(**************************************************************************) (* ___ *) (* ||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}% *) alias id "a" = "cic:/CoRN/reals/IVT/Nested_Intervals/a.var". alias id "b" = "cic:/CoRN/reals/IVT/Nested_Intervals/b.var". alias id "a_mon" = "cic:/CoRN/reals/IVT/Nested_Intervals/a_mon.var". alias id "b_mon" = "cic:/CoRN/reals/IVT/Nested_Intervals/b_mon.var". alias id "a_b" = "cic:/CoRN/reals/IVT/Nested_Intervals/a_b.var". alias id "b_a" = "cic:/CoRN/reals/IVT/Nested_Intervals/b_a.var". inline procedural "cic:/CoRN/reals/IVT/a_mon'.con". inline procedural "cic:/CoRN/reals/IVT/b_mon'.con". inline procedural "cic:/CoRN/reals/IVT/a_b'.con". inline procedural "cic:/CoRN/reals/IVT/intervals_cauchy.con". (* begin hide *) inline procedural "cic:/CoRN/reals/IVT/Nested_Intervals/a'.con" "Nested_Intervals__". (* end hide *) inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_limit.con". (*#* %\begin{convention}% Let [f] be a continuous real function. %\end{convention}% *) alias id "f" = "cic:/CoRN/reals/IVT/Nested_Intervals/f.var". alias id "f_contin" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_contin.var". inline procedural "cic:/CoRN/reals/IVT/f_contin_pos.con". inline procedural "cic:/CoRN/reals/IVT/f_contin_neg.con". (*#* Assume also that [forall i, f (a i) [<=] Zero [<=] f (b i)]. *) alias id "f_a" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_a.var". alias id "f_b" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_b.var". inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_zero.con". (* UNEXPORTED End Nested_Intervals *) (* UNEXPORTED Section Bisection *) (*#* ** Bissections *) alias id "f" = "cic:/CoRN/reals/IVT/Bisection/f.var". alias id "f_apzero_interval" = "cic:/CoRN/reals/IVT/Bisection/f_apzero_interval.var". alias id "a" = "cic:/CoRN/reals/IVT/Bisection/a.var". alias id "b" = "cic:/CoRN/reals/IVT/Bisection/b.var". alias id "a_b" = "cic:/CoRN/reals/IVT/Bisection/a_b.var". alias id "f_a" = "cic:/CoRN/reals/IVT/Bisection/f_a.var". alias id "f_b" = "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__". inline procedural "cic:/CoRN/reals/IVT/Bisection/lft.con" "Bisection__". inline procedural "cic:/CoRN/reals/IVT/Bisection/rht.con" "Bisection__". (* end hide *) inline procedural "cic:/CoRN/reals/IVT/a_lft.con". inline procedural "cic:/CoRN/reals/IVT/rht_b.con". inline procedural "cic:/CoRN/reals/IVT/lft_rht.con". inline procedural "cic:/CoRN/reals/IVT/smaller_lft.con". inline procedural "cic:/CoRN/reals/IVT/smaller_rht.con". (* UNEXPORTED Hint Resolve smaller_lft smaller_rht: algebra. *) inline procedural "cic:/CoRN/reals/IVT/Cbisect'.con". (* UNEXPORTED End Bisection *) (* UNEXPORTED Section Bisect_Interval *) alias id "f" = "cic:/CoRN/reals/IVT/Bisect_Interval/f.var". alias id "C_f_apzero_interval" = "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__". (* end hide *) inline procedural "cic:/CoRN/reals/IVT/bisect_interval.ind". inline procedural "cic:/CoRN/reals/IVT/Cbisect_exists.con". inline procedural "cic:/CoRN/reals/IVT/bisect.con". inline procedural "cic:/CoRN/reals/IVT/bisect_prop.con". (* UNEXPORTED End Bisect_Interval *) (* UNEXPORTED Section IVT_Op *) (*#* ** IVT for operations Same conventions as before. *) alias id "f" = "cic:/CoRN/reals/IVT/IVT_Op/f.var". alias id "f_contin" = "cic:/CoRN/reals/IVT/IVT_Op/f_contin.var". alias id "f_apzero_interval" = "cic:/CoRN/reals/IVT/IVT_Op/f_apzero_interval.var". alias id "a" = "cic:/CoRN/reals/IVT/IVT_Op/a.var". alias id "b" = "cic:/CoRN/reals/IVT/IVT_Op/b.var". alias id "a_b" = "cic:/CoRN/reals/IVT/IVT_Op/a_b.var". alias id "f_a" = "cic:/CoRN/reals/IVT/IVT_Op/f_a.var". alias id "f_b" = "cic:/CoRN/reals/IVT/IVT_Op/f_b.var". (* begin hide *) inline procedural "cic:/CoRN/reals/IVT/IVT_Op/Small.con" "IVT_Op__". (* end hide *) inline procedural "cic:/CoRN/reals/IVT/interval_sequence.con". inline procedural "cic:/CoRN/reals/IVT/IVT_Op/a_.con" "IVT_Op__". inline procedural "cic:/CoRN/reals/IVT/IVT_Op/b_.con" "IVT_Op__". inline procedural "cic:/CoRN/reals/IVT/intervals_smaller.con". inline procedural "cic:/CoRN/reals/IVT/intervals_small''.con". inline procedural "cic:/CoRN/reals/IVT/intervals_small'.con". inline procedural "cic:/CoRN/reals/IVT/intervals_small.con". inline procedural "cic:/CoRN/reals/IVT/Civt_op.con". (* UNEXPORTED End IVT_Op *) (* UNEXPORTED Section IVT_Poly *) (*#* ** IVT for polynomials *) inline procedural "cic:/CoRN/reals/IVT/Civt_poly.con". (* UNEXPORTED End IVT_Poly *)