src/HOL/Real/HahnBanach/Subspace.thy
author wenzelm
Sun Jun 04 19:39:29 2000 +0200 (2000-06-04)
changeset 9035 371f023d3dbd
parent 9013 9dd0274f76af
child 9370 cccba6147dae
permissions -rw-r--r--
removed explicit terminator (";");
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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 = -#1 (*) 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 = #1 (*) 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 = (#0::real) (*) 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" 
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    "u1 + v1 = u2 + v2" 
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  have eq: "u1 - u2 = v2 - v1" by (simp! add: vs_add_diff_swap)
wenzelm@9035
   333
  have u: "u1 - u2 : U" by (simp!) 
wenzelm@9035
   334
  with eq have v': "v2 - v1 : U" by simp 
wenzelm@9035
   335
  have v: "v2 - v1 : V" by (simp!) 
wenzelm@9035
   336
  with eq have u': "u1 - u2 : V" by simp
wenzelm@7656
   337
  
wenzelm@9035
   338
  show "u1 = u2"
wenzelm@9035
   339
  proof (rule vs_add_minus_eq)
wenzelm@9035
   340
    show "u1 - u2 = 00" by (rule Int_singletonD [OF _ u u']) 
wenzelm@9035
   341
    show "u1 : E" ..
wenzelm@9035
   342
    show "u2 : E" ..
wenzelm@9035
   343
  qed
wenzelm@7656
   344
wenzelm@9035
   345
  show "v1 = v2"
wenzelm@9035
   346
  proof (rule vs_add_minus_eq [RS sym])
wenzelm@9035
   347
    show "v2 - v1 = 00" by (rule Int_singletonD [OF _ v' v])
wenzelm@9035
   348
    show "v1 : E" ..
wenzelm@9035
   349
    show "v2 : E" ..
wenzelm@9035
   350
  qed
wenzelm@9035
   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@9035
   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@9035
   362
  ==> y1 = y2 & a1 = a2"
wenzelm@9035
   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" 
wenzelm@9035
   366
         "y1 + a1 (*) x0 = y2 + a2 (*) x0"
wenzelm@7535
   367
wenzelm@9035
   368
  have c: "y1 = y2 & a1 (*) x0 = a2 (*) x0"
wenzelm@9035
   369
  proof (rule decomp) 
wenzelm@9035
   370
    show "a1 (*) x0 : lin x0" .. 
wenzelm@9035
   371
    show "a2 (*) x0 : lin x0" ..
wenzelm@9035
   372
    show "H Int (lin x0) = {00}" 
wenzelm@9035
   373
    proof
wenzelm@9035
   374
      show "H Int lin x0 <= {00}" 
wenzelm@9035
   375
      proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2])
wenzelm@9035
   376
        fix x assume "x:H" "x : lin x0" 
wenzelm@9035
   377
        thus "x = 00"
wenzelm@9035
   378
        proof (unfold lin_def, elim CollectE exE conjE)
wenzelm@9035
   379
          fix a assume "x = a (*) x0"
wenzelm@9035
   380
          show ?thesis
wenzelm@9035
   381
          proof cases
wenzelm@9035
   382
            assume "a = (#0::real)" show ?thesis by (simp!)
wenzelm@9035
   383
          next
wenzelm@9035
   384
            assume "a ~= (#0::real)" 
wenzelm@9035
   385
            from h have "rinv a (*) a (*) x0 : H" 
wenzelm@9035
   386
              by (rule subspace_mult_closed) (simp!)
wenzelm@9035
   387
            also have "rinv a (*) a (*) x0 = x0" by (simp!)
wenzelm@9035
   388
            finally have "x0 : H" .
wenzelm@9035
   389
            thus ?thesis by contradiction
wenzelm@9035
   390
          qed
wenzelm@9035
   391
       qed
wenzelm@9035
   392
      qed
wenzelm@9035
   393
      show "{00} <= H Int lin x0"
wenzelm@9035
   394
      proof -
wenzelm@9035
   395
	have "00: H Int lin x0"
wenzelm@9035
   396
	proof (rule IntI)
wenzelm@9035
   397
	  show "00:H" ..
wenzelm@9035
   398
	  from lin_vs show "00 : lin x0" ..
wenzelm@9035
   399
	qed
wenzelm@9035
   400
	thus ?thesis by simp
wenzelm@9035
   401
      qed
wenzelm@9035
   402
    qed
wenzelm@9035
   403
    show "is_subspace (lin x0) E" ..
wenzelm@9035
   404
  qed
wenzelm@7656
   405
  
wenzelm@9035
   406
  from c show "y1 = y2" by simp
wenzelm@7656
   407
  
wenzelm@9035
   408
  show  "a1 = a2" 
wenzelm@9035
   409
  proof (rule vs_mult_right_cancel [RS iffD1])
wenzelm@9035
   410
    from c show "a1 (*) x0 = a2 (*) x0" by simp
wenzelm@9035
   411
  qed
wenzelm@9035
   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@9035
   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 |] 
wenzelm@9035
   422
  ==> (SOME (y, a). t = y + a (*) x0 & y : H) = (t, (#0::real))"
wenzelm@9035
   423
proof (rule, unfold split_paired_all)
wenzelm@7978
   424
  assume "is_vectorspace E" "is_subspace H E" "t:H" "x0 ~: H" "x0:E"
wenzelm@9035
   425
    "x0 ~= 00"
wenzelm@9035
   426
  have h: "is_vectorspace H" ..
wenzelm@9035
   427
  fix y a presume t1: "t = y + a (*) x0" and "y:H"
wenzelm@9035
   428
  have "y = t & a = (#0::real)" 
wenzelm@9035
   429
    by (rule decomp_H0) (assumption | (simp!))+
wenzelm@9035
   430
  thus "(y, a) = (t, (#0::real))" by (simp!)
wenzelm@9035
   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@9035
   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@9035
   442
  ==> h0 x = h y + a * xi"
wenzelm@9035
   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"
wenzelm@9035
   448
    "y:H" "x0 ~: H" "x0:E" "x0 ~= 00"
wenzelm@9035
   449
  have "x : H + (lin x0)" 
wenzelm@9035
   450
    by (simp! add: vs_sum_def lin_def) force+
wenzelm@9035
   451
  have "EX! xa. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)" 
wenzelm@9035
   452
  proof
wenzelm@9035
   453
    show "EX xa. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)"
wenzelm@9035
   454
      by (force!)
wenzelm@9035
   455
  next
wenzelm@9035
   456
    fix xa ya
nipkow@8703
   457
    assume "(\\<lambda>(y,a). x = y + a (*) x0 & y : H) xa"
wenzelm@9035
   458
           "(\\<lambda>(y,a). x = y + a (*) x0 & y : H) ya"
wenzelm@9035
   459
    show "xa = ya" 
wenzelm@9035
   460
    proof -
wenzelm@9035
   461
      show "fst xa = fst ya & snd xa = snd ya ==> xa = ya" 
wenzelm@9035
   462
        by (rule Pair_fst_snd_eq [RS iffD2])
wenzelm@9035
   463
      have x: "x = fst xa + snd xa (*) x0 & fst xa : H" 
wenzelm@9035
   464
        by (force!)
wenzelm@9035
   465
      have y: "x = fst ya + snd ya (*) x0 & fst ya : H" 
wenzelm@9035
   466
        by (force!)
wenzelm@9035
   467
      from x y show "fst xa = fst ya & snd xa = snd ya" 
wenzelm@9035
   468
        by (elim conjE) (rule decomp_H0, (simp!)+)
wenzelm@9035
   469
    qed
wenzelm@9035
   470
  qed
wenzelm@9035
   471
  hence eq: "(SOME (y, a). x = y + a (*) x0 & y:H) = (y, a)" 
wenzelm@9035
   472
    by (rule select1_equality) (force!)
wenzelm@9035
   473
  thus "h0 x = h y + a * xi" by (simp! add: Let_def)
wenzelm@9035
   474
qed
wenzelm@7535
   475
wenzelm@9035
   476
end