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 ACP @ CoCo 2023)

1 Non-Joinable Fork

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

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

t0 = +(s(p(0)),c_1)
+(p(s(0)),c_1)
+(0,c_1)
c_1
= t3

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