The rewrite relation of the following TRS is considered.
active(f(f(X))) | → | mark(c(f(g(f(X))))) | (1) |
active(c(X)) | → | mark(d(X)) | (2) |
active(h(X)) | → | mark(c(d(X))) | (3) |
mark(f(X)) | → | active(f(mark(X))) | (4) |
mark(c(X)) | → | active(c(X)) | (5) |
mark(g(X)) | → | active(g(X)) | (6) |
mark(d(X)) | → | active(d(X)) | (7) |
mark(h(X)) | → | active(h(mark(X))) | (8) |
f(mark(X)) | → | f(X) | (9) |
f(active(X)) | → | f(X) | (10) |
c(mark(X)) | → | c(X) | (11) |
c(active(X)) | → | c(X) | (12) |
g(mark(X)) | → | g(X) | (13) |
g(active(X)) | → | g(X) | (14) |
d(mark(X)) | → | d(X) | (15) |
d(active(X)) | → | d(X) | (16) |
h(mark(X)) | → | h(X) | (17) |
h(active(X)) | → | h(X) | (18) |
[active(x1)] | = | 1 · x1 |
[f(x1)] | = | 1 · x1 |
[mark(x1)] | = | 1 · x1 |
[c(x1)] | = | 1 · x1 |
[g(x1)] | = | 1 · x1 |
[d(x1)] | = | 1 · x1 |
[h(x1)] | = | 1 · x1 + 1 |
active(h(X)) | → | mark(c(d(X))) | (3) |
active#(f(f(X))) | → | mark#(c(f(g(f(X))))) | (19) |
active#(f(f(X))) | → | c#(f(g(f(X)))) | (20) |
active#(f(f(X))) | → | f#(g(f(X))) | (21) |
active#(f(f(X))) | → | g#(f(X)) | (22) |
active#(c(X)) | → | mark#(d(X)) | (23) |
active#(c(X)) | → | d#(X) | (24) |
mark#(f(X)) | → | active#(f(mark(X))) | (25) |
mark#(f(X)) | → | f#(mark(X)) | (26) |
mark#(f(X)) | → | mark#(X) | (27) |
mark#(c(X)) | → | active#(c(X)) | (28) |
mark#(g(X)) | → | active#(g(X)) | (29) |
mark#(d(X)) | → | active#(d(X)) | (30) |
mark#(h(X)) | → | active#(h(mark(X))) | (31) |
mark#(h(X)) | → | h#(mark(X)) | (32) |
mark#(h(X)) | → | mark#(X) | (33) |
f#(mark(X)) | → | f#(X) | (34) |
f#(active(X)) | → | f#(X) | (35) |
c#(mark(X)) | → | c#(X) | (36) |
c#(active(X)) | → | c#(X) | (37) |
g#(mark(X)) | → | g#(X) | (38) |
g#(active(X)) | → | g#(X) | (39) |
d#(mark(X)) | → | d#(X) | (40) |
d#(active(X)) | → | d#(X) | (41) |
h#(mark(X)) | → | h#(X) | (42) |
h#(active(X)) | → | h#(X) | (43) |
The dependency pairs are split into 6 components.
mark#(h(X)) | → | mark#(X) | (33) |
mark#(f(X)) | → | mark#(X) | (27) |
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) | (33) |
1 | > | 1 | |
mark#(f(X)) | → | mark#(X) | (27) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
f#(active(X)) | → | f#(X) | (35) |
f#(mark(X)) | → | f#(X) | (34) |
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(x0))) |
active(c(x0)) |
active(h(x0)) |
mark(f(x0)) |
mark(c(x0)) |
mark(g(x0)) |
mark(d(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) | (35) |
1 | > | 1 | |
f#(mark(X)) | → | f#(X) | (34) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
c#(active(X)) | → | c#(X) | (37) |
c#(mark(X)) | → | c#(X) | (36) |
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(x0))) |
active(c(x0)) |
active(h(x0)) |
mark(f(x0)) |
mark(c(x0)) |
mark(g(x0)) |
mark(d(x0)) |
mark(h(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
c#(active(X)) | → | c#(X) | (37) |
1 | > | 1 | |
c#(mark(X)) | → | c#(X) | (36) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
g#(active(X)) | → | g#(X) | (39) |
g#(mark(X)) | → | g#(X) | (38) |
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(x0))) |
active(c(x0)) |
active(h(x0)) |
mark(f(x0)) |
mark(c(x0)) |
mark(g(x0)) |
mark(d(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) | (39) |
1 | > | 1 | |
g#(mark(X)) | → | g#(X) | (38) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
d#(active(X)) | → | d#(X) | (41) |
d#(mark(X)) | → | d#(X) | (40) |
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(x0))) |
active(c(x0)) |
active(h(x0)) |
mark(f(x0)) |
mark(c(x0)) |
mark(g(x0)) |
mark(d(x0)) |
mark(h(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
d#(active(X)) | → | d#(X) | (41) |
1 | > | 1 | |
d#(mark(X)) | → | d#(X) | (40) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
h#(active(X)) | → | h#(X) | (43) |
h#(mark(X)) | → | h#(X) | (42) |
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(x0))) |
active(c(x0)) |
active(h(x0)) |
mark(f(x0)) |
mark(c(x0)) |
mark(g(x0)) |
mark(d(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) | (43) |
1 | > | 1 | |
h#(mark(X)) | → | h#(X) | (42) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.