The rewrite relation of the following TRS is considered.
active(zeros) | → | mark(cons(0,zeros)) | (1) |
active(tail(cons(X,XS))) | → | mark(XS) | (2) |
mark(zeros) | → | active(zeros) | (3) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (4) |
mark(0) | → | active(0) | (5) |
mark(tail(X)) | → | active(tail(mark(X))) | (6) |
cons(mark(X1),X2) | → | cons(X1,X2) | (7) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (8) |
cons(active(X1),X2) | → | cons(X1,X2) | (9) |
cons(X1,active(X2)) | → | cons(X1,X2) | (10) |
tail(mark(X)) | → | tail(X) | (11) |
tail(active(X)) | → | tail(X) | (12) |
[active(x1)] | = | 1 · x1 |
[zeros] | = | 0 |
[mark(x1)] | = | 2 · x1 |
[cons(x1, x2)] | = | 2 · x1 + 2 · x2 |
[0] | = | 0 |
[tail(x1)] | = | 1 + 2 · x1 |
active(tail(cons(X,XS))) | → | mark(XS) | (2) |
mark(tail(X)) | → | active(tail(mark(X))) | (6) |
active#(zeros) | → | mark#(cons(0,zeros)) | (13) |
active#(zeros) | → | cons#(0,zeros) | (14) |
mark#(zeros) | → | active#(zeros) | (15) |
mark#(cons(X1,X2)) | → | active#(cons(mark(X1),X2)) | (16) |
mark#(cons(X1,X2)) | → | cons#(mark(X1),X2) | (17) |
mark#(cons(X1,X2)) | → | mark#(X1) | (18) |
mark#(0) | → | active#(0) | (19) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (20) |
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (21) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (22) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (23) |
tail#(mark(X)) | → | tail#(X) | (24) |
tail#(active(X)) | → | tail#(X) | (25) |
The dependency pairs are split into 3 components.
mark#(cons(X1,X2)) | → | mark#(X1) | (18) |
mark#(zeros) | → | active#(zeros) | (15) |
active#(zeros) | → | mark#(cons(0,zeros)) | (13) |
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)) |
prec(zeros) | = | 2 | weight(zeros) | = | 1 | ||||
prec(active#) | = | 1 | weight(active#) | = | 1 | ||||
prec(0) | = | 0 | weight(0) | = | 1 |
π(mark#) | = | 1 |
π(cons) | = | 1 |
π(zeros) | = | [] |
π(active#) | = | [] |
π(0) | = | [] |
mark#(zeros) | → | active#(zeros) | (15) |
active#(zeros) | → | mark#(cons(0,zeros)) | (13) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) | → | mark#(X1) | (18) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (21) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (20) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (22) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (23) |
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(zeros) |
active(tail(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(tail(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) | (21) |
1 | ≥ | 1 | |
2 | > | 2 | |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (20) |
1 | > | 1 | |
2 | ≥ | 2 | |
cons#(active(X1),X2) | → | cons#(X1,X2) | (22) |
1 | > | 1 | |
2 | ≥ | 2 | |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (23) |
1 | ≥ | 1 | |
2 | > | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
tail#(active(X)) | → | tail#(X) | (25) |
tail#(mark(X)) | → | tail#(X) | (24) |
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(zeros) |
active(tail(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(tail(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
tail#(active(X)) | → | tail#(X) | (25) |
1 | > | 1 | |
tail#(mark(X)) | → | tail#(X) | (24) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.