Certification Problem

Input (COPS 13)

We consider the TRS containing the following rules:

f(f(x,y),z) f(x,f(y,z)) (1)
f(x,1) x (2)

The underlying signature is as follows:

{f/2, 1/0}

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 = f(f(f(c_1,1),1),c_2)
f(f(c_1,1),f(1,c_2))
f(c_1,f(1,c_2))
= t2

t0 = f(f(f(c_1,1),1),c_2)
f(f(c_1,1),c_2)
f(c_1,c_2)
= t2

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