Theory Transitive_Closure

(*  Title:      HOL/Transitive_Closure.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

section ‹Reflexive and Transitive closure of a relation›

theory Transitive_Closure
  imports Finite_Set
  abbrevs "^*" = "*" "**"
    and "^+" = "+" "++"
    and "^=" = "=" "=="

ML_file ‹~~/src/Provers/trancl.ML›

text rtrancl› is reflexive/transitive closure,
  trancl› is transitive closure,
  reflcl› is reflexive closure.

  These postfix operators have ‹maximum priority›, forcing their
  operands to be atomic.

context notes [[inductive_internals]]

inductive_set rtrancl :: "('a × 'a) set  ('a × 'a) set"  ("(_*)" [1000] 999)
  for r :: "('a × 'a) set"
    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 × 'a) set  ('a × 'a) set"  ("(_+)" [1000] 999)
  for r :: "('a × 'a) set"
    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+"

  rtranclp  ("(_**)" [1000] 1000) and
  tranclp  ("(_++)" [1000] 1000)

  rtrancl_def [nitpick_unfold del]
  rtranclp_def [nitpick_unfold del]
  trancl_def [nitpick_unfold del]
  tranclp_def [nitpick_unfold del]


abbreviation reflcl :: "('a × 'a) set  ('a × 'a) set"  ("(_=)" [1000] 999)
  where "r=  r  Id"

abbreviation reflclp :: "('a  'a  bool)  'a  'a  bool"  ("(_==)" [1000] 1000)
  where "r==  sup r (=)"

notation (ASCII)
  rtrancl  ("(_^*)" [1000] 999) and
  trancl  ("(_^+)" [1000] 999) and
  reflcl  ("(_^=)" [1000] 999) and
  rtranclp  ("(_^**)" [1000] 1000) and
  tranclp  ("(_^++)" [1000] 1000) and
  reflclp  ("(_^==)" [1000] 1000)

subsection ‹Reflexive closure›

lemma reflcl_set_eq [pred_set_conv]: "(sup (λx y. (x, y)  r) (=)) = (λx y. (x, y)  r  Id)"
  by (auto simp: fun_eq_iff)

lemma refl_reflcl[simp]: "refl (r=)"
  by (simp add: refl_on_def)

lemma reflp_on_reflclp[simp]: "reflp_on A R=="
  by (simp add: reflp_on_def)

lemma antisym_on_reflcl[simp]: "antisym_on A (r=)  antisym_on A r"
  by (simp add: antisym_on_def)

lemma antisymp_on_reflcp[simp]: "antisymp_on A R==  antisymp_on A R"
  by (rule antisym_on_reflcl[to_pred])

lemma trans_on_reflcl[simp]: "trans_on A r  trans_on A (r=)"
  by (auto intro: trans_onI dest: trans_onD)

lemma transp_on_reflclp[simp]: "transp_on A R  transp_on A R=="
  by (rule trans_on_reflcl[to_pred])

lemma reflclp_idemp [simp]: "(P==)== = P=="
  by blast

lemma reflclp_ident_if_reflp[simp]: "reflp R  R== = R"
  by (auto dest: reflpD)

subsection ‹Reflexive-transitive closure›

lemma r_into_rtrancl [intro]: "p. p  r  p  r*"
  ― ‹rtrancl› of r› contains r›
  by (simp add: split_tupled_all rtrancl_refl [THEN rtrancl_into_rtrancl])

lemma r_into_rtranclp [intro]: "r x y  r** x y"
  ― ‹rtrancl› of r› contains r›
  by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])

lemma rtranclp_mono: "r  s  r**  s**"
  ― ‹monotonicity of rtrancl›
proof (rule predicate2I)
  show "s** x y" if "r  s" "r** x y" for x y
    using r** x y r  s
    by (induction rule: rtranclp.induct) (blast intro: rtranclp.rtrancl_into_rtrancl)+

lemma mono_rtranclp[mono]: "(a b. x a b  y a b)  x** a b  y** a b"
   using rtranclp_mono[of x y] by auto

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 xa 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*)"
  unfolding refl_on_def by fast

text ‹Transitivity of transitive closure.›
lemma trans_rtrancl: "trans (r*)"
proof (rule transI)
  fix x y z
  assume "(x, y)  r*"
  assume "(y, z)  r*"
  then show "(x, z)  r*"
  proof induct
    case base
    show "(x, y)  r*" by fact
    case (step u v)
    from (x, u)  r* and (u, v)  r
    show "(x, v)  r*" ..

lemmas rtrancl_trans = trans_rtrancl [THEN transD]

lemma rtranclp_trans:
  assumes "r** x y"
    and "r** y z"
  shows "r** x z"
  using assms(2,1) by induct iprover+

lemma rtranclE [cases set: rtrancl]:
  fixes a b :: 'a
  assumes major: "(a, b)  r*"
    (base) "a = b"
  | (step) y where "(a, y)  r*" and "(y, b)  r"
  ― ‹elimination of rtrancl› -- by induction on a special formula›
proof -
  have "a = b  (y. (a, y)  r*  (y, b)  r)"
    by (rule major [THEN rtrancl_induct]) blast+
  then show ?thesis
    by (auto intro: base step)

lemma rtrancl_Int_subset: "Id  s  (r*  s) O r  s  r*  s"
  by (fastforce elim: rtrancl_induct)

lemma converse_rtranclp_into_rtranclp: "r a b  r** b c  r** a c"
  by (rule rtranclp_trans) iprover+

lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]

text  More termr* equations and inclusions.›

lemma rtranclp_idemp [simp]: "(r**)** = r**"
proof -
  have "r**** x y  r** x y" for x y
    by (induction rule: rtranclp_induct) (blast intro: rtranclp_trans)+
  then show ?thesis
    by (auto intro!: order_antisym)

lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]

lemma rtrancl_idemp_self_comp [simp]: "R* O R* = R*"
  by (force intro: rtrancl_trans)

lemma rtrancl_subset_rtrancl: "r  s*  r*  s*"
  by (drule rtrancl_mono, simp)

lemma rtranclp_subset: "R  S  S  R**  S** = R**"
  by (fastforce dest: rtranclp_mono)

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*"
  by (rule rtrancl_subset [symmetric]) auto

lemma rtranclp_r_diff_Id: "(inf r (≠))** = r**"
  by (rule rtranclp_subset [symmetric]) auto

theorem rtranclp_converseD:
  assumes "(r¯¯)** x y"
  shows "r** y x"
  using assms by induct (iprover intro: rtranclp_trans dest!: conversepD)+

lemmas rtrancl_converseD = rtranclp_converseD [to_set]

theorem rtranclp_converseI:
  assumes "r** y x"
  shows "(r¯¯)** x y"
  using assms by induct (iprover intro: rtranclp_trans conversepI)+

lemmas rtrancl_converseI = rtranclp_converseI [to_set]

lemma rtrancl_converse: "(r¯)* = (r*)¯"
  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
proof -
  have "x = z  (y. r x y  r** y z)"
    by (rule major [THEN converse_rtranclp_induct]) iprover+
  then show ?thesis
    by (auto intro: cases)

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  r* O r"
  by (auto intro: rtrancl_into_rtrancl elim: rtranclE)

lemma rtrancl_Un_separatorE:
  "(a, b)  (P  Q)*  x y. (a, x)  P*  (x, y)  Q  x = y  (a, b)  P*"
proof (induct rule: rtrancl.induct)
  case rtrancl_refl
  then show ?case by blast
  case rtrancl_into_rtrancl
  then show ?case by (blast intro: rtrancl_trans)

lemma rtrancl_Un_separator_converseE:
  "(a, b)  (P  Q)*  x y. (x, b)  P*  (y, x)  Q  y = x  (a, b)  P*"
proof (induct rule: converse_rtrancl_induct)
  case base
  then show ?case by blast
  case step
  then show ?case by (blast intro: rtrancl_trans)

lemma Image_closed_trancl:
  assumes "r `` X  X"
  shows "r* `` X = X"
proof -
  from assms have **: "{y. xX. (x, y)  r}  X"
    by auto
  have "x  X" if 1: "(y, x)  r*" and 2: "y  X" for x y
  proof -
    from 1 show "x  X"
    proof induct
      case base
      show ?case by (fact 2)
      case step
      with ** show ?case by auto
  then show ?thesis by auto

subsection ‹Transitive closure›

lemma totalp_on_tranclp: "totalp_on A R  totalp_on A (tranclp R)"
  by (auto intro: totalp_onI dest: totalp_onD)

lemma total_on_trancl: "total_on A r  total_on A (trancl r)"
  by (rule totalp_on_tranclp[to_set])

lemma trancl_mono:
  assumes "p  r+" "r  s"
  shows "p  s+"
proof -
  have "(a, b)  r+; r  s  (a, b)  s+" for a b
    by (induction rule: trancl.induct) (iprover dest: subsetD)+
  with assms show ?thesis
    by (cases p) force

lemma r_into_trancl': "p. p  r  p  r+"
  by (simp only: split_tupled_all) (erule r_into_trancl)

text  Conversions between trancl› and 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** a b"
  shows "r b c  r++ a c"
  using assms by (induct arbitrary: c) iprover+

lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]

lemma rtranclp_into_tranclp2:
  assumes "r a b" "r** b c" shows "r++ a c"
  ― ‹intro rule from r› and rtrancl›
  using r** b c
proof (cases rule: rtranclp.cases)
  case rtrancl_refl
  with assms show ?thesis
    by iprover
  case rtrancl_into_rtrancl
  with assms show ?thesis
    by (auto intro: rtranclp_trans [THEN rtranclp_into_tranclp1])

lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]

text ‹Nice induction rule for 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 xa 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+"
    (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  s  (r+  s) O r  s  r+  s"
  by (fastforce simp add: elim: trancl_induct)

lemma trancl_unfold: "r+ = r  r+ O r"
  by (auto intro: trancl_into_trancl elim: tranclE)

text ‹Transitivity of termr+
lemma trans_trancl [simp]: "trans (r+)"
proof (rule transI)
  fix x y z
  assume "(x, y)  r+"
  assume "(y, z)  r+"
  then show "(x, z)  r+"
  proof induct
    case (base u)
    from (x, y)  r+ and (y, u)  r
    show "(x, u)  r+" ..
    case (step u v)
    from (x, u)  r+ and (u, v)  r
    show "(x, v)  r+" ..

lemmas trancl_trans = trans_trancl [THEN transD]

lemma tranclp_trans:
  assumes "r++ x y"
    and "r++ y z"
  shows "r++ x z"
  using assms(2,1) by induct iprover+

lemma trancl_id [simp]: "trans r  r+ = r"
  unfolding trans_def by (fastforce simp add: elim: trancl_induct)

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 tranclp_converseI:
  assumes "(r++)¯¯ x y" shows "(r¯¯)++ x y"
  using conversepD [OF assms]
proof (induction rule: tranclp_induct)
  case (base y)
  then show ?case 
    by (iprover intro: conversepI)
  case (step y z)
  then show ?case
    by (iprover intro: conversepI tranclp_trans)

lemmas trancl_converseI = tranclp_converseI [to_set]

lemma tranclp_converseD:
  assumes "(r¯¯)++ x y" shows "(r++)¯¯ x y"
proof -
  have "r++ y x"
    using assms
    by (induction rule: tranclp_induct) (iprover dest: conversepD intro: tranclp_trans)+
  then show ?thesis
    by (rule conversepI)

lemmas trancl_converseD = tranclp_converseD [to_set]

lemma tranclp_converse: "(r¯¯)++ = (r++)¯¯"
  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"
proof -
  have "r¯¯++ b a"
    by (intro tranclp_converseI conversepI major)
  then show ?thesis
    by (induction rule: tranclp_induct) (blast intro: cases dest: tranclp_converseD)+

lemmas converse_trancl_induct = converse_tranclp_induct [to_set]

lemma tranclpD: "R++ x y  z. R x z  R** z y"
proof (induction rule: converse_tranclp_induct)
  case (step u v)
  then show ?case
    by (blast intro: rtranclp_trans)
qed auto

lemmas tranclD = tranclpD [to_set]

lemma converse_tranclpE:
  assumes major: "tranclp r x z"
    and base: "r x z  P"
    and step: "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
    case rtrancl_into_rtrancl
    then have "tranclp r y z"
      by (iprover intro: rtranclp_into_tranclp1)
    with r x y step show P
      by iprover

lemmas converse_tranclE = converse_tranclpE [to_set]

lemma tranclD2: "(x, y)  R+  z. (x, z)  R*  (z, y)  R"
  by (blast elim: tranclE intro: trancl_into_rtrancl)

lemma irrefl_tranclI: "r¯  r* = {}  (x, x)  r+"
  by (blast elim: tranclE dest: trancl_into_rtrancl)

lemma irrefl_trancl_rD: "x. (x, x)  r+  (x, y)  r  x  y"
  by (blast dest: r_into_trancl)

lemma trancl_subset_Sigma_aux: "(a, b)  r*  r  A × A  a = b  a  A"
  by (induct rule: rtrancl_induct) auto

lemma trancl_subset_Sigma:
  assumes "r  A × A" shows "r+  A × A"
proof (rule trancl_Int_subset [OF assms])
  show "(r+  A × A) O r  A × A"
    using assms by auto

lemma reflclp_tranclp [simp]: "(r++)== = r**"
  by (fast elim: rtranclp.cases tranclp_into_rtranclp dest: rtranclp_into_tranclp1)

lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]

lemma trancl_reflcl [simp]: "(r=)+ = r*"
proof -
  have "(a, b)  (r=)+  (a, b)  r*" for a b
    by (force dest: trancl_into_rtrancl)
  moreover have "(a, b)  (r=)+" if "(a, b)  r*" for a b
    using that
  proof (cases a b rule: rtranclE)
    case step
    show ?thesis
      by (rule rtrancl_into_trancl1) (use step in auto)
  qed auto
  ultimately show ?thesis
    by auto

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  a  b  R++ a b"
  by (force simp: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)

lemmas rtranclD = rtranclpD [to_set]

lemma rtrancl_eq_or_trancl: "(x,y)  R*  x = y  x  y  (x, y)  R+"
  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)

lemma trancl_insert: "(insert (y, x) r)+ = r+  {(a, b). (a, y)  r*  (x, b)  r*}"
  ― ‹primitive recursion for trancl› over finite relations›
proof -
  have "a b. (a, b)  (insert (y, x) r)+ 
           (a, b)  r+  {(a, b). (a, y)  r*  (x, b)  r*}"
    by (erule trancl_induct) (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)+
  moreover have "r+  {(a, b). (a, y)  r*  (x, b)  r*}   (insert (y, x) r)+"
    by (blast intro: trancl_mono rtrancl_mono [THEN [2] rev_subsetD]
                     rtrancl_trancl_trancl rtrancl_into_trancl2)
  ultimately show ?thesis
    by auto

lemma trancl_insert2:
  "(insert (a, b) r)+ = r+  {(x, y). ((x, a)  r+  x = a)  ((b, y)  r+  y = b)}"
  by (auto simp: trancl_insert rtrancl_eq_or_trancl)

lemma rtrancl_insert: "(insert (a,b) r)* = r*  {(x, y). (x, a)  r*  (b, y)  r*}"
  using trancl_insert[of a b r]
  by (simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast

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 Domain› and Range›

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

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

lemma rtrancl_Un_subset: "(R*  S*)  (R  S)*"
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast

lemma in_rtrancl_UnI: "x  R*  x  S*  x  (R  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:
  assumes "x  Domain R" shows "(x, y)  R*  x = y"
proof -
have "(x, y)  R*  x = y"
  by (erule rtrancl_induct) (use assms in auto)
  then show ?thesis
    by auto

lemma trancl_subset_Field2: "r+  Field r × Field r"
  by (rule trancl_Int_subset) (auto simp: Field_def)

lemma finite_trancl[simp]: "finite (r+) = finite r"
  show "finite (r+)  finite r"
    by (blast intro: r_into_trancl' finite_subset)
  show "finite r  finite (r+)"
    by (auto simp: finite_Field trancl_subset_Field2 [THEN finite_subset])

lemma finite_rtrancl_Image[simp]: assumes "finite R" "finite A" shows "finite (R* `` A)"
proof (rule ccontr)
  assume "infinite (R* `` A)"
  with assms show False
    by(simp add: rtrancl_trancl_reflcl Un_Image del: reflcl_trancl)

text ‹More about converse rtrancl› and trancl›, should
  be merged with main body.›

lemma single_valued_confluent:
  assumes "single_valued r" and xy: "(x, y)  r*" and xz: "(x, z)  r*"
  shows "(y, z)  r*  (z, y)  r*"
  using xy
proof (induction rule: rtrancl_induct)
  case base
  show ?case
    by (simp add: assms)
  case (step y z)
  with xz single_valued r show ?case
    by (auto elim: converse_rtranclE dest: single_valuedD intro: rtrancl_trans)

lemma r_r_into_trancl: "(a, b)  R  (b, c)  R  (a, c)  R+"
  by (fast intro: trancl_trans)

lemma trancl_into_trancl: "(a, b)  r+  (b, c)  r  (a, c)  r+"
  by (induct rule: trancl_induct) (fast intro: r_r_into_trancl trancl_trans)+

lemma tranclp_rtranclp_tranclp:
  assumes "r++ a b" "r** b c" shows "r++ a c"
proof -
  obtain z where "r a z" "r** z c"
    using assms by (iprover dest: tranclpD rtranclp_trans)
  then show ?thesis
    by (blast dest: rtranclp_into_tranclp2)

lemma rtranclp_conversep: "r¯¯** = r**¯¯"
  by(auto simp add: fun_eq_iff intro: rtranclp_converseI rtranclp_converseD)

lemmas symp_rtranclp = sym_rtrancl[to_pred]

lemmas symp_conv_conversep_eq = sym_conv_converse_eq[to_pred]

lemmas rtranclp_tranclp_absorb [simp] = rtrancl_trancl_absorb[to_pred]
lemmas tranclp_rtranclp_absorb [simp] = trancl_rtrancl_absorb[to_pred]
lemmas rtranclp_reflclp_absorb [simp] = rtrancl_reflcl_absorb[to_pred]

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 ‹Symmetric closure›

definition symclp :: "('a  'a  bool)  'a  'a  bool"
where "symclp r x y  r x y  r y x"

lemma symclpI [simp, intro?]:
  shows symclpI1: "r x y  symclp r x y"
    and symclpI2: "r y x  symclp r x y"
  by(simp_all add: symclp_def)

lemma symclpE [consumes 1, cases pred]:
  assumes "symclp r x y"
  obtains (base) "r x y" | (sym) "r y x"
  using assms by(auto simp add: symclp_def)

lemma symclp_pointfree: "symclp r = sup r r¯¯"
  by(auto simp add: symclp_def fun_eq_iff)

lemma symclp_greater: "r  symclp r"
  by(simp add: symclp_pointfree)

lemma symclp_conversep [simp]: "symclp r¯¯ = symclp r"
  by(simp add: symclp_pointfree sup.commute)

lemma symp_on_symclp [simp]: "symp_on A (symclp R)"
  by(auto simp add: symp_on_def elim: symclpE intro: symclpI)

lemma symp_symclp_eq: "symp r  symclp r = r"
  by(simp add: symclp_pointfree symp_conv_conversep_eq)

lemma symp_rtranclp_symclp [simp]: "symp (symclp r)**"
  by(simp add: symp_rtranclp)

lemma rtranclp_symclp_sym [sym]: "(symclp r)** x y  (symclp r)** y x"
  by(rule sympD[OF symp_rtranclp_symclp])

lemma symclp_idem [simp]: "symclp (symclp r) = symclp r"
  by(simp add: symclp_pointfree sup_commute converse_join)

lemma reflp_on_rtranclp [simp]: "reflp_on A R**"
  by (simp add: reflp_on_def)

subsection ‹The power operation on relations›

text R ^^ n = R O … O R›, the n-fold composition of R›

  relpow  "compow :: nat  ('a × 'a) set  ('a × 'a) set"
  relpowp  "compow :: nat  ('a  'a  bool)  ('a  'a  bool)"

primrec relpow :: "nat  ('a × 'a) set  ('a × 'a) set"
    "relpow 0 R = Id"
  | "relpow (Suc n) R = (R ^^ n) O R"

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


lemma relpowp_relpow_eq [pred_set_conv]:
  "(λx y. (x, y)  R) ^^ n = (λx y. (x, y)  R ^^ n)" for R :: "'a rel"
  by (induct n) (simp_all add: relcompp_relcomp_eq)

text ‹For code generation:›

definition relpow :: "nat  ('a × 'a) set  ('a × 'a) set"
  where relpow_code_def [code_abbrev]: "relpow = compow"

definition relpowp :: "nat  ('a  'a  bool)  ('a  'a  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]: "R ^^ 1 = R"
  for R :: "('a × 'a) set"
  by simp

lemma relpowp_1 [simp]: "P ^^ 1 = P"
  for P :: "'a  'a  bool"
  by (fact relpow_1 [to_pred])

lemma relpow_0_I: "(x, x)  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)   R ^^ n  (y, z)  R  (x, z)  R ^^ Suc n"
  by auto

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

lemma relpow_Suc_I2: "(x, y)  R  (y, z)  R ^^ n  (x, z)  R ^^ Suc n"
  by (induct n arbitrary: z) (simp, fastforce)

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

lemma relpow_0_E: "(x, y)  R ^^ 0  (x = y  P)  P"
  by simp

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

lemma relpow_Suc_E: "(x, z)  R ^^ Suc n  (y. (x, y)  R ^^ n  (y, z)  R  P)  P"
  by auto

lemma relpowp_Suc_E: "(P ^^ Suc n) x z  (y. (P ^^ n) x y  P y z  Q)  Q"
  by (fact relpow_Suc_E [to_pred])

lemma relpow_E:
  "(x, z)   R ^^ n 
    (n = 0  x = z  P) 
    (y m. n = Suc m  (x, y)   R ^^ m  (y, z)  R  P)  P"
  by (cases n) auto

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

lemma relpow_Suc_D2: "(x, z)  R ^^ Suc n  (y. (x, y)  R  (y, z)  R ^^ n)"
  by (induct n arbitrary: x z)
    (blast intro: relpow_0_I relpow_Suc_I elim: relpow_0_E relpow_Suc_E)+

lemma relpowp_Suc_D2: "(P ^^ Suc n) x z  y. P x y  (P ^^ n) y z"
  by (fact relpow_Suc_D2 [to_pred])

lemma relpow_Suc_E2: "(x, z)  R ^^ Suc n  (y. (x, y)  R  (y, z)  R ^^ n  P)  P"
  by (blast dest: relpow_Suc_D2)

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

lemma relpow_Suc_D2': "x y z. (x, y)  R ^^ n  (y, z)  R  (w. (x, w)  R  (w, z)  R ^^ n)"
  by (induct n) (simp_all, blast)

lemma relpowp_Suc_D2': "x y z. (P ^^ n) x y  P y z  (w. P x w  (P ^^ n) w z)"
  by (fact relpow_Suc_D2' [to_pred])

lemma relpow_E2:
  assumes "(x, z)  R ^^ n" "n = 0  x = z  P"
          "y m. n = Suc m  (x, y)  R  (y, z)  R ^^ m  P"
      shows "P"
proof (cases n)
  case 0
  with assms show ?thesis
    by simp
  case (Suc m)
  with assms relpow_Suc_D2' [of m R] show ?thesis
    by force

lemma relpowp_E2:
  "(P ^^ n) x z 
    (n = 0  x = z  Q) 
    (y m. n = Suc m  P x y  (P ^^ m) y z  Q)  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_all 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  ({} :: ('a × 'a) set) ^^ n = {}"
  by (cases n) auto

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

lemma rtrancl_imp_UN_relpow:
  assumes "p  R*"
  shows "p  (n. R ^^ n)"
proof (cases p)
  case (Pair x y)
  with assms have "(x, y)  R*" by simp
  then have "(x, y)  (n. R ^^ n)"
  proof induct
    case base
    show ?case by (blast intro: relpow_0_I)
    case step
    then show ?case by (blast intro: relpow_Suc_I)
  with Pair show ?thesis by simp

lemma rtranclp_imp_Sup_relpowp:
  assumes "(P**) x y"
  shows "(n. P ^^ n) x y"
  using assms and rtrancl_imp_UN_relpow [of "(x, y)", to_pred] by simp

lemma relpow_imp_rtrancl:
  assumes "p  R ^^ n"
  shows "p  R*"
proof (cases p)
  case (Pair x y)
  with assms have "(x, y)  R ^^ n" by simp
  then have "(x, y)  R*"
  proof (induct n arbitrary: x y)
    case 0
    then show ?case by simp
    case Suc
    then show ?case
      by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
  with Pair show ?thesis by simp

lemma relpowp_imp_rtranclp: "(P ^^ n) x y  (P**) x y"
  using relpow_imp_rtrancl [of "(x, y)", to_pred] by simp

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

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

lemma rtrancl_power: "p  R*  (n. p  R ^^ n)"
  by (simp add: rtrancl_is_UN_relpow)

lemma rtranclp_power: "(P**) x y  (n. (P ^^ n) x y)"
  by (simp add: rtranclp_is_Sup_relpowp)

lemma trancl_power: "p  R+  (n > 0. p  R ^^ n)"
proof -
  have "(a, b)  R+  (n>0. (a, b)  R ^^ n)" for a b
  proof safe
    show "(a, b)  R+  n>0. (a, b)  R ^^ n"
      by (fastforce simp: rtrancl_is_UN_relpow relcomp_unfold dest: tranclD2)
    show "(a, b)  R+" if "n > 0" "(a, b)  R ^^ n" for n
    proof (cases n)
      case (Suc m)
      with that show ?thesis
        by (auto simp: dest: relpow_imp_rtrancl rtrancl_into_trancl1)
    qed (use that in auto)
  then show ?thesis
    by (cases p) auto

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

lemma rtrancl_imp_relpow: "p  R*  n. p  R ^^ n"
  by (auto dest: rtrancl_imp_UN_relpow)

lemma rtranclp_imp_relpowp: "(P**) x y  n. (P ^^ n) x y"
  by (auto dest: rtranclp_imp_Sup_relpowp)

text ‹By Sternagel/Thiemann:›
lemma relpow_fun_conv: "(a, b)  R ^^ n  (f. f 0 = a  f n = b  (i<n. (f i, f (Suc i))  R))"
proof (induct n arbitrary: b)
  case 0
  show ?case by auto
  case (Suc n)
  show ?case
  proof -
    have "(y. (f. f 0 = a  f n = y  (i<n. (f i,f(Suc i))  R))  (y,b)  R) 
      (f. f 0 = a  f(Suc n) = b  (i<Suc n. (f i, f (Suc i))  R))"
      (is "?l  ?r")
      assume ?l
      then obtain c f
        where 1: "f 0 = a"  "f n = c"  "i. i < n  (f i, f (Suc i))  R"  "(c,b)  R"
        by auto
      let ?g = "λ m. if m = Suc n then b else f m"
      show ?r by (rule exI[of _ ?g]) (simp add: 1)
      assume ?r
      then obtain f where 1: "f 0 = a"  "b = f (Suc n)"  "i. i < Suc n  (f i, f (Suc i))  R"
        by auto
      show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], auto simp add: 1)
    then show ?thesis by (simp add: relcomp_unfold Suc)

lemma relpowp_fun_conv: "(P ^^ n) x y  (f. f 0 = x  f n = y  (i<n. P (f i) (f (Suc i))))"
  by (fact relpow_fun_conv [to_pred])

lemma relpow_finite_bounded1:
  fixes R :: "('a × 'a) set"
  assumes "finite R" and "k > 0"
  shows "R^^k  (n{n. 0 < n  n  card R}. R^^n)"
    (is "_  ?r")
proof -
  have "(a, b)  R^^(Suc k)  n. 0 < n  n  card R  (a, b)  R^^n" for a b k
  proof (induct k arbitrary: b)
    case 0
    then have "R  {}" by auto
    with card_0_eq[OF finite R] have "card R  Suc 0" by auto
    then show ?case using 0 by force
    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  card R" and "(a, a')  R ^^ n"
      by auto
    have "(a, b)  R^^(Suc n)"
      using (a, a')  R^^n and (a', b) R by auto
    from n  card R consider "n < card R" | "n = card R" by force
    then show ?case
    proof cases
      case 1
      then show ?thesis
        using (a, b)  R^^(Suc n) Suc_leI[OF n < card R] by blast
      case 2
      from (a, b)  R ^^ (Suc n) [unfolded relpow_fun_conv]
      obtain f where "f 0 = a" and "f (Suc n) = b"
        and steps: "i. i  n  (f i, f (Suc i))  R" by auto
      let ?p = "λi. (f i, f(Suc i))"
      let ?N = "{i. i  n}"
      have "?p ` ?N  R"
        using steps by auto
      from card_mono[OF assms(1) this] have "card (?p ` ?N)  card R" .
      also have " < 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  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 = "λl. if l  i then f l else f (l + (j - i))"
      let ?n = "Suc (n - (j - i))"
      have abl: "(a, b)  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
        fix k
        assume "k < ?n"
        show "(?g k, ?g (Suc k))  R"
        proof (cases "k < i")
          case True
          with i have "k  n"
            by auto
          from steps[OF this] show ?thesis
            using True by simp
          case False
          then have "i  k" by auto
          show ?thesis
          proof (cases "k = i")
            case True
            then show ?thesis
              using ij pij steps[OF i] by simp
            case False
            with i  k have "i < k" by auto
            then have small: "k + (j - i)  n"
              using k<?n by arith
            show ?thesis
              using steps[OF small] i<k by auto
      qed (simp add: f 0 = a)
      moreover have "?n  n"
        using i j ij by arith
      ultimately show ?thesis
        using n = card R by blast
  then show ?thesis
    using gr0_implies_Suc[OF k > 0] by auto

lemma relpow_finite_bounded:
  fixes R :: "('a × 'a) set"
  assumes "finite R"
  shows "R^^k  (n{n. n  card R}. R^^n)"
proof (cases k)
  case (Suc k')
  then show ?thesis
    using relpow_finite_bounded1[OF assms, of k] by auto
qed force

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

lemma trancl_finite_eq_relpow:
  assumes "finite R" shows "R+ = (n{n. 0 < n  n  card R}. R^^n)"
proof -
  have "a b n. 0 < n; (a, b)  R ^^ n  x>0. x  card R  (a, b)  R ^^ x"
    using assms by (auto dest: relpow_finite_bounded1)
  then show ?thesis
    by (auto simp: trancl_power)

lemma finite_relcomp[simp,intro]:
  assumes "finite R" and "finite S"
  shows "finite (R O S)"
  have "R O S = ((x, y)R. (u, v)S. if u = y then {(x, v)} else {})"
    by (force simp: split_def image_constant_conv split: if_splits)
  then show ?thesis
    using assms by clarsimp

lemma finite_relpow [simp, intro]:
  fixes R :: "('a × 'a) set"
  assumes "finite R"
  shows "n > 0  finite (R^^n)"
proof (induct n)
  case 0
  then show ?case by simp
  case (Suc n)
  then show ?case by (cases n) (use assms in simp_all)

lemma single_valued_relpow:
  fixes R :: "('a × 'a) set"
  shows "single_valued R  single_valued (R ^^ n)"
proof (induct n arbitrary: R)
  case 0
  then show ?case by simp
  case (Suc n)
  show ?case
    by (rule single_valuedI)
      (use Suc in fast dest: single_valuedD elim: relpow_Suc_E)

subsection ‹Bounded transitive closure›

definition ntrancl :: "nat  ('a × 'a) set  ('a × 'a) set"
  where "ntrancl n R = (i{i. 0 < i  i  Suc n}. R ^^ i)"

lemma ntrancl_Zero [simp, code]: "ntrancl 0 R = R"
  show "R  ntrancl 0 R"
    unfolding ntrancl_def by fastforce
  have "0 < i  i  Suc 0  i = 1" for i
    by auto
  then show "ntrancl 0 R  R"
    unfolding ntrancl_def by auto

lemma ntrancl_Suc [simp]: "ntrancl (Suc n) R = ntrancl n R O (Id  R)"
  have "(a, b)  ntrancl n R O (Id  R)" if "(a, b)  ntrancl (Suc n) R" for a b
  proof -
    from that obtain i where "0 < i" "i  Suc (Suc n)" "(a, b)  R ^^ i"
      unfolding ntrancl_def by auto
    show ?thesis
    proof (cases "i = 1")
      case True
      with (a, b)  R ^^ i show ?thesis
        by (auto simp: ntrancl_def)
      case False
      with 0 < i obtain j where j: "i = Suc j" "0 < j"
        by (cases i) auto
      with (a, b)  R ^^ i obtain c where c1: "(a, c)  R ^^ j" and c2: "(c, b)  R"
        by auto
      from c1 j i  Suc (Suc n) have "(a, c)  ntrancl n R"
        by (fastforce simp: ntrancl_def)
      with c2 show ?thesis by fastforce
  then show "ntrancl (Suc n) R  ntrancl n R O (Id  R)"
    by auto
  show "ntrancl n R O (Id  R)  ntrancl (Suc n) R"
    by (fastforce simp: ntrancl_def)

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

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

subsection ‹Acyclic relations›

definition acyclic :: "('a × 'a) set  bool"
  where "acyclic r  (x. (x,x)  r+)"

abbreviation acyclicP :: "('a  'a  bool)  bool"
  where "acyclicP r  acyclic {(x, y). r x y}"

lemma acyclic_irrefl [code]: "acyclic r  irrefl (r+)"
  by (simp add: acyclic_def irrefl_def)

lemma acyclicI: "x. (x, x)  r+  acyclic r"
  by (simp add: acyclic_def)

lemma (in preorder) acyclicI_order:
  assumes *: "a b. (a, b)  r  f b < f a"
  shows "acyclic r"
proof -
  have "f b < f a" if "(a, b)  r+" for a b
    using that by induct (auto intro: * less_trans)
  then show ?thesis
    by (auto intro!: acyclicI)

lemma acyclic_insert [iff]: "acyclic (insert (y, x) r)  acyclic r  (x, y)  r*"
  by (simp add: acyclic_def trancl_insert) (blast intro: rtrancl_trans)

lemma acyclic_converse [iff]: "acyclic (r¯)  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*)"
  by (simp add: acyclic_def antisym_def)
    (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)

(* 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"
  unfolding acyclic_def by (blast intro: trancl_mono)

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 for t =
          fun dec Const_Set.member _ for Const_Pair _ _ for a b rel =
                fun decr Const_rtrancl _ for r = (r,"r*")
                  | decr Const_trancl _ for 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 for t =
          fun dec (rel $ a $ b) =
              fun decr Const_rtranclp _ for r = (r,"r*")
                | decr Const_tranclp _ for 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))

lemma transp_rtranclp [simp]: "transp R**"
  by(auto simp add: transp_def)

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)›