YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

b(b(x0)) → a(a(a(x0)))
b(a(b(x0))) → a(x0)
b(a(a(x0))) → b(a(b(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
[a(x1)] =
0 1
0 0
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
1 1
0 0
· x1 +
-∞ -∞
-∞ -∞
the rules
b(b(x0)) → a(a(a(x0)))
b(a(a(x0))) → b(a(b(x0)))
remain.

1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
b(b(x0)) → a(a(a(x0)))
a(a(b(x0))) → b(a(b(x0)))

1.1.1 Bounds

The given TRS is match-bounded by 3. This is shown by the following automaton.