src/CCL/Trancl.thy
 author wenzelm Thu, 23 Jul 2009 21:59:56 +0200 changeset 32153 a0e57fb1b930 parent 24825 c4f13ab78f9d child 42156 df219e736a5d permissions -rw-r--r--
misc modernization: proper method setup instead of adhoc ML proofs;
```
(*  Title:      CCL/Trancl.thy
Author:     Martin Coen, Cambridge University Computer Laboratory
Copyright   1993  University of Cambridge
*)

header {* Transitive closure of a relation *}

theory Trancl
imports CCL
begin

consts
trans   :: "i set => o"                   (*transitivity predicate*)
id      :: "i set"
rtrancl :: "i set => i set"               ("(_^*)" [100] 100)
trancl  :: "i set => i set"               ("(_^+)" [100] 100)
relcomp :: "[i set,i set] => i set"       (infixr "O" 60)

axioms
trans_def:       "trans(r) == (ALL x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
relcomp_def:     (*composition of relations*)
"r O s == {xz. EX x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
id_def:          (*the identity relation*)
"id == {p. EX x. p = <x,x>}"
rtrancl_def:     "r^* == lfp(%s. id Un (r O s))"
trancl_def:      "r^+ == r O rtrancl(r)"

subsection {* Natural deduction for @{text "trans(r)"} *}

lemma transI:
"(!! x y z. [| <x,y>:r;  <y,z>:r |] ==> <x,z>:r) ==> trans(r)"
unfolding trans_def by blast

lemma transD: "[| trans(r);  <a,b>:r;  <b,c>:r |] ==> <a,c>:r"
unfolding trans_def by blast

subsection {* Identity relation *}

lemma idI: "<a,a> : id"
apply (unfold id_def)
apply (rule CollectI)
apply (rule exI)
apply (rule refl)
done

lemma idE:
"[| p: id;  !!x.[| p = <x,x> |] ==> P |] ==>  P"
apply (unfold id_def)
apply (erule CollectE)
apply blast
done

subsection {* Composition of two relations *}

lemma compI: "[| <a,b>:s; <b,c>:r |] ==> <a,c> : 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 = <x,z>;  <x,y>:s;  <y,z>:r |] ==> P
|] ==> P"
unfolding relcomp_def by blast

lemma compEpair:
"[| <a,c> : r O s;
!!y. [| <a,y>:s;  <y,c>:r |] ==> P
|] ==> 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: "[| 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: "<a,a> : r^*"
apply (subst rtrancl_unfold)
apply blast
done

(*Closure under composition with r*)
lemma rtrancl_into_rtrancl: "[| <a,b> : r^*;  <b,c> : r |] ==> <a,c> : r^*"
apply (subst rtrancl_unfold)
apply blast
done

(*rtrancl of r contains r*)
lemma r_into_rtrancl: "[| <a,b> : r |] ==> <a,b> : r^*"
apply (rule rtrancl_refl [THEN rtrancl_into_rtrancl])
apply assumption
done

subsection {* standard induction rule *}

lemma rtrancl_full_induct:
"[| <a,b> : 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:
"[| <a,b> : 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. <a,b> = <a,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:
"[| <a,b> : r^*;  (a = b) ==> P;
!!y.[| <a,y> : r^*; <y,b> : r |] ==> P |]
==> P"
apply (subgoal_tac "a = b | (EX y. <a,y> : r^* & <y,b> : 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: "[| <a,b> : r^+ |] ==> <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 |] ==> <a,b> : r^+"
unfolding trancl_def by (blast intro: rtrancl_refl)

(*intro rule by definition: from rtrancl and r*)
lemma rtrancl_into_trancl1: "[| <a,b> : r^*;  <b,c> : r |]   ==>  <a,c> : r^+"
unfolding trancl_def by blast

(*intro rule from r and rtrancl*)
lemma rtrancl_into_trancl2: "[| <a,b> : r;  <b,c> : r^* |]   ==>  <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:
"[| <a,b> : r^+;
<a,b> : r ==> P;
!!y.[| <a,y> : r^+;  <y,b> : r |] ==> P
|] ==> P"
apply (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+ & <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: "[| <a,b> : r;  <b,c> : r^+ |]   ==>  <a,c> : r^+"
apply (rule r_into_trancl [THEN trans_trancl [THEN transD]])
apply assumption+
done

end
```