X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FCoRN-Decl%2Fftc%2FFunctSequence.ma;h=34bd794f4bc6338a43c559cdc777395b5d91966f;hb=bf0cc84dcef9ae3d2145e79754bb39feb3985574;hp=c5c588036cb58fda9873de6e0de8fe2c039c7248;hpb=3199f2a8428b5d19a3a803c1b03d9f90e4120f74;p=helm.git diff --git a/helm/software/matita/contribs/CoRN-Decl/ftc/FunctSequence.ma b/helm/software/matita/contribs/CoRN-Decl/ftc/FunctSequence.ma index c5c588036..34bd794f4 100644 --- a/helm/software/matita/contribs/CoRN-Decl/ftc/FunctSequence.ma +++ b/helm/software/matita/contribs/CoRN-Decl/ftc/FunctSequence.ma @@ -16,18 +16,16 @@ 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 -Continuity -*) +include "ftc/Continuity.ma". -(* INCLUDE -PartInterval -*) +include "ftc/PartInterval.ma". (* UNEXPORTED -Section Definitions. +Section Definitions *) (*#* *Sequences of Functions @@ -54,101 +52,101 @@ be defined. For a discussion on the different notions of convergent see Bishop 1967. *) -inline cic:/CoRN/ftc/FunctSequence/a.var. +alias id "a" = "cic:/CoRN/ftc/FunctSequence/Definitions/a.var". -inline cic:/CoRN/ftc/FunctSequence/b.var. +alias id "b" = "cic:/CoRN/ftc/FunctSequence/Definitions/b.var". -inline cic:/CoRN/ftc/FunctSequence/Hab.var. +alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/Definitions/Hab.var". (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/I.con. +inline "cic:/CoRN/ftc/FunctSequence/Definitions/I.con" "Definitions__". (* end hide *) -inline cic:/CoRN/ftc/FunctSequence/f.var. +alias id "f" = "cic:/CoRN/ftc/FunctSequence/Definitions/f.var". -inline cic:/CoRN/ftc/FunctSequence/F.var. +alias id "F" = "cic:/CoRN/ftc/FunctSequence/Definitions/F.var". -inline cic:/CoRN/ftc/FunctSequence/contf.var. +alias id "contf" = "cic:/CoRN/ftc/FunctSequence/Definitions/contf.var". -inline cic:/CoRN/ftc/FunctSequence/contF.var. +alias id "contF" = "cic:/CoRN/ftc/FunctSequence/Definitions/contF.var". (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/incf.con. +inline "cic:/CoRN/ftc/FunctSequence/Definitions/incf.con" "Definitions__". -inline cic:/CoRN/ftc/FunctSequence/incF.con. +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/Cauchy_fun_seq.con". -inline cic:/CoRN/ftc/FunctSequence/conv_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/conv_norm_fun_seq.con". -inline cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1.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_seq'.con". -inline cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq2.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/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/conv_Cauchy_fun_seq'.con". -inline cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_seq2.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/Cauchy_fun_seq2_seq.con". -inline cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_norm.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/conv_fun_norm_seq.con". -inline cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1_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_seq_seq1.con. +inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_seq1.con". -inline cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1_seq.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. +inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_real.con". (* UNEXPORTED -End Definitions. +End Definitions *) (* UNEXPORTED -Section More_Definitions. +Section More_Definitions *) -inline cic:/CoRN/ftc/FunctSequence/a.var. +alias id "a" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/a.var". -inline cic:/CoRN/ftc/FunctSequence/b.var. +alias id "b" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/b.var". -inline cic:/CoRN/ftc/FunctSequence/Hab.var. +alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/Hab.var". (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/I.con. +inline "cic:/CoRN/ftc/FunctSequence/More_Definitions/I.con" "More_Definitions__". (* end hide *) -inline cic:/CoRN/ftc/FunctSequence/f.var. +alias id "f" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/f.var". -inline cic:/CoRN/ftc/FunctSequence/contf.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 @@ -156,7 +154,7 @@ 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. +inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq.con". (*#* It is useful to extract the limit as a partial function: @@ -164,18 +162,18 @@ It is useful to extract the limit as a partial function: (* begin show *) -inline cic:/CoRN/ftc/FunctSequence/H.var. +alias id "H" = "cic:/CoRN/ftc/FunctSequence/More_Definitions/H.var". (* end show *) -inline cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_Lim.con. +inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq_Lim.con". (* UNEXPORTED -End More_Definitions. +End More_Definitions *) (* UNEXPORTED -Section Irrelevance_of_Proofs. +Section Irrelevance_of_Proofs *) (*#* **Irrelevance of Proofs @@ -183,64 +181,64 @@ Section 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. *) -inline cic:/CoRN/ftc/FunctSequence/a.var. +alias id "a" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/a.var". -inline cic:/CoRN/ftc/FunctSequence/b.var. +alias id "b" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/b.var". -inline cic:/CoRN/ftc/FunctSequence/Hab.var. +alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/Hab.var". (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/I.con. +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/I.con" "Irrelevance_of_Proofs__". (* end hide *) -inline cic:/CoRN/ftc/FunctSequence/f.var. +alias id "f" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/f.var". (* begin show *) -inline cic:/CoRN/ftc/FunctSequence/contf.var. +alias id "contf" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contf.var". -inline cic:/CoRN/ftc/FunctSequence/contf0.var. +alias id "contf0" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contf0.var". (* end show *) -inline cic:/CoRN/ftc/FunctSequence/F.var. +alias id "F" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/F.var". (* begin show *) -inline cic:/CoRN/ftc/FunctSequence/contF.var. +alias id "contF" = "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contF.var". -inline cic:/CoRN/ftc/FunctSequence/contF0.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/conv_fun_seq'_wd.con". -inline cic:/CoRN/ftc/FunctSequence/Cauchy_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/Cauchy_fun_seq2_wd.con". -inline cic:/CoRN/ftc/FunctSequence/conv_norm_fun_seq_wd.con. +inline "cic:/CoRN/ftc/FunctSequence/conv_norm_fun_seq_wd.con". -inline cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1_wd.con. +inline "cic:/CoRN/ftc/FunctSequence/Cauchy_fun_seq1_wd.con". (* UNEXPORTED -End Irrelevance_of_Proofs. +End Irrelevance_of_Proofs *) (* UNEXPORTED -Section More_Proof_Irrelevance. +Section More_Proof_Irrelevance *) -inline cic:/CoRN/ftc/FunctSequence/conv_fun_seq_wd.con. +inline "cic:/CoRN/ftc/FunctSequence/conv_fun_seq_wd.con". (* UNEXPORTED -End More_Proof_Irrelevance. +End More_Proof_Irrelevance *) (* UNEXPORTED -Section More_Properties. +Section More_Properties *) (*#* **Other Properties @@ -250,82 +248,82 @@ limit; the limit is a continuous function; and convergence is well defined with respect to functional equality in the interval [[a,b]]. *) -inline cic:/CoRN/ftc/FunctSequence/a.var. +alias id "a" = "cic:/CoRN/ftc/FunctSequence/More_Properties/a.var". -inline cic:/CoRN/ftc/FunctSequence/b.var. +alias id "b" = "cic:/CoRN/ftc/FunctSequence/More_Properties/b.var". -inline cic:/CoRN/ftc/FunctSequence/Hab.var. +alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/More_Properties/Hab.var". (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/I.con. +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/I.con" "More_Properties__". (* end hide *) -inline cic:/CoRN/ftc/FunctSequence/f.var. +alias id "f" = "cic:/CoRN/ftc/FunctSequence/More_Properties/f.var". -inline cic:/CoRN/ftc/FunctSequence/g.var. +alias id "g" = "cic:/CoRN/ftc/FunctSequence/More_Properties/g.var". (* begin show *) -inline cic:/CoRN/ftc/FunctSequence/contf.var. +alias id "contf" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contf.var". -inline cic:/CoRN/ftc/FunctSequence/contf0.var. +alias id "contf0" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contf0.var". -inline cic:/CoRN/ftc/FunctSequence/contg.var. +alias id "contg" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contg.var". -inline cic:/CoRN/ftc/FunctSequence/contg0.var. +alias id "contg0" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contg0.var". (* end show *) -inline cic:/CoRN/ftc/FunctSequence/Cauchy_conv_fun_seq'.con. +inline "cic:/CoRN/ftc/FunctSequence/Cauchy_conv_fun_seq'.con". -inline cic:/CoRN/ftc/FunctSequence/F.var. +alias id "F" = "cic:/CoRN/ftc/FunctSequence/More_Properties/F.var". -inline cic:/CoRN/ftc/FunctSequence/G.var. +alias id "G" = "cic:/CoRN/ftc/FunctSequence/More_Properties/G.var". (* begin show *) -inline cic:/CoRN/ftc/FunctSequence/contF.var. +alias id "contF" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contF.var". -inline cic:/CoRN/ftc/FunctSequence/contF0.var. +alias id "contF0" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contF0.var". -inline cic:/CoRN/ftc/FunctSequence/contG.var. +alias id "contG" = "cic:/CoRN/ftc/FunctSequence/More_Properties/contG.var". -inline cic:/CoRN/ftc/FunctSequence/contG0.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'_wdl.con". -inline cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_wdr.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'_wdl'.con". -inline cic:/CoRN/ftc/FunctSequence/conv_fun_seq'_wdr'.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_fun_seq_wd.con". -inline cic:/CoRN/ftc/FunctSequence/Cauchy_cont_Lim.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/Cauchy_conv_fun_seq.con". -inline cic:/CoRN/ftc/FunctSequence/conv_Cauchy_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. +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. +inline "cic:/CoRN/ftc/FunctSequence/seq_conv_imp_fun_conv.con". (* UNEXPORTED -End More_Properties. +End More_Properties *) (* UNEXPORTED @@ -333,7 +331,7 @@ Hint Resolve Cauchy_cont_Lim: continuous. *) (* UNEXPORTED -Section Algebraic_Properties. +Section Algebraic_Properties *) (*#* **Algebraic Properties @@ -341,104 +339,104 @@ Section Algebraic_Properties. We now study how convergence is affected by algebraic operations, and some algebraic properties of the limit function. *) -inline cic:/CoRN/ftc/FunctSequence/a.var. +alias id "a" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/a.var". -inline cic:/CoRN/ftc/FunctSequence/b.var. +alias id "b" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/b.var". -inline cic:/CoRN/ftc/FunctSequence/Hab.var. +alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/Hab.var". (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/I.con. +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/I.con" "Algebraic_Properties__". (* end hide *) -inline cic:/CoRN/ftc/FunctSequence/f.var. +alias id "f" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/f.var". -inline cic:/CoRN/ftc/FunctSequence/g.var. +alias id "g" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/g.var". -inline cic:/CoRN/ftc/FunctSequence/contf.var. +alias id "contf" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contf.var". -inline cic:/CoRN/ftc/FunctSequence/contg.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. +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_Lim_seq_const.con". -inline cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_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. *) -inline cic:/CoRN/ftc/FunctSequence/F.var. +alias id "F" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/F.var". -inline cic:/CoRN/ftc/FunctSequence/G.var. +alias id "G" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/G.var". -inline cic:/CoRN/ftc/FunctSequence/contF.var. +alias id "contF" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contF.var". -inline cic:/CoRN/ftc/FunctSequence/contG.var. +alias id "contG" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contG.var". (* begin show *) -inline cic:/CoRN/ftc/FunctSequence/convF.var. +alias id "convF" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/convF.var". -inline cic:/CoRN/ftc/FunctSequence/convG.var. +alias id "convG" = "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/convG.var". (* end show *) (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/incf.con. +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/incf.con" "Algebraic_Properties__". -inline cic:/CoRN/ftc/FunctSequence/incg.con. +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/incg.con" "Algebraic_Properties__". -inline cic:/CoRN/ftc/FunctSequence/incF.con. +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/incF.con" "Algebraic_Properties__". -inline cic:/CoRN/ftc/FunctSequence/incG.con. +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_plus'.con". -inline cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_minus'.con. +inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_minus'.con". -inline cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_mult'.con. +inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_mult'.con". (* UNEXPORTED -End Algebraic_Properties. +End Algebraic_Properties *) (* UNEXPORTED -Section More_Algebraic_Properties. +Section More_Algebraic_Properties *) -inline cic:/CoRN/ftc/FunctSequence/a.var. +alias id "a" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/a.var". -inline cic:/CoRN/ftc/FunctSequence/b.var. +alias id "b" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/b.var". -inline cic:/CoRN/ftc/FunctSequence/Hab.var. +alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/Hab.var". (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/I.con. +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/I.con" "More_Algebraic_Properties__". (* end hide *) -inline cic:/CoRN/ftc/FunctSequence/f.var. +alias id "f" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/f.var". -inline cic:/CoRN/ftc/FunctSequence/g.var. +alias id "g" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/g.var". -inline cic:/CoRN/ftc/FunctSequence/contf.var. +alias id "contf" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/contf.var". -inline cic:/CoRN/ftc/FunctSequence/contg.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. @@ -446,60 +444,60 @@ The same is true if we don't make the limits explicit. (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/Hf.var. +alias id "Hf" = "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/Hf.var". -inline cic:/CoRN/ftc/FunctSequence/Hg.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_Lim_seq_plus.con". -inline cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_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_Lim_seq_minus.con". -inline cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_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_Lim_seq_mult.con". -inline cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_mult.con. +inline "cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_mult.con". (* UNEXPORTED -End More_Algebraic_Properties. +End More_Algebraic_Properties *) (* UNEXPORTED -Section Still_More_Algebraic_Properties. +Section Still_More_Algebraic_Properties *) -inline cic:/CoRN/ftc/FunctSequence/a.var. +alias id "a" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/a.var". -inline cic:/CoRN/ftc/FunctSequence/b.var. +alias id "b" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/b.var". -inline cic:/CoRN/ftc/FunctSequence/Hab.var. +alias id "Hab" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/Hab.var". (* begin hide *) -inline cic:/CoRN/ftc/FunctSequence/I.con. +inline "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/I.con" "Still_More_Algebraic_Properties__". (* end hide *) -inline cic:/CoRN/ftc/FunctSequence/f.var. +alias id "f" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/f.var". -inline cic:/CoRN/ftc/FunctSequence/contf.var. +alias id "contf" = "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/contf.var". -inline cic:/CoRN/ftc/FunctSequence/Hf.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_Lim_seq_inv.con". -inline cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_inv.con. +inline "cic:/CoRN/ftc/FunctSequence/fun_Cauchy_prop_inv.con". (* UNEXPORTED -End Still_More_Algebraic_Properties. +End Still_More_Algebraic_Properties *) (* UNEXPORTED