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 2022)

1 Non-Joinable Fork

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

t0 = f(f(x69,i(x69)),i(f(x69,i(x69))))
→ f(e,i(f(x69,i(x69))))
→ f(e,i(e))
→ i(e)
= t3

t0 = f(f(x69,i(x69)),i(f(x69,i(x69))))
→ e
= t1

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