src/HOL/Real.thy
author haftmann
Thu, 09 Apr 2015 09:12:47 +0200
changeset 59984 4f1eccec320c
parent 59867 58043346ca64
child 60162 645058aa9d6f
permissions -rw-r--r--
conversion between division on nat/int and division in archmedean fields
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(*  Title:      HOL/Real.thy
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    Author:     Jacques D. Fleuriot, University of Edinburgh, 1998
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    Author:     Larry Paulson, University of Cambridge
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    Author:     Jeremy Avigad, Carnegie Mellon University
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    Author:     Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
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    Construction of Cauchy Reals by Brian Huffman, 2010
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*)
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section {* Development of the Reals using Cauchy Sequences *}
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theory Real
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imports Rat Conditionally_Complete_Lattices
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begin
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text {*
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  This theory contains a formalization of the real numbers as
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  equivalence classes of Cauchy sequences of rationals.  See
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  @{file "~~/src/HOL/ex/Dedekind_Real.thy"} for an alternative
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  construction using Dedekind cuts.
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*}
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subsection {* Preliminary lemmas *}
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lemma add_diff_add:
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  fixes a b c d :: "'a::ab_group_add"
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  shows "(a + c) - (b + d) = (a - b) + (c - d)"
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  by simp
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lemma minus_diff_minus:
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  fixes a b :: "'a::ab_group_add"
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  shows "- a - - b = - (a - b)"
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  by simp
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lemma mult_diff_mult:
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  fixes x y a b :: "'a::ring"
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  shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
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  by (simp add: algebra_simps)
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lemma inverse_diff_inverse:
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  fixes a b :: "'a::division_ring"
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  assumes "a \<noteq> 0" and "b \<noteq> 0"
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  shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
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  using assms by (simp add: algebra_simps)
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lemma obtain_pos_sum:
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  fixes r :: rat assumes r: "0 < r"
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  obtains s t where "0 < s" and "0 < t" and "r = s + t"
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proof
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    from r show "0 < r/2" by simp
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    from r show "0 < r/2" by simp
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    show "r = r/2 + r/2" by simp
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qed
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subsection {* Sequences that converge to zero *}
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definition
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  vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
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where
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  "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
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lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X"
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  unfolding vanishes_def by simp
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lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
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  unfolding vanishes_def by simp
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lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
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  unfolding vanishes_def
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  apply (cases "c = 0", auto)
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  apply (rule exI [where x="\<bar>c\<bar>"], auto)
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  done
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lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"
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  unfolding vanishes_def by simp
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lemma vanishes_add:
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  assumes X: "vanishes X" and Y: "vanishes Y"
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  shows "vanishes (\<lambda>n. X n + Y n)"
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proof (rule vanishesI)
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  fix r :: rat assume "0 < r"
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  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
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    by (rule obtain_pos_sum)
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  obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
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    using vanishesD [OF X s] ..
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  obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
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    using vanishesD [OF Y t] ..
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  have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"
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  proof (clarsimp)
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    fix n assume n: "i \<le> n" "j \<le> n"
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    have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
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    also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
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    finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
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  qed
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  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
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qed
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lemma vanishes_diff:
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  assumes X: "vanishes X" and Y: "vanishes Y"
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  shows "vanishes (\<lambda>n. X n - Y n)"
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  unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus X Y)
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lemma vanishes_mult_bounded:
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  assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
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  assumes Y: "vanishes (\<lambda>n. Y n)"
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  shows "vanishes (\<lambda>n. X n * Y n)"
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proof (rule vanishesI)
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  fix r :: rat assume r: "0 < r"
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  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
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    using X by fast
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  obtain b where b: "0 < b" "r = a * b"
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  proof
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    show "0 < r / a" using r a by simp
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    show "r = a * (r / a)" using a by simp
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  qed
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  obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
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    using vanishesD [OF Y b(1)] ..
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  have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
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    by (simp add: b(2) abs_mult mult_strict_mono' a k)
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  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
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qed
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subsection {* Cauchy sequences *}
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definition
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  cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
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where
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  "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
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lemma cauchyI:
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  "(\<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"
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  unfolding cauchy_def by simp
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lemma cauchyD:
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  "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
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  unfolding cauchy_def by simp
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lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
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  unfolding cauchy_def by simp
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lemma cauchy_add [simp]:
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  assumes X: "cauchy X" and Y: "cauchy Y"
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  shows "cauchy (\<lambda>n. X n + Y n)"
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proof (rule cauchyI)
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  fix r :: rat assume "0 < r"
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  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
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    by (rule obtain_pos_sum)
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  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
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    using cauchyD [OF X s] ..
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  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
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    using cauchyD [OF Y t] ..
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  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r"
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  proof (clarsimp)
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    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
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    have "\<bar>(X m + Y m) - (X n + Y n)\<bar> \<le> \<bar>X m - X n\<bar> + \<bar>Y m - Y n\<bar>"
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      unfolding add_diff_add by (rule abs_triangle_ineq)
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    also have "\<dots> < s + t"
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      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
56541
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   218
    show "0 < v/b" using v b(1) by simp
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   219
    show "0 < u/a" using u a(1) by simp
51523
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
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56541
diff changeset
   285
    show "0 < b * r * b" by (simp add: `0 < r` b)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   286
    show "r = inverse b * (b * r * b) * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   287
      using b by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   288
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   289
  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
   290
    using cauchyD [OF X s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   291
  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
   292
  proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   293
    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
   294
    have "\<bar>inverse (X m) - inverse (X n)\<bar> =
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   295
          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
   296
      by (simp add: inverse_diff_inverse nz * abs_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   297
    also have "\<dots> < inverse b * s * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   298
      by (simp add: mult_strict_mono less_imp_inverse_less
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56541
diff changeset
   299
                    i j b * s)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   300
    finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   301
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   302
  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
   303
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   304
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   305
lemma vanishes_diff_inverse:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   306
  assumes X: "cauchy X" "\<not> vanishes X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   307
  assumes Y: "cauchy Y" "\<not> vanishes Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   308
  assumes XY: "vanishes (\<lambda>n. X n - Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   309
  shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   310
proof (rule vanishesI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   311
  fix r :: rat assume r: "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   312
  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
   313
    using cauchy_not_vanishes [OF X] by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   314
  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
   315
    using cauchy_not_vanishes [OF Y] by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   316
  obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   317
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   318
    show "0 < a * r * b"
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56541
diff changeset
   319
      using a r b by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   320
    show "inverse a * (a * r * b) * inverse b = r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   321
      using a r b by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   322
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   323
  obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   324
    using vanishesD [OF XY s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   325
  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
   326
  proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   327
    fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   328
    have "X n \<noteq> 0" and "Y n \<noteq> 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   329
      using i j a b n by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   330
    hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   331
        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
   332
      by (simp add: inverse_diff_inverse abs_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   333
    also have "\<dots> < inverse a * s * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   334
      apply (intro mult_strict_mono' less_imp_inverse_less)
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56217
diff changeset
   335
      apply (simp_all add: a b i j k n)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   336
      done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   337
    also note `inverse a * s * inverse b = r`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   338
    finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   339
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   340
  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
   341
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   342
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   343
subsection {* Equivalence relation on Cauchy sequences *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   344
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   345
definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   346
  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
   347
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   348
lemma realrelI [intro?]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   349
  assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   350
  shows "realrel X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   351
  using assms unfolding realrel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   352
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   353
lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   354
  unfolding realrel_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   355
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   356
lemma symp_realrel: "symp realrel"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   357
  unfolding realrel_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   358
  by (rule sympI, clarify, drule vanishes_minus, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   359
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   360
lemma transp_realrel: "transp realrel"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   361
  unfolding realrel_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   362
  apply (rule transpI, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   363
  apply (drule (1) vanishes_add)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   364
  apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   365
  done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   366
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   367
lemma part_equivp_realrel: "part_equivp realrel"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   368
  by (fast intro: part_equivpI symp_realrel transp_realrel
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   369
    realrel_refl cauchy_const)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   370
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   371
subsection {* The field of real numbers *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   372
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   373
quotient_type real = "nat \<Rightarrow> rat" / partial: realrel
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   374
  morphisms rep_real Real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   375
  by (rule part_equivp_realrel)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   376
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   377
lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   378
  unfolding real.pcr_cr_eq cr_real_def realrel_def by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   379
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   380
lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   381
  assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   382
proof (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   383
  case (1 X)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   384
  hence "cauchy X" by (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   385
  thus "P (Real X)" by (rule assms)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   386
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   387
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   388
lemma eq_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   389
  "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
   390
  using real.rel_eq_transfer
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54863
diff changeset
   391
  unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   392
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
   393
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
   394
by (simp add: real.domain_eq realrel_def)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   395
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59587
diff changeset
   396
instantiation real :: field
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   397
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   398
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   399
lift_definition zero_real :: "real" is "\<lambda>n. 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   400
  by (simp add: realrel_refl)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   401
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   402
lift_definition one_real :: "real" is "\<lambda>n. 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   403
  by (simp add: realrel_refl)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   404
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   405
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
   406
  unfolding realrel_def add_diff_add
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   407
  by (simp only: cauchy_add vanishes_add simp_thms)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   408
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   409
lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   410
  unfolding realrel_def minus_diff_minus
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   411
  by (simp only: cauchy_minus vanishes_minus simp_thms)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   412
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   413
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
   414
  unfolding realrel_def mult_diff_mult
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57447
diff changeset
   415
  by (subst (4) mult.commute, simp only: cauchy_mult vanishes_add
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   416
    vanishes_mult_bounded cauchy_imp_bounded simp_thms)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   417
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   418
lift_definition inverse_real :: "real \<Rightarrow> real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   419
  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
   420
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   421
  fix X Y assume "realrel X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   422
  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
   423
    unfolding realrel_def by simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   424
  have "vanishes X \<longleftrightarrow> vanishes Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   425
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   426
    assume "vanishes X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   427
    from vanishes_diff [OF this XY] show "vanishes Y" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   428
  next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   429
    assume "vanishes Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   430
    from vanishes_add [OF this XY] show "vanishes X" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   431
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   432
  thus "?thesis X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   433
    unfolding realrel_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   434
    by (simp add: vanishes_diff_inverse X Y XY)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   435
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   436
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   437
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   438
  "x - y = (x::real) + - y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   439
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   440
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   441
  "x / y = (x::real) * inverse y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   442
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   443
lemma add_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   444
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   445
  shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   446
  using assms plus_real.transfer
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54863
diff changeset
   447
  unfolding cr_real_eq rel_fun_def by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   448
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   449
lemma minus_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   450
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   451
  shows "- Real X = Real (\<lambda>n. - X n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   452
  using assms uminus_real.transfer
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54863
diff changeset
   453
  unfolding cr_real_eq rel_fun_def by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   454
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   455
lemma diff_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   456
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   457
  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
   458
  unfolding minus_real_def
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   459
  by (simp add: minus_Real add_Real X Y)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   460
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   461
lemma mult_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   462
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   463
  shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   464
  using assms times_real.transfer
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54863
diff changeset
   465
  unfolding cr_real_eq rel_fun_def by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   466
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   467
lemma inverse_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   468
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   469
  shows "inverse (Real X) =
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   470
    (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   471
  using assms inverse_real.transfer zero_real.transfer
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54863
diff changeset
   472
  unfolding cr_real_eq rel_fun_def by (simp split: split_if_asm, metis)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   473
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   474
instance proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   475
  fix a b c :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   476
  show "a + b = b + a"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   477
    by transfer (simp add: ac_simps realrel_def)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   478
  show "(a + b) + c = a + (b + c)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   479
    by transfer (simp add: ac_simps realrel_def)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   480
  show "0 + a = a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   481
    by transfer (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   482
  show "- a + a = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   483
    by transfer (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   484
  show "a - b = a + - b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   485
    by (rule minus_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   486
  show "(a * b) * c = a * (b * c)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   487
    by transfer (simp add: ac_simps realrel_def)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   488
  show "a * b = b * a"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   489
    by transfer (simp add: ac_simps realrel_def)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   490
  show "1 * a = a"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   491
    by transfer (simp add: ac_simps realrel_def)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   492
  show "(a + b) * c = a * c + b * c"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   493
    by transfer (simp add: distrib_right realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   494
  show "(0\<Colon>real) \<noteq> (1\<Colon>real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   495
    by transfer (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   496
  show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   497
    apply transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   498
    apply (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   499
    apply (rule vanishesI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   500
    apply (frule (1) cauchy_not_vanishes, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   501
    apply (rule_tac x=k in exI, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   502
    apply (drule_tac x=n in spec, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   503
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   504
  show "a / b = a * inverse b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   505
    by (rule divide_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   506
  show "inverse (0::real) = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   507
    by transfer (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   508
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   509
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   510
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   511
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   512
subsection {* Positive reals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   513
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   514
lift_definition positive :: "real \<Rightarrow> bool"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   515
  is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   516
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   517
  { fix X Y
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   518
    assume "realrel X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   519
    hence XY: "vanishes (\<lambda>n. X n - Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   520
      unfolding realrel_def by simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   521
    assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   522
    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
   523
      by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   524
    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
   525
      using `0 < r` by (rule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   526
    obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   527
      using vanishesD [OF XY s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   528
    have "\<forall>n\<ge>max i j. t < Y n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   529
    proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   530
      fix n assume n: "i \<le> n" "j \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   531
      have "\<bar>X n - Y n\<bar> < s" and "r < X n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   532
        using i j n by simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   533
      thus "t < Y n" unfolding r by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   534
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   535
    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
   536
  } note 1 = this
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   537
  fix X Y assume "realrel X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   538
  hence "realrel X Y" and "realrel Y X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   539
    using symp_realrel unfolding symp_def by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   540
  thus "?thesis X Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   541
    by (safe elim!: 1)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   542
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   543
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   544
lemma positive_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   545
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   546
  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
   547
  using assms positive.transfer
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54863
diff changeset
   548
  unfolding cr_real_eq rel_fun_def by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   549
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   550
lemma positive_zero: "\<not> positive 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   551
  by transfer auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   552
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   553
lemma positive_add:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   554
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   555
apply transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   556
apply (clarify, rename_tac a b i j)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   557
apply (rule_tac x="a + b" in exI, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   558
apply (rule_tac x="max i j" in exI, clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   559
apply (simp add: add_strict_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   560
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   561
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   562
lemma positive_mult:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   563
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   564
apply transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   565
apply (clarify, rename_tac a b i j)
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56541
diff changeset
   566
apply (rule_tac x="a * b" in exI, simp)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   567
apply (rule_tac x="max i j" in exI, clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   568
apply (rule mult_strict_mono, auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   569
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   570
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   571
lemma positive_minus:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   572
  "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   573
apply transfer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   574
apply (simp add: realrel_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   575
apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   576
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   577
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59587
diff changeset
   578
instantiation real :: linordered_field
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   579
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   580
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   581
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   582
  "x < y \<longleftrightarrow> positive (y - x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   583
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   584
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   585
  "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   586
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   587
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   588
  "abs (a::real) = (if a < 0 then - a else a)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   589
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   590
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   591
  "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
   592
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   593
instance proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   594
  fix a b c :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   595
  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   596
    by (rule abs_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   597
  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   598
    unfolding less_eq_real_def less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   599
    by (auto, drule (1) positive_add, simp_all add: positive_zero)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   600
  show "a \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   601
    unfolding less_eq_real_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   602
  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   603
    unfolding less_eq_real_def less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   604
    by (auto, drule (1) positive_add, simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   605
  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   606
    unfolding less_eq_real_def less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   607
    by (auto, drule (1) positive_add, simp add: positive_zero)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   608
  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
   609
    unfolding less_eq_real_def less_real_def by auto
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   610
    (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   611
    (* Should produce c + b - (c + a) \<equiv> b - a *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   612
  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
   613
    by (rule sgn_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   614
  show "a \<le> b \<or> b \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   615
    unfolding less_eq_real_def less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   616
    by (auto dest!: positive_minus)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   617
  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   618
    unfolding less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   619
    by (drule (1) positive_mult, simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   620
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   621
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   622
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   623
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   624
instantiation real :: distrib_lattice
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   625
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   626
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   627
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   628
  "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   629
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   630
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   631
  "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   632
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   633
instance proof
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54489
diff changeset
   634
qed (auto simp add: inf_real_def sup_real_def max_min_distrib2)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   635
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   636
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   637
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   638
lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   639
apply (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   640
apply (simp add: zero_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   641
apply (simp add: one_real_def add_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   642
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   643
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   644
lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   645
apply (cases x rule: int_diff_cases)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   646
apply (simp add: of_nat_Real diff_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   647
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   648
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   649
lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   650
apply (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   651
apply (simp add: Fract_of_int_quotient of_rat_divide)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   652
apply (simp add: of_int_Real divide_inverse)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   653
apply (simp add: inverse_Real mult_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   654
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   655
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   656
instance real :: archimedean_field
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   657
proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   658
  fix x :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   659
  show "\<exists>z. x \<le> of_int z"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   660
    apply (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   661
    apply (frule cauchy_imp_bounded, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   662
    apply (rule_tac x="ceiling b + 1" in exI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   663
    apply (rule less_imp_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   664
    apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   665
    apply (rule_tac x=1 in exI, simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   666
    apply (rule_tac x=0 in exI, clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   667
    apply (rule le_less_trans [OF abs_ge_self])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   668
    apply (rule less_le_trans [OF _ le_of_int_ceiling])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   669
    apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   670
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   671
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   672
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   673
instantiation real :: floor_ceiling
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   674
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   675
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   676
definition [code del]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   677
  "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
   678
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   679
instance proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   680
  fix x :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   681
  show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   682
    unfolding floor_real_def using floor_exists1 by (rule theI')
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   683
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   684
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   685
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   686
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   687
subsection {* Completeness *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   688
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   689
lemma not_positive_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   690
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   691
  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
   692
unfolding positive_Real [OF X]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   693
apply (auto, unfold not_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   694
apply (erule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   695
apply (drule_tac x=s in spec, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   696
apply (drule_tac r=t in cauchyD [OF X], clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   697
apply (drule_tac x=k in spec, clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   698
apply (rule_tac x=n in exI, clarify, rename_tac m)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   699
apply (drule_tac x=m in spec, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   700
apply (drule_tac x=n in spec, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   701
apply (drule spec, drule (1) mp, clarify, rename_tac i)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   702
apply (rule_tac x="max i k" in exI, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   703
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   704
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   705
lemma le_Real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   706
  assumes X: "cauchy X" and Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   707
  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
   708
unfolding not_less [symmetric, where 'a=real] less_real_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   709
apply (simp add: diff_Real not_positive_Real X Y)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   710
apply (simp add: diff_le_eq ac_simps)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   711
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   712
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   713
lemma le_RealI:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   714
  assumes Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   715
  shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   716
proof (induct x)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   717
  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
   718
  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
   719
    by (simp add: of_rat_Real le_Real)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   720
  {
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   721
    fix r :: rat assume "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   722
    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
   723
      by (rule obtain_pos_sum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   724
    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
   725
      using cauchyD [OF Y s] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   726
    obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   727
      using le [OF t] ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   728
    have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   729
    proof (clarsimp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   730
      fix n assume n: "i \<le> n" "j \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   731
      have "X n \<le> Y i + t" using n j by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   732
      moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   733
      ultimately show "X n \<le> Y n + r" unfolding r by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   734
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   735
    hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   736
  }
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   737
  thus "Real X \<le> Real Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   738
    by (simp add: of_rat_Real le_Real X Y)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   739
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   740
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   741
lemma Real_leI:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   742
  assumes X: "cauchy X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   743
  assumes le: "\<forall>n. of_rat (X n) \<le> y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   744
  shows "Real X \<le> y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   745
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   746
  have "- y \<le> - Real X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   747
    by (simp add: minus_Real X le_RealI of_rat_minus le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   748
  thus ?thesis by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   749
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   750
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   751
lemma less_RealD:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   752
  assumes Y: "cauchy Y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   753
  shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   754
by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   755
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   756
lemma of_nat_less_two_power:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   757
  "of_nat n < (2::'a::linordered_idom) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   758
apply (induct n)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   759
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   760
apply (subgoal_tac "(1::'a) \<le> 2 ^ n")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   761
apply (drule (1) add_le_less_mono, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   762
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   763
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   764
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   765
lemma complete_real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   766
  fixes S :: "real set"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   767
  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
   768
  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
   769
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   770
  obtain x where x: "x \<in> S" using assms(1) ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   771
  obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   772
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   773
  def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   774
  obtain a where a: "\<not> P a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   775
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   776
    have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   777
    also have "x - 1 < x" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   778
    finally have "of_int (floor (x - 1)) < x" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   779
    hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   780
    then show "\<not> P (of_int (floor (x - 1)))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   781
      unfolding P_def of_rat_of_int_eq using x by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   782
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   783
  obtain b where b: "P b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   784
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   785
    show "P (of_int (ceiling z))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   786
    unfolding P_def of_rat_of_int_eq
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   787
    proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   788
      fix y assume "y \<in> S"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   789
      hence "y \<le> z" using z by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   790
      also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   791
      finally show "y \<le> of_int (ceiling z)" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   792
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   793
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   794
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   795
  def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   796
  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
   797
  def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   798
  def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   799
  def C \<equiv> "\<lambda>n. avg (A n) (B n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   800
  have A_0 [simp]: "A 0 = a" unfolding A_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   801
  have B_0 [simp]: "B 0 = b" unfolding B_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   802
  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
   803
    unfolding A_def B_def C_def bisect_def split_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   804
  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
   805
    unfolding A_def B_def C_def bisect_def split_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   806
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   807
  have width: "\<And>n. B n - A n = (b - a) / 2^n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   808
    apply (simp add: eq_divide_eq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   809
    apply (induct_tac n, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   810
    apply (simp add: C_def avg_def algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   811
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   812
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   813
  have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   814
    apply (simp add: divide_less_eq)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57447
diff changeset
   815
    apply (subst mult.commute)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   816
    apply (frule_tac y=y in ex_less_of_nat_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   817
    apply clarify
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   818
    apply (rule_tac x=n in exI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   819
    apply (erule less_trans)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   820
    apply (rule mult_strict_right_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   821
    apply (rule le_less_trans [OF _ of_nat_less_two_power])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   822
    apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   823
    apply assumption
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   824
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   825
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   826
  have PA: "\<And>n. \<not> P (A n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   827
    by (induct_tac n, simp_all add: a)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   828
  have PB: "\<And>n. P (B n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   829
    by (induct_tac n, simp_all add: b)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   830
  have ab: "a < b"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   831
    using a b unfolding P_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   832
    apply (clarsimp simp add: not_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   833
    apply (drule (1) bspec)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   834
    apply (drule (1) less_le_trans)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   835
    apply (simp add: of_rat_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   836
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   837
  have AB: "\<And>n. A n < B n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   838
    by (induct_tac n, simp add: ab, simp add: C_def avg_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   839
  have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   840
    apply (auto simp add: le_less [where 'a=nat])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   841
    apply (erule less_Suc_induct)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   842
    apply (clarsimp simp add: C_def avg_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   843
    apply (simp add: add_divide_distrib [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   844
    apply (rule AB [THEN less_imp_le])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   845
    apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   846
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   847
  have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   848
    apply (auto simp add: le_less [where 'a=nat])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   849
    apply (erule less_Suc_induct)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   850
    apply (clarsimp simp add: C_def avg_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   851
    apply (simp add: add_divide_distrib [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   852
    apply (rule AB [THEN less_imp_le])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   853
    apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   854
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   855
  have cauchy_lemma:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   856
    "\<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
   857
    apply (rule cauchyI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   858
    apply (drule twos [where y="b - a"])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   859
    apply (erule exE)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   860
    apply (rule_tac x=n in exI, clarify, rename_tac i j)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   861
    apply (rule_tac y="B n - A n" in le_less_trans) defer
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   862
    apply (simp add: width)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   863
    apply (drule_tac x=n in spec)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   864
    apply (frule_tac x=i in spec, drule (1) mp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   865
    apply (frule_tac x=j in spec, drule (1) mp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   866
    apply (frule A_mono, drule B_mono)
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 arith
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   869
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   870
  have "cauchy A"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   871
    apply (rule cauchy_lemma [rule_format])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   872
    apply (simp add: A_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   873
    apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   874
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   875
  have "cauchy B"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   876
    apply (rule cauchy_lemma [rule_format])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   877
    apply (simp add: B_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   878
    apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   879
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   880
  have 1: "\<forall>x\<in>S. x \<le> Real B"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   881
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   882
    fix x assume "x \<in> S"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   883
    then show "x \<le> Real B"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   884
      using PB [unfolded P_def] `cauchy B`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   885
      by (simp add: le_RealI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   886
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   887
  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
   888
    apply clarify
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   889
    apply (erule contrapos_pp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   890
    apply (simp add: not_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   891
    apply (drule less_RealD [OF `cauchy A`], clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   892
    apply (subgoal_tac "\<not> P (A n)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   893
    apply (simp add: P_def not_le, clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   894
    apply (erule rev_bexI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   895
    apply (erule (1) less_trans)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   896
    apply (simp add: PA)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   897
    done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   898
  have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   899
  proof (rule vanishesI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   900
    fix r :: rat assume "0 < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   901
    then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   902
      using twos by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   903
    have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   904
    proof (clarify)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   905
      fix n assume n: "k \<le> n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   906
      have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   907
        by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   908
      also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56541
diff changeset
   909
        using n by (simp add: divide_left_mono)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   910
      also note k
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   911
      finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   912
    qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   913
    thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   914
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   915
  hence 3: "Real B = Real A"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   916
    by (simp add: eq_Real `cauchy A` `cauchy B` width)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   917
  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
   918
    using 1 2 3 by (rule_tac x="Real B" in exI, simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   919
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   920
51775
408d937c9486 revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents: 51773
diff changeset
   921
instantiation real :: linear_continuum
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   922
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   923
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   924
subsection{*Supremum of a set of reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   925
54281
b01057e72233 int and nat are conditionally_complete_lattices
hoelzl
parents: 54263
diff changeset
   926
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
   927
definition "Inf (X::real set) = - Sup (uminus ` X)"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   928
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   929
instance
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   930
proof
54258
adfc759263ab use bdd_above and bdd_below for conditionally complete lattices
hoelzl
parents: 54230
diff changeset
   931
  { fix x :: real and X :: "real set"
adfc759263ab use bdd_above and bdd_below for conditionally complete lattices
hoelzl
parents: 54230
diff changeset
   932
    assume x: "x \<in> X" "bdd_above X"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   933
    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
   934
      using complete_real[of X] unfolding bdd_above_def by blast
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   935
    then show "x \<le> Sup X"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   936
      unfolding Sup_real_def by (rule LeastI2_order) (auto simp: x) }
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   937
  note Sup_upper = this
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   938
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   939
  { fix z :: real and X :: "real set"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   940
    assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   941
    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
   942
      using complete_real[of X] by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   943
    then have "Sup X = s"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   944
      unfolding Sup_real_def by (best intro: Least_equality)  
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53076
diff changeset
   945
    also from s z have "... \<le> z"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   946
      by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   947
    finally show "Sup X \<le> z" . }
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   948
  note Sup_least = this
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   949
54281
b01057e72233 int and nat are conditionally_complete_lattices
hoelzl
parents: 54263
diff changeset
   950
  { 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
   951
      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
   952
  { 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
   953
      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
   954
  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
   955
    using zero_neq_one by blast
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   956
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   957
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   958
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   959
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   960
subsection {* Hiding implementation details *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   961
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   962
hide_const (open) vanishes cauchy positive Real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   963
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   964
declare Real_induct [induct del]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   965
declare Abs_real_induct [induct del]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   966
declare Abs_real_cases [cases del]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   967
53652
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
   968
lifting_update real.lifting
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
   969
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
   970
  
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   971
subsection{*More Lemmas*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   972
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   973
text {* BH: These lemmas should not be necessary; they should be
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   974
covered by existing simp rules and simplification procedures. *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   975
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   976
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   977
by simp (* solved by linordered_ring_less_cancel_factor simproc *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   978
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   979
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
   980
by simp (* solved by linordered_ring_le_cancel_factor simproc *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   981
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   982
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
   983
by simp (* solved by linordered_ring_le_cancel_factor simproc *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   984
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   985
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   986
subsection {* Embedding numbers into the Reals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   987
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   988
abbreviation
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   989
  real_of_nat :: "nat \<Rightarrow> real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   990
where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   991
  "real_of_nat \<equiv> of_nat"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   992
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   993
abbreviation
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   994
  real_of_int :: "int \<Rightarrow> real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   995
where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   996
  "real_of_int \<equiv> of_int"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   997
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   998
abbreviation
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
   999
  real_of_rat :: "rat \<Rightarrow> real"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1000
where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1001
  "real_of_rat \<equiv> of_rat"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1002
58042
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1003
class real_of =
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1004
  fixes real :: "'a \<Rightarrow> real"
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1005
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1006
instantiation nat :: real_of
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1007
begin
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1008
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1009
definition real_nat :: "nat \<Rightarrow> real" where real_of_nat_def [code_unfold]: "real \<equiv> of_nat" 
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1010
58042
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1011
instance ..
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1012
end
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1013
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1014
instantiation int :: real_of
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1015
begin
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1016
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1017
definition real_int :: "int \<Rightarrow> real" where real_of_int_def [code_unfold]: "real \<equiv> of_int" 
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1018
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1019
instance ..
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 58040
diff changeset
  1020
end
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1021
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1022
declare [[coercion_enabled]]
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1023
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1024
declare [[coercion "of_nat :: nat \<Rightarrow> int"]]
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1025
declare [[coercion "real   :: nat \<Rightarrow> real"]]
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1026
declare [[coercion "real   :: int \<Rightarrow> real"]]
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1027
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1028
(* We do not add rat to the coerced types, this has often unpleasant side effects when writing
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1029
inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1030
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1031
declare [[coercion_map map]]
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1032
declare [[coercion_map "\<lambda>f g h x. g (h (f x))"]]
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58983
diff changeset
  1033
declare [[coercion_map "\<lambda>f g (x,y). (f x, g y)"]]
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1034
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1035
lemma real_eq_of_nat: "real = of_nat"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1036
  unfolding real_of_nat_def ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1037
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1038
lemma real_eq_of_int: "real = of_int"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1039
  unfolding real_of_int_def ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1040
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1041
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1042
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1043
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1044
lemma real_of_one [simp]: "real (1::int) = (1::real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1045
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1046
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1047
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1048
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1049
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1050
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1051
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1052
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1053
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1054
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1055
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1056
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1057
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1058
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1059
lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1060
by (simp add: real_of_int_def of_int_power)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1061
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1062
lemmas power_real_of_int = real_of_int_power [symmetric]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1063
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1064
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
  1065
  apply (subst real_eq_of_int)+
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1066
  apply (rule of_int_setsum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1067
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1068
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1069
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1070
    (PROD x:A. real(f x))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1071
  apply (subst real_eq_of_int)+
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1072
  apply (rule of_int_setprod)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1073
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1074
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1075
lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1076
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1077
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1078
lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1079
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1080
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1081
lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1082
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1083
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1084
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
  1085
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1086
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1087
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
  1088
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1089
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1090
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
  1091
by (simp add: real_of_int_def) 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1092
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1093
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
  1094
by (simp add: real_of_int_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1095
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1096
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
  1097
by (simp add: real_of_int_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1098
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1099
lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1100
  unfolding real_of_one[symmetric] real_of_int_less_iff ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1101
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1102
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
  1103
  unfolding real_of_one[symmetric] real_of_int_le_iff ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1104
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1105
lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1106
  unfolding real_of_one[symmetric] real_of_int_less_iff ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1107
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1108
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
  1109
  unfolding real_of_one[symmetric] real_of_int_le_iff ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1110
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1111
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1112
by (auto simp add: abs_if)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1113
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1114
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1115
  apply (subgoal_tac "real n + 1 = real (n + 1)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1116
  apply (simp del: real_of_int_add)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1117
  apply auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1118
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1119
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1120
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1121
  apply (subgoal_tac "real m + 1 = real (m + 1)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1122
  apply (simp del: real_of_int_add)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1123
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1124
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1125
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1126
lemma real_of_int_div_aux: "(real (x::int)) / (real d) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1127
    real (x div d) + (real (x mod d)) / (real d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1128
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1129
  have "x = (x div d) * d + x mod d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1130
    by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1131
  then have "real x = real (x div d) * real d + real(x mod d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1132
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1133
  then have "real x / real d = ... / real d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1134
    by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1135
  then show ?thesis
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1136
    by (auto simp add: add_divide_distrib algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1137
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1138
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1139
lemma real_of_int_div:
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1140
  fixes d n :: int
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1141
  shows "d dvd n \<Longrightarrow> real (n div d) = real n / real d"
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1142
  by (simp add: real_of_int_div_aux)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1143
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1144
lemma real_of_int_div2:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1145
  "0 <= real (n::int) / real (x) - real (n div x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1146
  apply (case_tac "x = 0")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1147
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1148
  apply (case_tac "0 < x")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1149
  apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1150
  apply (subst real_of_int_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1151
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1152
  apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1153
  apply (subst real_of_int_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1154
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1155
  apply (subst zero_le_divide_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1156
  apply auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1157
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1158
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1159
lemma real_of_int_div3:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1160
  "real (n::int) / real (x) - real (n div x) <= 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1161
  apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1162
  apply (subst real_of_int_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1163
  apply (auto simp add: divide_le_eq intro: order_less_imp_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1164
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1165
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1166
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1167
by (insert real_of_int_div2 [of n x], simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1168
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1169
lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1170
unfolding real_of_int_def by (rule Ints_of_int)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1171
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1172
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1173
subsection{*Embedding the Naturals into the Reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1174
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1175
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1176
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1177
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1178
lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1179
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1180
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1181
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1182
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1183
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1184
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
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
(*Not for addsimps: often the LHS is used to represent a positive natural*)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1188
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
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_less_iff [iff]: 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1192
     "(real (n::nat) < real m) = (n < m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1193
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1194
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1195
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
  1196
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1197
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1198
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1199
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1200
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1201
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1202
by (simp add: real_of_nat_def del: of_nat_Suc)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1203
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1204
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1205
by (simp add: real_of_nat_def of_nat_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1206
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1207
lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1208
by (simp add: real_of_nat_def of_nat_power)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1209
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1210
lemmas power_real_of_nat = real_of_nat_power [symmetric]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1211
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1212
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1213
    (SUM x:A. real(f x))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1214
  apply (subst real_eq_of_nat)+
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1215
  apply (rule of_nat_setsum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1216
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1217
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1218
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1219
    (PROD x:A. real(f x))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1220
  apply (subst real_eq_of_nat)+
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1221
  apply (rule of_nat_setprod)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1222
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1223
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1224
lemma real_of_card: "real (card A) = setsum (%x.1) A"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1225
  apply (subst card_eq_setsum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1226
  apply (subst real_of_nat_setsum)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1227
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1228
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1229
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1230
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1231
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1232
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1233
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1234
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1235
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1236
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
  1237
by (simp add: add: real_of_nat_def of_nat_diff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1238
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1239
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1240
by (auto simp: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1241
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1242
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
  1243
by (simp add: add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1244
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1245
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1246
by (simp add: add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1247
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1248
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1249
  apply (subgoal_tac "real n + 1 = real (Suc n)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1250
  apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1251
  apply (auto simp add: real_of_nat_Suc)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1252
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1253
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1254
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1255
  apply (subgoal_tac "real m + 1 = real (Suc m)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1256
  apply (simp add: less_Suc_eq_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1257
  apply (simp add: real_of_nat_Suc)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1258
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1259
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1260
lemma real_of_nat_div_aux: "(real (x::nat)) / (real d) = 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1261
    real (x div d) + (real (x mod d)) / (real d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1262
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1263
  have "x = (x div d) * d + x mod d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1264
    by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1265
  then have "real x = real (x div d) * real d + real(x mod d)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1266
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1267
  then have "real x / real d = \<dots> / real d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1268
    by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1269
  then show ?thesis
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1270
    by (auto simp add: add_divide_distrib algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1271
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1272
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1273
lemma real_of_nat_div: "(d :: nat) dvd n ==>
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1274
    real(n div d) = real n / real d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1275
  by (subst real_of_nat_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1276
    (auto simp add: dvd_eq_mod_eq_0 [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1277
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1278
lemma real_of_nat_div2:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1279
  "0 <= real (n::nat) / real (x) - real (n div x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1280
apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1281
apply (subst real_of_nat_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1282
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1283
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1284
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1285
lemma real_of_nat_div3:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1286
  "real (n::nat) / real (x) - real (n div x) <= 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1287
apply(case_tac "x = 0")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1288
apply (simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1289
apply (simp add: algebra_simps)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1290
apply (subst real_of_nat_div_aux)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1291
apply simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1292
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1293
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1294
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1295
by (insert real_of_nat_div2 [of n x], simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1296
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1297
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1298
by (simp add: real_of_int_def real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1299
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1300
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1301
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1302
  apply force
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1303
  apply (simp only: real_of_int_of_nat_eq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1304
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1305
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1306
lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1307
unfolding real_of_nat_def by (rule of_nat_in_Nats)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1308
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1309
lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1310
unfolding real_of_nat_def by (rule Ints_of_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1311
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1312
subsection {* The Archimedean Property of the Reals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1313
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1314
theorem reals_Archimedean:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1315
  assumes x_pos: "0 < x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1316
  shows "\<exists>n. inverse (real (Suc n)) < x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1317
  unfolding real_of_nat_def using x_pos
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1318
  by (rule ex_inverse_of_nat_Suc_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1319
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1320
lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1321
  unfolding real_of_nat_def by (rule ex_less_of_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1322
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1323
lemma reals_Archimedean3:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1324
  assumes x_greater_zero: "0 < x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1325
  shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1326
  unfolding real_of_nat_def using `0 < x`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1327
  by (auto intro: ex_less_of_nat_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1328
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1329
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1330
subsection{* Rationals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1331
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1332
lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1333
by (simp add: real_eq_of_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1334
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1335
lemma Rats_eq_int_div_int:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1336
  "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1337
proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1338
  show "\<rat> \<subseteq> ?S"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1339
  proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1340
    fix x::real assume "x : \<rat>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1341
    then obtain r where "x = of_rat r" unfolding Rats_def ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1342
    have "of_rat r : ?S"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1343
      by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1344
    thus "x : ?S" using `x = of_rat r` by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1345
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1346
next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1347
  show "?S \<subseteq> \<rat>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1348
  proof(auto simp:Rats_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1349
    fix i j :: int assume "j \<noteq> 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1350
    hence "real i / real j = of_rat(Fract i j)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1351
      by (simp add:of_rat_rat real_eq_of_int)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1352
    thus "real i / real j \<in> range of_rat" by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1353
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1354
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1355
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1356
lemma Rats_eq_int_div_nat:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1357
  "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1358
proof(auto simp:Rats_eq_int_div_int)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1359
  fix i j::int assume "j \<noteq> 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1360
  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
  1361
  proof cases
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1362
    assume "j>0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1363
    hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1364
      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1365
    thus ?thesis by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1366
  next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1367
    assume "~ j>0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1368
    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
  1369
      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1370
    thus ?thesis by blast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1371
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1372
next
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1373
  fix i::int and n::nat assume "0 < n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1374
  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
  1375
  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
  1376
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1377
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1378
lemma Rats_abs_nat_div_natE:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1379
  assumes "x \<in> \<rat>"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1380
  obtains m n :: nat
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1381
  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
  1382
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1383
  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
  1384
    by(auto simp add: Rats_eq_int_div_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1385
  hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1386
  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
  1387
  let ?gcd = "gcd m n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1388
  from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1389
  let ?k = "m div ?gcd"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1390
  let ?l = "n div ?gcd"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1391
  let ?gcd' = "gcd ?k ?l"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1392
  have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1393
    by (rule dvd_mult_div_cancel)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1394
  have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1395
    by (rule dvd_mult_div_cancel)
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1396
  from `n \<noteq> 0` and gcd_l
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1397
  have "?gcd * ?l \<noteq> 0" by simp
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1398
  then have "?l \<noteq> 0" by (blast dest!: mult_not_zero) 
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1399
  moreover
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1400
  have "\<bar>x\<bar> = real ?k / real ?l"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1401
  proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1402
    from gcd have "real ?k / real ?l =
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1403
      real (?gcd * ?k) / real (?gcd * ?l)"
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58789
diff changeset
  1404
      by (simp only: real_of_nat_mult) simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1405
    also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1406
    also from x_rat have "\<dots> = \<bar>x\<bar>" ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1407
    finally show ?thesis ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1408
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1409
  moreover
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1410
  have "?gcd' = 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1411
  proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1412
    have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1413
      by (rule gcd_mult_distrib_nat)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1414
    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1415
    with gcd show ?thesis by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1416
  qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1417
  ultimately show ?thesis ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1418
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1419
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1420
subsection{*Density of the Rational Reals in the Reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1421
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1422
text{* This density proof is due to Stefan Richter and was ported by TN.  The
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1423
original source is \emph{Real Analysis} by H.L. Royden.
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1424
It employs the Archimedean property of the reals. *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1425
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1426
lemma Rats_dense_in_real:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1427
  fixes x :: real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1428
  assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1429
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1430
  from `x<y` have "0 < y-x" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1431
  with reals_Archimedean obtain q::nat 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1432
    where q: "inverse (real q) < y-x" and "0 < q" by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1433
  def p \<equiv> "ceiling (y * real q) - 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1434
  def r \<equiv> "of_int p / real q"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1435
  from q have "x < y - inverse (real q)" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1436
  also have "y - inverse (real q) \<le> r"
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: le_divide_eq left_diff_distrib le_of_int_ceiling `0 < q`)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1439
  finally have "x < r" .
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1440
  moreover have "r < y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1441
    unfolding r_def p_def
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1442
    by (simp add: divide_less_eq diff_less_eq `0 < q`
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1443
      less_ceiling_iff [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1444
  moreover from r_def have "r \<in> \<rat>" by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1445
  ultimately show ?thesis by fast
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1446
qed
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1447
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1448
lemma of_rat_dense:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1449
  fixes x y :: real
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1450
  assumes "x < y"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1451
  shows "\<exists>q :: rat. x < of_rat q \<and> of_rat q < y"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1452
using Rats_dense_in_real [OF `x < y`]
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1453
by (auto elim: Rats_cases)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1454
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1455
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1456
subsection{*Numerals and Arithmetic*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1457
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1458
lemma [code_abbrev]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1459
  "real_of_int (numeral k) = numeral k"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1460
  "real_of_int (- numeral k) = - numeral k"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1461
  by simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1462
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1463
text{*Collapse applications of @{const real} to @{const numeral}*}
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1464
lemma real_numeral [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1465
  "real (numeral v :: int) = numeral v"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1466
  "real (- numeral v :: int) = - numeral v"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1467
by (simp_all add: real_of_int_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1468
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1469
lemma  real_of_nat_numeral [simp]:
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1470
  "real (numeral v :: nat) = numeral v"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1471
by (simp add: real_of_nat_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1472
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1473
declaration {*
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1474
  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
  1475
    (* 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
  1476
  #> 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
  1477
    (* 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
  1478
  #> 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
  1479
      @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1480
      @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1481
      @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1482
      @{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)},
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1483
      @{thm real_of_int_def[symmetric]}, @{thm real_of_nat_def[symmetric]}]
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1484
  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1485
  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"})
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1486
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1487
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1488
*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1489
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1490
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1491
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1492
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1493
by arith
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1494
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1495
text {* FIXME: redundant with @{text add_eq_0_iff} below *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1496
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
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
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1500
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1501
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1502
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1503
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1504
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1505
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1506
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1507
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1508
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1509
by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1510
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1511
subsection {* Lemmas about powers *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1512
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1513
text {* FIXME: declare this in Rings.thy or not at all *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1514
declare abs_mult_self [simp]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1515
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1516
(* used by Import/HOL/real.imp *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1517
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1518
by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1519
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1520
lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1521
apply (induct "n")
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1522
apply (auto simp add: real_of_nat_Suc)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1523
apply (subst mult_2)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1524
apply (erule add_less_le_mono)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1525
apply (rule two_realpow_ge_one)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1526
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1527
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1528
text {* TODO: no longer real-specific; rename and move elsewhere *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1529
lemma realpow_Suc_le_self:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1530
  fixes r :: "'a::linordered_semidom"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1531
  shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1532
by (insert power_decreasing [of 1 "Suc n" r], simp)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1533
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1534
text {* TODO: no longer real-specific; rename and move elsewhere *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1535
lemma realpow_minus_mult:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1536
  fixes x :: "'a::monoid_mult"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1537
  shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1538
by (simp add: power_commutes split add: nat_diff_split)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1539
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1540
text {* FIXME: declare this [simp] for all types, or not at all *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1541
lemma real_two_squares_add_zero_iff [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1542
  "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1543
by (rule sum_squares_eq_zero_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1544
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1545
text {* FIXME: declare this [simp] for all types, or not at all *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1546
lemma realpow_two_sum_zero_iff [simp]:
53076
47c9aff07725 more symbols;
wenzelm
parents: 51956
diff changeset
  1547
     "(x\<^sup>2 + y\<^sup>2 = (0::real)) = (x = 0 & y = 0)"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1548
by (rule sum_power2_eq_zero_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1549
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1550
lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1551
by (rule_tac y = 0 in order_trans, auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1552
53076
47c9aff07725 more symbols;
wenzelm
parents: 51956
diff changeset
  1553
lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1554
by (auto simp add: power2_eq_square)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1555
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1556
58983
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1557
lemma numeral_power_eq_real_of_int_cancel_iff[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1558
  "numeral x ^ n = real (y::int) \<longleftrightarrow> numeral x ^ n = y"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1559
  by (metis real_numeral(1) real_of_int_inject real_of_int_power)
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1560
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1561
lemma real_of_int_eq_numeral_power_cancel_iff[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1562
  "real (y::int) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1563
  using numeral_power_eq_real_of_int_cancel_iff[of x n y]
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1564
  by metis
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1565
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1566
lemma numeral_power_eq_real_of_nat_cancel_iff[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1567
  "numeral x ^ n = real (y::nat) \<longleftrightarrow> numeral x ^ n = y"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1568
  by (metis of_nat_eq_iff of_nat_numeral real_of_int_eq_numeral_power_cancel_iff
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1569
    real_of_int_of_nat_eq zpower_int)
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1570
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1571
lemma real_of_nat_eq_numeral_power_cancel_iff[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1572
  "real (y::nat) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1573
  using numeral_power_eq_real_of_nat_cancel_iff[of x n y]
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1574
  by metis
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1575
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1576
lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1577
  "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1578
  unfolding real_of_nat_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1579
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1580
lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1581
  "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1582
  unfolding real_of_nat_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1583
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1584
lemma numeral_power_le_real_of_int_cancel_iff[simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1585
  "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1586
  unfolding real_of_int_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1587
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1588
lemma real_of_int_le_numeral_power_cancel_iff[simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1589
  "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1590
  unfolding real_of_int_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1591
58983
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1592
lemma numeral_power_less_real_of_nat_cancel_iff[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1593
  "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1594
  unfolding real_of_nat_less_iff[symmetric] by simp
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1595
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1596
lemma real_of_nat_less_numeral_power_cancel_iff[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1597
  "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1598
  unfolding real_of_nat_less_iff[symmetric] by simp
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1599
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1600
lemma numeral_power_less_real_of_int_cancel_iff[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1601
  "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::int) ^ n < a"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1602
  unfolding real_of_int_less_iff[symmetric] by simp
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1603
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1604
lemma real_of_int_less_numeral_power_cancel_iff[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1605
  "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1606
  unfolding real_of_int_less_iff[symmetric] by simp
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1607
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1608
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
  1609
  "(- numeral x::real) ^ n \<le> real a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1610
  unfolding real_of_int_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1611
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1612
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
  1613
  "real a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1614
  unfolding real_of_int_le_iff[symmetric] by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1615
56889
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1616
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1617
subsection{*Density of the Reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1618
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1619
lemma real_lbound_gt_zero:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1620
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1621
apply (rule_tac x = " (min d1 d2) /2" in exI)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1622
apply (simp add: min_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1623
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1624
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1625
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1626
text{*Similar results are proved in @{text Fields}*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1627
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1628
  by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1629
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1630
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1631
  by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1632
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1633
lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1634
  by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1635
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1636
subsection{*Absolute Value Function for the Reals*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1637
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1638
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1639
by (simp add: abs_if)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1640
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1641
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1642
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1643
by (force simp add: abs_le_iff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1644
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1645
lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1646
by (simp add: abs_if)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1647
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1648
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1649
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1650
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1651
lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1652
by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1653
 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1654
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1655
by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1656
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1657
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1658
subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1659
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1660
(* FIXME: theorems for negative numerals *)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1661
lemma numeral_less_real_of_int_iff [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1662
     "((numeral n) < real (m::int)) = (numeral n < m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1663
apply auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1664
apply (rule real_of_int_less_iff [THEN iffD1])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1665
apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1666
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1667
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1668
lemma numeral_less_real_of_int_iff2 [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1669
     "(real (m::int) < (numeral n)) = (m < numeral n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1670
apply auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1671
apply (rule real_of_int_less_iff [THEN iffD1])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1672
apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1673
done
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1674
56889
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1675
lemma real_of_nat_less_numeral_iff [simp]:
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1676
  "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1677
  using real_of_nat_less_iff[of n "numeral w"] by simp
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1678
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1679
lemma numeral_less_real_of_nat_iff [simp]:
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1680
  "numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1681
  using real_of_nat_less_iff[of "numeral w" n] by simp
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56571
diff changeset
  1682
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1683
lemma numeral_le_real_of_nat_iff[simp]:
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1684
  "(numeral n \<le> real(m::nat)) = (numeral n \<le> m)"
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1685
by (metis not_le real_of_nat_less_numeral_iff)
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1686
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1687
lemma numeral_le_real_of_int_iff [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1688
     "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1689
by (simp add: linorder_not_less [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1690
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1691
lemma numeral_le_real_of_int_iff2 [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1692
     "(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1693
by (simp add: linorder_not_less [symmetric])
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1694
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1695
lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1696
unfolding real_of_nat_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1697
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1698
lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1699
unfolding real_of_nat_def by (simp add: floor_minus)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1700
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1701
lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1702
unfolding real_of_int_def by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1703
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1704
lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1705
unfolding real_of_int_def by (simp add: floor_minus)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1706
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1707
lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1708
unfolding real_of_int_def by (rule floor_exists)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1709
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1710
lemma lemma_floor: "real m \<le> r \<Longrightarrow> r < real n + 1 \<Longrightarrow> m \<le> (n::int)"
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1711
  by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1712
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1713
lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1714
unfolding real_of_int_def by (rule of_int_floor_le)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1715
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1716
lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1717
  by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1718
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1719
lemma real_of_int_floor_cancel [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1720
    "(real (floor x) = x) = (\<exists>n::int. x = real n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1721
  using floor_real_of_int by metis
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1722
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1723
lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1724
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1725
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1726
lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1727
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1728
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1729
lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1730
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1731
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1732
lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1733
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1734
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1735
lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1736
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1737
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1738
lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1739
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1740
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1741
lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1742
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1743
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1744
lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1745
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1746
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1747
lemma le_floor: "real a <= x ==> a <= floor x"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1748
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1749
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1750
lemma real_le_floor: "a <= floor x ==> real a <= x"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1751
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1752
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1753
lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1754
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1755
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1756
lemma floor_less_eq: "(floor x < a) = (x < real a)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1757
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1758
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1759
lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1760
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1761
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1762
lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1763
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1764
58983
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1765
lemma floor_eq_iff: "floor x = b \<longleftrightarrow> real b \<le> x \<and> x < real (b + 1)"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1766
  by linarith
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1767
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1768
lemma floor_add [simp]: "floor (x + real a) = floor x + a"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1769
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1770
58983
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1771
lemma floor_add2[simp]: "floor (real a + x) = a + floor x"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1772
  by linarith
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1773
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1774
lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1775
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1776
58788
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1777
lemma floor_divide_real_eq_div: "0 \<le> b \<Longrightarrow> floor (a / real b) = floor a div b"
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1778
proof cases
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1779
  assume "0 < b"
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1780
  { fix i j :: int assume "real i \<le> a" "a < 1 + real i"
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1781
      "real j * real b \<le> a" "a < real b + real j * real b"
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1782
    then have "i < b + j * b" "j * b < 1 + i"
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1783
      unfolding real_of_int_less_iff[symmetric] by auto
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1784
    then have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b"
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1785
      by (auto simp: field_simps)
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1786
    then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1787
      using pos_mod_bound[OF `0<b`, of i] pos_mod_sign[OF `0<b`, of i] by linarith+
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1788
    then have "j = i div b"
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1789
      using `0 < b` unfolding mult_less_cancel_right by auto }
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1790
  with `0 < b` show ?thesis
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1791
    by (auto split: floor_split simp: field_simps)
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1792
qed auto
d17b3844b726 generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents: 58134
diff changeset
  1793
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1794
lemma floor_divide_eq_div:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1795
  "floor (real a / real b) = a div b"
59984
4f1eccec320c conversion between division on nat/int and division in archmedean fields
haftmann
parents: 59867
diff changeset
  1796
  using floor_divide_of_int_eq [of a b] real_eq_of_int by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1797
58097
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1798
lemma floor_divide_eq_div_numeral[simp]: "\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1799
  using floor_divide_eq_div[of "numeral a" "numeral b"] by simp
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1800
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1801
lemma floor_minus_divide_eq_div_numeral[simp]: "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1802
  using floor_divide_eq_div[of "- numeral a" "numeral b"] by simp
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1803
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1804
lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1805
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1806
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1807
lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1808
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1809
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1810
lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1811
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1812
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1813
lemma real_of_int_ceiling_cancel [simp]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1814
     "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1815
  using ceiling_real_of_int by metis
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1816
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1817
lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1818
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1819
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1820
lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1821
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1822
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1823
lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1824
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1825
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1826
lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1827
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1828
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1829
lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1830
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1831
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1832
lemma ceiling_le: "x <= real a ==> ceiling x <= a"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1833
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1834
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1835
lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1836
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1837
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1838
lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1839
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1840
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1841
lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1842
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1843
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1844
lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1845
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1846
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1847
lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1848
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1849
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1850
lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1851
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1852
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1853
lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1854
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1855
58097
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1856
lemma ceiling_divide_eq_div: "\<lceil>real a / real b\<rceil> = - (- a div b)"
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1857
  unfolding ceiling_def minus_divide_left real_of_int_minus[symmetric] floor_divide_eq_div by simp_all
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1858
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1859
lemma ceiling_divide_eq_div_numeral [simp]:
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1860
  "\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)"
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1861
  using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1862
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1863
lemma ceiling_minus_divide_eq_div_numeral [simp]:
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1864
  "\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (numeral a div numeral b)"
cfd3cff9387b add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents: 58061
diff changeset
  1865
  using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1866
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1867
text{* The following lemmas are remnants of the erstwhile functions natfloor
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1868
and natceiling. *}
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1869
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1870
lemma nat_floor_neg: "(x::real) <= 0 ==> nat(floor x) = 0"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1871
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1872
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1873
lemma le_nat_floor: "real x <= a ==> x <= nat(floor a)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1874
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1875
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1876
lemma le_mult_nat_floor:
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1877
  shows "nat(floor a) * nat(floor b) \<le> nat(floor (a * b))"
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1878
  by (cases "0 <= a & 0 <= b")
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1879
     (auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1880
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1881
lemma nat_ceiling_le_eq: "(nat(ceiling x) <= a) = (x <= real a)"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1882
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1883
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1884
lemma real_nat_ceiling_ge: "x <= real(nat(ceiling x))"
58040
9a867afaab5a better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents: 57514
diff changeset
  1885
  by linarith
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1886
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1887
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1888
lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1889
  by (auto intro!: bexI[of _ "of_nat (nat(ceiling x))"]) linarith
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1890
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1891
lemma Rats_no_bot_less: "\<exists> q \<in> \<rat>. q < (x :: real)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1892
  apply (auto intro!: bexI[of _ "of_int (floor x - 1)"])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1893
  apply (rule less_le_trans[OF _ of_int_floor_le])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1894
  apply simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1895
  done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
  1896
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1897
subsection {* Exponentiation with floor *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1898
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1899
lemma floor_power:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1900
  assumes "x = real (floor x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1901
  shows "floor (x ^ n) = floor x ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1902
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1903
  have *: "x ^ n = real (floor x ^ n)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1904
    using assms by (induct n arbitrary: x) simp_all
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1905
  show ?thesis unfolding real_of_int_inject[symmetric]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1906
    unfolding * floor_real_of_int ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1907
qed
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1908
(*
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1909
lemma natfloor_power:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1910
  assumes "x = real (natfloor x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1911
  shows "natfloor (x ^ n) = natfloor x ^ n"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1912
proof -
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1913
  from assms have "0 \<le> floor x" by auto
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1914
  note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1915
  from floor_power[OF this]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1916
  show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1917
    by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1918
qed
59587
8ea7b22525cb Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents: 59000
diff changeset
  1919
*)
58983
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1920
lemma floor_numeral_power[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1921
  "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1922
  by (metis floor_of_int of_int_numeral of_int_power)
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1923
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1924
lemma ceiling_numeral_power[simp]:
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1925
  "\<lceil>numeral x ^ n\<rceil> = numeral x ^ n"
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1926
  by (metis ceiling_of_int of_int_numeral of_int_power)
9c390032e4eb cancel real of power of numeral also for equality and strict inequality;
immler
parents: 58889
diff changeset
  1927
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1928
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1929
subsection {* Implementation of rational real numbers *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1930
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1931
text {* Formal constructor *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1932
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1933
definition Ratreal :: "rat \<Rightarrow> real" where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1934
  [code_abbrev, simp]: "Ratreal = of_rat"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1935
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1936
code_datatype Ratreal
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1937
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1938
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1939
text {* Numerals *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1940
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1941
lemma [code_abbrev]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1942
  "(of_rat (of_int a) :: real) = of_int a"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1943
  by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1944
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1945
lemma [code_abbrev]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1946
  "(of_rat 0 :: real) = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1947
  by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1948
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1949
lemma [code_abbrev]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1950
  "(of_rat 1 :: real) = 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1951
  by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1952
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1953
lemma [code_abbrev]:
58134
b563ec62d04e more convenient printing of real numbers after evaluation
haftmann
parents: 58097
diff changeset
  1954
  "(of_rat (- 1) :: real) = - 1"
b563ec62d04e more convenient printing of real numbers after evaluation
haftmann
parents: 58097
diff changeset
  1955
  by simp
b563ec62d04e more convenient printing of real numbers after evaluation
haftmann
parents: 58097
diff changeset
  1956
b563ec62d04e more convenient printing of real numbers after evaluation
haftmann
parents: 58097
diff changeset
  1957
lemma [code_abbrev]:
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1958
  "(of_rat (numeral k) :: real) = numeral k"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1959
  by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1960
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1961
lemma [code_abbrev]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1962
  "(of_rat (- numeral k) :: real) = - numeral k"
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1963
  by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1964
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1965
lemma [code_post]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1966
  "(of_rat (1 / numeral k) :: real) = 1 / numeral k"
58134
b563ec62d04e more convenient printing of real numbers after evaluation
haftmann
parents: 58097
diff changeset
  1967
  "(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"
b563ec62d04e more convenient printing of real numbers after evaluation
haftmann
parents: 58097
diff changeset
  1968
  "(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)"
b563ec62d04e more convenient printing of real numbers after evaluation
haftmann
parents: 58097
diff changeset
  1969
  "(of_rat (- (numeral k / numeral l)) :: real) = - (numeral k / numeral l)"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54281
diff changeset
  1970
  by (simp_all add: of_rat_divide of_rat_minus)
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1971
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1972
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1973
text {* Operations *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1974
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1975
lemma zero_real_code [code]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1976
  "0 = Ratreal 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1977
by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1978
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1979
lemma one_real_code [code]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1980
  "1 = Ratreal 1"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1981
by simp
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1982
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1983
instantiation real :: equal
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1984
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1985
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1986
definition "HOL.equal (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1987
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1988
instance proof
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1989
qed (simp add: equal_real_def)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1990
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1991
lemma real_equal_code [code]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1992
  "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1993
  by (simp add: equal_real_def equal)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1994
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1995
lemma [code nbe]:
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1996
  "HOL.equal (x::real) x \<longleftrightarrow> True"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1997
  by (rule equal_refl)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1998
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  1999
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2000
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2001
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2002
  by (simp add: of_rat_less_eq)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2003
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2004
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2005
  by (simp add: of_rat_less)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2006
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2007
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2008
  by (simp add: of_rat_add)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2009
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2010
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2011
  by (simp add: of_rat_mult)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2012
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2013
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2014
  by (simp add: of_rat_minus)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2015
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2016
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2017
  by (simp add: of_rat_diff)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2018
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2019
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2020
  by (simp add: of_rat_inverse)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2021
 
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2022
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2023
  by (simp add: of_rat_divide)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2024
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2025
lemma real_floor_code [code]: "floor (Ratreal x) = floor x"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2026
  by (metis Ratreal_def floor_le_iff floor_unique le_floor_iff of_int_floor_le of_rat_of_int_eq real_less_eq_code)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2027
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2028
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2029
text {* Quickcheck *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2030
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2031
definition (in term_syntax)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2032
  valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2033
  [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2034
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2035
notation fcomp (infixl "\<circ>>" 60)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2036
notation scomp (infixl "\<circ>\<rightarrow>" 60)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2037
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2038
instantiation real :: random
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2039
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2040
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2041
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2042
  "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2043
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2044
instance ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2045
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2046
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2047
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2048
no_notation fcomp (infixl "\<circ>>" 60)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2049
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2050
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2051
instantiation real :: exhaustive
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2052
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2053
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2054
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2055
  "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2056
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2057
instance ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2058
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2059
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2060
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2061
instantiation real :: full_exhaustive
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2062
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2063
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2064
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2065
  "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2066
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2067
instance ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2068
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2069
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2070
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2071
instantiation real :: narrowing
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2072
begin
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2073
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2074
definition
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2075
  "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2076
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2077
instance ..
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2078
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2079
end
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2080
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2081
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2082
subsection {* Setup for Nitpick *}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2083
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2084
declaration {*
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2085
  Nitpick_HOL.register_frac_type @{type_name real}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2086
   [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2087
    (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2088
    (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2089
    (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2090
    (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2091
    (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2092
    (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2093
    (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2094
*}
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2095
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2096
lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2097
    ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2098
    times_real_inst.times_real uminus_real_inst.uminus_real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2099
    zero_real_inst.zero_real
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2100
56078
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2101
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2102
subsection {* Setup for SMT *}
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2103
58061
3d060f43accb renamed new SMT module from 'SMT2' to 'SMT'
blanchet
parents: 58055
diff changeset
  2104
ML_file "Tools/SMT/smt_real.ML"
3d060f43accb renamed new SMT module from 'SMT2' to 'SMT'
blanchet
parents: 58055
diff changeset
  2105
ML_file "Tools/SMT/z3_real.ML"
56078
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2106
58061
3d060f43accb renamed new SMT module from 'SMT2' to 'SMT'
blanchet
parents: 58055
diff changeset
  2107
lemma [z3_rule]:
56078
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2108
  "0 + (x::real) = x"
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2109
  "x + 0 = x"
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2110
  "0 * x = 0"
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2111
  "1 * x = x"
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2112
  "x + y = y + x"
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 55945
diff changeset
  2113
  by auto
51523
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2114
97b5e8a1291c rename RealDef to Real
hoelzl
parents:
diff changeset
  2115
end