Reservoir

Toric varieties in Lean

This repository aims at formalising the theory of toric varieties.

What is formalisation?

The purpose of this repository is to digitise some mathematical definitions, theorem statements and theorem proofs. Digitisation, or formalisation, is a process where the source material, typically a mathematical textbook or a PDF file is transformed into definitions in a target system consisting of a computer implementation of a logical theory (such as set theory or type theory).

The source

The definitions, theorems and proofs in this repository are taken from Toric varieties by Cox, Little and Schenck.

The target

The formal system which we are using as a target is Lean 4. Lean is a dependently typed theorem prover and programming language based on the Calculus of Inductive Constructions. It is being developed at the Lean Focused Research Organization by Leonardo de Moura and his team.

Our project is backed by mathlib, the major classical maths library written in Lean 4.

Content

The Lean code is located within the Toric folder. Within it, one can find:

• A Mathlib subfolder for the prerequisites to be upstreamed to mathlib. Lemmas that belong in an existing mathlib file Mathlib.X will be located in Toric.Mathlib.X. We aim to preserve the property that Toric.Mathlib.X only imports Mathlib.X and files of the form Toric.Mathlib.Y where Mathlib.X (transitively) imports Mathlib.Y. Prerequisites that do not belong in any existing mathlib file are placed in subtheory folders. See below.

See the upstreaming dashboard for more information.

Interacting with the project

Getting the project

To build the Lean files of this project, you need to have a working version of Lean. See the installation instructions (under Regular install). Alternatively, click on the button below to open a Gitpod workspace containing the project.

In either case, run lake exe cache get and then lake build to build the project.

Contributing

This project is open to contribution.