set "baseuri" "cic:/matita/CoRN-Decl/ftc/Continuity".
+include "CoRN.ma".
+
(* $Id: Continuity.v,v 1.6 2004/04/23 10:00:58 lcf Exp $ *)
(*#* printing Norm_Funct %\ensuremath{\|\cdot\|}% *)
-(* INCLUDE
-NRootIR
-*)
+include "reals/NRootIR.ma".
-(* INCLUDE
-FunctSums
-*)
+include "ftc/FunctSums.ma".
(* UNEXPORTED
-Section Definitions_and_Basic_Results.
+Section Definitions_and_Basic_Results
*)
(*#* *Continuity
Here we define continuity and prove some basic properties of continuous functions.
*)
-inline cic:/CoRN/ftc/Continuity/a.var.
+alias id "a" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/a.var".
-inline cic:/CoRN/ftc/Continuity/b.var.
+alias id "b" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/b.var".
-inline cic:/CoRN/ftc/Continuity/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/Hab.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/I.con.
+inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/I.con" "Definitions_and_Basic_Results__".
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/F.var.
+alias id "F" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/F.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/P.con.
+inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/P.con" "Definitions_and_Basic_Results__".
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/Continuous_I.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I.con".
(*#*
For convenience, we distinguish the two properties of continuous functions.
*)
-inline cic:/CoRN/ftc/Continuity/contin_imp_inc.con.
+inline "cic:/CoRN/ftc/Continuity/contin_imp_inc.con".
-inline cic:/CoRN/ftc/Continuity/contin_prop.con.
+inline "cic:/CoRN/ftc/Continuity/contin_prop.con".
(*#*
Assume [F] to be continuous in [I]. Then it has a least upper bound and a greater lower bound on [I].
*)
-inline cic:/CoRN/ftc/Continuity/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/contF.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/Hinc'.con.
+inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/Hinc'.con" "Definitions_and_Basic_Results__".
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/Continuous_I_imp_tb_image.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_imp_tb_image.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_imp_lub.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_imp_lub.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_imp_glb.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_imp_glb.con".
(*#*
We now make this glb and lub into operators.
*)
-inline cic:/CoRN/ftc/Continuity/lub_funct.con.
+inline "cic:/CoRN/ftc/Continuity/lub_funct.con".
-inline cic:/CoRN/ftc/Continuity/glb_funct.con.
+inline "cic:/CoRN/ftc/Continuity/glb_funct.con".
(*#*
These operators have the expected properties.
*)
-inline cic:/CoRN/ftc/Continuity/lub_is_lub.con.
+inline "cic:/CoRN/ftc/Continuity/lub_is_lub.con".
-inline cic:/CoRN/ftc/Continuity/glb_is_glb.con.
+inline "cic:/CoRN/ftc/Continuity/glb_is_glb.con".
-inline cic:/CoRN/ftc/Continuity/glb_prop.con.
+inline "cic:/CoRN/ftc/Continuity/glb_prop.con".
-inline cic:/CoRN/ftc/Continuity/lub_prop.con.
+inline "cic:/CoRN/ftc/Continuity/lub_prop.con".
(*#*
The norm of a function is defined as being the supremum of its absolute value; that is equivalent to the following definition (which is often more convenient to use).
*)
-inline cic:/CoRN/ftc/Continuity/Norm_Funct.con.
+inline "cic:/CoRN/ftc/Continuity/Norm_Funct.con".
(*#*
The norm effectively bounds the absolute value of a function.
*)
-inline cic:/CoRN/ftc/Continuity/norm_bnd_AbsIR.con.
+inline "cic:/CoRN/ftc/Continuity/norm_bnd_AbsIR.con".
(*#*
The following is another way of characterizing the norm:
*)
-inline cic:/CoRN/ftc/Continuity/Continuous_I_imp_abs_lub.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_imp_abs_lub.con".
(*#*
We now prove some basic properties of the norm---namely that it is positive, and that it provides a least upper bound for the absolute value of its argument.
*)
-inline cic:/CoRN/ftc/Continuity/positive_norm.con.
+inline "cic:/CoRN/ftc/Continuity/positive_norm.con".
-inline cic:/CoRN/ftc/Continuity/norm_fun_lub.con.
+inline "cic:/CoRN/ftc/Continuity/norm_fun_lub.con".
-inline cic:/CoRN/ftc/Continuity/leEq_Norm_Funct.con.
+inline "cic:/CoRN/ftc/Continuity/leEq_Norm_Funct.con".
-inline cic:/CoRN/ftc/Continuity/less_Norm_Funct.con.
+inline "cic:/CoRN/ftc/Continuity/less_Norm_Funct.con".
(* UNEXPORTED
-End Definitions_and_Basic_Results.
+End Definitions_and_Basic_Results
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Local_Results.
+Section Local_Results
*)
(*#* **Algebraic Properties
We now state and prove some results about continuous functions. Assume that [I] is included in the domain of both [F] and [G].
*)
-inline cic:/CoRN/ftc/Continuity/a.var.
+alias id "a" = "cic:/CoRN/ftc/Continuity/Local_Results/a.var".
-inline cic:/CoRN/ftc/Continuity/b.var.
+alias id "b" = "cic:/CoRN/ftc/Continuity/Local_Results/b.var".
-inline cic:/CoRN/ftc/Continuity/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Local_Results/Hab.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/I.con.
+inline "cic:/CoRN/ftc/Continuity/Local_Results/I.con" "Local_Results__".
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/F.var.
+alias id "F" = "cic:/CoRN/ftc/Continuity/Local_Results/F.var".
-inline cic:/CoRN/ftc/Continuity/G.var.
+alias id "G" = "cic:/CoRN/ftc/Continuity/Local_Results/G.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/P.con.
+inline "cic:/CoRN/ftc/Continuity/Local_Results/P.con" "Local_Results__".
-inline cic:/CoRN/ftc/Continuity/Q.con.
+inline "cic:/CoRN/ftc/Continuity/Local_Results/Q.con" "Local_Results__".
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/incF.var.
+alias id "incF" = "cic:/CoRN/ftc/Continuity/Local_Results/incF.var".
-inline cic:/CoRN/ftc/Continuity/incG.var.
+alias id "incG" = "cic:/CoRN/ftc/Continuity/Local_Results/incG.var".
(*#*
The first result does not require the function to be continuous; however, its preconditions are easily verified by continuous functions, which justifies its inclusion in this section.
*)
-inline cic:/CoRN/ftc/Continuity/cont_no_sign_change.con.
+inline "cic:/CoRN/ftc/Continuity/cont_no_sign_change.con".
-inline cic:/CoRN/ftc/Continuity/cont_no_sign_change_pos.con.
+inline "cic:/CoRN/ftc/Continuity/cont_no_sign_change_pos.con".
-inline cic:/CoRN/ftc/Continuity/cont_no_sign_change_neg.con.
+inline "cic:/CoRN/ftc/Continuity/cont_no_sign_change_neg.con".
(*#*
Being continuous is an extensional property.
*)
-inline cic:/CoRN/ftc/Continuity/Continuous_I_wd.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_wd.con".
(*#*
A continuous function remains continuous if you restrict its domain.
*)
-inline cic:/CoRN/ftc/Continuity/included_imp_contin.con.
+inline "cic:/CoRN/ftc/Continuity/included_imp_contin.con".
(*#*
Constant functions and identity are continuous.
*)
-inline cic:/CoRN/ftc/Continuity/Continuous_I_const.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_const.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_id.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_id.con".
(*#*
Assume [F] and [G] are continuous in [I]. Then functions derived from these through algebraic operations are also continuous, provided (in the case of reciprocal and division) some extra conditions are met.
*)
-inline cic:/CoRN/ftc/Continuity/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Local_Results/contF.var".
-inline cic:/CoRN/ftc/Continuity/contG.var.
+alias id "contG" = "cic:/CoRN/ftc/Continuity/Local_Results/contG.var".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_plus.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_plus.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_inv.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_inv.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_mult.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_mult.con".
(* UNEXPORTED
Opaque AbsIR Max.
Transparent AbsIR Max.
*)
-inline cic:/CoRN/ftc/Continuity/Continuous_I_max.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_max.con".
(* begin show *)
-inline cic:/CoRN/ftc/Continuity/Hg'.var.
+alias id "Hg'" = "cic:/CoRN/ftc/Continuity/Local_Results/Hg'.var".
-inline cic:/CoRN/ftc/Continuity/Hg''.var.
+alias id "Hg''" = "cic:/CoRN/ftc/Continuity/Local_Results/Hg''.var".
(* end show *)
-inline cic:/CoRN/ftc/Continuity/Continuous_I_recip.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_recip.con".
(* UNEXPORTED
-End Local_Results.
+End Local_Results
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Corolaries.
+Section Corolaries
*)
-inline cic:/CoRN/ftc/Continuity/a.var.
+alias id "a" = "cic:/CoRN/ftc/Continuity/Corolaries/a.var".
-inline cic:/CoRN/ftc/Continuity/b.var.
+alias id "b" = "cic:/CoRN/ftc/Continuity/Corolaries/b.var".
-inline cic:/CoRN/ftc/Continuity/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Corolaries/Hab.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/I.con.
+inline "cic:/CoRN/ftc/Continuity/Corolaries/I.con" "Corolaries__".
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/F.var.
+alias id "F" = "cic:/CoRN/ftc/Continuity/Corolaries/F.var".
-inline cic:/CoRN/ftc/Continuity/G.var.
+alias id "G" = "cic:/CoRN/ftc/Continuity/Corolaries/G.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/P.con.
+inline "cic:/CoRN/ftc/Continuity/Corolaries/P.con" "Corolaries__".
-inline cic:/CoRN/ftc/Continuity/Q.con.
+inline "cic:/CoRN/ftc/Continuity/Corolaries/Q.con" "Corolaries__".
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Corolaries/contF.var".
-inline cic:/CoRN/ftc/Continuity/contG.var.
+alias id "contG" = "cic:/CoRN/ftc/Continuity/Corolaries/contG.var".
(*#*
The corresponding properties for subtraction, division and
product and constant functions.
*)
-inline cic:/CoRN/ftc/Continuity/Continuous_I_minus.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_minus.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_scal.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_scal.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_nth.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_nth.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_min.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_min.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_abs.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_abs.con".
-inline cic:/CoRN/ftc/Continuity/Hg'.var.
+alias id "Hg'" = "cic:/CoRN/ftc/Continuity/Corolaries/Hg'.var".
-inline cic:/CoRN/ftc/Continuity/Hg''.var.
+alias id "Hg''" = "cic:/CoRN/ftc/Continuity/Corolaries/Hg''.var".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_div.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_div.con".
(* UNEXPORTED
-End Corolaries.
+End Corolaries
*)
(* UNEXPORTED
-Section Other.
+Section Other
*)
(* UNEXPORTED
-Section Sums.
+Section Sums
*)
(*#*
We finally prove that the sum of an arbitrary family of continuous functions is still a continuous function.
*)
-inline cic:/CoRN/ftc/Continuity/a.var.
+alias id "a" = "cic:/CoRN/ftc/Continuity/Other/Sums/a.var".
-inline cic:/CoRN/ftc/Continuity/b.var.
+alias id "b" = "cic:/CoRN/ftc/Continuity/Other/Sums/b.var".
-inline cic:/CoRN/ftc/Continuity/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Other/Sums/Hab.var".
-inline cic:/CoRN/ftc/Continuity/Hab'.var.
+alias id "Hab'" = "cic:/CoRN/ftc/Continuity/Other/Sums/Hab'.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/I.con.
+inline "cic:/CoRN/ftc/Continuity/Other/Sums/I.con" "Other__Sums__".
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/Continuous_I_Sum0.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_Sum0.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_Sumx.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_Sumx.con".
-inline cic:/CoRN/ftc/Continuity/Continuous_I_Sum.con.
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_Sum.con".
(* UNEXPORTED
-End Sums.
+End Sums
*)
(*#*
For practical purposes, these characterization results are useful:
*)
-inline cic:/CoRN/ftc/Continuity/lub_charact.con.
+inline "cic:/CoRN/ftc/Continuity/lub_charact.con".
-inline cic:/CoRN/ftc/Continuity/glb_charact.con.
+inline "cic:/CoRN/ftc/Continuity/glb_charact.con".
(*#*
The following result is also extremely useful, as it allows us to set a lower bound on the glb of a function.
*)
-inline cic:/CoRN/ftc/Continuity/leEq_glb.con.
+inline "cic:/CoRN/ftc/Continuity/leEq_glb.con".
(*#*
The norm is also an extensional property.
*)
-inline cic:/CoRN/ftc/Continuity/Norm_Funct_wd.con.
+inline "cic:/CoRN/ftc/Continuity/Norm_Funct_wd.con".
(*#*
The value of the norm is covariant with the length of the interval.
*)
-inline cic:/CoRN/ftc/Continuity/included_imp_norm_leEq.con.
+inline "cic:/CoRN/ftc/Continuity/included_imp_norm_leEq.con".
(* UNEXPORTED
-End Other.
+End Other
*)
(* UNEXPORTED