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(x0)
b(a(b(x0))) → a(a(b(b(b(x0)))))

Proof

1 Dependency Pair Transformation

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

1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 0
-∞ 0
· x1 +
0 -∞
0 -∞
[b#(x1)] =
-∞ 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 0
1 0
· x1 +
0 -∞
0 -∞
[a#(x1)] =
0 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
together with the usable rules
a(x0) → x0
a(a(x0)) → b(x0)
b(a(b(x0))) → a(a(b(b(b(x0)))))
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(a(x0)) → b#(x0)
b#(a(b(x0))) → a#(a(b(b(b(x0)))))
remain.

1.1.1 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)] =
-∞ 0 0
-∞ 0 0
-∞ 0 0
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[b#(x1)] =
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[a(x1)] =
0 -∞ 0
1 0 0
0 0 0
· x1 +
0 -∞ -∞
-∞ -∞ -∞
0 -∞ -∞
[a#(x1)] =
0 -∞ 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
together with the usable rules
a(x0) → x0
a(a(x0)) → b(x0)
b(a(b(x0))) → a(a(b(b(b(x0)))))
(w.r.t. the implicit argument filter of the reduction pair), the pair
a#(a(x0)) → b#(x0)
remains.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.