Certification Problem

Input (COPS 246)

We consider the TRS containing the following rules:

a f(a) (1)
a g(h(a)) (2)
f(x) h(g(x)) (3)
h(x) f(g(x)) (4)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2023)

1 Non-Joinable Fork

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

t0 = a
g(h(a))
= t1

t0 = a
f(a)
= t1

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