Certification Problem

Input (TPDB TRS_Standard/Mixed_TRS/motivation)

The rewrite relation of the following TRS is considered.

g(h(g(x))) → g(x) (1)
g(g(x)) → g(h(g(x))) (2)
h(h(x)) → h(f(h(x),x)) (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[h(x1)] =
1 0 1
0 1 1
1 0 1
· x1 +
0 0 0
0 0 0
1 0 0
[g(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
[f(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 1
1 1 0
0 0 0
· x2 +
0 0 0
1 0 0
0 0 0
all of the following rules can be deleted.
g(h(g(x))) → g(x) (1)
h(h(x)) → h(f(h(x),x)) (3)

1.1 Rule Removal

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals
[h(x1)] =
1 0 0 0
0 0 0 0
1 1 0 0
0 0 0 0
· x1 +
0 0 0 0
0 0 0 0
1 0 0 0
1 0 0 0
[g(x1)] =
1 1 0 0
0 0 0 0
0 0 0 0
0 0 0 0
· x1 +
0 0 0 0
1 0 0 0
0 0 0 0
0 0 0 0
all of the following rules can be deleted.
g(g(x)) → g(h(g(x))) (2)

1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.