Require Import Rbase.
Require Import Rfunctions.
Require Export Rlimit.
Require Export Rderiv. Open Local Scope R_scope.
Implicit Type f : R -> R.
| Basic operations on functions |
Definition plus_fct f1 f2 (x:R) : R := f1 x + f2 x.
Definition opp_fct f (x:R) : R := - f x.
Definition mult_fct f1 f2 (x:R) : R := f1 x * f2 x.
Definition mult_real_fct (a:R) f (x:R) : R := a * f x.
Definition minus_fct f1 f2 (x:R) : R := f1 x - f2 x.
Definition div_fct f1 f2 (x:R) : R := f1 x / f2 x.
Definition div_real_fct (a:R) f (x:R) : R := a / f x.
Definition comp f1 f2 (x:R) : R := f1 (f2 x).
Definition inv_fct f (x:R) : R := / f x.
Infix "+" := plus_fct : Rfun_scope.
Notation "- x" := (opp_fct x) : Rfun_scope.
Infix "*" := mult_fct : Rfun_scope.
Infix "-" := minus_fct : Rfun_scope.
Infix "/" := div_fct : Rfun_scope.
Notation Local "f1 'o' f2" := (comp f1 f2)
(at level 20, right associativity) : Rfun_scope.
Notation "/ x" := (inv_fct x) : Rfun_scope.
Delimit Scope Rfun_scope with F.
Definition fct_cte (a x:R) : R := a.
Definition id (x:R) := x.
| Variations of functions |
Definition increasing f : Prop := forall x y:R, x <= y -> f x <= f y.
Definition decreasing f : Prop := forall x y:R, x <= y -> f y <= f x.
Definition strict_increasing f : Prop := forall x y:R, x < y -> f x < f y.
Definition strict_decreasing f : Prop := forall x y:R, x < y -> f y < f x.
Definition constant f : Prop := forall x y:R, f x = f y.
Definition no_cond (x:R) : Prop := True.
Definition constant_D_eq f (D:R -> Prop) (c:R) : Prop :=
forall x:R, D x -> f x = c.
| Definition of continuity as a limit |
Definition continuity_pt f (x0:R) : Prop := continue_in f no_cond x0.
Definition continuity f : Prop := forall x:R, continuity_pt f x.
Arguments Scope continuity_pt [Rfun_scope R_scope].
Arguments Scope continuity [Rfun_scope].
Lemma continuity_pt_plus :
forall f1 f2 (x0:R),
continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0.
unfold continuity_pt, plus_fct in |- *; unfold continue_in in |- *; intros;
apply limit_plus; assumption.
Qed.
Lemma continuity_pt_opp :
forall f (x0:R), continuity_pt f x0 -> continuity_pt (- f) x0.
unfold continuity_pt, opp_fct in |- *; unfold continue_in in |- *; intros;
apply limit_Ropp; assumption.
Qed.
Lemma continuity_pt_minus :
forall f1 f2 (x0:R),
continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0.
unfold continuity_pt, minus_fct in |- *; unfold continue_in in |- *; intros;
apply limit_minus; assumption.
Qed.
Lemma continuity_pt_mult :
forall f1 f2 (x0:R),
continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0.
unfold continuity_pt, mult_fct in |- *; unfold continue_in in |- *; intros;
apply limit_mul; assumption.
Qed.
Lemma continuity_pt_const : forall f (x0:R), constant f -> continuity_pt f x0.
unfold constant, continuity_pt in |- *; unfold continue_in in |- *;
unfold limit1_in in |- *; unfold limit_in in |- *;
intros; exists 1; split;
[ apply Rlt_0_1
| intros; generalize (H x x0); intro; rewrite H2; simpl in |- *;
rewrite R_dist_eq; assumption ].
Qed.
Lemma continuity_pt_scal :
forall f (a x0:R),
continuity_pt f x0 -> continuity_pt (mult_real_fct a f) x0.
unfold continuity_pt, mult_real_fct in |- *; unfold continue_in in |- *;
intros; apply (limit_mul (fun x:R => a) f (D_x no_cond x0) a (f x0) x0).
unfold limit1_in in |- *; unfold limit_in in |- *; intros; exists 1; split.
apply Rlt_0_1.
intros; rewrite R_dist_eq; assumption.
assumption.
Qed.
Lemma continuity_pt_inv :
forall f (x0:R), continuity_pt f x0 -> f x0 <> 0 -> continuity_pt (/ f) x0.
intros.
replace (/ f)%F with (fun x:R => / f x).
unfold continuity_pt in |- *; unfold continue_in in |- *; intros;
apply limit_inv; assumption.
unfold inv_fct in |- *; reflexivity.
Qed.
Lemma div_eq_inv : forall f1 f2, (f1 / f2)%F = (f1 * / f2)%F.
intros; reflexivity.
Qed.
Lemma continuity_pt_div :
forall f1 f2 (x0:R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 -> f2 x0 <> 0 -> continuity_pt (f1 / f2) x0.
intros; rewrite (div_eq_inv f1 f2); apply continuity_pt_mult;
[ assumption | apply continuity_pt_inv; assumption ].
Qed.
Lemma continuity_pt_comp :
forall f1 f2 (x:R),
continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x.
unfold continuity_pt in |- *; unfold continue_in in |- *; intros;
unfold comp in |- *.
cut
(limit1_in (fun x0:R => f2 (f1 x0))
(Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) (
f2 (f1 x)) x ->
limit1_in (fun x0:R => f2 (f1 x0)) (D_x no_cond x) (f2 (f1 x)) x).
intro; apply H1.
eapply limit_comp.
apply H.
apply H0.
unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
simpl in |- *; unfold R_dist in |- *; intros.
assert (H3:= H1 eps H2).
elim H3; intros.
exists x0.
split.
elim H4; intros; assumption.
intros; case (Req_dec (f1 x) (f1 x1)); intro.
rewrite H6; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
assumption.
elim H4; intros; apply H8.
split.
unfold Dgf, D_x, no_cond in |- *.
split.
split.
trivial.
elim H5; unfold D_x, no_cond in |- *; intros.
elim H9; intros; assumption.
split.
trivial.
assumption.
elim H5; intros; assumption.
Qed.
Lemma continuity_plus :
forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 + f2).
unfold continuity in |- *; intros;
apply (continuity_pt_plus f1 f2 x (H x) (H0 x)).
Qed.
Lemma continuity_opp : forall f, continuity f -> continuity (- f).
unfold continuity in |- *; intros; apply (continuity_pt_opp f x (H x)).
Qed.
Lemma continuity_minus :
forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 - f2).
unfold continuity in |- *; intros;
apply (continuity_pt_minus f1 f2 x (H x) (H0 x)).
Qed.
Lemma continuity_mult :
forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 * f2).
unfold continuity in |- *; intros;
apply (continuity_pt_mult f1 f2 x (H x) (H0 x)).
Qed.
Lemma continuity_const : forall f, constant f -> continuity f.
unfold continuity in |- *; intros; apply (continuity_pt_const f x H).
Qed.
Lemma continuity_scal :
forall f (a:R), continuity f -> continuity (mult_real_fct a f).
unfold continuity in |- *; intros; apply (continuity_pt_scal f a x (H x)).
Qed.
Lemma continuity_inv :
forall f, continuity f -> (forall x:R, f x <> 0) -> continuity (/ f).
unfold continuity in |- *; intros; apply (continuity_pt_inv f x (H x) (H0 x)).
Qed.
Lemma continuity_div :
forall f1 f2,
continuity f1 ->
continuity f2 -> (forall x:R, f2 x <> 0) -> continuity (f1 / f2).
unfold continuity in |- *; intros;
apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)).
Qed.
Lemma continuity_comp :
forall f1 f2, continuity f1 -> continuity f2 -> continuity (f2 o f1).
unfold continuity in |- *; intros.
apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))).
Qed.
| Derivative's definition using Landau's kernel |
Definition derivable_pt_lim f (x l:R) : Prop :=
forall eps:R,
0 < eps ->
exists delta : posreal,
(forall h:R,
h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - l) < eps).
Definition derivable_pt_abs f (x l:R) : Prop := derivable_pt_lim f x l.
Definition derivable_pt f (x:R) := sigT (derivable_pt_abs f x).
Definition derivable f := forall x:R, derivable_pt f x.
Definition derive_pt f (x:R) (pr:derivable_pt f x) := projT1 pr.
Definition derive f (pr:derivable f) (x:R) := derive_pt f x (pr x).
Arguments Scope derivable_pt_lim [Rfun_scope R_scope].
Arguments Scope derivable_pt_abs [Rfun_scope R_scope R_scope].
Arguments Scope derivable_pt [Rfun_scope R_scope].
Arguments Scope derivable [Rfun_scope].
Arguments Scope derive_pt [Rfun_scope R_scope _].
Arguments Scope derive [Rfun_scope _].
Definition antiderivative f (g:R -> R) (a b:R) : Prop :=
(forall x:R,
a <= x <= b -> exists pr : derivable_pt g x, f x = derive_pt g x pr) /\
a <= b.
| Class of differential functions |
Record Differential : Type := mkDifferential
{d1 :> R -> R; cond_diff : derivable d1}.
Record Differential_D2 : Type := mkDifferential_D2
{d2 :> R -> R;
cond_D1 : derivable d2;
cond_D2 : derivable (derive d2 cond_D1)}.
Lemma uniqueness_step1 :
forall f (x l1 l2:R),
limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l1 0 ->
limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l2 0 ->
l1 = l2.
intros;
apply
(single_limit (fun h:R => (f (x + h) - f x) / h) (
fun h:R => h <> 0) l1 l2 0); try assumption.
unfold adhDa in |- *; intros; exists (alp / 2).
split.
unfold Rdiv in |- *; apply prod_neq_R0.
red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1).
apply Rinv_neq_0_compat; discrR.
unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite Rabs_mult.
replace (Rabs (/ 2)) with (/ 2).
replace (Rabs alp) with alp.
apply Rmult_lt_reg_l with 2.
prove_sup0.
rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
[ idtac | discrR ]; rewrite Rmult_1_r; rewrite double;
pattern alp at 1 in |- *; replace alp with (alp + 0);
[ idtac | ring ]; apply Rplus_lt_compat_l; assumption.
symmetry in |- *; apply Rabs_right; left; assumption.
symmetry in |- *; apply Rabs_right; left; change (0 < / 2) in |- *;
apply Rinv_0_lt_compat; prove_sup0.
Qed.
Lemma uniqueness_step2 :
forall f (x l:R),
derivable_pt_lim f x l ->
limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0.
unfold derivable_pt_lim in |- *; intros; unfold limit1_in in |- *;
unfold limit_in in |- *; intros.
assert (H1:= H eps H0).
elim H1; intros.
exists (pos x0).
split.
apply (cond_pos x0).
simpl in |- *; unfold R_dist in |- *; intros.
elim H3; intros.
apply H2;
[ assumption
| unfold Rminus in H5; rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5;
assumption ].
Qed.
Lemma uniqueness_step3 :
forall f (x l:R),
limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 ->
derivable_pt_lim f x l.
unfold limit1_in, derivable_pt_lim in |- *; unfold limit_in in |- *;
unfold dist in |- *; simpl in |- *; intros.
elim (H eps H0).
intros; elim H1; intros.
exists (mkposreal x0 H2).
simpl in |- *; intros; unfold R_dist in H3; apply (H3 h).
split;
[ assumption
| unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; assumption ].
Qed.
Lemma uniqueness_limite :
forall f (x l1 l2:R),
derivable_pt_lim f x l1 -> derivable_pt_lim f x l2 -> l1 = l2.
intros.
assert (H1:= uniqueness_step2 _ _ _ H).
assert (H2:= uniqueness_step2 _ _ _ H0).
assert (H3:= uniqueness_step1 _ _ _ _ H1 H2).
assumption.
Qed.
Lemma derive_pt_eq :
forall f (x l:R) (pr:derivable_pt f x),
derive_pt f x pr = l <-> derivable_pt_lim f x l.
intros; split.
intro; assert (H1:= projT2 pr); unfold derive_pt in H; rewrite H in H1;
assumption.
intro; assert (H1:= projT2 pr); unfold derivable_pt_abs in H1.
assert (H2:= uniqueness_limite _ _ _ _ H H1).
unfold derive_pt in |- *; unfold derivable_pt_abs in |- *.
symmetry in |- *; assumption.
Qed.
Lemma derive_pt_eq_0 :
forall f (x l:R) (pr:derivable_pt f x),
derivable_pt_lim f x l -> derive_pt f x pr = l.
intros; elim (derive_pt_eq f x l pr); intros.
apply (H1 H).
Qed.
Lemma derive_pt_eq_1 :
forall f (x l:R) (pr:derivable_pt f x),
derive_pt f x pr = l -> derivable_pt_lim f x l.
intros; elim (derive_pt_eq f x l pr); intros.
apply (H0 H).
Qed.
| Equivalence of this definition with the one using limit concept |
Lemma derive_pt_D_in :
forall f (df:R -> R) (x:R) (pr:derivable_pt f x),
D_in f df no_cond x <-> derive_pt f x pr = df x.
intros; split.
unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *;
simpl in |- *; unfold R_dist in |- *; intros.
apply derive_pt_eq_0.
unfold derivable_pt_lim in |- *.
intros; elim (H eps H0); intros alpha H1; elim H1; intros;
exists (mkposreal alpha H2); intros; generalize (H3 (x + h));
intro; cut (x + h - x = h);
[ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha);
[ intro; generalize (H6 H8); rewrite H7; intro; assumption
| split;
[ unfold D_x in |- *; split;
[ unfold no_cond in |- *; trivial
| apply Rminus_not_eq_right; rewrite H7; assumption ]
| rewrite H7; assumption ] ]
| ring ].
intro.
assert (H0:= derive_pt_eq_1 f x (df x) pr H).
unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *;
unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
intros.
elim (H0 eps H1); intros alpha H2; exists (pos alpha); split.
apply (cond_pos alpha).
intros; elim H3; intros; unfold D_x in H4; elim H4; intros; cut (x0 - x <> 0).
intro; generalize (H2 (x0 - x) H8 H5); replace (x + (x0 - x)) with x0.
intro; assumption.
ring.
auto with real.
Qed.
Lemma derivable_pt_lim_D_in :
forall f (df:R -> R) (x:R),
D_in f df no_cond x <-> derivable_pt_lim f x (df x).
intros; split.
unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *;
simpl in |- *; unfold R_dist in |- *; intros.
unfold derivable_pt_lim in |- *.
intros; elim (H eps H0); intros alpha H1; elim H1; intros;
exists (mkposreal alpha H2); intros; generalize (H3 (x + h));
intro; cut (x + h - x = h);
[ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha);
[ intro; generalize (H6 H8); rewrite H7; intro; assumption
| split;
[ unfold D_x in |- *; split;
[ unfold no_cond in |- *; trivial
| apply Rminus_not_eq_right; rewrite H7; assumption ]
| rewrite H7; assumption ] ]
| ring ].
intro.
unfold derivable_pt_lim in H.
unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *;
unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
intros.
elim (H eps H0); intros alpha H2; exists (pos alpha); split.
apply (cond_pos alpha).
intros.
elim H1; intros; unfold D_x in H3; elim H3; intros; cut (x0 - x <> 0).
intro; generalize (H2 (x0 - x) H7 H4); replace (x + (x0 - x)) with x0.
intro; assumption.
ring.
auto with real.
Qed.
| derivability -> continuity |
Lemma derivable_derive :
forall f (x:R) (pr:derivable_pt f x), exists l : R, derive_pt f x pr = l.
intros; exists (projT1 pr).
unfold derive_pt in |- *; reflexivity.
Qed.
Theorem derivable_continuous_pt :
forall f (x:R), derivable_pt f x -> continuity_pt f x.
intros.
generalize (derivable_derive f x X); intro.
elim H; intros l H1.
cut (l = fct_cte l x).
intro.
rewrite H0 in H1.
generalize (derive_pt_D_in f (fct_cte l) x); intro.
elim (H2 X); intros.
generalize (H4 H1); intro.
unfold continuity_pt in |- *.
apply (cont_deriv f (fct_cte l) no_cond x H5).
unfold fct_cte in |- *; reflexivity.
Qed.
Theorem derivable_continuous : forall f, derivable f -> continuity f.
unfold derivable, continuity in |- *; intros.
apply (derivable_continuous_pt f x (X x)).
Qed.
| Main rules |
Lemma derivable_pt_lim_plus :
forall f1 f2 (x l1 l2:R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 + f2) x (l1 + l2).
intros.
apply uniqueness_step3.
assert (H1:= uniqueness_step2 _ _ _ H).
assert (H2:= uniqueness_step2 _ _ _ H0).
unfold plus_fct in |- *.
cut
(forall h:R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h).
intro.
generalize
(limit_plus (fun h':R => (f1 (x + h') - f1 x) / h')
(fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2).
unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
simpl in |- *; unfold R_dist in |- *; intros.
elim (H4 eps H5); intros.
exists x0.
elim H6; intros.
split.
assumption.
intros; rewrite H3; apply H8; assumption.
intro; unfold Rdiv in |- *; ring.
Qed.
Lemma derivable_pt_lim_opp :
forall f (x l:R), derivable_pt_lim f x l -> derivable_pt_lim (- f) x (- l).
intros.
apply uniqueness_step3.
assert (H1:= uniqueness_step2 _ _ _ H).
unfold opp_fct in |- *.
cut (forall h:R, (- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)).
intro.
generalize
(limit_Ropp (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 H1).
unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
simpl in |- *; unfold R_dist in |- *; intros.
elim (H2 eps H3); intros.
exists x0.
elim H4; intros.
split.
assumption.
intros; rewrite H0; apply H6; assumption.
intro; unfold Rdiv in |- *; ring.
Qed.
Lemma derivable_pt_lim_minus :
forall f1 f2 (x l1 l2:R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 - f2) x (l1 - l2).
intros.
apply uniqueness_step3.
assert (H1:= uniqueness_step2 _ _ _ H).
assert (H2:= uniqueness_step2 _ _ _ H0).
unfold minus_fct in |- *.
cut
(forall h:R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h).
intro.
generalize
(limit_minus (fun h':R => (f1 (x + h') - f1 x) / h')
(fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2).
unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
simpl in |- *; unfold R_dist in |- *; intros.
elim (H4 eps H5); intros.
exists x0.
elim H6; intros.
split.
assumption.
intros; rewrite <- H3; apply H8; assumption.
intro; unfold Rdiv in |- *; ring.
Qed.
Lemma derivable_pt_lim_mult :
forall f1 f2 (x l1 l2:R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2).
intros.
assert (H1:= derivable_pt_lim_D_in f1 (fun y:R => l1) x).
elim H1; intros.
assert (H4:= H3 H).
assert (H5:= derivable_pt_lim_D_in f2 (fun y:R => l2) x).
elim H5; intros.
assert (H8:= H7 H0).
clear H1 H2 H3 H5 H6 H7.
assert
(H1:=
derivable_pt_lim_D_in (f1 * f2)%F (fun y:R => l1 * f2 x + f1 x * l2) x).
elim H1; intros.
clear H1 H3.
apply H2.
unfold mult_fct in |- *.
apply (Dmult no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); assumption.
Qed.
Lemma derivable_pt_lim_const : forall a x:R, derivable_pt_lim (fct_cte a) x 0.
intros; unfold fct_cte, derivable_pt_lim in |- *.
intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus in |- *;
rewrite Rplus_opp_r; unfold Rdiv in |- *; rewrite Rmult_0_l;
rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
Qed.
Lemma derivable_pt_lim_scal :
forall f (a x l:R),
derivable_pt_lim f x l -> derivable_pt_lim (mult_real_fct a f) x (a * l).
intros.
assert (H0:= derivable_pt_lim_const a x).
replace (mult_real_fct a f) with (fct_cte a * f)%F.
replace (a * l) with (0 * f x + a * l); [ idtac | ring ].
apply (derivable_pt_lim_mult (fct_cte a) f x 0 l); assumption.
unfold mult_real_fct, mult_fct, fct_cte in |- *; reflexivity.
Qed.
Lemma derivable_pt_lim_id : forall x:R, derivable_pt_lim id x 1.
intro; unfold derivable_pt_lim in |- *.
intros eps Heps; exists (mkposreal eps Heps); intros h H1 H2;
unfold id in |- *; replace ((x + h - x) / h - 1) with 0.
rewrite Rabs_R0; apply Rle_lt_trans with (Rabs h).
apply Rabs_pos.
assumption.
unfold Rminus in |- *; rewrite Rplus_assoc; rewrite (Rplus_comm x);
rewrite Rplus_assoc.
rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv in |- *;
rewrite <- Rinv_r_sym.
symmetry in |- *; apply Rplus_opp_r.
assumption.
Qed.
Lemma derivable_pt_lim_Rsqr : forall x:R, derivable_pt_lim Rsqr x (2 * x).
intro; unfold derivable_pt_lim in |- *.
unfold Rsqr in |- *; intros eps Heps; exists (mkposreal eps Heps);
intros h H1 H2; replace (((x + h) * (x + h) - x * x) / h - 2 * x) with h.
assumption.
replace ((x + h) * (x + h) - x * x) with (2 * x * h + h * h);
[ idtac | ring ].
unfold Rdiv in |- *; rewrite Rmult_plus_distr_r.
repeat rewrite Rmult_assoc.
repeat rewrite <- Rinv_r_sym; [ idtac | assumption ].
ring.
Qed.
Lemma derivable_pt_lim_comp :
forall f1 f2 (x l1 l2:R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 (f1 x) l2 -> derivable_pt_lim (f2 o f1) x (l2 * l1).
intros; assert (H1:= derivable_pt_lim_D_in f1 (fun y:R => l1) x).
elim H1; intros.
assert (H4:= H3 H).
assert (H5:= derivable_pt_lim_D_in f2 (fun y:R => l2) (f1 x)).
elim H5; intros.
assert (H8:= H7 H0).
clear H1 H2 H3 H5 H6 H7.
assert (H1:= derivable_pt_lim_D_in (f2 o f1)%F (fun y:R => l2 * l1) x).
elim H1; intros.
clear H1 H3; apply H2.
unfold comp in |- *;
cut
(D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1)
(Dgf no_cond no_cond f1) x ->
D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1) no_cond x).
intro; apply H1.
rewrite Rmult_comm;
apply (Dcomp no_cond no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x);
assumption.
unfold Dgf, D_in, no_cond in |- *; unfold limit1_in in |- *;
unfold limit_in in |- *; unfold dist in |- *; simpl in |- *;
unfold R_dist in |- *; intros.
elim (H1 eps H3); intros.
exists x0; intros; split.
elim H5; intros; assumption.
intros; elim H5; intros; apply H9; split.
unfold D_x in |- *; split.
split; trivial.
elim H6; intros; unfold D_x in H10; elim H10; intros; assumption.
elim H6; intros; assumption.
Qed.
Lemma derivable_pt_plus :
forall f1 f2 (x:R),
derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 + f2) x.
unfold derivable_pt in |- *; intros.
elim X; intros.
elim X0; intros.
apply existT with (x0 + x1).
apply derivable_pt_lim_plus; assumption.
Qed.
Lemma derivable_pt_opp :
forall f (x:R), derivable_pt f x -> derivable_pt (- f) x.
unfold derivable_pt in |- *; intros.
elim X; intros.
apply existT with (- x0).
apply derivable_pt_lim_opp; assumption.
Qed.
Lemma derivable_pt_minus :
forall f1 f2 (x:R),
derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 - f2) x.
unfold derivable_pt in |- *; intros.
elim X; intros.
elim X0; intros.
apply existT with (x0 - x1).
apply derivable_pt_lim_minus; assumption.
Qed.
Lemma derivable_pt_mult :
forall f1 f2 (x:R),
derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 * f2) x.
unfold derivable_pt in |- *; intros.
elim X; intros.
elim X0; intros.
apply existT with (x0 * f2 x + f1 x * x1).
apply derivable_pt_lim_mult; assumption.
Qed.
Lemma derivable_pt_const : forall a x:R, derivable_pt (fct_cte a) x.
intros; unfold derivable_pt in |- *.
apply existT with 0.
apply derivable_pt_lim_const.
Qed.
Lemma derivable_pt_scal :
forall f (a x:R), derivable_pt f x -> derivable_pt (mult_real_fct a f) x.
unfold derivable_pt in |- *; intros.
elim X; intros.
apply existT with (a * x0).
apply derivable_pt_lim_scal; assumption.
Qed.
Lemma derivable_pt_id : forall x:R, derivable_pt id x.
unfold derivable_pt in |- *; intro.
exists 1.
apply derivable_pt_lim_id.
Qed.
Lemma derivable_pt_Rsqr : forall x:R, derivable_pt Rsqr x.
unfold derivable_pt in |- *; intro; apply existT with (2 * x).
apply derivable_pt_lim_Rsqr.
Qed.
Lemma derivable_pt_comp :
forall f1 f2 (x:R),
derivable_pt f1 x -> derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x.
unfold derivable_pt in |- *; intros.
elim X; intros.
elim X0; intros.
apply existT with (x1 * x0).
apply derivable_pt_lim_comp; assumption.
Qed.
Lemma derivable_plus :
forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 + f2).
unfold derivable in |- *; intros.
apply (derivable_pt_plus _ _ x (X _) (X0 _)).
Qed.
Lemma derivable_opp : forall f, derivable f -> derivable (- f).
unfold derivable in |- *; intros.
apply (derivable_pt_opp _ x (X _)).
Qed.
Lemma derivable_minus :
forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 - f2).
unfold derivable in |- *; intros.
apply (derivable_pt_minus _ _ x (X _) (X0 _)).
Qed.
Lemma derivable_mult :
forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 * f2).
unfold derivable in |- *; intros.
apply (derivable_pt_mult _ _ x (X _) (X0 _)).
Qed.
Lemma derivable_const : forall a:R, derivable (fct_cte a).
unfold derivable in |- *; intros.
apply derivable_pt_const.
Qed.
Lemma derivable_scal :
forall f (a:R), derivable f -> derivable (mult_real_fct a f).
unfold derivable in |- *; intros.
apply (derivable_pt_scal _ a x (X _)).
Qed.
Lemma derivable_id : derivable id.
unfold derivable in |- *; intro; apply derivable_pt_id.
Qed.
Lemma derivable_Rsqr : derivable Rsqr.
unfold derivable in |- *; intro; apply derivable_pt_Rsqr.
Qed.
Lemma derivable_comp :
forall f1 f2, derivable f1 -> derivable f2 -> derivable (f2 o f1).
unfold derivable in |- *; intros.
apply (derivable_pt_comp _ _ x (X _) (X0 _)).
Qed.
Lemma derive_pt_plus :
forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
derive_pt (f1 + f2) x (derivable_pt_plus _ _ _ pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2.
intros.
assert (H:= derivable_derive f1 x pr1).
assert (H0:= derivable_derive f2 x pr2).
assert
(H1:= derivable_derive (f1 + f2)%F x (derivable_pt_plus _ _ _ pr1 pr2)).
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
assert (H3:= projT2 pr1).
unfold derive_pt in H; rewrite H in H3.
assert (H4:= projT2 pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_plus; assumption.
Qed.
Lemma derive_pt_opp :
forall f (x:R) (pr1:derivable_pt f x),
derive_pt (- f) x (derivable_pt_opp _ _ pr1) = - derive_pt f x pr1.
intros.
assert (H:= derivable_derive f x pr1).
assert (H0:= derivable_derive (- f)%F x (derivable_pt_opp _ _ pr1)).
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
rewrite H; apply derive_pt_eq_0.
assert (H3:= projT2 pr1).
unfold derive_pt in H; rewrite H in H3.
apply derivable_pt_lim_opp; assumption.
Qed.
Lemma derive_pt_minus :
forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
derive_pt (f1 - f2) x (derivable_pt_minus _ _ _ pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2.
intros.
assert (H:= derivable_derive f1 x pr1).
assert (H0:= derivable_derive f2 x pr2).
assert
(H1:= derivable_derive (f1 - f2)%F x (derivable_pt_minus _ _ _ pr1 pr2)).
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
assert (H3:= projT2 pr1).
unfold derive_pt in H; rewrite H in H3.
assert (H4:= projT2 pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_minus; assumption.
Qed.
Lemma derive_pt_mult :
forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
derive_pt (f1 * f2) x (derivable_pt_mult _ _ _ pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2.
intros.
assert (H:= derivable_derive f1 x pr1).
assert (H0:= derivable_derive f2 x pr2).
assert
(H1:= derivable_derive (f1 * f2)%F x (derivable_pt_mult _ _ _ pr1 pr2)).
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
assert (H3:= projT2 pr1).
unfold derive_pt in H; rewrite H in H3.
assert (H4:= projT2 pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_mult; assumption.
Qed.
Lemma derive_pt_const :
forall a x:R, derive_pt (fct_cte a) x (derivable_pt_const a x) = 0.
intros.
apply derive_pt_eq_0.
apply derivable_pt_lim_const.
Qed.
Lemma derive_pt_scal :
forall f (a x:R) (pr:derivable_pt f x),
derive_pt (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr) =
a * derive_pt f x pr.
intros.
assert (H:= derivable_derive f x pr).
assert
(H0:= derivable_derive (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr)).
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
rewrite H; apply derive_pt_eq_0.
assert (H3:= projT2 pr).
unfold derive_pt in H; rewrite H in H3.
apply derivable_pt_lim_scal; assumption.
Qed.
Lemma derive_pt_id : forall x:R, derive_pt id x (derivable_pt_id _) = 1.
intros.
apply derive_pt_eq_0.
apply derivable_pt_lim_id.
Qed.
Lemma derive_pt_Rsqr :
forall x:R, derive_pt Rsqr x (derivable_pt_Rsqr _) = 2 * x.
intros.
apply derive_pt_eq_0.
apply derivable_pt_lim_Rsqr.
Qed.
Lemma derive_pt_comp :
forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 (f1 x)),
derive_pt (f2 o f1) x (derivable_pt_comp _ _ _ pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1.
intros.
assert (H:= derivable_derive f1 x pr1).
assert (H0:= derivable_derive f2 (f1 x) pr2).
assert
(H1:= derivable_derive (f2 o f1)%F x (derivable_pt_comp _ _ _ pr1 pr2)).
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
assert (H3:= projT2 pr1).
unfold derive_pt in H; rewrite H in H3.
assert (H4:= projT2 pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_comp; assumption.
Qed.
Definition pow_fct (n:nat) (y:R) : R := y ^ n.
Lemma derivable_pt_lim_pow_pos :
forall (x:R) (n:nat),
(0 < n)%nat -> derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n).
intros.
induction n as [| n Hrecn].
elim (lt_irrefl _ H).
cut (n = 0%nat \/ (0 < n)%nat).
intro; elim H0; intro.
rewrite H1; simpl in |- *.
replace (fun y:R => y * 1) with (id * fct_cte 1)%F.
replace (1 * 1) with (1 * fct_cte 1 x + id x * 0).
apply derivable_pt_lim_mult.
apply derivable_pt_lim_id.
apply derivable_pt_lim_const.
unfold fct_cte, id in |- *; ring.
reflexivity.
replace (fun y:R => y ^ S n) with (fun y:R => y * y ^ n).
replace (pred (S n)) with n; [ idtac | reflexivity ].
replace (fun y:R => y * y ^ n) with (id * (fun y:R => y ^ n))%F.
set (f:= fun y:R => y ^ n).
replace (INR (S n) * x ^ n) with (1 * f x + id x * (INR n * x ^ pred n)).
apply derivable_pt_lim_mult.
apply derivable_pt_lim_id.
unfold f in |- *; apply Hrecn; assumption.
unfold f in |- *.
pattern n at 1 5 in |- *; replace n with (S (pred n)).
unfold id in |- *; rewrite S_INR; simpl in |- *.
ring.
symmetry in |- *; apply S_pred with 0%nat; assumption.
unfold mult_fct, id in |- *; reflexivity.
reflexivity.
inversion H.
left; reflexivity.
right.
apply lt_le_trans with 1%nat.
apply lt_O_Sn.
assumption.
Qed.
Lemma derivable_pt_lim_pow :
forall (x:R) (n:nat),
derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n).
intros.
induction n as [| n Hrecn].
simpl in |- *.
rewrite Rmult_0_l.
replace (fun _:R => 1) with (fct_cte 1);
[ apply derivable_pt_lim_const | reflexivity ].
apply derivable_pt_lim_pow_pos.
apply lt_O_Sn.
Qed.
Lemma derivable_pt_pow :
forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x.
intros; unfold derivable_pt in |- *.
apply existT with (INR n * x ^ pred n).
apply derivable_pt_lim_pow.
Qed.
Lemma derivable_pow : forall n:nat, derivable (fun y:R => y ^ n).
intro; unfold derivable in |- *; intro; apply derivable_pt_pow.
Qed.
Lemma derive_pt_pow :
forall (n:nat) (x:R),
derive_pt (fun y:R => y ^ n) x (derivable_pt_pow n x) = INR n * x ^ pred n.
intros; apply derive_pt_eq_0.
apply derivable_pt_lim_pow.
Qed.
Lemma pr_nu :
forall f (x:R) (pr1 pr2:derivable_pt f x),
derive_pt f x pr1 = derive_pt f x pr2.
intros.
unfold derivable_pt in pr1.
unfold derivable_pt in pr2.
elim pr1; intros.
elim pr2; intros.
unfold derivable_pt_abs in p.
unfold derivable_pt_abs in p0.
simpl in |- *.
apply (uniqueness_limite f x x0 x1 p p0).
Qed.
| Local extremum's condition |
Theorem deriv_maximum :
forall f (a b c:R) (pr:derivable_pt f c),
a < c ->
c < b ->
(forall x:R, a < x -> x < b -> f x <= f c) -> derive_pt f c pr = 0.
intros; case (Rtotal_order 0 (derive_pt f c pr)); intro.
assert (H3:= derivable_derive f c pr).
elim H3; intros l H4; rewrite H4 in H2.
assert (H5:= derive_pt_eq_1 f c l pr H4).
cut (0 < l / 2);
[ intro
| unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
elim (H5 (l / 2) H6); intros delta H7.
cut (0 < (b - c) / 2).
intro; cut (Rmin (delta / 2) ((b - c) / 2) <> 0).
intro; cut (Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta).
intro.
assert (H11:= H7 (Rmin (delta / 2) ((b - c) / 2)) H9 H10).
cut (0 < Rmin (delta / 2) ((b - c) / 2)).
intro; cut (a < c + Rmin (delta / 2) ((b - c) / 2)).
intro; cut (c + Rmin (delta / 2) ((b - c) / 2) < b).
intro; assert (H15:= H1 (c + Rmin (delta / 2) ((b - c) / 2)) H13 H14).
cut
((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0).
intro; cut (- l < 0).
intro; unfold Rminus in H11.
cut
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + - l < 0).
intro;
cut
(Rabs
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + - l) < l / 2).
unfold Rabs in |- *;
case
(Rcase_abs
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + - l)); intro.
replace
(-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + - l)) with
(l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2))).
intro;
generalize
(Rplus_lt_compat_l (- l)
(l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2))) (l / 2) H19);
repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l;
rewrite Rplus_0_l; replace (- l + l / 2) with (- (l / 2)).
intro;
generalize
(Ropp_lt_gt_contravar
(-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2))) (- (l / 2)) H20);
repeat rewrite Ropp_involutive; intro;
generalize
(Rlt_trans 0 (l / 2)
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2)) H6 H21); intro;
elim
(Rlt_irrefl 0
(Rlt_le_trans 0
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2)) 0 H22 H16)).
pattern l at 2 in |- *; rewrite double_var.
ring.
ring.
intro.
assert
(H20:=
Rge_le
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + - l) 0 r).
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H20 H18)).
assumption.
rewrite <- Ropp_0;
replace
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + - l) with
(-
(l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
Rmin (delta / 2) ((b + - c) / 2)))).
apply Ropp_gt_lt_contravar;
change
(0 <
l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
Rmin (delta / 2) ((b + - c) / 2))) in |- *; apply Rplus_lt_le_0_compat;
[ assumption
| rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption ].
ring.
rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
replace
((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2)) with
(-
((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) /
Rmin (delta / 2) ((b - c) / 2))).
rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge;
unfold Rdiv in |- *; apply Rmult_le_pos;
[ generalize
(Rplus_le_compat_r (- f (c + Rmin (delta * / 2) ((b - c) * / 2)))
(f (c + Rmin (delta * / 2) ((b - c) * / 2))) (
f c) H15); rewrite Rplus_opp_r; intro; assumption
| left; apply Rinv_0_lt_compat; assumption ].
unfold Rdiv in |- *.
rewrite <- Ropp_mult_distr_l_reverse.
repeat rewrite <- (Rmult_comm (/ Rmin (delta * / 2) ((b - c) * / 2))).
apply Rmult_eq_reg_l with (Rmin (delta * / 2) ((b - c) * / 2)).
repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_r_sym.
repeat rewrite Rmult_1_l.
ring.
red in |- *; intro.
unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12).
red in |- *; intro.
unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12).
assert (H14:= Rmin_r (delta / 2) ((b - c) / 2)).
assert
(H15:=
Rplus_le_compat_l c (Rmin (delta / 2) ((b - c) / 2)) ((b - c) / 2) H14).
apply Rle_lt_trans with (c + (b - c) / 2).
assumption.
apply Rmult_lt_reg_l with 2.
prove_sup0.
replace (2 * (c + (b - c) / 2)) with (c + b).
replace (2 * b) with (b + b).
apply Rplus_lt_compat_r; assumption.
ring.
unfold Rdiv in |- *; rewrite Rmult_plus_distr_l.
repeat rewrite (Rmult_comm 2).
rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r.
ring.
discrR.
apply Rlt_trans with c.
assumption.
pattern c at 1 in |- *; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l;
assumption.
cut (0 < delta / 2).
intro;
apply
(Rmin_stable_in_posreal (mkposreal (delta / 2) H12)
(mkposreal ((b - c) / 2) H8)).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
unfold Rabs in |- *; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))).
intro.
cut (0 < delta / 2).
intro.
generalize
(Rmin_stable_in_posreal (mkposreal (delta / 2) H10)
(mkposreal ((b - c) / 2) H8)); simpl in |- *; intro;
elim (Rlt_irrefl 0 (Rlt_trans 0 (Rmin (delta / 2) ((b - c) / 2)) 0 H11 r)).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
intro; apply Rle_lt_trans with (delta / 2).
apply Rmin_l.
unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2.
prove_sup0.
rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
rewrite Rmult_1_l.
replace (2 * delta) with (delta + delta).
pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta);
apply Rplus_lt_compat_l.
rewrite Rplus_0_r; apply (cond_pos delta).
symmetry in |- *; apply double.
discrR.
cut (0 < delta / 2).
intro;
generalize
(Rmin_stable_in_posreal (mkposreal (delta / 2) H9)
(mkposreal ((b - c) / 2) H8)); simpl in |- *;
intro; red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
unfold Rdiv in |- *; apply Rmult_lt_0_compat.
generalize (Rplus_lt_compat_r (- c) c b H0); rewrite Rplus_opp_r; intro;
assumption.
apply Rinv_0_lt_compat; prove_sup0.
elim H2; intro.
symmetry in |- *; assumption.
generalize (derivable_derive f c pr); intro; elim H4; intros l H5.
rewrite H5 in H3; generalize (derive_pt_eq_1 f c l pr H5); intro;
cut (0 < - (l / 2)).
intro; elim (H6 (- (l / 2)) H7); intros delta H9.
cut (0 < (c - a) / 2).
intro; cut (Rmax (- (delta / 2)) ((a - c) / 2) < 0).
intro; cut (Rmax (- (delta / 2)) ((a - c) / 2) <> 0).
intro; cut (Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta).
intro; generalize (H9 (Rmax (- (delta / 2)) ((a - c) / 2)) H11 H12); intro;
cut (a < c + Rmax (- (delta / 2)) ((a - c) / 2)).
cut (c + Rmax (- (delta / 2)) ((a - c) / 2) < b).
intros; generalize (H1 (c + Rmax (- (delta / 2)) ((a - c) / 2)) H15 H14);
intro;
cut
(0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)).
intro; cut (0 < - l).
intro; unfold Rminus in H13;
cut
(0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + - l).
intro;
cut
(Rabs
((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + - l) <
- (l / 2)).
unfold Rabs in |- *;
case
(Rcase_abs
((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + - l)).
intro;
elim
(Rlt_irrefl 0
(Rlt_trans 0
((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + - l) 0 H19 r)).
intros;
generalize
(Rplus_lt_compat_r l
((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + - l) (
- (l / 2)) H20); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l;
rewrite Rplus_0_r; replace (- (l / 2) + l) with (l / 2).
cut (l / 2 < 0).
intros;
generalize
(Rlt_trans
((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2)) (l / 2) 0 H22 H21);
intro;
elim
(Rlt_irrefl 0
(Rle_lt_trans 0
((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)) 0 H17 H23)).
rewrite <- (Ropp_involutive (l / 2)); rewrite <- Ropp_0;
apply Ropp_lt_gt_contravar; assumption.
pattern l at 3 in |- *; rewrite double_var.
ring.
assumption.
apply Rplus_le_lt_0_compat; assumption.
rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
unfold Rdiv in |- *;
replace
((f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)) with
(- (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2)).
apply Rmult_le_pos.
generalize
(Rplus_le_compat_l (- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)))
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2))) (
f c) H16); rewrite Rplus_opp_l;
replace (- (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c)) with
(- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) + f c).
intro; assumption.
ring.
left; apply Rinv_0_lt_compat; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar;
assumption.
unfold Rdiv in |- *.
rewrite <- Ropp_inv_permute.
rewrite Rmult_opp_opp.
reflexivity.
unfold Rdiv in H11; assumption.
generalize (Rplus_lt_compat_l c (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H10);
rewrite Rplus_0_r; intro; apply Rlt_trans with c;
assumption.
generalize (RmaxLess2 (- (delta / 2)) ((a - c) / 2)); intro;
generalize
(Rplus_le_compat_l c ((a - c) / 2) (Rmax (- (delta / 2)) ((a - c) / 2)) H14);
intro; apply Rlt_le_trans with (c + (a - c) / 2).
apply Rmult_lt_reg_l with 2.
prove_sup0.
replace (2 * (c + (a - c) / 2)) with (a + c).
rewrite double.
apply Rplus_lt_compat_l; assumption.
ring.
rewrite <- Rplus_assoc.
rewrite <- double_var.
ring.
assumption.
unfold Rabs in |- *; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))).
intro; generalize (RmaxLess1 (- (delta / 2)) ((a - c) / 2)); intro;
generalize
(Ropp_le_ge_contravar (- (delta / 2)) (Rmax (- (delta / 2)) ((a - c) / 2))
H12); rewrite Ropp_involutive; intro;
generalize (Rge_le (delta / 2) (- Rmax (- (delta / 2)) ((a - c) / 2)) H13);
intro; apply Rle_lt_trans with (delta / 2).
assumption.
apply Rmult_lt_reg_l with 2.
prove_sup0.
unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l; rewrite double.
pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta);
apply Rplus_lt_compat_l; rewrite Rplus_0_r; apply (cond_pos delta).
discrR.
cut (- (delta / 2) < 0).
cut ((a - c) / 2 < 0).
intros;
generalize
(Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H13)
(mknegreal ((a - c) / 2) H12)); simpl in |- *;
intro; generalize (Rge_le (Rmax (- (delta / 2)) ((a - c) / 2)) 0 r);
intro;
elim
(Rlt_irrefl 0
(Rle_lt_trans 0 (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H15 H14)).
rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2));
apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2).
assumption.
unfold Rdiv in |- *.
rewrite <- Ropp_mult_distr_l_reverse.
rewrite (Ropp_minus_distr a c).
reflexivity.
rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *;
apply Rmult_lt_0_compat;
[ apply (cond_pos delta)
| assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).
cut ((a - c) / 2 < 0).
intro; cut (- (delta / 2) < 0).
intro;
apply
(Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H11)
(mknegreal ((a - c) / 2) H10)).
rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *;
apply Rmult_lt_0_compat;
[ apply (cond_pos delta)
| assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2));
apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2).
assumption.
unfold Rdiv in |- *.
rewrite <- Ropp_mult_distr_l_reverse.
rewrite (Ropp_minus_distr a c).
reflexivity.
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ generalize (Rplus_lt_compat_r (- a) a c H); rewrite Rplus_opp_r; intro;
assumption
| assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
replace (- (l / 2)) with (- l / 2).
unfold Rdiv in |- *; apply Rmult_lt_0_compat.
rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ].
unfold Rdiv in |- *; apply Ropp_mult_distr_l_reverse.
Qed.
Theorem deriv_minimum :
forall f (a b c:R) (pr:derivable_pt f c),
a < c ->
c < b ->
(forall x:R, a < x -> x < b -> f c <= f x) -> derive_pt f c pr = 0.
intros.
rewrite <- (Ropp_involutive (derive_pt f c pr)).
apply Ropp_eq_0_compat.
rewrite <- (derive_pt_opp f c pr).
cut (forall x:R, a < x -> x < b -> (- f)%F x <= (- f)%F c).
intro.
apply (deriv_maximum (- f)%F a b c (derivable_pt_opp _ _ pr) H H0 H2).
intros; unfold opp_fct in |- *; apply Ropp_ge_le_contravar; apply Rle_ge.
apply (H1 x H2 H3).
Qed.
Theorem deriv_constant2 :
forall f (a b c:R) (pr:derivable_pt f c),
a < c ->
c < b -> (forall x:R, a < x -> x < b -> f x = f c) -> derive_pt f c pr = 0.
intros.
eapply deriv_maximum with a b; try assumption.
intros; right; apply (H1 x H2 H3).
Qed.
Lemma nonneg_derivative_0 :
forall f (pr:derivable f),
increasing f -> forall x:R, 0 <= derive_pt f x (pr x).
intros; unfold increasing in H.
assert (H0:= derivable_derive f x (pr x)).
elim H0; intros l H1.
rewrite H1; case (Rtotal_order 0 l); intro.
left; assumption.
elim H2; intro.
right; assumption.
assert (H4:= derive_pt_eq_1 f x l (pr x) H1).
cut (0 < - (l / 2)).
intro; elim (H4 (- (l / 2)) H5); intros delta H6.
cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (delta / 2) < delta).
intro; decompose [and] H7; intros; generalize (H6 (delta / 2) H8 H11);
cut (0 <= (f (x + delta / 2) - f x) / (delta / 2)).
intro; cut (0 <= (f (x + delta / 2) - f x) / (delta / 2) - l).
intro; unfold Rabs in |- *;
case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)).
intro;
elim
(Rlt_irrefl 0
(Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2) - l) 0 H12 r)).
intros;
generalize
(Rplus_lt_compat_r l ((f (x + delta / 2) - f x) / (delta / 2) - l)
(- (l / 2)) H13); unfold Rminus in |- *;
replace (- (l / 2) + l) with (l / 2).
rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; intro;
generalize
(Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) (l / 2) H9 H14);
intro; cut (l / 2 < 0).
intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (l / 2) 0 H15 H16)).
rewrite <- Ropp_0 in H5;
generalize (Ropp_lt_gt_contravar (-0) (- (l / 2)) H5);
repeat rewrite Ropp_involutive; intro; assumption.
pattern l at 3 in |- *; rewrite double_var.
ring.
unfold Rminus in |- *; apply Rplus_le_le_0_compat.
unfold Rdiv in |- *; apply Rmult_le_pos.
cut (x <= x + delta * / 2).
intro; generalize (H x (x + delta * / 2) H12); intro;
generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H13);
rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
left; assumption.
left; apply Rinv_0_lt_compat; assumption.
left; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
unfold Rdiv in |- *; apply Rmult_le_pos.
cut (x <= x + delta * / 2).
intro; generalize (H x (x + delta * / 2) H9); intro;
generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H12);
rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
left; assumption.
left; apply Rinv_0_lt_compat; assumption.
split.
unfold Rdiv in |- *; apply prod_neq_R0.
generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H7;
elim (Rlt_irrefl 0 H7).
apply Rinv_neq_0_compat; discrR.
split.
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
replace (Rabs (delta / 2)) with (delta / 2).
unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2.
prove_sup0.
rewrite (Rmult_comm 2).
rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].
rewrite Rmult_1_r.
rewrite double.
pattern (pos delta) at 1 in |- *; rewrite <- Rplus_0_r.
apply Rplus_lt_compat_l; apply (cond_pos delta).
symmetry in |- *; apply Rabs_right.
left; change (0 < delta / 2) in |- *; unfold Rdiv in |- *;
apply Rmult_lt_0_compat;
[ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse;
apply Rmult_lt_0_compat.
apply Rplus_lt_reg_r with l.
unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption.
apply Rinv_0_lt_compat; prove_sup0.
Qed.