Library Coq.Reals.ArithProp

Require Import Rdefinitions Raxioms RIneq.
Require Import Rbasic_fun.
Require Import Even.
Require Import Div2.
Require Import ArithRing.

Local Open Scope Z_scope.
Local Open 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.

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.