Certification Problem

Input (COPS 1121)

We consider two TRSs R and S where R contains the rules

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

and S contains the following rules:

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

The underlying signature is as follows:

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

Prove or disprove commutation.

No.

The systems are not commuting due to the following forking derivations.

t₀	=	h(h(h(c,f(h(f(h(h(c,c),c)),b))),h(h(a,c),h(f(a),h(f(f(c)),h(c,b))))),h(f(a),c))
	→_S	h(h(h(c,f(h(f(h(h(c,c),c)),b))),h(h(f(f(h(a,c))),c),h(f(a),h(f(f(c)),h(c,b))))),h(f(a),c))
	=	s₁

and

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

There is no possibility to join s₁→_R^*·←_S^* t₁ 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.