src/CCL/Trancl.thy
changeset 1474 3f7d67927fe2
parent 0 a5a9c433f639
child 17456 bcf7544875b2
equal deleted inserted replaced
1473:e8d4606e6502 1474:3f7d67927fe2
     1 (*  Title: 	CCL/trancl.thy
     1 (*  Title:      CCL/trancl.thy
     2     ID:         $Id$
     2     ID:         $Id$
     3     Author: 	Martin Coen, Cambridge University Computer Laboratory
     3     Author:     Martin Coen, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     4     Copyright   1993  University of Cambridge
     5 
     5 
     6 Transitive closure of a relation
     6 Transitive closure of a relation
     7 *)
     7 *)
     8 
     8 
     9 Trancl = CCL +
     9 Trancl = CCL +
    10 
    10 
    11 consts
    11 consts
    12     trans   :: "i set => o" 	              (*transitivity predicate*)
    12     trans   :: "i set => o"                   (*transitivity predicate*)
    13     id	    :: "i set"
    13     id      :: "i set"
    14     rtrancl :: "i set => i set"	              ("(_^*)" [100] 100)
    14     rtrancl :: "i set => i set"               ("(_^*)" [100] 100)
    15     trancl  :: "i set => i set"	              ("(_^+)" [100] 100)  
    15     trancl  :: "i set => i set"               ("(_^+)" [100] 100)  
    16     O	    :: "[i set,i set] => i set"       (infixr 60)
    16     O       :: "[i set,i set] => i set"       (infixr 60)
    17 
    17 
    18 rules   
    18 rules   
    19 
    19 
    20 trans_def	"trans(r) == (ALL x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
    20 trans_def       "trans(r) == (ALL x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
    21 comp_def	(*composition of relations*)
    21 comp_def        (*composition of relations*)
    22 		"r O s == {xz. EX x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
    22                 "r O s == {xz. EX x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
    23 id_def		(*the identity relation*)
    23 id_def          (*the identity relation*)
    24 		"id == {p. EX x. p = <x,x>}"
    24                 "id == {p. EX x. p = <x,x>}"
    25 rtrancl_def	"r^* == lfp(%s. id Un (r O s))"
    25 rtrancl_def     "r^* == lfp(%s. id Un (r O s))"
    26 trancl_def	"r^+ == r O rtrancl(r)"
    26 trancl_def      "r^+ == r O rtrancl(r)"
    27 
    27 
    28 end
    28 end