The aim is to formalize in Mizar the theory from
A Compendium of Continuous Lattices (CCL)
by G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Springer-Verlag, 1980.

The theory already formalized in Mizar was done by 16 authors:
AuthorCCLMMLAuthorCCLMML
Grzegorz Bancerek 1991 Agnieszka J. Marasik 14
Czeslaw Bylinski 336 Robert Milewski 825
Noboru Endou216 Adam Naumowicz 311
Adam Grabowski 422 Piotr Rudnicki 343
Ewa Gradzka12 Yuji Sakai13
Jaroslaw Gryko36 Bartlomiej M. Skorulski24
Artur Kornilowicz 1145 Andrzej Trybulec 386
Beata Madras17 Mariusz Zynel 25
It is included in 58 articles and covers about 65% of main course of the book. Articles are stored in the Mizar Mathematical Library (MML). The formalization means the formalization of concepts and proofs mechanically checked for logical correctness.
The Mizar formalization of the theory of continuous lattices was supported by

The problem stated on QED Workshop II (Warsaw 1995) was:
Can we do formalization of advanced mathematics like that included in regular mathematical monographs in the current proof-checking systems?
Especially, in our case, we may ask:
Can we do such formalization in Mizar?
Why is the question so important? -- Former formalizations dealt with small and separated mathematical theories chosen often to be easy to formalize. There were, at least two exceptions: Jutting's formalization of Landau's Grundlagen in AUTOMATH and mizaring of Grzegorczyk's Arithmetic in Mizar 2. Both those systems did not required, let me call it, continuous formalization. That means that one may use statements which were not proved, e.g. mathlore. In recent Mizar any articles accepted to the Mizar Mathematical Library has to be a continuation of the Library.

In the MML there was (and still is) a quite big number of articles formalizing beginning of some theory and the theory has no continuation. However, those articles were, mainly, devoted to form the base for more advanced theories. The challenge consists in the fact that against such formalizations the mizaring of the monograph requires much bigger machinery of interrelated concepts and facts, and, then, needs more careful decisions on the way of formalization which may influence later.

Other question was if the MML was enough big to do such formalization.


Why the theory of continuous lattices? -- It seems to be a good compromise on, so called, well-definedness of mathematical field and advancement of it. It is relatively recent and combines different fields of mathematics (algebra, topology, categories).

Publications

Grzegorz Bancerek