YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 String Reversal

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

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

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 + 9
[b^#(x₁)]	=	5 · x₁ + 0
[a(x₁)]	=	4 · x₁ + 9
[a^#(x₁)]	=	0 · x₁ + 0

together with the usable rules

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

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

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

→

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

remains.

1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[b(x₁)]

=

-∞	-∞	-∞
1	0	-∞
1	-∞	-∞

· x₁ +

0	-∞	-∞
1	-∞	-∞
1	-∞	-∞

[b^#(x₁)]

=

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[a(x₁)]

=

-∞	-∞	0
-∞	-∞	0
0	-∞	-∞

· x₁ +

1	-∞	-∞
0	-∞	-∞
0	-∞	-∞

together with the usable rules

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