regexp: comparison Myhill

equal deleted inserted replaced

-:b04cc5e4e84c
+:7743d2ad71d1
 theory Myhill_1
-imports Main Folds Regular
+imports Regular
 "~~/src/HOL/Library/While_Combinator"
 begin
 section {* Direction @{text "finite partition \<Rightarrow> regular language"} *}
 section {* Equational systems *}
 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
-overloading L_rhs_trm \<equiv> "L:: rhs_trm \<Rightarrow> lang"
+L_trm::"trm \<Rightarrow> lang"
-begin
+where
-fun L_rhs_trm:: "rhs_trm \<Rightarrow> lang"
+"L_trm (Lam r) = L_rexp r"
-where
+| "L_trm (Trn X r) = X \<cdot> L_rexp r"
-"L_rhs_trm (Lam r) = L r"
-| "L_rhs_trm (Trn X r) = X ;; L r"
+fun
-end
+L_rhs::"trm set \<Rightarrow> lang"
+where
-overloading L_rhs \<equiv> "L:: rhs_trm set \<Rightarrow> lang"
+"L_rhs rhs = \<Union> (L_trm ` rhs)"
-begin
-fun L_rhs:: "rhs_trm set \<Rightarrow> lang"
-where
-"L_rhs rhs = \<Union> (L ` rhs)"
-end
 lemma L_rhs_set:
-shows "L {Trn X r | r. P r} = \<Union>{L (Trn X r) | r. P r}"
+shows "L_rhs {Trn X r | r. P r} = \<Union>{L_trm (Trn X r) | r. P r}"
-by (auto simp del: L_rhs_trm.simps)
+by (auto)
 lemma L_rhs_union_distrib:
-fixes A B::"rhs_trm set"
+fixes A B::"trm set"
-shows "L A \<union> L B = L (A \<union> B)"
+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 *}
 definition
 "Init_rhs CS X \<equiv>
 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)"
 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> 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}"
 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"
+"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"
 definition
 apply(erule finite_imageD)
 apply(auto simp add: inj_on_def)
 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}))"
+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_rhs_trm.simps) (auto simp add: Seq_def)
+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 {* Intial Equational System *}
-subsubsection {* Intialization *}
 lemma defined_by_str:
 assumes "s \<in> X" "X \<in> UNIV // \<approx>A"
 shows "X = \<approx>A `` {s}"
 using assms
 by auto
 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"
 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 ;; {[c]} \<subseteq> X"
+then have "Y \<cdot> {[c]} \<subseteq> X"
 unfolding Y_def Image_def Seq_def
 unfolding str_eq_rel_def
 by clarsimp
 moreover
 have "s \<in> Y" unfolding Y_def
 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"
+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
 }
 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 ;; {[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
 lemma Init_ES_satisfies_invariant:
 assumes finite_CS: "finite (UNIV // \<approx>A)"
 qed
 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
+have "X = L_rhs rhs" using l_eq_r by simp
-also have "\<dots> = L (b \<union> (rhs - b))" unfolding b_def by auto
+also have "\<dots> = L_rhs (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> = L_rhs b \<union> B" unfolding B_def by (simp only: L_rhs_union_distrib)
-also have "\<dots> = X ;; A \<union> B"
+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" .
+finally have "X = X \<cdot> A \<union> B" .
-then have "X = B ;; A\<star>"
+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:
 "finite rhs \<Longrightarrow> finite (Append_rexp_rhs rhs r)"
 by (auto simp:Append_rexp_rhs_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 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})"
+def A \<equiv> "L_rhs (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}))"
+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
 unfolding A_def
 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})"
+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 rhs_trm_soundness)
+using finite substor by (simp only: lang_of_append_rexp_rhs trm_soundness)
 ultimately show ?thesis by simp
 qed
 lemma Subst_keeps_finite_rhs:
 "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (Subst rhs Y yrhs)"
 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 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)
 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 (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)
 apply(simp_all)
 unfolding invariant_def ardenable_all_def ardenable_def
 qed
 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)}"
 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 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_rhs {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 have "X = L_rexp (\<Uplus>{r. Lam r \<in> A})" using fin by auto
-then show "\<exists>r::rexp. X = L r" by blast
+then show "\<exists>r. X = L_rexp r" by blast
 qed
 lemma bchoice_finite_set:
 assumes a: "\<forall>x \<in> S. \<exists>y. x = f y"
 and     b: "finite S"
 apply(auto)
 done
 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
 end

changeset 166	7743d2ad71d1
parent 162	e93760534354
child 170	b1258b7d2789