(* Here we have an unfortunate detour to develop a theory
     of many-hole contexts, which allows us to prove
     that relativization is injective on the left. *)

  Inductive Sctx : Set :=
  | NodeB : Sctx -> Sctx -> Sctx
  | NodeR : ShareTree -> Sctx -> Sctx
  | NodeL : Sctx -> ShareTree -> Sctx
  | Hole : Sctx
  .(* Here we have an unfortunate detour to develop a theory
     of many-hole contexts, which allows us to prove
     that relativization is injective on the left. *)

  Inductive Sctx : Set :=
  | NodeB : Sctx -> Sctx -> Sctx
  | NodeR : ShareTree -> Sctx -> Sctx
  | NodeL : Sctx -> ShareTree -> Sctx
  | Hole : Sctx
  .(* Here we have an unfortunate detour to develop a theory
     of many-hole contexts, which allows us to prove
     that relativization is injective on the left. *)

  Inductive Sctx : Set :=
  | NodeB : Sctx -> Sctx -> Sctx
  | NodeR : ShareTree -> Sctx -> Sctx
  | NodeL : Sctx -> ShareTree -> Sctx
  | Hole : Sctx
  .(* Here we have an unfortunate detour to develop a theory
     of many-hole contexts, which allows us to prove
     that relativization is injective on the left. *)

  Inductive Sctx : Set :=
  | NodeB : Sctx -> Sctx -> Sctx
  | NodeR : ShareTree -> Sctx -> Sctx
  | NodeL : Sctx -> ShareTree -> Sctx
  | Hole : Sctx
  .(* Here we have an unfortunate detour to develop a theory
     of many-hole contexts, which allows us to prove
     that relativization is injective on the left. *)

  Inductive Sctx : Set :=
  | NodeB : Sctx -> Sctx -> Sctx
  | NodeR : ShareTree -> Sctx -> Sctx
  | NodeL : Sctx -> ShareTree -> Sctx
  | Hole : Sctx
  .Fixpoint Sctx_app (c1 c2:Sctx) {struct c1} : Sctx :=
  match c1 with
  | NodeB l r => NodeB (Sctx_app l c2) (Sctx_app r c2)
  | NodeR l r => NodeR l (Sctx_app r c2)
  | NodeL l r => NodeL (Sctx_app l c2) r
  | Hole      => c2
  end.Fixpoint fill (c:Sctx) (x:ShareTree) {struct c} : ShareTree :=
  match c with
  | NodeR l r => Node l (fill r x)
  | NodeL l r => Node (fill l x) r
  | NodeB l r => Node (fill l x) (fill r x)
  | Hole      => x
  end.

Lemma fill_app : forall c1 c2 x, fill c1 (fill c2 x) = fill (Sctx_app c1 c2) x.