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