YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

[c(x₁)]

-∞	0	-∞
0	-∞	-∞
-∞	0	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
0	-∞	-∞

[a^#(x₁)]

-∞	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[a(x₁)]

0	-∞	0
1	0	0
1	0	0

· x₁ +

0	-∞	-∞
1	-∞	-∞
0	-∞	-∞

[b(x₁)]

-∞	0	-∞
0	1	0
0	0	0

· x₁ +

0	-∞	-∞
1	-∞	-∞
0	-∞	-∞

together with the usable rules

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

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

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

→

a^#(a(c(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

[c(x₁)]

-∞	0
0	-∞

· x₁ +

1	-∞
0	-∞

[a^#(x₁)]

-∞	0
-∞	-∞

· x₁ +

2	-∞
-∞	-∞

[a(x₁)]

0	2
-∞	0

· x₁ +

3	-∞
1	-∞

[b(x₁)]

2	0
0	-∞

· x₁ +

3	-∞
0	-∞

together with the usable rules

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