Certification Problem

Input (COPS 557)

We consider the TRS containing the following rules:

a b (1)
f(x,a) f(b,b) (2)
f(b,x) f(b,b) (3)
f(f(x,y),z) f(b,b) (4)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2020)

1 Non-Joinable Fork

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

t0 = f(c_1,a)
f(b,b)
= t1

t0 = f(c_1,a)
f(c_1,b)
= t1

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