Definition t := canonTree.Definition Ord (x y:canonTree) := shareTreeOrd (proj1_sig x) (proj1_sig y).Definition lub (x y:canonTree) : canonTree :=
      exist (fun t => canonicalTree t)
      (mkCanon (union_tree (proj1_sig x) (proj1_sig y)))
      (mkCanon_correct _).Definition glb (x y:canonTree) : canonTree :=
      exist (fun t => canonicalTree t)
      (mkCanon (intersect_tree (proj1_sig x) (proj1_sig y)))
      (mkCanon_correct _).Definition top : canonTree := exist (fun t => canonicalTree t) (Leaf true) I.Definition bot : canonTree := exist (fun t => canonicalTree t) (Leaf false) I.Definition comp (x:canonTree) : canonTree :=
      exist (fun t => canonicalTree t) (comp_tree (proj1_sig x))
      (comp_canonical _ (proj2_sig x)).

Lemma lub_upper2 : forall x y, Ord y (lub x y).