Certification Problem

Input (COPS 735)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{a/0, b/0, f/2}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2020)

1 Non-Joinable Fork

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

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

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

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