(****************************************************************************  IEEE754  :  Faux  Laurent Thery  ***************************************************************************** Auxillary properties about natural numbers, relative numbers and reals *)
Require Export Min.
Require Export Arith.
Require Export Reals.
Require Export Zpower.
Require Export ZArith.
Require Export Zcomplements.
Require Export sTactic.
Hint Resolve R1_neq_R0: real.
(*Missing rule for nat *)
intros a b H'.
apply sym_equal.
apply plus_minus; auto.
rewrite plus_comm; apply le_plus_minus; auto.
Qed.
intros p q r t H'; elim H'; simpl in |- *; auto with arith.
elim p; simpl in |- *; auto with arith.
intros n H m H0 H1 H2; apply plus_le_compat; auto with arith.
apply le_trans with (m := r + n * r); auto with arith.
Qed.
Hint Resolve lte_comp_mult: arith.
intros n m H'; rewrite H'; auto.
Qed.
intros n m H'; inversion H'; simpl in |- *; auto.
apply le_trans with (S n); auto.
Qed.
intros p; elim p; simpl in |- *.
auto with arith.
intros n0; case n0.
simpl in |- *; auto with arith.
intros n1 H' q r H'0 H'1.
apply lt_trans with (m := q + S n1 * r); auto with arith.
Qed.
Hint Resolve lt_comp_mult_l: arith.
intros; repeat rewrite (fun x : nat => mult_comm x p); auto with arith.
Qed.
Hint Resolve lt_comp_mult_r: arith.

Theorem lt_comp_mult : forall p q r s : nat, p < q -> r < s -> p * r < q * s.