Certification Problem

Input (COPS 654)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2021)

1 Non-Joinable Fork

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

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

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

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