Certification Problem

Input (COPS 590)

We consider the TRS containing the following rules:

+(0,0) 0 (1)
+(s(x),y) s(+(x,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 2023)

1 Non-Joinable Fork

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

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

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

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