Certification Problem

Input (COPS 650)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{c/0, h/2, a/0, f/1, b/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 = h(h(c,f(f(a))),f(b))
→ h(h(h(a,h(f(f(f(f(c)))),h(a,h(b,b)))),f(f(a))),f(b))
= t1

t0 = h(h(c,f(f(a))),f(b))
→ a
= t1

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