Certification Problem

Input (COPS 252)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{a/0, h/1, c/0, f/1, g/1, k/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 = a
f(c)
= t1

t0 = a
h(c)
= t1

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