(** More theorems about INR *)

(* Specifications: *)
Lemma INRp1_pos : forall n, 0 < INR n + 1.
Lemma Rlt_nat (x : R) : (exists n : nat, x = INR (S n)) -> 0 < x.
Lemma Rle_pow_lin (a : R) (n : nat) :
0 <= a -> 1 + INR n * a <= (1 + a) ^ n.
Lemma C_n_n: forall n, C n n = 1.
Lemma C_n_0: forall n, C n 0 = 1.
Fixpoint pow2 (n : nat) : nat :=
match n with
| O => 1%nat
| S n => (2 * pow2 n)%nat
end.
Lemma pow2_INR (n : nat) : INR (pow2 n) = 2^n.
Lemma pow2_pos (n : nat) : (0 < pow2 n)%nat.
(** * Rinv *)
Lemma Rinv_le_contravar :
forall x y, 0 < x -> x <= y -> / y <= / x.
Lemma Rinv_lt_cancel (x y : R) :
0 < y -> / y < / x -> x < y.