Certification Problem

Input (COPS 675)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

t0 = h(h(b,b),f(f(h(h(c,c),c))))
→ c
= t1

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