YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pairs

a^#(c(x0))	→	a^#(b(x0))
a^#(b(x0))	→	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

[a^#(x₁)]

0	0
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

0	0
0	-∞

· x₁ +

0	-∞
0	-∞

[c(x₁)]

1	1
0	1

· x₁ +

1	-∞
1	-∞

together with the usable rule

b(c(x0))

→

c(b(b(x0)))

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

a^#(b(x0))

→

a^#(x0)

remains.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pairs

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

1.1.2 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 +

0	-∞
-∞	-∞

[b(x₁)]

0	0
0	0

· x₁ +

-∞	-∞
-∞	-∞

[c(x₁)]

-∞	2
2	1

· x₁ +

3	-∞
2	-∞

together with the usable rule

b(c(x0))

→

c(b(b(x0)))

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

1.1.2.1 P is empty

There are no pairs anymore.