(* Title: HOL/Transitive_Closure.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
header {* Reflexive and Transitive closure of a relation *}
theory Transitive_Closure = Inductive:
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.
*}
consts
rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^*)" [1000] 999)
inductive "r^*"
intros
rtrancl_refl [intro!, CPure.intro!, simp]: "(a, a) : r^*"
rtrancl_into_rtrancl [CPure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
consts
trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^+)" [1000] 999)
inductive "r^+"
intros
r_into_trancl [intro, CPure.intro]: "(a, b) : r ==> (a, b) : r^+"
trancl_into_trancl [CPure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+"
syntax
"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999)
translations
"r^=" == "r \<union> Id"
syntax (xsymbols)
rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>*)" [1000] 999)
trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>+)" [1000] 999)
"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>=)" [1000] 999)
syntax (HTML output)
rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>*)" [1000] 999)
trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>+)" [1000] 999)
"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>=)" [1000] 999)
subsection {* Reflexive-transitive closure *}
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 rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*"
-- {* monotonicity of @{text rtrancl} *}
apply (rule subsetI)
apply (simp only: split_tupled_all)
apply (erule rtrancl.induct)
apply (rule_tac [2] rtrancl_into_rtrancl, blast+)
done
theorem rtrancl_induct [consumes 1, induct set: rtrancl]:
assumes a: "(a, b) : r^*"
and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z"
shows "P b"
proof -
from a have "a = a --> P b"
by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
thus ?thesis by rules
qed
lemmas rtrancl_induct2 =
rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names refl step]
lemma trans_rtrancl: "trans(r^*)"
-- {* transitivity of transitive closure!! -- by induction *}
proof (rule transI)
fix x y z
assume "(x, y) \<in> r\<^sup>*"
assume "(y, z) \<in> r\<^sup>*"
thus "(x, z) \<in> r\<^sup>*" by induct (rules!)+
qed
lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
lemma rtranclE:
"[| (a::'a,b) : r^*; (a = b) ==> P;
!!y.[| (a,y) : r^*; (y,b) : r |] ==> P
|] ==> P"
-- {* elimination of @{text rtrancl} -- by induction on a special formula *}
proof -
assume major: "(a::'a,b) : r^*"
case rule_context
show ?thesis
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 prems)+
done
qed
lemma converse_rtrancl_into_rtrancl:
"(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*"
by (rule rtrancl_trans) rules+
text {*
\medskip More @{term "r^*"} equations and inclusions.
*}
lemma rtrancl_idemp [simp]: "(r^*)^* = r^*"
apply auto
apply (erule rtrancl_induct)
apply (rule rtrancl_refl)
apply (blast intro: rtrancl_trans)
done
lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
apply (rule set_ext)
apply (simp only: split_tupled_all)
apply (blast intro: rtrancl_trans)
done
lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
by (drule rtrancl_mono, simp)
lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*"
apply (drule rtrancl_mono)
apply (drule rtrancl_mono, simp)
done
lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*"
by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD])
lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*"
by (blast intro!: rtrancl_subset intro: r_into_rtrancl)
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", blast)
apply (blast intro!: r_into_rtrancl)
done
theorem rtrancl_converseD:
assumes r: "(x, y) \<in> (r^-1)^*"
shows "(y, x) \<in> r^*"
proof -
from r show ?thesis
by induct (rules intro: rtrancl_trans dest!: converseD)+
qed
theorem rtrancl_converseI:
assumes r: "(y, x) \<in> r^*"
shows "(x, y) \<in> (r^-1)^*"
proof -
from r show ?thesis
by induct (rules intro: rtrancl_trans converseI)+
qed
lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
theorem converse_rtrancl_induct[consumes 1]:
assumes major: "(a, b) : r^*"
and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y"
shows "P a"
proof -
from rtrancl_converseI [OF major]
show ?thesis
by induct (rules intro: cases dest!: converseD rtrancl_converseD)+
qed
lemmas converse_rtrancl_induct2 =
converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names refl step]
lemma converse_rtranclE:
"[| (x,z):r^*;
x=z ==> P;
!!y. [| (x,y):r; (y,z):r^* |] ==> P
|] ==> P"
proof -
assume major: "(x,z):r^*"
case rule_context
show ?thesis
apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)")
apply (rule_tac [2] major [THEN converse_rtrancl_induct])
prefer 2 apply rules
prefer 2 apply rules
apply (erule asm_rl exE disjE conjE prems)+
done
qed
ML_setup {*
bind_thm ("converse_rtranclE2", split_rule
(read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE")));
*}
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)
subsection {* Transitive closure *}
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
apply (simp only: split_tupled_all)
apply (erule trancl.induct)
apply (rules 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 trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*"
by (erule trancl.induct) rules+
lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*"
shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r
by induct rules+
lemma rtrancl_into_trancl2: "[| (a,b) : r; (b,c) : r^* |] ==> (a,c) : r^+"
-- {* intro rule from @{text r} and @{text rtrancl} *}
apply (erule rtranclE, rules)
apply (rule rtrancl_trans [THEN rtrancl_into_trancl1])
apply (assumption | rule r_into_rtrancl)+
done
lemma trancl_induct [consumes 1, induct set: trancl]:
assumes a: "(a,b) : r^+"
and cases: "!!y. (a, y) : r ==> P y"
"!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z"
shows "P b"
-- {* Nice induction rule for @{text trancl} *}
proof -
from a have "a = a --> P b"
by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
thus ?thesis by rules
qed
lemma trancl_trans_induct:
"[| (x,y) : r^+;
!!x y. (x,y) : r ==> P x y;
!!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z
|] ==> P x y"
-- {* Another induction rule for trancl, incorporating transitivity *}
proof -
assume major: "(x,y) : r^+"
case rule_context
show ?thesis
by (rules intro: r_into_trancl major [THEN trancl_induct] prems)
qed
inductive_cases tranclE: "(a, b) : r^+"
lemma trans_trancl: "trans(r^+)"
-- {* Transitivity of @{term "r^+"} *}
proof (rule transI)
fix x y z
assume "(x, y) \<in> r^+"
assume "(y, z) \<in> r^+"
thus "(x, z) \<in> r^+" by induct (rules!)+
qed
lemmas trancl_trans = trans_trancl [THEN transD, standard]
lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*"
shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r
by induct (rules intro: trancl_trans)+
lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+"
by (erule transD [OF trans_trancl r_into_trancl])
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 r_into_trancl trancl_trans)
apply (rule subsetI)
apply (blast intro: trancl_mono rtrancl_mono
[THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
done
lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+"
apply (drule converseD)
apply (erule trancl.induct)
apply (rules intro: converseI trancl_trans)+
done
lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1"
apply (rule converseI)
apply (erule trancl.induct)
apply (rules dest: converseD intro: trancl_trans)+
done
lemma trancl_converse: "(r^-1)^+ = (r^+)^-1"
by (fastsimp simp add: split_tupled_all
intro!: trancl_converseI trancl_converseD)
lemma converse_trancl_induct:
"[| (a,b) : r^+; !!y. (y,b) : r ==> P(y);
!!y z.[| (y,z) : r; (z,b) : r^+; P(z) |] ==> P(y) |]
==> P(a)"
proof -
assume major: "(a,b) : r^+"
case rule_context
show ?thesis
apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]])
apply (rule prems)
apply (erule converseD)
apply (blast intro: prems dest!: trancl_converseD)
done
qed
lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*"
apply (erule converse_trancl_induct, auto)
apply (blast intro: rtrancl_trans)
done
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"
apply (erule rtrancl_induct, auto)
done
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_trancl [simp]: "(r^+)^= = r^*"
apply safe
apply (erule trancl_into_rtrancl)
apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
done
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 rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+"
by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl)
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"
by (simp add: Range_def trancl_converse [symmetric])
lemma Not_Domain_rtrancl:
"x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
apply auto
by (erule rev_mp, erule rtrancl_induct, auto)
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 trancl_rtrancl_trancl:
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+"
apply (drule tranclD)
apply (erule exE, erule conjE)
apply (drule rtrancl_trans, assumption)
apply (drule rtrancl_into_trancl2, assumption, assumption)
done
lemmas transitive_closure_trans [trans] =
r_r_into_trancl trancl_trans rtrancl_trans
trancl_into_trancl trancl_into_trancl2
rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
rtrancl_trancl_trancl trancl_rtrancl_trancl
declare trancl_into_rtrancl [elim]
declare rtranclE [cases set: rtrancl]
declare tranclE [cases set: trancl]
end