andrejbauer

#!/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

# 
METHOD: TABLE
ENCODE: Unicode
PROMPT: LaTeX
VERSION: 1.0
DELIMITER: ,
VALIDINPUTKEY: ^,.?!:;"'/\()[]{}<>$%&@*01234567890123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz
TERMINPUTKEY: 
BEGINCHARACTER
alpha	α

andrejbauer

andrejbauer

(* In Andromeda everything the user writes is "meta-level programming. *)

(* ML-level natural numbers *)
mltype rec mlnat =
  | zero
  | succ of mlnat
  end

(* We can use the meta-level numbers to define a program which computes n-fold iterations
  of f. Here we give an explicit type of iterate because the ML-type inference cannot tell

andrejbauer

{- Here is a short demonstration on how to use Agda's solvers that automatically
   prove equations. It seems quite impossible to find complete worked examples
   of how Agda solvers are used.

   Thanks go to @anormalform who helped out with these on Twitter, see
   https://twitter.com/andrejbauer/status/1549693506922364929?s=20&t=7cb1TbB2cspIgktKXU8rng

   I welcome improvements and additions to this file.

   This file works with Agda 2.6.2.2 and standard-library-1.7.1

andrejbauer

open import Data.Nat
open import Data.Fin as Fin

module Primrec where

  -- The datatype of primitive recursive function
  data PR : ∀ (k : ℕ) → Set where
    pr-zero : PR 0 -- zero
    pr-succ : PR 1 -- successor
    pr-proj : ∀ {k} (i : Fin k) → PR k -- projection

andrejbauer

(* Post's theorem in computability theory states that a subset S ⊆ ℕ is
   decidable if both S and its complement ℕ \ S are semidecidable.
   Just how much classical logic is needed to prove the theorem, though?
   In this note we show that Markov's principle suffices.

   Rather than working with subsets of ℕ we work synthetically, with
   semidecidable and decidable truth values. (This is more general than
   the traditional Post's theorem.)
*)

andrejbauer

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)

andrejbauer

(* 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

(*
   A formalization of first-order logic, theories and their models.
   As an example we formalize the language of ZF set theory and a a small fragment of ZF.
*)

(* A signature is given by function symbols, relation symbols, and their arities. *)
Structure Signature :=
  { funSym : Type (* names of function symbols *)
  ; funArity : funSym -> Type (* arities of function symbols *)
  ; relSym : Type (* names of relation symbols *)