https://github.com/bollu/diffgeo

A formalization of synthetic differential geometry in Coq using infinitesimal analysis
https://github.com/bollu/diffgeo

Last synced: 20 days ago
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A formalization of synthetic differential geometry in Coq using infinitesimal analysis

• Host: GitHub
• URL: https://github.com/bollu/diffgeo
• Owner: bollu
• Created: 2019-06-06T12:23:28.000Z (over 5 years ago)
• Default Branch: master
• Last Pushed: 2021-08-29T23:55:37.000Z (over 3 years ago)
• Last Synced: 2023-08-04T01:05:23.665Z (over 1 year ago)
• Language: Coq
• Homepage:
• Size: 3.26 MB
• Stars: 9
• Watchers: 3
• Forks: 1
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        
# Formalizing Diffgeo in Coq using infinitesimal analysis

Since a lot of the constructions in differential geometry arise from universal

properties, most of them can be done by "type chasing": If the types fit,

then the construction works out. Examples include:

- Pushforwards

- Pullbacks

- Directional Derivatives

It would be nice to see how much something like `sledgehammer` can automatically

fill in.

So, let's build Diffgeo in Coq! But do we _really_ want to go through the suffering

of constructing reals with Dedekind cuts or cauchy sequences? I for one don't

enjoy analysis nearly enough to put myself through that.

Enter __infinitesimal analysis___ - A clean axiomatisation of real numbers

that crucially relies on living in constructive logic to create infinitesimals

such as `dx`, and provides axioms to perform "physicist-style" proofs

rigorously!

Many thanks to [Arnaud Spiwack](http://assert-false.net/arnaud/)for showing me

this axiomatization.

## References

- [Synthetic Differential Geometry (Blog post by Andrej Bauer)][sgd blog]

- First steps in syntehtic computability (Math inside the effective topos).

- Axiomatize R and then move on.

[sgd blog]: http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/