Certification Problem

Input (COPS 661)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{b/0, f/1, h/2, c/0, a/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 = h(h(c,c),b)
→ h(h(c,c),f(h(f(f(b)),c)))
= t1

t0 = h(h(c,c),b)
→ c
= t1

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