Certification Problem

Input (COPS 654)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Prove or disprove confluence.

No.

The system is not confluent due to the following forking derivations.

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

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

When applying the cap-function on both terms (where variables may be treated like constants) then the resulting terms do not unify.