(** This is a weakened form of CRealLt_aboveSig which a special shape of eps needed below *)

(* Specifications: *)
Lemma CRealLt_aboveSig' : forall (x y : CReal) (n : Z),
(2 * 2^n < seq y n - seq x n)%Q
-> let (k, _) := QarchimedeanExp2_Z (/(seq y n - seq x n - (2 * 2 ^ n)%Q))
in forall p:Z,
(p <= n)%Z
-> (2 * 2^(Z.min (-k-1) n) < seq y p - seq x p)%Q.
Lemma CRealLt_above : forall (x y : CReal),
CRealLt x y
-> { n : Z | forall p : Z,
(p <= n)%Z -> (2 * 2 ^ n < seq y p - seq x p)%Q }.
(* The CRealLt index separates the Cauchy sequences *)
Lemma CRealLt_above_same : forall (x y : CReal) (n : Z),
(2 * 2 ^ n < seq y n - seq x n)%Q
-> forall p:Z, (p <= n)%Z -> Qlt (seq x p) (seq y p).
Lemma CRealLt_asym : forall x y : CReal, x < y -> x <= y.
Lemma CRealLt_irrefl : forall x:CReal, x < x -> False.
Lemma CRealLe_refl : forall x : CReal, x <= x.
Lemma CRealEq_refl : forall x : CReal, x == x.
Lemma CRealEq_sym : forall x y : CReal, CRealEq x y -> CRealEq y x.
Lemma CRealLt_dec : forall x y z : CReal,
x < y -> sum (x < z) (z < y).