Certification Problem

Input (COPS 658)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2021)

1 Non-Joinable Fork

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

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

t0 = f(f(f(f(a))))
→ a
= t1

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