YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Since only unary symbols occur, one can reverse all terms and obtains the TRS

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

The following set of initial dependency pairs has been identified.

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

b^#(a(a(a(x0))))	→	b^#(a(b(x0)))
b^#(a(a(a(x0))))	→	b^#(x0)

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the arctic semiring over the integers

[a(x₁)]

· x₁ +

[b(x₁)]

· x₁ +

[b^#(x₁)]

· x₁ +

together with the usable rules

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

(w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

There are no pairs anymore.