is also monotone. Let S be a non empty relational structure, let T be a reflexive non empty relational structure, and let a be an element of T. One can check that S ↦− → a is monotone. One can prove the following propositions: (2) Let S be a non empty relational structure and T be a lower

Summary. This paper is a continuation of [5] and concerns if-while algebras over integers. In these algebras the only elementary instructions are assignment instructions. The instruction assigns to a (program) variable a value which is calculated for the current state according to some arithmetic expression. The expression may include variables, constants, and a limited number of arithmetic operations. States are functions from a given set of locations into integers. A variable is a function from the states into the locations and an expression is a function from the states into integers. Additional conditions (computability) limit the set of variables and expressions and, simultaneously, allow to write algorithms in a natural way (and to prove their correctness). As examples the proofs of full correctness of two Euclid algorithms (with modulo operation and subtraction) and algorithm of exponentiation by squaring are given.

. The following proposition is true (2) 1 For every X such that 0 ∈ X and for every x such that x ∈ X holds x+1 ∈ X and for every k holds k ∈ X. Let n, k be natural numbers. Then n+k is a natural number. Let n, k be natural numbers. Note that n+k is natural. In this article we present several logical schemes

this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following

of conceptual tools that are directly useful in guiding the implementation and evaluation of work redesign projects. In the paragraphs to follow, we examine several existing theoretical approaches to work redesign, with a special eye toward the measurability of the concepts employed and the action implications

1. Every anti-chain in P has cardinality 1 =) every anti-chain in F has cardinality 1 2. There exists an anti-chain in P of cardinality 2, but no element of P is incomparable with two different elements =) every anti-chain in F has cardinality at most 2 3. There exists an element in P which is incomparable with two different elements =) there exists anti-chains of any cardinality in F.

〉). In the sequel A is a set. Let A, B be non empty sets, let C be a set, let f be a function from [:A, B:] into C, let a be an element of A, and let b be an element of B. Then f(a, b) is an element of C. The following proposition is true (2) 1 Let A, B, C be non empty sets and f1, f2 be functions from [:A, B

We wanted to prove the following proposition:

latter, the proof objects either have to be expressed directly in the program or extracted as obligations and verified separately. We now briefly reexamine the Curry-Howard isomorphism, when extended to the first-order level. We have the following correspondence:

In this note we call attention to the curious fact that the Fibonacci numbers arise when we look at that familiar example from group theory, the n X n nonsingular upper triangular matrices. Once incidence subgroups are defined the result follows quite easily. Let K be any field with more than two