src/HOL/Real/HahnBanach/Subspace.thy
author wenzelm
Fri Oct 22 20:14:31 1999 +0200 (1999-10-22)
changeset 7917 5e5b9813cce7
parent 7808 fd019ac3485f
child 7927 b50446a33c16
permissions -rw-r--r--
HahnBanach update by Gertrud Bauer;
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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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  "[| <0>: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 "<0>: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 follws from the non-emptyness 
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of the subspace and vector space laws.*};
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lemma zero_in_subspace [intro !!]:
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  "[| is_subspace U V; is_vectorspace V |] ==> <0>: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 "<0> = 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 "<0>: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 "<0> : 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 "<0> : 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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multiples of $x$. *};
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constdefs
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  lin :: "'a => 'a set"
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  "lin x == {y. EX a. y = a <*> x}";
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lemma linD: "x : lin v = (EX a::real. x = a <*> v)";
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  by (unfold lin_def) force;
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lemma linI [intro!!]: "a <*> x0 : lin x0";
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  by (unfold lin_def) force;
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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);
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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;
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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 "<0> : lin x"; 
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  proof (unfold lin_def, intro CollectI exI);
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    show "<0> = 0r <*> x"; by (simp!);
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  qed;
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  show "lin x <= V";
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  proof (unfold lin_def, intro subsetI, elim CollectE exE); 
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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, intro CollectI exI);
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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;
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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, intro CollectI exI);
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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;
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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 == {x. EX u:U. EX v:V. x = u + v}";
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(***
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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) simp;
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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, intro CollectI bexI); 
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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 "<0> : 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 + <0>"; by (simp!);
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    show "<0> : 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 "<0> : U + V";
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  proof (intro vs_sumI);
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    show "<0> : U"; ..;
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    show "<0> : V"; ..;
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    show "(<0>::'a) = <0> + <0>"; 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 $u:U$ and $v:V$ 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 = {<0>}; 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"  
wenzelm@7808
   329
    "U Int V = {<0>}" "u1:U" "u2:U" "v1:V" "v2:V" 
wenzelm@7917
   330
    "u1 + v1 = u2 + v2"; 
wenzelm@7917
   331
  have eq: "u1 - u2 = v2 - v1"; by (simp! add: vs_add_diff_swap);
wenzelm@7917
   332
  have u: "u1 - u2 : U"; by (simp!); 
wenzelm@7917
   333
  with eq; have v': "v2 - v1 : U"; by simp; 
wenzelm@7917
   334
  have v: "v2 - v1 : V"; by (simp!); 
wenzelm@7917
   335
  with eq; have u': "u1 - u2 : V"; by simp;
wenzelm@7656
   336
  
wenzelm@7656
   337
  show "u1 = u2";
wenzelm@7656
   338
  proof (rule vs_add_minus_eq);
wenzelm@7917
   339
    show "u1 - u2 = <0>"; by (rule Int_singletonD [OF _ u u']); 
wenzelm@7917
   340
    show "u1 : E"; ..;
wenzelm@7917
   341
    show "u2 : E"; ..;
wenzelm@7917
   342
  qed;
wenzelm@7656
   343
wenzelm@7656
   344
  show "v1 = v2";
wenzelm@7656
   345
  proof (rule vs_add_minus_eq [RS sym]);
wenzelm@7917
   346
    show "v2 - v1 = <0>"; by (rule Int_singletonD [OF _ v' v]);
wenzelm@7917
   347
    show "v1 : E"; ..;
wenzelm@7917
   348
    show "v2 : E"; ..;
wenzelm@7917
   349
  qed;
wenzelm@7656
   350
qed;
wenzelm@7656
   351
wenzelm@7917
   352
text {* An application of the previous lemma will be used in the 
wenzelm@7917
   353
proof of the Hahn-Banach theorem: for an element $y + a \mult x_0$ 
wenzelm@7917
   354
of the direct sum of a vectorspace $H$ and the linear closure of 
wenzelm@7917
   355
$x_0$ the components $y:H$ and $a$ are unique. *}; 
wenzelm@7917
   356
wenzelm@7917
   357
lemma decomp_H0: 
wenzelm@7917
   358
  "[| is_vectorspace E; is_subspace H E; y1 : H; y2 : H; 
wenzelm@7917
   359
  x0 ~: H; x0 : E; x0 ~= <0>; y1 + a1 <*> x0 = y2 + a2 <*> x0 |]
wenzelm@7535
   360
  ==> y1 = y2 & a1 = a2";
wenzelm@7535
   361
proof;
wenzelm@7656
   362
  assume "is_vectorspace E" and h: "is_subspace H E"
wenzelm@7656
   363
     and "y1 : H" "y2 : H" "x0 ~: H" "x0 : E" "x0 ~= <0>" 
wenzelm@7917
   364
         "y1 + a1 <*> x0 = y2 + a2 <*> x0";
wenzelm@7535
   365
wenzelm@7917
   366
  have c: "y1 = y2 & a1 <*> x0 = a2 <*> x0";
wenzelm@7656
   367
  proof (rule decomp); 
wenzelm@7917
   368
    show "a1 <*> x0 : lin x0"; ..; 
wenzelm@7917
   369
    show "a2 <*> x0 : lin x0"; ..;
wenzelm@7656
   370
    show "H Int (lin x0) = {<0>}"; 
wenzelm@7656
   371
    proof;
wenzelm@7656
   372
      show "H Int lin x0 <= {<0>}"; 
wenzelm@7656
   373
      proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2]);
wenzelm@7656
   374
        fix x; assume "x:H" "x:lin x0"; 
wenzelm@7656
   375
        thus "x = <0>";
wenzelm@7656
   376
        proof (unfold lin_def, elim CollectE exE);
wenzelm@7917
   377
          fix a; assume "x = a <*> x0";
wenzelm@7656
   378
          show ?thesis;
wenzelm@7917
   379
          proof (rule case_split);
wenzelm@7656
   380
            assume "a = 0r"; show ?thesis; by (simp!);
wenzelm@7656
   381
          next;
wenzelm@7656
   382
            assume "a ~= 0r"; 
wenzelm@7917
   383
            from h; have "rinv a <*> a <*> x0 : H"; 
wenzelm@7808
   384
              by (rule subspace_mult_closed) (simp!);
wenzelm@7917
   385
            also; have "rinv a <*> a <*> x0 = x0"; by (simp!);
wenzelm@7656
   386
            finally; have "x0 : H"; .;
wenzelm@7656
   387
            thus ?thesis; by contradiction;
wenzelm@7656
   388
          qed;
wenzelm@7656
   389
       qed;
wenzelm@7656
   390
      qed;
wenzelm@7656
   391
      show "{<0>} <= H Int lin x0";
wenzelm@7656
   392
      proof (intro subsetI, elim singletonE, intro IntI, simp+);
wenzelm@7656
   393
        show "<0> : H"; ..;
wenzelm@7656
   394
        from lin_vs; show "<0> : lin x0"; ..;
wenzelm@7656
   395
      qed;
wenzelm@7535
   396
    qed;
wenzelm@7656
   397
    show "is_subspace (lin x0) E"; ..;
wenzelm@7535
   398
  qed;
wenzelm@7656
   399
  
wenzelm@7656
   400
  from c; show "y1 = y2"; by simp;
wenzelm@7656
   401
  
wenzelm@7656
   402
  show  "a1 = a2"; 
wenzelm@7656
   403
  proof (rule vs_mult_right_cancel [RS iffD1]);
wenzelm@7917
   404
    from c; show "a1 <*> x0 = a2 <*> x0"; by simp;
wenzelm@7535
   405
  qed;
wenzelm@7535
   406
qed;
wenzelm@7535
   407
wenzelm@7917
   408
text {* Since for an element $y + a \mult x_0$ of the direct sum 
wenzelm@7917
   409
of a vectorspace $H$ and the linear closure of $x_0$ the components
wenzelm@7917
   410
$y\in H$ and $a$ are unique, follows from $y\in H$ the fact that 
wenzelm@7917
   411
$a = 0$.*}; 
wenzelm@7917
   412
wenzelm@7917
   413
lemma decomp_H0_H: 
wenzelm@7917
   414
  "[| is_vectorspace E; is_subspace H E; t:H; x0~:H; x0:E;
wenzelm@7917
   415
  x0 ~= <0> |] 
wenzelm@7917
   416
  ==> (SOME (y, a). t = y + a <*> x0 & y : H) = (t, 0r)";
wenzelm@7535
   417
proof (rule, unfold split_paired_all);
wenzelm@7917
   418
  assume "is_vectorspace E" "is_subspace H E" "t:H" "x0~:H" "x0:E"
wenzelm@7808
   419
    "x0 ~= <0>";
wenzelm@7566
   420
  have h: "is_vectorspace H"; ..;
wenzelm@7917
   421
  fix y a; presume t1: "t = y + a <*> x0" and "y : H";
wenzelm@7535
   422
  have "y = t & a = 0r"; 
wenzelm@7917
   423
    by (rule decomp_H0) (assumption | (simp!))+;
wenzelm@7566
   424
  thus "(y, a) = (t, 0r)"; by (simp!);
wenzelm@7566
   425
qed (simp!)+;
wenzelm@7535
   426
wenzelm@7917
   427
text {* The components $y\in H$ and $a$ in $y \plus a \mult x_0$ 
wenzelm@7917
   428
are unique, so the function $h_0$ defined by 
wenzelm@7917
   429
$h_0 (y \plus a \mult x_0) = h y + a * xi$ is definite. *};
wenzelm@7917
   430
wenzelm@7917
   431
lemma h0_definite:
wenzelm@7917
   432
  "[| h0 = (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a <*> x0 & y:H)
wenzelm@7566
   433
                in (h y) + a * xi);
wenzelm@7917
   434
  x = y + a <*> x0; is_vectorspace E; is_subspace H E;
wenzelm@7808
   435
  y:H; x0 ~: H; x0:E; x0 ~= <0> |]
wenzelm@7535
   436
  ==> h0 x = h y + a * xi";
wenzelm@7535
   437
proof -;  
wenzelm@7917
   438
  assume 
wenzelm@7917
   439
    "h0 = (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a <*> x0 & y:H)
wenzelm@7917
   440
               in (h y) + a * xi)"
wenzelm@7917
   441
    "x = y + a <*> x0" "is_vectorspace E" "is_subspace H E"
wenzelm@7917
   442
    "y:H" "x0 ~: H" "x0:E" "x0 ~= <0>";
wenzelm@7917
   443
  have "x : H + (lin x0)"; 
wenzelm@7917
   444
    by (simp! add: vs_sum_def lin_def) force+;
wenzelm@7917
   445
  have "EX! xa. ((\<lambda>(y, a). x = y + a <*> x0 & y:H) xa)"; 
wenzelm@7917
   446
  proof;
wenzelm@7917
   447
    show "EX xa. ((%(y, a). x = y + a <*> x0 & y:H) xa)";
wenzelm@7566
   448
      by (force!);
wenzelm@7535
   449
  next;
wenzelm@7535
   450
    fix xa ya;
wenzelm@7917
   451
    assume "(%(y,a). x = y + a <*> x0 & y : H) xa"
wenzelm@7917
   452
           "(%(y,a). x = y + a <*> x0 & y : H) ya";
wenzelm@7535
   453
    show "xa = ya"; ;
wenzelm@7535
   454
    proof -;
wenzelm@7535
   455
      show "fst xa = fst ya & snd xa = snd ya ==> xa = ya"; 
wenzelm@7566
   456
        by (rule Pair_fst_snd_eq [RS iffD2]);
wenzelm@7917
   457
      have x: "x = (fst xa) + (snd xa) <*> x0 & (fst xa) : H"; 
wenzelm@7808
   458
        by (force!);
wenzelm@7917
   459
      have y: "x = (fst ya) + (snd ya) <*> x0 & (fst ya) : H"; 
wenzelm@7808
   460
        by (force!);
wenzelm@7808
   461
      from x y; show "fst xa = fst ya & snd xa = snd ya"; 
wenzelm@7917
   462
        by (elim conjE) (rule decomp_H0, (simp!)+);
wenzelm@7535
   463
    qed;
wenzelm@7535
   464
  qed;
wenzelm@7917
   465
  hence eq: "(SOME (y, a). (x = y + a <*> x0 & y:H)) = (y, a)"; 
wenzelm@7808
   466
    by (rule select1_equality) (force!);
wenzelm@7656
   467
  thus "h0 x = h y + a * xi"; by (simp! add: Let_def);
wenzelm@7566
   468
qed;
wenzelm@7535
   469
wenzelm@7808
   470
end;