src/HOL/Real.thy
author haftmann
Wed, 25 Dec 2013 17:39:06 +0100
changeset 54863 82acc20ded73
parent 54489 03ff4d1e6784
child 55945 e96383acecf9
permissions -rw-r--r--
prefer more canonical names for lemmas on min/max
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     1
(*  Title:      HOL/Real.thy
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     2
    Author:     Jacques D. Fleuriot, University of Edinburgh, 1998
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     3
    Author:     Larry Paulson, University of Cambridge
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     4
    Author:     Jeremy Avigad, Carnegie Mellon University
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     5
    Author:     Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     6
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     7
    Construction of Cauchy Reals by Brian Huffman, 2010
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     8
*)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
     9
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    10
header {* Development of the Reals using Cauchy Sequences *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    11
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    12
theory Real
51773
9328c6681f3c spell conditional_ly_-complete lattices correct
hoelzl
parents: 51539
diff changeset
    13
imports Rat Conditionally_Complete_Lattices
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    14
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    15
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    16
text {*
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    17
  This theory contains a formalization of the real numbers as
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    18
  equivalence classes of Cauchy sequences of rationals.  See
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    19
  @{file "~~/src/HOL/ex/Dedekind_Real.thy"} for an alternative
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    20
  construction using Dedekind cuts.
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    21
*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    22
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    23
subsection {* Preliminary lemmas *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    24
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    25
lemma add_diff_add:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    26
  fixes a b c d :: "'a::ab_group_add"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    27
  shows "(a + c) - (b + d) = (a - b) + (c - d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    28
  by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    29
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    30
lemma minus_diff_minus:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    31
  fixes a b :: "'a::ab_group_add"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    32
  shows "- a - - b = - (a - b)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    33
  by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    34
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    35
lemma mult_diff_mult:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    36
  fixes x y a b :: "'a::ring"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    37
  shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    38
  by (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    39
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    40
lemma inverse_diff_inverse:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    41
  fixes a b :: "'a::division_ring"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    42
  assumes "a \<noteq> 0" and "b \<noteq> 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    43
  shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    44
  using assms by (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    45
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    46
lemma obtain_pos_sum:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    47
  fixes r :: rat assumes r: "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    48
  obtains s t where "0 < s" and "0 < t" and "r = s + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    49
proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    50
    from r show "0 < r/2" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    51
    from r show "0 < r/2" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    52
    show "r = r/2 + r/2" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    53
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    54
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    55
subsection {* Sequences that converge to zero *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    56
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    57
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    58
  vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    59
where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    60
  "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    61
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    62
lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    63
  unfolding vanishes_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    64
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    65
lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    66
  unfolding vanishes_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    67
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    68
lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    69
  unfolding vanishes_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    70
  apply (cases "c = 0", auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    71
  apply (rule exI [where x="\<bar>c\<bar>"], auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    72
  done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    73
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    74
lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    75
  unfolding vanishes_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    76
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    77
lemma vanishes_add:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    78
  assumes X: "vanishes X" and Y: "vanishes Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    79
  shows "vanishes (\<lambda>n. X n + Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    80
proof (rule vanishesI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    81
  fix r :: rat assume "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    82
  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    83
    by (rule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    84
  obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    85
    using vanishesD [OF X s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    86
  obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    87
    using vanishesD [OF Y t] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    88
  have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    89
  proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    90
    fix n assume n: "i \<le> n" "j \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    91
    have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    92
    also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    93
    finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    94
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    95
  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    96
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    97
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    98
lemma vanishes_diff:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
    99
  assumes X: "vanishes X" and Y: "vanishes Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   100
  shows "vanishes (\<lambda>n. X n - Y n)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53652
diff changeset
   101
  unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus X Y)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   102
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   103
lemma vanishes_mult_bounded:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   104
  assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   105
  assumes Y: "vanishes (\<lambda>n. Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   106
  shows "vanishes (\<lambda>n. X n * Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   107
proof (rule vanishesI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   108
  fix r :: rat assume r: "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   109
  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   110
    using X by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   111
  obtain b where b: "0 < b" "r = a * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   112
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   113
    show "0 < r / a" using r a by (simp add: divide_pos_pos)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   114
    show "r = a * (r / a)" using a by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   115
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   116
  obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   117
    using vanishesD [OF Y b(1)] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   118
  have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   119
    by (simp add: b(2) abs_mult mult_strict_mono' a k)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   120
  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   121
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   122
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   123
subsection {* Cauchy sequences *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   124
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   125
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   126
  cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   127
where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   128
  "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   129
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   130
lemma cauchyI:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   131
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r) \<Longrightarrow> cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   132
  unfolding cauchy_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   133
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   134
lemma cauchyD:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   135
  "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   136
  unfolding cauchy_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   137
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   138
lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   139
  unfolding cauchy_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   140
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   141
lemma cauchy_add [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   142
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   143
  shows "cauchy (\<lambda>n. X n + Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   144
proof (rule cauchyI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   145
  fix r :: rat assume "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   146
  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   147
    by (rule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   148
  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   149
    using cauchyD [OF X s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   150
  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   151
    using cauchyD [OF Y t] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   152
  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   153
  proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   154
    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   155
    have "\<bar>(X m + Y m) - (X n + Y n)\<bar> \<le> \<bar>X m - X n\<bar> + \<bar>Y m - Y n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   156
      unfolding add_diff_add by (rule abs_triangle_ineq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   157
    also have "\<dots> < s + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   158
      by (rule add_strict_mono, simp_all add: i j *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   159
    finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   160
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   161
  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   162
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   163
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   164
lemma cauchy_minus [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   165
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   166
  shows "cauchy (\<lambda>n. - X n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   167
using assms unfolding cauchy_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   168
unfolding minus_diff_minus abs_minus_cancel .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   169
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   170
lemma cauchy_diff [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   171
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   172
  shows "cauchy (\<lambda>n. X n - Y n)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53652
diff changeset
   173
  using assms unfolding diff_conv_add_uminus by (simp del: add_uminus_conv_diff)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   174
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   175
lemma cauchy_imp_bounded:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   176
  assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   177
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   178
  obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   179
    using cauchyD [OF assms zero_less_one] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   180
  show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   181
  proof (intro exI conjI allI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   182
    have "0 \<le> \<bar>X 0\<bar>" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   183
    also have "\<bar>X 0\<bar> \<le> Max (abs ` X ` {..k})" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   184
    finally have "0 \<le> Max (abs ` X ` {..k})" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   185
    thus "0 < Max (abs ` X ` {..k}) + 1" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   186
  next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   187
    fix n :: nat
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   188
    show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   189
    proof (rule linorder_le_cases)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   190
      assume "n \<le> k"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   191
      hence "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   192
      thus "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   193
    next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   194
      assume "k \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   195
      have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   196
      also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   197
        by (rule abs_triangle_ineq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   198
      also have "\<dots> < Max (abs ` X ` {..k}) + 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   199
        by (rule add_le_less_mono, simp, simp add: k `k \<le> n`)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   200
      finally show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   201
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   202
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   203
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   204
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   205
lemma cauchy_mult [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   206
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   207
  shows "cauchy (\<lambda>n. X n * Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   208
proof (rule cauchyI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   209
  fix r :: rat assume "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   210
  then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   211
    by (rule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   212
  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   213
    using cauchy_imp_bounded [OF X] by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   214
  obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   215
    using cauchy_imp_bounded [OF Y] by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   216
  obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   217
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   218
    show "0 < v/b" using v b(1) by (rule divide_pos_pos)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   219
    show "0 < u/a" using u a(1) by (rule divide_pos_pos)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   220
    show "r = a * (u/a) + (v/b) * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   221
      using a(1) b(1) `r = u + v` by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   222
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   223
  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   224
    using cauchyD [OF X s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   225
  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   226
    using cauchyD [OF Y t] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   227
  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m - X n * Y n\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   228
  proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   229
    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   230
    have "\<bar>X m * Y m - X n * Y n\<bar> = \<bar>X m * (Y m - Y n) + (X m - X n) * Y n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   231
      unfolding mult_diff_mult ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   232
    also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   233
      by (rule abs_triangle_ineq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   234
    also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   235
      unfolding abs_mult ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   236
    also have "\<dots> < a * t + s * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   237
      by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   238
    finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   239
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   240
  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   241
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   242
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   243
lemma cauchy_not_vanishes_cases:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   244
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   245
  assumes nz: "\<not> vanishes X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   246
  shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   247
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   248
  obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   249
    using nz unfolding vanishes_def by (auto simp add: not_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   250
  obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   251
    using `0 < r` by (rule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   252
  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   253
    using cauchyD [OF X s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   254
  obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   255
    using r by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   256
  have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   257
    using i `i \<le> k` by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   258
  have "X k \<le> - r \<or> r \<le> X k"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   259
    using `r \<le> \<bar>X k\<bar>` by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   260
  hence "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   261
    unfolding `r = s + t` using k by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   262
  hence "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   263
  thus "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   264
    using t by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   265
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   266
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   267
lemma cauchy_not_vanishes:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   268
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   269
  assumes nz: "\<not> vanishes X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   270
  shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   271
using cauchy_not_vanishes_cases [OF assms]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   272
by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   273
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   274
lemma cauchy_inverse [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   275
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   276
  assumes nz: "\<not> vanishes X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   277
  shows "cauchy (\<lambda>n. inverse (X n))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   278
proof (rule cauchyI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   279
  fix r :: rat assume "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   280
  obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   281
    using cauchy_not_vanishes [OF X nz] by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   282
  from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   283
  obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   284
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   285
    show "0 < b * r * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   286
      by (simp add: `0 < r` b mult_pos_pos)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   287
    show "r = inverse b * (b * r * b) * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   288
      using b by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   289
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   290
  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   291
    using cauchyD [OF X s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   292
  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   293
  proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   294
    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   295
    have "\<bar>inverse (X m) - inverse (X n)\<bar> =
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   296
          inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   297
      by (simp add: inverse_diff_inverse nz * abs_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   298
    also have "\<dots> < inverse b * s * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   299
      by (simp add: mult_strict_mono less_imp_inverse_less
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   300
                    mult_pos_pos i j b * s)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   301
    finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   302
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   303
  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   304
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   305
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   306
lemma vanishes_diff_inverse:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   307
  assumes X: "cauchy X" "\<not> vanishes X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   308
  assumes Y: "cauchy Y" "\<not> vanishes Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   309
  assumes XY: "vanishes (\<lambda>n. X n - Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   310
  shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   311
proof (rule vanishesI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   312
  fix r :: rat assume r: "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   313
  obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   314
    using cauchy_not_vanishes [OF X] by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   315
  obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   316
    using cauchy_not_vanishes [OF Y] by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   317
  obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   318
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   319
    show "0 < a * r * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   320
      using a r b by (simp add: mult_pos_pos)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   321
    show "inverse a * (a * r * b) * inverse b = r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   322
      using a r b by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   323
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   324
  obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   325
    using vanishesD [OF XY s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   326
  have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   327
  proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   328
    fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   329
    have "X n \<noteq> 0" and "Y n \<noteq> 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   330
      using i j a b n by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   331
    hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   332
        inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   333
      by (simp add: inverse_diff_inverse abs_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   334
    also have "\<dots> < inverse a * s * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   335
      apply (intro mult_strict_mono' less_imp_inverse_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   336
      apply (simp_all add: a b i j k n mult_nonneg_nonneg)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   337
      done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   338
    also note `inverse a * s * inverse b = r`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   339
    finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   340
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   341
  thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   342
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   343
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   344
subsection {* Equivalence relation on Cauchy sequences *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   345
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   346
definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   347
  where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   348
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   349
lemma realrelI [intro?]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   350
  assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   351
  shows "realrel X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   352
  using assms unfolding realrel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   353
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   354
lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   355
  unfolding realrel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   356
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   357
lemma symp_realrel: "symp realrel"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   358
  unfolding realrel_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   359
  by (rule sympI, clarify, drule vanishes_minus, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   360
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   361
lemma transp_realrel: "transp realrel"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   362
  unfolding realrel_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   363
  apply (rule transpI, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   364
  apply (drule (1) vanishes_add)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   365
  apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   366
  done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   367
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   368
lemma part_equivp_realrel: "part_equivp realrel"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   369
  by (fast intro: part_equivpI symp_realrel transp_realrel
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   370
    realrel_refl cauchy_const)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   371
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   372
subsection {* The field of real numbers *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   373
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   374
quotient_type real = "nat \<Rightarrow> rat" / partial: realrel
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   375
  morphisms rep_real Real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   376
  by (rule part_equivp_realrel)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   377
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   378
lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   379
  unfolding real.pcr_cr_eq cr_real_def realrel_def by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   380
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   381
lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   382
  assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   383
proof (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   384
  case (1 X)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   385
  hence "cauchy X" by (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   386
  thus "P (Real X)" by (rule assms)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   387
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   388
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   389
lemma eq_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   390
  "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   391
  using real.rel_eq_transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   392
  unfolding real.pcr_cr_eq cr_real_def fun_rel_def realrel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   393
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51775
diff changeset
   394
lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51775
diff changeset
   395
by (simp add: real.domain_eq realrel_def)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   396
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   397
instantiation real :: field_inverse_zero
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   398
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   399
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   400
lift_definition zero_real :: "real" is "\<lambda>n. 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   401
  by (simp add: realrel_refl)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   402
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   403
lift_definition one_real :: "real" is "\<lambda>n. 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   404
  by (simp add: realrel_refl)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   405
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   406
lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n + Y n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   407
  unfolding realrel_def add_diff_add
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   408
  by (simp only: cauchy_add vanishes_add simp_thms)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   409
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   410
lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   411
  unfolding realrel_def minus_diff_minus
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   412
  by (simp only: cauchy_minus vanishes_minus simp_thms)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   413
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   414
lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   415
  unfolding realrel_def mult_diff_mult
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   416
  by (subst (4) mult_commute, simp only: cauchy_mult vanishes_add
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   417
    vanishes_mult_bounded cauchy_imp_bounded simp_thms)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   418
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   419
lift_definition inverse_real :: "real \<Rightarrow> real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   420
  is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   421
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   422
  fix X Y assume "realrel X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   423
  hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   424
    unfolding realrel_def by simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   425
  have "vanishes X \<longleftrightarrow> vanishes Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   426
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   427
    assume "vanishes X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   428
    from vanishes_diff [OF this XY] show "vanishes Y" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   429
  next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   430
    assume "vanishes Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   431
    from vanishes_add [OF this XY] show "vanishes X" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   432
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   433
  thus "?thesis X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   434
    unfolding realrel_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   435
    by (simp add: vanishes_diff_inverse X Y XY)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   436
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   437
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   438
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   439
  "x - y = (x::real) + - y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   440
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   441
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   442
  "x / y = (x::real) * inverse y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   443
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   444
lemma add_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   445
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   446
  shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   447
  using assms plus_real.transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   448
  unfolding cr_real_eq fun_rel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   449
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   450
lemma minus_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   451
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   452
  shows "- Real X = Real (\<lambda>n. - X n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   453
  using assms uminus_real.transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   454
  unfolding cr_real_eq fun_rel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   455
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   456
lemma diff_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   457
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   458
  shows "Real X - Real Y = Real (\<lambda>n. X n - Y n)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53652
diff changeset
   459
  unfolding minus_real_def
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   460
  by (simp add: minus_Real add_Real X Y)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   461
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   462
lemma mult_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   463
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   464
  shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   465
  using assms times_real.transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   466
  unfolding cr_real_eq fun_rel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   467
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   468
lemma inverse_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   469
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   470
  shows "inverse (Real X) =
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   471
    (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   472
  using assms inverse_real.transfer zero_real.transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   473
  unfolding cr_real_eq fun_rel_def by (simp split: split_if_asm, metis)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   474
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   475
instance proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   476
  fix a b c :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   477
  show "a + b = b + a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   478
    by transfer (simp add: add_ac realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   479
  show "(a + b) + c = a + (b + c)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   480
    by transfer (simp add: add_ac realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   481
  show "0 + a = a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   482
    by transfer (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   483
  show "- a + a = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   484
    by transfer (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   485
  show "a - b = a + - b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   486
    by (rule minus_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   487
  show "(a * b) * c = a * (b * c)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   488
    by transfer (simp add: mult_ac realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   489
  show "a * b = b * a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   490
    by transfer (simp add: mult_ac realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   491
  show "1 * a = a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   492
    by transfer (simp add: mult_ac realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   493
  show "(a + b) * c = a * c + b * c"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   494
    by transfer (simp add: distrib_right realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   495
  show "(0\<Colon>real) \<noteq> (1\<Colon>real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   496
    by transfer (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   497
  show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   498
    apply transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   499
    apply (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   500
    apply (rule vanishesI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   501
    apply (frule (1) cauchy_not_vanishes, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   502
    apply (rule_tac x=k in exI, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   503
    apply (drule_tac x=n in spec, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   504
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   505
  show "a / b = a * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   506
    by (rule divide_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   507
  show "inverse (0::real) = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   508
    by transfer (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   509
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   510
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   511
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   512
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   513
subsection {* Positive reals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   514
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   515
lift_definition positive :: "real \<Rightarrow> bool"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   516
  is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   517
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   518
  { fix X Y
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   519
    assume "realrel X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   520
    hence XY: "vanishes (\<lambda>n. X n - Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   521
      unfolding realrel_def by simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   522
    assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   523
    then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   524
      by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   525
    obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   526
      using `0 < r` by (rule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   527
    obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   528
      using vanishesD [OF XY s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   529
    have "\<forall>n\<ge>max i j. t < Y n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   530
    proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   531
      fix n assume n: "i \<le> n" "j \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   532
      have "\<bar>X n - Y n\<bar> < s" and "r < X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   533
        using i j n by simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   534
      thus "t < Y n" unfolding r by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   535
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   536
    hence "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   537
  } note 1 = this
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   538
  fix X Y assume "realrel X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   539
  hence "realrel X Y" and "realrel Y X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   540
    using symp_realrel unfolding symp_def by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   541
  thus "?thesis X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   542
    by (safe elim!: 1)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   543
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   544
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   545
lemma positive_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   546
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   547
  shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   548
  using assms positive.transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   549
  unfolding cr_real_eq fun_rel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   550
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   551
lemma positive_zero: "\<not> positive 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   552
  by transfer auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   553
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   554
lemma positive_add:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   555
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   556
apply transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   557
apply (clarify, rename_tac a b i j)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   558
apply (rule_tac x="a + b" in exI, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   559
apply (rule_tac x="max i j" in exI, clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   560
apply (simp add: add_strict_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   561
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   562
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   563
lemma positive_mult:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   564
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   565
apply transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   566
apply (clarify, rename_tac a b i j)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   567
apply (rule_tac x="a * b" in exI, simp add: mult_pos_pos)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   568
apply (rule_tac x="max i j" in exI, clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   569
apply (rule mult_strict_mono, auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   570
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   571
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   572
lemma positive_minus:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   573
  "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   574
apply transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   575
apply (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   576
apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   577
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   578
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   579
instantiation real :: linordered_field_inverse_zero
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   580
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   581
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   582
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   583
  "x < y \<longleftrightarrow> positive (y - x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   584
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   585
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   586
  "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   587
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   588
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   589
  "abs (a::real) = (if a < 0 then - a else a)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   590
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   591
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   592
  "sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   593
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   594
instance proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   595
  fix a b c :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   596
  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   597
    by (rule abs_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   598
  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   599
    unfolding less_eq_real_def less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   600
    by (auto, drule (1) positive_add, simp_all add: positive_zero)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   601
  show "a \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   602
    unfolding less_eq_real_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   603
  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   604
    unfolding less_eq_real_def less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   605
    by (auto, drule (1) positive_add, simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   606
  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   607
    unfolding less_eq_real_def less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   608
    by (auto, drule (1) positive_add, simp add: positive_zero)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   609
  show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53652
diff changeset
   610
    unfolding less_eq_real_def less_real_def by auto
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   611
    (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   612
    (* Should produce c + b - (c + a) \<equiv> b - a *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   613
  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   614
    by (rule sgn_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   615
  show "a \<le> b \<or> b \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   616
    unfolding less_eq_real_def less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   617
    by (auto dest!: positive_minus)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   618
  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   619
    unfolding less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   620
    by (drule (1) positive_mult, simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   621
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   622
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   623
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   624
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   625
instantiation real :: distrib_lattice
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   626
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   627
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   628
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   629
  "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   630
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   631
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   632
  "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   633
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   634
instance proof
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54489
diff changeset
   635
qed (auto simp add: inf_real_def sup_real_def max_min_distrib2)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   636
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   637
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   638
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   639
lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   640
apply (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   641
apply (simp add: zero_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   642
apply (simp add: one_real_def add_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   643
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   644
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   645
lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   646
apply (cases x rule: int_diff_cases)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   647
apply (simp add: of_nat_Real diff_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   648
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   649
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   650
lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   651
apply (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   652
apply (simp add: Fract_of_int_quotient of_rat_divide)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   653
apply (simp add: of_int_Real divide_inverse)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   654
apply (simp add: inverse_Real mult_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   655
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   656
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   657
instance real :: archimedean_field
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   658
proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   659
  fix x :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   660
  show "\<exists>z. x \<le> of_int z"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   661
    apply (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   662
    apply (frule cauchy_imp_bounded, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   663
    apply (rule_tac x="ceiling b + 1" in exI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   664
    apply (rule less_imp_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   665
    apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   666
    apply (rule_tac x=1 in exI, simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   667
    apply (rule_tac x=0 in exI, clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   668
    apply (rule le_less_trans [OF abs_ge_self])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   669
    apply (rule less_le_trans [OF _ le_of_int_ceiling])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   670
    apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   671
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   672
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   673
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   674
instantiation real :: floor_ceiling
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   675
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   676
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   677
definition [code del]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   678
  "floor (x::real) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   679
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   680
instance proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   681
  fix x :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   682
  show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   683
    unfolding floor_real_def using floor_exists1 by (rule theI')
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   684
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   685
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   686
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   687
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   688
subsection {* Completeness *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   689
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   690
lemma not_positive_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   691
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   692
  shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   693
unfolding positive_Real [OF X]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   694
apply (auto, unfold not_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   695
apply (erule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   696
apply (drule_tac x=s in spec, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   697
apply (drule_tac r=t in cauchyD [OF X], clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   698
apply (drule_tac x=k in spec, clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   699
apply (rule_tac x=n in exI, clarify, rename_tac m)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   700
apply (drule_tac x=m in spec, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   701
apply (drule_tac x=n in spec, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   702
apply (drule spec, drule (1) mp, clarify, rename_tac i)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   703
apply (rule_tac x="max i k" in exI, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   704
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   705
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   706
lemma le_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   707
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   708
  shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   709
unfolding not_less [symmetric, where 'a=real] less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   710
apply (simp add: diff_Real not_positive_Real X Y)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   711
apply (simp add: diff_le_eq add_ac)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   712
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   713
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   714
lemma le_RealI:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   715
  assumes Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   716
  shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   717
proof (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   718
  fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   719
  hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   720
    by (simp add: of_rat_Real le_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   721
  {
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   722
    fix r :: rat assume "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   723
    then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   724
      by (rule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   725
    obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   726
      using cauchyD [OF Y s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   727
    obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   728
      using le [OF t] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   729
    have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   730
    proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   731
      fix n assume n: "i \<le> n" "j \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   732
      have "X n \<le> Y i + t" using n j by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   733
      moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   734
      ultimately show "X n \<le> Y n + r" unfolding r by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   735
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   736
    hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   737
  }
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   738
  thus "Real X \<le> Real Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   739
    by (simp add: of_rat_Real le_Real X Y)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   740
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   741
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   742
lemma Real_leI:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   743
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   744
  assumes le: "\<forall>n. of_rat (X n) \<le> y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   745
  shows "Real X \<le> y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   746
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   747
  have "- y \<le> - Real X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   748
    by (simp add: minus_Real X le_RealI of_rat_minus le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   749
  thus ?thesis by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   750
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   751
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   752
lemma less_RealD:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   753
  assumes Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   754
  shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   755
by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   756
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   757
lemma of_nat_less_two_power:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   758
  "of_nat n < (2::'a::linordered_idom) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   759
apply (induct n)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   760
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   761
apply (subgoal_tac "(1::'a) \<le> 2 ^ n")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   762
apply (drule (1) add_le_less_mono, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   763
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   764
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   765
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   766
lemma complete_real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   767
  fixes S :: "real set"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   768
  assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   769
  shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   770
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   771
  obtain x where x: "x \<in> S" using assms(1) ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   772
  obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   773
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   774
  def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   775
  obtain a where a: "\<not> P a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   776
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   777
    have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   778
    also have "x - 1 < x" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   779
    finally have "of_int (floor (x - 1)) < x" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   780
    hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   781
    then show "\<not> P (of_int (floor (x - 1)))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   782
      unfolding P_def of_rat_of_int_eq using x by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   783
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   784
  obtain b where b: "P b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   785
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   786
    show "P (of_int (ceiling z))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   787
    unfolding P_def of_rat_of_int_eq
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   788
    proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   789
      fix y assume "y \<in> S"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   790
      hence "y \<le> z" using z by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   791
      also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   792
      finally show "y \<le> of_int (ceiling z)" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   793
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   794
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   795
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   796
  def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   797
  def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   798
  def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   799
  def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   800
  def C \<equiv> "\<lambda>n. avg (A n) (B n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   801
  have A_0 [simp]: "A 0 = a" unfolding A_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   802
  have B_0 [simp]: "B 0 = b" unfolding B_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   803
  have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   804
    unfolding A_def B_def C_def bisect_def split_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   805
  have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   806
    unfolding A_def B_def C_def bisect_def split_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   807
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   808
  have width: "\<And>n. B n - A n = (b - a) / 2^n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   809
    apply (simp add: eq_divide_eq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   810
    apply (induct_tac n, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   811
    apply (simp add: C_def avg_def algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   812
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   813
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   814
  have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   815
    apply (simp add: divide_less_eq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   816
    apply (subst mult_commute)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   817
    apply (frule_tac y=y in ex_less_of_nat_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   818
    apply clarify
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   819
    apply (rule_tac x=n in exI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   820
    apply (erule less_trans)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   821
    apply (rule mult_strict_right_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   822
    apply (rule le_less_trans [OF _ of_nat_less_two_power])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   823
    apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   824
    apply assumption
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   825
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   826
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   827
  have PA: "\<And>n. \<not> P (A n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   828
    by (induct_tac n, simp_all add: a)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   829
  have PB: "\<And>n. P (B n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   830
    by (induct_tac n, simp_all add: b)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   831
  have ab: "a < b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   832
    using a b unfolding P_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   833
    apply (clarsimp simp add: not_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   834
    apply (drule (1) bspec)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   835
    apply (drule (1) less_le_trans)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   836
    apply (simp add: of_rat_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   837
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   838
  have AB: "\<And>n. A n < B n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   839
    by (induct_tac n, simp add: ab, simp add: C_def avg_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   840
  have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   841
    apply (auto simp add: le_less [where 'a=nat])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   842
    apply (erule less_Suc_induct)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   843
    apply (clarsimp simp add: C_def avg_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   844
    apply (simp add: add_divide_distrib [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   845
    apply (rule AB [THEN less_imp_le])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   846
    apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   847
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   848
  have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   849
    apply (auto simp add: le_less [where 'a=nat])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   850
    apply (erule less_Suc_induct)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   851
    apply (clarsimp simp add: C_def avg_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   852
    apply (simp add: add_divide_distrib [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   853
    apply (rule AB [THEN less_imp_le])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   854
    apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   855
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   856
  have cauchy_lemma:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   857
    "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   858
    apply (rule cauchyI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   859
    apply (drule twos [where y="b - a"])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   860
    apply (erule exE)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   861
    apply (rule_tac x=n in exI, clarify, rename_tac i j)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   862
    apply (rule_tac y="B n - A n" in le_less_trans) defer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   863
    apply (simp add: width)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   864
    apply (drule_tac x=n in spec)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   865
    apply (frule_tac x=i in spec, drule (1) mp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   866
    apply (frule_tac x=j in spec, drule (1) mp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   867
    apply (frule A_mono, drule B_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   868
    apply (frule A_mono, drule B_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   869
    apply arith
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   870
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   871
  have "cauchy A"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   872
    apply (rule cauchy_lemma [rule_format])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   873
    apply (simp add: A_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   874
    apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   875
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   876
  have "cauchy B"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   877
    apply (rule cauchy_lemma [rule_format])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   878
    apply (simp add: B_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   879
    apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   880
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   881
  have 1: "\<forall>x\<in>S. x \<le> Real B"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   882
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   883
    fix x assume "x \<in> S"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   884
    then show "x \<le> Real B"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   885
      using PB [unfolded P_def] `cauchy B`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   886
      by (simp add: le_RealI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   887
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   888
  have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   889
    apply clarify
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   890
    apply (erule contrapos_pp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   891
    apply (simp add: not_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   892
    apply (drule less_RealD [OF `cauchy A`], clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   893
    apply (subgoal_tac "\<not> P (A n)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   894
    apply (simp add: P_def not_le, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   895
    apply (erule rev_bexI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   896
    apply (erule (1) less_trans)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   897
    apply (simp add: PA)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   898
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   899
  have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   900
  proof (rule vanishesI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   901
    fix r :: rat assume "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   902
    then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   903
      using twos by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   904
    have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   905
    proof (clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   906
      fix n assume n: "k \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   907
      have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   908
        by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   909
      also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   910
        using n by (simp add: divide_left_mono mult_pos_pos)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   911
      also note k
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   912
      finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   913
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   914
    thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   915
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   916
  hence 3: "Real B = Real A"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   917
    by (simp add: eq_Real `cauchy A` `cauchy B` width)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   918
  show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   919
    using 1 2 3 by (rule_tac x="Real B" in exI, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   920
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   921
51775
408d937c9486 revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents: 51773
diff changeset
   922
instantiation real :: linear_continuum
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   923
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   924
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   925
subsection{*Supremum of a set of reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   926
54281
b01057e72233 int and nat are conditionally_complete_lattices
hoelzl
parents: 54263
diff changeset
   927
definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)"
b01057e72233 int and nat are conditionally_complete_lattices
hoelzl
parents: 54263
diff changeset
   928
definition "Inf (X::real set) = - Sup (uminus ` X)"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   929
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   930
instance
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   931
proof
54258
adfc759263ab use bdd_above and bdd_below for conditionally complete lattices
hoelzl
parents: 54230
diff changeset
   932
  { fix x :: real and X :: "real set"
adfc759263ab use bdd_above and bdd_below for conditionally complete lattices
hoelzl
parents: 54230
diff changeset
   933
    assume x: "x \<in> X" "bdd_above X"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   934
    then obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
54258
adfc759263ab use bdd_above and bdd_below for conditionally complete lattices
hoelzl
parents: 54230
diff changeset
   935
      using complete_real[of X] unfolding bdd_above_def by blast
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   936
    then show "x \<le> Sup X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   937
      unfolding Sup_real_def by (rule LeastI2_order) (auto simp: x) }
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   938
  note Sup_upper = this
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   939
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   940
  { fix z :: real and X :: "real set"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   941
    assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   942
    then obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   943
      using complete_real[of X] by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   944
    then have "Sup X = s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   945
      unfolding Sup_real_def by (best intro: Least_equality)  
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53076
diff changeset
   946
    also from s z have "... \<le> z"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   947
      by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   948
    finally show "Sup X \<le> z" . }
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   949
  note Sup_least = this
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   950
54281
b01057e72233 int and nat are conditionally_complete_lattices
hoelzl
parents: 54263
diff changeset
   951
  { fix x :: real and X :: "real set" assume x: "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
b01057e72233 int and nat are conditionally_complete_lattices
hoelzl
parents: 54263
diff changeset
   952
      using Sup_upper[of "-x" "uminus ` X"] by (auto simp: Inf_real_def) }
b01057e72233 int and nat are conditionally_complete_lattices
hoelzl
parents: 54263
diff changeset
   953
  { fix z :: real and X :: "real set" assume "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" then show "z \<le> Inf X"
b01057e72233 int and nat are conditionally_complete_lattices
hoelzl
parents: 54263
diff changeset
   954
      using Sup_least[of "uminus ` X" "- z"] by (force simp: Inf_real_def) }
51775
408d937c9486 revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents: 51773
diff changeset
   955
  show "\<exists>a b::real. a \<noteq> b"
408d937c9486 revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents: 51773
diff changeset
   956
    using zero_neq_one by blast
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   957
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   958
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   959
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   960
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   961
subsection {* Hiding implementation details *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   962
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   963
hide_const (open) vanishes cauchy positive Real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   964
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   965
declare Real_induct [induct del]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   966
declare Abs_real_induct [induct del]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   967
declare Abs_real_cases [cases del]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   968
53652
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
   969
lifting_update real.lifting
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
   970
lifting_forget real.lifting
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51775
diff changeset
   971
  
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   972
subsection{*More Lemmas*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   973
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   974
text {* BH: These lemmas should not be necessary; they should be
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   975
covered by existing simp rules and simplification procedures. *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   976
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   977
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   978
by simp (* redundant with mult_cancel_left *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   979
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   980
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   981
by simp (* redundant with mult_cancel_right *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   982
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   983
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   984
by simp (* solved by linordered_ring_less_cancel_factor simproc *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   985
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   986
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   987
by simp (* solved by linordered_ring_le_cancel_factor simproc *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   988
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   989
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   990
by simp (* solved by linordered_ring_le_cancel_factor simproc *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   991
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   992
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   993
subsection {* Embedding numbers into the Reals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   994
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   995
abbreviation
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   996
  real_of_nat :: "nat \<Rightarrow> real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   997
where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   998
  "real_of_nat \<equiv> of_nat"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   999
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1000
abbreviation
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1001
  real_of_int :: "int \<Rightarrow> real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1002
where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1003
  "real_of_int \<equiv> of_int"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1004
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1005
abbreviation
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1006
  real_of_rat :: "rat \<Rightarrow> real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1007
where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1008
  "real_of_rat \<equiv> of_rat"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1009
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1010
consts
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1011
  (*overloaded constant for injecting other types into "real"*)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1012
  real :: "'a => real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1013
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1014
defs (overloaded)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1015
  real_of_nat_def [code_unfold]: "real == real_of_nat"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1016
  real_of_int_def [code_unfold]: "real == real_of_int"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1017
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1018
declare [[coercion_enabled]]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1019
declare [[coercion "real::nat\<Rightarrow>real"]]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1020
declare [[coercion "real::int\<Rightarrow>real"]]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1021
declare [[coercion "int"]]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1022
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1023
declare [[coercion_map map]]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1024
declare [[coercion_map "% f g h x. g (h (f x))"]]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1025
declare [[coercion_map "% f g (x,y) . (f x, g y)"]]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1026
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1027
lemma real_eq_of_nat: "real = of_nat"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1028
  unfolding real_of_nat_def ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1029
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1030
lemma real_eq_of_int: "real = of_int"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1031
  unfolding real_of_int_def ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1032
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1033
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1034
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1035
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1036
lemma real_of_one [simp]: "real (1::int) = (1::real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1037
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1038
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1039
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1040
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1041
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1042
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1043
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1044
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1045
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1046
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1047
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1048
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1049
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1050
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1051
lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1052
by (simp add: real_of_int_def of_int_power)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1053
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1054
lemmas power_real_of_int = real_of_int_power [symmetric]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1055
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1056
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1057
  apply (subst real_eq_of_int)+
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1058
  apply (rule of_int_setsum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1059
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1060
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1061
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1062
    (PROD x:A. real(f x))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1063
  apply (subst real_eq_of_int)+
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1064
  apply (rule of_int_setprod)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1065
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1066
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1067
lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1068
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1069
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1070
lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1071
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1072
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1073
lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1074
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1075
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1076
lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1077
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1078
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1079
lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1080
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1081
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1082
lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1083
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1084
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1085
lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)" 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1086
by (simp add: real_of_int_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1087
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1088
lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1089
by (simp add: real_of_int_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1090
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1091
lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1092
  unfolding real_of_one[symmetric] real_of_int_less_iff ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1093
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1094
lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1095
  unfolding real_of_one[symmetric] real_of_int_le_iff ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1096
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1097
lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1098
  unfolding real_of_one[symmetric] real_of_int_less_iff ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1099
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1100
lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1101
  unfolding real_of_one[symmetric] real_of_int_le_iff ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1102
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1103
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1104
by (auto simp add: abs_if)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1105
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1106
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1107
  apply (subgoal_tac "real n + 1 = real (n + 1)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1108
  apply (simp del: real_of_int_add)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1109
  apply auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1110
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1111
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1112
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1113
  apply (subgoal_tac "real m + 1 = real (m + 1)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1114
  apply (simp del: real_of_int_add)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1115
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1116
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1117
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1118
lemma real_of_int_div_aux: "(real (x::int)) / (real d) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1119
    real (x div d) + (real (x mod d)) / (real d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1120
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1121
  have "x = (x div d) * d + x mod d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1122
    by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1123
  then have "real x = real (x div d) * real d + real(x mod d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1124
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1125
  then have "real x / real d = ... / real d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1126
    by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1127
  then show ?thesis
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1128
    by (auto simp add: add_divide_distrib algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1129
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1130
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1131
lemma real_of_int_div: "(d :: int) dvd n ==>
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1132
    real(n div d) = real n / real d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1133
  apply (subst real_of_int_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1134
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1135
  apply (simp add: dvd_eq_mod_eq_0)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1136
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1137
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1138
lemma real_of_int_div2:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1139
  "0 <= real (n::int) / real (x) - real (n div x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1140
  apply (case_tac "x = 0")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1141
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1142
  apply (case_tac "0 < x")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1143
  apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1144
  apply (subst real_of_int_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1145
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1146
  apply (subst zero_le_divide_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1147
  apply auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1148
  apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1149
  apply (subst real_of_int_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1150
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1151
  apply (subst zero_le_divide_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1152
  apply auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1153
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1154
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1155
lemma real_of_int_div3:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1156
  "real (n::int) / real (x) - real (n div x) <= 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1157
  apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1158
  apply (subst real_of_int_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1159
  apply (auto simp add: divide_le_eq intro: order_less_imp_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1160
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1161
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1162
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1163
by (insert real_of_int_div2 [of n x], simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1164
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1165
lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1166
unfolding real_of_int_def by (rule Ints_of_int)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1167
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1168
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1169
subsection{*Embedding the Naturals into the Reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1170
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1171
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1172
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1173
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1174
lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1175
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1176
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1177
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1178
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1179
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1180
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1181
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1182
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1183
(*Not for addsimps: often the LHS is used to represent a positive natural*)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1184
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1185
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1186
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1187
lemma real_of_nat_less_iff [iff]: 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1188
     "(real (n::nat) < real m) = (n < m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1189
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1190
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1191
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1192
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1193
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1194
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1195
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1196
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1197
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1198
by (simp add: real_of_nat_def del: of_nat_Suc)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1199
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1200
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1201
by (simp add: real_of_nat_def of_nat_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1202
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1203
lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1204
by (simp add: real_of_nat_def of_nat_power)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1205
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1206
lemmas power_real_of_nat = real_of_nat_power [symmetric]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1207
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1208
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1209
    (SUM x:A. real(f x))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1210
  apply (subst real_eq_of_nat)+
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1211
  apply (rule of_nat_setsum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1212
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1213
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1214
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1215
    (PROD x:A. real(f x))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1216
  apply (subst real_eq_of_nat)+
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1217
  apply (rule of_nat_setprod)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1218
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1219
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1220
lemma real_of_card: "real (card A) = setsum (%x.1) A"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1221
  apply (subst card_eq_setsum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1222
  apply (subst real_of_nat_setsum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1223
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1224
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1225
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1226
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1227
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1228
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1229
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1230
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1231
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1232
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1233
by (simp add: add: real_of_nat_def of_nat_diff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1234
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1235
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1236
by (auto simp: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1237
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1238
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1239
by (simp add: add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1240
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1241
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1242
by (simp add: add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1243
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1244
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1245
  apply (subgoal_tac "real n + 1 = real (Suc n)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1246
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1247
  apply (auto simp add: real_of_nat_Suc)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1248
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1249
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1250
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1251
  apply (subgoal_tac "real m + 1 = real (Suc m)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1252
  apply (simp add: less_Suc_eq_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1253
  apply (simp add: real_of_nat_Suc)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1254
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1255
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1256
lemma real_of_nat_div_aux: "(real (x::nat)) / (real d) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1257
    real (x div d) + (real (x mod d)) / (real d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1258
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1259
  have "x = (x div d) * d + x mod d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1260
    by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1261
  then have "real x = real (x div d) * real d + real(x mod d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1262
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1263
  then have "real x / real d = \<dots> / real d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1264
    by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1265
  then show ?thesis
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1266
    by (auto simp add: add_divide_distrib algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1267
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1268
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1269
lemma real_of_nat_div: "(d :: nat) dvd n ==>
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1270
    real(n div d) = real n / real d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1271
  by (subst real_of_nat_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1272
    (auto simp add: dvd_eq_mod_eq_0 [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1273
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1274
lemma real_of_nat_div2:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1275
  "0 <= real (n::nat) / real (x) - real (n div x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1276
apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1277
apply (subst real_of_nat_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1278
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1279
apply (subst zero_le_divide_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1280
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1281
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1282
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1283
lemma real_of_nat_div3:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1284
  "real (n::nat) / real (x) - real (n div x) <= 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1285
apply(case_tac "x = 0")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1286
apply (simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1287
apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1288
apply (subst real_of_nat_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1289
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1290
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1291
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1292
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1293
by (insert real_of_nat_div2 [of n x], simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1294
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1295
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1296
by (simp add: real_of_int_def real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1297
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1298
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1299
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1300
  apply force
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1301
  apply (simp only: real_of_int_of_nat_eq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1302
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1303
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1304
lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1305
unfolding real_of_nat_def by (rule of_nat_in_Nats)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1306
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1307
lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1308
unfolding real_of_nat_def by (rule Ints_of_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1309
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1310
subsection {* The Archimedean Property of the Reals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1311
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1312
theorem reals_Archimedean:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1313
  assumes x_pos: "0 < x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1314
  shows "\<exists>n. inverse (real (Suc n)) < x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1315
  unfolding real_of_nat_def using x_pos
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1316
  by (rule ex_inverse_of_nat_Suc_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1317
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1318
lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1319
  unfolding real_of_nat_def by (rule ex_less_of_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1320
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1321
lemma reals_Archimedean3:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1322
  assumes x_greater_zero: "0 < x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1323
  shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1324
  unfolding real_of_nat_def using `0 < x`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1325
  by (auto intro: ex_less_of_nat_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1326
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1327
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1328
subsection{* Rationals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1329
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1330
lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1331
by (simp add: real_eq_of_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1332
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1333
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1334
lemma Rats_eq_int_div_int:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1335
  "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1336
proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1337
  show "\<rat> \<subseteq> ?S"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1338
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1339
    fix x::real assume "x : \<rat>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1340
    then obtain r where "x = of_rat r" unfolding Rats_def ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1341
    have "of_rat r : ?S"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1342
      by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1343
    thus "x : ?S" using `x = of_rat r` by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1344
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1345
next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1346
  show "?S \<subseteq> \<rat>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1347
  proof(auto simp:Rats_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1348
    fix i j :: int assume "j \<noteq> 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1349
    hence "real i / real j = of_rat(Fract i j)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1350
      by (simp add:of_rat_rat real_eq_of_int)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1351
    thus "real i / real j \<in> range of_rat" by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1352
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1353
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1354
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1355
lemma Rats_eq_int_div_nat:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1356
  "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1357
proof(auto simp:Rats_eq_int_div_int)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1358
  fix i j::int assume "j \<noteq> 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1359
  show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1360
  proof cases
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1361
    assume "j>0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1362
    hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1363
      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1364
    thus ?thesis by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1365
  next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1366
    assume "~ j>0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1367
    hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1368
      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1369
    thus ?thesis by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1370
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1371
next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1372
  fix i::int and n::nat assume "0 < n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1373
  hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1374
  thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1375
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1376
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1377
lemma Rats_abs_nat_div_natE:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1378
  assumes "x \<in> \<rat>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1379
  obtains m n :: nat
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1380
  where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1381
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1382
  from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1383
    by(auto simp add: Rats_eq_int_div_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1384
  hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1385
  then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1386
  let ?gcd = "gcd m n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1387
  from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1388
  let ?k = "m div ?gcd"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1389
  let ?l = "n div ?gcd"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1390
  let ?gcd' = "gcd ?k ?l"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1391
  have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1392
    by (rule dvd_mult_div_cancel)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1393
  have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1394
    by (rule dvd_mult_div_cancel)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1395
  from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1396
  moreover
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1397
  have "\<bar>x\<bar> = real ?k / real ?l"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1398
  proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1399
    from gcd have "real ?k / real ?l =
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1400
        real (?gcd * ?k) / real (?gcd * ?l)" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1401
    also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1402
    also from x_rat have "\<dots> = \<bar>x\<bar>" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1403
    finally show ?thesis ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1404
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1405
  moreover
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1406
  have "?gcd' = 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1407
  proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1408
    have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1409
      by (rule gcd_mult_distrib_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1410
    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1411
    with gcd show ?thesis by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1412
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1413
  ultimately show ?thesis ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1414
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1415
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1416
subsection{*Density of the Rational Reals in the Reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1417
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1418
text{* This density proof is due to Stefan Richter and was ported by TN.  The
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1419
original source is \emph{Real Analysis} by H.L. Royden.
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1420
It employs the Archimedean property of the reals. *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1421
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1422
lemma Rats_dense_in_real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1423
  fixes x :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1424
  assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1425
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1426
  from `x<y` have "0 < y-x" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1427
  with reals_Archimedean obtain q::nat 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1428
    where q: "inverse (real q) < y-x" and "0 < q" by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1429
  def p \<equiv> "ceiling (y * real q) - 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1430
  def r \<equiv> "of_int p / real q"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1431
  from q have "x < y - inverse (real q)" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1432
  also have "y - inverse (real q) \<le> r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1433
    unfolding r_def p_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1434
    by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling `0 < q`)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1435
  finally have "x < r" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1436
  moreover have "r < y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1437
    unfolding r_def p_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1438
    by (simp add: divide_less_eq diff_less_eq `0 < q`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1439
      less_ceiling_iff [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1440
  moreover from r_def have "r \<in> \<rat>" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1441
  ultimately show ?thesis by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1442
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1443
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1444
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1445
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1446
subsection{*Numerals and Arithmetic*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1447
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1448
lemma [code_abbrev]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1449
  "real_of_int (numeral k) = numeral k"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1450
  "real_of_int (- numeral k) = - numeral k"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1451
  by simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1452
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1453
text{*Collapse applications of @{const real} to @{const numeral}*}
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1454
lemma real_numeral [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1455
  "real (numeral v :: int) = numeral v"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1456
  "real (- numeral v :: int) = - numeral v"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1457
by (simp_all add: real_of_int_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1458
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1459
lemma real_of_nat_numeral [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1460
  "real (numeral v :: nat) = numeral v"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1461
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1462
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1463
declaration {*
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1464
  K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1465
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1466
  #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1467
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1468
  #> Lin_Arith.add_simps [@{thm real_of_nat_zero}, @{thm real_of_nat_Suc}, @{thm real_of_nat_add},
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1469
      @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1470
      @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1471
      @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1472
      @{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)}]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1473
  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1474
  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"}))
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1475
*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1476
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1477
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1478
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1479
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1480
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1481
by arith
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1482
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1483
text {* FIXME: redundant with @{text add_eq_0_iff} below *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1484
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1485
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1486
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1487
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1488
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1489
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1490
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1491
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1492
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1493
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1494
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1495
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1496
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1497
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1498
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1499
subsection {* Lemmas about powers *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1500
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1501
text {* FIXME: declare this in Rings.thy or not at all *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1502
declare abs_mult_self [simp]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1503
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1504
(* used by Import/HOL/real.imp *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1505
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1506
by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1507
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1508
lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1509
apply (induct "n")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1510
apply (auto simp add: real_of_nat_Suc)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1511
apply (subst mult_2)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1512
apply (erule add_less_le_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1513
apply (rule two_realpow_ge_one)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1514
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1515
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1516
text {* TODO: no longer real-specific; rename and move elsewhere *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1517
lemma realpow_Suc_le_self:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1518
  fixes r :: "'a::linordered_semidom"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1519
  shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1520
by (insert power_decreasing [of 1 "Suc n" r], simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1521
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1522
text {* TODO: no longer real-specific; rename and move elsewhere *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1523
lemma realpow_minus_mult:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1524
  fixes x :: "'a::monoid_mult"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1525
  shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1526
by (simp add: power_commutes split add: nat_diff_split)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1527
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1528
text {* FIXME: declare this [simp] for all types, or not at all *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1529
lemma real_two_squares_add_zero_iff [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1530
  "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1531
by (rule sum_squares_eq_zero_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1532
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1533
text {* FIXME: declare this [simp] for all types, or not at all *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1534
lemma realpow_two_sum_zero_iff [simp]:
53076
47c9aff07725 more symbols;
wenzelm
parents: 51956
diff changeset
  1535
     "(x\<^sup>2 + y\<^sup>2 = (0::real)) = (x = 0 & y = 0)"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1536
by (rule sum_power2_eq_zero_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1537
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1538
lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1539
by (rule_tac y = 0 in order_trans, auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1540
53076
47c9aff07725 more symbols;
wenzelm
parents: 51956
diff changeset
  1541
lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1542
by (auto simp add: power2_eq_square)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1543
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1544
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1545
lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1546
  "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1547
  unfolding real_of_nat_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1548
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1549
lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1550
  "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1551
  unfolding real_of_nat_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1552
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1553
lemma numeral_power_le_real_of_int_cancel_iff[simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1554
  "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1555
  unfolding real_of_int_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1556
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1557
lemma real_of_int_le_numeral_power_cancel_iff[simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1558
  "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1559
  unfolding real_of_int_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1560
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1561
lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1562
  "(- numeral x::real) ^ n \<le> real a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1563
  unfolding real_of_int_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1564
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1565
lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1566
  "real a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1567
  unfolding real_of_int_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1568
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1569
subsection{*Density of the Reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1570
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff