include "ftc/PartInterval.ma".
(* UNEXPORTED
-Section Definitions.
+Section Definitions
*)
(*#* *Sequences of Functions
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_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
(* 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".
(* UNEXPORTED
-End More_Definitions.
+End More_Definitions
*)
(* UNEXPORTED
-Section Irrelevance_of_Proofs.
+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.
*)
-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/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
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/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/seq_conv_imp_fun_conv.con".
(* UNEXPORTED
-End More_Properties.
+End More_Properties
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Algebraic_Properties.
+Section Algebraic_Properties
*)
(*#* **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.
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_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.
(* 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_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_Cauchy_prop_inv.con".
(* UNEXPORTED
-End Still_More_Algebraic_Properties.
+End Still_More_Algebraic_Properties
*)
(* UNEXPORTED