Certification Problem

Input (COPS 647)

We consider the TRS containing the following rules:

f(b) → b (1)
b → h(h(c,f(b)),a) (2)

The underlying signature is as follows:

{f/1, b/0, h/2, c/0, a/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 = f(b)
→ f(h(h(c,f(b)),a))
→ f(h(h(c,f(h(h(c,f(b)),a))),a))
→ f(h(h(c,f(h(h(c,f(h(h(c,f(b)),a))),a))),a))
= t3

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

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