Certification Problem

Input (COPS 614)

We consider the TRS containing the following rules:

+(0,y) y (1)
+(s(x),y) s(+(x,y)) (2)
+(p(x),y) +(x,p(y)) (3)
p(s(x)) s(p(x)) (4)
s(p(x)) p(s(x)) (5)
p(s(0)) 0 (6)

The underlying signature is as follows:

{+/2, 0/0, s/1, p/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2020)

1 Non-Joinable Fork

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

t0 = +(p(s(0)),f5)
+(0,f5)
= t1

t0 = +(p(s(0)),f5)
+(s(0),p(f5))
= t1

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