp s r\<rbrakk> \<Longrightarrow> r = s"
-by (induction r s rule: le_rexp.induct) (auto dest: le_rexp_Zero)
-
-lemma le_rexp_trans: "\<lbrakk>le_rexp r s; le_rexp s t\<rbrakk> \<Longrightarrow> le_rexp r t"
-proof (induction r s arbitrary: t rule: le_rexp.induct)
- fix v t assume "le_rexp (Atom v) t" thus "le_rexp One t" by (cases t) auto
-next
- fix s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp One t" by (cases t) auto
-next
- fix s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp One t" by (cases t) auto
-next
- fix s t assume "le_rexp (Star s) t" thus "le_rexp One t" by (cases t) auto
-next
- fix v u t assume "le_rexp (Atom v) (Atom u)" "le_rexp (Atom u) t"
- thus "le_rexp (Atom v) t" by (cases t) auto
-next
- fix v s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp (Atom v) t" by (cases t) auto
-next
- fix v s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Atom v) t" by (cases t) auto
-next
- fix v s t assume "le_rexp (Star s) t" thus "le_rexp (Atom v) t" by (cases t) auto
-next
- fix r s t
- assume IH: "\<And>t. le_rexp r s \<Longrightarrow> le_rexp s t \<Longrightarrow> le_rexp r t"
- and "le_rexp (Star r) (Star s)" "le_rexp (Star s) t"
- thus "le_rexp (Star r) t" by (cases t) auto
-next
- fix r s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp (Star r) t" by (cases t) auto
-next
- fix r s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Star r) t" by (cases t) auto
-next
- fix r1 r2 s1 s2 t
- assume "\<And>t. r1 = s1 \<Longrightarrow> le_rexp r2 s2 \<Longrightarrow> le_rexp s2 t \<Longrightarrow> le_rexp r2 t"
- "\<And>t. r1 \<noteq> s1 \<Longrightarrow> le_rexp r1 s1 \<Longrightarrow> le_rexp s1 t \<Longrightarrow> le_rexp r1 t"
- "le_rexp (Plus r1 r2) (Plus s1 s2)" "le_rexp (Plus s1 s2) t"
- thus "le_rexp (Plus r1 r2) t" by (cases t) (auto split: split_if_asm intro: le_rexp_antisym)
-next
- fix r1 r2 s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Plus r1 r2) t" by (cases t) auto
-next
- fix r1 r2 s1 s2 t
- assume "\<And>t. r1 = s1 \<Longrightarrow> le_rexp r2 s2 \<Longrightarrow> le_rexp s2 t \<Longrightarrow> le_rexp r2 t"
- "\<And>t. r1 \<noteq> s1 \<Longrightarrow> le_rexp r1 s1 \<Longrightarrow> le_rexp s1 t \<Longrightarrow> le_rexp r1 t"
- "le_rexp (Times r1 r2) (Times s1 s2)" "le_rexp (Times s1 s2) t"
- thus "le_rexp (Times r1 r2) t" by (cases t) (auto split: split_if_asm intro: le_rexp_antisym)
-qed auto
-
-instance proof
-qed (auto simp add: less_eq_rexp_def less_rexp_def
- intro: le_rexp_refl le_rexp_antisym le_rexp_trans)
-
-end
-
-instantiation rexp :: (linorder) "{linorder}"
-begin
-
-lemma le_rexp_total: "le_rexp (r :: 'a :: linorder rexp) s \<or> le_rexp s r"
-by (induction r s rule: le_rexp.induct) auto
-
-instance proof
-qed (unfold less_eq_rexp_def less_rexp_def, rule le_rexp_total)
-
-end
-
-end