Certification Problem

Input (COPS 590)

We consider the TRS containing the following rules:

+(0,0) 0 (1)
+(s(x),y) s(+(x,y)) (2)
+(x,s(y)) s(+(y,x)) (3)
s(s(x)) 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 2022)

1 Non-Joinable Fork

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

t0 = +(s(x111),s(y))
s(+(x111,s(y)))
s(s(+(y,x111)))
+(y,x111)
= t3

t0 = +(s(x111),s(y))
s(+(y,s(x111)))
s(s(+(x111,y)))
+(x111,y)
= t3

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