(**************************************************************************) (* ___ *) (* ||M|| *) (* ||A|| A project by Andrea Asperti *) (* ||T|| *) (* ||I|| Developers: *) (* ||T|| The HELM team. *) (* ||A|| http://helm.cs.unibo.it *) (* \ / *) (* \ / This file is distributed under the terms of the *) (* v GNU General Public License Version 2 *) (* *) (**************************************************************************) (* This file was automatically generated: do not edit *********************) include "CoRN.ma". (* $Id: IVT.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *) include "reals/CPoly_Contin.ma". (* UNEXPORTED Section Nested_Intervals *) (*#* * Intermediate Value Theorem ** Nested intervals %\begin{convention}% Let [a,b:nat->IR] be sequences such that: - [a] is increasing; - [b] is decreasing; - [forall (i:nat), (a i) [<] (b i)]; - for every positive real number [eps], there is an [i] such that [(b i) [<] (a i) [+]eps]. %\end{convention}% *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/a.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/b.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/a_mon.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/b_mon.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/a_b.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/b_a.var *) inline procedural "cic:/CoRN/reals/IVT/a_mon'.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/b_mon'.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/a_b'.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/intervals_cauchy.con" as lemma. (* begin hide *) inline procedural "cic:/CoRN/reals/IVT/Nested_Intervals/a'.con" "Nested_Intervals__" as definition. (* end hide *) inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_limit.con" as lemma. (*#* %\begin{convention}% Let [f] be a continuous real function. %\end{convention}% *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/f.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/f_contin.var *) inline procedural "cic:/CoRN/reals/IVT/f_contin_pos.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/f_contin_neg.con" as lemma. (*#* Assume also that [forall i, f (a i) [<=] Zero [<=] f (b i)]. *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/f_a.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Nested_Intervals/f_b.var *) inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_zero.con" as lemma. (* UNEXPORTED End Nested_Intervals *) (* UNEXPORTED Section Bisection *) (*#* ** Bissections *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisection/f.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisection/f_apzero_interval.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisection/a.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisection/b.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisection/a_b.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisection/f_a.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisection/f_b.var *) (*#* %\begin{convention}% Let [Small] denote [Two[/]ThreeNZ], [lft] be [(Two[*]a[+]b) [/]ThreeNZ] and [rht] be [(a[+]Two[*]b) [/]ThreeNZ]. %\end{convention}% *) (* begin hide *) inline procedural "cic:/CoRN/reals/IVT/Bisection/Small.con" "Bisection__" as definition. inline procedural "cic:/CoRN/reals/IVT/Bisection/lft.con" "Bisection__" as definition. inline procedural "cic:/CoRN/reals/IVT/Bisection/rht.con" "Bisection__" as definition. (* end hide *) inline procedural "cic:/CoRN/reals/IVT/a_lft.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/rht_b.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/lft_rht.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/smaller_lft.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/smaller_rht.con" as lemma. (* UNEXPORTED Hint Resolve smaller_lft smaller_rht: algebra. *) inline procedural "cic:/CoRN/reals/IVT/Cbisect'.con" as lemma. (* UNEXPORTED End Bisection *) (* UNEXPORTED Section Bisect_Interval *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisect_Interval/f.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/Bisect_Interval/C_f_apzero_interval.var *) (* begin hide *) inline procedural "cic:/CoRN/reals/IVT/Bisect_Interval/Small.con" "Bisect_Interval__" as definition. (* end hide *) inline procedural "cic:/CoRN/reals/IVT/bisect_interval.ind". inline procedural "cic:/CoRN/reals/IVT/Cbisect_exists.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/bisect.con" as definition. inline procedural "cic:/CoRN/reals/IVT/bisect_prop.con" as lemma. (* UNEXPORTED End Bisect_Interval *) (* UNEXPORTED Section IVT_Op *) (*#* ** IVT for operations Same conventions as before. *) (* UNEXPORTED cic:/CoRN/reals/IVT/IVT_Op/f.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/IVT_Op/f_contin.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/IVT_Op/f_apzero_interval.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/IVT_Op/a.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/IVT_Op/b.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/IVT_Op/a_b.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/IVT_Op/f_a.var *) (* UNEXPORTED cic:/CoRN/reals/IVT/IVT_Op/f_b.var *) (* begin hide *) inline procedural "cic:/CoRN/reals/IVT/IVT_Op/Small.con" "IVT_Op__" as definition. (* end hide *) inline procedural "cic:/CoRN/reals/IVT/interval_sequence.con" as definition. inline procedural "cic:/CoRN/reals/IVT/IVT_Op/a_.con" "IVT_Op__" as definition. inline procedural "cic:/CoRN/reals/IVT/IVT_Op/b_.con" "IVT_Op__" as definition. inline procedural "cic:/CoRN/reals/IVT/intervals_smaller.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/intervals_small''.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/intervals_small'.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/intervals_small.con" as lemma. inline procedural "cic:/CoRN/reals/IVT/Civt_op.con" as lemma. (* UNEXPORTED End IVT_Op *) (* UNEXPORTED Section IVT_Poly *) (*#* ** IVT for polynomials *) inline procedural "cic:/CoRN/reals/IVT/Civt_poly.con" as lemma. (* UNEXPORTED End IVT_Poly *)