The Hitchiker's Guide to the Curry-Howard Correspondence

Chris Ford (@ctford)

Don't panic!

Types are propositions.

Construction is proof.

“Arthur Dent” : String
7 : Integer
String : Type
not : Bool -> Bool
(“foo”, 23) : (String, Integer)
Left 9 : Either Integer b

Values and types

identity : a -> a
identity x = x

a→a

Haskell Curry

William Howard

(a→b)→a→b

apply : (a -> b) -> a -> b
apply f x = f x
Idris> apply chr 65
'A' : Char

Application / implication

Idris> apply chr 66
'B' : Char
chr : Integer -> Char

(a→b)→(b→c)→(a→c)

comp : (a->b) -> (b->c) -> (a->c)
comp f g x = g (f x)
Idris> comp length (7==)
_ : List a -> Bool
length : List a -> Integer
(==3) : Integer -> Bool

Composition / modus ponens

a ∧ b

conjunction : a -> b -> (a, b)
conjunction x y = (x, y)

a ∨ b

disjunctionl : a -> Either a b
disjunctionl x = Left x
disjunctionr : b -> Either a b
disjunctionr y = Right y

a→a
(a→b)→a→b

a→b→(a, b)
(a, b) → b

a→b
(a→b)→(b→a)

Not : Type -> Type
Not p = p -> Falsity
Falsity : Type
Falsity = Void

⊥

second : Not (Either a b) -> (Not a, Not b) 
second f = (f . Left, f . Right)
first : Either (Not a) (Not b) -> Not (a, b) 
first (Left f) (x, _) = f x
first (Right g) (_, y) = g y
secondInverse : (Not a, Not b) -> Not (Either a b)
secondInverse (f, _) (Left x) = f x
secondInverse (_, g) (Right y) = g y

De Morgan's Laws

EverythingIsTrue : Type
EverythingIsTrue = (a : Type) -> a
CantProveEverything : Not EverythingIsTrue
CantProveEverything f = f Falsity

Not all propositions

Harmless

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)

Mostly harmless

Haskell> head []

*** Exception: Prelude.head:
*** empty list
Haskell> last [1..]

*** Interrupted.

Partial function

head : Vect (n + 1) a -> a
head (x::_) = x
Idris> head []
Can't unify Vect 0 a
with Vect (n + 1) iType

Total function

oops : a -> b
oops x = oops x

"Proving" absurdity

data Even : Nat -> Type where
  Zero : Even 0
  Next : Even n -> Even (n + 2)
Zero : Even 0
Next (Next (Next Zero)) : Even 6

Even naturals

add : Even m -> Even n -> Even (m + n)
add Zero y = y
add (Next x) y = Next (add x y)
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)

A provably even number

Even 3

¬Even 3

threeAintEven : Not (Even 3)
threeAintEven (Next e) with (e)
  | (Next _) impossible
  | Zero impossible

()

LifeTheUniverseAndEverything

→

Even 42

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

wadler.blogspot.co.uk

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