End PartialTreeboxRep_Ind_Pred_Ind_Hole.
Module PartialTreeRep_WandQFrame_Func_Hole.
Export PartialTree_WandQFrame_Func_Hole.
Definition partial_tree_rep := partialT tree_rep.

Lemma partial_tree_rep_singleton_left: forall (t2: tree val) k (v p p1 p2: val),
Int.min_signed <= Z.of_nat k <= Int.max_signed ->
is_pointer_or_null v ->
data_at Tsh t_struct_tree (Vint (Int.repr (Z.of_nat k)), (v, (p1, p2))) p *
tree_rep t2 p2
|-- partial_tree_rep (fun t1 => T t1 k v t2) p p1.