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(a(b(x0))) → Wait(Right3(x0))
Begin(b(x0)) → Wait(Right4(x0))
Right1(a(End(x0))) → Left(b(a(b(End(x0)))))
Right2(a(a(End(x0)))) → Left(b(a(b(End(x0)))))
Right3(b(End(x0))) → Left(a(b(a(End(x0)))))
Right4(b(a(End(x0)))) → Left(a(b(a(End(x0)))))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right1(b(x0)) → Ab(Right1(x0))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Aa(Left(x0)) → Left(a(x0))
Ab(Left(x0)) → Left(b(x0))
Wait(Left(x0)) → Begin(x0)
a(a(a(x0))) → b(a(b(x0)))
b(a(b(x0))) → a(b(a(x0)))

Proof

1 Loop

The following loop proves nontermination.

t0 = a(a(a(b(a(a(x185891))))))
→ε b(a(b(b(a(a(x185891))))))
→ε a(b(a(b(a(a(x185891))))))
→1 a(a(b(a(a(a(x185891))))))
→1.1.1 a(a(b(b(a(b(x185891))))))
→1.1.1 a(a(b(a(b(a(x185891))))))
→1.1 a(a(a(b(a(a(x185891))))))
= t6
where t6 = t0σ and σ = {x185891/x185891}