Certification Problem

Input (COPS 675)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2021)

1 Non-Joinable Fork

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

t0 = f(f(f(f(b))))
f(f(f(b)))
f(f(b))
f(b)
f(c)
= t4

t0 = f(f(f(f(b))))
f(f(f(b)))
f(f(b))
f(b)
b
= t4

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