--- a/Myhill_1.thy Tue May 31 20:32:49 2011 +0000
+++ b/Myhill_1.thy Thu Jun 02 16:44:35 2011 +0000
@@ -1,5 +1,5 @@
theory Myhill_1
-imports Main Folds Regular
+imports Regular
"~~/src/HOL/Library/While_Combinator"
begin
@@ -37,43 +37,37 @@
text {* The two kinds of terms in the rhs of equations. *}
-datatype rhs_trm =
+datatype trm =
Lam "rexp" (* Lambda-marker *)
| Trn "lang" "rexp" (* Transition *)
+fun
+ L_trm::"trm \<Rightarrow> lang"
+where
+ "L_trm (Lam r) = L_rexp r"
+| "L_trm (Trn X r) = X \<cdot> L_rexp r"
-overloading L_rhs_trm \<equiv> "L:: rhs_trm \<Rightarrow> lang"
-begin
- fun L_rhs_trm:: "rhs_trm \<Rightarrow> lang"
- where
- "L_rhs_trm (Lam r) = L r"
- | "L_rhs_trm (Trn X r) = X ;; L r"
-end
-
-overloading L_rhs \<equiv> "L:: rhs_trm set \<Rightarrow> lang"
-begin
- fun L_rhs:: "rhs_trm set \<Rightarrow> lang"
- where
- "L_rhs rhs = \<Union> (L ` rhs)"
-end
+fun
+ L_rhs::"trm set \<Rightarrow> lang"
+where
+ "L_rhs rhs = \<Union> (L_trm ` rhs)"
lemma L_rhs_set:
- shows "L {Trn X r | r. P r} = \<Union>{L (Trn X r) | r. P r}"
-by (auto simp del: L_rhs_trm.simps)
+ shows "L_rhs {Trn X r | r. P r} = \<Union>{L_trm (Trn X r) | r. P r}"
+by (auto)
lemma L_rhs_union_distrib:
- fixes A B::"rhs_trm set"
- shows "L A \<union> L B = L (A \<union> B)"
+ fixes A B::"trm set"
+ shows "L_rhs A \<union> L_rhs B = L_rhs (A \<union> B)"
by simp
-
text {* Transitions between equivalence classes *}
definition
transition :: "lang \<Rightarrow> char \<Rightarrow> lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)
where
- "Y \<Turnstile>c\<Rightarrow> X \<equiv> Y ;; {[c]} \<subseteq> X"
+ "Y \<Turnstile>c\<Rightarrow> X \<equiv> Y \<cdot> {[c]} \<subseteq> X"
text {* Initial equational system *}
@@ -91,7 +85,7 @@
section {* Arden Operation on equations *}
fun
- Append_rexp :: "rexp \<Rightarrow> rhs_trm \<Rightarrow> rhs_trm"
+ Append_rexp :: "rexp \<Rightarrow> trm \<Rightarrow> trm"
where
"Append_rexp r (Lam rexp) = Lam (SEQ rexp r)"
| "Append_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"
@@ -112,7 +106,7 @@
(rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> (Append_rexp_rhs xrhs (\<Uplus> {r. Trn X r \<in> rhs}))"
definition
- Subst_all :: "(lang \<times> rhs_trm set) set \<Rightarrow> lang \<Rightarrow> rhs_trm set \<Rightarrow> (lang \<times> rhs_trm set) set"
+ Subst_all :: "(lang \<times> trm set) set \<Rightarrow> lang \<Rightarrow> trm set \<Rightarrow> (lang \<times> trm set) set"
where
"Subst_all ES X xrhs \<equiv> {(Y, Subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
@@ -149,10 +143,10 @@
\<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
definition
- "soundness ES \<equiv> \<forall>(X, rhs) \<in> ES. X = L rhs"
+ "soundness ES \<equiv> \<forall>(X, rhs) \<in> ES. X = L_rhs rhs"
definition
- "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
+ "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L_rexp r)"
definition
"ardenable_all ES \<equiv> \<forall>(X, rhs) \<in> ES. ardenable rhs"
@@ -227,29 +221,28 @@
done
qed
-lemma rhs_trm_soundness:
+lemma trm_soundness:
assumes finite:"finite rhs"
- shows "L ({Trn X r| r. Trn X r \<in> rhs}) = X ;; (L (\<Uplus>{r. Trn X r \<in> rhs}))"
+ shows "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<Uplus>{r. Trn X r \<in> rhs}))"
proof -
have "finite {r. Trn X r \<in> rhs}"
by (rule finite_Trn[OF finite])
- then show "L ({Trn X r| r. Trn X r \<in> rhs}) = X ;; (L (\<Uplus>{r. Trn X r \<in> rhs}))"
- by (simp only: L_rhs_set L_rhs_trm.simps) (auto simp add: Seq_def)
+ then show "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<Uplus>{r. Trn X r \<in> rhs}))"
+ by (simp only: L_rhs_set L_trm.simps) (auto simp add: Seq_def)
qed
lemma lang_of_append_rexp:
- "L (Append_rexp r rhs_trm) = L rhs_trm ;; L r"
+ "L_trm (Append_rexp r trm) = L_trm trm \<cdot> L_rexp r"
by (induct rule: Append_rexp.induct)
(auto simp add: seq_assoc)
lemma lang_of_append_rexp_rhs:
- "L (Append_rexp_rhs rhs r) = L rhs ;; L r"
+ "L_rhs (Append_rexp_rhs rhs r) = L_rhs rhs \<cdot> L_rexp r"
unfolding Append_rexp_rhs_def
by (auto simp add: Seq_def lang_of_append_rexp)
-
-subsubsection {* Intialization *}
+subsubsection {* Intial Equational System *}
lemma defined_by_str:
assumes "s \<in> X" "X \<in> UNIV // \<approx>A"
@@ -261,7 +254,7 @@
lemma every_eqclass_has_transition:
assumes has_str: "s @ [c] \<in> X"
and in_CS: "X \<in> UNIV // \<approx>A"
- obtains Y where "Y \<in> UNIV // \<approx>A" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
+ obtains Y where "Y \<in> UNIV // \<approx>A" and "Y \<cdot> {[c]} \<subseteq> X" and "s \<in> Y"
proof -
def Y \<equiv> "\<approx>A `` {s}"
have "Y \<in> UNIV // \<approx>A"
@@ -269,7 +262,7 @@
moreover
have "X = \<approx>A `` {s @ [c]}"
using has_str in_CS defined_by_str by blast
- then have "Y ;; {[c]} \<subseteq> X"
+ then have "Y \<cdot> {[c]} \<subseteq> X"
unfolding Y_def Image_def Seq_def
unfolding str_eq_rel_def
by clarsimp
@@ -281,14 +274,14 @@
lemma l_eq_r_in_eqs:
assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
- shows "X = L rhs"
+ shows "X = L_rhs rhs"
proof
- show "X \<subseteq> L rhs"
+ show "X \<subseteq> L_rhs rhs"
proof
fix x
assume in_X: "x \<in> X"
{ assume empty: "x = []"
- then have "x \<in> L rhs" using X_in_eqs in_X
+ then have "x \<in> L_rhs rhs" using X_in_eqs in_X
unfolding Init_def Init_rhs_def
by auto
}
@@ -297,40 +290,40 @@
then obtain s c where decom: "x = s @ [c]"
using rev_cases by blast
have "X \<in> UNIV // \<approx>A" using X_in_eqs unfolding Init_def by auto
- then obtain Y where "Y \<in> UNIV // \<approx>A" "Y ;; {[c]} \<subseteq> X" "s \<in> Y"
+ then obtain Y where "Y \<in> UNIV // \<approx>A" "Y \<cdot> {[c]} \<subseteq> X" "s \<in> Y"
using decom in_X every_eqclass_has_transition by blast
- then have "x \<in> L {Trn Y (CHAR c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"
+ then have "x \<in> L_rhs {Trn Y (CHAR c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"
unfolding transition_def
using decom by (force simp add: Seq_def)
- then have "x \<in> L rhs" using X_in_eqs in_X
+ then have "x \<in> L_rhs rhs" using X_in_eqs in_X
unfolding Init_def Init_rhs_def by simp
}
- ultimately show "x \<in> L rhs" by blast
+ ultimately show "x \<in> L_rhs rhs" by blast
qed
next
- show "L rhs \<subseteq> X" using X_in_eqs
+ show "L_rhs rhs \<subseteq> X" using X_in_eqs
unfolding Init_def Init_rhs_def transition_def
by auto
qed
lemma test:
assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
- shows "X = \<Union> (L ` rhs)"
+ shows "X = \<Union> (L_trm ` rhs)"
using assms l_eq_r_in_eqs by (simp)
lemma finite_Init_rhs:
assumes finite: "finite CS"
shows "finite (Init_rhs CS X)"
proof-
- def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
+ def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}"
def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
have "finite (CS \<times> (UNIV::char set))" using finite by auto
then have "finite S" using S_def
by (rule_tac B = "CS \<times> UNIV" in finite_subset) (auto)
- moreover have "{Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X} = h ` S"
+ moreover have "{Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X} = h ` S"
unfolding S_def h_def image_def by auto
ultimately
- have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" by auto
+ have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}" by auto
then show "finite (Init_rhs CS X)" unfolding Init_rhs_def transition_def by simp
qed
@@ -359,32 +352,32 @@
subsubsection {* Interation step *}
lemma Arden_keeps_eq:
- assumes l_eq_r: "X = L rhs"
+ assumes l_eq_r: "X = L_rhs rhs"
and not_empty: "ardenable rhs"
and finite: "finite rhs"
- shows "X = L (Arden X rhs)"
+ shows "X = L_rhs (Arden X rhs)"
proof -
- def A \<equiv> "L (\<Uplus>{r. Trn X r \<in> rhs})"
+ def A \<equiv> "L_rexp (\<Uplus>{r. Trn X r \<in> rhs})"
def b \<equiv> "{Trn X r | r. Trn X r \<in> rhs}"
- def B \<equiv> "L (rhs - b)"
+ def B \<equiv> "L_rhs (rhs - b)"
have not_empty2: "[] \<notin> A"
using finite_Trn[OF finite] not_empty
unfolding A_def ardenable_def by simp
- have "X = L rhs" using l_eq_r by simp
- also have "\<dots> = L (b \<union> (rhs - b))" unfolding b_def by auto
- also have "\<dots> = L b \<union> B" unfolding B_def by (simp only: L_rhs_union_distrib)
- also have "\<dots> = X ;; A \<union> B"
+ have "X = L_rhs rhs" using l_eq_r by simp
+ also have "\<dots> = L_rhs (b \<union> (rhs - b))" unfolding b_def by auto
+ also have "\<dots> = L_rhs b \<union> B" unfolding B_def by (simp only: L_rhs_union_distrib)
+ also have "\<dots> = X \<cdot> A \<union> B"
unfolding b_def
- unfolding rhs_trm_soundness[OF finite]
+ unfolding trm_soundness[OF finite]
unfolding A_def
by blast
- finally have "X = X ;; A \<union> B" .
- then have "X = B ;; A\<star>"
+ finally have "X = X \<cdot> A \<union> B" .
+ then have "X = B \<cdot> A\<star>"
by (simp add: arden[OF not_empty2])
- also have "\<dots> = L (Arden X rhs)"
+ also have "\<dots> = L_rhs (Arden X rhs)"
unfolding Arden_def A_def B_def b_def
by (simp only: lang_of_append_rexp_rhs L_rexp.simps)
- finally show "X = L (Arden X rhs)" by simp
+ finally show "X = L_rhs (Arden X rhs)" by simp
qed
lemma Append_keeps_finite:
@@ -418,16 +411,16 @@
by (simp only: Subst_def Append_keeps_nonempty nonempty_set_union nonempty_set_sub)
lemma Subst_keeps_eq:
- assumes substor: "X = L xrhs"
+ assumes substor: "X = L_rhs xrhs"
and finite: "finite rhs"
- shows "L (Subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
+ shows "L_rhs (Subst rhs X xrhs) = L_rhs rhs" (is "?Left = ?Right")
proof-
- def A \<equiv> "L (rhs - {Trn X r | r. Trn X r \<in> rhs})"
- have "?Left = A \<union> L (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"
+ def A \<equiv> "L_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs})"
+ have "?Left = A \<union> L_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"
unfolding Subst_def
unfolding L_rhs_union_distrib[symmetric]
by (simp add: A_def)
- moreover have "?Right = A \<union> L ({Trn X r | r. Trn X r \<in> rhs})"
+ moreover have "?Right = A \<union> L_rhs {Trn X r | r. Trn X r \<in> rhs}"
proof-
have "rhs = (rhs - {Trn X r | r. Trn X r \<in> rhs}) \<union> ({Trn X r | r. Trn X r \<in> rhs})" by auto
thus ?thesis
@@ -435,8 +428,8 @@
unfolding L_rhs_union_distrib
by simp
qed
- moreover have "L (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = L ({Trn X r | r. Trn X r \<in> rhs})"
- using finite substor by (simp only: lang_of_append_rexp_rhs rhs_trm_soundness)
+ moreover have "L_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = L_rhs {Trn X r | r. Trn X r \<in> rhs}"
+ using finite substor by (simp only: lang_of_append_rexp_rhs trm_soundness)
ultimately show ?thesis by simp
qed
@@ -519,7 +512,7 @@
assumes invariant_ES: "invariant (ES \<union> {(Y, yrhs)})"
shows "invariant (Subst_all ES Y (Arden Y yrhs))"
proof (rule invariantI)
- have Y_eq_yrhs: "Y = L yrhs"
+ have Y_eq_yrhs: "Y = L_rhs yrhs"
using invariant_ES by (simp only:invariant_def soundness_def, blast)
have finite_yrhs: "finite yrhs"
using invariant_ES by (auto simp:invariant_def finite_rhs_def)
@@ -527,7 +520,7 @@
using invariant_ES by (auto simp:invariant_def ardenable_all_def)
show "soundness (Subst_all ES Y (Arden Y yrhs))"
proof -
- have "Y = L (Arden Y yrhs)"
+ have "Y = L_rhs (Arden Y yrhs)"
using Y_eq_yrhs invariant_ES finite_yrhs
using finite_Trn[OF finite_yrhs]
apply(rule_tac Arden_keeps_eq)
@@ -723,7 +716,7 @@
lemma every_eqcl_has_reg:
assumes finite_CS: "finite (UNIV // \<approx>A)"
and X_in_CS: "X \<in> (UNIV // \<approx>A)"
- shows "\<exists>r::rexp. X = L r"
+ shows "\<exists>r. X = L_rexp r"
proof -
from finite_CS X_in_CS
obtain xrhs where Inv_ES: "invariant {(X, xrhs)}"
@@ -742,14 +735,14 @@
using Arden_keeps_finite by auto
then have fin: "finite {r. Lam r \<in> A}" by (rule finite_Lam)
- have "X = L xrhs" using Inv_ES unfolding invariant_def soundness_def
+ have "X = L_rhs xrhs" using Inv_ES unfolding invariant_def soundness_def
by simp
- then have "X = L A" using Inv_ES
+ then have "X = L_rhs A" using Inv_ES
unfolding A_def invariant_def ardenable_all_def finite_rhs_def
by (rule_tac Arden_keeps_eq) (simp_all add: finite_Trn)
- then have "X = L {Lam r | r. Lam r \<in> A}" using eq by simp
- then have "X = L (\<Uplus>{r. Lam r \<in> A})" using fin by auto
- then show "\<exists>r::rexp. X = L r" by blast
+ then have "X = L_rhs {Lam r | r. Lam r \<in> A}" using eq by simp
+ then have "X = L_rexp (\<Uplus>{r. Lam r \<in> A})" using fin by auto
+ then show "\<exists>r. X = L_rexp r" by blast
qed
lemma bchoice_finite_set:
@@ -764,19 +757,19 @@
theorem Myhill_Nerode1:
assumes finite_CS: "finite (UNIV // \<approx>A)"
- shows "\<exists>r::rexp. A = L r"
+ shows "\<exists>r. A = L_rexp r"
proof -
have fin: "finite (finals A)"
using finals_in_partitions finite_CS by (rule finite_subset)
- have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r::rexp. X = L r"
+ have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r. X = L_rexp r"
using finite_CS every_eqcl_has_reg by blast
- then have a: "\<forall>X \<in> finals A. \<exists>r::rexp. X = L r"
+ then have a: "\<forall>X \<in> finals A. \<exists>r. X = L_rexp r"
using finals_in_partitions by auto
- then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L ` rs)" "finite rs"
+ then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L_rexp ` rs)" "finite rs"
using fin by (auto dest: bchoice_finite_set)
- then have "A = L (\<Uplus>rs)"
+ then have "A = L_rexp (\<Uplus>rs)"
unfolding lang_is_union_of_finals[symmetric] by simp
- then show "\<exists>r::rexp. A = L r" by blast
+ then show "\<exists>r. A = L_rexp r" by blast
qed