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 ACP @ CoCo 2023)

1 Non-Joinable Fork

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

t0 = f(g(g(g(c_1))))
g(f(f(g(g(c_1)))))
f(f(g(g(c_1))))
f(f(g(c_1)))
f(f(c_1))
= t4

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

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