The rewrite relation of the following TRS is considered.
active(and(true,X)) |
→ |
mark(X) |
(1) |
active(and(false,Y)) |
→ |
mark(false) |
(2) |
active(if(true,X,Y)) |
→ |
mark(X) |
(3) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(4) |
active(add(0,X)) |
→ |
mark(X) |
(5) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(6) |
active(first(0,X)) |
→ |
mark(nil) |
(7) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(8) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(9) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(10) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(11) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(12) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(13) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(14) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(15) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(16) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(17) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(18) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(19) |
proper(and(X1,X2)) |
→ |
and(proper(X1),proper(X2)) |
(20) |
proper(true) |
→ |
ok(true) |
(21) |
proper(false) |
→ |
ok(false) |
(22) |
proper(if(X1,X2,X3)) |
→ |
if(proper(X1),proper(X2),proper(X3)) |
(23) |
proper(add(X1,X2)) |
→ |
add(proper(X1),proper(X2)) |
(24) |
proper(0) |
→ |
ok(0) |
(25) |
proper(s(X)) |
→ |
s(proper(X)) |
(26) |
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(27) |
proper(nil) |
→ |
ok(nil) |
(28) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(29) |
proper(from(X)) |
→ |
from(proper(X)) |
(30) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(31) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(32) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(33) |
s(ok(X)) |
→ |
ok(s(X)) |
(34) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(35) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(36) |
from(ok(X)) |
→ |
ok(from(X)) |
(37) |
top(mark(X)) |
→ |
top(proper(X)) |
(38) |
top(ok(X)) |
→ |
top(active(X)) |
(39) |
active#(add(s(X),Y)) |
→ |
add#(X,Y) |
(40) |
active#(add(s(X),Y)) |
→ |
s#(add(X,Y)) |
(41) |
active#(first(s(X),cons(Y,Z))) |
→ |
first#(X,Z) |
(42) |
active#(first(s(X),cons(Y,Z))) |
→ |
cons#(Y,first(X,Z)) |
(43) |
active#(from(X)) |
→ |
s#(X) |
(44) |
active#(from(X)) |
→ |
from#(s(X)) |
(45) |
active#(from(X)) |
→ |
cons#(X,from(s(X))) |
(46) |
active#(and(X1,X2)) |
→ |
active#(X1) |
(47) |
active#(and(X1,X2)) |
→ |
and#(active(X1),X2) |
(48) |
active#(if(X1,X2,X3)) |
→ |
active#(X1) |
(49) |
active#(if(X1,X2,X3)) |
→ |
if#(active(X1),X2,X3) |
(50) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(51) |
active#(add(X1,X2)) |
→ |
add#(active(X1),X2) |
(52) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(53) |
active#(first(X1,X2)) |
→ |
first#(active(X1),X2) |
(54) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(55) |
active#(first(X1,X2)) |
→ |
first#(X1,active(X2)) |
(56) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(57) |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(58) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(59) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(60) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(61) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(62) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(63) |
proper#(and(X1,X2)) |
→ |
and#(proper(X1),proper(X2)) |
(64) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X3) |
(65) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X2) |
(66) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X1) |
(67) |
proper#(if(X1,X2,X3)) |
→ |
if#(proper(X1),proper(X2),proper(X3)) |
(68) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(69) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(70) |
proper#(add(X1,X2)) |
→ |
add#(proper(X1),proper(X2)) |
(71) |
proper#(s(X)) |
→ |
proper#(X) |
(72) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(73) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(74) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(75) |
proper#(first(X1,X2)) |
→ |
first#(proper(X1),proper(X2)) |
(76) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(77) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(78) |
proper#(cons(X1,X2)) |
→ |
cons#(proper(X1),proper(X2)) |
(79) |
proper#(from(X)) |
→ |
proper#(X) |
(80) |
proper#(from(X)) |
→ |
from#(proper(X)) |
(81) |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(82) |
if#(ok(X1),ok(X2),ok(X3)) |
→ |
if#(X1,X2,X3) |
(83) |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(84) |
s#(ok(X)) |
→ |
s#(X) |
(85) |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(86) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(87) |
from#(ok(X)) |
→ |
from#(X) |
(88) |
top#(mark(X)) |
→ |
proper#(X) |
(89) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(90) |
top#(ok(X)) |
→ |
active#(X) |
(91) |
top#(ok(X)) |
→ |
top#(active(X)) |
(92) |
The dependency pairs are split into 10
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(92) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(90) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(ok) |
= |
0 |
|
stat(ok) |
= |
lex
|
prec(proper) |
= |
0 |
|
stat(proper) |
= |
lex
|
prec(from) |
= |
3 |
|
stat(from) |
= |
lex
|
prec(cons) |
= |
0 |
|
stat(cons) |
= |
lex
|
prec(nil) |
= |
0 |
|
stat(nil) |
= |
lex
|
prec(first) |
= |
3 |
|
stat(first) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(add) |
= |
3 |
|
stat(add) |
= |
lex
|
prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
prec(if) |
= |
2 |
|
stat(if) |
= |
lex
|
prec(false) |
= |
0 |
|
stat(false) |
= |
lex
|
prec(mark) |
= |
1 |
|
stat(mark) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(and) |
= |
3 |
|
stat(and) |
= |
lex
|
prec(true) |
= |
0 |
|
stat(true) |
= |
lex
|
π(top#) |
= |
[1] |
π(ok) |
= |
1 |
π(proper) |
= |
1 |
π(from) |
= |
[] |
π(cons) |
= |
[] |
π(nil) |
= |
[] |
π(first) |
= |
[1,2] |
π(s) |
= |
[] |
π(add) |
= |
[1,2] |
π(0) |
= |
[] |
π(if) |
= |
[1,2,3] |
π(false) |
= |
[] |
π(mark) |
= |
[1] |
π(active) |
= |
1 |
π(and) |
= |
[1,2] |
π(true) |
= |
[] |
together with the usable
rules
active(and(true,X)) |
→ |
mark(X) |
(1) |
active(and(false,Y)) |
→ |
mark(false) |
(2) |
active(if(true,X,Y)) |
→ |
mark(X) |
(3) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(4) |
active(add(0,X)) |
→ |
mark(X) |
(5) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(6) |
active(first(0,X)) |
→ |
mark(nil) |
(7) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(8) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(9) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(10) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(11) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(12) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(13) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(14) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(17) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(33) |
s(ok(X)) |
→ |
ok(s(X)) |
(34) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(18) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(19) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(35) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(36) |
from(ok(X)) |
→ |
ok(from(X)) |
(37) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(15) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(31) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(16) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(32) |
proper(and(X1,X2)) |
→ |
and(proper(X1),proper(X2)) |
(20) |
proper(true) |
→ |
ok(true) |
(21) |
proper(false) |
→ |
ok(false) |
(22) |
proper(if(X1,X2,X3)) |
→ |
if(proper(X1),proper(X2),proper(X3)) |
(23) |
proper(add(X1,X2)) |
→ |
add(proper(X1),proper(X2)) |
(24) |
proper(0) |
→ |
ok(0) |
(25) |
proper(s(X)) |
→ |
s(proper(X)) |
(26) |
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(27) |
proper(nil) |
→ |
ok(nil) |
(28) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(29) |
proper(from(X)) |
→ |
from(proper(X)) |
(30) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(90) |
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(ok) |
= |
0 |
|
stat(ok) |
= |
lex
|
prec(from) |
= |
1 |
|
stat(from) |
= |
lex
|
prec(cons) |
= |
0 |
|
stat(cons) |
= |
lex
|
prec(nil) |
= |
0 |
|
stat(nil) |
= |
lex
|
prec(first) |
= |
9 |
|
stat(first) |
= |
lex
|
prec(s) |
= |
1 |
|
stat(s) |
= |
lex
|
prec(add) |
= |
2 |
|
stat(add) |
= |
lex
|
prec(0) |
= |
1 |
|
stat(0) |
= |
lex
|
prec(if) |
= |
1 |
|
stat(if) |
= |
lex
|
prec(false) |
= |
0 |
|
stat(false) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(and) |
= |
1 |
|
stat(and) |
= |
lex
|
prec(true) |
= |
8 |
|
stat(true) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
[1] |
π(from) |
= |
[] |
π(cons) |
= |
[] |
π(nil) |
= |
[] |
π(first) |
= |
[2] |
π(s) |
= |
[] |
π(add) |
= |
[1] |
π(0) |
= |
[] |
π(if) |
= |
[2] |
π(false) |
= |
[] |
π(mark) |
= |
[] |
π(active) |
= |
1 |
π(and) |
= |
[1] |
π(true) |
= |
[] |
together with the usable
rules
active(and(true,X)) |
→ |
mark(X) |
(1) |
active(and(false,Y)) |
→ |
mark(false) |
(2) |
active(if(true,X,Y)) |
→ |
mark(X) |
(3) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(4) |
active(add(0,X)) |
→ |
mark(X) |
(5) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(6) |
active(first(0,X)) |
→ |
mark(nil) |
(7) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(8) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(9) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(10) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(11) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(12) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(13) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(14) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(17) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(33) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(18) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(19) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(35) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(15) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(31) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(16) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(32) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(92) |
could be deleted.
1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(and(X1,X2)) |
→ |
active#(X1) |
(47) |
active#(if(X1,X2,X3)) |
→ |
active#(X1) |
(49) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(51) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(53) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(55) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(and(X1,X2)) |
→ |
active#(X1) |
(47) |
|
1 |
> |
1 |
active#(if(X1,X2,X3)) |
→ |
active#(X1) |
(49) |
|
1 |
> |
1 |
active#(add(X1,X2)) |
→ |
active#(X1) |
(51) |
|
1 |
> |
1 |
active#(first(X1,X2)) |
→ |
active#(X1) |
(53) |
|
1 |
> |
1 |
active#(first(X1,X2)) |
→ |
active#(X2) |
(55) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(62) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(63) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X3) |
(65) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X2) |
(66) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X1) |
(67) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(69) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(70) |
proper#(s(X)) |
→ |
proper#(X) |
(72) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(74) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(75) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(77) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(78) |
proper#(from(X)) |
→ |
proper#(X) |
(80) |
1.1.3 Subterm Criterion Processor
We use the projection
and remove the pairs:
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(62) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(63) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X3) |
(65) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X2) |
(66) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X1) |
(67) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(69) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(70) |
proper#(s(X)) |
→ |
proper#(X) |
(72) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(74) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(75) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(77) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(78) |
proper#(from(X)) |
→ |
proper#(X) |
(80) |
1.1.3.1 P is empty
There are no pairs anymore.
-
The
4th
component contains the
pair
from#(ok(X)) |
→ |
from#(X) |
(88) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
from#(ok(X)) |
→ |
from#(X) |
(88) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(87) |
1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(87) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(60) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(61) |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(86) |
1.1.6 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(60) |
|
2 |
≥ |
2 |
1 |
> |
1 |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(61) |
|
2 |
> |
2 |
1 |
≥ |
1 |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(86) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
1.1.7 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(ok(X)) |
→ |
s#(X) |
(85) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(59) |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(84) |
1.1.8 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(59) |
|
2 |
≥ |
2 |
1 |
> |
1 |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(84) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(58) |
if#(ok(X1),ok(X2),ok(X3)) |
→ |
if#(X1,X2,X3) |
(83) |
1.1.9 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(58) |
|
3 |
≥ |
3 |
2 |
≥ |
2 |
1 |
> |
1 |
if#(ok(X1),ok(X2),ok(X3)) |
→ |
if#(X1,X2,X3) |
(83) |
|
3 |
> |
3 |
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
10th
component contains the
pair
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(57) |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(82) |
1.1.10 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(57) |
|
2 |
≥ |
2 |
1 |
> |
1 |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(82) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.