Certification Problem

Input (COPS 716)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{a/0, h/2, f/1, b/0, c/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 = f(h(h(b,a),h(f(c),c)))
→ f(h(h(b,h(a,f(b))),h(f(c),c)))
= t1

t0 = f(h(h(b,a),h(f(c),c)))
→ c
= t1

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