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

Proof

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

c^#(a(x0)) → c^#(c(b(b(a(x0)))))

c^#(a(x0)) → c^#(b(b(a(x0))))

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

[c^#(x₁)] = 0 · x₁ + 0

[a(x₁)] = 7 · x₁ + 14

[b(x₁)] = -7 · x₁ + 0

[c(x₁)] = 7 · x₁ + 0

together with the usable rules

b(a(x0)) → x0

c(a(x0)) → a(c(c(b(b(a(x0))))))

c(b(x0)) → x0

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

There are no pairs anymore.