Certification Problem

Input (COPS 946)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{a/1, b/1}

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 = a(a(b(c_1)))
b(b(b(b(c_1))))
a(a(a(c_1)))
a(b(b(b(c_1))))
= t3

t0 = a(a(b(c_1)))
b(b(b(b(c_1))))
a(a(a(c_1)))
b(b(b(a(c_1))))
= t3

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