Require Import Rdefinitions Raxioms RIneq.
Require Import ZArith Ztac.
Local Open Scope R_scope.

Definition Int_part (r:R) : Z := (up r - 1)%Z.

Definition frac_part (r:R) : R := r - IZR (Int_part r).

Lemma tech_up : forall (r:R) (z:Z), r < IZR z -> IZR z <= r + 1 -> z = up r.

Lemma Int_part_spec : forall r z, r - 1 < IZR z <= r -> z = Int_part r.

Lemma up_tech :
  forall (r:R) (z:Z), IZR z <= r -> r < IZR (z + 1) -> (z + 1)%Z = up r.

Lemma fp_R0 : frac_part 0 = 0.

Lemma for_base_fp : forall r:R, IZR (up r) - r > 0 /\ IZR (up r) - r <= 1.

Lemma base_fp : forall r:R, frac_part r >= 0 /\ frac_part r < 1.

Lemma base_Int_part :
  forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1.

Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z.of_nat n.

Lemma fp_nat : forall r:R, frac_part r = 0 -> exists c : Z, r = IZR c.

Lemma R0_fp_O : forall r:R, 0 <> frac_part r -> 0 <> r.

Lemma Rplus_Int_part_frac_part : forall r, r = IZR (Int_part r) + frac_part r.

Lemma Int_part_frac_part_spec :
  forall r z f, 0 <= f < 1 -> r = (IZR z) + f -> z = Int_part r /\ f = frac_part r.
Lemma Rminus_Int_part1 :
  forall r1 r2:R,
    frac_part r1 >= frac_part r2 ->
    Int_part (r1 - r2) = (Int_part r1 - Int_part r2)%Z.

Lemma Rminus_Int_part2 :
  forall r1 r2:R,
    frac_part r1 < frac_part r2 ->
    Int_part (r1 - r2) = (Int_part r1 - Int_part r2 - 1)%Z.

Lemma Rminus_fp1 :
  forall r1 r2:R,
    frac_part r1 >= frac_part r2 ->
    frac_part (r1 - r2) = frac_part r1 - frac_part r2.

Lemma Rminus_fp2 :
  forall r1 r2:R,
    frac_part r1 < frac_part r2 ->
    frac_part (r1 - r2) = frac_part r1 - frac_part r2 + 1.

Lemma plus_Int_part1 :
  forall r1 r2:R,
    frac_part r1 + frac_part r2 >= 1 ->
    Int_part (r1 + r2) = (Int_part r1 + Int_part r2 + 1)%Z.

Lemma plus_Int_part2 :
  forall r1 r2:R,
    frac_part r1 + frac_part r2 < 1 ->
    Int_part (r1 + r2) = (Int_part r1 + Int_part r2)%Z.

Lemma plus_frac_part1 :
  forall r1 r2:R,
    frac_part r1 + frac_part r2 >= 1 ->
    frac_part (r1 + r2) = frac_part r1 + frac_part r2 - 1.

Lemma plus_frac_part2 :
  forall r1 r2:R,
    frac_part r1 + frac_part r2 < 1 ->
    frac_part (r1 + r2) = frac_part r1 + frac_part r2.