(* Author: Xingyuan Zhang, Chunhan Wu, Christian Urban *)

theory Myhill_1

imports "Folds"

        "~~/src/HOL/Library/While_Combinator" 

begin

section {* First direction of MN: @{text "finite partition \<Rightarrow> regular language"} *}

notation 

  conc (infixr "\<cdot>" 100) and

  star ("_\<star>" [101] 102)

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 :: "'a lang \<Rightarrow> ('a list \<times> 'a list) set" ("\<approx>_" [100] 100)

where

  "\<approx>A \<equiv> {(x, y).  (\<forall>z. x @ z \<in> A \<longleftrightarrow> y @ z \<in> A)}"

abbreviation

  str_eq_applied :: "'a list \<Rightarrow> 'a lang \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<approx>_ _")

where

  "x \<approx>A y \<equiv> (x, y) \<in> \<approx>A"

definition 

  finals :: "'a lang \<Rightarrow> 'a 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_def

by (auto) (metis append_Nil2)

lemma finals_in_partitions:

  shows "finals A \<subseteq> (UNIV // \<approx>A)"

unfolding finals_def quotient_def

by auto

subsection {* Equational systems *}

text {* The two kinds of terms in the rhs of equations. *}

datatype 'a trm = 

   Lam "'a rexp"            (* Lambda-marker *)

 | Trn "'a lang" "'a rexp"     (* Transition *)

fun 

  lang_trm::"'a trm \<Rightarrow> 'a lang"

where

  "lang_trm (Lam r) = lang r" 

| "lang_trm (Trn X r) = X \<cdot> lang r"

fun 

  lang_rhs::"('a trm) set \<Rightarrow> 'a lang"

where 

  "lang_rhs rhs = \<Union> (lang_trm ` rhs)"

lemma lang_rhs_set:

  shows "lang_rhs {Trn X r | r. P r} = \<Union>{lang_trm (Trn X r) | r. P r}"

by (auto)

lemma lang_rhs_union_distrib:

  shows "lang_rhs A \<union> lang_rhs B = lang_rhs (A \<union> B)"

by simp

text {* Transitions between equivalence classes *}

definition 

  transition :: "'a lang \<Rightarrow> 'a \<Rightarrow> 'a 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 One} \<union> {Trn Y (Atom c) | Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow> X}

      else 

          {Trn Y (Atom 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}"

subsection {* Arden Operation on equations *}

fun 

  Append_rexp :: "'a rexp \<Rightarrow> 'a trm \<Rightarrow> 'a trm"

where

  "Append_rexp r (Lam rexp)   = Lam (Times rexp r)"

| "Append_rexp r (Trn X rexp) = Trn X (Times 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}))"

subsection {* 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 :: "('a lang \<times> ('a trm) set) set \<Rightarrow> 'a lang \<Rightarrow> ('a trm) set \<Rightarrow> ('a lang \<times> ('a 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)"

subsection {* While-combinator and invariants *}

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"

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 = lang_rhs rhs"

definition 

  "ardenable rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> lang 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)

lemma finite_Trn:

  assumes fin: "finite rhs"

  shows "finite {r. Trn Y r \<in> rhs}"

using assms by (auto intro!: finite_vimageI simp add: inj_on_def)

lemma finite_Lam:

  assumes fin: "finite rhs"

  shows "finite {r. Lam r \<in> rhs}"

using assms by (auto intro!: finite_vimageI simp add: inj_on_def)

lemma trm_soundness:

  assumes finite:"finite rhs"

  shows "lang_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (lang (\<Uplus>{r. Trn X r \<in> rhs}))"

proof -

  have "finite {r. Trn X r \<in> rhs}" 

    by (rule finite_Trn[OF finite]) 

  then show "lang_rhs ({Trn X r| r. Trn X r \<in> rhs}) = X \<cdot> (lang (\<Uplus>{r. Trn X r \<in> rhs}))"

    by (simp only: lang_rhs_set lang_trm.simps) (auto simp add: conc_def)

qed

lemma lang_of_append_rexp:

  "lang_trm (Append_rexp r trm) = lang_trm trm \<cdot> lang r"

by (induct rule: Append_rexp.induct)

   (auto simp add: conc_assoc)

lemma lang_of_append_rexp_rhs:

  "lang_rhs (Append_rexp_rhs rhs r) = lang_rhs rhs \<cdot> lang r"

unfolding Append_rexp_rhs_def

by (auto simp add: conc_def lang_of_append_rexp)

subsection {* Intial Equational Systems *}

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_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 \<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 conc_def

    unfolding str_eq_def

    by clarsimp

  moreover

  have "s \<in> Y" unfolding Y_def 

    unfolding Image_def str_eq_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 = lang_rhs rhs"

proof 

  show "X \<subseteq> lang_rhs rhs"

  proof

    fix x

    assume in_X: "x \<in> X"

    { assume empty: "x = []"

      then have "x \<in> lang_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 metis

      then have "x \<in> lang_rhs {Trn Y (Atom c)| Y c. Y \<in> UNIV // \<approx>A \<and> Y \<Turnstile>c\<Rightarrow> X}"

        unfolding transition_def

        using decom by (force simp add: conc_def)

      then have "x \<in> lang_rhs rhs" using X_in_eqs in_X

        unfolding Init_def Init_rhs_def by simp

    }

    ultimately show "x \<in> lang_rhs rhs" by blast

  qed

next

  show "lang_rhs rhs \<subseteq> X" using X_in_eqs

    unfolding Init_def Init_rhs_def transition_def

    by auto 

qed

lemma finite_Init_rhs: 

  fixes CS::"(('a::finite) lang) set"

  assumes finite: "finite CS"

  shows "finite (Init_rhs CS X)"

using assms unfolding Init_rhs_def transition_def by simp

lemma Init_ES_satisfies_invariant:

  fixes A::"(('a::finite) lang)"

  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

subsection {* Interations *}

lemma Arden_preserves_soundness:

  assumes l_eq_r: "X = lang_rhs rhs"

  and not_empty: "ardenable rhs"

  and finite: "finite rhs"

  shows "X = lang_rhs (Arden X rhs)"

proof -

  def A \<equiv> "lang (\<Uplus>{r. Trn X r \<in> rhs})"

  def b \<equiv> "{Trn X r | r. Trn X r \<in> rhs}"

  def B \<equiv> "lang_rhs (rhs - b)"

  have not_empty2: "[] \<notin> A" 

    using finite_Trn[OF finite] not_empty

    unfolding A_def ardenable_def by simp

  have "X = lang_rhs rhs" using l_eq_r by simp

  also have "\<dots> = lang_rhs (b \<union> (rhs - b))" unfolding b_def by auto

  also have "\<dots> = lang_rhs b \<union> B" unfolding B_def by (simp only: lang_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: reversed_Arden[OF not_empty2])

  also have "\<dots> = lang_rhs (Arden X rhs)"

    unfolding Arden_def A_def B_def b_def

    by (simp only: lang_of_append_rexp_rhs lang.simps)

  finally show "X = lang_rhs (Arden X rhs)" by simp

qed 

lemma Append_preserves_finite:

  "finite rhs \<Longrightarrow> finite (Append_rexp_rhs rhs r)"

by (auto simp: Append_rexp_rhs_def)

lemma Arden_preserves_finite:

  "finite rhs \<Longrightarrow> finite (Arden X rhs)"

by (auto simp: Arden_def Append_preserves_finite)

lemma Append_preserves_ardenable:

  "ardenable rhs \<Longrightarrow> ardenable (Append_rexp_rhs rhs r)"

apply (auto simp: ardenable_def Append_rexp_rhs_def)

by (case_tac x, auto simp: conc_def)

lemma ardenable_set_sub:

  "ardenable rhs \<Longrightarrow> ardenable (rhs - A)"

by (auto simp:ardenable_def)

lemma ardenable_set_union:

  "\<lbrakk>ardenable rhs; ardenable rhs'\<rbrakk> \<Longrightarrow> ardenable (rhs \<union> rhs')"

by (auto simp:ardenable_def)

lemma Arden_preserves_ardenable:

  "ardenable rhs \<Longrightarrow> ardenable (Arden X rhs)"

by (simp only:Arden_def Append_preserves_ardenable ardenable_set_sub)

lemma Subst_preserves_ardenable:

  "\<lbrakk>ardenable rhs; ardenable xrhs\<rbrakk> \<Longrightarrow> ardenable (Subst rhs X xrhs)"

by (simp only: Subst_def Append_preserves_ardenable ardenable_set_union ardenable_set_sub)

lemma Subst_preserves_soundness:

  assumes substor: "X = lang_rhs xrhs"

  and finite: "finite rhs"

  shows "lang_rhs (Subst rhs X xrhs) = lang_rhs rhs" (is "?Left = ?Right")

proof-

  def A \<equiv> "lang_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs})"

  have "?Left = A \<union> lang_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs}))"

    unfolding Subst_def

    unfolding lang_rhs_union_distrib[symmetric]

    by (simp add: A_def)

  moreover have "?Right = A \<union> lang_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 lang_rhs_union_distrib

      by simp

  qed

  moreover 

  have "lang_rhs (Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})) = lang_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 simp

qed

lemma Subst_preserves_finite_rhs:

  "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (Subst rhs Y yrhs)"

by (auto simp: Subst_def Append_preserves_finite)

lemma Subst_all_preserves_finite:

  assumes finite: "finite ES"

  shows "finite (Subst_all ES Y yrhs)"

using assms unfolding Subst_all_def by simp

lemma Subst_all_preserves_finite_rhs:

  "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (Subst_all ES Y yrhs)"

by (auto intro:Subst_preserves_finite_rhs simp add:Subst_all_def finite_rhs_def)

lemma append_rhs_preserves_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 = "Times 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_preserves_cls)

by (auto simp: rhss_def)

lemma lhss_preserves_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_preserves_cls rhss_union_distrib)

by (auto simp: rhss_def)

lemma Subst_all_preserves_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 auto

          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)})"