# Library Coq.Reals.ArithProp

Require Import Rbase.
Require Import Rbasic_fun.
Require Import Even.
Require Import Div2.
Require Import ArithRing.
Open Local Scope Z_scope.
Open Local Scope R_scope.
Lemma minus_neq_O : forall n i:nat, (i < n)%nat -> (n - i)%nat <> 0%nat.
Lemma le_minusni_n : forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat.
Lemma lt_minus_O_lt : forall m n:nat, (m < n)%nat -> (0 < n - m)%nat.
Lemma even_odd_cor :
forall n:nat, exists p : nat , n = (2 * p)%nat \/ n = S (2 * p).
Lemma le_double : forall m n:nat, (2 * m <= 2 * n)%nat -> (m <= n)%nat.
Here, we have the euclidian division
This lemma is used in the proof of sin_eq_0 : (sin x)=0<->x=kPI
Lemma euclidian_division :
forall x y:R,
y <> 0 ->
exists k : Z , ( exists r : R , x = IZR k * y + r /\ 0 <= r < Rabs y ).
Lemma tech8 : forall n i:nat, (n <= S n + i)%nat.