YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

→

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1 Reduction Pair Processor

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

[c(x₁)]

=

0	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a^#(x₁)]

=

0	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

=

0	1	0
0	0	1
1	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[b(x₁)]

=

0	0	1
0	0	1
0	0	1

· 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.