The rewrite relation of the following TRS is considered.
| active(f(X)) | → | mark(g(h(f(X)))) | (1) |
| mark(f(X)) | → | active(f(mark(X))) | (2) |
| mark(g(X)) | → | active(g(X)) | (3) |
| mark(h(X)) | → | active(h(mark(X))) | (4) |
| f(mark(X)) | → | f(X) | (5) |
| f(active(X)) | → | f(X) | (6) |
| g(mark(X)) | → | g(X) | (7) |
| g(active(X)) | → | g(X) | (8) |
| h(mark(X)) | → | h(X) | (9) |
| h(active(X)) | → | h(X) | (10) |
| active#(f(X)) | → | mark#(g(h(f(X)))) | (11) |
| active#(f(X)) | → | g#(h(f(X))) | (12) |
| active#(f(X)) | → | h#(f(X)) | (13) |
| mark#(f(X)) | → | active#(f(mark(X))) | (14) |
| mark#(f(X)) | → | f#(mark(X)) | (15) |
| mark#(f(X)) | → | mark#(X) | (16) |
| mark#(g(X)) | → | active#(g(X)) | (17) |
| mark#(h(X)) | → | active#(h(mark(X))) | (18) |
| mark#(h(X)) | → | h#(mark(X)) | (19) |
| mark#(h(X)) | → | mark#(X) | (20) |
| f#(mark(X)) | → | f#(X) | (21) |
| f#(active(X)) | → | f#(X) | (22) |
| g#(mark(X)) | → | g#(X) | (23) |
| g#(active(X)) | → | g#(X) | (24) |
| h#(mark(X)) | → | h#(X) | (25) |
| h#(active(X)) | → | h#(X) | (26) |
The dependency pairs are split into 4 components.
| mark#(h(X)) | → | mark#(X) | (20) |
| mark#(f(X)) | → | mark#(X) | (16) |
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)) |
| f(active(x0)) |
| h(mark(x0)) |
| h(active(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| mark#(h(X)) | → | mark#(X) | (20) |
| 1 | > | 1 | |
| mark#(f(X)) | → | mark#(X) | (16) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| f#(active(X)) | → | f#(X) | (22) |
| f#(mark(X)) | → | f#(X) | (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(x0)) |
| mark(f(x0)) |
| mark(g(x0)) |
| mark(h(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| f#(active(X)) | → | f#(X) | (22) |
| 1 | > | 1 | |
| f#(mark(X)) | → | f#(X) | (21) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| g#(active(X)) | → | g#(X) | (24) |
| g#(mark(X)) | → | g#(X) | (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(f(x0)) |
| mark(f(x0)) |
| mark(g(x0)) |
| mark(h(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| g#(active(X)) | → | g#(X) | (24) |
| 1 | > | 1 | |
| g#(mark(X)) | → | g#(X) | (23) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| h#(active(X)) | → | h#(X) | (26) |
| h#(mark(X)) | → | h#(X) | (25) |
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(x0)) |
| mark(f(x0)) |
| mark(g(x0)) |
| mark(h(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| h#(active(X)) | → | h#(X) | (26) |
| 1 | > | 1 | |
| h#(mark(X)) | → | h#(X) | (25) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.