module Core.Environment

import Core.Context
import Core.Term
import Core.Thinning
import Core.Var

import Syntax.PreorderReasoning

%prefix_record_projections off

-- Definition ------------------------------------------------------------------

public export
data Env : Context -> Type where
  Lin : Env [<]
  Add : Env sx -> sy `Thins` sx -> Term sy -> Env (sx :< n)

%name Env env

-- Constructors ----------------------------------------------------------------

public export
(:<) : Env sx -> Term sx -> Env (sx :< n)
env :< t = Add env (id _) t

-- Operations ------------------------------------------------------------------

record IndexResult (sx : Context) where
  constructor MkRes
  {0 sy : Context}
  val : Term sy
  thin : sy `Thins` sx

shift : IndexResult sx -> IndexResult (sx :< n)
shift (MkRes val thin) = MkRes val (drop thin n)

expand : IndexResult sx -> Term sx
expand (MkRes val thin) = wkn val thin

doIndex : Env sx -> {0 i : Var sx} -> View i -> IndexResult sx
doIndex (Add env thin t) Here = shift $ MkRes t thin
doIndex (Add env thin t) (There i) = shift $ doIndex env $ view i

export
index : Env sx -> Var sx -> Term sx
index env i = expand $ doIndex env $ view i

-- Equality --------------------------------------------------------------------

namespace Eq
  public export
  data EnvEq : Env sx -> Env sx -> Type where
    Lin : EnvEq [<] [<]
    (:<) :
      EnvEq env1 env2 ->
      Term.wkn t thin1 = wkn u thin2 ->
      EnvEq (Add env1 thin1 t) (Add env2 thin2 u)

  %name EnvEq prf

expandShift : (val : IndexResult sx) -> expand (shift val) = wkn (expand val) (drop (id sx) n)
expandShift (MkRes val thin) = sym $ Calc $
  |~ wkn (wkn val thin) (drop (id sx) n)
  ~~ wkn val (drop (id sx) n . thin)     ...(wknHomo val thin $ drop (id sx) n)
  ~~ wkn val (drop (id sx . thin) n)     ...(cong (wkn val) $ compLeftDrop (id sx) thin n)
  ~~ wkn val (drop thin n)               ...(cong (\thin => wkn val (drop thin n)) $ compLeftIdentity thin)

doIndexEqIsEq :
  -- {env1, env2 : _} ->
  EnvEq env1 env2 ->
  {0 i : Var sx} ->
  (view : View i) ->
  expand (doIndex env1 view) = expand (doIndex env2 view)
doIndexEqIsEq {sx = sx :< n} ((:<) {t, u, thin1, thin2} prf eq) Here = Calc $
  |~ wkn t (drop thin1 n)
  ~~ wkn (wkn t thin1) (drop (id sx) n) ...(expandShift (MkRes t thin1))
  ~~ wkn (wkn u thin2) (drop (id sx) n) ...(cong (\t => wkn t (drop (id sx) n)) $ eq)
  ~~ wkn u (drop thin2 n)               ...(sym $ expandShift (MkRes u thin2))
doIndexEqIsEq {sx = sx :< n} ((:<) {env1, env2} prf eq) (There i) = Calc $
  |~ expand (shift $ doIndex env1 $ view i)
  ~~ wkn (expand $ doIndex env1 $ view i) (drop (id sx) n) ...(expandShift $ doIndex env1 $ view i)
  ~~ wkn (expand $ doIndex env2 $ view i) (drop (id sx) n) ...(cong (\t => wkn t (drop (id sx) n)) $ doIndexEqIsEq prf (view i))
  ~~ expand (shift $ doIndex env2 $ view i)                ...(sym $ expandShift $ doIndex env2 $ view i)

export
indexEqIsEq : EnvEq env1 env2 -> (i : Var sx) -> index env1 i = index env2 i
indexEqIsEq prf i = doIndexEqIsEq prf $ view i