內容大鋼
本卷收錄了吳文俊的Mechanical Theorem Proving in Geometries:Basic Principles 一書。 書中論述初等幾何機器證明的基本原理, 證明了奠基於各種公理系統的各種初等幾何, 只需相當於乘法交換律的某一公理成立, 大都可以機械化。 因此在理論上, 這些幾何的定理證明可以借肋于電腦來實施。 可以機械化的幾何包括了多種有序或無序的常用幾何、投影幾何、非歐幾何與圓幾何等。
全書共分六章。 前兩章是關於幾何機械化的預備知識, 集中介紹了常用幾何; 后四章致力於幾何的機械化問題。 第3 章為幾何定理證明的機械化與Hilbert 機械化定理, 第4, 5 章分別為(常用)無序幾何的機械化定理和(常用)有序幾何的機械化定理, 第6 章闡述各種幾何的機械化定理。
本書可供數學工作者和電腦科學工作者以及高等院校有關專業的師生參考。
目錄
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
[