Certification Problem

Input (COPS 800)

We consider the TRS containing the following rules:

a b (1)
f(g(a)) f(a) (2)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2020)

1 Non-Joinable Fork

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

t0 = f(g(a))
f(g(b))
= t1

t0 = f(g(a))
f(a)
= t1

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