(*  Title:      HOL/Hahn_Banach/Subspace.thy

    Author:     Gertrud Bauer, TU Munich

*)

header {* Subspaces *}

theory Subspace

imports Vector_Space

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 =

  fixes U :: "'a\<Colon>{minus, plus, zero, uminus} set" and V

  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 V: vectorspace V by fact

  have "U \<noteq> {}" by (rule 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 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