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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 rules: x, y, y 1 , y 2 , z, z 1 , z 2 denote sets, P , Q, X , X 0 , X 1 , X 2 , Y , Y 0 , Y 1 , Y 2 , V , Z denote sets, and C, D denote non empty sets. We now state three propositions: (1) If P ` [: X 1
Partial Functions
"... Summary. In the article we define partial functions. We also define the following notions related to partial functions and functions themselves: the empty function, the restriction of a function to a partial function from a set into a set, the set of all partial functions from a set into a set, the ..."
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Summary. In the article we define partial functions. We also define the following notions related to partial functions and functions themselves: the empty function, the restriction of a function to a partial function from a set into a set, the set of all partial functions from a set into a set
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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check that there exists a function which is empty. The following proposition is true (10) 1 rng # = #. Let us consider X, Y . Observe that there exists a relation between X and Y which is functionlike. Let us consider X, Y . A partial function from X to Y is a functionlike relation between X
Partial Functions in ACL2
 Journal of Automated Reasoning
"... We describe a macro for introducing \partial functions" into ACL2, i.e., functions not dened everywhere. The function \denitions" are actually admitted via the encapsulation principle. We discuss the basic issues surrounding partial functions in ACL2 and illustrate theorems that can be ..."
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Cited by 37 (7 self)
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We describe a macro for introducing \partial functions" into ACL2, i.e., functions not dened everywhere. The function \denitions" are actually admitted via the encapsulation principle. We discuss the basic issues surrounding partial functions in ACL2 and illustrate theorems that can
Partial Functions in Data Manipulation
"... In this paper we take a look at partial functions and their applications in computer science. The axiomatization from [1] has been introduced to formally derive equality between partial functions. We work with algebras satisfying this axiomatization and a finite generation assumption which is satisf ..."
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In this paper we take a look at partial functions and their applications in computer science. The axiomatization from [1] has been introduced to formally derive equality between partial functions. We work with algebras satisfying this axiomatization and a finite generation assumption which
Induction Proofs with Partial Functions
 Journal of Automated Reasoning
, 1998
"... In this paper we present a method for automated induction proofs about partial functions. We show that most wellknown techniques developed for (explicit) induction theorem proving are unsound when dealing with partial functions. But surprisingly, by slightly restricting the application of these te ..."
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Cited by 6 (4 self)
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In this paper we present a method for automated induction proofs about partial functions. We show that most wellknown techniques developed for (explicit) induction theorem proving are unsound when dealing with partial functions. But surprisingly, by slightly restricting the application
Proving partial correctness of partial functions
 PROC. CADEWORKSHOP MECHANIZATION OF PARTIAL FUNCTIONS
, 1996
"... We present a method for automated induction proofs about partial functions. This method cannot only be used to verify the partial correctness of functional programs, but it also solves some other challenge problems where reasoning about partial functions is necessary. For a further analysis of part ..."
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Cited by 2 (1 self)
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We present a method for automated induction proofs about partial functions. This method cannot only be used to verify the partial correctness of functional programs, but it also solves some other challenge problems where reasoning about partial functions is necessary. For a further analysis
Overview of the scalable video coding extension of the H.264/AVC standard
 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY IN CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY
, 2007
"... With the introduction of the H.264/AVC video coding standard, significant improvements have recently been demonstrated in video compression capability. The Joint Video Team of the ITUT VCEG and the ISO/IEC MPEG has now also standardized a Scalable Video Coding (SVC) extension of the H.264/AVC stand ..."
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Cited by 522 (6 self)
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standard. SVC enables the transmission and decoding of partial bit streams to provide video services with lower temporal or spatial resolutions or reduced fidelity while retaining a reconstruction quality that is high relative to the rate of the partial bit streams. Hence, SVC provides functionalities
Termination Analysis for Partial Functions
 IN PROCEEDINGS OF THE THIRD INTERNATIONAL STATIC ANALYSIS SYMPOSIUM (SAS'96
, 1996
"... This paper deals with automated termination analysis for partial functional programs, i.e. for functional programs which do not terminate for each input. We present a method to determine their domains (resp. nontrivial subsets of their domains) automatically. More precisely, for each functional p ..."
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Cited by 12 (6 self)
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This paper deals with automated termination analysis for partial functional programs, i.e. for functional programs which do not terminate for each input. We present a method to determine their domains (resp. nontrivial subsets of their domains) automatically. More precisely, for each functional
Subset Types and Partial Functions
 19th International Conference on Automated Deduction
, 2003
"... Abstract. A classical higherorder logic PFsub of partial functions is defined. The logic extends a version of Farmer’s logic PF by enriching the type system of the logic with subset types and dependent types. Validity in PFsub is then reduced to validity in PF by a translation. 1 ..."
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Cited by 1 (1 self)
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Abstract. A classical higherorder logic PFsub of partial functions is defined. The logic extends a version of Farmer’s logic PF by enriching the type system of the logic with subset types and dependent types. Validity in PFsub is then reduced to validity in PF by a translation. 1
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