Certification Problem

Input (COPS 963)

We consider the TRS containing the following rules:

b(a(a(x))) a(x) (1)
a(a(a(x))) b(b(b(x))) (2)
b(b(x)) a(b(a(x))) (3)

The underlying signature is as follows:

{b/1, a/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2023)

1 Non-Joinable Fork

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

t0 = b(a(a(a(c_1))))
b(b(b(b(c_1))))
a(b(a(b(b(c_1)))))
a(b(a(a(b(a(c_1))))))
a(a(b(a(c_1))))
= t4

t0 = b(a(a(a(c_1))))
a(a(c_1))
= t1

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