YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 String Reversal

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

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1.1 Reduction Pair Processor

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

[b(x₁)]

0	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[a(x₁)]

0	0	0
0	0	0
0	1	0

· x₁ +

1	0	0
1	0	0
0	0	0

[a^#(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

the pairs

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

remain.

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	-∞ · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[a^#(x₁)]	=	0 · x₁ + 2

together with the usable rules

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

(w.r.t. the implicit argument filter of the reduction pair), the pairs

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

remain.

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	-4 · x₁ + 8
[a(x₁)]	=	4 · x₁ + 8
[a^#(x₁)]	=	0 · x₁ + 0

together with the usable rules

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

(w.r.t. the implicit argument filter of the reduction pair), the pairs

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

remain.

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	-3 · x₁ + 0
[a(x₁)]	=	3 · x₁ + 2
[a^#(x₁)]	=	0 · x₁ + 4

together with the usable rules

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

(w.r.t. the implicit argument filter of the reduction pair), the pairs

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

remain.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the naturals

[b(x₁)]

1	0
0	0

· x₁ +

0	0
0	0

[a(x₁)]

1	1
0	0

· x₁ +

0	0
1	0

[a^#(x₁)]

4	0
0	0

· x₁ +

0	0
0	0

the pairs

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

remain.

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

a^#(b(x0))

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	4 · x₁ + 2
[a^#(x₁)]	=	0 · x₁ + -∞

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

1.1.1.1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.