# Complex: The Complex Numbers

This directory defines the type `complex` of the complex numbers, with numeric constants and some complex analysis. The development includes nonstandard analysis for the complex numbers. Note that the image `HOL-Complex` includes theories from the directories `HOL/Real` and `HOL/Hyperreal`. They define other types including `real` (the real numbers) and `hypreal` (the hyperreal or non-standard reals).

- CLim Limits, continuous functions, and derivatives for the complex numbers
- CSeries Finite summation and infinite series for the complex numbers
- CStar Star-transforms for the complex numbers, to form non-standard extensions of sets and functions
- Complex The complex numbers
- NSCA Nonstandard complex analysis
- NSComplex Ultrapower construction of the nonstandard complex numbers

## Real: Dedekind Cut Construction of the Real Line

- Lubs Definition of upper bounds, lubs and so on, to support completeness proofs.
- PReal The positive reals constructed using Dedekind cuts
- Rational The rational numbers constructed as equivalence classes of integers
- RComplete The reals are complete: they satisfy the supremum property. They also have the Archimedean property.
- RealDef The real numbers, their ordering properties, and embedding of the integers and the natural numbers
- RealPow Real numbers raised to natural number powers

## Hyperreal: Ultrafilter Construction of the Non-Standard Reals

See J. D. Fleuriot and L. C. Paulson. Mechanizing Nonstandard Real Analysis. LMS J. Computation and Mathematics 3 (2000), 140-190.

- Filter Theory of Filters and Ultrafilters. Main result is a version of the Ultrafilter Theorem proved using Zorn's Lemma.
- HLog Non-standard logarithms
- HSeries Non-standard theory of finite summation and infinite series
- HTranscendental Non-standard extensions of transcendental functions
- HyperDef Ultrapower construction of the hyperreals
- HyperNat Ultrapower construction of the hypernaturals
- HyperPow Powers theory for the hyperreals
- IntFloor Floor and Ceiling functions relating the reals and integers
- Integration Gage integrals
- Lim Theory of limits, continuous functions, and derivatives
- Log Logarithms for the reals
- MacLaurin MacLaurin series
- NatStar Star-transforms for the hypernaturals, to form non-standard extensions of sets and functions involving the naturals or reals
- NthRoot Existence of n-th roots of real numbers
- NSA Theory defining sets of infinite numbers, infinitesimals, the infinitely close relation, and their various algebraic properties.
- Poly Univariate real polynomials
- SEQ Convergence of sequences and series using standard and nonstandard analysis
- Series Finite summation and infinite series for the reals
- Star Nonstandard extensions of real sets and real functions
- Transcendental Power series and transcendental functions

Last modified $Date: 2005/03/07 18:19:12 $