YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

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)]

0	0	1
0	0	-∞
0	0	0

· x₁ +

-∞	-∞	-∞
0	-∞	-∞
0	-∞	-∞

[a^#(x₁)]

-∞	0	1
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[a(x₁)]

0	0	0
0	0	0
0	0	0

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[c(x₁)]

0	0	0
0	0	0
-∞	-∞	-∞

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

together with the usable rules

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

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

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

→

a^#(a(x0))

remains.

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	0
1	0

· x₁ +

3	-∞
3	-∞

[a^#(x₁)]

0	0
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

0	0
0	0

· x₁ +

3	-∞
1	-∞

[c(x₁)]

-∞	-∞
0	0

· x₁ +

0	-∞
1	-∞

together with the usable rules

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

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

1.1.1.1.1 P is empty

There are no pairs anymore.