Certification Problem

Input

The rewrite relation of the following TRS is considered.

f(f(x)) → f(g(f(x))) (1)
The evaluation strategy is innermost.

Property / Task

Determine bounds on the derivational complexity.

Answer / Result

An upperbound for the complexity is O(n).

Proof (by ttt2)

1 Rule Shifting

The rules
f(f(x)) → f(g(f(x))) (1)
are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1/64
[g(x1)] =
1 0
0 0
· x1 +
4 0
0 0
[f(x1)] =
1 5
0 0
· x1 +
0 0
4 0
which has the intended complexity.

1.1 R is empty

There are no rules in the TRS R. Hence, R/S has complexity O(1).