本卷收录了吴文俊的Mechanical Theorem Proving in Geometries:
Basic Principles 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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Contents
Author’s note to the English-language edition 1
1 Desarguesian geometry and the Desarguesian number system 13
1.1 Hilbert’s axiom system of ordinary geometry 13
1.2 The axiom of infinity and Desargues’ axioms 18
1.3 Rational points in a Desarguesian plane 25
1.4 The Desarguesian number system and rational number subsystem 30
1.5 The Desarguesian number system on a line 37
1.6 The Desarguesian number system associated with a Desarguesian plane 42
1.7 The coordinate system of Desarguesian plane geometry 55
2 Orthogonal geometry, metric geometry and ordinary geometry 63
2.1 The Pascalian axiom and commutative axiom of multiplication-(unordered) Pascalian geometry 63
2.2 Orthogonal axioms and (unordered) orthogonal geometry 70
2.3 The orthogonal coordinate system of (unordered) orthogonal geometry 80
2.4 (Unordered) metric geometry 91
2.5 The axioms of order and ordered metric geometry 102
2.6 Ordinary geometry and its subordinate geometries 109
3 Mechanization of theorem proving in geometry and Hilbert’s mechanization theorem 115
3.1 Comments on Euclidean proof method 115
3.2 The standardization of coordinate representation of geometric concepts 118
3.3 The mechanization of theorem proving and Hilbert’s mechanization theorem about pure point of intersection theorems in Pascalian geometry 124
3.4 Examples for Hilbert’s mechanical method 128
3.5 Proof of Hilbert’s mechanization theorem 139
4 The mechanization theorem of (ordinary) unordered geometry 149
4.1 Introduction 149
4.2 Factorization of polynomials 152
4.3 Well-ordering of polynomial sets 159
4.4 A constructive theory of algebraic varieties-irreducible ascending sets and irreducible algebraic varieties 169
4.5 A constructive theory of algebraic varieties-irreducible decomposition of algebraic varieties 178
4.6 A constructive theory of algebraic varieties-the notion of dimension and the dimension theorem 183
4.7 Proof of the mechanization theorem of unordered geometry 187
4.8 Examples for the mechanical method of unordered geometry 195
5 Mechanization theorems of (ordinary) ordered geometries 213
5.1 Introduction 213
5.2 Tarski’s theorem and Seidenberg’s method 220
5.3 Examples for the mechanical method of ordered geometries 228
6 Mechanization theorems of various geometries 235
6.1 Introduction 235
6.2 The mechanization of theorem proving in projective geometry 236
6.3 The mechanization of theorem proving in Bolyai-Lobachevsky’s hyperbolic non-Euclidean geometry 246
6.4 The mechanization of theorem proving in Riemann’s elliptic non-Euclidean geometry 258
6.5 The mechanization of theorem proving in two circle geometries 264
6.6 The mechanization of formula proving with transcendental functions 267
References 281
Subject index 285