--- a/src/HOL/Real/HahnBanach/Subspace.thy Fri Oct 08 16:18:51 1999 +0200
+++ b/src/HOL/Real/HahnBanach/Subspace.thy Fri Oct 08 16:40:27 1999 +0200
@@ -3,10 +3,13 @@
Author: Gertrud Bauer, TU Munich
*)
+
+header {* Subspaces *};
+
theory Subspace = LinearSpace:;
-section {* subspaces *};
+subsection {* Subspaces *};
constdefs
is_subspace :: "['a set, 'a set] => bool"
@@ -15,8 +18,9 @@
& 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";
+ "[| <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";
by (unfold is_subspace_def) simp;
lemma "is_subspace U V ==> U ~= {}";
@@ -28,23 +32,27 @@
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";
+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";
+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_diff_closed [simp, intro!!]: "[| is_subspace U V; x: U; y: U |] ==> x [-] y: U";
+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; x: U; y: U |] ==> x [-] y: U";
by (unfold diff_def negate_def) simp;
-lemma subspace_neg_closed [simp, intro!!]: "[| is_subspace U V; x: U |] ==> [-] x: U";
- by (unfold negate_def) simp;
+lemma subspace_neg_closed [simp, intro!!]:
+ "[| is_subspace U V; x: U |] ==> [-] x: U";
+ by (unfold negate_def) simp;
-
-theorem subspace_vs [intro!!]:
+lemma subspace_vs [intro!!]:
"[| is_subspace U V; is_vectorspace V |] ==> is_vectorspace U";
proof -;
assume "is_subspace U V" "is_vectorspace V";
@@ -65,7 +73,8 @@
show "ALL x:V. ALL a. a [*] x : V"; by (simp!);
qed;
-lemma subspace_trans: "[| is_subspace U V; is_subspace V W |] ==> is_subspace U W";
+lemma subspace_trans:
+ "[| is_subspace U V; is_subspace V W |] ==> is_subspace U W";
proof;
assume "is_subspace U V" "is_subspace V W";
show "<0> : U"; ..;
@@ -88,7 +97,9 @@
qed;
-section {* linear closure *};
+
+subsection {* Linear closure *};
+
constdefs
lin :: "'a => 'a set"
@@ -106,7 +117,8 @@
show "x = 1r [*] x"; by (simp!);
qed;
-lemma lin_subspace [intro!!]: "[| is_vectorspace V; x:V |] ==> is_subspace (lin x) V";
+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";
@@ -126,7 +138,8 @@
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);
+ show "x1 [+] x2 = (a1 + a2) [*] x";
+ by (simp! add: vs_add_mult_distrib2);
qed;
qed;
@@ -141,14 +154,17 @@
qed;
qed;
-
-lemma lin_vs [intro!!]: "[| is_vectorspace V; x:V |] ==> is_vectorspace (lin x)";
+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;
-section {* sum of two vectorspaces *};
+
+
+subsection {* Sum of two vectorspaces *};
+
constdefs
vectorspace_sum :: "['a set, 'a set] => 'a set"
@@ -159,11 +175,14 @@
lemmas vs_sumE = vs_sumD [RS iffD1, elimify];
-lemma vs_sumI [intro!!]: "[| x: U; y:V; (t::'a) = x [+] y |] ==> (t::'a) : vectorspace_sum U V";
+lemma vs_sumI [intro!!]:
+ "[| x: U; y:V; (t::'a) = x [+] y |]
+ ==> (t::'a) : vectorspace_sum U V";
by (unfold vectorspace_sum_def, intro CollectI bexI);
lemma subspace_vs_sum1 [intro!!]:
- "[| is_vectorspace U; is_vectorspace V |] ==> is_subspace U (vectorspace_sum U V)";
+ "[| is_vectorspace U; is_vectorspace V |]
+ ==> is_subspace U (vectorspace_sum U V)";
proof;
assume "is_vectorspace U" "is_vectorspace V";
show "<0> : U"; ..;
@@ -188,7 +207,6 @@
==> is_subspace (vectorspace_sum U V) E";
proof;
assume "is_subspace U E" "is_subspace V E" and e: "is_vectorspace E";
-
show "<0> : vectorspace_sum U V";
proof (intro vs_sumI);
show "<0> : U"; ..;
@@ -202,24 +220,28 @@
show "x:E"; by (simp!);
qed;
- show "ALL x:vectorspace_sum U V. ALL y:vectorspace_sum U V. x [+] y : vectorspace_sum U V";
+ show "ALL x:vectorspace_sum U V. ALL y:vectorspace_sum U V.
+ x [+] y : vectorspace_sum U V";
proof (intro ballI);
fix x y; assume "x:vectorspace_sum U V" "y:vectorspace_sum U V";
thus "x [+] y : vectorspace_sum U V";
proof (elim vs_sumE bexE, intro vs_sumI);
fix ux vx uy vy;
- assume "ux : U" "vx : V" "x = ux [+] vx" "uy : U" "vy : V" "y = uy [+] vy";
+ assume "ux : U" "vx : V" "x = ux [+] vx" "uy : U" "vy : V"
+ "y = uy [+] vy";
show "x [+] y = (ux [+] uy) [+] (vx [+] vy)"; by (simp!);
qed (simp!)+;
qed;
- show "ALL x:vectorspace_sum U V. ALL a. a [*] x : vectorspace_sum U V";
+ show "ALL x:vectorspace_sum U V. ALL a.
+ a [*] x : vectorspace_sum U V";
proof (intro ballI allI);
fix x a; assume "x:vectorspace_sum U V";
thus "a [*] x : vectorspace_sum 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);
+ show "a [*] x = (a [*] u) [+] (a [*] v)";
+ by (simp! add: vs_add_mult_distrib1);
qed (simp!)+;
qed;
qed;
@@ -233,17 +255,25 @@
qed;
-section {* special case: direct sum of a vectorspace and a linear closure of a vector *};
+
+subsection {* A special case *}
+
-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 |]
+text {* direct sum of a vectorspace and a linear closure of a vector
+*};
+
+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";
+ 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;
+ 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);
@@ -256,8 +286,8 @@
qed (rule);
qed;
-lemma decomp4: "[| is_vectorspace E; is_subspace H E; y1 : H; y2 : H; x0 ~: H; x0 :E;
- x0 ~= <0>; y1 [+] a1 [*] x0 = y2 [+] a2 [*] x0 |]
+lemma decomp4: "[| 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"
@@ -281,7 +311,8 @@
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!);
+ 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;
@@ -306,10 +337,11 @@
qed;
lemma decomp1:
- "[| is_vectorspace E; is_subspace H E; t:H; x0~:H; x0:E; x0 ~= <0> |]
+ "[| is_vectorspace E; is_subspace H E; t:H; x0~:H; x0:E; x0 ~= <0> |]
==> (@ (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>";
+ 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";
@@ -320,17 +352,17 @@
lemma decomp3:
"[| h0 = (%x. let (y, a) = @ (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> |]
+ 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 = (%x. let (y, a) = @ (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>";
-
+ "x = y [+] a [*] x0" "is_vectorspace E" "is_subspace H E"
+ "y:H" "x0 ~: H" "x0:E" "x0 ~= <0>";
have "x : vectorspace_sum H (lin x0)";
- by (simp! add: vectorspace_sum_def lin_def, intro bexI exI conjI) force+;
+ by (simp! add: vectorspace_sum_def lin_def, intro bexI exI conjI)
+ force+;
have "EX! xa. ((%(y, a). x = y [+] a [*] x0 & y:H) xa)";
proof%%;
show "EX xa. ((%(y, a). x = y [+] a [*] x0 & y:H) xa)";
@@ -343,15 +375,17 @@
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 decomp4, (simp!)+);
+ 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 decomp4, (simp!)+);
qed;
qed;
- hence eq: "(@ (y, a). (x = y [+] a [*] x0 & y:H)) = (y, a)"; by (rule select1_equality) (force!);
+ hence eq: "(@ (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;
-
-
+end;
\ No newline at end of file