new CoRN development, generated by transcript

[helm.git] / helm / software / matita / contribs / CoRN-Decl / ftc / Composition.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/ftc/Composition".
+
+(* $Id: Composition.v,v 1.4 2004/04/23 10:00:58 lcf Exp $ *)
+
+(* INCLUDE
+MoreFunctions
+*)
+
+(* UNEXPORTED
+Section Maps_into_Compacts.
+*)
+
+(* UNEXPORTED
+Section Part_Funct.
+*)
+
+(*#* *Composition
+
+Preservation results for functional composition are treated in this
+separate file.  We start by defining some auxiliary predicates, and
+then prove the preservation of continuity through composition and the
+chain rule for differentiation, both for compact and arbitrary
+intervals.
+
+%\begin{convention}% Throughout this section:
+- [a, b : IR] and [I] will denote [[a,b]];
+- [c, d : IR] and [J] will denote [[c,d]];
+- [F, F', G, G'] will be partial functions.
+
+%\end{convention}%
+
+** Maps into Compacts
+
+Both continuity and differentiability proofs require extra hypothesis
+on the functions involved---namely, that every compact interval is
+mapped into another compact interval.  We define this concept for
+partial functions, and prove some trivial results.
+*)
+
+inline cic:/CoRN/ftc/Composition/F.var.
+
+inline cic:/CoRN/ftc/Composition/G.var.
+
+inline cic:/CoRN/ftc/Composition/a.var.
+
+inline cic:/CoRN/ftc/Composition/b.var.
+
+inline cic:/CoRN/ftc/Composition/Hab.var.
+
+inline cic:/CoRN/ftc/Composition/c.var.
+
+inline cic:/CoRN/ftc/Composition/d.var.
+
+inline cic:/CoRN/ftc/Composition/Hcd.var.
+
+(* begin hide *)
+
+inline cic:/CoRN/ftc/Composition/I.con.
+
+(* end hide *)
+
+(* begin show *)
+
+inline cic:/CoRN/ftc/Composition/Hf.var.
+
+(* end show *)
+
+inline cic:/CoRN/ftc/Composition/maps_into_compacts.con.
+
+(* begin show *)
+
+inline cic:/CoRN/ftc/Composition/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_less.con.
+
+inline cic:/CoRN/ftc/Composition/maps_lemma_inc.con.
+
+(* UNEXPORTED
+End Part_Funct.
+*)
+
+(* UNEXPORTED
+End Maps_into_Compacts.
+*)
+
+(* UNEXPORTED
+Section Mapping.
+*)
+
+(*#*
+As was the case for division of partial functions, this condition
+completely characterizes the domain of the composite function.
+*)
+
+inline cic:/CoRN/ftc/Composition/F.var.
+
+inline cic:/CoRN/ftc/Composition/G.var.
+
+inline cic:/CoRN/ftc/Composition/a.var.
+
+inline cic:/CoRN/ftc/Composition/b.var.
+
+inline cic:/CoRN/ftc/Composition/Hab.var.
+
+inline cic:/CoRN/ftc/Composition/c.var.
+
+inline cic:/CoRN/ftc/Composition/d.var.
+
+inline cic:/CoRN/ftc/Composition/Hcd.var.
+
+(* begin show *)
+
+inline cic:/CoRN/ftc/Composition/Hf.var.
+
+inline cic:/CoRN/ftc/Composition/Hg.var.
+
+inline cic:/CoRN/ftc/Composition/maps.var.
+
+(* end show *)
+
+inline cic:/CoRN/ftc/Composition/included_comp.con.
+
+(* UNEXPORTED
+End Mapping.
+*)
+
+(* UNEXPORTED
+Section Interval_Continuity.
+*)
+
+(*#* **Continuity
+
+We now prove that the composition of two continuous partial functions is continuous.
+*)
+
+inline cic:/CoRN/ftc/Composition/a.var.
+
+inline cic:/CoRN/ftc/Composition/b.var.
+
+inline cic:/CoRN/ftc/Composition/Hab.var.
+
+(* begin hide *)
+
+inline cic:/CoRN/ftc/Composition/I.con.
+
+(* end hide *)
+
+inline cic:/CoRN/ftc/Composition/c.var.
+
+inline cic:/CoRN/ftc/Composition/d.var.
+
+inline cic:/CoRN/ftc/Composition/Hcd.var.
+
+inline cic:/CoRN/ftc/Composition/F.var.
+
+inline cic:/CoRN/ftc/Composition/G.var.
+
+(* begin show *)
+
+inline cic:/CoRN/ftc/Composition/contF.var.
+
+inline cic:/CoRN/ftc/Composition/contG.var.
+
+inline cic:/CoRN/ftc/Composition/Hmap.var.
+
+(* end show *)
+
+inline cic:/CoRN/ftc/Composition/Continuous_I_comp.con.
+
+(* UNEXPORTED
+End Interval_Continuity.
+*)
+
+(* UNEXPORTED
+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.
+
+inline cic:/CoRN/ftc/Composition/F'.var.
+
+inline cic:/CoRN/ftc/Composition/G.var.
+
+inline cic:/CoRN/ftc/Composition/G'.var.
+
+inline cic:/CoRN/ftc/Composition/a.var.
+
+inline cic:/CoRN/ftc/Composition/b.var.
+
+inline cic:/CoRN/ftc/Composition/Hab'.var.
+
+inline cic:/CoRN/ftc/Composition/c.var.
+
+inline cic:/CoRN/ftc/Composition/d.var.
+
+inline cic:/CoRN/ftc/Composition/Hcd'.var.
+
+(* begin hide *)
+
+inline cic:/CoRN/ftc/Composition/Hab.con.
+
+inline cic:/CoRN/ftc/Composition/Hcd.con.
+
+inline cic:/CoRN/ftc/Composition/I.con.
+
+(* end hide *)
+
+(* begin show *)
+
+inline cic:/CoRN/ftc/Composition/derF.var.
+
+inline cic:/CoRN/ftc/Composition/derG.var.
+
+inline cic:/CoRN/ftc/Composition/Hmap.var.
+
+(* end show *)
+
+inline cic:/CoRN/ftc/Composition/included_comp'.con.
+
+inline cic:/CoRN/ftc/Composition/maps'.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.
+
+(* UNEXPORTED
+End Derivative.
+*)
+
+(* UNEXPORTED
+Section Differentiability.
+*)
+
+(*#* **Differentiability
+
+Finally, we move on to differentiability.
+*)
+
+inline cic:/CoRN/ftc/Composition/F.var.
+
+inline cic:/CoRN/ftc/Composition/G.var.
+
+inline cic:/CoRN/ftc/Composition/a.var.
+
+inline cic:/CoRN/ftc/Composition/b.var.
+
+inline cic:/CoRN/ftc/Composition/Hab'.var.
+
+inline cic:/CoRN/ftc/Composition/c.var.
+
+inline cic:/CoRN/ftc/Composition/d.var.
+
+inline cic:/CoRN/ftc/Composition/Hcd'.var.
+
+(* begin hide *)
+
+inline cic:/CoRN/ftc/Composition/Hab.con.
+
+inline cic:/CoRN/ftc/Composition/Hcd.con.
+
+inline cic:/CoRN/ftc/Composition/I.con.
+
+(* end hide *)
+
+(* begin show *)
+
+inline cic:/CoRN/ftc/Composition/diffF.var.
+
+inline cic:/CoRN/ftc/Composition/diffG.var.
+
+inline cic:/CoRN/ftc/Composition/Hmap.var.
+
+(* end show *)
+
+inline cic:/CoRN/ftc/Composition/Diffble_I_comp.con.
+
+(* UNEXPORTED
+End Differentiability.
+*)
+
+(* UNEXPORTED
+Section Generalized_Intervals.
+*)
+
+(*#* **Generalizations
+
+We now generalize this results to arbitrary intervals.  We begin by generalizing the notion of mapping compacts into compacts.
+
+%\begin{convention}% We assume [I,J] to be proper intervals.
+%\end{convention}%
+*)
+
+inline cic:/CoRN/ftc/Composition/I.var.
+
+inline cic:/CoRN/ftc/Composition/J.var.
+
+inline cic:/CoRN/ftc/Composition/pI.var.
+
+inline cic:/CoRN/ftc/Composition/pJ.var.
+
+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/F.var.
+
+inline cic:/CoRN/ftc/Composition/F'.var.
+
+inline cic:/CoRN/ftc/Composition/G.var.
+
+inline cic:/CoRN/ftc/Composition/G'.var.
+
+(* begin show *)
+
+inline cic:/CoRN/ftc/Composition/Hmap.var.
+
+(* end show *)
+
+inline cic:/CoRN/ftc/Composition/Continuous_comp.con.
+
+(* begin show *)
+
+inline cic:/CoRN/ftc/Composition/Hmap'.var.
+
+(* end show *)
+
+inline cic:/CoRN/ftc/Composition/Derivative_comp.con.
+
+(* UNEXPORTED
+End Generalized_Intervals.
+*)
+
+(* UNEXPORTED
+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/negative_fun.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/Continuous_imp_maps_compacts_into.con.
+
+(*#*
+As a corollary, we get the generalization of differentiability property.
+*)
+
+inline cic:/CoRN/ftc/Composition/Diffble_comp.con.
+
+(* UNEXPORTED
+End Corollaries.
+*)
+
+(* UNEXPORTED
+Hint Immediate included_comp: included.
+*)
+
+(* UNEXPORTED
+Hint Immediate Continuous_I_comp Continuous_comp: continuous.
+*)
+