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 2023)

1 Non-Joinable Fork

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

t0 = +(+(s(x585),x586),s(y))
+(s(+(x585,x586)),s(y))
s(+(+(x585,x586),s(y)))
s(s(+(y,+(x585,x586))))
+(y,+(x585,x586))
= t4

t0 = +(+(s(x585),x586),s(y))
s(+(y,+(s(x585),x586)))
s(+(y,s(+(x585,x586))))
s(s(+(+(x585,x586),y)))
+(+(x585,x586),y)
= t4

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