The rewrite relation of the following TRS is considered.
active(first(0,X)) | → | mark(nil) | (1) |
active(first(s(X),cons(Y,Z))) | → | mark(cons(Y,first(X,Z))) | (2) |
active(from(X)) | → | mark(cons(X,from(s(X)))) | (3) |
mark(first(X1,X2)) | → | active(first(mark(X1),mark(X2))) | (4) |
mark(0) | → | active(0) | (5) |
mark(nil) | → | active(nil) | (6) |
mark(s(X)) | → | active(s(mark(X))) | (7) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (8) |
mark(from(X)) | → | active(from(mark(X))) | (9) |
first(mark(X1),X2) | → | first(X1,X2) | (10) |
first(X1,mark(X2)) | → | first(X1,X2) | (11) |
first(active(X1),X2) | → | first(X1,X2) | (12) |
first(X1,active(X2)) | → | first(X1,X2) | (13) |
s(mark(X)) | → | s(X) | (14) |
s(active(X)) | → | s(X) | (15) |
cons(mark(X1),X2) | → | cons(X1,X2) | (16) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (17) |
cons(active(X1),X2) | → | cons(X1,X2) | (18) |
cons(X1,active(X2)) | → | cons(X1,X2) | (19) |
from(mark(X)) | → | from(X) | (20) |
from(active(X)) | → | from(X) | (21) |
active#(first(0,X)) | → | mark#(nil) | (22) |
active#(first(s(X),cons(Y,Z))) | → | mark#(cons(Y,first(X,Z))) | (23) |
active#(first(s(X),cons(Y,Z))) | → | cons#(Y,first(X,Z)) | (24) |
active#(first(s(X),cons(Y,Z))) | → | first#(X,Z) | (25) |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (26) |
active#(from(X)) | → | cons#(X,from(s(X))) | (27) |
active#(from(X)) | → | from#(s(X)) | (28) |
active#(from(X)) | → | s#(X) | (29) |
mark#(first(X1,X2)) | → | active#(first(mark(X1),mark(X2))) | (30) |
mark#(first(X1,X2)) | → | first#(mark(X1),mark(X2)) | (31) |
mark#(first(X1,X2)) | → | mark#(X1) | (32) |
mark#(first(X1,X2)) | → | mark#(X2) | (33) |
mark#(0) | → | active#(0) | (34) |
mark#(nil) | → | active#(nil) | (35) |
mark#(s(X)) | → | active#(s(mark(X))) | (36) |
mark#(s(X)) | → | s#(mark(X)) | (37) |
mark#(s(X)) | → | mark#(X) | (38) |
mark#(cons(X1,X2)) | → | active#(cons(mark(X1),X2)) | (39) |
mark#(cons(X1,X2)) | → | cons#(mark(X1),X2) | (40) |
mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
mark#(from(X)) | → | active#(from(mark(X))) | (42) |
mark#(from(X)) | → | from#(mark(X)) | (43) |
mark#(from(X)) | → | mark#(X) | (44) |
first#(mark(X1),X2) | → | first#(X1,X2) | (45) |
first#(X1,mark(X2)) | → | first#(X1,X2) | (46) |
first#(active(X1),X2) | → | first#(X1,X2) | (47) |
first#(X1,active(X2)) | → | first#(X1,X2) | (48) |
s#(mark(X)) | → | s#(X) | (49) |
s#(active(X)) | → | s#(X) | (50) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (51) |
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (52) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (53) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (54) |
from#(mark(X)) | → | from#(X) | (55) |
from#(active(X)) | → | from#(X) | (56) |
The dependency pairs are split into 5 components.
active#(first(s(X),cons(Y,Z))) | → | mark#(cons(Y,first(X,Z))) | (23) |
mark#(cons(X1,X2)) | → | active#(cons(mark(X1),X2)) | (39) |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (26) |
mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
mark#(first(X1,X2)) | → | active#(first(mark(X1),mark(X2))) | (30) |
mark#(first(X1,X2)) | → | mark#(X1) | (32) |
mark#(first(X1,X2)) | → | mark#(X2) | (33) |
mark#(s(X)) | → | active#(s(mark(X))) | (36) |
mark#(s(X)) | → | mark#(X) | (38) |
mark#(from(X)) | → | active#(from(mark(X))) | (42) |
mark#(from(X)) | → | mark#(X) | (44) |
prec(first) | = | 1 | weight(first) | = | 2 | ||||
prec(0) | = | 0 | weight(0) | = | 2 | ||||
prec(nil) | = | 2 | weight(nil) | = | 4 |
π(active#) | = | 1 |
π(first) | = | [1,2] |
π(s) | = | 1 |
π(cons) | = | 1 |
π(mark#) | = | 1 |
π(mark) | = | 1 |
π(from) | = | 1 |
π(active) | = | 1 |
π(0) | = | [] |
π(nil) | = | [] |
active#(first(s(X),cons(Y,Z))) | → | mark#(cons(Y,first(X,Z))) | (23) |
mark#(first(X1,X2)) | → | mark#(X1) | (32) |
mark#(first(X1,X2)) | → | mark#(X2) | (33) |
prec(s) | = | 0 | weight(s) | = | 1 | ||||
prec(0) | = | 1 | weight(0) | = | 2 | ||||
prec(nil) | = | 2 | weight(nil) | = | 1 |
π(mark#) | = | 1 |
π(cons) | = | 1 |
π(active#) | = | 1 |
π(mark) | = | 1 |
π(from) | = | 1 |
π(first) | = | 2 |
π(s) | = | [1] |
π(active) | = | 1 |
π(0) | = | [] |
π(nil) | = | [] |
mark#(s(X)) | → | mark#(X) | (38) |
prec(from) | = | 2 | weight(from) | = | 1 | ||||
prec(s) | = | 0 | weight(s) | = | 2 | ||||
prec(0) | = | 1 | weight(0) | = | 2 | ||||
prec(nil) | = | 3 | weight(nil) | = | 1 |
π(mark#) | = | 1 |
π(cons) | = | 1 |
π(active#) | = | 1 |
π(mark) | = | 1 |
π(from) | = | [1] |
π(first) | = | 2 |
π(s) | = | [] |
π(active) | = | 1 |
π(0) | = | [] |
π(nil) | = | [] |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (26) |
mark#(from(X)) | → | mark#(X) | (44) |
The dependency pairs are split into 1 component.
mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
cons(mark(x0),x1) |
cons(x0,mark(x1)) |
cons(active(x0),x1) |
cons(x0,active(x1)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
first#(X1,mark(X2)) | → | first#(X1,X2) | (46) |
first#(mark(X1),X2) | → | first#(X1,X2) | (45) |
first#(active(X1),X2) | → | first#(X1,X2) | (47) |
first#(X1,active(X2)) | → | first#(X1,X2) | (48) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(first(0,x0)) |
active(first(s(x0),cons(x1,x2))) |
active(from(x0)) |
mark(first(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
first#(X1,mark(X2)) | → | first#(X1,X2) | (46) |
1 | ≥ | 1 | |
2 | > | 2 | |
first#(mark(X1),X2) | → | first#(X1,X2) | (45) |
1 | > | 1 | |
2 | ≥ | 2 | |
first#(active(X1),X2) | → | first#(X1,X2) | (47) |
1 | > | 1 | |
2 | ≥ | 2 | |
first#(X1,active(X2)) | → | first#(X1,X2) | (48) |
1 | ≥ | 1 | |
2 | > | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
s#(active(X)) | → | s#(X) | (50) |
s#(mark(X)) | → | s#(X) | (49) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(first(0,x0)) |
active(first(s(x0),cons(x1,x2))) |
active(from(x0)) |
mark(first(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
s#(active(X)) | → | s#(X) | (50) |
1 | > | 1 | |
s#(mark(X)) | → | s#(X) | (49) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (52) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (51) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (53) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (54) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(first(0,x0)) |
active(first(s(x0),cons(x1,x2))) |
active(from(x0)) |
mark(first(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (52) |
1 | ≥ | 1 | |
2 | > | 2 | |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (51) |
1 | > | 1 | |
2 | ≥ | 2 | |
cons#(active(X1),X2) | → | cons#(X1,X2) | (53) |
1 | > | 1 | |
2 | ≥ | 2 | |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (54) |
1 | ≥ | 1 | |
2 | > | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
from#(active(X)) | → | from#(X) | (56) |
from#(mark(X)) | → | from#(X) | (55) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(first(0,x0)) |
active(first(s(x0),cons(x1,x2))) |
active(from(x0)) |
mark(first(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
from#(active(X)) | → | from#(X) | (56) |
1 | > | 1 | |
from#(mark(X)) | → | from#(X) | (55) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.