Certification Problem

Input (COPS 670)

We consider the TRS containing the following rules:

c f(c) (1)
c c (2)
f(a) b (3)
a c (4)

The underlying signature is as follows:

{c/0, f/1, a/0, b/0}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2022)

1 Non-Joinable Fork

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

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

t0 = f(a)
b
= t1

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