Certification Problem

Input (COPS 588)

We consider the TRS containing the following rules:

+(0,y) y (1)
+(s(0),y) s(y) (2)
+(s(s(x)),y) s(s(+(y,x))) (3)
+(x,+(y,z)) +(+(z,y),x) (4)

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 = +(0,+(y,z))
+(y,z)
= t1

t0 = +(0,+(y,z))
+(+(z,y),0)
= t1

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