An open API service providing repository metadata for many open source software ecosystems.

Topic: "formalization"

EgbertRijke/HoTT-Intro 📦

An introductory course to Homotopy Type Theory

Language: Agda - Size: 7.11 MB - Last synced at: 17 days ago - Pushed at: almost 5 years ago - Stars: 365 - Forks: 28

GeoCoq/GeoCoq

A formalization of geometry in Coq based on Tarski's axiom system

Language: Coq - Size: 7.66 MB - Last synced at: 3 days ago - Pushed at: 3 days ago - Stars: 197 - Forks: 27

mgree/smoosh

The Symbolic, Mechanized, Observable, Operational SHell: an executable formalization of the POSIX shell standard.

Language: OCaml - Size: 3.66 MB - Last synced at: 27 days ago - Pushed at: about 2 years ago - Stars: 118 - Forks: 5

loganrjmurphy/LeanEuclid

LeanEuclid is a benchmark for autoformalization in the domain of Euclidean geometry, targeting the proof assistant Lean.

Language: Lean - Size: 3.57 MB - Last synced at: about 6 hours ago - Pushed at: about 7 hours ago - Stars: 98 - Forks: 8

TheoWinterhalter/formal-type-theory

Formalising Type Theory in a modular way for translations between type theories

Language: Coq - Size: 1.48 MB - Last synced at: about 16 hours ago - Pushed at: over 7 years ago - Stars: 94 - Forks: 4

Lysxia/system-F

Formalization of the polymorphic lambda calculus and its parametricity theorem

Language: Coq - Size: 12.7 KB - Last synced at: 30 days ago - Pushed at: about 2 months ago - Stars: 34 - Forks: 2

gallais/agda-presburger

Deciding Presburger arithmetic in agda

Language: Agda - Size: 155 KB - Last synced at: about 1 month ago - Pushed at: about 2 years ago - Stars: 31 - Forks: 3

artagnon/bonak

🧊 An indexed construction of semi-simplicial and semi-cubical sets

Language: TeX - Size: 4.06 MB - Last synced at: 7 days ago - Pushed at: 7 days ago - Stars: 28 - Forks: 3

annenkov/two-level

Two-Level Type Theory

Language: Lean - Size: 151 KB - Last synced at: 12 days ago - Pushed at: over 5 years ago - Stars: 28 - Forks: 1

jonsterling/coq-guarded-computational-type-theory

Language: Coq - Size: 421 KB - Last synced at: 14 days ago - Pushed at: about 7 years ago - Stars: 21 - Forks: 1

uds-psl/autosubst2

Official repository of the Autosubst 2 project.

Language: Haskell - Size: 128 KB - Last synced at: 11 days ago - Pushed at: almost 2 years ago - Stars: 20 - Forks: 5

ualib/ualib.github.io

The Agda Universal Algebra Library (UALib) is a library of types and programs (theorems and proofs) that formalizes the foundations of universal algebra in dependent type theory using the Agda proof assistant language.

Language: TeX - Size: 52.7 MB - Last synced at: about 2 years ago - Pushed at: over 3 years ago - Stars: 19 - Forks: 2

ivashkev/math-formalizations

Formalization of some elementary mathematical theories in Coq

Language: Coq - Size: 2.34 MB - Last synced at: 23 days ago - Pushed at: about 5 years ago - Stars: 14 - Forks: 0

superhaNds/cwfs

Formalization of Categories with Families

Language: Agda - Size: 1.45 MB - Last synced at: almost 2 years ago - Pushed at: over 3 years ago - Stars: 13 - Forks: 0

jonsterling/agda-effectful-forcing

Agda formalization of the paper, "Higher-Order Functions and Brouwer's Thesis". Deduces a Brouwer ordinal from a function ((nat -> nat) -> nat) in System T.

Language: Agda - Size: 1.24 MB - Last synced at: about 1 month ago - Pushed at: over 4 years ago - Stars: 12 - Forks: 0

felko/linear-algebra

Linear algebra formalization in Agda

Language: Agda - Size: 499 KB - Last synced at: about 2 years ago - Pushed at: over 5 years ago - Stars: 12 - Forks: 0

a2435191/lean-logic-formalization

Formalize "Logic Notes" by Lou van den Dries in Lean

Language: Lean - Size: 32.2 KB - Last synced at: 3 days ago - Pushed at: 3 days ago - Stars: 11 - Forks: 0

vltanh/lean4-analysis-tao

Formalization of "Analysis I" by Terence Tao

Language: Lean - Size: 67.4 KB - Last synced at: about 2 months ago - Pushed at: about 2 months ago - Stars: 11 - Forks: 0

harp-project/Core-Erlang-Formalization

Language: Coq - Size: 3.37 MB - Last synced at: 7 days ago - Pushed at: 8 days ago - Stars: 10 - Forks: 1

miculan/telegram-mtproto2-verification

Formal Verification of Telegram's MTProto 2.0

Language: Shell - Size: 150 KB - Last synced at: about 2 years ago - Pushed at: over 2 years ago - Stars: 10 - Forks: 0

lean-dojo/LeanMillenniumPrizeProblems

Formalization of the Millennium Problems in Lean4.

Language: C - Size: 3.17 MB - Last synced at: 21 days ago - Pushed at: about 1 month ago - Stars: 9 - Forks: 0

siraben/coq-wigderson

Formalization of Wigderson's graph coloring algorithm in Coq

Language: Coq - Size: 988 KB - Last synced at: 27 days ago - Pushed at: over 1 year ago - Stars: 9 - Forks: 3

capital-G/xenakis-workgroup

Material created during the Iannis Xenakis workgroup

Language: Jupyter Notebook - Size: 24.4 MB - Last synced at: 16 days ago - Pushed at: over 2 years ago - Stars: 9 - Forks: 2

tchajed/goedel-t

Formalization of termination of Gödel's System T

Language: Coq - Size: 70.3 KB - Last synced at: 24 days ago - Pushed at: almost 4 years ago - Stars: 9 - Forks: 0

Mbodin/CoqR

A Coq formalisation of the R programming language

Language: R - Size: 2.94 MB - Last synced at: over 1 year ago - Pushed at: over 2 years ago - Stars: 8 - Forks: 0

Lolirofle/stuff-in-agda

Formal proofs in mathematics/computer science/logic formalized in the Agda language. A hobby project I am working on in my free time.

Language: Agda - Size: 4.67 MB - Last synced at: about 2 years ago - Pushed at: over 2 years ago - Stars: 7 - Forks: 1

Lysxia/coq-mtl

Formalized laws for mtl

Language: Coq - Size: 56.6 KB - Last synced at: 27 days ago - Pushed at: over 5 years ago - Stars: 7 - Forks: 0

ajrouvoet/implicits.agda

Mechanized formalization of Implicit resolution in Agda

Language: Agda - Size: 510 KB - Last synced at: about 1 year ago - Pushed at: almost 8 years ago - Stars: 6 - Forks: 0

matematiflo/CompAssistedMath2024

GitHub repository for the seminar on Computer-assisted mathematics held at the University of Heidelberg during the Summer Semester of 2024.

Language: Jupyter Notebook - Size: 110 MB - Last synced at: 9 months ago - Pushed at: 9 months ago - Stars: 5 - Forks: 13

choukh/agda-veblen

veblen function in agda

Language: Agda - Size: 182 KB - Last synced at: about 2 months ago - Pushed at: 10 months ago - Stars: 5 - Forks: 0

marco10507/formalization-of-sorting-algorithms

some sorting algorithms' formalisation

Language: Isabelle - Size: 665 KB - Last synced at: about 1 year ago - Pushed at: over 5 years ago - Stars: 5 - Forks: 0

Gradual-Typing/LambdaIFCStar

The Agda mechanization of a gradual security-typed programming language with general mutable references.

Language: Agda - Size: 971 KB - Last synced at: 28 days ago - Pushed at: 28 days ago - Stars: 4 - Forks: 1

thery/minirubik

Solving the mini Rubik (2x2) in Coq

Language: Coq - Size: 165 KB - Last synced at: 27 days ago - Pushed at: 5 months ago - Stars: 4 - Forks: 0

vitalsong/bastard

more than just a package manager

Language: CMake - Size: 55.7 KB - Last synced at: about 1 year ago - Pushed at: about 1 year ago - Stars: 4 - Forks: 1

snow0815/tribAIn

tribAin - ontology for scientific experiments in the domain of tribology

Size: 1.36 MB - Last synced at: 4 months ago - Pushed at: over 4 years ago - Stars: 4 - Forks: 3

jjaassoonn/flat 📦

Equivalent definitions of flatness

Language: Lean - Size: 813 KB - Last synced at: 12 months ago - Pushed at: 12 months ago - Stars: 3 - Forks: 0

UniFormal/VSCode-MMT

MMT plugin for Visual Studio Code

Language: TypeScript - Size: 442 KB - Last synced at: about 1 year ago - Pushed at: about 1 year ago - Stars: 3 - Forks: 0

jeffreybakker/wsn

Calculating the Risk of Valve Failures when Maintaining Water Supply Networks

Language: Python - Size: 105 KB - Last synced at: over 1 year ago - Pushed at: over 2 years ago - Stars: 3 - Forks: 0

zchn/eth-acl2

An ACL2 formalization of the Ethereum VM, aiming to be both executable and suitable for proving interesting properties of EVM contracts.

Language: Common Lisp - Size: 249 KB - Last synced at: 6 months ago - Pushed at: over 2 years ago - Stars: 3 - Forks: 3

radeusgd/QuotedPatternMatchingProof

A mechanized proof of soundness of calculus defined in A Theory of Quoted Code Patterns which is a formalization of pattern matching on code available in Scala 3 as part of its new macro system.

Language: TeX - Size: 877 KB - Last synced at: 29 days ago - Pushed at: almost 5 years ago - Stars: 3 - Forks: 0

choukh/agda-googology

formalized googology in agda

Language: CSS - Size: 3.75 MB - Last synced at: 8 months ago - Pushed at: 8 months ago - Stars: 2 - Forks: 0

co-dan/BI-cutelim

Cut elimination for the logic of Bunched Implications (BI), and some extensions.

Language: Coq - Size: 769 KB - Last synced at: 10 days ago - Pushed at: almost 2 years ago - Stars: 2 - Forks: 1

siraben/senior-thesis

LaTeX sources for my undergraduate thesis

Language: TeX - Size: 791 KB - Last synced at: about 2 months ago - Pushed at: almost 2 years ago - Stars: 2 - Forks: 1

sacerdot/Crumbling-Abstract-Machines

This repository contains the formalization of part of the theory behind the Crumbling Abstract Machines, that has been introduced and partially developed in Beniamino Accattoli, Andrea Condoluci, Giulio Guerrieri, and Claudio Sacerdoti Coen. 2019. Crumbling Abstract Machines. In Proceedings of ACM Conference (Conference’17). ACM, New York, NY,USA,39 pages.

Language: Mathematica - Size: 854 KB - Last synced at: about 1 month ago - Pushed at: over 3 years ago - Stars: 2 - Forks: 1

lukaszcz/sortalgs

Various sorting algorithms formalised using the "sauto" component of CoqHammer 1.3.

Language: Coq - Size: 23.4 KB - Last synced at: 2 months ago - Pushed at: over 3 years ago - Stars: 2 - Forks: 0

zampino/ggt

A random collection of Agda facts around the Theory of Group Actions

Language: Agda - Size: 40 KB - Last synced at: about 1 month ago - Pushed at: over 4 years ago - Stars: 2 - Forks: 0

benhuds/Agda

Complexity analysis, type safety, soundness, etc.

Language: Agda - Size: 1.21 MB - Last synced at: about 2 months ago - Pushed at: almost 5 years ago - Stars: 2 - Forks: 0

kiranandcode/CamelClone

Automatically push changes to selected repositories - 0% tested yet 100% correct.

Language: Coq - Size: 13.7 KB - Last synced at: 12 days ago - Pushed at: over 5 years ago - Stars: 2 - Forks: 0

cogumbreiro/habanero-coq

Coq formalization of the Habanero programming model.

Language: Coq - Size: 1020 KB - Last synced at: about 2 years ago - Pushed at: almost 7 years ago - Stars: 2 - Forks: 0

thery/Selinger

Formalisation of Selinger proof about quantum gate

Language: Coq - Size: 208 KB - Last synced at: 12 days ago - Pushed at: 5 months ago - Stars: 1 - Forks: 0

thery/EdwardsEllipticCurve

Group Law for Elliptic Curves according to Tom Hales

Language: Coq - Size: 277 KB - Last synced at: 12 days ago - Pushed at: 5 months ago - Stars: 1 - Forks: 0

metinersin/FormalTextbookModelTheory

Language: HTML - Size: 869 KB - Last synced at: 8 months ago - Pushed at: 8 months ago - Stars: 1 - Forks: 0

themathqueen/monlib4

Formalising non-commutative graph theory in Lean

Language: Lean - Size: 917 KB - Last synced at: 10 months ago - Pushed at: 10 months ago - Stars: 1 - Forks: 0

pitmonticone/FLT3 Fork of riccardobrasca/flt3

A formalised proof of Fermat's Last Theorem for exponent 3 in the Lean proof assistant.

Language: TeX - Size: 550 KB - Last synced at: about 1 year ago - Pushed at: about 1 year ago - Stars: 1 - Forks: 1

bakseter/thesis

My master thesis (and related code) in Logic at the University of Bergen.

Language: Haskell - Size: 7.48 MB - Last synced at: 20 days ago - Pushed at: about 1 year ago - Stars: 1 - Forks: 0

ata-keskin/martingales

A Formalization of Martingales using Isabelle/HOL

Language: Isabelle - Size: 20 MB - Last synced at: 21 days ago - Pushed at: over 1 year ago - Stars: 1 - Forks: 0

choukh/MetaLogic

first-order logic and set theory

Language: Agda - Size: 340 KB - Last synced at: over 1 year ago - Pushed at: over 1 year ago - Stars: 1 - Forks: 0

jjaassoonn/twist

Definition of Serre's twisting sheaf (work in progress)

Language: Lean - Size: 62.5 KB - Last synced at: about 1 month ago - Pushed at: almost 3 years ago - Stars: 1 - Forks: 1

jjaassoonn/cc

Čech cohomology on topological space

Language: Lean - Size: 229 KB - Last synced at: about 1 month ago - Pushed at: almost 3 years ago - Stars: 1 - Forks: 0

imandra-ai/abstract-transition-systems

[alpha] An implementation of several classic transition systems that describe algorithms for SAT or SMT, for interactive exploration

Language: OCaml - Size: 229 KB - Last synced at: 3 days ago - Pushed at: about 3 years ago - Stars: 1 - Forks: 1

radeusgd/pDOT-GADT

Formal foundations for GADTs in Scala

Language: Scala - Size: 519 KB - Last synced at: about 1 month ago - Pushed at: almost 5 years ago - Stars: 1 - Forks: 0

fasapa/lambdaex

Specification, formalization and verification of the XC function from ex lambda calculus in Coq.

Language: Coq - Size: 135 KB - Last synced at: 9 months ago - Pushed at: about 5 years ago - Stars: 1 - Forks: 0

pilif0/bigraph

Formalization of Robin Milner's bigraphs in Isabelle/HOL

Language: Isabelle - Size: 12.7 KB - Last synced at: about 2 years ago - Pushed at: over 5 years ago - Stars: 1 - Forks: 0

nickgian/Bonsai-coq

A formalization of SRPs and Control Plane Compression in Coq.

Language: Coq - Size: 6.84 KB - Last synced at: about 2 years ago - Pushed at: almost 6 years ago - Stars: 1 - Forks: 0

UCL-PPLV/GCTransformations

Certified implementation of a parametrized framework for concurrent garbage collectors

Language: Coq - Size: 248 KB - Last synced at: 9 months ago - Pushed at: over 8 years ago - Stars: 1 - Forks: 1

Danelnov/Lambda-Calculus-Formalization-

This formalization uses the De Bruijn indices, the objective is to formalize the Church-Rosser theorem

Language: Lean - Size: 2.93 KB - Last synced at: 7 days ago - Pushed at: 7 days ago - Stars: 0 - Forks: 0

FMFI-UK-1-AIN-412/formalization-checker-backend

The back end of a tool for checking formalization exercises.

Language: JavaScript - Size: 247 KB - Last synced at: 18 days ago - Pushed at: 18 days ago - Stars: 0 - Forks: 5

UFrameIT/archives

All MMT archives of formalizations needed by the UFrameIT prototype in the FrameIT project.

Language: Shell - Size: 62.5 KB - Last synced at: 2 months ago - Pushed at: 2 months ago - Stars: 0 - Forks: 0

krr-up/flatland-formalization-notes

A space for the formalization of the underlying structure of Flatland.

Language: TeX - Size: 2.5 MB - Last synced at: 4 months ago - Pushed at: 4 months ago - Stars: 0 - Forks: 0

ewdlop/FormalML Fork of IBM/FormalML

Formalization of Machine Learning Theory with Applications to Program Synthesis

Language: Coq - Size: 8.84 MB - Last synced at: 4 months ago - Pushed at: 4 months ago - Stars: 0 - Forks: 0

FormalMathematicsLab/ICL_Course_FormalMathematics_2024 Fork of ImperialCollegeLondon/formalising-mathematics-2024

Repository hosting resources for the 2024 course in Formal Mathematics at @ImperialCollegeLondon taught by Kevin Buzzard (@kbuzzard).

Language: Lean - Size: 356 KB - Last synced at: 8 months ago - Pushed at: 8 months ago - Stars: 0 - Forks: 0

SDZZGNDRC/LADR

Formalizing "Linear Algebra Done Right" using Lean

Language: Lean - Size: 39.1 KB - Last synced at: 9 months ago - Pushed at: 9 months ago - Stars: 0 - Forks: 0

pitmonticone/carleson Fork of fpvandoorn/carleson

[WIP] A formalised proof of a generalised Carleson's Theorem in the Lean proof assistant.

Language: TeX - Size: 1.31 MB - Last synced at: 10 months ago - Pushed at: 10 months ago - Stars: 0 - Forks: 0

Seeker04/stlc-agda-elab

Agda formalisation of an elaborator for a simply typed language

Language: Agda - Size: 4.98 MB - Last synced at: 2 months ago - Pushed at: 10 months ago - Stars: 0 - Forks: 0

FormalMathematicsLab/LFTCM2024 Fork of riccardobrasca/LFTCM2024

Repository hosting resources for the 2024 workshop "Lean for the Curious Mathematician".

Language: Lean - Size: 693 KB - Last synced at: about 1 year ago - Pushed at: about 1 year ago - Stars: 0 - Forks: 0

FormalMathematicsLab/UniRome_Workshop_Lean_2024 Fork of fpvandoorn/LeanInRome

Repository hosting resources for the 2024 workshop "Computer-Verified Proofs: 48 Hours in Rome" organised by @oliver-butterley, @RafaelGreenblatt and @marcolenci.

Size: 2.6 MB - Last synced at: over 1 year ago - Pushed at: over 1 year ago - Stars: 0 - Forks: 0

ata-keskin/eudoxus-reals

An unusual construction of the real numbers using Isabelle/HOL

Language: Isabelle - Size: 814 KB - Last synced at: 4 months ago - Pushed at: over 1 year ago - Stars: 0 - Forks: 0

miguelpagano/formalmetatheory-stoughton Fork of ernius/formalmetatheory-stoughton

We prove the equivalence between three induction principles for the untyped lambda calculus.

Language: Agda - Size: 2.97 MB - Last synced at: over 1 year ago - Pushed at: over 1 year ago - Stars: 0 - Forks: 0

haztecaso/euclidean-geometry-lean

Trabajo de Fin de Grado de Matemáticas, en la Universidad Complutense de Madrid. Presentado en septiembre de 2023.

Language: TeX - Size: 40.4 MB - Last synced at: over 1 year ago - Pushed at: over 1 year ago - Stars: 0 - Forks: 0

bridgekat/leansubst

An Autosubst-like tool for handling de Bruijn indices / 一个和 Autosubst 类似的用于处理 de Bruijn 标号的工具

Language: Lean - Size: 54.7 KB - Last synced at: almost 2 years ago - Pushed at: almost 2 years ago - Stars: 0 - Forks: 0

FMFI-UK-1-AIN-412/formalization-checker

The front end of a tool for checking formalization exercises.

Language: JavaScript - Size: 5.91 MB - Last synced at: 23 days ago - Pushed at: almost 2 years ago - Stars: 0 - Forks: 5

dfirsov/easycrypt-zk-code

EasyCrypt formalization of definitions and derivations associated with zero-knowledge (sigma protocols).

Language: eC - Size: 759 KB - Last synced at: over 1 year ago - Pushed at: almost 2 years ago - Stars: 0 - Forks: 0

MathiasGarnier/formalean-inator

Aiming to have a personal well-founded math library written in lean. Hope it works...

Size: 62.5 KB - Last synced at: almost 2 years ago - Pushed at: over 2 years ago - Stars: 0 - Forks: 0

Paul-Lez/dedekind-kummer-theorem

This repository will eventually contain the code I wrote when formalising the Dedekind-Kummer Theorem

Size: 2.93 KB - Last synced at: over 1 year ago - Pushed at: over 2 years ago - Stars: 0 - Forks: 0

Kamirus/fine-grained-shift0-dollar

Coq Formalisation of "A Fine-Grained Evaluation Strategy for Delimited-Control Operators shift0/dollar"

Language: Coq - Size: 532 KB - Last synced at: 20 days ago - Pushed at: over 2 years ago - Stars: 0 - Forks: 0

BlueHuskyStudios/Blue-Husky-Stages-of-Product-Creation

The stages that Blue Husky Studios goes throguh when creating a product

Size: 7.81 KB - Last synced at: about 1 month ago - Pushed at: over 3 years ago - Stars: 0 - Forks: 0

pedroos/limits

Mathematical foundations exploration

Language: C# - Size: 268 KB - Last synced at: about 2 years ago - Pushed at: almost 4 years ago - Stars: 0 - Forks: 0

danielneubert/Reform.js 📦

Modern forms finally made easy.

Language: JavaScript - Size: 225 KB - Last synced at: 15 days ago - Pushed at: almost 4 years ago - Stars: 0 - Forks: 0

enshankar/BigProof

Collaborative repo for Big Proof Programme at the Isaac Newton Institute, Jun 26 to Aug 4, 2017

Size: 20.5 KB - Last synced at: over 1 year ago - Pushed at: almost 8 years ago - Stars: 0 - Forks: 0

ankitku/Twisted_Functor_Verification

Language: HTML - Size: 22.5 KB - Last synced at: about 2 years ago - Pushed at: almost 8 years ago - Stars: 0 - Forks: 0

ahovgaard/twelf-records

Formalization of a simple functional language with records and subtyping in Twelf.

Size: 12.7 KB - Last synced at: about 2 years ago - Pushed at: about 8 years ago - Stars: 0 - Forks: 0

zeionara/formal_languages_and_grammars

Homeworks for 'formal languages and grammars' discipline

Language: C++ - Size: 419 KB - Last synced at: 2 months ago - Pushed at: about 8 years ago - Stars: 0 - Forks: 0

Related Topics
coq 26 mathematics 18 agda 14 lean4 13 lambda-calculus 10 lean 6 formal-methods 6 logic 5 theorem-proving 5 type-theory 4 isabelle-hol 4 formal-proofs 4 mathematical-logic 4 formal-mathematics 4 proof-assistant 4 coq-formalization 4 formal-verification 3 ocaml 3 linear-algebra 3 dependent-types 3 hilbert-axioms 3 verification 3 homotopy-type-theory 2 agda-library 2 insertion-sort 2 geometry 2 euclid 2 elements 2 thesis 2 semantics 2 theorem-prover 2 interactive-theorem-proving 2 first-order-logic 2 euclidean-geometry 2 number-theory 2 scala 2 googology 2 proof 2 proofs 2 metatheory-language 2 mergesort 2 model-theory 1 noninterference 1 type-systems 1 async-finish 1 arithmetic-geometry 1 decision-procedure 1 textbook 1 presburger-arithmetic 1 language 1 workshop 1 conference 1 reflection 1 group-actions 1 stochastic 1 music-informatics 1 music-composition 1 ordinal-numbers 1 ordinal 1 group-theory 1 knowledge-representation 1 coq-library 1 grammar 1 formal 1 quicksort 1 selection-sort 1 mathlib 1 synchronization 1 phaser 1 homework 1 parallel-computing 1 sorting-algorithms 1 graph-algorithms 1 graph-coloring 1 graph-theory 1 habanero-coq 1 complexity-analysis 1 coercion 1 gradual-typing 1 information-flow 1 meta-theory 1 rubik-cube 1 blue-husky-software-platform 1 business-process 1 product-stages 1 flatland 1 martingales 1 stochastic-processes 1 proof-assistants 1 control-operators 1 delimited-continuations 1 proof-theory 1 cubical-type-theory 1 paper 1 research-project 1 millennium-problems 1 archimedes 1 continuity 1 desargues 1 pappus 1