Require Import Rbase.
Require Import R_Ifp.
Require Import Fourier.
Local Open Scope R_scope.

Implicit Type r : R.

Definition Rmin (x y:R) : R :=
  match Rle_dec x y with
    | left _ => x
    | right _ => y
  end.

Lemma Rmin_case : forall r1 r2 (P:R -> Type), P r1 -> P r2 -> P (Rmin r1 r2).

Lemma Rmin_case_strong : forall r1 r2 (P:R -> Type),
  (r1 <= r2 -> P r1) -> (r2 <= r1 -> P r2) -> P (Rmin r1 r2).

Lemma Rmin_Rgt_l : forall r1 r2 r, Rmin r1 r2 > r -> r1 > r /\ r2 > r.

Lemma Rmin_Rgt_r : forall r1 r2 r, r1 > r /\ r2 > r -> Rmin r1 r2 > r.

Lemma Rmin_Rgt : forall r1 r2 r, Rmin r1 r2 > r <-> r1 > r /\ r2 > r.

Lemma Rmin_l : forall x y:R, Rmin x y <= x.

Lemma Rmin_r : forall x y:R, Rmin x y <= y.

Lemma Rmin_left : forall x y, x <= y -> Rmin x y = x.

Lemma Rmin_right : forall x y, y <= x -> Rmin x y = y.

Lemma Rle_min_compat_r : forall x y z, x <= y -> Rmin x z <= Rmin y z.

Lemma Rle_min_compat_l : forall x y z, x <= y -> Rmin z x <= Rmin z y.

Lemma Rmin_comm : forall x y:R, Rmin x y = Rmin y x.

Lemma Rmin_stable_in_posreal : forall x y:posreal, 0 < Rmin x y.

Lemma Rmin_pos : forall x y:R, 0 < x -> 0 < y -> 0 < Rmin x y.

Lemma Rmin_glb : forall x y z:R, z <= x -> z <= y -> z <= Rmin x y.

Lemma Rmin_glb_lt : forall x y z:R, z < x -> z < y -> z < Rmin x y.

Definition Rmax (x y:R) : R :=
  match Rle_dec x y with
    | left _ => y
    | right _ => x
  end.

Lemma Rmax_case : forall r1 r2 (P:R -> Type), P r1 -> P r2 -> P (Rmax r1 r2).

Lemma Rmax_case_strong : forall r1 r2 (P:R -> Type),
  (r2 <= r1 -> P r1) -> (r1 <= r2 -> P r2) -> P (Rmax r1 r2).

Lemma Rmax_Rle : forall r1 r2 r, r <= Rmax r1 r2 <-> r <= r1 \/ r <= r2.

Lemma Rmax_comm : forall x y:R, Rmax x y = Rmax y x.


Lemma Rmax_l : forall x y:R, x <= Rmax x y.

Lemma Rmax_r : forall x y:R, y <= Rmax x y.


Lemma Rmax_left : forall x y, y <= x -> Rmax x y = x.

Lemma Rmax_right : forall x y, x <= y -> Rmax x y = y.

Lemma Rle_max_compat_r : forall x y z, x <= y -> Rmax x z <= Rmax y z.

Lemma Rle_max_compat_l : forall x y z, x <= y -> Rmax z x <= Rmax z y.

Lemma RmaxRmult :
  forall (p q:R) r, 0 <= r -> Rmax (r * p) (r * q) = r * Rmax p q.

Lemma Rmax_stable_in_negreal : forall x y:negreal, Rmax x y < 0.

Lemma Rmax_lub : forall x y z:R, x <= z -> y <= z -> Rmax x y <= z.

Lemma Rmax_lub_lt : forall x y z:R, x < z -> y < z -> Rmax x y < z.

Lemma Rmax_neg : forall x y:R, x < 0 -> y < 0 -> Rmax x y < 0.

Lemma Rcase_abs : forall r, {r < 0} + {r >= 0}.

Definition Rabs r : R :=
  match Rcase_abs r with
    | left _ => - r
    | right _ => r
  end.

Lemma Rabs_R0 : Rabs 0 = 0.

Lemma Rabs_R1 : Rabs 1 = 1.

Lemma Rabs_no_R0 : forall r, r <> 0 -> Rabs r <> 0.

Lemma Rabs_left : forall r, r < 0 -> Rabs r = - r.

Lemma Rabs_right : forall r, r >= 0 -> Rabs r = r.

Lemma Rabs_left1 : forall a:R, a <= 0 -> Rabs a = - a.

Lemma Rabs_pos : forall x:R, 0 <= Rabs x.

Lemma Rle_abs : forall x:R, x <= Rabs x.

Definition RRle_abs := Rle_abs.

Lemma Rabs_pos_eq : forall x:R, 0 <= x -> Rabs x = x.

Lemma Rabs_Rabsolu : forall x:R, Rabs (Rabs x) = Rabs x.

Lemma Rabs_pos_lt : forall x:R, x <> 0 -> 0 < Rabs x.

Lemma Rabs_minus_sym : forall x y:R, Rabs (x - y) = Rabs (y - x).

Lemma Rabs_mult : forall x y:R, Rabs (x * y) = Rabs x * Rabs y.

Lemma Rabs_Rinv : forall r, r <> 0 -> Rabs (/ r) = / Rabs r.

Lemma Rabs_Ropp : forall x:R, Rabs (- x) = Rabs x.

Lemma Rabs_triang : forall a b:R, Rabs (a + b) <= Rabs a + Rabs b.

Lemma Rabs_triang_inv : forall a b:R, Rabs a - Rabs b <= Rabs (a - b).

Lemma Rabs_triang_inv2 : forall a b:R, Rabs (Rabs a - Rabs b) <= Rabs (a - b).

Lemma Rabs_def1 : forall x a:R, x < a -> - a < x -> Rabs x < a.

Lemma Rabs_def2 : forall x a:R, Rabs x < a -> x < a /\ - a < x.

Lemma RmaxAbs :
  forall (p q:R) r, p <= q -> q <= r -> Rabs q <= Rmax (Rabs p) (Rabs r).

Lemma Rabs_Zabs : forall z:Z, Rabs (IZR z) = IZR (Z.abs z).

Lemma abs_IZR : forall z, IZR (Z.abs z) = Rabs (IZR z).