Certification Problem

Input (COPS 113)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

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

t0 = f(a,a)
c
= t1

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