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

theory Myhill_2

  imports Myhill_1 List_Prefix

begin

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

subsection {* Tagging functions *}

definition 

   tag_eq :: "('a list \<Rightarrow> 'b) \<Rightarrow> ('a list \<times> 'a list) set" ("=_=")

where

   "=tag= \<equiv> {(x, y). tag x = tag y}"

abbreviation

   tag_eq_applied :: "'a list \<Rightarrow> ('a list \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> bool" ("_ =_= _")

where

   "x =tag= y \<equiv> (x, y) \<in> =tag="

lemma [simp]:

  shows "(\<approx>A) `` {x} = (\<approx>A) `` {y} \<longleftrightarrow> x \<approx>A y"

unfolding str_eq_def by auto

lemma refined_intro:

  assumes "\<And>x y z. \<lbrakk>x =tag= y; x @ z \<in> A\<rbrakk> \<Longrightarrow> y @ z \<in> A"

  shows "=tag= \<subseteq> \<approx>A"

using assms unfolding str_eq_def tag_eq_def

apply(clarify, simp (no_asm_use))

by metis

lemma finite_eq_tag_rel:

  assumes rng_fnt: "finite (range tag)"

  shows "finite (UNIV // =tag=)"

proof -

  let "?f" =  "\<lambda>X. tag ` X" and ?A = "(UNIV // =tag=)"

  have "finite (?f ` ?A)" 

  proof -

    have "range ?f \<subseteq> (Pow (range tag))" unfolding Pow_def by auto

    moreover 

    have "finite (Pow (range tag))" using rng_fnt by simp

    ultimately 

    have "finite (range ?f)" unfolding image_def by (blast intro: finite_subset)

    moreover

    have "?f ` ?A \<subseteq> range ?f" by auto

    ultimately show "finite (?f ` ?A)" by (rule rev_finite_subset) 

  qed

  moreover

  have "inj_on ?f ?A"

  proof -

    { fix X Y

      assume X_in: "X \<in> ?A"

        and  Y_in: "Y \<in> ?A"

        and  tag_eq: "?f X = ?f Y"

      then obtain x y 

        where "x \<in> X" "y \<in> Y" "tag x = tag y"

        unfolding quotient_def Image_def image_def tag_eq_def

        by (simp) (blast)

      with X_in Y_in 

      have "X = Y"

        unfolding quotient_def tag_eq_def by auto

    } 

    then show "inj_on ?f ?A" unfolding inj_on_def by auto

  qed

  ultimately show "finite (UNIV // =tag=)" by (rule finite_imageD)

qed

lemma refined_partition_finite:

  assumes fnt: "finite (UNIV // R1)"

  and refined: "R1 \<subseteq> R2"

  and eq1: "equiv UNIV R1" and eq2: "equiv UNIV R2"

  shows "finite (UNIV // R2)"

proof -

  let ?f = "\<lambda>X. {R1 `` {x} | x. x \<in> X}" 

    and ?A = "UNIV // R2" and ?B = "UNIV // R1"

  have "?f ` ?A \<subseteq> Pow ?B"

    unfolding image_def Pow_def quotient_def by auto

  moreover

  have "finite (Pow ?B)" using fnt by simp

  ultimately  

  have "finite (?f ` ?A)" by (rule finite_subset)

  moreover

  have "inj_on ?f ?A"

  proof -

    { fix X Y

      assume X_in: "X \<in> ?A" and Y_in: "Y \<in> ?A" and eq_f: "?f X = ?f Y"

      from quotientE [OF X_in]

      obtain x where "X = R2 `` {x}" by blast

      with equiv_class_self[OF eq2] have x_in: "x \<in> X" by simp

      then have "R1 ``{x} \<in> ?f X" by auto

      with eq_f have "R1 `` {x} \<in> ?f Y" by simp

      then obtain y 

        where y_in: "y \<in> Y" and eq_r1_xy: "R1 `` {x} = R1 `` {y}" by auto

      with eq_equiv_class[OF _ eq1] 

      have "(x, y) \<in> R1" by blast

      with refined have "(x, y) \<in> R2" by auto

      with quotient_eqI [OF eq2 X_in Y_in x_in y_in]

      have "X = Y" .

    } 

    then show "inj_on ?f ?A" unfolding inj_on_def by blast 

  qed

  ultimately show "finite (UNIV // R2)" by (rule finite_imageD)

qed

lemma tag_finite_imageD:

  assumes rng_fnt: "finite (range tag)" 

  and     refined: "=tag=  \<subseteq> \<approx>A"

  shows "finite (UNIV // \<approx>A)"

proof (rule_tac refined_partition_finite [of "=tag="])

  show "finite (UNIV // =tag=)" by (rule finite_eq_tag_rel[OF rng_fnt])

next

  show "=tag= \<subseteq> \<approx>A" using refined .

next

  show "equiv UNIV =tag="

  and  "equiv UNIV (\<approx>A)" 

    unfolding equiv_def str_eq_def tag_eq_def refl_on_def sym_def trans_def

    by auto

qed

subsection {* Base cases: @{const Zero}, @{const One} and @{const Atom} *}

lemma quot_zero_eq:

  shows "UNIV // \<approx>{} = {UNIV}"

unfolding quotient_def Image_def str_eq_def by auto

lemma quot_zero_finiteI [intro]:

  shows "finite (UNIV // \<approx>{})"

unfolding quot_zero_eq by simp

lemma quot_one_subset:

  shows "UNIV // \<approx>{[]} \<subseteq> {{[]}, UNIV - {[]}}"

proof

  fix x

  assume "x \<in> UNIV // \<approx>{[]}"

  then obtain y where h: "x = {z. y \<approx>{[]} z}" 

    unfolding quotient_def Image_def by blast

  { assume "y = []"

    with h have "x = {[]}" by (auto simp: str_eq_def)

    then have "x \<in> {{[]}, UNIV - {[]}}" by simp }

  moreover

  { assume "y \<noteq> []"

    with h have "x = UNIV - {[]}" by (auto simp: str_eq_def)

    then have "x \<in> {{[]}, UNIV - {[]}}" by simp }

  ultimately show "x \<in> {{[]}, UNIV - {[]}}" by blast

qed

lemma quot_one_finiteI [intro]:

  shows "finite (UNIV // \<approx>{[]})"

by (rule finite_subset[OF quot_one_subset]) (simp)

lemma quot_atom_subset:

  "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"

proof 

  fix x 

  assume "x \<in> UNIV // \<approx>{[c]}"

  then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[c]}}" 

    unfolding quotient_def Image_def by blast

  show "x \<in> {{[]},{[c]}, UNIV - {[], [c]}}"

  proof -

    { assume "y = []" hence "x = {[]}" using h 

        by (auto simp: str_eq_def) } 

    moreover 

    { assume "y = [c]" hence "x = {[c]}" using h 

        by (auto dest!: spec[where x = "[]"] simp: str_eq_def) } 

    moreover 

    { assume "y \<noteq> []" and "y \<noteq> [c]"

      hence "\<forall> z. (y @ z) \<noteq> [c]" by (case_tac y, auto)

      moreover have "\<And> p. (p \<noteq> [] \<and> p \<noteq> [c]) = (\<forall> q. p @ q \<noteq> [c])" 

        by (case_tac p, auto)

      ultimately have "x = UNIV - {[],[c]}" using h

        by (auto simp add: str_eq_def)

    } 

    ultimately show ?thesis by blast

  qed

qed

lemma quot_atom_finiteI [intro]:

  shows "finite (UNIV // \<approx>{[c]})"

by (rule finite_subset[OF quot_atom_subset]) (simp)

subsection {* Case for @{const Plus} *}

definition 

  tag_Plus :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a list \<Rightarrow> ('a lang \<times> 'a lang)"

where

  "tag_Plus A B \<equiv> \<lambda>x. (\<approx>A `` {x}, \<approx>B `` {x})"

lemma quot_plus_finiteI [intro]:

  assumes finite1: "finite (UNIV // \<approx>A)"

  and     finite2: "finite (UNIV // \<approx>B)"

  shows "finite (UNIV // \<approx>(A \<union> B))"

proof (rule_tac tag = "tag_Plus A B" in tag_finite_imageD)

  have "finite ((UNIV // \<approx>A) \<times> (UNIV // \<approx>B))" 

    using finite1 finite2 by auto

  then show "finite (range (tag_Plus A B))"

    unfolding tag_Plus_def quotient_def

    by (rule rev_finite_subset) (auto)

next

  show "=tag_Plus A B= \<subseteq> \<approx>(A \<union> B)"

    unfolding tag_eq_def tag_Plus_def str_eq_def by auto

qed

subsection {* Case for @{text "Times"} *}

definition

  "Partitions x \<equiv> {(x\<^sub>p, x\<^sub>s). x\<^sub>p @ x\<^sub>s = x}"

lemma conc_partitions_elim:

  assumes "x \<in> A \<cdot> B"

  shows "\<exists>(u, v) \<in> Partitions x. u \<in> A \<and> v \<in> B"

using assms unfolding conc_def Partitions_def

by auto

lemma conc_partitions_intro:

  assumes "(u, v) \<in> Partitions x \<and> u \<in> A \<and>  v \<in> B"

  shows "x \<in> A \<cdot> B"

using assms unfolding conc_def Partitions_def

by auto

lemma equiv_class_member:

  assumes "x \<in> A"

  and "\<approx>A `` {x} = \<approx>A `` {y}" 

  shows "y \<in> A"

using assms

apply(simp)

apply(simp add: str_eq_def)

apply(metis append_Nil2)

done

definition 

  tag_Times :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a list \<Rightarrow> 'a lang \<times> 'a lang set"

where

  "tag_Times A B \<equiv> \<lambda>x. (\<approx>A `` {x}, {(\<approx>B `` {x\<^sub>s}) | x\<^sub>p x\<^sub>s. x\<^sub>p \<in> A \<and> (x\<^sub>p, x\<^sub>s) \<in> Partitions x})"

lemma tag_Times_injI:

  assumes a: "tag_Times A B x = tag_Times A B y"

  and     c: "x @ z \<in> A \<cdot> B"

  shows "y @ z \<in> A \<cdot> B"

proof -

  from c obtain u v where 

    h1: "(u, v) \<in> Partitions (x @ z)" and

    h2: "u \<in> A" and

    h3: "v \<in> B" by (auto dest: conc_partitions_elim)

  from h1 have "x @ z = u @ v" unfolding Partitions_def by simp

  then obtain us 

    where "(x = u @ us \<and> us @ z = v) \<or> (x @ us = u \<and> z = us @ v)"

    by (auto simp add: append_eq_append_conv2)

  moreover

  { assume eq: "x = u @ us" "us @ z = v"

    have "(\<approx>B `` {us}) \<in> snd (tag_Times A B x)"

      unfolding Partitions_def tag_Times_def using h2 eq 

      by (auto simp add: str_eq_def)

    then have "(\<approx>B `` {us}) \<in> snd (tag_Times A B y)"

      using a by simp

    then obtain u' us' where

      q1: "u' \<in> A" and

      q2: "\<approx>B `` {us} = \<approx>B `` {us'}" and

      q3: "(u', us') \<in> Partitions y" 

      unfolding tag_Times_def by auto

    from q2 h3 eq 

    have "us' @ z \<in> B"

      unfolding Image_def str_eq_def by auto

    then have "y @ z \<in> A \<cdot> B" using q1 q3 

      unfolding Partitions_def by auto

  }

  moreover

  { assume eq: "x @ us = u" "z = us @ v"

    have "(\<approx>A `` {x}) = fst (tag_Times A B x)" 

      by (simp add: tag_Times_def)

    then have "(\<approx>A `` {x}) = fst (tag_Times A B y)"

      using a by simp

    then have "\<approx>A `` {x} = \<approx>A `` {y}" 

      by (simp add: tag_Times_def)

    moreover 

    have "x @ us \<in> A" using h2 eq by simp

    ultimately 

    have "y @ us \<in> A" using equiv_class_member 

      unfolding Image_def str_eq_def by blast

    then have "(y @ us) @ v \<in> A \<cdot> B" 

      using h3 unfolding conc_def by blast

    then have "y @ z \<in> A \<cdot> B" using eq by simp 

  }

  ultimately show "y @ z \<in> A \<cdot> B" by blast

qed

lemma quot_conc_finiteI [intro]:

  assumes fin1: "finite (UNIV // \<approx>A)" 

  and     fin2: "finite (UNIV // \<approx>B)" 

  shows "finite (UNIV // \<approx>(A \<cdot> B))"

proof (rule_tac tag = "tag_Times A B" in tag_finite_imageD)

  have "\<And>x y z. \<lbrakk>tag_Times A B x = tag_Times A B y; x @ z \<in> A \<cdot> B\<rbrakk> \<Longrightarrow> y @ z \<in> A \<cdot> B"

    by (rule tag_Times_injI)

       (auto simp add: tag_Times_def tag_eq_def)

  then show "=tag_Times A B= \<subseteq> \<approx>(A \<cdot> B)"

    by (rule refined_intro)

       (auto simp add: tag_eq_def)

next

  have *: "finite ((UNIV // \<approx>A) \<times> (Pow (UNIV // \<approx>B)))" 

    using fin1 fin2 by auto

  show "finite (range (tag_Times A B))" 

    unfolding tag_Times_def

    apply(rule finite_subset[OF _ *])

    unfolding quotient_def

    by auto

qed

subsection {* Case for @{const "Star"} *}

lemma star_partitions_elim:

  assumes "x @ z \<in> A\<star>" "x \<noteq> []"

  shows "\<exists>(u, v) \<in> Partitions (x @ z). u < x \<and> u \<in> A\<star> \<and> v \<in> A\<star>"

proof -

  have "([], x @ z) \<in> Partitions (x @ z)" "[] < x" "[] \<in> A\<star>" "x @ z \<in> A\<star>"

    using assms by (auto simp add: Partitions_def strict_prefix_def)

  then show "\<exists>(u, v) \<in> Partitions (x @ z). u < x \<and> u \<in> A\<star> \<and> v \<in> A\<star>"

    by blast

qed

lemma finite_set_has_max2: 

  "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists> max \<in> A. \<forall> a \<in> A. length a \<le> length max"

apply(induct rule:finite.induct)

apply(simp)

by (metis (hide_lams, no_types) all_not_in_conv insert_iff linorder_le_cases order_trans)

lemma finite_strict_prefix_set: 

  shows "finite {xa. xa < (x::'a list)}"

apply (induct x rule:rev_induct, simp)

apply (subgoal_tac "{xa. xa < xs @ [x]} = {xa. xa < xs} \<union> {xs}")

by (auto simp:strict_prefix_def)

lemma append_eq_cases:

  assumes a: "x @ y = m @ n" "m \<noteq> []"  

  shows "x \<le> m \<or> m < x"

unfolding prefix_def strict_prefix_def using a

by (auto simp add: append_eq_append_conv2)

lemma star_spartitions_elim2:

  assumes a: "x @ z \<in> A\<star>" 

  and     b: "x \<noteq> []"

  shows "\<exists>(u, v) \<in> Partitions x. \<exists> (u', v') \<in> Partitions z. u < x \<and> u \<in> A\<star> \<and> v @ u' \<in> A \<and> v' \<in> A\<star>"

proof -

  def S \<equiv> "{u | u v. (u, v) \<in> Partitions x \<and> u < x \<and> u \<in> A\<star> \<and> v @ z \<in> A\<star>}"

  have "finite {u. u < x}" by (rule finite_strict_prefix_set)

  then have "finite S" unfolding S_def

    by (rule rev_finite_subset) (auto)

  moreover 

  have "S \<noteq> {}" using a b unfolding S_def Partitions_def

    by (auto simp: strict_prefix_def)

  ultimately have "\<exists> u_max \<in> S. \<forall> u \<in> S. length u \<le> length u_max"  

    using finite_set_has_max2 by blast

  then obtain u_max v 

    where h0: "(u_max, v) \<in> Partitions x"

    and h1: "u_max < x" 

    and h2: "u_max \<in> A\<star>" 

    and h3: "v @ z \<in> A\<star>"  

    and h4: "\<forall> u v. (u, v) \<in> Partitions x \<and> u < x \<and> u \<in> A\<star> \<and> v @ z \<in> A\<star> \<longrightarrow> length u \<le> length u_max"

    unfolding S_def Partitions_def by blast

  have q: "v \<noteq> []" using h0 h1 b unfolding Partitions_def by auto

  from h3 obtain a b

    where i1: "(a, b) \<in> Partitions (v @ z)"

    and   i2: "a \<in> A"

    and   i3: "b \<in> A\<star>"

    and   i4: "a \<noteq> []"

    unfolding Partitions_def

    using q by (auto dest: star_decom)

  have "v \<le> a"

  proof (rule ccontr)

    assume a: "\<not>(v \<le> a)"

    from i1 have i1': "a @ b = v @ z" unfolding Partitions_def by simp

    then have "a \<le> v \<or> v < a" using append_eq_cases q by blast

    then have q: "a < v" using a unfolding strict_prefix_def prefix_def by auto

    then obtain as where eq: "a @ as = v" unfolding strict_prefix_def prefix_def by auto

    have "(u_max @ a, as) \<in> Partitions x" using eq h0 unfolding Partitions_def by auto

    moreover

    have "u_max @ a < x" using h0 eq q unfolding Partitions_def strict_prefix_def prefix_def by auto

    moreover

    have "u_max @ a \<in> A\<star>" using i2 h2 by simp

    moreover

    have "as @ z \<in> A\<star>" using i1' i2 i3 eq by auto

    ultimately have "length (u_max @ a) \<le> length u_max" using h4 by blast

    with i4 show "False" by auto

  qed

  with i1 obtain za zb

    where k1: "v @ za = a"

    and   k2: "(za, zb) \<in> Partitions z" 

    and   k4: "zb = b" 

    unfolding Partitions_def prefix_def

    by (auto simp add: append_eq_append_conv2)

  show "\<exists> (u, v) \<in> Partitions x. \<exists> (u', v') \<in> Partitions z. u < x \<and> u \<in> A\<star> \<and> v @ u' \<in> A \<and> v' \<in> A\<star>"

    using h0 h1 h2 i2 i3 k1 k2 k4 unfolding Partitions_def by blast

qed

definition 

  tag_Star :: "'a lang \<Rightarrow> 'a list \<Rightarrow> ('a lang) set"

where

  "tag_Star A \<equiv> \<lambda>x. {\<approx>A `` {v} | u v. u < x \<and> u \<in> A\<star> \<and> (u, v) \<in> Partitions x}"

lemma tag_Star_non_empty_injI:

  assumes a: "tag_Star A x = tag_Star A y"

  and     c: "x @ z \<in> A\<star>"

  and     d: "x \<noteq> []"

  shows "y @ z \<in> A\<star>"

proof -

  obtain u v u' v' 

    where a1: "(u,  v) \<in> Partitions x" "(u', v')\<in> Partitions z"

    and   a2: "u < x"

    and   a3: "u \<in> A\<star>"

    and   a4: "v @ u' \<in> A" 

    and   a5: "v' \<in> A\<star>"

    using c d by (auto dest: star_spartitions_elim2)

  have "(\<approx>A) `` {v} \<in> tag_Star A x" 

    apply(simp add: tag_Star_def Partitions_def str_eq_def)

    using a1 a2 a3 by (auto simp add: Partitions_def)

  then have "(\<approx>A) `` {v} \<in> tag_Star A y" using a by simp

  then obtain u1 v1 

    where b1: "v \<approx>A v1"

    and   b3: "u1 \<in> A\<star>"

    and   b4: "(u1, v1) \<in> Partitions y"

    unfolding tag_Star_def by auto

  have c: "v1 @ u' \<in> A\<star>" using b1 a4 unfolding str_eq_def by simp

  have "u1 @ (v1 @ u') @ v' \<in> A\<star>"

    using b3 c a5 by (simp only: append_in_starI)

  then show "y @ z \<in> A\<star>" using b4 a1 

    unfolding Partitions_def by auto

qed

lemma tag_Star_empty_injI:

  assumes a: "tag_Star A x = tag_Star A y"

  and     c: "x @ z \<in> A\<star>"

  and     d: "x = []"

  shows "y @ z \<in> A\<star>"

proof -

  from a have "{} = tag_Star A y" unfolding tag_Star_def using d by auto 

  then have "y = []"

    unfolding tag_Star_def Partitions_def strict_prefix_def prefix_def

    by (auto) (metis Nil_in_star append_self_conv2)

  then show "y @ z \<in> A\<star>" using c d by simp

qed

lemma quot_star_finiteI [intro]:

  assumes finite1: "finite (UNIV // \<approx>A)"

  shows "finite (UNIV // \<approx>(A\<star>))"

proof (rule_tac tag = "tag_Star A" in tag_finite_imageD)

  have "\<And>x y z. \<lbrakk>tag_Star A x = tag_Star A y; x @ z \<in> A\<star>\<rbrakk> \<Longrightarrow> y @ z \<in> A\<star>"

    by (case_tac "x = []") (blast intro: tag_Star_empty_injI tag_Star_non_empty_injI)+

  then show "=(tag_Star A)= \<subseteq> \<approx>(A\<star>)"

    by (rule refined_intro) (auto simp add: tag_eq_def)

next

  have *: "finite (Pow (UNIV // \<approx>A))" 

     using finite1 by auto

  show "finite (range (tag_Star A))"

    unfolding tag_Star_def 

    by (rule finite_subset[OF _ *])

       (auto simp add: quotient_def)

qed

subsection {* The conclusion of the second direction *}

lemma Myhill_Nerode2:

  fixes r::"'a rexp"

  shows "finite (UNIV // \<approx>(lang r))"

by (induct r) (auto)

end