YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

→

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[a(x₁)]

=

0	1	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

=

0	1	0
0	0	1
1	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[b^#(x₁)]

=

0	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

all pairs could be removed.

1.1.1.1 P is empty

There are no pairs anymore.