--- /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/FunctSequence".
+
+include "CoRN.ma".
+
+(* $Id: FunctSequence.v,v 1.5 2004/04/23 10:00:58 lcf Exp $ *)
+
+include "ftc/Continuity.ma".
+
+include "ftc/PartInterval.ma".
+
+(* UNEXPORTED
+Section Definitions
+*)
+
+(*#* *Sequences of Functions
+
+In this file we define some more operators on functions, namely
+sequences and limits. These concepts are defined only for continuous
+functions.
+
+%\begin{convention}% Throughout this section:
+ - [a] and [b] will be real numbers and the interval [[a,b]]
+will be denoted by [I];
+ - [f, g] and [h] will denote sequences of continuous functions;
+ - [F, G] and [H] will denote continuous functions.
+
+%\end{convention}%
+
+**Definitions
+
+A sequence of functions is simply an object of type [nat->PartIR].
+However, we will be interested mostly in convergent and Cauchy
+sequences. Several definitions of these concepts will be formalized;
+they mirror the several different ways in which a Cauchy sequence can
+be defined. For a discussion on the different notions of convergent
+see Bishop 1967.
+*)
+
+alias id "a" = "cic:/CoRN/ftc/FunctSequence/Definitions/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/FunctSequence/Definitions/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/Definitions/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Definitions/I.con" "Definitions__".
+
+(* end hide *)
+
+alias id "f" = "cic:/CoRN/ftc/FunctSequence/Definitions/f.var".
+
+alias id "F" = "cic:/CoRN/ftc/FunctSequence/Definitions/F.var".
+
+alias id "contf" = "cic:/CoRN/ftc/FunctSequence/Definitions/contf.var".
+
+alias id "contF" = "cic:/CoRN/ftc/FunctSequence/Definitions/contF.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Definitions/incf.con" "Definitions__".
+
+inline "cic:/CoRN/ftc/FunctSequence/Definitions/incF.con" "Definitions__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_norm_fun_seq.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq2.con".
+
+(*#*
+These definitions are all shown to be equivalent.
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_seq'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq'_seq.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_Cauchy_fun_seq'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_seq2.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq2_seq.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_norm.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_norm_seq.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1_seq'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq'_seq1.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_seq1.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1_seq.con".
+
+(*#*
+A Cauchy sequence of functions is pointwise a Cauchy sequence.
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_real.con".
+
+(* UNEXPORTED
+End Definitions
+*)
+
+(* UNEXPORTED
+Section More_Definitions
+*)
+
+alias id "a" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/More_Definitions/I.con" "More_Definitions__".
+
+(* end hide *)
+
+alias id "f" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/f.var".
+
+alias id "contf" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/contf.var".
+
+(*#*
+We can also say that [f] is simply convergent if it converges to some
+continuous function. Notice that we do not quantify directly over
+partial functions, for reasons which were already explained.
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq.con".
+
+(*#*
+It is useful to extract the limit as a partial function:
+*)
+
+(* begin show *)
+
+alias id "H" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/H.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_Lim.con".
+
+(* UNEXPORTED
+End More_Definitions
+*)
+
+(* UNEXPORTED
+Section Irrelevance_of_Proofs
+*)
+
+(*#* **Irrelevance of Proofs
+
+This section contains a number of technical results stating mainly that being a Cauchy sequence or converging to some limit is a property of the sequence itself and independent of the proofs we supply of its continuity or the continuity of its limit.
+*)
+
+alias id "a" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/I.con" "Irrelevance_of_Proofs__".
+
+(* end hide *)
+
+alias id "f" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/f.var".
+
+(* begin show *)
+
+alias id "contf" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contf.var".
+
+alias id "contf0" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contf0.var".
+
+(* end show *)
+
+alias id "F" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/F.var".
+
+(* begin show *)
+
+alias id "contF" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contF.var".
+
+alias id "contF0" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contF0.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_wd.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq'_wd.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq2_wd.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_norm_fun_seq_wd.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1_wd.con".
+
+(* UNEXPORTED
+End Irrelevance_of_Proofs
+*)
+
+(* UNEXPORTED
+Section More_Proof_Irrelevance
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq_wd.con".
+
+(* UNEXPORTED
+End More_Proof_Irrelevance
+*)
+
+(* UNEXPORTED
+Section More_Properties
+*)
+
+(*#* **Other Properties
+
+Still more technical details---a convergent sequence converges to its
+limit; the limit is a continuous function; and convergence is well
+defined with respect to functional equality in the interval [[a,b]].
+*)
+
+alias id "a" = "cic:/CoRN/ftc/FunctSequence/More_Properties/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/FunctSequence/More_Properties/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/More_Properties/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/More_Properties/I.con" "More_Properties__".
+
+(* end hide *)
+
+alias id "f" = "cic:/CoRN/ftc/FunctSequence/More_Properties/f.var".
+
+alias id "g" = "cic:/CoRN/ftc/FunctSequence/More_Properties/g.var".
+
+(* begin show *)
+
+alias id "contf" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contf.var".
+
+alias id "contf0" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contf0.var".
+
+alias id "contg" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contg.var".
+
+alias id "contg0" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contg0.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_conv_fun_seq'.con".
+
+alias id "F" = "cic:/CoRN/ftc/FunctSequence/More_Properties/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/FunctSequence/More_Properties/G.var".
+
+(* begin show *)
+
+alias id "contF" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contF.var".
+
+alias id "contF0" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contF0.var".
+
+alias id "contG" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contG.var".
+
+alias id "contG0" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contG0.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_wdl.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_wdr.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_wdl'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_wdr'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_wd.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_cont_Lim.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/Cauchy_conv_fun_seq.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/conv_Cauchy_fun_seq.con".
+
+(*#*
+More interesting is the fact that a convergent sequence of functions converges pointwise as a sequence of real numbers.
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_conv_imp_seq_conv.con".
+
+(*#*
+And a sequence of real numbers converges iff the corresponding sequence of constant functions converges to the corresponding constant function.
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/seq_conv_imp_fun_conv.con".
+
+(* UNEXPORTED
+End More_Properties
+*)
+
+(* UNEXPORTED
+Hint Resolve Cauchy_cont_Lim: continuous.
+*)
+
+(* UNEXPORTED
+Section Algebraic_Properties
+*)
+
+(*#* **Algebraic Properties
+
+We now study how convergence is affected by algebraic operations, and some algebraic properties of the limit function.
+*)
+
+alias id "a" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/I.con" "Algebraic_Properties__".
+
+(* end hide *)
+
+alias id "f" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/f.var".
+
+alias id "g" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/g.var".
+
+alias id "contf" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contf.var".
+
+alias id "contg" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contg.var".
+
+(*#*
+First, the limit function is unique.
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/FLim_unique.con".
+
+(*#* Constant sequences (not sequences of constant functions!) always converge.
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_const.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_const.con".
+
+(*#*
+We now prove that if two sequences converge than their sum (difference, product) also converge to the sum (difference, product) of their limits.
+*)
+
+alias id "F" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/G.var".
+
+alias id "contF" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contF.var".
+
+alias id "contG" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contG.var".
+
+(* begin show *)
+
+alias id "convF" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/convF.var".
+
+alias id "convG" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/convG.var".
+
+(* end show *)
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/incf.con" "Algebraic_Properties__".
+
+inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/incg.con" "Algebraic_Properties__".
+
+inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/incF.con" "Algebraic_Properties__".
+
+inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/incG.con" "Algebraic_Properties__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_plus'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_minus'.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_mult'.con".
+
+(* UNEXPORTED
+End Algebraic_Properties
+*)
+
+(* UNEXPORTED
+Section More_Algebraic_Properties
+*)
+
+alias id "a" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/I.con" "More_Algebraic_Properties__".
+
+(* end hide *)
+
+alias id "f" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/f.var".
+
+alias id "g" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/g.var".
+
+alias id "contf" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/contf.var".
+
+alias id "contg" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/contg.var".
+
+(*#*
+The same is true if we don't make the limits explicit.
+*)
+
+(* begin hide *)
+
+alias id "Hf" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/Hf.var".
+
+alias id "Hg" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/Hg.var".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_plus.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_plus.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_minus.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_minus.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_mult.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_mult.con".
+
+(* UNEXPORTED
+End More_Algebraic_Properties
+*)
+
+(* UNEXPORTED
+Section Still_More_Algebraic_Properties
+*)
+
+alias id "a" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/I.con" "Still_More_Algebraic_Properties__".
+
+(* end hide *)
+
+alias id "f" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/f.var".
+
+alias id "contf" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/contf.var".
+
+alias id "Hf" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/Hf.var".
+
+(*#*
+As a corollary, we get the analogous property for the sequence of algebraic inverse functions.
+*)
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_inv.con".
+
+inline "cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_inv.con".
+
+(* UNEXPORTED
+End Still_More_Algebraic_Properties
+*)
+
+(* UNEXPORTED
+Hint Resolve Continuous_I_Sum Continuous_I_Sumx Continuous_I_Sum0: continuous.
+*)
+
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