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 csi @ CoCo 2020)

1 Non-Joinable Fork

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

t0 = b(b(c(c(x109))))
b(b(x109))
= t1

t0 = b(b(c(c(x109))))
c(a(c(a(a(c(x109))))))
c(a(c(a(b(c(x109))))))
c(a(c(b(b(c(x109))))))
c(b(c(b(b(c(x109))))))
c(b(c(c(a(c(a(a(x109))))))))
c(b(c(c(a(c(a(b(x109))))))))
c(b(c(c(a(c(b(b(x109))))))))
c(b(c(c(b(c(b(b(x109))))))))
c(b(b(c(b(b(x109))))))
c(c(a(c(a(a(b(b(x109))))))))
c(c(a(c(a(b(b(b(x109))))))))
c(c(a(c(b(b(b(b(x109))))))))
c(c(b(c(b(b(b(b(x109))))))))
b(c(b(b(b(b(x109))))))
= t14

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