# nLab constructive analysis

Contents

### Context

#### Constructivism, Realizability, Computability

intuitionistic mathematics

# Contents

## Idea

Constructive analysis is the incarnation of analysis in constructive mathematics. It is related to, but distinct from, computable analysis; the latter is developed in classical logic explicitly using computability theory, whereas constructive analysis is developed in constructive logic where the computation is implicitly built-in. One can compile results in constructive analysis to computable analysis using realizability.

In applications in computer science one uses for instance the completion monad for exact computations with real numbers (as opposed to floating point arithmetic. Therefore one also sometimes speaks of exact analysis. See also at computable real number.

computability

type I computabilitytype II computability
typical domainnatural numbers $\mathbb{N}$Baire space of infinite sequences $\mathbb{B} = \mathbb{N}^{\mathbb{N}}$
computable functionspartial recursive functioncomputable function (analysis)
type of computable mathematicsrecursive mathematicscomputable analysis, Type Two Theory of Effectivity
type of realizabilitynumber realizabilityfunction realizability
partial combinatory algebraKleene's first partial combinatory algebraKleene's second partial combinatory algebra

## References

The formulation of analysis in constructive mathematics was maybe inititated in

• Errett Bishop, Foundations of constructive analysis. McGraw-Hill, (1967)

together with the basic notion of Bishop set/setoid.

A survey is in

• Herman Geuvers, Milad Niqui, Bas Spitters, Freek Wiedijk, Constructive analysis, types and exact real numbers, Science Mathematical Structures in Computer Science / Volume 17 / Issue 01 / February 2007, pp 3-36 (publisher)

An undergraduate real analysis textbook taking a constructive approach using interval analysis is

• Mark Bridger, Real Analysis: A Constructive Approach Through Interval Arithmetic, Pure and Applied Undergraduate Texts 38, American Mathematical Society, 2019.

Implementations of constructive real number analysis in type theory implemented in Coq via the completion monad are discussed in

• R. O’Connor, A Monadic, Functional Implementation of Real Numbers. MSCS, 17(1):129{159, 2007.

• R. O’Connor, Certied exact transcendental real number computation in Coq, In TPHOLs 2008, LNCS 5170, pages 246–261, 2008.

• R. O’Connor, Incompleteness and Completeness: Formalizing Logic and Analysis in Type Theory, PhD thesis, Radboud University Nijmegen, 2009.

• Robbert Krebbers, Bas Spitters, Type classes for efficient exact real arithmetic in Coq (arXiv:1106.3448)

• Bas Spitters, Verified Implementation of Exact Real Arithmetic in Type Theory, talk at Computable Analysis and Rigorous Numeric (pdf)

With emphasis on the use of the univalence axiom:

• Auke Booij, Constructive analysis in univalent type theory, 2017 (pdf)

• Auke Booij, Extensional constructive real analysis via locators (arXiv:1805.06781)