# Curry-Howard-Lambek correspondence

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== The Answer == | == The Answer == | ||

As is well established by now, | As is well established by now, | ||

− | |||

− | |||

<haskell>theAnswer :: Integer | <haskell>theAnswer :: Integer | ||

theAnswer = 42</haskell> | theAnswer = 42</haskell> | ||

+ | The logical interpretation of the program is that the type <hask>Integer</hask> is inhibited (by the value <hask>42</hask>), so the existence of this program ''proves'' the proposition <hask>Integer</hask> (a type without any value is the "bottom" type, a proposition with no proof). | ||

− | The | + | == Theorems for free! == |

+ | Things get interesting when polymorphism comes in. The composition operator in Haskell proves a very simple theorem. | ||

+ | <haskell>(.) :: (a -> b) -> (b -> c) -> (a -> c) | ||

+ | (.) f g x = f (g x)</haskell> | ||

+ | The type is, actually, <hask>forall a b c. (a -> b) -> (b -> c) -> (a -> c)</hask>, to be a bit verbose, which says, logically speaking, for all propositions <hask>a, b</hask> and <hask>c</hask>, if from <hask>a</hask>, <hask>b</hask> can be proven, and if from <hask>b</hask>, <hask>c</hask> can be proven, then from <hask>a</hask>, <hask>c</hask> can be proven (the program says how to go about proving: just compose the given proofs!) |

## Revision as of 03:08, 2 November 2006

**Curry-Howard Isomorphism** is an isomorphism between types (in programming languages) and propositions (in logic). Interestingly, the isomorphism maps programs (functions in Haskell) to (constructive) proofs (and *vice versa*).

## Contents |

## 1 The Answer

As is well established by now,

theAnswer :: Integer theAnswer = 42

Integer

42

*proves*the proposition

Integer

## 2 Theorems for free!

Things get interesting when polymorphism comes in. The composition operator in Haskell proves a very simple theorem.

(.) :: (a -> b) -> (b -> c) -> (a -> c) (.) f g x = f (g x)

forall a b c. (a -> b) -> (b -> c) -> (a -> c)

a, b

c

a

b

b

c

a

c