src/HOL/Complex/Complex.thy
author huffman
Tue, 08 May 2007 05:30:10 +0200
changeset 22861 8ec47039614e
parent 22852 2490d4b4671a
child 22883 005be8dafce0
permissions -rw-r--r--
clean up complex norm proofs, remove redundant lemmas
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
     1
(*  Title:       Complex.thy
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
     2
    ID:      $Id$
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
     3
    Author:      Jacques D. Fleuriot
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
     4
    Copyright:   2001 University of Edinburgh
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
     5
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
     6
*)
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
     7
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
     8
header {* Complex Numbers: Rectangular and Polar Representations *}
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
     9
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15085
diff changeset
    10
theory Complex
22655
83878e551c8c minimize imports
huffman
parents: 21404
diff changeset
    11
imports "../Hyperreal/Transcendental"
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15085
diff changeset
    12
begin
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    13
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    14
datatype complex = Complex real real
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    15
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14443
diff changeset
    16
instance complex :: "{zero, one, plus, times, minus, inverse, power}" ..
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    17
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    18
consts
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    19
  "ii"    :: complex    ("\<i>")
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    20
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    21
consts Re :: "complex => real"
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
    22
primrec Re: "Re (Complex x y) = x"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    23
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    24
consts Im :: "complex => real"
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
    25
primrec Im: "Im (Complex x y) = y"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    26
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    27
lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    28
  by (induct z) simp
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    29
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    30
defs (overloaded)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    31
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    32
  complex_zero_def:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    33
  "0 == Complex 0 0"
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    34
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    35
  complex_one_def:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    36
  "1 == Complex 1 0"
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    37
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    38
  i_def: "ii == Complex 0 1"
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    39
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    40
  complex_minus_def: "- z == Complex (- Re z) (- Im z)"
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    41
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    42
  complex_inverse_def:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    43
   "inverse z ==
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    44
    Complex (Re z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>)) (- Im z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>))"
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    45
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    46
  complex_add_def:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    47
    "z + w == Complex (Re z + Re w) (Im z + Im w)"
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    48
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    49
  complex_diff_def:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    50
    "z - w == z + - (w::complex)"
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    51
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    52
  complex_mult_def:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    53
    "z * w == Complex (Re z * Re w - Im z * Im w) (Re z * Im w + Im z * Re w)"
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    54
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    55
  complex_divide_def: "w / (z::complex) == w * inverse z"
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    56
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
    57
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    58
lemma complex_equality [intro?]: "Re z = Re w ==> Im z = Im w ==> z = w"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    59
  by (induct z, induct w) simp
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    60
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    61
lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    62
by (induct w, induct z, simp)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    63
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    64
lemma complex_Re_zero [simp]: "Re 0 = 0"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    65
by (simp add: complex_zero_def)
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    66
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    67
lemma complex_Im_zero [simp]: "Im 0 = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    68
by (simp add: complex_zero_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    69
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
    70
lemma complex_zero_iff [simp]: "(Complex x y = 0) = (x = 0 \<and> y = 0)"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
    71
unfolding complex_zero_def by simp
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
    72
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    73
lemma complex_Re_one [simp]: "Re 1 = 1"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    74
by (simp add: complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    75
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    76
lemma complex_Im_one [simp]: "Im 1 = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    77
by (simp add: complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    78
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    79
lemma complex_Re_i [simp]: "Re(ii) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    80
by (simp add: i_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    81
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    82
lemma complex_Im_i [simp]: "Im(ii) = 1"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    83
by (simp add: i_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    84
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    85
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    86
subsection{*Unary Minus*}
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    87
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
    88
lemma complex_minus [simp]: "- (Complex x y) = Complex (-x) (-y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    89
by (simp add: complex_minus_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    90
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    91
lemma complex_Re_minus [simp]: "Re (-z) = - Re z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
    92
by (simp add: complex_minus_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    93
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    94
lemma complex_Im_minus [simp]: "Im (-z) = - Im z"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
    95
by (simp add: complex_minus_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    96
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    97
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    98
subsection{*Addition*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
    99
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   100
lemma complex_add [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   101
     "Complex x1 y1 + Complex x2 y2 = Complex (x1+x2) (y1+y2)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   102
by (simp add: complex_add_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   103
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   104
lemma complex_Re_add [simp]: "Re(x + y) = Re(x) + Re(y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   105
by (simp add: complex_add_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   106
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   107
lemma complex_Im_add [simp]: "Im(x + y) = Im(x) + Im(y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   108
by (simp add: complex_add_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   109
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   110
lemma complex_add_commute: "(u::complex) + v = v + u"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   111
by (simp add: complex_add_def add_commute)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   112
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   113
lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   114
by (simp add: complex_add_def add_assoc)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   115
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   116
lemma complex_add_zero_left: "(0::complex) + z = z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   117
by (simp add: complex_add_def complex_zero_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   118
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   119
lemma complex_add_zero_right: "z + (0::complex) = z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   120
by (simp add: complex_add_def complex_zero_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   121
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   122
lemma complex_add_minus_left: "-z + z = (0::complex)"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   123
by (simp add: complex_add_def complex_minus_def complex_zero_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   124
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   125
lemma complex_diff:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   126
      "Complex x1 y1 - Complex x2 y2 = Complex (x1-x2) (y1-y2)"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   127
by (simp add: complex_add_def complex_minus_def complex_diff_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   128
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   129
lemma complex_Re_diff [simp]: "Re(x - y) = Re(x) - Re(y)"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   130
by (simp add: complex_diff_def)
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   131
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   132
lemma complex_Im_diff [simp]: "Im(x - y) = Im(x) - Im(y)"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   133
by (simp add: complex_diff_def)
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   134
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   135
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   136
subsection{*Multiplication*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   137
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   138
lemma complex_mult [simp]:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   139
     "Complex x1 y1 * Complex x2 y2 = Complex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   140
by (simp add: complex_mult_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   141
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   142
lemma complex_mult_commute: "(w::complex) * z = z * w"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   143
by (simp add: complex_mult_def mult_commute add_commute)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   144
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   145
lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   146
by (simp add: complex_mult_def mult_ac add_ac
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   147
              right_diff_distrib right_distrib left_diff_distrib left_distrib)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   148
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   149
lemma complex_mult_one_left: "(1::complex) * z = z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   150
by (simp add: complex_mult_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   151
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   152
lemma complex_mult_one_right: "z * (1::complex) = z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   153
by (simp add: complex_mult_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   154
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   155
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   156
subsection{*Inverse*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   157
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   158
lemma complex_inverse [simp]:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   159
     "inverse (Complex x y) = Complex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   160
by (simp add: complex_inverse_def)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   161
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   162
lemma complex_mult_inv_left: "z \<noteq> (0::complex) ==> inverse(z) * z = 1"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   163
apply (induct z)
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   164
apply (rename_tac x y)
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   165
apply (auto simp add:
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15140
diff changeset
   166
             complex_one_def complex_zero_def add_divide_distrib [symmetric] 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15140
diff changeset
   167
             power2_eq_square mult_ac)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15140
diff changeset
   168
apply (simp_all add: real_sum_squares_not_zero real_sum_squares_not_zero2) 
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   169
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   170
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   171
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   172
subsection {* The field of complex numbers *}
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   173
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   174
instance complex :: field
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   175
proof
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   176
  fix z u v w :: complex
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   177
  show "(u + v) + w = u + (v + w)"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   178
    by (rule complex_add_assoc)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   179
  show "z + w = w + z"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   180
    by (rule complex_add_commute)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   181
  show "0 + z = z"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   182
    by (rule complex_add_zero_left)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   183
  show "-z + z = 0"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   184
    by (rule complex_add_minus_left)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   185
  show "z - w = z + -w"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   186
    by (simp add: complex_diff_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   187
  show "(u * v) * w = u * (v * w)"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   188
    by (rule complex_mult_assoc)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   189
  show "z * w = w * z"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   190
    by (rule complex_mult_commute)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   191
  show "1 * z = z"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   192
    by (rule complex_mult_one_left)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14335
diff changeset
   193
  show "0 \<noteq> (1::complex)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   194
    by (simp add: complex_zero_def complex_one_def)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   195
  show "(u + v) * w = u * w + v * w"
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14387
diff changeset
   196
    by (simp add: complex_mult_def complex_add_def left_distrib 
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14387
diff changeset
   197
                  diff_minus add_ac)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   198
  show "z / w = z * inverse w"
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   199
    by (simp add: complex_divide_def)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   200
  assume "w \<noteq> 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   201
  thus "inverse w * w = 1"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   202
    by (simp add: complex_mult_inv_left)
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   203
qed
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   204
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   205
instance complex :: division_by_zero
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   206
proof
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   207
  show "inverse 0 = (0::complex)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   208
    by (simp add: complex_inverse_def complex_zero_def)
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   209
qed
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14334
diff changeset
   210
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   211
20556
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   212
subsection{*The real algebra of complex numbers*}
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   213
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   214
instance complex :: scaleR ..
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   215
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   216
defs (overloaded)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   217
  complex_scaleR_def: "r *# x == Complex r 0 * x"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   218
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   219
instance complex :: real_field
20556
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   220
proof
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   221
  fix a b :: real
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   222
  fix x y :: complex
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   223
  show "a *# (x + y) = a *# x + a *# y"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   224
    by (simp add: complex_scaleR_def right_distrib)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   225
  show "(a + b) *# x = a *# x + b *# x"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   226
    by (simp add: complex_scaleR_def left_distrib [symmetric])
20763
052b348a98a9 rearranged axioms and simp rules for scaleR
huffman
parents: 20725
diff changeset
   227
  show "a *# b *# x = (a * b) *# x"
20556
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   228
    by (simp add: complex_scaleR_def mult_assoc [symmetric])
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   229
  show "1 *# x = x"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   230
    by (simp add: complex_scaleR_def complex_one_def [symmetric])
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   231
  show "a *# x * y = a *# (x * y)"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   232
    by (simp add: complex_scaleR_def mult_assoc)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   233
  show "x * a *# y = a *# (x * y)"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   234
    by (simp add: complex_scaleR_def mult_left_commute)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   235
qed
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   236
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   237
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   238
subsection{*Embedding Properties for @{term complex_of_real} Map*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   239
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   240
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   241
  complex_of_real :: "real => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   242
  "complex_of_real == of_real"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   243
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   244
lemma complex_of_real_def: "complex_of_real r = Complex r 0"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   245
by (simp add: of_real_def complex_scaleR_def)
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   246
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   247
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   248
by (simp add: complex_of_real_def)
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   249
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   250
lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   251
by (simp add: complex_of_real_def)
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   252
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   253
lemma Complex_add_complex_of_real [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   254
     "Complex x y + complex_of_real r = Complex (x+r) y"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   255
by (simp add: complex_of_real_def)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   256
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   257
lemma complex_of_real_add_Complex [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   258
     "complex_of_real r + Complex x y = Complex (r+x) y"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   259
by (simp add: i_def complex_of_real_def)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   260
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   261
lemma Complex_mult_complex_of_real:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   262
     "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   263
by (simp add: complex_of_real_def)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   264
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   265
lemma complex_of_real_mult_Complex:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   266
     "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   267
by (simp add: i_def complex_of_real_def)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   268
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   269
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   270
by (simp add: i_def complex_of_real_def)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   271
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   272
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   273
by (simp add: i_def complex_of_real_def)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   274
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   275
lemma complex_of_real_inverse:
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   276
     "complex_of_real(inverse x) = inverse(complex_of_real x)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   277
by (rule of_real_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   278
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   279
lemma complex_of_real_divide:
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   280
      "complex_of_real(x/y) = complex_of_real x / complex_of_real y"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   281
by (rule of_real_divide)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   282
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   283
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   284
subsection{*The Functions @{term Re} and @{term Im}*}
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   285
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   286
lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   287
by (induct z, induct w, simp)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   288
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   289
lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   290
by (induct z, induct w, simp)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   291
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   292
lemma Re_i_times [simp]: "Re(ii * z) = - Im z"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   293
by (simp add: complex_Re_mult_eq)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   294
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   295
lemma Re_times_i [simp]: "Re(z * ii) = - Im z"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   296
by (simp add: complex_Re_mult_eq)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   297
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   298
lemma Im_i_times [simp]: "Im(ii * z) = Re z"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   299
by (simp add: complex_Im_mult_eq)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   300
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   301
lemma Im_times_i [simp]: "Im(z * ii) = Re z"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   302
by (simp add: complex_Im_mult_eq)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   303
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   304
lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   305
by (simp add: complex_Re_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   306
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   307
lemma complex_Re_mult_complex_of_real [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   308
     "Re (z * complex_of_real c) = Re(z) * c"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   309
by (simp add: complex_Re_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   310
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   311
lemma complex_Im_mult_complex_of_real [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   312
     "Im (z * complex_of_real c) = Im(z) * c"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   313
by (simp add: complex_Im_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   314
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   315
lemma complex_Re_mult_complex_of_real2 [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   316
     "Re (complex_of_real c * z) = c * Re(z)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   317
by (simp add: complex_Re_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   318
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   319
lemma complex_Im_mult_complex_of_real2 [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   320
     "Im (complex_of_real c * z) = c * Im(z)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   321
by (simp add: complex_Im_mult_eq)
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   322
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   323
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   324
subsection{*Conjugation is an Automorphism*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   325
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   326
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   327
  cnj :: "complex => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   328
  "cnj z = Complex (Re z) (-Im z)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   329
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   330
lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   331
by (simp add: cnj_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   332
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   333
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   334
by (simp add: cnj_def complex_Re_Im_cancel_iff)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   335
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   336
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   337
by (simp add: cnj_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   338
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   339
lemma complex_cnj_complex_of_real [simp]:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   340
     "cnj (complex_of_real x) = complex_of_real x"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   341
by (simp add: complex_of_real_def complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   342
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   343
lemma complex_cnj_minus: "cnj (-z) = - cnj z"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   344
by (simp add: cnj_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   345
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   346
lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   347
by (induct z, simp add: complex_cnj power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   348
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   349
lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   350
by (induct w, induct z, simp add: complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   351
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   352
lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   353
by (simp add: diff_minus complex_cnj_add complex_cnj_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   354
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   355
lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   356
by (induct w, induct z, simp add: complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   357
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   358
lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   359
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   360
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   361
lemma complex_cnj_one [simp]: "cnj 1 = 1"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   362
by (simp add: cnj_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   363
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   364
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   365
by (induct z, simp add: complex_cnj complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   366
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   367
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   368
apply (induct z)
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   369
apply (simp add: complex_add complex_cnj complex_of_real_def diff_minus
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   370
                 complex_minus i_def complex_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   371
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   372
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   373
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   374
by (simp add: cnj_def complex_zero_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   375
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   376
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   377
by (induct z, simp add: complex_zero_def complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   378
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   379
lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   380
by (induct z, simp add: complex_cnj complex_of_real_def power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   381
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   382
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   383
subsection{*Modulus*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   384
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   385
instance complex :: norm
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   386
  complex_norm_def: "norm z \<equiv> sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" ..
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   387
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   388
abbreviation
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   389
  cmod :: "complex \<Rightarrow> real" where
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   390
  "cmod \<equiv> norm"
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   391
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   392
lemmas cmod_def = complex_norm_def
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   393
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   394
lemma complex_mod [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   395
by (simp add: cmod_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   396
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   397
lemma complex_mod_triangle_ineq [simp]: "cmod (x + y) \<le> cmod x + cmod y"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   398
apply (simp add: cmod_def)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   399
apply (rule real_sqrt_sum_squares_triangle_ineq)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   400
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   401
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   402
lemma complex_mod_mult: "cmod (x * y) = cmod x * cmod y"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   403
apply (induct x, induct y)
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   404
apply (simp add: real_sqrt_mult_distrib [symmetric])
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   405
apply (rule_tac f=sqrt in arg_cong)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   406
apply (simp add: power2_sum power2_diff power_mult_distrib ring_distrib)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   407
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   408
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   409
lemma complex_mod_complex_of_real: "cmod (complex_of_real x) = \<bar>x\<bar>"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   410
by (simp add: complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   411
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22655
diff changeset
   412
lemma complex_norm_scaleR:
2490d4b4671a clean up RealVector classes
huffman
parents: 22655
diff changeset
   413
  "norm (scaleR a x) = \<bar>a\<bar> * norm (x::complex)"
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   414
unfolding scaleR_conv_of_real
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   415
by (simp only: complex_mod_mult complex_mod_complex_of_real)
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22655
diff changeset
   416
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   417
instance complex :: real_normed_field
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   418
proof
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   419
  fix r :: real
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   420
  fix x y :: complex
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   421
  show "0 \<le> cmod x"
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   422
    by (induct x) simp
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   423
  show "(cmod x = 0) = (x = 0)"
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   424
    by (induct x) simp
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   425
  show "cmod (x + y) \<le> cmod x + cmod y"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   426
    by (rule complex_mod_triangle_ineq)
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22655
diff changeset
   427
  show "cmod (scaleR r x) = \<bar>r\<bar> * cmod x"
2490d4b4671a clean up RealVector classes
huffman
parents: 22655
diff changeset
   428
    by (rule complex_norm_scaleR)
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   429
  show "cmod (x * y) = cmod x * cmod y"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   430
    by (rule complex_mod_mult)
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   431
qed
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   432
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   433
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   434
by (induct z, simp add: complex_cnj)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   435
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   436
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   437
by (simp add: complex_mod_mult power2_eq_square)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   438
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   439
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   440
by simp
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   441
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   442
lemma cmod_complex_polar [simp]:
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   443
     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   444
apply (simp only: cmod_unit_one complex_mod_mult)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   445
apply (simp add: complex_mod_complex_of_real)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   446
done
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   447
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   448
lemma complex_Re_le_cmod: "Re x \<le> cmod x"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   449
unfolding complex_norm_def
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   450
by (rule real_sqrt_sum_squares_ge1)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   451
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   452
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   453
by (rule order_trans [OF _ norm_ge_zero], simp)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   454
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   455
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   456
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   457
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   458
lemma complex_mod_add_less:
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   459
     "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   460
by (auto intro: order_le_less_trans complex_mod_triangle_ineq)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   461
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   462
lemma complex_mod_mult_less:
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   463
     "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   464
by (auto intro: real_mult_less_mono' simp add: complex_mod_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   465
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   466
lemma complex_mod_diff_ineq [simp]: "cmod(a) - cmod(b) \<le> cmod(a + b)"
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   467
(* TODO: generalize *)
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   468
proof -
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   469
  have "cmod a - cmod b = cmod a - cmod (- b)" by simp
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   470
  also have "\<dots> \<le> cmod (a - - b)" by (rule norm_triangle_ineq2)
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   471
  also have "\<dots> = cmod (a + b)" by simp
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   472
  finally show ?thesis .
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   473
qed
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   474
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   475
lemmas real_sum_squared_expand = power2_sum [where 'a=real]
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   476
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   477
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   478
subsection{*Exponentiation*}
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   479
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   480
primrec
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   481
     complexpow_0:   "z ^ 0       = 1"
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   482
     complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)"
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   483
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   484
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14691
diff changeset
   485
instance complex :: recpower
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   486
proof
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   487
  fix z :: complex
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   488
  fix n :: nat
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   489
  show "z^0 = 1" by simp
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   490
  show "z^(Suc n) = z * (z^n)" by simp
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   491
qed
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   492
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   493
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   494
lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   495
by (rule of_real_power)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   496
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   497
lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   498
apply (induct_tac "n")
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   499
apply (auto simp add: complex_cnj_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   500
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   501
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   502
lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   503
by (rule norm_power)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   504
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   505
lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   506
by (simp add: i_def complex_one_def numeral_2_eq_2)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   507
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   508
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   509
by (simp add: i_def complex_zero_def)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   510
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   511
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   512
subsection{*The Function @{term sgn}*}
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   513
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   514
definition
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   515
  (*------------ Argand -------------*)
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   516
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   517
  sgn :: "complex => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   518
  "sgn z = z / complex_of_real(cmod z)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   519
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   520
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   521
  arg :: "complex => real" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   522
  "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   523
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   524
lemma sgn_zero [simp]: "sgn 0 = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   525
by (simp add: sgn_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   526
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   527
lemma sgn_one [simp]: "sgn 1 = 1"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   528
by (simp add: sgn_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   529
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   530
lemma sgn_minus: "sgn (-z) = - sgn(z)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   531
by (simp add: sgn_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   532
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   533
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   534
by (simp add: sgn_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   535
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   536
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   537
by (simp add: i_def complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   538
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   539
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   540
by (simp add: i_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   541
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   542
lemma complex_eq_cancel_iff2 [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   543
     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   544
by (simp add: complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   545
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   546
lemma Complex_eq_0 [simp]: "(Complex x y = 0) = (x = 0 & y = 0)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   547
by (simp add: complex_zero_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   548
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   549
lemma Complex_eq_1 [simp]: "(Complex x y = 1) = (x = 1 & y = 0)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   550
by (simp add: complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   551
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   552
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 & y = 1)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   553
by (simp add: i_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   554
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   555
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   556
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   557
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   558
proof (induct z)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   559
  case (Complex x y)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   560
    have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   561
      by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   562
    thus "Re (sgn (Complex x y)) = Re (Complex x y) /cmod (Complex x y)"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   563
       by (simp add: sgn_def complex_of_real_def divide_inverse)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   564
qed
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   565
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   566
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   567
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   568
proof (induct z)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   569
  case (Complex x y)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   570
    have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   571
      by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   572
    thus "Im (sgn (Complex x y)) = Im (Complex x y) /cmod (Complex x y)"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   573
       by (simp add: sgn_def complex_of_real_def divide_inverse)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   574
qed
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   575
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   576
lemma complex_inverse_complex_split:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   577
     "inverse(complex_of_real x + ii * complex_of_real y) =
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   578
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   579
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   580
by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   581
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   582
(*----------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   583
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   584
(* many of the theorems are not used - so should they be kept?                *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   585
(*----------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   586
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   587
lemma complex_of_real_zero_iff: "(complex_of_real y = 0) = (y = 0)"
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   588
by (rule of_real_eq_0_iff)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   589
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   590
lemma cos_arg_i_mult_zero_pos:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   591
   "0 < y ==> cos (arg(Complex 0 y)) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   592
apply (simp add: arg_def abs_if)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   593
apply (rule_tac a = "pi/2" in someI2, auto)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   594
apply (rule order_less_trans [of _ 0], auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   595
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   596
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   597
lemma cos_arg_i_mult_zero_neg:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   598
   "y < 0 ==> cos (arg(Complex 0 y)) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   599
apply (simp add: arg_def abs_if)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   600
apply (rule_tac a = "- pi/2" in someI2, auto)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   601
apply (rule order_trans [of _ 0], auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   602
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   603
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   604
lemma cos_arg_i_mult_zero [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   605
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   606
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   607
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   608
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   609
subsection{*Finally! Polar Form for Complex Numbers*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   610
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   611
definition
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   612
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   613
  (* abbreviation for (cos a + i sin a) *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   614
  cis :: "real => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   615
  "cis a = Complex (cos a) (sin a)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   616
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   617
definition
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   618
  (* abbreviation for r*(cos a + i sin a) *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   619
  rcis :: "[real, real] => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   620
  "rcis r a = complex_of_real r * cis a"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   621
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   622
definition
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   623
  (* e ^ (x + iy) *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   624
  expi :: "complex => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   625
  "expi z = complex_of_real(exp (Re z)) * cis (Im z)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   626
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   627
lemma complex_split_polar:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   628
     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   629
apply (induct z)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   630
apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   631
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   632
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   633
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   634
apply (induct z)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   635
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   636
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   637
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   638
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   639
by (simp add: rcis_def cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   640
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   641
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   642
by (simp add: rcis_def cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   643
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   644
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   645
proof -
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   646
  have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   647
    by (simp only: power_mult_distrib right_distrib)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   648
  thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   649
qed
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   650
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   651
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   652
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   653
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   654
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   655
apply (simp add: cmod_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   656
apply (rule real_sqrt_eq_iff [THEN iffD2])
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   657
apply (auto simp add: complex_mult_cnj
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   658
            simp del: of_real_add)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   659
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   660
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   661
lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   662
by (induct z, simp add: complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   663
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   664
lemma complex_Im_cnj [simp]: "Im(cnj z) = - Im z"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   665
by (induct z, simp add: complex_cnj)
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   666
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   667
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   668
by (induct z, simp add: complex_cnj complex_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   669
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   670
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   671
(*---------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   672
(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   673
(*---------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   674
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   675
lemma cis_rcis_eq: "cis a = rcis 1 a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   676
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   677
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   678
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   679
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   680
              complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   681
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   682
lemma cis_mult: "cis a * cis b = cis (a + b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   683
by (simp add: cis_rcis_eq rcis_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   684
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   685
lemma cis_zero [simp]: "cis 0 = 1"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   686
by (simp add: cis_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   687
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   688
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   689
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   690
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   691
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   692
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   693
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   694
lemma complex_of_real_minus_one:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   695
   "complex_of_real (-(1::real)) = -(1::complex)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   696
by (simp add: complex_of_real_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   697
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   698
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   699
by (simp add: complex_mult_assoc [symmetric])
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   700
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   701
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   702
lemma cis_real_of_nat_Suc_mult:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   703
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   704
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   705
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   706
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   707
apply (induct_tac "n")
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   708
apply (auto simp add: cis_real_of_nat_Suc_mult)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   709
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   710
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   711
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   712
by (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   713
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   714
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   715
by (simp add: cis_def complex_inverse_complex_split diff_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   716
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   717
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   718
by (simp add: divide_inverse rcis_def complex_of_real_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   719
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   720
lemma cis_divide: "cis a / cis b = cis (a - b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   721
by (simp add: complex_divide_def cis_mult real_diff_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   722
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   723
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   724
apply (simp add: complex_divide_def)
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   725
apply (case_tac "r2=0", simp)
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   726
apply (simp add: rcis_inverse rcis_mult real_diff_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   727
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   728
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   729
lemma Re_cis [simp]: "Re(cis a) = cos a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   730
by (simp add: cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   731
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   732
lemma Im_cis [simp]: "Im(cis a) = sin a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   733
by (simp add: cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   734
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   735
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   736
by (auto simp add: DeMoivre)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   737
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   738
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   739
by (auto simp add: DeMoivre)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   740
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   741
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   742
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   743
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   744
lemma expi_zero [simp]: "expi (0::complex) = 1"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   745
by (simp add: expi_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   746
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   747
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   748
apply (insert rcis_Ex [of z])
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   749
apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] of_real_mult)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   750
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   751
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   752
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   753
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   754
subsection{*Numerals and Arithmetic*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   755
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   756
instance complex :: number ..
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   757
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   758
defs (overloaded)
20485
3078fd2eec7b got rid of Numeral.bin type
haftmann
parents: 19765
diff changeset
   759
  complex_number_of_def: "(number_of w :: complex) == of_int w"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   760
    --{*the type constraint is essential!*}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   761
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   762
instance complex :: number_ring
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   763
by (intro_classes, simp add: complex_number_of_def)
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   764
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   765
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   766
text{*Collapse applications of @{term complex_of_real} to @{term number_of}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   767
lemma complex_number_of [simp]: "complex_of_real (number_of w) = number_of w"
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   768
by (rule of_real_number_of_eq)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   769
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   770
text{*This theorem is necessary because theorems such as
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   771
   @{text iszero_number_of_0} only hold for ordered rings. They cannot
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   772
   be generalized to fields in general because they fail for finite fields.
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   773
   They work for type complex because the reals can be embedded in them.*}
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   774
(* TODO: generalize and move to Real/RealVector.thy *)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   775
lemma iszero_complex_number_of [simp]:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   776
     "iszero (number_of w :: complex) = iszero (number_of w :: real)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   777
by (simp only: complex_of_real_zero_iff complex_number_of [symmetric] 
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   778
               iszero_def)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   779
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   780
lemma complex_number_of_cnj [simp]: "cnj(number_of v :: complex) = number_of v"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15234
diff changeset
   781
by (simp only: complex_number_of [symmetric] complex_cnj_complex_of_real)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   782
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   783
lemma complex_number_of_cmod: 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   784
      "cmod(number_of v :: complex) = abs (number_of v :: real)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   785
by (simp only: complex_number_of [symmetric] complex_mod_complex_of_real)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   786
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   787
lemma complex_number_of_Re [simp]: "Re(number_of v :: complex) = number_of v"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   788
by (simp only: complex_number_of [symmetric] Re_complex_of_real)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   789
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   790
lemma complex_number_of_Im [simp]: "Im(number_of v :: complex) = 0"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   791
by (simp only: complex_number_of [symmetric] Im_complex_of_real)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   792
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   793
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   794
by (simp add: expi_def complex_Re_mult_eq complex_Im_mult_eq cis_def)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   795
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   796
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   797
(*examples:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   798
print_depth 22
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   799
set timing;
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   800
set trace_simp;
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   801
fun test s = (Goal s, by (Simp_tac 1)); 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   802
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   803
test "23 * ii + 45 * ii= (x::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   804
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   805
test "5 * ii + 12 - 45 * ii= (x::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   806
test "5 * ii + 40 - 12 * ii + 9 = (x::complex) + 89 * ii";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   807
test "5 * ii + 40 - 12 * ii + 9 - 78 = (x::complex) + 89 * ii";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   808
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   809
test "l + 10 * ii + 90 + 3*l +  9 + 45 * ii= (x::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   810
test "87 + 10 * ii + 90 + 3*7 +  9 + 45 * ii= (x::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   811
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   812
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   813
fun test s = (Goal s; by (Asm_simp_tac 1)); 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   814
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   815
test "x*k = k*(y::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   816
test "k = k*(y::complex)"; 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   817
test "a*(b*c) = (b::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   818
test "a*(b*c) = d*(b::complex)*(x*a)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   819
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   820
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   821
test "(x*k) / (k*(y::complex)) = (uu::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   822
test "(k) / (k*(y::complex)) = (uu::complex)"; 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   823
test "(a*(b*c)) / ((b::complex)) = (uu::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   824
test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   825
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14691
diff changeset
   826
FIXME: what do we do about this?
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   827
test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   828
*)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   829
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
   830
end