NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(f(x0)) → Wait(Right1(x0))
Begin(a(x0)) → Wait(Right2(x0))
Begin(c(x0)) → Wait(Right3(x0))
Begin(s(x0)) → Wait(Right4(x0))
Begin(f(x0)) → Wait(Right5(x0))
Right1(c(End(x0))) → Left(f(n(a(c(End(x0))))))
Right2(n(End(x0))) → Left(c(End(x0)))
Right3(c(End(x0))) → Left(c(End(x0)))
Right4(n(End(x0))) → Left(f(s(s(End(x0)))))
Right5(n(End(x0))) → Left(f(n(End(x0))))
Right1(f(x0)) → Af(Right1(x0))
Right2(f(x0)) → Af(Right2(x0))
Right3(f(x0)) → Af(Right3(x0))
Right4(f(x0)) → Af(Right4(x0))
Right5(f(x0)) → Af(Right5(x0))
Right1(n(x0)) → An(Right1(x0))
Right2(n(x0)) → An(Right2(x0))
Right3(n(x0)) → An(Right3(x0))
Right4(n(x0)) → An(Right4(x0))
Right5(n(x0)) → An(Right5(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))
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))
Right1(s(x0)) → As(Right1(x0))
Right2(s(x0)) → As(Right2(x0))
Right3(s(x0)) → As(Right3(x0))
Right4(s(x0)) → As(Right4(x0))
Right5(s(x0)) → As(Right5(x0))
Af(Left(x0)) → Left(f(x0))
An(Left(x0)) → Left(n(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
As(Left(x0)) → Left(s(x0))
Wait(Left(x0)) → Begin(x0)
f(x0) → n(c(n(a(x0))))
c(f(x0)) → f(n(a(c(x0))))
n(a(x0)) → c(x0)
c(c(x0)) → c(x0)
n(s(x0)) → f(s(s(x0)))
n(f(x0)) → f(n(x0))

Proof

1 Loop

The following loop proves nontermination.

t0 = Begin(f(n(End(x9269))))
→ε Wait(Right5(n(End(x9269))))
→1 Wait(Left(f(n(End(x9269)))))
→ε Begin(f(n(End(x9269))))
= t3
where t3 = t0σ and σ = {x9269/x9269}