module Quox.Syntax.Term.Subst

import Quox.No
import Quox.Syntax.Term.Base
import Data.Vect

%default total

namespace CanDSubst
  public export
  interface CanDSubst (0 tm : Nat -> Nat -> Type) where
    (//) : tm dfrom n -> Lazy (DSubst dfrom dto) -> tm dto n

||| does the minimal reasonable work:
||| - deletes the closure around an atomic constant like `TYPE`
||| - deletes an identity substitution
||| - composes (lazily) with an existing top-level dim-closure
||| - otherwise, wraps in a new closure
export
CanDSubst (Term q) where
  s          // Shift SZ = s
  TYPE l     // _        = TYPE l
  DCloT s ph // th       = DCloT s $ ph . th
  s          // th       = DCloT s th

private
subDArgs : Elim q dfrom n -> DSubst dfrom dto -> Elim q dto n
subDArgs (f :% d) th = subDArgs f th :% (d // th)
subDArgs e        th = DCloE e th

||| does the minimal reasonable work:
||| - deletes the closure around a term variable
||| - deletes an identity substitution
||| - composes (lazily) with an existing top-level dim-closure
||| - immediately looks up bound variables in a
|||   top-level sequence of dimension applications
||| - otherwise, wraps in a new closure
export
CanDSubst (Elim q) where
  e          // Shift SZ = e
  F x        // _        = F x
  B i        // _        = B i
  f :% d     // th       = subDArgs (f :% d) th
  DCloE e ph // th       = DCloE e $ ph . th
  e          // th       = DCloE e th

namespace DSubst.ScopeTermN
  export %inline
  (//) : ScopeTermN s q dfrom n -> Lazy (DSubst dfrom dto) ->
         ScopeTermN s q dto n
  S ns (Y body) // th = S ns $ Y $ body // th
  S ns (N body) // th = S ns $ N $ body // th

namespace DSubst.DScopeTermN
  export %inline
  (//) : {s : Nat} ->
         DScopeTermN s q dfrom n -> Lazy (DSubst dfrom dto) ->
         DScopeTermN s q dto n
  S ns (Y body) // th = S ns $ Y $ body // pushN s th
  S ns (N body) // th = S ns $ N $ body // th


export %inline FromVar (Elim q d) where fromVar = B
export %inline FromVar (Term q d) where fromVar = E . fromVar


||| does the minimal reasonable work:
||| - deletes the closure around a *free* name
||| - deletes an identity substitution
||| - composes (lazily) with an existing top-level closure
||| - immediately looks up a bound variable
||| - otherwise, wraps in a new closure
export
CanSubstSelf (Elim q d) where
  F x       // _  = F x
  B i       // th = th !! i
  CloE e ph // th = assert_total CloE e $ ph . th
  e         // th = case force th of
                         Shift SZ => e
                         th       => CloE e th

namespace CanTSubst
  public export
  interface CanTSubst (0 tm : TermLike) where
    (//) : tm q d from -> Lazy (TSubst q d from to) -> tm q d to

||| does the minimal reasonable work:
||| - deletes the closure around an atomic constant like `TYPE`
||| - deletes an identity substitution
||| - composes (lazily) with an existing top-level closure
||| - goes inside `E` in case it is a simple variable or something
||| - otherwise, wraps in a new closure
export
CanTSubst Term where
  TYPE l    // _  = TYPE l
  E e       // th = E $ e // th
  CloT s ph // th = CloT s $ ph . th
  s         // th = case force th of
                         Shift SZ => s
                         th       => CloT s th

namespace ScopeTermN
  export %inline
  (//) : {s : Nat} ->
         ScopeTermN s q d from -> Lazy (TSubst q d from to) ->
         ScopeTermN s q d to
  S ns (Y body) // th = S ns $ Y $ body // pushN s th
  S ns (N body) // th = S ns $ N $ body // th

namespace DScopeTermN
  export %inline
  (//) : {s : Nat} ->
         DScopeTermN s q d from -> Lazy (TSubst q d from to) ->
         DScopeTermN s q d to
  S ns (Y body) // th = S ns $ Y $ body // map (// shift s) th
  S ns (N body) // th = S ns $ N $ body // th

export %inline CanShift (Term q d) where s // by = s // Shift by
export %inline CanShift (Elim q d) where e // by = e // Shift by

export %inline
{s : Nat} -> CanShift (ScopeTermN s q d) where
  b // by = b // Shift by


export %inline
comp : DSubst dfrom dto -> TSubst q dfrom from mid -> TSubst q dto mid to ->
       TSubst q dto from to
comp th ps ph = map (// th) ps . ph


public export %inline
dweakT : {by : Nat} -> Term q d n -> Term q (by + d) n
dweakT t = t // shift by

public export %inline
dweakE : {by : Nat} -> Elim q d n -> Elim q (by + d) n
dweakE t = t // shift by


public export %inline
weakT : {default 1 by : Nat} -> Term q d n -> Term q d (by + n)
weakT t = t // shift by

public export %inline
weakE : {default 1 by : Nat} -> Elim q d n -> Elim q d (by + n)
weakE t = t // shift by


parameters {s : Nat}
  namespace ScopeTermBody
    export %inline
    (.term) : ScopedBody s (Term q d) n -> Term q d (s + n)
    (Y b).term = b
    (N b).term = weakT b {by = s}

  namespace ScopeTermN
    export %inline
    (.term) : ScopeTermN s q d n -> Term q d (s + n)
    t.term = t.body.term

  namespace DScopeTermBody
    export %inline
    (.term) : ScopedBody s (\d => Term q d n) d -> Term q (s + d) n
    (Y b).term = b
    (N b).term = dweakT b {by = s}

  namespace DScopeTermN
    export %inline
    (.term) : DScopeTermN s q d n -> Term q (s + d) n
    t.term = t.body.term


export %inline
subN : ScopeTermN s q d n -> Vect s (Elim q d n) -> Term q d n
subN (S _ (Y body)) es = body // fromVect es
subN (S _ (N body)) _  = body

export %inline
sub1 : ScopeTerm q d n -> Elim q d n -> Term q d n
sub1 t e = subN t [e]

export %inline
dsubN : DScopeTermN s q d n -> Vect s (Dim d) -> Term q d n
dsubN (S _ (Y body)) ps = body // fromVect ps
dsubN (S _ (N body)) _  = body

export %inline
dsub1 : DScopeTerm q d n -> Dim d -> Term q d n
dsub1 t p = dsubN t [p]


public export %inline
(.zero) : DScopeTerm q d n -> Term q d n
body.zero = dsub1 body $ K Zero

public export %inline
(.one) : DScopeTerm q d n -> Term q d n
body.one = dsub1 body $ K One


public export
0 CloTest : TermLike -> Type
CloTest tm = forall q, d, n. tm q d n -> Bool

interface PushSubsts (0 tm : TermLike) (0 isClo : CloTest tm) | tm where
  pushSubstsWith : DSubst dfrom dto -> TSubst q dto from to ->
                   tm q dfrom from -> Subset (tm q dto to) (No . isClo)

public export
0 NotClo : {isClo : CloTest tm} -> PushSubsts tm isClo => Pred (tm q d n)
NotClo = No . isClo

public export
0 NonClo : (tm : TermLike) -> {isClo : CloTest tm} ->
           PushSubsts tm isClo => TermLike
NonClo tm q d n = Subset (tm q d n) NotClo

public export %inline
nclo : {isClo : CloTest tm} -> (0 _ : PushSubsts tm isClo) =>
       (t : tm q d n) -> (0 nc : NotClo t) => NonClo tm q d n
nclo t = Element t nc

parameters {0 isClo : CloTest tm} {auto _ : PushSubsts tm isClo}
  ||| if the input term has any top-level closures, push them under one layer of
  ||| syntax
  export %inline
  pushSubsts : tm q d n -> NonClo tm q d n
  pushSubsts s = pushSubstsWith id id s

  export %inline
  pushSubstsWith' : DSubst dfrom dto -> TSubst q dto from to ->
                    tm q dfrom from -> tm q dto to
  pushSubstsWith' th ph x = fst $ pushSubstsWith th ph x

mutual
  public export
  isCloT : CloTest Term
  isCloT (CloT  {}) = True
  isCloT (DCloT {}) = True
  isCloT (E e)      = isCloE e
  isCloT _          = False

  public export
  isCloE : CloTest Elim
  isCloE (CloE  {}) = True
  isCloE (DCloE {}) = True
  isCloE _          = False

mutual
  export
  PushSubsts Term Subst.isCloT where
    pushSubstsWith th ph (TYPE l) =
      nclo $ TYPE l
    pushSubstsWith th ph (Pi qty a body) =
      nclo $ Pi qty (a // th // ph) (body // th // ph)
    pushSubstsWith th ph (Lam body) =
      nclo $ Lam (body // th // ph)
    pushSubstsWith th ph (Sig a b) =
      nclo $ Sig (a // th // ph) (b // th // ph)
    pushSubstsWith th ph (Pair s t) =
      nclo $ Pair (s // th // ph) (t // th // ph)
    pushSubstsWith th ph (Enum tags) =
      nclo $ Enum tags
    pushSubstsWith th ph (Tag tag) =
      nclo $ Tag tag
    pushSubstsWith th ph (Eq ty l r) =
      nclo $ Eq (ty // th // ph) (l // th // ph) (r // th // ph)
    pushSubstsWith th ph (DLam body) =
      nclo $ DLam (body // th // ph)
    pushSubstsWith th ph (E e) =
      let Element e nc = pushSubstsWith th ph e in nclo $ E e
    pushSubstsWith th ph (CloT s ps) =
      pushSubstsWith th (comp th ps ph) s
    pushSubstsWith th ph (DCloT s ps) =
      pushSubstsWith (ps . th) ph s

  export
  PushSubsts Elim Subst.isCloE where
    pushSubstsWith th ph (F x) =
      nclo $ F x
    pushSubstsWith th ph (B i) =
      let res = ph !! i in
      case nchoose $ isCloE res of
        Left  yes => assert_total pushSubsts res
        Right no  => Element res no
    pushSubstsWith th ph (f :@ s) =
      nclo $ (f // th // ph) :@ (s // th // ph)
    pushSubstsWith th ph (CasePair pi p r b) =
      nclo $ CasePair pi (p // th // ph) (r // th // ph) (b // th // ph)
    pushSubstsWith th ph (CaseEnum pi t r arms) =
      nclo $ CaseEnum pi (t // th // ph) (r // th // ph)
               (map (\b => b // th // ph) arms)
    pushSubstsWith th ph (f :% d) =
      nclo $ (f // th // ph) :% (d // th)
    pushSubstsWith th ph (s :# a) =
      nclo $ (s // th // ph) :# (a // th // ph)
    pushSubstsWith th ph (CloE e ps) =
      pushSubstsWith th (comp th ps ph) e
    pushSubstsWith th ph (DCloE e ps) =
      pushSubstsWith (ps . th) ph e


parameters {0 d', n' : Nat}
 mutual
  namespace Term
    export
    inject : Term q d n -> Term q (d + d') (n + n')
    inject (TYPE l) = TYPE l
    inject (Pi qty arg res) = Pi qty (inject arg) (inject res)
    inject (Lam body) = Lam (inject body)
    inject (Sig fst snd) = Sig (inject fst) (inject snd)
    inject (Pair fst snd) = Pair (inject fst) (inject snd)
    inject (Enum cases) = Enum cases
    inject (Tag tag) = Tag tag
    inject (Eq ty l r) = Eq (inject ty) (inject l) (inject r)
    inject (DLam body) = DLam (inject body)
    inject (E e) = E (inject e)
    inject (CloT tm th) = CloT (inject tm) (inject th)
    inject (DCloT tm th) = DCloT (inject tm) (inject th)

  namespace Elim
    export
    inject : Elim q d n -> Elim q (d + d') (n + n')
    inject (F x) = F x
    inject (B i) = B $ inject i
    inject (fun :@ arg) = (inject fun) :@ (inject arg)
    inject (CasePair qty pair ret body) =
      CasePair qty (inject pair) (inject ret) (inject body)
    inject (CaseEnum qty tag ret arms) =
      CaseEnum qty (inject tag) (inject ret)
        (assert_total $ map inject arms)
    inject (fun :% arg) = (inject fun) :% (inject arg)
    inject (tm :# ty) = (inject tm) :# (inject ty)
    inject (CloE el th) = CloE (inject el) (inject th)
    inject (DCloE el th) = DCloE (inject el) (inject th)

  namespace ScopeTerm
    export
    inject : ScopeTermN s q d n -> ScopeTermN s q (d + d') (n + n')
    inject (S names (N body)) = S names (N (inject body))
    inject (S names (Y body)) {s, n} =
      S names $ Y $ rewrite plusAssociative s n n' in inject body

  namespace DScopeTerm
    export
    inject : DScopeTermN s q d n -> DScopeTermN s q (d + d') (n + n')
    inject (S names (N body)) = S names (N (inject body))
    inject (S names (Y body)) {s, d} =
      S names $ Y $ rewrite plusAssociative s d d' in inject body

  namespace TSubst
    export
    inject : TSubst q d from to -> TSubst q (d + d') (from + n') (to + n')
    inject (Shift by) = Shift $ inject by
    inject (t ::: th) = inject t ::: inject th

  namespace DSubst
    export
    inject : DSubst from to -> DSubst (from + d') (to + d')
    inject (Shift by) = Shift $ inject by
    inject (p ::: th) = inject p ::: inject th


-- [fixme]
-- Error: Linearity checking failed on metavar Quox.Syntax.Term.Subst.{b:7362}
-- (Int not a function type)
{-
%transform "Term.inject" Term.inject t = believe_me t
%transform "Elim.inject" Elim.inject e = believe_me e
%transform "ScopeTerm.inject" ScopeTerm.inject t = believe_me t
%transform "DScopeTerm.inject" DScopeTerm.inject t = believe_me t
%transform "TSubst.inject" TSubst.inject th = believe_me th
%transform "DSubst.inject" DSubst.inject th = believe_me th
-}