(* Title: CCL/Trancl.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section
‹Transitive closure of a relation
›
theory Trancl
imports CCL
begin
definition trans ::
"i set \ o" (*transitivity predicate*)
where "trans(r) == (ALL x y z. :r \ :r \ :r)"
definition id ::
"i set" (*the identity relation*)
where "id == {p. EX x. p = }"
definition relcomp ::
"[i set,i set] \ i set" (
infixr ‹O
› 60)
(*composition of relations*)
where "r O s == {xz. EX x y z. xz = \ :s \ :r}"
definition rtrancl ::
"i set \ i set" (
‹(
‹notation=
‹postfix ^*
››_^*)
› [100] 100)
where "r^* == lfp(\s. id Un (r O s))"
definition trancl ::
"i set \ i set" (
‹(
‹notation=
‹postfix ^+
››_^+)
› [100] 100)
where "r^+ == r O rtrancl(r)"
subsection
‹Natural deduction
for ‹trans(r)
››
lemma transI:
"(\x y z. \:r; :r\ \ :r) \ trans(r)"
unfolding trans_def
by blast
lemma transD:
"\trans(r); :r; :r\ \ :r"
unfolding trans_def
by blast
subsection
‹Identity relation
›
lemma idI:
" : id"
apply (unfold id_def)
apply (rule CollectI)
apply (rule exI)
apply (rule refl)
done
lemma idE:
"\p: id; \x. p = \ P\ \ P"
apply (unfold id_def)
apply (erule CollectE)
apply blast
done
subsection
‹Composition of two relations
›
lemma compI:
"\:s; :r\ \ : r O s"
unfolding relcomp_def
by blast
(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
lemma compE:
"\xz : r O s; \x y z. \xz = ; :s; :r\ \ P\ \ P"
unfolding relcomp_def
by blast
lemma compEpair:
"\ : r O s; \y. \:s; :r\ \ P\ \ P"
apply (erule compE)
apply (simp add: pair_inject)
done
lemmas [intro] = compI idI
and [elim] = compE idE
lemma comp_mono:
"\r'<=r; s'<=s\ \ (r' O s') <= (r O s)"
by blast
subsection
‹The relation rtrancl
›
lemma rtrancl_fun_mono:
"mono(\s. id Un (r O s))"
apply (rule monoI)
apply (rule monoI subset_refl comp_mono Un_mono)+
apply assumption
done
lemma rtrancl_unfold:
"r^* = id Un (r O r^*)"
by (rule rtrancl_fun_mono [
THEN rtrancl_def [
THEN def_lfp_Tarski]])
(*Reflexivity of rtrancl*)
lemma rtrancl_refl:
" : r^*"
apply (subst rtrancl_unfold)
apply blast
done
(*Closure under composition with r*)
lemma rtrancl_into_rtrancl:
"\ : r^*; : r\ \ : r^*"
apply (subst rtrancl_unfold)
apply blast
done
(*rtrancl of r contains r*)
lemma r_into_rtrancl:
" : r \ : r^*"
apply (rule rtrancl_refl [
THEN rtrancl_into_rtrancl])
apply assumption
done
subsection
‹standard
induction rule
›
lemma rtrancl_full_induct:
"\ : r^*;
∧x. P(<x,x>);
∧x y z.
[P(<x,y>); <x,y>: r^*; <y,z>: r
] ==> P(<x,z>)
]
==> P(<a,b>)
"
apply (erule def_induct [OF rtrancl_def])
apply (rule rtrancl_fun_mono)
apply blast
done
(*nice induction rule*)
lemma rtrancl_induct:
"\ : r^*;
P(a);
∧y z.
[<a,y> : r^*; <y,z> : r; P(y)
] ==> P(z)
]
==> P(b)
"
(*by induction on this formula*)
apply (subgoal_tac
"ALL y. = \ P(y)")
(*now solve first subgoal: this formula is sufficient*)
apply blast
(*now do the induction*)
apply (erule rtrancl_full_induct)
apply blast
apply blast
done
(*transitivity of transitive closure!! -- by induction.*)
lemma trans_rtrancl:
"trans(r^*)"
apply (rule transI)
apply (rule_tac b = z
in rtrancl_induct)
apply (fast elim: rtrancl_into_rtrancl)+
done
(*elimination of rtrancl -- by induction on a special formula*)
lemma rtranclE:
"\ : r^*; a = b \ P; \y. \ : r^*; : r\ \ P\ \ P"
apply (subgoal_tac
"a = b | (EX y. : r^* \ : r)")
prefer 2
apply (erule rtrancl_induct)
apply blast
apply blast
apply blast
done
subsection
‹The relation trancl
›
subsubsection
‹Conversions between trancl
and rtrancl
›
lemma trancl_into_rtrancl:
" : r^+ \ : r^*"
apply (unfold trancl_def)
apply (erule compEpair)
apply (erule rtrancl_into_rtrancl)
apply assumption
done
(*r^+ contains r*)
lemma r_into_trancl:
" : r \ : r^+"
unfolding trancl_def
by (blast intro: rtrancl_refl)
(*intro rule by definition: from rtrancl and r*)
lemma rtrancl_into_trancl1:
"\ : r^*; : r\ \ : r^+"
unfolding trancl_def
by blast
(*intro rule from r and rtrancl*)
lemma rtrancl_into_trancl2:
"\ : r; : r^*\ \ : r^+"
apply (erule rtranclE)
apply (erule subst)
apply (erule r_into_trancl)
apply (rule trans_rtrancl [
THEN transD,
THEN rtrancl_into_trancl1])
apply (assumption | rule r_into_rtrancl)+
done
(*elimination of r^+ -- NOT an induction rule*)
lemma tranclE:
"\ : r^+;
<a,b> : r
==> P;
∧y.
[<a,y> : r^+; <y,b> : r
] ==> P
] ==> P
"
apply (subgoal_tac
" : r | (EX y. : r^+ \ : r)")
apply blast
apply (unfold trancl_def)
apply (erule compEpair)
apply (erule rtranclE)
apply blast
apply (blast intro!: rtrancl_into_trancl1)
done
(*Transitivity of r^+.
Proved by unfolding since it uses transitivity of rtrancl. *)
lemma trans_trancl:
"trans(r^+)"
apply (unfold trancl_def)
apply (rule transI)
apply (erule compEpair)+
apply (erule rtrancl_into_rtrancl [
THEN trans_rtrancl [
THEN transD,
THEN compI]])
apply assumption+
done
lemma trancl_into_trancl2:
"\ : r; : r^+\ \ : r^+"
by (rule r_into_trancl [
THEN trans_trancl [
THEN transD]])
end
