Certification Problem

Input (COPS 589)

We consider the TRS containing the following rules:

+(0,y)	→	y	(1)
+(s(x),y)	→	s(+(y,x))	(2)
+(+(x,x),x)	→	+(x,+(x,x))	(3)

The underlying signature is as follows:

{+/2, 0/0, s/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t₀	=	+(+(s(x217),s(x217)),s(x217))
	→	+(s(+(s(x217),x217)),s(x217))
	→	+(s(s(+(x217,x217))),s(x217))
	→	s(+(s(x217),s(+(x217,x217))))
	→	s(s(+(s(+(x217,x217)),x217)))
	→	s(s(s(+(x217,+(x217,x217)))))
	=	t₅

t₀	=	+(+(s(x217),s(x217)),s(x217))
	→	+(s(x217),+(s(x217),s(x217)))
	→	+(s(x217),s(+(s(x217),x217)))
	→	+(s(x217),s(s(+(x217,x217))))
	→	s(+(s(s(+(x217,x217))),x217))
	→	s(s(+(x217,s(+(x217,x217)))))
	=	t₅

The two resulting terms cannot be joined for the following reason:

When applying the cap-function on both terms (where variables may be treated like constants) then the resulting terms do not unify.