YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

[b(x₁)]

3	3
3	0

· x₁ +

-∞	-∞
-∞	-∞

[c(x₁)]

0	0
3	0

· x₁ +

-∞	-∞
-∞	-∞

[a(x₁)]

0	-∞
0	-∞

· x₁ +

-∞	-∞
-∞	-∞

the rules

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

remain.

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1	1
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

0	1
0	0

· x₁ +

3	-∞
1	-∞

[a^#(x₁)]

0	-∞
-∞	-∞

· x₁ +

-∞	-∞
-∞	-∞

[c(x₁)]

0	1
-∞	0

· x₁ +

3	-∞
0	-∞

[a(x₁)]

0	1
-∞	0

· x₁ +

0	-∞
1	-∞

together with the usable rules

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

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

a^#(a(b(c(x0))))	→	b^#(a(a(x0)))
a^#(a(b(c(x0))))	→	b^#(b(a(a(x0))))
b^#(x0)	→	a^#(a(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	0
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

0	2
0	0

· x₁ +

3	-∞
1	-∞

[a^#(x₁)]

-∞	0
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[c(x₁)]

0	-∞
0	0

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

0	0
0	0

· x₁ +

1	-∞
0	-∞

together with the usable rules

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

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

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

remain.

1.1.1.1.1 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)]

-∞	2
-∞	-∞

· x₁ +

1	-∞
-∞	-∞

[b(x₁)]

-∞	2
2	2

· x₁ +

2	-∞
2	-∞

[a^#(x₁)]

-∞	0
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[c(x₁)]

-∞	2
0	0

· x₁ +

2	-∞
-4	-∞

[a(x₁)]

-∞	0
-∞	0

· x₁ +

0	-∞
0	-∞

together with the usable rules

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

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

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

→

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

remains.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.