Sketch-as-Proof

This paper presents an extension of Gentzen's LK, called LPGK, which is suitable for expressing projective geometry and for deducing theorems of plane projective geometry. The properties of this calculus are investigated and the cut elimination theorem for LPGK is proven. A formulization of sketches is presented and the equivalence between sketches and formal proofs is demonstrated

Sketch-as-proof ⋆

Abstract. This paper presents an extension of Gentzen’s LK, called LPGK, which is suitable for expressing projective geometry and for deducing theorems of plane projective geometry. The properties of this calculus are investigated and the cut elimination theorem for LPGK is proven. A formulization of sketches is presented and the equivalence between sketches and formal proofs is demonstrated.

Sketch-as-proof: A proof-theoretic analysis of axiomatic projective geometry

Contents Contents iii List of Figures v 1 Introduction 1 2 Projective Geometry 2 2.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.1 The Euclidean Axiom of Parallelism . . . . . . . . . . . . . 3 2.1.2 Hilbert and the new approach to Geometry . . . . . . . . . 3 2.2 What is Projective Geometry . . . . . . . . . . . . . . . . . . . . . 5 2.3 Examples for Projective Planes . . . . . . . . . . . . . . . . . . . . 5 2.3.1 The projective closed Euclidean plane \Pi EP . . . . . . . . . 5 2.3.2 The projective Desargues-Plane . . . . . . . . . . . . . . . . 6 2.3.3 The minimal Projective Plane . . . . . . . . . . . . . . . . . 6 2.4 The Connection between iRealityj and the Axiomatic Method . . . 7 2.5 Some Consequences of the Axioms . . . . . . . . . . . . . . . . . . 8 3 Proof Theory 10 3.1 Introduction to Proof Theory . . . . . . . . . . . . . . . . . . . . . 10 3.2 What is Gentzen-like Proof Theory? . . . . . . . . . . . . . . . . . 11 3.3 Example Proofs i