Certification Problem

Input (COPS 732)

We consider the TRS containing the following rules:

a a (1)
a b (2)
a f(a,a) (3)
f(b,b) f(a,a) (4)
f(x0,a) f(a,b) (5)

The underlying signature is as follows:

{a/0, b/0, f/2}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2023)

1 Non-Joinable Fork

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

t0 = f(f3,a)
f(f3,b)
= t1

t0 = f(f3,a)
f(a,b)
= t1

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