src/HOL/Real/HahnBanach/Subspace.thy
changeset 9374 153853af318b
parent 9370 cccba6147dae
child 9408 d3d56e1d2ec1
     1.1 --- a/src/HOL/Real/HahnBanach/Subspace.thy	Sun Jul 16 21:00:32 2000 +0200
     1.2 +++ b/src/HOL/Real/HahnBanach/Subspace.thy	Mon Jul 17 13:58:18 2000 +0200
     1.3 @@ -16,39 +16,39 @@
     1.4  scalar multiplication. *}
     1.5  
     1.6  constdefs 
     1.7 -  is_subspace ::  "['a::{minus, plus} set, 'a set] => bool"
     1.8 -  "is_subspace U V == U ~= {} & U <= V 
     1.9 -     & (ALL x:U. ALL y:U. ALL a. x + y : U & a (*) x : U)"
    1.10 +  is_subspace ::  "['a::{plus, minus, zero} set, 'a set] => bool"
    1.11 +  "is_subspace U V == U \<noteq> {} \<and> U <= V 
    1.12 +     \<and> (\<forall>x \<in> U. \<forall>y \<in> U. \<forall>a. x + y \<in> U \<and> a \<cdot> x\<in> U)"
    1.13  
    1.14  lemma subspaceI [intro]: 
    1.15 -  "[| 00 : U; U <= V; ALL x:U. ALL y:U. (x + y : U); 
    1.16 -  ALL x:U. ALL a. a (*) x : U |]
    1.17 +  "[| 0 \<in> U; U <= V; \<forall>x \<in> U. \<forall>y \<in> U. (x + y \<in> U); 
    1.18 +  \<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U |]
    1.19    ==> is_subspace U V"
    1.20  proof (unfold is_subspace_def, intro conjI) 
    1.21 -  assume "00 : U" thus "U ~= {}" by fast
    1.22 +  assume "0 \<in> U" thus "U \<noteq> {}" by fast
    1.23  qed (simp+)
    1.24  
    1.25 -lemma subspace_not_empty [intro??]: "is_subspace U V ==> U ~= {}"
    1.26 +lemma subspace_not_empty [intro??]: "is_subspace U V ==> U \<noteq> {}"
    1.27    by (unfold is_subspace_def) simp 
    1.28  
    1.29  lemma subspace_subset [intro??]: "is_subspace U V ==> U <= V"
    1.30    by (unfold is_subspace_def) simp
    1.31  
    1.32  lemma subspace_subsetD [simp, intro??]: 
    1.33 -  "[| is_subspace U V; x:U |] ==> x:V"
    1.34 +  "[| is_subspace U V; x \<in> U |] ==> x \<in> V"
    1.35    by (unfold is_subspace_def) force
    1.36  
    1.37  lemma subspace_add_closed [simp, intro??]: 
    1.38 -  "[| is_subspace U V; x:U; y:U |] ==> x + y : U"
    1.39 +  "[| is_subspace U V; x \<in> U; y \<in> U |] ==> x + y \<in> U"
    1.40    by (unfold is_subspace_def) simp
    1.41  
    1.42  lemma subspace_mult_closed [simp, intro??]: 
    1.43 -  "[| is_subspace U V; x:U |] ==> a (*) x : U"
    1.44 +  "[| is_subspace U V; x \<in> U |] ==> a \<cdot> x \<in> U"
    1.45    by (unfold is_subspace_def) simp
    1.46  
    1.47  lemma subspace_diff_closed [simp, intro??]: 
    1.48 -  "[| is_subspace U V; is_vectorspace V; x:U; y:U |] 
    1.49 -  ==> x - y : U"
    1.50 +  "[| is_subspace U V; is_vectorspace V; x \<in> U; y \<in> U |] 
    1.51 +  ==> x - y \<in> U"
    1.52    by (simp! add: diff_eq1 negate_eq1)
    1.53  
    1.54  text {* Similar as for linear spaces, the existence of the 
    1.55 @@ -56,23 +56,23 @@
    1.56  of the carrier set and by vector space laws.*}
    1.57  
    1.58  lemma zero_in_subspace [intro??]:
    1.59 -  "[| is_subspace U V; is_vectorspace V |] ==> 00 : U"
    1.60 +  "[| is_subspace U V; is_vectorspace V |] ==> 0 \<in> U"
    1.61  proof - 
    1.62    assume "is_subspace U V" and v: "is_vectorspace V"
    1.63 -  have "U ~= {}" ..
    1.64 -  hence "EX x. x:U" by force
    1.65 +  have "U \<noteq> {}" ..
    1.66 +  hence "\<exists>x. x \<in> U" by force
    1.67    thus ?thesis 
    1.68    proof 
    1.69 -    fix x assume u: "x:U" 
    1.70 -    hence "x:V" by (simp!)
    1.71 -    with v have "00 = x - x" by (simp!)
    1.72 -    also have "... : U" by (rule subspace_diff_closed)
    1.73 +    fix x assume u: "x \<in> U" 
    1.74 +    hence "x \<in> V" by (simp!)
    1.75 +    with v have "0 = x - x" by (simp!)
    1.76 +    also have "... \<in> U" by (rule subspace_diff_closed)
    1.77      finally show ?thesis .
    1.78    qed
    1.79  qed
    1.80  
    1.81  lemma subspace_neg_closed [simp, intro??]: 
    1.82 -  "[| is_subspace U V; is_vectorspace V; x:U |] ==> - x : U"
    1.83 +  "[| is_subspace U V; is_vectorspace V; x \<in> U |] ==> - x \<in> U"
    1.84    by (simp add: negate_eq1)
    1.85  
    1.86  text_raw {* \medskip *}
    1.87 @@ -84,11 +84,11 @@
    1.88    assume "is_subspace U V" "is_vectorspace V"
    1.89    show ?thesis
    1.90    proof 
    1.91 -    show "00 : U" ..
    1.92 -    show "ALL x:U. ALL a. a (*) x : U" by (simp!)
    1.93 -    show "ALL x:U. ALL y:U. x + y : U" by (simp!)
    1.94 -    show "ALL x:U. - x = -#1 (*) x" by (simp! add: negate_eq1)
    1.95 -    show "ALL x:U. ALL y:U. x - y =  x + - y" 
    1.96 +    show "0 \<in> U" ..
    1.97 +    show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" by (simp!)
    1.98 +    show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" by (simp!)
    1.99 +    show "\<forall>x \<in> U. - x = -#1 \<cdot> x" by (simp! add: negate_eq1)
   1.100 +    show "\<forall>x \<in> U. \<forall>y \<in> U. x - y =  x + - y" 
   1.101        by (simp! add: diff_eq1)
   1.102    qed (simp! add: vs_add_mult_distrib1 vs_add_mult_distrib2)+
   1.103  qed
   1.104 @@ -98,10 +98,10 @@
   1.105  lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V"
   1.106  proof 
   1.107    assume "is_vectorspace V"
   1.108 -  show "00 : V" ..
   1.109 +  show "0 \<in> V" ..
   1.110    show "V <= V" ..
   1.111 -  show "ALL x:V. ALL y:V. x + y : V" by (simp!)
   1.112 -  show "ALL x:V. ALL a. a (*) x : V" by (simp!)
   1.113 +  show "\<forall>x \<in> V. \<forall>y \<in> V. x + y \<in> V" by (simp!)
   1.114 +  show "\<forall>x \<in> V. \<forall>a. a \<cdot> x \<in> V" by (simp!)
   1.115  qed
   1.116  
   1.117  text {* The subspace relation is transitive. *}
   1.118 @@ -111,22 +111,22 @@
   1.119    ==> is_subspace U W"
   1.120  proof 
   1.121    assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
   1.122 -  show "00 : U" ..
   1.123 +  show "0 \<in> U" ..
   1.124  
   1.125    have "U <= V" ..
   1.126    also have "V <= W" ..
   1.127    finally show "U <= W" .
   1.128  
   1.129 -  show "ALL x:U. ALL y:U. x + y : U" 
   1.130 +  show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" 
   1.131    proof (intro ballI)
   1.132 -    fix x y assume "x:U" "y:U"
   1.133 -    show "x + y : U" by (simp!)
   1.134 +    fix x y assume "x \<in> U" "y \<in> U"
   1.135 +    show "x + y \<in> U" by (simp!)
   1.136    qed
   1.137  
   1.138 -  show "ALL x:U. ALL a. a (*) x : U"
   1.139 +  show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
   1.140    proof (intro ballI allI)
   1.141 -    fix x a assume "x:U"
   1.142 -    show "a (*) x : U" by (simp!)
   1.143 +    fix x a assume "x \<in> U"
   1.144 +    show "a \<cdot> x \<in> U" by (simp!)
   1.145    qed
   1.146  qed
   1.147  
   1.148 @@ -138,60 +138,60 @@
   1.149  scalar multiples of $x$. *}
   1.150  
   1.151  constdefs
   1.152 -  lin :: "'a => 'a set"
   1.153 -  "lin x == {a (*) x | a. True}" 
   1.154 +  lin :: "('a::{minus,plus,zero}) => 'a set"
   1.155 +  "lin x == {a \<cdot> x | a. True}" 
   1.156  
   1.157 -lemma linD: "x : lin v = (EX a::real. x = a (*) v)"
   1.158 +lemma linD: "x \<in> lin v = (\<exists>a::real. x = a \<cdot> v)"
   1.159    by (unfold lin_def) fast
   1.160  
   1.161 -lemma linI [intro??]: "a (*) x0 : lin x0"
   1.162 +lemma linI [intro??]: "a \<cdot> x0 \<in> lin x0"
   1.163    by (unfold lin_def) fast
   1.164  
   1.165  text {* Every vector is contained in its linear closure. *}
   1.166  
   1.167 -lemma x_lin_x: "[| is_vectorspace V; x:V |] ==> x : lin x"
   1.168 +lemma x_lin_x: "[| is_vectorspace V; x \<in> V |] ==> x \<in> lin x"
   1.169  proof (unfold lin_def, intro CollectI exI conjI)
   1.170 -  assume "is_vectorspace V" "x:V"
   1.171 -  show "x = #1 (*) x" by (simp!)
   1.172 +  assume "is_vectorspace V" "x \<in> V"
   1.173 +  show "x = #1 \<cdot> x" by (simp!)
   1.174  qed simp
   1.175  
   1.176  text {* Any linear closure is a subspace. *}
   1.177  
   1.178  lemma lin_subspace [intro??]: 
   1.179 -  "[| is_vectorspace V; x:V |] ==> is_subspace (lin x) V"
   1.180 +  "[| is_vectorspace V; x \<in> V |] ==> is_subspace (lin x) V"
   1.181  proof
   1.182 -  assume "is_vectorspace V" "x:V"
   1.183 -  show "00 : lin x" 
   1.184 +  assume "is_vectorspace V" "x \<in> V"
   1.185 +  show "0 \<in> lin x" 
   1.186    proof (unfold lin_def, intro CollectI exI conjI)
   1.187 -    show "00 = (#0::real) (*) x" by (simp!)
   1.188 +    show "0 = (#0::real) \<cdot> x" by (simp!)
   1.189    qed simp
   1.190  
   1.191    show "lin x <= V"
   1.192    proof (unfold lin_def, intro subsetI, elim CollectE exE conjE) 
   1.193 -    fix xa a assume "xa = a (*) x" 
   1.194 -    show "xa:V" by (simp!)
   1.195 +    fix xa a assume "xa = a \<cdot> x" 
   1.196 +    show "xa \<in> V" by (simp!)
   1.197    qed
   1.198  
   1.199 -  show "ALL x1 : lin x. ALL x2 : lin x. x1 + x2 : lin x" 
   1.200 +  show "\<forall>x1 \<in> lin x. \<forall>x2 \<in> lin x. x1 + x2 \<in> lin x" 
   1.201    proof (intro ballI)
   1.202 -    fix x1 x2 assume "x1 : lin x" "x2 : lin x" 
   1.203 -    thus "x1 + x2 : lin x"
   1.204 +    fix x1 x2 assume "x1 \<in> lin x" "x2 \<in> lin x" 
   1.205 +    thus "x1 + x2 \<in> lin x"
   1.206      proof (unfold lin_def, elim CollectE exE conjE, 
   1.207        intro CollectI exI conjI)
   1.208 -      fix a1 a2 assume "x1 = a1 (*) x" "x2 = a2 (*) x"
   1.209 -      show "x1 + x2 = (a1 + a2) (*) x" 
   1.210 +      fix a1 a2 assume "x1 = a1 \<cdot> x" "x2 = a2 \<cdot> x"
   1.211 +      show "x1 + x2 = (a1 + a2) \<cdot> x" 
   1.212          by (simp! add: vs_add_mult_distrib2)
   1.213      qed simp
   1.214    qed
   1.215  
   1.216 -  show "ALL xa:lin x. ALL a. a (*) xa : lin x" 
   1.217 +  show "\<forall>xa \<in> lin x. \<forall>a. a \<cdot> xa \<in> lin x" 
   1.218    proof (intro ballI allI)
   1.219 -    fix x1 a assume "x1 : lin x" 
   1.220 -    thus "a (*) x1 : lin x"
   1.221 +    fix x1 a assume "x1 \<in> lin x" 
   1.222 +    thus "a \<cdot> x1 \<in> lin x"
   1.223      proof (unfold lin_def, elim CollectE exE conjE,
   1.224        intro CollectI exI conjI)
   1.225 -      fix a1 assume "x1 = a1 (*) x"
   1.226 -      show "a (*) x1 = (a * a1) (*) x" by (simp!)
   1.227 +      fix a1 assume "x1 = a1 \<cdot> x"
   1.228 +      show "a \<cdot> x1 = (a * a1) \<cdot> x" by (simp!)
   1.229      qed simp
   1.230    qed 
   1.231  qed
   1.232 @@ -199,9 +199,9 @@
   1.233  text {* Any linear closure is a vector space. *}
   1.234  
   1.235  lemma lin_vs [intro??]: 
   1.236 -  "[| is_vectorspace V; x:V |] ==> is_vectorspace (lin x)"
   1.237 +  "[| is_vectorspace V; x \<in> V |] ==> is_vectorspace (lin x)"
   1.238  proof (rule subspace_vs)
   1.239 -  assume "is_vectorspace V" "x:V"
   1.240 +  assume "is_vectorspace V" "x \<in> V"
   1.241    show "is_subspace (lin x) V" ..
   1.242  qed
   1.243  
   1.244 @@ -215,22 +215,22 @@
   1.245  instance set :: (plus) plus by intro_classes
   1.246  
   1.247  defs vs_sum_def:
   1.248 -  "U + V == {u + v | u v. u:U & v:V}" (***
   1.249 +  "U + V == {u + v | u v. u \<in> U \<and> v \<in> V}" (***
   1.250  
   1.251  constdefs 
   1.252    vs_sum :: 
   1.253 -  "['a::{minus, plus} set, 'a set] => 'a set"         (infixl "+" 65)
   1.254 -  "vs_sum U V == {x. EX u:U. EX v:V. x = u + v}";
   1.255 +  "['a::{plus, minus, zero} set, 'a set] => 'a set"         (infixl "+" 65)
   1.256 +  "vs_sum U V == {x. \<exists>u \<in> U. \<exists>v \<in> V. x = u + v}";
   1.257  ***)
   1.258  
   1.259  lemma vs_sumD: 
   1.260 -  "x: U + V = (EX u:U. EX v:V. x = u + v)"
   1.261 +  "x \<in> U + V = (\<exists>u \<in> U. \<exists>v \<in> V. x = u + v)"
   1.262      by (unfold vs_sum_def) fast
   1.263  
   1.264  lemmas vs_sumE = vs_sumD [RS iffD1, elimify]
   1.265  
   1.266  lemma vs_sumI [intro??]: 
   1.267 -  "[| x:U; y:V; t= x + y |] ==> t : U + V"
   1.268 +  "[| x \<in> U; y \<in> V; t= x + y |] ==> t \<in> U + V"
   1.269    by (unfold vs_sum_def) fast
   1.270  
   1.271  text{* $U$ is a subspace of $U + V$. *}
   1.272 @@ -240,20 +240,20 @@
   1.273    ==> is_subspace U (U + V)"
   1.274  proof 
   1.275    assume "is_vectorspace U" "is_vectorspace V"
   1.276 -  show "00 : U" ..
   1.277 +  show "0 \<in> U" ..
   1.278    show "U <= U + V"
   1.279    proof (intro subsetI vs_sumI)
   1.280 -  fix x assume "x:U"
   1.281 -    show "x = x + 00" by (simp!)
   1.282 -    show "00 : V" by (simp!)
   1.283 +  fix x assume "x \<in> U"
   1.284 +    show "x = x + 0" by (simp!)
   1.285 +    show "0 \<in> V" by (simp!)
   1.286    qed
   1.287 -  show "ALL x:U. ALL y:U. x + y : U" 
   1.288 +  show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" 
   1.289    proof (intro ballI)
   1.290 -    fix x y assume "x:U" "y:U" show "x + y : U" by (simp!)
   1.291 +    fix x y assume "x \<in> U" "y \<in> U" show "x + y \<in> U" by (simp!)
   1.292    qed
   1.293 -  show "ALL x:U. ALL a. a (*) x : U" 
   1.294 +  show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" 
   1.295    proof (intro ballI allI)
   1.296 -    fix x a assume "x:U" show "a (*) x : U" by (simp!)
   1.297 +    fix x a assume "x \<in> U" show "a \<cdot> x \<in> U" by (simp!)
   1.298    qed
   1.299  qed
   1.300  
   1.301 @@ -264,38 +264,38 @@
   1.302    ==> is_subspace (U + V) E"
   1.303  proof 
   1.304    assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
   1.305 -  show "00 : U + V"
   1.306 +  show "0 \<in> U + V"
   1.307    proof (intro vs_sumI)
   1.308 -    show "00 : U" ..
   1.309 -    show "00 : V" ..
   1.310 -    show "(00::'a) = 00 + 00" by (simp!)
   1.311 +    show "0 \<in> U" ..
   1.312 +    show "0 \<in> V" ..
   1.313 +    show "(0::'a) = 0 + 0" by (simp!)
   1.314    qed
   1.315    
   1.316    show "U + V <= E"
   1.317    proof (intro subsetI, elim vs_sumE bexE)
   1.318 -    fix x u v assume "u : U" "v : V" "x = u + v"
   1.319 -    show "x:E" by (simp!)
   1.320 +    fix x u v assume "u \<in> U" "v \<in> V" "x = u + v"
   1.321 +    show "x \<in> E" by (simp!)
   1.322    qed
   1.323    
   1.324 -  show "ALL x: U + V. ALL y: U + V. x + y : U + V"
   1.325 +  show "\<forall>x \<in> U + V. \<forall>y \<in> U + V. x + y \<in> U + V"
   1.326    proof (intro ballI)
   1.327 -    fix x y assume "x : U + V" "y : U + V"
   1.328 -    thus "x + y : U + V"
   1.329 +    fix x y assume "x \<in> U + V" "y \<in> U + V"
   1.330 +    thus "x + y \<in> U + V"
   1.331      proof (elim vs_sumE bexE, intro vs_sumI)
   1.332        fix ux vx uy vy 
   1.333 -      assume "ux : U" "vx : V" "x = ux + vx" 
   1.334 -	and "uy : U" "vy : V" "y = uy + vy"
   1.335 +      assume "ux \<in> U" "vx \<in> V" "x = ux + vx" 
   1.336 +	and "uy \<in> U" "vy \<in> V" "y = uy + vy"
   1.337        show "x + y = (ux + uy) + (vx + vy)" by (simp!)
   1.338      qed (simp!)+
   1.339    qed
   1.340  
   1.341 -  show "ALL x : U + V. ALL a. a (*) x : U + V"
   1.342 +  show "\<forall>x \<in> U + V. \<forall>a. a \<cdot> x \<in> U + V"
   1.343    proof (intro ballI allI)
   1.344 -    fix x a assume "x : U + V"
   1.345 -    thus "a (*) x : U + V"
   1.346 +    fix x a assume "x \<in> U + V"
   1.347 +    thus "a \<cdot> x \<in> U + V"
   1.348      proof (elim vs_sumE bexE, intro vs_sumI)
   1.349 -      fix a x u v assume "u : U" "v : V" "x = u + v"
   1.350 -      show "a (*) x = (a (*) u) + (a (*) v)" 
   1.351 +      fix a x u v assume "u \<in> U" "v \<in> V" "x = u + v"
   1.352 +      show "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" 
   1.353          by (simp! add: vs_add_mult_distrib1)
   1.354      qed (simp!)+
   1.355    qed
   1.356 @@ -323,154 +323,154 @@
   1.357  
   1.358  lemma decomp: 
   1.359    "[| is_vectorspace E; is_subspace U E; is_subspace V E; 
   1.360 -  U Int V = {00}; u1:U; u2:U; v1:V; v2:V; u1 + v1 = u2 + v2 |] 
   1.361 -  ==> u1 = u2 & v1 = v2" 
   1.362 +  U \<inter> V = {0}; u1 \<in> U; u2 \<in> U; v1 \<in> V; v2 \<in> V; u1 + v1 = u2 + v2 |] 
   1.363 +  ==> u1 = u2 \<and> v1 = v2" 
   1.364  proof 
   1.365    assume "is_vectorspace E" "is_subspace U E" "is_subspace V E"  
   1.366 -    "U Int V = {00}" "u1:U" "u2:U" "v1:V" "v2:V" 
   1.367 +    "U \<inter> V = {0}" "u1 \<in> U" "u2 \<in> U" "v1 \<in> V" "v2 \<in> V" 
   1.368      "u1 + v1 = u2 + v2" 
   1.369    have eq: "u1 - u2 = v2 - v1" by (simp! add: vs_add_diff_swap)
   1.370 -  have u: "u1 - u2 : U" by (simp!) 
   1.371 -  with eq have v': "v2 - v1 : U" by simp 
   1.372 -  have v: "v2 - v1 : V" by (simp!) 
   1.373 -  with eq have u': "u1 - u2 : V" by simp
   1.374 +  have u: "u1 - u2 \<in> U" by (simp!) 
   1.375 +  with eq have v': "v2 - v1 \<in> U" by simp 
   1.376 +  have v: "v2 - v1 \<in> V" by (simp!) 
   1.377 +  with eq have u': "u1 - u2 \<in> V" by simp
   1.378    
   1.379    show "u1 = u2"
   1.380    proof (rule vs_add_minus_eq)
   1.381 -    show "u1 - u2 = 00" by (rule Int_singletonD [OF _ u u']) 
   1.382 -    show "u1 : E" ..
   1.383 -    show "u2 : E" ..
   1.384 +    show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u']) 
   1.385 +    show "u1 \<in> E" ..
   1.386 +    show "u2 \<in> E" ..
   1.387    qed
   1.388  
   1.389    show "v1 = v2"
   1.390    proof (rule vs_add_minus_eq [RS sym])
   1.391 -    show "v2 - v1 = 00" by (rule Int_singletonD [OF _ v' v])
   1.392 -    show "v1 : E" ..
   1.393 -    show "v2 : E" ..
   1.394 +    show "v2 - v1 = 0" by (rule Int_singletonD [OF _ v' v])
   1.395 +    show "v1 \<in> E" ..
   1.396 +    show "v2 \<in> E" ..
   1.397    qed
   1.398  qed
   1.399  
   1.400  text {* An application of the previous lemma will be used in the proof
   1.401 -of the Hahn-Banach Theorem (see page \pageref{decomp-H0-use}): for any
   1.402 +of the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any
   1.403  element $y + a \mult x_0$ of the direct sum of a vectorspace $H$ and
   1.404  the linear closure of $x_0$ the components $y \in H$ and $a$ are
   1.405  uniquely determined. *}
   1.406  
   1.407 -lemma decomp_H0: 
   1.408 -  "[| is_vectorspace E; is_subspace H E; y1 : H; y2 : H; 
   1.409 -  x0 ~: H; x0 : E; x0 ~= 00; y1 + a1 (*) x0 = y2 + a2 (*) x0 |]
   1.410 -  ==> y1 = y2 & a1 = a2"
   1.411 +lemma decomp_H': 
   1.412 +  "[| is_vectorspace E; is_subspace H E; y1 \<in> H; y2 \<in> H; 
   1.413 +  x' \<notin> H; x' \<in> E; x' \<noteq> 0; y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x' |]
   1.414 +  ==> y1 = y2 \<and> a1 = a2"
   1.415  proof
   1.416    assume "is_vectorspace E" and h: "is_subspace H E"
   1.417 -     and "y1 : H" "y2 : H" "x0 ~: H" "x0 : E" "x0 ~= 00" 
   1.418 -         "y1 + a1 (*) x0 = y2 + a2 (*) x0"
   1.419 +     and "y1 \<in> H" "y2 \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" 
   1.420 +         "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
   1.421  
   1.422 -  have c: "y1 = y2 & a1 (*) x0 = a2 (*) x0"
   1.423 +  have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
   1.424    proof (rule decomp) 
   1.425 -    show "a1 (*) x0 : lin x0" .. 
   1.426 -    show "a2 (*) x0 : lin x0" ..
   1.427 -    show "H Int (lin x0) = {00}" 
   1.428 +    show "a1 \<cdot> x' \<in> lin x'" .. 
   1.429 +    show "a2 \<cdot> x' \<in> lin x'" ..
   1.430 +    show "H \<inter> (lin x') = {0}" 
   1.431      proof
   1.432 -      show "H Int lin x0 <= {00}" 
   1.433 +      show "H \<inter> lin x' <= {0}" 
   1.434        proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2])
   1.435 -        fix x assume "x:H" "x : lin x0" 
   1.436 -        thus "x = 00"
   1.437 +        fix x assume "x \<in> H" "x \<in> lin x'" 
   1.438 +        thus "x = 0"
   1.439          proof (unfold lin_def, elim CollectE exE conjE)
   1.440 -          fix a assume "x = a (*) x0"
   1.441 +          fix a assume "x = a \<cdot> x'"
   1.442            show ?thesis
   1.443            proof cases
   1.444              assume "a = (#0::real)" show ?thesis by (simp!)
   1.445            next
   1.446 -            assume "a ~= (#0::real)" 
   1.447 -            from h have "rinv a (*) a (*) x0 : H" 
   1.448 +            assume "a \<noteq> (#0::real)" 
   1.449 +            from h have "rinv a \<cdot> a \<cdot> x' \<in> H" 
   1.450                by (rule subspace_mult_closed) (simp!)
   1.451 -            also have "rinv a (*) a (*) x0 = x0" by (simp!)
   1.452 -            finally have "x0 : H" .
   1.453 +            also have "rinv a \<cdot> a \<cdot> x' = x'" by (simp!)
   1.454 +            finally have "x' \<in> H" .
   1.455              thus ?thesis by contradiction
   1.456            qed
   1.457         qed
   1.458        qed
   1.459 -      show "{00} <= H Int lin x0"
   1.460 +      show "{0} <= H \<inter> lin x'"
   1.461        proof -
   1.462 -	have "00: H Int lin x0"
   1.463 +	have "0 \<in> H \<inter> lin x'"
   1.464  	proof (rule IntI)
   1.465 -	  show "00:H" ..
   1.466 -	  from lin_vs show "00 : lin x0" ..
   1.467 +	  show "0 \<in> H" ..
   1.468 +	  from lin_vs show "0 \<in> lin x'" ..
   1.469  	qed
   1.470  	thus ?thesis by simp
   1.471        qed
   1.472      qed
   1.473 -    show "is_subspace (lin x0) E" ..
   1.474 +    show "is_subspace (lin x') E" ..
   1.475    qed
   1.476    
   1.477    from c show "y1 = y2" by simp
   1.478    
   1.479    show  "a1 = a2" 
   1.480    proof (rule vs_mult_right_cancel [RS iffD1])
   1.481 -    from c show "a1 (*) x0 = a2 (*) x0" by simp
   1.482 +    from c show "a1 \<cdot> x' = a2 \<cdot> x'" by simp
   1.483    qed
   1.484  qed
   1.485  
   1.486 -text {* Since for any element $y + a \mult x_0$ of the direct sum 
   1.487 -of a vectorspace $H$ and the linear closure of $x_0$ the components
   1.488 +text {* Since for any element $y + a \mult x'$ of the direct sum 
   1.489 +of a vectorspace $H$ and the linear closure of $x'$ the components
   1.490  $y\in H$ and $a$ are unique, it follows from $y\in H$ that 
   1.491  $a = 0$.*} 
   1.492  
   1.493 -lemma decomp_H0_H: 
   1.494 -  "[| is_vectorspace E; is_subspace H E; t:H; x0 ~: H; x0:E;
   1.495 -  x0 ~= 00 |] 
   1.496 -  ==> (SOME (y, a). t = y + a (*) x0 & y : H) = (t, (#0::real))"
   1.497 +lemma decomp_H'_H: 
   1.498 +  "[| is_vectorspace E; is_subspace H E; t \<in> H; x' \<notin> H; x' \<in> E;
   1.499 +  x' \<noteq> 0 |] 
   1.500 +  ==> (SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, (#0::real))"
   1.501  proof (rule, unfold split_tupled_all)
   1.502 -  assume "is_vectorspace E" "is_subspace H E" "t:H" "x0 ~: H" "x0:E"
   1.503 -    "x0 ~= 00"
   1.504 +  assume "is_vectorspace E" "is_subspace H E" "t \<in> H" "x' \<notin> H" "x' \<in> E"
   1.505 +    "x' \<noteq> 0"
   1.506    have h: "is_vectorspace H" ..
   1.507 -  fix y a presume t1: "t = y + a (*) x0" and "y:H"
   1.508 -  have "y = t & a = (#0::real)" 
   1.509 -    by (rule decomp_H0) (assumption | (simp!))+
   1.510 +  fix y a presume t1: "t = y + a \<cdot> x'" and "y \<in> H"
   1.511 +  have "y = t \<and> a = (#0::real)" 
   1.512 +    by (rule decomp_H') (assumption | (simp!))+
   1.513    thus "(y, a) = (t, (#0::real))" by (simp!)
   1.514  qed (simp!)+
   1.515  
   1.516 -text {* The components $y\in H$ and $a$ in $y \plus a \mult x_0$ 
   1.517 -are unique, so the function $h_0$ defined by 
   1.518 -$h_0 (y \plus a \mult x_0) = h y + a \cdot \xi$ is definite. *}
   1.519 +text {* The components $y\in H$ and $a$ in $y \plus a \mult x'$ 
   1.520 +are unique, so the function $h'$ defined by 
   1.521 +$h' (y \plus a \mult x') = h y + a \cdot \xi$ is definite. *}
   1.522  
   1.523 -lemma h0_definite:
   1.524 -  "[| h0 == (\\<lambda>x. let (y, a) = SOME (y, a). (x = y + a (*) x0 & y:H)
   1.525 +lemma h'_definite:
   1.526 +  "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
   1.527                  in (h y) + a * xi);
   1.528 -  x = y + a (*) x0; is_vectorspace E; is_subspace H E;
   1.529 -  y:H; x0 ~: H; x0:E; x0 ~= 00 |]
   1.530 -  ==> h0 x = h y + a * xi"
   1.531 +  x = y + a \<cdot> x'; is_vectorspace E; is_subspace H E;
   1.532 +  y \<in> H; x' \<notin> H; x' \<in> E; x' \<noteq> 0 |]
   1.533 +  ==> h' x = h y + a * xi"
   1.534  proof -  
   1.535    assume 
   1.536 -    "h0 == (\\<lambda>x. let (y, a) = SOME (y, a). (x = y + a (*) x0 & y:H)
   1.537 +    "h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
   1.538                 in (h y) + a * xi)"
   1.539 -    "x = y + a (*) x0" "is_vectorspace E" "is_subspace H E"
   1.540 -    "y:H" "x0 ~: H" "x0:E" "x0 ~= 00"
   1.541 -  have "x : H + (lin x0)" 
   1.542 +    "x = y + a \<cdot> x'" "is_vectorspace E" "is_subspace H E"
   1.543 +    "y \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
   1.544 +  have "x \<in> H + (lin x')" 
   1.545      by (simp! add: vs_sum_def lin_def) force+
   1.546 -  have "EX! xa. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)" 
   1.547 +  have "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)" 
   1.548    proof
   1.549 -    show "EX xa. ((\\<lambda>(y, a). x = y + a (*) x0 & y:H) xa)"
   1.550 +    show "\<exists>xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
   1.551        by (force!)
   1.552    next
   1.553      fix xa ya
   1.554 -    assume "(\\<lambda>(y,a). x = y + a (*) x0 & y : H) xa"
   1.555 -           "(\\<lambda>(y,a). x = y + a (*) x0 & y : H) ya"
   1.556 +    assume "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) xa"
   1.557 +           "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) ya"
   1.558      show "xa = ya" 
   1.559      proof -
   1.560 -      show "fst xa = fst ya & snd xa = snd ya ==> xa = ya" 
   1.561 +      show "fst xa = fst ya \<and> snd xa = snd ya ==> xa = ya" 
   1.562          by (simp add: Pair_fst_snd_eq)
   1.563 -      have x: "x = fst xa + snd xa (*) x0 & fst xa : H" 
   1.564 +      have x: "x = fst xa + snd xa \<cdot> x' \<and> fst xa \<in> H" 
   1.565          by (force!)
   1.566 -      have y: "x = fst ya + snd ya (*) x0 & fst ya : H" 
   1.567 +      have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H" 
   1.568          by (force!)
   1.569 -      from x y show "fst xa = fst ya & snd xa = snd ya" 
   1.570 -        by (elim conjE) (rule decomp_H0, (simp!)+)
   1.571 +      from x y show "fst xa = fst ya \<and> snd xa = snd ya" 
   1.572 +        by (elim conjE) (rule decomp_H', (simp!)+)
   1.573      qed
   1.574    qed
   1.575 -  hence eq: "(SOME (y, a). x = y + a (*) x0 & y:H) = (y, a)" 
   1.576 +  hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)" 
   1.577      by (rule select1_equality) (force!)
   1.578 -  thus "h0 x = h y + a * xi" by (simp! add: Let_def)
   1.579 +  thus "h' x = h y + a * xi" by (simp! add: Let_def)
   1.580  qed
   1.581  
   1.582  end
   1.583 \ No newline at end of file