Dieses Buch ist eine Übersetzung des epochemachenden Werkes über maschinelles geometrisches Beweisen von Professor Wu, das 1984 auf Chinesisch erschienen ist. Inzwischen ist Wus Methode für das maschinelle geometrische Beweisen, die auf dem Konzept der charakteristischen Mengen basiert, eine der fundamentalen, auch praktisch erfolgreichen Methoden in diesem Gebiet geworden. In der Tat hat diese Methode den Bereich dessen, was automatisch bewiesen werden kann, drastisch erweitert.
Dieses Buch ist eine Übersetzung des epochemachenden Werkes über maschinelles geometrisches Beweisen von Professor Wu, das 1984 auf Chinesisch erschienen ist. Inzwischen ist Wus Methode für das maschinelle geometrische Beweisen, die auf dem Konzept der charakteristischen Mengen basiert, eine der fundamentalen, auch praktisch erfolgreichen Methoden in diesem Gebiet geworden. In der Tat hat diese Methode den Bereich dessen, was automatisch bewiesen werden kann, drastisch erweitert.
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.- 2 Orthogonal geometry, metric geometry and ordinary geometry.- 2.1 The Pascalian axiom and commutative axiom of multiplication - (unordered) Pascalian geometry.- 2.2 Orthogonal 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 theoryof 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.
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.- 2 Orthogonal geometry, metric geometry and ordinary geometry.- 2.1 The Pascalian axiom and commutative axiom of multiplication - (unordered) Pascalian geometry.- 2.2 Orthogonal 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 theoryof 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.
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