Certification Problem

Input (COPS 720)

We consider the TRS containing the following rules:

b → f(f(b)) (1)
f(a) → c (2)
f(a) → b (3)
b → f(h(a,b)) (4)

The underlying signature is as follows:

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

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 = b
→ f(f(b))
→ f(f(f(h(a,b))))
→ f(f(f(h(a,f(h(a,b))))))
= t3

t0 = b
→ f(h(a,b))
→ f(h(a,f(h(a,b))))
→ f(h(a,f(h(a,f(h(a,b))))))
= t3

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