YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers
[c(x1)] =
0 1 -∞
0 1 -∞
-∞ 0 -∞
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[a(x1)] =
0 0 0
-∞ -∞ 0
0 1 -∞
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[b(x1)] =
0 0 1
0 0 1
-∞ -∞ 0
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
the rules
a(a(x0)) → b(b(c(x0)))
c(b(x0)) → a(a(x0))
remain.

1.1 Bounds

The given TRS is match-bounded by 1. This is shown by the following automaton.