Certification Problem

Input (COPS 6)

We consider the TRS containing the following rules:

f(f(x,y),z) f(x,f(y,z)) (1)
f(i(x1),x1) e (2)

The underlying signature is as follows:

{f/2, i/1, e/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(i(i(c_1)),i(c_1)),c_1)
f(i(i(c_1)),f(i(c_1),c_1))
f(i(i(c_1)),e)
= t2

t0 = f(f(i(i(c_1)),i(c_1)),c_1)
f(e,c_1)
= t1

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