YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 2
2 0
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
2 0
0 -∞
· x1 +
-∞ -∞
-∞ -∞
the rules
b(b(x0)) → a(a(x0))
a(a(x0)) → a(b(a(x0)))
remain.

1.1 Rule Removal

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
1 3
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 -∞
0 -∞
· x1 +
-∞ -∞
-∞ -∞
the rule
a(a(x0)) → a(b(a(x0)))
remains.

1.1.1 Rule Removal

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
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 1
0 1
· x1 +
-∞ -∞
-∞ -∞
all rules could be removed.

1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.