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 2023)

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: