src/HOL/Real/HahnBanach/VectorSpace.thy
 author kleing Wed Apr 14 14:13:05 2004 +0200 (2004-04-14) changeset 14565 c6dc17aab88a parent 13515 a6a7025fd7e8 child 14710 247615bfffb8 permissions -rw-r--r--
use more symbols in HTML output
     1 (*  Title:      HOL/Real/HahnBanach/VectorSpace.thy

     2     ID:         $Id$

     3     Author:     Gertrud Bauer, TU Munich

     4 *)

     5

     6 header {* Vector spaces *}

     7

     8 theory VectorSpace = Aux:

     9

    10 subsection {* Signature *}

    11

    12 text {*

    13   For the definition of real vector spaces a type @{typ 'a} of the

    14   sort @{text "{plus, minus, zero}"} is considered, on which a real

    15   scalar multiplication @{text \<cdot>} is declared.

    16 *}

    17

    18 consts

    19   prod  :: "real \<Rightarrow> 'a::{plus, minus, zero} \<Rightarrow> 'a"     (infixr "'(*')" 70)

    20

    21 syntax (xsymbols)

    22   prod  :: "real \<Rightarrow> 'a \<Rightarrow> 'a"                          (infixr "\<cdot>" 70)

    23 syntax (HTML output)

    24   prod  :: "real \<Rightarrow> 'a \<Rightarrow> 'a"                          (infixr "\<cdot>" 70)

    25

    26

    27 subsection {* Vector space laws *}

    28

    29 text {*

    30   A \emph{vector space} is a non-empty set @{text V} of elements from

    31   @{typ 'a} with the following vector space laws: The set @{text V} is

    32   closed under addition and scalar multiplication, addition is

    33   associative and commutative; @{text "- x"} is the inverse of @{text

    34   x} w.~r.~t.~addition and @{text 0} is the neutral element of

    35   addition.  Addition and multiplication are distributive; scalar

    36   multiplication is associative and the real number @{text "1"} is

    37   the neutral element of scalar multiplication.

    38 *}

    39

    40 locale vectorspace = var V +

    41   assumes non_empty [iff, intro?]: "V \<noteq> {}"

    42     and add_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y \<in> V"

    43     and mult_closed [iff]: "x \<in> V \<Longrightarrow> a \<cdot> x \<in> V"

    44     and add_assoc: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y) + z = x + (y + z)"

    45     and add_commute: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = y + x"

    46     and diff_self [simp]: "x \<in> V \<Longrightarrow> x - x = 0"

    47     and add_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 + x = x"

    48     and add_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"

    49     and add_mult_distrib2: "x \<in> V \<Longrightarrow> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"

    50     and mult_assoc: "x \<in> V \<Longrightarrow> (a * b) \<cdot> x = a \<cdot> (b \<cdot> x)"

    51     and mult_1 [simp]: "x \<in> V \<Longrightarrow> 1 \<cdot> x = x"

    52     and negate_eq1: "x \<in> V \<Longrightarrow> - x = (- 1) \<cdot> x"

    53     and diff_eq1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = x + - y"

    54

    55 lemma (in vectorspace) negate_eq2: "x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"

    56   by (rule negate_eq1 [symmetric])

    57

    58 lemma (in vectorspace) negate_eq2a: "x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"

    59   by (simp add: negate_eq1)

    60

    61 lemma (in vectorspace) diff_eq2: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"

    62   by (rule diff_eq1 [symmetric])

    63

    64 lemma (in vectorspace) diff_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"

    65   by (simp add: diff_eq1 negate_eq1)

    66

    67 lemma (in vectorspace) neg_closed [iff]: "x \<in> V \<Longrightarrow> - x \<in> V"

    68   by (simp add: negate_eq1)

    69

    70 lemma (in vectorspace) add_left_commute:

    71   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> x + (y + z) = y + (x + z)"

    72 proof -

    73   assume xyz: "x \<in> V"  "y \<in> V"  "z \<in> V"

    74   hence "x + (y + z) = (x + y) + z"

    75     by (simp only: add_assoc)

    76   also from xyz have "... = (y + x) + z" by (simp only: add_commute)

    77   also from xyz have "... = y + (x + z)" by (simp only: add_assoc)

    78   finally show ?thesis .

    79 qed

    80

    81 theorems (in vectorspace) add_ac =

    82   add_assoc add_commute add_left_commute

    83

    84

    85 text {* The existence of the zero element of a vector space

    86   follows from the non-emptiness of carrier set. *}

    87

    88 lemma (in vectorspace) zero [iff]: "0 \<in> V"

    89 proof -

    90   from non_empty obtain x where x: "x \<in> V" by blast

    91   then have "0 = x - x" by (rule diff_self [symmetric])

    92   also from x have "... \<in> V" by (rule diff_closed)

    93   finally show ?thesis .

    94 qed

    95

    96 lemma (in vectorspace) add_zero_right [simp]:

    97   "x \<in> V \<Longrightarrow>  x + 0 = x"

    98 proof -

    99   assume x: "x \<in> V"

   100   from this and zero have "x + 0 = 0 + x" by (rule add_commute)

   101   also from x have "... = x" by (rule add_zero_left)

   102   finally show ?thesis .

   103 qed

   104

   105 lemma (in vectorspace) mult_assoc2:

   106     "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"

   107   by (simp only: mult_assoc)

   108

   109 lemma (in vectorspace) diff_mult_distrib1:

   110     "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"

   111   by (simp add: diff_eq1 negate_eq1 add_mult_distrib1 mult_assoc2)

   112

   113 lemma (in vectorspace) diff_mult_distrib2:

   114   "x \<in> V \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"

   115 proof -

   116   assume x: "x \<in> V"

   117   have " (a - b) \<cdot> x = (a + - b) \<cdot> x"

   118     by (simp add: real_diff_def)

   119   also have "... = a \<cdot> x + (- b) \<cdot> x"

   120     by (rule add_mult_distrib2)

   121   also from x have "... = a \<cdot> x + - (b \<cdot> x)"

   122     by (simp add: negate_eq1 mult_assoc2)

   123   also from x have "... = a \<cdot> x - (b \<cdot> x)"

   124     by (simp add: diff_eq1)

   125   finally show ?thesis .

   126 qed

   127

   128 lemmas (in vectorspace) distrib =

   129   add_mult_distrib1 add_mult_distrib2

   130   diff_mult_distrib1 diff_mult_distrib2

   131

   132

   133 text {* \medskip Further derived laws: *}

   134

   135 lemma (in vectorspace) mult_zero_left [simp]:

   136   "x \<in> V \<Longrightarrow> 0 \<cdot> x = 0"

   137 proof -

   138   assume x: "x \<in> V"

   139   have "0 \<cdot> x = (1 - 1) \<cdot> x" by simp

   140   also have "... = (1 + - 1) \<cdot> x" by simp

   141   also have "... =  1 \<cdot> x + (- 1) \<cdot> x"

   142     by (rule add_mult_distrib2)

   143   also from x have "... = x + (- 1) \<cdot> x" by simp

   144   also from x have "... = x + - x" by (simp add: negate_eq2a)

   145   also from x have "... = x - x" by (simp add: diff_eq2)

   146   also from x have "... = 0" by simp

   147   finally show ?thesis .

   148 qed

   149

   150 lemma (in vectorspace) mult_zero_right [simp]:

   151   "a \<cdot> 0 = (0::'a)"

   152 proof -

   153   have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))" by simp

   154   also have "... =  a \<cdot> 0 - a \<cdot> 0"

   155     by (rule diff_mult_distrib1) simp_all

   156   also have "... = 0" by simp

   157   finally show ?thesis .

   158 qed

   159

   160 lemma (in vectorspace) minus_mult_cancel [simp]:

   161     "x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"

   162   by (simp add: negate_eq1 mult_assoc2)

   163

   164 lemma (in vectorspace) add_minus_left_eq_diff:

   165   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + y = y - x"

   166 proof -

   167   assume xy: "x \<in> V"  "y \<in> V"

   168   hence "- x + y = y + - x" by (simp add: add_commute)

   169   also from xy have "... = y - x" by (simp add: diff_eq1)

   170   finally show ?thesis .

   171 qed

   172

   173 lemma (in vectorspace) add_minus [simp]:

   174     "x \<in> V \<Longrightarrow> x + - x = 0"

   175   by (simp add: diff_eq2)

   176

   177 lemma (in vectorspace) add_minus_left [simp]:

   178     "x \<in> V \<Longrightarrow> - x + x = 0"

   179   by (simp add: diff_eq2 add_commute)

   180

   181 lemma (in vectorspace) minus_minus [simp]:

   182     "x \<in> V \<Longrightarrow> - (- x) = x"

   183   by (simp add: negate_eq1 mult_assoc2)

   184

   185 lemma (in vectorspace) minus_zero [simp]:

   186     "- (0::'a) = 0"

   187   by (simp add: negate_eq1)

   188

   189 lemma (in vectorspace) minus_zero_iff [simp]:

   190   "x \<in> V \<Longrightarrow> (- x = 0) = (x = 0)"

   191 proof

   192   assume x: "x \<in> V"

   193   {

   194     from x have "x = - (- x)" by (simp add: minus_minus)

   195     also assume "- x = 0"

   196     also have "- ... = 0" by (rule minus_zero)

   197     finally show "x = 0" .

   198   next

   199     assume "x = 0"

   200     then show "- x = 0" by simp

   201   }

   202 qed

   203

   204 lemma (in vectorspace) add_minus_cancel [simp]:

   205     "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"

   206   by (simp add: add_assoc [symmetric] del: add_commute)

   207

   208 lemma (in vectorspace) minus_add_cancel [simp]:

   209     "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"

   210   by (simp add: add_assoc [symmetric] del: add_commute)

   211

   212 lemma (in vectorspace) minus_add_distrib [simp]:

   213     "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - (x + y) = - x + - y"

   214   by (simp add: negate_eq1 add_mult_distrib1)

   215

   216 lemma (in vectorspace) diff_zero [simp]:

   217     "x \<in> V \<Longrightarrow> x - 0 = x"

   218   by (simp add: diff_eq1)

   219

   220 lemma (in vectorspace) diff_zero_right [simp]:

   221     "x \<in> V \<Longrightarrow> 0 - x = - x"

   222   by (simp add: diff_eq1)

   223

   224 lemma (in vectorspace) add_left_cancel:

   225   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y = x + z) = (y = z)"

   226 proof

   227   assume x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"

   228   {

   229     from y have "y = 0 + y" by simp

   230     also from x y have "... = (- x + x) + y" by simp

   231     also from x y have "... = - x + (x + y)"

   232       by (simp add: add_assoc neg_closed)

   233     also assume "x + y = x + z"

   234     also from x z have "- x + (x + z) = - x + x + z"

   235       by (simp add: add_assoc [symmetric] neg_closed)

   236     also from x z have "... = z" by simp

   237     finally show "y = z" .

   238   next

   239     assume "y = z"

   240     then show "x + y = x + z" by (simp only:)

   241   }

   242 qed

   243

   244 lemma (in vectorspace) add_right_cancel:

   245     "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (y + x = z + x) = (y = z)"

   246   by (simp only: add_commute add_left_cancel)

   247

   248 lemma (in vectorspace) add_assoc_cong:

   249   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x' \<in> V \<Longrightarrow> y' \<in> V \<Longrightarrow> z \<in> V

   250     \<Longrightarrow> x + y = x' + y' \<Longrightarrow> x + (y + z) = x' + (y' + z)"

   251   by (simp only: add_assoc [symmetric])

   252

   253 lemma (in vectorspace) mult_left_commute:

   254     "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"

   255   by (simp add: real_mult_commute mult_assoc2)

   256

   257 lemma (in vectorspace) mult_zero_uniq:

   258   "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> a \<cdot> x = 0 \<Longrightarrow> a = 0"

   259 proof (rule classical)

   260   assume a: "a \<noteq> 0"

   261   assume x: "x \<in> V"  "x \<noteq> 0" and ax: "a \<cdot> x = 0"

   262   from x a have "x = (inverse a * a) \<cdot> x" by simp

   263   also have "... = inverse a \<cdot> (a \<cdot> x)" by (rule mult_assoc)

   264   also from ax have "... = inverse a \<cdot> 0" by simp

   265   also have "... = 0" by simp

   266   finally have "x = 0" .

   267   thus "a = 0" by contradiction

   268 qed

   269

   270 lemma (in vectorspace) mult_left_cancel:

   271   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (a \<cdot> x = a \<cdot> y) = (x = y)"

   272 proof

   273   assume x: "x \<in> V" and y: "y \<in> V" and a: "a \<noteq> 0"

   274   from x have "x = 1 \<cdot> x" by simp

   275   also from a have "... = (inverse a * a) \<cdot> x" by simp

   276   also from x have "... = inverse a \<cdot> (a \<cdot> x)"

   277     by (simp only: mult_assoc)

   278   also assume "a \<cdot> x = a \<cdot> y"

   279   also from a y have "inverse a \<cdot> ... = y"

   280     by (simp add: mult_assoc2)

   281   finally show "x = y" .

   282 next

   283   assume "x = y"

   284   then show "a \<cdot> x = a \<cdot> y" by (simp only:)

   285 qed

   286

   287 lemma (in vectorspace) mult_right_cancel:

   288   "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> (a \<cdot> x = b \<cdot> x) = (a = b)"

   289 proof

   290   assume x: "x \<in> V" and neq: "x \<noteq> 0"

   291   {

   292     from x have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"

   293       by (simp add: diff_mult_distrib2)

   294     also assume "a \<cdot> x = b \<cdot> x"

   295     with x have "a \<cdot> x - b \<cdot> x = 0" by simp

   296     finally have "(a - b) \<cdot> x = 0" .

   297     with x neq have "a - b = 0" by (rule mult_zero_uniq)

   298     thus "a = b" by simp

   299   next

   300     assume "a = b"

   301     then show "a \<cdot> x = b \<cdot> x" by (simp only:)

   302   }

   303 qed

   304

   305 lemma (in vectorspace) eq_diff_eq:

   306   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x = z - y) = (x + y = z)"

   307 proof

   308   assume x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"

   309   {

   310     assume "x = z - y"

   311     hence "x + y = z - y + y" by simp

   312     also from y z have "... = z + - y + y"

   313       by (simp add: diff_eq1)

   314     also have "... = z + (- y + y)"

   315       by (rule add_assoc) (simp_all add: y z)

   316     also from y z have "... = z + 0"

   317       by (simp only: add_minus_left)

   318     also from z have "... = z"

   319       by (simp only: add_zero_right)

   320     finally show "x + y = z" .

   321   next

   322     assume "x + y = z"

   323     hence "z - y = (x + y) - y" by simp

   324     also from x y have "... = x + y + - y"

   325       by (simp add: diff_eq1)

   326     also have "... = x + (y + - y)"

   327       by (rule add_assoc) (simp_all add: x y)

   328     also from x y have "... = x" by simp

   329     finally show "x = z - y" ..

   330   }

   331 qed

   332

   333 lemma (in vectorspace) add_minus_eq_minus:

   334   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = 0 \<Longrightarrow> x = - y"

   335 proof -

   336   assume x: "x \<in> V" and y: "y \<in> V"

   337   from x y have "x = (- y + y) + x" by simp

   338   also from x y have "... = - y + (x + y)" by (simp add: add_ac)

   339   also assume "x + y = 0"

   340   also from y have "- y + 0 = - y" by simp

   341   finally show "x = - y" .

   342 qed

   343

   344 lemma (in vectorspace) add_minus_eq:

   345   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = 0 \<Longrightarrow> x = y"

   346 proof -

   347   assume x: "x \<in> V" and y: "y \<in> V"

   348   assume "x - y = 0"

   349   with x y have eq: "x + - y = 0" by (simp add: diff_eq1)

   350   with _ _ have "x = - (- y)"

   351     by (rule add_minus_eq_minus) (simp_all add: x y)

   352   with x y show "x = y" by simp

   353 qed

   354

   355 lemma (in vectorspace) add_diff_swap:

   356   "a \<in> V \<Longrightarrow> b \<in> V \<Longrightarrow> c \<in> V \<Longrightarrow> d \<in> V \<Longrightarrow> a + b = c + d

   357     \<Longrightarrow> a - c = d - b"

   358 proof -

   359   assume vs: "a \<in> V"  "b \<in> V"  "c \<in> V"  "d \<in> V"

   360     and eq: "a + b = c + d"

   361   then have "- c + (a + b) = - c + (c + d)"

   362     by (simp add: add_left_cancel)

   363   also have "... = d" by (rule minus_add_cancel)

   364   finally have eq: "- c + (a + b) = d" .

   365   from vs have "a - c = (- c + (a + b)) + - b"

   366     by (simp add: add_ac diff_eq1)

   367   also from vs eq have "...  = d + - b"

   368     by (simp add: add_right_cancel)

   369   also from vs have "... = d - b" by (simp add: diff_eq2)

   370   finally show "a - c = d - b" .

   371 qed

   372

   373 lemma (in vectorspace) vs_add_cancel_21:

   374   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> u \<in> V

   375     \<Longrightarrow> (x + (y + z) = y + u) = (x + z = u)"

   376 proof

   377   assume vs: "x \<in> V"  "y \<in> V"  "z \<in> V"  "u \<in> V"

   378   {

   379     from vs have "x + z = - y + y + (x + z)" by simp

   380     also have "... = - y + (y + (x + z))"

   381       by (rule add_assoc) (simp_all add: vs)

   382     also from vs have "y + (x + z) = x + (y + z)"

   383       by (simp add: add_ac)

   384     also assume "x + (y + z) = y + u"

   385     also from vs have "- y + (y + u) = u" by simp

   386     finally show "x + z = u" .

   387   next

   388     assume "x + z = u"

   389     with vs show "x + (y + z) = y + u"

   390       by (simp only: add_left_commute [of x])

   391   }

   392 qed

   393

   394 lemma (in vectorspace) add_cancel_end:

   395   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + (y + z) = y) = (x = - z)"

   396 proof

   397   assume vs: "x \<in> V"  "y \<in> V"  "z \<in> V"

   398   {

   399     assume "x + (y + z) = y"

   400     with vs have "(x + z) + y = 0 + y"

   401       by (simp add: add_ac)

   402     with vs have "x + z = 0"

   403       by (simp only: add_right_cancel add_closed zero)

   404     with vs show "x = - z" by (simp add: add_minus_eq_minus)

   405   next

   406     assume eq: "x = - z"

   407     hence "x + (y + z) = - z + (y + z)" by simp

   408     also have "... = y + (- z + z)"

   409       by (rule add_left_commute) (simp_all add: vs)

   410     also from vs have "... = y"  by simp

   411     finally show "x + (y + z) = y" .

   412   }

   413 qed

   414

   415 end