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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 *)
(************************************************************************)
(** Definition of a square root function for N. *)
Require Import BinPos BinNat.
Local Open Scope N_scope.
Definition Nsqrtrem n :=
match n with
| N0 => (N0, N0)
| Npos p =>
match Pos.sqrtrem p with
| (s, IsPos r) => (Npos s, Npos r)
| (s, _) => (Npos s, N0)
end
end.
Definition Nsqrt n :=
match n with
| N0 => N0
| Npos p => Npos (Pos.sqrt p)
end.
Lemma Nsqrtrem_spec : forall n,
let (s,r) := Nsqrtrem n in n = s*s + r /\ r <= 2*s.
Proof.
destruct n. now split.
generalize (Pos.sqrtrem_spec p). simpl.
destruct 1; simpl; subst; now split.
Qed.
Lemma Nsqrt_spec : forall n,
let s := Nsqrt n in s*s <= n < (Nsucc s)*(Nsucc s).
Proof.
destruct n. now split. apply (Pos.sqrt_spec p).
Qed.
Lemma Nsqrtrem_sqrt : forall n, fst (Nsqrtrem n) = Nsqrt n.
Proof.
destruct n. reflexivity.
unfold Nsqrtrem, Nsqrt, Pos.sqrt.
destruct (Pos.sqrtrem p) as (s,r). now destruct r.
Qed.
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