End PartialTreeboxRep_WandFrame.
Module PartialTreeboxRep_WandQFrame_Ind_Hole.
Definition partial_treebox_rep (pt: partial_tree val) (p_root p_in: val): mpred :=
ALL t: tree val, treebox_rep t p_in -* treebox_rep (partial_tree_tree pt t) p_root.

Lemma partial_treebox_rep_singleton_left: forall (t2: tree val) k (v p b: val),
Int.min_signed <= Z.of_nat k <= Int.max_signed ->
is_pointer_or_null v ->
data_at Tsh (tptr t_struct_tree) p b *
field_at Tsh t_struct_tree [StructField _key] (Vint (Int.repr (Z.of_nat k))) p *
field_at Tsh t_struct_tree [StructField _value] v p *
treebox_rep t2 (field_address t_struct_tree [StructField _right] p)
|-- partial_treebox_rep (L H k v t2) b (field_address t_struct_tree [StructField _left] p).