Certification Problem

Input (COPS 980)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{a/1, b/1, c/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 = b(b(c(c(c(c(c_1))))))
c(a(c(a(a(c(c(c(c_1))))))))
c(a(c(a(a(c(c_1))))))
= t2

t0 = b(b(c(c(c(c(c_1))))))
b(b(c(c(c_1))))
b(b(c_1))
= t2

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