(* Title: HOL/Transitive_Closure.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
section \<open>Reflexive and Transitive closure of a relation\<close>
theory Transitive_Closure
imports Relation
abbrevs "^*" = "\<^sup>*" "\<^sup>*\<^sup>*"
and "^+" = "\<^sup>+" "\<^sup>+\<^sup>+"
and "^=" = "\<^sup>=" "\<^sup>=\<^sup>="
begin
ML_file \<open>~~/src/Provers/trancl.ML\<close>
text \<open>
\<open>rtrancl\<close> is reflexive/transitive closure,
\<open>trancl\<close> is transitive closure,
\<open>reflcl\<close> is reflexive closure.
These postfix operators have \<^emph>\<open>maximum priority\<close>, forcing their
operands to be atomic.
\<close>
context notes [[inductive_internals]]
begin
inductive_set rtrancl :: "('a \ 'a) set \ ('a \ 'a) set" ("(_\<^sup>*)" [1000] 999)
for r :: "('a \ 'a) set"
where
rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) \ r\<^sup>*"
| rtrancl_into_rtrancl [Pure.intro]: "(a, b) \ r\<^sup>* \ (b, c) \ r \ (a, c) \ r\<^sup>*"
inductive_set trancl :: "('a \ 'a) set \ ('a \ 'a) set" ("(_\<^sup>+)" [1000] 999)
for r :: "('a \ 'a) set"
where
r_into_trancl [intro, Pure.intro]: "(a, b) \ r \ (a, b) \ r\<^sup>+"
| trancl_into_trancl [Pure.intro]: "(a, b) \ r\<^sup>+ \ (b, c) \ r \ (a, c) \ r\<^sup>+"
notation
rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000)
declare
rtrancl_def [nitpick_unfold del]
rtranclp_def [nitpick_unfold del]
trancl_def [nitpick_unfold del]
tranclp_def [nitpick_unfold del]
end
abbreviation reflcl :: "('a \ 'a) set \ ('a \ 'a) set" ("(_\<^sup>=)" [1000] 999)
where "r\<^sup>= \ r \ Id"
abbreviation reflclp :: "('a \ 'a \ bool) \ 'a \ 'a \ bool" ("(_\<^sup>=\<^sup>=)" [1000] 1000)
where "r\<^sup>=\<^sup>= \ 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 \<open>Reflexive closure\<close>
lemma refl_reflcl[simp]: "refl (r\<^sup>=)"
by (simp add: refl_on_def)
lemma antisym_reflcl[simp]: "antisym (r\<^sup>=) = antisym r"
by (simp add: antisym_def)
lemma trans_reflclI[simp]: "trans r \ trans (r\<^sup>=)"
unfolding trans_def by blast
lemma reflclp_idemp [simp]: "(P\<^sup>=\<^sup>=)\<^sup>=\<^sup>= = P\<^sup>=\<^sup>="
by blast
subsection \<open>Reflexive-transitive closure\<close>
lemma reflcl_set_eq [pred_set_conv]: "(sup (\x y. (x, y) \ r) (=)) = (\x y. (x, y) \ r \ Id)"
by (auto simp: fun_eq_iff)
lemma r_into_rtrancl [intro]: "\p. p \ r \ p \ r\<^sup>*"
\<comment> \<open>\<open>rtrancl\<close> of \<open>r\<close> contains \<open>r\<close>\<close>
by (simp add: split_tupled_all rtrancl_refl [THEN rtrancl_into_rtrancl])
lemma r_into_rtranclp [intro]: "r x y \ r\<^sup>*\<^sup>* x y"
\<comment> \<open>\<open>rtrancl\<close> of \<open>r\<close> contains \<open>r\<close>\<close>
by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
lemma rtranclp_mono: "r \ s \ r\<^sup>*\<^sup>* \ s\<^sup>*\<^sup>*"
\<comment> \<open>monotonicity of \<open>rtrancl\<close>\<close>
proof (rule predicate2I)
show "s\<^sup>*\<^sup>* x y" if "r \ s" "r\<^sup>*\<^sup>* x y" for x y
using \<open>r\<^sup>*\<^sup>* x y\<close> \<open>r \<le> s\<close>
by (induction rule: rtranclp.induct) (blast intro: rtranclp.rtrancl_into_rtrancl)+
qed
lemma mono_rtranclp[mono]: "(\a b. x a b \ y a b) \ x\<^sup>*\<^sup>* a b \ y\<^sup>*\<^sup>* 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\<^sup>*\<^sup>* a b"
and cases: "P a" "\y z. r\<^sup>*\<^sup>* 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\<^sup>*)"
unfolding refl_on_def by fast
text \<open>Transitivity of transitive closure.\<close>
lemma trans_rtrancl: "trans (r\<^sup>*)"
proof (rule transI)
fix x y z
assume "(x, y) \ r\<^sup>*"
assume "(y, z) \ r\<^sup>*"
then show "(x, z) \ r\<^sup>*"
proof induct
case base
show "(x, y) \ r\<^sup>*" by fact
next
case (step u v)
from \<open>(x, u) \<in> r\<^sup>*\<close> and \<open>(u, v) \<in> r\<close>
show "(x, v) \ r\<^sup>*" ..
qed
qed
lemmas rtrancl_trans = trans_rtrancl [THEN transD]
lemma rtranclp_trans:
assumes "r\<^sup>*\<^sup>* x y"
and "r\<^sup>*\<^sup>* y z"
shows "r\<^sup>*\<^sup>* x z"
using assms(2,1) by induct iprover+
lemma rtranclE [cases set: rtrancl]:
fixes a b :: 'a
assumes major: "(a, b) \ r\<^sup>*"
obtains
(base) "a = b"
| (step) y where "(a, y) \ r\<^sup>*" and "(y, b) \ r"
\<comment> \<open>elimination of \<open>rtrancl\<close> -- by induction on a special formula\<close>
proof -
have "a = b \ (\y. (a, y) \ r\<^sup>* \ (y, b) \ r)"
by (rule major [THEN rtrancl_induct]) blast+
then show ?thesis
by (auto intro: base step)
qed
lemma rtrancl_Int_subset: "Id \ s \ (r\<^sup>* \ s) O r \ s \ r\<^sup>* \ s"
by (fastforce elim: rtrancl_induct)
lemma converse_rtranclp_into_rtranclp: "r a b \ r\<^sup>*\<^sup>* b c \ r\<^sup>*\<^sup>* a c"
by (rule rtranclp_trans) iprover+
lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]
text \<open>\<^medskip> More \<^term>\<open>r\<^sup>*\<close> equations and inclusions.\<close>
lemma rtranclp_idemp [simp]: "(r\<^sup>*\<^sup>*)\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
proof -
have "r\<^sup>*\<^sup>*\<^sup>*\<^sup>* x y \ r\<^sup>*\<^sup>* x y" for x y
by (induction rule: rtranclp_induct) (blast intro: rtranclp_trans)+
then show ?thesis
by (auto intro!: order_antisym)
qed
lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
lemma rtrancl_idemp_self_comp [simp]: "R\<^sup>* O R\<^sup>* = R\<^sup>*"
by (force intro: rtrancl_trans)
lemma rtrancl_subset_rtrancl: "r \ s\<^sup>* \ r\<^sup>* \ s\<^sup>*"
by (drule rtrancl_mono, simp)
lemma rtranclp_subset: "R \ S \ S \ R\<^sup>*\<^sup>* \ S\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
by (fastforce dest: rtranclp_mono)
lemmas rtrancl_subset = rtranclp_subset [to_set]
lemma rtranclp_sup_rtranclp: "(sup (R\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*))\<^sup>*\<^sup>* = (sup R S)\<^sup>*\<^sup>*"
by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])
lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]
lemma rtranclp_reflclp [simp]: "(R\<^sup>=\<^sup>=)\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
by (blast intro!: rtranclp_subset)
lemmas rtrancl_reflcl [simp] = rtranclp_reflclp [to_set]
lemma rtrancl_r_diff_Id: "(r - Id)\<^sup>* = r\<^sup>*"
by (rule rtrancl_subset [symmetric]) auto
lemma rtranclp_r_diff_Id: "(inf r (\))\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
by (rule rtranclp_subset [symmetric]) auto
theorem rtranclp_converseD:
assumes "(r\\)\<^sup>*\<^sup>* x y"
shows "r\<^sup>*\<^sup>* y x"
using assms by induct (iprover intro: rtranclp_trans dest!: conversepD)+
lemmas rtrancl_converseD = rtranclp_converseD [to_set]
theorem rtranclp_converseI:
assumes "r\<^sup>*\<^sup>* y x"
shows "(r\\)\<^sup>*\<^sup>* x y"
using assms by induct (iprover intro: rtranclp_trans conversepI)+
lemmas rtrancl_converseI = rtranclp_converseI [to_set]
lemma rtrancl_converse: "(r\)\<^sup>* = (r\<^sup>*)\"
by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
lemma sym_rtrancl: "sym r \ sym (r\<^sup>*)"
by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
theorem converse_rtranclp_induct [consumes 1, case_names base step]:
assumes major: "r\<^sup>*\<^sup>* a b"
and cases: "P b" "\y z. r y z \ r\<^sup>*\<^sup>* 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\<^sup>*\<^sup>* x z"
and cases: "x = z \ P" "\y. r x y \ r\<^sup>*\<^sup>* y z \ P"
shows P
proof -
have "x = z \ (\y. r x y \ r\<^sup>*\<^sup>* y z)"
by (rule_tac major [THEN converse_rtranclp_induct]) iprover+
then show ?thesis
by (auto intro: cases)
qed
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\<^sup>* = r\<^sup>* O r"
by (blast elim: rtranclE converse_rtranclE
intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
lemma rtrancl_unfold: "r\<^sup>* = Id \ r\<^sup>* O r"
by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
lemma rtrancl_Un_separatorE:
"(a, b) \ (P \ Q)\<^sup>* \ \x y. (a, x) \ P\<^sup>* \ (x, y) \ Q \ x = y \ (a, b) \ P\<^sup>*"
proof (induct rule: rtrancl.induct)
case rtrancl_refl
then show ?case by blast
next
case rtrancl_into_rtrancl
then show ?case by (blast intro: rtrancl_trans)
qed
lemma rtrancl_Un_separator_converseE:
"(a, b) \ (P \ Q)\<^sup>* \ \x y. (x, b) \ P\<^sup>* \ (y, x) \ Q \ y = x \ (a, b) \ P\<^sup>*"
proof (induct rule: converse_rtrancl_induct)
case base
then show ?case by blast
next
case step
then show ?case by (blast intro: rtrancl_trans)
qed
lemma Image_closed_trancl:
assumes "r `` X \ X"
shows "r\<^sup>* `` X = X"
proof -
from assms have **: "{y. \x\X. (x, y) \ r} \ X"
by auto
have "x \ X" if 1: "(y, x) \ r\<^sup>*" and 2: "y \ X" for x y
proof -
from 1 show "x \ X"
proof induct
case base
show ?case by (fact 2)
next
case step
with ** show ?case by auto
qed
qed
then show ?thesis by auto
qed
subsection \<open>Transitive closure\<close>
lemma trancl_mono:
assumes "p \ r\<^sup>+" "r \ s"
shows "p \ s\<^sup>+"
proof -
have "\(a, b) \ r\<^sup>+; r \ s\ \ (a, b) \ s\<^sup>+" for a b
by (induction rule: trancl.induct) (iprover dest: subsetD)+
with assms show ?thesis
by (cases p) force
qed
lemma r_into_trancl': "\p. p \ r \ p \ r\<^sup>+"
by (simp only: split_tupled_all) (erule r_into_trancl)
text \<open>\<^medskip> Conversions between \<open>trancl\<close> and \<open>rtrancl\<close>.\<close>
lemma tranclp_into_rtranclp: "r\<^sup>+\<^sup>+ a b \ r\<^sup>*\<^sup>* a b"
by (erule tranclp.induct) iprover+
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
lemma rtranclp_into_tranclp1:
assumes "r\<^sup>*\<^sup>* a b"
shows "r b c \ r\<^sup>+\<^sup>+ 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\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
\<comment> \<open>intro rule from \<open>r\<close> and \<open>rtrancl\<close>\<close>
using \<open>r\<^sup>*\<^sup>* b c\<close>
proof (cases rule: rtranclp.cases)
case rtrancl_refl
with assms show ?thesis
by iprover
next
case rtrancl_into_rtrancl
with assms show ?thesis
by (auto intro: rtranclp_trans [THEN rtranclp_into_tranclp1])
qed
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
text \<open>Nice induction rule for \<open>trancl\<close>\<close>
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
assumes a: "r\<^sup>+\<^sup>+ a b"
and cases: "\y. r a y \ P y" "\y z. r\<^sup>+\<^sup>+ 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\<^sup>+\<^sup>+ x y"
and cases: "\x y. r x y \ P x y" "\x y z. r\<^sup>+\<^sup>+ x y \ P x y \ r\<^sup>+\<^sup>+ y z \ P y z \ P x z"
shows "P x y"
\<comment> \<open>Another induction rule for trancl, incorporating transitivity\<close>
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\<^sup>+"
obtains
(base) "(a, b) \ r"
| (step) c where "(a, c) \ r\<^sup>+" and "(c, b) \ r"
using assms by cases simp_all
lemma trancl_Int_subset: "r \ s \ (r\<^sup>+ \ s) O r \ s \ r\<^sup>+ \ s"
by (fastforce simp add: elim: trancl_induct)
lemma trancl_unfold: "r\<^sup>+ = r \ r\<^sup>+ O r"
by (auto intro: trancl_into_trancl elim: tranclE)
text \<open>Transitivity of \<^term>\<open>r\<^sup>+\<close>\<close>
lemma trans_trancl [simp]: "trans (r\<^sup>+)"
proof (rule transI)
fix x y z
assume "(x, y) \ r\<^sup>+"
assume "(y, z) \ r\<^sup>+"
then show "(x, z) \ r\<^sup>+"
proof induct
case (base u)
from \<open>(x, y) \<in> r\<^sup>+\<close> and \<open>(y, u) \<in> r\<close>
show "(x, u) \ r\<^sup>+" ..
next
case (step u v)
from \<open>(x, u) \<in> r\<^sup>+\<close> and \<open>(u, v) \<in> r\<close>
show "(x, v) \ r\<^sup>+" ..
qed
qed
lemmas trancl_trans = trans_trancl [THEN transD]
lemma tranclp_trans:
assumes "r\<^sup>+\<^sup>+ x y"
and "r\<^sup>+\<^sup>+ y z"
shows "r\<^sup>+\<^sup>+ x z"
using assms(2,1) by induct iprover+
lemma trancl_id [simp]: "trans r \ r\<^sup>+ = r"
unfolding trans_def by (fastforce simp add: elim: trancl_induct)
lemma rtranclp_tranclp_tranclp:
assumes "r\<^sup>*\<^sup>* x y"
shows "\z. r\<^sup>+\<^sup>+ y z \ r\<^sup>+\<^sup>+ 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\<^sup>+\<^sup>+ b c \ r\<^sup>+\<^sup>+ 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\<^sup>+\<^sup>+)\\ x y" shows "(r\\)\<^sup>+\<^sup>+ x y"
using conversepD [OF assms]
proof (induction rule: tranclp_induct)
case (base y)
then show ?case
by (iprover intro: conversepI)
next
case (step y z)
then show ?case
by (iprover intro: conversepI tranclp_trans)
qed
lemmas trancl_converseI = tranclp_converseI [to_set]
lemma tranclp_converseD:
assumes "(r\\)\<^sup>+\<^sup>+ x y" shows "(r\<^sup>+\<^sup>+)\\ x y"
proof -
have "r\<^sup>+\<^sup>+ y x"
using assms
by (induction rule: tranclp_induct) (iprover dest: conversepD intro: tranclp_trans)+
then show ?thesis
by (rule conversepI)
qed
lemmas trancl_converseD = tranclp_converseD [to_set]
lemma tranclp_converse: "(r\\)\<^sup>+\<^sup>+ = (r\<^sup>+\<^sup>+)\\"
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\<^sup>+)"
by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
lemma converse_tranclp_induct [consumes 1, case_names base step]:
assumes major: "r\<^sup>+\<^sup>+ a b"
and cases: "\y. r y b \ P y" "\y z. r y z \ r\<^sup>+\<^sup>+ z b \ P z \ P y"
shows "P a"
proof -
have "r\\\<^sup>+\<^sup>+ b a"
by (intro tranclp_converseI conversepI major)
then show ?thesis
by (induction rule: tranclp_induct) (blast intro: cases dest: tranclp_converseD)+
qed
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
lemma tranclpD: "R\<^sup>+\<^sup>+ x y \ \z. R x z \ R\<^sup>*\<^sup>* 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 \<open>r x y\<close> base show P
by iprover
next
case rtrancl_into_rtrancl
then have "tranclp r y z"
by (iprover intro: rtranclp_into_tranclp1)
with \<open>r x y\<close> step show P
by iprover
qed
qed
lemmas converse_tranclE = converse_tranclpE [to_set]
lemma tranclD2: "(x, y) \ R\<^sup>+ \ \z. (x, z) \ R\<^sup>* \ (z, y) \ R"
by (blast elim: tranclE intro: trancl_into_rtrancl)
lemma irrefl_tranclI: "r\ \ r\<^sup>* = {} \ (x, x) \ r\<^sup>+"
by (blast elim: tranclE dest: trancl_into_rtrancl)
lemma irrefl_trancl_rD: "\x. (x, x) \ r\<^sup>+ \ (x, y) \ r \ x \ y"
by (blast dest: r_into_trancl)
lemma trancl_subset_Sigma_aux: "(a, b) \ r\<^sup>* \ r \ A \ A \ a = b \ a \ A"
by (induct rule: rtrancl_induct) auto
lemma trancl_subset_Sigma:
assumes "r \ A \ A" shows "r\<^sup>+ \ A \ A"
proof (rule trancl_Int_subset [OF assms])
show "(r\<^sup>+ \ A \ A) O r \ A \ A"
using assms by auto
qed
lemma reflclp_tranclp [simp]: "(r\<^sup>+\<^sup>+)\<^sup>=\<^sup>= = r\<^sup>*\<^sup>*"
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\<^sup>=)\<^sup>+ = r\<^sup>*"
proof -
have "(a, b) \ (r\<^sup>=)\<^sup>+ \ (a, b) \ r\<^sup>*" for a b
by (force dest: trancl_into_rtrancl)
moreover have "(a, b) \ (r\<^sup>=)\<^sup>+" if "(a, b) \ r\<^sup>*" 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
qed
lemma rtrancl_trancl_reflcl [code]: "r\<^sup>* = (r\<^sup>+)\<^sup>="
by simp
lemma trancl_empty [simp]: "{}\<^sup>+ = {}"
by (auto elim: trancl_induct)
lemma rtrancl_empty [simp]: "{}\<^sup>* = Id"
by (rule subst [OF reflcl_trancl]) simp
lemma rtranclpD: "R\<^sup>*\<^sup>* a b \ a = b \ a \ b \ R\<^sup>+\<^sup>+ 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\<^sup>* \ x = y \ x \ y \ (x, y) \ R\<^sup>+"
by (fast elim: trancl_into_rtrancl dest: rtranclD)
lemma trancl_unfold_right: "r\<^sup>+ = r\<^sup>* O r"
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
lemma trancl_unfold_left: "r\<^sup>+ = r O r\<^sup>*"
by (auto dest: tranclD intro: rtrancl_into_trancl2)
lemma trancl_insert: "(insert (y, x) r)\<^sup>+ = r\<^sup>+ \ {(a, b). (a, y) \ r\<^sup>* \ (x, b) \ r\<^sup>*}"
\<comment> \<open>primitive recursion for \<open>trancl\<close> over finite relations\<close>
proof -
have "\a b. (a, b) \ (insert (y, x) r)\<^sup>+ \
(a, b) \<in> r\<^sup>+ \<union> {(a, b). (a, y) \<in> r\<^sup>* \<and> (x, b) \<in> r\<^sup>*}"
by (erule trancl_induct) (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)+
moreover have "r\<^sup>+ \ {(a, b). (a, y) \ r\<^sup>* \ (x, b) \ r\<^sup>*} \ (insert (y, x) r)\<^sup>+"
by (blast intro: trancl_mono rtrancl_mono [THEN [2] rev_subsetD]
rtrancl_trancl_trancl rtrancl_into_trancl2)
ultimately show ?thesis
by auto
qed
lemma trancl_insert2:
"(insert (a, b) r)\<^sup>+ = r\<^sup>+ \ {(x, y). ((x, a) \ r\<^sup>+ \ x = a) \ ((b, y) \ r\<^sup>+ \ y = b)}"
by (auto simp: trancl_insert rtrancl_eq_or_trancl)
lemma rtrancl_insert: "(insert (a,b) r)\<^sup>* = r\<^sup>* \ {(x, y). (x, a) \ r\<^sup>* \ (b, y) \ r\<^sup>*}"
using trancl_insert[of a b r]
by (simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast
text \<open>Simplifying nested closures\<close>
lemma rtrancl_trancl_absorb[simp]: "(R\<^sup>*)\<^sup>+ = R\<^sup>*"
by (simp add: trans_rtrancl)
lemma trancl_rtrancl_absorb[simp]: "(R\<^sup>+)\<^sup>* = R\<^sup>*"
by (subst reflcl_trancl[symmetric]) simp
lemma rtrancl_reflcl_absorb[simp]: "(R\<^sup>*)\<^sup>= = R\<^sup>*"
by auto
text \<open>\<open>Domain\<close> and \<open>Range\<close>\<close>
lemma Domain_rtrancl [simp]: "Domain (R\<^sup>*) = UNIV"
by blast
lemma Range_rtrancl [simp]: "Range (R\<^sup>*) = UNIV"
by blast
lemma rtrancl_Un_subset: "(R\<^sup>* \ S\<^sup>*) \ (R \ S)\<^sup>*"
by (rule rtrancl_Un_rtrancl [THEN subst]) fast
lemma in_rtrancl_UnI: "x \ R\<^sup>* \ x \ S\<^sup>* \ x \ (R \ S)\<^sup>*"
by (blast intro: subsetD [OF rtrancl_Un_subset])
lemma trancl_domain [simp]: "Domain (r\<^sup>+) = Domain r"
by (unfold Domain_unfold) (blast dest: tranclD)
lemma trancl_range [simp]: "Range (r\<^sup>+) = Range r"
unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
lemma Not_Domain_rtrancl:
assumes "x \ Domain R" shows "(x, y) \ R\<^sup>* \ x = y"
proof -
have "(x, y) \ R\<^sup>* \ x = y"
by (erule rtrancl_induct) (use assms in auto)
then show ?thesis
by auto
qed
lemma trancl_subset_Field2: "r\<^sup>+ \ Field r \ Field r"
by (rule trancl_Int_subset) (auto simp: Field_def)
lemma finite_trancl[simp]: "finite (r\<^sup>+) = finite r"
proof
show "finite (r\<^sup>+) \ finite r"
by (blast intro: r_into_trancl' finite_subset)
show "finite r \ finite (r\<^sup>+)"
by (auto simp: finite_Field trancl_subset_Field2 [THEN finite_subset])
qed
lemma finite_rtrancl_Image[simp]: assumes "finite R" "finite A" shows "finite (R\<^sup>* `` A)"
proof (rule ccontr)
assume "infinite (R\<^sup>* `` A)"
with assms show False
by(simp add: rtrancl_trancl_reflcl Un_Image del: reflcl_trancl)
qed
text \<open>More about converse \<open>rtrancl\<close> and \<open>trancl\<close>, should
be merged with main body.\<close>
lemma single_valued_confluent:
assumes "single_valued r" and xy: "(x, y) \ r\<^sup>*" and xz: "(x, z) \ r\<^sup>*"
shows "(y, z) \ r\<^sup>* \ (z, y) \ r\<^sup>*"
using xy
proof (induction rule: rtrancl_induct)
case base
show ?case
by (simp add: assms)
next
case (step y z)
with xz \<open>single_valued r\<close> show ?case
by (auto elim: converse_rtranclE dest: single_valuedD intro: rtrancl_trans)
qed
lemma r_r_into_trancl: "(a, b) \ R \ (b, c) \ R \ (a, c) \ R\<^sup>+"
by (fast intro: trancl_trans)
lemma trancl_into_trancl: "(a, b) \ r\<^sup>+ \ (b, c) \ r \ (a, c) \ r\<^sup>+"
by (induct rule: trancl_induct) (fast intro: r_r_into_trancl trancl_trans)+
lemma tranclp_rtranclp_tranclp:
assumes "r\<^sup>+\<^sup>+ a b" "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
proof -
obtain z where "r a z" "r\<^sup>*\<^sup>* z c"
using assms by (iprover dest: tranclpD rtranclp_trans)
then show ?thesis
by (blast dest: rtranclp_into_tranclp2)
qed
lemma rtranclp_conversep: "r\\\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*\\"
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 \<open>Symmetric closure\<close>
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_symclp [simp]: "symp (symclp r)"
by(auto simp add: symp_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)\<^sup>*\<^sup>*"
by(simp add: symp_rtranclp)
lemma rtranclp_symclp_sym [sym]: "(symclp r)\<^sup>*\<^sup>* x y \ (symclp r)\<^sup>*\<^sup>* 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_rtranclp [simp]: "reflp R\<^sup>*\<^sup>*"
using refl_rtrancl[to_pred, of R] reflp_refl_eq[of "{(x, y). R\<^sup>*\<^sup>* x y}"] by simp
subsection \<open>The power operation on relations\<close>
text \<open>\<open>R ^^ n = R O \<dots> O R\<close>, the n-fold composition of \<open>R\<close>\<close>
overloading
relpow \<equiv> "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
relpowp \<equiv> "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
begin
primrec relpow :: "nat \ ('a \ 'a) set \ ('a \ 'a) set"
where
"relpow 0 R = Id"
| "relpow (Suc n) R = (R ^^ n) O R"
primrec relpowp :: "nat \ ('a \ 'a \ bool) \ ('a \ 'a \ bool)"
where
"relpowp 0 R = HOL.eq"
| "relpowp (Suc n) R = (R ^^ n) OO R"
end
lemma relpowp_relpow_eq [pred_set_conv]:
"(\x y. (x, y) \ R) ^^ n = (\x y. (x, y) \ R ^^ n)" for R :: "'a rel"
by (induct n) (simp_all add: relcompp_relcomp_eq)
text \<open>For code generation:\<close>
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 \<Longrightarrow> x = z \<Longrightarrow> P) \<Longrightarrow>
(\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
by (cases n) auto
lemma relpowp_E:
"(P ^^ n) x z \
(n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q) \<Longrightarrow>
(\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
by (fact relpow_E [to_pred])
lemma relpow_Suc_D2: "(x, z) \ 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
next
case (Suc m)
with assms relpow_Suc_D2' [of m R] show ?thesis
by force
qed
lemma relpowp_E2:
"(P ^^ n) x z \
(n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q) \<Longrightarrow>
(\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
by (fact relpow_E2 [to_pred])
lemma relpow_add: "R ^^ (m + n) = R^^m O R^^n"
by (induct n) auto
lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
by (fact relpow_add [to_pred])
lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
by (induct n) (simp_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\<^sup>*"
shows "p \ (\n. R ^^ n)"
proof (cases p)
case (Pair x y)
with assms have "(x, y) \ R\<^sup>*" by simp
then have "(x, y) \ (\n. R ^^ n)"
proof induct
case base
show ?case by (blast intro: relpow_0_I)
next
case step
then show ?case by (blast intro: relpow_Suc_I)
qed
with Pair show ?thesis by simp
qed
lemma rtranclp_imp_Sup_relpowp:
assumes "(P\<^sup>*\<^sup>*) 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\<^sup>*"
proof (cases p)
case (Pair x y)
with assms have "(x, y) \ R ^^ n" by simp
then have "(x, y) \ R\<^sup>*"
proof (induct n arbitrary: x y)
case 0
then show ?case by simp
next
case Suc
then show ?case
by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
qed
with Pair show ?thesis by simp
qed
lemma relpowp_imp_rtranclp: "(P ^^ n) x y \ (P\<^sup>*\<^sup>*) x y"
using relpow_imp_rtrancl [of "(x, y)", to_pred] by simp
lemma rtrancl_is_UN_relpow: "R\<^sup>* = (\n. R ^^ n)"
by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)
lemma rtranclp_is_Sup_relpowp: "P\<^sup>*\<^sup>* = (\n. P ^^ n)"
using rtrancl_is_UN_relpow [to_pred, of P] by auto
lemma rtrancl_power: "p \ R\<^sup>* \ (\n. p \ R ^^ n)"
by (simp add: rtrancl_is_UN_relpow)
lemma rtranclp_power: "(P\<^sup>*\<^sup>*) x y \ (\n. (P ^^ n) x y)"
by (simp add: rtranclp_is_Sup_relpowp)
lemma trancl_power: "p \ R\<^sup>+ \ (\n > 0. p \ R ^^ n)"
proof -
have "(a, b) \ R\<^sup>+ \ (\n>0. (a, b) \ R ^^ n)" for a b
proof safe
show "(a, b) \ R\<^sup>+ \ \n>0. (a, b) \ R ^^ n"
by (fastforce simp: rtrancl_is_UN_relpow relcomp_unfold dest: tranclD2)
show "(a, b) \ R\<^sup>+" 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)
qed
then show ?thesis
by (cases p) auto
qed
lemma tranclp_power: "(P\<^sup>+\<^sup>+) 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\<^sup>* \ \n. p \ R ^^ n"
by (auto dest: rtrancl_imp_UN_relpow)
lemma rtranclp_imp_relpowp: "(P\<^sup>*\<^sup>*) x y \ \n. (P ^^ n) x y"
by (auto dest: rtranclp_imp_Sup_relpowp)
text \<open>By Sternagel/Thiemann:\<close>
lemma relpow_fun_conv: "(a, b) \ R ^^ n \ (\f. f 0 = a \ f n = b \ (\i R))"
proof (induct n arbitrary: b)
case 0
show ?case by auto
next
case (Suc n)
show ?case
proof (simp add: relcomp_unfold Suc)
show "(\y. (\f. f 0 = a \ f n = y \ (\i R)) \ (y,b) \ 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" "\i. i < n \ (f i, f (Suc i)) \ R" "(c,b) \ R"
by auto
let ?g = "\ 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)" "\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], insert 1, auto)
qed
qed
qed
lemma relpowp_fun_conv: "(P ^^ n) x y \ (\f. f 0 = x \ f n = y \ (\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 \<open>finite R\<close>] have "card R \<ge> Suc 0" by auto
then show ?case using 0 by force
next
case (Suc k)
then obtain a' where "(a, a') \<in> R^^(Suc k)" and "(a', b) \<in> R"
by auto
from Suc(1)[OF \<open>(a, a') \<in> R^^(Suc k)\<close>] obtain n where "n \<le> card R" and "(a, a') \<in> R ^^ n"
by auto
have "(a, b) \ R^^(Suc n)"
using \<open>(a, a') \<in> R^^n\<close> and \<open>(a', b)\<in> R\<close> by auto
from \<open>n \<le> card R\<close> consider "n < card R" | "n = card R" by force
then show ?case
proof cases
case 1
then show ?thesis
using \<open>(a, b) \<in> R^^(Suc n)\<close> Suc_leI[OF \<open>n < card R\<close>] by blast
next
case 2
from \<open>(a, b) \<in> R ^^ (Suc n)\<close> [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 = "\i. (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 \<open>n = card R\<close> 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 = "\l. 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 \<open>f(Suc n) = b\<close> j ij by auto
next
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
next
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
next
case False
with \<open>i \<le> k\<close> have "i < k" by auto
then have small: "k + (j - i) \ n"
using \<open>k<?n\<close> by arith
show ?thesis
using steps[OF small] \<open>i<k\<close> by auto
qed
qed
qed (simp add: \<open>f 0 = a\<close>)
moreover have "?n \ n"
using i j ij by arith
ultimately show ?thesis
using \<open>n = card R\<close> by blast
qed
qed
then show ?thesis
using gr0_implies_Suc[OF \<open>k > 0\<close>] by auto
qed
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\<^sup>* = (\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\<^sup>+ = (\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)
qed
lemma finite_relcomp[simp,intro]:
assumes "finite R" and "finite S"
shows "finite (R O S)"
proof-
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
qed
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
next
case (Suc n)
then show ?case by (cases n) (use assms in simp_all)
qed
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
next
case (Suc n)
show ?case
by (rule single_valuedI)
(use Suc in \<open>fast dest: single_valuedD elim: relpow_Suc_E\<close>)
qed
subsection \<open>Bounded transitive closure\<close>
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"
proof
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
qed
lemma ntrancl_Suc [simp]: "ntrancl (Suc n) R = ntrancl n R O (Id \ R)"
proof
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 \<open>(a, b) \<in> R ^^ i\<close> show ?thesis
by (auto simp: ntrancl_def)
next
case False
with \<open>0 < i\<close> obtain j where j: "i = Suc j" "0 < j"
by (cases i) auto
with \<open>(a, b) \<in> R ^^ i\<close> obtain c where c1: "(a, c) \<in> R ^^ j" and c2: "(c, b) \<in> R"
by auto
from c1 j \<open>i \<le> Suc (Suc n)\<close> have "(a, c) \<in> ntrancl n R"
by (fastforce simp: ntrancl_def)
with c2 show ?thesis by fastforce
qed
qed
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)
qed
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 \<open>Acyclic relations\<close>
definition acyclic :: "('a \ 'a) set \ bool"
where "acyclic r \ (\x. (x,x) \ r\<^sup>+)"
abbreviation acyclicP :: "('a \ 'a \ bool) \ bool"
where "acyclicP r \ acyclic {(x, y). r x y}"
lemma acyclic_irrefl [code]: "acyclic r \ irrefl (r\<^sup>+)"
by (simp add: acyclic_def irrefl_def)
lemma acyclicI: "\x. (x, x) \ r\<^sup>+ \ 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\<^sup>+" for a b
using that by induct (auto intro: * less_trans)
then show ?thesis
by (auto intro!: acyclicI)
qed
lemma acyclic_insert [iff]: "acyclic (insert (y, x) r) \ acyclic r \ (x, y) \ r\<^sup>*"
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\<^sup>*)"
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\<^sup>* ) \<Longrightarrow> acyclic(r - Id)
\<Longrightarrow> antisym( r\<^sup>* ) = acyclic(r - Id)";
*)
lemma acyclic_subset: "acyclic s \ r \ s \ acyclic r"
unfolding acyclic_def by (blast intro: trancl_mono)
subsection \<open>Setup of transitivity reasoner\<close>
ML \<open>
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>\<open>Trueprop\<close> $ t) =
let
fun dec (Const (\<^const_name>\<open>Set.member\<close>, _) $ (Const (\<^const_name>\<open>Pair\<close>, _) $ a $ b) $ rel) =
let
fun decr (Const (\<^const_name>\<open>rtrancl\<close>, _ ) $ r) = (r,"r*")
| decr (Const (\<^const_name>\<open>trancl\<close>, _ ) $ 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>\<open>Trueprop\<close> $ t) =
let
fun dec (rel $ a $ b) =
let
fun decr (Const (\<^const_name>\<open>rtranclp\<close>, _ ) $ r) = (r,"r*")
| decr (Const (\<^const_name>\<open>tranclp\<close>, _ ) $ 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;
);
\<close>
setup \<open>
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))
\<close>
lemma transp_rtranclp [simp]: "transp R\<^sup>*\<^sup>*"
by(auto simp add: transp_def)
text \<open>Optional methods.\<close>
method_setup trancl =
\<open>Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac)\<close>
\<open>simple transitivity reasoner\<close>
method_setup rtrancl =
\<open>Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac)\<close>
\<open>simple transitivity reasoner\<close>
method_setup tranclp =
\<open>Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac)\<close>
\<open>simple transitivity reasoner (predicate version)\<close>
method_setup rtranclp =
\<open>Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac)\<close>
\<open>simple transitivity reasoner (predicate version)\<close>
end
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