In this paper, we report on the formal proof that Hilbert's axiom system can be derived from Tarski's system. For this purpose we mechanized the proofs of the first twelve chapters of Schwabäuser, Szmielew and Tarski's book: Metamathematische Methoden in der Geometrie. The proofs are checked formally within classical logic using the Coq proof assistant. The goal of this development is to provide clear foundations for other formalizations of geometry and implementations of decision procedures.

In this paper, we report on the formalization of a synthetic proof of Pappus' theorem. We provide two versions of the theorem: the first one is proved in neutral geometry (without assuming the parallel postulate), the second (usual) version is proved in Euclidean geometry. The proof that we formalize is the one presented by Hilbert in The Foundations of Geometry, which has been described in detail by Schwabhäuser, Szmielew and Tarski in part I of Metamathematische Methoden in der Geometrie. We highlight the steps that are still missing in this later version. The proofs are checked formally using the Coq proof assistant. Our proofs are based on Tarski's axiom system for geometry without any continuity axiom. This theorem is an important milestone toward obtaining the arithmetization of geometry, which will allow us to provide a connection between analytic and synthetic geometry.

This paper describes the formalization of the arithmetization of Euclidean plane geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of the book by Schwabhäuser Szmielew and Tarski Metamathematische Methoden in der Geometrie. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a "back-translation" from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method. Moreover, we solve a challenge proposed by Beeson: we prove that, given two points, an equilateral triangle based on these two points can be constructed in Euclidean Hilbert planes. Finally, we derive the axioms for another automated deduction method: the area method.

This paper describes the formalization of the arithmetization of Euclidean geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of [SST83]. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a "back-translation" from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method.

Toxic and pathologic protein aggregation is a common feature of neurodegenerative diseases from Alzheimer’s and Parkinson’s diseases to amyotrophic lateral sclerosis (ALS). Chaperone proteins such as the metazoan heat shock protein 70 (Hsp70), Hsp40, and Hsp110 complex can identify these misfolded aggregates and restore them to their functional native states. However, previous studies have used a wide range of chaperone complex components and proportions. Using aggregated firefly luciferase as a model substrate, we set out to optimize the stoichiometric ratio for the disaggregation of these aggregates in vitro. We tested ratios of Hsp70:40:110 from 1:10:1 to 1000:100:1 and used the luminescence of firefly luciferin as a measure of disaggregation activity. With this data, we proceeded to refine the canonical Hsp70/40/110 disaggregase network against fused in sarcoma (FUS), a protein that forms pathologic protein aggregates in ALS and frontotemporal dementia (FTD). We used absorbance measurements at 395 nm from a high‐throughput plate reader to determine the turbidity levels of FUS aggregates with various combinations of Hsp70/40/110. Together, our findings support the existence of an ideal Hsp70/40/110 system which provides a framework to optimize the disassembly of protein aggregates in other neurodegenerative diseases.