diff options
Diffstat (limited to 'theories/ZArith/Zsqrt.v')
| -rw-r--r-- | theories/ZArith/Zsqrt.v | 53 |
1 files changed, 52 insertions, 1 deletions
diff --git a/theories/ZArith/Zsqrt.v b/theories/ZArith/Zsqrt.v index 3f475a63..6ea952e6 100644 --- a/theories/ZArith/Zsqrt.v +++ b/theories/ZArith/Zsqrt.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(* $Id: Zsqrt.v 9551 2007-01-29 15:13:35Z bgregoir $ *) +(* $Id: Zsqrt.v 10295 2007-11-06 22:46:21Z letouzey $ *) Require Import ZArithRing. Require Import Omega. @@ -148,6 +148,7 @@ Definition Zsqrt_plain (x:Z) : Z := end. (** A basic theorem about Zsqrt_plain *) + Theorem Zsqrt_interval : forall n:Z, 0 <= n -> @@ -162,3 +163,53 @@ Proof. intros p Hle; elim Hle; auto. Qed. +(** Positivity *) + +Theorem Zsqrt_plain_is_pos: forall n, 0 <= n -> 0 <= Zsqrt_plain n. +Proof. + intros n m; case (Zsqrt_interval n); auto with zarith. + intros H1 H2; case (Zle_or_lt 0 (Zsqrt_plain n)); auto. + intros H3; contradict H2; auto; apply Zle_not_lt. + apply Zle_trans with ( 2 := H1 ). + replace ((Zsqrt_plain n + 1) * (Zsqrt_plain n + 1)) + with (Zsqrt_plain n * Zsqrt_plain n + (2 * Zsqrt_plain n + 1)); + auto with zarith. + ring. +Qed. + +(** Direct correctness on squares. *) + +Theorem Zsqrt_square_id: forall a, 0 <= a -> Zsqrt_plain (a * a) = a. +Proof. + intros a H. + generalize (Zsqrt_plain_is_pos (a * a)); auto with zarith; intros Haa. + case (Zsqrt_interval (a * a)); auto with zarith. + intros H1 H2. + case (Zle_or_lt a (Zsqrt_plain (a * a))); intros H3; auto. + case Zle_lt_or_eq with (1:=H3); auto; clear H3; intros H3. + contradict H1; auto; apply Zlt_not_le; auto with zarith. + apply Zle_lt_trans with (a * Zsqrt_plain (a * a)); auto with zarith. + apply Zmult_lt_compat_r; auto with zarith. + contradict H2; auto; apply Zle_not_lt; auto with zarith. + apply Zmult_le_compat; auto with zarith. +Qed. + +(** [Zsqrt_plain] is increasing *) + +Theorem Zsqrt_le: + forall p q, 0 <= p <= q -> Zsqrt_plain p <= Zsqrt_plain q. +Proof. + intros p q [H1 H2]; case Zle_lt_or_eq with (1:=H2); clear H2; intros H2; + [ | subst q; auto with zarith]. + case (Zle_or_lt (Zsqrt_plain p) (Zsqrt_plain q)); auto; intros H3. + assert (Hp: (0 <= Zsqrt_plain q)). + apply Zsqrt_plain_is_pos; auto with zarith. + absurd (q <= p); auto with zarith. + apply Zle_trans with ((Zsqrt_plain q + 1) * (Zsqrt_plain q + 1)). + case (Zsqrt_interval q); auto with zarith. + apply Zle_trans with (Zsqrt_plain p * Zsqrt_plain p); auto with zarith. + apply Zmult_le_compat; auto with zarith. + case (Zsqrt_interval p); auto with zarith. +Qed. + + |