Re: completeness of the relational lattice

On 30 jun, 00:00, Vadim Tropashko <vadimtro_inva..._at_yahoo.com> wrote:
> On Jun 29, 1:40 pm, Jan Hidders <hidd..._at_gmail.com> wrote:
>
>
>
> > On 29 jun, 18:20, Vadim Tropashko <vadimtro_inva..._at_yahoo.com> wrote:
>
> > > On Jun 28, 3:41 pm, Jan Hidders <hidd..._at_gmail.com> wrote:
>
> > > > > > I invite you to challenge me to show that it can prove an equation
> > > > > > that holds. Of course you should also check if all these equations can
> > > > > > be derived by you.
>
> > > One more challenge:
>
> > > <xy> + S + Q = <xy> + S
>
> > > where S * [] = [z], and Q header is unconstrained. (I don't see axioms
> > > about the lattice bottom element:-)
>
> > Sorry. The rules axiomatize the algebra for relations with finite
> > headers. [...]
>
> I'm OK with no W.

Ok. I misunderstood "unconstrained". You mean it is an arbitrary header.

It seems I forgot a rule :-( (it turns out I needed it in a part of the proof I had no worked out yet)
This rule is of course:

(62) r + <> = <>

And then it is easy:

         <xy> + S + Q

(20) = <xy> + S + (S * []) + Q
(22) = <xy> + S + [z] + Q
(26) = <> + S + Q
(62) = <>
(62) = <> + S
(26) = <xy> + S + [z]
(22) = <xy> + S + (S * [])
(20) = <xy> + S

• Jan Hidders