Certification Problem

Input (COPS 682)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2023)

1 Non-Joinable Fork

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

t0 = f(f(h(b,a)))
f(f(h(b,b)))
= t1

t0 = f(f(h(b,a)))
a
= t1

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