Certification Problem

Input (COPS 715)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

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

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