Certification Problem

Input (COPS 713)

We consider the TRS containing the following rules:

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

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 = f(h(h(b,f(a)),a))
f(h(h(b,b),a))
f(a)
f(c)
= t3

t0 = f(h(h(b,f(a)),a))
f(h(h(b,b),a))
f(a)
b
= t3

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