NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(a(a(x0))) → Wait(Right1(x0))
Begin(a(x0)) → Wait(Right2(x0))
Begin(c(x0)) → Wait(Right3(x0))
Right1(a(End(x0))) → Left(b(End(x0)))
Right2(a(a(End(x0)))) → Left(b(End(x0)))
Right3(b(End(x0))) → Left(c(a(c(b(End(x0))))))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right1(b(x0)) → Ab(Right1(x0))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right1(c(x0)) → Ac(Right1(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Aa(Left(x0)) → Left(a(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Wait(Left(x0)) → Begin(x0)
a(x0) → x0
a(a(a(x0))) → b(x0)
b(c(x0)) → c(a(c(b(x0))))
c(x0) → x0

Proof

1 Loop

The following loop proves nontermination.

t0 = Begin(c(b(End(x45914))))
→ε Wait(Right3(b(End(x45914))))
→1 Wait(Left(c(a(c(b(End(x45914)))))))
→ε Begin(c(a(c(b(End(x45914))))))
→1 Begin(a(c(b(End(x45914)))))
→1 Begin(c(b(End(x45914))))
= t5
where t5 = t0σ and σ = {x45914/x45914}