Certification Problem

Input (COPS 708)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

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(f(f(c)),h(c,h(c,h(h(f(b),f(f(f(f(b))))),b))))
→ h(f(f(f(a))),h(c,h(c,h(h(f(b),f(f(f(f(b))))),b))))
→ h(f(f(f(f(f(b))))),h(c,h(c,h(h(f(b),f(f(f(f(b))))),b))))
= t2

t0 = h(f(f(c)),h(c,h(c,h(h(f(b),f(f(f(f(b))))),b))))
→ h(h(a,a),f(c))
→ h(h(f(f(b)),a),f(c))
→ h(h(f(f(b)),f(f(b))),f(c))
= t3

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