The rewrite relation of the following TRS is considered.
active(f(X,g(X),Y)) | → | mark(f(Y,Y,Y)) | (1) |
active(g(b)) | → | mark(c) | (2) |
active(b) | → | mark(c) | (3) |
mark(f(X1,X2,X3)) | → | active(f(X1,X2,X3)) | (4) |
mark(g(X)) | → | active(g(mark(X))) | (5) |
mark(b) | → | active(b) | (6) |
mark(c) | → | active(c) | (7) |
f(mark(X1),X2,X3) | → | f(X1,X2,X3) | (8) |
f(X1,mark(X2),X3) | → | f(X1,X2,X3) | (9) |
f(X1,X2,mark(X3)) | → | f(X1,X2,X3) | (10) |
f(active(X1),X2,X3) | → | f(X1,X2,X3) | (11) |
f(X1,active(X2),X3) | → | f(X1,X2,X3) | (12) |
f(X1,X2,active(X3)) | → | f(X1,X2,X3) | (13) |
g(mark(X)) | → | g(X) | (14) |
g(active(X)) | → | g(X) | (15) |
active#(f(X,g(X),Y)) | → | mark#(f(Y,Y,Y)) | (16) |
active#(f(X,g(X),Y)) | → | f#(Y,Y,Y) | (17) |
active#(g(b)) | → | mark#(c) | (18) |
active#(b) | → | mark#(c) | (19) |
mark#(f(X1,X2,X3)) | → | active#(f(X1,X2,X3)) | (20) |
mark#(g(X)) | → | active#(g(mark(X))) | (21) |
mark#(g(X)) | → | g#(mark(X)) | (22) |
mark#(g(X)) | → | mark#(X) | (23) |
mark#(b) | → | active#(b) | (24) |
mark#(c) | → | active#(c) | (25) |
f#(mark(X1),X2,X3) | → | f#(X1,X2,X3) | (26) |
f#(X1,mark(X2),X3) | → | f#(X1,X2,X3) | (27) |
f#(X1,X2,mark(X3)) | → | f#(X1,X2,X3) | (28) |
f#(active(X1),X2,X3) | → | f#(X1,X2,X3) | (29) |
f#(X1,active(X2),X3) | → | f#(X1,X2,X3) | (30) |
f#(X1,X2,active(X3)) | → | f#(X1,X2,X3) | (31) |
g#(mark(X)) | → | g#(X) | (32) |
g#(active(X)) | → | g#(X) | (33) |
The dependency pairs are split into 4 components.
mark#(g(X)) | → | mark#(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.
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 |
As there is no critical graph in the transitive closure, there are no infinite chains.
mark#(f(X1,X2,X3)) | → | active#(f(X1,X2,X3)) | (20) |
active#(f(X,g(X),Y)) | → | mark#(f(Y,Y,Y)) | (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),x1,x2) |
f(x0,mark(x1),x2) |
f(x0,x1,mark(x2)) |
f(active(x0),x1,x2) |
f(x0,active(x1),x2) |
f(x0,x1,active(x2)) |
g(mark(x0)) |
g(active(x0)) |
mark#(f(z1,z1,z1)) | → | active#(f(z1,z1,z1)) | (34) |
The dependency pairs are split into 0 components.
f#(X1,mark(X2),X3) | → | f#(X1,X2,X3) | (27) |
f#(mark(X1),X2,X3) | → | f#(X1,X2,X3) | (26) |
f#(X1,X2,mark(X3)) | → | f#(X1,X2,X3) | (28) |
f#(active(X1),X2,X3) | → | f#(X1,X2,X3) | (29) |
f#(X1,active(X2),X3) | → | f#(X1,X2,X3) | (30) |
f#(X1,X2,active(X3)) | → | f#(X1,X2,X3) | (31) |
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,g(x0),x1)) |
active(g(b)) |
active(b) |
mark(f(x0,x1,x2)) |
mark(g(x0)) |
mark(b) |
mark(c) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
f#(X1,mark(X2),X3) | → | f#(X1,X2,X3) | (27) |
1 | ≥ | 1 | |
2 | > | 2 | |
3 | ≥ | 3 | |
f#(mark(X1),X2,X3) | → | f#(X1,X2,X3) | (26) |
1 | > | 1 | |
2 | ≥ | 2 | |
3 | ≥ | 3 | |
f#(X1,X2,mark(X3)) | → | f#(X1,X2,X3) | (28) |
1 | ≥ | 1 | |
2 | ≥ | 2 | |
3 | > | 3 | |
f#(active(X1),X2,X3) | → | f#(X1,X2,X3) | (29) |
1 | > | 1 | |
2 | ≥ | 2 | |
3 | ≥ | 3 | |
f#(X1,active(X2),X3) | → | f#(X1,X2,X3) | (30) |
1 | ≥ | 1 | |
2 | > | 2 | |
3 | ≥ | 3 | |
f#(X1,X2,active(X3)) | → | f#(X1,X2,X3) | (31) |
1 | ≥ | 1 | |
2 | ≥ | 2 | |
3 | > | 3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
g#(active(X)) | → | g#(X) | (33) |
g#(mark(X)) | → | g#(X) | (32) |
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,g(x0),x1)) |
active(g(b)) |
active(b) |
mark(f(x0,x1,x2)) |
mark(g(x0)) |
mark(b) |
mark(c) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
g#(active(X)) | → | g#(X) | (33) |
1 | > | 1 | |
g#(mark(X)) | → | g#(X) | (32) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.