NO Non-Confluence Proof

Non-Confluence Proof

by Hakusan

Input

The rewrite relation of the following TRS is considered.

a	→	f(f(h(a,c)))
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

Proof

1 Non-Joinable Fork

The system is not confluent 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))

→_1.2.1.1 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))

= t₁

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))

→_ε c

= 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(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))
	→_1.2.1.1	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))
	=	t₁

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))
	→_ε	c
	=	t₁