Certification Problem

Input (COPS 666)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2020)

1 Non-Joinable Fork

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

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

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

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