Certification Problem

Input (TPDB TRS_Outermost/Strategy_outermost_added_08/OvConsOS_nosorts-noand_Z)

The rewrite relation of the following TRS is considered.

zeros	→	cons(0,n__zeros)	(1)
U11(tt,L)	→	U12(tt,activate(L))	(2)
U12(tt,L)	→	s(length(activate(L)))	(3)
U21(tt,IL,M,N)	→	U22(tt,activate(IL),activate(M),activate(N))	(4)
U22(tt,IL,M,N)	→	U23(tt,activate(IL),activate(M),activate(N))	(5)
U23(tt,IL,M,N)	→	cons(activate(N),n__take(activate(M),activate(IL)))	(6)
length(nil)	→	0	(7)
length(cons(N,L))	→	U11(tt,activate(L))	(8)
take(0,IL)	→	nil	(9)
take(s(M),cons(N,IL))	→	U21(tt,activate(IL),M,N)	(10)
zeros	→	n__zeros	(11)
take(X1,X2)	→	n__take(X1,X2)	(12)
activate(n__zeros)	→	zeros	(13)
activate(n__take(X1,X2))	→	take(X1,X2)	(14)
activate(X)	→	X	(15)

The evaluation strategy is outermost

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by AProVE @ termCOMP 2023)

1 Loop

The following loop proves nontermination.

t₀	=	U12(tt,activate(activate(n__zeros)))
	→	s(length(activate(activate(activate(n__zeros)))))
	→	s(length(activate(activate(n__zeros))))
	→	s(length(activate(n__zeros)))
	→	s(length(zeros))
	→	s(length(cons(0,n__zeros)))
	→	s(U11(tt,activate(n__zeros)))
	→	s(U12(tt,activate(activate(n__zeros))))
	=	t₇

where t₇ = C[t₀] and C = s(☐)