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 csi @ CoCo 2023)

1 Non-Joinable Fork

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

t0 = a(a(a(x37)))
a(b(c(x37)))
= t1

t0 = a(a(a(x37)))
b(c(a(x37)))
= t1

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