Certification Problem

Input (COPS 659)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2023)

1 Non-Joinable Fork

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

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

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

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