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CPoly_Contin1.v
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CPoly_Contin1.v
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(* Copyright © 1998-2006
* Henk Barendregt
* Luís Cruz-Filipe
* Herman Geuvers
* Mariusz Giero
* Rik van Ginneken
* Dimitri Hendriks
* Sébastien Hinderer
* Bart Kirkels
* Pierre Letouzey
* Iris Loeb
* Lionel Mamane
* Milad Niqui
* Russell O’Connor
* Randy Pollack
* Nickolay V. Shmyrev
* Bas Spitters
* Dan Synek
* Freek Wiedijk
* Jan Zwanenburg
*
* This work is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This work is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this work; if not, write to the Free Software Foundation, Inc.,
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*)
Require Export CC_Props.
(**
* Continuity of complex polynomials
*)
Section Mult_CC_Continuous.
Lemma mult_absCC : forall (x y : CC) (X Y : IR),
AbsCC x [<=] X -> AbsCC y [<=] Y -> AbsCC (x[*]y) [<=] X[*]Y.
Proof.
intros.
astepl (AbsCC x[*]AbsCC y).
apply mult_resp_leEq_both.
apply AbsCC_nonneg. apply AbsCC_nonneg. auto. auto.
Qed.
Lemma estimate_absCC : forall x : CC, {X : IR | [0] [<] X | AbsCC x [<=] X}.
Proof.
intros.
exists (AbsCC x[+][1]).
astepl ([0][+]ZeroR).
apply plus_resp_leEq_less. apply AbsCC_nonneg. apply pos_one.
astepl (AbsCC x[+][0]).
apply less_leEq.
apply plus_resp_less_lft. apply pos_one.
Qed.
Lemma mult_CC_contin : forall (x y : CC) (e : IR),
[0] [<] e -> {c : IR | [0] [<] c | {d : IR | [0] [<] d | forall x' y',
AbsCC (x[-]x') [<=] c -> AbsCC (y[-]y') [<=] d -> AbsCC (x[*]y[-]x'[*]y') [<=] e}}.
Proof.
do 3 intro. intro H.
cut ([0] [<] e [/]TwoNZ). intro H0.
elim (estimate_absCC x). intro X. intros H1 H2.
elim (estimate_absCC y). intro Y. intros H3 H4.
cut (Y[#][0]). intro H5.
exists (e [/]TwoNZ[/] Y[//]H5).
apply div_resp_pos. auto. auto.
cut ([0] [<] X[+](e [/]TwoNZ[/] Y[//]H5)). intro.
cut (X[+](e [/]TwoNZ[/] Y[//]H5)[#][0]). intro H7.
exists (e [/]TwoNZ[/] X[+](e [/]TwoNZ[/] Y[//]H5)[//]H7).
apply div_resp_pos. auto. auto.
intros.
apply leEq_wdl with (AbsCC ((x[-]x')[*]y[+]x'[*](y[-]y'))).
apply leEq_transitive with (AbsCC ((x[-]x')[*]y)[+]AbsCC (x'[*](y[-]y'))).
apply triangle.
rstepr (e [/]TwoNZ[+]e [/]TwoNZ).
apply plus_resp_leEq_both.
apply leEq_wdr with ((e [/]TwoNZ[/] Y[//]H5)[*]Y).
apply mult_absCC; auto.
rational.
apply leEq_wdr with ((X[+](e [/]TwoNZ[/] Y[//]H5))[*]
(e [/]TwoNZ[/] X[+](e [/]TwoNZ[/] Y[//]H5)[//]H7)).
apply mult_absCC; auto.
apply leEq_wdl with (AbsCC (x[+](x'[-]x))).
apply leEq_transitive with (AbsCC x[+]AbsCC (x'[-]x)).
apply triangle.
apply plus_resp_leEq_both. auto.
astepl (AbsCC [--](x'[-]x)).
apply leEq_wdl with (AbsCC (x[-]x')). auto.
apply AbsCC_wd. rational.
apply AbsCC_wd. rational.
rational.
apply AbsCC_wd. rational.
apply Greater_imp_ap. auto.
apply plus_resp_pos; auto.
apply div_resp_pos; auto.
apply Greater_imp_ap. auto.
apply pos_div_two. auto.
Qed.
End Mult_CC_Continuous.
Section CPoly_CC_Continuous.
(**
%\begin{convention}% Let [g] be a polynomial over the complex numbers.
%\end{convention}%
*)
Variable g : CCX.
Lemma cpoly_CC_contin : forall (x : CC) (e : IR), [0] [<] e ->
{d : IR | [0] [<] d | forall x', AbsCC (x[-]x') [<=] d -> AbsCC (g ! x[-]g ! x') [<=] e}.
Proof.
elim g.
intros.
exists OneR. intros. apply pos_one. intros.
apply leEq_wdl with ZeroR. apply less_leEq. auto.
cut ([0] [=] AbsCC ([0][-][0])). auto.
Step_final (AbsCC [0]).
intros a f. intro H. do 2 intro. intro H0.
elim (mult_CC_contin x f ! x e H0). intro d1. intros H1 H2.
elim H2. clear H2. intro c. intros H2 H3.
elim (H x c H2). clear H. intro d2. intros H H4.
exists (Min d1 d2). apply less_Min; auto. intros.
simpl in |- *.
cut (AbsCC (a[+]x[*]f ! x[-](a[+]x'[*]f ! x')) [<=] e). auto.
apply leEq_wdl with (AbsCC (x[*]f ! x[-]x'[*]f ! x')).
apply H3. clear H3.
apply leEq_transitive with (Min d1 d2); auto. apply Min_leEq_lft.
apply H4. clear H4.
apply leEq_transitive with (Min d1 d2); auto. apply Min_leEq_rht.
apply AbsCC_wd.
rational.
Qed.
Lemma contin_polyCC : CCcontin (fun x => g ! x).
Proof.
unfold CCcontin in |- *. unfold CCcontinAt in |- *. unfold CCfunLim in |- *.
intros.
elim (cpoly_CC_contin x e); auto.
intro d. intros H0 H1.
exists d. auto. intros.
apply H1; auto.
Qed.
End CPoly_CC_Continuous.