Certification Problem

Input (COPS 591)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2022)

1 Non-Joinable Fork

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

t0 = *(s(c_1),s(c_2))
+(*(c_1,s(c_2)),s(c_2))
+(+(c_1,*(c_1,c_2)),s(c_2))
= t2

t0 = *(s(c_1),s(c_2))
+(s(c_1),*(s(c_1),c_2))
s(+(c_1,*(s(c_1),c_2)))
= t2

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