Certification Problem

Input (COPS 651)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Non-Joinable Fork

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

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

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

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