TypeClasses.lean

/-
  Type Classes via Dictionary Passing
  
  This implements verified type class compilation:
  1. Define type classes as record types (dictionaries)
  2. Instances are values of dictionary type
  3. Constrained functions take dictionary arguments
  4. Prove laws are preserved through compilation
-/

-- ═══════════════════════════════════════════════════════════════════════════════
-- TYPES
-- ═══════════════════════════════════════════════════════════════════════════════

inductive Ty where
  | unit : Ty
  | bool : Ty
  | int : Ty
  | arr : Ty → Ty → Ty
  | prod : Ty → Ty → Ty  -- For dictionary records
  deriving Repr, DecidableEq

notation:50 α " ⟶ " β => Ty.arr α β
notation:60 α " × " β => Ty.prod α β

-- ═══════════════════════════════════════════════════════════════════════════════
-- SEMANTIC INTERPRETATION  
-- ═══════════════════════════════════════════════════════════════════════════════

def interpTy : Ty → Type
  | .unit => Unit
  | .bool => Bool
  | .int => Int
  | .arr α β => interpTy α → interpTy β
  | .prod α β => interpTy α × interpTy β

-- ═══════════════════════════════════════════════════════════════════════════════
-- TYPE CLASSES AS DICTIONARIES
-- ═══════════════════════════════════════════════════════════════════════════════

-- Eq class: { eq : a → a → Bool }
def EqDict (τ : Ty) : Ty := τ ⟶ τ ⟶ .bool

-- Ord class: { eq : a → a → Bool, compare : a → a → Ordering }
-- For simplicity, Ordering = Int (-1, 0, 1)
def OrdDict (τ : Ty) : Ty := (τ ⟶ τ ⟶ .bool) × (τ ⟶ τ ⟶ .int)

-- Show class: { show : a → String }
-- We'll use Int as a stand-in for String
def ShowDict (τ : Ty) : Ty := τ ⟶ .int

-- Semigroup class: { append : a → a → a }
def SemigroupDict (τ : Ty) : Ty := τ ⟶ τ ⟶ τ

-- Monoid class: { append : a → a → a, mempty : a }
def MonoidDict (τ : Ty) : Ty := (τ ⟶ τ ⟶ τ) × τ

-- ═══════════════════════════════════════════════════════════════════════════════
-- SEMANTIC TYPE CLASSES (with laws!)
-- ═══════════════════════════════════════════════════════════════════════════════

-- Semantic Eq: eq must be reflexive, symmetric, transitive
structure SemEq (α : Type) where
  eq : α → α → Bool
  refl : ∀ x, eq x x = true
  symm : ∀ x y, eq x y = eq y x
  trans : ∀ x y z, eq x y = true → eq y z = true → eq x z = true

-- Semantic Semigroup: append must be associative
structure SemSemigroup (α : Type) where
  append : α → α → α
  assoc : ∀ x y z, append (append x y) z = append x (append y z)

-- Semantic Monoid: extends Semigroup with identity
structure SemMonoid (α : Type) extends SemSemigroup α where
  mempty : α
  left_id : ∀ x, append mempty x = x
  right_id : ∀ x, append x mempty = x

-- Semantic Functor
structure SemFunctor (f : Type → Type) where
  map : {α β : Type} → (α → β) → f α → f β
  map_id : ∀ {α} (x : f α), map id x = x
  map_comp : ∀ {α β γ} (g : β → γ) (h : α → β) (x : f α), map (g ∘ h) x = map g (map h x)

-- Semantic Monad
structure SemMonad (m : Type → Type) extends SemFunctor m where
  pure : {α : Type} → α → m α
  bind : {α β : Type} → m α → (α → m β) → m β
  left_id : ∀ {α β} (a : α) (f : α → m β), bind (pure a) f = f a
  right_id : ∀ {α} (ma : m α), bind ma pure = ma
  assoc : ∀ {α β γ} (ma : m α) (f : α → m β) (g : β → m γ),
    bind (bind ma f) g = bind ma (fun x => bind (f x) g)

-- ═══════════════════════════════════════════════════════════════════════════════
-- DICTIONARY INTERPRETATION
-- ═══════════════════════════════════════════════════════════════════════════════

-- Interpret an Eq dictionary as a SemEq (forgetting laws)
def interpEqDict (d : interpTy (EqDict τ)) : interpTy τ → interpTy τ → Bool := d

-- Interpret a Monoid dictionary
def interpMonoidDict (d : interpTy (MonoidDict τ)) : (interpTy τ → interpTy τ → interpTy τ) × interpTy τ :=
  (d.1, d.2)

-- ═══════════════════════════════════════════════════════════════════════════════
-- INSTANCES (dictionaries with proofs)
-- ═══════════════════════════════════════════════════════════════════════════════

-- Bool has decidable equality
def eqBool : SemEq Bool where
  eq := fun a b => a == b
  refl := by intro x; cases x <;> rfl
  symm := by intro x y; cases x <;> cases y <;> rfl
  trans := by intro x y z hxy hyz; cases x <;> cases y <;> cases z <;> simp_all

-- Int has decidable equality (proofs standard)
def eqInt : SemEq Int where
  eq := fun a b => decide (a = b)
  refl := by intro x; simp
  symm := by intro x y; cases h : decide (x = y) <;> cases h' : decide (y = x) <;> simp_all
  trans := by intro x y z hxy hyz; simp_all

-- List is a semigroup under append
def semigroupList (α : Type) : SemSemigroup (List α) where
  append := List.append
  assoc := List.append_assoc

-- List is a monoid
def monoidList (α : Type) : SemMonoid (List α) where
  toSemSemigroup := semigroupList α
  mempty := []
  left_id := List.nil_append
  right_id := List.append_nil

-- Option is a functor
def functorOption : SemFunctor Option where
  map := Option.map
  map_id := by intro α x; cases x <;> rfl
  map_comp := by intro α β γ g h x; cases x <;> rfl

-- Option is a monad
def monadOption : SemMonad Option where
  toSemFunctor := functorOption
  pure := some
  bind := Option.bind
  left_id := by intros; rfl
  right_id := by intro α ma; cases ma <;> rfl
  assoc := by intro α β γ ma f g; cases ma <;> rfl

-- Define our own bind to have full control
def listBind {α β : Type} (xs : List α) (f : α → List β) : List β :=
  match xs with
  | [] => []
  | h :: t => f h ++ listBind t f

-- Helper: listBind distributes over append
theorem listBindAppend {α β : Type} (xs ys : List α) (f : α → List β) :
    listBind (xs ++ ys) f = listBind xs f ++ listBind ys f := by
  induction xs with
  | nil => rfl
  | cons h t ih => 
      -- Goal: listBind ((h :: t) ++ ys) f = listBind (h :: t) f ++ listBind ys f
      -- i.e., listBind (h :: (t ++ ys)) f = (f h ++ listBind t f) ++ listBind ys f
      -- i.e., f h ++ listBind (t ++ ys) f = (f h ++ listBind t f) ++ listBind ys f
      show f h ++ listBind (t ++ ys) f = (f h ++ listBind t f) ++ listBind ys f
      rw [ih, List.append_assoc]

-- List is a monad (fully verified!)
def monadList : SemMonad List where
  toSemFunctor := {
    map := List.map
    map_id := fun xs => by induction xs <;> simp_all [List.map]
    map_comp := fun g h xs => by induction xs <;> simp_all [List.map, Function.comp]
  }
  pure := fun x => [x]
  bind := listBind
  left_id := fun _ _ => by simp [listBind, List.append_nil]
  right_id := fun ma => by 
    induction ma with
    | nil => rfl
    | cons h t ih => simp [listBind]; exact ih
  assoc := fun ma f g => by
    induction ma with
    | nil => rfl
    | cons h t ih => 
        -- Goal: listBind (listBind (h :: t) f) g = listBind (h :: t) (fun x => listBind (f x) g)
        show listBind (f h ++ listBind t f) g = listBind (f h) g ++ listBind t (fun x => listBind (f x) g)
        rw [listBindAppend]
        simp only [ih]

-- ═══════════════════════════════════════════════════════════════════════════════
-- CONSTRAINED FUNCTIONS
-- ═══════════════════════════════════════════════════════════════════════════════

-- Before type class resolution: polymorphic with explicit dictionary
-- notEq : ∀a. EqDict a → a → a → Bool
-- notEq dict x y = not (dict.eq x y)

def notEq (eq : α → α → Bool) (x y : α) : Bool := !(eq x y)

-- With SemEq, we can prove properties:
theorem notEq_irrefl (E : SemEq α) (x : α) : notEq E.eq x x = false := by
  simp [notEq, E.refl]

theorem notEq_symm (E : SemEq α) (x y : α) : notEq E.eq x y = notEq E.eq y x := by
  simp [notEq, E.symm]

-- Monoid power function
def power (M : SemMonoid α) (x : α) (n : Nat) : α :=
  match n with
  | 0 => M.mempty
  | n + 1 => M.append x (power M x n)

theorem power_zero (M : SemMonoid α) (x : α) : power M x 0 = M.mempty := rfl

theorem power_succ (M : SemMonoid α) (x : α) (n : Nat) : 
    power M x (n + 1) = M.append x (power M x n) := rfl

theorem power_add (M : SemMonoid α) (x : α) (m n : Nat) :
    power M x (m + n) = M.append (power M x m) (power M x n) := by
  induction m with
  | zero => simp only [power, Nat.zero_add, M.left_id]
  | succ m ih => 
      simp only [power, Nat.succ_add]
      rw [ih, M.assoc]

-- ═══════════════════════════════════════════════════════════════════════════════
-- FUNCTOR/MONAD COMBINATORS WITH PROOFS
-- ═══════════════════════════════════════════════════════════════════════════════

-- void : Functor f => f a → f Unit
def void (F : SemFunctor f) (fa : f α) : f Unit :=
  F.map (fun _ => ()) fa

-- join : Monad m => m (m a) → m a
def join (M : SemMonad m) (mma : m (m α)) : m α :=
  M.bind mma id

-- Kleisli composition
def kleisli (M : SemMonad m) (f : α → m β) (g : β → m γ) (a : α) : m γ :=
  M.bind (f a) g

-- Prove join in terms of bind
theorem join_bind (M : SemMonad m) (mma : m (m α)) :
    join M mma = M.bind mma id := rfl

-- Prove Kleisli associativity
theorem kleisli_assoc (M : SemMonad m) (f : α → m β) (g : β → m γ) (h : γ → m δ) (a : α) :
    kleisli M (kleisli M f g) h a = kleisli M f (kleisli M g h) a := by
  simp only [kleisli]
  rw [M.assoc]

-- ═══════════════════════════════════════════════════════════════════════════════
-- CODE GENERATION
-- ═══════════════════════════════════════════════════════════════════════════════

-- Generate PureScript for notEq using dictionary passing
def codegenNotEq : String :=
  "notEq eqDict x y = not (eqDict x y)"

-- Generate PureScript for power
def codegenPower : String :=
  "power monoidDict x n = case n of\n" ++
  "  0 -> monoidDict.mempty\n" ++
  "  _ -> monoidDict.append x (power monoidDict x (n - 1))"

-- Generate PureScript for join
def codegenJoin : String :=
  "join monadDict mma = monadDict.bind mma identity"

-- Generate PureScript for Kleisli
def codegenKleisli : String :=
  "kleisli monadDict f g a = monadDict.bind (f a) g"

#eval codegenNotEq
#eval codegenPower
#eval codegenJoin
#eval codegenKleisli

-- ═══════════════════════════════════════════════════════════════════════════════
-- THE KEY INSIGHT
-- ═══════════════════════════════════════════════════════════════════════════════

/-
  What we've achieved:
  
  1. TYPE CLASSES AS STRUCTURES
     - SemEq, SemSemigroup, SemMonoid, SemFunctor, SemMonad
     - Each bundles operations WITH their laws
     
  2. VERIFIED INSTANCES
     - eqBool, eqInt: prove reflexivity, symmetry, transitivity
     - monoidList: proves identity and associativity
     - monadOption, monadList: prove all three monad laws
     
  3. GENERIC FUNCTIONS WITH PROOFS
     - notEq: works for any Eq, prove ¬(x = x) = false
     - power: works for any Monoid, prove power_add
     - kleisli: works for any Monad, prove associativity
     
  4. CODE GENERATION
     - Shows how to emit dictionary-passing PureScript
     
  THE CONNECTION:
  
  - We write `notEq E.eq x x` in Lean
  - We prove `notEq_irrefl : notEq E.eq x x = false`
  - We generate `notEq eqDict x y = not (eqDict x y)`
  - The generated code has the same structure
  - Therefore the property holds in PureScript too!
  
  THE GAP:
  
  - We still emit strings, not typed terms
  - A full solution would have Term constructors for dictionary operations
  - Then codegen would be a verified transformation
-/

-- ═══════════════════════════════════════════════════════════════════════════════
-- WHAT'S STILL HARD
-- ═══════════════════════════════════════════════════════════════════════════════

/-
  1. DICTIONARY INFERENCE
     PureScript/Haskell find dictionaries automatically.
     We pass them explicitly (which is what the compiler does internally).
     
  2. SUPERCLASS CONSTRAINTS  
     class Eq a => Ord a
     Need to thread Eq dictionary through Ord dictionary.
     
  3. HIGHER-KINDED POLYMORPHISM
     Functor (f : Type → Type) requires kind polymorphism.
     Our Ty doesn't have kinds.
     
  4. OVERLAPPING INSTANCES
     Multiple applicable instances need priority rules.
     
  5. ASSOCIATED TYPES / TYPE FAMILIES
     class Collection c where
       type Elem c :: Type
     Requires more sophisticated type representation.
     
  These are all solvable but each requires significant work.
  The architecture (dictionaries + proofs) is sound.
-/