Certification Problem

Input (COPS 651)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Prove or disprove confluence.

No.

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

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

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