quox/lib/Quox/Typechecker.idr

module Quox.Typechecker

import public Quox.Syntax
import public Quox.Typing
import public Quox.Equal
import public Control.Monad.Either
import Decidable.Decidable
import Data.SnocVect

%default total


public export
0 CanTC' : (q : Type) -> (q -> Type) -> (Type -> Type) -> Type
CanTC' q isGlobal m = (HasErr q m, MonadReader (Definitions' q isGlobal) m)

public export
0 CanTC : (q : Type) -> IsQty q => (Type -> Type) -> Type
CanTC q = CanTC' q IsGlobal


private covering %inline
expectTYPE : CanTC' q _ m => Term q d n -> m Universe
expectTYPE s =
  case whnfD !ask s of
    Element (TYPE l) _ => pure l
    _ => throwError $ ExpectedTYPE s

private covering %inline
expectPi : CanTC' q _ m => Term q d n ->
           m (q, Term q d n, ScopeTerm q d n)
expectPi ty =
  case whnfD !ask ty of
    Element (Pi qty _ arg res) _ => pure (qty, arg, res)
    _ => throwError $ ExpectedPi ty

private covering %inline
expectSig : CanTC' q _ m => Term q d n ->
            m (Term q d n, ScopeTerm q d n)
expectSig ty =
  case whnfD !ask ty of
    Element (Sig _ arg res) _ => pure (arg, res)
    _ => throwError $ ExpectedSig ty

private covering %inline
expectEq : CanTC' q _ m => Term q d n ->
           m (DScopeTerm q d n, Term q d n, Term q d n)
expectEq ty =
  case whnfD !ask ty of
    Element (Eq _ ty l r) _ => pure (ty, l, r)
    _ => throwError $ ExpectedEq ty


private
popQs : HasErr q m => IsQty q =>
        SnocVect s q -> QOutput q (s + n) -> m (QOutput q n)
popQs [<]         qctx         = pure qctx
popQs (pis :< pi) (qctx :< rh) = do expectCompatQ rh pi; popQs pis qctx

private %inline
popQ : HasErr q m => IsQty q => q -> QOutput q (S n) -> m (QOutput q n)
popQ pi = popQs [< pi]



private %inline
tail : TyContext q d (S n) -> TyContext q d n
tail = {tctx $= tail, qctx $= tail}


private %inline
weakI : IsQty q => InferResult q d n -> InferResult q d (S n)
weakI = {type $= weakT, qout $= (:< zero)}

private
lookupBound : IsQty q => q -> Var n -> TyContext q d n -> InferResult q d n
lookupBound pi VZ (MkTyContext {tctx = tctx :< ty, _}) =
  InfRes {type = weakT ty, qout = (zero <$ tctx) :< pi}
lookupBound pi (VS i) ctx =
  weakI $ lookupBound pi i (tail ctx)

private
lookupFree : CanTC' q g m => Name -> m (Definition' q g)
lookupFree x = lookupFree' !ask x

private %inline
subjMult : IsQty q => (sg : SQty q) -> q -> SQty q
subjMult sg qty = if isYes $ isZero qty then szero else sg


export
makeDimEq : DContext d -> DimEq d
makeDimEq DNil           = zeroEq
makeDimEq (DBind dctx)   = makeDimEq dctx :<? Nothing
makeDimEq (DEq p q dctx) = set p q $ makeDimEq dctx


parameters {auto _ : IsQty q} {auto _ : CanTC q m}
 mutual
  -- [todo] it seems like the options here for dealing with substitutions are
  -- to either push them or parametrise the whole typechecker over ambient
  -- substitutions. both of them seem like the same amount of work for the
  -- computer but pushing is less work for the me

  ||| `check ctx sg subj ty` checks that in the context `ctx`, the term
  ||| `subj` has the type `ty`, with quantity `sg`. if so, returns the
  ||| quantities of all bound variables that it used.
  export covering %inline
  check : TyContext q d n -> SQty q -> Term q d n -> Term q d n ->
          m (CheckResult q n)
  check ctx sg subj ty =
    let Element subj nc = pushSubsts subj in
    check' ctx sg subj nc ty

  ||| `check0 ctx subj ty` checks a term in a zero context.
  export covering %inline
  check0 : TyContext q d n -> Term q d n -> Term q d n -> m (CheckResult q n)
  check0 ctx = check (zeroed ctx) szero

  ||| `infer ctx sg subj` infers the type of `subj` in the context `ctx`,
  ||| and returns its type and the bound variables it used.
  export covering %inline
  infer : TyContext q d n -> SQty q -> Elim q d n -> m (InferResult q d n)
  infer ctx sg subj =
    let Element subj nc = pushSubsts subj in
    infer' ctx sg subj nc


  export covering
  check' : TyContext q d n -> SQty q ->
           (subj : Term q d n) -> (0 nc : NotClo subj) -> Term q d n ->
           m (CheckResult q n)

  check' ctx sg (TYPE l) _ ty = do
    -- if ℓ < ℓ' then Ψ | Γ ⊢ Type ℓ · 0 ⇐ Type ℓ' ⊳ 𝟎
    l' <- expectTYPE ty
    expectEqualQ zero sg.fst
    unless (l < l') $ throwError $ BadUniverse l l'
    pure $ zeroFor ctx

  check' ctx sg (Pi qty _ arg res) _ ty = do
    l <- expectTYPE ty
    expectEqualQ zero sg.fst
    -- if Ψ | Γ ⊢ A · 0 ⇐ Type ℓ ⊳ 𝟎
    ignore $ check0 ctx arg (TYPE l)
    -- if Ψ | Γ, x · 0 : A ⊢ B · 0 ⇐ Type ℓ ⊳ 𝟎
    case res of
      TUsed   res => ignore $ check0 (extendTy arg zero ctx) res (TYPE l)
      TUnused res => ignore $ check0 ctx res (TYPE l)
    -- then Ψ | Γ ⊢ (x : A) → B · 0 ⇐ Type ℓ ⊳ 𝟎
    pure $ zeroFor ctx

  check' ctx sg (Lam _ body) _ ty = do
    (qty, arg, res) <- expectPi ty
    -- if Ψ | Γ, x · πσ : A ⊢ t · σ ⇐ B ⊳ Σ, x · σπ
    qout <- check (extendTy arg (sg.fst * qty) ctx) sg body.term res.term
    -- then Ψ | Γ ⊢ λx. t · σ ⇐ (x · π : A) → B ⊳ Σ
    popQ (sg.fst * qty) qout

  check' ctx sg (Sig _ fst snd) _ ty = do
    l <- expectTYPE ty
    expectEqualQ zero sg.fst
    -- if Ψ | Γ ⊢ A · 0 ⇐ Type ℓ ⊳ 𝟎
    ignore $ check0 ctx fst (TYPE l)
    -- if Ψ | Γ, x · 0 : A ⊢ B · 0 ⇐ Type ℓ ⊳ 𝟎
    case snd of
      TUsed   snd => ignore $ check0 (extendTy fst zero ctx) snd (TYPE l)
      TUnused snd => ignore $ check0 ctx snd (TYPE l)
    -- then Ψ | Γ ⊢ (x : A) × B · 0 ⇐ Type ℓ ⊳ 𝟎
    pure $ zeroFor ctx

  check' ctx sg (Pair fst snd) _ ty = do
    (tfst, tsnd) <- expectSig ty
    -- if Ψ | Γ ⊢ s · σ ⇐ A ⊳ Σ₁
    qfst <- check ctx sg fst tfst
    let tsnd = sub1 tsnd (fst :# tfst)
    -- if Ψ | Γ ⊢ t · σ ⇐ B[s] ⊳ Σ₂
    qsnd <- check ctx sg snd tsnd
    -- then Ψ | Γ ⊢ (s, t) · σ ⇐ (x : A) × B ⊳ Σ₁ + Σ₂
    pure $ qfst + qsnd

  check' ctx sg (Eq i t l r) _ ty = do
    u <- expectTYPE ty
    expectEqualQ zero sg.fst
    -- if Ψ, i | Γ ⊢ A · 0 ⇐ Type ℓ ⊳ 𝟎
    case t of
      DUsed   t => ignore $ check0 (extendDim ctx) t (TYPE u)
      DUnused t => ignore $ check0 ctx t (TYPE u)
    -- if Ψ | Γ ⊢ l · 0 ⇐ A‹0› ⊳ 𝟎
    ignore $ check0 ctx t.zero l
    -- if Ψ | Γ ⊢ r · 0 ⇐ A‹1› ⊳ 𝟎
    ignore $ check0 ctx t.one  r
    -- then Ψ | Γ ⊢ Eq [i ⇒ A] l r ⇐ Type ℓ ⊳ 𝟎
    pure $ zeroFor ctx

  check' ctx sg (DLam i body) _ ty = do
    (ty, l, r) <- expectEq ty
    -- if Ψ, i | Γ ⊢ t · σ ⇐ A ⊳ Σ
    qout <- check (extendDim ctx) sg body.term ty.term
    let eqs = makeDimEq ctx.dctx
    -- if Ψ ⊢ t‹0› = l
    equal eqs ctx.tctx ty.zero body.zero l
    -- if Ψ ⊢ t‹1› = r
    equal eqs ctx.tctx ty.one  body.one  r
    -- then Ψ | Γ ⊢ (λᴰi ⇒ t) · σ ⇐ Eq [i ⇒ A] l r ⊳ Σ
    pure qout

  check' ctx sg (E e) _ ty = do
    -- if Ψ | Γ ⊢ e · σ ⇒ A' ⊳ Σ
    infres <- infer ctx sg e
    -- if Ψ ⊢ A' <: A
    subtype (makeDimEq ctx.dctx) ctx.tctx infres.type ty
    -- then Ψ | Γ ⊢ e · σ ⇐ A ⊳ Σ
    pure infres.qout

  export covering
  infer' : TyContext q d n -> SQty q ->
           (subj : Elim q d n) -> (0 nc : NotClo subj) ->
           m (InferResult q d n)

  infer' ctx sg (F x) _ = do
    -- if x · π : A {≔ s} in global context
    g <- lookupFree x
    -- if σ ≤ π
    expectCompatQ sg.fst g.qty
    -- then Ψ | Γ ⊢ x ⇒ A ⊳ 𝟎
    pure $ InfRes {type = g.type.get, qout = zeroFor ctx}

  infer' ctx sg (B i) _ =
    -- if x : A ∈ Γ
    -- then Ψ | Γ ⊢ x · σ ⇒ A ⊳ (𝟎, σ · x, 𝟎)
    pure $ lookupBound sg.fst i ctx

  infer' ctx sg (fun :@ arg) _ = do
    -- if Ψ | Γ ⊢ f · σ ⇒ (x · π : A) → B ⊳ Σ₁
    funres <- infer ctx sg fun
    (qty, argty, res) <- expectPi funres.type
    -- if Ψ | Γ ⊢ s · σ∧π ⇐ A ⊳ Σ₂
    --   (0∧π = σ∧0 = 0; σ∧π = σ otherwise)
    argout <- check ctx (subjMult sg qty) arg argty
    -- then Ψ | Γ ⊢ f s · σ ⇒ B[s] ⊳ Σ₁ + Σ₂
    pure $ InfRes {
      type = sub1 res $ arg :# argty,
      qout = funres.qout + argout
    }

  infer' ctx sg (CasePair pi pair _ ret _ _ body) _ = do
    -- if 1 ≤ π
    expectCompatQ one pi
    -- if Ψ | Γ ⊢ pair · 1 ⇒ (x : A) × B ⊳ Σ₁
    pairres <- infer ctx sone pair
    ignore $ check0 (extendTy pairres.type zero ctx) ret.term (TYPE UAny)
    (tfst, tsnd) <- expectSig pairres.type
    -- if Ψ | Γ, x · π : A, y · π : B ⊢ σ body ⇐ ret[(x, y)]
    --      ⊳ Σ₂, x · π₁, y · π₂
    -- if π₁, π₂ ≤ π
    let bodyctx = extendTyN [< (tfst, pi), (tsnd.term, pi)] ctx
        bodyty  = substCasePairRet pairres.type ret
    bodyout <- check bodyctx sg body.term bodyty >>= popQs [< pi, pi]
    -- then Ψ | Γ ⊢ σ case ⋯ ⇒ ret[pair] ⊳ πΣ₁ + Σ₂
    pure $ InfRes {
      type = sub1 ret pair,
      qout = pi * pairres.qout + bodyout
    }


  infer' ctx sg (fun :% dim) _ = do
    -- if Ψ | Γ ⊢ f · σ ⇒ Eq [i ⇒ A] l r ⊳ Σ
    InfRes {type, qout} <- infer ctx sg fun
    (ty, _, _) <- expectEq type
    -- then Ψ | Γ ⊢ f p · σ ⇒ A‹p› ⊳ Σ
    pure $ InfRes {type = dsub1 ty dim, qout}

  infer' ctx sg (term :# type) _ = do
    -- if Ψ | Γ ⊢ A · 0 ⇐ Type ℓ ⊳ 𝟎
    ignore $ check0 ctx type (TYPE UAny)
    -- if Ψ | Γ ⊢ s · σ ⇐ A ⊳ Σ
    qout <- check ctx sg term type
    -- then Ψ | Γ ⊢ (s ∷ A) · σ ⇒ A ⊳ Σ
    pure $ InfRes {type, qout}