Certification Problem

Input (COPS 80)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2021)

1 Non-Joinable Fork

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

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

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

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