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theory Re1
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imports "Main"
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begin
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section {* Sequential Composition of Sets *}
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definition
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Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
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where
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"A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"
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text {* Two Simple Properties about Sequential Composition *}
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lemma seq_empty [simp]:
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shows "A ;; {[]} = A"
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and "{[]} ;; A = A"
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by (simp_all add: Sequ_def)
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lemma seq_null [simp]:
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shows "A ;; {} = {}"
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and "{} ;; A = {}"
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by (simp_all add: Sequ_def)
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section {* Regular Expressions *}
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datatype rexp =
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NULL
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| EMPTY
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| CHAR char
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| SEQ rexp rexp
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| ALT rexp rexp
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section {* Semantics of Regular Expressions *}
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fun
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L :: "rexp \<Rightarrow> string set"
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where
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"L (NULL) = {}"
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| "L (EMPTY) = {[]}"
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| "L (CHAR c) = {[c]}"
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| "L (SEQ r1 r2) = (L r1) ;; (L r2)"
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| "L (ALT r1 r2) = (L r1) \<union> (L r2)"
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value "L(CHAR c)"
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value "L(SEQ(CHAR c)(CHAR b))"
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section {* Values *}
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datatype val =
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Void
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| Char char
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| Seq val val
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| Right val
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| Left val
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section {* Relation between values and regular expressions *}
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inductive Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100)
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where
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"\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : SEQ r1 r2"
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| "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : ALT r1 r2"
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| "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : ALT r1 r2"
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| "\<turnstile> Void : EMPTY"
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| "\<turnstile> Char c : CHAR c"
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section {* The string behind a value *}
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fun flat :: "val \<Rightarrow> string"
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where
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"flat(Void) = []"
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| "flat(Char c) = [c]"
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| "flat(Left v) = flat(v)"
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| "flat(Right v) = flat(v)"
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| "flat(Seq v1 v2) = flat(v1) @ flat(v2)"
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value "flat(Seq(Char c)(Char b))"
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value "flat(Right(Void))"
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fun flats :: "val \<Rightarrow> string list"
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where
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"flats(Void) = [[]]"
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| "flats(Char c) = [[c]]"
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| "flats(Left v) = flats(v)"
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| "flats(Right v) = flats(v)"
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| "flats(Seq v1 v2) = (flats v1) @ (flats v2)"
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value "flats(Seq(Char c)(Char b))"
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lemma Prf_flat_L:
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assumes "\<turnstile> v : r" shows "flat v \<in> L r"
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using assms
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apply(induct)
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apply(auto simp add: Sequ_def)
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done
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lemma L_flat_Prf:
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"L(r) = {flat v | v. \<turnstile> v : r}"
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apply(induct r)
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apply(auto dest: Prf_flat_L simp add: Sequ_def)
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apply (metis Prf.intros(4) flat.simps(1))
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apply (metis Prf.intros(5) flat.simps(2))
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apply (metis Prf.intros(1) flat.simps(5))
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apply (metis Prf.intros(2) flat.simps(3))
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apply (metis Prf.intros(3) flat.simps(4))
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apply(erule Prf.cases)
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apply(auto)
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done
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definition definition prefix :: :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubset> _" [100, 100] 100)
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where
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"s1 \<sqsubset> s2 \<equiv> \<exists>s3. s1 @ s3 = s2"
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section {* Ordering of values *}
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inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<succ>_ _" [100, 100, 100] 100)
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where
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"\<lbrakk>v1 = v1'; v2 \<succ>r2 v2'\<rbrakk> \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')"
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| "v1 \<succ>r1 v1' \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')"
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| "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Right v2)"
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| "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Left v1)"
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| "v2 \<succ>r2 v2' \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Right v2')"
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| "v1 \<succ>r1 v1' \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Left v1')"
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| "Void \<succ>EMPTY Void"
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| "(Char c) \<succ>(CHAR c) (Char c)"
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section {* The ordering is reflexive *}
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lemma ValOrd_refl:
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assumes "\<turnstile> v : r"
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shows "v \<succ>r v"
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using assms
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apply(induct)
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apply(auto intro: ValOrd.intros)
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done
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lemma ValOrd_flats:
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assumes "v1 \<succ>r v2"
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shows "hd (flats v2) = hd (flats v1)"
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using assms
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apply(induct)
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apply(auto)
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oops
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section {* Posix definition *}
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definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool"
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where
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"POSIX v r \<equiv> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v \<succ>r v')"
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(*
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an alternative definition: might cause problems
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with theorem mkeps_POSIX
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*)
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definition POSIX2 :: "val \<Rightarrow> rexp \<Rightarrow> bool"
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where
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"POSIX2 v r \<equiv> \<turnstile> v : r \<and> (\<forall>v'. \<turnstile> v' : r \<longrightarrow> v \<succ>r v')"
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definition POSIX3 :: "val \<Rightarrow> rexp \<Rightarrow> bool"
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where
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"POSIX3 v r \<equiv> \<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> length (flat v') \<le> length(flat v)) \<longrightarrow> v \<succ>r v')"
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lemma POSIX_SEQ:
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assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2"
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shows "POSIX v1 r1 \<and> POSIX v2 r2"
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using assms
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unfolding POSIX_def
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apply(auto)
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apply(drule_tac x="Seq v' v2" in spec)
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apply(simp)
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apply(erule impE)
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apply(rule Prf.intros)
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apply(simp)
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apply(simp)
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apply(erule ValOrd.cases)
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apply(simp_all)
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apply(clarify)
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defer
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apply(drule_tac x="Seq v1 v'" in spec)
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apply(simp)
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apply(erule impE)
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apply(rule Prf.intros)
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apply(simp)
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apply(simp)
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apply(erule ValOrd.cases)
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apply(simp_all)
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apply(clarify)
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oops (*not true*)
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lemma POSIX_SEQ_I:
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assumes "POSIX v1 r1" "POSIX v2 r2"
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shows "POSIX (Seq v1 v2) (SEQ r1 r2)"
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using assms
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unfolding POSIX_def
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apply(auto)
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apply(rotate_tac 2)
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apply(erule Prf.cases)
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apply(simp_all)[5]
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apply(auto)[1]
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apply(rule ValOrd.intros)
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oops (* maybe also not true *)
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lemma POSIX3_SEQ_I:
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assumes "POSIX3 v1 r1" "POSIX3 v2 r2"
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shows "POSIX3 (Seq v1 v2) (SEQ r1 r2)"
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using assms
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unfolding POSIX3_def
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apply(auto)
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apply (metis Prf.intros(1))
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apply(rotate_tac 4)
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apply(erule Prf.cases)
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apply(simp_all)[5]
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apply(auto)[1]
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apply(case_tac "v1 = v1a")
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apply(auto)
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apply (metis ValOrd.intros(1))
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apply (rule ValOrd.intros(2))
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oops
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lemma POSIX_ALT2:
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assumes "POSIX (Left v1) (ALT r1 r2)"
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shows "POSIX v1 r1"
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using assms
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unfolding POSIX_def
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apply(auto)
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apply(drule_tac x="Left v'" in spec)
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apply(simp)
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apply(drule mp)
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apply(rule Prf.intros)
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apply(auto)
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apply(erule ValOrd.cases)
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apply(simp_all)
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done
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lemma POSIX2_ALT:
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assumes "POSIX2 (Left v1) (ALT r1 r2)"
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shows "POSIX2 v1 r1"
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using assms
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unfolding POSIX2_def
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apply(auto)
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oops
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lemma POSIX_ALT:
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assumes "POSIX (Left v1) (ALT r1 r2)"
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shows "POSIX v1 r1"
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using assms
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unfolding POSIX_def
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apply(auto)
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apply(drule_tac x="Left v'" in spec)
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apply(simp)
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apply(drule mp)
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apply(rule Prf.intros)
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apply(auto)
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apply(erule ValOrd.cases)
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apply(simp_all)
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done
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lemma POSIX2_ALT:
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assumes "POSIX2 (Left v1) (ALT r1 r2)"
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shows "POSIX2 v1 r1"
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using assms
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apply(simp add: POSIX2_def)
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apply(auto)
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apply(erule Prf.cases)
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apply(simp_all)[5]
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apply(drule_tac x="Left v'" in spec)
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apply(drule mp)
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apply(rule Prf.intros)
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apply(auto)
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apply(erule ValOrd.cases)
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apply(simp_all)
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done
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lemma POSIX_ALT1a:
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assumes "POSIX (Right v2) (ALT r1 r2)"
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shows "POSIX v2 r2"
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using assms
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unfolding POSIX_def
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apply(auto)
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apply(drule_tac x="Right v'" in spec)
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apply(simp)
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apply(drule mp)
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apply(rule Prf.intros)
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apply(auto)
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apply(erule ValOrd.cases)
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apply(simp_all)
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done
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lemma POSIX2_ALT1a:
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assumes "POSIX2 (Right v2) (ALT r1 r2)"
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shows "POSIX2 v2 r2"
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using assms
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unfolding POSIX2_def
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apply(auto)
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apply(erule Prf.cases)
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apply(simp_all)[5]
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apply(drule_tac x="Right v'" in spec)
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apply(drule mp)
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apply(rule Prf.intros)
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apply(auto)
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apply(erule ValOrd.cases)
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apply(simp_all)
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done
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lemma POSIX_ALT1b:
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assumes "POSIX (Right v2) (ALT r1 r2)"
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shows "(\<forall>v'. (\<turnstile> v' : r2 \<and> flat v' = flat v2) \<longrightarrow> v2 \<succ>r2 v')"
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using assms
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apply(drule_tac POSIX_ALT1a)
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unfolding POSIX_def
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apply(auto)
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done
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lemma POSIX_ALT_I1:
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assumes "POSIX v1 r1"
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shows "POSIX (Left v1) (ALT r1 r2)"
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using assms
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unfolding POSIX_def
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apply(auto)
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apply(rotate_tac 3)
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apply(erule Prf.cases)
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apply(simp_all)[5]
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apply(auto)
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apply(rule ValOrd.intros)
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apply(auto)
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apply(rule ValOrd.intros)
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by simp
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lemma POSIX2_ALT_I1:
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assumes "POSIX2 v1 r1"
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shows "POSIX2 (Left v1) (ALT r1 r2)"
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using assms
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unfolding POSIX2_def
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apply(auto)
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apply(rule Prf.intros)
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apply(simp)
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apply(rotate_tac 2)
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apply(erule Prf.cases)
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apply(simp_all)[5]
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apply(auto)
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347 |
apply(rule ValOrd.intros)
|
|
|
348 |
apply(auto)
|
|
|
349 |
apply(rule ValOrd.intros)
|
|
|
350 |
oops
|
|
|
351 |
|
|
|
352 |
lemma POSIX_ALT_I2:
|
|
|
353 |
assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')"
|
|
|
354 |
shows "POSIX (Right v2) (ALT r1 r2)"
|
|
|
355 |
using assms
|
|
|
356 |
unfolding POSIX_def
|
|
|
357 |
apply(auto)
|
|
|
358 |
apply(rotate_tac 3)
|
|
|
359 |
apply(erule Prf.cases)
|
|
|
360 |
apply(simp_all)[5]
|
|
|
361 |
apply(auto)
|
|
|
362 |
apply(rule ValOrd.intros)
|
|
|
363 |
apply metis
|
|
|
364 |
done
|
|
|
365 |
|
|
|
366 |
|
|
|
367 |
|
|
|
368 |
|
|
|
369 |
|
|
|
370 |
section {* The Matcher *}
|
|
|
371 |
|
|
|
372 |
fun
|
|
|
373 |
nullable :: "rexp \<Rightarrow> bool"
|
|
|
374 |
where
|
|
|
375 |
"nullable (NULL) = False"
|
|
|
376 |
| "nullable (EMPTY) = True"
|
|
|
377 |
| "nullable (CHAR c) = False"
|
|
|
378 |
| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
|
|
|
379 |
| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
|
|
|
380 |
|
|
|
381 |
lemma nullable_correctness:
|
|
|
382 |
shows "nullable r \<longleftrightarrow> [] \<in> (L r)"
|
|
|
383 |
apply (induct r)
|
|
|
384 |
apply(auto simp add: Sequ_def)
|
|
|
385 |
done
|
|
|
386 |
|
|
|
387 |
fun mkeps :: "rexp \<Rightarrow> val"
|
|
|
388 |
where
|
|
|
389 |
"mkeps(EMPTY) = Void"
|
|
|
390 |
| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"
|
|
|
391 |
| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
|
|
|
392 |
|
|
|
393 |
lemma mkeps_nullable:
|
|
|
394 |
assumes "nullable(r)" shows "\<turnstile> mkeps r : r"
|
|
|
395 |
using assms
|
|
|
396 |
apply(induct rule: nullable.induct)
|
|
|
397 |
apply(auto intro: Prf.intros)
|
|
|
398 |
done
|
|
|
399 |
|
|
|
400 |
lemma mkeps_flat:
|
|
|
401 |
assumes "nullable(r)" shows "flat (mkeps r) = []"
|
|
|
402 |
using assms
|
|
|
403 |
apply(induct rule: nullable.induct)
|
|
|
404 |
apply(auto)
|
|
|
405 |
done
|
|
|
406 |
|
|
|
407 |
text {*
|
|
|
408 |
The value mkeps returns is always the correct POSIX
|
|
|
409 |
value.
|
|
|
410 |
*}
|
|
|
411 |
|
|
|
412 |
lemma mkeps_POSIX2:
|
|
|
413 |
assumes "nullable r"
|
|
|
414 |
shows "POSIX2 (mkeps r) r"
|
|
|
415 |
using assms
|
|
|
416 |
apply(induct r)
|
|
|
417 |
apply(auto)[1]
|
|
|
418 |
apply(simp add: POSIX2_def)
|
|
|
419 |
oops
|
|
|
420 |
|
|
|
421 |
lemma mkeps_POSIX3:
|
|
|
422 |
assumes "nullable r"
|
|
|
423 |
shows "POSIX3 (mkeps r) r"
|
|
|
424 |
using assms
|
|
|
425 |
apply(induct r)
|
|
|
426 |
apply(auto)[1]
|
|
|
427 |
apply(simp add: POSIX3_def)
|
|
|
428 |
apply(auto)[1]
|
|
|
429 |
apply (metis Prf.intros(4))
|
|
|
430 |
apply(erule Prf.cases)
|
|
|
431 |
apply(simp_all)[5]
|
|
|
432 |
apply (metis ValOrd.intros)
|
|
|
433 |
apply(simp add: POSIX3_def)
|
|
|
434 |
apply(auto)[1]
|
|
|
435 |
apply(simp add: POSIX3_def)
|
|
|
436 |
apply(auto)[1]
|
|
|
437 |
apply (metis mkeps.simps(2) mkeps_nullable nullable.simps(5))
|
|
|
438 |
apply(rotate_tac 6)
|
|
|
439 |
apply(erule Prf.cases)
|
|
|
440 |
apply(simp_all)[5]
|
|
|
441 |
apply (metis ValOrd.intros(2) add_leE gen_length_code(1) gen_length_def mkeps_flat)
|
|
|
442 |
apply(auto)
|
|
|
443 |
apply(simp add: POSIX3_def)
|
|
|
444 |
apply(auto)
|
|
|
445 |
apply (metis Prf.intros(2))
|
|
|
446 |
apply(rotate_tac 4)
|
|
|
447 |
apply(erule Prf.cases)
|
|
|
448 |
apply(simp_all)[5]
|
|
|
449 |
apply (metis ValOrd.intros(6))
|
|
|
450 |
apply(auto)[1]
|
|
|
451 |
apply (metis ValOrd.intros(3))
|
|
|
452 |
apply(simp add: POSIX3_def)
|
|
|
453 |
apply(auto)
|
|
|
454 |
apply (metis Prf.intros(2))
|
|
|
455 |
apply(rotate_tac 6)
|
|
|
456 |
apply(erule Prf.cases)
|
|
|
457 |
apply(simp_all)[5]
|
|
|
458 |
apply (metis ValOrd.intros(6))
|
|
|
459 |
apply (metis ValOrd.intros(3))
|
|
|
460 |
apply(simp add: POSIX3_def)
|
|
|
461 |
apply(auto)
|
|
|
462 |
apply (metis Prf.intros(3))
|
|
|
463 |
apply(rotate_tac 5)
|
|
|
464 |
apply(erule Prf.cases)
|
|
|
465 |
apply(simp_all)[5]
|
|
|
466 |
apply (metis Prf_flat_L drop_0 drop_all list.size(3) mkeps_flat nullable_correctness)
|
|
|
467 |
by (metis ValOrd.intros(5))
|
|
|
468 |
|
|
|
469 |
|
|
|
470 |
lemma mkeps_POSIX:
|
|
|
471 |
assumes "nullable r"
|
|
|
472 |
shows "POSIX (mkeps r) r"
|
|
|
473 |
using assms
|
|
|
474 |
apply(induct r)
|
|
|
475 |
apply(auto)[1]
|
|
|
476 |
apply(simp add: POSIX_def)
|
|
|
477 |
apply(auto)[1]
|
|
|
478 |
apply(erule Prf.cases)
|
|
|
479 |
apply(simp_all)[5]
|
|
|
480 |
apply (metis ValOrd.intros)
|
|
|
481 |
apply(simp add: POSIX_def)
|
|
|
482 |
apply(auto)[1]
|
|
|
483 |
apply(simp add: POSIX_def)
|
|
|
484 |
apply(auto)[1]
|
|
|
485 |
apply(erule Prf.cases)
|
|
|
486 |
apply(simp_all)[5]
|
|
|
487 |
apply(auto)
|
|
|
488 |
apply (simp add: ValOrd.intros(2) mkeps_flat)
|
|
|
489 |
apply(simp add: POSIX_def)
|
|
|
490 |
apply(auto)[1]
|
|
|
491 |
apply(erule Prf.cases)
|
|
|
492 |
apply(simp_all)[5]
|
|
|
493 |
apply(auto)
|
|
|
494 |
apply (simp add: ValOrd.intros(6))
|
|
|
495 |
apply (simp add: ValOrd.intros(3))
|
|
|
496 |
apply(simp add: POSIX_def)
|
|
|
497 |
apply(auto)[1]
|
|
|
498 |
apply(erule Prf.cases)
|
|
|
499 |
apply(simp_all)[5]
|
|
|
500 |
apply(auto)
|
|
|
501 |
apply (simp add: ValOrd.intros(6))
|
|
|
502 |
apply (simp add: ValOrd.intros(3))
|
|
|
503 |
apply(simp add: POSIX_def)
|
|
|
504 |
apply(auto)[1]
|
|
|
505 |
apply(erule Prf.cases)
|
|
|
506 |
apply(simp_all)[5]
|
|
|
507 |
apply(auto)
|
|
|
508 |
apply (metis Prf_flat_L mkeps_flat nullable_correctness)
|
|
|
509 |
by (simp add: ValOrd.intros(5))
|
|
|
510 |
|
|
|
511 |
|
|
|
512 |
lemma mkeps_POSIX2:
|
|
|
513 |
assumes "nullable r"
|
|
|
514 |
shows "POSIX2 (mkeps r) r"
|
|
|
515 |
using assms
|
|
|
516 |
apply(induct r)
|
|
|
517 |
apply(simp)
|
|
|
518 |
apply(simp)
|
|
|
519 |
apply(simp add: POSIX2_def)
|
|
|
520 |
apply(rule conjI)
|
|
|
521 |
apply(rule Prf.intros)
|
|
|
522 |
apply(auto)[1]
|
|
|
523 |
apply(erule Prf.cases)
|
|
|
524 |
apply(simp_all)[5]
|
|
|
525 |
apply(rule ValOrd.intros)
|
|
|
526 |
apply(simp)
|
|
|
527 |
apply(simp)
|
|
|
528 |
apply(simp add: POSIX2_def)
|
|
|
529 |
apply(rule conjI)
|
|
|
530 |
apply(rule Prf.intros)
|
|
|
531 |
apply(simp add: mkeps_nullable)
|
|
|
532 |
apply(simp add: mkeps_nullable)
|
|
|
533 |
apply(auto)[1]
|
|
|
534 |
apply(rotate_tac 6)
|
|
|
535 |
apply(erule Prf.cases)
|
|
|
536 |
apply(simp_all)[5]
|
|
|
537 |
apply(rule ValOrd.intros(2))
|
|
|
538 |
apply(simp)
|
|
|
539 |
apply(simp only: nullable.simps)
|
|
|
540 |
apply(erule disjE)
|
|
|
541 |
apply(simp)
|
|
|
542 |
thm POSIX2_ALT1a
|
|
|
543 |
apply(rule POSIX2_ALT)
|
|
|
544 |
apply(simp add: POSIX2_def)
|
|
|
545 |
apply(rule conjI)
|
|
|
546 |
apply(rule Prf.intros)
|
|
|
547 |
apply(simp add: mkeps_nullable)
|
|
|
548 |
oops
|
|
|
549 |
|
|
|
550 |
|
|
|
551 |
section {* Derivatives *}
|
|
|
552 |
|
|
|
553 |
fun
|
|
|
554 |
der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
|
|
|
555 |
where
|
|
|
556 |
"der c (NULL) = NULL"
|
|
|
557 |
| "der c (EMPTY) = NULL"
|
|
|
558 |
| "der c (CHAR c') = (if c = c' then EMPTY else NULL)"
|
|
|
559 |
| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
|
|
|
560 |
| "der c (SEQ r1 r2) =
|
|
|
561 |
(if nullable r1
|
|
|
562 |
then ALT (SEQ (der c r1) r2) (der c r2)
|
|
|
563 |
else SEQ (der c r1) r2)"
|
|
|
564 |
|
|
|
565 |
fun
|
|
|
566 |
ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
|
|
|
567 |
where
|
|
|
568 |
"ders [] r = r"
|
|
|
569 |
| "ders (c # s) r = ders s (der c r)"
|
|
|
570 |
|
|
|
571 |
section {* Injection function *}
|
|
|
572 |
|
|
|
573 |
fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
|
|
|
574 |
where
|
|
|
575 |
"injval (CHAR d) c Void = Char d"
|
|
|
576 |
| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"
|
|
|
577 |
| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"
|
|
|
578 |
| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
|
|
|
579 |
| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
|
|
|
580 |
| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
|
|
|
581 |
|
|
|
582 |
section {* Projection function *}
|
|
|
583 |
|
|
|
584 |
fun projval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
|
|
|
585 |
where
|
|
|
586 |
"projval (CHAR d) c _ = Void"
|
|
|
587 |
| "projval (ALT r1 r2) c (Left v1) = Left(projval r1 c v1)"
|
|
|
588 |
| "projval (ALT r1 r2) c (Right v2) = Right(projval r2 c v2)"
|
|
|
589 |
| "projval (SEQ r1 r2) c (Seq v1 v2) =
|
|
|
590 |
(if flat v1 = [] then Right(projval r2 c v2)
|
|
|
591 |
else if nullable r1 then Left (Seq (projval r1 c v1) v2)
|
|
|
592 |
else Seq (projval r1 c v1) v2)"
|
|
|
593 |
|
|
|
594 |
text {*
|
|
|
595 |
Injection value is related to r
|
|
|
596 |
*}
|
|
|
597 |
|
|
|
598 |
lemma v3:
|
|
|
599 |
assumes "\<turnstile> v : der c r" shows "\<turnstile> (injval r c v) : r"
|
|
|
600 |
using assms
|
|
|
601 |
apply(induct arbitrary: v rule: der.induct)
|
|
|
602 |
apply(simp)
|
|
|
603 |
apply(erule Prf.cases)
|
|
|
604 |
apply(simp_all)[5]
|
|
|
605 |
apply(simp)
|
|
|
606 |
apply(erule Prf.cases)
|
|
|
607 |
apply(simp_all)[5]
|
|
|
608 |
apply(case_tac "c = c'")
|
|
|
609 |
apply(simp)
|
|
|
610 |
apply(erule Prf.cases)
|
|
|
611 |
apply(simp_all)[5]
|
|
|
612 |
apply (metis Prf.intros(5))
|
|
|
613 |
apply(simp)
|
|
|
614 |
apply(erule Prf.cases)
|
|
|
615 |
apply(simp_all)[5]
|
|
|
616 |
apply(simp)
|
|
|
617 |
apply(erule Prf.cases)
|
|
|
618 |
apply(simp_all)[5]
|
|
|
619 |
apply (metis Prf.intros(2))
|
|
|
620 |
apply (metis Prf.intros(3))
|
|
|
621 |
apply(simp)
|
|
|
622 |
apply(case_tac "nullable r1")
|
|
|
623 |
apply(simp)
|
|
|
624 |
apply(erule Prf.cases)
|
|
|
625 |
apply(simp_all)[5]
|
|
|
626 |
apply(auto)[1]
|
|
|
627 |
apply(erule Prf.cases)
|
|
|
628 |
apply(simp_all)[5]
|
|
|
629 |
apply(auto)[1]
|
|
|
630 |
apply (metis Prf.intros(1))
|
|
|
631 |
apply(auto)[1]
|
|
|
632 |
apply (metis Prf.intros(1) mkeps_nullable)
|
|
|
633 |
apply(simp)
|
|
|
634 |
apply(erule Prf.cases)
|
|
|
635 |
apply(simp_all)[5]
|
|
|
636 |
apply(auto)[1]
|
|
|
637 |
apply(rule Prf.intros)
|
|
|
638 |
apply(auto)[2]
|
|
|
639 |
done
|
|
|
640 |
|
|
|
641 |
text {*
|
|
|
642 |
The string behin the injection value is an added c
|
|
|
643 |
*}
|
|
|
644 |
|
|
|
645 |
lemma v4:
|
|
|
646 |
assumes "\<turnstile> v : der c r" shows "flat (injval r c v) = c # (flat v)"
|
|
|
647 |
using assms
|
|
|
648 |
apply(induct arbitrary: v rule: der.induct)
|
|
|
649 |
apply(simp)
|
|
|
650 |
apply(erule Prf.cases)
|
|
|
651 |
apply(simp_all)[5]
|
|
|
652 |
apply(simp)
|
|
|
653 |
apply(erule Prf.cases)
|
|
|
654 |
apply(simp_all)[5]
|
|
|
655 |
apply(simp)
|
|
|
656 |
apply(case_tac "c = c'")
|
|
|
657 |
apply(simp)
|
|
|
658 |
apply(auto)[1]
|
|
|
659 |
apply(erule Prf.cases)
|
|
|
660 |
apply(simp_all)[5]
|
|
|
661 |
apply(simp)
|
|
|
662 |
apply(erule Prf.cases)
|
|
|
663 |
apply(simp_all)[5]
|
|
|
664 |
apply(simp)
|
|
|
665 |
apply(erule Prf.cases)
|
|
|
666 |
apply(simp_all)[5]
|
|
|
667 |
apply(simp)
|
|
|
668 |
apply(case_tac "nullable r1")
|
|
|
669 |
apply(simp)
|
|
|
670 |
apply(erule Prf.cases)
|
|
|
671 |
apply(simp_all)[5]
|
|
|
672 |
apply(auto)[1]
|
|
|
673 |
apply(erule Prf.cases)
|
|
|
674 |
apply(simp_all)[5]
|
|
|
675 |
apply(auto)[1]
|
|
|
676 |
apply (metis mkeps_flat)
|
|
|
677 |
apply(simp)
|
|
|
678 |
apply(erule Prf.cases)
|
|
|
679 |
apply(simp_all)[5]
|
|
|
680 |
done
|
|
|
681 |
|
|
|
682 |
text {*
|
|
|
683 |
Injection followed by projection is the identity.
|
|
|
684 |
*}
|
|
|
685 |
|
|
|
686 |
lemma proj_inj_id:
|
|
|
687 |
assumes "\<turnstile> v : der c r"
|
|
|
688 |
shows "projval r c (injval r c v) = v"
|
|
|
689 |
using assms
|
|
|
690 |
apply(induct r arbitrary: c v rule: rexp.induct)
|
|
|
691 |
apply(simp)
|
|
|
692 |
apply(erule Prf.cases)
|
|
|
693 |
apply(simp_all)[5]
|
|
|
694 |
apply(simp)
|
|
|
695 |
apply(erule Prf.cases)
|
|
|
696 |
apply(simp_all)[5]
|
|
|
697 |
apply(simp)
|
|
|
698 |
apply(case_tac "c = char")
|
|
|
699 |
apply(simp)
|
|
|
700 |
apply(erule Prf.cases)
|
|
|
701 |
apply(simp_all)[5]
|
|
|
702 |
apply(simp)
|
|
|
703 |
apply(erule Prf.cases)
|
|
|
704 |
apply(simp_all)[5]
|
|
|
705 |
defer
|
|
|
706 |
apply(simp)
|
|
|
707 |
apply(erule Prf.cases)
|
|
|
708 |
apply(simp_all)[5]
|
|
|
709 |
apply(simp)
|
|
|
710 |
apply(case_tac "nullable rexp1")
|
|
|
711 |
apply(simp)
|
|
|
712 |
apply(erule Prf.cases)
|
|
|
713 |
apply(simp_all)[5]
|
|
|
714 |
apply(auto)[1]
|
|
|
715 |
apply(erule Prf.cases)
|
|
|
716 |
apply(simp_all)[5]
|
|
|
717 |
apply(auto)[1]
|
|
|
718 |
apply (metis list.distinct(1) v4)
|
|
|
719 |
apply(auto)[1]
|
|
|
720 |
apply (metis mkeps_flat)
|
|
|
721 |
apply(auto)
|
|
|
722 |
apply(erule Prf.cases)
|
|
|
723 |
apply(simp_all)[5]
|
|
|
724 |
apply(auto)[1]
|
|
|
725 |
apply(simp add: v4)
|
|
|
726 |
done
|
|
|
727 |
|
|
|
728 |
lemma "L r \<noteq> {} \<Longrightarrow> \<exists>v. POSIX3 v r"
|
|
|
729 |
apply(induct r)
|
|
|
730 |
apply(simp)
|
|
|
731 |
apply(simp add: POSIX3_def)
|
|
|
732 |
apply(rule_tac x="Void" in exI)
|
|
|
733 |
apply(auto)[1]
|
|
|
734 |
apply (metis Prf.intros(4))
|
|
|
735 |
apply (metis POSIX3_def flat.simps(1) mkeps.simps(1) mkeps_POSIX3 nullable.simps(2) order_refl)
|
|
|
736 |
apply(simp add: POSIX3_def)
|
|
|
737 |
apply(rule_tac x="Char char" in exI)
|
|
|
738 |
apply(auto)[1]
|
|
|
739 |
apply (metis Prf.intros(5))
|
|
|
740 |
apply(erule Prf.cases)
|
|
|
741 |
apply(simp_all)[5]
|
|
|
742 |
apply (metis ValOrd.intros(8))
|
|
|
743 |
apply(simp add: Sequ_def)
|
|
|
744 |
apply(auto)[1]
|
|
|
745 |
apply(drule meta_mp)
|
|
|
746 |
apply(auto)[2]
|
|
|
747 |
apply(drule meta_mp)
|
|
|
748 |
apply(auto)[2]
|
|
|
749 |
apply(rule_tac x="Seq v va" in exI)
|
|
|
750 |
apply(simp (no_asm) add: POSIX3_def)
|
|
|
751 |
apply(auto)[1]
|
|
|
752 |
apply (metis POSIX3_def Prf.intros(1))
|
|
|
753 |
apply(erule Prf.cases)
|
|
|
754 |
apply(simp_all)[5]
|
|
|
755 |
apply(clarify)
|
|
|
756 |
apply(case_tac "v \<succ>r1a v1")
|
|
|
757 |
apply(rule ValOrd.intros(2))
|
|
|
758 |
apply(simp)
|
|
|
759 |
apply(case_tac "v = v1")
|
|
|
760 |
apply(rule ValOrd.intros(1))
|
|
|
761 |
apply(simp)
|
|
|
762 |
apply(simp)
|
|
|
763 |
apply (metis ValOrd_refl)
|
|
|
764 |
apply(simp add: POSIX3_def)
|
|
|
765 |
|
|
|
766 |
|
|
|
767 |
lemma "\<exists>v. POSIX v r"
|
|
|
768 |
apply(induct r)
|
|
|
769 |
apply(rule exI)
|
|
|
770 |
apply(simp add: POSIX_def)
|
|
|
771 |
apply (metis (full_types) Prf_flat_L der.simps(1) der.simps(2) der.simps(3) flat.simps(1) nullable.simps(1) nullable_correctness proj_inj_id projval.simps(1) v3 v4)
|
|
|
772 |
apply(rule_tac x = "Void" in exI)
|
|
|
773 |
apply(simp add: POSIX_def)
|
|
|
774 |
apply (metis POSIX_def flat.simps(1) mkeps.simps(1) mkeps_POSIX nullable.simps(2))
|
|
|
775 |
apply(rule_tac x = "Char char" in exI)
|
|
|
776 |
apply(simp add: POSIX_def)
|
|
|
777 |
apply(auto) [1]
|
|
|
778 |
apply(erule Prf.cases)
|
|
|
779 |
apply(simp_all) [5]
|
|
|
780 |
apply (metis ValOrd.intros(8))
|
|
|
781 |
defer
|
|
|
782 |
apply(auto)
|
|
|
783 |
apply (metis POSIX_ALT_I1)
|
|
|
784 |
(* maybe it is too early to instantiate this existential quantifier *)
|
|
|
785 |
(* potentially this is the wrong POSIX value *)
|
|
|
786 |
apply(case_tac "r1 = NULL")
|
|
|
787 |
apply(simp add: POSIX_def)
|
|
|
788 |
apply(auto)[1]
|
|
|
789 |
apply (metis L.simps(1) L.simps(4) Prf_flat_L mkeps_flat nullable.simps(1) nullable.simps(2) nullable_correctness seq_null(2))
|
|
|
790 |
apply(case_tac "r1 = EMPTY")
|
|
|
791 |
apply(rule_tac x = "Seq Void va" in exI )
|
|
|
792 |
apply(simp (no_asm) add: POSIX_def)
|
|
|
793 |
apply(auto)
|
|
|
794 |
apply(erule Prf.cases)
|
|
|
795 |
apply(simp_all)
|
|
|
796 |
apply(auto)[1]
|
|
|
797 |
apply(erule Prf.cases)
|
|
|
798 |
apply(simp_all)
|
|
|
799 |
apply(rule ValOrd.intros(2))
|
|
|
800 |
apply(rule ValOrd.intros)
|
|
|
801 |
apply(case_tac "\<exists>c. r1 = CHAR c")
|
|
|
802 |
apply(auto)
|
|
|
803 |
apply(rule_tac x = "Seq (Char c) va" in exI )
|
|
|
804 |
apply(simp (no_asm) add: POSIX_def)
|
|
|
805 |
apply(auto)
|
|
|
806 |
apply(erule Prf.cases)
|
|
|
807 |
apply(simp_all)
|
|
|
808 |
apply(auto)[1]
|
|
|
809 |
apply(erule Prf.cases)
|
|
|
810 |
apply(simp_all)
|
|
|
811 |
apply(auto)[1]
|
|
|
812 |
apply(rule ValOrd.intros(2))
|
|
|
813 |
apply(rule ValOrd.intros)
|
|
|
814 |
apply(case_tac "\<exists>r1a r1b. r1 = ALT r1a r1b")
|
|
|
815 |
apply(auto)
|
|
|
816 |
oops (* not sure if this can be proved by induction *)
|
|
|
817 |
|
|
|
818 |
text {*
|
|
|
819 |
|
|
|
820 |
HERE: Crucial lemma that does not go through in the sequence case.
|
|
|
821 |
|
|
|
822 |
*}
|
|
|
823 |
lemma v5:
|
|
|
824 |
assumes "\<turnstile> v : der c r" "POSIX v (der c r)"
|
|
|
825 |
shows "POSIX (injval r c v) r"
|
|
|
826 |
using assms
|
|
|
827 |
apply(induct arbitrary: v rule: der.induct)
|
|
|
828 |
apply(simp)
|
|
|
829 |
apply(erule Prf.cases)
|
|
|
830 |
apply(simp_all)[5]
|
|
|
831 |
apply(simp)
|
|
|
832 |
apply(erule Prf.cases)
|
|
|
833 |
apply(simp_all)[5]
|
|
|
834 |
apply(simp)
|
|
|
835 |
apply(case_tac "c = c'")
|
|
|
836 |
apply(auto simp add: POSIX_def)[1]
|
|
|
837 |
apply(erule Prf.cases)
|
|
|
838 |
apply(simp_all)[5]
|
|
|
839 |
apply(erule Prf.cases)
|
|
|
840 |
apply(simp_all)[5]
|
|
|
841 |
using ValOrd.simps apply blast
|
|
|
842 |
apply(auto)
|
|
|
843 |
apply(erule Prf.cases)
|
|
|
844 |
apply(simp_all)[5]
|
|
|
845 |
(* base cases done *)
|
|
|
846 |
(* ALT case *)
|
|
|
847 |
apply(erule Prf.cases)
|
|
|
848 |
apply(simp_all)[5]
|
|
|
849 |
using POSIX_ALT POSIX_ALT_I1 apply blast
|
|
|
850 |
apply(clarify)
|
|
|
851 |
apply(subgoal_tac "POSIX v2 (der c r2)")
|
|
|
852 |
prefer 2
|
|
|
853 |
apply(auto simp add: POSIX_def)[1]
|
|
|
854 |
apply (metis POSIX_ALT1a POSIX_def flat.simps(4))
|
|
|
855 |
apply(frule POSIX_ALT1a)
|
|
|
856 |
apply(drule POSIX_ALT1b)
|
|
|
857 |
apply(rule POSIX_ALT_I2)
|
|
|
858 |
apply(rotate_tac 1)
|
|
|
859 |
apply(drule_tac x="v2" in meta_spec)
|
|
|
860 |
apply(simp)
|
|
|
861 |
apply(subgoal_tac "\<turnstile> Right (injval r2 c v2) : (ALT r1 r2)")
|
|
|
862 |
prefer 2
|
|
|
863 |
apply (metis Prf.intros(3) v3)
|
|
|
864 |
|
|
|
865 |
apply auto[1]
|
|
|
866 |
apply(subst v4)
|
|
|
867 |
apply(auto)[2]
|
|
|
868 |
apply(subst (asm) (4) POSIX_def)
|
|
|
869 |
apply(subst (asm) v4)
|
|
|
870 |
apply(drule_tac x="v2" in meta_spec)
|
|
|
871 |
apply(simp)
|
|
|
872 |
|
|
|
873 |
apply(auto)[2]
|
|
|
874 |
|
|
|
875 |
thm POSIX_ALT_I2
|
|
|
876 |
apply(rule POSIX_ALT_I2)
|
|
|
877 |
|
|
|
878 |
apply(rule ccontr)
|
|
|
879 |
apply(auto simp add: POSIX_def)[1]
|
|
|
880 |
|
|
|
881 |
apply(rule allI)
|
|
|
882 |
apply(rule impI)
|
|
|
883 |
apply(erule conjE)
|
|
|
884 |
thm POSIX_ALT_I2
|
|
|
885 |
apply(frule POSIX_ALT1a)
|
|
|
886 |
apply(drule POSIX_ALT1b)
|
|
|
887 |
apply(rule POSIX_ALT_I2)
|
|
|
888 |
apply auto[1]
|
|
|
889 |
apply(subst v4)
|
|
|
890 |
apply(auto)[2]
|
|
|
891 |
apply(rotate_tac 1)
|
|
|
892 |
apply(drule_tac x="v2" in meta_spec)
|
|
|
893 |
apply(simp)
|
|
|
894 |
apply(subst (asm) (4) POSIX_def)
|
|
|
895 |
apply(subst (asm) v4)
|
|
|
896 |
apply(auto)[2]
|
|
|
897 |
(* stuck in the ALT case *)
|