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 2023)

1 Non-Joinable Fork

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

t0 = +(s(x),+(0,x41))
+(s(x),x41)
s(+(x,x41))
= t2

t0 = +(s(x),+(0,x41))
s(+(x,+(0,x41)))
s(+(+(x,0),x41))
= t2

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