--- a/src/HOL/Real/HahnBanach/Subspace.thy Sun Jul 16 21:00:32 2000 +0200
+++ b/src/HOL/Real/HahnBanach/Subspace.thy Mon Jul 17 13:58:18 2000 +0200
@@ -16,39 +16,39 @@
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)"
+ is_subspace :: "['a::{plus, minus, zero} set, 'a set] => bool"
+ "is_subspace U V == U \<noteq> {} \<and> U <= V
+ \<and> (\<forall>x \<in> U. \<forall>y \<in> U. \<forall>a. x + y \<in> U \<and> a \<cdot> x\<in> 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 |]
+ "[| 0 \<in> U; U <= V; \<forall>x \<in> U. \<forall>y \<in> U. (x + y \<in> U);
+ \<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U |]
==> is_subspace U V"
proof (unfold is_subspace_def, intro conjI)
- assume "00 : U" thus "U ~= {}" by fast
+ assume "0 \<in> U" thus "U \<noteq> {}" by fast
qed (simp+)
-lemma subspace_not_empty [intro??]: "is_subspace U V ==> U ~= {}"
+lemma subspace_not_empty [intro??]: "is_subspace U V ==> U \<noteq> {}"
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"
+ "[| is_subspace U V; x \<in> U |] ==> x \<in> V"
by (unfold is_subspace_def) force
lemma subspace_add_closed [simp, intro??]:
- "[| is_subspace U V; x:U; y:U |] ==> x + y : U"
+ "[| is_subspace U V; x \<in> U; y \<in> U |] ==> x + y \<in> U"
by (unfold is_subspace_def) simp
lemma subspace_mult_closed [simp, intro??]:
- "[| is_subspace U V; x:U |] ==> a (*) x : U"
+ "[| is_subspace U V; x \<in> U |] ==> a \<cdot> x \<in> 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"
+ "[| is_subspace U V; is_vectorspace V; x \<in> U; y \<in> U |]
+ ==> x - y \<in> U"
by (simp! add: diff_eq1 negate_eq1)
text {* Similar as for linear spaces, the existence of the
@@ -56,23 +56,23 @@
of the carrier set and by vector space laws.*}
lemma zero_in_subspace [intro??]:
- "[| is_subspace U V; is_vectorspace V |] ==> 00 : U"
+ "[| is_subspace U V; is_vectorspace V |] ==> 0 \<in> U"
proof -
assume "is_subspace U V" and v: "is_vectorspace V"
- have "U ~= {}" ..
- hence "EX x. x:U" by force
+ have "U \<noteq> {}" ..
+ hence "\<exists>x. x \<in> 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)
+ fix x assume u: "x \<in> U"
+ hence "x \<in> V" by (simp!)
+ with v have "0 = x - x" by (simp!)
+ also have "... \<in> 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"
+ "[| is_subspace U V; is_vectorspace V; x \<in> U |] ==> - x \<in> U"
by (simp add: negate_eq1)
text_raw {* \medskip *}
@@ -84,11 +84,11 @@
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"
+ show "0 \<in> U" ..
+ show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" by (simp!)
+ show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" by (simp!)
+ show "\<forall>x \<in> U. - x = -#1 \<cdot> x" by (simp! add: negate_eq1)
+ show "\<forall>x \<in> U. \<forall>y \<in> U. x - y = x + - y"
by (simp! add: diff_eq1)
qed (simp! add: vs_add_mult_distrib1 vs_add_mult_distrib2)+
qed
@@ -98,10 +98,10 @@
lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V"
proof
assume "is_vectorspace V"
- show "00 : V" ..
+ show "0 \<in> 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!)
+ show "\<forall>x \<in> V. \<forall>y \<in> V. x + y \<in> V" by (simp!)
+ show "\<forall>x \<in> V. \<forall>a. a \<cdot> x \<in> V" by (simp!)
qed
text {* The subspace relation is transitive. *}
@@ -111,22 +111,22 @@
==> is_subspace U W"
proof
assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
- show "00 : U" ..
+ show "0 \<in> U" ..
have "U <= V" ..
also have "V <= W" ..
finally show "U <= W" .
- show "ALL x:U. ALL y:U. x + y : U"
+ show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
proof (intro ballI)
- fix x y assume "x:U" "y:U"
- show "x + y : U" by (simp!)
+ fix x y assume "x \<in> U" "y \<in> U"
+ show "x + y \<in> U" by (simp!)
qed
- show "ALL x:U. ALL a. a (*) x : U"
+ show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
proof (intro ballI allI)
- fix x a assume "x:U"
- show "a (*) x : U" by (simp!)
+ fix x a assume "x \<in> U"
+ show "a \<cdot> x \<in> U" by (simp!)
qed
qed
@@ -138,60 +138,60 @@
scalar multiples of $x$. *}
constdefs
- lin :: "'a => 'a set"
- "lin x == {a (*) x | a. True}"
+ lin :: "('a::{minus,plus,zero}) => 'a set"
+ "lin x == {a \<cdot> x | a. True}"
-lemma linD: "x : lin v = (EX a::real. x = a (*) v)"
+lemma linD: "x \<in> lin v = (\<exists>a::real. x = a \<cdot> v)"
by (unfold lin_def) fast
-lemma linI [intro??]: "a (*) x0 : lin x0"
+lemma linI [intro??]: "a \<cdot> x0 \<in> lin x0"
by (unfold lin_def) fast
text {* Every vector is contained in its linear closure. *}
-lemma x_lin_x: "[| is_vectorspace V; x:V |] ==> x : lin x"
+lemma x_lin_x: "[| is_vectorspace V; x \<in> V |] ==> x \<in> lin x"
proof (unfold lin_def, intro CollectI exI conjI)
- assume "is_vectorspace V" "x:V"
- show "x = #1 (*) x" by (simp!)
+ assume "is_vectorspace V" "x \<in> V"
+ show "x = #1 \<cdot> x" by (simp!)
qed simp
text {* Any linear closure is a subspace. *}
lemma lin_subspace [intro??]:
- "[| is_vectorspace V; x:V |] ==> is_subspace (lin x) V"
+ "[| is_vectorspace V; x \<in> V |] ==> is_subspace (lin x) V"
proof
- assume "is_vectorspace V" "x:V"
- show "00 : lin x"
+ assume "is_vectorspace V" "x \<in> V"
+ show "0 \<in> lin x"
proof (unfold lin_def, intro CollectI exI conjI)
- show "00 = (#0::real) (*) x" by (simp!)
+ show "0 = (#0::real) \<cdot> 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!)
+ fix xa a assume "xa = a \<cdot> x"
+ show "xa \<in> V" by (simp!)
qed
- show "ALL x1 : lin x. ALL x2 : lin x. x1 + x2 : lin x"
+ show "\<forall>x1 \<in> lin x. \<forall>x2 \<in> lin x. x1 + x2 \<in> lin x"
proof (intro ballI)
- fix x1 x2 assume "x1 : lin x" "x2 : lin x"
- thus "x1 + x2 : lin x"
+ fix x1 x2 assume "x1 \<in> lin x" "x2 \<in> lin x"
+ thus "x1 + x2 \<in> 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"
+ fix a1 a2 assume "x1 = a1 \<cdot> x" "x2 = a2 \<cdot> x"
+ show "x1 + x2 = (a1 + a2) \<cdot> x"
by (simp! add: vs_add_mult_distrib2)
qed simp
qed
- show "ALL xa:lin x. ALL a. a (*) xa : lin x"
+ show "\<forall>xa \<in> lin x. \<forall>a. a \<cdot> xa \<in> lin x"
proof (intro ballI allI)
- fix x1 a assume "x1 : lin x"
- thus "a (*) x1 : lin x"
+ fix x1 a assume "x1 \<in> lin x"
+ thus "a \<cdot> x1 \<in> 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!)
+ fix a1 assume "x1 = a1 \<cdot> x"
+ show "a \<cdot> x1 = (a * a1) \<cdot> x" by (simp!)
qed simp
qed
qed
@@ -199,9 +199,9 @@
text {* Any linear closure is a vector space. *}
lemma lin_vs [intro??]:
- "[| is_vectorspace V; x:V |] ==> is_vectorspace (lin x)"
+ "[| is_vectorspace V; x \<in> V |] ==> is_vectorspace (lin x)"
proof (rule subspace_vs)
- assume "is_vectorspace V" "x:V"
+ assume "is_vectorspace V" "x \<in> V"
show "is_subspace (lin x) V" ..
qed
@@ -215,22 +215,22 @@
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 \<in> U \<and> v \<in> 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}";
+ "['a::{plus, minus, zero} set, 'a set] => 'a set" (infixl "+" 65)
+ "vs_sum U V == {x. \<exists>u \<in> U. \<exists>v \<in> V. x = u + v}";
***)
lemma vs_sumD:
- "x: U + V = (EX u:U. EX v:V. x = u + v)"
+ "x \<in> U + V = (\<exists>u \<in> U. \<exists>v \<in> V. x = u + v)"
by (unfold vs_sum_def) fast
lemmas vs_sumE = vs_sumD [RS iffD1, elimify]
lemma vs_sumI [intro??]:
- "[| x:U; y:V; t= x + y |] ==> t : U + V"
+ "[| x \<in> U; y \<in> V; t= x + y |] ==> t \<in> U + V"
by (unfold vs_sum_def) fast
text{* $U$ is a subspace of $U + V$. *}
@@ -240,20 +240,20 @@
==> is_subspace U (U + V)"
proof
assume "is_vectorspace U" "is_vectorspace V"
- show "00 : U" ..
+ show "0 \<in> 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!)
+ fix x assume "x \<in> U"
+ show "x = x + 0" by (simp!)
+ show "0 \<in> V" by (simp!)
qed
- show "ALL x:U. ALL y:U. x + y : U"
+ show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
proof (intro ballI)
- fix x y assume "x:U" "y:U" show "x + y : U" by (simp!)
+ fix x y assume "x \<in> U" "y \<in> U" show "x + y \<in> U" by (simp!)
qed
- show "ALL x:U. ALL a. a (*) x : U"
+ show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
proof (intro ballI allI)
- fix x a assume "x:U" show "a (*) x : U" by (simp!)
+ fix x a assume "x \<in> U" show "a \<cdot> x \<in> U" by (simp!)
qed
qed
@@ -264,38 +264,38 @@
==> is_subspace (U + V) E"
proof
assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
- show "00 : U + V"
+ show "0 \<in> U + V"
proof (intro vs_sumI)
- show "00 : U" ..
- show "00 : V" ..
- show "(00::'a) = 00 + 00" by (simp!)
+ show "0 \<in> U" ..
+ show "0 \<in> 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!)
+ fix x u v assume "u \<in> U" "v \<in> V" "x = u + v"
+ show "x \<in> E" by (simp!)
qed
- show "ALL x: U + V. ALL y: U + V. x + y : U + V"
+ show "\<forall>x \<in> U + V. \<forall>y \<in> U + V. x + y \<in> U + V"
proof (intro ballI)
- fix x y assume "x : U + V" "y : U + V"
- thus "x + y : U + V"
+ fix x y assume "x \<in> U + V" "y \<in> U + V"
+ thus "x + y \<in> 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"
+ assume "ux \<in> U" "vx \<in> V" "x = ux + vx"
+ and "uy \<in> U" "vy \<in> 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"
+ show "\<forall>x \<in> U + V. \<forall>a. a \<cdot> x \<in> U + V"
proof (intro ballI allI)
- fix x a assume "x : U + V"
- thus "a (*) x : U + V"
+ fix x a assume "x \<in> U + V"
+ thus "a \<cdot> x \<in> 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)"
+ fix a x u v assume "u \<in> U" "v \<in> V" "x = u + v"
+ show "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)"
by (simp! add: vs_add_mult_distrib1)
qed (simp!)+
qed
@@ -323,154 +323,154 @@
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"
+ U \<inter> V = {0}; u1 \<in> U; u2 \<in> U; v1 \<in> V; v2 \<in> V; u1 + v1 = u2 + v2 |]
+ ==> u1 = u2 \<and> 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"
+ "U \<inter> V = {0}" "u1 \<in> U" "u2 \<in> U" "v1 \<in> V" "v2 \<in> 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 \<in> U" by (simp!)
+ with eq have v': "v2 - v1 \<in> U" by simp
+ have v: "v2 - v1 \<in> V" by (simp!)
+ with eq have u': "u1 - u2 \<in> 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" ..
+ show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u'])
+ show "u1 \<in> E" ..
+ show "u2 \<in> 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" ..
+ show "v2 - v1 = 0" by (rule Int_singletonD [OF _ v' v])
+ show "v1 \<in> E" ..
+ show "v2 \<in> 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
+of the Hahn-Banach Theorem (see page \pageref{decomp-H-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. *}
-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"
+lemma decomp_H':
+ "[| is_vectorspace E; is_subspace H E; y1 \<in> H; y2 \<in> H;
+ x' \<notin> H; x' \<in> E; x' \<noteq> 0; y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x' |]
+ ==> y1 = y2 \<and> 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"
+ and "y1 \<in> H" "y2 \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
+ "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
- have c: "y1 = y2 & a1 (*) x0 = a2 (*) x0"
+ have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
proof (rule decomp)
- show "a1 (*) x0 : lin x0" ..
- show "a2 (*) x0 : lin x0" ..
- show "H Int (lin x0) = {00}"
+ show "a1 \<cdot> x' \<in> lin x'" ..
+ show "a2 \<cdot> x' \<in> lin x'" ..
+ show "H \<inter> (lin x') = {0}"
proof
- show "H Int lin x0 <= {00}"
+ show "H \<inter> lin x' <= {0}"
proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2])
- fix x assume "x:H" "x : lin x0"
- thus "x = 00"
+ fix x assume "x \<in> H" "x \<in> lin x'"
+ thus "x = 0"
proof (unfold lin_def, elim CollectE exE conjE)
- fix a assume "x = a (*) x0"
+ fix a assume "x = a \<cdot> x'"
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"
+ assume "a \<noteq> (#0::real)"
+ from h have "rinv a \<cdot> a \<cdot> x' \<in> H"
by (rule subspace_mult_closed) (simp!)
- also have "rinv a (*) a (*) x0 = x0" by (simp!)
- finally have "x0 : H" .
+ also have "rinv a \<cdot> a \<cdot> x' = x'" by (simp!)
+ finally have "x' \<in> H" .
thus ?thesis by contradiction
qed
qed
qed
- show "{00} <= H Int lin x0"
+ show "{0} <= H \<inter> lin x'"
proof -
- have "00: H Int lin x0"
+ have "0 \<in> H \<inter> lin x'"
proof (rule IntI)
- show "00:H" ..
- from lin_vs show "00 : lin x0" ..
+ show "0 \<in> H" ..
+ from lin_vs show "0 \<in> lin x'" ..
qed
thus ?thesis by simp
qed
qed
- show "is_subspace (lin x0) E" ..
+ show "is_subspace (lin x') 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
+ from c show "a1 \<cdot> x' = a2 \<cdot> x'" 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
+text {* Since for any element $y + a \mult x'$ of the direct sum
+of a vectorspace $H$ and the linear closure of $x'$ the components
$y\in H$ and $a$ are unique, it follows from $y\in H$ that
$a = 0$.*}
-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))"
+lemma decomp_H'_H:
+ "[| is_vectorspace E; is_subspace H E; t \<in> H; x' \<notin> H; x' \<in> E;
+ x' \<noteq> 0 |]
+ ==> (SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, (#0::real))"
proof (rule, unfold split_tupled_all)
- assume "is_vectorspace E" "is_subspace H E" "t:H" "x0 ~: H" "x0:E"
- "x0 ~= 00"
+ assume "is_vectorspace E" "is_subspace H E" "t \<in> H" "x' \<notin> H" "x' \<in> E"
+ "x' \<noteq> 0"
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!))+
+ fix y a presume t1: "t = y + a \<cdot> x'" and "y \<in> H"
+ have "y = t \<and> a = (#0::real)"
+ by (rule decomp_H') (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. *}
+text {* The components $y\in H$ and $a$ in $y \plus a \mult x'$
+are unique, so the function $h'$ defined by
+$h' (y \plus a \mult x') = h y + a \cdot \xi$ is definite. *}
-lemma h0_definite:
- "[| h0 == (\\<lambda>x. let (y, a) = SOME (y, a). (x = y + a (*) x0 & y:H)
+lemma h'_definite:
+ "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> 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"
+ x = y + a \<cdot> x'; is_vectorspace E; is_subspace H E;
+ y \<in> H; x' \<notin> H; x' \<in> E; x' \<noteq> 0 |]
+ ==> h' x = h y + a * xi"
proof -
assume
- "h0 == (\\<lambda>x. let (y, a) = SOME (y, a). (x = y + a (*) x0 & y:H)
+ "h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> 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)"
+ "x = y + a \<cdot> x'" "is_vectorspace E" "is_subspace H E"
+ "y \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
+ have "x \<in> H + (lin x')"
by (simp! add: vs_sum_def lin_def) force+
- have "EX! xa. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)"
+ have "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
proof
- show "EX xa. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)"
+ show "\<exists>xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> 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"
+ assume "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) xa"
+ "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) ya"
show "xa = ya"
proof -
- show "fst xa = fst ya & snd xa = snd ya ==> xa = ya"
+ show "fst xa = fst ya \<and> snd xa = snd ya ==> xa = ya"
by (simp add: Pair_fst_snd_eq)
- have x: "x = fst xa + snd xa (*) x0 & fst xa : H"
+ have x: "x = fst xa + snd xa \<cdot> x' \<and> fst xa \<in> H"
by (force!)
- have y: "x = fst ya + snd ya (*) x0 & fst ya : H"
+ have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H"
by (force!)
- from x y show "fst xa = fst ya & snd xa = snd ya"
- by (elim conjE) (rule decomp_H0, (simp!)+)
+ from x y show "fst xa = fst ya \<and> snd xa = snd ya"
+ by (elim conjE) (rule decomp_H', (simp!)+)
qed
qed
- hence eq: "(SOME (y, a). x = y + a (*) x0 & y:H) = (y, a)"
+ hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
by (rule select1_equality) (force!)
- thus "h0 x = h y + a * xi" by (simp! add: Let_def)
+ thus "h' x = h y + a * xi" by (simp! add: Let_def)
qed
end
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