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<H1>Complex: The Complex Numbers</H1> |
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<P>This directory defines the type <KBD>complex</KBD> of the complex numbers, |
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with numeric constants and some complex analysis. The development includes |
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nonstandard analysis for the complex numbers. Note that the image |
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<KBD>HOL-Complex</KBD> includes theories from the directories |
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<KBD><a href="#Anchor-Real">HOL/Real</a></KBD> and <KBD><a href="#Anchor-Hyperreal">HOL/Hyperreal</a></KBD>. They define other types including <kbd>real</kbd> (the real numbers) and <kbd>hypreal</kbd> (the hyperreal or non-standard reals). |
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<ul> |
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<li><a href="CLim.html">CLim</a> Limits, continuous functions, and derivatives for the complex numbers |
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<li><a href="CSeries.html">CSeries</a> Finite summation and infinite series for the complex numbers |
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<li><a href="CStar.html">CStar</a> Star-transforms for the complex numbers, to form non-standard extensions of sets and functions |
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<li><a href="Complex.html">Complex</a> The complex numbers |
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<li><a href="NSCA.html">NSCA</a> Nonstandard complex analysis |
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<li><a href="NSComplex.html">NSComplex</a> Ultrapower construction of the nonstandard complex numbers |
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</ul> |
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<h2><a name="Anchor-Real" id="Anchor-Real"></a>Real: Dedekind Cut Construction of the Real Line</h2> |
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<ul> |
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<li><a href="Lubs.html">Lubs</a> Definition of upper bounds, lubs and so on, to support completeness proofs. |
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<li><a href="PReal.html">PReal</a> The positive reals constructed using Dedekind cuts |
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<li><a href="Rational.html">Rational</a> The rational numbers constructed as equivalence classes of integers |
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<li><a href="RComplete.html">RComplete</a> The reals are complete: they satisfy the supremum property. They also have the Archimedean property. |
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<li><a href="RealDef.html">RealDef</a> The real numbers, their ordering properties, and embedding of the integers and the natural numbers |
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<li><a href="RealPow.html">RealPow</a> Real numbers raised to natural number powers |
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</ul> |
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<h2><a name="Anchor-Hyperreal" id="Anchor-Hyperreal"></a>Hyperreal: Ultrafilter Construction of the Non-Standard Reals</h2> |
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See J. D. Fleuriot and L. C. Paulson. Mechanizing Nonstandard Real Analysis. LMS J. Computation and Mathematics 3 (2000), 140-190. |
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<ul> |
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<li><a href="Filter.html">Filter</a> Theory of Filters and Ultrafilters. Main result is a version of the Ultrafilter Theorem proved using Zorn's Lemma. |
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<li><a href="HLog.html">HLog</a> Non-standard logarithms |
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<li><a href="HSeries.html">HSeries</a> Non-standard theory of finite summation and infinite series |
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<li><a href="HTranscendental.html">HTranscendental</a> Non-standard extensions of transcendental functions |
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<li><a href="HyperDef.html">HyperDef</a> Ultrapower construction of the hyperreals |
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<li><a href="HyperNat.html">HyperNat</a> Ultrapower construction of the hypernaturals |
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<li><a href="HyperPow.html">HyperPow</a> Powers theory for the hyperreals |
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<!-- <li><a href="IntFloor.html">IntFloor</a> Floor and Ceiling functions relating the reals and integers --> |
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<li><a href="Integration.html">Integration</a> Gage integrals |
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<li><a href="Lim.html">Lim</a> Theory of limits, continuous functions, and derivatives |
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<li><a href="Log.html">Log</a> Logarithms for the reals |
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<li><a href="MacLaurin.html">MacLaurin</a> MacLaurin series |
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<li><a href="NatStar.html">NatStar</a> Star-transforms for the hypernaturals, to form non-standard extensions of sets and functions involving the naturals or reals |
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<li><a href="NthRoot.html">NthRoot</a> Existence of n-th roots of real numbers |
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<li><a href="NSA.html">NSA</a> Theory defining sets of infinite numbers, infinitesimals, the infinitely close relation, and their various algebraic properties. |
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<li><a href="Poly.html">Poly</a> Univariate real polynomials |
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<li><a href="SEQ.html">SEQ</a> Convergence of sequences and series using standard and nonstandard analysis |
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<li><a href="Series.html">Series</a> Finite summation and infinite series for the reals |
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<li><a href="Star.html">Star</a> Nonstandard extensions of real sets and real functions |
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<li><a href="Transcendental.html">Transcendental</a> Power series and transcendental functions |
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</ul> |
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<P>Last modified $Date$ |
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