(* 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 $u \in U$ and $v \in V$ 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 $y \in H$ 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
$h_0 (y \plus a \mult x_0) = h y + a \cdot \xi$ 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;