andrejbauer

(** Prop is a complete lattice. We can ask which propositions are finite. *)

Definition directed (D : Prop -> Prop) :=
  (exists p , D p) /\ forall p q, D p -> D q -> exists r, D r /\ (p -> r) /\  (q -> r).

(** A proposition p is finite when it has the following property:
    for any directed D, if p ≤ sup D then there is q ∈ D such that p ≤ q. *)
Definition finite (p : Prop) :=
  forall D, directed D ->
    (p -> exists q, D q /\ q) -> exists q, D q /\ (p -> q).

andrejbauer

-- empty type
data 𝟘 : Set where

absurd : {A : Set} → 𝟘 → A
absurd ()

-- booleans
data 𝟚 : Set where
  false true : 𝟚

andrejbauer

(* If all subsets of a singleton set are finite, then excluded middle holds.*)
Require Import List.

(* We present the subsets of X as characteristic maps S : X -> Prop.
   Thus to say that x : X is an element of S : Powerset X we write S x. *)
Definition Powerset X := X -> Prop.

(* Auxiliary relation "l lists the elements of S". *)
Definition lists {X} (l : list X) (S : Powerset X) :=
  forall x, S x <-> In x l.

andrejbauer

-- Counterexample by Chung Kil Hur, improved by Andrej Bauer

-- We show that it is inconsistent to have an injection I : (Set → Set) → Set and excluded middle.
-- Indeed, excluded middle and I together give a surjection J : Set → (Set → Set),
-- which by Lawvere's fixed point theorem begets a fixed point operator on Set.
-- However, negation does not have a fixed point.

module cantor where

-- generalities

andrejbauer

(* 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

(* 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

#!/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

(** * General support for creation of dynamic effects *)

(** We show how to use the multicore Ocaml effects to dynamically generate local
   effects. Such effects are akin to the Eff resources, and they can be used to
   implement ML references.

   The code is based on "Eff directly in OCaml" by Oleg Kiselyov and KC
   Sivaramakrishnan (http://kcsrk.info/papers/caml-eff17.pdf). It was written by
   Andrej Bauer, Oleg Kiselyov, and Stephen Dolan at the Dagstuhl seminar
   "Algebraic Effect Handlers go Mainstream". *)

andrejbauer

(** This code is compatible with Eff 5.0, see http://www.eff-lang.org *)

(** We show that with algebraic effects and handlers a total functional
    [(int -> bool) -> bool] has a tree representation. *)

(* A tree representation of a functional. *)
type tree =
  | Answer of bool
  | Question of int * tree * tree

andrejbauer

(** 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