src/HOL/Transitive_Closure.thy

prefer explicit code symbol type over ad-hoc name mangling

(*  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 Relation
begin

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

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^== \<equiv> sup r op ="

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

lemma reflclp_idemp [simp]: "(P^==)^==  =  P^=="
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]

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_reflclp [simp]: "(R^==)^** = R^**"
by (blast intro!: rtranclp_subset)

lemmas rtrancl_reflcl [simp] = rtranclp_reflclp [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
  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]

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 (fastforce 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 reflclp_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] = reflclp_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 rtrancl_trancl_reflcl [code]: "r^* = (r^+)^="
  by simp

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: reflclp_tranclp [symmetric] simp del: reflclp_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_unfold) (blast dest: tranclD)

lemma trancl_range [simp]: "Range (r^+) = Range r"
  unfolding Domain_converse [symmetric] 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"
  relpowp == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
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"

primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
    "relpowp 0 R = HOL.eq"
  | "relpowp (Suc n) R = (R ^^ n) OO R"

end

lemma relpowp_relpow_eq [pred_set_conv]:
  fixes R :: "'a rel"
  shows "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)"
  by (induct n) (simp_all add: relcompp_relcomp_eq)

text {* for code generation *}

definition relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
  relpow_code_def [code_abbrev]: "relpow = compow"

definition relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
  relpowp_code_def [code_abbrev]: "relpowp = compow"

lemma [code]:
  "relpow (Suc n) R = (relpow n R) O R"
  "relpow 0 R = Id"
  by (simp_all add: relpow_code_def)

lemma [code]:
  "relpowp (Suc n) R = (R ^^ n) OO R"
  "relpowp 0 R = HOL.eq"
  by (simp_all add: relpowp_code_def)

hide_const (open) relpow
hide_const (open) relpowp

lemma relpow_1 [simp]:
  fixes R :: "('a \<times> 'a) set"
  shows "R ^^ 1 = R"
  by simp

lemma relpowp_1 [simp]:
  fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
  shows "P ^^ 1 = P"
  by (fact relpow_1 [to_pred])

lemma relpow_0_I: 
  "(x, x) \<in> R ^^ 0"
  by simp

lemma relpowp_0_I:
  "(P ^^ 0) x x"
  by (fact relpow_0_I [to_pred])

lemma relpow_Suc_I:
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
  by auto

lemma relpowp_Suc_I:
  "(P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> (P ^^ Suc n) x z"
  by (fact relpow_Suc_I [to_pred])

lemma relpow_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, fastforce)

lemma relpowp_Suc_I2:
  "P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> (P ^^ Suc n) x z"
  by (fact relpow_Suc_I2 [to_pred])

lemma relpow_0_E:
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
  by simp

lemma relpowp_0_E:
  "(P ^^ 0) x y \<Longrightarrow> (x = y \<Longrightarrow> Q) \<Longrightarrow> Q"
  by (fact relpow_0_E [to_pred])

lemma relpow_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 relpowp_Suc_E:
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. (P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
  by (fact relpow_Suc_E [to_pred])

lemma relpow_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 relpowp_E:
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
  \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q)
  \<Longrightarrow> Q"
  by (fact relpow_E [to_pred])

lemma relpow_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: relpow_0_I elim: relpow_0_E relpow_Suc_E)
  apply (blast intro: relpow_Suc_I elim: relpow_0_E relpow_Suc_E)
  done

lemma relpowp_Suc_D2:
  "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
  by (fact relpow_Suc_D2 [to_pred])

lemma relpow_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: relpow_Suc_D2)

lemma relpowp_Suc_E2:
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
  by (fact relpow_Suc_E2 [to_pred])

lemma relpow_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 relpowp_Suc_D2':
  "\<forall>x y z. (P ^^ n) x y \<and> P y z \<longrightarrow> (\<exists>w. P x w \<and> (P ^^ n) w z)"
  by (fact relpow_Suc_D2' [to_pred])

lemma relpow_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 relpow_Suc_D2', simp, blast)
  done

lemma relpowp_E2:
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
    \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q)
  \<Longrightarrow> Q"
  by (fact relpow_E2 [to_pred])

lemma relpow_add: "R ^^ (m+n) = R^^m O R^^n"
  by (induct n) auto

lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
  by (fact relpow_add [to_pred])

lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
  by (induct n) (simp, simp add: O_assoc [symmetric])

lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
  by (fact relpow_commute [to_pred])

lemma relpow_empty:
  "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
  by (cases n) auto

lemma relpowp_bot:
  "0 < n \<Longrightarrow> (\<bottom> :: 'a \<Rightarrow> 'a \<Rightarrow> bool) ^^ n = \<bottom>"
  by (fact relpow_empty [to_pred])

lemma rtrancl_imp_UN_relpow:
  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: relpow_0_I)
  next
    case step then show ?case by (blast intro: relpow_Suc_I)
  qed
  with Pair show ?thesis by simp
qed

lemma rtranclp_imp_Sup_relpowp:
  assumes "(P^**) x y"
  shows "(\<Squnion>n. P ^^ n) x y"
  using assms and rtrancl_imp_UN_relpow [to_pred] by blast

lemma relpow_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: relpow_Suc_E intro: rtrancl_into_rtrancl)
  qed
  with Pair show ?thesis by simp
qed

lemma relpowp_imp_rtranclp:
  assumes "(P ^^ n) x y"
  shows "(P^**) x y"
  using assms and relpow_imp_rtrancl [to_pred] by blast

lemma rtrancl_is_UN_relpow:
  "R^* = (\<Union>n. R ^^ n)"
  by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)

lemma rtranclp_is_Sup_relpowp:
  "P^** = (\<Squnion>n. P ^^ n)"
  using rtrancl_is_UN_relpow [to_pred, of P] by auto

lemma rtrancl_power:
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
  by (simp add: rtrancl_is_UN_relpow)

lemma rtranclp_power:
  "(P^**) x y \<longleftrightarrow> (\<exists>n. (P ^^ n) x y)"
  by (simp add: rtranclp_is_Sup_relpowp)

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_relpow)
   apply (rule_tac x="Suc n" in exI)
   apply (clarsimp simp: relcomp_unfold)
   apply fastforce
  apply clarsimp
  apply (case_tac n, simp)
  apply clarsimp
  apply (drule relpow_imp_rtrancl)
  apply (drule rtrancl_into_trancl1) apply auto
  done

lemma tranclp_power:
  "(P^++) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
  using trancl_power [to_pred, of P "(x, y)"] by simp

lemma rtrancl_imp_relpow:
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
  by (auto dest: rtrancl_imp_UN_relpow)

lemma rtranclp_imp_relpowp:
  "(P^**) x y \<Longrightarrow> \<exists>n. (P ^^ n) x y"
  by (auto dest: rtranclp_imp_Sup_relpowp)

text{* By Sternagel/Thiemann: *}
lemma relpow_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: relcomp_unfold 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 relpowp_fun_conv:
  "(P ^^ n) x y \<longleftrightarrow> (\<exists>f. f 0 = x \<and> f n = y \<and> (\<forall>i<n. P (f i) (f (Suc i))))"
  by (fact relpow_fun_conv [to_pred])

lemma relpow_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 relpow_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 relpow_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 relpow_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 relpow_finite_bounded1[OF assms, of k] by auto

lemma rtrancl_finite_eq_relpow:
  "finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"
by(fastforce simp: rtrancl_power dest: relpow_finite_bounded)

lemma trancl_finite_eq_relpow:
  "finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"
apply(auto simp add: trancl_power)
apply(auto dest: relpow_finite_bounded1)
done

lemma finite_relcomp[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_relpow:
  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: relpow_Suc_E)
done


subsection {* Bounded transitive closure *}

definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
where
  "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"

lemma ntrancl_Zero [simp, code]:
  "ntrancl 0 R = R"
proof
  show "R \<subseteq> ntrancl 0 R"
    unfolding ntrancl_def by fastforce
next
  { 
    fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto
  }
  from this show "ntrancl 0 R \<le> R"
    unfolding ntrancl_def by auto
qed

lemma ntrancl_Suc [simp]:
  "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
proof
  {
    fix a b
    assume "(a, b) \<in> ntrancl (Suc n) R"
    from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
      unfolding ntrancl_def by auto
    have "(a, b) \<in> ntrancl n R O (Id \<union> R)"
    proof (cases "i = 1")
      case True
      from this `(a, b) \<in> R ^^ i` show ?thesis
        unfolding ntrancl_def by auto
    next
      case False
      from this `0 < i` obtain j where j: "i = Suc j" "0 < j"
        by (cases i) auto
      from this `(a, b) \<in> R ^^ i` obtain c where c1: "(a, c) \<in> R ^^ j" and c2:"(c, b) \<in> R"
        by auto
      from c1 j `i \<le> Suc (Suc n)` have "(a, c) \<in> ntrancl n R"
        unfolding ntrancl_def by fastforce
      from this c2 show ?thesis by fastforce
    qed
  }
  from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
    by auto
next
  show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
    unfolding ntrancl_def by fastforce
qed

lemma [code]:
  "ntrancl (Suc n) r = (let r' = ntrancl n r in r' Un r' O r)"
unfolding Let_def by auto

lemma finite_trancl_ntranl:
  "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
  by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)


subsection {* Acyclic relations *}

definition acyclic :: "('a * 'a) set => bool" where
  "acyclic r \<longleftrightarrow> (!x. (x,x) ~: r^+)"

abbreviation acyclicP :: "('a => 'a => bool) => bool" where
  "acyclicP r \<equiv> acyclic {(x, y). r x y}"

lemma acyclic_irrefl [code]:
  "acyclic r \<longleftrightarrow> irrefl (r^+)"
  by (simp add: acyclic_def irrefl_def)

lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
  by (simp add: acyclic_def)

lemma (in order) acyclicI_order:
  assumes *: "\<And>a b. (a, b) \<in> r \<Longrightarrow> f b < f a"
  shows "acyclic r"
proof -
  { fix a b assume "(a, b) \<in> r\<^sup>+"
    then have "f b < f a"
      by induct (auto intro: * less_trans) }
  then show ?thesis
    by (auto intro!: acyclicI)
qed

lemma acyclic_insert [iff]:
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
apply (simp add: acyclic_def trancl_insert)
apply (blast intro: rtrancl_trans)
done

lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
by (simp add: acyclic_def trancl_converse)

lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]

lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
apply (simp add: acyclic_def antisym_def)
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
done

(* Other direction:
acyclic = no loops
antisym = only self loops
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
==> antisym( r^* ) = acyclic(r - Id)";
*)

lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
apply (simp add: acyclic_def)
apply (blast intro: trancl_mono)
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};

  fun decomp (@{const Trueprop} $ t) =
    let fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel ) =
        let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
              | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
              | decr r = (r,"r");
            val (rel,r) = decr (Envir.beta_eta_contract rel);
        in SOME (a,b,rel,r) end
      | dec _ =  NONE
    in dec t end
    | decomp _ = NONE;
);

structure Tranclp_Tac = Trancl_Tac
(
  val r_into_trancl = @{thm tranclp.r_into_trancl};
  val trancl_trans  = @{thm tranclp_trans};
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
  val r_into_rtrancl = @{thm r_into_rtranclp};
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
  val rtrancl_trans = @{thm rtranclp_trans};

  fun decomp (@{const Trueprop} $ t) =
    let fun dec (rel $ a $ b) =
        let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*")
              | decr (Const ("Transitive_Closure.tranclp", _ ) $ r)  = (r,"r+")
              | decr r = (r,"r");
            val (rel,r) = decr rel;
        in SOME (a, b, rel, r) end
      | dec _ =  NONE
    in dec t end
    | decomp _ = NONE;
);
*}

setup {*
  map_theory_simpset (fn ctxt => ctxt
    addSolver (mk_solver "Trancl" Trancl_Tac.trancl_tac)
    addSolver (mk_solver "Rtrancl" Trancl_Tac.rtrancl_tac)
    addSolver (mk_solver "Tranclp" Tranclp_Tac.trancl_tac)
    addSolver (mk_solver "Rtranclp" Tranclp_Tac.rtrancl_tac))
*}


text {* Optional methods. *}

method_setup trancl =
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac) *}
  {* simple transitivity reasoner *}
method_setup rtrancl =
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac) *}
  {* simple transitivity reasoner *}
method_setup tranclp =
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac) *}
  {* simple transitivity reasoner (predicate version) *}
method_setup rtranclp =
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac) *}
  {* simple transitivity reasoner (predicate version) *}

end