set "baseuri" "cic:/matita/CoRN-Decl/ftc/Composition".
+include "CoRN.ma".
+
(* $Id: Composition.v,v 1.4 2004/04/23 10:00:58 lcf Exp $ *)
-(* INCLUDE
-MoreFunctions
-*)
+include "ftc/MoreFunctions.ma".
(* UNEXPORTED
-Section Maps_into_Compacts.
+Section Maps_into_Compacts
*)
(* UNEXPORTED
-Section Part_Funct.
+Section Part_Funct
*)
(*#* *Composition
partial functions, and prove some trivial results.
*)
-inline cic:/CoRN/ftc/Composition/F.var.
+alias id "F" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/F.var".
-inline cic:/CoRN/ftc/Composition/G.var.
+alias id "G" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/G.var".
-inline cic:/CoRN/ftc/Composition/a.var.
+alias id "a" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/a.var".
-inline cic:/CoRN/ftc/Composition/b.var.
+alias id "b" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/b.var".
-inline cic:/CoRN/ftc/Composition/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/Hab.var".
-inline cic:/CoRN/ftc/Composition/c.var.
+alias id "c" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/c.var".
-inline cic:/CoRN/ftc/Composition/d.var.
+alias id "d" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/d.var".
-inline cic:/CoRN/ftc/Composition/Hcd.var.
+alias id "Hcd" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/Hcd.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Composition/I.con.
+inline "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/I.con" "Maps_into_Compacts__Part_Funct__".
(* end hide *)
(* begin show *)
-inline cic:/CoRN/ftc/Composition/Hf.var.
+alias id "Hf" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/Hf.var".
(* end show *)
-inline cic:/CoRN/ftc/Composition/maps_into_compacts.con.
+inline "cic:/CoRN/ftc/Composition/maps_into_compacts.con".
(* begin show *)
-inline cic:/CoRN/ftc/Composition/maps.var.
+alias id "maps" = "cic:/CoRN/ftc/Composition/Maps_into_Compacts/Part_Funct/maps.var".
(* end show *)
-inline cic:/CoRN/ftc/Composition/maps_lemma'.con.
+inline "cic:/CoRN/ftc/Composition/maps_lemma'.con".
-inline cic:/CoRN/ftc/Composition/maps_lemma.con.
+inline "cic:/CoRN/ftc/Composition/maps_lemma.con".
-inline cic:/CoRN/ftc/Composition/maps_lemma_less.con.
+inline "cic:/CoRN/ftc/Composition/maps_lemma_less.con".
-inline cic:/CoRN/ftc/Composition/maps_lemma_inc.con.
+inline "cic:/CoRN/ftc/Composition/maps_lemma_inc.con".
(* UNEXPORTED
-End Part_Funct.
+End Part_Funct
*)
(* UNEXPORTED
-End Maps_into_Compacts.
+End Maps_into_Compacts
*)
(* UNEXPORTED
-Section Mapping.
+Section Mapping
*)
(*#*
completely characterizes the domain of the composite function.
*)
-inline cic:/CoRN/ftc/Composition/F.var.
+alias id "F" = "cic:/CoRN/ftc/Composition/Mapping/F.var".
-inline cic:/CoRN/ftc/Composition/G.var.
+alias id "G" = "cic:/CoRN/ftc/Composition/Mapping/G.var".
-inline cic:/CoRN/ftc/Composition/a.var.
+alias id "a" = "cic:/CoRN/ftc/Composition/Mapping/a.var".
-inline cic:/CoRN/ftc/Composition/b.var.
+alias id "b" = "cic:/CoRN/ftc/Composition/Mapping/b.var".
-inline cic:/CoRN/ftc/Composition/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/Composition/Mapping/Hab.var".
-inline cic:/CoRN/ftc/Composition/c.var.
+alias id "c" = "cic:/CoRN/ftc/Composition/Mapping/c.var".
-inline cic:/CoRN/ftc/Composition/d.var.
+alias id "d" = "cic:/CoRN/ftc/Composition/Mapping/d.var".
-inline cic:/CoRN/ftc/Composition/Hcd.var.
+alias id "Hcd" = "cic:/CoRN/ftc/Composition/Mapping/Hcd.var".
(* begin show *)
-inline cic:/CoRN/ftc/Composition/Hf.var.
+alias id "Hf" = "cic:/CoRN/ftc/Composition/Mapping/Hf.var".
-inline cic:/CoRN/ftc/Composition/Hg.var.
+alias id "Hg" = "cic:/CoRN/ftc/Composition/Mapping/Hg.var".
-inline cic:/CoRN/ftc/Composition/maps.var.
+alias id "maps" = "cic:/CoRN/ftc/Composition/Mapping/maps.var".
(* end show *)
-inline cic:/CoRN/ftc/Composition/included_comp.con.
+inline "cic:/CoRN/ftc/Composition/included_comp.con".
(* UNEXPORTED
-End Mapping.
+End Mapping
*)
(* UNEXPORTED
-Section Interval_Continuity.
+Section Interval_Continuity
*)
(*#* **Continuity
We now prove that the composition of two continuous partial functions is continuous.
*)
-inline cic:/CoRN/ftc/Composition/a.var.
+alias id "a" = "cic:/CoRN/ftc/Composition/Interval_Continuity/a.var".
-inline cic:/CoRN/ftc/Composition/b.var.
+alias id "b" = "cic:/CoRN/ftc/Composition/Interval_Continuity/b.var".
-inline cic:/CoRN/ftc/Composition/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/Composition/Interval_Continuity/Hab.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Composition/I.con.
+inline "cic:/CoRN/ftc/Composition/Interval_Continuity/I.con" "Interval_Continuity__".
(* end hide *)
-inline cic:/CoRN/ftc/Composition/c.var.
+alias id "c" = "cic:/CoRN/ftc/Composition/Interval_Continuity/c.var".
-inline cic:/CoRN/ftc/Composition/d.var.
+alias id "d" = "cic:/CoRN/ftc/Composition/Interval_Continuity/d.var".
-inline cic:/CoRN/ftc/Composition/Hcd.var.
+alias id "Hcd" = "cic:/CoRN/ftc/Composition/Interval_Continuity/Hcd.var".
-inline cic:/CoRN/ftc/Composition/F.var.
+alias id "F" = "cic:/CoRN/ftc/Composition/Interval_Continuity/F.var".
-inline cic:/CoRN/ftc/Composition/G.var.
+alias id "G" = "cic:/CoRN/ftc/Composition/Interval_Continuity/G.var".
(* begin show *)
-inline cic:/CoRN/ftc/Composition/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/Composition/Interval_Continuity/contF.var".
-inline cic:/CoRN/ftc/Composition/contG.var.
+alias id "contG" = "cic:/CoRN/ftc/Composition/Interval_Continuity/contG.var".
-inline cic:/CoRN/ftc/Composition/Hmap.var.
+alias id "Hmap" = "cic:/CoRN/ftc/Composition/Interval_Continuity/Hmap.var".
(* end show *)
-inline cic:/CoRN/ftc/Composition/Continuous_I_comp.con.
+inline "cic:/CoRN/ftc/Composition/Continuous_I_comp.con".
(* UNEXPORTED
-End Interval_Continuity.
+End Interval_Continuity
*)
(* UNEXPORTED
-Section Derivative.
+Section Derivative
*)
(*#* **Derivative
We now work with the derivative relation and prove the chain rule for partial functions.
*)
-inline cic:/CoRN/ftc/Composition/F.var.
+alias id "F" = "cic:/CoRN/ftc/Composition/Derivative/F.var".
-inline cic:/CoRN/ftc/Composition/F'.var.
+alias id "F'" = "cic:/CoRN/ftc/Composition/Derivative/F'.var".
-inline cic:/CoRN/ftc/Composition/G.var.
+alias id "G" = "cic:/CoRN/ftc/Composition/Derivative/G.var".
-inline cic:/CoRN/ftc/Composition/G'.var.
+alias id "G'" = "cic:/CoRN/ftc/Composition/Derivative/G'.var".
-inline cic:/CoRN/ftc/Composition/a.var.
+alias id "a" = "cic:/CoRN/ftc/Composition/Derivative/a.var".
-inline cic:/CoRN/ftc/Composition/b.var.
+alias id "b" = "cic:/CoRN/ftc/Composition/Derivative/b.var".
-inline cic:/CoRN/ftc/Composition/Hab'.var.
+alias id "Hab'" = "cic:/CoRN/ftc/Composition/Derivative/Hab'.var".
-inline cic:/CoRN/ftc/Composition/c.var.
+alias id "c" = "cic:/CoRN/ftc/Composition/Derivative/c.var".
-inline cic:/CoRN/ftc/Composition/d.var.
+alias id "d" = "cic:/CoRN/ftc/Composition/Derivative/d.var".
-inline cic:/CoRN/ftc/Composition/Hcd'.var.
+alias id "Hcd'" = "cic:/CoRN/ftc/Composition/Derivative/Hcd'.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Composition/Hab.con.
+inline "cic:/CoRN/ftc/Composition/Derivative/Hab.con" "Derivative__".
-inline cic:/CoRN/ftc/Composition/Hcd.con.
+inline "cic:/CoRN/ftc/Composition/Derivative/Hcd.con" "Derivative__".
-inline cic:/CoRN/ftc/Composition/I.con.
+inline "cic:/CoRN/ftc/Composition/Derivative/I.con" "Derivative__".
(* end hide *)
(* begin show *)
-inline cic:/CoRN/ftc/Composition/derF.var.
+alias id "derF" = "cic:/CoRN/ftc/Composition/Derivative/derF.var".
-inline cic:/CoRN/ftc/Composition/derG.var.
+alias id "derG" = "cic:/CoRN/ftc/Composition/Derivative/derG.var".
-inline cic:/CoRN/ftc/Composition/Hmap.var.
+alias id "Hmap" = "cic:/CoRN/ftc/Composition/Derivative/Hmap.var".
(* end show *)
-inline cic:/CoRN/ftc/Composition/included_comp'.con.
+inline "cic:/CoRN/ftc/Composition/included_comp'.con".
-inline cic:/CoRN/ftc/Composition/maps'.con.
+inline "cic:/CoRN/ftc/Composition/maps'.con".
-inline cic:/CoRN/ftc/Composition/Derivative_I_comp.con.
+inline "cic:/CoRN/ftc/Composition/Derivative_I_comp.con".
(*#*
The next lemma will be useful when we move on to differentiability.
*)
-inline cic:/CoRN/ftc/Composition/Diffble_I_comp_aux.con.
+inline "cic:/CoRN/ftc/Composition/Diffble_I_comp_aux.con".
(* UNEXPORTED
-End Derivative.
+End Derivative
*)
(* UNEXPORTED
-Section Differentiability.
+Section Differentiability
*)
(*#* **Differentiability
Finally, we move on to differentiability.
*)
-inline cic:/CoRN/ftc/Composition/F.var.
+alias id "F" = "cic:/CoRN/ftc/Composition/Differentiability/F.var".
-inline cic:/CoRN/ftc/Composition/G.var.
+alias id "G" = "cic:/CoRN/ftc/Composition/Differentiability/G.var".
-inline cic:/CoRN/ftc/Composition/a.var.
+alias id "a" = "cic:/CoRN/ftc/Composition/Differentiability/a.var".
-inline cic:/CoRN/ftc/Composition/b.var.
+alias id "b" = "cic:/CoRN/ftc/Composition/Differentiability/b.var".
-inline cic:/CoRN/ftc/Composition/Hab'.var.
+alias id "Hab'" = "cic:/CoRN/ftc/Composition/Differentiability/Hab'.var".
-inline cic:/CoRN/ftc/Composition/c.var.
+alias id "c" = "cic:/CoRN/ftc/Composition/Differentiability/c.var".
-inline cic:/CoRN/ftc/Composition/d.var.
+alias id "d" = "cic:/CoRN/ftc/Composition/Differentiability/d.var".
-inline cic:/CoRN/ftc/Composition/Hcd'.var.
+alias id "Hcd'" = "cic:/CoRN/ftc/Composition/Differentiability/Hcd'.var".
(* begin hide *)
-inline cic:/CoRN/ftc/Composition/Hab.con.
+inline "cic:/CoRN/ftc/Composition/Differentiability/Hab.con" "Differentiability__".
-inline cic:/CoRN/ftc/Composition/Hcd.con.
+inline "cic:/CoRN/ftc/Composition/Differentiability/Hcd.con" "Differentiability__".
-inline cic:/CoRN/ftc/Composition/I.con.
+inline "cic:/CoRN/ftc/Composition/Differentiability/I.con" "Differentiability__".
(* end hide *)
(* begin show *)
-inline cic:/CoRN/ftc/Composition/diffF.var.
+alias id "diffF" = "cic:/CoRN/ftc/Composition/Differentiability/diffF.var".
-inline cic:/CoRN/ftc/Composition/diffG.var.
+alias id "diffG" = "cic:/CoRN/ftc/Composition/Differentiability/diffG.var".
-inline cic:/CoRN/ftc/Composition/Hmap.var.
+alias id "Hmap" = "cic:/CoRN/ftc/Composition/Differentiability/Hmap.var".
(* end show *)
-inline cic:/CoRN/ftc/Composition/Diffble_I_comp.con.
+inline "cic:/CoRN/ftc/Composition/Diffble_I_comp.con".
(* UNEXPORTED
-End Differentiability.
+End Differentiability
*)
(* UNEXPORTED
-Section Generalized_Intervals.
+Section Generalized_Intervals
*)
(*#* **Generalizations
%\end{convention}%
*)
-inline cic:/CoRN/ftc/Composition/I.var.
+alias id "I" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/I.var".
-inline cic:/CoRN/ftc/Composition/J.var.
+alias id "J" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/J.var".
-inline cic:/CoRN/ftc/Composition/pI.var.
+alias id "pI" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/pI.var".
-inline cic:/CoRN/ftc/Composition/pJ.var.
+alias id "pJ" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/pJ.var".
-inline cic:/CoRN/ftc/Composition/maps_compacts_into.con.
+inline "cic:/CoRN/ftc/Composition/maps_compacts_into.con".
(*#*
Now everything comes naturally:
*)
-inline cic:/CoRN/ftc/Composition/comp_inc_lemma.con.
+inline "cic:/CoRN/ftc/Composition/comp_inc_lemma.con".
-inline cic:/CoRN/ftc/Composition/F.var.
+alias id "F" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/F.var".
-inline cic:/CoRN/ftc/Composition/F'.var.
+alias id "F'" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/F'.var".
-inline cic:/CoRN/ftc/Composition/G.var.
+alias id "G" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/G.var".
-inline cic:/CoRN/ftc/Composition/G'.var.
+alias id "G'" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/G'.var".
(* begin show *)
-inline cic:/CoRN/ftc/Composition/Hmap.var.
+alias id "Hmap" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/Hmap.var".
(* end show *)
-inline cic:/CoRN/ftc/Composition/Continuous_comp.con.
+inline "cic:/CoRN/ftc/Composition/Continuous_comp.con".
(* begin show *)
-inline cic:/CoRN/ftc/Composition/Hmap'.var.
+alias id "Hmap'" = "cic:/CoRN/ftc/Composition/Generalized_Intervals/Hmap'.var".
(* end show *)
-inline cic:/CoRN/ftc/Composition/Derivative_comp.con.
+inline "cic:/CoRN/ftc/Composition/Derivative_comp.con".
(* UNEXPORTED
-End Generalized_Intervals.
+End Generalized_Intervals
*)
(* UNEXPORTED
-Section Corollaries.
+Section Corollaries
*)
(*#*
Finally, some criteria to prove that a function with a specific domain maps compacts into compacts:
*)
-inline cic:/CoRN/ftc/Composition/positive_fun.con.
+inline "cic:/CoRN/ftc/Composition/positive_fun.con".
-inline cic:/CoRN/ftc/Composition/negative_fun.con.
+inline "cic:/CoRN/ftc/Composition/negative_fun.con".
-inline cic:/CoRN/ftc/Composition/positive_imp_maps_compacts_into.con.
+inline "cic:/CoRN/ftc/Composition/positive_imp_maps_compacts_into.con".
-inline cic:/CoRN/ftc/Composition/negative_imp_maps_compacts_into.con.
+inline "cic:/CoRN/ftc/Composition/negative_imp_maps_compacts_into.con".
-inline cic:/CoRN/ftc/Composition/Continuous_imp_maps_compacts_into.con.
+inline "cic:/CoRN/ftc/Composition/Continuous_imp_maps_compacts_into.con".
(*#*
As a corollary, we get the generalization of differentiability property.
*)
-inline cic:/CoRN/ftc/Composition/Diffble_comp.con.
+inline "cic:/CoRN/ftc/Composition/Diffble_comp.con".
(* UNEXPORTED
-End Corollaries.
+End Corollaries
*)
(* UNEXPORTED