Certification Problem

Input (COPS 32)

We consider the TRS containing the following rules:

f(g(x),h(x)) a (1)
g(b) d (2)
h(c) d (3)

The underlying signature is as follows:

{f/2, g/1, h/1, a/0, b/0, d/0, c/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(g(b),h(b))
f(d,h(b))
= t1

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

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