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 -