Certification Problem

Input (COPS 958)

We consider the TRS containing the following rules:

b(b(b(x))) a(a(a(x))) (1)
a(a(a(x))) b(a(b(x))) (2)

The underlying signature is as follows:

{b/1, a/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2022)

1 Non-Joinable Fork

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

t0 = b(b(b(a(c_1))))
a(a(a(a(c_1))))
a(b(a(b(c_1))))
= t2

t0 = b(b(b(a(c_1))))
a(a(a(a(c_1))))
b(a(b(a(c_1))))
= t2

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