Certification Problem

Input (COPS 226)

We consider the TRS containing the following rules:

c → f(c,c) (1)
c → g(c,c) (2)
f(x,f(x,x)) → c (3)

The underlying signature is as follows:

{c/0, f/2, g/2}

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(c,f(c,c))
→ f(f(c,c),f(c,c))
→ f(f(c,g(c,c)),f(c,c))
→ f(f(c,g(c,c)),f(c,g(c,c)))
= t3

t0 = f(c,f(c,c))
→ c
→ g(c,c)
→ g(g(c,c),c)
= t3

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