Certification Problem

Input (COPS 701)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

t0 = h(f(a),a)
→ h(c,a)
= t1

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