Certification Problem

Input (COPS 668)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

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

t0 = h(b,f(a))
h(b,f(c))
b
= t2

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