YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(a(a(b(x0)))) → a#(x0)
a#(a(a(b(x0)))) → b#(a(x0))
a#(a(a(b(x0)))) → a#(b(a(x0)))
a#(a(a(b(x0)))) → b#(a(b(a(x0))))
b#(b(a(x0))) → a#(b(a(x0)))
b#(b(a(x0))) → a#(a(b(a(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
[a(x1)] =
0 0 0
-∞ -∞ 0
0 -∞ -∞
· x1 +
1 -∞ -∞
-∞ -∞ -∞
0 -∞ -∞
[b#(x1)] =
-∞ 1 -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[b(x1)] =
-∞ 0 -∞
-∞ 0 1
-∞ 0 -∞
· x1 +
0 -∞ -∞
0 -∞ -∞
-∞ -∞ -∞
[a#(x1)] =
-∞ -∞ 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
together with the usable rules
a(a(a(b(x0)))) → b(a(b(a(x0))))
b(b(a(x0))) → a(a(b(a(x0))))
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(a(a(b(x0)))) → b#(a(x0))
a#(a(a(b(x0)))) → b#(a(b(a(x0))))
remain.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.