Certification Problem

Input (TPDB TRS_Standard/Secret_07_TRS/7)

The rewrite relation of the following TRS is considered.

f(a,f(a,f(b,f(x,y))))	→	f(b,f(c,f(b,f(a,f(a,f(a,f(x,y)))))))	(1)
f(a,f(c,f(x,y)))	→	f(b,f(x,y))	(2)

Prove or disprove termination.

No.

The following set of initial dependency pairs has been identified.

It remains to prove infiniteness of the resulting DP problem.

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

and the following rules have been deleted.

We consider narrowings of the pair below position 2 to get the following set of pairs

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

f^#(a,f(a,f(b,f(b,f(x0,x1)))))

→

f^#(a,f(b,f(c,f(b,f(a,f(a,f(a,f(x0,x1))))))))

(10)

and the following rules have been deleted.

The pairs are instantiated to the following pairs.

The following loop proves infiniteness of the DP problem.

t₀	=	f^#(a,f(a,f(a,f(a,f(a,f(c,f(b,f(x,y))))))))
	→_R	f^#(a,f(a,f(a,f(a,f(b,f(b,f(x,y)))))))
	→_R	f^#(a,f(a,f(b,f(c,f(b,f(a,f(a,f(a,f(b,f(x,y))))))))))
	→_R	f^#(a,f(a,f(b,f(c,f(b,f(a,f(b,f(c,f(b,f(a,f(a,f(a,f(x,y)))))))))))))
	→_P	f^#(a,f(a,f(b,f(b,f(a,f(b,f(c,f(b,f(a,f(a,f(a,f(x,y))))))))))))
	→_P	f^#(a,f(a,f(b,f(a,f(b,f(c,f(b,f(a,f(a,f(a,f(x,y)))))))))))
	→_P	f^#(a,f(a,f(b,f(c,f(b,f(a,f(a,f(a,f(c,f(b,f(a,f(a,f(a,f(x,y))))))))))))))
	→_P	f^#(a,f(a,f(b,f(b,f(a,f(a,f(a,f(c,f(b,f(a,f(a,f(a,f(x,y)))))))))))))
	→_P	f^#(a,f(a,f(b,f(a,f(a,f(a,f(c,f(b,f(a,f(a,f(a,f(x,y))))))))))))
	→_P	f^#(a,f(a,f(a,f(a,f(a,f(c,f(b,f(a,f(a,f(a,f(x,y)))))))))))
	=	t₉

where t₉ = t₀σ and σ = {x/a, y/f(a,f(a,f(x,y)))}