YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1 Reduction Pair Processor with Usable Rules

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

[b(x₁)]

0	1
0	0

· x₁ +

2	-∞
0	-∞

[b^#(x₁)]

1	3
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

0	0
0	0

· x₁ +

-∞	-∞
-∞	-∞

[a^#(x₁)]

2	0
-∞	-∞

· x₁ +

-∞	-∞
-∞	-∞

together with the usable rules

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

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

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

remain.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(x0))

→

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

1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[b(x₁)]

-∞	0
-∞	-∞

· x₁ +

0	-∞
3	-∞

[b^#(x₁)]

-∞	2
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

0	0
-∞	-∞

· x₁ +

2	-∞
-∞	-∞

together with the usable rules

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

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

1.1.1.1.1.1 P is empty

There are no pairs anymore.