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(a(x0)) b(c(x0))
a(b(b(x0))) b(b(a(a(x0))))

Proof

1 Dependency Pair Transformation

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

1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the arctic semiring over the integers
[c(x1)] =
0 0 0 0
0 0 0 0
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
· x1 +
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
[a#(x1)] =
0 0 -∞ 0
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
· x1 +
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
[a(x1)] =
0 0 0 -∞
0 0 0 0
0 0 0 0
0 0 0 0
· x1 +
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
0 -∞ -∞ -∞
[b(x1)] =
0 0 1 0
-∞ -∞ 0 1
-∞ 0 0 0
0 -∞ 0 0
· x1 +
-∞ -∞ -∞ -∞
0 -∞ -∞ -∞
0 -∞ -∞ -∞
0 -∞ -∞ -∞
together with the usable rules
a(x0) x0
a(a(x0)) b(c(x0))
a(b(b(x0))) b(b(a(a(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.