src/HOL/Real/HahnBanach/Subspace.thy
author nipkow
Thu Apr 13 15:01:50 2000 +0200 (2000-04-13)
changeset 8703 816d8f6513be
parent 8280 259073d16f84
child 9013 9dd0274f76af
permissions -rw-r--r--
Times -> <*>
** -> <*lex*>
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(*  Title:      HOL/Real/HahnBanach/Subspace.thy
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    ID:         $Id$
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    Author:     Gertrud Bauer, TU Munich
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*)
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header {* Subspaces *};
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theory Subspace = VectorSpace:;
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subsection {* Definition *};
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text {* A non-empty subset $U$ of a vector space $V$ is a 
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\emph{subspace} of $V$, iff $U$ is closed under addition and 
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scalar multiplication. *};
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constdefs 
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  is_subspace ::  "['a::{minus, plus} set, 'a set] => bool"
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  "is_subspace U V == U ~= {} & U <= V 
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     & (ALL x:U. ALL y:U. ALL a. x + y : U & a (*) x : U)";
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lemma subspaceI [intro]: 
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  "[| 00 : U; U <= V; ALL x:U. ALL y:U. (x + y : U); 
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  ALL x:U. ALL a. a (*) x : U |]
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  ==> is_subspace U V";
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proof (unfold is_subspace_def, intro conjI); 
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  assume "00 : U"; thus "U ~= {}"; by fast;
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qed (simp+);
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lemma subspace_not_empty [intro??]: "is_subspace U V ==> U ~= {}";
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  by (unfold is_subspace_def) simp; 
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lemma subspace_subset [intro??]: "is_subspace U V ==> U <= V";
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  by (unfold is_subspace_def) simp;
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lemma subspace_subsetD [simp, intro??]: 
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  "[| is_subspace U V; x:U |] ==> x:V";
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  by (unfold is_subspace_def) force;
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lemma subspace_add_closed [simp, intro??]: 
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  "[| is_subspace U V; x:U; y:U |] ==> x + y : U";
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  by (unfold is_subspace_def) simp;
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lemma subspace_mult_closed [simp, intro??]: 
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  "[| is_subspace U V; x:U |] ==> a (*) x : U";
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  by (unfold is_subspace_def) simp;
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lemma subspace_diff_closed [simp, intro??]: 
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  "[| is_subspace U V; is_vectorspace V; x:U; y:U |] 
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  ==> x - y : U";
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  by (simp! add: diff_eq1 negate_eq1);
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text {* Similar as for linear spaces, the existence of the 
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zero element in every subspace follows from the non-emptiness 
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of the carrier set and by vector space laws.*};
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lemma zero_in_subspace [intro??]:
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  "[| is_subspace U V; is_vectorspace V |] ==> 00 : U";
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proof -; 
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  assume "is_subspace U V" and v: "is_vectorspace V";
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  have "U ~= {}"; ..;
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  hence "EX x. x:U"; by force;
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  thus ?thesis; 
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  proof; 
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    fix x; assume u: "x:U"; 
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    hence "x:V"; by (simp!);
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    with v; have "00 = x - x"; by (simp!);
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    also; have "... : U"; by (rule subspace_diff_closed);
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    finally; show ?thesis; .;
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  qed;
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qed;
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lemma subspace_neg_closed [simp, intro??]: 
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  "[| is_subspace U V; is_vectorspace V; x:U |] ==> - x : U";
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  by (simp add: negate_eq1);
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text_raw {* \medskip *};
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text {* Further derived laws: every subspace is a vector space. *};
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lemma subspace_vs [intro??]:
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  "[| is_subspace U V; is_vectorspace V |] ==> is_vectorspace U";
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proof -;
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  assume "is_subspace U V" "is_vectorspace V";
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  show ?thesis;
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  proof; 
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    show "00 : U"; ..;
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    show "ALL x:U. ALL a. a (*) x : U"; by (simp!);
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    show "ALL x:U. ALL y:U. x + y : U"; by (simp!);
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    show "ALL x:U. - x = -1r (*) x"; by (simp! add: negate_eq1);
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    show "ALL x:U. ALL y:U. x - y =  x + - y"; 
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      by (simp! add: diff_eq1);
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  qed (simp! add: vs_add_mult_distrib1 vs_add_mult_distrib2)+;
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qed;
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text {* The subspace relation is reflexive. *};
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lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V";
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proof; 
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  assume "is_vectorspace V";
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  show "00 : V"; ..;
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  show "V <= V"; ..;
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  show "ALL x:V. ALL y:V. x + y : V"; by (simp!);
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  show "ALL x:V. ALL a. a (*) x : V"; by (simp!);
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qed;
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text {* The subspace relation is transitive. *};
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lemma subspace_trans: 
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  "[| is_subspace U V; is_vectorspace V; is_subspace V W |] 
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  ==> is_subspace U W";
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proof; 
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  assume "is_subspace U V" "is_subspace V W" "is_vectorspace V";
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  show "00 : U"; ..;
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  have "U <= V"; ..;
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  also; have "V <= W"; ..;
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  finally; show "U <= W"; .;
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  show "ALL x:U. ALL y:U. x + y : U"; 
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  proof (intro ballI);
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    fix x y; assume "x:U" "y:U";
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    show "x + y : U"; by (simp!);
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  qed;
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  show "ALL x:U. ALL a. a (*) x : U";
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  proof (intro ballI allI);
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    fix x a; assume "x:U";
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    show "a (*) x : U"; by (simp!);
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  qed;
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qed;
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subsection {* Linear closure *};
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text {* The \emph{linear closure} of a vector $x$ is the set of all
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scalar multiples of $x$. *};
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constdefs
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  lin :: "'a => 'a set"
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  "lin x == {a (*) x | a. True}"; 
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lemma linD: "x : lin v = (EX a::real. x = a (*) v)";
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  by (unfold lin_def) fast;
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lemma linI [intro??]: "a (*) x0 : lin x0";
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  by (unfold lin_def) fast;
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text {* Every vector is contained in its linear closure. *};
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lemma x_lin_x: "[| is_vectorspace V; x:V |] ==> x : lin x";
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proof (unfold lin_def, intro CollectI exI conjI);
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  assume "is_vectorspace V" "x:V";
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  show "x = 1r (*) x"; by (simp!);
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qed simp;
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text {* Any linear closure is a subspace. *};
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lemma lin_subspace [intro??]: 
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  "[| is_vectorspace V; x:V |] ==> is_subspace (lin x) V";
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proof;
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  assume "is_vectorspace V" "x:V";
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  show "00 : lin x"; 
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  proof (unfold lin_def, intro CollectI exI conjI);
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    show "00 = 0r (*) x"; by (simp!);
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  qed simp;
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  show "lin x <= V";
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  proof (unfold lin_def, intro subsetI, elim CollectE exE conjE); 
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    fix xa a; assume "xa = a (*) x"; 
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    show "xa:V"; by (simp!);
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  qed;
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  show "ALL x1 : lin x. ALL x2 : lin x. x1 + x2 : lin x"; 
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  proof (intro ballI);
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    fix x1 x2; assume "x1 : lin x" "x2 : lin x"; 
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    thus "x1 + x2 : lin x";
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    proof (unfold lin_def, elim CollectE exE conjE, 
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      intro CollectI exI conjI);
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      fix a1 a2; assume "x1 = a1 (*) x" "x2 = a2 (*) x";
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      show "x1 + x2 = (a1 + a2) (*) x"; 
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        by (simp! add: vs_add_mult_distrib2);
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    qed simp;
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  qed;
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  show "ALL xa:lin x. ALL a. a (*) xa : lin x"; 
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  proof (intro ballI allI);
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    fix x1 a; assume "x1 : lin x"; 
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    thus "a (*) x1 : lin x";
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    proof (unfold lin_def, elim CollectE exE conjE,
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      intro CollectI exI conjI);
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      fix a1; assume "x1 = a1 (*) x";
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      show "a (*) x1 = (a * a1) (*) x"; by (simp!);
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    qed simp;
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  qed; 
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qed;
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text {* Any linear closure is a vector space. *};
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lemma lin_vs [intro??]: 
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  "[| is_vectorspace V; x:V |] ==> is_vectorspace (lin x)";
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proof (rule subspace_vs);
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  assume "is_vectorspace V" "x:V";
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  show "is_subspace (lin x) V"; ..;
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qed;
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subsection {* Sum of two vectorspaces *};
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text {* The \emph{sum} of two vectorspaces $U$ and $V$ is the set of
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all sums of elements from $U$ and $V$. *};
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instance set :: (plus) plus; by intro_classes;
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defs vs_sum_def:
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  "U + V == {u + v | u v. u:U & v:V}"; (***
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constdefs 
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  vs_sum :: 
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  "['a::{minus, plus} set, 'a set] => 'a set"         (infixl "+" 65)
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  "vs_sum U V == {x. EX u:U. EX v:V. x = u + v}";
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***)
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lemma vs_sumD: 
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  "x: U + V = (EX u:U. EX v:V. x = u + v)";
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    by (unfold vs_sum_def) fast;
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lemmas vs_sumE = vs_sumD [RS iffD1, elimify];
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lemma vs_sumI [intro??]: 
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  "[| x:U; y:V; t= x + y |] ==> t : U + V";
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  by (unfold vs_sum_def) fast;
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text{* $U$ is a subspace of $U + V$. *};
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lemma subspace_vs_sum1 [intro??]: 
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  "[| is_vectorspace U; is_vectorspace V |]
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  ==> is_subspace U (U + V)";
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proof; 
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  assume "is_vectorspace U" "is_vectorspace V";
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  show "00 : U"; ..;
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  show "U <= U + V";
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  proof (intro subsetI vs_sumI);
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  fix x; assume "x:U";
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    show "x = x + 00"; by (simp!);
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    show "00 : V"; by (simp!);
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  qed;
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  show "ALL x:U. ALL y:U. x + y : U"; 
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  proof (intro ballI);
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    fix x y; assume "x:U" "y:U"; show "x + y : U"; by (simp!);
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  qed;
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  show "ALL x:U. ALL a. a (*) x : U"; 
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  proof (intro ballI allI);
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    fix x a; assume "x:U"; show "a (*) x : U"; by (simp!);
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  qed;
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qed;
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text{* The sum of two subspaces is again a subspace.*};
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lemma vs_sum_subspace [intro??]: 
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  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
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  ==> is_subspace (U + V) E";
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proof; 
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  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E";
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  show "00 : U + V";
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  proof (intro vs_sumI);
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    show "00 : U"; ..;
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    show "00 : V"; ..;
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    show "(00::'a) = 00 + 00"; by (simp!);
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  qed;
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  show "U + V <= E";
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  proof (intro subsetI, elim vs_sumE bexE);
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    fix x u v; assume "u : U" "v : V" "x = u + v";
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    show "x:E"; by (simp!);
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  qed;
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  show "ALL x: U + V. ALL y: U + V. x + y : U + V";
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  proof (intro ballI);
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    fix x y; assume "x : U + V" "y : U + V";
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    thus "x + y : U + V";
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    proof (elim vs_sumE bexE, intro vs_sumI);
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      fix ux vx uy vy; 
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      assume "ux : U" "vx : V" "x = ux + vx" 
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	and "uy : U" "vy : V" "y = uy + vy";
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      show "x + y = (ux + uy) + (vx + vy)"; by (simp!);
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    qed (simp!)+;
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  qed;
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  show "ALL x : U + V. ALL a. a (*) x : U + V";
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  proof (intro ballI allI);
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    fix x a; assume "x : U + V";
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    thus "a (*) x : U + V";
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    proof (elim vs_sumE bexE, intro vs_sumI);
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      fix a x u v; assume "u : U" "v : V" "x = u + v";
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      show "a (*) x = (a (*) u) + (a (*) v)"; 
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        by (simp! add: vs_add_mult_distrib1);
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    qed (simp!)+;
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  qed;
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qed;
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text{* The sum of two subspaces is a vectorspace. *};
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lemma vs_sum_vs [intro??]: 
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  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
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  ==> is_vectorspace (U + V)";
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proof (rule subspace_vs);
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  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E";
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  show "is_subspace (U + V) E"; ..;
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qed;
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subsection {* Direct sums *};
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text {* The sum of $U$ and $V$ is called \emph{direct}, iff the zero 
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element is the only common element of $U$ and $V$. For every element
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$x$ of the direct sum of $U$ and $V$ the decomposition in
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$x = u + v$ with $u \in U$ and $v \in V$ is unique.*}; 
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lemma decomp: 
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  "[| is_vectorspace E; is_subspace U E; is_subspace V E; 
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  U Int V = {00}; u1:U; u2:U; v1:V; v2:V; u1 + v1 = u2 + v2 |] 
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  ==> u1 = u2 & v1 = v2"; 
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proof; 
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  assume "is_vectorspace E" "is_subspace U E" "is_subspace V E"  
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    "U Int V = {00}" "u1:U" "u2:U" "v1:V" "v2:V" 
wenzelm@7917
   331
    "u1 + v1 = u2 + v2"; 
wenzelm@7917
   332
  have eq: "u1 - u2 = v2 - v1"; by (simp! add: vs_add_diff_swap);
wenzelm@7917
   333
  have u: "u1 - u2 : U"; by (simp!); 
wenzelm@7917
   334
  with eq; have v': "v2 - v1 : U"; by simp; 
wenzelm@7917
   335
  have v: "v2 - v1 : V"; by (simp!); 
wenzelm@7917
   336
  with eq; have u': "u1 - u2 : V"; by simp;
wenzelm@7656
   337
  
wenzelm@7656
   338
  show "u1 = u2";
wenzelm@7656
   339
  proof (rule vs_add_minus_eq);
nipkow@8703
   340
    show "u1 - u2 = 00"; by (rule Int_singletonD [OF _ u u']); 
wenzelm@7917
   341
    show "u1 : E"; ..;
wenzelm@7917
   342
    show "u2 : E"; ..;
wenzelm@7917
   343
  qed;
wenzelm@7656
   344
wenzelm@7656
   345
  show "v1 = v2";
wenzelm@7656
   346
  proof (rule vs_add_minus_eq [RS sym]);
nipkow@8703
   347
    show "v2 - v1 = 00"; by (rule Int_singletonD [OF _ v' v]);
wenzelm@7917
   348
    show "v1 : E"; ..;
wenzelm@7917
   349
    show "v2 : E"; ..;
wenzelm@7917
   350
  qed;
wenzelm@7656
   351
qed;
wenzelm@7656
   352
wenzelm@7978
   353
text {* An application of the previous lemma will be used in the proof
wenzelm@7978
   354
of the Hahn-Banach Theorem (see page \pageref{decomp-H0-use}): for any
wenzelm@7978
   355
element $y + a \mult x_0$ of the direct sum of a vectorspace $H$ and
wenzelm@7978
   356
the linear closure of $x_0$ the components $y \in H$ and $a$ are
wenzelm@7978
   357
uniquely determined. *};
wenzelm@7917
   358
wenzelm@7917
   359
lemma decomp_H0: 
wenzelm@7917
   360
  "[| is_vectorspace E; is_subspace H E; y1 : H; y2 : H; 
nipkow@8703
   361
  x0 ~: H; x0 : E; x0 ~= 00; y1 + a1 (*) x0 = y2 + a2 (*) x0 |]
wenzelm@7535
   362
  ==> y1 = y2 & a1 = a2";
wenzelm@7535
   363
proof;
wenzelm@7656
   364
  assume "is_vectorspace E" and h: "is_subspace H E"
nipkow@8703
   365
     and "y1 : H" "y2 : H" "x0 ~: H" "x0 : E" "x0 ~= 00" 
nipkow@8703
   366
         "y1 + a1 (*) x0 = y2 + a2 (*) x0";
wenzelm@7535
   367
nipkow@8703
   368
  have c: "y1 = y2 & a1 (*) x0 = a2 (*) x0";
wenzelm@7656
   369
  proof (rule decomp); 
nipkow@8703
   370
    show "a1 (*) x0 : lin x0"; ..; 
nipkow@8703
   371
    show "a2 (*) x0 : lin x0"; ..;
nipkow@8703
   372
    show "H Int (lin x0) = {00}"; 
wenzelm@7656
   373
    proof;
nipkow@8703
   374
      show "H Int lin x0 <= {00}"; 
wenzelm@7656
   375
      proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2]);
wenzelm@7978
   376
        fix x; assume "x:H" "x : lin x0"; 
nipkow@8703
   377
        thus "x = 00";
wenzelm@7978
   378
        proof (unfold lin_def, elim CollectE exE conjE);
nipkow@8703
   379
          fix a; assume "x = a (*) x0";
wenzelm@7656
   380
          show ?thesis;
wenzelm@8280
   381
          proof cases;
wenzelm@7656
   382
            assume "a = 0r"; show ?thesis; by (simp!);
wenzelm@7656
   383
          next;
wenzelm@7656
   384
            assume "a ~= 0r"; 
nipkow@8703
   385
            from h; have "rinv a (*) a (*) x0 : H"; 
wenzelm@7808
   386
              by (rule subspace_mult_closed) (simp!);
nipkow@8703
   387
            also; have "rinv a (*) a (*) x0 = x0"; by (simp!);
wenzelm@7656
   388
            finally; have "x0 : H"; .;
wenzelm@7656
   389
            thus ?thesis; by contradiction;
wenzelm@7656
   390
          qed;
wenzelm@7656
   391
       qed;
wenzelm@7656
   392
      qed;
nipkow@8703
   393
      show "{00} <= H Int lin x0";
wenzelm@8169
   394
      proof -;
nipkow@8703
   395
	have "00: H Int lin x0";
wenzelm@8169
   396
	proof (rule IntI);
nipkow@8703
   397
	  show "00:H"; ..;
nipkow@8703
   398
	  from lin_vs; show "00 : lin x0"; ..;
wenzelm@8169
   399
	qed;
wenzelm@8169
   400
	thus ?thesis; by simp;
wenzelm@7656
   401
      qed;
wenzelm@7535
   402
    qed;
wenzelm@7656
   403
    show "is_subspace (lin x0) E"; ..;
wenzelm@7535
   404
  qed;
wenzelm@7656
   405
  
wenzelm@7656
   406
  from c; show "y1 = y2"; by simp;
wenzelm@7656
   407
  
wenzelm@7656
   408
  show  "a1 = a2"; 
wenzelm@7656
   409
  proof (rule vs_mult_right_cancel [RS iffD1]);
nipkow@8703
   410
    from c; show "a1 (*) x0 = a2 (*) x0"; by simp;
wenzelm@7535
   411
  qed;
wenzelm@7535
   412
qed;
wenzelm@7535
   413
wenzelm@7978
   414
text {* Since for any element $y + a \mult x_0$ of the direct sum 
wenzelm@7917
   415
of a vectorspace $H$ and the linear closure of $x_0$ the components
wenzelm@7978
   416
$y\in H$ and $a$ are unique, it follows from $y\in H$ that 
wenzelm@7917
   417
$a = 0$.*}; 
wenzelm@7917
   418
wenzelm@7917
   419
lemma decomp_H0_H: 
wenzelm@7978
   420
  "[| is_vectorspace E; is_subspace H E; t:H; x0 ~: H; x0:E;
nipkow@8703
   421
  x0 ~= 00 |] 
nipkow@8703
   422
  ==> (SOME (y, a). t = y + a (*) x0 & y : H) = (t, 0r)";
wenzelm@7535
   423
proof (rule, unfold split_paired_all);
wenzelm@7978
   424
  assume "is_vectorspace E" "is_subspace H E" "t:H" "x0 ~: H" "x0:E"
nipkow@8703
   425
    "x0 ~= 00";
wenzelm@7566
   426
  have h: "is_vectorspace H"; ..;
nipkow@8703
   427
  fix y a; presume t1: "t = y + a (*) x0" and "y:H";
wenzelm@7535
   428
  have "y = t & a = 0r"; 
wenzelm@7917
   429
    by (rule decomp_H0) (assumption | (simp!))+;
wenzelm@7566
   430
  thus "(y, a) = (t, 0r)"; by (simp!);
wenzelm@7566
   431
qed (simp!)+;
wenzelm@7535
   432
wenzelm@7917
   433
text {* The components $y\in H$ and $a$ in $y \plus a \mult x_0$ 
wenzelm@7917
   434
are unique, so the function $h_0$ defined by 
wenzelm@7927
   435
$h_0 (y \plus a \mult x_0) = h y + a \cdot \xi$ is definite. *};
wenzelm@7917
   436
wenzelm@7917
   437
lemma h0_definite:
nipkow@8703
   438
  "[| h0 == (\\<lambda>x. let (y, a) = SOME (y, a). (x = y + a (*) x0 & y:H)
wenzelm@7566
   439
                in (h y) + a * xi);
nipkow@8703
   440
  x = y + a (*) x0; is_vectorspace E; is_subspace H E;
nipkow@8703
   441
  y:H; x0 ~: H; x0:E; x0 ~= 00 |]
wenzelm@7535
   442
  ==> h0 x = h y + a * xi";
wenzelm@7535
   443
proof -;  
wenzelm@7917
   444
  assume 
nipkow@8703
   445
    "h0 == (\\<lambda>x. let (y, a) = SOME (y, a). (x = y + a (*) x0 & y:H)
wenzelm@7917
   446
               in (h y) + a * xi)"
nipkow@8703
   447
    "x = y + a (*) x0" "is_vectorspace E" "is_subspace H E"
nipkow@8703
   448
    "y:H" "x0 ~: H" "x0:E" "x0 ~= 00";
wenzelm@7917
   449
  have "x : H + (lin x0)"; 
wenzelm@7917
   450
    by (simp! add: vs_sum_def lin_def) force+;
nipkow@8703
   451
  have "EX! xa. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)"; 
wenzelm@7917
   452
  proof;
nipkow@8703
   453
    show "EX xa. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)";
wenzelm@7566
   454
      by (force!);
wenzelm@7535
   455
  next;
wenzelm@7535
   456
    fix xa ya;
nipkow@8703
   457
    assume "(\\<lambda>(y,a). x = y + a (*) x0 & y : H) xa"
nipkow@8703
   458
           "(\\<lambda>(y,a). x = y + a (*) x0 & y : H) ya";
wenzelm@7535
   459
    show "xa = ya"; ;
wenzelm@7535
   460
    proof -;
wenzelm@7535
   461
      show "fst xa = fst ya & snd xa = snd ya ==> xa = ya"; 
wenzelm@7566
   462
        by (rule Pair_fst_snd_eq [RS iffD2]);
nipkow@8703
   463
      have x: "x = fst xa + snd xa (*) x0 & fst xa : H"; 
wenzelm@7808
   464
        by (force!);
nipkow@8703
   465
      have y: "x = fst ya + snd ya (*) x0 & fst ya : H"; 
wenzelm@7808
   466
        by (force!);
wenzelm@7808
   467
      from x y; show "fst xa = fst ya & snd xa = snd ya"; 
wenzelm@7917
   468
        by (elim conjE) (rule decomp_H0, (simp!)+);
wenzelm@7535
   469
    qed;
wenzelm@7535
   470
  qed;
nipkow@8703
   471
  hence eq: "(SOME (y, a). x = y + a (*) x0 & y:H) = (y, a)"; 
wenzelm@7808
   472
    by (rule select1_equality) (force!);
wenzelm@7656
   473
  thus "h0 x = h y + a * xi"; by (simp! add: Let_def);
wenzelm@7566
   474
qed;
wenzelm@7535
   475
wenzelm@7808
   476
end;