Certification Problem

Input (COPS 31)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2021)

1 Non-Joinable Fork

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

t0 = f(g(g(b)),h(g(b),d))
f(g(c),h(g(b),d))
f(g(c),e)
= t2

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

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