https://github.com/Lix0120/eudoxus

Formalise Eudoxus reals in lean and prove they form a complete Archimedean ordered field.
https://github.com/Lix0120/eudoxus

lean theorem-proving

Last synced: about 2 months ago
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Formalise Eudoxus reals in lean and prove they form a complete Archimedean ordered field.

• Host: GitHub
• URL: https://github.com/Lix0120/eudoxus
• Owner: Lix0120
• Created: 2020-07-31T20:57:36.000Z (over 4 years ago)
• Default Branch: master
• Last Pushed: 2021-02-14T11:51:41.000Z (almost 4 years ago)
• Last Synced: 2024-08-10T14:12:31.949Z (6 months ago)
• Topics: lean, theorem-proving
• Language: Lean
• Homepage:
• Size: 123 KB
• Stars: 6
• Watchers: 1
• Forks: 1
• Open Issues: 1
• Metadata Files:
  • Readme: README.md

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README

        
# The Eudoxus Real Numbers

The Eudoxus real number is a construction of reals directly from integers. Every number is a set of functions ```ℤ → ℤ``` growing "almost linearly" 

(called almost homomorphisms) in which

the difference between each two functions is bounded. 


We defined addition, multiplication, inverse, less than, etc between the numbers, proving they form an additive group, a ring, a total order, etc.

Finally, we verified that the construction is a complete ordered (called ```conditionally_complete_linear_order``` in lean) field 

-- every such field is isomorphic to ℝ. 


The formalizing idea is from the paper by R. D. Arthan which can be found at https://arxiv.org/abs/math/0405454.