YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(b(b(a(x0)))) → a#(b(b(x0)))
a#(b(b(a(x0)))) → a#(b(a(b(b(x0)))))
a#(b(b(a(x0)))) → a#(a(b(a(b(b(x0))))))

1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
-∞ 1 -∞
-∞ -∞ -∞
0 0 -∞
· x1 +
0 -∞ -∞
0 -∞ -∞
0 -∞ -∞
[a(x1)] =
0 0 0
-∞ 1 0
-∞ 0 0
· x1 +
0 -∞ -∞
1 -∞ -∞
0 -∞ -∞
[a#(x1)] =
-∞ 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
together with the usable rules
a(x0) → x0
a(b(b(a(x0)))) → a(a(b(a(b(b(x0))))))
(w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

1.1.1 P is empty

There are no pairs anymore.