Certification Problem

Input (COPS 943)

We consider the TRS containing the following rules:

a(a(x)) b(c(x)) (1)
b(b(x)) c(d(x)) (2)
c(c(x)) d(d(d(x))) (3)
d(d(d(x))) a(c(x)) (4)

The underlying signature is as follows:

{a/1, b/1, c/1, d/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 = c(b(b(c_1)))
c(c(d(c_1)))
d(d(d(d(c_1))))
a(c(d(c_1)))
= t3

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

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