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 csi @ CoCo 2022)

1 Non-Joinable Fork

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

t0 = f(f(x,1),z)
f(x,z)
= t1

t0 = f(f(x,1),z)
f(x,f(1,z))
= t1

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