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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

-∞	0
0	-∞

· x₁ +

0	-∞
1	-∞

[a^#(x₁)]

1	0
-∞	-∞

· x₁ +

2	-∞
-∞	-∞

[a(x₁)]

1	0
0	-∞

· x₁ +

3	-∞
2	-∞

[c(x₁)]

-∞	0
0	1

· x₁ +

2	-∞
3	-∞

together with the usable rules

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

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

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

remain.

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	3
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

-∞	0
0	-∞

· x₁ +

0	-∞
-∞	-∞

[a^#(x₁)]

0	3
-∞	-∞

· x₁ +

3	-∞
-∞	-∞

[a(x₁)]

-∞	0
0	2

· x₁ +

0	-∞
3	-∞

[c(x₁)]

2	0
0	-∞

· x₁ +

3	-∞
0	-∞

together with the usable rules

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

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

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

remain.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.