src/HOL/Real/RealVector.thy
author huffman
Tue, 26 Aug 2008 23:49:46 +0200
changeset 28009 e93b121074fb
parent 27553 d315a513a150
child 28029 4c55cdec4ce7
permissions -rw-r--r--
move real_vector class proofs into vector_space and group_hom locales
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
27552
15cf4ed9c2a1 re-removed subclass relation real_field < field_char_0: coregularity violation in NSA/HyperDef
haftmann
parents: 27515
diff changeset
     1
(*  Title:      RealVector.thy
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
     2
    ID:         $Id$
27552
15cf4ed9c2a1 re-removed subclass relation real_field < field_char_0: coregularity violation in NSA/HyperDef
haftmann
parents: 27515
diff changeset
     3
    Author:     Brian Huffman
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
     4
*)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
     5
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
     6
header {* Vector Spaces and Algebras over the Reals *}
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
     7
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
     8
theory RealVector
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
     9
imports RealPow
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
    10
begin
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
    11
28009
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    12
subsection {* Group homomorphisms *}
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    13
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    14
locale group_hom =
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    15
  ab_group_add
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    16
    mA (infixl "-\<^sub>A" 65)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    17
    uA ("-\<^sub>A _" [81] 80)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    18
    zA ("0\<^sub>A")
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    19
    pA (infixl "+\<^sub>A" 65) +
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    20
  ab_group_add
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    21
    mB (infixl "-\<^sub>B" 65)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    22
    uB ("-\<^sub>B _" [81] 80)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    23
    zB ("0\<^sub>B")
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    24
    pB (infixl "+\<^sub>B" 65) +
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    25
  fixes f :: "'a \<Rightarrow> 'b"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    26
  assumes plus: "f (x +\<^sub>A y) = f x +\<^sub>B f y"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    27
begin
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    28
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    29
lemma zero: "f 0\<^sub>A = 0\<^sub>B"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    30
proof -
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    31
  have "f 0\<^sub>A +\<^sub>B f 0\<^sub>A = f (0\<^sub>A +\<^sub>A 0\<^sub>A)" by (rule plus [symmetric])
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    32
  also have "f (0\<^sub>A +\<^sub>A 0\<^sub>A) = 0\<^sub>B +\<^sub>B f 0\<^sub>A" by simp
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    33
  finally show "f 0\<^sub>A = 0\<^sub>B" by (rule pB.add_right_imp_eq)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    34
qed
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    35
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    36
lemma uminus: "f (-\<^sub>A x) = -\<^sub>B f x"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    37
proof -
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    38
  have "f (-\<^sub>A x) +\<^sub>B f x = f (-\<^sub>A x +\<^sub>A x)" by (rule plus [symmetric])
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    39
  also have "\<dots> = -\<^sub>B f x +\<^sub>B f x" by (simp add: zero)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    40
  finally show "f (-\<^sub>A x) = -\<^sub>B f x" by (rule pB.add_right_imp_eq)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    41
qed
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    42
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    43
lemma diff: "f (x -\<^sub>A y) = f x -\<^sub>B f y"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    44
by (simp add: mA_uA_zA_pA.diff_minus mB_uB_zB_pB.diff_minus plus uminus)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    45
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    46
text {* TODO:
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    47
Locale-ize definition of setsum, so we can prove a lemma for it *}
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    48
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    49
end
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    50
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    51
subsection {* Vector spaces *}
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    52
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    53
locale vector_space =
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    54
  field
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    55
    inverse
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    56
    divide (infixl "'/\<^sub>F" 70)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    57
    one ("1\<^sub>F")
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    58
    times (infixl "*\<^sub>F" 70)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    59
    mF (infixl "-\<^sub>F" 65)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    60
    uF ("-\<^sub>F _" [81] 80)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    61
    zF ("0\<^sub>F")
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    62
    pF (infixl "+\<^sub>F" 65) +
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    63
  ab_group_add
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    64
    mV (infixl "-\<^sub>V" 65)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    65
    uV ("-\<^sub>V _" [81] 80)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    66
    zV ("0\<^sub>V")
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    67
    pV (infixl "+\<^sub>V" 65) +
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    68
  fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" (infixr "%*" 75)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    69
  assumes scale_right_distrib: "scale a (x +\<^sub>V y) = scale a x +\<^sub>V scale a y"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    70
  and scale_left_distrib: "scale (a +\<^sub>F b) x = scale a x +\<^sub>V scale b x"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    71
  and scale_scale [simp]: "scale a (scale b x) = scale (a *\<^sub>F b) x"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    72
  and scale_one [simp]: "scale 1\<^sub>F x = x"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    73
begin
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    74
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    75
lemma scale_left_commute:
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    76
  "scale a (scale b x) = scale b (scale a x)"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    77
by (simp add: mult_commute)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    78
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    79
lemma scale_zero_left [simp]: "scale 0\<^sub>F x = 0\<^sub>V"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    80
  and scale_minus_left [simp]: "scale (-\<^sub>F a) x = -\<^sub>V (scale a x)"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    81
  and scale_left_diff_distrib: "scale (a -\<^sub>F b) x = scale a x -\<^sub>V scale b x"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    82
proof -
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    83
  interpret s: group_hom
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    84
    [mF uF zF pF mV uV zV pV "\<lambda>a. scale a x"]
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    85
    by unfold_locales (rule scale_left_distrib)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    86
  show "scale 0\<^sub>F x = 0\<^sub>V" by (rule s.zero)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    87
  show "scale (-\<^sub>F a) x = -\<^sub>V (scale a x)" by (rule s.uminus)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    88
  show "scale (a -\<^sub>F b) x = scale a x -\<^sub>V scale b x" by (rule s.diff)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    89
qed
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    90
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    91
lemma scale_zero_right [simp]: "scale a 0\<^sub>V = 0\<^sub>V"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    92
  and scale_minus_right [simp]: "scale a (-\<^sub>V x) = -\<^sub>V (scale a x)"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    93
  and scale_right_diff_distrib: "scale a (x -\<^sub>V y) = scale a x -\<^sub>V scale a y"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    94
proof -
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    95
  interpret s: group_hom
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    96
    [mV uV zV pV mV uV zV pV "\<lambda>x. scale a x"]
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    97
    by unfold_locales (rule scale_right_distrib)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    98
  show "scale a 0\<^sub>V = 0\<^sub>V" by (rule s.zero)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
    99
  show "scale a (-\<^sub>V x) = -\<^sub>V (scale a x)" by (rule s.uminus)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   100
  show "scale a (x -\<^sub>V y) = scale a x -\<^sub>V scale a y" by (rule s.diff)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   101
qed
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   102
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   103
lemma scale_eq_0_iff [simp]:
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   104
  "scale a x = 0\<^sub>V \<longleftrightarrow> a = 0\<^sub>F \<or> x = 0\<^sub>V"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   105
proof cases
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   106
  assume "a = 0\<^sub>F" thus ?thesis by simp
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   107
next
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   108
  assume anz [simp]: "a \<noteq> 0\<^sub>F"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   109
  { assume "scale a x = 0\<^sub>V"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   110
    hence "scale (inverse a) (scale a x) = 0\<^sub>V" by simp
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   111
    hence "x = 0\<^sub>V" by simp }
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   112
  thus ?thesis by force
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   113
qed
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   114
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   115
lemma scale_left_imp_eq:
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   116
  "\<lbrakk>a \<noteq> 0\<^sub>F; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   117
proof -
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   118
  assume nonzero: "a \<noteq> 0\<^sub>F"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   119
  assume "scale a x = scale a y"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   120
  hence "scale a (x -\<^sub>V y) = 0\<^sub>V"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   121
     by (simp add: scale_right_diff_distrib)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   122
  hence "x -\<^sub>V y = 0\<^sub>V" by (simp add: nonzero)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   123
  thus "x = y" by (simp only: mV_uV_zV_pV.right_minus_eq)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   124
qed
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   125
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   126
lemma scale_right_imp_eq:
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   127
  "\<lbrakk>x \<noteq> 0\<^sub>V; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   128
proof -
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   129
  assume nonzero: "x \<noteq> 0\<^sub>V"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   130
  assume "scale a x = scale b x"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   131
  hence "scale (a -\<^sub>F b) x = 0\<^sub>V"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   132
     by (simp add: scale_left_diff_distrib)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   133
  hence "a -\<^sub>F b = 0\<^sub>F" by (simp add: nonzero)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   134
  thus "a = b" by (simp only: mF_uF_zF_pF.right_minus_eq)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   135
qed
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   136
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   137
lemma scale_cancel_left:
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   138
  "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0\<^sub>F"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   139
by (auto intro: scale_left_imp_eq)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   140
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   141
lemma scale_cancel_right:
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   142
  "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0\<^sub>V"
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   143
by (auto intro: scale_right_imp_eq)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   144
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   145
end
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   146
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   147
(* TODO: locale additive is superseded by group_hom; remove *)
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   148
subsection {* Locale for additive functions *}
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   149
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   150
locale additive =
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   151
  fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   152
  assumes add: "f (x + y) = f x + f y"
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   153
begin
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   154
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   155
lemma zero: "f 0 = 0"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   156
proof -
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   157
  have "f 0 = f (0 + 0)" by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   158
  also have "\<dots> = f 0 + f 0" by (rule add)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   159
  finally show "f 0 = 0" by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   160
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   161
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   162
lemma minus: "f (- x) = - f x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   163
proof -
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   164
  have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   165
  also have "\<dots> = - f x + f x" by (simp add: zero)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   166
  finally show "f (- x) = - f x" by (rule add_right_imp_eq)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   167
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   168
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   169
lemma diff: "f (x - y) = f x - f y"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   170
by (simp add: diff_def add minus)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   171
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   172
lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
22942
bf718970e5ef add lemma additive.setsum
huffman
parents: 22912
diff changeset
   173
apply (cases "finite A")
bf718970e5ef add lemma additive.setsum
huffman
parents: 22912
diff changeset
   174
apply (induct set: finite)
bf718970e5ef add lemma additive.setsum
huffman
parents: 22912
diff changeset
   175
apply (simp add: zero)
bf718970e5ef add lemma additive.setsum
huffman
parents: 22912
diff changeset
   176
apply (simp add: add)
bf718970e5ef add lemma additive.setsum
huffman
parents: 22912
diff changeset
   177
apply (simp add: zero)
bf718970e5ef add lemma additive.setsum
huffman
parents: 22912
diff changeset
   178
done
bf718970e5ef add lemma additive.setsum
huffman
parents: 22912
diff changeset
   179
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   180
end
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   181
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   182
subsection {* Real vector spaces *}
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   183
22636
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   184
class scaleR = type +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   185
  fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
24748
ee0a0eb6b738 proper syntax during class specification
haftmann
parents: 24588
diff changeset
   186
begin
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   187
20763
052b348a98a9 rearranged axioms and simp rules for scaleR
huffman
parents: 20722
diff changeset
   188
abbreviation
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   189
  divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
24748
ee0a0eb6b738 proper syntax during class specification
haftmann
parents: 24588
diff changeset
   190
where
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   191
  "x /\<^sub>R r == scaleR (inverse r) x"
24748
ee0a0eb6b738 proper syntax during class specification
haftmann
parents: 24588
diff changeset
   192
ee0a0eb6b738 proper syntax during class specification
haftmann
parents: 24588
diff changeset
   193
end
ee0a0eb6b738 proper syntax during class specification
haftmann
parents: 24588
diff changeset
   194
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   195
instantiation real :: scaleR
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   196
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   197
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   198
definition
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   199
  real_scaleR_def [simp]: "scaleR a x = a * x"
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   200
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   201
instance ..
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   202
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   203
end
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   204
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   205
class real_vector = scaleR + ab_group_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   206
  assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   207
  and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   208
  and scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x"
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   209
  and scaleR_one [simp]: "scaleR 1 x = x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   210
28009
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   211
interpretation real_vector: vector_space
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   212
  [inverse divide "1" times minus uminus "0" plus
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   213
   minus uminus "0" plus "scaleR::real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"]
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   214
apply unfold_locales
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   215
apply (rule scaleR_right_distrib)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   216
apply (rule scaleR_left_distrib)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   217
apply (rule scaleR_scaleR)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   218
apply (rule scaleR_one)
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   219
done
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   220
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   221
text {* Recover original theorem names *}
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   222
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   223
lemmas scaleR_left_commute = real_vector.scale_left_commute
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   224
lemmas scaleR_zero_left = real_vector.scale_zero_left
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   225
lemmas scaleR_minus_left = real_vector.scale_minus_left
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   226
lemmas scaleR_left_diff_distrib = real_vector.scale_left_diff_distrib
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   227
lemmas scaleR_zero_right = real_vector.scale_zero_right
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   228
lemmas scaleR_minus_right = real_vector.scale_minus_right
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   229
lemmas scaleR_right_diff_distrib = real_vector.scale_right_diff_distrib
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   230
lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   231
lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   232
lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   233
lemmas scaleR_cancel_left = real_vector.scale_cancel_left
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   234
lemmas scaleR_cancel_right = real_vector.scale_cancel_right
e93b121074fb move real_vector class proofs into vector_space and group_hom locales
huffman
parents: 27553
diff changeset
   235
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   236
class real_algebra = real_vector + ring +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   237
  assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   238
  and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   239
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   240
class real_algebra_1 = real_algebra + ring_1
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   241
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   242
class real_div_algebra = real_algebra_1 + division_ring
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   243
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   244
class real_field = real_div_algebra + field
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   245
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   246
instance real :: real_field
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   247
apply (intro_classes, unfold real_scaleR_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   248
apply (rule right_distrib)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   249
apply (rule left_distrib)
20763
052b348a98a9 rearranged axioms and simp rules for scaleR
huffman
parents: 20722
diff changeset
   250
apply (rule mult_assoc [symmetric])
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   251
apply (rule mult_1_left)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   252
apply (rule mult_assoc)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   253
apply (rule mult_left_commute)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   254
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   255
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   256
interpretation scaleR_left: additive ["(\<lambda>a. scaleR a x::'a::real_vector)"]
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   257
by unfold_locales (rule scaleR_left_distrib)
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   258
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   259
interpretation scaleR_right: additive ["(\<lambda>x. scaleR a x::'a::real_vector)"]
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   260
by unfold_locales (rule scaleR_right_distrib)
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   261
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   262
lemma nonzero_inverse_scaleR_distrib:
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   263
  fixes x :: "'a::real_div_algebra" shows
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   264
  "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
20763
052b348a98a9 rearranged axioms and simp rules for scaleR
huffman
parents: 20722
diff changeset
   265
by (rule inverse_unique, simp)
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   266
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   267
lemma inverse_scaleR_distrib:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   268
  fixes x :: "'a::{real_div_algebra,division_by_zero}"
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   269
  shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   270
apply (case_tac "a = 0", simp)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   271
apply (case_tac "x = 0", simp)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   272
apply (erule (1) nonzero_inverse_scaleR_distrib)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   273
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   274
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   275
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   276
subsection {* Embedding of the Reals into any @{text real_algebra_1}:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   277
@{term of_real} *}
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   278
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   279
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21210
diff changeset
   280
  of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   281
  "of_real r = scaleR r 1"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   282
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   283
lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
20763
052b348a98a9 rearranged axioms and simp rules for scaleR
huffman
parents: 20722
diff changeset
   284
by (simp add: of_real_def)
052b348a98a9 rearranged axioms and simp rules for scaleR
huffman
parents: 20722
diff changeset
   285
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   286
lemma of_real_0 [simp]: "of_real 0 = 0"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   287
by (simp add: of_real_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   288
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   289
lemma of_real_1 [simp]: "of_real 1 = 1"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   290
by (simp add: of_real_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   291
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   292
lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   293
by (simp add: of_real_def scaleR_left_distrib)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   294
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   295
lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   296
by (simp add: of_real_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   297
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   298
lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   299
by (simp add: of_real_def scaleR_left_diff_distrib)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   300
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   301
lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
20763
052b348a98a9 rearranged axioms and simp rules for scaleR
huffman
parents: 20722
diff changeset
   302
by (simp add: of_real_def mult_commute)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   303
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   304
lemma nonzero_of_real_inverse:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   305
  "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   306
   inverse (of_real x :: 'a::real_div_algebra)"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   307
by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   308
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   309
lemma of_real_inverse [simp]:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   310
  "of_real (inverse x) =
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   311
   inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   312
by (simp add: of_real_def inverse_scaleR_distrib)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   313
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   314
lemma nonzero_of_real_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   315
  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   316
   (of_real x / of_real y :: 'a::real_field)"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   317
by (simp add: divide_inverse nonzero_of_real_inverse)
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   318
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   319
lemma of_real_divide [simp]:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   320
  "of_real (x / y) =
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   321
   (of_real x / of_real y :: 'a::{real_field,division_by_zero})"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   322
by (simp add: divide_inverse)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   323
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   324
lemma of_real_power [simp]:
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   325
  "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n"
20772
7a51ed817ec7 tuned definitions/proofs;
wenzelm
parents: 20763
diff changeset
   326
by (induct n) (simp_all add: power_Suc)
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   327
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   328
lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   329
by (simp add: of_real_def scaleR_cancel_right)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   330
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   331
lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   332
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   333
lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   334
proof
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   335
  fix r
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   336
  show "of_real r = id r"
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   337
    by (simp add: of_real_def)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   338
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   339
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   340
text{*Collapse nested embeddings*}
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   341
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
20772
7a51ed817ec7 tuned definitions/proofs;
wenzelm
parents: 20763
diff changeset
   342
by (induct n) auto
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   343
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   344
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   345
by (cases z rule: int_diff_cases, simp)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   346
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   347
lemma of_real_number_of_eq:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   348
  "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   349
by (simp add: number_of_eq)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   350
22912
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   351
text{*Every real algebra has characteristic zero*}
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   352
instance real_algebra_1 < ring_char_0
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   353
proof
23282
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   354
  fix m n :: nat
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   355
  have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   356
    by (simp only: of_real_eq_iff of_nat_eq_iff)
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   357
  thus "(of_nat m = (of_nat n::'a)) = (m = n)"
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   358
    by (simp only: of_real_of_nat_eq)
22912
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   359
qed
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   360
27553
d315a513a150 instance real_field < field_char_0;
huffman
parents: 27552
diff changeset
   361
instance real_field < field_char_0 ..
d315a513a150 instance real_field < field_char_0;
huffman
parents: 27552
diff changeset
   362
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   363
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   364
subsection {* The Set of Real Numbers *}
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   365
20772
7a51ed817ec7 tuned definitions/proofs;
wenzelm
parents: 20763
diff changeset
   366
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21210
diff changeset
   367
  Reals :: "'a::real_algebra_1 set" where
27435
b3f8e9bdf9a7 cleaned up some code generator configuration
haftmann
parents: 25571
diff changeset
   368
  [code func del]: "Reals \<equiv> range of_real"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   369
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20828
diff changeset
   370
notation (xsymbols)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   371
  Reals  ("\<real>")
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   372
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   373
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   374
by (simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   375
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   376
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   377
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
20718
4c4869e4ddb7 add lemmas of_int_in_Reals, of_nat_in_Reals
huffman
parents: 20694
diff changeset
   378
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   379
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   380
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   381
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   382
lemma Reals_number_of [simp]:
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   383
  "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   384
by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
20718
4c4869e4ddb7 add lemmas of_int_in_Reals, of_nat_in_Reals
huffman
parents: 20694
diff changeset
   385
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   386
lemma Reals_0 [simp]: "0 \<in> Reals"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   387
apply (unfold Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   388
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   389
apply (rule of_real_0 [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   390
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   391
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   392
lemma Reals_1 [simp]: "1 \<in> Reals"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   393
apply (unfold Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   394
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   395
apply (rule of_real_1 [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   396
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   397
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   398
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   399
apply (auto simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   400
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   401
apply (rule of_real_add [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   402
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   403
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   404
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   405
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   406
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   407
apply (rule of_real_minus [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   408
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   409
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   410
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   411
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   412
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   413
apply (rule of_real_diff [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   414
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   415
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   416
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   417
apply (auto simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   418
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   419
apply (rule of_real_mult [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   420
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   421
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   422
lemma nonzero_Reals_inverse:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   423
  fixes a :: "'a::real_div_algebra"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   424
  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   425
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   426
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   427
apply (erule nonzero_of_real_inverse [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   428
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   429
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   430
lemma Reals_inverse [simp]:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   431
  fixes a :: "'a::{real_div_algebra,division_by_zero}"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   432
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   433
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   434
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   435
apply (rule of_real_inverse [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   436
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   437
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   438
lemma nonzero_Reals_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   439
  fixes a b :: "'a::real_field"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   440
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   441
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   442
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   443
apply (erule nonzero_of_real_divide [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   444
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   445
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   446
lemma Reals_divide [simp]:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   447
  fixes a b :: "'a::{real_field,division_by_zero}"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   448
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   449
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   450
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   451
apply (rule of_real_divide [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   452
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   453
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   454
lemma Reals_power [simp]:
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   455
  fixes a :: "'a::{real_algebra_1,recpower}"
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   456
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   457
apply (auto simp add: Reals_def)
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   458
apply (rule range_eqI)
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   459
apply (rule of_real_power [symmetric])
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   460
done
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   461
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   462
lemma Reals_cases [cases set: Reals]:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   463
  assumes "q \<in> \<real>"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   464
  obtains (of_real) r where "q = of_real r"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   465
  unfolding Reals_def
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   466
proof -
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   467
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   468
  then obtain r where "q = of_real r" ..
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   469
  then show thesis ..
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   470
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   471
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   472
lemma Reals_induct [case_names of_real, induct set: Reals]:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   473
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   474
  by (rule Reals_cases) auto
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   475
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   476
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   477
subsection {* Real normed vector spaces *}
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   478
22636
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   479
class norm = type +
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   480
  fixes norm :: "'a \<Rightarrow> real"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   481
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   482
instantiation real :: norm
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   483
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   484
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   485
definition
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   486
  real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>"
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   487
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   488
instance ..
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   489
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25062
diff changeset
   490
end
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   491
24520
40b220403257 fix sgn_div_norm class
huffman
parents: 24513
diff changeset
   492
class sgn_div_norm = scaleR + norm + sgn +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   493
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   494
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   495
class real_normed_vector = real_vector + sgn_div_norm +
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   496
  assumes norm_ge_zero [simp]: "0 \<le> norm x"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   497
  and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   498
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   499
  and norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   500
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   501
class real_normed_algebra = real_algebra + real_normed_vector +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   502
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   503
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   504
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   505
  assumes norm_one [simp]: "norm 1 = 1"
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   506
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   507
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   508
  assumes norm_mult: "norm (x * y) = norm x * norm y"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   509
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   510
class real_normed_field = real_field + real_normed_div_algebra
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   511
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   512
instance real_normed_div_algebra < real_normed_algebra_1
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   513
proof
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   514
  fix x y :: 'a
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   515
  show "norm (x * y) \<le> norm x * norm y"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   516
    by (simp add: norm_mult)
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   517
next
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   518
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   519
    by (rule norm_mult)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   520
  thus "norm (1::'a) = 1" by simp
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   521
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   522
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   523
instance real :: real_normed_field
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   524
apply (intro_classes, unfold real_norm_def real_scaleR_def)
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   525
apply (simp add: real_sgn_def)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   526
apply (rule abs_ge_zero)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   527
apply (rule abs_eq_0)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   528
apply (rule abs_triangle_ineq)
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   529
apply (rule abs_mult)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   530
apply (rule abs_mult)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   531
done
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   532
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   533
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   534
by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   535
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   536
lemma zero_less_norm_iff [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   537
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   538
  shows "(0 < norm x) = (x \<noteq> 0)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   539
by (simp add: order_less_le)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   540
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   541
lemma norm_not_less_zero [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   542
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   543
  shows "\<not> norm x < 0"
20828
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   544
by (simp add: linorder_not_less)
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   545
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   546
lemma norm_le_zero_iff [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   547
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   548
  shows "(norm x \<le> 0) = (x = 0)"
20828
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   549
by (simp add: order_le_less)
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   550
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   551
lemma norm_minus_cancel [simp]:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   552
  fixes x :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   553
  shows "norm (- x) = norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   554
proof -
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   555
  have "norm (- x) = norm (scaleR (- 1) x)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   556
    by (simp only: scaleR_minus_left scaleR_one)
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   557
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   558
    by (rule norm_scaleR)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   559
  finally show ?thesis by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   560
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   561
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   562
lemma norm_minus_commute:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   563
  fixes a b :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   564
  shows "norm (a - b) = norm (b - a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   565
proof -
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   566
  have "norm (- (b - a)) = norm (b - a)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   567
    by (rule norm_minus_cancel)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   568
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   569
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   570
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   571
lemma norm_triangle_ineq2:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   572
  fixes a b :: "'a::real_normed_vector"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   573
  shows "norm a - norm b \<le> norm (a - b)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   574
proof -
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   575
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   576
    by (rule norm_triangle_ineq)
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   577
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   578
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   579
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   580
lemma norm_triangle_ineq3:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   581
  fixes a b :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   582
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   583
apply (subst abs_le_iff)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   584
apply auto
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   585
apply (rule norm_triangle_ineq2)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   586
apply (subst norm_minus_commute)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   587
apply (rule norm_triangle_ineq2)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   588
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   589
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   590
lemma norm_triangle_ineq4:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   591
  fixes a b :: "'a::real_normed_vector"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   592
  shows "norm (a - b) \<le> norm a + norm b"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   593
proof -
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   594
  have "norm (a + - b) \<le> norm a + norm (- b)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   595
    by (rule norm_triangle_ineq)
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   596
  thus ?thesis
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   597
    by (simp only: diff_minus norm_minus_cancel)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   598
qed
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   599
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   600
lemma norm_diff_ineq:
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   601
  fixes a b :: "'a::real_normed_vector"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   602
  shows "norm a - norm b \<le> norm (a + b)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   603
proof -
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   604
  have "norm a - norm (- b) \<le> norm (a - - b)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   605
    by (rule norm_triangle_ineq2)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   606
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   607
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   608
20551
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   609
lemma norm_diff_triangle_ineq:
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   610
  fixes a b c d :: "'a::real_normed_vector"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   611
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   612
proof -
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   613
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   614
    by (simp add: diff_minus add_ac)
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   615
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   616
    by (rule norm_triangle_ineq)
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   617
  finally show ?thesis .
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   618
qed
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   619
22857
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   620
lemma abs_norm_cancel [simp]:
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   621
  fixes a :: "'a::real_normed_vector"
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   622
  shows "\<bar>norm a\<bar> = norm a"
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   623
by (rule abs_of_nonneg [OF norm_ge_zero])
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   624
22880
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   625
lemma norm_add_less:
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   626
  fixes x y :: "'a::real_normed_vector"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   627
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   628
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   629
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   630
lemma norm_mult_less:
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   631
  fixes x y :: "'a::real_normed_algebra"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   632
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   633
apply (rule order_le_less_trans [OF norm_mult_ineq])
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   634
apply (simp add: mult_strict_mono')
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   635
done
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   636
22857
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   637
lemma norm_of_real [simp]:
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   638
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   639
unfolding of_real_def by (simp add: norm_scaleR)
20560
49996715bc6e norm_one is now proved from other class axioms
huffman
parents: 20554
diff changeset
   640
22876
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   641
lemma norm_number_of [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   642
  "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   643
    = \<bar>number_of w\<bar>"
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   644
by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   645
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   646
lemma norm_of_int [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   647
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   648
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   649
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   650
lemma norm_of_nat [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   651
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   652
apply (subst of_real_of_nat_eq [symmetric])
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   653
apply (subst norm_of_real, simp)
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   654
done
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   655
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   656
lemma nonzero_norm_inverse:
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   657
  fixes a :: "'a::real_normed_div_algebra"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   658
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   659
apply (rule inverse_unique [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   660
apply (simp add: norm_mult [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   661
done
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   662
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   663
lemma norm_inverse:
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   664
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   665
  shows "norm (inverse a) = inverse (norm a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   666
apply (case_tac "a = 0", simp)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   667
apply (erule nonzero_norm_inverse)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   668
done
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   669
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   670
lemma nonzero_norm_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   671
  fixes a b :: "'a::real_normed_field"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   672
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   673
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   674
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   675
lemma norm_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   676
  fixes a b :: "'a::{real_normed_field,division_by_zero}"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   677
  shows "norm (a / b) = norm a / norm b"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   678
by (simp add: divide_inverse norm_mult norm_inverse)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   679
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   680
lemma norm_power_ineq:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   681
  fixes x :: "'a::{real_normed_algebra_1,recpower}"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   682
  shows "norm (x ^ n) \<le> norm x ^ n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   683
proof (induct n)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   684
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   685
next
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   686
  case (Suc n)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   687
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   688
    by (rule norm_mult_ineq)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   689
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   690
    using norm_ge_zero by (rule mult_left_mono)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   691
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   692
    by (simp add: power_Suc)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   693
qed
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   694
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   695
lemma norm_power:
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   696
  fixes x :: "'a::{real_normed_div_algebra,recpower}"
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   697
  shows "norm (x ^ n) = norm x ^ n"
20772
7a51ed817ec7 tuned definitions/proofs;
wenzelm
parents: 20763
diff changeset
   698
by (induct n) (simp_all add: power_Suc norm_mult)
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   699
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   700
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   701
subsection {* Sign function *}
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   702
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   703
lemma norm_sgn:
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   704
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   705
by (simp add: sgn_div_norm norm_scaleR)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   706
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   707
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   708
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   709
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   710
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   711
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   712
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   713
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   714
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   715
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   716
lemma sgn_scaleR:
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   717
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   718
by (simp add: sgn_div_norm norm_scaleR mult_ac)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   719
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   720
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   721
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   722
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   723
lemma sgn_of_real:
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   724
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   725
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   726
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   727
lemma sgn_mult:
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   728
  fixes x y :: "'a::real_normed_div_algebra"
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   729
  shows "sgn (x * y) = sgn x * sgn y"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   730
by (simp add: sgn_div_norm norm_mult mult_commute)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   731
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   732
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   733
by (simp add: sgn_div_norm divide_inverse)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   734
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   735
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   736
unfolding real_sgn_eq by simp
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   737
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   738
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   739
unfolding real_sgn_eq by simp
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   740
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   741
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   742
subsection {* Bounded Linear and Bilinear Operators *}
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   743
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   744
locale bounded_linear = additive +
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   745
  constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   746
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   747
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   748
begin
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   749
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   750
lemma pos_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   751
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   752
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   753
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   754
    using bounded by fast
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   755
  show ?thesis
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   756
  proof (intro exI impI conjI allI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   757
    show "0 < max 1 K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   758
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   759
  next
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   760
    fix x
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   761
    have "norm (f x) \<le> norm x * K" using K .
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   762
    also have "\<dots> \<le> norm x * max 1 K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   763
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   764
    finally show "norm (f x) \<le> norm x * max 1 K" .
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   765
  qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   766
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   767
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   768
lemma nonneg_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   769
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   770
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   771
  from pos_bounded
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   772
  show ?thesis by (auto intro: order_less_imp_le)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   773
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   774
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   775
end
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   776
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   777
locale bounded_bilinear =
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   778
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   779
                 \<Rightarrow> 'c::real_normed_vector"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   780
    (infixl "**" 70)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   781
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   782
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   783
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   784
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   785
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   786
begin
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   787
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   788
lemma pos_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   789
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   790
apply (cut_tac bounded, erule exE)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   791
apply (rule_tac x="max 1 K" in exI, safe)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   792
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   793
apply (drule spec, drule spec, erule order_trans)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   794
apply (rule mult_left_mono [OF le_maxI2])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   795
apply (intro mult_nonneg_nonneg norm_ge_zero)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   796
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   797
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   798
lemma nonneg_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   799
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   800
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   801
  from pos_bounded
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   802
  show ?thesis by (auto intro: order_less_imp_le)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   803
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   804
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   805
lemma additive_right: "additive (\<lambda>b. prod a b)"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   806
by (rule additive.intro, rule add_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   807
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   808
lemma additive_left: "additive (\<lambda>a. prod a b)"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   809
by (rule additive.intro, rule add_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   810
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   811
lemma zero_left: "prod 0 b = 0"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   812
by (rule additive.zero [OF additive_left])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   813
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   814
lemma zero_right: "prod a 0 = 0"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   815
by (rule additive.zero [OF additive_right])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   816
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   817
lemma minus_left: "prod (- a) b = - prod a b"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   818
by (rule additive.minus [OF additive_left])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   819
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   820
lemma minus_right: "prod a (- b) = - prod a b"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   821
by (rule additive.minus [OF additive_right])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   822
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   823
lemma diff_left:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   824
  "prod (a - a') b = prod a b - prod a' b"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   825
by (rule additive.diff [OF additive_left])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   826
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   827
lemma diff_right:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   828
  "prod a (b - b') = prod a b - prod a b'"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   829
by (rule additive.diff [OF additive_right])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   830
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   831
lemma bounded_linear_left:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   832
  "bounded_linear (\<lambda>a. a ** b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   833
apply (unfold_locales)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   834
apply (rule add_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   835
apply (rule scaleR_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   836
apply (cut_tac bounded, safe)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   837
apply (rule_tac x="norm b * K" in exI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   838
apply (simp add: mult_ac)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   839
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   840
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   841
lemma bounded_linear_right:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   842
  "bounded_linear (\<lambda>b. a ** b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   843
apply (unfold_locales)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   844
apply (rule add_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   845
apply (rule scaleR_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   846
apply (cut_tac bounded, safe)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   847
apply (rule_tac x="norm a * K" in exI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   848
apply (simp add: mult_ac)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   849
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   850
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   851
lemma prod_diff_prod:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   852
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   853
by (simp add: diff_left diff_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   854
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   855
end
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   856
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   857
interpretation mult:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   858
  bounded_bilinear ["op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"]
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   859
apply (rule bounded_bilinear.intro)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   860
apply (rule left_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   861
apply (rule right_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   862
apply (rule mult_scaleR_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   863
apply (rule mult_scaleR_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   864
apply (rule_tac x="1" in exI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   865
apply (simp add: norm_mult_ineq)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   866
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   867
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   868
interpretation mult_left:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   869
  bounded_linear ["(\<lambda>x::'a::real_normed_algebra. x * y)"]
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   870
by (rule mult.bounded_linear_left)
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   871
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   872
interpretation mult_right:
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   873
  bounded_linear ["(\<lambda>y::'a::real_normed_algebra. x * y)"]
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   874
by (rule mult.bounded_linear_right)
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   875
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   876
interpretation divide:
23120
a34f204e9c88 interpretation bounded_linear_divide
huffman
parents: 23113
diff changeset
   877
  bounded_linear ["(\<lambda>x::'a::real_normed_field. x / y)"]
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   878
unfolding divide_inverse by (rule mult.bounded_linear_left)
23120
a34f204e9c88 interpretation bounded_linear_divide
huffman
parents: 23113
diff changeset
   879
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   880
interpretation scaleR: bounded_bilinear ["scaleR"]
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   881
apply (rule bounded_bilinear.intro)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   882
apply (rule scaleR_left_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   883
apply (rule scaleR_right_distrib)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   884
apply simp
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   885
apply (rule scaleR_left_commute)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   886
apply (rule_tac x="1" in exI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   887
apply (simp add: norm_scaleR)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   888
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   889
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   890
interpretation scaleR_left: bounded_linear ["\<lambda>r. scaleR r x"]
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   891
by (rule scaleR.bounded_linear_left)
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   892
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   893
interpretation scaleR_right: bounded_linear ["\<lambda>x. scaleR r x"]
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   894
by (rule scaleR.bounded_linear_right)
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   895
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   896
interpretation of_real: bounded_linear ["\<lambda>r. of_real r"]
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
   897
unfolding of_real_def by (rule scaleR.bounded_linear_left)
22625
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
   898
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   899
end