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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

0	3
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

0	1
1	2

· x₁ +

0	-∞
1	-∞

[a^#(x₁)]

0	2
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

0	1
-∞	0

· x₁ +

0	-∞
-∞	-∞

[c(x₁)]

0	0
0	-∞

· x₁ +

0	-∞
0	-∞

together with the usable rules

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

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

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

3	2
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

2	1
1	0

· x₁ +

1	-∞
0	-∞

[a^#(x₁)]

3	-∞
-∞	-∞

· x₁ +

-∞	-∞
-∞	-∞

[a(x₁)]

0	-∞
1	0

· x₁ +

-∞	-∞
0	-∞

[c(x₁)]

-∞	0
0	1

· x₁ +

0	-∞
0	-∞

together with the usable rules

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

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

b^#(x0)

→

a^#(x0)

remains.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.