Displaying similar documents to “Preliminaries to Classical First Order Model Theory”

The Sum and Product of Finite Sequences of Complex Numbers

Keiichi Miyajima, Takahiro Kato (2010)

Formalized Mathematics

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This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11].

Relational Formal Characterization of Rough Sets

Adam Grabowski (2013)

Formalized Mathematics

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The notion of a rough set, developed by Pawlak [10], is an important tool to describe situation of incomplete or partially unknown information. In this article, which is essentially the continuation of [6], we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library). Here we drop the classical equivalence- and tolerance-based...

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

Marco Caminati (2011)

Formalized Mathematics

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Fourth of a series of articles laying down the bases for classical first order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined.The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts...

Model Checking. Part II

Kazuhisa Ishida (2008)

Formalized Mathematics

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This article provides the definition of linear temporal logic (LTL) and its properties relevant to model checking based on [9]. Mizar formalization of LTL language and satisfiability is based on [2, 3].

First Order Languages: Further Syntax and Semantics

Marco Caminati (2011)

Formalized Mathematics

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Third of a series of articles laying down the bases for classical first order model theory. Interpretation of a language in a universe set. Evaluation of a term in a universe. Truth evaluation of an atomic formula. Reassigning the value of a symbol in a given interpretation. Syntax and semantics of a non atomic formula are then defined concurrently (this point is explained in [16], 4.2.1). As a consequence, the evaluation of any w.f.f. string and the relation of logical implication are...

Representation Theorem for Stacks

Grzegorz Bancerek (2011)

Formalized Mathematics

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In the paper the concept of stacks is formalized. As the main result the Theorem of Representation for Stacks is given. Formalization is done according to [13].

The Vector Space of Subsets of a Set Based on Symmetric Difference

Jesse Alama (2008)

Formalized Mathematics

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For each set X, the power set of X forms a vector space over the field Z2 (the two-element field {0, 1} with addition and multiplication done modulo 2): vector addition is disjoint union, and scalar multiplication is defined by the two equations (1 · x:= x, 0 · x := ∅ for subsets x of X). See [10], Exercise 2.K, for more information.MML identifier: BSPACE, version: 7.8.05 4.89.993

Sorting by Exchanging

Grzegorz Bancerek (2011)

Formalized Mathematics

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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 Sort, and Quicksort.