src/HOL/Real/HahnBanach/Subspace.thy

 (*  Title:      HOL/Real/HahnBanach/Subspace.thy
     ID:         $Id$
     Author:     Gertrud Bauer, TU Munich
 *)
 
-
 header {* Subspaces *}
 
 theory Subspace = VectorSpace:
 
 
 subsection {* Definition *}
 
-text {* A non-empty subset $U$ of a vector space $V$ is a 
-\emph{subspace} of $V$, iff $U$ is closed under addition and 
-scalar multiplication. *}
-
-constdefs 
-  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]: 
-  "[| 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) 
+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.
+*}
+
+constdefs
+  is_subspace ::  "'a::{plus, minus, zero} set \<Rightarrow> 'a set \<Rightarrow> bool"
+  "is_subspace U V \<equiv> U \<noteq> {} \<and> U \<subseteq> 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]:
+  "0 \<in> U \<Longrightarrow> U \<subseteq> V \<Longrightarrow> \<forall>x \<in> U. \<forall>y \<in> U. (x + y \<in> U) \<Longrightarrow>
+  \<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U
+  \<Longrightarrow> is_subspace U V"
+proof (unfold is_subspace_def, intro conjI)
   assume "0 \<in> U" thus "U \<noteq> {}" by fast
 qed (simp+)
 
-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 \<in> U |] ==> x \<in> V"
-  by (unfold is_subspace_def) force
-
-lemma subspace_add_closed [simp, intro?]: 
-  "[| 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 \<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 \<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 
-zero element in every subspace follows from the non-emptiness 
+lemma subspace_not_empty [intro?]: "is_subspace U V \<Longrightarrow> U \<noteq> {}"
+  by (unfold is_subspace_def) blast
+
+lemma subspace_subset [intro?]: "is_subspace U V \<Longrightarrow> U \<subseteq> V"
+  by (unfold is_subspace_def) blast
+
+lemma subspace_subsetD [simp, intro?]:
+  "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
+  by (unfold is_subspace_def) blast
+
+lemma subspace_add_closed [simp, intro?]:
+  "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"
+  by (unfold is_subspace_def) blast
+
+lemma subspace_mult_closed [simp, intro?]:
+  "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"
+  by (unfold is_subspace_def) blast
+
+lemma subspace_diff_closed [simp, intro?]:
+  "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> x \<in> U \<Longrightarrow> y \<in> U
+  \<Longrightarrow> x - y \<in> U"
+  by (simp add: diff_eq1 negate_eq1)
+
+text {* 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 zero_in_subspace [intro?]:
-  "[| is_subspace U V; is_vectorspace V |] ==> 0 \<in> U"
-proof - 
+  "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> 0 \<in> U"
+proof -
   assume "is_subspace U V" and v: "is_vectorspace V"
   have "U \<noteq> {}" ..
-  hence "\<exists>x. x \<in> U" by force
-  thus ?thesis 
-  proof 
-    fix x assume u: "x \<in> U" 
+  hence "\<exists>x. x \<in> U" by blast
+  thus ?thesis
+  proof
+    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 \<in> U |] ==> - x \<in> U"
+lemma subspace_neg_closed [simp, intro?]:
+  "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> x \<in> U \<Longrightarrow> - x \<in> U"
   by (simp add: negate_eq1)
 
-text_raw {* \medskip *}
-text {* Further derived laws: every subspace is a vector space. *}
+text {* \medskip Further derived laws: every subspace is a vector space. *}
 
 lemma subspace_vs [intro?]:
-  "[| is_subspace U V; is_vectorspace V |] ==> is_vectorspace U"
+  "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> is_vectorspace U"
 proof -
-  assume "is_subspace U V" "is_vectorspace V"
+  assume "is_subspace U V"  "is_vectorspace V"
   show ?thesis
-  proof 
+  proof
     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" 
+    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
 
 text {* The subspace relation is reflexive. *}
 
-lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V"
-proof 
+lemma subspace_refl [intro]: "is_vectorspace V \<Longrightarrow> is_subspace V V"
+proof
   assume "is_vectorspace V"
   show "0 \<in> V" ..
-  show "V <= V" ..
+  show "V \<subseteq> V" ..
   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. *}
 
-lemma subspace_trans: 
-  "[| is_subspace U V; is_vectorspace V; is_subspace V W |] 
-  ==> is_subspace U W"
-proof 
-  assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
+lemma subspace_trans:
+  "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> is_subspace V W
+  \<Longrightarrow> is_subspace U W"
+proof
+  assume "is_subspace U V"  "is_subspace V W"  "is_vectorspace V"
   show "0 \<in> U" ..
 
-  have "U <= V" ..
-  also have "V <= W" ..
-  finally show "U <= W" .
-
-  show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" 
+  have "U \<subseteq> V" ..
+  also have "V \<subseteq> W" ..
+  finally show "U \<subseteq> W" .
+
+  show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
   proof (intro ballI)
-    fix x y assume "x \<in> U" "y \<in> U"
+    fix x y assume "x \<in> U"  "y \<in> U"
     show "x + y \<in> U" by (simp!)
   qed
 
   show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
   proof (intro ballI allI)

 
 
 
 subsection {* Linear closure *}
 
-text {* The \emph{linear closure} of a vector $x$ is the set of all
-scalar multiples of $x$. *}
+text {*
+  The \emph{linear closure} of a vector @{text x} is the set of all
+  scalar multiples of @{text x}.
+*}
 
 constdefs
-  lin :: "('a::{minus,plus,zero}) => 'a set"
-  "lin x == {a \<cdot> x | a. True}" 
+  lin :: "('a::{minus,plus,zero}) \<Rightarrow> 'a set"
+  "lin x \<equiv> {a \<cdot> x | a. True}"
 
 lemma linD: "x \<in> lin v = (\<exists>a::real. x = a \<cdot> v)"
   by (unfold lin_def) fast
 
 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 \<in> V |] ==> x \<in> lin x"
+lemma x_lin_x: "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x \<in> lin x"
 proof (unfold lin_def, intro CollectI exI conjI)
-  assume "is_vectorspace V" "x \<in> V"
+  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 \<in> V |] ==> is_subspace (lin x) V"
-proof
-  assume "is_vectorspace V" "x \<in> V"
-  show "0 \<in> lin x" 
+lemma lin_subspace [intro?]:
+  "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> is_subspace (lin x) V"
+proof
+  assume "is_vectorspace V"  "x \<in> V"
+  show "0 \<in> lin x"
   proof (unfold lin_def, intro CollectI exI conjI)
     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 \<cdot> x" 
+  show "lin x \<subseteq> V"
+  proof (unfold lin_def, intro subsetI, elim CollectE exE conjE)
+    fix xa a assume "xa = a \<cdot> x"
     show "xa \<in> V" by (simp!)
   qed
 
-  show "\<forall>x1 \<in> lin x. \<forall>x2 \<in> lin x. x1 + x2 \<in> 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 \<in> lin x" "x2 \<in> 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, 
+    proof (unfold lin_def, elim CollectE exE conjE,
       intro CollectI exI conjI)
-      fix a1 a2 assume "x1 = a1 \<cdot> x" "x2 = a2 \<cdot> x"
-      show "x1 + x2 = (a1 + a2) \<cdot> 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 "\<forall>xa \<in> lin x. \<forall>a. a \<cdot> xa \<in> 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 \<in> 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 \<cdot> x"
       show "a \<cdot> x1 = (a * a1) \<cdot> x" by (simp!)
     qed simp
-  qed 
+  qed
 qed
 
 text {* Any linear closure is a vector space. *}
 
-lemma lin_vs [intro?]: 
-  "[| is_vectorspace V; x \<in> V |] ==> is_vectorspace (lin x)"
+lemma lin_vs [intro?]:
+  "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> is_vectorspace (lin x)"
 proof (rule subspace_vs)
-  assume "is_vectorspace V" "x \<in> V"
+  assume "is_vectorspace V"  "x \<in> V"
   show "is_subspace (lin x) V" ..
 qed
 
 
 
 subsection {* Sum of two vectorspaces *}
 
-text {* The \emph{sum} of two vectorspaces $U$ and $V$ is the set of
-all sums of elements from $U$ and $V$. *}
+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}.
+*}
 
 instance set :: (plus) plus ..
 
-defs vs_sum_def:
-  "U + V == {u + v | u v. u \<in> U \<and> v \<in> V}" (***
-
-constdefs 
-  vs_sum :: 
-  "['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: 
+defs (overloaded)
+  vs_sum_def: "U + V \<equiv> {u + v | u v. u \<in> U \<and> v \<in> V}"
+
+lemma vs_sumD:
   "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 [THEN iffD1, elim_format, standard]
 
-lemma vs_sumI [intro?]: 
-  "[| x \<in> U; y \<in> V; t= x + y |] ==> t \<in> U + V"
+lemma vs_sumI [intro?]:
+  "x \<in> U \<Longrightarrow> y \<in> V \<Longrightarrow> t = x + y \<Longrightarrow> t \<in> U + V"
   by (unfold vs_sum_def) fast
 
-text{* $U$ is a subspace of $U + V$. *}
-
-lemma subspace_vs_sum1 [intro?]: 
-  "[| is_vectorspace U; is_vectorspace V |]
-  ==> is_subspace U (U + V)"
-proof 
-  assume "is_vectorspace U" "is_vectorspace V"
+text {* @{text U} is a subspace of @{text "U + V"}. *}
+
+lemma subspace_vs_sum1 [intro?]:
+  "is_vectorspace U \<Longrightarrow> is_vectorspace V
+  \<Longrightarrow> is_subspace U (U + V)"
+proof
+  assume "is_vectorspace U"  "is_vectorspace V"
   show "0 \<in> U" ..
-  show "U <= U + V"
+  show "U \<subseteq> U + V"
   proof (intro subsetI vs_sumI)
   fix x assume "x \<in> U"
     show "x = x + 0" by (simp!)
     show "0 \<in> V" by (simp!)
   qed
-  show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" 
+  show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
   proof (intro ballI)
-    fix x y assume "x \<in> U" "y \<in> U" show "x + y \<in> U" by (simp!)
-  qed
-  show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" 
+    fix x y assume "x \<in> U"  "y \<in> U" show "x + y \<in> U" by (simp!)
+  qed
+  show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
   proof (intro ballI allI)
     fix x a assume "x \<in> U" show "a \<cdot> x \<in> U" by (simp!)
   qed
 qed
 
 text{* The sum of two subspaces is again a subspace.*}
 
-lemma vs_sum_subspace [intro?]: 
-  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
-  ==> is_subspace (U + V) E"
-proof 
-  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
+lemma vs_sum_subspace [intro?]:
+  "is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow> is_vectorspace E
+  \<Longrightarrow> is_subspace (U + V) E"
+proof
+  assume "is_subspace U E"  "is_subspace V E"  "is_vectorspace E"
   show "0 \<in> U + V"
   proof (intro vs_sumI)
     show "0 \<in> U" ..
     show "0 \<in> V" ..
     show "(0::'a) = 0 + 0" by (simp!)
   qed
-  
-  show "U + V <= E"
+
+  show "U + V \<subseteq> E"
   proof (intro subsetI, elim vs_sumE bexE)
-    fix x u v assume "u \<in> U" "v \<in> V" "x = u + v"
+    fix x u v assume "u \<in> U"  "v \<in> V"  "x = u + v"
     show "x \<in> E" by (simp!)
   qed
-  
+
   show "\<forall>x \<in> U + V. \<forall>y \<in> U + V. x + y \<in> U + V"
   proof (intro ballI)
-    fix x y assume "x \<in> U + V" "y \<in> 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 \<in> U" "vx \<in> V" "x = ux + vx" 
-	and "uy \<in> U" "vy \<in> V" "y = uy + vy"
+      fix ux vx 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 (simp_all!)
   qed
 
   show "\<forall>x \<in> U + V. \<forall>a. a \<cdot> x \<in> U + V"
   proof (intro ballI allI)
     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 \<in> U" "v \<in> V" "x = u + v"
-      show "a \<cdot> x = (a \<cdot> u) + (a \<cdot> 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 (simp_all!)
   qed
 qed
 
 text{* The sum of two subspaces is a vectorspace. *}
 
-lemma vs_sum_vs [intro?]: 
-  "[| is_subspace U E; is_subspace V E; is_vectorspace E |] 
-  ==> is_vectorspace (U + V)"
+lemma vs_sum_vs [intro?]:
+  "is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow> is_vectorspace E
+  \<Longrightarrow> is_vectorspace (U + V)"
 proof (rule subspace_vs)
-  assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
+  assume "is_subspace U E"  "is_subspace V E"  "is_vectorspace E"
   show "is_subspace (U + V) E" ..
 qed
 
 
 
 subsection {* Direct sums *}
 
 
-text {* The sum of $U$ and $V$ is called \emph{direct}, iff the zero 
-element is the only common element of $U$ and $V$. For every element
-$x$ of the direct sum of $U$ and $V$ the decomposition in
-$x = u + v$ with $u \in U$ and $v \in V$ is unique.*} 
-
-lemma decomp: 
-  "[| is_vectorspace E; is_subspace U E; is_subspace V E; 
-  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 \<inter> V = {0}" "u1 \<in> U" "u2 \<in> U" "v1 \<in> V" "v2 \<in> V" 
-    "u1 + v1 = u2 + v2" 
+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:
+  "is_vectorspace E \<Longrightarrow> is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow>
+  U \<inter> V = {0} \<Longrightarrow> u1 \<in> U \<Longrightarrow> u2 \<in> U \<Longrightarrow> v1 \<in> V \<Longrightarrow> v2 \<in> V \<Longrightarrow>
+  u1 + v1 = u2 + v2 \<Longrightarrow> u1 = u2 \<and> v1 = v2"
+proof
+  assume "is_vectorspace E"  "is_subspace U E"  "is_subspace V E"
+    "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 \<in> U" by (simp!) 
-  with eq have v': "v2 - v1 \<in> U" by simp 
-  have v: "v2 - v1 \<in> 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 = 0" by (rule Int_singletonD [OF _ u u']) 
+    show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u'])
     show "u1 \<in> E" ..
     show "u2 \<in> E" ..
   qed
 
   show "v1 = v2"

     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-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_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"
+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':
+  "is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow> y1 \<in> H \<Longrightarrow> y2 \<in> H \<Longrightarrow>
+  x' \<notin> H \<Longrightarrow> x' \<in> E \<Longrightarrow> x' \<noteq> 0 \<Longrightarrow> y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'
+  \<Longrightarrow> y1 = y2 \<and> a1 = a2"
 proof
   assume "is_vectorspace E" and h: "is_subspace H E"
-     and "y1 \<in> H" "y2 \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" 
+     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 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
-  proof (rule decomp) 
-    show "a1 \<cdot> x' \<in> lin x'" .. 
+  proof (rule decomp)
+    show "a1 \<cdot> x' \<in> lin x'" ..
     show "a2 \<cdot> x' \<in> lin x'" ..
-    show "H \<inter> (lin x') = {0}" 
+    show "H \<inter> (lin x') = {0}"
     proof
-      show "H \<inter> lin x' <= {0}" 
+      show "H \<inter> lin x' \<subseteq> {0}"
       proof (intro subsetI, elim IntE, rule singleton_iff [THEN iffD2])
-        fix x assume "x \<in> H" "x \<in> lin x'" 
+        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 \<cdot> x'"
           show ?thesis
           proof cases
             assume "a = (#0::real)" show ?thesis by (simp!)
           next
-            assume "a \<noteq> (#0::real)" 
-            from h have "inverse a \<cdot> a \<cdot> x' \<in> H" 
+            assume "a \<noteq> (#0::real)"
+            from h have "inverse a \<cdot> a \<cdot> x' \<in> H"
               by (rule subspace_mult_closed) (simp!)
             also have "inverse a \<cdot> a \<cdot> x' = x'" by (simp!)
             finally have "x' \<in> H" .
             thus ?thesis by contradiction
           qed
        qed
       qed
-      show "{0} <= H \<inter> lin x'"
+      show "{0} \<subseteq> H \<inter> lin x'"
       proof -
-	have "0 \<in> H \<inter> lin x'"
-	proof (rule IntI)
-	  show "0 \<in> H" ..
-	  from lin_vs show "0 \<in> lin x'" ..
-	qed
-	thus ?thesis by simp
+        have "0 \<in> H \<inter> lin x'"
+        proof (rule IntI)
+          show "0 \<in> H" ..
+          from lin_vs show "0 \<in> lin x'" ..
+        qed
+        thus ?thesis by simp
       qed
     qed
     show "is_subspace (lin x') E" ..
   qed
-  
+
   from c show "y1 = y2" by simp
-  
-  show  "a1 = a2" 
+
+  show  "a1 = a2"
   proof (rule vs_mult_right_cancel [THEN iffD1])
     from c show "a1 \<cdot> x' = a2 \<cdot> x'" by simp
   qed
 qed
 
-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_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))"
+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:
+  "is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow> t \<in> H \<Longrightarrow> x' \<notin> H \<Longrightarrow> x' \<in> E
+  \<Longrightarrow> x' \<noteq> 0
+  \<Longrightarrow> (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 \<in> H" "x' \<notin> H" "x' \<in> E"
+  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 \<cdot> x'" and "y \<in> H"
-  have "y = t \<and> a = (#0::real)" 
-    by (rule decomp_H') (assumption | (simp!))+
+  have "y = t \<and> a = (#0::real)"
+    by (rule decomp_H') (auto!)
   thus "(y, a) = (t, (#0::real))" by (simp!)
-qed (simp!)+
-
-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. *}
+qed (simp_all!)
+
+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:
-  "[| 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 \<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 
-    "h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
+  "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
+                in (h y) + a * xi) \<Longrightarrow>
+  x = y + a \<cdot> x' \<Longrightarrow> is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow>
+  y \<in> H \<Longrightarrow> x' \<notin> H \<Longrightarrow> x' \<in> E \<Longrightarrow> x' \<noteq> 0
+  \<Longrightarrow> h' x = h y + a * xi"
+proof -
+  assume
+    "h' \<equiv> (\<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 \<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 "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)" 
+    "x = y + a \<cdot> x'"  "is_vectorspace E"  "is_subspace H E"
+    "y \<in> H"  "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"
+  hence "x \<in> H + (lin x')"
+    by (auto simp add: vs_sum_def lin_def)
+  have "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
   proof
     show "\<exists>xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
-      by (force!)
+      by (blast!)
   next
     fix xa 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" 
+    show "xa = ya"
     proof -
-      show "fst xa = fst ya \<and> snd xa = snd ya ==> xa = ya" 
+      show "fst xa = fst ya \<and> snd xa = snd ya \<Longrightarrow> xa = ya"
         by (simp add: Pair_fst_snd_eq)
-      have x: "x = fst xa + snd xa \<cdot> x' \<and> fst xa \<in> H" 
-        by (force!)
-      have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H" 
-        by (force!)
-      from x y show "fst xa = fst ya \<and> snd xa = snd ya" 
+      have x: "x = fst xa + snd xa \<cdot> x' \<and> fst xa \<in> H"
+        by (auto!)
+      have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H"
+        by (auto!)
+      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 \<cdot> x' \<and> y \<in> H) = (y, a)" 
-    by (rule some1_equality) (force!)
+  hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
+    by (rule some1_equality) (blast!)
   thus "h' x = h y + a * xi" by (simp! add: Let_def)
 qed
 
 end