theory Myhill_1imports Regular "~~/src/HOL/Library/While_Combinator" beginsection {* 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 simptext {* 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_defunfolding Image_defunfolding str_eq_rel_defby (auto) (metis append_Nil2)lemma finals_in_partitions: shows "finals A \<subseteq> (UNIV // \<approx>A)"unfolding finals_def quotient_defby autosection {* Equational systems *}text {* The two kinds of terms in the rhs of equations. *}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"fun L_rhs::"trm set \<Rightarrow> lang"where "L_rhs rhs = \<Union> (L_trm ` rhs)"lemma L_rhs_set: 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::"trm set" shows "L_rhs A \<union> L_rhs B = L_rhs (A \<union> B)"by simptext {* 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 \<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} 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> 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 "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> 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> 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 assmsby (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 rhs"definition "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 "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 autodefinition "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)qedlemma 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) doneqedlemma 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}))"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)qedlemma lang_of_append_rexp: "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_rhs (Append_rexp_rhs rhs r) = L_rhs rhs \<cdot> L_rexp r"unfolding Append_rexp_rhs_defby (auto simp add: Seq_def lang_of_append_rexp)subsubsection {* Intial Equational System *}lemma defined_by_str: assumes "s \<in> X" "X \<in> UNIV // \<approx>A" shows "X = \<approx>A `` {s}"using assmsunfolding quotient_def Image_def str_eq_rel_defby autolemma 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 \<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 \<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 unfolding Image_def str_eq_rel_def by simp ultimately show thesis using that by blastqedlemma l_eq_r_in_eqs: assumes X_in_eqs: "(X, rhs) \<in> Init (UNIV // \<approx>A)" shows "X = L_rhs rhs"proof show "X \<subseteq> L_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 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 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 rhs" using X_in_eqs in_X unfolding Init_def Init_rhs_def by simp } ultimately show "x \<in> L_rhs rhs" by blast qednext show "L_rhs rhs \<subseteq> X" using X_in_eqs unfolding Init_def Init_rhs_def transition_def by auto qedlemma 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: 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 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" 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 then show "finite (Init_rhs CS X)" unfolding Init_rhs_def transition_def by simpqedlemma 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 autoqedsubsubsection {* Interation step *}lemma Arden_keeps_eq: assumes l_eq_r: "X = L_rhs rhs" and not_empty: "ardenable rhs" and finite: "finite rhs" shows "X = L_rhs (Arden X rhs)"proof - 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 (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) 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)" 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 simpqed 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_rhs xrhs" and finite: "finite rhs" shows "L_rhs (Subst rhs X xrhs) = L_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}))" unfolding Subst_def unfolding L_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}" 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_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 simpqedlemma 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 simpqedlemma 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)qedlemma 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" 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)" 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)qedlemma 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 simpqedlemma 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 autoqedlemma 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) doneqedlemma 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) qedqedlemma 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) doneqedsubsubsection {* 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)qedlemma every_eqcl_has_reg: assumes finite_CS: "finite (UNIV // \<approx>A)" and X_in_CS: "X \<in> (UNIV // \<approx>A)" 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 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_rhs xrhs" using Inv_ES unfolding invariant_def soundness_def by simp 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_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 blastqedlemma 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] bapply(erule_tac exE)apply(rule_tac x="fa ` S" in exI)apply(auto)donetheorem Myhill_Nerode1: assumes finite_CS: "finite (UNIV // \<approx>A)" 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. X = L_rexp r" using finite_CS every_eqcl_has_reg by blast 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_rexp ` rs)" "finite rs" using fin by (auto dest: bchoice_finite_set) then have "A = L_rexp (\<Uplus>rs)" unfolding lang_is_union_of_finals[symmetric] by simp then show "\<exists>r. A = L_rexp r" by blastqed end