Certification Problem

Input (COPS 706)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

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(h(c,c))
f(h(c,f(c)))
= t1

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

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