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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
Require Export ArithRing.
Require Export Compare_dec.
Require Export Wf_nat.
Require Export Arith.
Require Export Omega.
intros a; elim a; auto.
intros n' Hrec b; case b; auto.
Qed.
Remark expand_mult2 : forall x : nat, 2 * x = x + x.
intros x; ring.
Qed.
unfold not in |- *; intros x y H H1; elim (lt_irrefl x);
pattern x at 2 in |- *; rewrite H1; auto.
Qed.
Hint Resolve lt_neq.
intros f Hmon x y Hlt; case (le_gt_dec y x); auto.
intros Hle; elim (le_lt_or_eq _ _ Hle).
intros Hlt'; elim (lt_asym _ _ Hlt); apply Hmon; auto.
intros Heq; elim (lt_neq _ _ Hlt); rewrite Heq; auto.
Qed.
intros a b c; elim c.
intros H; elim H; auto.
intros c'; case c'.
intros; omega.
intros c'' Hrec Hneq Hlt;
repeat rewrite <- (fun x : nat => mult_n_Sm x (S c'')).
auto with *.
Qed.
Remark add_sub_square_identity :
forall a b : nat,
(b + a - b) * (b + a - b) = (b + a) * (b + a) + b * b - 2 * ((b + a) * b).
intros a b; rewrite minus_plus.
repeat rewrite mult_plus_distr_r || rewrite <- (mult_comm (b + a)).
replace (b * b + a * b + (b * a + a * a) + b * b) with
(b * b + a * b + (b * b + a * b + a * a)); try (ring; fail).
rewrite expand_mult2; repeat rewrite minus_plus; auto with *.
Qed.

Theorem sub_square_identity :
forall a b : nat, b <= a -> (a - b) * (a - b) = a * a + b * b - 2 * (a * b).