Certification Problem

Input (COPS 738)

We consider the TRS containing the following rules:

a a (1)
f(b,a) f(a,b) (2)
f(x,a) b (3)

The underlying signature is as follows:

{a/0, f/2, 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(b,a)
f(a,b)
= t1

t0 = f(b,a)
b
= t1

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