
haftmann@28952: (*  Title       : HOL/RealDef.thy
huffman@36795:     Author      : Jacques D. Fleuriot, 1998
paulson@14387:     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
avigad@16819:     Additional contributions by Jeremy Avigad
huffman@36795:     Construction of Cauchy Reals by Brian Huffman, 2010
paulson@14269: *)
paulson@14269: 
huffman@36795: header {* Development of the Reals using Cauchy Sequences *}
paulson@14387: 
nipkow@15131: theory RealDef
huffman@36795: imports Rat
nipkow@15131: begin
paulson@5588: 
huffman@36795: text {*
huffman@36795:   This theory contains a formalization of the real numbers as
huffman@36795:   equivalence classes of Cauchy sequences of rationals.  See
huffman@40810:   @{file "~~/src/HOL/ex/Dedekind_Real.thy"} for an alternative
huffman@40810:   construction using Dedekind cuts.
huffman@36795: *}
huffman@36795: 
huffman@36795: subsection {* Preliminary lemmas *}
huffman@36795: 
huffman@36795: lemma add_diff_add:
huffman@36795:   fixes a b c d :: "'a::ab_group_add"
huffman@36795:   shows "(a + c) - (b + d) = (a - b) + (c - d)"
huffman@36795:   by simp
huffman@36795: 
huffman@36795: lemma minus_diff_minus:
huffman@36795:   fixes a b :: "'a::ab_group_add"
huffman@36795:   shows "- a - - b = - (a - b)"
huffman@36795:   by simp
huffman@36795: 
huffman@36795: lemma mult_diff_mult:
huffman@36795:   fixes x y a b :: "'a::ring"
huffman@36795:   shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
huffman@36795:   by (simp add: algebra_simps)
huffman@36795: 
huffman@36795: lemma inverse_diff_inverse:
huffman@36795:   fixes a b :: "'a::division_ring"
huffman@36795:   assumes "a \<noteq> 0" and "b \<noteq> 0"
huffman@36795:   shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
huffman@36795:   using assms by (simp add: algebra_simps)
huffman@36795: 
huffman@36795: lemma obtain_pos_sum:
huffman@36795:   fixes r :: rat assumes r: "0 < r"
huffman@36795:   obtains s t where "0 < s" and "0 < t" and "r = s + t"
huffman@36795: proof
huffman@36795:     from r show "0 < r/2" by simp
huffman@36795:     from r show "0 < r/2" by simp
huffman@36795:     show "r = r/2 + r/2" by simp
huffman@36795: qed
huffman@36795: 
huffman@36795: subsection {* Sequences that converge to zero *}
huffman@36795: 
wenzelm@19765: definition
huffman@36795:   vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
huffman@36795: where
huffman@36795:   "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
huffman@36795: 
huffman@36795: lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X"
huffman@36795:   unfolding vanishes_def by simp
huffman@36795: 
huffman@36795: lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
huffman@36795:   unfolding vanishes_def by simp
huffman@36795: 
huffman@36795: lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
huffman@36795:   unfolding vanishes_def
huffman@36795:   apply (cases "c = 0", auto)
huffman@36795:   apply (rule exI [where x="\<bar>c\<bar>"], auto)
huffman@36795:   done
huffman@36795: 
huffman@36795: lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"
huffman@36795:   unfolding vanishes_def by simp
paulson@14269: 
huffman@36795: lemma vanishes_add:
huffman@36795:   assumes X: "vanishes X" and Y: "vanishes Y"
huffman@36795:   shows "vanishes (\<lambda>n. X n + Y n)"
huffman@36795: proof (rule vanishesI)
huffman@36795:   fix r :: rat assume "0 < r"
huffman@36795:   then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
huffman@36795:     by (rule obtain_pos_sum)
huffman@36795:   obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
huffman@36795:     using vanishesD [OF X s] ..
huffman@36795:   obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
huffman@36795:     using vanishesD [OF Y t] ..
huffman@36795:   have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"
huffman@36795:   proof (clarsimp)
huffman@36795:     fix n assume n: "i \<le> n" "j \<le> n"
huffman@36795:     have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
huffman@36795:     also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
huffman@36795:     finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
huffman@36795:   qed
huffman@36795:   thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
huffman@36795: qed
huffman@36795: 
huffman@36795: lemma vanishes_diff:
huffman@36795:   assumes X: "vanishes X" and Y: "vanishes Y"
huffman@36795:   shows "vanishes (\<lambda>n. X n - Y n)"
huffman@36795: unfolding diff_minus by (intro vanishes_add vanishes_minus X Y)
huffman@36795: 
huffman@36795: lemma vanishes_mult_bounded:
huffman@36795:   assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
huffman@36795:   assumes Y: "vanishes (\<lambda>n. Y n)"
huffman@36795:   shows "vanishes (\<lambda>n. X n * Y n)"
huffman@36795: proof (rule vanishesI)
huffman@36795:   fix r :: rat assume r: "0 < r"
huffman@36795:   obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
huffman@36795:     using X by fast
huffman@36795:   obtain b where b: "0 < b" "r = a * b"
huffman@36795:   proof
huffman@36795:     show "0 < r / a" using r a by (simp add: divide_pos_pos)
huffman@36795:     show "r = a * (r / a)" using a by simp
huffman@36795:   qed
huffman@36795:   obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
huffman@36795:     using vanishesD [OF Y b(1)] ..
huffman@36795:   have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
huffman@36795:     by (simp add: b(2) abs_mult mult_strict_mono' a k)
huffman@36795:   thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
huffman@36795: qed
huffman@36795: 
huffman@36795: subsection {* Cauchy sequences *}
paulson@5588: 
wenzelm@19765: definition
haftmann@44278:   cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
huffman@36795: where
huffman@36795:   "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
huffman@36795: 
huffman@36795: lemma cauchyI:
huffman@36795:   "(\<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"
huffman@36795:   unfolding cauchy_def by simp
huffman@36795: 
huffman@36795: lemma cauchyD:
huffman@36795:   "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
huffman@36795:   unfolding cauchy_def by simp
huffman@36795: 
huffman@36795: lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
huffman@36795:   unfolding cauchy_def by simp
huffman@36795: 
huffman@36795: lemma cauchy_add [simp]:
huffman@36795:   assumes X: "cauchy X" and Y: "cauchy Y"
huffman@36795:   shows "cauchy (\<lambda>n. X n + Y n)"
huffman@36795: proof (rule cauchyI)
huffman@36795:   fix r :: rat assume "0 < r"
huffman@36795:   then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
huffman@36795:     by (rule obtain_pos_sum)
huffman@36795:   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
huffman@36795:     using cauchyD [OF X s] ..
huffman@36795:   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
huffman@36795:     using cauchyD [OF Y t] ..
huffman@36795:   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r"
huffman@36795:   proof (clarsimp)
huffman@36795:     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
huffman@36795:     have "\<bar>(X m + Y m) - (X n + Y n)\<bar> \<le> \<bar>X m - X n\<bar> + \<bar>Y m - Y n\<bar>"
huffman@36795:       unfolding add_diff_add by (rule abs_triangle_ineq)
huffman@36795:     also have "\<dots> < s + t"
huffman@36795:       by (rule add_strict_mono, simp_all add: i j *)
huffman@36795:     finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .
huffman@36795:   qed
huffman@36795:   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
huffman@36795: qed
huffman@36795: 
huffman@36795: lemma cauchy_minus [simp]:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   shows "cauchy (\<lambda>n. - X n)"
huffman@36795: using assms unfolding cauchy_def
huffman@36795: unfolding minus_diff_minus abs_minus_cancel .
huffman@36795: 
huffman@36795: lemma cauchy_diff [simp]:
huffman@36795:   assumes X: "cauchy X" and Y: "cauchy Y"
huffman@36795:   shows "cauchy (\<lambda>n. X n - Y n)"
huffman@36795: using assms unfolding diff_minus by simp
huffman@36795: 
huffman@36795: lemma cauchy_imp_bounded:
huffman@36795:   assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
huffman@36795: proof -
huffman@36795:   obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1"
huffman@36795:     using cauchyD [OF assms zero_less_one] ..
huffman@36795:   show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
huffman@36795:   proof (intro exI conjI allI)
huffman@36795:     have "0 \<le> \<bar>X 0\<bar>" by simp
huffman@36795:     also have "\<bar>X 0\<bar> \<le> Max (abs ` X ` {..k})" by simp
huffman@36795:     finally have "0 \<le> Max (abs ` X ` {..k})" .
huffman@36795:     thus "0 < Max (abs ` X ` {..k}) + 1" by simp
huffman@36795:   next
huffman@36795:     fix n :: nat
huffman@36795:     show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1"
huffman@36795:     proof (rule linorder_le_cases)
huffman@36795:       assume "n \<le> k"
huffman@36795:       hence "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp
huffman@36795:       thus "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" by simp
huffman@36795:     next
huffman@36795:       assume "k \<le> n"
huffman@36795:       have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp
huffman@36795:       also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>"
huffman@36795:         by (rule abs_triangle_ineq)
huffman@36795:       also have "\<dots> < Max (abs ` X ` {..k}) + 1"
huffman@36795:         by (rule add_le_less_mono, simp, simp add: k `k \<le> n`)
huffman@36795:       finally show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" .
huffman@36795:     qed
huffman@36795:   qed
huffman@36795: qed
huffman@36795: 
huffman@36795: lemma cauchy_mult [simp]:
huffman@36795:   assumes X: "cauchy X" and Y: "cauchy Y"
huffman@36795:   shows "cauchy (\<lambda>n. X n * Y n)"
huffman@36795: proof (rule cauchyI)
huffman@36795:   fix r :: rat assume "0 < r"
huffman@36795:   then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"
huffman@36795:     by (rule obtain_pos_sum)
huffman@36795:   obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
huffman@36795:     using cauchy_imp_bounded [OF X] by fast
huffman@36795:   obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"
huffman@36795:     using cauchy_imp_bounded [OF Y] by fast
huffman@36795:   obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"
huffman@36795:   proof
huffman@36795:     show "0 < v/b" using v b(1) by (rule divide_pos_pos)
huffman@36795:     show "0 < u/a" using u a(1) by (rule divide_pos_pos)
huffman@36795:     show "r = a * (u/a) + (v/b) * b"
huffman@36795:       using a(1) b(1) `r = u + v` by simp
huffman@36795:   qed
huffman@36795:   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
huffman@36795:     using cauchyD [OF X s] ..
huffman@36795:   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
huffman@36795:     using cauchyD [OF Y t] ..
huffman@36795:   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m - X n * Y n\<bar> < r"
huffman@36795:   proof (clarsimp)
huffman@36795:     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
huffman@36795:     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>"
huffman@36795:       unfolding mult_diff_mult ..
huffman@36795:     also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"
huffman@36795:       by (rule abs_triangle_ineq)
huffman@36795:     also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"
huffman@36795:       unfolding abs_mult ..
huffman@36795:     also have "\<dots> < a * t + s * b"
huffman@36795:       by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)
huffman@36795:     finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .
huffman@36795:   qed
huffman@36795:   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
huffman@36795: qed
huffman@36795: 
huffman@36795: lemma cauchy_not_vanishes_cases:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   assumes nz: "\<not> vanishes X"
huffman@36795:   shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"
huffman@36795: proof -
huffman@36795:   obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"
huffman@36795:     using nz unfolding vanishes_def by (auto simp add: not_less)
huffman@36795:   obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t"
huffman@36795:     using `0 < r` by (rule obtain_pos_sum)
huffman@36795:   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
huffman@36795:     using cauchyD [OF X s] ..
huffman@36795:   obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>"
huffman@36795:     using r by fast
huffman@36795:   have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
huffman@36795:     using i `i \<le> k` by auto
huffman@36795:   have "X k \<le> - r \<or> r \<le> X k"
huffman@36795:     using `r \<le> \<bar>X k\<bar>` by auto
huffman@36795:   hence "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
huffman@36795:     unfolding `r = s + t` using k by auto
huffman@36795:   hence "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
huffman@36795:   thus "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
huffman@36795:     using t by auto
huffman@36795: qed
huffman@36795: 
huffman@36795: lemma cauchy_not_vanishes:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   assumes nz: "\<not> vanishes X"
huffman@36795:   shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
huffman@36795: using cauchy_not_vanishes_cases [OF assms]
huffman@36795: by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto)
huffman@36795: 
huffman@36795: lemma cauchy_inverse [simp]:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   assumes nz: "\<not> vanishes X"
huffman@36795:   shows "cauchy (\<lambda>n. inverse (X n))"
huffman@36795: proof (rule cauchyI)
huffman@36795:   fix r :: rat assume "0 < r"
huffman@36795:   obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
huffman@36795:     using cauchy_not_vanishes [OF X nz] by fast
huffman@36795:   from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto
huffman@36795:   obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"
huffman@36795:   proof
huffman@36795:     show "0 < b * r * b"
huffman@36795:       by (simp add: `0 < r` b mult_pos_pos)
huffman@36795:     show "r = inverse b * (b * r * b) * inverse b"
huffman@36795:       using b by simp
huffman@36795:   qed
huffman@36795:   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
huffman@36795:     using cauchyD [OF X s] ..
huffman@36795:   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
huffman@36795:   proof (clarsimp)
huffman@36795:     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
huffman@36795:     have "\<bar>inverse (X m) - inverse (X n)\<bar> =
huffman@36795:           inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"
huffman@36795:       by (simp add: inverse_diff_inverse nz * abs_mult)
huffman@36795:     also have "\<dots> < inverse b * s * inverse b"
huffman@36795:       by (simp add: mult_strict_mono less_imp_inverse_less
huffman@36795:                     mult_pos_pos i j b * s)
huffman@36795:     finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
huffman@36795:   qed
huffman@36795:   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
huffman@36795: qed
huffman@36795: 
huffman@47901: lemma vanishes_diff_inverse:
huffman@47901:   assumes X: "cauchy X" "\<not> vanishes X"
huffman@47901:   assumes Y: "cauchy Y" "\<not> vanishes Y"
huffman@47901:   assumes XY: "vanishes (\<lambda>n. X n - Y n)"
huffman@47901:   shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
huffman@47901: proof (rule vanishesI)
huffman@47901:   fix r :: rat assume r: "0 < r"
huffman@47901:   obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
huffman@47901:     using cauchy_not_vanishes [OF X] by fast
huffman@47901:   obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
huffman@47901:     using cauchy_not_vanishes [OF Y] by fast
huffman@47901:   obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
huffman@47901:   proof
huffman@47901:     show "0 < a * r * b"
huffman@47901:       using a r b by (simp add: mult_pos_pos)
huffman@47901:     show "inverse a * (a * r * b) * inverse b = r"
huffman@47901:       using a r b by simp
huffman@47901:   qed
huffman@47901:   obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
huffman@47901:     using vanishesD [OF XY s] ..
huffman@47901:   have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
huffman@47901:   proof (clarsimp)
huffman@47901:     fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
huffman@47901:     have "X n \<noteq> 0" and "Y n \<noteq> 0"
huffman@47901:       using i j a b n by auto
huffman@47901:     hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
huffman@47901:         inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
huffman@47901:       by (simp add: inverse_diff_inverse abs_mult)
huffman@47901:     also have "\<dots> < inverse a * s * inverse b"
huffman@47901:       apply (intro mult_strict_mono' less_imp_inverse_less)
huffman@47901:       apply (simp_all add: a b i j k n mult_nonneg_nonneg)
huffman@47901:       done
huffman@47901:     also note `inverse a * s * inverse b = r`
huffman@47901:     finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
huffman@47901:   qed
huffman@47901:   thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
huffman@47901: qed
huffman@47901: 
huffman@36795: subsection {* Equivalence relation on Cauchy sequences *}
huffman@36795: 
huffman@47902: definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
huffman@47902:   where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"
huffman@36795: 
huffman@47902: lemma realrelI [intro?]:
huffman@47902:   assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"
huffman@47902:   shows "realrel X Y"
huffman@47902:   using assms unfolding realrel_def by simp
huffman@36795: 
huffman@47902: lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
huffman@47902:   unfolding realrel_def by simp
paulson@14484: 
huffman@47902: lemma symp_realrel: "symp realrel"
huffman@36795:   unfolding realrel_def
huffman@47902:   by (rule sympI, clarify, drule vanishes_minus, simp)
huffman@47902: 
huffman@47902: lemma transp_realrel: "transp realrel"
huffman@47902:   unfolding realrel_def
huffman@47902:   apply (rule transpI, clarify)
huffman@36795:   apply (drule (1) vanishes_add)
huffman@36795:   apply (simp add: algebra_simps)
huffman@36795:   done
huffman@36795: 
huffman@47902: lemma part_equivp_realrel: "part_equivp realrel"
huffman@47902:   by (fast intro: part_equivpI symp_realrel transp_realrel
huffman@47902:     realrel_refl cauchy_const)
huffman@36795: 
huffman@36795: subsection {* The field of real numbers *}
huffman@36795: 
huffman@47902: quotient_type real = "nat \<Rightarrow> rat" / partial: realrel
huffman@47902:   morphisms rep_real Real
huffman@47902:   by (rule part_equivp_realrel)
huffman@36795: 
huffman@47902: lemma cr_real_eq: "cr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
huffman@47902:   unfolding cr_real_def realrel_def by simp
huffman@36795: 
huffman@47902: lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
huffman@47902:   assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"
huffman@47902: proof (induct x)
huffman@47902:   case (1 X)
huffman@47902:   hence "cauchy X" by (simp add: realrel_def)
huffman@47902:   thus "P (Real X)" by (rule assms)
huffman@47902: qed
huffman@36795: 
huffman@36795: lemma eq_Real:
huffman@36795:   "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
huffman@47902:   using real.rel_eq_transfer
huffman@47902:   unfolding cr_real_def fun_rel_def realrel_def by simp
huffman@47902: 
huffman@47902: declare real.forall_transfer [transfer_rule del]
huffman@36795: 
huffman@47902: lemma forall_real_transfer [transfer_rule]: (* TODO: generate automatically *)
huffman@47902:   "(fun_rel (fun_rel cr_real op =) op =)
huffman@47902:     (transfer_bforall cauchy) transfer_forall"
huffman@47902:   using Quotient_forall_transfer [OF Quotient_real]
huffman@47902:   by (simp add: realrel_def)
huffman@47902: 
huffman@47902: instantiation real :: field_inverse_zero
huffman@47902: begin
huffman@47902: 
huffman@47902: lift_definition zero_real :: "real" is "\<lambda>n. 0"
huffman@47902:   by (simp add: realrel_refl)
huffman@36795: 
huffman@47902: lift_definition one_real :: "real" is "\<lambda>n. 1"
huffman@47902:   by (simp add: realrel_refl)
huffman@47902: 
huffman@47902: lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n + Y n"
huffman@47902:   unfolding realrel_def add_diff_add
huffman@47902:   by (simp only: cauchy_add vanishes_add simp_thms)
huffman@36795: 
huffman@47902: lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n"
huffman@47902:   unfolding realrel_def minus_diff_minus
huffman@47902:   by (simp only: cauchy_minus vanishes_minus simp_thms)
huffman@36795: 
huffman@47902: lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
huffman@47902:   unfolding realrel_def mult_diff_mult
huffman@47902:   by (subst (4) mult_commute, simp only: cauchy_mult vanishes_add
huffman@47902:     vanishes_mult_bounded cauchy_imp_bounded simp_thms)
huffman@47902: 
huffman@47902: lift_definition inverse_real :: "real \<Rightarrow> real"
huffman@47902:   is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"
huffman@47902: proof -
huffman@47902:   fix X Y assume "realrel X Y"
huffman@36795:   hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
huffman@36795:     unfolding realrel_def by simp_all
huffman@36795:   have "vanishes X \<longleftrightarrow> vanishes Y"
huffman@36795:   proof
huffman@36795:     assume "vanishes X"
huffman@36795:     from vanishes_diff [OF this XY] show "vanishes Y" by simp
huffman@36795:   next
huffman@36795:     assume "vanishes Y"
huffman@36795:     from vanishes_add [OF this XY] show "vanishes X" by simp
huffman@36795:   qed
huffman@47902:   thus "?thesis X Y"
huffman@47902:     unfolding realrel_def
huffman@47902:     by (simp add: vanishes_diff_inverse X Y XY)
huffman@36795: qed
huffman@36795: 
huffman@36795: definition
huffman@36795:   "x - y = (x::real) + - y"
bauerg@10606: 
haftmann@25571: definition
huffman@36795:   "x / y = (x::real) * inverse y"
huffman@36795: 
huffman@36795: lemma add_Real:
huffman@36795:   assumes X: "cauchy X" and Y: "cauchy Y"
huffman@36795:   shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
huffman@47902:   using assms plus_real.transfer
huffman@47902:   unfolding cr_real_eq fun_rel_def by simp
huffman@36795: 
huffman@36795: lemma minus_Real:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   shows "- Real X = Real (\<lambda>n. - X n)"
huffman@47902:   using assms uminus_real.transfer
huffman@47902:   unfolding cr_real_eq fun_rel_def by simp
paulson@5588: 
huffman@36795: lemma diff_Real:
huffman@36795:   assumes X: "cauchy X" and Y: "cauchy Y"
huffman@36795:   shows "Real X - Real Y = Real (\<lambda>n. X n - Y n)"
huffman@36795:   unfolding minus_real_def diff_minus
huffman@36795:   by (simp add: minus_Real add_Real X Y)
haftmann@25571: 
huffman@36795: lemma mult_Real:
huffman@36795:   assumes X: "cauchy X" and Y: "cauchy Y"
huffman@36795:   shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
huffman@47902:   using assms times_real.transfer
huffman@47902:   unfolding cr_real_eq fun_rel_def by simp
huffman@36795: 
huffman@36795: lemma inverse_Real:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   shows "inverse (Real X) =
huffman@36795:     (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
huffman@47902:   using assms inverse_real.transfer zero_real.transfer
huffman@47902:   unfolding cr_real_eq fun_rel_def by (simp split: split_if_asm, metis)
paulson@14269: 
huffman@36795: instance proof
huffman@36795:   fix a b c :: real
huffman@36795:   show "a + b = b + a"
huffman@47902:     by transfer (simp add: add_ac realrel_def)
huffman@36795:   show "(a + b) + c = a + (b + c)"
huffman@47902:     by transfer (simp add: add_ac realrel_def)
huffman@36795:   show "0 + a = a"
huffman@47902:     by transfer (simp add: realrel_def)
huffman@36795:   show "- a + a = 0"
huffman@47902:     by transfer (simp add: realrel_def)
huffman@36795:   show "a - b = a + - b"
huffman@36795:     by (rule minus_real_def)
huffman@36795:   show "(a * b) * c = a * (b * c)"
huffman@47902:     by transfer (simp add: mult_ac realrel_def)
huffman@36795:   show "a * b = b * a"
huffman@47902:     by transfer (simp add: mult_ac realrel_def)
huffman@36795:   show "1 * a = a"
huffman@47902:     by transfer (simp add: mult_ac realrel_def)
huffman@36795:   show "(a + b) * c = a * c + b * c"
huffman@47902:     by transfer (simp add: left_distrib realrel_def)
huffman@36795:   show "(0\<Colon>real) \<noteq> (1\<Colon>real)"
huffman@47902:     by transfer (simp add: realrel_def)
huffman@36795:   show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
huffman@47902:     apply transfer
huffman@47902:     apply (simp add: realrel_def)
huffman@36795:     apply (rule vanishesI)
huffman@36795:     apply (frule (1) cauchy_not_vanishes, clarify)
huffman@36795:     apply (rule_tac x=k in exI, clarify)
huffman@36795:     apply (drule_tac x=n in spec, simp)
huffman@36795:     done
huffman@36795:   show "a / b = a * inverse b"
huffman@36795:     by (rule divide_real_def)
huffman@36795:   show "inverse (0::real) = 0"
huffman@47902:     by transfer (simp add: realrel_def)
huffman@36795: qed
haftmann@25571: 
haftmann@25571: end
paulson@14334: 
huffman@36795: subsection {* Positive reals *}
paulson@14269: 
huffman@47902: lift_definition positive :: "real \<Rightarrow> bool"
huffman@47902:   is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
huffman@47902: proof -
huffman@47902:   { fix X Y
huffman@47902:     assume "realrel X Y"
huffman@47902:     hence XY: "vanishes (\<lambda>n. X n - Y n)"
huffman@47902:       unfolding realrel_def by simp_all
huffman@47902:     assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
huffman@47902:     then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
huffman@47902:       by fast
huffman@47902:     obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
huffman@47902:       using `0 < r` by (rule obtain_pos_sum)
huffman@47902:     obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
huffman@47902:       using vanishesD [OF XY s] ..
huffman@47902:     have "\<forall>n\<ge>max i j. t < Y n"
huffman@47902:     proof (clarsimp)
huffman@47902:       fix n assume n: "i \<le> n" "j \<le> n"
huffman@47902:       have "\<bar>X n - Y n\<bar> < s" and "r < X n"
huffman@47902:         using i j n by simp_all
huffman@47902:       thus "t < Y n" unfolding r by simp
huffman@47902:     qed
huffman@47902:     hence "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast
huffman@47902:   } note 1 = this
huffman@47902:   fix X Y assume "realrel X Y"
huffman@47902:   hence "realrel X Y" and "realrel Y X"
huffman@47902:     using symp_realrel unfolding symp_def by auto
huffman@47902:   thus "?thesis X Y"
huffman@47902:     by (safe elim!: 1)
paulson@14484: qed
paulson@14269: 
huffman@36795: lemma positive_Real:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
huffman@47902:   using assms positive.transfer
huffman@47902:   unfolding cr_real_eq fun_rel_def by simp
huffman@23287: 
huffman@36795: lemma positive_zero: "\<not> positive 0"
huffman@47902:   by transfer auto
paulson@14269: 
huffman@36795: lemma positive_add:
huffman@36795:   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
huffman@47902: apply transfer
huffman@36795: apply (clarify, rename_tac a b i j)
huffman@36795: apply (rule_tac x="a + b" in exI, simp)
huffman@36795: apply (rule_tac x="max i j" in exI, clarsimp)
huffman@36795: apply (simp add: add_strict_mono)
paulson@14269: done
paulson@14269: 
huffman@36795: lemma positive_mult:
huffman@36795:   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
huffman@47902: apply transfer
huffman@36795: apply (clarify, rename_tac a b i j)
huffman@36795: apply (rule_tac x="a * b" in exI, simp add: mult_pos_pos)
huffman@36795: apply (rule_tac x="max i j" in exI, clarsimp)
huffman@36795: apply (rule mult_strict_mono, auto)
huffman@36795: done
huffman@36795: 
huffman@36795: lemma positive_minus:
huffman@36795:   "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
huffman@47902: apply transfer
huffman@47902: apply (simp add: realrel_def)
huffman@36795: apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
paulson@14269: done
paulson@14334: 
huffman@36795: instantiation real :: linordered_field_inverse_zero
huffman@36795: begin
paulson@14341: 
huffman@36795: definition
huffman@36795:   "x < y \<longleftrightarrow> positive (y - x)"
paulson@14341: 
huffman@36795: definition
huffman@36795:   "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"
paulson@14334: 
huffman@36795: definition
huffman@36795:   "abs (a::real) = (if a < 0 then - a else a)"
paulson@14269: 
huffman@36795: definition
huffman@36795:   "sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
paulson@14269: 
huffman@36795: instance proof
huffman@36795:   fix a b c :: real
huffman@36795:   show "\<bar>a\<bar> = (if a < 0 then - a else a)"
huffman@36795:     by (rule abs_real_def)
huffman@36795:   show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
huffman@36795:     unfolding less_eq_real_def less_real_def
huffman@36795:     by (auto, drule (1) positive_add, simp_all add: positive_zero)
huffman@36795:   show "a \<le> a"
huffman@36795:     unfolding less_eq_real_def by simp
huffman@36795:   show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
huffman@36795:     unfolding less_eq_real_def less_real_def
huffman@36795:     by (auto, drule (1) positive_add, simp add: algebra_simps)
huffman@36795:   show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
huffman@36795:     unfolding less_eq_real_def less_real_def
huffman@36795:     by (auto, drule (1) positive_add, simp add: positive_zero)
huffman@36795:   show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
huffman@47108:     unfolding less_eq_real_def less_real_def by (auto simp: diff_minus) (* by auto *)
huffman@47108:     (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)
huffman@47108:     (* Should produce c + b - (c + a) \<equiv> b - a *)
huffman@36795:   show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
huffman@36795:     by (rule sgn_real_def)
huffman@36795:   show "a \<le> b \<or> b \<le> a"
huffman@36795:     unfolding less_eq_real_def less_real_def
huffman@36795:     by (auto dest!: positive_minus)
huffman@36795:   show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
huffman@36795:     unfolding less_real_def
huffman@36795:     by (drule (1) positive_mult, simp add: algebra_simps)
huffman@23288: qed
paulson@14378: 
huffman@36795: end
paulson@14334: 
haftmann@25571: instantiation real :: distrib_lattice
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
huffman@36795:   "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
haftmann@25571: 
haftmann@25571: definition
huffman@36795:   "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
huffman@36795: 
huffman@36795: instance proof
huffman@36795: qed (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
huffman@36795: 
huffman@36795: end
haftmann@25571: 
huffman@36795: lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
huffman@36795: apply (induct x)
huffman@36795: apply (simp add: zero_real_def)
huffman@36795: apply (simp add: one_real_def add_Real)
huffman@36795: done
paulson@14378: 
huffman@36795: lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
huffman@36795: apply (cases x rule: int_diff_cases)
huffman@36795: apply (simp add: of_nat_Real diff_Real)
huffman@36795: done
paulson@14334: 
huffman@36795: lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
huffman@36795: apply (induct x)
huffman@36795: apply (simp add: Fract_of_int_quotient of_rat_divide)
huffman@36795: apply (simp add: of_int_Real divide_inverse)
huffman@36795: apply (simp add: inverse_Real mult_Real)
huffman@36795: done
huffman@36795: 
huffman@36795: instance real :: archimedean_field
paulson@14334: proof
huffman@36795:   fix x :: real
huffman@36795:   show "\<exists>z. x \<le> of_int z"
huffman@36795:     apply (induct x)
huffman@36795:     apply (frule cauchy_imp_bounded, clarify)
huffman@36795:     apply (rule_tac x="ceiling b + 1" in exI)
huffman@36795:     apply (rule less_imp_le)
huffman@36795:     apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
huffman@36795:     apply (rule_tac x=1 in exI, simp add: algebra_simps)
huffman@36795:     apply (rule_tac x=0 in exI, clarsimp)
huffman@36795:     apply (rule le_less_trans [OF abs_ge_self])
huffman@36795:     apply (rule less_le_trans [OF _ le_of_int_ceiling])
huffman@36795:     apply simp
huffman@36795:     done
paulson@14334: qed
paulson@14334: 
bulwahn@43732: instantiation real :: floor_ceiling
bulwahn@43732: begin
bulwahn@43732: 
bulwahn@43732: definition [code del]:
bulwahn@43732:   "floor (x::real) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
bulwahn@43732: 
bulwahn@43732: instance proof
bulwahn@43732:   fix x :: real
bulwahn@43732:   show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
bulwahn@43732:     unfolding floor_real_def using floor_exists1 by (rule theI')
bulwahn@43732: qed
bulwahn@43732: 
bulwahn@43732: end
bulwahn@43732: 
huffman@36795: subsection {* Completeness *}
paulson@14365: 
huffman@36795: lemma not_positive_Real:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
huffman@36795: unfolding positive_Real [OF X]
huffman@36795: apply (auto, unfold not_less)
huffman@36795: apply (erule obtain_pos_sum)
huffman@36795: apply (drule_tac x=s in spec, simp)
huffman@36795: apply (drule_tac r=t in cauchyD [OF X], clarify)
huffman@36795: apply (drule_tac x=k in spec, clarsimp)
huffman@36795: apply (rule_tac x=n in exI, clarify, rename_tac m)
huffman@36795: apply (drule_tac x=m in spec, simp)
huffman@36795: apply (drule_tac x=n in spec, simp)
huffman@36795: apply (drule spec, drule (1) mp, clarify, rename_tac i)
huffman@36795: apply (rule_tac x="max i k" in exI, simp)
huffman@36795: done
huffman@36795: 
huffman@36795: lemma le_Real:
huffman@36795:   assumes X: "cauchy X" and Y: "cauchy Y"
huffman@36795:   shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
huffman@36795: unfolding not_less [symmetric, where 'a=real] less_real_def
huffman@36795: apply (simp add: diff_Real not_positive_Real X Y)
huffman@36795: apply (simp add: diff_le_eq add_ac)
huffman@36795: done
paulson@14365: 
huffman@36795: lemma le_RealI:
huffman@36795:   assumes Y: "cauchy Y"
huffman@36795:   shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
huffman@36795: proof (induct x)
huffman@36795:   fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
huffman@36795:   hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
huffman@36795:     by (simp add: of_rat_Real le_Real)
huffman@36795:   {
huffman@36795:     fix r :: rat assume "0 < r"
huffman@36795:     then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
huffman@36795:       by (rule obtain_pos_sum)
huffman@36795:     obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
huffman@36795:       using cauchyD [OF Y s] ..
huffman@36795:     obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
huffman@36795:       using le [OF t] ..
huffman@36795:     have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
huffman@36795:     proof (clarsimp)
huffman@36795:       fix n assume n: "i \<le> n" "j \<le> n"
huffman@36795:       have "X n \<le> Y i + t" using n j by simp
huffman@36795:       moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
huffman@36795:       ultimately show "X n \<le> Y n + r" unfolding r by simp
huffman@36795:     qed
huffman@36795:     hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
huffman@36795:   }
huffman@36795:   thus "Real X \<le> Real Y"
huffman@36795:     by (simp add: of_rat_Real le_Real X Y)
huffman@36795: qed
paulson@14365: 
huffman@36795: lemma Real_leI:
huffman@36795:   assumes X: "cauchy X"
huffman@36795:   assumes le: "\<forall>n. of_rat (X n) \<le> y"
huffman@36795:   shows "Real X \<le> y"
huffman@36795: proof -
huffman@36795:   have "- y \<le> - Real X"
huffman@36795:     by (simp add: minus_Real X le_RealI of_rat_minus le)
huffman@36795:   thus ?thesis by simp
huffman@36795: qed
huffman@36795: 
huffman@36795: lemma less_RealD:
huffman@36795:   assumes Y: "cauchy Y"
huffman@36795:   shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
huffman@36795: by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
huffman@36795: 
huffman@36795: lemma of_nat_less_two_power:
huffman@47108:   "of_nat n < (2::'a::linordered_idom) ^ n"
huffman@36795: apply (induct n)
huffman@36795: apply simp
huffman@36795: apply (subgoal_tac "(1::'a) \<le> 2 ^ n")
huffman@36795: apply (drule (1) add_le_less_mono, simp)
huffman@36795: apply simp
paulson@14365: done
paulson@14365: 
huffman@36795: lemma complete_real:
huffman@36795:   fixes S :: "real set"
huffman@36795:   assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
huffman@36795:   shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
huffman@36795: proof -
huffman@36795:   obtain x where x: "x \<in> S" using assms(1) ..
huffman@36795:   obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
paulson@14365: 
huffman@36795:   def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"
huffman@36795:   obtain a where a: "\<not> P a"
huffman@36795:   proof
huffman@36795:     have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)
huffman@36795:     also have "x - 1 < x" by simp
huffman@36795:     finally have "of_int (floor (x - 1)) < x" .
huffman@36795:     hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)
huffman@36795:     then show "\<not> P (of_int (floor (x - 1)))"
huffman@36795:       unfolding P_def of_rat_of_int_eq using x by fast
huffman@36795:   qed
huffman@36795:   obtain b where b: "P b"
huffman@36795:   proof
huffman@36795:     show "P (of_int (ceiling z))"
huffman@36795:     unfolding P_def of_rat_of_int_eq
huffman@36795:     proof
huffman@36795:       fix y assume "y \<in> S"
huffman@36795:       hence "y \<le> z" using z by simp
huffman@36795:       also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)
huffman@36795:       finally show "y \<le> of_int (ceiling z)" .
huffman@36795:     qed
huffman@36795:   qed
paulson@14365: 
huffman@36795:   def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"
huffman@36795:   def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)"
huffman@36795:   def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"
huffman@36795:   def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"
huffman@36795:   def C \<equiv> "\<lambda>n. avg (A n) (B n)"
huffman@36795:   have A_0 [simp]: "A 0 = a" unfolding A_def by simp
huffman@36795:   have B_0 [simp]: "B 0 = b" unfolding B_def by simp
huffman@36795:   have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
huffman@36795:     unfolding A_def B_def C_def bisect_def split_def by simp
huffman@36795:   have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
huffman@36795:     unfolding A_def B_def C_def bisect_def split_def by simp
paulson@14365: 
huffman@36795:   have width: "\<And>n. B n - A n = (b - a) / 2^n"
huffman@36795:     apply (simp add: eq_divide_eq)
huffman@36795:     apply (induct_tac n, simp)
huffman@36795:     apply (simp add: C_def avg_def algebra_simps)
huffman@36795:     done
huffman@36795: 
huffman@36795:   have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
huffman@36795:     apply (simp add: divide_less_eq)
huffman@36795:     apply (subst mult_commute)
huffman@36795:     apply (frule_tac y=y in ex_less_of_nat_mult)
huffman@36795:     apply clarify
huffman@36795:     apply (rule_tac x=n in exI)
huffman@36795:     apply (erule less_trans)
huffman@36795:     apply (rule mult_strict_right_mono)
huffman@36795:     apply (rule le_less_trans [OF _ of_nat_less_two_power])
huffman@36795:     apply simp
huffman@36795:     apply assumption
huffman@36795:     done
paulson@14365: 
huffman@36795:   have PA: "\<And>n. \<not> P (A n)"
huffman@36795:     by (induct_tac n, simp_all add: a)
huffman@36795:   have PB: "\<And>n. P (B n)"
huffman@36795:     by (induct_tac n, simp_all add: b)
huffman@36795:   have ab: "a < b"
huffman@36795:     using a b unfolding P_def
huffman@36795:     apply (clarsimp simp add: not_le)
huffman@36795:     apply (drule (1) bspec)
huffman@36795:     apply (drule (1) less_le_trans)
huffman@36795:     apply (simp add: of_rat_less)
huffman@36795:     done
huffman@36795:   have AB: "\<And>n. A n < B n"
huffman@36795:     by (induct_tac n, simp add: ab, simp add: C_def avg_def)
huffman@36795:   have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
huffman@36795:     apply (auto simp add: le_less [where 'a=nat])
huffman@36795:     apply (erule less_Suc_induct)
huffman@36795:     apply (clarsimp simp add: C_def avg_def)
huffman@36795:     apply (simp add: add_divide_distrib [symmetric])
huffman@36795:     apply (rule AB [THEN less_imp_le])
huffman@36795:     apply simp
huffman@36795:     done
huffman@36795:   have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
huffman@36795:     apply (auto simp add: le_less [where 'a=nat])
huffman@36795:     apply (erule less_Suc_induct)
huffman@36795:     apply (clarsimp simp add: C_def avg_def)
huffman@36795:     apply (simp add: add_divide_distrib [symmetric])
huffman@36795:     apply (rule AB [THEN less_imp_le])
huffman@36795:     apply simp
huffman@36795:     done
huffman@36795:   have cauchy_lemma:
huffman@36795:     "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
huffman@36795:     apply (rule cauchyI)
huffman@36795:     apply (drule twos [where y="b - a"])
huffman@36795:     apply (erule exE)
huffman@36795:     apply (rule_tac x=n in exI, clarify, rename_tac i j)
huffman@36795:     apply (rule_tac y="B n - A n" in le_less_trans) defer
huffman@36795:     apply (simp add: width)
huffman@36795:     apply (drule_tac x=n in spec)
huffman@36795:     apply (frule_tac x=i in spec, drule (1) mp)
huffman@36795:     apply (frule_tac x=j in spec, drule (1) mp)
huffman@36795:     apply (frule A_mono, drule B_mono)
huffman@36795:     apply (frule A_mono, drule B_mono)
huffman@36795:     apply arith
huffman@36795:     done
huffman@36795:   have "cauchy A"
huffman@36795:     apply (rule cauchy_lemma [rule_format])
huffman@36795:     apply (simp add: A_mono)
huffman@36795:     apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
huffman@36795:     done
huffman@36795:   have "cauchy B"
huffman@36795:     apply (rule cauchy_lemma [rule_format])
huffman@36795:     apply (simp add: B_mono)
huffman@36795:     apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
huffman@36795:     done
huffman@36795:   have 1: "\<forall>x\<in>S. x \<le> Real B"
huffman@36795:   proof
huffman@36795:     fix x assume "x \<in> S"
huffman@36795:     then show "x \<le> Real B"
huffman@36795:       using PB [unfolded P_def] `cauchy B`
huffman@36795:       by (simp add: le_RealI)
huffman@36795:   qed
huffman@36795:   have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
huffman@36795:     apply clarify
huffman@36795:     apply (erule contrapos_pp)
huffman@36795:     apply (simp add: not_le)
huffman@36795:     apply (drule less_RealD [OF `cauchy A`], clarify)
huffman@36795:     apply (subgoal_tac "\<not> P (A n)")
huffman@36795:     apply (simp add: P_def not_le, clarify)
huffman@36795:     apply (erule rev_bexI)
huffman@36795:     apply (erule (1) less_trans)
huffman@36795:     apply (simp add: PA)
huffman@36795:     done
huffman@36795:   have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
huffman@36795:   proof (rule vanishesI)
huffman@36795:     fix r :: rat assume "0 < r"
huffman@36795:     then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
huffman@36795:       using twos by fast
huffman@36795:     have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
huffman@36795:     proof (clarify)
huffman@36795:       fix n assume n: "k \<le> n"
huffman@36795:       have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
huffman@36795:         by simp
huffman@36795:       also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
huffman@36795:         using n by (simp add: divide_left_mono mult_pos_pos)
huffman@36795:       also note k
huffman@36795:       finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
huffman@36795:     qed
huffman@36795:     thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
huffman@36795:   qed
huffman@36795:   hence 3: "Real B = Real A"
huffman@36795:     by (simp add: eq_Real `cauchy A` `cauchy B` width)
huffman@36795:   show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
huffman@36795:     using 1 2 3 by (rule_tac x="Real B" in exI, simp)
paulson@14484: qed
paulson@14365: 
huffman@36795: subsection {* Hiding implementation details *}
paulson@14365: 
huffman@47902: hide_const (open) vanishes cauchy positive Real
paulson@14365: 
huffman@36795: declare Real_induct [induct del]
huffman@36795: declare Abs_real_induct [induct del]
huffman@36795: declare Abs_real_cases [cases del]
huffman@36795: 
huffman@47902: lemmas [transfer_rule del] =
huffman@47902:   real.All_transfer real.Ex_transfer real.rel_eq_transfer forall_real_transfer
huffman@47902:   zero_real.transfer one_real.transfer plus_real.transfer uminus_real.transfer
huffman@47902:   times_real.transfer inverse_real.transfer positive.transfer
huffman@47902: 
paulson@14334: subsection{*More Lemmas*}
paulson@14334: 
huffman@36776: text {* BH: These lemmas should not be necessary; they should be
huffman@36776: covered by existing simp rules and simplification procedures. *}
huffman@36776: 
paulson@14334: lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
huffman@36776: by simp (* redundant with mult_cancel_left *)
paulson@14334: 
paulson@14334: lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
huffman@36776: by simp (* redundant with mult_cancel_right *)
paulson@14334: 
paulson@14334: lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
huffman@36776: by simp (* solved by linordered_ring_less_cancel_factor simproc *)
paulson@14334: 
paulson@14334: lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
huffman@36776: by simp (* solved by linordered_ring_le_cancel_factor simproc *)
paulson@14334: 
paulson@14334: lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
huffman@47428: by simp (* solved by linordered_ring_le_cancel_factor simproc *)
paulson@14334: 
paulson@14334: 
haftmann@24198: subsection {* Embedding numbers into the Reals *}
haftmann@24198: 
haftmann@24198: abbreviation
haftmann@24198:   real_of_nat :: "nat \<Rightarrow> real"
haftmann@24198: where
haftmann@24198:   "real_of_nat \<equiv> of_nat"
haftmann@24198: 
haftmann@24198: abbreviation
haftmann@24198:   real_of_int :: "int \<Rightarrow> real"
haftmann@24198: where
haftmann@24198:   "real_of_int \<equiv> of_int"
haftmann@24198: 
haftmann@24198: abbreviation
haftmann@24198:   real_of_rat :: "rat \<Rightarrow> real"
haftmann@24198: where
haftmann@24198:   "real_of_rat \<equiv> of_rat"
haftmann@24198: 
haftmann@24198: consts
haftmann@24198:   (*overloaded constant for injecting other types into "real"*)
haftmann@24198:   real :: "'a => real"
paulson@14365: 
paulson@14378: defs (overloaded)
haftmann@31998:   real_of_nat_def [code_unfold]: "real == real_of_nat"
haftmann@31998:   real_of_int_def [code_unfold]: "real == real_of_int"
paulson@14365: 
wenzelm@40939: declare [[coercion_enabled]]
nipkow@40864: declare [[coercion "real::nat\<Rightarrow>real"]]
nipkow@40864: declare [[coercion "real::int\<Rightarrow>real"]]
nipkow@41022: declare [[coercion "int"]]
nipkow@40864: 
hoelzl@41024: declare [[coercion_map map]]
noschinl@42112: declare [[coercion_map "% f g h x. g (h (f x))"]]
hoelzl@41024: declare [[coercion_map "% f g (x,y) . (f x, g y)"]]
hoelzl@41024: 
avigad@16819: lemma real_eq_of_nat: "real = of_nat"
haftmann@24198:   unfolding real_of_nat_def ..
avigad@16819: 
avigad@16819: lemma real_eq_of_int: "real = of_int"
haftmann@24198:   unfolding real_of_int_def ..
avigad@16819: 
paulson@14365: lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378: by (simp add: real_of_int_def) 
paulson@14365: 
paulson@14365: lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378: by (simp add: real_of_int_def) 
paulson@14334: 
avigad@16819: lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
paulson@14378: by (simp add: real_of_int_def) 
paulson@14365: 
avigad@16819: lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
paulson@14378: by (simp add: real_of_int_def) 
avigad@16819: 
avigad@16819: lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
avigad@16819: by (simp add: real_of_int_def) 
paulson@14365: 
avigad@16819: lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
paulson@14378: by (simp add: real_of_int_def) 
paulson@14334: 
huffman@35344: lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
huffman@35344: by (simp add: real_of_int_def of_int_power)
huffman@35344: 
huffman@35344: lemmas power_real_of_int = real_of_int_power [symmetric]
huffman@35344: 
avigad@16819: lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
avigad@16819:   apply (subst real_eq_of_int)+
avigad@16819:   apply (rule of_int_setsum)
avigad@16819: done
avigad@16819: 
avigad@16819: lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
avigad@16819:     (PROD x:A. real(f x))"
avigad@16819:   apply (subst real_eq_of_int)+
avigad@16819:   apply (rule of_int_setprod)
avigad@16819: done
paulson@14365: 
chaieb@27668: lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
paulson@14378: by (simp add: real_of_int_def) 
paulson@14365: 
chaieb@27668: lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
paulson@14378: by (simp add: real_of_int_def) 
paulson@14365: 
chaieb@27668: lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
paulson@14378: by (simp add: real_of_int_def) 
paulson@14365: 
chaieb@27668: lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378: by (simp add: real_of_int_def) 
paulson@14365: 
chaieb@27668: lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
avigad@16819: by (simp add: real_of_int_def) 
avigad@16819: 
chaieb@27668: lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
avigad@16819: by (simp add: real_of_int_def) 
avigad@16819: 
chaieb@27668: lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)" 
avigad@16819: by (simp add: real_of_int_def)
avigad@16819: 
chaieb@27668: lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
avigad@16819: by (simp add: real_of_int_def)
avigad@16819: 
hoelzl@47597: lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"
hoelzl@47597:   unfolding real_of_one[symmetric] real_of_int_less_iff ..
hoelzl@47597: 
hoelzl@47597: lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"
hoelzl@47597:   unfolding real_of_one[symmetric] real_of_int_le_iff ..
hoelzl@47597: 
hoelzl@47597: lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"
hoelzl@47597:   unfolding real_of_one[symmetric] real_of_int_less_iff ..
hoelzl@47597: 
hoelzl@47597: lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"
hoelzl@47597:   unfolding real_of_one[symmetric] real_of_int_le_iff ..
hoelzl@47597: 
avigad@16888: lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
avigad@16888: by (auto simp add: abs_if)
avigad@16888: 
avigad@16819: lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
avigad@16819:   apply (subgoal_tac "real n + 1 = real (n + 1)")
avigad@16819:   apply (simp del: real_of_int_add)
avigad@16819:   apply auto
avigad@16819: done
avigad@16819: 
avigad@16819: lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
avigad@16819:   apply (subgoal_tac "real m + 1 = real (m + 1)")
avigad@16819:   apply (simp del: real_of_int_add)
avigad@16819:   apply simp
avigad@16819: done
avigad@16819: 
bulwahn@46670: lemma real_of_int_div_aux: "(real (x::int)) / (real d) = 
avigad@16819:     real (x div d) + (real (x mod d)) / (real d)"
avigad@16819: proof -
avigad@16819:   have "x = (x div d) * d + x mod d"
avigad@16819:     by auto
avigad@16819:   then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819:     by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
avigad@16819:   then have "real x / real d = ... / real d"
avigad@16819:     by simp
avigad@16819:   then show ?thesis
bulwahn@46670:     by (auto simp add: add_divide_distrib algebra_simps)
avigad@16819: qed
avigad@16819: 
bulwahn@46670: lemma real_of_int_div: "(d :: int) dvd n ==>
avigad@16819:     real(n div d) = real n / real d"
bulwahn@46670:   apply (subst real_of_int_div_aux)
avigad@16819:   apply simp
nipkow@30042:   apply (simp add: dvd_eq_mod_eq_0)
avigad@16819: done
avigad@16819: 
avigad@16819: lemma real_of_int_div2:
avigad@16819:   "0 <= real (n::int) / real (x) - real (n div x)"
avigad@16819:   apply (case_tac "x = 0")
avigad@16819:   apply simp
avigad@16819:   apply (case_tac "0 < x")
nipkow@29667:   apply (simp add: algebra_simps)
avigad@16819:   apply (subst real_of_int_div_aux)
avigad@16819:   apply simp
avigad@16819:   apply (subst zero_le_divide_iff)
avigad@16819:   apply auto
nipkow@29667:   apply (simp add: algebra_simps)
avigad@16819:   apply (subst real_of_int_div_aux)
avigad@16819:   apply simp
avigad@16819:   apply (subst zero_le_divide_iff)
avigad@16819:   apply auto
avigad@16819: done
avigad@16819: 
avigad@16819: lemma real_of_int_div3:
avigad@16819:   "real (n::int) / real (x) - real (n div x) <= 1"
nipkow@29667:   apply (simp add: algebra_simps)
avigad@16819:   apply (subst real_of_int_div_aux)
bulwahn@46670:   apply (auto simp add: divide_le_eq intro: order_less_imp_le)
avigad@16819: done
avigad@16819: 
avigad@16819: lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
nipkow@27964: by (insert real_of_int_div2 [of n x], simp)
nipkow@27964: 
huffman@35635: lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"
huffman@35635: unfolding real_of_int_def by (rule Ints_of_int)
huffman@35635: 
nipkow@27964: 
paulson@14365: subsection{*Embedding the Naturals into the Reals*}
paulson@14365: 
paulson@14334: lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365: by (simp add: real_of_nat_def)
paulson@14334: 
huffman@30082: lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
huffman@30082: by (simp add: real_of_nat_def)
huffman@30082: 
paulson@14334: lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365: by (simp add: real_of_nat_def)
paulson@14334: 
paulson@14365: lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378: by (simp add: real_of_nat_def)
paulson@14334: 
paulson@14334: (*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334: lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378: by (simp add: real_of_nat_def)
paulson@14334: 
paulson@14334: lemma real_of_nat_less_iff [iff]: 
paulson@14334:      "(real (n::nat) < real m) = (n < m)"
paulson@14365: by (simp add: real_of_nat_def)
paulson@14334: 
paulson@14334: lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378: by (simp add: real_of_nat_def)
paulson@14334: 
paulson@14334: lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
huffman@47489: by (simp add: real_of_nat_def)
paulson@14334: 
paulson@14365: lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378: by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365: 
paulson@14334: lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
huffman@23431: by (simp add: real_of_nat_def of_nat_mult)
paulson@14334: 
huffman@35344: lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
huffman@35344: by (simp add: real_of_nat_def of_nat_power)
huffman@35344: 
huffman@35344: lemmas power_real_of_nat = real_of_nat_power [symmetric]
huffman@35344: 
avigad@16819: lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
avigad@16819:     (SUM x:A. real(f x))"
avigad@16819:   apply (subst real_eq_of_nat)+
avigad@16819:   apply (rule of_nat_setsum)
avigad@16819: done
avigad@16819: 
avigad@16819: lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
avigad@16819:     (PROD x:A. real(f x))"
avigad@16819:   apply (subst real_eq_of_nat)+
avigad@16819:   apply (rule of_nat_setprod)
avigad@16819: done
avigad@16819: 
avigad@16819: lemma real_of_card: "real (card A) = setsum (%x.1) A"
avigad@16819:   apply (subst card_eq_setsum)
avigad@16819:   apply (subst real_of_nat_setsum)
avigad@16819:   apply simp
avigad@16819: done
avigad@16819: 
paulson@14334: lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378: by (simp add: real_of_nat_def)
paulson@14334: 
paulson@14387: lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
paulson@14378: by (simp add: real_of_nat_def)
paulson@14334: 
paulson@14365: lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
huffman@23438: by (simp add: add: real_of_nat_def of_nat_diff)
paulson@14334: 
nipkow@25162: lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
nipkow@25140: by (auto simp: real_of_nat_def)
paulson@14365: 
paulson@14365: lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378: by (simp add: add: real_of_nat_def)
paulson@14334: 
paulson@14365: lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378: by (simp add: add: real_of_nat_def)
paulson@14334: 
avigad@16819: lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
avigad@16819:   apply (subgoal_tac "real n + 1 = real (Suc n)")
avigad@16819:   apply simp
avigad@16819:   apply (auto simp add: real_of_nat_Suc)
avigad@16819: done
avigad@16819: 
avigad@16819: lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
avigad@16819:   apply (subgoal_tac "real m + 1 = real (Suc m)")
avigad@16819:   apply (simp add: less_Suc_eq_le)
avigad@16819:   apply (simp add: real_of_nat_Suc)
avigad@16819: done
avigad@16819: 
bulwahn@46670: lemma real_of_nat_div_aux: "(real (x::nat)) / (real d) = 
avigad@16819:     real (x div d) + (real (x mod d)) / (real d)"
avigad@16819: proof -
avigad@16819:   have "x = (x div d) * d + x mod d"
avigad@16819:     by auto
avigad@16819:   then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819:     by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
avigad@16819:   then have "real x / real d = \<dots> / real d"
avigad@16819:     by simp
avigad@16819:   then show ?thesis
bulwahn@46670:     by (auto simp add: add_divide_distrib algebra_simps)
avigad@16819: qed
avigad@16819: 
bulwahn@46670: lemma real_of_nat_div: "(d :: nat) dvd n ==>
avigad@16819:     real(n div d) = real n / real d"
bulwahn@46670:   by (subst real_of_nat_div_aux)
bulwahn@46670:     (auto simp add: dvd_eq_mod_eq_0 [symmetric])
avigad@16819: 
avigad@16819: lemma real_of_nat_div2:
avigad@16819:   "0 <= real (n::nat) / real (x) - real (n div x)"
nipkow@29667: apply (simp add: algebra_simps)
nipkow@25134: apply (subst real_of_nat_div_aux)
nipkow@25134: apply simp
nipkow@25134: apply (subst zero_le_divide_iff)
nipkow@25134: apply simp
avigad@16819: done
avigad@16819: 
avigad@16819: lemma real_of_nat_div3:
avigad@16819:   "real (n::nat) / real (x) - real (n div x) <= 1"
nipkow@25134: apply(case_tac "x = 0")
nipkow@25134: apply (simp)
nipkow@29667: apply (simp add: algebra_simps)
nipkow@25134: apply (subst real_of_nat_div_aux)
nipkow@25134: apply simp
avigad@16819: done
avigad@16819: 
avigad@16819: lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
nipkow@29667: by (insert real_of_nat_div2 [of n x], simp)
avigad@16819: 
paulson@14426: lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
paulson@14426: by (simp add: real_of_int_def real_of_nat_def)
paulson@14334: 
avigad@16819: lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
avigad@16819:   apply (subgoal_tac "real(int(nat x)) = real(nat x)")
avigad@16819:   apply force
huffman@44822:   apply (simp only: real_of_int_of_nat_eq)
avigad@16819: done
paulson@14387: 
huffman@35635: lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
huffman@35635: unfolding real_of_nat_def by (rule of_nat_in_Nats)
huffman@35635: 
huffman@35635: lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
huffman@35635: unfolding real_of_nat_def by (rule Ints_of_nat)
huffman@35635: 
nipkow@28001: 
nipkow@28001: subsection{* Rationals *}
nipkow@28001: 
nipkow@28091: lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
nipkow@28091: by (simp add: real_eq_of_nat)
nipkow@28091: 
nipkow@28091: 
nipkow@28001: lemma Rats_eq_int_div_int:
nipkow@28091:   "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
nipkow@28001: proof
nipkow@28091:   show "\<rat> \<subseteq> ?S"
nipkow@28001:   proof
nipkow@28091:     fix x::real assume "x : \<rat>"
nipkow@28001:     then obtain r where "x = of_rat r" unfolding Rats_def ..
nipkow@28001:     have "of_rat r : ?S"
nipkow@28001:       by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
nipkow@28001:     thus "x : ?S" using `x = of_rat r` by simp
nipkow@28001:   qed
nipkow@28001: next
nipkow@28091:   show "?S \<subseteq> \<rat>"
nipkow@28001:   proof(auto simp:Rats_def)
nipkow@28001:     fix i j :: int assume "j \<noteq> 0"
nipkow@28001:     hence "real i / real j = of_rat(Fract i j)"
nipkow@28001:       by (simp add:of_rat_rat real_eq_of_int)
nipkow@28001:     thus "real i / real j \<in> range of_rat" by blast
nipkow@28001:   qed
nipkow@28001: qed
nipkow@28001: 
nipkow@28001: lemma Rats_eq_int_div_nat:
nipkow@28091:   "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
nipkow@28001: proof(auto simp:Rats_eq_int_div_int)
nipkow@28001:   fix i j::int assume "j \<noteq> 0"
nipkow@28001:   show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
nipkow@28001:   proof cases
nipkow@28001:     assume "j>0"
nipkow@28001:     hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
nipkow@28001:       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
nipkow@28001:     thus ?thesis by blast
nipkow@28001:   next
nipkow@28001:     assume "~ j>0"
nipkow@28001:     hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0`
nipkow@28001:       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
nipkow@28001:     thus ?thesis by blast
nipkow@28001:   qed
nipkow@28001: next
nipkow@28001:   fix i::int and n::nat assume "0 < n"
nipkow@28001:   hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
nipkow@28001:   thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
nipkow@28001: qed
nipkow@28001: 
nipkow@28001: lemma Rats_abs_nat_div_natE:
nipkow@28001:   assumes "x \<in> \<rat>"
huffman@31706:   obtains m n :: nat
huffman@31706:   where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
nipkow@28001: proof -
nipkow@28001:   from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
nipkow@28001:     by(auto simp add: Rats_eq_int_div_nat)
nipkow@28001:   hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
nipkow@28001:   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
nipkow@28001:   let ?gcd = "gcd m n"
huffman@31706:   from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by simp
nipkow@28001:   let ?k = "m div ?gcd"
nipkow@28001:   let ?l = "n div ?gcd"
nipkow@28001:   let ?gcd' = "gcd ?k ?l"
nipkow@28001:   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
nipkow@28001:     by (rule dvd_mult_div_cancel)
nipkow@28001:   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
nipkow@28001:     by (rule dvd_mult_div_cancel)
nipkow@28001:   from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
nipkow@28001:   moreover
nipkow@28001:   have "\<bar>x\<bar> = real ?k / real ?l"
nipkow@28001:   proof -
nipkow@28001:     from gcd have "real ?k / real ?l =
nipkow@28001:         real (?gcd * ?k) / real (?gcd * ?l)" by simp
nipkow@28001:     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
nipkow@28001:     also from x_rat have "\<dots> = \<bar>x\<bar>" ..
nipkow@28001:     finally show ?thesis ..
nipkow@28001:   qed
nipkow@28001:   moreover
nipkow@28001:   have "?gcd' = 1"
nipkow@28001:   proof -
nipkow@28001:     have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
nipkow@31952:       by (rule gcd_mult_distrib_nat)
nipkow@28001:     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
huffman@31706:     with gcd show ?thesis by auto
nipkow@28001:   qed
nipkow@28001:   ultimately show ?thesis ..
nipkow@28001: qed
nipkow@28001: 
nipkow@28001: 
paulson@14387: subsection{*Numerals and Arithmetic*}
paulson@14387: 
haftmann@46028: lemma [code_abbrev]:
huffman@47108:   "real_of_int (numeral k) = numeral k"
huffman@47108:   "real_of_int (neg_numeral k) = neg_numeral k"
huffman@47108:   by simp_all
paulson@14387: 
paulson@14387: text{*Collapse applications of @{term real} to @{term number_of}*}
huffman@47108: lemma real_numeral [simp]:
huffman@47108:   "real (numeral v :: int) = numeral v"
huffman@47108:   "real (neg_numeral v :: int) = neg_numeral v"
huffman@47108: by (simp_all add: real_of_int_def)
paulson@14387: 
huffman@47108: lemma real_of_nat_numeral [simp]:
huffman@47108:   "real (numeral v :: nat) = numeral v"
huffman@47108: by (simp add: real_of_nat_def)
paulson@14387: 
haftmann@31100: declaration {*
haftmann@31100:   K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
haftmann@31100:     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
haftmann@31100:   #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
haftmann@31100:     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
haftmann@31100:   #> Lin_Arith.add_simps [@{thm real_of_nat_zero}, @{thm real_of_nat_Suc}, @{thm real_of_nat_add},
haftmann@31100:       @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
haftmann@31100:       @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
haftmann@31100:       @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
huffman@47108:       @{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)}]
huffman@36795:   #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
huffman@36795:   #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"}))
haftmann@31100: *}
paulson@14387: 
kleing@19023: 
paulson@14387: subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
paulson@14387: 
paulson@14387: lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
paulson@14387: by arith
paulson@14387: 
huffman@36839: text {* FIXME: redundant with @{text add_eq_0_iff} below *}
paulson@15085: lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
paulson@14387: by auto
paulson@14387: 
paulson@15085: lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
paulson@14387: by auto
paulson@14387: 
paulson@15085: lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
paulson@14387: by auto
paulson@14387: 
paulson@15085: lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
paulson@14387: by auto
paulson@14387: 
paulson@15085: lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
paulson@14387: by auto
paulson@14387: 
huffman@36839: subsection {* Lemmas about powers *}
paulson@14387: 
huffman@36839: text {* FIXME: declare this in Rings.thy or not at all *}
huffman@36839: declare abs_mult_self [simp]
huffman@36839: 
huffman@36839: (* used by Import/HOL/real.imp *)
huffman@36839: lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
huffman@36839: by simp
huffman@36839: 
huffman@36839: lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
huffman@36839: apply (induct "n")
huffman@36839: apply (auto simp add: real_of_nat_Suc)
huffman@36839: apply (subst mult_2)
huffman@36839: apply (erule add_less_le_mono)
huffman@36839: apply (rule two_realpow_ge_one)
huffman@36839: done
huffman@36839: 
huffman@36839: text {* TODO: no longer real-specific; rename and move elsewhere *}
huffman@36839: lemma realpow_Suc_le_self:
huffman@36839:   fixes r :: "'a::linordered_semidom"
huffman@36839:   shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
huffman@36839: by (insert power_decreasing [of 1 "Suc n" r], simp)
huffman@36839: 
huffman@36839: text {* TODO: no longer real-specific; rename and move elsewhere *}
huffman@36839: lemma realpow_minus_mult:
huffman@36839:   fixes x :: "'a::monoid_mult"
huffman@36839:   shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
huffman@36839: by (simp add: power_commutes split add: nat_diff_split)
huffman@36839: 
huffman@36839: text {* FIXME: declare this [simp] for all types, or not at all *}
huffman@36839: lemma real_two_squares_add_zero_iff [simp]:
huffman@36839:   "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
huffman@36839: by (rule sum_squares_eq_zero_iff)
huffman@36839: 
huffman@36839: text {* FIXME: declare this [simp] for all types, or not at all *}
huffman@36839: lemma realpow_two_sum_zero_iff [simp]:
huffman@36839:      "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)"
huffman@36839: by (rule sum_power2_eq_zero_iff)
huffman@36839: 
huffman@36839: lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
huffman@36839: by (rule_tac y = 0 in order_trans, auto)
huffman@36839: 
huffman@36839: lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
huffman@36839: by (auto simp add: power2_eq_square)
huffman@36839: 
huffman@36839: 
hoelzl@47598: lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
hoelzl@47598:   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
hoelzl@47598:   unfolding real_of_nat_le_iff[symmetric] by simp
hoelzl@47598: 
hoelzl@47598: lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
hoelzl@47598:   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
hoelzl@47598:   unfolding real_of_nat_le_iff[symmetric] by simp
hoelzl@47598: 
hoelzl@47598: lemma numeral_power_le_real_of_int_cancel_iff[simp]:
hoelzl@47598:   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
hoelzl@47598:   unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47598: 
hoelzl@47598: lemma real_of_int_le_numeral_power_cancel_iff[simp]:
hoelzl@47598:   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
hoelzl@47598:   unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47598: 
hoelzl@47598: lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
hoelzl@47598:   "(neg_numeral x::real) ^ n \<le> real a \<longleftrightarrow> (neg_numeral x::int) ^ n \<le> a"
hoelzl@47598:   unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47598: 
hoelzl@47598: lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
hoelzl@47598:   "real a \<le> (neg_numeral x::real) ^ n \<longleftrightarrow> a \<le> (neg_numeral x::int) ^ n"
hoelzl@47598:   unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47598: 
huffman@36839: subsection{*Density of the Reals*}
paulson@14387: 
paulson@14387: lemma real_lbound_gt_zero:
paulson@14387:      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
paulson@14387: apply (rule_tac x = " (min d1 d2) /2" in exI)
paulson@14387: apply (simp add: min_def)
paulson@14387: done
paulson@14387: 
paulson@14387: 
haftmann@35050: text{*Similar results are proved in @{text Fields}*}
paulson@14387: lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
paulson@14387:   by auto
paulson@14387: 
paulson@14387: lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
paulson@14387:   by auto
paulson@14387: 
paulson@14387: 
paulson@14387: subsection{*Absolute Value Function for the Reals*}
paulson@14387: 
paulson@14387: lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
paulson@15003: by (simp add: abs_if)
paulson@14387: 
huffman@23289: (* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
paulson@14387: lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
haftmann@35050: by (force simp add: abs_le_iff)
paulson@14387: 
huffman@44344: lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"
paulson@15003: by (simp add: abs_if)
paulson@14387: 
paulson@14387: lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
huffman@22958: by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
paulson@14387: 
huffman@44344: lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"
webertj@20217: by simp
paulson@14387:  
paulson@14387: lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
webertj@20217: by simp
paulson@14387: 
berghofe@24534: 
haftmann@27544: subsection {* Implementation of rational real numbers *}
berghofe@24534: 
huffman@47108: text {* Formal constructor *}
huffman@47108: 
haftmann@27544: definition Ratreal :: "rat \<Rightarrow> real" where
huffman@47108:   [code_abbrev, simp]: "Ratreal = of_rat"
berghofe@24534: 
haftmann@24623: code_datatype Ratreal
berghofe@24534: 
huffman@47108: 
huffman@47108: text {* Numerals *}
huffman@47108: 
huffman@47108: lemma [code_abbrev]:
huffman@47108:   "(of_rat (of_int a) :: real) = of_int a"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma [code_abbrev]:
huffman@47108:   "(of_rat 0 :: real) = 0"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma [code_abbrev]:
huffman@47108:   "(of_rat 1 :: real) = 1"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma [code_abbrev]:
huffman@47108:   "(of_rat (numeral k) :: real) = numeral k"
huffman@47108:   by simp
berghofe@24534: 
huffman@47108: lemma [code_abbrev]:
huffman@47108:   "(of_rat (neg_numeral k) :: real) = neg_numeral k"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma [code_post]:
huffman@47108:   "(of_rat (0 / r)  :: real) = 0"
huffman@47108:   "(of_rat (r / 0)  :: real) = 0"
huffman@47108:   "(of_rat (1 / 1)  :: real) = 1"
huffman@47108:   "(of_rat (numeral k / 1) :: real) = numeral k"
huffman@47108:   "(of_rat (neg_numeral k / 1) :: real) = neg_numeral k"
huffman@47108:   "(of_rat (1 / numeral k) :: real) = 1 / numeral k"
huffman@47108:   "(of_rat (1 / neg_numeral k) :: real) = 1 / neg_numeral k"
huffman@47108:   "(of_rat (numeral k / numeral l)  :: real) = numeral k / numeral l"
huffman@47108:   "(of_rat (numeral k / neg_numeral l)  :: real) = numeral k / neg_numeral l"
huffman@47108:   "(of_rat (neg_numeral k / numeral l)  :: real) = neg_numeral k / numeral l"
huffman@47108:   "(of_rat (neg_numeral k / neg_numeral l)  :: real) = neg_numeral k / neg_numeral l"
huffman@47108:   by (simp_all add: of_rat_divide)
huffman@47108: 
huffman@47108: 
huffman@47108: text {* Operations *}
huffman@47108: 
huffman@47108: lemma zero_real_code [code]:
haftmann@27544:   "0 = Ratreal 0"
haftmann@27544: by simp
berghofe@24534: 
huffman@47108: lemma one_real_code [code]:
haftmann@27544:   "1 = Ratreal 1"
haftmann@27544: by simp
haftmann@27544: 
haftmann@38857: instantiation real :: equal
haftmann@26513: begin
haftmann@26513: 
haftmann@38857: definition "HOL.equal (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
haftmann@26513: 
haftmann@38857: instance proof
haftmann@38857: qed (simp add: equal_real_def)
berghofe@24534: 
haftmann@38857: lemma real_equal_code [code]:
haftmann@38857:   "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
haftmann@38857:   by (simp add: equal_real_def equal)
haftmann@26513: 
haftmann@38857: lemma [code nbe]:
haftmann@38857:   "HOL.equal (x::real) x \<longleftrightarrow> True"
haftmann@38857:   by (rule equal_refl)
haftmann@28351: 
haftmann@26513: end
berghofe@24534: 
haftmann@27544: lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
haftmann@27652:   by (simp add: of_rat_less_eq)
berghofe@24534: 
haftmann@27544: lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
haftmann@27652:   by (simp add: of_rat_less)
berghofe@24534: 
haftmann@27544: lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
haftmann@27544:   by (simp add: of_rat_add)
berghofe@24534: 
haftmann@27544: lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
haftmann@27544:   by (simp add: of_rat_mult)
haftmann@27544: 
haftmann@27544: lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
haftmann@27544:   by (simp add: of_rat_minus)
berghofe@24534: 
haftmann@27544: lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
haftmann@27544:   by (simp add: of_rat_diff)
berghofe@24534: 
haftmann@27544: lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
haftmann@27544:   by (simp add: of_rat_inverse)
haftmann@27544:  
haftmann@27544: lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
haftmann@27544:   by (simp add: of_rat_divide)
berghofe@24534: 
bulwahn@43733: lemma real_floor_code [code]: "floor (Ratreal x) = floor x"
bulwahn@43733:   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)
bulwahn@43733: 
huffman@47108: 
huffman@47108: text {* Quickcheck *}
huffman@47108: 
haftmann@31203: definition (in term_syntax)
haftmann@32657:   valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
haftmann@32657:   [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
haftmann@31203: 
haftmann@37751: notation fcomp (infixl "\<circ>>" 60)
haftmann@37751: notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203: 
haftmann@31203: instantiation real :: random
haftmann@31203: begin
haftmann@31203: 
haftmann@31203: definition
haftmann@37751:   "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
haftmann@31203: 
haftmann@31203: instance ..
haftmann@31203: 
haftmann@31203: end
haftmann@31203: 
haftmann@37751: no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751: no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203: 
bulwahn@41920: instantiation real :: exhaustive
bulwahn@41231: begin
bulwahn@41231: 
bulwahn@41231: definition
bulwahn@45818:   "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"
bulwahn@42311: 
bulwahn@42311: instance ..
bulwahn@42311: 
bulwahn@42311: end
bulwahn@42311: 
bulwahn@42311: instantiation real :: full_exhaustive
bulwahn@42311: begin
bulwahn@42311: 
bulwahn@42311: definition
bulwahn@45818:   "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"
bulwahn@41231: 
bulwahn@41231: instance ..
bulwahn@41231: 
bulwahn@41231: end
bulwahn@41231: 
bulwahn@43887: instantiation real :: narrowing
bulwahn@43887: begin
bulwahn@43887: 
bulwahn@43887: definition
bulwahn@43887:   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"
bulwahn@43887: 
bulwahn@43887: instance ..
bulwahn@43887: 
bulwahn@43887: end
bulwahn@43887: 
bulwahn@43887: 
bulwahn@45184: subsection {* Setup for Nitpick *}
berghofe@24534: 
blanchet@38287: declaration {*
blanchet@38287:   Nitpick_HOL.register_frac_type @{type_name real}
wenzelm@33209:    [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
wenzelm@33209:     (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
wenzelm@33209:     (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
wenzelm@33209:     (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
wenzelm@33209:     (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
wenzelm@33209:     (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
blanchet@45859:     (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),
wenzelm@33209:     (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
blanchet@33197: *}
blanchet@33197: 
huffman@47108: lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real
blanchet@37397:     ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real
blanchet@33197:     times_real_inst.times_real uminus_real_inst.uminus_real
blanchet@33197:     zero_real_inst.zero_real
blanchet@33197: 
paulson@5588: end




