93 research outputs found

### Type classes for efficient exact real arithmetic in Coq

Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275

### Integrals and Valuations

We construct a homeomorphism between the compact regular locale of integrals
on a Riesz space and the locale of (valuations) on its spectrum. In fact, we
construct two geometric theories and show that they are biinterpretable. The
constructions are elementary and tightly connected to the Riesz space
structure.Comment: Submitted for publication 15/05/0

### Constructive Theory of Banach algebras

We present a way to organize a constructive development of the theory of
Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We
illustrate this by giving elementary proofs of Wiener's result on the inverse
of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we
show how this can be used in a localic, or point-free, description of the
spectrum of a Banach algebra

### A constructive proof of Simpson's Rule

For most purposes, one can replace the use of Rolle's theorem and the mean
value theorem, which are not constructively valid, by the law of bounded
change. The proof of two basic results in numerical analysis, the error term
for Lagrange interpolation and Simpson's rule, however seem to require the full
strength of the classical Rolle's Theorem. The goal of this note is to justify
these two results constructively, using ideas going back to Amp\`ere and
Genocchi

### Sets in homotopy type theory

Homotopy Type Theory may be seen as an internal language for the
$\infty$-category of weak $\infty$-groupoids which in particular models the
univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids
as a new foundation for mathematics called the Univalent Foundations of
Mathematics. It includes the sets as weak $\infty$-groupoids with contractible
connected components, and thereby it includes (much of) the traditional set
theoretical foundations as a special case. We thus wonder whether those
`discrete' groupoids do in fact form a (predicative) topos. More generally,
homotopy type theory is conjectured to be the internal language of `elementary'
$\infty$-toposes. We prove that sets in homotopy type theory form a $\Pi
W$-pretopos. This is similar to the fact that the $0$-truncation of an
$\infty$-topos is a topos. We show that both a subobject classifier and a
$0$-object classifier are available for the type theoretical universe of sets.
However, both of these are large and moreover, the $0$-object classifier for
sets is a function between $1$-types (i.e. groupoids) rather than between sets.
Assuming an impredicative propositional resizing rule we may render the
subobject classifier small and then we actually obtain a topos of sets

### Modalities in homotopy type theory

Univalent homotopy type theory (HoTT) may be seen as a language for the
category of $\infty$-groupoids. It is being developed as a new foundation for
mathematics and as an internal language for (elementary) higher toposes. We
develop the theory of factorization systems, reflective subuniverses, and
modalities in homotopy type theory, including their construction using a
"localization" higher inductive type. This produces in particular the
($n$-connected, $n$-truncated) factorization system as well as internal
presentations of subtoposes, through lex modalities. We also develop the
semantics of these constructions

### Bohrification

New foundations for quantum logic and quantum spaces are constructed by
merging algebraic quantum theory and topos theory. Interpreting Bohr's
"doctrine of classical concepts" mathematically, given a quantum theory
described by a noncommutative C*-algebra A, we construct a topos T(A), which
contains the "Bohrification" B of A as an internal commutative C*-algebra. Then
B has a spectrum, a locale internal to T(A), the external description S(A) of
which we interpret as the "Bohrified" phase space of the physical system. As in
classical physics, the open subsets of S(A) correspond to (atomic)
propositions, so that the "Bohrified" quantum logic of A is given by the
Heyting algebra structure of S(A). The key difference between this logic and
its classical counterpart is that the former does not satisfy the law of the
excluded middle, and hence is intuitionistic. When A contains sufficiently many
projections (e.g. when A is a von Neumann algebra, or, more generally, a
Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be
compared with the traditional quantum logic, i.e. the orthomodular lattice of
projections in A. This time, the main difference is that the former is
distributive (even when A is noncommutative), while the latter is not.
This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in
"Deep Beauty" (ed. H. Halvorson

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