Results 1 
9 of
9
Packaging mathematical structures
 THEOREM PROVING IN HIGHER ORDER LOGICS 5674
, 2009
"... This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.
Organizing numerical theories using axiomatic type classes
 Journal of Automated Reasoning
, 2004
"... Mathematical reasoning may involve several arithmetic types, including those of the natural, integer, rational, real and complex numbers. These types satisfy many of the same algebraic laws. These laws need to be made available to users, uniformly and preferably without repetition, but with due acco ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
Mathematical reasoning may involve several arithmetic types, including those of the natural, integer, rational, real and complex numbers. These types satisfy many of the same algebraic laws. These laws need to be made available to users, uniformly and preferably without repetition, but with due account for the peculiarities of each type. Subtyping, where a type inherits properties from a supertype, can eliminate repetition only for a fixed type hierarchy set up in advance by implementors. The approach recently adopted for Isabelle uses axiomatic type classes, an established approach to overloading. Abstractions such as semirings, rings, fields and their ordered counterparts are defined and theorems are proved algebraically. Types that meet the abstractions inherit the appropriate theorems. 1
Defining functions on equivalence classes
 ACM Transactions on Computational Logic
"... A quotient construction defines an abstract type from a concrete type, using an equivalence relation to identify elements of the concrete type that are to be regarded as indistinguishable. The elements of a quotient type are equivalence classes: sets of equivalent concrete values. Simple techniques ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
A quotient construction defines an abstract type from a concrete type, using an equivalence relation to identify elements of the concrete type that are to be regarded as indistinguishable. The elements of a quotient type are equivalence classes: sets of equivalent concrete values. Simple techniques are presented for defining and reasoning about quotient construction, based on a general lemma library concerning functions that operate on equivalence classes. The techniques are applied to a definition of the integers from the natural numbers, and then to the definition of a recursive datatype satisfying equational constraints.
Modeling Inheritance as Coercion in the Kenzo System
"... Abstract: In this paper the analysis of the data structures used in a symbolic computation system, called Kenzo, is undertaken. We deal with the specification of the inheritance relationship since Kenzo is an objectoriented system, written in CLOS, the Common Lisp Object System. We show how the ord ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Abstract: In this paper the analysis of the data structures used in a symbolic computation system, called Kenzo, is undertaken. We deal with the specification of the inheritance relationship since Kenzo is an objectoriented system, written in CLOS, the Common Lisp Object System. We show how the ordersorted algebraic specification formalism can be adapted, through the “inheritance as coercion ” metaphor, in order to model the simple inheritance between structures in Kenzo.
Bridging the gap between formal specification and bitlevel floatingpoint arithmetic
"... Floatingpoint arithmetic is defined by the IEEE754 standard and has often been
formalized. We propose a new Coq formalization based on the bitlevel representation of the standard and we prove strong links between this new formalization and
a previous highlevel one. In this process, we have defin ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Floatingpoint arithmetic is defined by the IEEE754 standard and has often been
formalized. We propose a new Coq formalization based on the bitlevel representation of the standard and we prove strong links between this new formalization and
a previous highlevel one. In this process, we have defined functions for any rounding mode described by the standard. Our library can now be applied to certify
both software and hardware. Developing results in those two dramatically different
directions, like no other formal development so far, guarantees that nothing was
forgotten or poorly specified in our formalization. It also lets us compare our work
with the existing bitlevel formalizations developed with other proof assistants.
Project Title Systems for ComputerSupported Mathematical Knowledge Evolution Specific Programme Structuring the European Research Area Activity Human Resources and Mobility Activities
, 2003
"... Abstract The longterm goal of the Calculemus interest group is to foster the allembracing integration of symbolic reasoning into mathematical research, mathematics education, and formal methods in computer science. A new generation of mathematical software systems is currently under development tha ..."
Abstract
 Add to MetaCart
Abstract The longterm goal of the Calculemus interest group is to foster the allembracing integration of symbolic reasoning into mathematical research, mathematics education, and formal methods in computer science. A new generation of mathematical software systems is currently under development that provides integrated computerbased support for most work tasks of a mathematician — including computation and reasoning as well as search in large mathematical data bases. Calculemus anticipates that in the long run these systems will change mathematical practice and that they will have a strong societal impact, not least in the sense that powerful infrastructure for mathematical research and education will become better accessible. Mathematical reasoning systems have a strong impact on other fields, most notably in computer science for the verification of safety and security properties — and it is in these areas where a severe shortage of trained engineers exists. CalculemusII will address this training and education problem via an integrated programme of distributed PhD supervision, post PhD training, industrial internships, international seminars, and lectures as well as an international Calculemus Summer School.
Journal of Automated Reasoning manuscript No. (will be inserted by the editor) Computing with Classical Real Numbers
, 809
"... Abstract There are two incompatible Coq libraries that have a theory of the real numbers; the Coq standard library gives an axiomatic treatment of classical real numbers, while the CoRN library from Nijmegen defines constructively valid real numbers. Unfortunately, this means results about one struc ..."
Abstract
 Add to MetaCart
Abstract There are two incompatible Coq libraries that have a theory of the real numbers; the Coq standard library gives an axiomatic treatment of classical real numbers, while the CoRN library from Nijmegen defines constructively valid real numbers. Unfortunately, this means results about one structure cannot easily be used in the other structure. We present a way interfacing these two libraries by showing that their real number structures are isomorphic assuming the classical axioms already present in the standard library reals. This allows us to use O’Connor’s decision procedure for solving ground inequalities present in CoRN to solve inequalities about the reals from the Coq standard library, and it allows theorems from the Coq standard library to apply to problem about the CoRN reals.