Tic Tac Type: Dependent Types with Idris

Functional programming provides a fundamental basis for reasoning about our programs and building out principled abstractions. However, it is not enough on its own and we also rely on other programming tools to aid in constructing programs correctly and efficiently. An advanced example of such a tool is dependent types; that is, types that depend on values. Dependent types can be used to provide strong guarantees about a program's behaviour that are difficult or impossible to express with traditional type-systems.

Dependent types are gaining traction in mainstream FP languages such as Haskell (via language extensions) and Scala (via path-dependent sub-typing), but disappointingly, they are still less accessible to everyday programming than they can and should be. To help address this problem, Idris is a new programming language built from the ground up with the explicit goal of having better support for dependent types (as well as a number of other useful tools such as totality checking and tactic based theorem proving).

This talk will step through an end-to-end example in Idris, from fundamentals like building type-safe APIs, and performing IO through to building user interfaces. By the end of the talk we will have identified the strengths of this programming model, the practicalities of using them in Idris (in contrast to Scala and Haskell) and produced a battle-hardened, multi-player, tic-tac-toe game ready for production.

Transcript

1. Tic
   @markhibberd
   Type
   Tac
   

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2. View Slide

3. Data is validated with respect to other data
   !
   If types are to capture valid data precisely,
   we must let them depend on terms”
   - Conor McBride, Winging It
   “
   

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4. View Slide

5. IDRIS
   a primer
   

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6. haskell like syntax
   

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7. but…
   types and values co-exist in the
   same term language
   

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8. idris
   haskell
   !
   1 infixr 7 :.
   2
   3 data List a
   4 = Nil
   5 | a :. List a
   6
   7 headOr :: a -> List a -> a
   8 headOr x Nil = x
   9 headOr _ (h :. _) = h
   10
   11 type String =
   12 List Char
   1 infixr 7 :.
   2
   3 data List a
   4 = Nil
   5 | (:.) a (List a)
   6
   7 headOr : a -> List a -> a
   8 headOr x Nil = x
   9 headOr _ (h :. _) = h
   10
   11 String : Type
   12 String = List Char
   

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9. idris
   haskell
   !
   1 infixr 7 :.
   2
   3 data List a
   4 = Nil
   5 | a :. List a
   6
   7 headOr :: a -> List a -> a
   8 headOr x Nil = x
   9 headOr _ (h :. _) = h
   10
   11 type String =
   12 List Char
   1 infixr 7 :.
   2
   3 data List a
   4 = Nil
   5 | (:.) a (List a)
   6
   7 headOr : {a: Type} -> (x: a)
   -> (xs: List a) -> a
   8 headOr {a} x Nil = x
   9 headOr {a} _ (h :. _) = h
   10
   11 String : Type
   12 String = List Char
   

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10. idris
    haskell
    !
    1 data Nat
    2 = Z
    3 | S Nat
    !
    1 data Nat
    2 = Z
    3 | S Nat
    

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11. idris
    haskell
    !
    1 {-# LANGUAGE GADTs #-}
    2
    3 data Nat where
    4 Z :: Nat
    5 S :: Nat -> Nat
    !
    1 data Nat : Type where
    2 Z : Nat
    3 S : Nat -> Nat
    !
    !
    

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12. idris
    haskell
    !
    1 {-# LANGUAGE GADTs #-}
    2 {-# LANGUAGE KindSignatures #-}
    3
    4 data Nat :: * where
    5 Z :: Nat
    6 S :: Nat -> Nat
    !
    1 data Nat : Type where
    2 Z : Nat
    3 S : Nat -> Nat
    !
    !
    !
    

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13. idris
    haskell
    1 {-# LANGUAGE GADTs #-}
    2 {-# LANGUAGE KindSignatures #-}
    3
    4 data Z
    5 data S a = S a
    6
    7 data Nat a :: * -> * where
    8 Zero :: Nat Z
    9 Succ :: Nat n -> Nat (S n)
    10
    11 data Fin n :: * -> * where
    12 FZero :: Nat n -> Fin (S n)
    13 FSucc :: Nat n -> Fin n
    14 -> Fin (S n)
    !
    1 data Nat : Type where
    2 Z : Nat
    3 S : Nat -> Nat
    4
    5 data Fin : Nat -> Type where
    6 fZ : Fin (S k)
    7 fS : Fin k -> Fin (S k)
    !
    !
    !
    !
    !
    !
    !
    

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14. idris
    haskell
    !
    1 {-# LANGUAGE GADTs #-}
    2 {-# LANGUAGE DataKinds #-}
    3 {-# LANGUAGE KindSignatures #-}
    4
    5 data Nat :: * where
    6 Z :: Nat
    7 S :: Nat -> Nat
    8
    9 data Fin :: Nat -> * where
    10 FZ :: Fin Z
    11 FS :: Fin n -> Fin (S n)
    !
    1 data Nat : Type where
    2 Z : Nat
    3 S : Nat -> Nat
    4
    5 data Fin : Nat -> Type where
    6 fZ : Fin (S k)
    7 fS : Fin k -> Fin (S k)
    !
    !
    !
    !
    

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15. “Stick a number in my type. What is that?”
    - Manuel Chakravarty
    

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16. !
    1 data Vect : Nat -> Type -> Type where
    2 Nil : Vect Z a
    3 (::) : (x : a) -> (xs : Vect n a) -> Vect (S n) a
    !
    !
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17. !
    1 data Vect : Nat -> Type -> Type where
    2 Nil : Vect Z a
    3 (::) : (x : a) -> (xs : Vect n a) -> Vect (S n) a
    4
    5 v2 : Vect 2 String
    6 v2 = "hello" :: "idris" :: Nil
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18. 1 data Vect : Nat -> Type -> Type where
    2 Nil : Vect Z a
    3 (::) : (x : a) -> (xs : Vect n a) -> Vect (S n) a
    4
    5 v2 : Vect 2 String
    6 v2 = "hello" :: "idris" :: Nil
    7
    8 v3 : Vect ?len String
    9 v3 = "tic" :: "tac" :: "type" :: Nil
    !
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19. 1 data Vect : Nat -> Type -> Type where
    2 Nil : Vect Z a
    3 (::) : (x : a) -> (xs : Vect n a) -> Vect (S n) a
    4
    5 v2 : Vect 2 String
    6 v2 = "hello" :: "idris" :: Nil
    7
    8 v3 : Vect ?len String
    9 v3 = "tic" :: "tac" :: "type" :: Nil
    10
    11 len = proof search
    !
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20. !
    1 data Vect : Nat -> Type -> Type where
    2 Nil : Vect Z a
    3 (::) : (x : a) -> (xs : Vect n a) -> Vect (S n) a
    4
    5 index : Fin n -> Vect n a -> a
    6 index fZ (x::xs) = x
    7 index (fS k) (x::xs) = index k xs
    8 index fZ [] impossible
    !
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21. 1 (++) : Vect n a -> Vect m a -> Vect (n + m) a
    2 (++) Nil m = m
    3 (++) (h :: t) m = h :: (t ++ m)
    !
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22. 1 (++) : Vect n a -> Vect m a -> Vect (n + m) a
    2 (++) Nil m = m
    3 (++) (h :: t) m = h :: (t ++ m)
    4
    5 filter : (a -> Bool) -> Vect n a -> (p ** Vect p a)
    6 filter p [] = ( _ ** [] )
    7 filter p (x::xs) with (filter' p xs)
    8 | (n ** tail) =
    9 if p x then
    10 ((S n) ** x::tail)
    11 else
    12 (n ** tail)
    !
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23. what did I just see?
    

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24. dependent product types
    aka ∏ types
    

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25. dependent sum types
    aka Σ types
    

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26. Curry–Howard correspondence
    

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27. (Intuitionistic) Logic Programming
    FALSE ⊥
    TRUE ()
    Disjunction (or) Sum Types
    Conjunction (and) Product Types
    Implication Function Types
    

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28. (Predicate) Logic Programming
    FALSE ⊥
    TRUE ()
    Disjunction (or) Sum Types
    Conjunction (and) Product Types
    Implication Function Types
    Existential Quantification (exists.) Σ Types
    Universal Quantification (forall.) ∏ Types
    

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29. So how do we go about
    constructing programs?
    

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30. Start With Value Level Primitives
    

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31. 1 data Player = X | O
    2 data Cell = Occupied Player | Unoccupied
    3
    4 instance Show Player where {...}
    5 instance Eq Player where {...}
    6 instance Show Cell where {...}
    7 instance Eq Cell where {…}
    !
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32. north-east
    north-west north
    center east
    west
    south-east
    south-west south
    

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33. 1 Position : Type
    2 Position = Fin 9
    3
    4 nw : Position
    5 nw = 0
    6 {...}
    7 se : Position
    8 se = 8
    9
    10 parse : String -> Maybe Position
    11 parse "nw" = Just nw
    12 parse "n" = Just n
    13 parse "ne" = Just ne
    14 parse "e" = Just e
    15 parse "c" = Just c
    16 parse "w" = Just w
    17 parse "sw" = Just sw
    18 parse "s" = Just s
    19 parse "se" = Just se
    20 parse _ = Nothing
    

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34. 1 data Board =
    2 B (Vect 9 Cell)
    3
    4 instance Eq Board where {...}
    5 instance Show Board where {...}
    6
    7 -- How many positions are occupied ?
    8 occupied : Board -> Nat
    9
    10 -- Who's turn is it?
    11 turn : Board -> Player
    12
    13 -- Is this position free?
    14 free : Position -> Board -> Bool
    15
    16 -- Is there a winner on the board?
    17 winner : Board -> Maybe Player
    18
    19 -- Is this a valid board?
    20 isValidBoard : Board -> Bool
    

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35. Types and Proofs
    

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36. !
    1 data so : Bool -> Type where
    2 oh : so True
    !
    !
    !
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37. !
    1 data so : Bool -> Type where
    2 oh : so True
    3
    4 printIf3 : (n: Nat)
    5 -> {default oh prf : so (n == 3)}
    6 -> IO ()
    7 printIf3 _ =
    8 print "3s are good."
    !
    !
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38. *> :x printIf3 3
    "3s are good."
    !
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39. *> :x printIf3 3
    "3s are good."
    !
    *> :x printIf3 3 { prf = oh }
    "3s are good.”
    !
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40. *> :x printIf3 3
    "3s are good."
    !
    *> :x printIf3 3 { prf = oh }
    "3s are good.”
    !
    *> :x printIf3 2
    (input):1:13:When elaborating argument prf to
    function Main.printIf3:
    Can't unify
    IsJust (Just x)
    with
    IsJust (fromInteger 2 == (fromInteger 3))
    !
    Specifically:
    Can't unify
    True
    with
    False
    !
    

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41. 1 natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
    2 natToFin Z (S j) = Just fZ
    3 natToFin (S k) (S j) with (natToFin k j)
    4 | Just k' = Just (fS k')
    5 | Nothing = Nothing
    !
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42. 1 natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
    2 natToFin Z (S j) = Just fZ
    3 natToFin (S k) (S j) with (natToFin k j)
    4 | Just k' = Just (fS k')
    5 | Nothing = Nothing
    6
    7 data IsJust : Maybe a -> Type where
    8 ItIsJust : IsJust {a} (Just x)
    !
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43. 1 natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
    2 natToFin Z (S j) = Just fZ
    3 natToFin (S k) (S j) with (natToFin k j)
    4 | Just k' = Just (fS k')
    5 | Nothing = Nothing
    6
    7 data IsJust : Maybe a -> Type where
    8 ItIsJust : IsJust {a} (Just x)
    9
    10 fromNat : (x: Nat)
    11 -> {default ItIsJust
    12 prf : (IsJust (natToFin x n))}
    13 -> Fin n
    !
    !
    !
    !
    !
    !
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44. 1 natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
    2 natToFin Z (S j) = Just fZ
    3 natToFin (S k) (S j) with (natToFin k j)
    4 | Just k' = Just (fS k')
    5 | Nothing = Nothing
    6
    7 data IsJust : Maybe a -> Type where
    8 ItIsJust : IsJust {a} (Just x)
    9
    10 fromNat : (x: Nat)
    11 -> {default ItIsJust
    12 prf : (IsJust (natToFin x n))}
    13 -> Fin n
    14 fromNat {n} x {prf} with (natToFin x n)
    !
    !
    !
    !
    !
    

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45. 1 natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
    2 natToFin Z (S j) = Just fZ
    3 natToFin (S k) (S j) with (natToFin k j)
    4 | Just k' = Just (fS k')
    5 | Nothing = Nothing
    6
    7 data IsJust : Maybe a -> Type where
    8 ItIsJust : IsJust {a} (Just x)
    9
    10 fromNat : (x: Nat)
    11 -> {default ItIsJust
    12 prf : (IsJust (natToFin x n))}
    13 -> Fin n
    14 fromNat {n} x {prf} with (natToFin x n)
    15 fromNat {n} x {prf = ItIsJust} | Just y = y
    !
    !
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46. !
    1 *> fromNat 7
    2 Can't resolve type class Enum iType
    !
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47. !
    1 *> fromNat 7
    2 Can't resolve type class Enum iType
    3
    4 *FinProof> :t the
    5 Prelude.Basics.the : (a : Type) -> a -> a
    !
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48. !
    1 *> fromNat 7
    2 Can't resolve type class Enum iType
    3
    4 *FinProof> :t the
    5 Prelude.Basics.the : (a : Type) -> a -> a
    6
    7 *FinProof> the (Fin 10) (fromNat 7)
    8 fS (fS (fS (fS (fS (fS (fS fZ)))))) : Fin 10
    !
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49. !
    1 *> fromNat 7
    2 Can't resolve type class Enum iType
    3
    4 *FinProof> :t the
    5 Prelude.Basics.the : (a : Type) -> a -> a
    6
    7 *FinProof> the (Fin 10) (fromNat 7)
    8 fS (fS (fS (fS (fS (fS (fS fZ)))))) : Fin 10
    9
    10 *FinProof> the (Fin 10) (fromNat 10)
    11 (input):1:23:When elaborating argument prf
    12 Can't unify
    13 IsJust (Just x)
    14 with
    15 IsJust (natToFin 10 10)
    16
    17 Specifically:
    18 Can't unify
    19 Just x
    20 with
    21 Nothing
    

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50. 1 data ValidMove : Board -> Type where
    2 IsValidMove : Position -> Player
    3 -> (b : Board) -> ValidMove b
    !
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51. 1 data ValidMove : Board -> Type where
    2 IsValidMove : Position -> Player
    3 -> (b : Board) -> ValidMove b
    4
    5 tryValidMove : Position
    6 -> Player
    7 -> (b : Board)
    8 -> Maybe (ValidMove b)
    !
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52. 1 data ValidMove : Board -> Type where
    2 IsValidMove : Position -> Player
    3 -> (b : Board) -> ValidMove b
    4
    5 tryValidMove : Position
    6 -> Player
    7 -> (b : Board)
    8 -> Maybe (ValidMove b)
    9
    10 validMove : (pos : Position)
    11 -> (p : Player)
    12 -> (b : Board)
    13 -> {default ItIsJust
    14 prf : IsJust (tryValidMove pos p b)}
    15 -> ValidMove
    !
    !
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53. 1 data ValidMove : Board -> Type where
    2 IsValidMove : Position -> Player
    3 -> (b : Board) -> ValidMove b
    4
    5 tryValidMove : Position
    6 -> Player
    7 -> (b : Board)
    8 -> Maybe (ValidMove b)
    9
    10 validMove : (pos : Position)
    11 -> (p : Player)
    12 -> (b : Board)
    13 -> {default ItIsJust
    14 prf : IsJust (tryValidMove pos p b)}
    15 -> ValidMove
    16 validMove pos p b {prf} with (tryValidMove pos p b)
    !
    !
    !
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54. 1 data ValidMove : Board -> Type where
    2 IsValidMove : Position -> Player
    3 -> (b : Board) -> ValidMove b
    4
    5 tryValidMove : Position
    6 -> Player
    7 -> (b : Board)
    8 -> Maybe (ValidMove b)
    9
    10 validMove : (pos : Position)
    11 -> (p : Player)
    12 -> (b : Board)
    13 -> {default ItIsJust
    14 prf : IsJust (tryValidMove pos p b)}
    15 -> ValidMove
    16 validMove pos p b {prf} with (tryValidMove pos p b)
    17 validMove pos p b {prf = ItIsJust} | Just y = y
    !
    !
    !
    

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55. Break on Through to the Other Side
    

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56. 1 data Game : Board -> Type where
    2
    3 -- start a new game with an empty board
    4 start : Game empty
    5
    6 -- make a (guaranteed to be valid) move
    7 move' : {b : Board} -> (m : ValidMove b)
    8 -> Game b -> Game (runMove m)
    !
    !
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57. !
    1 -- Make a valid move
    2 runMove : ValidMove board -> Board
    3 runMove (IsValidMove position player (B board)) =
    4 B $ replaceAt position (Occupied player) board
    5
    !
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58. !
    1 -- Make a valid move
    2 runMove : ValidMove board -> Board
    3 runMove (IsValidMove position player (B board)) =
    4 B $ replaceAt position (Occupied player) board
    5
    6 -- Make a valid move if you can prove it
    7 move : {b : Board}
    8 -> (pos : Position)
    9 -> (p : Player)
    10 -> (g : Game board)
    11 -> {default ItIsJust
    12 prf : (IsJust (tryValidMove pos p b))}
    13 -> Game (runMove $ validMove pos p b {prf})
    !
    !
    !
    !
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59. !
    1 -- Make a valid move
    2 runMove : ValidMove board -> Board
    3 runMove (IsValidMove position player (B board)) =
    4 B $ replaceAt position (Occupied player) board
    5
    6 -- Make a valid move if you can prove it
    7 move : {b : Board}
    8 -> (pos : Position)
    9 -> (p : Player)
    10 -> (g : Game board)
    11 -> {default ItIsJust
    12 prf : (IsJust (tryValidMove pos p b))}
    13 -> Game (runMove $ validMove pos p b {prf})
    14 move {b} pos p g {prf} =
    15 move' (validMove pos p b {prf}) game
    !
    !
    !
    !
    !
    

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60. 1 data Prev : Game b -> Type where
    2 HasPrev : {bb : Board}
    3 -> {m : ValidMove bb}
    4 -> {g : Game bb}
    5 -> Prev (move' m g)
    !
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61. 1 data Prev : Game b -> Type where
    2 HasPrev : {bb : Board}
    3 -> {m : ValidMove bb}
    4 -> {g : Game bb}
    5 -> Prev (move' m g)
    6
    7 -- What was the previous board?
    8 previousBoard : (gg : Game b)
    9 -> {default HasPrev prf : (Prev gg)}
    10 -> Board
    !
    !
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62. 1 data Prev : Game b -> Type where
    2 HasPrev : {bb : Board}
    3 -> {m : ValidMove bb}
    4 -> {g : Game bb}
    5 -> Prev (move' m g)
    6
    7 -- What was the previous board?
    8 previousBoard : (gg : Game b)
    9 -> {default HasPrev prf : (Prev gg)}
    10 -> Board
    11 previousBoard (move' {b} m g) {prf = HasPrev} = b
    !
    !
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63. 1 data Prev : Game b -> Type where
    2 HasPrev : {bb : Board}
    3 -> {m : ValidMove bb}
    4 -> {g : Game bb}
    5 -> Prev (move' m g)
    6
    7 -- What was the previous board?
    8 previousBoard : (gg : Game b)
    9 -> {default HasPrev prf : (Prev gg)}
    10 -> Board
    11 previousBoard (move' {b} m g) {prf = HasPrev} = b
    12
    13 -- Take back the last move?
    14 takeBack : (gg : Game b)
    15 -> {default HasPrev prf : (Prev gg)}
    16 -> Game (previousBoard gg {prf})
    !
    !
    !
    !
    

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64. 1 data Prev : Game b -> Type where
    2 HasPrev : {bb : Board}
    3 -> {m : ValidMove bb}
    4 -> {g : Game bb}
    5 -> Prev (move' m g)
    6
    7 -- What was the previous board?
    8 previousBoard : (gg : Game b)
    9 -> {default HasPrev prf : (Prev gg)}
    10 -> Board
    11 previousBoard (move' {b} m g) {prf = HasPrev} = b
    12
    13 -- Take back the last move?
    14 takeBack : (gg : Game b)
    15 -> {default HasPrev prf : (Prev gg)}
    16 -> Game (previousBoard gg {prf})
    17 takeBack (move' {b} m g) {prf = HasPrev} = g
    !
    !
    !
    

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65. 1 game0 : ?t1
    2 game0 = start
    3 t1 = proof search
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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66. 1 game0 : ?t1
    2 game0 = start
    3 t1 = proof search
    4
    5 --game0x : ?t2
    6 --game0x = takeBack game0
    7 --t3 = proof search
    8
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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67. 1 game0 : ?t1
    2 game0 = start
    3 t1 = proof search
    4
    5 --game0x : ?t2
    6 --game0x = takeBack game0
    7 --t3 = proof search
    8
    9 game1 : ?t3
    10 game1 = move ne X game0
    11 t3 = proof search
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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68. 1 game0 : ?t1
    2 game0 = start
    3 t1 = proof search
    4
    5 --game0x : ?t2
    6 --game0x = takeBack game0
    7 --t3 = proof search
    8
    9 game1 : ?t3
    10 game1 = move ne X game0
    11 t3 = proof search
    12
    13 --game1x : ?t4
    14 --game1x = move ne O game1
    15 --t4 = proof search
    !
    !
    !
    !
    !
    

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69. 1 game0 : ?t1
    2 game0 = start
    3 t1 = proof search
    4
    5 --game0x : ?t2
    6 --game0x = takeBack game0
    7 --t3 = proof search
    8
    9 game1 : ?t3
    10 game1 = move ne X game0
    11 t3 = proof search
    12
    13 --game1x : ?t4
    14 --game1x = move ne O game1
    15 --t4 = proof search
    16
    17 game2 : ?t5
    18 game2 = move sw O game1
    19 t5 = proof search
    !
    

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70. Existentials for Free
    

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71. 1 data TicTacToeState =
    2 InPlay | Done
    3
    4 toState : Board -> TicTacToeState
    5 toState b =
    6 case (winner b, complete b) of
    7 (Just p, _) => Done
    8 (Nothing, True) => Done
    9 (Nothing, False) => InPlay
    10
    11 data TicTacToe : TicTacToeState -> Type where
    12 T : (Game b) -> TicTacToe (toState b)
    !
    !
    !
    !
    !
    !
    !
    !
    

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72. Effects
    

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73. !
    1 Eff : (m : Type -> Type)
    2 -> (x : Type)
    3 -> List EFFECT
    4 -> (x -> List EFFECT)
    5 -> Type
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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74. !
    1 Eff : (m : Type -> Type)
    2 -> (x : Type)
    3 -> List EFFECT
    4 -> (x -> List EFFECT)
    5 -> Type
    6
    7
    8 STATE : Type -> EFFECT
    9 EXCEPTION : Type -> EFFECT
    10 FILEIO : Type -> EFFECT
    11 STDIO : EFFECT
    12 RND : EFFECT
    !
    !
    !
    !
    !
    !
    !
    !
    

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75. 1 data TicTacToeRules : Effect where
    2 Move :
    3 (position : Position)
    4 -> (player : Player)
    5 -> { TicTacToe st ==> {st'} (TicTacToe st') }
    6 TicTacToeRules TicTacToeState
    7
    8 Get : { g } TicTacToeRules g
    9
    10 GetBoard : { TicTacToe st } TicTacToeRules Board
    11
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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76. 1 data TicTacToeRules : Effect where
    2 Move :
    3 (position : Position)
    4 -> (player : Player)
    5 -> { TicTacToe st ==> {st'} (TicTacToe st') }
    6 TicTacToeRules TicTacToeState
    7
    8 Get : { g } TicTacToeRules g
    9
    10 GetBoard : { TicTacToe st } TicTacToeRules Board
    11
    12 TICTACTOE : TicTacToeState -> EFFECT
    13 TICTACTOE s = MkEff (TicTacToe s) TicTacToeRules
    !
    !
    !
    !
    !
    !
    !
    

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77. 1 using (m : Type -> Type)
    2 instance Handler TicTacToeRules m where
    3 handle (T game) (Move position player) k
    4 handle t Get k
    5 handle (T game) GetBoard k
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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78. 1 using (m : Type -> Type)
    2 instance Handler TicTacToeRules m where
    3 handle (T game) (Move position player) k =
    4 let b = toBoard game in
    5 case tryValidMove position player b of
    6 Just m' =>
    7 let updated = move' m' game in
    8 k (toState . toBoard $ updated) (T updated)
    9 Nothing =>
    10 k (toState . toBoard $ game) (T game)
    11
    12 handle t Get k =
    13 k t t
    14
    15 handle (T game) GetBoard k =
    16 k (toBoard game) (T game)
    !
    !
    !
    !
    

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79. 1 game : { [TICTACTOE InPlay, STDIO] ==>
    2 [TICTACTOE Done, STDIO] } Eff IO ()
    3 game = do
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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80. 1 game : { [TICTACTOE InPlay, STDIO] ==>
    2 [TICTACTOE Done, STDIO] } Eff IO ()
    3 game = do
    4 let current = !GetBoard
    5 let player = turn current
    6 let _ = !(printState current player)
    7 let input = !getStr
    8 case parse (trim input) of
    9 Nothing => '
    10 do putStrLn $ "Invalid move: " ++ input
    11 pure !game
    12 Just position =>
    13 do InPlay <- Move position player
    14 | Done => putStrLn "Done"
    15 game
    !
    !
    !
    !
    !
    

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81. Constraints are Context Dependent
    

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82. !
    1 data CurrentGame : Type where
    2 Current : (TicTacToe state) -> CurrentGame
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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83. !
    1 data CurrentGame : Type where
    2 Current : (TicTacToe state) -> CurrentGame
    3
    4 receive' : { [STDIO,
    5 STATE CurrentGame,
    6 TCPSERVERCLIENT ClientConnected] ==>
    7 [STDIO,
    8 STATE CurrentGame,
    9 TCPSERVERCLIENT ()] } Eff IO ()
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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84. !
    1 data CurrentGame : Type where
    2 Current : (TicTacToe state) -> CurrentGame
    3
    4 receive' : { [STDIO,
    5 STATE CurrentGame,
    6 TCPSERVERCLIENT ClientConnected] ==>
    7 [STDIO,
    8 STATE CurrentGame,
    9 TCPSERVERCLIENT ()] } Eff IO ()
    10
    11 game : CurrentGame -> Game x
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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85. !
    1 data CurrentGame : Type where
    2 Current : (TicTacToe state) -> CurrentGame
    3
    4 receive' : { [STDIO,
    5 STATE CurrentGame,
    6 TCPSERVERCLIENT ClientConnected] ==>
    7 [STDIO,
    8 STATE CurrentGame,
    9 TCPSERVERCLIENT ()] } Eff IO ()
    10
    11 game : CurrentGame -> Game x —??????????
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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86. !
    1 data CurrentGame : Type where
    2 Current : (TicTacToe state) -> CurrentGame
    3
    4 receive' : { [STDIO,
    5 STATE CurrentGame,
    6 TCPSERVERCLIENT ClientConnected] ==>
    7 [STDIO,
    8 STATE CurrentGame,
    9 TCPSERVERCLIENT ()] } Eff IO ()
    10
    11 game : CurrentGame -> (x ** Game x)
    12 game (Current (T z)) =
    13 (toBoard z ** z)
    !
    !
    !
    !
    !
    !
    !
    

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87. !
    1 data CurrentGame : Type where
    2 Current : (TicTacToe state) -> CurrentGame
    3
    4 receive' : { [STDIO,
    5 STATE CurrentGame,
    6 TCPSERVERCLIENT ClientConnected] ==>
    7 [STDIO,
    8 STATE CurrentGame,
    9 TCPSERVERCLIENT ()] } Eff IO ()
    10
    11 game : CurrentGame -> (x ** Game x)
    !
    !
    !
    !
    !
    !
    !
    !
    !
    

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88. “the most popular and best
    established lightweight formal
    methods are type systems”
    - Benjamin Pierce, Types and Programming Languages
    

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89. Tic
    @markhibberd
    Type
    Tac
    

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