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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the arctic semiring over the integers

[b^#(x₁)]

0	0	0	0
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

0	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

[c(x₁)]

0	0	0	0
-∞	-∞	-∞	-∞
-∞	0	-∞	-∞
0	0	0	0

· x₁ +

-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

[a^#(x₁)]

0	0	1	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

0	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

[a(x₁)]

0	0	1	0
0	0	1	0
0	0	-∞	0
0	0	1	0

· x₁ +

-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
0	-∞	-∞	-∞
-∞	-∞	-∞	-∞

[b(x₁)]

0	0	0
0	0	0
-∞	0	-∞
0	1	0

· x₁ +

-∞	-∞	-∞	-∞
0	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

together with the usable rules

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

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

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

→

b^#(c(x0))

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the arctic semiring over the integers

[b^#(x₁)]

-∞	0	0	0
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

0	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

[c(x₁)]

0	0	-∞
-∞	0	0
-∞	-∞	0
-∞	0	-∞

· x₁ +

1	-∞	-∞	-∞
0	-∞	-∞	-∞
0	-∞	-∞	-∞
0	-∞	-∞	-∞

[b(x₁)]

0	-∞	-∞
1	0	-∞
0	1	-∞
1	0	1

· x₁ +

-∞	-∞	-∞	-∞
1	-∞	-∞	-∞
0	-∞	-∞	-∞
-∞	-∞	-∞	-∞

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 P is empty

There are no pairs anymore.