Certification Problem

Input (COPS 9)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{f/1, g/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 = f(f(f(g(f(f(f(g(f(c_1)))))))))
f(f(g(f(f(g(f(c_1)))))))
f(g(f(g(f(c_1)))))
f(g(g(c_1)))
= t3

t0 = f(f(f(g(f(f(f(g(f(c_1)))))))))
f(f(g(f(f(g(f(c_1)))))))
f(g(f(g(f(c_1)))))
g(g(f(c_1)))
= t3

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