(*  Title:      HOL/Transitive_Closure.thy

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1992  University of Cambridge

*)

header {* Reflexive and Transitive closure of a relation *}

theory Transitive_Closure

imports Predicate

uses "~~/src/Provers/trancl.ML"

begin

text {*

  @{text rtrancl} is reflexive/transitive closure,

  @{text trancl} is transitive closure,

  @{text reflcl} is reflexive closure.

  These postfix operators have \emph{maximum priority}, forcing their

  operands to be atomic.

*}

inductive_set

  rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   ("(_^*)" [1000] 999)

  for r :: "('a \<times> 'a) set"

where

    rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"

  | rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"

inductive_set

  trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_^+)" [1000] 999)

  for r :: "('a \<times> 'a) set"

where

    r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"

  | trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a, c) : r^+"

declare rtrancl_def [nitpick_unfold del]

        rtranclp_def [nitpick_unfold del]

        trancl_def [nitpick_unfold del]

        tranclp_def [nitpick_unfold del]

notation

  rtranclp  ("(_^**)" [1000] 1000) and

  tranclp  ("(_^++)" [1000] 1000)

abbreviation

  reflclp :: "('a => 'a => bool) => 'a => 'a => bool"  ("(_^==)" [1000] 1000) where

  "r^== == sup r op ="

abbreviation

  reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set"  ("(_^=)" [1000] 999) where

  "r^= == r \<union> Id"

notation (xsymbols)

  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and

  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and

  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and

  rtrancl  ("(_\<^sup>*)" [1000] 999) and

  trancl  ("(_\<^sup>+)" [1000] 999) and

  reflcl  ("(_\<^sup>=)" [1000] 999)

notation (HTML output)

  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and

  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and

  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and

  rtrancl  ("(_\<^sup>*)" [1000] 999) and

  trancl  ("(_\<^sup>+)" [1000] 999) and

  reflcl  ("(_\<^sup>=)" [1000] 999)

subsection {* Reflexive closure *}

lemma refl_reflcl[simp]: "refl(r^=)"

by(simp add:refl_on_def)

lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"

by(simp add:antisym_def)

lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"

unfolding trans_def by blast

subsection {* Reflexive-transitive closure *}

lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r \<union> Id)"

  by (auto simp add: fun_eq_iff)

lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"

  -- {* @{text rtrancl} of @{text r} contains @{text r} *}

  apply (simp only: split_tupled_all)

  apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])

  done

lemma r_into_rtranclp [intro]: "r x y ==> r^** x y"

  -- {* @{text rtrancl} of @{text r} contains @{text r} *}

  by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])

lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**"

  -- {* monotonicity of @{text rtrancl} *}

  apply (rule predicate2I)

  apply (erule rtranclp.induct)

   apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)

  done

lemmas rtrancl_mono = rtranclp_mono [to_set]

theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:

  assumes a: "r^** a b"

    and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"

  shows "P b" using a

  by (induct x\<equiv>a b) (rule cases)+

lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]

lemmas rtranclp_induct2 =

  rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule,

                 consumes 1, case_names refl step]

lemmas rtrancl_induct2 =

  rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),

                 consumes 1, case_names refl step]

lemma refl_rtrancl: "refl (r^*)"

by (unfold refl_on_def) fast

text {* Transitivity of transitive closure. *}

lemma trans_rtrancl: "trans (r^*)"

proof (rule transI)

  fix x y z

  assume "(x, y) \<in> r\<^sup>*"

  assume "(y, z) \<in> r\<^sup>*"

  then show "(x, z) \<in> r\<^sup>*"

  proof induct

    case base

    show "(x, y) \<in> r\<^sup>*" by fact

  next

    case (step u v)

    from `(x, u) \<in> r\<^sup>*` and `(u, v) \<in> r`

    show "(x, v) \<in> r\<^sup>*" ..

  qed

qed

lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]

lemma rtranclp_trans:

  assumes xy: "r^** x y"

  and yz: "r^** y z"

  shows "r^** x z" using yz xy

  by induct iprover+

lemma rtranclE [cases set: rtrancl]:

  assumes major: "(a::'a, b) : r^*"

  obtains

    (base) "a = b"

  | (step) y where "(a, y) : r^*" and "(y, b) : r"

  -- {* elimination of @{text rtrancl} -- by induction on a special formula *}

  apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")

   apply (rule_tac [2] major [THEN rtrancl_induct])

    prefer 2 apply blast

   prefer 2 apply blast

  apply (erule asm_rl exE disjE conjE base step)+

  done

lemma rtrancl_Int_subset: "[| Id \<subseteq> s; (r^* \<inter> s) O r \<subseteq> s|] ==> r^* \<subseteq> s"

  apply (rule subsetI)

  apply (rule_tac p="x" in PairE, clarify)

  apply (erule rtrancl_induct, auto) 

  done

lemma converse_rtranclp_into_rtranclp:

  "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"

  by (rule rtranclp_trans) iprover+

lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]

text {*

  \medskip More @{term "r^*"} equations and inclusions.

*}

lemma rtranclp_idemp [simp]: "(r^**)^** = r^**"

  apply (auto intro!: order_antisym)

  apply (erule rtranclp_induct)

   apply (rule rtranclp.rtrancl_refl)

  apply (blast intro: rtranclp_trans)

  done

lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]

lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"

  apply (rule set_eqI)

  apply (simp only: split_tupled_all)

  apply (blast intro: rtrancl_trans)

  done

lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"

  apply (drule rtrancl_mono)

  apply simp

  done

lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**"

  apply (drule rtranclp_mono)

  apply (drule rtranclp_mono)

  apply simp

  done

lemmas rtrancl_subset = rtranclp_subset [to_set]

lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**"

  by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])

lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]

lemma rtranclp_reflcl [simp]: "(R^==)^** = R^**"

  by (blast intro!: rtranclp_subset)

lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set]

lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"

  apply (rule sym)

  apply (rule rtrancl_subset, blast, clarify)

  apply (rename_tac a b)

  apply (case_tac "a = b")

   apply blast

  apply (blast intro!: r_into_rtrancl)

  done

lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**"

  apply (rule sym)

  apply (rule rtranclp_subset)

   apply blast+

  done

theorem rtranclp_converseD:

  assumes r: "(r^--1)^** x y"

  shows "r^** y x"

proof -

  from r show ?thesis

    by induct (iprover intro: rtranclp_trans dest!: conversepD)+

qed

lemmas rtrancl_converseD = rtranclp_converseD [to_set]

theorem rtranclp_converseI:

  assumes "r^** y x"

  shows "(r^--1)^** x y"

  using assms

  by induct (iprover intro: rtranclp_trans conversepI)+

lemmas rtrancl_converseI = rtranclp_converseI [to_set]

lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"

  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)

lemma sym_rtrancl: "sym r ==> sym (r^*)"

  by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])

theorem converse_rtranclp_induct [consumes 1, case_names base step]:

  assumes major: "r^** a b"

    and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"

  shows "P a"

  using rtranclp_converseI [OF major]

  by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+

lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]

lemmas converse_rtranclp_induct2 =

  converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,

                 consumes 1, case_names refl step]

lemmas converse_rtrancl_induct2 =

  converse_rtrancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),

                 consumes 1, case_names refl step]

lemma converse_rtranclpE [consumes 1, case_names base step]:

  assumes major: "r^** x z"

    and cases: "x=z ==> P"

      "!!y. [| r x y; r^** y z |] ==> P"

  shows P

  apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")

   apply (rule_tac [2] major [THEN converse_rtranclp_induct])

    prefer 2 apply iprover

   prefer 2 apply iprover

  apply (erule asm_rl exE disjE conjE cases)+

  done

lemmas converse_rtranclE = converse_rtranclpE [to_set]

lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]

lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]

lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"

  by (blast elim: rtranclE converse_rtranclE

    intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)

lemma rtrancl_unfold: "r^* = Id Un r^* O r"

  by (auto intro: rtrancl_into_rtrancl elim: rtranclE)

lemma rtrancl_Un_separatorE:

  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (a,x) : P^* \<longrightarrow> (x,y) : Q \<longrightarrow> x=y \<Longrightarrow> (a,b) : P^*"

apply (induct rule:rtrancl.induct)

 apply blast

apply (blast intro:rtrancl_trans)

done

lemma rtrancl_Un_separator_converseE:

  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (x,b) : P^* \<longrightarrow> (y,x) : Q \<longrightarrow> y=x \<Longrightarrow> (a,b) : P^*"

apply (induct rule:converse_rtrancl_induct)

 apply blast

apply (blast intro:rtrancl_trans)

done

lemma Image_closed_trancl:

  assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"

proof -

  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto

  have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X"

  proof -

    fix x y

    assume *: "y \<in> X"

    assume "(y, x) \<in> r\<^sup>*"

    then show "x \<in> X"

    proof induct

      case base show ?case by (fact *)

    next

      case step with ** show ?case by auto

    qed

  qed

  then show ?thesis by auto

qed

subsection {* Transitive closure *}

lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"

  apply (simp add: split_tupled_all)

  apply (erule trancl.induct)

   apply (iprover dest: subsetD)+

  done

lemma r_into_trancl': "!!p. p : r ==> p : r^+"

  by (simp only: split_tupled_all) (erule r_into_trancl)

text {*

  \medskip Conversions between @{text trancl} and @{text rtrancl}.

*}

lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"

  by (erule tranclp.induct) iprover+

lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]

lemma rtranclp_into_tranclp1: assumes r: "r^** a b"

  shows "!!c. r b c ==> r^++ a c" using r

  by induct iprover+

lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]

lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"

  -- {* intro rule from @{text r} and @{text rtrancl} *}

  apply (erule rtranclp.cases)

   apply iprover

  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])

    apply (simp | rule r_into_rtranclp)+

  done

lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]

text {* Nice induction rule for @{text trancl} *}

lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:

  assumes a: "r^++ a b"

  and cases: "!!y. r a y ==> P y"

    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"

  shows "P b" using a

  by (induct x\<equiv>a b) (iprover intro: cases)+

lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]

lemmas tranclp_induct2 =

  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,

    consumes 1, case_names base step]

lemmas trancl_induct2 =

  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),

    consumes 1, case_names base step]

lemma tranclp_trans_induct:

  assumes major: "r^++ x y"

    and cases: "!!x y. r x y ==> P x y"

      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"

  shows "P x y"

  -- {* Another induction rule for trancl, incorporating transitivity *}

  by (iprover intro: major [THEN tranclp_induct] cases)

lemmas trancl_trans_induct = tranclp_trans_induct [to_set]

lemma tranclE [cases set: trancl]:

  assumes "(a, b) : r^+"

  obtains

    (base) "(a, b) : r"

  | (step) c where "(a, c) : r^+" and "(c, b) : r"

  using assms by cases simp_all

lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"

  apply (rule subsetI)

  apply (rule_tac p = x in PairE)

  apply clarify

  apply (erule trancl_induct)

   apply auto

  done

lemma trancl_unfold: "r^+ = r Un r^+ O r"

  by (auto intro: trancl_into_trancl elim: tranclE)

text {* Transitivity of @{term "r^+"} *}

lemma trans_trancl [simp]: "trans (r^+)"

proof (rule transI)

  fix x y z

  assume "(x, y) \<in> r^+"

  assume "(y, z) \<in> r^+"

  then show "(x, z) \<in> r^+"

  proof induct

    case (base u)

    from `(x, y) \<in> r^+` and `(y, u) \<in> r`

    show "(x, u) \<in> r^+" ..

  next

    case (step u v)

    from `(x, u) \<in> r^+` and `(u, v) \<in> r`

    show "(x, v) \<in> r^+" ..

  qed

qed

lemmas trancl_trans = trans_trancl [THEN transD, standard]

lemma tranclp_trans:

  assumes xy: "r^++ x y"

  and yz: "r^++ y z"

  shows "r^++ x z" using yz xy

  by induct iprover+

lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"

  apply auto

  apply (erule trancl_induct)

   apply assumption

  apply (unfold trans_def)

  apply blast

  done

lemma rtranclp_tranclp_tranclp:

  assumes "r^** x y"

  shows "!!z. r^++ y z ==> r^++ x z" using assms

  by induct (iprover intro: tranclp_trans)+

lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]

lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"

  by (erule tranclp_trans [OF tranclp.r_into_trancl])

lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]

lemma trancl_insert:

  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"

  -- {* primitive recursion for @{text trancl} over finite relations *}

  apply (rule equalityI)

   apply (rule subsetI)

   apply (simp only: split_tupled_all)

   apply (erule trancl_induct, blast)

   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)

  apply (rule subsetI)

  apply (blast intro: trancl_mono rtrancl_mono

    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)

  done

lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"

  apply (drule conversepD)

  apply (erule tranclp_induct)

  apply (iprover intro: conversepI tranclp_trans)+

  done

lemmas trancl_converseI = tranclp_converseI [to_set]

lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"

  apply (rule conversepI)

  apply (erule tranclp_induct)

  apply (iprover dest: conversepD intro: tranclp_trans)+

  done

lemmas trancl_converseD = tranclp_converseD [to_set]

lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"

  by (fastsimp simp add: fun_eq_iff

    intro!: tranclp_converseI dest!: tranclp_converseD)

lemmas trancl_converse = tranclp_converse [to_set]

lemma sym_trancl: "sym r ==> sym (r^+)"

  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])

lemma converse_tranclp_induct [consumes 1, case_names base step]:

  assumes major: "r^++ a b"

    and cases: "!!y. r y b ==> P(y)"

      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"

  shows "P a"

  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])

   apply (rule cases)

   apply (erule conversepD)

  apply (blast intro: assms dest!: tranclp_converseD)

  done

lemmas converse_trancl_induct = converse_tranclp_induct [to_set]

lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"

  apply (erule converse_tranclp_induct)

   apply auto

  apply (blast intro: rtranclp_trans)

  done

lemmas tranclD = tranclpD [to_set]

lemma converse_tranclpE:

  assumes major: "tranclp r x z"

  assumes base: "r x z ==> P"

  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"

  shows P

proof -

  from tranclpD[OF major]

  obtain y where "r x y" and "rtranclp r y z" by iprover

  from this(2) show P

  proof (cases rule: rtranclp.cases)

    case rtrancl_refl

    with `r x y` base show P by iprover

  next

    case rtrancl_into_rtrancl

    from this have "tranclp r y z"

      by (iprover intro: rtranclp_into_tranclp1)

    with `r x y` step show P by iprover

  qed

qed

lemmas converse_tranclE = converse_tranclpE [to_set]

lemma tranclD2:

  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"

  by (blast elim: tranclE intro: trancl_into_rtrancl)

lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"

  by (blast elim: tranclE dest: trancl_into_rtrancl)

lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"

  by (blast dest: r_into_trancl)

lemma trancl_subset_Sigma_aux:

    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"

  by (induct rule: rtrancl_induct) auto

lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"

  apply (rule subsetI)

  apply (simp only: split_tupled_all)

  apply (erule tranclE)

   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+

  done

lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"

  apply (safe intro!: order_antisym)

   apply (erule tranclp_into_rtranclp)

  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)

  done

lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]

lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"

  apply safe

   apply (drule trancl_into_rtrancl, simp)

  apply (erule rtranclE, safe)

   apply (rule r_into_trancl, simp)

  apply (rule rtrancl_into_trancl1)

   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)

  done

lemma trancl_empty [simp]: "{}^+ = {}"

  by (auto elim: trancl_induct)

lemma rtrancl_empty [simp]: "{}^* = Id"

  by (rule subst [OF reflcl_trancl]) simp

lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"

  by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)

lemmas rtranclD = rtranclpD [to_set]

lemma rtrancl_eq_or_trancl:

  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"

  by (fast elim: trancl_into_rtrancl dest: rtranclD)

lemma trancl_unfold_right: "r^+ = r^* O r"

by (auto dest: tranclD2 intro: rtrancl_into_trancl1)

lemma trancl_unfold_left: "r^+ = r O r^*"

by (auto dest: tranclD intro: rtrancl_into_trancl2)

text {* Simplifying nested closures *}

lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"

by (simp add: trans_rtrancl)

lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"

by (subst reflcl_trancl[symmetric]) simp

lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"

by auto

text {* @{text Domain} and @{text Range} *}

lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"

  by blast

lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"

  by blast

lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"

  by (rule rtrancl_Un_rtrancl [THEN subst]) fast

lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"

  by (blast intro: subsetD [OF rtrancl_Un_subset])

lemma trancl_domain [simp]: "Domain (r^+) = Domain r"

  by (unfold Domain_def) (blast dest: tranclD)

lemma trancl_range [simp]: "Range (r^+) = Range r"

unfolding Range_def by(simp add: trancl_converse [symmetric])

lemma Not_Domain_rtrancl:

    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"

  apply auto

  apply (erule rev_mp)

  apply (erule rtrancl_induct)

   apply auto

  done

lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"

  apply clarify

  apply (erule trancl_induct)

   apply (auto simp add: Field_def)

  done

lemma finite_trancl[simp]: "finite (r^+) = finite r"

  apply auto

   prefer 2

   apply (rule trancl_subset_Field2 [THEN finite_subset])

   apply (rule finite_SigmaI)

    prefer 3

    apply (blast intro: r_into_trancl' finite_subset)

   apply (auto simp add: finite_Field)

  done

text {* More about converse @{text rtrancl} and @{text trancl}, should

  be merged with main body. *}

lemma single_valued_confluent:

  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>

  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"

  apply (erule rtrancl_induct)

  apply simp

  apply (erule disjE)

   apply (blast elim:converse_rtranclE dest:single_valuedD)

  apply(blast intro:rtrancl_trans)

  done

lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"

  by (fast intro: trancl_trans)

lemma trancl_into_trancl [rule_format]:

    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"

  apply (erule trancl_induct)

   apply (fast intro: r_r_into_trancl)

  apply (fast intro: r_r_into_trancl trancl_trans)

  done

lemma tranclp_rtranclp_tranclp:

    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"

  apply (drule tranclpD)

  apply (elim exE conjE)

  apply (drule rtranclp_trans, assumption)

  apply (drule rtranclp_into_tranclp2, assumption, assumption)

  done

lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]

lemmas transitive_closure_trans [trans] =

  r_r_into_trancl trancl_trans rtrancl_trans

  trancl.trancl_into_trancl trancl_into_trancl2

  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl

  rtrancl_trancl_trancl trancl_rtrancl_trancl

lemmas transitive_closurep_trans' [trans] =

  tranclp_trans rtranclp_trans

  tranclp.trancl_into_trancl tranclp_into_tranclp2

  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp

  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp

declare trancl_into_rtrancl [elim]

subsection {* The power operation on relations *}

text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}

overloading

  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"

begin

primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where

    "relpow 0 R = Id"

  | "relpow (Suc n) R = (R ^^ n) O R"

end

lemma rel_pow_1 [simp]:

  fixes R :: "('a \<times> 'a) set"

  shows "R ^^ 1 = R"

  by simp

lemma rel_pow_0_I: 

  "(x, x) \<in> R ^^ 0"

  by simp

lemma rel_pow_Suc_I:

  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"

  by auto

lemma rel_pow_Suc_I2:

  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"

  by (induct n arbitrary: z) (simp, fastsimp)

lemma rel_pow_0_E:

  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"

  by simp

lemma rel_pow_Suc_E:

  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"

  by auto

lemma rel_pow_E:

  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)

   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)

   \<Longrightarrow> P"

  by (cases n) auto

lemma rel_pow_Suc_D2:

  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"

  apply (induct n arbitrary: x z)

   apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)

  apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)

  done

lemma rel_pow_Suc_E2:

  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"

  by (blast dest: rel_pow_Suc_D2)

lemma rel_pow_Suc_D2':

  "\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"

  by (induct n) (simp_all, blast)

lemma rel_pow_E2:

  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)

     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)

   \<Longrightarrow> P"

  apply (cases n, simp)

  apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)

  done

lemma rel_pow_add: "R ^^ (m+n) = R^^m O R^^n"

by(induct n) auto

lemma rel_pow_commute: "R O R ^^ n = R ^^ n O R"

by (induct n) (simp, simp add: O_assoc [symmetric])

lemma rtrancl_imp_UN_rel_pow:

  assumes "p \<in> R^*"

  shows "p \<in> (\<Union>n. R ^^ n)"

proof (cases p)

  case (Pair x y)

  with assms have "(x, y) \<in> R^*" by simp

  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct

    case base show ?case by (blast intro: rel_pow_0_I)

  next

    case step then show ?case by (blast intro: rel_pow_Suc_I)

  qed

  with Pair show ?thesis by simp

qed

lemma rel_pow_imp_rtrancl:

  assumes "p \<in> R ^^ n"

  shows "p \<in> R^*"

proof (cases p)

  case (Pair x y)

  with assms have "(x, y) \<in> R ^^ n" by simp

  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)

    case 0 then show ?case by simp

  next

    case Suc then show ?case

      by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)

  qed

  with Pair show ?thesis by simp

qed

lemma rtrancl_is_UN_rel_pow:

  "R^* = (\<Union>n. R ^^ n)"

  by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)

lemma rtrancl_power:

  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"

  by (simp add: rtrancl_is_UN_rel_pow)

lemma trancl_power:

  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"

  apply (cases p)

  apply simp

  apply (rule iffI)

   apply (drule tranclD2)

   apply (clarsimp simp: rtrancl_is_UN_rel_pow)

   apply (rule_tac x="Suc n" in exI)

   apply (clarsimp simp: rel_comp_def)

   apply fastsimp

  apply clarsimp

  apply (case_tac n, simp)

  apply clarsimp

  apply (drule rel_pow_imp_rtrancl)

  apply (drule rtrancl_into_trancl1) apply auto

  done

lemma rtrancl_imp_rel_pow:

  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"

  by (auto dest: rtrancl_imp_UN_rel_pow)

text{* By Sternagel/Thiemann: *}

lemma rel_pow_fun_conv:

  "((a,b) \<in> R ^^ n) = (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f(Suc i)) \<in> R))"

proof (induct n arbitrary: b)

  case 0 show ?case by auto

next

  case (Suc n)

  show ?case

  proof (simp add: rel_comp_def Suc)

    show "(\<exists>y. (\<exists>f. f 0 = a \<and> f n = y \<and> (\<forall>i<n. (f i,f(Suc i)) \<in> R)) \<and> (y,b) \<in> R)

     = (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"

    (is "?l = ?r")

    proof

      assume ?l

      then obtain c f where 1: "f 0 = a"  "f n = c"  "\<And>i. i < n \<Longrightarrow> (f i, f (Suc i)) \<in> R"  "(c,b) \<in> R" by auto

      let ?g = "\<lambda> m. if m = Suc n then b else f m"

      show ?r by (rule exI[of _ ?g], simp add: 1)

    next

      assume ?r

      then obtain f where 1: "f 0 = a"  "b = f (Suc n)"  "\<And>i. i < Suc n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto

      show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)

    qed

  qed

qed

lemma rel_pow_finite_bounded1:

assumes "finite(R :: ('a*'a)set)" and "k>0"

shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r")

proof-

  { fix a b k

    have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n"

    proof(induct k arbitrary: b)

      case 0

      hence "R \<noteq> {}" by auto

      with card_0_eq[OF `finite R`] have "card R >= Suc 0" by auto

      thus ?case using 0 by force

    next

      case (Suc k)

      then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto

      from Suc(1)[OF `(a,a') : R^^(Suc k)`]

      obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto

      have "(a,b) : R^^(Suc n)" using `(a,a') \<in> R^^n` and `(a',b)\<in> R` by auto

      { assume "n < card R"

        hence ?case using `(a,b): R^^(Suc n)` Suc_leI[OF `n < card R`] by blast

      } moreover

      { assume "n = card R"

        from `(a,b) \<in> R ^^ (Suc n)`[unfolded rel_pow_fun_conv]

        obtain f where "f 0 = a" and "f(Suc n) = b"

          and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto

        let ?p = "%i. (f i, f(Suc i))"

        let ?N = "{i. i \<le> n}"

        have "?p ` ?N <= R" using steps by auto

        from card_mono[OF assms(1) this]

        have "card(?p ` ?N) <= card R" .

        also have "\<dots> < card ?N" using `n = card R` by simp

        finally have "~ inj_on ?p ?N" by(rule pigeonhole)

        then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and

          pij: "?p i = ?p j" by(auto simp: inj_on_def)

        let ?i = "min i j" let ?j = "max i j"

        have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j" 

          and ij: "?i < ?j"

          using i j ij pij unfolding min_def max_def by auto

        from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j"

          and pij: "?p i = ?p j" by blast

        let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j - i))"

        let ?n = "Suc(n - (j - i))"

        have abl: "(a,b) \<in> R ^^ ?n" unfolding rel_pow_fun_conv

        proof (rule exI[of _ ?g], intro conjI impI allI)

          show "?g ?n = b" using `f(Suc n) = b` j ij by auto

        next

          fix k assume "k < ?n"

          show "(?g k, ?g (Suc k)) \<in> R"

          proof (cases "k < i")

            case True

            with i have "k <= n" by auto

            from steps[OF this] show ?thesis using True by simp

          next

            case False

            hence "i \<le> k" by auto

            show ?thesis

            proof (cases "k = i")

              case True

              thus ?thesis using ij pij steps[OF i] by simp

            next

              case False

              with `i \<le> k` have "i < k" by auto

              hence small: "k + (j - i) <= n" using `k<?n` by arith

              show ?thesis using steps[OF small] `i<k` by auto

            qed

          qed

        qed (simp add: `f 0 = a`)

        moreover have "?n <= n" using i j ij by arith

        ultimately have ?case using `n = card R` by blast

      }

      ultimately show ?case using `n \<le> card R` by force

    qed

  }

  thus ?thesis using gr0_implies_Suc[OF `k>0`] by auto

qed

lemma rel_pow_finite_bounded:

assumes "finite(R :: ('a*'a)set)"

shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)"

apply(cases k)

 apply force

using rel_pow_finite_bounded1[OF assms, of k] by auto

lemma rtrancl_finite_eq_rel_pow:

  "finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"

by(fastsimp simp: rtrancl_power dest: rel_pow_finite_bounded)

lemma trancl_finite_eq_rel_pow:

  "finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"

apply(auto simp add: trancl_power)

apply(auto dest: rel_pow_finite_bounded1)

done

lemma finite_rel_comp[simp,intro]:

assumes "finite R" and "finite S"

shows "finite(R O S)"

proof-

  have "R O S = (UN (x,y) : R. \<Union>((%(u,v). if u=y then {(x,v)} else {}) ` S))"

    by(force simp add: split_def)

  thus ?thesis using assms by(clarsimp)

qed

lemma finite_relpow[simp,intro]:

  assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)"

apply(induct n)

 apply simp

apply(case_tac n)

 apply(simp_all add: assms)

done

lemma single_valued_rel_pow:

  fixes R :: "('a * 'a) set"

  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"

apply (induct n arbitrary: R)

apply simp_all

apply (rule single_valuedI)

apply (fast dest: single_valuedD elim: rel_pow_Suc_E)

done

subsection {* Setup of transitivity reasoner *}

ML {*

structure Trancl_Tac = Trancl_Tac

(

  val r_into_trancl = @{thm trancl.r_into_trancl};

  val trancl_trans  = @{thm trancl_trans};

  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};

  val r_into_rtrancl = @{thm r_into_rtrancl};

  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};

  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};

  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};

  val rtrancl_trans = @{thm rtrancl_trans};