The Hitchiker's Guide to the Curry-Howard Correspondence

Functions are proofs.

Um, I can see how that might work. Go on.

Types are propositions. Really? In what sense?

In fact, a function is the proof of the proposition its type represents.

Woah, you've lost me now.

Don't Panic!

The Curry-Howard Correspondence is an elegant bridge between the planet of logic and the planet of programming, and it's not actually that hard to understand.

In this talk I'll use the Idris dependently-typed functional programming language for examples, as its type system is sophisticated enough to construct interesting automated proofs simply by writing functions. This talk is not designed to convert you into a theoretical computer scientist, but to share with you a wonderful sight in your journey through the vast and peculiar universe of programming.

A familiarity with functional programming would be useful for appreciating this talk, but it will not require any significant prior study of theoretical computer science.

Transcript

1. The Hitchhiker's guide to the Curry-Howard correspondence Chris Ford (@ctford) ThoughtWorks
2. Number of papers published today by the foremost expert on the Curry-Howard correspondence...
3. Number of papers published today by the foremost expert on the Curry-Howard correspondence... 1
4. Don't panic!
5. a→a
6. a→a identity : a ­> a identity x = x
7. Idris> identity “Bruce Durling” “Bruce Durling” : String Idris> identity 3 3 : Integer Idris> identity String String : Type
8. Haskell Curry William A. Howard
10. (a→b)→a→b
11. (a→b)→a→b apply : (a ­> b) ­> a ­> b apply f x = f x
12. chr : Integer ­> Char Idris> apply chr 65 'A' : Char (3==) : Integer ­> Bool Idris> apply (3==) 4 False : Bool
13. (a→b)→(b→c)→(a→c)
14. (a→b)→(b→c)→(a→c) comp : (a­>b) ­> (b­>c) ­> (a­>c) comp f g x = g (f x)
15. length : List a ­> Integer (3==) : Integer ­> Bool Idris> comp length (3==) : List a ­> Bool
16. a→a (a→b)→a→b a→b→(a, b) (a, b) → b a→b (a→b)→(b→a)
17. ⊥
18. ⊥ AnythingGoes : Type AnythingGoes = (a : Type) ­> a cantProveItAll : AnythingGoes ­> _|_ cantProveItAll f = f _|_
19. Harmless.
20. NullPointerException council.office.Lock.acquire (Lock.java:42) council.office.FilingCabinet.find (FilingCabinet.java:42) council.office.Leopard.beware (Leopard.java:42) council.office.Lavatory.find (Lavatory.java:42) council.office.Cellar.find (Cellar.java:42) council.office.Consultation.post (Consultation.java:42) council.policy.Bypass.plan (Bypass.java:42) council.policy.ExpansionManager.execute (ExpansionManager.java:42) council.policy.Budget.spend (Budget.java:42)
21. Mostly harmless.
22. Haskell> head [] *** Exception: Prelude.head: *** empty list
23. Haskell> last [1..] *** Interrupted.
24. Edwin Brady, creator of Idris
25. Edwin Brady, creator of Idris and Whitespace
26. Type Nat = Z or (S Nat)
27. Type Nat = Z or (S Nat) Type List = [] or (x :: List)
28. Type Nat = Z or (S Nat) Type List = [] or (x :: List) data Vect : Nat ­> Type ­> Type where Nil : Vect Z a (::) : (x : a) ­> (xs : Vect n a) ­> Vect (S n) a
29. List : Type ­> Type [3, 4] : List Integer Vect : Nat ­> Type ­> Type [3, 4] : Vect 2 Integer
30. head : Vect (S n) a ­> a head (x::_) = x
31. head : Vect (S n) a ­> a head (x::_) = x Idris> head [] Can't unify Vect 0 a with Vect (S n) iType
32. concat : Vect m a ­> Vect n a ­> Vect (m + n) a concat [] ys = ys concat (x::xs) ys = concat xs ys
33. concat : Vect m a ­> Vect n a ­> Vect (m + n) a concat [] ys = ys concat (x::xs) ys = x::concat xs ys
34. sort : Ord a => Vect m a ­> Vect m a
35. data Even : Nat ­> Type where Zero : Even Z Next : Even n ­> Even (S (S n)) Zero : Even Z Next (Next (Next Zero)) : Even 6
36. total add : Even m ­> Even n ­> Even (m + n) add Zero y = y add (Next x) y = Next (add x y)
37. total add : Multiple m a ­> Multiple m b ­> Multiple m (a + b) add (NoneOf _) y = y add (Next x) y ?= add x (Next y) Main.add_lemma_1 = proof intros rewrite plusAssociative n m b rewrite sym $ plusCommutative m b trivial
38. total fortyTwoIsEven : Even 42 fortyTwoIsEven = mul 21 (Next Zero) where mul : (n:Nat) ­> Even m ­> Even (n*m) mul Z _ = Zero mul (S n) e = add e (mul n e)
39. Even 3
40. Even 3 Even 3→⊥
41. total threeAintEven : Even 3 ­> _|_ threeAintEven (Next e) with (e) | (Next _) impossible | Zero impossible
42. ()
43. LifeTheUniverseAndEverything → Even 42
44. References Edwin Brady Programming in Idris: A Tutorial idris-lang.org Brian McKenna EvenOdd in Agda, Idris, Haskell, Scala brianmckenna.org Philip Wadler Propositions as Types – updated today! wadler.blogspot.co.uk