Certification Problem

Input (COPS 573)

We consider the TRS containing the following rules:

+(0,y) y (1)
+(s(x),y) s(+(x,y)) (2)
+(x,0) +(0,x) (3)
+(x,s(y)) +(s(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 = +(s(x306),s(y))
s(+(x306,s(y)))
s(+(s(y),x306))
s(s(+(y,x306)))
= t3

t0 = +(s(x306),s(y))
+(s(y),s(x306))
s(+(y,s(x306)))
s(+(s(x306),y))
s(s(+(x306,y)))
= t4

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