X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FCoRN-Decl%2Fftc%2FFunctSequence.ma;fp=helm%2Fsoftware%2Fmatita%2Fcontribs%2FCoRN-Decl%2Fftc%2FFunctSequence.ma;h=23918bcbab5e57d0eb9bffe01fcb174d688d46b1;hb=5e01cba364607e7937aec2e359c34f049bb0f108;hp=768b71285917bea80f9068c4e26f705bc9867bb3;hpb=137a822662f81efbbeac7ddc833fc9ffe252a70e;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 768b71285..23918bcba 100644 --- a/helm/software/matita/contribs/CoRN-Decl/ftc/FunctSequence.ma +++ b/helm/software/matita/contribs/CoRN-Decl/ftc/FunctSequence.ma @@ -25,7 +25,7 @@ include "ftc/Continuity.ma". include "ftc/PartInterval.ma". (* UNEXPORTED -Section Definitions. +Section Definitions *) (*#* *Sequences of Functions @@ -52,31 +52,31 @@ be defined. For a discussion on the different notions of convergent see Bishop 1967. *) -inline "cic:/CoRN/ftc/FunctSequence/a.var". +inline "cic:/CoRN/ftc/FunctSequence/Definitions/a.var" "Definitions__". -inline "cic:/CoRN/ftc/FunctSequence/b.var". +inline "cic:/CoRN/ftc/FunctSequence/Definitions/b.var" "Definitions__". -inline "cic:/CoRN/ftc/FunctSequence/Hab.var". +inline "cic:/CoRN/ftc/FunctSequence/Definitions/Hab.var" "Definitions__". (* 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". +inline "cic:/CoRN/ftc/FunctSequence/Definitions/f.var" "Definitions__". -inline "cic:/CoRN/ftc/FunctSequence/F.var". +inline "cic:/CoRN/ftc/FunctSequence/Definitions/F.var" "Definitions__". -inline "cic:/CoRN/ftc/FunctSequence/contf.var". +inline "cic:/CoRN/ftc/FunctSequence/Definitions/contf.var" "Definitions__". -inline "cic:/CoRN/ftc/FunctSequence/contF.var". +inline "cic:/CoRN/ftc/FunctSequence/Definitions/contF.var" "Definitions__". (* 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 *) @@ -125,28 +125,28 @@ A Cauchy sequence of functions is pointwise a Cauchy sequence. 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". +inline "cic:/CoRN/ftc/FunctSequence/More_Definitions/a.var" "More_Definitions__". -inline "cic:/CoRN/ftc/FunctSequence/b.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Definitions/b.var" "More_Definitions__". -inline "cic:/CoRN/ftc/FunctSequence/Hab.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Definitions/Hab.var" "More_Definitions__". (* 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". +inline "cic:/CoRN/ftc/FunctSequence/More_Definitions/f.var" "More_Definitions__". -inline "cic:/CoRN/ftc/FunctSequence/contf.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Definitions/contf.var" "More_Definitions__". (*#* We can also say that [f] is simply convergent if it converges to some @@ -162,18 +162,18 @@ It is useful to extract the limit as a partial function: (* begin show *) -inline "cic:/CoRN/ftc/FunctSequence/H.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Definitions/H.var" "More_Definitions__". (* end show *) 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 @@ -181,35 +181,35 @@ 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". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/a.var" "Irrelevance_of_Proofs__". -inline "cic:/CoRN/ftc/FunctSequence/b.var". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/b.var" "Irrelevance_of_Proofs__". -inline "cic:/CoRN/ftc/FunctSequence/Hab.var". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/Hab.var" "Irrelevance_of_Proofs__". (* 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". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/f.var" "Irrelevance_of_Proofs__". (* begin show *) -inline "cic:/CoRN/ftc/FunctSequence/contf.var". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contf.var" "Irrelevance_of_Proofs__". -inline "cic:/CoRN/ftc/FunctSequence/contf0.var". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contf0.var" "Irrelevance_of_Proofs__". (* end show *) -inline "cic:/CoRN/ftc/FunctSequence/F.var". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/F.var" "Irrelevance_of_Proofs__". (* begin show *) -inline "cic:/CoRN/ftc/FunctSequence/contF.var". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contF.var" "Irrelevance_of_Proofs__". -inline "cic:/CoRN/ftc/FunctSequence/contF0.var". +inline "cic:/CoRN/ftc/FunctSequence/Irrelevance_of_Proofs/contF0.var" "Irrelevance_of_Proofs__". (* end show *) @@ -224,21 +224,21 @@ 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. +End Irrelevance_of_Proofs *) (* UNEXPORTED -Section More_Proof_Irrelevance. +Section More_Proof_Irrelevance *) 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 @@ -248,49 +248,49 @@ 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". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/a.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/b.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/b.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/Hab.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/Hab.var" "More_Properties__". (* 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". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/f.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/g.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/g.var" "More_Properties__". (* begin show *) -inline "cic:/CoRN/ftc/FunctSequence/contf.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/contf.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contf0.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/contf0.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contg.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/contg.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contg0.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/contg0.var" "More_Properties__". (* end show *) inline "cic:/CoRN/ftc/FunctSequence/Cauchy_conv_fun_seq'.con". -inline "cic:/CoRN/ftc/FunctSequence/F.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/F.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/G.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/G.var" "More_Properties__". (* begin show *) -inline "cic:/CoRN/ftc/FunctSequence/contF.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/contF.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contF0.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/contF0.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contG.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/contG.var" "More_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contG0.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Properties/contG0.var" "More_Properties__". (* end show *) @@ -323,7 +323,7 @@ And a sequence of real numbers converges iff the corresponding sequence of const inline "cic:/CoRN/ftc/FunctSequence/seq_conv_imp_fun_conv.con". (* UNEXPORTED -End More_Properties. +End More_Properties *) (* UNEXPORTED @@ -331,7 +331,7 @@ Hint Resolve Cauchy_cont_Lim: continuous. *) (* UNEXPORTED -Section Algebraic_Properties. +Section Algebraic_Properties *) (*#* **Algebraic Properties @@ -339,25 +339,25 @@ 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". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/a.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/b.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/b.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/Hab.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/Hab.var" "Algebraic_Properties__". (* 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". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/f.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/g.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/g.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contf.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contf.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contg.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contg.var" "Algebraic_Properties__". (*#* First, the limit function is unique. @@ -376,31 +376,31 @@ 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". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/F.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/G.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/G.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contF.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contF.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contG.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/contG.var" "Algebraic_Properties__". (* begin show *) -inline "cic:/CoRN/ftc/FunctSequence/convF.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/convF.var" "Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/convG.var". +inline "cic:/CoRN/ftc/FunctSequence/Algebraic_Properties/convG.var" "Algebraic_Properties__". (* 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 *) @@ -411,32 +411,32 @@ inline "cic:/CoRN/ftc/FunctSequence/fun_Lim_seq_minus'.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". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/a.var" "More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/b.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/b.var" "More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/Hab.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/Hab.var" "More_Algebraic_Properties__". (* 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". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/f.var" "More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/g.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/g.var" "More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contf.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/contf.var" "More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contg.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/contg.var" "More_Algebraic_Properties__". (*#* The same is true if we don't make the limits explicit. @@ -444,9 +444,9 @@ The same is true if we don't make the limits explicit. (* begin hide *) -inline "cic:/CoRN/ftc/FunctSequence/Hf.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/Hf.var" "More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/Hg.var". +inline "cic:/CoRN/ftc/FunctSequence/More_Algebraic_Properties/Hg.var" "More_Algebraic_Properties__". (* end hide *) @@ -463,30 +463,30 @@ 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. +End More_Algebraic_Properties *) (* UNEXPORTED -Section Still_More_Algebraic_Properties. +Section Still_More_Algebraic_Properties *) -inline "cic:/CoRN/ftc/FunctSequence/a.var". +inline "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/a.var" "Still_More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/b.var". +inline "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/b.var" "Still_More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/Hab.var". +inline "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/Hab.var" "Still_More_Algebraic_Properties__". (* 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". +inline "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/f.var" "Still_More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/contf.var". +inline "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/contf.var" "Still_More_Algebraic_Properties__". -inline "cic:/CoRN/ftc/FunctSequence/Hf.var". +inline "cic:/CoRN/ftc/FunctSequence/Still_More_Algebraic_Properties/Hf.var" "Still_More_Algebraic_Properties__". (*#* As a corollary, we get the analogous property for the sequence of algebraic inverse functions. @@ -497,7 +497,7 @@ 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. +End Still_More_Algebraic_Properties *) (* UNEXPORTED