src/HOL/Hahn_Banach/Vector_Space.thy
author wenzelm
Sat Jan 05 17:24:33 2019 +0100 (10 months ago)
changeset 69597 ff784d5a5bfb
parent 61879 e4f9d8f094fe
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isabelle update -u control_cartouches;
     1 (*  Title:      HOL/Hahn_Banach/Vector_Space.thy
     2     Author:     Gertrud Bauer, TU Munich
     3 *)
     4 
     5 section \<open>Vector spaces\<close>
     6 
     7 theory Vector_Space
     8 imports Complex_Main Bounds
     9 begin
    10 
    11 subsection \<open>Signature\<close>
    12 
    13 text \<open>
    14   For the definition of real vector spaces a type \<^typ>\<open>'a\<close> of the sort
    15   \<open>{plus, minus, zero}\<close> is considered, on which a real scalar multiplication
    16   \<open>\<cdot>\<close> is declared.
    17 \<close>
    18 
    19 consts
    20   prod :: "real \<Rightarrow> 'a::{plus,minus,zero} \<Rightarrow> 'a"  (infixr "\<cdot>" 70)
    21 
    22 
    23 subsection \<open>Vector space laws\<close>
    24 
    25 text \<open>
    26   A \<^emph>\<open>vector space\<close> is a non-empty set \<open>V\<close> of elements from \<^typ>\<open>'a\<close> with the
    27   following vector space laws: The set \<open>V\<close> is closed under addition and scalar
    28   multiplication, addition is associative and commutative; \<open>- x\<close> is the
    29   inverse of \<open>x\<close> wrt.\ addition and \<open>0\<close> is the neutral element of addition.
    30   Addition and multiplication are distributive; scalar multiplication is
    31   associative and the real number \<open>1\<close> is the neutral element of scalar
    32   multiplication.
    33 \<close>
    34 
    35 locale vectorspace =
    36   fixes V
    37   assumes non_empty [iff, intro?]: "V \<noteq> {}"
    38     and add_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y \<in> V"
    39     and mult_closed [iff]: "x \<in> V \<Longrightarrow> a \<cdot> x \<in> V"
    40     and add_assoc: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y) + z = x + (y + z)"
    41     and add_commute: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = y + x"
    42     and diff_self [simp]: "x \<in> V \<Longrightarrow> x - x = 0"
    43     and add_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 + x = x"
    44     and add_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
    45     and add_mult_distrib2: "x \<in> V \<Longrightarrow> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
    46     and mult_assoc: "x \<in> V \<Longrightarrow> (a * b) \<cdot> x = a \<cdot> (b \<cdot> x)"
    47     and mult_1 [simp]: "x \<in> V \<Longrightarrow> 1 \<cdot> x = x"
    48     and negate_eq1: "x \<in> V \<Longrightarrow> - x = (- 1) \<cdot> x"
    49     and diff_eq1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = x + - y"
    50 begin
    51 
    52 lemma negate_eq2: "x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"
    53   by (rule negate_eq1 [symmetric])
    54 
    55 lemma negate_eq2a: "x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"
    56   by (simp add: negate_eq1)
    57 
    58 lemma diff_eq2: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"
    59   by (rule diff_eq1 [symmetric])
    60 
    61 lemma diff_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"
    62   by (simp add: diff_eq1 negate_eq1)
    63 
    64 lemma neg_closed [iff]: "x \<in> V \<Longrightarrow> - x \<in> V"
    65   by (simp add: negate_eq1)
    66 
    67 lemma add_left_commute:
    68   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> x + (y + z) = y + (x + z)"
    69 proof -
    70   assume xyz: "x \<in> V"  "y \<in> V"  "z \<in> V"
    71   then have "x + (y + z) = (x + y) + z"
    72     by (simp only: add_assoc)
    73   also from xyz have "\<dots> = (y + x) + z" by (simp only: add_commute)
    74   also from xyz have "\<dots> = y + (x + z)" by (simp only: add_assoc)
    75   finally show ?thesis .
    76 qed
    77 
    78 lemmas add_ac = add_assoc add_commute add_left_commute
    79 
    80 
    81 text \<open>
    82   The existence of the zero element of a vector space follows from the
    83   non-emptiness of carrier set.
    84 \<close>
    85 
    86 lemma zero [iff]: "0 \<in> V"
    87 proof -
    88   from non_empty obtain x where x: "x \<in> V" by blast
    89   then have "0 = x - x" by (rule diff_self [symmetric])
    90   also from x x have "\<dots> \<in> V" by (rule diff_closed)
    91   finally show ?thesis .
    92 qed
    93 
    94 lemma add_zero_right [simp]: "x \<in> V \<Longrightarrow>  x + 0 = x"
    95 proof -
    96   assume x: "x \<in> V"
    97   from this and zero have "x + 0 = 0 + x" by (rule add_commute)
    98   also from x have "\<dots> = x" by (rule add_zero_left)
    99   finally show ?thesis .
   100 qed
   101 
   102 lemma mult_assoc2: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
   103   by (simp only: mult_assoc)
   104 
   105 lemma diff_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
   106   by (simp add: diff_eq1 negate_eq1 add_mult_distrib1 mult_assoc2)
   107 
   108 lemma diff_mult_distrib2: "x \<in> V \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
   109 proof -
   110   assume x: "x \<in> V"
   111   have " (a - b) \<cdot> x = (a + - b) \<cdot> x"
   112     by simp
   113   also from x have "\<dots> = a \<cdot> x + (- b) \<cdot> x"
   114     by (rule add_mult_distrib2)
   115   also from x have "\<dots> = a \<cdot> x + - (b \<cdot> x)"
   116     by (simp add: negate_eq1 mult_assoc2)
   117   also from x have "\<dots> = a \<cdot> x - (b \<cdot> x)"
   118     by (simp add: diff_eq1)
   119   finally show ?thesis .
   120 qed
   121 
   122 lemmas distrib =
   123   add_mult_distrib1 add_mult_distrib2
   124   diff_mult_distrib1 diff_mult_distrib2
   125 
   126 
   127 text \<open>\<^medskip> Further derived laws:\<close>
   128 
   129 lemma mult_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 \<cdot> x = 0"
   130 proof -
   131   assume x: "x \<in> V"
   132   have "0 \<cdot> x = (1 - 1) \<cdot> x" by simp
   133   also have "\<dots> = (1 + - 1) \<cdot> x" by simp
   134   also from x have "\<dots> =  1 \<cdot> x + (- 1) \<cdot> x"
   135     by (rule add_mult_distrib2)
   136   also from x have "\<dots> = x + (- 1) \<cdot> x" by simp
   137   also from x have "\<dots> = x + - x" by (simp add: negate_eq2a)
   138   also from x have "\<dots> = x - x" by (simp add: diff_eq2)
   139   also from x have "\<dots> = 0" by simp
   140   finally show ?thesis .
   141 qed
   142 
   143 lemma mult_zero_right [simp]: "a \<cdot> 0 = (0::'a)"
   144 proof -
   145   have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))" by simp
   146   also have "\<dots> =  a \<cdot> 0 - a \<cdot> 0"
   147     by (rule diff_mult_distrib1) simp_all
   148   also have "\<dots> = 0" by simp
   149   finally show ?thesis .
   150 qed
   151 
   152 lemma minus_mult_cancel [simp]: "x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"
   153   by (simp add: negate_eq1 mult_assoc2)
   154 
   155 lemma add_minus_left_eq_diff: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + y = y - x"
   156 proof -
   157   assume xy: "x \<in> V"  "y \<in> V"
   158   then have "- x + y = y + - x" by (simp add: add_commute)
   159   also from xy have "\<dots> = y - x" by (simp add: diff_eq1)
   160   finally show ?thesis .
   161 qed
   162 
   163 lemma add_minus [simp]: "x \<in> V \<Longrightarrow> x + - x = 0"
   164   by (simp add: diff_eq2)
   165 
   166 lemma add_minus_left [simp]: "x \<in> V \<Longrightarrow> - x + x = 0"
   167   by (simp add: diff_eq2 add_commute)
   168 
   169 lemma minus_minus [simp]: "x \<in> V \<Longrightarrow> - (- x) = x"
   170   by (simp add: negate_eq1 mult_assoc2)
   171 
   172 lemma minus_zero [simp]: "- (0::'a) = 0"
   173   by (simp add: negate_eq1)
   174 
   175 lemma minus_zero_iff [simp]:
   176   assumes x: "x \<in> V"
   177   shows "(- x = 0) = (x = 0)"
   178 proof
   179   from x have "x = - (- x)" by simp
   180   also assume "- x = 0"
   181   also have "- \<dots> = 0" by (rule minus_zero)
   182   finally show "x = 0" .
   183 next
   184   assume "x = 0"
   185   then show "- x = 0" by simp
   186 qed
   187 
   188 lemma add_minus_cancel [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"
   189   by (simp add: add_assoc [symmetric])
   190 
   191 lemma minus_add_cancel [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"
   192   by (simp add: add_assoc [symmetric])
   193 
   194 lemma minus_add_distrib [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - (x + y) = - x + - y"
   195   by (simp add: negate_eq1 add_mult_distrib1)
   196 
   197 lemma diff_zero [simp]: "x \<in> V \<Longrightarrow> x - 0 = x"
   198   by (simp add: diff_eq1)
   199 
   200 lemma diff_zero_right [simp]: "x \<in> V \<Longrightarrow> 0 - x = - x"
   201   by (simp add: diff_eq1)
   202 
   203 lemma add_left_cancel:
   204   assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
   205   shows "(x + y = x + z) = (y = z)"
   206 proof
   207   from y have "y = 0 + y" by simp
   208   also from x y have "\<dots> = (- x + x) + y" by simp
   209   also from x y have "\<dots> = - x + (x + y)" by (simp add: add.assoc)
   210   also assume "x + y = x + z"
   211   also from x z have "- x + (x + z) = - x + x + z" by (simp add: add.assoc)
   212   also from x z have "\<dots> = z" by simp
   213   finally show "y = z" .
   214 next
   215   assume "y = z"
   216   then show "x + y = x + z" by (simp only:)
   217 qed
   218 
   219 lemma add_right_cancel:
   220     "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (y + x = z + x) = (y = z)"
   221   by (simp only: add_commute add_left_cancel)
   222 
   223 lemma add_assoc_cong:
   224   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x' \<in> V \<Longrightarrow> y' \<in> V \<Longrightarrow> z \<in> V
   225     \<Longrightarrow> x + y = x' + y' \<Longrightarrow> x + (y + z) = x' + (y' + z)"
   226   by (simp only: add_assoc [symmetric])
   227 
   228 lemma mult_left_commute: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
   229   by (simp add: mult.commute mult_assoc2)
   230 
   231 lemma mult_zero_uniq:
   232   assumes x: "x \<in> V"  "x \<noteq> 0" and ax: "a \<cdot> x = 0"
   233   shows "a = 0"
   234 proof (rule classical)
   235   assume a: "a \<noteq> 0"
   236   from x a have "x = (inverse a * a) \<cdot> x" by simp
   237   also from \<open>x \<in> V\<close> have "\<dots> = inverse a \<cdot> (a \<cdot> x)" by (rule mult_assoc)
   238   also from ax have "\<dots> = inverse a \<cdot> 0" by simp
   239   also have "\<dots> = 0" by simp
   240   finally have "x = 0" .
   241   with \<open>x \<noteq> 0\<close> show "a = 0" by contradiction
   242 qed
   243 
   244 lemma mult_left_cancel:
   245   assumes x: "x \<in> V" and y: "y \<in> V" and a: "a \<noteq> 0"
   246   shows "(a \<cdot> x = a \<cdot> y) = (x = y)"
   247 proof
   248   from x have "x = 1 \<cdot> x" by simp
   249   also from a have "\<dots> = (inverse a * a) \<cdot> x" by simp
   250   also from x have "\<dots> = inverse a \<cdot> (a \<cdot> x)"
   251     by (simp only: mult_assoc)
   252   also assume "a \<cdot> x = a \<cdot> y"
   253   also from a y have "inverse a \<cdot> \<dots> = y"
   254     by (simp add: mult_assoc2)
   255   finally show "x = y" .
   256 next
   257   assume "x = y"
   258   then show "a \<cdot> x = a \<cdot> y" by (simp only:)
   259 qed
   260 
   261 lemma mult_right_cancel:
   262   assumes x: "x \<in> V" and neq: "x \<noteq> 0"
   263   shows "(a \<cdot> x = b \<cdot> x) = (a = b)"
   264 proof
   265   from x have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
   266     by (simp add: diff_mult_distrib2)
   267   also assume "a \<cdot> x = b \<cdot> x"
   268   with x have "a \<cdot> x - b \<cdot> x = 0" by simp
   269   finally have "(a - b) \<cdot> x = 0" .
   270   with x neq have "a - b = 0" by (rule mult_zero_uniq)
   271   then show "a = b" by simp
   272 next
   273   assume "a = b"
   274   then show "a \<cdot> x = b \<cdot> x" by (simp only:)
   275 qed
   276 
   277 lemma eq_diff_eq:
   278   assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
   279   shows "(x = z - y) = (x + y = z)"
   280 proof
   281   assume "x = z - y"
   282   then have "x + y = z - y + y" by simp
   283   also from y z have "\<dots> = z + - y + y"
   284     by (simp add: diff_eq1)
   285   also have "\<dots> = z + (- y + y)"
   286     by (rule add_assoc) (simp_all add: y z)
   287   also from y z have "\<dots> = z + 0"
   288     by (simp only: add_minus_left)
   289   also from z have "\<dots> = z"
   290     by (simp only: add_zero_right)
   291   finally show "x + y = z" .
   292 next
   293   assume "x + y = z"
   294   then have "z - y = (x + y) - y" by simp
   295   also from x y have "\<dots> = x + y + - y"
   296     by (simp add: diff_eq1)
   297   also have "\<dots> = x + (y + - y)"
   298     by (rule add_assoc) (simp_all add: x y)
   299   also from x y have "\<dots> = x" by simp
   300   finally show "x = z - y" ..
   301 qed
   302 
   303 lemma add_minus_eq_minus:
   304   assumes x: "x \<in> V" and y: "y \<in> V" and xy: "x + y = 0"
   305   shows "x = - y"
   306 proof -
   307   from x y have "x = (- y + y) + x" by simp
   308   also from x y have "\<dots> = - y + (x + y)" by (simp add: add_ac)
   309   also note xy
   310   also from y have "- y + 0 = - y" by simp
   311   finally show "x = - y" .
   312 qed
   313 
   314 lemma add_minus_eq:
   315   assumes x: "x \<in> V" and y: "y \<in> V" and xy: "x - y = 0"
   316   shows "x = y"
   317 proof -
   318   from x y xy have eq: "x + - y = 0" by (simp add: diff_eq1)
   319   with _ _ have "x = - (- y)"
   320     by (rule add_minus_eq_minus) (simp_all add: x y)
   321   with x y show "x = y" by simp
   322 qed
   323 
   324 lemma add_diff_swap:
   325   assumes vs: "a \<in> V"  "b \<in> V"  "c \<in> V"  "d \<in> V"
   326     and eq: "a + b = c + d"
   327   shows "a - c = d - b"
   328 proof -
   329   from assms have "- c + (a + b) = - c + (c + d)"
   330     by (simp add: add_left_cancel)
   331   also have "\<dots> = d" using \<open>c \<in> V\<close> \<open>d \<in> V\<close> by (rule minus_add_cancel)
   332   finally have eq: "- c + (a + b) = d" .
   333   from vs have "a - c = (- c + (a + b)) + - b"
   334     by (simp add: add_ac diff_eq1)
   335   also from vs eq have "\<dots>  = d + - b"
   336     by (simp add: add_right_cancel)
   337   also from vs have "\<dots> = d - b" by (simp add: diff_eq2)
   338   finally show "a - c = d - b" .
   339 qed
   340 
   341 lemma vs_add_cancel_21:
   342   assumes vs: "x \<in> V"  "y \<in> V"  "z \<in> V"  "u \<in> V"
   343   shows "(x + (y + z) = y + u) = (x + z = u)"
   344 proof
   345   from vs have "x + z = - y + y + (x + z)" by simp
   346   also have "\<dots> = - y + (y + (x + z))"
   347     by (rule add_assoc) (simp_all add: vs)
   348   also from vs have "y + (x + z) = x + (y + z)"
   349     by (simp add: add_ac)
   350   also assume "x + (y + z) = y + u"
   351   also from vs have "- y + (y + u) = u" by simp
   352   finally show "x + z = u" .
   353 next
   354   assume "x + z = u"
   355   with vs show "x + (y + z) = y + u"
   356     by (simp only: add_left_commute [of x])
   357 qed
   358 
   359 lemma add_cancel_end:
   360   assumes vs: "x \<in> V"  "y \<in> V"  "z \<in> V"
   361   shows "(x + (y + z) = y) = (x = - z)"
   362 proof
   363   assume "x + (y + z) = y"
   364   with vs have "(x + z) + y = 0 + y" by (simp add: add_ac)
   365   with vs have "x + z = 0" by (simp only: add_right_cancel add_closed zero)
   366   with vs show "x = - z" by (simp add: add_minus_eq_minus)
   367 next
   368   assume eq: "x = - z"
   369   then have "x + (y + z) = - z + (y + z)" by simp
   370   also have "\<dots> = y + (- z + z)" by (rule add_left_commute) (simp_all add: vs)
   371   also from vs have "\<dots> = y"  by simp
   372   finally show "x + (y + z) = y" .
   373 qed
   374 
   375 end
   376 
   377 end