--- a/Myhill_1.thy Fri Jun 03 13:59:21 2011 +0000
+++ b/Myhill_1.thy Mon Jul 25 13:33:38 2011 +0000
@@ -1,5 +1,5 @@
theory Myhill_1
-imports Regular
+imports More_Regular_Set
"~~/src/HOL/Library/While_Combinator"
begin
@@ -12,12 +12,12 @@
text {* Myhill-Nerode relation *}
definition
- str_eq_rel :: "lang \<Rightarrow> (string \<times> string) set" ("\<approx>_" [100] 100)
+ str_eq_rel :: "'a lang \<Rightarrow> ('a list \<times> 'a list) set" ("\<approx>_" [100] 100)
where
"\<approx>A \<equiv> {(x, y). (\<forall>z. x @ z \<in> A \<longleftrightarrow> y @ z \<in> A)}"
definition
- finals :: "lang \<Rightarrow> lang set"
+ finals :: "'a lang \<Rightarrow> 'a lang set"
where
"finals A \<equiv> {\<approx>A `` {s} | s . s \<in> A}"
@@ -37,35 +37,35 @@
text {* The two kinds of terms in the rhs of equations. *}
-datatype trm =
- Lam "rexp" (* Lambda-marker *)
- | Trn "lang" "rexp" (* Transition *)
+datatype 'a trm =
+ Lam "'a rexp" (* Lambda-marker *)
+ | Trn "'a lang" "'a rexp" (* Transition *)
fun
- L_trm::"trm \<Rightarrow> lang"
+ lang_trm::"'a trm \<Rightarrow> 'a lang"
where
- "L_trm (Lam r) = L_rexp r"
-| "L_trm (Trn X r) = X \<cdot> L_rexp r"
+ "lang_trm (Lam r) = lang r"
+| "lang_trm (Trn X r) = X \<cdot> lang r"
fun
- L_rhs::"trm set \<Rightarrow> lang"
+ lang_rhs::"('a trm) set \<Rightarrow> 'a lang"
where
- "L_rhs rhs = \<Union> (L_trm ` rhs)"
+ "lang_rhs rhs = \<Union> (lang_trm ` rhs)"
-lemma L_rhs_set:
- shows "L_rhs {Trn X r | r. P r} = \<Union>{L_trm (Trn X r) | r. P r}"
+lemma lang_rhs_set:
+ shows "lang_rhs {Trn X r | r. P r} = \<Union>{lang_trm (Trn X r) | r. P r}"
by (auto)
-lemma L_rhs_union_distrib:
- fixes A B::"trm set"
- shows "L_rhs A \<union> L_rhs B = L_rhs (A \<union> B)"
+lemma lang_rhs_union_distrib:
+ fixes A B::"('a trm) set"
+ shows "lang_rhs A \<union> lang_rhs B = lang_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)
+ transition :: "'a lang \<Rightarrow> 'a \<Rightarrow> 'a lang \<Rightarrow> bool" ("_ \<Turnstile>_\<Rightarrow>_" [100,100,100] 100)
where
"Y \<Turnstile>c\<Rightarrow> X \<equiv> Y \<cdot> {[c]} \<subseteq> X"
@@ -74,9 +74,9 @@
definition
"Init_rhs CS X \<equiv>
if ([] \<in> X) then
- {Lam EMPTY} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}
+ {Lam One} \<union> {Trn Y (Atom c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}
else
- {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}"
+ {Trn Y (Atom c)| Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}"
definition
"Init CS \<equiv> {(X, Init_rhs CS X) | X. X \<in> CS}"
@@ -85,10 +85,10 @@
section {* Arden Operation on equations *}
fun
- Append_rexp :: "rexp \<Rightarrow> trm \<Rightarrow> trm"
+ Append_rexp :: "'a rexp \<Rightarrow> 'a trm \<Rightarrow> 'a trm"
where
- "Append_rexp r (Lam rexp) = Lam (SEQ rexp r)"
-| "Append_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"
+ "Append_rexp r (Lam rexp) = Lam (Times rexp r)"
+| "Append_rexp r (Trn X rexp) = Trn X (Times rexp r)"
definition
@@ -96,7 +96,7 @@
definition
"Arden X rhs \<equiv>
- Append_rexp_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs}) (STAR (\<Uplus> {r. Trn X r \<in> rhs}))"
+ Append_rexp_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs}) (Star (\<Uplus> {r. Trn X r \<in> rhs}))"
section {* Substitution Operation on equations *}
@@ -106,7 +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> trm set) set \<Rightarrow> lang \<Rightarrow> trm set \<Rightarrow> (lang \<times> trm set) set"
+ Subst_all :: "('a lang \<times> ('a trm) set) set \<Rightarrow> 'a lang \<Rightarrow> ('a trm) set \<Rightarrow> ('a lang \<times> ('a trm) set) set"
where
"Subst_all ES X xrhs \<equiv> {(Y, Subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
@@ -143,10 +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 rhs"
+ "soundness ES \<equiv> \<forall>(X, rhs) \<in> ES. X = lang_rhs rhs"
definition
- "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L_rexp r)"
+ "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> lang r)"
definition
"ardenable_all ES \<equiv> \<forall>(X, rhs) \<in> ES. ardenable rhs"
@@ -223,23 +223,23 @@
lemma trm_soundness:
assumes finite:"finite rhs"
- shows "L_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (L_rexp (\<Uplus>{r. Trn X r \<in> rhs}))"
+ shows "lang_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (lang (\<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_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)
+ then show "lang_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (lang (\<Uplus>{r. Trn X r \<in> rhs}))"
+ by (simp only: lang_rhs_set lang_trm.simps) (auto simp add: conc_def)
qed
lemma lang_of_append_rexp:
- "L_trm (Append_rexp r trm) = L_trm trm \<cdot> L_rexp r"
+ "lang_trm (Append_rexp r trm) = lang_trm trm \<cdot> lang r"
by (induct rule: Append_rexp.induct)
- (auto simp add: seq_assoc)
+ (auto simp add: conc_assoc)
lemma lang_of_append_rexp_rhs:
- "L_rhs (Append_rexp_rhs rhs r) = L_rhs rhs \<cdot> L_rexp r"
+ "lang_rhs (Append_rexp_rhs rhs r) = lang_rhs rhs \<cdot> lang r"
unfolding Append_rexp_rhs_def
-by (auto simp add: Seq_def lang_of_append_rexp)
+by (auto simp add: conc_def lang_of_append_rexp)
subsubsection {* Intial Equational System *}
@@ -263,7 +263,7 @@
have "X = \<approx>A `` {s @ [c]}"
using has_str in_CS defined_by_str by blast
then have "Y \<cdot> {[c]} \<subseteq> X"
- unfolding Y_def Image_def Seq_def
+ unfolding Y_def Image_def conc_def
unfolding str_eq_rel_def
by clarsimp
moreover
@@ -274,14 +274,14 @@
lemma l_eq_r_in_eqs:
assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)"
- shows "X = L_rhs rhs"
+ shows "X = lang_rhs rhs"
proof
- show "X \<subseteq> L_rhs rhs"
+ show "X \<subseteq> lang_rhs rhs"
proof
fix x
assume in_X: "x \<in> X"
{ assume empty: "x = []"
- then have "x \<in> L_rhs rhs" using X_in_eqs in_X
+ then have "x \<in> lang_rhs rhs" using X_in_eqs in_X
unfolding Init_def Init_rhs_def
by auto
}
@@ -291,43 +291,42 @@
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 \<cdot> {[c]} \<subseteq> X" "s \<in> Y"
- using decom in_X every_eqclass_has_transition by blast
- then have "x \<in> L_rhs {Trn Y (CHAR c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"
+ using decom in_X every_eqclass_has_transition by metis
+ then have "x \<in> lang_rhs {Trn Y (Atom 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 rhs" using X_in_eqs in_X
+ using decom by (force simp add: conc_def)
+ then have "x \<in> lang_rhs rhs" using X_in_eqs in_X
unfolding Init_def Init_rhs_def by simp
}
- ultimately show "x \<in> L_rhs rhs" by blast
+ ultimately show "x \<in> lang_rhs rhs" by blast
qed
next
- show "L_rhs rhs \<subseteq> X" using X_in_eqs
+ show "lang_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_trm ` rhs)"
-using assms l_eq_r_in_eqs by (simp)
lemma finite_Init_rhs:
+ fixes CS::"(('a::finite) lang) set"
assumes finite: "finite CS"
shows "finite (Init_rhs CS X)"
proof-
- 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
+ def S \<equiv> "{(Y, c)| Y c::'a. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}"
+ def h \<equiv> "\<lambda> (Y, c::'a). Trn Y (Atom c)"
+ have "finite (CS \<times> (UNIV::('a::finite) 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 \<cdot> {[c]} \<subseteq> X} = h ` S"
+ moreover have "{Trn Y (Atom c) |Y c::'a. 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 \<cdot> {[c]} \<subseteq> X}" by auto
+ have "finite {Trn Y (Atom 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
+
lemma Init_ES_satisfies_invariant:
+ fixes A::"(('a::finite) lang)"
assumes finite_CS: "finite (UNIV // \<approx>A)"
shows "invariant (Init (UNIV // \<approx>A))"
proof (rule invariantI)
@@ -352,20 +351,20 @@
subsubsection {* Interation step *}
lemma Arden_keeps_eq:
- assumes l_eq_r: "X = L_rhs rhs"
+ assumes l_eq_r: "X = lang_rhs rhs"
and not_empty: "ardenable rhs"
and finite: "finite rhs"
- shows "X = L_rhs (Arden X rhs)"
+ shows "X = lang_rhs (Arden X rhs)"
proof -
- def A \<equiv> "L_rexp (\<Uplus>{r. Trn X r \<in> rhs})"
+ def A \<equiv> "lang (\<Uplus>{r. Trn X r \<in> rhs})"
def b \<equiv> "{Trn X r | r. Trn X r \<in> rhs}"
- def B \<equiv> "L_rhs (rhs - b)"
+ def B \<equiv> "lang_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 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)
+ have "X = lang_rhs rhs" using l_eq_r by simp
+ also have "\<dots> = lang_rhs (b \<union> (rhs - b))" unfolding b_def by auto
+ also have "\<dots> = lang_rhs b \<union> B" unfolding B_def by (simp only: lang_rhs_union_distrib)
also have "\<dots> = X \<cdot> A \<union> B"
unfolding b_def
unfolding trm_soundness[OF finite]
@@ -374,24 +373,24 @@
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_rhs (Arden X rhs)"
+ also have "\<dots> = lang_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_rhs (Arden X rhs)" by simp
+ by (simp only: lang_of_append_rexp_rhs lang.simps)
+ finally show "X = lang_rhs (Arden X rhs)" by simp
qed
lemma Append_keeps_finite:
"finite rhs \<Longrightarrow> finite (Append_rexp_rhs rhs r)"
-by (auto simp:Append_rexp_rhs_def)
+by (auto simp: Append_rexp_rhs_def)
lemma Arden_keeps_finite:
"finite rhs \<Longrightarrow> finite (Arden X rhs)"
-by (auto simp:Arden_def Append_keeps_finite)
+by (auto simp: Arden_def Append_keeps_finite)
lemma Append_keeps_nonempty:
"ardenable rhs \<Longrightarrow> ardenable (Append_rexp_rhs rhs r)"
-apply (auto simp:ardenable_def Append_rexp_rhs_def)
-by (case_tac x, auto simp:Seq_def)
+apply (auto simp: ardenable_def Append_rexp_rhs_def)
+by (case_tac x, auto simp: conc_def)
lemma nonempty_set_sub:
"ardenable rhs \<Longrightarrow> ardenable (rhs - A)"
@@ -411,24 +410,25 @@
by (simp only: Subst_def Append_keeps_nonempty nonempty_set_union nonempty_set_sub)
lemma Subst_keeps_eq:
- assumes substor: "X = L_rhs xrhs"
+ assumes substor: "X = lang_rhs xrhs"
and finite: "finite rhs"
- shows "L_rhs (Subst rhs X xrhs) = L_rhs rhs" (is "?Left = ?Right")
+ shows "lang_rhs (Subst rhs X xrhs) = lang_rhs rhs" (is "?Left = ?Right")
proof-
- 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}))"
+ def A \<equiv> "lang_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs})"
+ have "?Left = A \<union> lang_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"
unfolding Subst_def
- unfolding L_rhs_union_distrib[symmetric]
+ unfolding lang_rhs_union_distrib[symmetric]
by (simp add: A_def)
- moreover have "?Right = A \<union> L_rhs {Trn X r | r. Trn X r \<in> rhs}"
+ moreover have "?Right = A \<union> lang_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
unfolding A_def
- unfolding L_rhs_union_distrib
+ unfolding lang_rhs_union_distrib
by simp
qed
- 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}"
+ moreover
+ have "lang_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = lang_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
@@ -441,8 +441,8 @@
assumes finite: "finite ES"
shows "finite (Subst_all ES Y yrhs)"
proof -
- def eqns \<equiv> "{(X::lang, rhs) |X rhs. (X, rhs) \<in> ES}"
- def h \<equiv> "\<lambda>(X::lang, rhs). (X, Subst rhs Y yrhs)"
+ def eqns \<equiv> "{(X::'a lang, rhs) |X rhs. (X, rhs) \<in> ES}"
+ def h \<equiv> "\<lambda>(X::'a lang, rhs). (X, Subst rhs Y yrhs)"
have "finite (h ` eqns)" using finite h_def eqns_def by auto
moreover
have "Subst_all ES Y yrhs = h ` eqns" unfolding h_def eqns_def Subst_all_def by auto
@@ -456,24 +456,24 @@
lemma append_rhs_keeps_cls:
"rhss (Append_rexp_rhs rhs r) = rhss rhs"
-apply (auto simp:rhss_def Append_rexp_rhs_def)
-apply (case_tac xa, auto simp:image_def)
-by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
+apply (auto simp: rhss_def Append_rexp_rhs_def)
+apply (case_tac xa, auto simp: image_def)
+by (rule_tac x = "Times ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
lemma Arden_removes_cl:
"rhss (Arden Y yrhs) = rhss yrhs - {Y}"
apply (simp add:Arden_def append_rhs_keeps_cls)
-by (auto simp:rhss_def)
+by (auto simp: rhss_def)
lemma lhss_keeps_cls:
"lhss (Subst_all ES Y yrhs) = lhss ES"
-by (auto simp:lhss_def Subst_all_def)
+by (auto simp: lhss_def Subst_all_def)
lemma Subst_updates_cls:
"X \<notin> rhss xrhs \<Longrightarrow>
rhss (Subst rhs X xrhs) = rhss rhs \<union> rhss xrhs - {X}"
apply (simp only:Subst_def append_rhs_keeps_cls rhss_union_distrib)
-by (auto simp:rhss_def)
+by (auto simp: rhss_def)
lemma Subst_all_keeps_validity:
assumes sc: "validity (ES \<union> {(Y, yrhs)})" (is "validity ?A")
@@ -490,17 +490,17 @@
moreover have "rhss xrhs' \<subseteq> lhss ES"
proof-
have "rhss xrhs' \<subseteq> rhss xrhs \<union> rhss (Arden Y yrhs) - {Y}"
- proof-
+ proof -
have "Y \<notin> rhss (Arden Y yrhs)"
- using Arden_removes_cl by simp
- thus ?thesis using xrhs_xrhs' by (auto simp:Subst_updates_cls)
+ using Arden_removes_cl by auto
+ thus ?thesis using xrhs_xrhs' by (auto simp: Subst_updates_cls)
qed
moreover have "rhss xrhs \<subseteq> lhss ES \<union> {Y}" using X_in sc
apply (simp only:validity_def lhss_union_distrib)
by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lhss_def)
moreover have "rhss (Arden Y yrhs) \<subseteq> lhss ES \<union> {Y}"
using sc
- by (auto simp add:Arden_removes_cl validity_def lhss_def)
+ by (auto simp add: Arden_removes_cl validity_def lhss_def)
ultimately show ?thesis by auto
qed
ultimately show ?thesis by simp
@@ -512,7 +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_rhs yrhs"
+ have Y_eq_yrhs: "Y = lang_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)
@@ -520,7 +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_rhs (Arden Y yrhs)"
+ have "Y = lang_rhs (Arden Y yrhs)"
using Y_eq_yrhs invariant_ES finite_yrhs
using finite_Trn[OF finite_yrhs]
apply(rule_tac Arden_keeps_eq)
@@ -530,7 +530,7 @@
done
thus ?thesis using invariant_ES
unfolding invariant_def finite_rhs_def2 soundness_def Subst_all_def
- by (auto simp add: Subst_keeps_eq simp del: L_rhs.simps)
+ by (auto simp add: Subst_keeps_eq simp del: lang_rhs.simps)
qed
show "finite (Subst_all ES Y (Arden Y yrhs))"
using invariant_ES by (simp add:invariant_def Subst_all_keeps_finite)
@@ -557,13 +557,13 @@
proof -
have "finite yrhs" using invariant_ES
by (auto simp:invariant_def finite_rhs_def)
- thus ?thesis using Arden_keeps_finite by simp
+ thus ?thesis using Arden_keeps_finite by auto
qed
ultimately show ?thesis
by (simp add:Subst_all_keeps_finite_rhs)
qed
show "validity (Subst_all ES Y (Arden Y yrhs))"
- using invariant_ES Subst_all_keeps_validity by (simp add:invariant_def)
+ using invariant_ES Subst_all_keeps_validity by (auto simp add: invariant_def)
qed
lemma Remove_in_card_measure:
@@ -571,7 +571,7 @@
and in_ES: "(X, rhs) \<in> ES"
shows "(Remove ES X rhs, ES) \<in> measure card"
proof -
- def f \<equiv> "\<lambda> x. ((fst x)::lang, Subst (snd x) X (Arden X rhs))"
+ def f \<equiv> "\<lambda> x. ((fst x)::'a lang, Subst (snd x) X (Arden X rhs))"
def ES' \<equiv> "ES - {(X, rhs)}"
have "Subst_all ES' X (Arden X rhs) = f ` ES'"
apply (auto simp: Subst_all_def f_def image_def)
@@ -674,6 +674,7 @@
subsubsection {* Conclusion of the proof *}
lemma Solve:
+ fixes A::"('a::finite) lang"
assumes fin: "finite (UNIV // \<approx>A)"
and X_in: "X \<in> (UNIV // \<approx>A)"
shows "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}"
@@ -714,9 +715,10 @@
qed
lemma every_eqcl_has_reg:
+ fixes A::"('a::finite) lang"
assumes finite_CS: "finite (UNIV // \<approx>A)"
and X_in_CS: "X \<in> (UNIV // \<approx>A)"
- shows "\<exists>r. X = L_rexp r"
+ shows "\<exists>r. X = lang r"
proof -
from finite_CS X_in_CS
obtain xrhs where Inv_ES: "invariant {(X, xrhs)}"
@@ -735,14 +737,14 @@
using Arden_keeps_finite by auto
then have fin: "finite {r. Lam r \<in> A}" by (rule finite_Lam)
- have "X = L_rhs xrhs" using Inv_ES unfolding invariant_def soundness_def
+ have "X = lang_rhs xrhs" using Inv_ES unfolding invariant_def soundness_def
by simp
- then have "X = L_rhs A" using Inv_ES
+ then have "X = lang_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_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
+ then have "X = lang_rhs {Lam r | r. Lam r \<in> A}" using eq by simp
+ then have "X = lang (\<Uplus>{r. Lam r \<in> A})" using fin by auto
+ then show "\<exists>r. X = lang r" by blast
qed
lemma bchoice_finite_set:
@@ -756,20 +758,21 @@
done
theorem Myhill_Nerode1:
+ fixes A::"('a::finite) lang"
assumes finite_CS: "finite (UNIV // \<approx>A)"
- shows "\<exists>r. A = L_rexp r"
+ shows "\<exists>r. A = lang 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. X = L_rexp r"
+ have "\<forall>X \<in> (UNIV // \<approx>A). \<exists>r. X = lang r"
using finite_CS every_eqcl_has_reg by blast
- then have a: "\<forall>X \<in> finals A. \<exists>r. X = L_rexp r"
+ then have a: "\<forall>X \<in> finals A. \<exists>r. X = lang r"
using finals_in_partitions by auto
- then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L_rexp ` rs)" "finite rs"
+ then obtain rs::"('a rexp) set" where "\<Union> (finals A) = \<Union>(lang ` rs)" "finite rs"
using fin by (auto dest: bchoice_finite_set)
- then have "A = L_rexp (\<Uplus>rs)"
+ then have "A = lang (\<Uplus>rs)"
unfolding lang_is_union_of_finals[symmetric] by simp
- then show "\<exists>r. A = L_rexp r" by blast
+ then show "\<exists>r. A = lang r" by blast
qed