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The following proposition is true 1. PRELIMINARIES
"... (1) Let S, T be upcomplete Scott toplattices and M be a subset of SCMaps(S,T). Then SCMaps(S,T) M is a continuous map from S into T. Let S be a non empty relational structure and let T be a non empty reflexive relational structure. One can verify that every map from S into T which is constant is a ..."
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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
One can prove the following proposition
"... Summary. This paper is a continuation of [5] and concerns ifwhile 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 ex ..."
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Summary. This paper is a continuation of [5] and concerns ifwhile 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 fundamental properties of natural numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.h ..."
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Cited by 688 (73 self)
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. 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
Partial Functions
"... 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 rules ..."
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Cited by 492 (10 self)
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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
Motivation through the Design of Work: Test of a Theory. Organizational Behavior and Human Performance,
, 1976
"... A model is proposed that specifies the conditions under which individuals will become internally motivated to perform effectively on their jobs. The model focuses on the interaction among three classes of variables: (a) the psychological states of employees that must be present for internally motiv ..."
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Cited by 622 (2 self)
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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
The following proposition answers the question of the relation between cardinalities of antichains of P and antichains of F? Proposition 1
"... 1. Every antichain in P has cardinality 1 =) every antichain in F has cardinality 1 2. There exists an antichain in P of cardinality 2, but no element of P is incomparable with two different elements =) every antichain in F has cardinality at most 2 3. There exists an element in P which is incom ..."
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1. Every antichain in P has cardinality 1 =) every antichain in F has cardinality 1 2. There exists an antichain in P of cardinality 2, but no element of P is incomparable with two different elements =) every antichain in F has cardinality at most 2 3. There exists an element in P which is incomparable with two different elements =) there exists antichains of any cardinality in F.
Binary operations
 Journal of Formalized Mathematics
, 1989
"... Summary. In this paper we define binary and unary operations on domains. We also define the following predicates concerning the operations:... is commutative,... is associative,... is the unity of..., and... is distributive wrt.... A number of schemes useful in justifying the existence of the operat ..."
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Cited by 363 (6 self)
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〉). 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
Propositions ∧ ⊃ ⊤ ∨ ⊥ ∀ ∃
"... 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 CurryHoward isomorphism, when extended to the firstorder level. We have the following correspondence: ..."
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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 CurryHoward isomorphism, when extended to the firstorder level. We have the following correspondence:
Proposition
"... 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 el ..."
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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
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