Certification Problem

Input (COPS 983)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{a/1, b/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2021)

1 Non-Joinable Fork

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

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

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

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