muri takes a Haskell type, and generates a Haskell term with that type (or a more general one), if possible.

Equivalently, under the propositions-as-types correspondence, muri is a theorem prover for intuitionistic propositional logic. It takes a proposition as input, and produces a proof of that proposition, if one exists.

Input types are limited to type variables, product types (a, b), sum types Either a b and function types a -> b.

Output terms are built using lambda abstraction, function application, let and case expressions, pair construction and the Left and Right type constructors.

muri was inspired by Lennart Augustsson's Djinn, and uses the LJT calculus from "Contraction-Free Sequent Calculi for Intuitionistic Logic" by Roy Dyckhoff to ensure termination.

muri TYPE

$ muri 'a -> (b, c) -> (b, (a, c))'
\a -> \(b, c) -> (b, (a, c))

$ muri '(a, b) -> Either a b'
\(a, b) -> Left a

$ muri 'Either (a, c) (b, c) -> (Either a b, c)'
\a -> case a of { Left (b, c) -> (Left b, c); Right (b, c) -> (Right b, c) }

$ muri '(Either (a -> f) a -> f) -> f'
\a -> a (Left (\c -> a (Right c)))

$ muri '(a -> b) -> (Either (a -> f) b -> f) -> f'
\a -> \b -> b (Left (\c -> b (Right (a c))))

$ muri '(Either (a -> f) (b -> f) -> f) -> ((a, b) -> f) -> f'
\a -> \b -> a (Left (\e -> a (Right (\f -> b (e, f)))))

$ muri '((a -> b) -> c) -> ((a, b -> c) -> b) -> c'
\a -> \b -> a (\f -> b (f, \e -> a (\_ -> e)))

$ muri 'a -> b'
Impossible.