Certification Problem

Input (COPS 16)

We consider the TRS containing the following rules:

f(x,x) g(x) (1)
f(x,g(x)) b (2)
h(c,y) f(h(y,c),h(y,y)) (3)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2023)

1 Non-Joinable Fork

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

t0 = h(c,c)
f(h(c,c),h(c,c))
g(h(c,c))
g(f(h(c,c),h(c,c)))
g(f(h(c,c),f(h(c,c),h(c,c))))
g(f(h(c,c),g(h(c,c))))
g(b)
= t6

t0 = h(c,c)
f(h(c,c),h(c,c))
f(h(c,c),f(h(c,c),h(c,c)))
f(h(c,c),g(h(c,c)))
b
= t4

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