(* Title: HOL/Transitive_Closure.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
header {* Reflexive and Transitive closure of a relation *}
theory Transitive_Closure
imports Predicate
uses "~~/src/Provers/trancl.ML"
begin
text {*
@{text rtrancl} is reflexive/transitive closure,
@{text trancl} is transitive closure,
@{text reflcl} is reflexive closure.
These postfix operators have \emph{maximum priority}, forcing their
operands to be atomic.
*}
inductive_set
rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_^*)" [1000] 999)
for r :: "('a \<times> 'a) set"
where
rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
| rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
inductive_set
trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_^+)" [1000] 999)
for r :: "('a \<times> 'a) set"
where
r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
| trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a, c) : r^+"
declare rtrancl_def [nitpick_unfold del]
rtranclp_def [nitpick_unfold del]
trancl_def [nitpick_unfold del]
tranclp_def [nitpick_unfold del]
notation
rtranclp ("(_^**)" [1000] 1000) and
tranclp ("(_^++)" [1000] 1000)
abbreviation
reflclp :: "('a => 'a => bool) => 'a => 'a => bool" ("(_^==)" [1000] 1000) where
"r^== == sup r op ="
abbreviation
reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) where
"r^= == r \<union> Id"
notation (xsymbols)
rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
reflclp ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
rtrancl ("(_\<^sup>*)" [1000] 999) and
trancl ("(_\<^sup>+)" [1000] 999) and
reflcl ("(_\<^sup>=)" [1000] 999)
notation (HTML output)
rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
reflclp ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
rtrancl ("(_\<^sup>*)" [1000] 999) and
trancl ("(_\<^sup>+)" [1000] 999) and
reflcl ("(_\<^sup>=)" [1000] 999)
subsection {* Reflexive closure *}
lemma refl_reflcl[simp]: "refl(r^=)"
by(simp add:refl_on_def)
lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"
by(simp add:antisym_def)
lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"
unfolding trans_def by blast
subsection {* Reflexive-transitive closure *}
lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r \<union> Id)"
by (auto simp add: fun_eq_iff)
lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
-- {* @{text rtrancl} of @{text r} contains @{text r} *}
apply (simp only: split_tupled_all)
apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
done
lemma r_into_rtranclp [intro]: "r x y ==> r^** x y"
-- {* @{text rtrancl} of @{text r} contains @{text r} *}
by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**"
-- {* monotonicity of @{text rtrancl} *}
apply (rule predicate2I)
apply (erule rtranclp.induct)
apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)
done
lemmas rtrancl_mono = rtranclp_mono [to_set]
theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
assumes a: "r^** a b"
and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"
shows "P b" using a
by (induct x\<equiv>a b) (rule cases)+
lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]
lemmas rtranclp_induct2 =
rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule,
consumes 1, case_names refl step]
lemmas rtrancl_induct2 =
rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names refl step]
lemma refl_rtrancl: "refl (r^*)"
by (unfold refl_on_def) fast
text {* Transitivity of transitive closure. *}
lemma trans_rtrancl: "trans (r^*)"
proof (rule transI)
fix x y z
assume "(x, y) \<in> r\<^sup>*"
assume "(y, z) \<in> r\<^sup>*"
then show "(x, z) \<in> r\<^sup>*"
proof induct
case base
show "(x, y) \<in> r\<^sup>*" by fact
next
case (step u v)
from `(x, u) \<in> r\<^sup>*` and `(u, v) \<in> r`
show "(x, v) \<in> r\<^sup>*" ..
qed
qed
lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
lemma rtranclp_trans:
assumes xy: "r^** x y"
and yz: "r^** y z"
shows "r^** x z" using yz xy
by induct iprover+
lemma rtranclE [cases set: rtrancl]:
assumes major: "(a::'a, b) : r^*"
obtains
(base) "a = b"
| (step) y where "(a, y) : r^*" and "(y, b) : r"
-- {* elimination of @{text rtrancl} -- by induction on a special formula *}
apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
apply (rule_tac [2] major [THEN rtrancl_induct])
prefer 2 apply blast
prefer 2 apply blast
apply (erule asm_rl exE disjE conjE base step)+
done
lemma rtrancl_Int_subset: "[| Id \<subseteq> s; (r^* \<inter> s) O r \<subseteq> s|] ==> r^* \<subseteq> s"
apply (rule subsetI)
apply (rule_tac p="x" in PairE, clarify)
apply (erule rtrancl_induct, auto)
done
lemma converse_rtranclp_into_rtranclp:
"r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
by (rule rtranclp_trans) iprover+
lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]
text {*
\medskip More @{term "r^*"} equations and inclusions.
*}
lemma rtranclp_idemp [simp]: "(r^**)^** = r^**"
apply (auto intro!: order_antisym)
apply (erule rtranclp_induct)
apply (rule rtranclp.rtrancl_refl)
apply (blast intro: rtranclp_trans)
done
lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
apply (rule set_eqI)
apply (simp only: split_tupled_all)
apply (blast intro: rtrancl_trans)
done
lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
apply (drule rtrancl_mono)
apply simp
done
lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**"
apply (drule rtranclp_mono)
apply (drule rtranclp_mono)
apply simp
done
lemmas rtrancl_subset = rtranclp_subset [to_set]
lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**"
by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])
lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]
lemma rtranclp_reflcl [simp]: "(R^==)^** = R^**"
by (blast intro!: rtranclp_subset)
lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set]
lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
apply (rule sym)
apply (rule rtrancl_subset, blast, clarify)
apply (rename_tac a b)
apply (case_tac "a = b")
apply blast
apply (blast intro!: r_into_rtrancl)
done
lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**"
apply (rule sym)
apply (rule rtranclp_subset)
apply blast+
done
theorem rtranclp_converseD:
assumes r: "(r^--1)^** x y"
shows "r^** y x"
proof -
from r show ?thesis
by induct (iprover intro: rtranclp_trans dest!: conversepD)+
qed
lemmas rtrancl_converseD = rtranclp_converseD [to_set]
theorem rtranclp_converseI:
assumes "r^** y x"
shows "(r^--1)^** x y"
using assms
by induct (iprover intro: rtranclp_trans conversepI)+
lemmas rtrancl_converseI = rtranclp_converseI [to_set]
lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
lemma sym_rtrancl: "sym r ==> sym (r^*)"
by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
theorem converse_rtranclp_induct [consumes 1, case_names base step]:
assumes major: "r^** a b"
and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"
shows "P a"
using rtranclp_converseI [OF major]
by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+
lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
lemmas converse_rtranclp_induct2 =
converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
consumes 1, case_names refl step]
lemmas converse_rtrancl_induct2 =
converse_rtrancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names refl step]
lemma converse_rtranclpE [consumes 1, case_names base step]:
assumes major: "r^** x z"
and cases: "x=z ==> P"
"!!y. [| r x y; r^** y z |] ==> P"
shows P
apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")
apply (rule_tac [2] major [THEN converse_rtranclp_induct])
prefer 2 apply iprover
prefer 2 apply iprover
apply (erule asm_rl exE disjE conjE cases)+
done
lemmas converse_rtranclE = converse_rtranclpE [to_set]
lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]
lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
by (blast elim: rtranclE converse_rtranclE
intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
lemma rtrancl_unfold: "r^* = Id Un r^* O r"
by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
lemma rtrancl_Un_separatorE:
"(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (a,x) : P^* \<longrightarrow> (x,y) : Q \<longrightarrow> x=y \<Longrightarrow> (a,b) : P^*"
apply (induct rule:rtrancl.induct)
apply blast
apply (blast intro:rtrancl_trans)
done
lemma rtrancl_Un_separator_converseE:
"(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (x,b) : P^* \<longrightarrow> (y,x) : Q \<longrightarrow> y=x \<Longrightarrow> (a,b) : P^*"
apply (induct rule:converse_rtrancl_induct)
apply blast
apply (blast intro:rtrancl_trans)
done
lemma Image_closed_trancl:
assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
proof -
from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto
have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X"
proof -
fix x y
assume *: "y \<in> X"
assume "(y, x) \<in> r\<^sup>*"
then show "x \<in> X"
proof induct
case base show ?case by (fact *)
next
case step with ** show ?case by auto
qed
qed
then show ?thesis by auto
qed
subsection {* Transitive closure *}
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
apply (simp add: split_tupled_all)
apply (erule trancl.induct)
apply (iprover dest: subsetD)+
done
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
by (simp only: split_tupled_all) (erule r_into_trancl)
text {*
\medskip Conversions between @{text trancl} and @{text rtrancl}.
*}
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
by (erule tranclp.induct) iprover+
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
shows "!!c. r b c ==> r^++ a c" using r
by induct iprover+
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
-- {* intro rule from @{text r} and @{text rtrancl} *}
apply (erule rtranclp.cases)
apply iprover
apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
apply (simp | rule r_into_rtranclp)+
done
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
text {* Nice induction rule for @{text trancl} *}
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
assumes a: "r^++ a b"
and cases: "!!y. r a y ==> P y"
"!!y z. r^++ a y ==> r y z ==> P y ==> P z"
shows "P b" using a
by (induct x\<equiv>a b) (iprover intro: cases)+
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
lemmas tranclp_induct2 =
tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
consumes 1, case_names base step]
lemmas trancl_induct2 =
trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names base step]
lemma tranclp_trans_induct:
assumes major: "r^++ x y"
and cases: "!!x y. r x y ==> P x y"
"!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
shows "P x y"
-- {* Another induction rule for trancl, incorporating transitivity *}
by (iprover intro: major [THEN tranclp_induct] cases)
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
lemma tranclE [cases set: trancl]:
assumes "(a, b) : r^+"
obtains
(base) "(a, b) : r"
| (step) c where "(a, c) : r^+" and "(c, b) : r"
using assms by cases simp_all
lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
apply (rule subsetI)
apply (rule_tac p = x in PairE)
apply clarify
apply (erule trancl_induct)
apply auto
done
lemma trancl_unfold: "r^+ = r Un r^+ O r"
by (auto intro: trancl_into_trancl elim: tranclE)
text {* Transitivity of @{term "r^+"} *}
lemma trans_trancl [simp]: "trans (r^+)"
proof (rule transI)
fix x y z
assume "(x, y) \<in> r^+"
assume "(y, z) \<in> r^+"
then show "(x, z) \<in> r^+"
proof induct
case (base u)
from `(x, y) \<in> r^+` and `(y, u) \<in> r`
show "(x, u) \<in> r^+" ..
next
case (step u v)
from `(x, u) \<in> r^+` and `(u, v) \<in> r`
show "(x, v) \<in> r^+" ..
qed
qed
lemmas trancl_trans = trans_trancl [THEN transD, standard]
lemma tranclp_trans:
assumes xy: "r^++ x y"
and yz: "r^++ y z"
shows "r^++ x z" using yz xy
by induct iprover+
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
apply auto
apply (erule trancl_induct)
apply assumption
apply (unfold trans_def)
apply blast
done
lemma rtranclp_tranclp_tranclp:
assumes "r^** x y"
shows "!!z. r^++ y z ==> r^++ x z" using assms
by induct (iprover intro: tranclp_trans)+
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
by (erule tranclp_trans [OF tranclp.r_into_trancl])
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
lemma trancl_insert:
"(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
-- {* primitive recursion for @{text trancl} over finite relations *}
apply (rule equalityI)
apply (rule subsetI)
apply (simp only: split_tupled_all)
apply (erule trancl_induct, blast)
apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
apply (rule subsetI)
apply (blast intro: trancl_mono rtrancl_mono
[THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
done
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
apply (drule conversepD)
apply (erule tranclp_induct)
apply (iprover intro: conversepI tranclp_trans)+
done
lemmas trancl_converseI = tranclp_converseI [to_set]
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
apply (rule conversepI)
apply (erule tranclp_induct)
apply (iprover dest: conversepD intro: tranclp_trans)+
done
lemmas trancl_converseD = tranclp_converseD [to_set]
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
by (fastsimp simp add: fun_eq_iff
intro!: tranclp_converseI dest!: tranclp_converseD)
lemmas trancl_converse = tranclp_converse [to_set]
lemma sym_trancl: "sym r ==> sym (r^+)"
by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
lemma converse_tranclp_induct [consumes 1, case_names base step]:
assumes major: "r^++ a b"
and cases: "!!y. r y b ==> P(y)"
"!!y z.[| r y z; r^++ z b; P(z) |] ==> P(y)"
shows "P a"
apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
apply (rule cases)
apply (erule conversepD)
apply (blast intro: assms dest!: tranclp_converseD)
done
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
apply (erule converse_tranclp_induct)
apply auto
apply (blast intro: rtranclp_trans)
done
lemmas tranclD = tranclpD [to_set]
lemma converse_tranclpE:
assumes major: "tranclp r x z"
assumes base: "r x z ==> P"
assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
shows P
proof -
from tranclpD[OF major]
obtain y where "r x y" and "rtranclp r y z" by iprover
from this(2) show P
proof (cases rule: rtranclp.cases)
case rtrancl_refl
with `r x y` base show P by iprover
next
case rtrancl_into_rtrancl
from this have "tranclp r y z"
by (iprover intro: rtranclp_into_tranclp1)
with `r x y` step show P by iprover
qed
qed
lemmas converse_tranclE = converse_tranclpE [to_set]
lemma tranclD2:
"(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
by (blast elim: tranclE intro: trancl_into_rtrancl)
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
by (blast elim: tranclE dest: trancl_into_rtrancl)
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
by (blast dest: r_into_trancl)
lemma trancl_subset_Sigma_aux:
"(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
by (induct rule: rtrancl_induct) auto
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
apply (rule subsetI)
apply (simp only: split_tupled_all)
apply (erule tranclE)
apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
done
lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"
apply (safe intro!: order_antisym)
apply (erule tranclp_into_rtranclp)
apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
done
lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
apply safe
apply (drule trancl_into_rtrancl, simp)
apply (erule rtranclE, safe)
apply (rule r_into_trancl, simp)
apply (rule rtrancl_into_trancl1)
apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
done
lemma trancl_empty [simp]: "{}^+ = {}"
by (auto elim: trancl_induct)
lemma rtrancl_empty [simp]: "{}^* = Id"
by (rule subst [OF reflcl_trancl]) simp
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)
lemmas rtranclD = rtranclpD [to_set]
lemma rtrancl_eq_or_trancl:
"(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
by (fast elim: trancl_into_rtrancl dest: rtranclD)
lemma trancl_unfold_right: "r^+ = r^* O r"
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
lemma trancl_unfold_left: "r^+ = r O r^*"
by (auto dest: tranclD intro: rtrancl_into_trancl2)
text {* Simplifying nested closures *}
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
by (simp add: trans_rtrancl)
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
by (subst reflcl_trancl[symmetric]) simp
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
by auto
text {* @{text Domain} and @{text Range} *}
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
by blast
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
by blast
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
by (rule rtrancl_Un_rtrancl [THEN subst]) fast
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
by (blast intro: subsetD [OF rtrancl_Un_subset])
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
by (unfold Domain_def) (blast dest: tranclD)
lemma trancl_range [simp]: "Range (r^+) = Range r"
unfolding Range_def by(simp add: trancl_converse [symmetric])
lemma Not_Domain_rtrancl:
"x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
apply auto
apply (erule rev_mp)
apply (erule rtrancl_induct)
apply auto
done
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
apply clarify
apply (erule trancl_induct)
apply (auto simp add: Field_def)
done
lemma finite_trancl[simp]: "finite (r^+) = finite r"
apply auto
prefer 2
apply (rule trancl_subset_Field2 [THEN finite_subset])
apply (rule finite_SigmaI)
prefer 3
apply (blast intro: r_into_trancl' finite_subset)
apply (auto simp add: finite_Field)
done
text {* More about converse @{text rtrancl} and @{text trancl}, should
be merged with main body. *}
lemma single_valued_confluent:
"\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
\<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
apply (erule rtrancl_induct)
apply simp
apply (erule disjE)
apply (blast elim:converse_rtranclE dest:single_valuedD)
apply(blast intro:rtrancl_trans)
done
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
by (fast intro: trancl_trans)
lemma trancl_into_trancl [rule_format]:
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
apply (erule trancl_induct)
apply (fast intro: r_r_into_trancl)
apply (fast intro: r_r_into_trancl trancl_trans)
done
lemma tranclp_rtranclp_tranclp:
"r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
apply (drule tranclpD)
apply (elim exE conjE)
apply (drule rtranclp_trans, assumption)
apply (drule rtranclp_into_tranclp2, assumption, assumption)
done
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
lemmas transitive_closure_trans [trans] =
r_r_into_trancl trancl_trans rtrancl_trans
trancl.trancl_into_trancl trancl_into_trancl2
rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
rtrancl_trancl_trancl trancl_rtrancl_trancl
lemmas transitive_closurep_trans' [trans] =
tranclp_trans rtranclp_trans
tranclp.trancl_into_trancl tranclp_into_tranclp2
rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
declare trancl_into_rtrancl [elim]
subsection {* The power operation on relations *}
text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
overloading
relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
begin
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
"relpow 0 R = Id"
| "relpow (Suc n) R = (R ^^ n) O R"
end
lemma rel_pow_1 [simp]:
fixes R :: "('a \<times> 'a) set"
shows "R ^^ 1 = R"
by simp
lemma rel_pow_0_I:
"(x, x) \<in> R ^^ 0"
by simp
lemma rel_pow_Suc_I:
"(x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
by auto
lemma rel_pow_Suc_I2:
"(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
by (induct n arbitrary: z) (simp, fastsimp)
lemma rel_pow_0_E:
"(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
by simp
lemma rel_pow_Suc_E:
"(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
by auto
lemma rel_pow_E:
"(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
\<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
\<Longrightarrow> P"
by (cases n) auto
lemma rel_pow_Suc_D2:
"(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
apply (induct n arbitrary: x z)
apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
done
lemma rel_pow_Suc_E2:
"(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
by (blast dest: rel_pow_Suc_D2)
lemma rel_pow_Suc_D2':
"\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"
by (induct n) (simp_all, blast)
lemma rel_pow_E2:
"(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
\<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
\<Longrightarrow> P"
apply (cases n, simp)
apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
done
lemma rel_pow_add: "R ^^ (m+n) = R^^m O R^^n"
by(induct n) auto
lemma rel_pow_commute: "R O R ^^ n = R ^^ n O R"
by (induct n) (simp, simp add: O_assoc [symmetric])
lemma rtrancl_imp_UN_rel_pow:
assumes "p \<in> R^*"
shows "p \<in> (\<Union>n. R ^^ n)"
proof (cases p)
case (Pair x y)
with assms have "(x, y) \<in> R^*" by simp
then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
case base show ?case by (blast intro: rel_pow_0_I)
next
case step then show ?case by (blast intro: rel_pow_Suc_I)
qed
with Pair show ?thesis by simp
qed
lemma rel_pow_imp_rtrancl:
assumes "p \<in> R ^^ n"
shows "p \<in> R^*"
proof (cases p)
case (Pair x y)
with assms have "(x, y) \<in> R ^^ n" by simp
then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
case 0 then show ?case by simp
next
case Suc then show ?case
by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
qed
with Pair show ?thesis by simp
qed
lemma rtrancl_is_UN_rel_pow:
"R^* = (\<Union>n. R ^^ n)"
by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
lemma rtrancl_power:
"p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
by (simp add: rtrancl_is_UN_rel_pow)
lemma trancl_power:
"p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
apply (cases p)
apply simp
apply (rule iffI)
apply (drule tranclD2)
apply (clarsimp simp: rtrancl_is_UN_rel_pow)
apply (rule_tac x="Suc n" in exI)
apply (clarsimp simp: rel_comp_def)
apply fastsimp
apply clarsimp
apply (case_tac n, simp)
apply clarsimp
apply (drule rel_pow_imp_rtrancl)
apply (drule rtrancl_into_trancl1) apply auto
done
lemma rtrancl_imp_rel_pow:
"p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
by (auto dest: rtrancl_imp_UN_rel_pow)
text{* By Sternagel/Thiemann: *}
lemma rel_pow_fun_conv:
"((a,b) \<in> R ^^ n) = (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f(Suc i)) \<in> R))"
proof (induct n arbitrary: b)
case 0 show ?case by auto
next
case (Suc n)
show ?case
proof (simp add: rel_comp_def Suc)
show "(\<exists>y. (\<exists>f. f 0 = a \<and> f n = y \<and> (\<forall>i<n. (f i,f(Suc i)) \<in> R)) \<and> (y,b) \<in> R)
= (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
(is "?l = ?r")
proof
assume ?l
then obtain c f where 1: "f 0 = a" "f n = c" "\<And>i. i < n \<Longrightarrow> (f i, f (Suc i)) \<in> R" "(c,b) \<in> R" by auto
let ?g = "\<lambda> m. if m = Suc n then b else f m"
show ?r by (rule exI[of _ ?g], simp add: 1)
next
assume ?r
then obtain f where 1: "f 0 = a" "b = f (Suc n)" "\<And>i. i < Suc n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
qed
qed
qed
lemma rel_pow_finite_bounded1:
assumes "finite(R :: ('a*'a)set)" and "k>0"
shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r")
proof-
{ fix a b k
have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n"
proof(induct k arbitrary: b)
case 0
hence "R \<noteq> {}" by auto
with card_0_eq[OF `finite R`] have "card R >= Suc 0" by auto
thus ?case using 0 by force
next
case (Suc k)
then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto
from Suc(1)[OF `(a,a') : R^^(Suc k)`]
obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto
have "(a,b) : R^^(Suc n)" using `(a,a') \<in> R^^n` and `(a',b)\<in> R` by auto
{ assume "n < card R"
hence ?case using `(a,b): R^^(Suc n)` Suc_leI[OF `n < card R`] by blast
} moreover
{ assume "n = card R"
from `(a,b) \<in> R ^^ (Suc n)`[unfolded rel_pow_fun_conv]
obtain f where "f 0 = a" and "f(Suc n) = b"
and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
let ?p = "%i. (f i, f(Suc i))"
let ?N = "{i. i \<le> n}"
have "?p ` ?N <= R" using steps by auto
from card_mono[OF assms(1) this]
have "card(?p ` ?N) <= card R" .
also have "\<dots> < card ?N" using `n = card R` by simp
finally have "~ inj_on ?p ?N" by(rule pigeonhole)
then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and
pij: "?p i = ?p j" by(auto simp: inj_on_def)
let ?i = "min i j" let ?j = "max i j"
have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j"
and ij: "?i < ?j"
using i j ij pij unfolding min_def max_def by auto
from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j"
and pij: "?p i = ?p j" by blast
let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j - i))"
let ?n = "Suc(n - (j - i))"
have abl: "(a,b) \<in> R ^^ ?n" unfolding rel_pow_fun_conv
proof (rule exI[of _ ?g], intro conjI impI allI)
show "?g ?n = b" using `f(Suc n) = b` j ij by auto
next
fix k assume "k < ?n"
show "(?g k, ?g (Suc k)) \<in> R"
proof (cases "k < i")
case True
with i have "k <= n" by auto
from steps[OF this] show ?thesis using True by simp
next
case False
hence "i \<le> k" by auto
show ?thesis
proof (cases "k = i")
case True
thus ?thesis using ij pij steps[OF i] by simp
next
case False
with `i \<le> k` have "i < k" by auto
hence small: "k + (j - i) <= n" using `k<?n` by arith
show ?thesis using steps[OF small] `i<k` by auto
qed
qed
qed (simp add: `f 0 = a`)
moreover have "?n <= n" using i j ij by arith
ultimately have ?case using `n = card R` by blast
}
ultimately show ?case using `n \<le> card R` by force
qed
}
thus ?thesis using gr0_implies_Suc[OF `k>0`] by auto
qed
lemma rel_pow_finite_bounded:
assumes "finite(R :: ('a*'a)set)"
shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)"
apply(cases k)
apply force
using rel_pow_finite_bounded1[OF assms, of k] by auto
lemma rtrancl_finite_eq_rel_pow:
"finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"
by(fastsimp simp: rtrancl_power dest: rel_pow_finite_bounded)
lemma trancl_finite_eq_rel_pow:
"finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"
apply(auto simp add: trancl_power)
apply(auto dest: rel_pow_finite_bounded1)
done
lemma finite_rel_comp[simp,intro]:
assumes "finite R" and "finite S"
shows "finite(R O S)"
proof-
have "R O S = (UN (x,y) : R. \<Union>((%(u,v). if u=y then {(x,v)} else {}) ` S))"
by(force simp add: split_def)
thus ?thesis using assms by(clarsimp)
qed
lemma finite_relpow[simp,intro]:
assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)"
apply(induct n)
apply simp
apply(case_tac n)
apply(simp_all add: assms)
done
lemma single_valued_rel_pow:
fixes R :: "('a * 'a) set"
shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
apply (induct n arbitrary: R)
apply simp_all
apply (rule single_valuedI)
apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
done
subsection {* Setup of transitivity reasoner *}
ML {*
structure Trancl_Tac = Trancl_Tac
(
val r_into_trancl = @{thm trancl.r_into_trancl};
val trancl_trans = @{thm trancl_trans};
val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
val r_into_rtrancl = @{thm r_into_rtrancl};
val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
val rtrancl_trans = @{thm rtrancl_trans};
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 {*
Simplifier.map_simpset_global (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