Certification Problem

Input (TPDB TRS_Standard/AProVE_06/nonterm)

The rewrite relation of the following TRS is considered.

f(s(s(s(s(s(s(s(s(x)))))))),y,y)	→	f(id(s(s(s(s(s(s(s(s(x))))))))),y,y)	(1)
id(s(x))	→	s(id(x))	(2)
id(0)	→	0	(3)

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by ttt2 @ termCOMP 2023)

1 Loop

The following loop proves nontermination.

t₀	=	f(s(s(s(s(s(s(s(s(x2029)))))))),y,y)
	→	f(id(s(s(s(s(s(s(s(s(x2029))))))))),y,y)
	→	f(s(id(s(s(s(s(s(s(s(x2029))))))))),y,y)
	→	f(s(s(id(s(s(s(s(s(s(x2029))))))))),y,y)
	→	f(s(s(s(id(s(s(s(s(s(x2029))))))))),y,y)
	→	f(s(s(s(s(id(s(s(s(s(x2029))))))))),y,y)
	→	f(s(s(s(s(s(id(s(s(s(x2029))))))))),y,y)
	→	f(s(s(s(s(s(s(id(s(s(x2029))))))))),y,y)
	→	f(s(s(s(s(s(s(s(id(s(x2029))))))))),y,y)
	→	f(s(s(s(s(s(s(s(s(id(x2029))))))))),y,y)
	=	t₉

where t₉ = t₀σ and σ = {x2029/id(x2029), y/y}