Theory Transitive_Closure

Up to index of Isabelle/HOL-Proofs

theory Transitive_Closure
imports Predicate
uses $ISABELLE_HOME/src/Provers/trancl.ML
(*  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 × 'a) set => ('a × 'a) set" ("(_^*)" [1000] 999)
for r :: "('a × '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 × 'a) set => ('a × 'a) set" ("(_^+)" [1000] 999)
for r :: "('a × '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_def del]
rtranclp_def [nitpick_def del]
trancl_def [nitpick_def del]
tranclp_def [nitpick_def 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 × 'a) set => ('a × 'a) set" ("(_^=)" [1000] 999) where
"r^= == r ∪ Id"


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


notation (HTML output)
rtranclp ("(_**)" [1000] 1000) and
tranclp ("(_++)" [1000] 1000) and
reflclp ("(_==)" [1000] 1000) and
rtrancl ("(_*)" [1000] 999) and
trancl ("(_+)" [1000] 999) and
reflcl ("(_=)" [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 ==> trans(r^=)"
unfolding trans_def by blast


subsection {* Reflexive-transitive closure *}

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

lemma r_into_rtrancl [intro]: "!!p. p ∈ r ==> p ∈ 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 ≤ s ==> r^** ≤ 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 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^*)"
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) ∈ r*"
assume "(y, z) ∈ r*"
then show "(x, z) ∈ r*"
proof induct
case base
show "(x, y) ∈ r*" by fact
next
case (step u v)
from `(x, u) ∈ r*` and `(u, v) ∈ r`
show "(x, v) ∈ r*" ..
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 ⊆ s; (r^* ∩ s) O r ⊆ s|] ==> r^* ⊆ 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 ==> r** b c ==> r** 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 ⊆ s^* ==> r^* ⊆ s^*"
apply (drule rtrancl_mono)
apply simp
done

lemma rtranclp_subset: "R ≤ S ==> S ≤ 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 ∪ Q)^* ==> ∀x y. (a,x) : P^* --> (x,y) : Q --> x=y ==> (a,b) : P^*"

apply (induct rule:rtrancl.induct)
apply blast
apply (blast intro:rtrancl_trans)
done

lemma rtrancl_Un_separator_converseE:
"(a,b) : (P ∪ Q)^* ==> ∀x y. (x,b) : P^* --> (y,x) : Q --> y=x ==> (a,b) : P^*"

apply (induct rule:converse_rtrancl_induct)
apply blast
apply (blast intro:rtrancl_trans)
done

lemma Image_closed_trancl:
assumes "r `` X ⊆ X" shows "r* `` X = X"

proof -
from assms have **: "{y. ∃x∈X. (x, y) ∈ r} ⊆ X" by auto
have "!!x y. (y, x) ∈ r* ==> y ∈ X ==> x ∈ X"
proof -
fix x y
assume *: "y ∈ X"
assume "(y, x) ∈ r*"
then show "x ∈ 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 ∈ r^+ ==> r ⊆ s ==> p ∈ 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 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^+"
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 ⊆ s; (r^+ ∩ s) O r ⊆ s|] ==> r^+ ⊆ 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) ∈ 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^+" ..
next
case (step u v)
from `(x, u) ∈ r^+` and `(u, v) ∈ r`
show "(x, v) ∈ 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 ==> 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^+ ∪ {(a, b). (a, y) ∈ r^* ∧ (x, b) ∈ 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 ∧ 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: "!! 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) ∈ R+ ==> ∃z. (x, z) ∈ R* ∧ (z, y) ∈ R"

by (blast elim: tranclE intro: trancl_into_rtrancl)

lemma irrefl_tranclI: "r^-1 ∩ r^* = {} ==> (x, x) ∉ r^+"
by (blast elim: tranclE dest: trancl_into_rtrancl)

lemma irrefl_trancl_rD: "!!X. ALL 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: "r ⊆ A × A ==> r^+ ⊆ A × 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 ∨ a ≠ b ∧ 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) ∈ 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)


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^* ∪ S^*) ⊆ (R Un 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_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 × Field r"
apply clarify
apply (erule trancl_induct)
apply (auto simp add: Field_def)
done

lemma finite_trancl: "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:
"[| single_valued r; (x,y) ∈ r^*; (x,z) ∈ r^* |]
==> (y,z) ∈ r^* ∨ (z,y) ∈ 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) ∈ R ==> (b, c) ∈ R ==> (a, c) ∈ R^+"
by (fast intro: trancl_trans)

lemma trancl_into_trancl [rule_format]:
"(a, b) ∈ r+ ==> (b, c) ∈ r --> (a,c) ∈ r+"

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++ a b ==> r** b c ==> r++ 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 => ('a × 'a) set => ('a × 'a) set"
begin


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


end

lemma rel_pow_1 [simp]:
fixes R :: "('a × 'a) set"
shows "R ^^ 1 = R"

by simp

lemma rel_pow_0_I:
"(x, x) ∈ R ^^ 0"

by simp

lemma rel_pow_Suc_I:
"(x, y) ∈ R ^^ n ==> (y, z) ∈ R ==> (x, z) ∈ R ^^ Suc n"

by auto

lemma rel_pow_Suc_I2:
"(x, y) ∈ R ==> (y, z) ∈ R ^^ n ==> (x, z) ∈ R ^^ Suc n"

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

lemma rel_pow_0_E:
"(x, y) ∈ R ^^ 0 ==> (x = y ==> P) ==> P"

by simp

lemma rel_pow_Suc_E:
"(x, z) ∈ R ^^ Suc n ==> (!!y. (x, y) ∈ R ^^ n ==> (y, z) ∈ R ==> P) ==> P"

by auto

lemma rel_pow_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 rel_pow_Suc_D2:
"(x, z) ∈ R ^^ Suc n ==> (∃y. (x, y) ∈ R ∧ (y, z) ∈ 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) ∈ R ^^ Suc n ==> (!!y. (x, y) ∈ R ==> (y, z) ∈ R ^^ n ==> P) ==> P"

by (blast dest: rel_pow_Suc_D2)

lemma rel_pow_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 rel_pow_E2:
"(x, z) ∈ R ^^ n ==> (n = 0 ==> x = z ==> P)
==> (!!y m. n = Suc m ==> (x, y) ∈ R ==> (y, z) ∈ R ^^ m ==> P)
==> 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 ∈ R^*"
shows "p ∈ (\<Union>n. R ^^ n)"

proof (cases p)
case (Pair x y)
with assms have "(x, y) ∈ R^*" by simp
then have "(x, y) ∈ (\<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 ∈ 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
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 ∈ R^* <-> (∃n. p ∈ R ^^ n)"

by (simp add: rtrancl_is_UN_rel_pow)

lemma trancl_power:
"p ∈ R^+ <-> (∃n > 0. p ∈ 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 ∈ R^* ==> ∃n. p ∈ R ^^ n"

by (auto dest: rtrancl_imp_UN_rel_pow)

lemma single_valued_rel_pow:
fixes R :: "('a * 'a) set"
shows "single_valued R ==> 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};

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;
);
*}


declaration {* fn _ =>
Simplifier.map_ss (fn ss => ss
addSolver (mk_solver' "Trancl" (Trancl_Tac.trancl_tac o Simplifier.the_context))
addSolver (mk_solver' "Rtrancl" (Trancl_Tac.rtrancl_tac o Simplifier.the_context))
addSolver (mk_solver' "Tranclp" (Tranclp_Tac.trancl_tac o Simplifier.the_context))
addSolver (mk_solver' "Rtranclp" (Tranclp_Tac.rtrancl_tac o Simplifier.the_context)))
*}



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