Library Coq.Reals.R_sqr
Rsqr : some results
Ltac ring_Rsqr := unfold Rsqr; ring.
Lemma Rsqr_neg : forall x:R, Rsqr x = Rsqr (- x).
Proof.
intros; ring_Rsqr.
Qed.
Lemma Rsqr_mult : forall x y:R, Rsqr (x * y) = Rsqr x * Rsqr y.
Proof.
intros; ring_Rsqr.
Qed.
Lemma Rsqr_plus : forall x y:R, Rsqr (x + y) = Rsqr x + Rsqr y + 2 * x * y.
Proof.
intros; ring_Rsqr.
Qed.
Lemma Rsqr_minus : forall x y:R, Rsqr (x - y) = Rsqr x + Rsqr y - 2 * x * y.
Proof.
intros; ring_Rsqr.
Qed.
Lemma Rsqr_neg_minus : forall x y:R, Rsqr (x - y) = Rsqr (y - x).
Proof.
intros; ring_Rsqr.
Qed.
Lemma Rsqr_1 : Rsqr 1 = 1.
Proof.
ring_Rsqr.
Qed.
Lemma Rsqr_gt_0_0 : forall x:R, 0 < Rsqr x -> x <> 0.
Proof.
intros; red; intro; rewrite H0 in H; rewrite Rsqr_0 in H;
elim (Rlt_irrefl 0 H).
Qed.
Lemma Rsqr_pos_lt : forall x:R, x <> 0 -> 0 < Rsqr x.
Proof.
intros; case (Rtotal_order 0 x); intro;
[ unfold Rsqr; apply Rmult_lt_0_compat; assumption
| elim H0; intro;
[ elim H; symmetry ; exact H1
| rewrite Rsqr_neg; generalize (Ropp_lt_gt_contravar x 0 H1);
rewrite Ropp_0; intro; unfold Rsqr;
apply Rmult_lt_0_compat; assumption ] ].
Qed.
Lemma Rsqr_div : forall x y:R, y <> 0 -> Rsqr (x / y) = Rsqr x / Rsqr y.
Proof.
intros; unfold Rsqr.
unfold Rdiv.
rewrite Rinv_mult_distr.
repeat rewrite Rmult_assoc.
apply Rmult_eq_compat_l.
rewrite Rmult_comm.
repeat rewrite Rmult_assoc.
apply Rmult_eq_compat_l.
reflexivity.
assumption.
assumption.
Qed.
Lemma Rsqr_eq_0 : forall x:R, Rsqr x = 0 -> x = 0.
Proof.
unfold Rsqr; intros; generalize (Rmult_integral x x H); intro;
elim H0; intro; assumption.
Qed.
Lemma Rsqr_minus_plus : forall a b:R, (a - b) * (a + b) = Rsqr a - Rsqr b.
Proof.
intros; ring_Rsqr.
Qed.
Lemma Rsqr_plus_minus : forall a b:R, (a + b) * (a - b) = Rsqr a - Rsqr b.
Proof.
intros; ring_Rsqr.
Qed.
Lemma Rsqr_incr_0 :
forall x y:R, Rsqr x <= Rsqr y -> 0 <= x -> 0 <= y -> x <= y.
Proof.
intros; destruct (Rle_dec x y) as [Hle|Hnle];
[ assumption
| cut (y < x);
[ intro; unfold Rsqr in H;
generalize (Rmult_le_0_lt_compat y x y x H1 H1 H2 H2);
intro; generalize (Rle_lt_trans (x * x) (y * y) (x * x) H H3);
intro; elim (Rlt_irrefl (x * x) H4)
| auto with real ] ].
Qed.
Lemma Rsqr_incr_0_var : forall x y:R, Rsqr x <= Rsqr y -> 0 <= y -> x <= y.
Proof.
intros; destruct (Rle_dec x y) as [Hle|Hnle];
[ assumption
| cut (y < x);
[ intro; unfold Rsqr in H;
generalize (Rmult_le_0_lt_compat y x y x H0 H0 H1 H1);
intro; generalize (Rle_lt_trans (x * x) (y * y) (x * x) H H2);
intro; elim (Rlt_irrefl (x * x) H3)
| auto with real ] ].
Qed.
Lemma Rsqr_incr_1 :
forall x y:R, x <= y -> 0 <= x -> 0 <= y -> Rsqr x <= Rsqr y.
Proof.
intros; unfold Rsqr; apply Rmult_le_compat; assumption.
Qed.
Lemma Rsqr_incrst_0 :
forall x y:R, Rsqr x < Rsqr y -> 0 <= x -> 0 <= y -> x < y.
Proof.
intros; case (Rtotal_order x y); intro;
[ assumption
| elim H2; intro;
[ rewrite H3 in H; elim (Rlt_irrefl (Rsqr y) H)
| generalize (Rmult_le_0_lt_compat y x y x H1 H1 H3 H3); intro;
unfold Rsqr in H; generalize (Rlt_trans (x * x) (y * y) (x * x) H H4);
intro; elim (Rlt_irrefl (x * x) H5) ] ].
Qed.
Lemma Rsqr_incrst_1 :
forall x y:R, x < y -> 0 <= x -> 0 <= y -> Rsqr x < Rsqr y.
Proof.
intros; unfold Rsqr; apply Rmult_le_0_lt_compat; assumption.
Qed.
Lemma Rsqr_neg_pos_le_0 :
forall x y:R, Rsqr x <= Rsqr y -> 0 <= y -> - y <= x.
Proof.
intros; destruct (Rcase_abs x) as [Hlt|Hle].
generalize (Ropp_lt_gt_contravar x 0 Hlt); rewrite Ropp_0; intro;
generalize (Rlt_le 0 (- x) H1); intro; rewrite (Rsqr_neg x) in H;
generalize (Rsqr_incr_0 (- x) y H H2 H0); intro;
rewrite <- (Ropp_involutive x); apply Ropp_ge_le_contravar;
apply Rle_ge; assumption.
apply Rle_trans with 0;
[ rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption
| apply Rge_le; assumption ].
Qed.
Lemma Rsqr_neg_pos_le_1 :
forall x y:R, - y <= x -> x <= y -> 0 <= y -> Rsqr x <= Rsqr y.
Proof.
intros x y H H0 H1; destruct (Rcase_abs x) as [Hlt|Hle].
apply Ropp_lt_gt_contravar, Rlt_le in Hlt; rewrite Ropp_0 in Hlt;
apply Ropp_le_ge_contravar, Rge_le in H; rewrite Ropp_involutive in H;
rewrite (Rsqr_neg x); apply Rsqr_incr_1; assumption.
apply Rge_le in Hle; apply Rsqr_incr_1; assumption.
Qed.
Lemma neg_pos_Rsqr_le : forall x y:R, - y <= x -> x <= y -> Rsqr x <= Rsqr y.
Proof.
intros x y H H0; destruct (Rcase_abs x) as [Hlt|Hle].
apply Ropp_lt_gt_contravar, Rlt_le in Hlt; rewrite Ropp_0 in Hlt;
apply Ropp_le_ge_contravar, Rge_le in H; rewrite Ropp_involutive in H.
assert (0 <= y) by (apply Rle_trans with (-x); assumption).
rewrite (Rsqr_neg x); apply Rsqr_incr_1; assumption.
apply Rge_le in Hle;
assert (0 <= y) by (apply Rle_trans with x; assumption).
apply Rsqr_incr_1; assumption.
Qed.
Lemma Rsqr_abs : forall x:R, Rsqr x = Rsqr (Rabs x).
Proof.
intro; unfold Rabs; case (Rcase_abs x); intro;
[ apply Rsqr_neg | reflexivity ].
Qed.
Lemma Rsqr_le_abs_0 : forall x y:R, Rsqr x <= Rsqr y -> Rabs x <= Rabs y.
Proof.
intros; apply Rsqr_incr_0; repeat rewrite <- Rsqr_abs;
[ assumption | apply Rabs_pos | apply Rabs_pos ].
Qed.
Lemma Rsqr_le_abs_1 : forall x y:R, Rabs x <= Rabs y -> Rsqr x <= Rsqr y.
Proof.
intros; rewrite (Rsqr_abs x); rewrite (Rsqr_abs y);
apply (Rsqr_incr_1 (Rabs x) (Rabs y) H (Rabs_pos x) (Rabs_pos y)).
Qed.
Lemma Rsqr_lt_abs_0 : forall x y:R, Rsqr x < Rsqr y -> Rabs x < Rabs y.
Proof.
intros; apply Rsqr_incrst_0; repeat rewrite <- Rsqr_abs;
[ assumption | apply Rabs_pos | apply Rabs_pos ].
Qed.
Lemma Rsqr_lt_abs_1 : forall x y:R, Rabs x < Rabs y -> Rsqr x < Rsqr y.
Proof.
intros; rewrite (Rsqr_abs x); rewrite (Rsqr_abs y);
apply (Rsqr_incrst_1 (Rabs x) (Rabs y) H (Rabs_pos x) (Rabs_pos y)).
Qed.
Lemma Rsqr_inj : forall x y:R, 0 <= x -> 0 <= y -> Rsqr x = Rsqr y -> x = y.
Proof.
intros; generalize (Rle_le_eq (Rsqr x) (Rsqr y)); intro; elim H2; intros _ H3;
generalize (H3 H1); intro; elim H4; intros; apply Rle_antisym;
apply Rsqr_incr_0; assumption.
Qed.
Lemma Rsqr_eq_abs_0 : forall x y:R, Rsqr x = Rsqr y -> Rabs x = Rabs y.
Proof.
intros; unfold Rabs; case (Rcase_abs x) as [Hltx|Hgex];
case (Rcase_abs y) as [Hlty|Hgey].
rewrite (Rsqr_neg x), (Rsqr_neg y) in H;
generalize (Ropp_lt_gt_contravar y 0 Hlty);
generalize (Ropp_lt_gt_contravar x 0 Hltx); rewrite Ropp_0;
intros; generalize (Rlt_le 0 (- x) H0); generalize (Rlt_le 0 (- y) H1);
intros; apply Rsqr_inj; assumption.
rewrite (Rsqr_neg x) in H; generalize (Rge_le y 0 Hgey); intro;
generalize (Ropp_lt_gt_contravar x 0 Hltx); rewrite Ropp_0;
intro; generalize (Rlt_le 0 (- x) H1); intro; apply Rsqr_inj;
assumption.
rewrite (Rsqr_neg y) in H; generalize (Rge_le x 0 Hgex); intro;
generalize (Ropp_lt_gt_contravar y 0 Hlty); rewrite Ropp_0;
intro; generalize (Rlt_le 0 (- y) H1); intro; apply Rsqr_inj;
assumption.
apply Rsqr_inj; auto using Rge_le.
Qed.
Lemma Rsqr_eq_asb_1 : forall x y:R, Rabs x = Rabs y -> Rsqr x = Rsqr y.
Proof.
intros; cut (Rsqr (Rabs x) = Rsqr (Rabs y)).
intro; repeat rewrite <- Rsqr_abs in H0; assumption.
rewrite H; reflexivity.
Qed.
Lemma triangle_rectangle :
forall x y z:R,
0 <= z -> Rsqr x + Rsqr y <= Rsqr z -> - z <= x <= z /\ - z <= y <= z.
Proof.
intros;
generalize (plus_le_is_le (Rsqr x) (Rsqr y) (Rsqr z) (Rle_0_sqr y) H0);
rewrite Rplus_comm in H0;
generalize (plus_le_is_le (Rsqr y) (Rsqr x) (Rsqr z) (Rle_0_sqr x) H0);
intros; split;
[ split;
[ apply Rsqr_neg_pos_le_0; assumption
| apply Rsqr_incr_0_var; assumption ]
| split;
[ apply Rsqr_neg_pos_le_0; assumption
| apply Rsqr_incr_0_var; assumption ] ].
Qed.
Lemma triangle_rectangle_lt :
forall x y z:R,
Rsqr x + Rsqr y < Rsqr z -> Rabs x < Rabs z /\ Rabs y < Rabs z.
Proof.
intros; split;
[ generalize (plus_lt_is_lt (Rsqr x) (Rsqr y) (Rsqr z) (Rle_0_sqr y) H);
intro; apply Rsqr_lt_abs_0; assumption
| rewrite Rplus_comm in H;
generalize (plus_lt_is_lt (Rsqr y) (Rsqr x) (Rsqr z) (Rle_0_sqr x) H);
intro; apply Rsqr_lt_abs_0; assumption ].
Qed.
Lemma triangle_rectangle_le :
forall x y z:R,
Rsqr x + Rsqr y <= Rsqr z -> Rabs x <= Rabs z /\ Rabs y <= Rabs z.
Proof.
intros; split;
[ generalize (plus_le_is_le (Rsqr x) (Rsqr y) (Rsqr z) (Rle_0_sqr y) H);
intro; apply Rsqr_le_abs_0; assumption
| rewrite Rplus_comm in H;
generalize (plus_le_is_le (Rsqr y) (Rsqr x) (Rsqr z) (Rle_0_sqr x) H);
intro; apply Rsqr_le_abs_0; assumption ].
Qed.
Lemma Rsqr_inv : forall x:R, x <> 0 -> Rsqr (/ x) = / Rsqr x.
Proof.
intros; unfold Rsqr.
rewrite Rinv_mult_distr; try reflexivity || assumption.
Qed.
Lemma canonical_Rsqr :
forall (a:nonzeroreal) (b c x:R),
a * Rsqr x + b * x + c =
a * Rsqr (x + b / (2 * a)) + (4 * a * c - Rsqr b) / (4 * a).
Proof.
intros.
unfold Rsqr.
field.
apply a.
Qed.
Lemma Rsqr_eq : forall x y:R, Rsqr x = Rsqr y -> x = y \/ x = - y.
Proof.
intros; unfold Rsqr in H;
generalize (Rplus_eq_compat_l (- (y * y)) (x * x) (y * y) H);
rewrite Rplus_opp_l; replace (- (y * y) + x * x) with ((x - y) * (x + y)).
intro; generalize (Rmult_integral (x - y) (x + y) H0); intro; elim H1; intros.
left; apply Rminus_diag_uniq; assumption.
right; apply Rminus_diag_uniq; unfold Rminus; rewrite Ropp_involutive;
assumption.
ring.
Qed.