NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(0(x0)) Wait(Right1(x0))
Begin(s(0(x0))) Wait(Right2(x0))
Begin(0(x0)) Wait(Right3(x0))
Begin(s(s(x0))) Wait(Right4(x0))
Begin(s(x0)) Wait(Right5(x0))
Begin(s(x0)) Wait(Right6(x0))
Begin(i(x0)) Wait(Right7(x0))
Begin(s(x0)) Wait(Right8(x0))
Right1(p(End(x0))) Left(s(s(0(s(s(p(End(x0))))))))
Right2(p(End(x0))) Left(0(End(x0)))
Right3(p(s(End(x0)))) Left(0(End(x0)))
Right4(p(End(x0))) Left(s(p(s(End(x0)))))
Right5(p(s(End(x0)))) Left(s(p(s(End(x0)))))
Right6(f(End(x0))) Left(g(q(i(End(x0)))))
Right7(q(End(x0))) Left(q(s(End(x0))))
Right8(q(End(x0))) Left(s(s(End(x0))))
Right1(p(x0)) Ap(Right1(x0))
Right2(p(x0)) Ap(Right2(x0))
Right3(p(x0)) Ap(Right3(x0))
Right4(p(x0)) Ap(Right4(x0))
Right5(p(x0)) Ap(Right5(x0))
Right6(p(x0)) Ap(Right6(x0))
Right7(p(x0)) Ap(Right7(x0))
Right8(p(x0)) Ap(Right8(x0))
Right1(0(x0)) A0(Right1(x0))
Right2(0(x0)) A0(Right2(x0))
Right3(0(x0)) A0(Right3(x0))
Right4(0(x0)) A0(Right4(x0))
Right5(0(x0)) A0(Right5(x0))
Right6(0(x0)) A0(Right6(x0))
Right7(0(x0)) A0(Right7(x0))
Right8(0(x0)) A0(Right8(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))
Right6(s(x0)) As(Right6(x0))
Right7(s(x0)) As(Right7(x0))
Right8(s(x0)) As(Right8(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))
Right6(f(x0)) Af(Right6(x0))
Right7(f(x0)) Af(Right7(x0))
Right8(f(x0)) Af(Right8(x0))
Right1(g(x0)) Ag(Right1(x0))
Right2(g(x0)) Ag(Right2(x0))
Right3(g(x0)) Ag(Right3(x0))
Right4(g(x0)) Ag(Right4(x0))
Right5(g(x0)) Ag(Right5(x0))
Right6(g(x0)) Ag(Right6(x0))
Right7(g(x0)) Ag(Right7(x0))
Right8(g(x0)) Ag(Right8(x0))
Right1(q(x0)) Aq(Right1(x0))
Right2(q(x0)) Aq(Right2(x0))
Right3(q(x0)) Aq(Right3(x0))
Right4(q(x0)) Aq(Right4(x0))
Right5(q(x0)) Aq(Right5(x0))
Right6(q(x0)) Aq(Right6(x0))
Right7(q(x0)) Aq(Right7(x0))
Right8(q(x0)) Aq(Right8(x0))
Right1(i(x0)) Ai(Right1(x0))
Right2(i(x0)) Ai(Right2(x0))
Right3(i(x0)) Ai(Right3(x0))
Right4(i(x0)) Ai(Right4(x0))
Right5(i(x0)) Ai(Right5(x0))
Right6(i(x0)) Ai(Right6(x0))
Right7(i(x0)) Ai(Right7(x0))
Right8(i(x0)) Ai(Right8(x0))
Ap(Left(x0)) Left(p(x0))
A0(Left(x0)) Left(0(x0))
As(Left(x0)) Left(s(x0))
Af(Left(x0)) Left(f(x0))
Ag(Left(x0)) Left(g(x0))
Aq(Left(x0)) Left(q(x0))
Ai(Left(x0)) Left(i(x0))
Wait(Left(x0)) Begin(x0)
p(0(x0)) s(s(0(s(s(p(x0))))))
p(s(0(x0))) 0(x0)
p(s(s(x0))) s(p(s(x0)))
f(s(x0)) g(q(i(x0)))
g(x0) f(p(p(x0)))
q(i(x0)) q(s(x0))
q(s(x0)) s(s(x0))
i(x0) s(x0)

Proof

1 Loop

The following loop proves nontermination.

t0 = Begin(s(p(s(End(x27909)))))
ε Wait(Right5(p(s(End(x27909)))))
1 Wait(Left(s(p(s(End(x27909))))))
ε Begin(s(p(s(End(x27909)))))
= t3
where t3 = t0σ and σ = {x27909/x27909}