Parametricity @parametricity
Theorems for Free! homepages.inf.ed.ac.uk/wadler/topics/… Parametrikistan Joined January 2015-
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I wonder what this function does. flip :: Functor f => f (a -> b) -> a -> f b Wait, no I don't.
f :: Lens a b -> a -> b -> a This does one of two things. ∀ l x y. x /= y => f l x y x /= x Now I know which.
@rightfold is thinking of a function with the type (a -> b) -> a -> b The identity function specialized to functions.
@IlanGodik @StewOConnor Unfortunately, no reasonable way to fit the full premise of the conclusion in 140.
length :: Foldable f => f a -> Int ∴ length (1, '2', "3") must return 1.
@Tuplet @chris__martin @puffnfresh Fast & Loose Reasoning is all about discarding ⊥ and ∴ head.
@Tuplet @chris__martin @puffnfresh What is this thing, head? Remember, no ⊥. Fast & Loose Reasoning.
Generalize an abstraction because removing nonessential information clarifies its independent meaning. Re-use is accidental. @fronx
@TheUwap There are (Integer ^ Integer) implementations of Integer -> Integer.
a -> (a, a, a, a) I definitely know what this function does. (a, a, a, a) -> a I almost know what this function does.
@catnaroek Fast & Loose Reasoning is Morally Correct.
f :: Monad f=>f T -> f T The result might depend on previous effect values g :: Applicative f=>f T -> f T This definitely doesn't though
Proof `fmap f.fmap g=fmap(f.g)` follows from `fmap id=id` and the free theorem: fpcomplete.com/user/edwardk/s… /cc @BartoszMilewski @parametricity
Using Data.Foldable and parametricity, one can be sure that: foldMap f &&& foldMap g = foldMap (f &&& g)
@mwotton @st58 @Symbo1ics You are correct. With the given information, reverse is the only candidate.
@mwotton @st58 @Symbo1ics The type guarantees f [] = []
f :: [a] -> [a] This type tells us some, but not all, things. ∀ x. f [x] = [x] ∀ x y. f (x++y) = f y++f x These tests tell us the rest.
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