module Core.Var

import Core.Context
import Core.Thinning

%prefix_record_projections off

-- Variables -------------------------------------------------------------------

data IsVar : Nat -> Context -> Type where
  Here : IsVar 0 (sx :< n)
  There : IsVar k sx -> IsVar (S k) (sx :< n)

export
record Var (sx : Context) where
  constructor MakeVar
  index : Nat
  0 prf : IsVar index sx

%name IsVar prf
%name Var i, j, k

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

export
here : Var (sx :< n)
here = MakeVar 0 Here

export
there : Var sx -> Var (sx :< n)
there (MakeVar index prf) = MakeVar (S index) (There prf)

-- Views -----------------------------------------------------------------------

namespace View
  public export
  data View : Var sx -> Type where
    Here : View Core.Var.here
    There : (i : Var sx) -> View (there i)

  %name View.View view

export
view : (i : Var sx) -> View i
view (MakeVar 0 Here) = Here
view (MakeVar (S k) (There prf)) = There (MakeVar k prf)

-- Views are Unique

viewInverse : {0 i : Var sx} -> View i -> Var sx
viewInverse Here = here
viewInverse (There i) = there i

viewInversePrf1 : (view : View i) -> viewInverse view = i
viewInversePrf1 Here = Refl
viewInversePrf1 (There i) = Refl

viewInversePrf2 :
  {0 i : Var sx} ->
  (v : View i) ->
  view (viewInverse v) = (rewrite viewInversePrf1 v in v)
viewInversePrf2 Here = Refl
viewInversePrf2 (There (MakeVar index prf)) = Refl

export
viewUnique : {0 i : Var sx} -> (view1, view2 : View i) -> view1 = view2
viewUnique view1 view2 =
  rewrite sym $ viewInversePrf2 view1 in
  rewrite sym $ viewInversePrf2 view2 in
  rewrite viewInversePrf1 view1 in
  rewrite viewInversePrf1 view2 in
  Refl

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

doWkn : Var sx -> {0 thin : sx `Thins` sy} -> View thin -> Var sy
doWkn i (Id sy) = i
doWkn i (Drop thin n) = there $ doWkn i (view thin)
doWkn (MakeVar 0 Here) (Keep thin n) = here
doWkn (MakeVar (S k) (There prf)) (Keep thin n) = there $ doWkn (MakeVar k prf) (view thin)

export
wkn : Var sx -> sx `Thins` sy -> Var sy
wkn i thin = doWkn i (view thin)