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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 *)
(************************************************************************)
(* $Id: Zsqrt.v 13323 2010-07-24 15:57:30Z herbelin $ *)
Require Import ZArithRing.
Require Import Omega.
Require Export ZArith_base.
Open Local Scope Z_scope.
(**********************************************************************)
(** 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.
|
