--- a/src/HOL/Real/HahnBanach/Subspace.thy Sun Jun 04 00:09:04 2000 +0200
+++ b/src/HOL/Real/HahnBanach/Subspace.thy Sun Jun 04 19:39:29 2000 +0200
@@ -4,218 +4,218 @@
*)
-header {* Subspaces *};
+header {* Subspaces *}
-theory Subspace = VectorSpace:;
+theory Subspace = VectorSpace:
-subsection {* Definition *};
+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. *};
+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)";
+ & (ALL x:U. ALL y:U. ALL a. x + y : U & a (*) x : U)"
lemma subspaceI [intro]:
"[| 00 : 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 "00 : U"; thus "U ~= {}"; by fast;
-qed (simp+);
+ ==> is_subspace U V"
+proof (unfold is_subspace_def, intro conjI)
+ assume "00 : 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_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_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;
+ "[| 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;
+ "[| 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;
+ "[| 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);
+ ==> 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 follows from the non-emptiness
-of the carrier set and by vector space laws.*};
+of the carrier set and by vector space laws.*}
lemma zero_in_subspace [intro??]:
- "[| is_subspace U V; is_vectorspace V |] ==> 00 : 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 "00 = x - x"; by (simp!);
- also; have "... : U"; by (rule subspace_diff_closed);
- finally; show ?thesis; .;
- qed;
-qed;
+ "[| is_subspace U V; is_vectorspace V |] ==> 00 : 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 "00 = 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);
+ "[| 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. *};
+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 "00 : 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 = -#1 (*) 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;
+ "[| is_subspace U V; is_vectorspace V |] ==> is_vectorspace U"
+proof -
+ assume "is_subspace U V" "is_vectorspace V"
+ show ?thesis
+ proof
+ show "00 : 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 = -#1 (*) 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. *};
+text {* The subspace relation is reflexive. *}
-lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V";
-proof;
- assume "is_vectorspace V";
- show "00 : 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;
+lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V"
+proof
+ assume "is_vectorspace V"
+ show "00 : 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. *};
+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 "00 : U"; ..;
+ ==> is_subspace U W"
+proof
+ assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
+ show "00 : U" ..
- have "U <= V"; ..;
- also; have "V <= W"; ..;
- finally; show "U <= W"; .;
+ 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 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;
+ 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 *};
+subsection {* Linear closure *}
text {* The \emph{linear closure} of a vector $x$ is the set of all
-scalar multiples of $x$. *};
+scalar multiples of $x$. *}
constdefs
lin :: "'a => 'a set"
- "lin x == {a (*) x | a. True}";
+ "lin x == {a (*) x | a. True}"
-lemma linD: "x : lin v = (EX a::real. x = a (*) v)";
- by (unfold lin_def) fast;
+lemma linD: "x : lin v = (EX a::real. x = a (*) v)"
+ by (unfold lin_def) fast
-lemma linI [intro??]: "a (*) x0 : lin x0";
- by (unfold lin_def) fast;
+lemma linI [intro??]: "a (*) x0 : lin x0"
+ by (unfold lin_def) fast
-text {* Every vector is contained in its linear closure. *};
+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 conjI);
- assume "is_vectorspace V" "x:V";
- show "x = #1 (*) x"; by (simp!);
-qed simp;
+lemma x_lin_x: "[| is_vectorspace V; x:V |] ==> x : lin x"
+proof (unfold lin_def, intro CollectI exI conjI)
+ assume "is_vectorspace V" "x:V"
+ show "x = #1 (*) x" by (simp!)
+qed simp
-text {* Any linear closure is a subspace. *};
+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 "00 : lin x";
- proof (unfold lin_def, intro CollectI exI conjI);
- show "00 = (#0::real) (*) x"; by (simp!);
- qed simp;
+ "[| is_vectorspace V; x:V |] ==> is_subspace (lin x) V"
+proof
+ assume "is_vectorspace V" "x:V"
+ show "00 : lin x"
+ proof (unfold lin_def, intro CollectI exI conjI)
+ show "00 = (#0::real) (*) x" by (simp!)
+ qed simp
- show "lin x <= V";
- proof (unfold lin_def, intro subsetI, elim CollectE exE conjE);
- fix xa a; assume "xa = a (*) x";
- show "xa:V"; by (simp!);
- qed;
+ show "lin x <= V"
+ proof (unfold lin_def, intro subsetI, elim CollectE exE conjE)
+ 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";
+ 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 conjE,
- intro CollectI exI conjI);
- fix a1 a2; assume "x1 = a1 (*) x" "x2 = a2 (*) x";
- show "x1 + x2 = (a1 + a2) (*) x";
- by (simp! add: vs_add_mult_distrib2);
- qed simp;
- qed;
+ intro CollectI exI conjI)
+ fix a1 a2 assume "x1 = a1 (*) x" "x2 = a2 (*) x"
+ show "x1 + x2 = (a1 + a2) (*) x"
+ by (simp! add: vs_add_mult_distrib2)
+ qed simp
+ 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";
+ 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 conjE,
- intro CollectI exI conjI);
- fix a1; assume "x1 = a1 (*) x";
- show "a (*) x1 = (a * a1) (*) x"; by (simp!);
- qed simp;
- qed;
-qed;
+ intro CollectI exI conjI)
+ fix a1 assume "x1 = a1 (*) x"
+ show "a (*) x1 = (a * a1) (*) x" by (simp!)
+ qed simp
+ qed
+qed
-text {* Any linear closure is a vector space. *};
+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;
+ "[| 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 *};
+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 $V$. *};
+all sums of elements from $U$ and $V$. *}
-instance set :: (plus) plus; by intro_classes;
+instance set :: (plus) plus by intro_classes
defs vs_sum_def:
- "U + V == {u + v | u v. u:U & v:V}"; (***
+ "U + V == {u + v | u v. u:U & v:V}" (***
constdefs
vs_sum ::
@@ -224,253 +224,253 @@
***)
lemma vs_sumD:
- "x: U + V = (EX u:U. EX v:V. x = u + v)";
- by (unfold vs_sum_def) fast;
+ "x: U + V = (EX u:U. EX v:V. x = u + v)"
+ by (unfold vs_sum_def) fast
-lemmas vs_sumE = vs_sumD [RS iffD1, elimify];
+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) fast;
+ "[| x:U; y:V; t= x + y |] ==> t : U + V"
+ by (unfold vs_sum_def) fast
-text{* $U$ is a subspace of $U + V$. *};
+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 "00 : U"; ..;
- show "U <= U + V";
- proof (intro subsetI vs_sumI);
- fix x; assume "x:U";
- show "x = x + 00"; by (simp!);
- show "00 : 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;
+ ==> is_subspace U (U + V)"
+proof
+ assume "is_vectorspace U" "is_vectorspace V"
+ show "00 : U" ..
+ show "U <= U + V"
+ proof (intro subsetI vs_sumI)
+ fix x assume "x:U"
+ show "x = x + 00" by (simp!)
+ show "00 : 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.*};
+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 "00 : U + V";
- proof (intro vs_sumI);
- show "00 : U"; ..;
- show "00 : V"; ..;
- show "(00::'a) = 00 + 00"; by (simp!);
- qed;
+ ==> is_subspace (U + V) E"
+proof
+ assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
+ show "00 : U + V"
+ proof (intro vs_sumI)
+ show "00 : U" ..
+ show "00 : V" ..
+ show "(00::'a) = 00 + 00" 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 "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;
+ 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;
+ 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;
+ 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. *};
+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;
+ ==> 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 *};
+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.*};
+$x = u + v$ 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 = {00}; u1:U; u2:U; v1:V; v2:V; u1 + v1 = u2 + v2 |]
- ==> u1 = u2 & v1 = v2";
-proof;
+ ==> u1 = u2 & v1 = v2"
+proof
assume "is_vectorspace E" "is_subspace U E" "is_subspace V E"
"U Int V = {00}" "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;
+ "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 = 00"; by (rule Int_singletonD [OF _ u u']);
- show "u1 : E"; ..;
- show "u2 : E"; ..;
- qed;
+ show "u1 = u2"
+ proof (rule vs_add_minus_eq)
+ show "u1 - u2 = 00" 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 = 00"; by (rule Int_singletonD [OF _ v' v]);
- show "v1 : E"; ..;
- show "v2 : E"; ..;
- qed;
-qed;
+ show "v1 = v2"
+ proof (rule vs_add_minus_eq [RS sym])
+ show "v2 - v1 = 00" 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 (see page \pageref{decomp-H0-use}): for any
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
-uniquely determined. *};
+uniquely determined. *}
lemma decomp_H0:
"[| is_vectorspace E; is_subspace H E; y1 : H; y2 : H;
x0 ~: H; x0 : E; x0 ~= 00; y1 + a1 (*) x0 = y2 + a2 (*) x0 |]
- ==> y1 = y2 & a1 = a2";
-proof;
+ ==> 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 ~= 00"
- "y1 + a1 (*) x0 = y2 + a2 (*) x0";
+ "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) = {00}";
- proof;
- show "H Int lin x0 <= {00}";
- proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2]);
- fix x; assume "x:H" "x : lin x0";
- thus "x = 00";
- proof (unfold lin_def, elim CollectE exE conjE);
- fix a; assume "x = a (*) x0";
- show ?thesis;
- proof cases;
- assume "a = (#0::real)"; show ?thesis; by (simp!);
- next;
- assume "a ~= (#0::real)";
- 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 "{00} <= H Int lin x0";
- proof -;
- have "00: H Int lin x0";
- proof (rule IntI);
- show "00:H"; ..;
- from lin_vs; show "00 : lin x0"; ..;
- qed;
- thus ?thesis; by simp;
- qed;
- qed;
- show "is_subspace (lin x0) E"; ..;
- qed;
+ 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) = {00}"
+ proof
+ show "H Int lin x0 <= {00}"
+ proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2])
+ fix x assume "x:H" "x : lin x0"
+ thus "x = 00"
+ proof (unfold lin_def, elim CollectE exE conjE)
+ fix a assume "x = a (*) x0"
+ show ?thesis
+ proof cases
+ assume "a = (#0::real)" show ?thesis by (simp!)
+ next
+ assume "a ~= (#0::real)"
+ 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 "{00} <= H Int lin x0"
+ proof -
+ have "00: H Int lin x0"
+ proof (rule IntI)
+ show "00:H" ..
+ from lin_vs show "00 : lin x0" ..
+ qed
+ thus ?thesis by simp
+ qed
+ qed
+ show "is_subspace (lin x0) E" ..
+ qed
- from c; show "y1 = y2"; by simp;
+ 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;
+ 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 any 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, it follows from $y\in H$ that
-$a = 0$.*};
+$a = 0$.*}
lemma decomp_H0_H:
"[| is_vectorspace E; is_subspace H E; t:H; x0 ~: H; x0:E;
x0 ~= 00 |]
- ==> (SOME (y, a). t = y + a (*) x0 & y : H) = (t, (#0::real))";
-proof (rule, unfold split_paired_all);
+ ==> (SOME (y, a). t = y + a (*) x0 & y : H) = (t, (#0::real))"
+proof (rule, unfold split_paired_all)
assume "is_vectorspace E" "is_subspace H E" "t:H" "x0 ~: H" "x0:E"
- "x0 ~= 00";
- have h: "is_vectorspace H"; ..;
- fix y a; presume t1: "t = y + a (*) x0" and "y:H";
- have "y = t & a = (#0::real)";
- by (rule decomp_H0) (assumption | (simp!))+;
- thus "(y, a) = (t, (#0::real))"; by (simp!);
-qed (simp!)+;
+ "x0 ~= 00"
+ have h: "is_vectorspace H" ..
+ fix y a presume t1: "t = y + a (*) x0" and "y:H"
+ have "y = t & a = (#0::real)"
+ by (rule decomp_H0) (assumption | (simp!))+
+ thus "(y, a) = (t, (#0::real))" 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. *};
+$h_0 (y \plus a \mult x_0) = h y + a \cdot \xi$ 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 ~= 00 |]
- ==> h0 x = h y + a * xi";
-proof -;
+ ==> 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 ~= 00";
- 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. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)";
- by (force!);
- next;
- fix xa ya;
+ "y:H" "x0 ~: H" "x0:E" "x0 ~= 00"
+ 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. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)"
+ by (force!)
+ next
+ fix xa ya
assume "(\\<lambda>(y,a). x = y + a (*) x0 & y : H) xa"
- "(\\<lambda>(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;
+ "(\\<lambda>(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;
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+end
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