Certification Problem

Input (COPS 580)

We consider the TRS containing the following rules:

+(0,0) 0 (1)
+(s(0),y) s(+(0,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 ACP @ CoCo 2020)

1 Non-Joinable Fork

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

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

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

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