regexp: comparison Myhill

equal deleted inserted replaced

-:b794db0b79db
+:b1258b7d2789
 theory Myhill_1
-imports Regular
+imports More_Regular_Set
 "~~/src/HOL/Library/While_Combinator"
 begin
 section {* Direction @{text "finite partition \<Rightarrow> regular language"} *}
 by simp
 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}"
 lemma lang_is_union_of_finals:
 shows "A = \<Union> finals A"
 section {* Equational systems *}
 text {* The two kinds of terms in the rhs of equations. *}
-datatype trm =
+datatype 'a trm =
-Lam "rexp"            (* Lambda-marker *)
+Lam "'a rexp"            (* Lambda-marker *)
-| Trn "lang" "rexp"     (* Transition *)
+| 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"
+"lang_trm (Lam r) = lang r"
-| "L_trm (Trn X r) = X \<cdot> L_rexp 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:
+lemma lang_rhs_set:
-shows "L_rhs {Trn X r | r. P r} = \<Union>{L_trm (Trn X r) | r. P r}"
+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:
+lemma lang_rhs_union_distrib:
-fixes A B::"trm set"
+fixes A B::"('a trm) set"
-shows "L_rhs A \<union> L_rhs B = L_rhs (A \<union> B)"
+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"
 text {* Initial equational system *}
 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}"
 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 (Lam rexp)   = Lam (Times rexp r)"
-| "Append_rexp r (Trn X rexp) = Trn X (SEQ rexp r)"
+| "Append_rexp r (Trn X rexp) = Trn X (Times rexp r)"
 definition
 "Append_rexp_rhs rhs rexp \<equiv> (Append_rexp rexp) ` rhs"
 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 *}
 definition
 "Subst rhs X xrhs \<equiv>
 (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}"
 definition
 "Remove ES X xrhs \<equiv>
 definition
 "distinctness ES \<equiv>
 \<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"
 definition
 done
 qed
 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}))"
+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: L_rhs_set L_trm.simps) (auto simp add: Seq_def)
+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 *}
 lemma defined_by_str:
 unfolding Y_def quotient_def by auto
 moreover
 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
 have "s \<in> Y" unfolding Y_def
 unfolding Image_def str_eq_rel_def by simp
 ultimately show thesis using that by blast
 qed
 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
 }
 moreover
 { assume not_empty: "x \<noteq> []"
 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 \<cdot> {[c]} \<subseteq> X" "s \<in> Y"
-using decom in_X every_eqclass_has_transition by blast
+using decom in_X every_eqclass_has_transition by metis
-then have "x \<in> L_rhs {Trn Y (CHAR c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"
+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)
+	using decom by (force simp add: conc_def)
-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 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 S \<equiv> "{(Y, c)| Y c::'a. Y \<in> CS \<and> Y \<cdot> {[c]} \<subseteq> X}"
-def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
+def h \<equiv> "\<lambda> (Y, c::'a). Trn Y (Atom c)"
-have "finite (CS \<times> (UNIV::char set))" using finite by auto
+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)
 show "soundness (Init (UNIV // \<approx>A))"
 unfolding soundness_def
 qed
 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
+have "X = lang_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> = lang_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> = 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]
 unfolding A_def
 by blast
 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)
+by (simp only: lang_of_append_rexp_rhs lang.simps)
-finally show "X = L_rhs (Arden X rhs)" by simp
+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)
+apply (auto simp: ardenable_def Append_rexp_rhs_def)
-by (case_tac x, auto simp:Seq_def)
+by (case_tac x, auto simp: conc_def)
 lemma nonempty_set_sub:
 "ardenable rhs \<Longrightarrow> ardenable (rhs - A)"
 by (auto simp:ardenable_def)
 lemma Subst_keeps_nonempty:
 "\<lbrakk>ardenable rhs; ardenable xrhs\<rbrakk> \<Longrightarrow> ardenable (Subst rhs X xrhs)"
 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})"
+def A \<equiv> "lang_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}))"
+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
 lemma Subst_keeps_finite_rhs:
 lemma Subst_all_keeps_finite:
 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 eqns \<equiv> "{(X::'a lang, rhs) |X rhs. (X, rhs) \<in> ES}"
-def h \<equiv> "\<lambda>(X::lang, rhs). (X, Subst rhs Y yrhs)"
+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
 ultimately
 show "finite (Subst_all ES Y yrhs)" by simp
 "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (Subst_all ES Y yrhs)"
 by (auto intro:Subst_keeps_finite_rhs simp add:Subst_all_def finite_rhs_def)
 lemma append_rhs_keeps_cls:
 "rhss (Append_rexp_rhs rhs r) = rhss rhs"
-apply (auto simp:rhss_def Append_rexp_rhs_def)
+apply (auto simp: rhss_def Append_rexp_rhs_def)
-apply (case_tac xa, auto simp:image_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+)
+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")
 shows "validity (Subst_all ES Y (Arden Y yrhs))"  (is "validity ?B")
 proof -
 proof-
 have "lhss ?B = lhss ES" by (auto simp add:lhss_def Subst_all_def)
 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
+using Arden_removes_cl by auto
-thus ?thesis using xrhs_xrhs' by (auto simp:Subst_updates_cls)
+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
 qed
 } thus ?thesis by (auto simp only:Subst_all_def validity_def)
 lemma Subst_all_satisfies_invariant:
 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)
 have nonempty_yrhs: "ardenable yrhs"
 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)
 apply(simp_all)
 unfolding invariant_def ardenable_all_def ardenable_def
 apply(auto)
 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)
 show "distinctness (Subst_all ES Y (Arden Y yrhs))"
 using invariant_ES
 by (simp add:invariant_def finite_rhs_def)
 moreover have "finite (Arden Y yrhs)"
 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:
 assumes finite: "finite ES"
 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)
 by (rule_tac x = "(Y, yrhs)" in bexI, simp+)
 then have "card (Subst_all ES' X (Arden X rhs)) \<le> card ES'"
 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)}"
 proof -
 def Inv \<equiv> "\<lambda>ES. invariant ES \<and> (\<exists>rhs. (X, rhs) \<in> ES)"
 show "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}"
 unfolding Solve_def by (rule while_rule)
 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)}"
 using Solve by metis
 have "finite A" using Inv_ES unfolding A_def invariant_def finite_rhs_def
 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 = lang_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 have "X = lang (\<Uplus>{r. Lam r \<in> A})" using fin by auto
-then show "\<exists>r. X = L_rexp r" by blast
+then show "\<exists>r. X = lang r" by blast
 qed
 lemma bchoice_finite_set:
 assumes a: "\<forall>x \<in> S. \<exists>y. x = f y"
 and     b: "finite S"
 apply(rule_tac x="fa ` S" in exI)
 apply(auto)
 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
 end

changeset 170	b1258b7d2789
parent 166	7743d2ad71d1
child 179	edacc141060f