This document provides a summary of the Curry-Howard correspondence in practice. It includes 3 code examples demonstrating issues that can arise when combining traits in Scala in unintended ways, resulting in runtime exceptions. It also includes links to online resources about functional programming and proving sorting algorithms correct using proof assistants.

1. Curry-Howard w praktyce
Jan Pustelnik
Segfault Wrocław 2018

2. Dwa słowa o mnie...
●
Twitter & github: gosubpl
●
Zawód wyuczony: matematyk
●
Zawód wykonywany: Softare Engineer @ Actyx

10. Photo by Nathon Oski
on Unsplash

16. Źródło: Wikipedia

17. Źródło: Wikipedia

30. def f[X, Y]: X ⇒ Y ⇒ X = implement
println(f(123)("abc"))
123
https://github.com/Chymyst/curryhotard/blob/master/docs/Tutorial.md

33. object world extends App {
trait A { type L <: Nothing }
trait B { type L >: Any}
def toL(b: B)(x: Any): b.L = x
val p: B with A = null
// we can create a value of type Nothing
println(toL(p)("hello"): Nothing)
// at runtime: java.lang.ClassCastException: java.lang.String cannot be cast
to scala.runtime.Nothing$
// variation on a theme...
def cast[T,U](x: T): U = toL(p)(x)
println(cast[Int,String](1) + "hello")
// at runtime: java.lang.ClassCastException: java.lang.Integer cannot be
cast to java.lang.String
println(cast[String,Int => Int]("hello")(1))
// at runtime: java.lang.ClassCastException: java.lang.String cannot be cast
to scala.Function1
}
https://issues.scala-lang.org/brotse/SI-9633

36. https://softarefoundations.cis.upenn.edu/current/vfa-
current/Sort.html

41. (** val insert : nat -> nat list -> nat list **)
let rec insert i = function
| Nil -> Cons (i, Nil)
| Cons (h, t) ->
(match Nat.leb i h with
| True -> Cons (i, (Cons (h, t)))
| False -> Cons (h, (insert i t)))
(** val insertion_sort : nat list -> nat list **)
let insertion_sort =
fold_right insert Nil
https://osa1.net/posts/2014-09-08-proving-sorting-correct.html