Certification Problem

Input (COPS 613)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2021)

1 Non-Joinable Fork

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

t0 = +(p(s(c_1)),0)
+(s(p(c_1)),0)
+(c_1,0)
= t2

t0 = +(p(s(c_1)),0)
p(+(0,s(c_1)))
p(s(c_1))
s(p(c_1))
c_1
= t4

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