Certification Problem

Input (COPS 703)

We consider the TRS containing the following rules:

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

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 2020)

1 Non-Joinable Fork

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

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

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

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