Certification Problem

Input (COPS 568)

We consider the TRS containing the following rules:

f(x,x) x (1)
f(x,e) x (2)
f(e,x) x (3)
f(x,i(x)) e (4)
f(i(x),x) e (5)

The underlying signature is as follows:

{f/2, e/0, i/1}

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(i(e),e)
i(e)
= t1

t0 = f(i(e),e)
e
= t1

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