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本卷收录了吴文俊的MechanicalTheoremProvinginGeometries:BasicPrinciples一书.书中论述初等几何机器证明的基本原理,证明了奠基于各种公理系统的各种初等几何,只需相当于乘法交换律的某一公理成立,大都可以机械化.因此在理论上,这些几何的定理证明可以借肋于计算机来实施.可以机械化的几何包括了多种有序或无序的常用几何、投影几何、非欧几何与圆几何等.全书共分六章.前两章是关于几何机械化的预备知识,集中介绍了常用几何;后四章致力于几何的机械化问题.第3章为几何定理证明的机械化与Hilbert机械化定理,第4,5章分别为(常用)无序几何的机械化定理和(常用)有序几何的机械化定理,第6章阐述各种几何的机械化定理.
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Author's note to the English-language edition
1 Desarguesian geometry and the Desarguesian number system
1.1 Hilbert's axiom system of ordinary geometry
1.2 The axiom of infinity and Desargues' axioms
1.3 Rational points in a Desarguesian plane
1.4 The Desarguesian number system and rational number subsystem
1.5 The Desarguesian number system on a line
1.6 The Desarguesian number system associated with a Desarguesian plane
1.7 The coordinate system of Desarguesian plane geometry
20rthogonal geometry, metric geometry and ordinary geometry
2.1 The Pascalian axiom and commutative axiom of multiplication- (unordered) Pascalian geometry
2.20 rthogonal axioms and (unordered) orthogonal geometry
2.3 The orthogonal coordinate system of (unordered) orthogonal geometry
2.4 (Unordered) metric geometry
2.5 The axioms of order and ordered metric geometry
2.6 Ordinary geometry and its subordinate geometries
3 Mechanization of theorem proving in geometry and Hilbert's mechanization theorem
3.1 Comments on Euclidean proof method
3.2 The standardization of coordinate representation of geometric concepts
3.3 The mechanization of theorem proving and Hilbert's mechanization theorem about pure point of intersection theorems in Pascalian geometry
3.4 Examples for Hilbert's mechanical method
3.5 Proof of Hilbert's mechanization theorem
4 The mechanization theorem of (ordinary) unordered geometry
4.1 Introduction
4.2 Factorization of polynomials
4.3 Well-ordering of polynomial sets
4.4 A constructive theory of algebraic varieties -irreducible ascending sets and irreducible algebraic varieties
4.5 A constructive theory of algebraic varieties -irreducible decomposition of algebraic varieties
4.6 A constructive theory of algebraic varieties -the notion of dimension and the dimension theorem
4.7 Proof of the mechanization theorem of unordered geometry
4.8 Examples for the mechanical method of unordered geometry
5 Mechanization theorems of (ordinary) ordered geometries
5.1 Introduction
5.2 Tarski's theorem and Seidenberg's method
5.3 Examples for the mechanical method of ordered geometries
6 Mechanization theorems of various geometries
6.1 Introduction
6.2 The mechanization of theorem proving in projective geometry
6.3 The mechanization of theorem proving in Bolyai-Lobachevsky's hyperbolic non-Euclidean geometry
6.4 The mechanization of theorem proving in Riemann's elliptic non-Euclidean geometry
6.5 The mechanization of theorem proving in two circle geometries
6.6 The mechanization of formula proving with transcendental functions
References
Subject index
