Certification Problem

Input (COPS 802)

We consider the TRS containing the following rules:

f(a,b,x) f(x,x,x) (1)
a c (2)

The underlying signature is as follows:

{f/3, a/0, b/0, c/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(a,b,a)
f(a,a,a)
f(a,c,a)
= t2

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

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