YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

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

1.1 Reduction Pair Processor with Usable Rules

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

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.