The rewrite relation of the following TRS is considered.
active(f(f(a))) | → | mark(c(f(g(f(a))))) | (1) |
mark(f(X)) | → | active(f(mark(X))) | (2) |
mark(a) | → | active(a) | (3) |
mark(c(X)) | → | active(c(X)) | (4) |
mark(g(X)) | → | active(g(mark(X))) | (5) |
f(mark(X)) | → | f(X) | (6) |
f(active(X)) | → | f(X) | (7) |
c(mark(X)) | → | c(X) | (8) |
c(active(X)) | → | c(X) | (9) |
g(mark(X)) | → | g(X) | (10) |
g(active(X)) | → | g(X) | (11) |
active#(f(f(a))) | → | mark#(c(f(g(f(a))))) | (12) |
active#(f(f(a))) | → | c#(f(g(f(a)))) | (13) |
active#(f(f(a))) | → | f#(g(f(a))) | (14) |
active#(f(f(a))) | → | g#(f(a)) | (15) |
mark#(f(X)) | → | active#(f(mark(X))) | (16) |
mark#(f(X)) | → | f#(mark(X)) | (17) |
mark#(f(X)) | → | mark#(X) | (18) |
mark#(a) | → | active#(a) | (19) |
mark#(c(X)) | → | active#(c(X)) | (20) |
mark#(g(X)) | → | active#(g(mark(X))) | (21) |
mark#(g(X)) | → | g#(mark(X)) | (22) |
mark#(g(X)) | → | mark#(X) | (23) |
f#(mark(X)) | → | f#(X) | (24) |
f#(active(X)) | → | f#(X) | (25) |
c#(mark(X)) | → | c#(X) | (26) |
c#(active(X)) | → | c#(X) | (27) |
g#(mark(X)) | → | g#(X) | (28) |
g#(active(X)) | → | g#(X) | (29) |
The dependency pairs are split into 4 components.
mark#(g(X)) | → | mark#(X) | (23) |
mark#(f(X)) | → | mark#(X) | (18) |
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)) |
g(mark(x0)) |
g(active(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(g(X)) | → | mark#(X) | (23) |
1 | > | 1 | |
mark#(f(X)) | → | mark#(X) | (18) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
f#(active(X)) | → | f#(X) | (25) |
f#(mark(X)) | → | f#(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(f(f(a))) |
mark(f(x0)) |
mark(a) |
mark(c(x0)) |
mark(g(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
f#(active(X)) | → | f#(X) | (25) |
1 | > | 1 | |
f#(mark(X)) | → | f#(X) | (24) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
c#(active(X)) | → | c#(X) | (27) |
c#(mark(X)) | → | c#(X) | (26) |
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(f(a))) |
mark(f(x0)) |
mark(a) |
mark(c(x0)) |
mark(g(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
c#(active(X)) | → | c#(X) | (27) |
1 | > | 1 | |
c#(mark(X)) | → | c#(X) | (26) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
g#(active(X)) | → | g#(X) | (29) |
g#(mark(X)) | → | g#(X) | (28) |
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(f(a))) |
mark(f(x0)) |
mark(a) |
mark(c(x0)) |
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) | (29) |
1 | > | 1 | |
g#(mark(X)) | → | g#(X) | (28) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.