YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

b^#(c(x0)) → b^#(b(x0))

b^#(c(x0)) → b^#(x0)

1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers

[b^#(x₁)] = 3 · x₁ + 0

[a(x₁)] = 1 · x₁ + 0

[b(x₁)] = -1 · x₁ + 0

[c(x₁)] = 1 · x₁ + 8

together with the usable rules

a(b(x0)) → x0

a(c(x0)) → c(c(x0))

b(c(x0)) → a(a(b(b(x0))))

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

There are no pairs anymore.