YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(a(a(x0))) → b(b(b(x0)))
b(a(a(b(x0)))) → x0
b(a(a(b(x0)))) → b(a(a(a(b(x0)))))

Proof

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 -∞ 0
0 0 0
-∞ 0 0
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[a(x1)] =
0 0 0
0 0 -∞
0 3 -∞
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
the rules
a(a(a(x0))) → b(b(b(x0)))
b(a(a(b(x0)))) → b(a(a(a(b(x0)))))
remain.

1.1 Bounds

The given TRS is match-bounded by 1. This is shown by the following automaton.