Certification Problem

Input (COPS 762)

We consider the TRS containing the following rules:

a b (1)
f(a,b) f(a,a) (2)
f(b,a) f(a,a) (3)
f(a,a) c (4)
g(x) f(x,x) (5)

The underlying signature is as follows:

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

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 = g(a)
g(b)
f(b,b)
= t2

t0 = g(a)
f(a,a)
c
= t2

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