--- a/src/HOL/Real/HahnBanach/Subspace.thy Mon Dec 29 13:23:53 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,514 +0,0 @@
-(* Title: HOL/Real/HahnBanach/Subspace.thy
- ID: $Id$
- Author: Gertrud Bauer, TU Munich
-*)
-
-header {* Subspaces *}
-
-theory Subspace
-imports VectorSpace
-begin
-
-subsection {* Definition *}
-
-text {*
- A non-empty subset @{text U} of a vector space @{text V} is a
- \emph{subspace} of @{text V}, iff @{text U} is closed under addition
- and scalar multiplication.
-*}
-
-locale subspace = var U + var V +
- constrains U :: "'a\<Colon>{minus, plus, zero, uminus} set"
- assumes non_empty [iff, intro]: "U \<noteq> {}"
- and subset [iff]: "U \<subseteq> V"
- and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"
- and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"
-
-notation (symbols)
- subspace (infix "\<unlhd>" 50)
-
-declare vectorspace.intro [intro?] subspace.intro [intro?]
-
-lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V"
- by (rule subspace.subset)
-
-lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V"
- using subset by blast
-
-lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
- by (rule subspace.subsetD)
-
-lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V"
- by (rule subspace.subsetD)
-
-lemma (in subspace) diff_closed [iff]:
- assumes "vectorspace V"
- assumes x: "x \<in> U" and y: "y \<in> U"
- shows "x - y \<in> U"
-proof -
- interpret vectorspace [V] by fact
- from x y show ?thesis by (simp add: diff_eq1 negate_eq1)
-qed
-
-text {*
- \medskip 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.
-*}
-
-lemma (in subspace) zero [intro]:
- assumes "vectorspace V"
- shows "0 \<in> U"
-proof -
- interpret vectorspace [V] by fact
- have "U \<noteq> {}" by (rule U_V.non_empty)
- then obtain x where x: "x \<in> U" by blast
- then have "x \<in> V" .. then have "0 = x - x" by simp
- also from `vectorspace V` x x have "\<dots> \<in> U" by (rule U_V.diff_closed)
- finally show ?thesis .
-qed
-
-lemma (in subspace) neg_closed [iff]:
- assumes "vectorspace V"
- assumes x: "x \<in> U"
- shows "- x \<in> U"
-proof -
- interpret vectorspace [V] by fact
- from x show ?thesis by (simp add: negate_eq1)
-qed
-
-text {* \medskip Further derived laws: every subspace is a vector space. *}
-
-lemma (in subspace) vectorspace [iff]:
- assumes "vectorspace V"
- shows "vectorspace U"
-proof -
- interpret vectorspace [V] by fact
- show ?thesis
- proof
- show "U \<noteq> {}" ..
- fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U"
- fix a b :: real
- from x y show "x + y \<in> U" by simp
- from x show "a \<cdot> x \<in> U" by simp
- from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac)
- from x y show "x + y = y + x" by (simp add: add_ac)
- from x show "x - x = 0" by simp
- from x show "0 + x = x" by simp
- from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib)
- from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib)
- from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc)
- from x show "1 \<cdot> x = x" by simp
- from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1)
- from x y show "x - y = x + - y" by (simp add: diff_eq1)
- qed
-qed
-
-
-text {* The subspace relation is reflexive. *}
-
-lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V"
-proof
- show "V \<noteq> {}" ..
- show "V \<subseteq> V" ..
- fix x y assume x: "x \<in> V" and y: "y \<in> V"
- fix a :: real
- from x y show "x + y \<in> V" by simp
- from x show "a \<cdot> x \<in> V" by simp
-qed
-
-text {* The subspace relation is transitive. *}
-
-lemma (in vectorspace) subspace_trans [trans]:
- "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W"
-proof
- assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W"
- from uv show "U \<noteq> {}" by (rule subspace.non_empty)
- show "U \<subseteq> W"
- proof -
- from uv have "U \<subseteq> V" by (rule subspace.subset)
- also from vw have "V \<subseteq> W" by (rule subspace.subset)
- finally show ?thesis .
- qed
- fix x y assume x: "x \<in> U" and y: "y \<in> U"
- from uv and x y show "x + y \<in> U" by (rule subspace.add_closed)
- from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed)
-qed
-
-
-subsection {* Linear closure *}
-
-text {*
- The \emph{linear closure} of a vector @{text x} is the set of all
- scalar multiples of @{text x}.
-*}
-
-definition
- lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set" where
- "lin x = {a \<cdot> x | a. True}"
-
-lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"
- unfolding lin_def by blast
-
-lemma linI' [iff]: "a \<cdot> x \<in> lin x"
- unfolding lin_def by blast
-
-lemma linE [elim]: "x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C"
- unfolding lin_def by blast
-
-
-text {* Every vector is contained in its linear closure. *}
-
-lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x"
-proof -
- assume "x \<in> V"
- then have "x = 1 \<cdot> x" by simp
- also have "\<dots> \<in> lin x" ..
- finally show ?thesis .
-qed
-
-lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x"
-proof
- assume "x \<in> V"
- then show "0 = 0 \<cdot> x" by simp
-qed
-
-text {* Any linear closure is a subspace. *}
-
-lemma (in vectorspace) lin_subspace [intro]:
- "x \<in> V \<Longrightarrow> lin x \<unlhd> V"
-proof
- assume x: "x \<in> V"
- then show "lin x \<noteq> {}" by (auto simp add: x_lin_x)
- show "lin x \<subseteq> V"
- proof
- fix x' assume "x' \<in> lin x"
- then obtain a where "x' = a \<cdot> x" ..
- with x show "x' \<in> V" by simp
- qed
- fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x"
- show "x' + x'' \<in> lin x"
- proof -
- from x' obtain a' where "x' = a' \<cdot> x" ..
- moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" ..
- ultimately have "x' + x'' = (a' + a'') \<cdot> x"
- using x by (simp add: distrib)
- also have "\<dots> \<in> lin x" ..
- finally show ?thesis .
- qed
- fix a :: real
- show "a \<cdot> x' \<in> lin x"
- proof -
- from x' obtain a' where "x' = a' \<cdot> x" ..
- with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc)
- also have "\<dots> \<in> lin x" ..
- finally show ?thesis .
- qed
-qed
-
-
-text {* Any linear closure is a vector space. *}
-
-lemma (in vectorspace) lin_vectorspace [intro]:
- assumes "x \<in> V"
- shows "vectorspace (lin x)"
-proof -
- from `x \<in> V` have "subspace (lin x) V"
- by (rule lin_subspace)
- from this and vectorspace_axioms show ?thesis
- by (rule subspace.vectorspace)
-qed
-
-
-subsection {* Sum of two vectorspaces *}
-
-text {*
- The \emph{sum} of two vectorspaces @{text U} and @{text V} is the
- set of all sums of elements from @{text U} and @{text V}.
-*}
-
-instantiation "fun" :: (type, type) plus
-begin
-
-definition
- sum_def: "plus_fun U V = {u + v | u v. u \<in> U \<and> v \<in> V}" (* FIXME not fully general!? *)
-
-instance ..
-
-end
-
-lemma sumE [elim]:
- "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
- unfolding sum_def by blast
-
-lemma sumI [intro]:
- "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"
- unfolding sum_def by blast
-
-lemma sumI' [intro]:
- "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"
- unfolding sum_def by blast
-
-text {* @{text U} is a subspace of @{text "U + V"}. *}
-
-lemma subspace_sum1 [iff]:
- assumes "vectorspace U" "vectorspace V"
- shows "U \<unlhd> U + V"
-proof -
- interpret vectorspace [U] by fact
- interpret vectorspace [V] by fact
- show ?thesis
- proof
- show "U \<noteq> {}" ..
- show "U \<subseteq> U + V"
- proof
- fix x assume x: "x \<in> U"
- moreover have "0 \<in> V" ..
- ultimately have "x + 0 \<in> U + V" ..
- with x show "x \<in> U + V" by simp
- qed
- fix x y assume x: "x \<in> U" and "y \<in> U"
- then show "x + y \<in> U" by simp
- from x show "\<And>a. a \<cdot> x \<in> U" by simp
- qed
-qed
-
-text {* The sum of two subspaces is again a subspace. *}
-
-lemma sum_subspace [intro?]:
- assumes "subspace U E" "vectorspace E" "subspace V E"
- shows "U + V \<unlhd> E"
-proof -
- interpret subspace [U E] by fact
- interpret vectorspace [E] by fact
- interpret subspace [V E] by fact
- show ?thesis
- proof
- have "0 \<in> U + V"
- proof
- show "0 \<in> U" using `vectorspace E` ..
- show "0 \<in> V" using `vectorspace E` ..
- show "(0::'a) = 0 + 0" by simp
- qed
- then show "U + V \<noteq> {}" by blast
- show "U + V \<subseteq> E"
- proof
- fix x assume "x \<in> U + V"
- then obtain u v where "x = u + v" and
- "u \<in> U" and "v \<in> V" ..
- then show "x \<in> E" by simp
- qed
- fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"
- show "x + y \<in> U + V"
- proof -
- from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..
- moreover
- from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" ..
- ultimately
- have "ux + uy \<in> U"
- and "vx + vy \<in> V"
- and "x + y = (ux + uy) + (vx + vy)"
- using x y by (simp_all add: add_ac)
- then show ?thesis ..
- qed
- fix a show "a \<cdot> x \<in> U + V"
- proof -
- from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..
- then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"
- and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib)
- then show ?thesis ..
- qed
- qed
-qed
-
-text{* The sum of two subspaces is a vectorspace. *}
-
-lemma sum_vs [intro?]:
- "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"
- by (rule subspace.vectorspace) (rule sum_subspace)
-
-
-subsection {* Direct sums *}
-
-text {*
- The sum of @{text U} and @{text V} is called \emph{direct}, iff the
- zero element is the only common element of @{text U} and @{text
- V}. For every element @{text x} of the direct sum of @{text U} and
- @{text V} the decomposition in @{text "x = u + v"} with
- @{text "u \<in> U"} and @{text "v \<in> V"} is unique.
-*}
-
-lemma decomp:
- assumes "vectorspace E" "subspace U E" "subspace V E"
- assumes direct: "U \<inter> V = {0}"
- and u1: "u1 \<in> U" and u2: "u2 \<in> U"
- and v1: "v1 \<in> V" and v2: "v2 \<in> V"
- and sum: "u1 + v1 = u2 + v2"
- shows "u1 = u2 \<and> v1 = v2"
-proof -
- interpret vectorspace [E] by fact
- interpret subspace [U E] by fact
- interpret subspace [V E] by fact
- show ?thesis
- proof
- have U: "vectorspace U" (* FIXME: use interpret *)
- using `subspace U E` `vectorspace E` by (rule subspace.vectorspace)
- have V: "vectorspace V"
- using `subspace V E` `vectorspace E` by (rule subspace.vectorspace)
- from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"
- by (simp add: add_diff_swap)
- from u1 u2 have u: "u1 - u2 \<in> U"
- by (rule vectorspace.diff_closed [OF U])
- with eq have v': "v2 - v1 \<in> U" by (simp only:)
- from v2 v1 have v: "v2 - v1 \<in> V"
- by (rule vectorspace.diff_closed [OF V])
- with eq have u': " u1 - u2 \<in> V" by (simp only:)
-
- show "u1 = u2"
- proof (rule add_minus_eq)
- from u1 show "u1 \<in> E" ..
- from u2 show "u2 \<in> E" ..
- from u u' and direct show "u1 - u2 = 0" by blast
- qed
- show "v1 = v2"
- proof (rule add_minus_eq [symmetric])
- from v1 show "v1 \<in> E" ..
- from v2 show "v2 \<in> E" ..
- from v v' and direct show "v2 - v1 = 0" by blast
- qed
- qed
-qed
-
-text {*
- An application of the previous lemma will be used in the proof of
- the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any
- element @{text "y + a \<cdot> x\<^sub>0"} of the direct sum of a
- vectorspace @{text H} and the linear closure of @{text "x\<^sub>0"}
- the components @{text "y \<in> H"} and @{text a} are uniquely
- determined.
-*}
-
-lemma decomp_H':
- assumes "vectorspace E" "subspace H E"
- assumes y1: "y1 \<in> H" and y2: "y2 \<in> H"
- and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
- and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
- shows "y1 = y2 \<and> a1 = a2"
-proof -
- interpret vectorspace [E] by fact
- interpret subspace [H E] by fact
- show ?thesis
- proof
- have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
- proof (rule decomp)
- show "a1 \<cdot> x' \<in> lin x'" ..
- show "a2 \<cdot> x' \<in> lin x'" ..
- show "H \<inter> lin x' = {0}"
- proof
- show "H \<inter> lin x' \<subseteq> {0}"
- proof
- fix x assume x: "x \<in> H \<inter> lin x'"
- then obtain a where xx': "x = a \<cdot> x'"
- by blast
- have "x = 0"
- proof cases
- assume "a = 0"
- with xx' and x' show ?thesis by simp
- next
- assume a: "a \<noteq> 0"
- from x have "x \<in> H" ..
- with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp
- with a and x' have "x' \<in> H" by (simp add: mult_assoc2)
- with `x' \<notin> H` show ?thesis by contradiction
- qed
- then show "x \<in> {0}" ..
- qed
- show "{0} \<subseteq> H \<inter> lin x'"
- proof -
- have "0 \<in> H" using `vectorspace E` ..
- moreover have "0 \<in> lin x'" using `x' \<in> E` ..
- ultimately show ?thesis by blast
- qed
- qed
- show "lin x' \<unlhd> E" using `x' \<in> E` ..
- qed (rule `vectorspace E`, rule `subspace H E`, rule y1, rule y2, rule eq)
- then show "y1 = y2" ..
- from c have "a1 \<cdot> x' = a2 \<cdot> x'" ..
- with x' show "a1 = a2" by (simp add: mult_right_cancel)
- qed
-qed
-
-text {*
- Since for any element @{text "y + a \<cdot> x'"} of the direct sum of a
- vectorspace @{text H} and the linear closure of @{text x'} the
- components @{text "y \<in> H"} and @{text a} are unique, it follows from
- @{text "y \<in> H"} that @{text "a = 0"}.
-*}
-
-lemma decomp_H'_H:
- assumes "vectorspace E" "subspace H E"
- assumes t: "t \<in> H"
- and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
- shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
-proof -
- interpret vectorspace [E] by fact
- interpret subspace [H E] by fact
- show ?thesis
- proof (rule, simp_all only: split_paired_all split_conv)
- from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp
- fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H"
- have "y = t \<and> a = 0"
- proof (rule decomp_H')
- from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp
- from ya show "y \<in> H" ..
- qed (rule `vectorspace E`, rule `subspace H E`, rule t, (rule x')+)
- with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp
- qed
-qed
-
-text {*
- The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"}
- are unique, so the function @{text h'} defined by
- @{text "h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>"} is definite.
-*}
-
-lemma h'_definite:
- fixes H
- assumes h'_def:
- "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
- in (h y) + a * xi)"
- and x: "x = y + a \<cdot> x'"
- assumes "vectorspace E" "subspace H E"
- assumes y: "y \<in> H"
- and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
- shows "h' x = h y + a * xi"
-proof -
- interpret vectorspace [E] by fact
- interpret subspace [H E] by fact
- from x y x' have "x \<in> H + lin x'" by auto
- have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")
- proof (rule ex_ex1I)
- from x y show "\<exists>p. ?P p" by blast
- fix p q assume p: "?P p" and q: "?P q"
- show "p = q"
- proof -
- from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H"
- by (cases p) simp
- from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H"
- by (cases q) simp
- have "fst p = fst q \<and> snd p = snd q"
- proof (rule decomp_H')
- from xp show "fst p \<in> H" ..
- from xq show "fst q \<in> H" ..
- from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'"
- by simp
- qed (rule `vectorspace E`, rule `subspace H E`, (rule x')+)
- then show ?thesis by (cases p, cases q) simp
- qed
- qed
- then have eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
- by (rule some1_equality) (simp add: x y)
- with h'_def show "h' x = h y + a * xi" by (simp add: Let_def)
-qed
-
-end