NO Non-Confluence Proof

Non-Confluence Proof

by Hakusan

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Non-Joinable Fork

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))))

→_{2.2.1.1.1.2.1.1} 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

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

→_ε h(b,b)

= 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.

Tool configuration

Hakusan

version: 0.10

t₀	=	h(f(f(b)),h(c,h(f(f(h(h(b,h(c,c)),h(f(a),c)))),f(a))))
	→_{2.2.1.1.1.2.1.1}	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

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