Certification Problem

Input (COPS 663)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Non-Joinable Fork

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

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

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

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