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
- 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: 2005/10/04 08:58:46 $