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andrejbauer / algebraic.v
Last active Jan 15, 2020
Unions of algebraic sets are algebraic
View algebraic.v
(* A Coq formalization of the theorem that the the union of algebraic sets are algebraic.
The file is self-contained, so we start with some general definitions
and facts from logic and sets, and basic algebraic definitions.
It should be straight-forward to translate it to any setting that has
a decent library of basic facts of logic, set theory and algebra.
*)
(* We formalize the following "paper" proof.
@andrejbauer
andrejbauer / Bijection.md
Last active Jan 15, 2020
A bijection between numbers and pairs of numbers.
View Bijection.md

A bijection between numbers and pairs of numbers

Every once in a while I am faced with someone who denies that the rational numbers (or fractions, or pairs of integers) can be put into a bijective correspondence with natural numbers. To deal with the situation, I coded up the bijection. So now I can just say: "Really? Interesting. Please provide a pair of numbers (i,j) which is not enumerated by f, as defined in my gist." I am still waiting for a valid counter-example.

Anyhow, here is a demo of f and g at work. I am using the Python version, but a Haskell variant is included as well.

The 100-th pair is:

>>> f(100)
(10, 4)
View beamer-slides-with-notes.pdf
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@andrejbauer
andrejbauer / topology.v
Last active Oct 22, 2019
How to get started with point-set topology in Coq. This is not actually how one would do it, but it is an intuitive setup for a classical mathematician.
View topology.v
(* How do to topology in Coq if you are secretly an HOL fan.
We will not use type classes or canonical structures because they
count as "advanced" technology. But we will use notations.
*)
(* We think of subsets as propositional functions.
Thus, if [A] is a type [x : A] and [U] is a subset of [A],
[U x] means "[x] is an element of [U]".
*)
Definition P (A : Type) := A -> Prop.
@andrejbauer
andrejbauer / algebra.v
Last active Oct 5, 2019
Uinversal algebra in Coq
View algebra.v
(* An initial attempt at universal algebra in Coq.
Author: Andrej Bauer <Andrej.Bauer@andrej.com>
If someone knows of a less painful way of doing this, please let me know.
We would like to define the notion of an algebra with given operations satisfying given
equations. For example, a group has of three operations (unit, multiplication, inverse)
and five equations (associativity, unit left, unit right, inverse left, inverse right).
*)
@andrejbauer
andrejbauer / ConstructibleNumbers.nb
Last active Oct 3, 2019
Generation of constructible numbers in Mathematica
View ConstructibleNumbers.nb
(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 11.2' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
@andrejbauer
andrejbauer / algebraic.py
Created Aug 14, 2019
Pictures of algebraic numbers
View algebraic.py
#!/usr/local/bin/python3
# Compute algebraic numbers in the complex plane and draw a nice picture
import numpy
import sys
import argparse
import math
import cairo
import pickle
@andrejbauer
andrejbauer / mandelbrot.c
Created Dec 11, 2013
A simple program for computing the Mandelbrot set.
View mandelbrot.c
/*
This program is an adaptation of the Mandelbrot program
from the Programming Rosetta Stone, see
http://rosettacode.org/wiki/Mandelbrot_set
Compile the program with:
gcc -o mandelbrot -O4 mandelbrot.c
Usage:
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andrejbauer / Graph.v
Last active Jan 28, 2019
Graph theory in Coq
View Graph.v
(** An attempt to formalize graphs. *)
Require Import Arith.
(** In order to avoid the intricacies of constructive mathematics,
we consider finite simple graphs whose sets of vertices are
natural numbers 0, 1, , ..., n-1 and the edges form a decidable
relation. *)
(** We shall work a lot with statements of the form
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andrejbauer / README.md
Last active Sep 26, 2018
How to formulate and prove the statement "all functions are continuous" in an effectful functional language?
View README.md

Are all functions continuous?

Mathematical background

Brouwer's statement "all functions are continuous" can be formulated without reference to topology as follows.

Definition: A functional f : (N → N) → N is continuous at a : N → N when there exists m : N such that, for all b : N → N, if ∀ k < m, a k = b k then f a = f b.

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