Certification Problem

Input (COPS 242)

We consider the TRS containing the following rules:

a h(c) (1)
a h(f(c)) (2)
g(x) g(x) (3)
h(f(x)) h(f(g(x))) (4)

The underlying signature is as follows:

{a/0, h/1, c/0, f/1, g/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2021)

1 Non-Joinable Fork

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

t0 = a
h(f(c))
= t1

t0 = a
h(c)
= t1

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