src/HOL/Trancl.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Trancl.thy	Fri Mar 03 12:02:25 1995 +0100
@@ -0,0 +1,26 @@
+(*  Title: 	HOL/trancl.thy
+    ID:         $Id$
+    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+Transitive closure of a relation
+
+rtrancl is refl/transitive closure;  trancl is transitive closure
+*)
+
+Trancl = Lfp + Prod + 
+consts
+    trans   :: "('a * 'a)set => bool" 	(*transitivity predicate*)
+    id	    :: "('a * 'a)set"
+    rtrancl :: "('a * 'a)set => ('a * 'a)set"	("(_^*)" [100] 100)
+    trancl  :: "('a * 'a)set => ('a * 'a)set"	("(_^+)" [100] 100)  
+    O	    :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)
+defs   
+trans_def	"trans(r) == (!x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
+comp_def	(*composition of relations*)
+		"r O s == {xz. ? x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
+id_def		(*the identity relation*)
+		"id == {p. ? x. p = <x,x>}"
+rtrancl_def	"r^* == lfp(%s. id Un (r O s))"
+trancl_def	"r^+ == r O rtrancl(r)"
+end