src/HOL/Real/HahnBanach/Subspace.thy
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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 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 |] ==> <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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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 "<0> : lin x"; 
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  proof (unfold lin_def, intro CollectI exI conjI);
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    show "<0> = 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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   198
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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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   219
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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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   226
lemma vs_sumD: 
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  "x: U + V = (EX u:U. EX v:V. x = u + v)";
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   228
    by (unfold vs_sum_def) fast;
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   229
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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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   234
  by (unfold vs_sum_def) fast;
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text{* $U$ is a subspace of $U + V$. *};
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   237
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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   238
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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   241
proof; 
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  assume "is_vectorspace U" "is_vectorspace V";
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  show "<0> : U"; ..;
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   244
  show "U <= U + V";
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   245
  proof (intro subsetI vs_sumI);
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   246
  fix x; assume "x:U";
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   247
    show "x = x + <0>"; by (simp!);
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   248
    show "<0> : V"; by (simp!);
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   249
  qed;
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  show "ALL x:U. ALL y:U. x + y : U"; 
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   251
  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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   253
  qed;
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   254
  show "ALL x:U. ALL a. a <*> x : U"; 
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   255
  proof (intro ballI allI);
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   256
    fix x a; assume "x:U"; show "a <*> x : U"; by (simp!);
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   257
  qed;
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   258
qed;
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   259
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text{* The sum of two subspaces is again a subspace.*};
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   261
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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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   264
  ==> is_subspace (U + V) E";
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   265
proof; 
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  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E";
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   267
  show "<0> : U + V";
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   268
  proof (intro vs_sumI);
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   269
    show "<0> : U"; ..;
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   270
    show "<0> : V"; ..;
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   271
    show "(<0>::'a) = <0> + <0>"; by (simp!);
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   272
  qed;
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   273
  
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   274
  show "U + V <= E";
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   275
  proof (intro subsetI, elim vs_sumE bexE);
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   276
    fix x u v; assume "u : U" "v : V" "x = u + v";
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   277
    show "x:E"; by (simp!);
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   278
  qed;
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   279
  
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  show "ALL x: U + V. ALL y: U + V. x + y : U + V";
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   281
  proof (intro ballI);
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   282
    fix x y; assume "x : U + V" "y : U + V";
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   283
    thus "x + y : U + V";
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   284
    proof (elim vs_sumE bexE, intro vs_sumI);
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   285
      fix ux vx uy vy; 
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   286
      assume "ux : U" "vx : V" "x = ux + vx" 
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   287
	and "uy : U" "vy : V" "y = uy + vy";
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   288
      show "x + y = (ux + uy) + (vx + vy)"; by (simp!);
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   289
    qed (simp!)+;
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   290
  qed;
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diff changeset
   291
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   292
  show "ALL x : U + V. ALL a. a <*> x : U + V";
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   293
  proof (intro ballI allI);
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   294
    fix x a; assume "x : U + V";
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   295
    thus "a <*> x : U + V";
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   296
    proof (elim vs_sumE bexE, intro vs_sumI);
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   297
      fix a x u v; assume "u : U" "v : V" "x = u + v";
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   298
      show "a <*> x = (a <*> u) + (a <*> v)"; 
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   299
        by (simp! add: vs_add_mult_distrib1);
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   300
    qed (simp!)+;
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   301
  qed;
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   302
qed;
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   303
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   304
text{* The sum of two subspaces is a vectorspace. *};
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   305
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   306
lemma vs_sum_vs [intro!!]: 
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   307
  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   308
  ==> is_vectorspace (U + V)";
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   309
proof (rule subspace_vs);
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   310
  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E";
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   311
  show "is_subspace (U + V) E"; ..;
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   312
qed;
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   313
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   314
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   315
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   316
subsection {* Direct sums *};
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   317
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   318
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   319
text {* The sum of $U$ and $V$ is called \emph{direct}, iff the zero 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   320
element is the only common element of $U$ and $V$. For every element
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   321
$x$ of the direct sum of $U$ and $V$ the decomposition in
7927
b50446a33c16 update by Gertrud Bauer;
wenzelm
parents: 7917
diff changeset
   322
$x = u + v$ with $u \in U$ and $v \in V$ is unique.*}; 
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   323
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   324
lemma decomp: 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   325
  "[| is_vectorspace E; is_subspace U E; is_subspace V E; 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   326
  U Int V = {<0>}; u1:U; u2:U; v1:V; v2:V; u1 + v1 = u2 + v2 |] 
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   327
  ==> u1 = u2 & v1 = v2"; 
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   328
proof; 
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   329
  assume "is_vectorspace E" "is_subspace U E" "is_subspace V E"  
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   330
    "U Int V = {<0>}" "u1:U" "u2:U" "v1:V" "v2:V" 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   331
    "u1 + v1 = u2 + v2"; 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   332
  have eq: "u1 - u2 = v2 - v1"; by (simp! add: vs_add_diff_swap);
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   333
  have u: "u1 - u2 : U"; by (simp!); 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   334
  with eq; have v': "v2 - v1 : U"; by simp; 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   335
  have v: "v2 - v1 : V"; by (simp!); 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   336
  with eq; have u': "u1 - u2 : V"; by simp;
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   337
  
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   338
  show "u1 = u2";
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   339
  proof (rule vs_add_minus_eq);
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   340
    show "u1 - u2 = <0>"; by (rule Int_singletonD [OF _ u u']); 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   341
    show "u1 : E"; ..;
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   342
    show "u2 : E"; ..;
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   343
  qed;
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   344
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   345
  show "v1 = v2";
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   346
  proof (rule vs_add_minus_eq [RS sym]);
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   347
    show "v2 - v1 = <0>"; by (rule Int_singletonD [OF _ v' v]);
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   348
    show "v1 : E"; ..;
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   349
    show "v2 : E"; ..;
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   350
  qed;
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   351
qed;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   352
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   353
text {* An application of the previous lemma will be used in the proof
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   354
of the Hahn-Banach Theorem (see page \pageref{decomp-H0-use}): for any
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   355
element $y + a \mult x_0$ of the direct sum of a vectorspace $H$ and
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   356
the linear closure of $x_0$ the components $y \in H$ and $a$ are
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   357
uniquely determined. *};
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   358
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   359
lemma decomp_H0: 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   360
  "[| is_vectorspace E; is_subspace H E; y1 : H; y2 : H; 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   361
  x0 ~: H; x0 : E; x0 ~= <0>; y1 + a1 <*> x0 = y2 + a2 <*> x0 |]
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   362
  ==> y1 = y2 & a1 = a2";
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   363
proof;
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   364
  assume "is_vectorspace E" and h: "is_subspace H E"
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   365
     and "y1 : H" "y2 : H" "x0 ~: H" "x0 : E" "x0 ~= <0>" 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   366
         "y1 + a1 <*> x0 = y2 + a2 <*> x0";
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   367
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   368
  have c: "y1 = y2 & a1 <*> x0 = a2 <*> x0";
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   369
  proof (rule decomp); 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   370
    show "a1 <*> x0 : lin x0"; ..; 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   371
    show "a2 <*> x0 : lin x0"; ..;
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   372
    show "H Int (lin x0) = {<0>}"; 
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   373
    proof;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   374
      show "H Int lin x0 <= {<0>}"; 
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   375
      proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2]);
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   376
        fix x; assume "x:H" "x : lin x0"; 
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   377
        thus "x = <0>";
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   378
        proof (unfold lin_def, elim CollectE exE conjE);
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   379
          fix a; assume "x = a <*> x0";
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   380
          show ?thesis;
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   381
          proof (rule case_split);
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   382
            assume "a = 0r"; show ?thesis; by (simp!);
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   383
          next;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   384
            assume "a ~= 0r"; 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   385
            from h; have "rinv a <*> a <*> x0 : H"; 
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   386
              by (rule subspace_mult_closed) (simp!);
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   387
            also; have "rinv a <*> a <*> x0 = x0"; by (simp!);
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   388
            finally; have "x0 : H"; .;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   389
            thus ?thesis; by contradiction;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   390
          qed;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   391
       qed;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   392
      qed;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   393
      show "{<0>} <= H Int lin x0";
8169
77b3bc101de5 eliminated proof script;
wenzelm
parents: 7978
diff changeset
   394
      proof -;
77b3bc101de5 eliminated proof script;
wenzelm
parents: 7978
diff changeset
   395
	have "<0>: H Int lin x0";
77b3bc101de5 eliminated proof script;
wenzelm
parents: 7978
diff changeset
   396
	proof (rule IntI);
77b3bc101de5 eliminated proof script;
wenzelm
parents: 7978
diff changeset
   397
	  show "<0>:H"; ..;
77b3bc101de5 eliminated proof script;
wenzelm
parents: 7978
diff changeset
   398
	  from lin_vs; show "<0> : lin x0"; ..;
77b3bc101de5 eliminated proof script;
wenzelm
parents: 7978
diff changeset
   399
	qed;
77b3bc101de5 eliminated proof script;
wenzelm
parents: 7978
diff changeset
   400
	thus ?thesis; by simp;
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   401
      qed;
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   402
    qed;
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   403
    show "is_subspace (lin x0) E"; ..;
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   404
  qed;
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   405
  
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   406
  from c; show "y1 = y2"; by simp;
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   407
  
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   408
  show  "a1 = a2"; 
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   409
  proof (rule vs_mult_right_cancel [RS iffD1]);
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   410
    from c; show "a1 <*> x0 = a2 <*> x0"; by simp;
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   411
  qed;
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   412
qed;
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   413
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   414
text {* Since for any element $y + a \mult x_0$ of the direct sum 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   415
of a vectorspace $H$ and the linear closure of $x_0$ the components
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   416
$y\in H$ and $a$ are unique, it follows from $y\in H$ that 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   417
$a = 0$.*}; 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   418
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   419
lemma decomp_H0_H: 
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   420
  "[| is_vectorspace E; is_subspace H E; t:H; x0 ~: H; x0:E;
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   421
  x0 ~= <0> |] 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   422
  ==> (SOME (y, a). t = y + a <*> x0 & y : H) = (t, 0r)";
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   423
proof (rule, unfold split_paired_all);
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   424
  assume "is_vectorspace E" "is_subspace H E" "t:H" "x0 ~: H" "x0:E"
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   425
    "x0 ~= <0>";
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   426
  have h: "is_vectorspace H"; ..;
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   427
  fix y a; presume t1: "t = y + a <*> x0" and "y:H";
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   428
  have "y = t & a = 0r"; 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   429
    by (rule decomp_H0) (assumption | (simp!))+;
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   430
  thus "(y, a) = (t, 0r)"; by (simp!);
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   431
qed (simp!)+;
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   432
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   433
text {* The components $y\in H$ and $a$ in $y \plus a \mult x_0$ 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   434
are unique, so the function $h_0$ defined by 
7927
b50446a33c16 update by Gertrud Bauer;
wenzelm
parents: 7917
diff changeset
   435
$h_0 (y \plus a \mult x_0) = h y + a \cdot \xi$ is definite. *};
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   436
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   437
lemma h0_definite:
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   438
  "[| h0 == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a <*> x0 & y:H)
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   439
                in (h y) + a * xi);
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   440
  x = y + a <*> x0; is_vectorspace E; is_subspace H E;
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   441
  y:H; x0 ~: H; x0:E; x0 ~= <0> |]
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   442
  ==> h0 x = h y + a * xi";
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   443
proof -;  
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   444
  assume 
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   445
    "h0 == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a <*> x0 & y:H)
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   446
               in (h y) + a * xi)"
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   447
    "x = y + a <*> x0" "is_vectorspace E" "is_subspace H E"
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   448
    "y:H" "x0 ~: H" "x0:E" "x0 ~= <0>";
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   449
  have "x : H + (lin x0)"; 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   450
    by (simp! add: vs_sum_def lin_def) force+;
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   451
  have "EX! xa. ((\<lambda>(y, a). x = y + a <*> x0 & y:H) xa)"; 
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   452
  proof;
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   453
    show "EX xa. ((\<lambda>(y, a). x = y + a <*> x0 & y:H) xa)";
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   454
      by (force!);
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   455
  next;
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   456
    fix xa ya;
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   457
    assume "(\<lambda>(y,a). x = y + a <*> x0 & y : H) xa"
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   458
           "(\<lambda>(y,a). x = y + a <*> x0 & y : H) ya";
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   459
    show "xa = ya"; ;
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   460
    proof -;
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   461
      show "fst xa = fst ya & snd xa = snd ya ==> xa = ya"; 
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   462
        by (rule Pair_fst_snd_eq [RS iffD2]);
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   463
      have x: "x = fst xa + snd xa <*> x0 & fst xa : H"; 
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   464
        by (force!);
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   465
      have y: "x = fst ya + snd ya <*> x0 & fst ya : H"; 
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   466
        by (force!);
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   467
      from x y; show "fst xa = fst ya & snd xa = snd ya"; 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   468
        by (elim conjE) (rule decomp_H0, (simp!)+);
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   469
    qed;
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   470
  qed;
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   471
  hence eq: "(SOME (y, a). x = y + a <*> x0 & y:H) = (y, a)"; 
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   472
    by (rule select1_equality) (force!);
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   473
  thus "h0 x = h y + a * xi"; by (simp! add: Let_def);
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   474
qed;
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   475
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   476
end;