--- a/thys/LexerExt.thy Thu Oct 05 12:45:13 2017 +0100
+++ b/thys/LexerExt.thy Sat Oct 07 22:16:16 2017 +0100
@@ -1,613 +1,23 @@
theory LexerExt
- imports Main
+ imports SpecExt
begin
-section {* Sequential Composition of Languages *}
-
-definition
- Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
-where
- "A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"
-
-text {* Two Simple Properties about Sequential Composition *}
-
-lemma Sequ_empty [simp]:
- shows "A ;; {[]} = A"
- and "{[]} ;; A = A"
-by (simp_all add: Sequ_def)
-
-lemma Sequ_null [simp]:
- shows "A ;; {} = {}"
- and "{} ;; A = {}"
-by (simp_all add: Sequ_def)
-
-lemma Sequ_assoc:
- shows "A ;; (B ;; C) = (A ;; B) ;; C"
-apply(auto simp add: Sequ_def)
-apply(metis append_assoc)
-apply(metis)
-done
-
-lemma Sequ_UNION:
- shows "(\<Union>x\<in>A. B ;; C x) = B ;; (\<Union>x\<in>A. C x)"
-by (auto simp add: Sequ_def)
-
-
-section {* Semantic Derivative (Left Quotient) of Languages *}
-
-definition
- Der :: "char \<Rightarrow> string set \<Rightarrow> string set"
-where
- "Der c A \<equiv> {s. c # s \<in> A}"
-
-definition
- Ders :: "string \<Rightarrow> string set \<Rightarrow> string set"
-where
- "Ders s A \<equiv> {s'. s @ s' \<in> A}"
-
-lemma Der_null [simp]:
- shows "Der c {} = {}"
-unfolding Der_def
-by auto
-
-lemma Der_empty [simp]:
- shows "Der c {[]} = {}"
-unfolding Der_def
-by auto
-
-lemma Der_char [simp]:
- shows "Der c {[d]} = (if c = d then {[]} else {})"
-unfolding Der_def
-by auto
-
-lemma Der_Sequ [simp]:
- shows "Der c (A ;; B) = (Der c A) ;; B \<union> (if [] \<in> A then Der c B else {})"
-unfolding Der_def Sequ_def
-by (auto simp add: Cons_eq_append_conv)
-
-lemma Der_union [simp]:
- shows "Der c (A \<union> B) = Der c A \<union> Der c B"
-unfolding Der_def
-by auto
-
-lemma Der_UNION:
- shows "Der c (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. Der c (B x))"
-by (auto simp add: Der_def)
-
-
-section {* Power operation for Sets *}
-
-fun
- Pow :: "string set \<Rightarrow> nat \<Rightarrow> string set" ("_ \<up> _" [101, 102] 101)
-where
- "A \<up> 0 = {[]}"
-| "A \<up> (Suc n) = A ;; (A \<up> n)"
-
-lemma Pow_empty [simp]:
- shows "[] \<in> A \<up> n \<longleftrightarrow> (n = 0 \<or> [] \<in> A)"
-by(induct n) (auto simp add: Sequ_def)
-
-lemma Der_Pow [simp]:
- shows "Der c (A \<up> (Suc n)) =
- (Der c A) ;; (A \<up> n) \<union> (if [] \<in> A then Der c (A \<up> n) else {})"
-unfolding Der_def Sequ_def
-by(auto simp add: Cons_eq_append_conv Sequ_def)
-
-lemma Der_Pow_subseteq:
- assumes "[] \<in> A"
- shows "Der c (A \<up> n) \<subseteq> (Der c A) ;; (A \<up> n)"
-using assms
-apply(induct n)
-apply(simp add: Sequ_def Der_def)
-apply(simp)
-apply(rule conjI)
-apply (smt Sequ_def append_Nil2 mem_Collect_eq Sequ_assoc subsetI)
-apply(subgoal_tac "((Der c A) ;; (A \<up> n)) \<subseteq> ((Der c A) ;; (A ;; (A \<up> n)))")
-apply(auto)[1]
-by (smt Sequ_def append_Nil2 mem_Collect_eq Sequ_assoc subsetI)
-
-lemma test:
- shows "(\<Union>x\<le>Suc n. Der c (A \<up> x)) = (\<Union>x\<le>n. Der c A ;; A \<up> x)"
-apply(induct n)
-apply(simp)
-apply(auto)[1]
-apply(case_tac xa)
-apply(simp)
-apply(simp)
-apply(auto)[1]
-apply(case_tac "[] \<in> A")
-apply(simp)
-apply(simp)
-by (smt Der_Pow Der_Pow_subseteq UN_insert atMost_Suc sup.orderE sup_bot.right_neutral)
-
-lemma test2:
- shows "(\<Union>x\<in>(Suc ` A). Der c (B \<up> x)) = (\<Union>x\<in>A. Der c B ;; B \<up> x)"
-apply(auto)[1]
-apply(case_tac "[] \<in> B")
-apply(simp)
-using Der_Pow_subseteq apply blast
-apply(simp)
-done
-
-
-section {* Kleene Star for Languages *}
-
-definition
- Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102) where
- "A\<star> = (\<Union>n. A \<up> n)"
-
-lemma Star_empty [intro]:
- shows "[] \<in> A\<star>"
-unfolding Star_def
-by auto
-
-lemma Star_step [intro]:
- assumes "s1 \<in> A" "s2 \<in> A\<star>"
- shows "s1 @ s2 \<in> A\<star>"
-proof -
- from assms obtain n where "s1 \<in>A" "s2 \<in> A \<up> n"
- unfolding Star_def by auto
- then have "s1 @ s2 \<in> A ;; (A \<up> n)"
- by (auto simp add: Sequ_def)
- then have "s1 @ s2 \<in> A \<up> (Suc n)"
- by simp
- then show "s1 @ s2 \<in> A\<star>"
- unfolding Star_def
- by (auto simp del: Pow.simps)
-qed
-
-lemma star_cases:
- shows "A\<star> = {[]} \<union> A ;; A\<star>"
-unfolding Star_def
-apply(simp add: Sequ_def)
-apply(auto)
-apply(case_tac xa)
-apply(auto simp add: Sequ_def)
-apply(rule_tac x="Suc xa" in exI)
-apply(auto simp add: Sequ_def)
-done
-
-lemma Der_Star1:
- shows "Der c (A ;; A\<star>) = (Der c A) ;; A\<star>"
-proof -
- have "(Der c A) ;; A\<star> = (Der c A) ;; (\<Union>n. A \<up> n)"
- unfolding Star_def by simp
- also have "... = (\<Union>n. Der c A ;; A \<up> n)"
- unfolding Sequ_UNION by simp
- also have "... = (\<Union>x\<in>(Suc ` UNIV). Der c (A \<up> x))"
- unfolding test2 by simp
- also have "... = (\<Union>n. Der c (A \<up> (Suc n)))"
- by (simp)
- also have "... = Der c (\<Union>n. A \<up> (Suc n))"
- unfolding Der_UNION by simp
- also have "... = Der c (A ;; (\<Union>n. A \<up> n))"
- by (simp only: Pow.simps Sequ_UNION)
- finally show "Der c (A ;; A\<star>) = (Der c A) ;; A\<star>"
- unfolding Star_def[symmetric] by blast
-qed
-
-lemma Der_star [simp]:
- shows "Der c (A\<star>) = (Der c A) ;; A\<star>"
-proof -
- have "Der c (A\<star>) = Der c ({[]} \<union> A ;; A\<star>)"
- by (simp only: star_cases[symmetric])
- also have "... = Der c (A ;; A\<star>)"
- by (simp only: Der_union Der_empty) (simp)
- also have "... = (Der c A) ;; A\<star>"
- using Der_Star1 by simp
- finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .
-qed
-
-
-
-
-section {* Regular Expressions *}
-
-datatype rexp =
- ZERO
-| ONE
-| CHAR char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-| UPNTIMES rexp nat
-| NTIMES rexp nat
-| FROMNTIMES rexp nat
-| NMTIMES rexp nat nat
-| PLUS rexp
-
-section {* Semantics of Regular Expressions *}
-
-fun
- L :: "rexp \<Rightarrow> string set"
-where
- "L (ZERO) = {}"
-| "L (ONE) = {[]}"
-| "L (CHAR c) = {[c]}"
-| "L (SEQ r1 r2) = (L r1) ;; (L r2)"
-| "L (ALT r1 r2) = (L r1) \<union> (L r2)"
-| "L (STAR r) = (L r)\<star>"
-| "L (UPNTIMES r n) = (\<Union>i\<in> {..n} . (L r) \<up> i)"
-| "L (NTIMES r n) = (L r) \<up> n"
-| "L (FROMNTIMES r n) = (\<Union>i\<in> {n..} . (L r) \<up> i)"
-| "L (NMTIMES r n m) = (\<Union>i\<in>{n..m} . (L r) \<up> i)"
-| "L (PLUS r) = (\<Union>i\<in> {1..} . (L r) \<up> i)"
-
-section {* Nullable, Derivatives *}
-
-fun
- nullable :: "rexp \<Rightarrow> bool"
-where
- "nullable (ZERO) = False"
-| "nullable (ONE) = True"
-| "nullable (CHAR c) = False"
-| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
-| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
-| "nullable (STAR r) = True"
-| "nullable (UPNTIMES r n) = True"
-| "nullable (NTIMES r n) = (if n = 0 then True else nullable r)"
-| "nullable (FROMNTIMES r n) = (if n = 0 then True else nullable r)"
-| "nullable (NMTIMES r n m) = (if m < n then False else (if n = 0 then True else nullable r))"
-| "nullable (PLUS r) = (nullable r)"
-
-fun
- der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
-where
- "der c (ZERO) = ZERO"
-| "der c (ONE) = ZERO"
-| "der c (CHAR d) = (if c = d then ONE else ZERO)"
-| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
-| "der c (SEQ r1 r2) =
- (if nullable r1
- then ALT (SEQ (der c r1) r2) (der c r2)
- else SEQ (der c r1) r2)"
-| "der c (STAR r) = SEQ (der c r) (STAR r)"
-| "der c (UPNTIMES r 0) = ZERO"
-| "der c (UPNTIMES r (Suc n)) = SEQ (der c r) (UPNTIMES r n)"
-| "der c (NTIMES r 0) = ZERO"
-| "der c (NTIMES r (Suc n)) = SEQ (der c r) (NTIMES r n)"
-| "der c (FROMNTIMES r 0) = SEQ (der c r) (FROMNTIMES r 0)"
-| "der c (FROMNTIMES r (Suc n)) = SEQ (der c r) (FROMNTIMES r n)"
-| "der c (NMTIMES r n m) =
- (if m < n then ZERO
- else (if n = 0 then (if m = 0 then ZERO else
- SEQ (der c r) (UPNTIMES r (m - 1))) else
- SEQ (der c r) (NMTIMES r (n - 1) (m - 1))))"
-| "der c (PLUS r) = SEQ (der c r) (STAR r)"
-
-fun
- ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
-where
- "ders [] r = r"
-| "ders (c # s) r = ders s (der c r)"
-
-
-lemma nullable_correctness:
- shows "nullable r \<longleftrightarrow> [] \<in> (L r)"
-apply(induct r)
-apply(auto simp add: Sequ_def)
-done
-
-lemma Der_Pow_notin:
- assumes "[] \<notin> A"
- shows "Der c (A \<up> (Suc n)) = (Der c A) ;; (A \<up> n)"
-using assms
-by(simp)
-
-lemma der_correctness:
- shows "L (der c r) = Der c (L r)"
-apply(induct c r rule: der.induct)
-apply(simp_all add: nullable_correctness)[7]
-apply(simp only: der.simps L.simps)
-apply(simp only: Der_UNION)
-apply(simp only: Sequ_UNION[symmetric])
-apply(simp add: test)
-apply(simp)
-(* NTIMES *)
-apply(simp only: L.simps der.simps)
-apply(simp)
-apply(rule impI)
-apply (simp add: Der_Pow_subseteq sup_absorb1)
-(* FROMNTIMES *)
-apply(simp only: der.simps)
-apply(simp only: L.simps)
-apply(simp)
-using Der_star Star_def apply auto[1]
-apply(simp only: der.simps)
-apply(simp only: L.simps)
-apply(simp add: Der_UNION)
-apply(subst Sequ_UNION[symmetric])
-apply(subst test2[symmetric])
-apply(subgoal_tac "(Suc ` {n..}) = {Suc n..}")
-apply simp
-apply(auto simp add: image_def)[1]
-using Suc_le_D apply blast
-(* NMTIMES *)
-apply(case_tac "n \<le> m")
-prefer 2
-apply(simp only: der.simps if_True)
-apply(simp only: L.simps)
-apply(simp)
-apply(auto)
-apply(subst (asm) Sequ_UNION[symmetric])
-apply(subst (asm) test[symmetric])
-apply(auto simp add: Der_UNION)[1]
-apply(auto simp add: Der_UNION)[1]
-apply(subst Sequ_UNION[symmetric])
-apply(subst test[symmetric])
-apply(auto)[1]
-apply(subst (asm) Sequ_UNION[symmetric])
-apply(subst (asm) test2[symmetric])
-apply(auto simp add: Der_UNION)[1]
-apply(subst Sequ_UNION[symmetric])
-apply(subst test2[symmetric])
-apply(auto simp add: Der_UNION)[1]
-(* PLUS *)
-apply(auto simp add: Sequ_def Star_def)[1]
-apply(simp add: Der_UNION)
-apply(rule_tac x="Suc xa" in bexI)
-apply(auto simp add: Sequ_def Der_def)[2]
-apply (metis append_Cons)
-apply(simp add: Der_UNION)
-apply(erule_tac bexE)
-apply(case_tac xa)
-apply(simp)
-apply(simp)
-apply(auto)
-apply(auto simp add: Sequ_def Der_def)[1]
-using Star_def apply auto[1]
-apply(case_tac "[] \<in> L r")
-apply(auto)
-using Der_UNION Der_star Star_def by fastforce
-
-
-lemma ders_correctness:
- shows "L (ders s r) = Ders s (L r)"
-apply(induct s arbitrary: r)
-apply(simp_all add: Ders_def der_correctness Der_def)
-done
-
-lemma ders_ZERO:
- shows "ders s (ZERO) = ZERO"
-apply(induct s)
-apply(simp_all)
-done
-
-lemma ders_ONE:
- shows "ders s (ONE) = (if s = [] then ONE else ZERO)"
-apply(induct s)
-apply(simp_all add: ders_ZERO)
-done
-
-lemma ders_CHAR:
- shows "ders s (CHAR c) = (if s = [c] then ONE else
- (if s = [] then (CHAR c) else ZERO))"
-apply(induct s)
-apply(simp_all add: ders_ZERO ders_ONE)
-done
-
-lemma ders_ALT:
- shows "ders s (ALT r1 r2) = ALT (ders s r1) (ders s r2)"
-apply(induct s arbitrary: r1 r2)
-apply(simp_all)
-done
-
-section {* Values *}
-
-datatype val =
- Void
-| Char char
-| Seq val val
-| Right val
-| Left val
-| Stars "val list"
-
-
-section {* The string behind a value *}
-
-fun
- flat :: "val \<Rightarrow> string"
-where
- "flat (Void) = []"
-| "flat (Char c) = [c]"
-| "flat (Left v) = flat v"
-| "flat (Right v) = flat v"
-| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"
-| "flat (Stars []) = []"
-| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))"
-
-lemma flat_Stars [simp]:
- "flat (Stars vs) = concat (map flat vs)"
-by (induct vs) (auto)
-
-
-section {* Relation between values and regular expressions *}
-
-inductive
- Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100)
-where
- "\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : SEQ r1 r2"
-| "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : ALT r1 r2"
-| "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : ALT r1 r2"
-| "\<turnstile> Void : ONE"
-| "\<turnstile> Char c : CHAR c"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : STAR r"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<le> n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : UPNTIMES r n"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs = n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : NTIMES r n"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> n\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : FROMNTIMES r n"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> n; length vs \<le> m\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : NMTIMES r n m"
-| "\<lbrakk>\<forall>v \<in> set vs. \<turnstile> v : r; length vs \<ge> 1\<rbrakk> \<Longrightarrow> \<turnstile> Stars vs : PLUS r"
-
-
-inductive_cases Prf_elims:
- "\<turnstile> v : ZERO"
- "\<turnstile> v : SEQ r1 r2"
- "\<turnstile> v : ALT r1 r2"
- "\<turnstile> v : ONE"
- "\<turnstile> v : CHAR c"
-(* "\<turnstile> vs : STAR r"*)
-
-lemma Prf_STAR:
- assumes "\<forall>v\<in>set vs. \<turnstile> v : r \<and> flat v \<in> L r"
- shows "concat (map flat vs) \<in> L r\<star>"
-using assms
-apply(induct vs)
-apply(auto)
-done
-
-lemma Prf_Pow:
- assumes "\<forall>v\<in>set vs. \<turnstile> v : r \<and> flat v \<in> L r"
- shows "concat (map flat vs) \<in> L r \<up> length vs"
-using assms
-apply(induct vs)
-apply(auto simp add: Sequ_def)
-done
-
-lemma Prf_flat_L:
- assumes "\<turnstile> v : r" shows "flat v \<in> L r"
-using assms
-apply(induct v r rule: Prf.induct)
-apply(auto simp add: Sequ_def)
-apply(rule Prf_STAR)
-apply(simp)
-apply(rule_tac x="length vs" in bexI)
-apply(rule Prf_Pow)
-apply(simp)
-apply(simp)
-apply(rule Prf_Pow)
-apply(simp)
-apply(rule_tac x="length vs" in bexI)
-apply(rule Prf_Pow)
-apply(simp)
-apply(simp)
-apply(rule_tac x="length vs" in bexI)
-apply(rule Prf_Pow)
-apply(simp)
-apply(simp)
-using Prf_Pow by blast
-
-lemma Star_Pow:
- assumes "s \<in> A \<up> n"
- shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A) \<and> (length ss = n)"
-using assms
-apply(induct n arbitrary: s)
-apply(auto simp add: Sequ_def)
-apply(drule_tac x="s2" in meta_spec)
-apply(auto)
-apply(rule_tac x="s1#ss" in exI)
-apply(simp)
-done
-
-lemma Star_string:
- assumes "s \<in> A\<star>"
- shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A)"
-using assms
-apply(auto simp add: Star_def)
-using Star_Pow by blast
-
-
-lemma Star_val:
- assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
- shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
-using assms
-apply(induct ss)
-apply(auto)
-apply (metis empty_iff list.set(1))
-by (metis concat.simps(2) list.simps(9) set_ConsD)
-
-lemma Star_val_length:
- assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
- shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r) \<and> (length vs) = (length ss)"
-using assms
-apply(induct ss)
-apply(auto)
-by (metis List.bind_def bind_simps(2) length_Suc_conv set_ConsD)
-
-
-
-
-
-lemma L_flat_Prf2:
- assumes "s \<in> L r" shows "\<exists>v. \<turnstile> v : r \<and> flat v = s"
-using assms
-apply(induct r arbitrary: s)
-apply(auto simp add: Sequ_def intro: Prf.intros)
-using Prf.intros(1) flat.simps(5) apply blast
-using Prf.intros(2) flat.simps(3) apply blast
-using Prf.intros(3) flat.simps(4) apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-using Star_string Star_val apply force
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x2)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> (length vs = x)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply(rule Prf.intros)
-apply(simp)
-apply(simp)
-using Star_Pow Star_val_length apply blast
-done
-
-lemma L_flat_Prf:
- "L(r) = {flat v | v. \<turnstile> v : r}"
-using Prf_flat_L L_flat_Prf2 by blast
-
-
-section {* Sulzmann and Lu functions *}
+section {* The Lexer Functions by Sulzmann and Lu *}
fun
mkeps :: "rexp \<Rightarrow> val"
where
"mkeps(ONE) = Void"
| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"
-| "mkeps(ALT r1 r2) = (if nullable r1 then Left (mkeps r1) else Right (mkeps r2))"
+| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
| "mkeps(STAR r) = Stars []"
| "mkeps(UPNTIMES r n) = Stars []"
| "mkeps(NTIMES r n) = Stars (replicate n (mkeps r))"
| "mkeps(FROMNTIMES r n) = Stars (replicate n (mkeps r))"
| "mkeps(NMTIMES r n m) = Stars (replicate n (mkeps r))"
-| "mkeps(PLUS r) = Stars ([mkeps r])"
-
-
+
fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
where
"injval (CHAR d) c Void = Char d"
@@ -617,364 +27,157 @@
| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (UPNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (FROMNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (NMTIMES r n m) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-| "injval (PLUS r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
-
-section {* Mkeps, injval *}
+| "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+| "injval (FROMNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+| "injval (UPNTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+| "injval (NMTIMES r n m) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+
+fun
+ lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"
+where
+ "lexer r [] = (if nullable r then Some(mkeps r) else None)"
+| "lexer r (c#s) = (case (lexer (der c r) s) of
+ None \<Rightarrow> None
+ | Some(v) \<Rightarrow> Some(injval r c v))"
-lemma mkeps_nullable:
- assumes "nullable(r)"
- shows "\<turnstile> mkeps r : r"
-using assms
-apply(induct r rule: mkeps.induct)
-apply(auto intro: Prf.intros)
-by (metis Prf.intros(10) in_set_replicate length_replicate not_le order_refl)
+
+
+section {* Mkeps, Injval Properties *}
lemma mkeps_flat:
assumes "nullable(r)"
shows "flat (mkeps r) = []"
using assms
-apply (induct rule: nullable.induct)
-apply(auto)
-by meson
-
-
-lemma Prf_injval:
- assumes "\<turnstile> v : der c r"
- shows "\<turnstile> (injval r c v) : r"
+ apply(induct rule: nullable.induct)
+ apply(auto)
+ by presburger
+
+
+lemma mkeps_nullable:
+ assumes "nullable(r)"
+ shows "\<Turnstile> mkeps r : r"
using assms
-apply(induct r arbitrary: c v rule: rexp.induct)
-apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)
-(* STAR *)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(auto)
-using Prf.intros(6) apply auto[1]
-(* UPNTIMES *)
-apply(case_tac x2)
-apply(simp)
-using Prf_elims(1) apply auto[1]
-apply(simp)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-using Prf.intros(7) apply auto[1]
-(* NTIMES *)
-apply(case_tac x2)
-apply(simp)
-using Prf_elims(1) apply auto[1]
-apply(simp)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-using Prf.intros(8) apply auto[1]
-(* FROMNTIMES *)
-apply(case_tac x2)
-apply(simp)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-using Prf.intros(9) apply auto[1]
-apply(rotate_tac 1)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-using Prf.intros(9) apply auto
-(* NMTIMES *)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rule Prf.intros)
-apply(auto)[2]
-apply simp
-apply(rotate_tac 4)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(clarify)
-apply(rule Prf.intros)
-apply(auto)[2]
-apply simp
-(* PLUS *)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(auto)
-using Prf.intros apply auto[1]
+apply(induct rule: nullable.induct)
+ apply(auto intro: Prf.intros split: if_splits)
+ using Prf.intros(8) apply force
+ apply(subst append.simps(1)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp)
+ apply(simp)
+ apply (simp add: mkeps_flat)
+ apply(simp)
+ using Prf.intros(9) apply force
+ apply(subst append.simps(1)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp)
+ apply(simp)
+ apply (simp add: mkeps_flat)
+ apply(simp)
+ using Prf.intros(11) apply force
+ apply(subst append.simps(1)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp)
+ apply(simp)
+ apply (simp add: mkeps_flat)
+ apply(simp)
+ apply(simp)
done
-
+
lemma Prf_injval_flat:
- assumes "\<turnstile> v : der c r"
+ assumes "\<Turnstile> v : der c r"
shows "flat (injval r c v) = c # (flat v)"
using assms
apply(induct arbitrary: v rule: der.induct)
-apply(auto elim!: Prf_elims split: if_splits)
-apply(metis mkeps_flat)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 4)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)
+apply(auto elim!: Prf_elims intro: mkeps_flat split: if_splits)
+done
+
+lemma Prf_injval:
+ assumes "\<Turnstile> v : der c r"
+ shows "\<Turnstile> (injval r c v) : r"
+using assms
+apply(induct r arbitrary: c v rule: rexp.induct)
+apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)[6]
+ apply(simp add: Prf_injval_flat)
+ apply(simp)
+ apply(case_tac x2)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ using Prf.intros(7) Prf_injval_flat apply auto[1]
+ apply(simp)
+ apply(case_tac x2)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(subst append.simps(2)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp add: Prf_injval_flat)
+ apply(simp)
+ apply(simp)
+ prefer 2
+ apply(simp)
+ apply(case_tac "x3a < x2")
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(case_tac x2)
+ apply(simp)
+ apply(case_tac x3a)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ using Prf.intros(12) Prf_injval_flat apply auto[1]
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(subst append.simps(2)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp add: Prf_injval_flat)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ using Prf.intros(12) Prf_injval_flat apply auto[1]
+ apply(case_tac x2)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp_all)
+ apply (simp add: Prf.intros(10) Prf_injval_flat)
+ using Prf.intros(10) Prf_injval_flat apply auto[1]
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp_all)
+ apply(subst append.simps(2)[symmetric])
+ apply(rule Prf.intros)
+ apply(simp add: Prf_injval_flat)
+ apply(simp)
+ apply(simp)
+ using Prf.intros(10) Prf_injval_flat apply auto
done
-section {* Our Alternative Posix definition *}
-inductive
- Posix :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
-where
- Posix_ONE: "[] \<in> ONE \<rightarrow> Void"
-| Posix_CHAR: "[c] \<in> (CHAR c) \<rightarrow> (Char c)"
-| Posix_ALT1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"
-| Posix_ALT2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"
-| Posix_SEQ: "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)\<rbrakk> \<Longrightarrow>
- (s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"
-| Posix_STAR1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>
- \<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"
-| Posix_STAR2: "[] \<in> STAR r \<rightarrow> Stars []"
-| Posix_UPNTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> UPNTIMES r n \<rightarrow> Stars vs; flat v \<noteq> [];
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r n))\<rbrakk>
- \<Longrightarrow> (s1 @ s2) \<in> UPNTIMES r (Suc n) \<rightarrow> Stars (v # vs)"
-| Posix_UPNTIMES2: "[] \<in> UPNTIMES r n \<rightarrow> Stars []"
-| Posix_NTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NTIMES r n \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (Stars vs) = [];
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r n))\<rbrakk>
- \<Longrightarrow> (s1 @ s2) \<in> NTIMES r (Suc n) \<rightarrow> Stars (v # vs)"
-| Posix_NTIMES2: "[] \<in> NTIMES r 0 \<rightarrow> Stars []"
-| Posix_FROMNTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> FROMNTIMES r n \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (Stars vs) = [];
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r n))\<rbrakk>
- \<Longrightarrow> (s1 @ s2) \<in> FROMNTIMES r (Suc n) \<rightarrow> Stars (v # vs)"
-| Posix_FROMNTIMES2: "s \<in> STAR r \<rightarrow> Stars vs \<Longrightarrow> s \<in> FROMNTIMES r 0 \<rightarrow> Stars vs"
-| Posix_NMTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NMTIMES r n m \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (Stars vs) = [];
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (NMTIMES r n m))\<rbrakk>
- \<Longrightarrow> (s1 @ s2) \<in> NMTIMES r (Suc n) (Suc m) \<rightarrow> Stars (v # vs)"
-| Posix_NMTIMES2: "s \<in> UPNTIMES r m \<rightarrow> Stars vs \<Longrightarrow> s \<in> NMTIMES r 0 m \<rightarrow> Stars vs"
-| Posix_PLUS1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v = [] \<Longrightarrow> flat (Stars vs) = [];
- \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>
- \<Longrightarrow> (s1 @ s2) \<in> PLUS r \<rightarrow> Stars (v # vs)"
-
-inductive_cases Posix_elims:
- "s \<in> ZERO \<rightarrow> v"
- "s \<in> ONE \<rightarrow> v"
- "s \<in> CHAR c \<rightarrow> v"
- "s \<in> ALT r1 r2 \<rightarrow> v"
- "s \<in> SEQ r1 r2 \<rightarrow> v"
- "s \<in> STAR r \<rightarrow> v"
-
-lemma Posix1:
- assumes "s \<in> r \<rightarrow> v"
- shows "s \<in> L r" "flat v = s"
-using assms
-apply (induct s r v rule: Posix.induct)
-apply(auto simp add: Sequ_def)
-apply(rule_tac x="Suc x" in bexI)
-apply(auto simp add: Sequ_def)
-apply(rule_tac x="Suc x" in bexI)
-using Sequ_def apply auto[2]
-apply(simp add: Star_def)
-apply(rule_tac x="Suc x" in bexI)
-apply(auto simp add: Sequ_def)
-apply(simp add: Star_def)
-apply(clarify)
-apply(rule_tac x="Suc x" in bexI)
-apply(auto simp add: Sequ_def)
-done
-
-
-lemma
- "([] @ []) \<in> PLUS (ONE) \<rightarrow> Stars ([Void])"
-apply(rule Posix_PLUS1)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(simp)
-apply(simp)
-done
-
-lemma
- assumes "s \<in> r \<rightarrow> v" "flat v \<noteq> []" "\<forall>s' \<in> L r. length s' < length s"
- shows "([] @ (s @ [])) \<in> PLUS (ALT ONE r) \<rightarrow> Stars ([Left Void, Right v])"
-using assms
-oops
-
-lemma
- "([] @ ([] @ [])) \<in> FROMNTIMES (ONE) (Suc (Suc 0)) \<rightarrow> Stars ([Void, Void])"
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(auto)
-done
-
-
-lemma
- assumes "s \<in> r \<rightarrow> v" "flat v \<noteq> []"
- "s \<in> PLUS (ALT ONE r) \<rightarrow> Stars ([Left(Void), Right(v)])"
- shows "False"
-using assms
-apply(rotate_tac 2)
-apply(erule_tac Posix.cases)
-apply(simp_all)
-apply(auto)
-oops
-
-
-
-
-
-lemma Posix1a:
- assumes "s \<in> r \<rightarrow> v"
- shows "\<turnstile> v : r"
-using assms
-apply(induct s r v rule: Posix.induct)
-apply(auto intro: Prf.intros)
-apply(rule Prf.intros)
-apply(auto)[1]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[1]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[1]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[1]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[2]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[3]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)[3]
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rotate_tac 3)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(rule Prf.intros)
-apply(auto)
-done
-
-
-lemma Posix_NTIMES_mkeps:
- assumes "[] \<in> r \<rightarrow> mkeps r"
- shows "[] \<in> NTIMES r n \<rightarrow> Stars (replicate n (mkeps r))"
-apply(induct n)
-apply(simp)
-apply (rule Posix_NTIMES2)
-apply(simp)
-apply(subst append_Nil[symmetric])
-apply (rule Posix_NTIMES1)
-apply(auto)
-apply(rule assms)
-done
-
-lemma Posix_FROMNTIMES_mkeps:
- assumes "[] \<in> r \<rightarrow> mkeps r"
- shows "[] \<in> FROMNTIMES r n \<rightarrow> Stars (replicate n (mkeps r))"
-apply(induct n)
-apply(simp)
-apply (rule Posix_FROMNTIMES2)
-apply (rule Posix.intros)
-apply(simp)
-apply(subst append_Nil[symmetric])
-apply (rule Posix_FROMNTIMES1)
-apply(auto)
-apply(rule assms)
-done
-
-lemma Posix_NMTIMES_mkeps:
- assumes "[] \<in> r \<rightarrow> mkeps r" "n \<le> m"
- shows "[] \<in> NMTIMES r n m \<rightarrow> Stars (replicate n (mkeps r))"
-using assms(2)
-apply(induct n arbitrary: m)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(case_tac m)
-apply(simp)
-apply(simp)
-apply(subst append_Nil[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule assms)
-done
-
-
+text {*
+ Mkeps and injval produce, or preserve, Posix values.
+*}
lemma Posix_mkeps:
assumes "nullable r"
@@ -984,261 +187,21 @@
apply(auto intro: Posix.intros simp add: nullable_correctness Sequ_def)
apply(subst append.simps(1)[symmetric])
apply(rule Posix.intros)
-apply(auto)
-apply(case_tac n)
-apply(simp)
-apply (simp add: Posix_NTIMES2)
-apply(simp)
-apply(subst append.simps(1)[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule Posix_NTIMES_mkeps)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(case_tac n)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(simp)
-apply(subst append.simps(1)[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule Posix_FROMNTIMES_mkeps)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(case_tac n)
-apply(simp)
-apply(rule Posix.intros)
-apply(rule Posix.intros)
-apply(simp)
-apply(case_tac m)
-apply(simp)
-apply(simp)
-apply(subst append_Nil[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule Posix_NMTIMES_mkeps)
-apply(auto)
-apply(subst append_Nil[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-apply(rule Posix.intros)
-done
-
-
-lemma Posix_determ:
- assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
- shows "v1 = v2"
-using assms
-proof (induct s r v1 arbitrary: v2 rule: Posix.induct)
- case (Posix_ONE v2)
- have "[] \<in> ONE \<rightarrow> v2" by fact
- then show "Void = v2" by cases auto
-next
- case (Posix_CHAR c v2)
- have "[c] \<in> CHAR c \<rightarrow> v2" by fact
- then show "Char c = v2" by cases auto
-next
- case (Posix_ALT1 s r1 v r2 v2)
- have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact
- moreover
- have "s \<in> r1 \<rightarrow> v" by fact
- then have "s \<in> L r1" by (simp add: Posix1)
- ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto
- moreover
- have IH: "\<And>v2. s \<in> r1 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
- ultimately have "v = v'" by simp
- then show "Left v = v2" using eq by simp
-next
- case (Posix_ALT2 s r2 v r1 v2)
- have "s \<in> ALT r1 r2 \<rightarrow> v2" by fact
- moreover
- have "s \<notin> L r1" by fact
- ultimately obtain v' where eq: "v2 = Right v'" "s \<in> r2 \<rightarrow> v'"
- by cases (auto simp add: Posix1)
- moreover
- have IH: "\<And>v2. s \<in> r2 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
- ultimately have "v = v'" by simp
- then show "Right v = v2" using eq by simp
-next
- case (Posix_SEQ s1 r1 v1 s2 r2 v2 v')
- have "(s1 @ s2) \<in> SEQ r1 r2 \<rightarrow> v'"
- "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact+
- then obtain v1' v2' where "v' = Seq v1' v2'" "s1 \<in> r1 \<rightarrow> v1'" "s2 \<in> r2 \<rightarrow> v2'"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) by fastforce+
- moreover
- have IHs: "\<And>v1'. s1 \<in> r1 \<rightarrow> v1' \<Longrightarrow> v1 = v1'"
- "\<And>v2'. s2 \<in> r2 \<rightarrow> v2' \<Longrightarrow> v2 = v2'" by fact+
- ultimately show "Seq v1 v2 = v'" by simp
-next
- case (Posix_STAR1 s1 r v s2 vs v2)
- have "(s1 @ s2) \<in> STAR r \<rightarrow> v2"
- "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" "flat v \<noteq> []"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact+
- then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (STAR r) \<rightarrow> (Stars vs')"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) apply fastforce
- apply (metis Posix1(1) Posix_STAR1.hyps(6) append_Nil append_Nil2)
- using Posix1(2) by blast
- moreover
- have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
- "\<And>v2. s2 \<in> STAR r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
- ultimately show "Stars (v # vs) = v2" by auto
-next
- case (Posix_STAR2 r v2)
- have "[] \<in> STAR r \<rightarrow> v2" by fact
- then show "Stars [] = v2" by cases (auto simp add: Posix1)
-next
- case (Posix_UPNTIMES1 s1 r v s2 n vs v2)
- have "(s1 @ s2) \<in> UPNTIMES r (Suc n) \<rightarrow> v2"
- "s1 \<in> r \<rightarrow> v" "s2 \<in> (UPNTIMES r n) \<rightarrow> Stars vs" "flat v \<noteq> []"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r n))" by fact+
- then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (UPNTIMES r n) \<rightarrow> (Stars vs')"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) apply fastforce
- apply (metis Posix1(1) Posix_UPNTIMES1.hyps(6) append_Nil append_Nil2)
- using Posix1(2) by blast
- moreover
- have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
- "\<And>v2. s2 \<in> UPNTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
- ultimately show "Stars (v # vs) = v2" by auto
-next
- case (Posix_UPNTIMES2 r n v2)
- have "[] \<in> UPNTIMES r n \<rightarrow> v2" by fact
- then show "Stars [] = v2" by cases (auto simp add: Posix1)
-next
- case (Posix_NTIMES2 r v2)
- have "[] \<in> NTIMES r 0 \<rightarrow> v2" by fact
- then show "Stars [] = v2" by cases (auto simp add: Posix1)
-next
- case (Posix_NTIMES1 s1 r v s2 n vs v2)
- have "(s1 @ s2) \<in> NTIMES r (Suc n) \<rightarrow> v2" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"
- "s1 \<in> r \<rightarrow> v" "s2 \<in> (NTIMES r n) \<rightarrow> Stars vs"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r n))" by fact+
- then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (NTIMES r n) \<rightarrow> (Stars vs')"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) apply fastforce
- by (metis Posix1(1) Posix_NTIMES1.hyps(6) append_Nil append_Nil2)
- moreover
- have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
- "\<And>v2. s2 \<in> NTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
- ultimately show "Stars (v # vs) = v2" by auto
-next
- case (Posix_FROMNTIMES2 s r v1 v2)
- have "s \<in> FROMNTIMES r 0 \<rightarrow> v2" "s \<in> STAR r \<rightarrow> Stars v1"
- "\<And>v3. s \<in> STAR r \<rightarrow> v3 \<Longrightarrow> Stars v1 = v3" by fact+
- then show ?case
- apply(cases)
+ apply(auto)
+ done
+
+lemma test:
+ assumes "s \<in> der c (FROMNTIMES r 0) \<rightarrow> v"
+ shows "XXX"
+using assms
+ apply(simp)
+ apply(erule Posix_elims)
+ apply(drule FROMNTIMES_0)
apply(auto)
- done
-next
- case (Posix_FROMNTIMES1 s1 r v s2 n vs v2)
- have "(s1 @ s2) \<in> FROMNTIMES r (Suc n) \<rightarrow> v2"
- "s1 \<in> r \<rightarrow> v" "s2 \<in> (FROMNTIMES r n) \<rightarrow> Stars vs" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r n))" by fact+
- then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (FROMNTIMES r n) \<rightarrow> (Stars vs')"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) apply fastforce
- by (metis Posix1(1) Posix_FROMNTIMES1.hyps(6) append_Nil append_Nil2)
- moreover
- have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
- "\<And>v2. s2 \<in> FROMNTIMES r n \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
- ultimately show "Stars (v # vs) = v2" by auto
-next
- case (Posix_NMTIMES2 s r m v1 v2)
- have "s \<in> NMTIMES r 0 m \<rightarrow> v2" "s \<in> UPNTIMES r m \<rightarrow> Stars v1"
- "\<And>v3. s \<in> UPNTIMES r m \<rightarrow> v3 \<Longrightarrow> Stars v1 = v3" by fact+
- then show ?case
- apply(cases)
- apply(auto)
- done
-next
- case (Posix_NMTIMES1 s1 r v s2 n m vs v2)
- have "(s1 @ s2) \<in> NMTIMES r (Suc n) (Suc m) \<rightarrow> v2"
- "s1 \<in> r \<rightarrow> v" "s2 \<in> (NMTIMES r n m) \<rightarrow> Stars vs" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NMTIMES r n m))" by fact+
- then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (NMTIMES r n m) \<rightarrow> (Stars vs')"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) apply fastforce
- by (metis Posix1(1) Posix_NMTIMES1.hyps(6) self_append_conv self_append_conv2)
- moreover
- have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
- "\<And>v2. s2 \<in> NMTIMES r n m \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
- ultimately show "Stars (v # vs) = v2" by auto
-next
- case (Posix_PLUS1 s1 r v s2 vs v2)
- have "(s1 @ s2) \<in> PLUS r \<rightarrow> v2"
- "s1 \<in> r \<rightarrow> v" "s2 \<in> (STAR r) \<rightarrow> Stars vs" (*"flat v = [] \<Longrightarrow> flat (Stars vs) = []"*)
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact+
- then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (STAR r) \<rightarrow> (Stars vs')"
- apply(cases) apply (auto simp add: append_eq_append_conv2)
- using Posix1(1) apply fastforce
- by (metis Posix1(1) Posix_PLUS1.hyps(6) append_self_conv append_self_conv2)
- moreover
- have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
- "\<And>v2. s2 \<in> STAR r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
- ultimately show "Stars (v # vs) = v2" by auto
-qed
-
-lemma NTIMES_Stars:
- assumes "\<turnstile> v : NTIMES r n"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> length vs = n"
-using assms
-apply(induct n arbitrary: v)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-done
-
-lemma UPNTIMES_Stars:
- assumes "\<turnstile> v : UPNTIMES r n"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> length vs \<le> n"
-using assms
-apply(induct n arbitrary: v)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-done
-
-lemma FROMNTIMES_Stars:
- assumes "\<turnstile> v : FROMNTIMES r n"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> n \<le> length vs"
-using assms
-apply(induct n arbitrary: v)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-done
-
-lemma PLUS_Stars:
- assumes "\<turnstile> v : PLUS r"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> 1 \<le> length vs"
-using assms
-apply(erule_tac Prf.cases)
-apply(simp_all)
-done
-
-lemma NMTIMES_Stars:
- assumes "\<turnstile> v : NMTIMES r n m"
- shows "\<exists>vs. v = Stars vs \<and> (\<forall>v \<in> set vs. \<turnstile> v : r) \<and> n \<le> length vs \<and> length vs \<le> m"
-using assms
-apply(induct n arbitrary: m v)
-apply(erule Prf.cases)
-apply(simp_all)
-apply(erule Prf.cases)
-apply(simp_all)
-done
-
+oops
lemma Posix_injval:
- assumes "s \<in> (der c r) \<rightarrow> v"
+ assumes "s \<in> (der c r) \<rightarrow> v"
shows "(c # s) \<in> r \<rightarrow> (injval r c v)"
using assms
proof(induct r arbitrary: s v rule: rexp.induct)
@@ -1352,7 +315,57 @@
then show "(c # s) \<in> SEQ r1 r2 \<rightarrow> injval (SEQ r1 r2) c v" using not_nullable by simp
qed
next
- case (STAR r)
+case (UPNTIMES r n s v)
+ have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
+ have "s \<in> der c (UPNTIMES r n) \<rightarrow> v" by fact
+ then consider
+ (cons) v1 vs s1 s2 where
+ "v = Seq v1 (Stars vs)" "s = s1 @ s2"
+ "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (UPNTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+ "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r (n - 1)))"
+ (* here *)
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ apply(erule Posix_elims)
+ apply(simp)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply(drule_tac x="v1" in meta_spec)
+ apply(drule_tac x="vss" in meta_spec)
+ apply(drule_tac x="s1" in meta_spec)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp add: der_correctness Der_def)
+ apply(erule Posix_elims)
+ apply(auto)
+ done
+ then show "(c # s) \<in> (UPNTIMES r n) \<rightarrow> injval (UPNTIMES r n) c v"
+ proof (cases)
+ case cons
+ have "s1 \<in> der c r \<rightarrow> v1" by fact
+ then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
+ moreover
+ have "s2 \<in> (UPNTIMES r (n - 1)) \<rightarrow> Stars vs" by fact
+ moreover
+ have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact
+ then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
+ then have "flat (injval r c v1) \<noteq> []" by simp
+ moreover
+ have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r (n - 1)))" by fact
+ then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r (n - 1)))"
+ by (simp add: der_correctness Der_def)
+ ultimately
+ have "((c # s1) @ s2) \<in> UPNTIMES r n \<rightarrow> Stars (injval r c v1 # vs)"
+ thm Posix.intros
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ apply(case_tac n)
+ apply(simp)
+ using Posix_elims(1) UPNTIMES.prems apply auto[1]
+ apply(simp)
+ done
+ then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" using cons by(simp)
+ qed
+ next
+ case (STAR r)
have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
have "s \<in> der c (STAR r) \<rightarrow> v" by fact
then consider
@@ -1384,242 +397,146 @@
have "((c # s1) @ s2) \<in> STAR r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros)
then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp)
qed
-next
- case (UPNTIMES r n)
- have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
- have "s \<in> der c (UPNTIMES r n) \<rightarrow> v" by fact
+ next
+ case (NTIMES r n s v)
+ have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
+ have "s \<in> der c (NTIMES r n) \<rightarrow> v" by fact
then consider
- (cons) m v1 vs s1 s2 where
- "n = Suc m"
+ (cons) v1 vs s1 s2 where
"v = Seq v1 (Stars vs)" "s = s1 @ s2"
- "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (UPNTIMES r m) \<rightarrow> (Stars vs)"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r m))"
- apply(case_tac n)
- apply(simp)
- using Posix_elims(1) apply blast
- apply(simp)
- apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)
- by (metis Posix1a UPNTIMES_Stars)
- then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v"
+ "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+ "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))"
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ apply(erule Posix_elims)
+ apply(simp)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply(drule_tac x="v1" in meta_spec)
+ apply(drule_tac x="vss" in meta_spec)
+ apply(drule_tac x="s1" in meta_spec)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp add: der_correctness Der_def)
+ apply(erule Posix_elims)
+ apply(auto)
+ done
+ then show "(c # s) \<in> (NTIMES r n) \<rightarrow> injval (NTIMES r n) c v"
proof (cases)
case cons
- have "n = Suc m" by fact
- moreover
have "s1 \<in> der c r \<rightarrow> v1" by fact
then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
moreover
- have "s2 \<in> UPNTIMES r m \<rightarrow> Stars vs" by fact
+ have "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> Stars vs" by fact
moreover
have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact
then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
then have "flat (injval r c v1) \<noteq> []" by simp
moreover
- have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (UPNTIMES r m))" by fact
- then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r m))"
+ have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))" by fact
+ then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))"
by (simp add: der_correctness Der_def)
ultimately
- have "((c # s1) @ s2) \<in> UPNTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)"
- apply(rule_tac Posix.intros(8))
- apply(simp_all)
- done
- then show "(c # s) \<in> UPNTIMES r n \<rightarrow> injval (UPNTIMES r n) c v" using cons by(simp)
- qed
-next
- case (NTIMES r n)
- have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
- have "s \<in> der c (NTIMES r n) \<rightarrow> v" by fact
- then consider
- (cons) m v1 vs s1 s2 where
- "n = Suc m"
- "v = Seq v1 (Stars vs)" "s = s1 @ s2" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"
- "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NTIMES r m) \<rightarrow> (Stars vs)"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r m))"
- apply(case_tac n)
- apply(simp)
- using Posix_elims(1) apply blast
- apply(simp)
- apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)
- by (metis NTIMES_Stars Posix1a)
- then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v"
- proof (cases)
- case cons
- have "n = Suc m" by fact
- moreover
- have "s1 \<in> der c r \<rightarrow> v1" by fact
- then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
- moreover
- have "flat v = [] \<Longrightarrow> flat (Stars vs) = []" by fact
- moreover
- have "s2 \<in> NTIMES r m \<rightarrow> Stars vs" by fact
- moreover
- have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r m))" by fact
- then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r m))"
- by (simp add: der_correctness Der_def)
- ultimately
- have "((c # s1) @ s2) \<in> NTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)"
- apply(rule_tac Posix.intros(10))
- apply(simp_all)
- by (simp add: Posix1(2))
+ have "((c # s1) @ s2) \<in> NTIMES r n \<rightarrow> Stars (injval r c v1 # vs)"
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ apply(case_tac n)
+ apply(simp)
+ using Posix_elims(1) NTIMES.prems apply auto[1]
+ apply(simp)
+ done
then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" using cons by(simp)
- qed
-next
- case (FROMNTIMES r n)
+ qed
+ next
+ case (FROMNTIMES r n s v)
have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
have "s \<in> der c (FROMNTIMES r n) \<rightarrow> v" by fact
then consider
- (null) v1 vs s1 s2 where
- "n = 0"
+ (cons) v1 vs s1 s2 where
"v = Seq v1 (Stars vs)" "s = s1 @ s2"
- "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r 0) \<rightarrow> (Stars vs)"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r 0))"
- | (cons) m v1 vs s1 s2 where
- "n = Suc m"
- "v = Seq v1 (Stars vs)" "s = s1 @ s2" "flat v = [] \<Longrightarrow> flat (Stars vs) = []"
- "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r m) \<rightarrow> (Stars vs)"
- "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r m))"
- apply(case_tac n)
- apply(simp)
- apply(auto elim!: Posix_elims(1-5) simp add: der_correctness Der_def intro: Posix.intros)
- defer
- apply (metis FROMNTIMES_Stars Posix1a)
- apply(rotate_tac 5)
- apply(erule Posix.cases)
- apply(simp_all)
- apply(clarify)
- by (simp add: Posix_FROMNTIMES2)
- then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v"
+ "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (FROMNTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+ "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r (n - 1)))"
+ | (null) v1 where
+ "v = Seq v1 (Stars [])"
+ "s \<in> der c r \<rightarrow> v1" "[] \<in> (FROMNTIMES r 0) \<rightarrow> (Stars [])" "n = 0"
+ (* here *)
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ apply(erule Posix_elims)
+ apply(simp)
+ apply(case_tac "n = 0")
+ apply(simp)
+ apply(drule FROMNTIMES_0)
+ apply(simp)
+ using FROMNTIMES_0 Posix_mkeps apply force
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply(drule_tac x="v1" in meta_spec)
+ apply(drule_tac x="vss" in meta_spec)
+ apply(drule_tac x="s1" in meta_spec)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp add: der_correctness Der_def)
+ apply(case_tac "n = 0")
+ apply(simp)
+ apply(simp)
+ apply(rotate_tac 4)
+ apply(erule Posix_elims)
+ apply(auto)[2]
+ done
+ then show "(c # s) \<in> (FROMNTIMES r n) \<rightarrow> injval (FROMNTIMES r n) c v"
proof (cases)
case cons
- have "n = Suc m" by fact
- moreover
have "s1 \<in> der c r \<rightarrow> v1" by fact
then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
moreover
- have "s2 \<in> FROMNTIMES r m \<rightarrow> Stars vs" by fact
- moreover
- have "flat v = [] \<Longrightarrow> flat (Stars vs) = []" by fact
+ have "s2 \<in> (FROMNTIMES r (n - 1)) \<rightarrow> Stars vs" by fact
moreover
- have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r m))" by fact
- then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r m))"
+ have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact
+ then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
+ then have "flat (injval r c v1) \<noteq> []" by simp
+ moreover
+ have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (FROMNTIMES r (n - 1)))" by fact
+ then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r (n - 1)))"
by (simp add: der_correctness Der_def)
ultimately
- have "((c # s1) @ s2) \<in> FROMNTIMES r (Suc m) \<rightarrow> Stars (injval r c v1 # vs)"
- apply(rule_tac Posix.intros(12))
- apply(simp_all)
- by (simp add: Posix1(2))
+ have "((c # s1) @ s2) \<in> FROMNTIMES r n \<rightarrow> Stars (injval r c v1 # vs)"
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ apply(case_tac n)
+ apply(simp)
+ using Posix_elims(1) FROMNTIMES.prems apply auto[1]
+ using cons(5) apply blast
+ apply(simp)
+ done
then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using cons by(simp)
- next
- case null
- then have "((c # s1) @ s2) \<in> FROMNTIMES r 0 \<rightarrow> Stars (injval r c v1 # vs)"
- apply(rule_tac Posix.intros)
- apply(rule_tac Posix.intros)
- apply(rule IH)
- apply(simp)
- apply(rotate_tac 4)
- apply(erule Posix.cases)
- apply(simp_all)
- apply (simp add: Posix1a Prf_injval_flat)
- apply(simp only: Star_def)
- apply(simp only: der_correctness Der_def)
- apply(simp)
- done
- then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using null by simp
- qed
-next
- case (NMTIMES r n m)
- have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
- have "s \<in> der c (NMTIMES r n m) \<rightarrow> v" by fact
- then show "(c # s) \<in> (NMTIMES r n m) \<rightarrow> injval (NMTIMES r n m) c v"
- apply(case_tac "m < n")
- apply(simp)
- using Posix_elims(1) apply blast
- apply(case_tac n)
- apply(simp_all)
- apply(case_tac m)
- apply(simp)
- apply(erule Posix_elims(1))
- apply(simp)
- apply(erule Posix.cases)
- apply(simp_all)
- apply(clarify)
- apply(rotate_tac 5)
- apply(frule Posix1a)
- apply(drule UPNTIMES_Stars)
- apply(clarify)
- apply(simp)
- apply(subst append_Cons[symmetric])
- apply(rule Posix.intros)
- apply(rule Posix.intros)
- apply(auto)
- apply(rule IH)
- apply blast
- using Posix1a Prf_injval_flat apply auto[1]
- apply(simp add: der_correctness Der_def)
- apply blast
- apply(case_tac m)
- apply(simp)
- apply(simp)
- apply(erule Posix.cases)
- apply(simp_all)
- apply(clarify)
- apply(rotate_tac 6)
- apply(frule Posix1a)
- apply(drule NMTIMES_Stars)
- apply(clarify)
- apply(simp)
- apply(subst append_Cons[symmetric])
- apply(rule Posix.intros)
- apply(rule IH)
- apply(blast)
- apply(simp)
- apply(simp add: der_correctness Der_def)
- using Posix1a Prf_injval_flat list.distinct(1) apply auto[1]
- apply(simp add: der_correctness Der_def)
- done
-next
- case (PLUS r)
- have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
- have "s \<in> der c (PLUS r) \<rightarrow> v" by fact
- then show "(c # s) \<in> PLUS r \<rightarrow> injval (PLUS r) c v"
- apply -
- apply(erule Posix.cases)
- apply(simp_all)
- apply(auto)
- apply(rotate_tac 3)
- apply(erule Posix.cases)
- apply(simp_all)
- apply(subst append_Cons[symmetric])
- apply(rule Posix.intros)
- using PLUS.hyps apply auto[1]
- apply(rule Posix.intros)
- apply(simp)
- apply(simp)
- apply(simp)
- apply(simp)
- apply(simp)
- using Posix1a Prf_injval_flat apply auto[1]
- apply(simp add: der_correctness Der_def)
- apply(subst append_Nil2[symmetric])
- apply(rule Posix.intros)
- using PLUS.hyps apply auto[1]
- apply(rule Posix.intros)
- apply(simp)
- apply(simp)
- done
+ next
+ case null
+ have "s \<in> der c r \<rightarrow> v1" by fact
+ then have "(c # s) \<in> r \<rightarrow> injval r c v1" using IH by simp
+ moreover
+ have "[] \<in> (FROMNTIMES r 0) \<rightarrow> Stars []" by fact
+ moreover
+ have "(c # s) \<in> r \<rightarrow> injval r c v1" by fact
+ then have "flat (injval r c v1) = (c # s)" by (rule Posix1)
+ then have "flat (injval r c v1) \<noteq> []" by simp
+ ultimately
+ have "((c # s) @ []) \<in> FROMNTIMES r 1 \<rightarrow> Stars [injval r c v1]"
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ done
+ then show "(c # s) \<in> FROMNTIMES r n \<rightarrow> injval (FROMNTIMES r n) c v" using null
+ apply(simp)
+ apply(erule Posix_elims)
+ apply(auto)
+ apply(rotate_tac 6)
+ apply(drule FROMNTIMES_0)
+ apply(simp)
+ sorry
+ qed
+ next
+ case (NMTIMES x1 x2 m s v)
+ then show ?case sorry
qed
-
-section {* The Lexer by Sulzmann and Lu *}
-
-fun
- lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"
-where
- "lexer r [] = (if nullable r then Some(mkeps r) else None)"
-| "lexer r (c#s) = (case (lexer (der c r) s) of
- None \<Rightarrow> None
- | Some(v) \<Rightarrow> Some(injval r c v))"
-
+section {* Lexer Correctness *}
lemma lexer_correct_None:
shows "s \<notin> L r \<longleftrightarrow> lexer r s = None"
@@ -1645,8 +562,4 @@
using Posix1(1) Posix_determ lexer_correct_None lexer_correct_Some apply fastforce
using Posix1(1) lexer_correct_None lexer_correct_Some by blast
-value "lexer (PLUS (ALT ONE (SEQ (CHAR (CHR ''a'')) (CHAR (CHR ''b''))))) [CHR ''a'', CHR ''b'']"
-
-
-unused_thms
end
\ No newline at end of file