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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import ZArithRing.
Require Import Omega.
Require Export ZArith_base.
Open Local Scope Z_scope.
(** THIS FILE IS DEPRECATED
Instead of the various [Zsqrt] defined here, please use rather
[Z.sqrt] (or [Z.sqrtrem]). The latter are pure functions without
proof parts, and more results are available about them.
Some equivalence proofs between the old and the new versions
can be found below. Importing ZArith will provides by default
the new versions.
*)
(**********************************************************************)
(** Definition and properties of square root on Z *)
(** The following tactic replaces all instances of (POS (xI ...)) by
`2*(POS ...)+1`, but only when ... is not made only with xO, XI, or xH. *)
Ltac compute_POS :=
match goal with
| |- context [(Zpos (xI ?X1))] =>
match constr:X1 with
| context [1%positive] => fail 1
| _ => rewrite (BinInt.Zpos_xI X1)
end
| |- context [(Zpos (xO ?X1))] =>
match constr:X1 with
| context [1%positive] => fail 1
| _ => rewrite (BinInt.Zpos_xO X1)
end
end.
Inductive sqrt_data (n:Z) : Set :=
c_sqrt : forall s r:Z, n = s * s + r -> 0 <= r <= 2 * s -> sqrt_data n.
Definition sqrtrempos : forall p:positive, sqrt_data (Zpos p).
refine
(fix sqrtrempos (p:positive) : sqrt_data (Zpos p) :=
match p return sqrt_data (Zpos p) with
| xH => c_sqrt 1 1 0 _ _
| xO xH => c_sqrt 2 1 1 _ _
| xI xH => c_sqrt 3 1 2 _ _
| xO (xO p') =>
match sqrtrempos p' with
| c_sqrt s' r' Heq Hint =>
match Z_le_gt_dec (4 * s' + 1) (4 * r') with
| left Hle =>
c_sqrt (Zpos (xO (xO p'))) (2 * s' + 1)
(4 * r' - (4 * s' + 1)) _ _
| right Hgt => c_sqrt (Zpos (xO (xO p'))) (2 * s') (4 * r') _ _
end
end
| xO (xI p') =>
match sqrtrempos p' with
| c_sqrt s' r' Heq Hint =>
match Z_le_gt_dec (4 * s' + 1) (4 * r' + 2) with
| left Hle =>
c_sqrt (Zpos (xO (xI p'))) (2 * s' + 1)
(4 * r' + 2 - (4 * s' + 1)) _ _
| right Hgt =>
c_sqrt (Zpos (xO (xI p'))) (2 * s') (4 * r' + 2) _ _
end
end
| xI (xO p') =>
match sqrtrempos p' with
| c_sqrt s' r' Heq Hint =>
match Z_le_gt_dec (4 * s' + 1) (4 * r' + 1) with
| left Hle =>
c_sqrt (Zpos (xI (xO p'))) (2 * s' + 1)
(4 * r' + 1 - (4 * s' + 1)) _ _
| right Hgt =>
c_sqrt (Zpos (xI (xO p'))) (2 * s') (4 * r' + 1) _ _
end
end
| xI (xI p') =>
match sqrtrempos p' with
| c_sqrt s' r' Heq Hint =>
match Z_le_gt_dec (4 * s' + 1) (4 * r' + 3) with
| left Hle =>
c_sqrt (Zpos (xI (xI p'))) (2 * s' + 1)
(4 * r' + 3 - (4 * s' + 1)) _ _
| right Hgt =>
c_sqrt (Zpos (xI (xI p'))) (2 * s') (4 * r' + 3) _ _
end
end
end); clear sqrtrempos; repeat compute_POS;
try (try rewrite Heq; ring); try omega.
Defined.
(** Define with integer input, but with a strong (readable) specification. *)
Definition Zsqrt :
forall x:Z,
0 <= x ->
{s : Z & {r : Z | x = s * s + r /\ s * s <= x < (s + 1) * (s + 1)}}.
refine
(fun x =>
match
x
return
0 <= x ->
{s : Z & {r : Z | x = s * s + r /\ s * s <= x < (s + 1) * (s + 1)}}
with
| Zpos p =>
fun h =>
match sqrtrempos p with
| c_sqrt s r Heq Hint =>
existS
(fun s:Z =>
{r : Z |
Zpos p = s * s + r /\ s * s <= Zpos p < (s + 1) * (s + 1)})
s
(exist
(fun r:Z =>
Zpos p = s * s + r /\
s * s <= Zpos p < (s + 1) * (s + 1)) r _)
end
| Zneg p =>
fun h =>
False_rec
{s : Z &
{r : Z |
Zneg p = s * s + r /\ s * s <= Zneg p < (s + 1) * (s + 1)}}
(h (refl_equal Datatypes.Gt))
| Z0 =>
fun h =>
existS
(fun s:Z =>
{r : Z | 0 = s * s + r /\ s * s <= 0 < (s + 1) * (s + 1)}) 0
(exist
(fun r:Z => 0 = 0 * 0 + r /\ 0 * 0 <= 0 < (0 + 1) * (0 + 1)) 0
_)
end); try omega.
split; [ omega | rewrite Heq; ring_simplify (s*s) ((s + 1) * (s + 1)); omega ].
Defined.
(** Define a function of type Z->Z that computes the integer square root,
but only for positive numbers, and 0 for others. *)
Definition Zsqrt_plain (x:Z) : Z :=
match x with
| Zpos p =>
match Zsqrt (Zpos p) (Zorder.Zle_0_pos p) with
| existS s _ => s
end
| Zneg p => 0
| Z0 => 0
end.
(** A basic theorem about Zsqrt_plain *)
Theorem Zsqrt_interval :
forall n:Z,
0 <= n ->
Zsqrt_plain n * Zsqrt_plain n <= n <
(Zsqrt_plain n + 1) * (Zsqrt_plain n + 1).
Proof.
intros x; case x.
unfold Zsqrt_plain in |- *; omega.
intros p; unfold Zsqrt_plain in |- *;
case (Zsqrt (Zpos p) (Zorder.Zle_0_pos p)).
intros s [r [Heq Hint]] Hle; assumption.
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.
(** Equivalence between Zsqrt_plain and [Z.sqrt] *)
Lemma Zsqrt_equiv : forall n, Zsqrt_plain n = Z.sqrt n.
Proof.
intros. destruct (Z_le_gt_dec 0 n).
symmetry. apply Z.sqrt_unique; trivial.
now apply Zsqrt_interval.
now destruct n.
Qed.
|
