author | wenzelm |
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(* Title: HOL/Real/HahnBanach/FunctionOrder.thy |
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ID: $Id$ |
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Author: Gertrud Bauer, TU Munich |
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*) |
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theory FunctionOrder = Subspace + Linearform:; |
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section {* Order on functions *}; |
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types 'a graph = "('a * real) set"; |
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constdefs |
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graph :: "['a set, 'a => real] => 'a graph " |
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"graph F f == {p. EX x. p = (x, f x) & x:F}"; (* == {(x, f x). x:F} *) |
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constdefs |
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domain :: "'a graph => 'a set" |
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"domain g == {x. EX y. (x, y):g}"; |
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constdefs |
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funct :: "'a graph => ('a => real)" |
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"funct g == %x. (@ y. (x, y):g)"; |
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lemma graphI [intro!!]: "x:F ==> (x, f x) : graph F f"; |
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by (unfold graph_def, intro CollectI exI) force; |
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lemma graphI2 [intro!!]: "x:F ==> EX t: (graph F f). t = (x, f x)"; |
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by (unfold graph_def, force); |
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lemma graphD1 [intro!!]: "(x, y): graph F f ==> x:F"; |
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by (unfold graph_def, elim CollectE exE) force; |
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lemma graphD2 [intro!!]: "(x, y): graph H h ==> y = h x"; |
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by (unfold graph_def, elim CollectE exE) force; |
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lemma graph_extD1 [intro!!]: "[| graph H h <= graph H' h'; x:H |] ==> h x = h' x"; |
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by (unfold graph_def, force); |
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lemma graph_extD2 [intro!!]: "[| graph H h <= graph H' h' |] ==> H <= H'"; |
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by (unfold graph_def, force); |
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lemma graph_extI: "[| !! x. x: H ==> h x = h' x; H <= H'|] ==> graph H h <= graph H' h'"; |
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by (unfold graph_def, force); |
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lemma graph_domain_funct: |
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"(!!x y z. (x, y):g ==> (x, z):g ==> z = y) ==> graph (domain g) (funct g) = g"; |
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proof (unfold domain_def, unfold funct_def, unfold graph_def, auto); |
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fix a b; assume "(a, b) : g"; |
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show "(a, SOME y. (a, y) : g) : g"; by (rule selectI2); |
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show "EX y. (a, y) : g"; ..; |
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assume uniq: "!!x y z. (x, y):g ==> (x, z):g ==> z = y"; |
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show "b = (SOME y. (a, y) : g)"; |
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proof (rule select_equality [RS sym]); |
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fix y; assume "(a, y):g"; show "y = b"; by (rule uniq); |
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qed; |
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qed; |
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constdefs |
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norm_prev_extensions :: |
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"['a set, 'a => real, 'a set, 'a => real] => 'a graph set" |
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"norm_prev_extensions E p F f == {g. EX H h. graph H h = g |
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& is_linearform H h |
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& is_subspace H E |
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& is_subspace F H |
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& (graph F f <= graph H h) |
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& (ALL x:H. h x <= p x)}"; |
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lemma norm_prev_extension_D: |
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"(g: norm_prev_extensions E p F f) ==> (EX H h. graph H h = g |
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& is_linearform H h |
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& is_subspace H E |
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& is_subspace F H |
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& (graph F f <= graph H h) |
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& (ALL x:H. h x <= p x))"; |
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by (unfold norm_prev_extensions_def) force; |
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lemma norm_prev_extensionI2 [intro]: |
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"[| is_linearform H h; |
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is_subspace H E; |
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is_subspace F H; |
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(graph F f <= graph H h); |
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(ALL x:H. h x <= p x) |] |
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==> (graph H h : norm_prev_extensions E p F f)"; |
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by (unfold norm_prev_extensions_def) force; |
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lemma norm_prev_extensionI [intro]: |
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"(EX H h. graph H h = g |
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& is_linearform H h |
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& is_subspace H E |
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& is_subspace F H |
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& (graph F f <= graph H h) |
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& (ALL x:H. h x <= p x)) |
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==> (g: norm_prev_extensions E p F f) "; |
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by (unfold norm_prev_extensions_def) force; |
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end; |
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