Definition cPairPi1 (a : nat) := a - sumToN (searchXY a).Definition cPairPi2 (a : nat) := searchXY a - cPairPi1 a.

(* Specifications: *)
Lemma cPairProjectionsHelp : forall a b : nat, b < sumToN (S a) -> sumToN a <= b -> searchXY b = a.
Lemma cPairProjections : forall a : nat, cPair (cPairPi1 a) (cPairPi2 a) = a.