src/HOL/Real/HahnBanach/Subspace.thy
author wenzelm
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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::{plus, minus, zero} set, 'a set] => bool"
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  "is_subspace U V == U \<noteq> {} \<and> U <= V 
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     \<and> (\<forall>x \<in> U. \<forall>y \<in> U. \<forall>a. x + y \<in> U \<and> a \<cdot> x\<in> U)"
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lemma subspaceI [intro]: 
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  "[| 0 \<in> U; U <= V; \<forall>x \<in> U. \<forall>y \<in> U. (x + y \<in> U); 
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  \<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> 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 \<in> U" thus "U \<noteq> {}" by fast
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qed (simp+)
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lemma subspace_not_empty [intro?]: "is_subspace U V ==> U \<noteq> {}"
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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 \<in> U |] ==> x \<in> 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 \<in> U; y \<in> U |] ==> x + y \<in> 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 \<in> U |] ==> a \<cdot> x \<in> 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 \<in> U; y \<in> U |] 
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  ==> x - y \<in> 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 \<in> 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 \<noteq> {}" ..
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  hence "\<exists>x. x \<in> U" by force
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  thus ?thesis 
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  proof 
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    fix x assume u: "x \<in> U" 
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    hence "x \<in> V" by (simp!)
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    with v have "0 = x - x" by (simp!)
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    also have "... \<in> 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 \<in> U |] ==> - x \<in> 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 \<in> U" ..
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    show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" by (simp!)
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    show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" by (simp!)
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    show "\<forall>x \<in> U. - x = -#1 \<cdot> x" by (simp! add: negate_eq1)
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    show "\<forall>x \<in> U. \<forall>y \<in> 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 \<in> V" ..
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  show "V <= V" ..
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  show "\<forall>x \<in> V. \<forall>y \<in> V. x + y \<in> V" by (simp!)
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  show "\<forall>x \<in> V. \<forall>a. a \<cdot> x \<in> 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 \<in> 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 "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" 
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  proof (intro ballI)
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    fix x y assume "x \<in> U" "y \<in> U"
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    show "x + y \<in> U" by (simp!)
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  qed
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  show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
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  proof (intro ballI allI)
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    fix x a assume "x \<in> U"
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    show "a \<cdot> x \<in> U" by (simp!)
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  qed
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qed
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599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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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::{minus,plus,zero}) => 'a set"
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  "lin x == {a \<cdot> x | a. True}" 
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lemma linD: "x \<in> lin v = (\<exists>a::real. x = a \<cdot> v)"
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  by (unfold lin_def) fast
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lemma linI [intro?]: "a \<cdot> x0 \<in> 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 \<in> V |] ==> x \<in> lin x"
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   153
proof (unfold lin_def, intro CollectI exI conjI)
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  assume "is_vectorspace V" "x \<in> V"
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  show "x = #1 \<cdot> x" by (simp!)
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qed simp
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text {* Any linear closure is a subspace. *}
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   159
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lemma lin_subspace [intro?]: 
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  "[| is_vectorspace V; x \<in> V |] ==> is_subspace (lin x) V"
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   162
proof
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   163
  assume "is_vectorspace V" "x \<in> V"
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  show "0 \<in> lin x" 
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   165
  proof (unfold lin_def, intro CollectI exI conjI)
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parents: 9370
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   166
    show "0 = (#0::real) \<cdot> x" by (simp!)
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  qed simp
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   168
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   169
  show "lin x <= V"
371f023d3dbd removed explicit terminator (";");
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parents: 9013
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   170
  proof (unfold lin_def, intro subsetI, elim CollectE exE conjE) 
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parents: 9370
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   171
    fix xa a assume "xa = a \<cdot> x" 
153853af318b - xsymbols for
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parents: 9370
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   172
    show "xa \<in> V" by (simp!)
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   173
  qed
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c5a3f980a7af accomodate refined facts handling;
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parents: 7535
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   174
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   175
  show "\<forall>x1 \<in> lin x. \<forall>x2 \<in> lin x. x1 + x2 \<in> lin x" 
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parents: 9013
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   176
  proof (intro ballI)
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   177
    fix x1 x2 assume "x1 \<in> lin x" "x2 \<in> lin x" 
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   178
    thus "x1 + x2 \<in> lin x"
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parents: 7927
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   179
    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 \<cdot> x" "x2 = a2 \<cdot> x"
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      show "x1 + x2 = (a1 + a2) \<cdot> x" 
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        by (simp! add: vs_add_mult_distrib2)
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    qed simp
371f023d3dbd removed explicit terminator (";");
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  qed
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   186
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   187
  show "\<forall>xa \<in> lin x. \<forall>a. a \<cdot> xa \<in> lin x" 
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wenzelm
parents: 9013
diff changeset
   188
  proof (intro ballI allI)
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parents: 9370
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   189
    fix x1 a assume "x1 \<in> lin x" 
153853af318b - xsymbols for
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parents: 9370
diff changeset
   190
    thus "a \<cdot> x1 \<in> lin x"
7978
1b99ee57d131 final update by Gertrud Bauer;
wenzelm
parents: 7927
diff changeset
   191
    proof (unfold lin_def, elim CollectE exE conjE,
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   192
      intro CollectI exI conjI)
9374
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   193
      fix a1 assume "x1 = a1 \<cdot> x"
153853af318b - xsymbols for
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parents: 9370
diff changeset
   194
      show "a \<cdot> x1 = (a * a1) \<cdot> x" by (simp!)
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wenzelm
parents: 9013
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   195
    qed simp
371f023d3dbd removed explicit terminator (";");
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   196
  qed 
371f023d3dbd removed explicit terminator (";");
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   197
qed
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599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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parents:
diff changeset
   198
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   199
text {* Any linear closure is a vector space. *}
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   200
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lemma lin_vs [intro?]: 
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   202
  "[| is_vectorspace V; x \<in> V |] ==> is_vectorspace (lin x)"
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parents: 9013
diff changeset
   203
proof (rule subspace_vs)
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   204
  assume "is_vectorspace V" "x \<in> V"
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parents: 9013
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   205
  show "is_subspace (lin x) V" ..
371f023d3dbd removed explicit terminator (";");
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parents: 9013
diff changeset
   206
qed
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599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
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   207
7808
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
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   208
fd019ac3485f update from Gertrud;
wenzelm
parents: 7656
diff changeset
   209
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   210
subsection {* Sum of two vectorspaces *}
7808
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   211
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   212
text {* The \emph{sum} of two vectorspaces $U$ and $V$ is the set of
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   213
all sums of elements from $U$ and $V$. *}
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parents:
diff changeset
   214
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   215
instance set :: (plus) plus by intro_classes
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   216
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
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diff changeset
   217
defs vs_sum_def:
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   218
  "U + V == {u + v | u v. u \<in> U \<and> v \<in> V}" (***
7917
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wenzelm
parents: 7808
diff changeset
   219
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599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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parents:
diff changeset
   220
constdefs 
7917
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parents: 7808
diff changeset
   221
  vs_sum :: 
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parents: 9370
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   222
  "['a::{plus, minus, zero} set, 'a set] => 'a set"         (infixl "+" 65)
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diff changeset
   223
  "vs_sum U V == {x. \<exists>u \<in> U. \<exists>v \<in> V. x = u + v}";
7917
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wenzelm
parents: 7808
diff changeset
   224
***)
7535
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parents:
diff changeset
   225
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
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   226
lemma vs_sumD: 
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diff changeset
   227
  "x \<in> U + V = (\<exists>u \<in> U. \<exists>v \<in> V. x = u + v)"
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wenzelm
parents: 9013
diff changeset
   228
    by (unfold vs_sum_def) fast
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   229
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   230
lemmas vs_sumE = vs_sumD [RS iffD1, elimify]
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diff changeset
   231
9408
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   232
lemma vs_sumI [intro?]: 
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   233
  "[| x \<in> U; y \<in> V; t= x + y |] ==> t \<in> U + V"
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parents: 9013
diff changeset
   234
  by (unfold vs_sum_def) fast
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   235
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   236
text{* $U$ is a subspace of $U + V$. *}
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wenzelm
parents:
diff changeset
   237
9408
d3d56e1d2ec1 classical atts now intro! / intro / intro?;
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diff changeset
   238
lemma subspace_vs_sum1 [intro?]: 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   239
  "[| is_vectorspace U; is_vectorspace V |]
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   240
  ==> is_subspace U (U + V)"
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   241
proof 
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   242
  assume "is_vectorspace U" "is_vectorspace V"
9374
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parents: 9370
diff changeset
   243
  show "0 \<in> U" ..
9035
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wenzelm
parents: 9013
diff changeset
   244
  show "U <= U + V"
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   245
  proof (intro subsetI vs_sumI)
9374
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parents: 9370
diff changeset
   246
  fix x assume "x \<in> U"
153853af318b - xsymbols for
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parents: 9370
diff changeset
   247
    show "x = x + 0" by (simp!)
153853af318b - xsymbols for
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parents: 9370
diff changeset
   248
    show "0 \<in> V" by (simp!)
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wenzelm
parents: 9013
diff changeset
   249
  qed
9374
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parents: 9370
diff changeset
   250
  show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" 
9035
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wenzelm
parents: 9013
diff changeset
   251
  proof (intro ballI)
9374
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parents: 9370
diff changeset
   252
    fix x y assume "x \<in> U" "y \<in> U" show "x + y \<in> U" by (simp!)
9035
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wenzelm
parents: 9013
diff changeset
   253
  qed
9374
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parents: 9370
diff changeset
   254
  show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   255
  proof (intro ballI allI)
9374
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parents: 9370
diff changeset
   256
    fix x a assume "x \<in> U" show "a \<cdot> x \<in> U" by (simp!)
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   257
  qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   258
qed
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   259
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   260
text{* The sum of two subspaces is again a subspace.*}
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   261
9408
d3d56e1d2ec1 classical atts now intro! / intro / intro?;
wenzelm
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   262
lemma vs_sum_subspace [intro?]: 
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   263
  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
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wenzelm
parents: 9013
diff changeset
   264
  ==> is_subspace (U + V) E"
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   265
proof 
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   266
  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
9374
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parents: 9370
diff changeset
   267
  show "0 \<in> U + V"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   268
  proof (intro vs_sumI)
9374
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parents: 9370
diff changeset
   269
    show "0 \<in> U" ..
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   270
    show "0 \<in> V" ..
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   271
    show "(0::'a) = 0 + 0" by (simp!)
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wenzelm
parents: 9013
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   272
  qed
7566
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diff changeset
   273
  
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wenzelm
parents: 9013
diff changeset
   274
  show "U + V <= E"
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   275
  proof (intro subsetI, elim vs_sumE bexE)
9374
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bauerg
parents: 9370
diff changeset
   276
    fix x u v assume "u \<in> U" "v \<in> V" "x = u + v"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   277
    show "x \<in> E" by (simp!)
9035
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wenzelm
parents: 9013
diff changeset
   278
  qed
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   279
  
9374
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parents: 9370
diff changeset
   280
  show "\<forall>x \<in> U + V. \<forall>y \<in> U + V. x + y \<in> U + V"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   281
  proof (intro ballI)
9374
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bauerg
parents: 9370
diff changeset
   282
    fix x y assume "x \<in> U + V" "y \<in> U + V"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   283
    thus "x + y \<in> U + V"
9035
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wenzelm
parents: 9013
diff changeset
   284
    proof (elim vs_sumE bexE, intro vs_sumI)
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   285
      fix ux vx uy vy 
9374
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bauerg
parents: 9370
diff changeset
   286
      assume "ux \<in> U" "vx \<in> V" "x = ux + vx" 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   287
	and "uy \<in> U" "vy \<in> V" "y = uy + vy"
9035
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wenzelm
parents: 9013
diff changeset
   288
      show "x + y = (ux + uy) + (vx + vy)" by (simp!)
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   289
    qed (simp!)+
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   290
  qed
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   291
9374
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bauerg
parents: 9370
diff changeset
   292
  show "\<forall>x \<in> U + V. \<forall>a. a \<cdot> x \<in> U + V"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   293
  proof (intro ballI allI)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   294
    fix x a assume "x \<in> U + V"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   295
    thus "a \<cdot> x \<in> U + V"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   296
    proof (elim vs_sumE bexE, intro vs_sumI)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   297
      fix a x u v assume "u \<in> U" "v \<in> V" "x = u + v"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   298
      show "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   299
        by (simp! add: vs_add_mult_distrib1)
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   300
    qed (simp!)+
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   301
  qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   302
qed
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   303
9035
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wenzelm
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diff changeset
   304
text{* The sum of two subspaces is a vectorspace. *}
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   305
9408
d3d56e1d2ec1 classical atts now intro! / intro / intro?;
wenzelm
parents: 9374
diff changeset
   306
lemma vs_sum_vs [intro?]: 
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   307
  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   308
  ==> is_vectorspace (U + V)"
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   309
proof (rule subspace_vs)
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   310
  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   311
  show "is_subspace (U + V) E" ..
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
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
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
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
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
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; 
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   326
  U \<inter> V = {0}; u1 \<in> U; u2 \<in> U; v1 \<in> V; v2 \<in> V; u1 + v1 = u2 + v2 |] 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   327
  ==> u1 = u2 \<and> v1 = v2" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
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"  
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   330
    "U \<inter> V = {0}" "u1 \<in> U" "u2 \<in> U" "v1 \<in> V" "v2 \<in> V" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   331
    "u1 + v1 = u2 + v2" 
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   332
  have eq: "u1 - u2 = v2 - v1" by (simp! add: vs_add_diff_swap)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   333
  have u: "u1 - u2 \<in> U" by (simp!) 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   334
  with eq have v': "v2 - v1 \<in> U" by simp 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   335
  have v: "v2 - v1 \<in> V" by (simp!) 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   336
  with eq have u': "u1 - u2 \<in> V" by simp
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   337
  
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   338
  show "u1 = u2"
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   339
  proof (rule vs_add_minus_eq)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   340
    show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u']) 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   341
    show "u1 \<in> E" ..
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   342
    show "u2 \<in> E" ..
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   343
  qed
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   344
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   345
  show "v1 = v2"
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   346
  proof (rule vs_add_minus_eq [RS sym])
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   347
    show "v2 - v1 = 0" by (rule Int_singletonD [OF _ v' v])
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   348
    show "v1 \<in> E" ..
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   349
    show "v2 \<in> E" ..
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   350
  qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   351
qed
7656
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
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   354
of the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any
7978
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
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   357
uniquely determined. *}
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   358
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   359
lemma decomp_H': 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   360
  "[| is_vectorspace E; is_subspace H E; y1 \<in> H; y2 \<in> H; 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   361
  x' \<notin> H; x' \<in> E; x' \<noteq> 0; y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x' |]
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   362
  ==> y1 = y2 \<and> a1 = a2"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
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"
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   365
     and "y1 \<in> H" "y2 \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   366
         "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   367
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   368
  have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   369
  proof (rule decomp) 
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   370
    show "a1 \<cdot> x' \<in> lin x'" .. 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   371
    show "a2 \<cdot> x' \<in> lin x'" ..
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   372
    show "H \<inter> (lin x') = {0}" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   373
    proof
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   374
      show "H \<inter> lin x' <= {0}" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   375
      proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2])
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   376
        fix x assume "x \<in> H" "x \<in> lin x'" 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   377
        thus "x = 0"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   378
        proof (unfold lin_def, elim CollectE exE conjE)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   379
          fix a assume "x = a \<cdot> x'"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   380
          show ?thesis
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   381
          proof cases
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   382
            assume "a = (#0::real)" show ?thesis by (simp!)
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   383
          next
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   384
            assume "a \<noteq> (#0::real)" 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   385
            from h have "rinv a \<cdot> a \<cdot> x' \<in> H" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   386
              by (rule subspace_mult_closed) (simp!)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   387
            also have "rinv a \<cdot> a \<cdot> x' = x'" by (simp!)
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   388
            finally have "x' \<in> H" .
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   389
            thus ?thesis by contradiction
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   390
          qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   391
       qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   392
      qed
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   393
      show "{0} <= H \<inter> lin x'"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   394
      proof -
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   395
	have "0 \<in> H \<inter> lin x'"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   396
	proof (rule IntI)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   397
	  show "0 \<in> H" ..
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   398
	  from lin_vs show "0 \<in> lin x'" ..
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   399
	qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   400
	thus ?thesis by simp
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   401
      qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   402
    qed
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   403
    show "is_subspace (lin x') E" ..
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   404
  qed
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   405
  
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   406
  from c show "y1 = y2" by simp
7656
2f18c0ffc348 update from Gertrud;
wenzelm
parents: 7567
diff changeset
   407
  
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   408
  show  "a1 = a2" 
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   409
  proof (rule vs_mult_right_cancel [RS iffD1])
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   410
    from c show "a1 \<cdot> x' = a2 \<cdot> x'" by simp
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   411
  qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   412
qed
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   413
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   414
text {* Since for any element $y + a \mult x'$ of the direct sum 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   415
of a vectorspace $H$ and the linear closure of $x'$ 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 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   417
$a = 0$.*} 
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   418
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   419
lemma decomp_H'_H: 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   420
  "[| is_vectorspace E; is_subspace H E; t \<in> H; x' \<notin> H; x' \<in> E;
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   421
  x' \<noteq> 0 |] 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   422
  ==> (SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, (#0::real))"
9370
cccba6147dae use split_tupled_all;
wenzelm
parents: 9035
diff changeset
   423
proof (rule, unfold split_tupled_all)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   424
  assume "is_vectorspace E" "is_subspace H E" "t \<in> H" "x' \<notin> H" "x' \<in> E"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   425
    "x' \<noteq> 0"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   426
  have h: "is_vectorspace H" ..
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   427
  fix y a presume t1: "t = y + a \<cdot> x'" and "y \<in> H"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   428
  have "y = t \<and> a = (#0::real)" 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   429
    by (rule decomp_H') (assumption | (simp!))+
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   430
  thus "(y, a) = (t, (#0::real))" by (simp!)
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   431
qed (simp!)+
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   432
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   433
text {* The components $y\in H$ and $a$ in $y \plus a \mult x'$ 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   434
are unique, so the function $h'$ defined by 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   435
$h' (y \plus a \mult x') = h y + a \cdot \xi$ is definite. *}
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   436
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   437
lemma h'_definite:
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   438
  "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
7566
c5a3f980a7af accomodate refined facts handling;
wenzelm
parents: 7535
diff changeset
   439
                in (h y) + a * xi);
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   440
  x = y + a \<cdot> x'; is_vectorspace E; is_subspace H E;
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   441
  y \<in> H; x' \<notin> H; x' \<in> E; x' \<noteq> 0 |]
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   442
  ==> h' x = h y + a * xi"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   443
proof -  
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   444
  assume 
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   445
    "h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents: 7808
diff changeset
   446
               in (h y) + a * xi)"
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   447
    "x = y + a \<cdot> x'" "is_vectorspace E" "is_subspace H E"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   448
    "y \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   449
  have "x \<in> H + (lin x')" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   450
    by (simp! add: vs_sum_def lin_def) force+
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   451
  have "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   452
  proof
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   453
    show "\<exists>xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   454
      by (force!)
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   455
  next
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   456
    fix xa ya
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   457
    assume "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) xa"
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   458
           "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) ya"
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   459
    show "xa = ya" 
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   460
    proof -
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   461
      show "fst xa = fst ya \<and> snd xa = snd ya ==> xa = ya" 
9370
cccba6147dae use split_tupled_all;
wenzelm
parents: 9035
diff changeset
   462
        by (simp add: Pair_fst_snd_eq)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   463
      have x: "x = fst xa + snd xa \<cdot> x' \<and> fst xa \<in> H" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   464
        by (force!)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   465
      have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   466
        by (force!)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   467
      from x y show "fst xa = fst ya \<and> snd xa = snd ya" 
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   468
        by (elim conjE) (rule decomp_H', (simp!)+)
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   469
    qed
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   470
  qed
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   471
  hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)" 
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   472
    by (rule select1_equality) (force!)
9374
153853af318b - xsymbols for
bauerg
parents: 9370
diff changeset
   473
  thus "h' x = h y + a * xi" by (simp! add: Let_def)
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   474
qed
7535
599d3414b51d The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff changeset
   475
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   476
end