When I’m writing Haskell, quite often I come across type signatures that are just a mishmash of types. For example, take the definitely not made up function foo:

foo :: a -> b -> c -> (b, d -> c)

My go-to strategy for filling out the implementation of a function like this is to just start by plopping down a hole, and refining down the solution. However, this process feels incredibly mechanical. Almost so mechanical that it could be programmed…

Let’s try to give our mechanical process a type. First, we know it’s going to be a function that starts when given a type. After all, what is a process but a function?

type MechanicalProcess goal = goal -> ???

Next, we know that at some points, we are going to need to make some subgoals. For example, consider trying to fill in a hole _h :: (a, b). The expression you create will look something like (_h1 :: a, _h2 :: b), which obviously yields 2 subgoals.

type MechanicalProcess goal = goal -> ([goal], ???)

Now, what do we do with our solutions? Well, all we need is a way to map the solutions of all of the subgoals into a single solution. This gives us the final type:

type MechanicalProcess g solution = g -> ([g], [solution] -> solution)

Finally, MechanicalProcess is a terrible name, so lets give it a better one! Let’s call it a Tactic, because this seems essentially like a “tactic” for filling out implementations for types!

newtype Tactic g s = Tactic { unTactic :: g -> ([g], [s] -> s) }

## Some Simple Tactics

From here on out, we will be using the refinery library, because our Tactic type has a few issues that make it not super friendly to work with. Don’t worry though, the refinery version of Tactic is morally equivalent, though it does have a few extra features. Let’s take a look at it.

newtype TacticT g s m a = ...

It still has the goal and solution type parameters, but now it’s been turned into a monad transformer! This lets us handle things like errors/environments/etc much easier. Also, the a parameter is pretty much just there to make the Monad gods happy, and is set to () all of the time.

When discussing how to actually use these things, it will help to be able to look at actual, concrete examples. To do this, let’s define a super simple subset of Haskell so that we can all be on the same page.

import Control.Monad.Except
import Data.List (find)
import Refinery.Tactic

type Var = String
data Expr
= Var Var
| Hole      -- ^ An incomplete expression
| Lambda Var Expr
| App Expr Expr
| Pair Expr Expr
| ProjL Expr -- ^ Equivalent to fst
| ProjR Expr -- ^ Equivalent to snd
deriving Show

data Type
= TVar Var
| Type :-> Type -- ^ Function type
| Type :*: Type -- ^ Product/Pair type
deriving (Show, Eq)

infixr :->

-- A Goal is a context containing variables bound to types,
-- along with a type that we are trying to fill.
data Goal = Goal [(Var, Type)] Type
deriving (Show)

-- We are gonna need errors
data TacticError
= GoalMismatch String Type
| UnsolvedSubgoals [Goal]
deriving (Show)

type Tactic = TacticT Goal Expr (StateT Int (Except TacticError)) ()
-- To be discussed shortly
type Rule a = RuleT Goal Expr (StateT Int (Except TacticError)) a

-- Create a fresh variable
fresh :: Rule Var
fresh = do
i <- get
modify (+1)
return $"x" ++ show i -- Sometimes, we need a monad to make holes, but in this case we don't instance MonadExtract Expr Identity where hole = return$ Hole

runTactic :: Type -> Tactic -> Either TacticError Expr
runTactic ty tac = runExcept $flip evalStateT 0$ runTacticT tac (Goal [] ty) >>= \case
(t, []) -> return t
(_, sg) -> throwError $UnsolvedSubgoals sg Now, lets define a few super simple Tactics so that we can play around a bit! First, let’s implement the pair tactic that takes a goal of type (a,b) and yields 2 subgoals of type a and b. In refinery, we use the rule function to create these sort of domain-specific tactics. It’s type (when specialized to our case) is rule :: Goal -> Rule Expr -> Tactic Now, here’s the implementation of pair pair :: Tactic pair = rule$ \(Goal ctx t) -> case t of
(a :*: b) -> do
l <- subgoal (Goal ctx a)
r <- subgoal (Goal ctx b)
return $Pair l r t -> throwError$ GoalMismatch "pair" t

Pretty straightforward, isn’t it! As an aside, all the Rule/RuleT portion of rule does is give us access to the subgoal :: Goal -> Rule Expr function.

As a slightly more complex example, heres the function tactic:

function :: Tactic
function = rule $\(Goal ctx t) -> case t of (a :-> b) -> do x <- fresh body <- subgoal (Goal ((x,a):ctx) b) return$ Lambda x body
t -> throwError $GoalMismatch "function" t This is also super simple! All this does is bind the fresh variable x to the type a in the context, and create a subgoal of type b. Now, for one last tactic: assumption. assumption :: Tactic assumption = rule$ \(Goal ctx t) -> case find ((== t) . snd) ctx of
Just (x, _) -> return $Var x Nothing -> throwError$ GoalMismatch "assumption" t

All assumption does is see if any tactic has put anything in the context that could match the goal, and if there is, it just creates an expression that references that variable, along with no new subgoals.

## Composing Tactics and Tactic Combinators

Now that we’ve got our simple tactics, we can start to stick them together! The simplest way to do this is to compose 2 Tactics using >>. Doing so creates a new Tactic that uses the 2nd Tactic to solve the subgoals of the 1st. For example:

pair >> pair

This will create a Tactic that, when presented with a goal of type ((a,b), (c,d)), will create 4 subgoals a, b, c, and d.

As useful as this is, what if we have a goal that looks like ((a,c), d -> d)? In this case, we can use:

<@> :: Tactic -> [Tactic] -> Tactic

This combinator takes a tactic, and a list of tactics to apply to the subgoals in order. In our example, the tactic we want would be:

pair <@> [pair, function]

Now, this is great and all, but we’ve just traded in 1 way to write programs for a slightly more complicated way to write programs. What we really want is some tactic auto that can just solve our problems for us. To accomplish this, we are going to need to introduct 1 more combinator:

<!> :: Tactic -> Tactic -> Tactic

This combinator will try to run the 1st tactic, and, if it fails, will run the 2nd tactic. With this power, we can finally write our auto tactic:

auto :: Tactic
auto = do
function <!> pair <!> assumption
auto

That’s all there is to it! Let’s try it out on our function foo from the very beginning!

λ runTactic (TVar "a" :-> TVar "b" :-> TVar "c" :-> (TVar "b" :*: (TVar "d" :->
TVar "c"))) auto

Right (Lambda "x0" (Lambda "x1" (Lambda "x2" (Pair (Var "x1") (Lambda "x3" (Var "x2"))))))

Prettying that up a bit, we can see that \x0 x1 x2 -> (x1, \x3 -> x2) really does have the type we want!

## Extending The Horizons

Obviously, there is a ton more work required to make this fully general. However, I’ve started that work in tactic-haskell, which uses this approach to work with actual Haskell types, and to generate actual Haskell code. In the future, I plan on writing a bit on the actual guts of refinery itself, and perhaps a tutorial on tactic-haskell itself once it matures a bit more. Until then, happy hacking!