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(x0)) → Wait(Right3(x0))
Begin(c(a(x0))) → Wait(Right4(x0))
Begin(a(x0)) → Wait(Right5(x0))
Begin(b(x0)) → Wait(Right6(x0))
Begin(c(b(x0))) → Wait(Right7(x0))
Begin(b(x0)) → Wait(Right8(x0))
Right1(b(End(x0))) → Left(a(b(c(End(x0)))))
Right2(b(a(End(x0)))) → Left(a(b(c(End(x0)))))
Right3(c(End(x0))) → Left(a(c(End(x0))))
Right4(b(End(x0))) → Left(a(b(c(End(x0)))))
Right5(b(c(End(x0)))) → Left(a(b(c(End(x0)))))
Right6(c(End(x0))) → Left(b(a(End(x0))))
Right7(a(End(x0))) → Left(c(b(a(End(x0)))))
Right8(a(c(End(x0)))) → Left(c(b(a(End(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))
Right5(b(x0)) → Ab(Right5(x0))
Right6(b(x0)) → Ab(Right6(x0))
Right7(b(x0)) → Ab(Right7(x0))
Right8(b(x0)) → Ab(Right8(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))
Right5(a(x0)) → Aa(Right5(x0))
Right6(a(x0)) → Aa(Right6(x0))
Right7(a(x0)) → Aa(Right7(x0))
Right8(a(x0)) → Aa(Right8(x0))
Right1(c(x0)) → Ac(Right1(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right6(c(x0)) → Ac(Right6(x0))
Right7(c(x0)) → Ac(Right7(x0))
Right8(c(x0)) → Ac(Right8(x0))
Ab(Left(x0)) → Left(b(x0))
Aa(Left(x0)) → Left(a(x0))
Ac(Left(x0)) → Left(c(x0))
Wait(Left(x0)) → Begin(x0)
b(a(a(x0))) → a(b(c(x0)))
c(a(x0)) → a(c(x0))
b(c(a(x0))) → a(b(c(x0)))
c(b(x0)) → b(a(x0))
a(c(b(x0))) → c(b(a(x0)))

Proof

1 Loop

The following loop proves nontermination.

t0 = Begin(a(c(End(x7704))))
→ε Wait(Right3(c(End(x7704))))
→1 Wait(Left(a(c(End(x7704)))))
→ε Begin(a(c(End(x7704))))
= t3
where t3 = t0σ and σ = {x7704/x7704}