Parallel Postulates and Continuity Axioms: A Mechanized Study in Intuitionistic Logic Using Coq

  title={Parallel Postulates and Continuity Axioms: A Mechanized Study in Intuitionistic Logic Using Coq},
  author={Pierre Boutry and Charly Gries and Julien Narboux and Pascal Schreck},
  journal={Journal of Automated Reasoning},
In this paper we focus on the formalization of the proofs of equivalence between different versions of Euclid’s 5th postulate. Our study is performed in the context of Tarski’s neutral geometry, or equivalently in Hilbert’s geometry defined by the first three groups of axioms, and uses an intuitionistic logic, assuming excluded-middle only for point equality. Our formalization provides a clarification of the conditions under which different versions of the postulates are equivalent. Following… 
On the formalization of foundations of geometry
A new proof is exposed that Euclid’s parallel postulate is not derivable from the other axioms of first-order Euclidean geometry, and Pejas’ classification of parallel postulates is refined.
The Ubiquitous Axiom
This paper starts with a survey of what is known regarding an axiom, referred to as the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect. Several statements
Tarski Geometry Axioms. Part IV – Right Angle
The Mizar system is used to systematically formalize Chapter 8 of the SST book, defining the notion of right angle and proving some of its basic properties, a theory of intersecting lines (including orthogonality).
Proof-checking Euclid
We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely
Implementing Euclid’s straightedge and compass constructions in type theory
This paper outlines the implementation of Euclidean geometry based on straightedge and compass constructions in the intuitionistic type theory of the Nuprl proof assistant, which enables a concise and intuitive expression of Euclid’s constructions.
Euclid after Computer Proof-Checking
  • M. Beeson
  • Philosophy
    The American Mathematical Monthly
  • 2022
Euclid pioneered the concept of a mathematical theory developed from axioms by a series of justified proof steps. From the outset there were critics and improvers. In this century the use of
A machine-checked direct proof of the Steiner-lehmus theorem
This paper has formalized a constructive axiom set for Euclidean plane geometry in a proof assistant that implements a constructive logic and has built the proof of the Steiner-Lehmus theorem on this constructive foundation.
Formalization of the Poincaré Disc Model of Hyperbolic Geometry
The Poincaré disc model of hyperbolic geometry within the Isabelle/HOL proof assistant is described and is shown to satisfy Tarski’s axioms except for Euclid's axiom.
Theorem Proving as Constraint Solving with Coherent Logic


From Hilbert to Tarski
In this paper, we describe the formal proof using the Coq proof assistant that Tarski's axioms for plane neutral geometry (excluding continuity axioms) can be derived from the corresponding Hilbert's
A constructive version of Tarski's geometry
  • M. Beeson
  • Mathematics
    Ann. Pure Appl. Log.
  • 2015
A Mechanical Verification of  the Independence of Tarski's  Euclidean Axiom
This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's
Constructive Geometry and the Parallel postulate
This paper completely settle the questions about implications between the three versions of the parallel postulates: the strong parallel postulate easily implies Euclid 5, and in fact Euclid5 also implies the strong Parallel Postulate, although the proof is lengthy, depending on the verification that Euclid 4 suffices to define multiplication geometrically.
A Synthetic Proof of Pappus’ Theorem in Tarski’s Geometry
This paper reports on the formalization of a synthetic proof of Pappus’ theorem, which is an important milestone toward obtaining the arithmetization of geometry, which will allow to provide a connection between analytic and synthetic geometry.
A short note about case distinctions in Tarski's geometry
In this paper we study some decidability properties in the context of Tarski's axiom system for geometry. We removed excluded middle from our assumptions and studied how our formal proof of the first
Brouwer and Euclid
Axiomatizations of Hyperbolic and Absolute Geometries
A survey of finite first-order axiomatizations for hyperbolic and absolute geometries. 1. Hyperbolic Geometry Elementary Hyperbolic Geometry as conceived by Hilbert To axiomatize a geometry one needs
From Tarski to Descartes: Formalization of the Arithmetization of Euclidean Geometry
The formalization of the arithmetization of Euclidean geometry in the Coq proof assistant is described, derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Grobner basis method.
Mechanical Theorem Proving in Tarski's Geometry
The mechanization of the proofs of the first height chapters of Schwabauser, Szmielew and Tarski's book: Meta-mathematische Methoden in der Geometrie is described to provide foundations for other formalizations of geometry and implementations of decision procedures.