--- a/Theories/Myhill_1.thy Tue May 31 20:32:49 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,783 +0,0 @@
-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