--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Theories/Myhill_1.thy Wed Mar 23 12:17:30 2011 +0000
@@ -0,0 +1,783 @@
+theory Myhill_1
+imports Main Folds Regular
+ "~~/src/HOL/Library/While_Combinator"
+begin
+
+section {* Direction @{text "finite partition \<Rightarrow> regular language"} *}
+
+lemma Pair_Collect[simp]:
+ shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
+by simp
+
+text {* Myhill-Nerode relation *}
+
+definition
+ str_eq_rel :: "lang \<Rightarrow> (string \<times> string) 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"
+where
+ "finals A \<equiv> {\<approx>A `` {s} | s . s \<in> A}"
+
+lemma lang_is_union_of_finals:
+ shows "A = \<Union> finals A"
+unfolding finals_def
+unfolding Image_def
+unfolding str_eq_rel_def
+by (auto) (metis append_Nil2)
+
+lemma finals_in_partitions:
+ shows "finals A \<subseteq> (UNIV // \<approx>A)"
+unfolding finals_def quotient_def
+by auto
+
+section {* Equational systems *}
+
+text {* The two kinds of terms in the rhs of equations. *}
+
+datatype rhs_trm =
+ Lam "rexp" (* Lambda-marker *)
+ | Trn "lang" "rexp" (* Transition *)
+
+
+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
+
+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)
+
+lemma L_rhs_union_distrib:
+ fixes A B::"rhs_trm set"
+ shows "L A \<union> L B = L (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"
+
+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}
+ else
+ {Trn Y (CHAR 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> rhs_trm \<Rightarrow> rhs_trm"
+where
+ "Append_rexp r (Lam rexp) = Lam (SEQ rexp r)"
+| "Append_rexp r (Trn X rexp) = Trn X (SEQ 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}))"
+
+
+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> rhs_trm set) set \<Rightarrow> lang \<Rightarrow> rhs_trm set \<Rightarrow> (lang \<times> rhs_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>
+ Subst_all (ES - {(X, xrhs)}) X (Arden X xrhs)"
+
+
+section {* While-combinator *}
+
+definition
+ "Iter X ES \<equiv> (let (Y, yrhs) = SOME (Y, yrhs). (Y, yrhs) \<in> ES \<and> X \<noteq> Y
+ in Remove ES Y yrhs)"
+
+lemma IterI2:
+ assumes "(Y, yrhs) \<in> ES"
+ and "X \<noteq> Y"
+ and "\<And>Y yrhs. \<lbrakk>(Y, yrhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> Q (Remove ES Y yrhs)"
+ shows "Q (Iter X ES)"
+unfolding Iter_def using assms
+by (rule_tac a="(Y, yrhs)" in someI2) (auto)
+
+abbreviation
+ "Cond ES \<equiv> card ES \<noteq> 1"
+
+definition
+ "Solve X ES \<equiv> while Cond (Iter X) ES"
+
+
+section {* Invariants *}
+
+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"
+
+definition
+ "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
+
+definition
+ "ardenable_all ES \<equiv> \<forall>(X, rhs) \<in> ES. ardenable rhs"
+
+definition
+ "finite_rhs ES \<equiv> \<forall>(X, rhs) \<in> ES. finite rhs"
+
+lemma finite_rhs_def2:
+ "finite_rhs ES = (\<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs)"
+unfolding finite_rhs_def by auto
+
+definition
+ "rhss rhs \<equiv> {X | X r. Trn X r \<in> rhs}"
+
+definition
+ "lhss ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
+
+definition
+ "validity ES \<equiv> \<forall>(X, rhs) \<in> ES. rhss rhs \<subseteq> lhss ES"
+
+lemma rhss_union_distrib:
+ shows "rhss (A \<union> B) = rhss A \<union> rhss B"
+by (auto simp add: rhss_def)
+
+lemma lhss_union_distrib:
+ shows "lhss (A \<union> B) = lhss A \<union> lhss B"
+by (auto simp add: lhss_def)
+
+
+definition
+ "invariant ES \<equiv> finite ES
+ \<and> finite_rhs ES
+ \<and> soundness ES
+ \<and> distinctness ES
+ \<and> ardenable_all ES
+ \<and> validity ES"
+
+
+lemma invariantI:
+ assumes "soundness ES" "finite ES" "distinctness ES" "ardenable_all ES"
+ "finite_rhs ES" "validity ES"
+ shows "invariant ES"
+using assms by (simp add: invariant_def)
+
+
+subsection {* The proof of this direction *}
+
+lemma finite_Trn:
+ assumes fin: "finite rhs"
+ shows "finite {r. Trn Y r \<in> rhs}"
+proof -
+ have "finite {Trn Y r | Y r. Trn Y r \<in> rhs}"
+ by (rule rev_finite_subset[OF fin]) (auto)
+ then have "finite ((\<lambda>(Y, r). Trn Y r) ` {(Y, r) | Y r. Trn Y r \<in> rhs})"
+ by (simp add: image_Collect)
+ then have "finite {(Y, r) | Y r. Trn Y r \<in> rhs}"
+ by (erule_tac finite_imageD) (simp add: inj_on_def)
+ then show "finite {r. Trn Y r \<in> rhs}"
+ by (erule_tac f="snd" in finite_surj) (auto simp add: image_def)
+qed
+
+lemma finite_Lam:
+ assumes fin: "finite rhs"
+ shows "finite {r. Lam r \<in> rhs}"
+proof -
+ have "finite {Lam r | r. Lam r \<in> rhs}"
+ by (rule rev_finite_subset[OF fin]) (auto)
+ then show "finite {r. Lam r \<in> rhs}"
+ apply(simp add: image_Collect[symmetric])
+ apply(erule finite_imageD)
+ apply(auto simp add: inj_on_def)
+ done
+qed
+
+lemma rhs_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}))"
+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)
+qed
+
+lemma lang_of_append_rexp:
+ "L (Append_rexp r rhs_trm) = L rhs_trm ;; L 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"
+unfolding Append_rexp_rhs_def
+by (auto simp add: Seq_def lang_of_append_rexp)
+
+
+
+subsubsection {* Intialization *}
+
+lemma defined_by_str:
+ assumes "s \<in> X" "X \<in> UNIV // \<approx>A"
+ shows "X = \<approx>A `` {s}"
+using assms
+unfolding quotient_def Image_def str_eq_rel_def
+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"
+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"
+ unfolding Y_def Image_def Seq_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"
+proof
+ show "X \<subseteq> L 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
+ 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"
+ 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}"
+ unfolding transition_def
+ using decom by (force simp add: Seq_def)
+ then have "x \<in> L rhs" using X_in_eqs in_X
+ unfolding Init_def Init_rhs_def by simp
+ }
+ ultimately show "x \<in> L rhs" by blast
+ qed
+next
+ show "L 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)"
+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 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"
+ 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
+ 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)"
+ shows "invariant (Init (UNIV // \<approx>A))"
+proof (rule invariantI)
+ show "soundness (Init (UNIV // \<approx>A))"
+ unfolding soundness_def
+ using l_eq_r_in_eqs by auto
+ show "finite (Init (UNIV // \<approx>A))" using finite_CS
+ unfolding Init_def by simp
+ show "distinctness (Init (UNIV // \<approx>A))"
+ unfolding distinctness_def Init_def by simp
+ show "ardenable_all (Init (UNIV // \<approx>A))"
+ unfolding ardenable_all_def Init_def Init_rhs_def ardenable_def
+ by auto
+ show "finite_rhs (Init (UNIV // \<approx>A))"
+ using finite_Init_rhs[OF finite_CS]
+ unfolding finite_rhs_def Init_def by auto
+ show "validity (Init (UNIV // \<approx>A))"
+ unfolding validity_def Init_def Init_rhs_def rhss_def lhss_def
+ by auto
+qed
+
+subsubsection {* Interation step *}
+
+lemma Arden_keeps_eq:
+ assumes l_eq_r: "X = L rhs"
+ and not_empty: "ardenable rhs"
+ and finite: "finite rhs"
+ shows "X = L (Arden X rhs)"
+proof -
+ def A \<equiv> "L (\<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)"
+ 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"
+ unfolding b_def
+ unfolding rhs_trm_soundness[OF finite]
+ unfolding A_def
+ by blast
+ finally have "X = X ;; A \<union> B" .
+ then have "X = B ;; A\<star>"
+ by (simp add: arden[OF not_empty2])
+ also have "\<dots> = L (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
+qed
+
+lemma Append_keeps_finite:
+ "finite rhs \<Longrightarrow> finite (Append_rexp_rhs rhs r)"
+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)
+
+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)
+
+lemma nonempty_set_sub:
+ "ardenable rhs \<Longrightarrow> ardenable (rhs - A)"
+by (auto simp:ardenable_def)
+
+lemma nonempty_set_union:
+ "\<lbrakk>ardenable rhs; ardenable rhs'\<rbrakk> \<Longrightarrow> ardenable (rhs \<union> rhs')"
+by (auto simp:ardenable_def)
+
+lemma Arden_keeps_nonempty:
+ "ardenable rhs \<Longrightarrow> ardenable (Arden X rhs)"
+by (simp only:Arden_def Append_keeps_nonempty nonempty_set_sub)
+
+
+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"
+ and finite: "finite rhs"
+ shows "L (Subst rhs X xrhs) = L 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}))"
+ 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})"
+ 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})"
+ using finite substor by (simp only: lang_of_append_rexp_rhs 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)"
+by (auto simp: Subst_def Append_keeps_finite)
+
+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 h \<equiv> "\<lambda>(X::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
+qed
+
+lemma Subst_all_keeps_finite_rhs:
+ "\<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 (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+)
+
+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)
+
+lemma lhss_keeps_cls:
+ "lhss (Subst_all ES Y yrhs) = lhss ES"
+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)
+
+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 -
+ { fix X xrhs'
+ assume "(X, xrhs') \<in> ?B"
+ then obtain xrhs
+ where xrhs_xrhs': "xrhs' = Subst xrhs Y (Arden Y yrhs)"
+ and X_in: "(X, xrhs) \<in> ES" by (simp add:Subst_all_def, blast)
+ have "rhss xrhs' \<subseteq> lhss ?B"
+ 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-
+ have "Y \<notin> rhss (Arden Y yrhs)"
+ using Arden_removes_cl by simp
+ 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)
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by simp
+ qed
+ } thus ?thesis by (auto simp only:Subst_all_def validity_def)
+qed
+
+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"
+ 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)"
+ 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)
+ 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
+ unfolding distinctness_def Subst_all_def invariant_def by auto
+ show "ardenable_all (Subst_all ES Y (Arden Y yrhs))"
+ proof -
+ { fix X rhs
+ assume "(X, rhs) \<in> ES"
+ hence "ardenable rhs" using invariant_ES
+ by (auto simp add:invariant_def ardenable_all_def)
+ with nonempty_yrhs
+ have "ardenable (Subst rhs Y (Arden Y yrhs))"
+ by (simp add:nonempty_yrhs
+ Subst_keeps_nonempty Arden_keeps_nonempty)
+ } thus ?thesis by (auto simp add:ardenable_all_def Subst_all_def)
+ qed
+ show "finite_rhs (Subst_all ES Y (Arden Y yrhs))"
+ proof-
+ have "finite_rhs ES" 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
+ 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)
+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 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'"
+ unfolding ES'_def using finite by (auto intro: card_image_le)
+ also have "\<dots> < card ES" unfolding ES'_def
+ using in_ES finite by (rule_tac card_Diff1_less)
+ finally show "(Remove ES X rhs, ES) \<in> measure card"
+ unfolding Remove_def ES'_def by simp
+qed
+
+
+lemma Subst_all_cls_remains:
+ "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (Subst_all ES Y yrhs)"
+by (auto simp: Subst_all_def)
+
+lemma card_noteq_1_has_more:
+ assumes card:"Cond ES"
+ and e_in: "(X, xrhs) \<in> ES"
+ and finite: "finite ES"
+ shows "\<exists>(Y, yrhs) \<in> ES. (X, xrhs) \<noteq> (Y, yrhs)"
+proof-
+ have "card ES > 1" using card e_in finite
+ by (cases "card ES") (auto)
+ then have "card (ES - {(X, xrhs)}) > 0"
+ using finite e_in by auto
+ then have "(ES - {(X, xrhs)}) \<noteq> {}" using finite by (rule_tac notI, simp)
+ then show "\<exists>(Y, yrhs) \<in> ES. (X, xrhs) \<noteq> (Y, yrhs)"
+ by auto
+qed
+
+lemma iteration_step_measure:
+ assumes Inv_ES: "invariant ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and Cnd: "Cond ES "
+ shows "(Iter X ES, ES) \<in> measure card"
+proof -
+ have fin: "finite ES" using Inv_ES unfolding invariant_def by simp
+ then obtain Y yrhs
+ where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
+ using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+ then have "(Y, yrhs) \<in> ES " "X \<noteq> Y"
+ using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+ by auto
+ then show "(Iter X ES, ES) \<in> measure card"
+ apply(rule IterI2)
+ apply(rule Remove_in_card_measure)
+ apply(simp_all add: fin)
+ done
+qed
+
+lemma iteration_step_invariant:
+ assumes Inv_ES: "invariant ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and Cnd: "Cond ES"
+ shows "invariant (Iter X ES)"
+proof -
+ have finite_ES: "finite ES" using Inv_ES by (simp add: invariant_def)
+ then obtain Y yrhs
+ where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
+ using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+ then have "(Y, yrhs) \<in> ES" "X \<noteq> Y"
+ using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+ by auto
+ then show "invariant (Iter X ES)"
+ proof(rule IterI2)
+ fix Y yrhs
+ assume h: "(Y, yrhs) \<in> ES" "X \<noteq> Y"
+ then have "ES - {(Y, yrhs)} \<union> {(Y, yrhs)} = ES" by auto
+ then show "invariant (Remove ES Y yrhs)" unfolding Remove_def
+ using Inv_ES
+ by (rule_tac Subst_all_satisfies_invariant) (simp)
+ qed
+qed
+
+lemma iteration_step_ex:
+ assumes Inv_ES: "invariant ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and Cnd: "Cond ES"
+ shows "\<exists>xrhs'. (X, xrhs') \<in> (Iter X ES)"
+proof -
+ have finite_ES: "finite ES" using Inv_ES by (simp add: invariant_def)
+ then obtain Y yrhs
+ where "(Y, yrhs) \<in> ES" "(X, xrhs) \<noteq> (Y, yrhs)"
+ using Cnd X_in_ES by (drule_tac card_noteq_1_has_more) (auto)
+ then have "(Y, yrhs) \<in> ES " "X \<noteq> Y"
+ using X_in_ES Inv_ES unfolding invariant_def distinctness_def
+ by auto
+ then show "\<exists>xrhs'. (X, xrhs') \<in> (Iter X ES)"
+ apply(rule IterI2)
+ unfolding Remove_def
+ apply(rule Subst_all_cls_remains)
+ using X_in_ES
+ apply(auto)
+ done
+qed
+
+
+subsubsection {* Conclusion of the proof *}
+
+lemma Solve:
+ 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)"
+ have "Inv (Init (UNIV // \<approx>A))" unfolding Inv_def
+ using fin X_in by (simp add: Init_ES_satisfies_invariant, simp add: Init_def)
+ moreover
+ { fix ES
+ assume inv: "Inv ES" and crd: "Cond ES"
+ then have "Inv (Iter X ES)"
+ unfolding Inv_def
+ by (auto simp add: iteration_step_invariant iteration_step_ex) }
+ moreover
+ { fix ES
+ assume inv: "Inv ES" and not_crd: "\<not>Cond ES"
+ from inv obtain rhs where "(X, rhs) \<in> ES" unfolding Inv_def by auto
+ moreover
+ from not_crd have "card ES = 1" by simp
+ ultimately
+ have "ES = {(X, rhs)}" by (auto simp add: card_Suc_eq)
+ then have "\<exists>rhs'. ES = {(X, rhs')} \<and> invariant {(X, rhs')}" using inv
+ unfolding Inv_def by auto }
+ moreover
+ have "wf (measure card)" by simp
+ moreover
+ { fix ES
+ assume inv: "Inv ES" and crd: "Cond ES"
+ then have "(Iter X ES, ES) \<in> measure card"
+ unfolding Inv_def
+ apply(clarify)
+ apply(rule_tac iteration_step_measure)
+ apply(auto)
+ done }
+ ultimately
+ 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:
+ assumes finite_CS: "finite (UNIV // \<approx>A)"
+ and X_in_CS: "X \<in> (UNIV // \<approx>A)"
+ shows "\<exists>r::rexp. X = L r"
+proof -
+ from finite_CS X_in_CS
+ obtain xrhs where Inv_ES: "invariant {(X, xrhs)}"
+ using Solve by metis
+
+ def A \<equiv> "Arden X xrhs"
+ have "rhss xrhs \<subseteq> {X}" using Inv_ES
+ unfolding validity_def invariant_def rhss_def lhss_def
+ by auto
+ then have "rhss A = {}" unfolding A_def
+ by (simp add: Arden_removes_cl)
+ then have eq: "{Lam r | r. Lam r \<in> A} = A" unfolding rhss_def
+ by (auto, case_tac x, auto)
+
+ 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
+ by simp
+ then have "X = L 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
+qed
+
+lemma bchoice_finite_set:
+ assumes a: "\<forall>x \<in> S. \<exists>y. x = f y"
+ and b: "finite S"
+ shows "\<exists>ys. (\<Union> S) = \<Union>(f ` ys) \<and> finite ys"
+using bchoice[OF a] b
+apply(erule_tac exE)
+apply(rule_tac x="fa ` S" in exI)
+apply(auto)
+done
+
+theorem Myhill_Nerode1:
+ assumes finite_CS: "finite (UNIV // \<approx>A)"
+ shows "\<exists>r::rexp. A = L 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"
+ using finite_CS every_eqcl_has_reg by blast
+ then have a: "\<forall>X \<in> finals A. \<exists>r::rexp. X = L r"
+ using finals_in_partitions by auto
+ then obtain rs::"rexp set" where "\<Union> (finals A) = \<Union>(L ` rs)" "finite rs"
+ using fin by (auto dest: bchoice_finite_set)
+ then have "A = L (\<Uplus>rs)"
+ unfolding lang_is_union_of_finals[symmetric] by simp
+ then show "\<exists>r::rexp. A = L r" by blast
+qed
+
+
+end
\ No newline at end of file