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Your idea of types is similar to Predicate Dispatch[1]. I believe it's also similar to Racket's contracts[2].
This is an extremely powerful type system in the sense that you can express literally anything with it. However, I believe that a more restrictive type system can actually be better, because it provides more guarantees (especially at compile-time).
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I don't think it's possible to pattern match on a function's arguments, body, and closure, simply because Racket doesn't expose those things.
