--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/ftc/TaylorLemma".
+
+include "CoRN.ma".
+
+(* $Id: TaylorLemma.v,v 1.8 2004/04/23 10:01:01 lcf Exp $ *)
+
+include "ftc/Rolle.ma".
+
+(* UNEXPORTED
+Opaque Min.
+*)
+
+(* UNEXPORTED
+Section Taylor_Defs
+*)
+
+(*#* *Taylor's Theorem
+
+We now prove Taylor's theorem for the remainder of the Taylor
+series. This proof is done in two steps: first, we prove the lemma
+for a proper compact interval; next we generalize the result to two
+arbitrary (eventually equal) points in a proper interval.
+
+** First case
+
+We assume two different points [a] and [b] in the domain of [F] and
+define the nth order derivative of [F] in the interval
+[[Min(a,b),Max(a,b)]].
+*)
+
+alias id "a" = "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/b.var".
+
+alias id "Hap" = "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/Hap.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/Hab'.con" "Taylor_Defs__".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/Hab.con" "Taylor_Defs__".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/I.con" "Taylor_Defs__".
+
+(* end hide *)
+
+alias id "F" = "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/F.var".
+
+alias id "Ha" = "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/Ha.var".
+
+alias id "Hb" = "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/Hb.var".
+
+(* begin show *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/fi.con" "Taylor_Defs__".
+
+(* end show *)
+
+(*#*
+This last local definition is simply:
+$f_i=f^{(i)}$#f<sub>i</sub>=f<sup>(i)</sup>#.
+*)
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma1.con".
+
+(* end hide *)
+
+(*#*
+Now we can define the Taylor sequence around [a]. The auxiliary
+definition gives, for any [i], the function expressed by the rule
+%\[g(x)=\frac{f^{(i)}
+(a)}{i!}*(x-a)^i.\]%#g(x)=f<sup>(i)</sup>(a)/i!*(x-a)<sup>i</sup>.#
+We denote by [A] and [B] the elements of [[Min(a,b),Max(a,b)]]
+corresponding to [a] and [b].
+*)
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/TL_compact_a.con" "Taylor_Defs__".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/TL_compact_b.con" "Taylor_Defs__".
+
+(* NOTATION
+Notation A := (Build_subcsetoid_crr IR _ _ TL_compact_a).
+*)
+
+(* NOTATION
+Notation B := (Build_subcsetoid_crr IR _ _ TL_compact_b).
+*)
+
+(* end hide *)
+
+(* begin show *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/funct_i.con" "Taylor_Defs__".
+
+(* end show *)
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/funct_i'.con" "Taylor_Defs__".
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_a_i.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_b_i.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_x_i.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_a_i'.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_b_i'.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_x_i'.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma2.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma2'.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma3.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma3'.con".
+
+(* end hide *)
+
+(*#*
+Adding the previous expressions up to a given bound [n] gives us the
+Taylor sum of order [n].
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_seq'.con".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/Taylor_seq'_aux.con" "Taylor_Defs__".
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_lemma_a.con".
+
+(* end hide *)
+
+(*#*
+It is easy to show that [b] is in the domain of this series, which allows us to write down the Taylor remainder around [b].
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_lemma_b.con".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_lemma_a'.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/TL_lemma_b'.con".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_rem.con".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/g.con" "Taylor_Defs__".
+
+(* UNEXPORTED
+Opaque Taylor_seq'_aux Taylor_rem.
+*)
+
+(* UNEXPORTED
+Transparent Taylor_rem.
+*)
+
+(* UNEXPORTED
+Opaque Taylor_seq'.
+*)
+
+(* UNEXPORTED
+Transparent Taylor_seq' Taylor_seq'_aux.
+*)
+
+(* UNEXPORTED
+Opaque funct_i'.
+*)
+
+(* UNEXPORTED
+Opaque funct_i.
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma4.con".
+
+(* UNEXPORTED
+Transparent funct_i funct_i'.
+*)
+
+(* UNEXPORTED
+Opaque Taylor_seq'_aux.
+*)
+
+(* UNEXPORTED
+Transparent Taylor_seq'_aux.
+*)
+
+(* UNEXPORTED
+Opaque FSumx.
+*)
+
+(* UNEXPORTED
+Opaque funct_i'.
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma5.con".
+
+(* UNEXPORTED
+Transparent funct_i' FSumx.
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/funct_aux.con" "Taylor_Defs__".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma6.con".
+
+(* UNEXPORTED
+Ltac Lazy_Included :=
+ repeat first
+ [ apply included_IR
+ | apply included_FPlus
+ | apply included_FInv
+ | apply included_FMinus
+ | apply included_FMult
+ | apply included_FNth
+ | apply included_refl ].
+*)
+
+(* UNEXPORTED
+Ltac Lazy_Eq :=
+ repeat first
+ [ apply bin_op_wd_unfolded
+ | apply un_op_wd_unfolded
+ | apply cg_minus_wd
+ | apply div_wd
+ | apply csf_wd_unfolded ]; Algebra.
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma7.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma8.con".
+
+(* UNEXPORTED
+Opaque funct_aux.
+*)
+
+(* UNEXPORTED
+Transparent funct_aux.
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma9.con".
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/g'.con" "Taylor_Defs__".
+
+(* UNEXPORTED
+Opaque Taylor_rem funct_aux.
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma10.con".
+
+(* UNEXPORTED
+Transparent Taylor_rem funct_aux.
+*)
+
+(* end hide *)
+
+(*#*
+Now Taylor's theorem.
+
+%\begin{convention}% Let [e] be a positive real number.
+%\end{convention}%
+*)
+
+alias id "e" = "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/e.var".
+
+alias id "He" = "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/He.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma11.con".
+
+(* end hide *)
+
+(* begin show *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_Defs/deriv_Sn'.con" "Taylor_Defs__".
+
+(* end show *)
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/TaylorLemma/TLH.con".
+
+(* end hide *)
+
+(* UNEXPORTED
+Opaque funct_aux.
+*)
+
+(* UNEXPORTED
+Opaque Taylor_rem.
+*)
+
+(* UNEXPORTED
+Transparent Taylor_rem funct_aux.
+*)
+
+inline "cic:/CoRN/ftc/TaylorLemma/Taylor_lemma.con".
+
+(* UNEXPORTED
+End Taylor_Defs
+*)
+
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