Certification Problem

Input (COPS 685)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

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

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

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