(* Title: CCL/Trancl.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section \<open>Transitive closure of a relation\<close>
theory Trancl
imports CCL
begin
definition trans :: "i set \<Rightarrow> o" (*transitivity predicate*)
where "trans(r) == (ALL x y z. <x,y>:r \<longrightarrow> <y,z>:r \<longrightarrow> <x,z>:r)"
definition id :: "i set" (*the identity relation*)
where "id == {p. EX x. p = <x,x>}"
definition relcomp :: "[i set,i set] \<Rightarrow> i set" (infixr "O" 60) (*composition of relations*)
where "r O s == {xz. EX x y z. xz = <x,z> \<and> <x,y>:s \<and> <y,z>:r}"
definition rtrancl :: "i set \<Rightarrow> i set" ("(_^*)" [100] 100)
where "r^* == lfp(\<lambda>s. id Un (r O s))"
definition trancl :: "i set \<Rightarrow> i set" ("(_^+)" [100] 100)
where "r^+ == r O rtrancl(r)"
subsection \<open>Natural deduction for \<open>trans(r)\<close>\<close>
lemma transI: "(\<And>x y z. \<lbrakk><x,y>:r; <y,z>:r\<rbrakk> \<Longrightarrow> <x,z>:r) \<Longrightarrow> trans(r)"
unfolding trans_def by blast
lemma transD: "\<lbrakk>trans(r); <a,b>:r; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c>:r"
unfolding trans_def by blast
subsection \<open>Identity relation\<close>
lemma idI: "<a,a> : id"
apply (unfold id_def)
apply (rule CollectI)
apply (rule exI)
apply (rule refl)
done
lemma idE: "\<lbrakk>p: id; \<And>x. p = <x,x> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
apply (unfold id_def)
apply (erule CollectE)
apply blast
done
subsection \<open>Composition of two relations\<close>
lemma compI: "\<lbrakk><a,b>:s; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c> : r O s"
unfolding relcomp_def by blast
(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
lemma compE: "\<lbrakk>xz : r O s; \<And>x y z. \<lbrakk>xz = <x,z>; <x,y>:s; <y,z>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
unfolding relcomp_def by blast
lemma compEpair: "\<lbrakk><a,c> : r O s; \<And>y. \<lbrakk><a,y>:s; <y,c>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
apply (erule compE)
apply (simp add: pair_inject)
done
lemmas [intro] = compI idI
and [elim] = compE idE
and [elim!] = pair_inject
lemma comp_mono: "\<lbrakk>r'<=r; s'<=s\<rbrakk> \<Longrightarrow> (r' O s') <= (r O s)"
by blast
subsection \<open>The relation rtrancl\<close>
lemma rtrancl_fun_mono: "mono(\<lambda>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: "<a,a> : r^*"
apply (subst rtrancl_unfold)
apply blast
done
(*Closure under composition with r*)
lemma rtrancl_into_rtrancl: "\<lbrakk><a,b> : r^*; <b,c> : r\<rbrakk> \<Longrightarrow> <a,c> : r^*"
apply (subst rtrancl_unfold)
apply blast
done
(*rtrancl of r contains r*)
lemma r_into_rtrancl: "<a,b> : r \<Longrightarrow> <a,b> : r^*"
apply (rule rtrancl_refl [THEN rtrancl_into_rtrancl])
apply assumption
done
subsection \<open>standard induction rule\<close>
lemma rtrancl_full_induct:
"\<lbrakk><a,b> : r^*;
\<And>x. P(<x,x>);
\<And>x y z. \<lbrakk>P(<x,y>); <x,y>: r^*; <y,z>: r\<rbrakk> \<Longrightarrow> P(<x,z>)\<rbrakk>
\<Longrightarrow> P(<a,b>)"
apply (erule def_induct [OF rtrancl_def])
apply (rule rtrancl_fun_mono)
apply blast
done
(*nice induction rule*)
lemma rtrancl_induct:
"\<lbrakk><a,b> : r^*;
P(a);
\<And>y z. \<lbrakk><a,y> : r^*; <y,z> : r; P(y)\<rbrakk> \<Longrightarrow> P(z) \<rbrakk>
\<Longrightarrow> P(b)"
(*by induction on this formula*)
apply (subgoal_tac "ALL y. <a,b> = <a,y> \<longrightarrow> 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:
"\<lbrakk><a,b> : r^*; a = b \<Longrightarrow> P; \<And>y. \<lbrakk><a,y> : r^*; <y,b> : r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
apply (subgoal_tac "a = b | (EX y. <a,y> : r^* \<and> <y,b> : r)")
prefer 2
apply (erule rtrancl_induct)
apply blast
apply blast
apply blast
done
subsection \<open>The relation trancl\<close>
subsubsection \<open>Conversions between trancl and rtrancl\<close>
lemma trancl_into_rtrancl: "<a,b> : r^+ \<Longrightarrow> <a,b> : r^*"
apply (unfold trancl_def)
apply (erule compEpair)
apply (erule rtrancl_into_rtrancl)
apply assumption
done
(*r^+ contains r*)
lemma r_into_trancl: "<a,b> : r \<Longrightarrow> <a,b> : r^+"
unfolding trancl_def by (blast intro: rtrancl_refl)
(*intro rule by definition: from rtrancl and r*)
lemma rtrancl_into_trancl1: "\<lbrakk><a,b> : r^*; <b,c> : r\<rbrakk> \<Longrightarrow> <a,c> : r^+"
unfolding trancl_def by blast
(*intro rule from r and rtrancl*)
lemma rtrancl_into_trancl2: "\<lbrakk><a,b> : r; <b,c> : r^*\<rbrakk> \<Longrightarrow> <a,c> : 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:
"\<lbrakk><a,b> : r^+;
<a,b> : r \<Longrightarrow> P;
\<And>y. \<lbrakk><a,y> : r^+; <y,b> : r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
apply (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+ \<and> <y,b> : 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: "\<lbrakk><a,b> : r; <b,c> : r^+\<rbrakk> \<Longrightarrow> <a,c> : r^+"
apply (rule r_into_trancl [THEN trans_trancl [THEN transD]])
apply assumption+
done
end