Module Qcert.cNRAEnv.Context.cNRAEnvContext

Require Import Equivalence.
Require Import Morphisms.
Require Import Setoid.
Require Import EquivDec.
Require Import Program.
Require Import List.
Require Import Permutation.
Require Import String.
Require Import NPeano.
Require Import Arith.
Require Import Bool.
Require Import Utils.
Require Import BindingsNat.
Require Import DataRuntime.
Require Import cNRAEnv.
Require Import cNRAEnvEq.

Declare Scope nraenv_core_ctxt_scope.

Section cNRAEnvContext.
  Local Open Scope nraenv_core_scope.

  Context {fruntime:foreign_runtime}.

  Inductive nraenv_core_ctxt : Set :=
  | CcNRAEnvHole : nat -> nraenv_core_ctxt
  | CcNRAEnvPlug : nraenv_core -> nraenv_core_ctxt
  | CcNRAEnvBinop : binary_op -> nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvUnop : unary_op -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvMap : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvMapProduct : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvProduct : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvSelect : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvDefault : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvEither : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvEitherConcat : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvApp : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvAppEnv : nraenv_core_ctxt -> nraenv_core_ctxt -> nraenv_core_ctxt
  | CcNRAEnvMapEnv : nraenv_core_ctxt -> nraenv_core_ctxt
  .

  Definition CcNRAEnvID : nraenv_core_ctxt
    := CcNRAEnvPlug cNRAEnvID.

    Definition CcNRAEnvEnv : nraenv_core_ctxt
    := CcNRAEnvPlug cNRAEnvEnv.

    Definition CcNRAEnvGetConstant s : nraenv_core_ctxt
    := CcNRAEnvPlug (cNRAEnvGetConstant s).

  Definition CcNRAEnvConst : data -> nraenv_core_ctxt
    := fun d => CcNRAEnvPlug (cNRAEnvConst d).

  Fixpoint aec_holes (c:nraenv_core_ctxt) : list nat :=
    match c with
      | CcNRAEnvHole x => x::nil
      | CcNRAEnvPlug a => nil
      | CcNRAEnvBinop b c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvUnop u c' => aec_holes c'
      | CcNRAEnvMap c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvMapProduct c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvProduct c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvSelect c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvDefault c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvEither c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvEitherConcat c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvApp c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvAppEnv c1 c2 => aec_holes c1 ++ aec_holes c2
      | CcNRAEnvMapEnv c1 => aec_holes c1
    end.

  Fixpoint aec_simplify (c:nraenv_core_ctxt) : nraenv_core_ctxt :=
    match c with
      | CcNRAEnvHole x => CcNRAEnvHole x
      | CcNRAEnvPlug a => CcNRAEnvPlug a
      | CcNRAEnvBinop b c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvBinop b a1 a2)
          | c1', c2' => CcNRAEnvBinop b c1' c2'
        end
      | CcNRAEnvUnop u c =>
        match aec_simplify c with
          | CcNRAEnvPlug a => CcNRAEnvPlug (cNRAEnvUnop u a)
          | c' => CcNRAEnvUnop u c'
        end
      | CcNRAEnvMap c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvMap a1 a2)
          | c1', c2' => CcNRAEnvMap c1' c2'
        end
      | CcNRAEnvMapProduct c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvMapProduct a1 a2)
          | c1', c2' => CcNRAEnvMapProduct c1' c2'
        end
      | CcNRAEnvProduct c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvProduct a1 a2)
          | c1', c2' => CcNRAEnvProduct c1' c2'
        end
      | CcNRAEnvSelect c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvSelect a1 a2)
          | c1', c2' => CcNRAEnvSelect c1' c2'
        end
      | CcNRAEnvDefault c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvDefault a1 a2)
          | c1', c2' => CcNRAEnvDefault c1' c2'
        end
      | CcNRAEnvEither c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvEither a1 a2)
          | c1', c2' => CcNRAEnvEither c1' c2'
        end
      | CcNRAEnvEitherConcat c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvEitherConcat a1 a2)
          | c1', c2' => CcNRAEnvEitherConcat c1' c2'
        end
      | CcNRAEnvApp c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvApp a1 a2)
          | c1', c2' => CcNRAEnvApp c1' c2'
        end
      | CcNRAEnvAppEnv c1 c2 =>
        match aec_simplify c1, aec_simplify c2 with
          | (CcNRAEnvPlug a1), (CcNRAEnvPlug a2) => CcNRAEnvPlug (cNRAEnvAppEnv a1 a2)
          | c1', c2' => CcNRAEnvAppEnv c1' c2'
        end
      | CcNRAEnvMapEnv c1 =>
        match aec_simplify c1 with
          | (CcNRAEnvPlug a1) => CcNRAEnvPlug (cNRAEnvMapEnv a1)
          | c1' => CcNRAEnvMapEnv c1'
        end
    end.

  Lemma aec_simplify_holes_preserved c :
    aec_holes (aec_simplify c) = aec_holes c.

  Definition aec_nraenv_core_of_ctxt c
    := match (aec_simplify c) with
         | CcNRAEnvPlug a => Some a
         | _ => None
       end.

  Lemma aec_simplify_nholes c :
    aec_holes c = nil -> {a | aec_simplify c = CcNRAEnvPlug a}.

  Lemma aec_nraenv_core_of_ctxt_nholes c :
    aec_holes c = nil -> {a | aec_nraenv_core_of_ctxt c = Some a}.

  Lemma aec_simplify_idempotent c :
    aec_simplify (aec_simplify c) = aec_simplify c.

  Fixpoint aec_subst (c:nraenv_core_ctxt) (x:nat) (p:nraenv_core) : nraenv_core_ctxt :=
    match c with
      | CcNRAEnvHole x'
        => if x == x' then CcNRAEnvPlug p else CcNRAEnvHole x'
      | CcNRAEnvPlug a
        => CcNRAEnvPlug a
      | CcNRAEnvBinop b c1 c2
        => CcNRAEnvBinop b (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvUnop u c
        => CcNRAEnvUnop u (aec_subst c x p)
      | CcNRAEnvMap c1 c2
        => CcNRAEnvMap (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvMapProduct c1 c2
        => CcNRAEnvMapProduct (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvProduct c1 c2
        => CcNRAEnvProduct (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvSelect c1 c2
        => CcNRAEnvSelect (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvDefault c1 c2
        => CcNRAEnvDefault (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvEither c1 c2
        => CcNRAEnvEither (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvEitherConcat c1 c2
        => CcNRAEnvEitherConcat (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvApp c1 c2
        => CcNRAEnvApp (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvAppEnv c1 c2
        => CcNRAEnvAppEnv (aec_subst c1 x p) (aec_subst c2 x p)
      | CcNRAEnvMapEnv c1
        => CcNRAEnvMapEnv (aec_subst c1 x p)
    end.

  Definition aec_substp (c:nraenv_core_ctxt) xp
    := let '(x, p) := xp in aec_subst c x p.
    
  Definition aec_substs c ps :=
    fold_left aec_substp ps c.

  
   Lemma aec_substs_app c ps1 ps2 :
     aec_substs c (ps1 ++ ps2) =
     aec_substs (aec_substs c ps1) ps2.

  Lemma aec_subst_holes_nin c x p :
    ~ In x (aec_holes c) -> aec_subst c x p = c.

  Lemma aec_subst_holes_remove c x p :
    aec_holes (aec_subst c x p) = remove_all x (aec_holes c).

    Lemma aec_substp_holes_remove c xp :
      aec_holes (aec_substp c xp) = remove_all (fst xp) (aec_holes c).

 Lemma aec_substs_holes_remove c ps :
    aec_holes (aec_substs c ps) =
    fold_left (fun h xy => remove_all (fst xy) h) ps (aec_holes c).
  
  Lemma aec_substs_Plug a ps :
    aec_substs (CcNRAEnvPlug a) ps = CcNRAEnvPlug a.
  
  Lemma aec_substs_Binop b c1 c2 ps :
    aec_substs (CcNRAEnvBinop b c1 c2) ps = CcNRAEnvBinop b (aec_substs c1 ps) (aec_substs c2 ps).

  Lemma aec_substs_Unop u c ps :
      aec_substs (CcNRAEnvUnop u c) ps = CcNRAEnvUnop u (aec_substs c ps).

  Lemma aec_substs_Map c1 c2 ps :
    aec_substs ( CcNRAEnvMap c1 c2) ps =
    CcNRAEnvMap (aec_substs c1 ps) (aec_substs c2 ps).

  Lemma aec_substs_MapProduct c1 c2 ps :
    aec_substs ( CcNRAEnvMapProduct c1 c2) ps =
    CcNRAEnvMapProduct (aec_substs c1 ps) (aec_substs c2 ps).
  
  Lemma aec_substs_Product c1 c2 ps :
    aec_substs ( CcNRAEnvProduct c1 c2) ps =
    CcNRAEnvProduct (aec_substs c1 ps) (aec_substs c2 ps).
  
  Lemma aec_substs_Select c1 c2 ps :
    aec_substs ( CcNRAEnvSelect c1 c2) ps =
    CcNRAEnvSelect (aec_substs c1 ps) (aec_substs c2 ps).
  
  Lemma aec_substs_Default c1 c2 ps :
    aec_substs ( CcNRAEnvDefault c1 c2) ps =
    CcNRAEnvDefault (aec_substs c1 ps) (aec_substs c2 ps).
  
  Lemma aec_substs_Either c1 c2 ps :
    aec_substs ( CcNRAEnvEither c1 c2) ps =
    CcNRAEnvEither (aec_substs c1 ps) (aec_substs c2 ps).
  
  Lemma aec_substs_EitherConcat c1 c2 ps :
    aec_substs ( CcNRAEnvEitherConcat c1 c2) ps =
    CcNRAEnvEitherConcat (aec_substs c1 ps) (aec_substs c2 ps).
  
  Lemma aec_substs_App c1 c2 ps :
    aec_substs ( CcNRAEnvApp c1 c2) ps =
    CcNRAEnvApp (aec_substs c1 ps) (aec_substs c2 ps).
  
  Lemma aec_substs_AppEnv c1 c2 ps :
    aec_substs ( CcNRAEnvAppEnv c1 c2) ps =
    CcNRAEnvAppEnv (aec_substs c1 ps) (aec_substs c2 ps).
  
  Lemma aec_substs_MapEnv c1 ps :
    aec_substs ( CcNRAEnvMapEnv c1) ps =
    CcNRAEnvMapEnv (aec_substs c1 ps).

  Local Hint Rewrite
       aec_substs_Plug
       aec_substs_Binop
       aec_substs_Unop
       aec_substs_Map
       aec_substs_MapProduct
       aec_substs_Product
       aec_substs_Select
       aec_substs_Default
       aec_substs_Either
       aec_substs_EitherConcat
       aec_substs_App
       aec_substs_AppEnv
       aec_substs_MapEnv : aec_substs.
  
  Lemma aec_simplify_holes_binary_op b c1 c2:
    aec_holes (CcNRAEnvBinop b c1 c2) <> nil ->
    aec_simplify (CcNRAEnvBinop b c1 c2) = CcNRAEnvBinop b (aec_simplify c1) (aec_simplify c2).

  Lemma aec_simplify_holes_unary_op u c:
    aec_holes (CcNRAEnvUnop u c ) <> nil ->
    aec_simplify (CcNRAEnvUnop u c) = CcNRAEnvUnop u (aec_simplify c).

  Lemma aec_simplify_holes_map c1 c2:
    aec_holes (CcNRAEnvMap c1 c2) <> nil ->
    aec_simplify (CcNRAEnvMap c1 c2) = CcNRAEnvMap (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_mapconcat c1 c2:
    aec_holes (CcNRAEnvMapProduct c1 c2) <> nil ->
    aec_simplify (CcNRAEnvMapProduct c1 c2) = CcNRAEnvMapProduct (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_product c1 c2:
    aec_holes (CcNRAEnvProduct c1 c2) <> nil ->
    aec_simplify (CcNRAEnvProduct c1 c2) = CcNRAEnvProduct (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_select c1 c2:
    aec_holes (CcNRAEnvSelect c1 c2) <> nil ->
    aec_simplify (CcNRAEnvSelect c1 c2) = CcNRAEnvSelect (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_default c1 c2:
    aec_holes (CcNRAEnvDefault c1 c2) <> nil ->
    aec_simplify (CcNRAEnvDefault c1 c2) = CcNRAEnvDefault (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_either c1 c2:
    aec_holes (CcNRAEnvEither c1 c2) <> nil ->
    aec_simplify (CcNRAEnvEither c1 c2) = CcNRAEnvEither (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_eitherconcat c1 c2:
    aec_holes (CcNRAEnvEitherConcat c1 c2) <> nil ->
    aec_simplify (CcNRAEnvEitherConcat c1 c2) = CcNRAEnvEitherConcat (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_app c1 c2:
    aec_holes (CcNRAEnvApp c1 c2) <> nil ->
    aec_simplify (CcNRAEnvApp c1 c2) = CcNRAEnvApp (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_appenv c1 c2:
    aec_holes (CcNRAEnvAppEnv c1 c2) <> nil ->
    aec_simplify (CcNRAEnvAppEnv c1 c2) = CcNRAEnvAppEnv (aec_simplify c1) (aec_simplify c2).

    Lemma aec_simplify_holes_mapenv c1 :
    aec_holes (CcNRAEnvMapEnv c1) <> nil ->
    aec_simplify (CcNRAEnvMapEnv c1) = CcNRAEnvMapEnv (aec_simplify c1).

   Lemma aec_subst_nholes c x p :
     (aec_holes c) = nil -> aec_subst c x p = c.

   Lemma aec_subst_simplify_nholes c x p :
     (aec_holes c) = nil ->
     aec_subst (aec_simplify c) x p = aec_simplify c.

  Lemma aec_simplify_subst_simplify1 c x p :
    aec_simplify (aec_subst (aec_simplify c) x p) =
    aec_simplify (aec_subst c x p).

  Lemma aec_simplify_substs_simplify1 c ps :
    aec_simplify (aec_substs (aec_simplify c) ps) =
    aec_simplify (aec_substs c ps).

    Section equivs.
    Context (base_equiv:nraenv_core->nraenv_core->Prop).

    Definition nraenv_core_ctxt_equiv (c1 c2 : nraenv_core_ctxt)
      := forall (ps:list (nat * nraenv_core)),
           match aec_simplify (aec_substs c1 ps),
                 aec_simplify (aec_substs c2 ps)
           with
             | CcNRAEnvPlug a1, CcNRAEnvPlug a2 => base_equiv a1 a2
             | _, _ => True
           end.

   Definition nraenv_core_ctxt_equiv_strict (c1 c2 : nraenv_core_ctxt)
     := forall (ps:list (nat * nraenv_core)),
          is_list_sorted lt_dec (domain ps) = true ->
          equivlist (domain ps) (aec_holes c1 ++ aec_holes c2) ->
          match aec_simplify (aec_substs c1 ps),
                aec_simplify (aec_substs c2 ps)
          with
            | CcNRAEnvPlug a1, CcNRAEnvPlug a2 => base_equiv a1 a2
            | _, _ => True
          end.

   Global Instance aec_simplify_proper :
     Proper (nraenv_core_ctxt_equiv ==> nraenv_core_ctxt_equiv) aec_simplify.
  
  Lemma aec_simplify_proper_inv x y:
    nraenv_core_ctxt_equiv (aec_simplify x) (aec_simplify y) -> nraenv_core_ctxt_equiv x y.

 Instance aec_subst_proper_part1 :
   Proper (nraenv_core_ctxt_equiv ==> eq ==> eq ==> nraenv_core_ctxt_equiv) aec_subst.

  Global Instance aec_substs_proper_part1: Proper (nraenv_core_ctxt_equiv ==> eq ==> nraenv_core_ctxt_equiv) aec_substs.

  Definition nraenv_core_ctxt_equiv_strict1 (c1 c2 : nraenv_core_ctxt)
     := forall (ps:list (nat * nraenv_core)),
          NoDup (domain ps) ->
          equivlist (domain ps) (aec_holes c1 ++ aec_holes c2) ->
          match aec_simplify (aec_substs c1 ps),
                aec_simplify (aec_substs c2 ps)
          with
            | CcNRAEnvPlug a1, CcNRAEnvPlug a2 => base_equiv a1 a2
            | _, _ => True
          end.

   Lemma aec_subst_swap_neq c x1 x2 y1 y2 :
     x1 <> x2 ->
   aec_subst (aec_subst c x1 y1) x2 y2 =
   aec_subst (aec_subst c x2 y2) x1 y1.
   
   Lemma aec_subst_swap_eq c x1 y1 y2 :
     aec_subst (aec_subst c x1 y1) x1 y2 =
     aec_subst c x1 y1.
           
   Lemma aec_substs_subst_swap_simpl x c y ps :
     ~ In x (domain ps) ->
      aec_substs (aec_subst c x y) ps
      =
      (aec_subst (aec_substs c ps) x y).
   
   Lemma aec_substs_perm c ps1 ps2 :
     NoDup (domain ps1) ->
     Permutation ps1 ps2 ->
     (aec_substs c ps1) =
     (aec_substs c ps2).
       
   Lemma nraenv_core_ctxt_equiv_strict_equiv1 (c1 c2 : nraenv_core_ctxt) :
     nraenv_core_ctxt_equiv_strict1 c1 c2 <-> nraenv_core_ctxt_equiv_strict c1 c2.

   Definition nraenv_core_ctxt_equiv_strict2 (c1 c2 : nraenv_core_ctxt)
     := forall (ps:list (nat * nraenv_core)),
          equivlist (domain ps) (aec_holes c1 ++ aec_holes c2) ->
          match aec_simplify (aec_substs c1 ps),
                aec_simplify (aec_substs c2 ps)
          with
            | CcNRAEnvPlug a1, CcNRAEnvPlug a2 => base_equiv a1 a2
            | _, _ => True
          end.

   Lemma aec_substs_in_nholes c x ps :
         In x (domain ps) ->
      ~ In x (aec_holes (aec_substs c ps)).
    
   Lemma substs_bdistinct_domain_rev c ps :
    (aec_substs c (bdistinct_domain (rev ps)))
    =
    (aec_substs c ps).
  
   Lemma nraenv_core_ctxt_equiv_strict1_equiv2 (c1 c2 : nraenv_core_ctxt) :
     nraenv_core_ctxt_equiv_strict2 c1 c2 <-> nraenv_core_ctxt_equiv_strict1 c1 c2.

   Definition nraenv_core_ctxt_equiv_strict3 (c1 c2 : nraenv_core_ctxt)
     := forall (ps:list (nat * nraenv_core)),
          incl (aec_holes c1 ++ aec_holes c2) (domain ps) ->
          match aec_simplify (aec_substs c1 ps),
                aec_simplify (aec_substs c2 ps)
          with
            | CcNRAEnvPlug a1, CcNRAEnvPlug a2 => base_equiv a1 a2
            | _, _ => True
          end.
   
   Lemma cut_down_to_substs c ps cut :
     incl (aec_holes c) cut ->
     (aec_substs c ps) = (aec_substs c (cut_down_to ps cut)).
         
   Lemma nraenv_core_ctxt_equiv_strict2_equiv3 (c1 c2 : nraenv_core_ctxt) :
     nraenv_core_ctxt_equiv_strict3 c1 c2 <-> nraenv_core_ctxt_equiv_strict2 c1 c2.

   Lemma nraenv_core_ctxt_equiv_strict3_equiv (c1 c2 : nraenv_core_ctxt) :
     nraenv_core_ctxt_equiv c1 c2 <-> nraenv_core_ctxt_equiv_strict3 c1 c2.

   Theorem nraenv_core_ctxt_equiv_strict_equiv (c1 c2 : nraenv_core_ctxt) :
     nraenv_core_ctxt_equiv c1 c2 <-> nraenv_core_ctxt_equiv_strict c1 c2.

   Lemma aec_holes_saturated_subst {B} f c ps :
      incl (aec_holes c) (domain ps) ->
      aec_holes
        (aec_substs c
                   (map (fun xy : nat * B => (fst xy, (f (snd xy)))) ps)) = nil.

  Global Instance nraenv_core_ctxt_equiv_refl {refl:Reflexive base_equiv}: Reflexive nraenv_core_ctxt_equiv.

  Global Instance nraenv_core_ctxt_equiv_sym {sym:Symmetric base_equiv}: Symmetric nraenv_core_ctxt_equiv.

  Global Instance nraenv_core_ctxt_equiv_trans {trans:Transitive base_equiv}: Transitive nraenv_core_ctxt_equiv.

  Global Instance nraenv_core_ctxt_equiv_equivalence {equiv:Equivalence base_equiv}: Equivalence nraenv_core_ctxt_equiv.

  Global Instance nraenv_core_ctxt_equiv_preorder {pre:PreOrder base_equiv} : PreOrder nraenv_core_ctxt_equiv.

  Global Instance nraenv_core_ctxt_equiv_strict_refl {refl:Reflexive base_equiv}: Reflexive nraenv_core_ctxt_equiv_strict.

  Global Instance nraenv_core_ctxt_equiv_strict_sym {sym:Symmetric base_equiv}: Symmetric nraenv_core_ctxt_equiv_strict.

  Global Instance nraenv_core_ctxt_equiv_strict_trans {trans:Transitive base_equiv}: Transitive nraenv_core_ctxt_equiv_strict.
  
  Global Instance nraenv_core_ctxt_equiv_strict_equivalence {equiv:Equivalence base_equiv}: Equivalence nraenv_core_ctxt_equiv_strict.

  Global Instance nraenv_core_ctxt_equiv_strict_preorder {pre:PreOrder base_equiv} : PreOrder nraenv_core_ctxt_equiv_strict.

  Global Instance CcNRAEnvPlug_proper :
    Proper (base_equiv ==> nraenv_core_ctxt_equiv) CcNRAEnvPlug.

  Global Instance CcNRAEnvPlug_proper_strict :
    Proper (base_equiv ==> nraenv_core_ctxt_equiv_strict) CcNRAEnvPlug.

  End equivs.
  
End cNRAEnvContext.


Delimit Scope nraenv_core_ctxt_scope with nraenv_core_ctxt.

Notation "'ID'" := (CcNRAEnvID) (at level 50) : nraenv_core_ctxt_scope.
Notation "'ENV'" := (CcNRAEnvEnv) (at level 50) : nraenv_core_ctxt_scope.

Notation "‵‵ c" := (CcNRAEnvConst (dconst c)) (at level 0) : nraenv_core_ctxt_scope.
Notation "‵ c" := (CcNRAEnvConst c) (at level 0) : nraenv_core_ctxt_scope.
Notation "‵{||}" := (CcNRAEnvConst (dcoll nil)) (at level 0) : nraenv_core_ctxt_scope.
Notation "‵[||]" := (CcNRAEnvConst (drec nil)) (at level 50) : nraenv_core_ctxt_scope.

Notation "r1 ∧ r2" := (CcNRAEnvBinop OpAnd r1 r2) (right associativity, at level 65): nraenv_core_ctxt_scope.
Notation "r1 ∨ r2" := (CcNRAEnvBinop OpOr r1 r2) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "r1 ≐ r2" := (CcNRAEnvBinop OpEqual r1 r2) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "r1 ≤ r2" := (CcNRAEnvBinop OpLe r1 r2) (no associativity, at level 70): nraenv_core_ctxt_scope.
Notation "r1 ⋃ r2" := (CcNRAEnvBinop OpBagUnion r1 r2) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "r1 − r2" := (CcNRAEnvBinop OpBagDiff r1 r2) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "r1 ♯min r2" := (CcNRAEnvBinop OpBagMin r1 r2) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "r1 ♯max r2" := (CcNRAEnvBinop OpBagMax r1 r2) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "p ⊕ r" := ((CcNRAEnvBinop OpRecConcat) p r) (at level 70) : nraenv_core_ctxt_scope.
Notation "p ⊗ r" := ((CcNRAEnvBinop OpRecMerge) p r) (at level 70) : nraenv_core_ctxt_scope.

Notation "¬( r1 )" := (CcNRAEnvUnop OpNeg r1) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "ε( r1 )" := (CcNRAEnvUnop OpDistinct r1) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "♯count( r1 )" := (CcNRAEnvUnop OpCount r1) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "♯flatten( d )" := (CcNRAEnvUnop OpFlatten d) (at level 50) : nraenv_core_ctxt_scope.
Notation "‵{| d |}" := ((CcNRAEnvUnop OpBag) d) (at level 50) : nraenv_core_ctxt_scope.
Notation "‵[| ( s , r ) |]" := ((CcNRAEnvUnop (OpRec s)) r) (at level 50) : nraenv_core_ctxt_scope.
Notation "¬π[ s1 ]( r )" := ((CcNRAEnvUnop (OpRecRemove s1)) r) (at level 50) : nraenv_core_ctxt_scope.
Notation "p · r" := ((CcNRAEnvUnop (OpDot r)) p) (left associativity, at level 40): nraenv_core_ctxt_scope.

Notation "χ⟨ p ⟩( r )" := (CcNRAEnvMap p r) (at level 70) : nraenv_core_ctxt_scope.
Notation "⋈ᵈ⟨ e2 ⟩( e1 )" := (CcNRAEnvMapProduct e2 e1) (at level 70) : nraenv_core_ctxt_scope.
Notation "r1 × r2" := (CcNRAEnvProduct r1 r2) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "σ⟨ p ⟩( r )" := (CcNRAEnvSelect p r) (at level 70) : nraenv_core_ctxt_scope.
Notation "r1 ∥ r2" := (CcNRAEnvDefault r1 r2) (right associativity, at level 70): nraenv_core_ctxt_scope.
Notation "r1 ◯ r2" := (CcNRAEnvApp r1 r2) (right associativity, at level 60): nraenv_core_ctxt_scope.

Notation "r1 ◯ₑ r2" := (CcNRAEnvAppEnv r1 r2) (right associativity, at level 60): nraenv_core_ctxt_scope.

Notation "χᵉ⟨ p ⟩( r )" := (CcNRAEnvMapEnv p r) (at level 70) : nraenv_core_ctxt_scope.

Notation "$ n" := (CcNRAEnvHole n) (at level 50) : nraenv_core_ctxt_scope.

Notation "X ≡ₑ Y" := (nraenv_core_ctxt_equiv nraenv_core_eq X Y) (at level 90) : nraenv_core_ctxt_scope.

Global Hint Rewrite
       @aec_substs_Plug
       @aec_substs_Binop
       @aec_substs_Unop
       @aec_substs_Map
       @aec_substs_MapProduct
       @aec_substs_Product
       @aec_substs_Select
       @aec_substs_Default
       @aec_substs_Either
       @aec_substs_EitherConcat
       @aec_substs_App
       @aec_substs_AppEnv
       @aec_substs_MapEnv : aec_substs.