src/HOL/Real/HahnBanach/Subspace.thy

HahnBanach update by Gertrud Bauer;

(*  Title:      HOL/Real/HahnBanach/Subspace.thy
    ID:         $Id$
    Author:     Gertrud Bauer, TU Munich
*)


header {* Subspaces *};

theory Subspace = VectorSpace:;


subsection {* Definition *};

text {* A non-empty subset $U$ of a vector space $V$ is a 
\emph{subspace} of $V$, iff $U$ is closed under addition and 
scalar multiplication. *};

constdefs 
  is_subspace ::  "['a::{minus, plus} set, 'a set] => bool"
  "is_subspace U V ==  U ~= {}  & U <= V 
     &  (ALL x:U. ALL y:U. ALL a. x + y : U & a <*> x : U)";

lemma subspaceI [intro]: 
  "[| <0>:U; U <= V; ALL x:U. ALL y:U. (x + y : U); 
  ALL x:U. ALL a. a <*> x : U |]
  ==> is_subspace U V";
proof (unfold is_subspace_def, intro conjI); 
  assume "<0>:U"; thus "U ~= {}"; by fast;
qed (simp+);

lemma subspace_not_empty [intro!!]: "is_subspace U V ==> U ~= {}";
  by (unfold is_subspace_def) simp; 

lemma subspace_subset [intro !!]: "is_subspace U V ==> U <= V";
  by (unfold is_subspace_def) simp;

lemma subspace_subsetD [simp, intro!!]: 
  "[| is_subspace U V; x:U |]==> x:V";
  by (unfold is_subspace_def) force;

lemma subspace_add_closed [simp, intro!!]: 
  "[| is_subspace U V; x: U; y: U |] ==> x + y : U";
  by (unfold is_subspace_def) simp;

lemma subspace_mult_closed [simp, intro!!]: 
  "[| is_subspace U V; x: U |] ==> a <*> x: U";
  by (unfold is_subspace_def) simp;

lemma subspace_diff_closed [simp, intro!!]: 
  "[| is_subspace U V; is_vectorspace V; x: U; y: U |] 
  ==> x - y: U";
  by (simp! add: diff_eq1 negate_eq1);

text {* Similar as for linear spaces, the existence of the 
zero element in every subspace follws from the non-emptyness 
of the subspace and vector space laws.*};

lemma zero_in_subspace [intro !!]:
  "[| is_subspace U V; is_vectorspace V |] ==> <0>:U";
proof -; 
  assume "is_subspace U V" and v: "is_vectorspace V";
  have "U ~= {}"; ..;
  hence "EX x. x:U"; by force;
  thus ?thesis; 
  proof; 
    fix x; assume u: "x:U"; 
    hence "x:V"; by (simp!);
    with v; have "<0> = x - x"; by (simp!);
    also; have "... : U"; by (rule subspace_diff_closed);
    finally; show ?thesis; .;
  qed;
qed;

lemma subspace_neg_closed [simp, intro!!]: 
  "[| is_subspace U V; is_vectorspace V; x: U |] ==> - x: U";
  by (simp add: negate_eq1);

text_raw {* \medskip *};
text {* Further derived laws: Every subspace is a vector space. *};

lemma subspace_vs [intro!!]:
  "[| is_subspace U V; is_vectorspace V |] ==> is_vectorspace U";
proof -;
  assume "is_subspace U V" "is_vectorspace V";
  show ?thesis;
  proof; 
    show "<0>:U"; ..;
    show "ALL x:U. ALL a. a <*> x : U"; by (simp!);
    show "ALL x:U. ALL y:U. x + y : U"; by (simp!);
    show "ALL x:U. - x = -1r <*> x"; by (simp! add: negate_eq1);
    show "ALL x:U. ALL y:U. x - y =  x + - y"; 
      by (simp! add: diff_eq1);
  qed (simp! add: vs_add_mult_distrib1 vs_add_mult_distrib2)+;
qed;

text {* The subspace relation is reflexive. *};

lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V";
proof; 
  assume "is_vectorspace V";
  show "<0> : V"; ..;
  show "V <= V"; ..;
  show "ALL x:V. ALL y:V. x + y : V"; by (simp!);
  show "ALL x:V. ALL a. a <*> x : V"; by (simp!);
qed;

text {* The subspace relation is transitive. *};

lemma subspace_trans: 
  "[| is_subspace U V; is_vectorspace V; is_subspace V W |] 
  ==> is_subspace U W";
proof; 
  assume "is_subspace U V" "is_subspace V W" "is_vectorspace V";
  show "<0> : U"; ..;

  have "U <= V"; ..;
  also; have "V <= W"; ..;
  finally; show "U <= W"; .;

  show "ALL x:U. ALL y:U. x + y : U"; 
  proof (intro ballI);
    fix x y; assume "x:U" "y:U";
    show "x + y : U"; by (simp!);
  qed;

  show "ALL x:U. ALL a. a <*> x : U";
  proof (intro ballI allI);
    fix x a; assume "x:U";
    show "a <*> x : U"; by (simp!);
  qed;
qed;



subsection {* Linear closure *};

text {* The \emph{linear closure} of a vector $x$ is the set of all 
multiples of $x$. *};

constdefs
  lin :: "'a => 'a set"
  "lin x == {y. EX a. y = a <*> x}";

lemma linD: "x : lin v = (EX a::real. x = a <*> v)";
  by (unfold lin_def) force;

lemma linI [intro!!]: "a <*> x0 : lin x0";
  by (unfold lin_def) force;

text {* Every vector is contained in its linear closure. *};

lemma x_lin_x: "[| is_vectorspace V; x:V |] ==> x:lin x";
proof (unfold lin_def, intro CollectI exI);
  assume "is_vectorspace V" "x:V";
  show "x = 1r <*> x"; by (simp!);
qed;

text {* Any linear closure is a subspace. *};

lemma lin_subspace [intro!!]: 
  "[| is_vectorspace V; x:V |] ==> is_subspace (lin x) V";
proof;
  assume "is_vectorspace V" "x:V";
  show "<0> : lin x"; 
  proof (unfold lin_def, intro CollectI exI);
    show "<0> = 0r <*> x"; by (simp!);
  qed;

  show "lin x <= V";
  proof (unfold lin_def, intro subsetI, elim CollectE exE); 
    fix xa a; assume "xa = a <*> x"; 
    show "xa:V"; by (simp!);
  qed;

  show "ALL x1 : lin x. ALL x2 : lin x. x1 + x2 : lin x"; 
  proof (intro ballI);
    fix x1 x2; assume "x1 : lin x" "x2 : lin x"; 
    thus "x1 + x2 : lin x";
    proof (unfold lin_def, elim CollectE exE, intro CollectI exI);
      fix a1 a2; assume "x1 = a1 <*> x" "x2 = a2 <*> x";
      show "x1 + x2 = (a1 + a2) <*> x"; 
        by (simp! add: vs_add_mult_distrib2);
    qed;
  qed;

  show "ALL xa:lin x. ALL a. a <*> xa : lin x"; 
  proof (intro ballI allI);
    fix x1 a; assume "x1 : lin x"; 
    thus "a <*> x1 : lin x";
    proof (unfold lin_def, elim CollectE exE, intro CollectI exI);
      fix a1; assume "x1 = a1 <*> x";
      show "a <*> x1 = (a * a1) <*> x"; by (simp!);
    qed;
  qed; 
qed;

text {* Any linear closure is a vector space. *};

lemma lin_vs [intro!!]: 
  "[| is_vectorspace V; x:V |] ==> is_vectorspace (lin x)";
proof (rule subspace_vs);
  assume "is_vectorspace V" "x:V";
  show "is_subspace (lin x) V"; ..;
qed;



subsection {* Sum of two vectorspaces *};

text {* The \emph{sum} of two vectorspaces $U$ and $V$ is the set of
all sums of elements from $U$ and $V$. *};

instance set :: (plus) plus; by intro_classes;

defs vs_sum_def:
  "U + V == {x. EX u:U. EX v:V. x = u + v}";

(***
constdefs 
  vs_sum :: 
  "['a::{minus, plus} set, 'a set] => 'a set"         (infixl "+" 65)
  "vs_sum U V == {x. EX u:U. EX v:V. x = u + v}";
***)

lemma vs_sumD: 
  "x: U + V = (EX u:U. EX v:V. x = u + v)";
  by (unfold vs_sum_def) simp;

lemmas vs_sumE = vs_sumD [RS iffD1, elimify];

lemma vs_sumI [intro!!]: 
  "[| x:U; y:V; t= x + y |] ==> t : U + V";
  by (unfold vs_sum_def, intro CollectI bexI); 

text{* $U$ is a subspace of $U + V$. *};

lemma subspace_vs_sum1 [intro!!]: 
  "[| is_vectorspace U; is_vectorspace V |]
  ==> is_subspace U (U + V)";
proof; 
  assume "is_vectorspace U" "is_vectorspace V";
  show "<0> : U"; ..;
  show "U <= U + V";
  proof (intro subsetI vs_sumI);
  fix x; assume "x:U";
    show "x = x + <0>"; by (simp!);
    show "<0> : V"; by (simp!);
  qed;
  show "ALL x:U. ALL y:U. x + y : U"; 
  proof (intro ballI);
    fix x y; assume "x:U" "y:U"; show "x + y : U"; by (simp!);
  qed;
  show "ALL x:U. ALL a. a <*> x : U"; 
  proof (intro ballI allI);
    fix x a; assume "x:U"; show "a <*> x : U"; by (simp!);
  qed;
qed;

text{* The sum of two subspaces is again a subspace.*};

lemma vs_sum_subspace [intro!!]: 
  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
  ==> is_subspace (U + V) E";
proof; 
  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E";
  show "<0> : U + V";
  proof (intro vs_sumI);
    show "<0> : U"; ..;
    show "<0> : V"; ..;
    show "(<0>::'a) = <0> + <0>"; by (simp!);
  qed;
  
  show "U + V <= E";
  proof (intro subsetI, elim vs_sumE bexE);
    fix x u v; assume "u : U" "v : V" "x = u + v";
    show "x:E"; by (simp!);
  qed;
  
  show "ALL x: U + V. ALL y: U + V. x + y : U + V";
  proof (intro ballI);
    fix x y; assume "x : U + V" "y : U + V";
    thus "x + y : U + V";
    proof (elim vs_sumE bexE, intro vs_sumI);
      fix ux vx uy vy; 
      assume "ux : U" "vx : V" "x = ux + vx" 
	and "uy : U" "vy : V" "y = uy + vy";
      show "x + y = (ux + uy) + (vx + vy)"; by (simp!);
    qed (simp!)+;
  qed;

  show "ALL x: U + V. ALL a. a <*> x : U + V";
  proof (intro ballI allI);
    fix x a; assume "x : U + V";
    thus "a <*> x : U + V";
    proof (elim vs_sumE bexE, intro vs_sumI);
      fix a x u v; assume "u : U" "v : V" "x = u + v";
      show "a <*> x = (a <*> u) + (a <*> v)"; 
        by (simp! add: vs_add_mult_distrib1);
    qed (simp!)+;
  qed;
qed;

text{* The sum of two subspaces is a vectorspace. *};

lemma vs_sum_vs [intro!!]: 
  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
  ==> is_vectorspace (U + V)";
proof (rule subspace_vs);
  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E";
  show "is_subspace (U + V) E"; ..;
qed;



subsection {* Direct sums *};


text {* The sum of $U$ and $V$ is called \emph{direct}, iff the zero 
element is the only common element of $U$ and $V$. For every element
$x$ of the direct sum of $U$ and $V$ the decomposition in
$x = u + v$ with $u:U$ and $v:V$ is unique.*}; 

lemma decomp: 
  "[| is_vectorspace E; is_subspace U E; is_subspace V E; 
  U Int V = {<0>}; u1:U; u2:U; v1:V; v2:V; u1 + v1 = u2 + v2 |] 
  ==> u1 = u2 & v1 = v2"; 
proof; 
  assume "is_vectorspace E" "is_subspace U E" "is_subspace V E"  
    "U Int V = {<0>}" "u1:U" "u2:U" "v1:V" "v2:V" 
    "u1 + v1 = u2 + v2"; 
  have eq: "u1 - u2 = v2 - v1"; by (simp! add: vs_add_diff_swap);
  have u: "u1 - u2 : U"; by (simp!); 
  with eq; have v': "v2 - v1 : U"; by simp; 
  have v: "v2 - v1 : V"; by (simp!); 
  with eq; have u': "u1 - u2 : V"; by simp;
  
  show "u1 = u2";
  proof (rule vs_add_minus_eq);
    show "u1 - u2 = <0>"; by (rule Int_singletonD [OF _ u u']); 
    show "u1 : E"; ..;
    show "u2 : E"; ..;
  qed;

  show "v1 = v2";
  proof (rule vs_add_minus_eq [RS sym]);
    show "v2 - v1 = <0>"; by (rule Int_singletonD [OF _ v' v]);
    show "v1 : E"; ..;
    show "v2 : E"; ..;
  qed;
qed;

text {* An application of the previous lemma will be used in the 
proof of the Hahn-Banach theorem: for an element $y + a \mult x_0$ 
of the direct sum of a vectorspace $H$ and the linear closure of 
$x_0$ the components $y:H$ and $a$ are unique. *}; 

lemma decomp_H0: 
  "[| is_vectorspace E; is_subspace H E; y1 : H; y2 : H; 
  x0 ~: H; x0 : E; x0 ~= <0>; y1 + a1 <*> x0 = y2 + a2 <*> x0 |]
  ==> y1 = y2 & a1 = a2";
proof;
  assume "is_vectorspace E" and h: "is_subspace H E"
     and "y1 : H" "y2 : H" "x0 ~: H" "x0 : E" "x0 ~= <0>" 
         "y1 + a1 <*> x0 = y2 + a2 <*> x0";

  have c: "y1 = y2 & a1 <*> x0 = a2 <*> x0";
  proof (rule decomp); 
    show "a1 <*> x0 : lin x0"; ..; 
    show "a2 <*> x0 : lin x0"; ..;
    show "H Int (lin x0) = {<0>}"; 
    proof;
      show "H Int lin x0 <= {<0>}"; 
      proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2]);
        fix x; assume "x:H" "x:lin x0"; 
        thus "x = <0>";
        proof (unfold lin_def, elim CollectE exE);
          fix a; assume "x = a <*> x0";
          show ?thesis;
          proof (rule case_split);
            assume "a = 0r"; show ?thesis; by (simp!);
          next;
            assume "a ~= 0r"; 
            from h; have "rinv a <*> a <*> x0 : H"; 
              by (rule subspace_mult_closed) (simp!);
            also; have "rinv a <*> a <*> x0 = x0"; by (simp!);
            finally; have "x0 : H"; .;
            thus ?thesis; by contradiction;
          qed;
       qed;
      qed;
      show "{<0>} <= H Int lin x0";
      proof (intro subsetI, elim singletonE, intro IntI, simp+);
        show "<0> : H"; ..;
        from lin_vs; show "<0> : lin x0"; ..;
      qed;
    qed;
    show "is_subspace (lin x0) E"; ..;
  qed;
  
  from c; show "y1 = y2"; by simp;
  
  show  "a1 = a2"; 
  proof (rule vs_mult_right_cancel [RS iffD1]);
    from c; show "a1 <*> x0 = a2 <*> x0"; by simp;
  qed;
qed;

text {* Since for an element $y + a \mult x_0$ of the direct sum 
of a vectorspace $H$ and the linear closure of $x_0$ the components
$y\in H$ and $a$ are unique, follows from $y\in H$ the fact that 
$a = 0$.*}; 

lemma decomp_H0_H: 
  "[| is_vectorspace E; is_subspace H E; t:H; x0~:H; x0:E;
  x0 ~= <0> |] 
  ==> (SOME (y, a). t = y + a <*> x0 & y : H) = (t, 0r)";
proof (rule, unfold split_paired_all);
  assume "is_vectorspace E" "is_subspace H E" "t:H" "x0~:H" "x0:E"
    "x0 ~= <0>";
  have h: "is_vectorspace H"; ..;
  fix y a; presume t1: "t = y + a <*> x0" and "y : H";
  have "y = t & a = 0r"; 
    by (rule decomp_H0) (assumption | (simp!))+;
  thus "(y, a) = (t, 0r)"; by (simp!);
qed (simp!)+;

text {* The components $y\in H$ and $a$ in $y \plus a \mult x_0$ 
are unique, so the function $h_0$ defined by 
$h_0 (y \plus a \mult x_0) = h y + a * xi$ is definite. *};

lemma h0_definite:
  "[| h0 = (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a <*> x0 & y:H)
                in (h y) + a * xi);
  x = y + a <*> x0; is_vectorspace E; is_subspace H E;
  y:H; x0 ~: H; x0:E; x0 ~= <0> |]
  ==> h0 x = h y + a * xi";
proof -;  
  assume 
    "h0 = (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a <*> x0 & y:H)
               in (h y) + a * xi)"
    "x = y + a <*> x0" "is_vectorspace E" "is_subspace H E"
    "y:H" "x0 ~: H" "x0:E" "x0 ~= <0>";
  have "x : H + (lin x0)"; 
    by (simp! add: vs_sum_def lin_def) force+;
  have "EX! xa. ((\<lambda>(y, a). x = y + a <*> x0 & y:H) xa)"; 
  proof;
    show "EX xa. ((%(y, a). x = y + a <*> x0 & y:H) xa)";
      by (force!);
  next;
    fix xa ya;
    assume "(%(y,a). x = y + a <*> x0 & y : H) xa"
           "(%(y,a). x = y + a <*> x0 & y : H) ya";
    show "xa = ya"; ;
    proof -;
      show "fst xa = fst ya & snd xa = snd ya ==> xa = ya"; 
        by (rule Pair_fst_snd_eq [RS iffD2]);
      have x: "x = (fst xa) + (snd xa) <*> x0 & (fst xa) : H"; 
        by (force!);
      have y: "x = (fst ya) + (snd ya) <*> x0 & (fst ya) : H"; 
        by (force!);
      from x y; show "fst xa = fst ya & snd xa = snd ya"; 
        by (elim conjE) (rule decomp_H0, (simp!)+);
    qed;
  qed;
  hence eq: "(SOME (y, a). (x = y + a <*> x0 & y:H)) = (y, a)"; 
    by (rule select1_equality) (force!);
  thus "h0 x = h y + a * xi"; by (simp! add: Let_def);
qed;

end;