Certification Problem

Input (COPS 573)

We consider the TRS containing the following rules:

+(0,y) y (1)
+(s(x),y) s(+(x,y)) (2)
+(x,0) +(0,x) (3)
+(x,s(y)) +(s(y),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 2021)

1 Non-Joinable Fork

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

t0 = +(s(c_1),s(c_2))
+(s(c_2),s(c_1))
s(+(c_2,s(c_1)))
s(+(s(c_1),c_2))
s(s(+(c_1,c_2)))
= t4

t0 = +(s(c_1),s(c_2))
s(+(c_1,s(c_2)))
s(+(s(c_2),c_1))
s(s(+(c_2,c_1)))
= t3

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