Certification Problem

Input (COPS 719)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{b/0, f/1, c/0, a/0, h/2}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Non-Joinable Fork

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

t0 = f(f(f(c)))
f(f(f(b)))
= t1

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

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