wenzelm@31795: (*  Title:      HOL/Hahn_Banach/Subspace.thy
wenzelm@7566:     Author:     Gertrud Bauer, TU Munich
wenzelm@7566: *)
wenzelm@7535: 
wenzelm@9035: header {* Subspaces *}
wenzelm@7808: 
wenzelm@27612: theory Subspace
huffman@44190: imports Vector_Space "~~/src/HOL/Library/Set_Algebras"
wenzelm@27612: begin
wenzelm@7535: 
wenzelm@9035: subsection {* Definition *}
wenzelm@7535: 
wenzelm@10687: text {*
wenzelm@10687:   A non-empty subset @{text U} of a vector space @{text V} is a
wenzelm@10687:   \emph{subspace} of @{text V}, iff @{text U} is closed under addition
wenzelm@10687:   and scalar multiplication.
wenzelm@10687: *}
wenzelm@7917: 
ballarin@29234: locale subspace =
ballarin@29234:   fixes U :: "'a\<Colon>{minus, plus, zero, uminus} set" and V
wenzelm@13515:   assumes non_empty [iff, intro]: "U \<noteq> {}"
wenzelm@13515:     and subset [iff]: "U \<subseteq> V"
wenzelm@13515:     and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"
wenzelm@13515:     and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"
wenzelm@7535: 
wenzelm@21210: notation (symbols)
wenzelm@19736:   subspace  (infix "\<unlhd>" 50)
ballarin@14254: 
wenzelm@19736: declare vectorspace.intro [intro?] subspace.intro [intro?]
wenzelm@7535: 
wenzelm@13515: lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V"
wenzelm@13515:   by (rule subspace.subset)
wenzelm@7566: 
wenzelm@13515: lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V"
wenzelm@13515:   using subset by blast
wenzelm@7566: 
wenzelm@13515: lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
wenzelm@13515:   by (rule subspace.subsetD)
wenzelm@7535: 
wenzelm@13515: lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V"
wenzelm@13515:   by (rule subspace.subsetD)
wenzelm@13515: 
wenzelm@13547: lemma (in subspace) diff_closed [iff]:
ballarin@27611:   assumes "vectorspace V"
wenzelm@27612:   assumes x: "x \<in> U" and y: "y \<in> U"
wenzelm@27612:   shows "x - y \<in> U"
ballarin@27611: proof -
ballarin@29234:   interpret vectorspace V by fact
wenzelm@27612:   from x y show ?thesis by (simp add: diff_eq1 negate_eq1)
ballarin@27611: qed
wenzelm@7917: 
wenzelm@13515: text {*
wenzelm@13515:   \medskip Similar as for linear spaces, the existence of the zero
wenzelm@13515:   element in every subspace follows from the non-emptiness of the
wenzelm@13515:   carrier set and by vector space laws.
wenzelm@13515: *}
wenzelm@13515: 
wenzelm@13547: lemma (in subspace) zero [intro]:
ballarin@27611:   assumes "vectorspace V"
wenzelm@13547:   shows "0 \<in> U"
wenzelm@10687: proof -
wenzelm@30729:   interpret V: vectorspace V by fact
ballarin@29234:   have "U \<noteq> {}" by (rule non_empty)
wenzelm@13515:   then obtain x where x: "x \<in> U" by blast
wenzelm@27612:   then have "x \<in> V" .. then have "0 = x - x" by simp
ballarin@29234:   also from `vectorspace V` x x have "\<dots> \<in> U" by (rule diff_closed)
wenzelm@13515:   finally show ?thesis .
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@13547: lemma (in subspace) neg_closed [iff]:
ballarin@27611:   assumes "vectorspace V"
wenzelm@27612:   assumes x: "x \<in> U"
wenzelm@27612:   shows "- x \<in> U"
ballarin@27611: proof -
ballarin@29234:   interpret vectorspace V by fact
wenzelm@27612:   from x show ?thesis by (simp add: negate_eq1)
ballarin@27611: qed
wenzelm@13515: 
wenzelm@10687: text {* \medskip Further derived laws: every subspace is a vector space. *}
wenzelm@7535: 
wenzelm@13547: lemma (in subspace) vectorspace [iff]:
ballarin@27611:   assumes "vectorspace V"
wenzelm@13547:   shows "vectorspace U"
ballarin@27611: proof -
ballarin@29234:   interpret vectorspace V by fact
wenzelm@27612:   show ?thesis
wenzelm@27612:   proof
ballarin@27611:     show "U \<noteq> {}" ..
ballarin@27611:     fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U"
ballarin@27611:     fix a b :: real
ballarin@27611:     from x y show "x + y \<in> U" by simp
ballarin@27611:     from x show "a \<cdot> x \<in> U" by simp
ballarin@27611:     from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac)
ballarin@27611:     from x y show "x + y = y + x" by (simp add: add_ac)
ballarin@27611:     from x show "x - x = 0" by simp
ballarin@27611:     from x show "0 + x = x" by simp
ballarin@27611:     from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib)
ballarin@27611:     from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib)
ballarin@27611:     from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc)
ballarin@27611:     from x show "1 \<cdot> x = x" by simp
ballarin@27611:     from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1)
ballarin@27611:     from x y show "x - y = x + - y" by (simp add: diff_eq1)
ballarin@27611:   qed
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@13515: 
wenzelm@9035: text {* The subspace relation is reflexive. *}
wenzelm@7917: 
wenzelm@13515: lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V"
wenzelm@10687: proof
wenzelm@13515:   show "V \<noteq> {}" ..
wenzelm@10687:   show "V \<subseteq> V" ..
wenzelm@44887: next
wenzelm@13515:   fix x y assume x: "x \<in> V" and y: "y \<in> V"
wenzelm@13515:   fix a :: real
wenzelm@13515:   from x y show "x + y \<in> V" by simp
wenzelm@13515:   from x show "a \<cdot> x \<in> V" by simp
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@9035: text {* The subspace relation is transitive. *}
wenzelm@7917: 
wenzelm@13515: lemma (in vectorspace) subspace_trans [trans]:
wenzelm@13515:   "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W"
wenzelm@10687: proof
wenzelm@13515:   assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W"
wenzelm@13515:   from uv show "U \<noteq> {}" by (rule subspace.non_empty)
wenzelm@13515:   show "U \<subseteq> W"
wenzelm@13515:   proof -
wenzelm@13515:     from uv have "U \<subseteq> V" by (rule subspace.subset)
wenzelm@13515:     also from vw have "V \<subseteq> W" by (rule subspace.subset)
wenzelm@13515:     finally show ?thesis .
wenzelm@9035:   qed
wenzelm@13515:   fix x y assume x: "x \<in> U" and y: "y \<in> U"
wenzelm@13515:   from uv and x y show "x + y \<in> U" by (rule subspace.add_closed)
wenzelm@13515:   from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed)
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@7535: 
wenzelm@9035: subsection {* Linear closure *}
wenzelm@7808: 
wenzelm@10687: text {*
wenzelm@10687:   The \emph{linear closure} of a vector @{text x} is the set of all
wenzelm@10687:   scalar multiples of @{text x}.
wenzelm@10687: *}
wenzelm@7535: 
wenzelm@44887: definition lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set"
wenzelm@44887:   where "lin x = {a \<cdot> x | a. True}"
wenzelm@7535: 
wenzelm@13515: lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"
wenzelm@27612:   unfolding lin_def by blast
wenzelm@7535: 
wenzelm@13515: lemma linI' [iff]: "a \<cdot> x \<in> lin x"
wenzelm@27612:   unfolding lin_def by blast
wenzelm@13515: 
wenzelm@27612: lemma linE [elim]: "x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C"
wenzelm@27612:   unfolding lin_def by blast
wenzelm@13515: 
wenzelm@7656: 
wenzelm@9035: text {* Every vector is contained in its linear closure. *}
wenzelm@7917: 
wenzelm@13515: lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x"
wenzelm@13515: proof -
wenzelm@13515:   assume "x \<in> V"
wenzelm@27612:   then have "x = 1 \<cdot> x" by simp
wenzelm@13515:   also have "\<dots> \<in> lin x" ..
wenzelm@13515:   finally show ?thesis .
wenzelm@13515: qed
wenzelm@13515: 
wenzelm@13515: lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x"
wenzelm@13515: proof
wenzelm@13515:   assume "x \<in> V"
wenzelm@27612:   then show "0 = 0 \<cdot> x" by simp
wenzelm@13515: qed
wenzelm@7535: 
wenzelm@9035: text {* Any linear closure is a subspace. *}
wenzelm@7917: 
wenzelm@13515: lemma (in vectorspace) lin_subspace [intro]:
wenzelm@44887:   assumes x: "x \<in> V"
wenzelm@44887:   shows "lin x \<unlhd> V"
wenzelm@9035: proof
wenzelm@44887:   from x show "lin x \<noteq> {}" by auto
wenzelm@44887: next
wenzelm@10687:   show "lin x \<subseteq> V"
wenzelm@13515:   proof
wenzelm@13515:     fix x' assume "x' \<in> lin x"
wenzelm@13515:     then obtain a where "x' = a \<cdot> x" ..
wenzelm@13515:     with x show "x' \<in> V" by simp
wenzelm@9035:   qed
wenzelm@44887: next
wenzelm@13515:   fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x"
wenzelm@13515:   show "x' + x'' \<in> lin x"
wenzelm@13515:   proof -
wenzelm@13515:     from x' obtain a' where "x' = a' \<cdot> x" ..
wenzelm@13515:     moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" ..
wenzelm@13515:     ultimately have "x' + x'' = (a' + a'') \<cdot> x"
wenzelm@13515:       using x by (simp add: distrib)
wenzelm@13515:     also have "\<dots> \<in> lin x" ..
wenzelm@13515:     finally show ?thesis .
wenzelm@9035:   qed
wenzelm@13515:   fix a :: real
wenzelm@13515:   show "a \<cdot> x' \<in> lin x"
wenzelm@13515:   proof -
wenzelm@13515:     from x' obtain a' where "x' = a' \<cdot> x" ..
wenzelm@13515:     with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc)
wenzelm@13515:     also have "\<dots> \<in> lin x" ..
wenzelm@13515:     finally show ?thesis .
wenzelm@10687:   qed
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@13515: 
wenzelm@9035: text {* Any linear closure is a vector space. *}
wenzelm@7917: 
wenzelm@13515: lemma (in vectorspace) lin_vectorspace [intro]:
wenzelm@23378:   assumes "x \<in> V"
wenzelm@23378:   shows "vectorspace (lin x)"
wenzelm@23378: proof -
wenzelm@23378:   from `x \<in> V` have "subspace (lin x) V"
wenzelm@23378:     by (rule lin_subspace)
wenzelm@26199:   from this and vectorspace_axioms show ?thesis
wenzelm@23378:     by (rule subspace.vectorspace)
wenzelm@23378: qed
wenzelm@7808: 
wenzelm@7808: 
wenzelm@9035: subsection {* Sum of two vectorspaces *}
wenzelm@7808: 
wenzelm@10687: text {*
wenzelm@10687:   The \emph{sum} of two vectorspaces @{text U} and @{text V} is the
wenzelm@10687:   set of all sums of elements from @{text U} and @{text V}.
wenzelm@10687: *}
wenzelm@7535: 
krauss@47445: lemma sum_def: "U + V = {u + v | u v. u \<in> U \<and> v \<in> V}"
huffman@44190:   unfolding set_plus_def by auto
wenzelm@7917: 
wenzelm@13515: lemma sumE [elim]:
krauss@47445:     "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
wenzelm@27612:   unfolding sum_def by blast
wenzelm@7535: 
wenzelm@13515: lemma sumI [intro]:
krauss@47445:     "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"
wenzelm@27612:   unfolding sum_def by blast
wenzelm@7566: 
wenzelm@13515: lemma sumI' [intro]:
krauss@47445:     "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"
wenzelm@27612:   unfolding sum_def by blast
wenzelm@7917: 
krauss@47445: text {* @{text U} is a subspace of @{text "U + V"}. *}
wenzelm@7535: 
wenzelm@13515: lemma subspace_sum1 [iff]:
ballarin@27611:   assumes "vectorspace U" "vectorspace V"
krauss@47445:   shows "U \<unlhd> U + V"
ballarin@27611: proof -
ballarin@29234:   interpret vectorspace U by fact
ballarin@29234:   interpret vectorspace V by fact
wenzelm@27612:   show ?thesis
wenzelm@27612:   proof
ballarin@27611:     show "U \<noteq> {}" ..
krauss@47445:     show "U \<subseteq> U + V"
ballarin@27611:     proof
ballarin@27611:       fix x assume x: "x \<in> U"
ballarin@27611:       moreover have "0 \<in> V" ..
krauss@47445:       ultimately have "x + 0 \<in> U + V" ..
krauss@47445:       with x show "x \<in> U + V" by simp
ballarin@27611:     qed
ballarin@27611:     fix x y assume x: "x \<in> U" and "y \<in> U"
wenzelm@27612:     then show "x + y \<in> U" by simp
ballarin@27611:     from x show "\<And>a. a \<cdot> x \<in> U" by simp
wenzelm@9035:   qed
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@13515: text {* The sum of two subspaces is again a subspace. *}
wenzelm@7917: 
wenzelm@13515: lemma sum_subspace [intro?]:
ballarin@27611:   assumes "subspace U E" "vectorspace E" "subspace V E"
krauss@47445:   shows "U + V \<unlhd> E"
ballarin@27611: proof -
ballarin@29234:   interpret subspace U E by fact
ballarin@29234:   interpret vectorspace E by fact
ballarin@29234:   interpret subspace V E by fact
wenzelm@27612:   show ?thesis
wenzelm@27612:   proof
krauss@47445:     have "0 \<in> U + V"
ballarin@27611:     proof
ballarin@27611:       show "0 \<in> U" using `vectorspace E` ..
ballarin@27611:       show "0 \<in> V" using `vectorspace E` ..
ballarin@27611:       show "(0::'a) = 0 + 0" by simp
ballarin@27611:     qed
krauss@47445:     then show "U + V \<noteq> {}" by blast
krauss@47445:     show "U + V \<subseteq> E"
ballarin@27611:     proof
krauss@47445:       fix x assume "x \<in> U + V"
ballarin@27611:       then obtain u v where "x = u + v" and
wenzelm@32960:         "u \<in> U" and "v \<in> V" ..
ballarin@27611:       then show "x \<in> E" by simp
ballarin@27611:     qed
wenzelm@44887:   next
krauss@47445:     fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"
krauss@47445:     show "x + y \<in> U + V"
ballarin@27611:     proof -
ballarin@27611:       from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..
ballarin@27611:       moreover
ballarin@27611:       from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" ..
ballarin@27611:       ultimately
ballarin@27611:       have "ux + uy \<in> U"
wenzelm@32960:         and "vx + vy \<in> V"
wenzelm@32960:         and "x + y = (ux + uy) + (vx + vy)"
wenzelm@32960:         using x y by (simp_all add: add_ac)
wenzelm@27612:       then show ?thesis ..
ballarin@27611:     qed
krauss@47445:     fix a show "a \<cdot> x \<in> U + V"
ballarin@27611:     proof -
ballarin@27611:       from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..
wenzelm@27612:       then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"
wenzelm@32960:         and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib)
wenzelm@27612:       then show ?thesis ..
ballarin@27611:     qed
wenzelm@9035:   qed
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@9035: text{* The sum of two subspaces is a vectorspace. *}
wenzelm@7917: 
wenzelm@13515: lemma sum_vs [intro?]:
krauss@47445:     "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"
wenzelm@13547:   by (rule subspace.vectorspace) (rule sum_subspace)
wenzelm@7535: 
wenzelm@7808: 
wenzelm@9035: subsection {* Direct sums *}
wenzelm@7808: 
wenzelm@10687: text {*
wenzelm@10687:   The sum of @{text U} and @{text V} is called \emph{direct}, iff the
wenzelm@10687:   zero element is the only common element of @{text U} and @{text
wenzelm@10687:   V}. For every element @{text x} of the direct sum of @{text U} and
wenzelm@10687:   @{text V} the decomposition in @{text "x = u + v"} with
wenzelm@10687:   @{text "u \<in> U"} and @{text "v \<in> V"} is unique.
wenzelm@10687: *}
wenzelm@7808: 
wenzelm@10687: lemma decomp:
ballarin@27611:   assumes "vectorspace E" "subspace U E" "subspace V E"
wenzelm@13515:   assumes direct: "U \<inter> V = {0}"
wenzelm@13515:     and u1: "u1 \<in> U" and u2: "u2 \<in> U"
wenzelm@13515:     and v1: "v1 \<in> V" and v2: "v2 \<in> V"
wenzelm@13515:     and sum: "u1 + v1 = u2 + v2"
wenzelm@13515:   shows "u1 = u2 \<and> v1 = v2"
ballarin@27611: proof -
ballarin@29234:   interpret vectorspace E by fact
ballarin@29234:   interpret subspace U E by fact
ballarin@29234:   interpret subspace V E by fact
wenzelm@27612:   show ?thesis
wenzelm@27612:   proof
ballarin@27611:     have U: "vectorspace U"  (* FIXME: use interpret *)
ballarin@27611:       using `subspace U E` `vectorspace E` by (rule subspace.vectorspace)
ballarin@27611:     have V: "vectorspace V"
ballarin@27611:       using `subspace V E` `vectorspace E` by (rule subspace.vectorspace)
ballarin@27611:     from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"
ballarin@27611:       by (simp add: add_diff_swap)
ballarin@27611:     from u1 u2 have u: "u1 - u2 \<in> U"
ballarin@27611:       by (rule vectorspace.diff_closed [OF U])
ballarin@27611:     with eq have v': "v2 - v1 \<in> U" by (simp only:)
ballarin@27611:     from v2 v1 have v: "v2 - v1 \<in> V"
ballarin@27611:       by (rule vectorspace.diff_closed [OF V])
ballarin@27611:     with eq have u': " u1 - u2 \<in> V" by (simp only:)
ballarin@27611:     
ballarin@27611:     show "u1 = u2"
ballarin@27611:     proof (rule add_minus_eq)
ballarin@27611:       from u1 show "u1 \<in> E" ..
ballarin@27611:       from u2 show "u2 \<in> E" ..
ballarin@27611:       from u u' and direct show "u1 - u2 = 0" by blast
ballarin@27611:     qed
ballarin@27611:     show "v1 = v2"
ballarin@27611:     proof (rule add_minus_eq [symmetric])
ballarin@27611:       from v1 show "v1 \<in> E" ..
ballarin@27611:       from v2 show "v2 \<in> E" ..
ballarin@27611:       from v v' and direct show "v2 - v1 = 0" by blast
ballarin@27611:     qed
wenzelm@9035:   qed
wenzelm@9035: qed
wenzelm@7656: 
wenzelm@10687: text {*
wenzelm@10687:   An application of the previous lemma will be used in the proof of
wenzelm@10687:   the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any
wenzelm@10687:   element @{text "y + a \<cdot> x\<^sub>0"} of the direct sum of a
wenzelm@10687:   vectorspace @{text H} and the linear closure of @{text "x\<^sub>0"}
wenzelm@10687:   the components @{text "y \<in> H"} and @{text a} are uniquely
wenzelm@10687:   determined.
wenzelm@10687: *}
wenzelm@7917: 
wenzelm@10687: lemma decomp_H':
ballarin@27611:   assumes "vectorspace E" "subspace H E"
wenzelm@13515:   assumes y1: "y1 \<in> H" and y2: "y2 \<in> H"
wenzelm@13515:     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"
wenzelm@13515:     and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
wenzelm@13515:   shows "y1 = y2 \<and> a1 = a2"
ballarin@27611: proof -
ballarin@29234:   interpret vectorspace E by fact
ballarin@29234:   interpret subspace H E by fact
wenzelm@27612:   show ?thesis
wenzelm@27612:   proof
ballarin@27611:     have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
ballarin@27611:     proof (rule decomp)
ballarin@27611:       show "a1 \<cdot> x' \<in> lin x'" ..
ballarin@27611:       show "a2 \<cdot> x' \<in> lin x'" ..
ballarin@27611:       show "H \<inter> lin x' = {0}"
wenzelm@13515:       proof
wenzelm@32960:         show "H \<inter> lin x' \<subseteq> {0}"
wenzelm@32960:         proof
ballarin@27611:           fix x assume x: "x \<in> H \<inter> lin x'"
ballarin@27611:           then obtain a where xx': "x = a \<cdot> x'"
ballarin@27611:             by blast
ballarin@27611:           have "x = 0"
ballarin@27611:           proof cases
ballarin@27611:             assume "a = 0"
ballarin@27611:             with xx' and x' show ?thesis by simp
ballarin@27611:           next
ballarin@27611:             assume a: "a \<noteq> 0"
ballarin@27611:             from x have "x \<in> H" ..
ballarin@27611:             with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp
ballarin@27611:             with a and x' have "x' \<in> H" by (simp add: mult_assoc2)
ballarin@27611:             with `x' \<notin> H` show ?thesis by contradiction
ballarin@27611:           qed
wenzelm@27612:           then show "x \<in> {0}" ..
wenzelm@32960:         qed
wenzelm@32960:         show "{0} \<subseteq> H \<inter> lin x'"
wenzelm@32960:         proof -
ballarin@27611:           have "0 \<in> H" using `vectorspace E` ..
ballarin@27611:           moreover have "0 \<in> lin x'" using `x' \<in> E` ..
ballarin@27611:           ultimately show ?thesis by blast
wenzelm@32960:         qed
wenzelm@9035:       qed
ballarin@27611:       show "lin x' \<unlhd> E" using `x' \<in> E` ..
ballarin@27611:     qed (rule `vectorspace E`, rule `subspace H E`, rule y1, rule y2, rule eq)
wenzelm@27612:     then show "y1 = y2" ..
ballarin@27611:     from c have "a1 \<cdot> x' = a2 \<cdot> x'" ..
ballarin@27611:     with x' show "a1 = a2" by (simp add: mult_right_cancel)
ballarin@27611:   qed
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@10687: text {*
wenzelm@10687:   Since for any element @{text "y + a \<cdot> x'"} of the direct sum of a
wenzelm@10687:   vectorspace @{text H} and the linear closure of @{text x'} the
wenzelm@10687:   components @{text "y \<in> H"} and @{text a} are unique, it follows from
wenzelm@10687:   @{text "y \<in> H"} that @{text "a = 0"}.
wenzelm@10687: *}
wenzelm@7917: 
wenzelm@10687: lemma decomp_H'_H:
ballarin@27611:   assumes "vectorspace E" "subspace H E"
wenzelm@13515:   assumes t: "t \<in> H"
wenzelm@13515:     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"
wenzelm@13515:   shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
ballarin@27611: proof -
ballarin@29234:   interpret vectorspace E by fact
ballarin@29234:   interpret subspace H E by fact
wenzelm@27612:   show ?thesis
wenzelm@27612:   proof (rule, simp_all only: split_paired_all split_conv)
ballarin@27611:     from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp
ballarin@27611:     fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H"
ballarin@27611:     have "y = t \<and> a = 0"
ballarin@27611:     proof (rule decomp_H')
ballarin@27611:       from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp
ballarin@27611:       from ya show "y \<in> H" ..
ballarin@27611:     qed (rule `vectorspace E`, rule `subspace H E`, rule t, (rule x')+)
ballarin@27611:     with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp
ballarin@27611:   qed
wenzelm@13515: qed
wenzelm@7535: 
wenzelm@10687: text {*
wenzelm@10687:   The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"}
wenzelm@10687:   are unique, so the function @{text h'} defined by
wenzelm@10687:   @{text "h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>"} is definite.
wenzelm@10687: *}
wenzelm@7917: 
bauerg@9374: lemma h'_definite:
ballarin@27611:   fixes H
wenzelm@13515:   assumes h'_def:
wenzelm@44887:     "h' \<equiv> \<lambda>x.
wenzelm@44887:       let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
wenzelm@44887:       in (h y) + a * xi"
wenzelm@13515:     and x: "x = y + a \<cdot> x'"
ballarin@27611:   assumes "vectorspace E" "subspace H E"
wenzelm@13515:   assumes y: "y \<in> H"
wenzelm@13515:     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"
wenzelm@13515:   shows "h' x = h y + a * xi"
wenzelm@10687: proof -
ballarin@29234:   interpret vectorspace E by fact
ballarin@29234:   interpret subspace H E by fact
krauss@47445:   from x y x' have "x \<in> H + lin x'" by auto
wenzelm@13515:   have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")
wenzelm@18689:   proof (rule ex_ex1I)
wenzelm@13515:     from x y show "\<exists>p. ?P p" by blast
wenzelm@13515:     fix p q assume p: "?P p" and q: "?P q"
wenzelm@13515:     show "p = q"
wenzelm@9035:     proof -
wenzelm@13515:       from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H"
wenzelm@13515:         by (cases p) simp
wenzelm@13515:       from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H"
wenzelm@13515:         by (cases q) simp
wenzelm@13515:       have "fst p = fst q \<and> snd p = snd q"
wenzelm@13515:       proof (rule decomp_H')
wenzelm@13515:         from xp show "fst p \<in> H" ..
wenzelm@13515:         from xq show "fst q \<in> H" ..
wenzelm@13515:         from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'"
wenzelm@13515:           by simp
wenzelm@23378:       qed (rule `vectorspace E`, rule `subspace H E`, (rule x')+)
wenzelm@27612:       then show ?thesis by (cases p, cases q) simp
wenzelm@9035:     qed
wenzelm@9035:   qed
wenzelm@27612:   then have eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
wenzelm@13515:     by (rule some1_equality) (simp add: x y)
wenzelm@13515:   with h'_def show "h' x = h y + a * xi" by (simp add: Let_def)
wenzelm@9035: qed
wenzelm@7535: 
wenzelm@10687: end