Four statements equivalent to well-foundedness (well-founded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omega-chains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies well-foundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.

The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from [14] and [13]. The theory here presented is an abstract of the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The base idea behind the formalization is dependence on variables which is determined by variable-dependence (variables may depend on other variables). The dependence constitutes a Galois connection between opposite poset of dependence-closed set of variables and the sup-semilattice of widening of Mizar types (smooth type widening). In the paper the concepts strictly connected with Mizar expressions are formalized. Among them are quasi-loci, quasi-terms, quasi-adjectives, and quasitypes. The structural induction and operation of substitution are also introduced. The prefix quasi is used to indicate that some rules of construction of Mizar expressions may not be fulfilled. For example, variables, quasi-loci, and quasiterms have no assigned types and, in result, there is no possibility to conduct type-checking of arguments. The other gaps concern inconsistent and out-ofcontext clusters of adjectives in types. Those rules are required in the Mizar identification process. However, the expression appearing in later processes of Mizar checker may not satisfy the rules. So, introduced apparatus is enough and adequate to describe data structures and algorithms from the Mizar checker (like equational classes).

The following theorem is due to Dilworth [8]: Let P be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of P is k, then P is the union of k chains (cliques). In this article we formalize an elegant proof of the above theorem for finite posets by Perles [13]. The result is then used in proving the case of infinite posets following the original proof of Dilworth [8]. A dual of Dilworth’s theorem also holds: a poset with maximum clique m is a union of m independent sets. The proof of this dual fact is considerably easier; we follow the proof by Mirsky [11]. Mirsky states also a corollary that a poset of r × s + 1 elements possesses a clique of size r + 1 or an independent set of size s + 1, or both. This corollary is then used to prove the result of Erdős and Szekeres [9]. Instead of using posets, we drop reflexivity and state the facts about antisymmetric and transitive relations.

Summary. The main result of the article is the solution to the problem of short axiomatizations of orthomodular ortholattices. Based on EQP/Otter results [10], we gave a set of three equations which is equivalent to the classical, much longer equational basis of such a class. Also the basic example of the lattice which is not orthomodular, i.e. benzene (or B6) is defined in two settings – as a relational structure (poset) and as a lattice. As a preliminary work, we present the proofs of the dependence of other axiomatizations of ortholattices. The formalization of the properties of orthomodular lattices follows [4].

Summary. The following theorem is due to Dilworth [12]: Let P be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of P is k, then P is the union of k chains (cliques). In this article we formalize an elegant proof of the above theorem for finite posets by Perles [17]. The result is then used in proving the case of infinite posets following the original proof of Dilworth [12]. A dual of Dilworth’s theorem also holds: a poset with maximum clique m is a union of m independent sets. The proof of this dual fact is considerably easier; we follow the proof by Mirsky [16]. Mirsky states also a corollary that a poset of r × s + 1 elements possesses a clique of size r + 1 or an independent set of size s + 1, or both. This corollary is then used to prove the result of Erdős and Szekeres [13]. Instead of using posets, we drop reflexivity and state the facts about antisymmetric and transitive relations.

Summary. This Mizar paper presents the definition of a “Preordered Coherent Space ” (PCS). Furthermore, the paper defines a number of operations on PCS’s and states and proves a number of elementary lemmas about these operations. PCS’s have many useful properties which could qualify them for mathematical study in their own right. PCS’s were invented, however, to construct Scott domains, to solve domain equations, and to construct models of various versions of lambda calculus. For more on PCS’s, see [11]. The present Mizar paper defines the operations

Summary. This text includes definition of chain-complete poset, fix-point theorem on it, and definition of the function space of continuous functions on chain-complete posets [10].

Summary. We show that exchanging of pairs in an array which are in incorrect order leads to sorted array. It justifies correctness of Bubble Sort, Insertion

Let ω(G) and χ(G) be the clique number and the chromatic number of a graph G. Mycielski [11] presented a construction that for any n creates a graph Mn which is triangle-free (ω(G) = 2) with χ(G)> n. The starting point is the complete graph of two vertices (K2). M(n+1) is obtained from Mn through the operation µ(G) called the Mycielskian of a graph G. We first define the operation µ(G) and then show that ω(µ(G)) = ω(G) and χ(µ(G)) = χ(G) + 1. This is done for arbitrary graph G, see also [10]. Then we define the sequence of graphs Mn each of exponential size in n and give their clique and chromatic numbers.