# type-settheory: Type-level sets and functions expressed as types

[ bsd3, language, library, math, type-system ] [ Propose Tags ]

Type classes can express sets and functions on the type level, but they are not first-class citizens. Here we take the approach of expressing type-level sets and functions as types. The instance system is replaced by value-level proofs which can be directly manipulated. In this way the Haskell type level can support a quite expressive constructive set theory; for example, we have:

• Subsets and extensional set equality

• Unions (binary or of sets of sets), intersections, cartesian products, powersets, and a kind of dependent sum and product

• Functions and their composition, images, preimages, injectivity

The meaning of the proposition-types here is not purely by convention; it is actually grounded in GHC "reality": A proof of A :=: B gives us a safe coercion operator A -> B (while the logic is inconsistent at compile-time due to the fact that Haskell has general recursion, we still have that proofs of falsities are undefined or non-terminating programs, so for example if Refl is successfully pattern-matched, the proof must have been correct).

Versions 0.1, 0.1.1, 0.1.2, 0.1.3, 0.1.3.1 base (==4.*), containers, mtl, syb, template-haskell, type-equality [details] BSD-3-Clause Daniel Schüssler daniels@community.haskell.org Math, Language head: darcs get http://code.haskell.org/~daniels/type-settheory by DanielSchuessler at Tue Oct 27 18:53:41 UTC 2009 NixOS:0.1.3.1 1696 total (12 in the last 30 days) (no votes yet) [estimated by rule of succession] λ λ λ Docs uploaded by userBuild status unknown Hackage Matrix CI

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