Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.ZArith.Zsqrt_compat

Require Import ZArithRing.
Require Import Omega.
Require Export ZArith_base.
Local Open 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 (Pos2Z.inj_xI X1)
end
| |- context [(Zpos (xO ?X1))] =>
match constr:(X1) with
| context [1%positive] => fail 1
| _ => rewrite (Pos2Z.inj_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).
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)}}.
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) (Pos2Z.is_nonneg p) with
| existT _ 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).
Positivity
Theorem Zsqrt_plain_is_pos: forall n, 0 <= n -> 0 <= Zsqrt_plain n.
Direct correctness on squares.
Theorem Zsqrt_square_id: forall a, 0 <= a -> Zsqrt_plain (a * a) = a.
Zsqrt_plain is increasing
Theorem Zsqrt_le:
forall p q, 0 <= p <= q -> Zsqrt_plain p <= Zsqrt_plain q.
Equivalence between Zsqrt_plain and Z.sqrt
Lemma Zsqrt_equiv : forall n, Zsqrt_plain n = Z.sqrt n.