(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 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.
intros a b H; rewrite (le_plus_minus b a H); apply add_sub_square_identity.
Qed.
intros x; case x.
intros y; case y; simpl in |- *; auto with *.
intros x' y Hlt; apply lt_trans with (S x' * y).
rewrite (mult_comm (S x') y); apply mult_lt; auto.
apply mult_lt; omega.
Qed.

Theorem root_monotonic : forall x y : nat, x * x < y * y -> x < y.