Add Zfloor/Zceil to Reals

Author: thery

doc/changelog/01-added/89-Zfloor.rst

@@ -0,0 +1,10 @@
+- in `Reals`
+
+  + new file `Zfloor.v` with definitions of `Zfloor` and `Zceil`
+    and corresponding lemmas `up_Zfloor`, `IZR_up_Zfloor`,
+    `Zfloor_bound`, `Zfloor_lub`, `Zfloor_eq`, `Zfloor_le`,
+    `Zfloor_addz`, `ZfloorZ`, `ZfloorNZ`, `ZfloorD_cond`, `Zceil_eq`,
+    `Zceil_le`, `Zceil_addz`, `ZceilD_cond`, `ZfloorB_cond`,
+    `ZceilB_cond`
+    (`#89 <https://github.com/coq/stdlib/pull/89>`_,
+    by Laurent Théry).

theories/Reals/Reals.v

@@ -26,6 +26,7 @@
 
 Require Export Rbase.
 Require Export Rfunctions.
+Require Export Zfloor.
 Require Export SeqSeries.
 Require Export Rtrigo.
 Require Export Ranalysis.

theories/Reals/Zfloor.v

@@ -0,0 +1,129 @@
+(******************************************************************************)
+(*                                                                            *)
+(*  Zfloor x        == floor function : R -> Z                                *)
+(*  Zceil x         == ceil function : R -> Z                                 *)
+(*                                                                            *)
+(******************************************************************************)
+
+Require Import Rbase Rfunctions Lra Lia.
+
+Local Open Scope R_scope.
+
+Definition Zfloor (x : R) := (up x - 1)%Z.
+
+Lemma up_Zfloor x : up x = (Zfloor x + 1)%Z.
+Proof. unfold Zfloor; lia. Qed.
+
+Lemma IZR_up_Zfloor x : IZR (up x) = IZR (Zfloor x) + 1.
+Proof. now rewrite up_Zfloor, plus_IZR. Qed.
+
+Lemma Zfloor_bound x : IZR (Zfloor x) <= x < IZR (Zfloor x) + 1.
+Proof.
+unfold Zfloor; rewrite minus_IZR.
+generalize (archimed x); lra.
+Qed.
+
+Lemma Zfloor_lub (z : Z) x : IZR z <= x -> (z <= Zfloor x)%Z.
+Proof.
+intro H.
+assert (H1 : (z < Zfloor x + 1)%Z);[| lia].
+apply lt_IZR; rewrite plus_IZR.
+now generalize (Zfloor_bound x); lra.
+Qed.
+
+Lemma Zfloor_eq (z : Z) x :  IZR z <= x < IZR z + 1 -> Zfloor x = z.
+Proof.
+intro xB.
+assert (ZxB := Zfloor_bound x).
+assert (B : (Zfloor x < z + 1 /\ z <= Zfloor x)%Z) ; [| lia].
+split; [|apply Zfloor_lub; lra].
+apply lt_IZR; rewrite plus_IZR; lra.
+Qed.
+
+Lemma Zfloor_le x y : x <= y -> (Zfloor x <= Zfloor y)%Z.
+Proof.
+intro H; apply Zfloor_lub; generalize (Zfloor_bound x); lra.
+Qed.
+
+Lemma Zfloor_addz (z: Z) x : Zfloor (x + IZR z) = (Zfloor x + z)%Z.
+Proof.
+assert (ZB := Zfloor_bound x).
+now apply Zfloor_eq; rewrite plus_IZR; lra.
+Qed.
+
+Lemma ZfloorZ (z : Z) : Zfloor (IZR z) = z.
+Proof. now apply Zfloor_eq; lra. Qed.
+
+
+Lemma ZfloorNZ (z : Z) : Zfloor (- IZR z) = (- z)%Z.
+Proof. now rewrite <- opp_IZR, ZfloorZ. Qed.
+
+Lemma ZfloorD_cond r1 r2 :
+  if Rle_dec (IZR (Zfloor r1) + IZR (Zfloor r2) + 1) (r1 + r2)
+  then Zfloor (r1 + r2) = (Zfloor r1 + Zfloor r2 + 1)%Z
+  else  Zfloor (r1 + r2) = (Zfloor r1 + Zfloor r2)%Z.
+Proof.
+destruct (Zfloor_bound r1, Zfloor_bound r2) as [H1 H2].
+case Rle_dec; intro H.
+  now apply Zfloor_eq; rewrite plus_IZR, plus_IZR; lra.
+now apply Zfloor_eq; rewrite plus_IZR; lra.
+Qed.
+
+Definition Zceil (x : R) := (- Zfloor (- x))%Z.
+
+Theorem Zceil_bound x : (IZR (Zceil x) - 1 < x <= IZR (Zceil x))%R.
+Proof.
+now unfold Zceil; generalize (Zfloor_bound (- x)); rewrite !opp_IZR; lra.
+Qed.
+
+Theorem Zfloor_ceil_bound x : (IZR (Zfloor x) <= x <= IZR (Zceil x))%R.
+Proof. now generalize (Zfloor_bound x) (Zceil_bound x); lra. Qed.
+
+Theorem ZceilN x : (Zceil (- x) = - Zfloor x)%Z.
+Proof. now unfold Zceil; rewrite Ropp_involutive. Qed.
+
+Theorem ZfloorN x : (Zfloor (- x) = - Zceil x)%Z.
+Proof. unfold Zceil; lia. Qed.
+
+Lemma Zceil_eq (z : Z) x :  IZR z - 1 < x <= IZR z -> Zceil x = z.
+Proof.
+intro xB; assert (H : Zfloor (- x) = (- z)%Z); [|unfold Zceil; lia].
+now apply Zfloor_eq; rewrite opp_IZR; lra.
+Qed.
+
+Lemma Zceil_le x y : x <= y -> (Zceil x <= Zceil y)%Z.
+Proof.
+intro xLy; apply Z.opp_le_mono; unfold Zceil; rewrite !Z.opp_involutive.
+now apply Zfloor_le; lra.
+Qed.
+
+Lemma Zceil_addz (z: Z) x : Zceil (x + IZR z) = (Zceil x + z)%Z.
+Proof.
+now unfold Zceil; rewrite Ropp_plus_distr, <- opp_IZR, Zfloor_addz; lia.
+Qed.
+
+Lemma ZceilD_cond r1 r2 :
+  if Rle_dec (r1 + r2) (IZR (Zceil r1) + IZR (Zceil r2) - 1)
+  then Zceil (r1 + r2) = (Zceil r1 + Zceil r2 - 1)%Z
+  else Zceil (r1 + r2) = (Zceil r1 + Zceil r2)%Z.
+Proof.
+generalize (ZfloorD_cond (- r1) (- r2)).
+now unfold Zceil; rewrite !opp_IZR; do 2 case Rle_dec; try lra;
+    rewrite Ropp_plus_distr; lia.
+Qed.
+
+Lemma ZfloorB_cond r1 r2 :
+  if Rle_dec (IZR (Zfloor r1) - IZR (Zceil r2) + 1) (r1 - r2)
+  then Zfloor (r1 - r2) = (Zfloor r1 - Zceil r2 + 1)%Z
+  else  Zfloor (r1 - r2) = (Zfloor r1 - Zceil r2)%Z.
+Proof.
+now generalize (ZfloorD_cond r1 (- r2)); rewrite !ZfloorN, !opp_IZR.
+Qed.
+
+Lemma ZceilB_cond r1 r2 :
+  if Rle_dec (r1 - r2) (IZR (Zceil r1) - IZR (Zfloor r2) - 1)
+  then Zceil (r1 - r2) = (Zceil r1 - Zfloor r2 - 1)%Z
+  else Zceil (r1 - r2) = (Zceil r1 - Zfloor r2)%Z.
+Proof.
+now generalize (ZceilD_cond r1 (- r2)); rewrite !ZceilN, !opp_IZR.
+Qed.