Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Reals.Ranalysis2

Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
Require Import Lra.
Local Open Scope R_scope.
Lemma formule :
forall (x h l1 l2:R) (f1 f2:R -> R),
h <> 0 ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
(l1 * f2 x - l2 * f1 x) / Rsqr (f2 x) =
/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2) +
l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x).
Lemma maj_term1 :
forall (x h eps l1 alp_f2:R) (eps_f2 alp_f1d:posreal)
(f1 f2:R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall h:R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) < Rabs (eps * f2 x / 8)) ->
(forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f1d ->
Rabs h < Rmin eps_f2 alp_f2 ->
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4.
Lemma maj_term2 :
forall (x h eps l1 alp_f2 alp_f2t2:R) (eps_f2:posreal)
(f2:R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall a:R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))) ->
(forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2t2 ->
Rabs h < Rmin eps_f2 alp_f2 ->
l1 <> 0 -> Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) < eps / 4.
Lemma maj_term3 :
forall (x h eps l2 alp_f2:R) (eps_f2 alp_f2d:posreal)
(f1 f2:R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall h:R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs (Rsqr (f2 x) * eps / (8 * f1 x))) ->
(forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2d ->
Rabs h < Rmin eps_f2 alp_f2 ->
f1 x <> 0 ->
Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) <
eps / 4.
Lemma maj_term4 :
forall (x h eps l2 alp_f2 alp_f2c:R) (eps_f2:posreal)
(f1 f2:R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall a:R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) ->
(forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2c ->
Rabs h < Rmin eps_f2 alp_f2 ->
f1 x <> 0 ->
l2 <> 0 ->
Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x)) <
eps / 4.
Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a).
Lemma Rabs_4 :
forall a b c d:R, Rabs (a + b + c + d) <= Rabs a + Rabs b + Rabs c + Rabs d.
Lemma Rlt_4 :
forall a b c d e f g h:R,
a < b -> c < d -> e < f -> g < h -> a + c + e + g < b + d + f + h.
Lemma quadruple : forall x:R, 4 * x = x + x + x + x.
Lemma quadruple_var : forall x:R, x = x / 4 + x / 4 + x / 4 + x / 4.
Lemma continuous_neq_0 :
forall (f:R -> R) (x0:R),
continuity_pt f x0 ->
f x0 <> 0 ->
exists eps : posreal, (forall h:R, Rabs h < eps -> f (x0 + h) <> 0).