Proofs are programs. Formal verification of programs is a huge research area. (See, for example, the group at Carnegie Mellon.) Many complex programs have been verified; for example, see this kernel written in Haskell (repaired 404 link is for seL4, see also the moved to location and the project’s website).
I’ve only scanned through the paper, but here’s a rough summary of how it all hangs together. From page 86 of the paper. … polynomial time algorithms succeed by successively “breaking up” the problem into smaller subproblems that are joined to each other through conditional independence. Consequently, polynomial time algorithms cannot solve problems in regimes … Read more
@~ is coercion application. The angle brackets denote a reflexive coercion of their contained type with role given by the underscored letter. Thus <Nat.Flip x_ap0H>_N is an equality proof that Nat.Flip x_apH is equal to Nat.Flip x_apH nominally (as equal types not just equal representations). PS has a lot of arguments. We look at the … Read more
The pumping lemma is a simple proof to show that a language is not regular, meaning that a Finite State Machine cannot be built for it. The canonical example is the language (a^n)(b^n). This is the simple language which is just any number of as, followed by the same number of bs. So the strings … Read more
Consider this monad which is isomorphic to the (Bool ->) monad: data Pair a = P a a instance Functor Pair where fmap f (P x y) = P (f x) (f y) instance Monad Pair where return x = P x x P a b >>= f = P x y where P x … Read more