Certification Problem

Input (COPS 667)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

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

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