module Quox.Syntax.Term.Base

import public Quox.Syntax.Var
import public Quox.Syntax.Shift
import public Quox.Syntax.Subst
import public Quox.Syntax.Universe
import public Quox.Syntax.Qty
import public Quox.Syntax.Dim
import public Quox.Name
import public Quox.OPE

import Quox.Pretty

import public Data.DPair
import Data.List
import Data.Maybe
import Data.Nat
import public Data.So
import Data.String
import Data.Vect

%default total


infixl 8 :#
infixl 9 :@
mutual
  public export
  TSubst : Nat -> Nat -> Nat -> Type
  TSubst d = Subst (\n => Elim d n)

  ||| first argument `d` is dimension scope size, second `n` is term scope size
  public export
  data Term : (d, n : Nat) -> Type where
    ||| type of types
    TYPE : (l : Universe) -> Term d n

    ||| function type
    Pi  : (qty : Qty) -> (x : Name) ->
          (arg : Term d n) -> (res : ScopeTerm d n) -> Term d n
    ||| function term
    Lam : (x : Name) -> (body : ScopeTerm d n) -> Term d n

    ||| elimination
    E : (e : Elim d n) -> Term d n

    ||| term closure/suspended substitution
    CloT  : (tm : Term d from)  -> (th : Lazy (TSubst d from to)) -> Term d to
    ||| dimension closure/suspended substitution
    DCloT : (tm : Term dfrom n) -> (th : Lazy (DSubst dfrom dto)) -> Term dto n

  ||| first argument `d` is dimension scope size, second `n` is term scope size
  public export
  data Elim : (d, n : Nat) -> Type where
    ||| free variable
    F : (x : Name) -> Elim d n
    ||| bound variable
    B : (i : Var n) -> Elim d n

    ||| term application
    (:@) : (fun : Elim d n) -> (arg : Term d n) -> Elim d n

    ||| type-annotated term
    (:#) : (tm, ty : Term d n) -> Elim d n

    ||| term closure/suspended substitution
    CloE  : (el : Elim d from)  -> (th : Lazy (TSubst d from to)) -> Elim d to
    ||| dimension closure/suspended substitution
    DCloE : (el : Elim dfrom n) -> (th : Lazy (DSubst dfrom dto)) -> Elim dto n

  ||| a scope with one more bound variable
  public export
  data ScopeTerm : (d, n : Nat) -> Type where
    ||| variable is used
    TUsed : (body : Term d (S n)) -> ScopeTerm d n
    ||| variable is unused
    TUnused : (body : Term d n) -> ScopeTerm d n

  ||| a scope with one more bound dimension variable
  public export
  data DScopeTerm : (d, n : Nat) -> Type where
    ||| variable is used
    DUsed : (body : Term (S d) n) -> DScopeTerm d n
    ||| variable is unused
    DUnused : (body : Term d n) -> DScopeTerm d n

%name Term s, t, r
%name Elim e, f
%name ScopeTerm body
%name DScopeTerm body

public export %inline
Arr : (qty : Qty) -> (arg, res : Term d n) -> Term d n
Arr {qty, arg, res} = Pi {qty, x = "_", arg, res = TUnused res}

||| same as `F` but as a term
public export %inline
FT : Name -> Term d n
FT = E . F

||| abbreviation for a bound variable like `BV 4` instead of
||| `B (VS (VS (VS (VS VZ))))`
public export %inline
BV : (i : Nat) -> (0 _ : LT i n) => Elim d n
BV i = B $ V i

||| same as `BV` but as a term
public export %inline
BVT : (i : Nat) -> (0 _ : LT i n) => Term d n
BVT i = E $ BV i