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Andrej Bauer andrejbauer

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andrejbauer / finite.v
Created Feb 5, 2021
The finite truth values are precisely the decidable truth values.
View finite.v
(** 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 / doubleNegationInitialAlgebra.agda
Created Jan 4, 2021
The initial algebra for the functor X ↦ (X → ∅) → ∅ is ∅.
View doubleNegationInitialAlgebra.agda
-- empty type
data 𝟘 : Set where
absurd : {A : Set} 𝟘 A
absurd ()
-- booleans
data 𝟚 : Set where
false true : 𝟚
andrejbauer / finite_subsets_and_LEM.v
Created Nov 15, 2020
The proof that excluded middle follows if all subsets of a singleton set are finite.
View finite_subsets_and_LEM.v
(* 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 / UniverseInjection.agda
Last active Oct 11, 2020
Formalization of the fact that a dependent type theory with excluded middle cannot have a universe Set such that Set → Set injects into Set.
View UniverseInjection.agda
-- 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 / 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 / ConstructibleNumbers.nb
Last active Sep 25, 2020
Generation of constructible numbers in Mathematica
View ConstructibleNumbers.nb
(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* *)
(* CreatedBy='Mathematica 11.2' *)
(*CacheID: 234*)
(* Internal cache information:
andrejbauer /
Created Aug 14, 2019
Pictures of algebraic numbers
# 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 /
Last active May 2, 2018
Experiments in using multicore OCaml effects to simulate dynamically created local effects.
(** * 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 ( It was written by
Andrej Bauer, Oleg Kiselyov, and Stephen Dolan at the Dagstuhl seminar
"Algebraic Effect Handlers go Mainstream". *)
andrejbauer / functional2tree.eff
Created Apr 19, 2018
Use of effects and handlers to compute the tree representation of a functional.
View functional2tree.eff
(** This code is compatible with Eff 5.0, see *)
(** 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 / 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