Certification Problem

Input (COPS 655)

We consider the TRS containing the following rules:

b a (1)
f(h(b,f(a))) f(c) (2)
c h(c,c) (3)
h(f(f(b)),a) c (4)
h(a,a) a (5)

The underlying signature is as follows:

{b/0, a/0, f/1, h/2, c/0}

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 = f(h(b,f(a)))
f(h(a,f(a)))
= t1

t0 = f(h(b,f(a)))
f(c)
= t1

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