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theory FunctionOrder(* Title: HOL/Real/HahnBanach/FunctionOrder.thy ID: $Id: FunctionOrder.thy,v 1.23 2008/01/02 14:14:12 haftmann Exp $ Author: Gertrud Bauer, TU Munich *) header {* An order on functions *} theory FunctionOrder imports Subspace Linearform begin subsection {* The graph of a function *} text {* We define the \emph{graph} of a (real) function @{text f} with domain @{text F} as the set \begin{center} @{text "{(x, f x). x ∈ F}"} \end{center} So we are modeling partial functions by specifying the domain and the mapping function. We use the term ``function'' also for its graph. *} types 'a graph = "('a × real) set" definition graph :: "'a set => ('a => real) => 'a graph" where "graph F f = {(x, f x) | x. x ∈ F}" lemma graphI [intro]: "x ∈ F ==> (x, f x) ∈ graph F f" by (unfold graph_def) blast lemma graphI2 [intro?]: "x ∈ F ==> ∃t ∈ graph F f. t = (x, f x)" by (unfold graph_def) blast lemma graphE [elim?]: "(x, y) ∈ graph F f ==> (x ∈ F ==> y = f x ==> C) ==> C" by (unfold graph_def) blast subsection {* Functions ordered by domain extension *} text {* A function @{text h'} is an extension of @{text h}, iff the graph of @{text h} is a subset of the graph of @{text h'}. *} lemma graph_extI: "(!!x. x ∈ H ==> h x = h' x) ==> H ⊆ H' ==> graph H h ⊆ graph H' h'" by (unfold graph_def) blast lemma graph_extD1 [dest?]: "graph H h ⊆ graph H' h' ==> x ∈ H ==> h x = h' x" by (unfold graph_def) blast lemma graph_extD2 [dest?]: "graph H h ⊆ graph H' h' ==> H ⊆ H'" by (unfold graph_def) blast subsection {* Domain and function of a graph *} text {* The inverse functions to @{text graph} are @{text domain} and @{text funct}. *} definition "domain" :: "'a graph => 'a set" where "domain g = {x. ∃y. (x, y) ∈ g}" definition funct :: "'a graph => ('a => real)" where "funct g = (λx. (SOME y. (x, y) ∈ g))" text {* The following lemma states that @{text g} is the graph of a function if the relation induced by @{text g} is unique. *} lemma graph_domain_funct: assumes uniq: "!!x y z. (x, y) ∈ g ==> (x, z) ∈ g ==> z = y" shows "graph (domain g) (funct g) = g" proof (unfold domain_def funct_def graph_def, auto) (* FIXME !? *) fix a b assume g: "(a, b) ∈ g" from g show "(a, SOME y. (a, y) ∈ g) ∈ g" by (rule someI2) from g show "∃y. (a, y) ∈ g" .. from g show "b = (SOME y. (a, y) ∈ g)" proof (rule some_equality [symmetric]) fix y assume "(a, y) ∈ g" with g show "y = b" by (rule uniq) qed qed subsection {* Norm-preserving extensions of a function *} text {* Given a linear form @{text f} on the space @{text F} and a seminorm @{text p} on @{text E}. The set of all linear extensions of @{text f}, to superspaces @{text H} of @{text F}, which are bounded by @{text p}, is defined as follows. *} definition norm_pres_extensions :: "'a::{plus, minus, uminus, zero} set => ('a => real) => 'a set => ('a => real) => 'a graph set" where "norm_pres_extensions E p F f = {g. ∃H h. g = graph H h ∧ linearform H h ∧ H \<unlhd> E ∧ F \<unlhd> H ∧ graph F f ⊆ graph H h ∧ (∀x ∈ H. h x ≤ p x)}" lemma norm_pres_extensionE [elim]: "g ∈ norm_pres_extensions E p F f ==> (!!H h. g = graph H h ==> linearform H h ==> H \<unlhd> E ==> F \<unlhd> H ==> graph F f ⊆ graph H h ==> ∀x ∈ H. h x ≤ p x ==> C) ==> C" by (unfold norm_pres_extensions_def) blast lemma norm_pres_extensionI2 [intro]: "linearform H h ==> H \<unlhd> E ==> F \<unlhd> H ==> graph F f ⊆ graph H h ==> ∀x ∈ H. h x ≤ p x ==> graph H h ∈ norm_pres_extensions E p F f" by (unfold norm_pres_extensions_def) blast lemma norm_pres_extensionI: (* FIXME ? *) "∃H h. g = graph H h ∧ linearform H h ∧ H \<unlhd> E ∧ F \<unlhd> H ∧ graph F f ⊆ graph H h ∧ (∀x ∈ H. h x ≤ p x) ==> g ∈ norm_pres_extensions E p F f" by (unfold norm_pres_extensions_def) blast end
lemma graphI:
x ∈ F ==> (x, f x) ∈ graph F f
lemma graphI2:
x ∈ F ==> ∃t∈graph F f. t = (x, f x)
lemma graphE:
[| (x, y) ∈ graph F f; [| x ∈ F; y = f x |] ==> C |] ==> C
lemma graph_extI:
[| !!x. x ∈ H ==> h x = h' x; H ⊆ H' |] ==> graph H h ⊆ graph H' h'
lemma graph_extD1:
[| graph H h ⊆ graph H' h'; x ∈ H |] ==> h x = h' x
lemma graph_extD2:
graph H h ⊆ graph H' h' ==> H ⊆ H'
lemma graph_domain_funct:
(!!x y z. [| (x, y) ∈ g; (x, z) ∈ g |] ==> z = y)
==> graph (domain g) (funct g) = g
lemma norm_pres_extensionE:
[| g ∈ norm_pres_extensions E p F f;
!!H h. [| g = graph H h; linearform H h; subspace H E; subspace F H;
graph F f ⊆ graph H h; ∀x∈H. h x ≤ p x |]
==> C |]
==> C
lemma norm_pres_extensionI2:
[| linearform H h; subspace H E; subspace F H; graph F f ⊆ graph H h;
∀x∈H. h x ≤ p x |]
==> graph H h ∈ norm_pres_extensions E p F f
lemma norm_pres_extensionI:
∃H h. g = graph H h ∧
linearform H h ∧
subspace H E ∧ subspace F H ∧ graph F f ⊆ graph H h ∧ (∀x∈H. h x ≤ p x)
==> g ∈ norm_pres_extensions E p F f