Certification Problem

Input (COPS 1279)

We consider the TRS containing the following rules:

f(g(x)) g(f(f(x))) (1)
g(x) x (2)

The underlying signature is as follows:

{f/1, g/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2023)

1 Non-Joinable Fork

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

t0 = f(g(x))
f(x)
= t1

t0 = f(g(x))
g(f(f(x)))
f(f(x))
= t2

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