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(* Specifications: *)
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Lemma mul_factor_wd : forall l l':R, 1 + (Rabs l + Rabs l') <> 0.
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Lemma mul_factor_gt : forall eps l l':R, eps > 0 -> eps * mul_factor l l' > 0.
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Lemma mul_factor_gt_f :
forall eps l l':R, eps > 0 -> Rmin 1 (eps * mul_factor l l') > 0.
Lemma limit_inv :
forall (f:R -> R) (D:R -> Prop) (l x0:R),
limit1_in f D l x0 -> l <> 0 -> limit1_in (fun x:R => / f x) D (/ l) x0.
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Lemma Rabs_R0 : Rabs 0 = 0.
Lemma Rabs_R1 : Rabs 1 = 1.
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Lemma Rabs_no_R0 : forall r, r <> 0 -> Rabs r <> 0.
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Lemma Rabs_left : forall r, r < 0 -> Rabs r = - r.
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Lemma Rabs_right : forall r, r >= 0 -> Rabs r = r.
Lemma Rabs_left1 : forall a:R, a <= 0 -> Rabs a = - a.
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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_le : forall a b, -b <= a <= b -> Rabs a <= b.
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Lemma Rabs_pos_eq : forall x:R, 0 <= x -> Rabs x = x.
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Lemma Rabs_Rabsolu : forall x:R, Rabs (Rabs x) = Rabs x.
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Lemma Rabs_pos_lt : forall x:R, x <> 0 -> 0 < Rabs x.
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Lemma Rabs_minus_sym : forall x y:R, Rabs (x - y) = Rabs (y - x).
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Lemma Rabs_mult : forall x y:R, Rabs (x * y) = Rabs x * Rabs y.
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Lemma Rabs_inv r : Rabs (/ r) = / Rabs r.
Lemma Rabs_Rinv_depr : forall r, r <> 0 -> Rabs (/ r) = / Rabs r.
#[deprecated(since="8.16",note="Use Rabs_inv.")]
Notation Rabs_Rinv := Rabs_Rinv_depr.
Lemma Rabs_Ropp : forall x:R, Rabs (- x) = Rabs x.
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Lemma Rabs_triang : forall a b:R, Rabs (a + b) <= Rabs a + Rabs b.
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Lemma Rabs_triang_inv : forall a b:R, Rabs a - Rabs b <= Rabs (a - b).
(* ||a|-|b||<=|a-b| *)
Lemma Rabs_triang_inv2 : forall a b:R, Rabs (Rabs a - Rabs b) <= Rabs (a - b).
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Lemma Rabs_def1 : forall x a:R, x < a -> - a < x -> Rabs x < a.
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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).
Lemma Ropp_Rmax : forall x y, - Rmax x y = Rmin (-x) (-y).
intros x y; apply Rmax_case_strong.