NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(a(a(a(x0)))) → Wait(Right1(x0))
Begin(a(a(x0))) → Wait(Right2(x0))
Begin(a(x0)) → Wait(Right3(x0))
Begin(b(a(a(x0)))) → Wait(Right4(x0))
Begin(a(a(x0))) → Wait(Right5(x0))
Begin(a(x0)) → Wait(Right6(x0))
Begin(b(b(b(c(c(x0)))))) → Wait(Right7(x0))
Begin(b(b(c(c(x0))))) → Wait(Right8(x0))
Begin(b(c(c(x0)))) → Wait(Right9(x0))
Begin(c(c(x0))) → Wait(Right10(x0))
Begin(c(x0)) → Wait(Right11(x0))
Begin(b(b(b(x0)))) → Wait(Right12(x0))
Begin(b(b(x0))) → Wait(Right13(x0))
Begin(b(x0)) → Wait(Right14(x0))
Begin(c(a(a(x0)))) → Wait(Right15(x0))
Begin(a(a(x0))) → Wait(Right16(x0))
Begin(a(x0)) → Wait(Right17(x0))
Right1(a(End(x0))) → Left(b(b(End(x0))))
Right2(a(a(End(x0)))) → Left(b(b(End(x0))))
Right3(a(a(a(End(x0))))) → Left(b(b(End(x0))))
Right4(b(End(x0))) → Left(a(a(b(b(End(x0))))))
Right5(b(b(End(x0)))) → Left(a(a(b(b(End(x0))))))
Right6(b(b(a(End(x0))))) → Left(a(a(b(b(End(x0))))))
Right7(b(End(x0))) → Left(c(c(a(a(End(x0))))))
Right8(b(b(End(x0)))) → Left(c(c(a(a(End(x0))))))
Right9(b(b(b(End(x0))))) → Left(c(c(a(a(End(x0))))))
Right10(b(b(b(b(End(x0)))))) → Left(c(c(a(a(End(x0))))))
Right11(b(b(b(b(c(End(x0))))))) → Left(c(c(a(a(End(x0))))))
Right12(b(End(x0))) → Left(a(a(a(a(a(a(End(x0))))))))
Right13(b(b(End(x0)))) → Left(a(a(a(a(a(a(End(x0))))))))
Right14(b(b(b(End(x0))))) → Left(a(a(a(a(a(a(End(x0))))))))
Right15(c(End(x0))) → Left(b(b(a(a(c(c(End(x0))))))))
Right16(c(c(End(x0)))) → Left(b(b(a(a(c(c(End(x0))))))))
Right17(c(c(a(End(x0))))) → Left(b(b(a(a(c(c(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))
Right5(a(x0)) → Aa(Right5(x0))
Right6(a(x0)) → Aa(Right6(x0))
Right7(a(x0)) → Aa(Right7(x0))
Right8(a(x0)) → Aa(Right8(x0))
Right9(a(x0)) → Aa(Right9(x0))
Right10(a(x0)) → Aa(Right10(x0))
Right11(a(x0)) → Aa(Right11(x0))
Right12(a(x0)) → Aa(Right12(x0))
Right13(a(x0)) → Aa(Right13(x0))
Right14(a(x0)) → Aa(Right14(x0))
Right15(a(x0)) → Aa(Right15(x0))
Right16(a(x0)) → Aa(Right16(x0))
Right17(a(x0)) → Aa(Right17(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))
Right9(b(x0)) → Ab(Right9(x0))
Right10(b(x0)) → Ab(Right10(x0))
Right11(b(x0)) → Ab(Right11(x0))
Right12(b(x0)) → Ab(Right12(x0))
Right13(b(x0)) → Ab(Right13(x0))
Right14(b(x0)) → Ab(Right14(x0))
Right15(b(x0)) → Ab(Right15(x0))
Right16(b(x0)) → Ab(Right16(x0))
Right17(b(x0)) → Ab(Right17(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))
Right9(c(x0)) → Ac(Right9(x0))
Right10(c(x0)) → Ac(Right10(x0))
Right11(c(x0)) → Ac(Right11(x0))
Right12(c(x0)) → Ac(Right12(x0))
Right13(c(x0)) → Ac(Right13(x0))
Right14(c(x0)) → Ac(Right14(x0))
Right15(c(x0)) → Ac(Right15(x0))
Right16(c(x0)) → Ac(Right16(x0))
Right17(c(x0)) → Ac(Right17(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(a(a(a(x0)))) → b(b(x0))
b(b(a(a(x0)))) → a(a(b(b(x0))))
b(b(b(b(c(c(x0)))))) → c(c(a(a(x0))))
b(b(b(b(x0)))) → a(a(a(a(a(a(x0))))))
c(c(a(a(x0)))) → b(b(a(a(c(c(x0))))))

Proof

1 Loop

The following loop proves nontermination.

t0 = Begin(a(a(b(b(End(x31042))))))
→ε Wait(Right5(b(b(End(x31042)))))
→1 Wait(Left(a(a(b(b(End(x31042)))))))
→ε Begin(a(a(b(b(End(x31042))))))
= t3
where t3 = t0σ and σ = {x31042/x31042}