Certification Problem

Input (COPS 587)

We consider the TRS containing the following rules:

+(0,y) y (1)
+(s(x),y) s(+(x,y)) (2)
+(x,+(y,z)) +(+(x,y),z) (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 = +(x,+(0,z))
+(x,z)
= t1

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

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