(* Title: HOL/Real.thy Author: Jacques D. Fleuriot, University of Edinburgh, 1998 Author: Larry Paulson, University of Cambridge Author: Jeremy Avigad, Carnegie Mellon University Author: Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 Construction of Cauchy Reals by Brian Huffman, 2010 *) section ‹Development of the Reals using Cauchy Sequences› theory Real imports Rat begin text ‹ This theory contains a formalization of the real numbers as equivalence classes of Cauchy sequences of rationals. See the AFP entry @{text Dedekind_Real} for an alternative construction using Dedekind cuts. › subsection ‹Preliminary lemmas› text‹Useful in convergence arguments› lemma inverse_of_nat_le: fixes n::nat shows "⟦n ≤ m; n≠0⟧ ⟹ 1 / of_nat m ≤ (1::'a::linordered_field) / of_nat n" by (simp add: frac_le) lemma add_diff_add: "(a + c) - (b + d) = (a - b) + (c - d)" for a b c d :: "'a::ab_group_add" by simp lemma minus_diff_minus: "- a - - b = - (a - b)" for a b :: "'a::ab_group_add" by simp lemma mult_diff_mult: "(x * y - a * b) = x * (y - b) + (x - a) * b" for x y a b :: "'a::ring" by (simp add: algebra_simps) lemma inverse_diff_inverse: fixes a b :: "'a::division_ring" assumes "a ≠ 0" and "b ≠ 0" shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)" using assms by (simp add: algebra_simps) lemma obtain_pos_sum: fixes r :: rat assumes r: "0 < r" obtains s t where "0 < s" and "0 < t" and "r = s + t" proof from r show "0 < r/2" by simp from r show "0 < r/2" by simp show "r = r/2 + r/2" by simp qed subsection ‹Sequences that converge to zero› definition vanishes :: "(nat ⇒ rat) ⇒ bool" where "vanishes X ⟷ (∀r>0. ∃k. ∀n≥k. ¦X n¦ < r)" lemma vanishesI: "(⋀r. 0 < r ⟹ ∃k. ∀n≥k. ¦X n¦ < r) ⟹ vanishes X" unfolding vanishes_def by simp lemma vanishesD: "vanishes X ⟹ 0 < r ⟹ ∃k. ∀n≥k. ¦X n¦ < r" unfolding vanishes_def by simp lemma vanishes_const [simp]: "vanishes (λn. c) ⟷ c = 0" proof (cases "c = 0") case True then show ?thesis by (simp add: vanishesI) next case False then show ?thesis unfolding vanishes_def using zero_less_abs_iff by blast qed lemma vanishes_minus: "vanishes X ⟹ vanishes (λn. - X n)" unfolding vanishes_def by simp lemma vanishes_add: assumes X: "vanishes X" and Y: "vanishes Y" shows "vanishes (λn. X n + Y n)" proof (rule vanishesI) fix r :: rat assume "0 < r" then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" by (rule obtain_pos_sum) obtain i where i: "∀n≥i. ¦X n¦ < s" using vanishesD [OF X s] .. obtain j where j: "∀n≥j. ¦Y n¦ < t" using vanishesD [OF Y t] .. have "∀n≥max i j. ¦X n + Y n¦ < r" proof clarsimp fix n assume n: "i ≤ n" "j ≤ n" have "¦X n + Y n¦ ≤ ¦X n¦ + ¦Y n¦" by (rule abs_triangle_ineq) also have "… < s + t" by (simp add: add_strict_mono i j n) finally show "¦X n + Y n¦ < r" by (simp only: r) qed then show "∃k. ∀n≥k. ¦X n + Y n¦ < r" .. qed lemma vanishes_diff: assumes "vanishes X" "vanishes Y" shows "vanishes (λn. X n - Y n)" unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus assms) lemma vanishes_mult_bounded: assumes X: "∃a>0. ∀n. ¦X n¦ < a" assumes Y: "vanishes (λn. Y n)" shows "vanishes (λn. X n * Y n)" proof (rule vanishesI) fix r :: rat assume r: "0 < r" obtain a where a: "0 < a" "∀n. ¦X n¦ < a" using X by blast obtain b where b: "0 < b" "r = a * b" proof show "0 < r / a" using r a by simp show "r = a * (r / a)" using a by simp qed obtain k where k: "∀n≥k. ¦Y n¦ < b" using vanishesD [OF Y b(1)] .. have "∀n≥k. ¦X n * Y n¦ < r" by (simp add: b(2) abs_mult mult_strict_mono' a k) then show "∃k. ∀n≥k. ¦X n * Y n¦ < r" .. qed subsection ‹Cauchy sequences› definition cauchy :: "(nat ⇒ rat) ⇒ bool" where "cauchy X ⟷ (∀r>0. ∃k. ∀m≥k. ∀n≥k. ¦X m - X n¦ < r)" lemma cauchyI: "(⋀r. 0 < r ⟹ ∃k. ∀m≥k. ∀n≥k. ¦X m - X n¦ < r) ⟹ cauchy X" unfolding cauchy_def by simp lemma cauchyD: "cauchy X ⟹ 0 < r ⟹ ∃k. ∀m≥k. ∀n≥k. ¦X m - X n¦ < r" unfolding cauchy_def by simp lemma cauchy_const [simp]: "cauchy (λn. x)" unfolding cauchy_def by simp lemma cauchy_add [simp]: assumes X: "cauchy X" and Y: "cauchy Y" shows "cauchy (λn. X n + Y n)" proof (rule cauchyI) fix r :: rat assume "0 < r" then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" by (rule obtain_pos_sum) obtain i where i: "∀m≥i. ∀n≥i. ¦X m - X n¦ < s" using cauchyD [OF X s] .. obtain j where j: "∀m≥j. ∀n≥j. ¦Y m - Y n¦ < t" using cauchyD [OF Y t] .. have "∀m≥max i j. ∀n≥max i j. ¦(X m + Y m) - (X n + Y n)¦ < r" proof clarsimp fix m n assume *: "i ≤ m" "j ≤ m" "i ≤ n" "j ≤ n" have "¦(X m + Y m) - (X n + Y n)¦ ≤ ¦X m - X n¦ + ¦Y m - Y n¦" unfolding add_diff_add by (rule abs_triangle_ineq) also have "… < s + t" by (rule add_strict_mono) (simp_all add: i j *) finally show "¦(X m + Y m) - (X n + Y n)¦ < r" by (simp only: r) qed then show "∃k. ∀m≥k. ∀n≥k. ¦(X m + Y m) - (X n + Y n)¦ < r" .. qed lemma cauchy_minus [simp]: assumes X: "cauchy X" shows "cauchy (λn. - X n)" using assms unfolding cauchy_def unfolding minus_diff_minus abs_minus_cancel . lemma cauchy_diff [simp]: assumes "cauchy X" "cauchy Y" shows "cauchy (λn. X n - Y n)" using assms unfolding diff_conv_add_uminus by (simp del: add_uminus_conv_diff) lemma cauchy_imp_bounded: assumes "cauchy X" shows "∃b>0. ∀n. ¦X n¦ < b" proof - obtain k where k: "∀m≥k. ∀n≥k. ¦X m - X n¦ < 1" using cauchyD [OF assms zero_less_one] .. show "∃b>0. ∀n. ¦X n¦ < b" proof (intro exI conjI allI) have "0 ≤ ¦X 0¦" by simp also have "¦X 0¦ ≤ Max (abs ` X ` {..k})" by simp finally have "0 ≤ Max (abs ` X ` {..k})" . then show "0 < Max (abs ` X ` {..k}) + 1" by simp next fix n :: nat show "¦X n¦ < Max (abs ` X ` {..k}) + 1" proof (rule linorder_le_cases) assume "n ≤ k" then have "¦X n¦ ≤ Max (abs ` X ` {..k})" by simp then show "¦X n¦ < Max (abs ` X ` {..k}) + 1" by simp next assume "k ≤ n" have "¦X n¦ = ¦X k + (X n - X k)¦" by simp also have "¦X k + (X n - X k)¦ ≤ ¦X k¦ + ¦X n - X k¦" by (rule abs_triangle_ineq) also have "… < Max (abs ` X ` {..k}) + 1" by (rule add_le_less_mono) (simp_all add: k ‹k ≤ n›) finally show "¦X n¦ < Max (abs ` X ` {..k}) + 1" . qed qed qed lemma cauchy_mult [simp]: assumes X: "cauchy X" and Y: "cauchy Y" shows "cauchy (λn. X n * Y n)" proof (rule cauchyI) fix r :: rat assume "0 < r" then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v" by (rule obtain_pos_sum) obtain a where a: "0 < a" "∀n. ¦X n¦ < a" using cauchy_imp_bounded [OF X] by blast obtain b where b: "0 < b" "∀n. ¦Y n¦ < b" using cauchy_imp_bounded [OF Y] by blast obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b" proof show "0 < v/b" using v b(1) by simp show "0 < u/a" using u a(1) by simp show "r = a * (u/a) + (v/b) * b" using a(1) b(1) ‹r = u + v› by simp qed obtain i where i: "∀m≥i. ∀n≥i. ¦X m - X n¦ < s" using cauchyD [OF X s] .. obtain j where j: "∀m≥j. ∀n≥j. ¦Y m - Y n¦ < t" using cauchyD [OF Y t] .. have "∀m≥max i j. ∀n≥max i j. ¦X m * Y m - X n * Y n¦ < r" proof clarsimp fix m n assume *: "i ≤ m" "j ≤ m" "i ≤ n" "j ≤ n" have "¦X m * Y m - X n * Y n¦ = ¦X m * (Y m - Y n) + (X m - X n) * Y n¦" unfolding mult_diff_mult .. also have "… ≤ ¦X m * (Y m - Y n)¦ + ¦(X m - X n) * Y n¦" by (rule abs_triangle_ineq) also have "… = ¦X m¦ * ¦Y m - Y n¦ + ¦X m - X n¦ * ¦Y n¦" unfolding abs_mult .. also have "… < a * t + s * b" by (simp_all add: add_strict_mono mult_strict_mono' a b i j *) finally show "¦X m * Y m - X n * Y n¦ < r" by (simp only: r) qed then show "∃k. ∀m≥k. ∀n≥k. ¦X m * Y m - X n * Y n¦ < r" .. qed lemma cauchy_not_vanishes_cases: assumes X: "cauchy X" assumes nz: "¬ vanishes X" shows "∃b>0. ∃k. (∀n≥k. b < - X n) ∨ (∀n≥k. b < X n)" proof - obtain r where "0 < r" and r: "∀k. ∃n≥k. r ≤ ¦X n¦" using nz unfolding vanishes_def by (auto simp add: not_less) obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t" using ‹0 < r› by (rule obtain_pos_sum) obtain i where i: "∀m≥i. ∀n≥i. ¦X m - X n¦ < s" using cauchyD [OF X s] .. obtain k where "i ≤ k" and "r ≤ ¦X k¦" using r by blast have k: "∀n≥k. ¦X n - X k¦ < s" using i ‹i ≤ k› by auto have "X k ≤ - r ∨ r ≤ X k" using ‹r ≤ ¦X k¦› by auto then have "(∀n≥k. t < - X n) ∨ (∀n≥k. t < X n)" unfolding ‹r = s + t› using k by auto then have "∃k. (∀n≥k. t < - X n) ∨ (∀n≥k. t < X n)" .. then show "∃t>0. ∃k. (∀n≥k. t < - X n) ∨ (∀n≥k. t < X n)" using t by auto qed lemma cauchy_not_vanishes: assumes X: "cauchy X" and nz: "¬ vanishes X" shows "∃b>0. ∃k. ∀n≥k. b < ¦X n¦" using cauchy_not_vanishes_cases [OF assms] by (elim ex_forward conj_forward asm_rl) auto lemma cauchy_inverse [simp]: assumes X: "cauchy X" and nz: "¬ vanishes X" shows "cauchy (λn. inverse (X n))" proof (rule cauchyI) fix r :: rat assume "0 < r" obtain b i where b: "0 < b" and i: "∀n≥i. b < ¦X n¦" using cauchy_not_vanishes [OF X nz] by blast from b i have nz: "∀n≥i. X n ≠ 0" by auto obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b" proof show "0 < b * r * b" by (simp add: ‹0 < r› b) show "r = inverse b * (b * r * b) * inverse b" using b by simp qed obtain j where j: "∀m≥j. ∀n≥j. ¦X m - X n¦ < s" using cauchyD [OF X s] .. have "∀m≥max i j. ∀n≥max i j. ¦inverse (X m) - inverse (X n)¦ < r" proof clarsimp fix m n assume *: "i ≤ m" "j ≤ m" "i ≤ n" "j ≤ n" have "¦inverse (X m) - inverse (X n)¦ = inverse ¦X m¦ * ¦X m - X n¦ * inverse ¦X n¦" by (simp add: inverse_diff_inverse nz * abs_mult) also have "… < inverse b * s * inverse b" by (simp add: mult_strict_mono less_imp_inverse_less i j b * s) finally show "¦inverse (X m) - inverse (X n)¦ < r" by (simp only: r) qed then show "∃k. ∀m≥k. ∀n≥k. ¦inverse (X m) - inverse (X n)¦ < r" .. qed lemma vanishes_diff_inverse: assumes X: "cauchy X" "¬ vanishes X" and Y: "cauchy Y" "¬ vanishes Y" and XY: "vanishes (λn. X n - Y n)" shows "vanishes (λn. inverse (X n) - inverse (Y n))" proof (rule vanishesI) fix r :: rat assume r: "0 < r" obtain a i where a: "0 < a" and i: "∀n≥i. a < ¦X n¦" using cauchy_not_vanishes [OF X] by blast obtain b j where b: "0 < b" and j: "∀n≥j. b < ¦Y n¦" using cauchy_not_vanishes [OF Y] by blast obtain s where s: "0 < s" and "inverse a * s * inverse b = r" proof show "0 < a * r * b" using a r b by simp show "inverse a * (a * r * b) * inverse b = r" using a r b by simp qed obtain k where k: "∀n≥k. ¦X n - Y n¦ < s" using vanishesD [OF XY s] .. have "∀n≥max (max i j) k. ¦inverse (X n) - inverse (Y n)¦ < r" proof clarsimp fix n assume n: "i ≤ n" "j ≤ n" "k ≤ n" with i j a b have "X n ≠ 0" and "Y n ≠ 0" by auto then have "¦inverse (X n) - inverse (Y n)¦ = inverse ¦X n¦ * ¦X n - Y n¦ * inverse ¦Y n¦" by (simp add: inverse_diff_inverse abs_mult) also have "… < inverse a * s * inverse b" by (intro mult_strict_mono' less_imp_inverse_less) (simp_all add: a b i j k n) also note ‹inverse a * s * inverse b = r› finally show "¦inverse (X n) - inverse (Y n)¦ < r" . qed then show "∃k. ∀n≥k. ¦inverse (X n) - inverse (Y n)¦ < r" .. qed subsection ‹Equivalence relation on Cauchy sequences› definition realrel :: "(nat ⇒ rat) ⇒ (nat ⇒ rat) ⇒ bool" where "realrel = (λX Y. cauchy X ∧ cauchy Y ∧ vanishes (λn. X n - Y n))" lemma realrelI [intro?]: "cauchy X ⟹ cauchy Y ⟹ vanishes (λn. X n - Y n) ⟹ realrel X Y" by (simp add: realrel_def) lemma realrel_refl: "cauchy X ⟹ realrel X X" by (simp add: realrel_def) lemma symp_realrel: "symp realrel" by (simp add: abs_minus_commute realrel_def symp_def vanishes_def) lemma transp_realrel: "transp realrel" unfolding realrel_def by (rule transpI) (force simp add: dest: vanishes_add) lemma part_equivp_realrel: "part_equivp realrel" by (blast intro: part_equivpI symp_realrel transp_realrel realrel_refl cauchy_const) subsection ‹The field of real numbers› quotient_type real = "nat ⇒ rat" / partial: realrel morphisms rep_real Real by (rule part_equivp_realrel) lemma cr_real_eq: "pcr_real = (λx y. cauchy x ∧ Real x = y)" unfolding real.pcr_cr_eq cr_real_def realrel_def by auto lemma Real_induct [induct type: real]: (* TODO: generate automatically *) assumes "⋀X. cauchy X ⟹ P (Real X)" shows "P x" proof (induct x) case (1 X) then have "cauchy X" by (simp add: realrel_def) then show "P (Real X)" by (rule assms) qed lemma eq_Real: "cauchy X ⟹ cauchy Y ⟹ Real X = Real Y ⟷ vanishes (λn. X n - Y n)" using real.rel_eq_transfer unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy" by (simp add: real.domain_eq realrel_def) instantiation real :: field begin lift_definition zero_real :: "real" is "λn. 0" by (simp add: realrel_refl) lift_definition one_real :: "real" is "λn. 1" by (simp add: realrel_refl) lift_definition plus_real :: "real ⇒ real ⇒ real" is "λX Y n. X n + Y n" unfolding realrel_def add_diff_add by (simp only: cauchy_add vanishes_add simp_thms) lift_definition uminus_real :: "real ⇒ real" is "λX n. - X n" unfolding realrel_def minus_diff_minus by (simp only: cauchy_minus vanishes_minus simp_thms) lift_definition times_real :: "real ⇒ real ⇒ real" is "λX Y n. X n * Y n" proof - fix f1 f2 f3 f4 have "⟦cauchy f1; cauchy f4; vanishes (λn. f1 n - f2 n); vanishes (λn. f3 n - f4 n)⟧ ⟹ vanishes (λn. f1 n * (f3 n - f4 n) + f4 n * (f1 n - f2 n))" by (simp add: vanishes_add vanishes_mult_bounded cauchy_imp_bounded) then show "⟦realrel f1 f2; realrel f3 f4⟧ ⟹ realrel (λn. f1 n * f3 n) (λn. f2 n * f4 n)" by (simp add: mult.commute realrel_def mult_diff_mult) qed lift_definition inverse_real :: "real ⇒ real" is "λX. if vanishes X then (λn. 0) else (λn. inverse (X n))" proof - fix X Y assume "realrel X Y" then have X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (λn. X n - Y n)" by (simp_all add: realrel_def) have "vanishes X ⟷ vanishes Y" proof assume "vanishes X" from vanishes_diff [OF this XY] show "vanishes Y" by simp next assume "vanishes Y" from vanishes_add [OF this XY] show "vanishes X" by simp qed then show "?thesis X Y" by (simp add: vanishes_diff_inverse X Y XY realrel_def) qed definition "x - y = x + - y" for x y :: real definition "x div y = x * inverse y" for x y :: real lemma add_Real: "cauchy X ⟹ cauchy Y ⟹ Real X + Real Y = Real (λn. X n + Y n)" using plus_real.transfer by (simp add: cr_real_eq rel_fun_def) lemma minus_Real: "cauchy X ⟹ - Real X = Real (λn. - X n)" using uminus_real.transfer by (simp add: cr_real_eq rel_fun_def) lemma diff_Real: "cauchy X ⟹ cauchy Y ⟹ Real X - Real Y = Real (λn. X n - Y n)" by (simp add: minus_Real add_Real minus_real_def) lemma mult_Real: "cauchy X ⟹ cauchy Y ⟹ Real X * Real Y = Real (λn. X n * Y n)" using times_real.transfer by (simp add: cr_real_eq rel_fun_def) lemma inverse_Real: "cauchy X ⟹ inverse (Real X) = (if vanishes X then 0 else Real (λn. inverse (X n)))" using inverse_real.transfer zero_real.transfer unfolding cr_real_eq rel_fun_def by (simp split: if_split_asm, metis) instance proof fix a b c :: real show "a + b = b + a" by transfer (simp add: ac_simps realrel_def) show "(a + b) + c = a + (b + c)" by transfer (simp add: ac_simps realrel_def) show "0 + a = a" by transfer (simp add: realrel_def) show "- a + a = 0" by transfer (simp add: realrel_def) show "a - b = a + - b" by (rule minus_real_def) show "(a * b) * c = a * (b * c)" by transfer (simp add: ac_simps realrel_def) show "a * b = b * a" by transfer (simp add: ac_simps realrel_def) show "1 * a = a" by transfer (simp add: ac_simps realrel_def) show "(a + b) * c = a * c + b * c" by transfer (simp add: distrib_right realrel_def) show "(0::real) ≠ (1::real)" by transfer (simp add: realrel_def) have "vanishes (λn. inverse (X n) * X n - 1)" if X: "cauchy X" "¬ vanishes X" for X proof (rule vanishesI) fix r::rat assume "0 < r" obtain b k where "b>0" "∀n≥k. b < ¦X n¦" using X cauchy_not_vanishes by blast then show "∃k. ∀n≥k. ¦inverse (X n) * X n - 1¦ < r" using ‹0 < r› by force qed then show "a ≠ 0 ⟹ inverse a * a = 1" by transfer (simp add: realrel_def) show "a div b = a * inverse b" by (rule divide_real_def) show "inverse (0::real) = 0" by transfer (simp add: realrel_def) qed end subsection ‹Positive reals› lift_definition positive :: "real ⇒ bool" is "λX. ∃r>0. ∃k. ∀n≥k. r < X n" proof - have 1: "∃r>0. ∃k. ∀n≥k. r < Y n" if *: "realrel X Y" and **: "∃r>0. ∃k. ∀n≥k. r < X n" for X Y proof - from * have XY: "vanishes (λn. X n - Y n)" by (simp_all add: realrel_def) from ** obtain r i where "0 < r" and i: "∀n≥i. r < X n" by blast obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" using ‹0 < r› by (rule obtain_pos_sum) obtain j where j: "∀n≥j. ¦X n - Y n¦ < s" using vanishesD [OF XY s] .. have "∀n≥max i j. t < Y n" proof clarsimp fix n assume n: "i ≤ n" "j ≤ n" have "¦X n - Y n¦ < s" and "r < X n" using i j n by simp_all then show "t < Y n" by (simp add: r) qed then show ?thesis using t by blast qed fix X Y assume "realrel X Y" then have "realrel X Y" and "realrel Y X" using symp_realrel by (auto simp: symp_def) then show "?thesis X Y" by (safe elim!: 1) qed lemma positive_Real: "cauchy X ⟹ positive (Real X) ⟷ (∃r>0. ∃k. ∀n≥k. r < X n)" using positive.transfer by (simp add: cr_real_eq rel_fun_def) lemma positive_zero: "¬ positive 0" by transfer auto lemma positive_add: assumes "positive x" "positive y" shows "positive (x + y)" proof - have *: "⟦∀n≥i. a < x n; ∀n≥j. b < y n; 0 < a; 0 < b; n ≥ max i j⟧ ⟹ a+b < x n + y n" for x y and a b::rat and i j n::nat by (simp add: add_strict_mono) show ?thesis using assms by transfer (blast intro: * pos_add_strict) qed lemma positive_mult: assumes "positive x" "positive y" shows "positive (x * y)" proof - have *: "⟦∀n≥i. a < x n; ∀n≥j. b < y n; 0 < a; 0 < b; n ≥ max i j⟧ ⟹ a*b < x n * y n" for x y and a b::rat and i j n::nat by (simp add: mult_strict_mono') show ?thesis using assms by transfer (blast intro: * mult_pos_pos) qed lemma positive_minus: "¬ positive x ⟹ x ≠ 0 ⟹ positive (- x)" apply transfer apply (simp add: realrel_def) apply (blast dest: cauchy_not_vanishes_cases) done instantiation real :: linordered_field begin definition "x < y ⟷ positive (y - x)" definition "x ≤ y ⟷ x < y ∨ x = y" for x y :: real definition "¦a¦ = (if a < 0 then - a else a)" for a :: real definition "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)" for a :: real instance proof fix a b c :: real show "¦a¦ = (if a < 0 then - a else a)" by (rule abs_real_def) show "a < b ⟷ a ≤ b ∧ ¬ b ≤ a" "a ≤ b ⟹ b ≤ c ⟹ a ≤ c" "a ≤ a" "a ≤ b ⟹ b ≤ a ⟹ a = b" "a ≤ b ⟹ c + a ≤ c + b" unfolding less_eq_real_def less_real_def by (force simp add: positive_zero dest: positive_add)+ show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)" by (rule sgn_real_def) show "a ≤ b ∨ b ≤ a" by (auto dest!: positive_minus simp: less_eq_real_def less_real_def) show "a < b ⟹ 0 < c ⟹ c * a < c * b" unfolding less_real_def by (force simp add: algebra_simps dest: positive_mult) qed end instantiation real :: distrib_lattice begin definition "(inf :: real ⇒ real ⇒ real) = min" definition "(sup :: real ⇒ real ⇒ real) = max" instance by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2) end lemma of_nat_Real: "of_nat x = Real (λn. of_nat x)" by (induct x) (simp_all add: zero_real_def one_real_def add_Real) lemma of_int_Real: "of_int x = Real (λn. of_int x)" by (cases x rule: int_diff_cases) (simp add: of_nat_Real diff_Real) lemma of_rat_Real: "of_rat x = Real (λn. x)" proof (induct x) case (Fract a b) then show ?case apply (simp add: Fract_of_int_quotient of_rat_divide) apply (simp add: of_int_Real divide_inverse inverse_Real mult_Real) done qed instance real :: archimedean_field proof show "∃z. x ≤ of_int z" for x :: real proof (induct x) case (1 X) then obtain b where "0 < b" and b: "⋀n. ¦X n¦ < b" by (blast dest: cauchy_imp_bounded) then have "Real X < of_int (⌈b⌉ + 1)" using 1 apply (simp add: of_int_Real less_real_def diff_Real positive_Real) apply (rule_tac x=1 in exI) apply (simp add: algebra_simps) by (metis abs_ge_self le_less_trans le_of_int_ceiling less_le) then show ?case using less_eq_real_def by blast qed qed instantiation real :: floor_ceiling begin definition [code del]: "⌊x::real⌋ = (THE z. of_int z ≤ x ∧ x < of_int (z + 1))" instance proof show "of_int ⌊x⌋ ≤ x ∧ x < of_int (⌊x⌋ + 1)" for x :: real unfolding floor_real_def using floor_exists1 by (rule theI') qed end subsection ‹Completeness› lemma not_positive_Real: assumes "cauchy X" shows "¬ positive (Real X) ⟷ (∀r>0. ∃k. ∀n≥k. X n ≤ r)" (is "?lhs = ?rhs") unfolding positive_Real [OF assms] proof (intro iffI allI notI impI) show "∃k. ∀n≥k. X n ≤ r" if r: "¬ (∃r>0. ∃k. ∀n≥k. r < X n)" and "0 < r" for r proof - obtain s t where "s > 0" "t > 0" "r = s+t" using ‹r > 0› obtain_pos_sum by blast obtain k where k: "⋀m n. ⟦m≥k; n≥k⟧ ⟹ ¦X m - X n¦ < t" using cauchyD [OF assms ‹t > 0›] by blast obtain n where "n ≥ k" "X n ≤ s" by (meson r ‹0 < s› not_less) then have "X l ≤ r" if "l ≥ n" for l using k [OF ‹n ≥ k›, of l] that ‹r = s+t› by linarith then show ?thesis by blast qed qed (meson le_cases not_le) lemma le_Real: assumes "cauchy X" "cauchy Y" shows "Real X ≤ Real Y = (∀r>0. ∃k. ∀n≥k. X n ≤ Y n + r)" unfolding not_less [symmetric, where 'a=real] less_real_def apply (simp add: diff_Real not_positive_Real assms) apply (simp add: diff_le_eq ac_simps) done lemma le_RealI: assumes Y: "cauchy Y" shows "∀n. x ≤ of_rat (Y n) ⟹ x ≤ Real Y" proof (induct x) fix X assume X: "cauchy X" and "∀n. Real X ≤ of_rat (Y n)" then have le: "⋀m r. 0 < r ⟹ ∃k. ∀n≥k. X n ≤ Y m + r" by (simp add: of_rat_Real le_Real) then have "∃k. ∀n≥k. X n ≤ Y n + r" if "0 < r" for r :: rat proof - from that obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" by (rule obtain_pos_sum) obtain i where i: "∀m≥i. ∀n≥i. ¦Y m - Y n¦ < s" using cauchyD [OF Y s] .. obtain j where j: "∀n≥j. X n ≤ Y i + t" using le [OF t] .. have "∀n≥max i j. X n ≤ Y n + r" proof clarsimp fix n assume n: "i ≤ n" "j ≤ n" have "X n ≤ Y i + t" using n j by simp moreover have "¦Y i - Y n¦ < s" using n i by simp ultimately show "X n ≤ Y n + r" unfolding r by simp qed then show ?thesis .. qed then show "Real X ≤ Real Y" by (simp add: of_rat_Real le_Real X Y) qed lemma Real_leI: assumes X: "cauchy X" assumes le: "∀n. of_rat (X n) ≤ y" shows "Real X ≤ y" proof - have "- y ≤ - Real X" by (simp add: minus_Real X le_RealI of_rat_minus le) then show ?thesis by simp qed lemma less_RealD: assumes "cauchy Y" shows "x < Real Y ⟹ ∃n. x < of_rat (Y n)" apply (erule contrapos_pp) apply (simp add: not_less) apply (erule Real_leI [OF assms]) done lemma of_nat_less_two_power [simp]: "of_nat n < (2::'a::linordered_idom) ^ n" apply (induct n) apply simp apply (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc) done lemma complete_real: fixes S :: "real set" assumes "∃x. x ∈ S" and "∃z. ∀x∈S. x ≤ z" shows "∃y. (∀x∈S. x ≤ y) ∧ (∀z. (∀x∈S. x ≤ z) ⟶ y ≤ z)" proof - obtain x where x: "x ∈ S" using assms(1) .. obtain z where z: "∀x∈S. x ≤ z" using assms(2) .. define P where "P x ⟷ (∀y∈S. y ≤ of_rat x)" for x obtain a where a: "¬ P a" proof have "of_int ⌊x - 1⌋ ≤ x - 1" by (rule of_int_floor_le) also have "x - 1 < x" by simp finally have "of_int ⌊x - 1⌋ < x" . then have "¬ x ≤ of_int ⌊x - 1⌋" by (simp only: not_le) then show "¬ P (of_int ⌊x - 1⌋)" unfolding P_def of_rat_of_int_eq using x by blast qed obtain b where b: "P b" proof show "P (of_int ⌈z⌉)" unfolding P_def of_rat_of_int_eq proof fix y assume "y ∈ S" then have "y ≤ z" using z by simp also have "z ≤ of_int ⌈z⌉" by (rule le_of_int_ceiling) finally show "y ≤ of_int ⌈z⌉" . qed qed define avg where "avg x y = x/2 + y/2" for x y :: rat define bisect where "bisect = (λ(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y))" define A where "A n = fst ((bisect ^^ n) (a, b))" for n define B where "B n = snd ((bisect ^^ n) (a, b))" for n define C where "C n = avg (A n) (B n)" for n have A_0 [simp]: "A 0 = a" unfolding A_def by simp have B_0 [simp]: "B 0 = b" unfolding B_def by simp have A_Suc [simp]: "⋀n. A (Suc n) = (if P (C n) then A n else C n)" unfolding A_def B_def C_def bisect_def split_def by simp have B_Suc [simp]: "⋀n. B (Suc n) = (if P (C n) then C n else B n)" unfolding A_def B_def C_def bisect_def split_def by simp have width: "B n - A n = (b - a) / 2^n" for n proof (induct n) case (Suc n) then show ?case by (simp add: C_def eq_divide_eq avg_def algebra_simps) qed simp have twos: "∃n. y / 2 ^ n < r" if "0 < r" for y r :: rat proof - obtain n where "y / r < rat_of_nat n" using ‹0 < r› reals_Archimedean2 by blast then have "∃n. y < r * 2 ^ n" by (metis divide_less_eq less_trans mult.commute of_nat_less_two_power that) then show ?thesis by (simp add: field_split_simps) qed have PA: "¬ P (A n)" for n by (induct n) (simp_all add: a) have PB: "P (B n)" for n by (induct n) (simp_all add: b) have ab: "a < b" using a b unfolding P_def by (meson leI less_le_trans of_rat_less) have AB: "A n < B n" for n by (induct n) (simp_all add: ab C_def avg_def) have "A i ≤ A j ∧ B j ≤ B i" if "i < j" for i j using that proof (induction rule: less_Suc_induct) case (1 i) then show ?case apply (clarsimp simp add: C_def avg_def add_divide_distrib [symmetric]) apply (rule AB [THEN less_imp_le]) done qed simp then have A_mono: "A i ≤ A j" and B_mono: "B j ≤ B i" if "i ≤ j" for i j by (metis eq_refl le_neq_implies_less that)+ have cauchy_lemma: "cauchy X" if *: "⋀n i. i≥n ⟹ A n ≤ X i ∧ X i ≤ B n" for X proof (rule cauchyI) fix r::rat assume "0 < r" then obtain k where k: "(b - a) / 2 ^ k < r" using twos by blast have "¦X m - X n¦ < r" if "m≥k" "n≥k" for m n proof - have "¦X m - X n¦ ≤ B k - A k" by (simp add: * abs_rat_def diff_mono that) also have "... < r" by (simp add: k width) finally show ?thesis . qed then show "∃k. ∀m≥k. ∀n≥k. ¦X m - X n¦ < r" by blast qed have "cauchy A" by (rule cauchy_lemma) (meson AB A_mono B_mono dual_order.strict_implies_order less_le_trans) have "cauchy B" by (rule cauchy_lemma) (meson AB A_mono B_mono dual_order.strict_implies_order le_less_trans) have "∀x∈S. x ≤ Real B" proof fix x assume "x ∈ S" then show "x ≤ Real B" using PB [unfolded P_def] ‹cauchy B› by (simp add: le_RealI) qed moreover have "∀z. (∀x∈S. x ≤ z) ⟶ Real A ≤ z" by (meson PA Real_leI P_def ‹cauchy A› le_cases order.trans) moreover have "vanishes (λn. (b - a) / 2 ^ n)" proof (rule vanishesI) fix r :: rat assume "0 < r" then obtain k where k: "¦b - a¦ / 2 ^ k < r" using twos by blast have "∀n≥k. ¦(b - a) / 2 ^ n¦ < r" proof clarify fix n assume n: "k ≤ n" have "¦(b - a) / 2 ^ n¦ = ¦b - a¦ / 2 ^ n" by simp also have "… ≤ ¦b - a¦ / 2 ^ k" using n by (simp add: divide_left_mono) also note k finally show "¦(b - a) / 2 ^ n¦ < r" . qed then show "∃k. ∀n≥k. ¦(b - a) / 2 ^ n¦ < r" .. qed then have "Real B = Real A" by (simp add: eq_Real ‹cauchy A› ‹cauchy B› width) ultimately show "∃y. (∀x∈S. x ≤ y) ∧ (∀z. (∀x∈S. x ≤ z) ⟶ y ≤ z)" by force qed instantiation real :: linear_continuum begin subsection ‹Supremum of a set of reals› definition "Sup X = (LEAST z::real. ∀x∈X. x ≤ z)" definition "Inf X = - Sup (uminus ` X)" for X :: "real set" instance proof show Sup_upper: "x ≤ Sup X" if "x ∈ X" "bdd_above X" for x :: real and X :: "real set" proof - from that obtain s where s: "∀y∈X. y ≤ s" "⋀z. ∀y∈X. y ≤ z ⟹ s ≤ z" using complete_real[of X] unfolding bdd_above_def by blast then show ?thesis unfolding Sup_real_def by (rule LeastI2_order) (auto simp: that) qed show Sup_least: "Sup X ≤ z" if "X ≠ {}" and z: "⋀x. x ∈ X ⟹ x ≤ z" for z :: real and X :: "real set" proof - from that obtain s where s: "∀y∈X. y ≤ s" "⋀z. ∀y∈X. y ≤ z ⟹ s ≤ z" using complete_real [of X] by blast then have "Sup X = s" unfolding Sup_real_def by (best intro: Least_equality) also from s z have "… ≤ z" by blast finally show ?thesis . qed show "Inf X ≤ x" if "x ∈ X" "bdd_below X" for x :: real and X :: "real set" using Sup_upper [of "-x" "uminus ` X"] by (auto simp: Inf_real_def that) show "z ≤ Inf X" if "X ≠ {}" "⋀x. x ∈ X ⟹ z ≤ x" for z :: real and X :: "real set" using Sup_least [of "uminus ` X" "- z"] by (force simp: Inf_real_def that) show "∃a b::real. a ≠ b" using zero_neq_one by blast qed end subsection ‹Hiding implementation details› hide_const (open) vanishes cauchy positive Real declare Real_induct [induct del] declare Abs_real_induct [induct del] declare Abs_real_cases [cases del] lifting_update real.lifting lifting_forget real.lifting subsection ‹Embedding numbers into the Reals› abbreviation real_of_nat :: "nat ⇒ real" where "real_of_nat ≡ of_nat" abbreviation real :: "nat ⇒ real" where "real ≡ of_nat" abbreviation real_of_int :: "int ⇒ real" where "real_of_int ≡ of_int" abbreviation real_of_rat :: "rat ⇒ real" where "real_of_rat ≡ of_rat" declare [[coercion_enabled]] declare [[coercion "of_nat :: nat ⇒ int"]] declare [[coercion "of_nat :: nat ⇒ real"]] declare [[coercion "of_int :: int ⇒ real"]] (* We do not add rat to the coerced types, this has often unpleasant side effects when writing inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *) declare [[coercion_map map]] declare [[coercion_map "λf g h x. g (h (f x))"]] declare [[coercion_map "λf g (x,y). (f x, g y)"]] declare of_int_eq_0_iff [algebra, presburger] declare of_int_eq_1_iff [algebra, presburger] declare of_int_eq_iff [algebra, presburger] declare of_int_less_0_iff [algebra, presburger] declare of_int_less_1_iff [algebra, presburger] declare of_int_less_iff [algebra, presburger] declare of_int_le_0_iff [algebra, presburger] declare of_int_le_1_iff [algebra, presburger] declare of_int_le_iff [algebra, presburger] declare of_int_0_less_iff [algebra, presburger] declare of_int_0_le_iff [algebra, presburger] declare of_int_1_less_iff [algebra, presburger] declare of_int_1_le_iff [algebra, presburger] lemma int_less_real_le: "n < m ⟷ real_of_int n + 1 ≤ real_of_int m" proof - have "(0::real) ≤ 1" by (metis less_eq_real_def zero_less_one) then show ?thesis by (metis floor_of_int less_floor_iff) qed lemma int_le_real_less: "n ≤ m ⟷ real_of_int n < real_of_int m + 1" by (meson int_less_real_le not_le) lemma real_of_int_div_aux: "(real_of_int x) / (real_of_int d) = real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)" proof - have "x = (x div d) * d + x mod d" by auto then have "real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)" by (metis of_int_add of_int_mult) then have "real_of_int x / real_of_int d = … / real_of_int d" by simp then show ?thesis by (auto simp add: add_divide_distrib algebra_simps) qed lemma real_of_int_div: "d dvd n ⟹ real_of_int (n div d) = real_of_int n / real_of_int d" for d n :: int by (simp add: real_of_int_div_aux) lemma real_of_int_div2: "0 ≤ real_of_int n / real_of_int x - real_of_int (n div x)" proof (cases "x = 0") case False then show ?thesis by (metis diff_ge_0_iff_ge floor_divide_of_int_eq of_int_floor_le) qed simp lemma real_of_int_div3: "real_of_int n / real_of_int x - real_of_int (n div x) ≤ 1" apply (simp add: algebra_simps) by (metis add.commute floor_correct floor_divide_of_int_eq less_eq_real_def of_int_1 of_int_add) lemma real_of_int_div4: "real_of_int (n div x) ≤ real_of_int n / real_of_int x" using real_of_int_div2 [of n x] by simp subsection ‹Embedding the Naturals into the Reals› lemma real_of_card: "real (card A) = sum (λx. 1) A" by simp lemma nat_less_real_le: "n < m ⟷ real n + 1 ≤ real m" by (metis discrete of_nat_1 of_nat_add of_nat_le_iff) lemma nat_le_real_less: "n ≤ m ⟷ real n < real m + 1" for m n :: nat by (meson nat_less_real_le not_le) lemma real_of_nat_div_aux: "real x / real d = real (x div d) + real (x mod d) / real d" proof - have "x = (x div d) * d + x mod d" by auto then have "real x = real (x div d) * real d + real(x mod d)" by (metis of_nat_add of_nat_mult) then have "real x / real d = … / real d" by simp then show ?thesis by (auto simp add: add_divide_distrib algebra_simps) qed lemma real_of_nat_div: "d dvd n ⟹ real(n div d) = real n / real d" by (subst real_of_nat_div_aux) (auto simp add: dvd_eq_mod_eq_0 [symmetric]) lemma real_of_nat_div2: "0 ≤ real n / real x - real (n div x)" for n x :: nat apply (simp add: algebra_simps) by (metis floor_divide_of_nat_eq of_int_floor_le of_int_of_nat_eq) lemma real_of_nat_div3: "real n / real x - real (n div x) ≤ 1" for n x :: nat by (metis of_int_of_nat_eq real_of_int_div3 of_nat_div) lemma real_of_nat_div4: "real (n div x) ≤ real n / real x" for n x :: nat using real_of_nat_div2 [of n x] by simp lemma real_binomial_eq_mult_binomial_Suc: assumes "k ≤ n" shows "real(n choose k) = (n + 1 - k) / (n + 1) * (Suc n choose k)" using assms by (simp add: of_nat_binomial_eq_mult_binomial_Suc [of k n] add.commute of_nat_diff) subsection ‹The Archimedean Property of the Reals› lemma real_arch_inverse: "0 < e ⟷ (∃n::nat. n ≠ 0 ∧ 0 < inverse (real n) ∧ inverse (real n) < e)" using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat] by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc) lemma reals_Archimedean3: "0 < x ⟹ ∀y. ∃n. y < real n * x" by (auto intro: ex_less_of_nat_mult) lemma real_archimedian_rdiv_eq_0: assumes x0: "x ≥ 0" and c: "c ≥ 0" and xc: "⋀m::nat. m > 0 ⟹ real m * x ≤ c" shows "x = 0" by (metis reals_Archimedean3 dual_order.order_iff_strict le0 le_less_trans not_le x0 xc) lemma inverse_Suc: "inverse (Suc n) > 0" using of_nat_0_less_iff positive_imp_inverse_positive zero_less_Suc by blast lemma Archimedean_eventually_inverse: fixes ε::real shows "(∀⇩_{F}n in sequentially. inverse (real (Suc n)) < ε) ⟷ 0 < ε" (is "?lhs=?rhs") proof assume ?lhs then show ?rhs unfolding eventually_at_top_dense using inverse_Suc order_less_trans by blast next assume ?rhs then obtain N where "inverse (Suc N) < ε" using reals_Archimedean by blast moreover have "inverse (Suc n) ≤ inverse (Suc N)" if "n ≥ N" for n using inverse_Suc that by fastforce ultimately show ?lhs unfolding eventually_sequentially using order_le_less_trans by blast qed (*HOL Light's FORALL_POS_MONO_1_EQ*) text ‹On the relationship between two different ways of converting to 0› lemma Inter_eq_Inter_inverse_Suc: assumes "⋀r' r. r' < r ⟹ A r' ⊆ A r" shows "⋂ (A ` {0<..}) = (⋂n. A(inverse(Suc n)))" proof have "x ∈ A ε" if x: "∀n. x ∈ A (inverse (Suc n))" and "ε>0" for x and ε :: real proof - obtain n where "inverse (Suc n) < ε" using ‹ε>0› reals_Archimedean by blast with assms x show ?thesis by blast qed then show "(⋂n. A(inverse(Suc n))) ⊆ (⋂ε∈{0<..}. A ε)" by auto qed (use inverse_Suc in fastforce) subsection ‹Rationals› lemma Rats_abs_iff[simp]: "¦(x::real)¦ ∈ ℚ ⟷ x ∈ ℚ" by(simp add: abs_real_def split: if_splits) lemma Rats_eq_int_div_int: "ℚ = {real_of_int i / real_of_int j | i j. j ≠ 0}" (is "_ = ?S") proof show "ℚ ⊆ ?S" proof fix x :: real assume "x ∈ ℚ" then obtain r where "x = of_rat r" unfolding Rats_def .. have "of_rat r ∈ ?S" by (cases r) (auto simp add: of_rat_rat) then show "x ∈ ?S" using ‹x = of_rat r› by simp qed next show "?S ⊆ ℚ" proof (auto simp: Rats_def) fix i j :: int assume "j ≠ 0" then have "real_of_int i / real_of_int j = of_rat (Fract i j)" by (simp add: of_rat_rat) then show "real_of_int i / real_of_int j ∈ range of_rat" by blast qed qed lemma Rats_eq_int_div_nat: "ℚ = { real_of_int i / real n | i n. n ≠ 0}" proof (auto simp: Rats_eq_int_div_int) fix i j :: int assume "j ≠ 0" show "∃(i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i' / real n ∧ 0 < n" proof (cases "j > 0") case True then have "real_of_int i / real_of_int j = real_of_int i / real (nat j) ∧ 0 < nat j" by simp then show ?thesis by blast next case False with ‹j ≠ 0› have "real_of_int i / real_of_int j = real_of_int (- i) / real (nat (- j)) ∧ 0 < nat (- j)" by simp then show ?thesis by blast qed next fix i :: int and n :: nat assume "0 < n" then have "real_of_int i / real n = real_of_int i / real_of_int(int n) ∧ int n ≠ 0" by simp then show "∃i' j. real_of_int i / real n = real_of_int i' / real_of_int j ∧ j ≠ 0" by blast qed lemma Rats_abs_nat_div_natE: assumes "x ∈ ℚ" obtains m n :: nat where "n ≠ 0" and "¦x¦ = real m / real n" and "coprime m n" proof - from ‹x ∈ ℚ› obtain i :: int and n :: nat where "n ≠ 0" and "x = real_of_int i / real n" by (auto simp add: Rats_eq_int_div_nat) then have "¦x¦ = real (nat ¦i¦) / real n" by simp then obtain m :: nat where x_rat: "¦x¦ = real m / real n" by blast let ?gcd = "gcd m n" from ‹n ≠ 0› have gcd: "?gcd ≠ 0" by simp let ?k = "m div ?gcd" let ?l = "n div ?gcd" let ?gcd' = "gcd ?k ?l" have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m" by (rule dvd_mult_div_cancel) have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n" by (rule dvd_mult_div_cancel) from ‹n ≠ 0› and gcd_l have "?gcd * ?l ≠ 0" by simp then have "?l ≠ 0" by (blast dest!: mult_not_zero) moreover have "¦x¦ = real ?k / real ?l" proof - from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)" by (simp add: real_of_nat_div) also from gcd_k and gcd_l have "… = real m / real n" by simp also from x_rat have "… = ¦x¦" .. finally show ?thesis .. qed moreover have "?gcd' = 1" proof - have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)" by (rule gcd_mult_distrib_nat) with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp with gcd show ?thesis by auto qed then have "coprime ?k ?l" by (simp only: coprime_iff_gcd_eq_1) ultimately show ?thesis .. qed subsection ‹Density of the Rational Reals in the Reals› text ‹ This density proof is due to Stefan Richter and was ported by TN. The original source is ∗‹Real Analysis› by H.L. Royden. It employs the Archimedean property of the reals.› lemma Rats_dense_in_real: fixes x :: real assumes "x < y" shows "∃r∈ℚ. x < r ∧ r < y" proof - from ‹x < y› have "0 < y - x" by simp with reals_Archimedean obtain q :: nat where q: "inverse (real q) < y - x" and "0 < q" by blast define p where "p = ⌈y * real q⌉ - 1" define r where "r = of_int p / real q" from q have "x < y - inverse (real q)" by simp also from ‹0 < q› have "y - inverse (real q) ≤ r" by (simp add: r_def p_def le_divide_eq left_diff_distrib) finally have "x < r" . moreover from ‹0 < q› have "r < y" by (simp add: r_def p_def divide_less_eq diff_less_eq less_ceiling_iff [symmetric]) moreover have "r ∈ ℚ" by (simp add: r_def) ultimately show ?thesis by blast qed lemma of_rat_dense: fixes x y :: real assumes "x < y" shows "∃q :: rat. x < of_rat q ∧ of_rat q < y" using Rats_dense_in_real [OF ‹x < y›] by (auto elim: Rats_cases) subsection ‹Numerals and Arithmetic› declaration ‹ K (Lin_Arith.add_inj_const (\<^const_name>‹of_nat›, \<^typ>‹nat ⇒ real›) #> Lin_Arith.add_inj_const (\<^const_name>‹of_int›, \<^typ>‹int ⇒ real›)) › subsection ‹Simprules combining ‹x + y› and ‹0›› (* FIXME ARE THEY NEEDED? *) lemma real_add_minus_iff [simp]: "x + - a = 0 ⟷ x = a" for x a :: real by arith lemma real_add_less_0_iff: "x + y < 0 ⟷ y < - x" for x y :: real by auto lemma real_0_less_add_iff: "0 < x + y ⟷ - x < y" for x y :: real by auto lemma real_add_le_0_iff: "x + y ≤ 0 ⟷ y ≤ - x" for x y :: real by auto lemma real_0_le_add_iff: "0 ≤ x + y ⟷ - x ≤ y" for x y :: real by auto subsection ‹Lemmas about powers› lemma two_realpow_ge_one: "(1::real) ≤ 2 ^ n" by simp (* FIXME: declare this [simp] for all types, or not at all *) declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp] lemma real_minus_mult_self_le [simp]: "- (u * u) ≤ x * x" for u x :: real by (rule order_trans [where y = 0]) auto lemma realpow_square_minus_le [simp]: "- u⇧^{2}≤ x⇧^{2}" for u x :: real by (auto simp add: power2_eq_square) subsection ‹Density of the Reals› lemma field_lbound_gt_zero: "0 < d1 ⟹ 0 < d2 ⟹ ∃e. 0 < e ∧ e < d1 ∧ e < d2" for d1 d2 :: "'a::linordered_field" by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def) lemma field_less_half_sum: "x < y ⟹ x < (x + y) / 2" for x y :: "'a::linordered_field" by auto lemma field_sum_of_halves: "x / 2 + x / 2 = x" for x :: "'a::linordered_field" by simp subsection ‹Archimedean properties and useful consequences› text‹Bernoulli's inequality› proposition Bernoulli_inequality: fixes x :: real assumes "-1 ≤ x" shows "1 + n * x ≤ (1 + x) ^ n" proof (induct n) case 0 then show ?case by simp next case (Suc n) have "1 + Suc n * x ≤ 1 + (Suc n)*x + n * x^2" by (simp add: algebra_simps) also have "... = (1 + x) * (1 + n*x)" by (auto simp: power2_eq_square algebra_simps) also have "... ≤ (1 + x) ^ Suc n" using Suc.hyps assms mult_left_mono by fastforce finally show ?case . qed corollary Bernoulli_inequality_even: fixes x :: real assumes "even n" shows "1 + n * x ≤ (1 + x) ^ n" proof (cases "-1 ≤ x ∨ n=0") case True then show ?thesis by (auto simp: Bernoulli_inequality) next case False then have "real n ≥ 1" by simp with False have "n * x ≤ -1" by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one) then have "1 + n * x ≤ 0" by auto also have "... ≤ (1 + x) ^ n" using assms using zero_le_even_power by blast finally show ?thesis . qed corollary real_arch_pow: fixes x :: real assumes x: "1 < x" shows "∃n. y < x^n" proof - from x have x0: "x - 1 > 0" by arith from reals_Archimedean3[OF x0, rule_format, of y] obtain n :: nat where n: "y < real n * (x - 1)" by metis from x0 have x00: "x- 1 ≥ -1" by arith from Bernoulli_inequality[OF x00, of n] n have "y < x^n" by auto then show ?thesis by metis qed corollary real_arch_pow_inv: fixes x y :: real assumes y: "y > 0" and x1: "x < 1" shows "∃n. x^n < y" proof (cases "x > 0") case True with x1 have ix: "1 < 1/x" by (simp add: field_simps) from real_arch_pow[OF ix, of "1/y"] obtain n where n: "1/y < (1/x)^n" by blast then show ?thesis using y ‹x > 0› by (auto simp add: field_simps) next case False with y x1 show ?thesis by (metis less_le_trans not_less power_one_right) qed lemma forall_pos_mono: "(⋀d e::real. d < e ⟹ P d ⟹ P e) ⟹ (⋀n::nat. n ≠ 0 ⟹ P (inverse (real n))) ⟹ (⋀e. 0 < e ⟹ P e)" by (metis real_arch_inverse) lemma forall_pos_mono_1: "(⋀d e::real. d < e ⟹ P d ⟹ P e) ⟹ (⋀n. P (inverse (real (Suc n)))) ⟹ 0 < e ⟹ P e" using reals_Archimedean by blast lemma Archimedean_eventually_pow: fixes x::real assumes "1 < x" shows "∀⇩_{F}n in sequentially. b < x ^ n" proof - obtain N where "⋀n. n≥N ⟹ b < x ^ n" by (metis assms le_less order_less_trans power_strict_increasing_iff real_arch_pow) then show ?thesis using eventually_sequentially by blast qed lemma Archimedean_eventually_pow_inverse: fixes x::real assumes "¦x¦ < 1" "ε > 0" shows "∀⇩_{F}n in sequentially. ¦x^n¦ < ε" proof (cases "x = 0") case True then show ?thesis by (simp add: assms eventually_at_top_dense zero_power) next case False then have "∀⇩_{F}n in sequentially. inverse ε < inverse ¦x¦ ^ n" by (simp add: Archimedean_eventually_pow assms(1) one_less_inverse) then show ?thesis by eventually_elim (metis ‹ε > 0› inverse_less_imp_less power_abs power_inverse) qed subsection ‹Floor and Ceiling Functions from the Reals to the Integers› (* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *) lemma real_of_nat_less_numeral_iff [simp]: "real n < numeral w ⟷ n < numeral w" for n :: nat by (metis of_nat_less_iff of_nat_numeral) lemma numeral_less_real_of_nat_iff [simp]: "numeral w < real n ⟷ numeral w < n" for n :: nat by (metis of_nat_less_iff of_nat_numeral) lemma numeral_le_real_of_nat_iff [simp]: "numeral n ≤ real m ⟷ numeral n ≤ m" for m :: nat by (metis not_le real_of_nat_less_numeral_iff) lemma of_int_floor_cancel [simp]: "of_int ⌊x⌋ = x ⟷ (∃n::int. x = of_int n)" by (metis floor_of_int) lemma of_int_floor [simp]: "a ∈ ℤ ⟹ of_int (floor a) = a" by (metis Ints_cases of_int_floor_cancel) lemma floor_frac [simp]: "⌊frac r⌋ = 0" by (simp add: frac_def) lemma frac_1 [simp]: "frac 1 = 0" by (simp add: frac_def) lemma frac_in_Rats_iff [simp]: fixes r::"'a::{floor_ceiling,field_char_0}" shows "frac r ∈ ℚ ⟷ r ∈ ℚ" by (metis Rats_add Rats_diff Rats_of_int diff_add_cancel frac_def) lemma floor_eq: "real_of_int n < x ⟹ x < real_of_int n + 1 ⟹ ⌊x⌋ = n" by linarith lemma floor_eq2: "real_of_int n ≤ x ⟹ x < real_of_int n + 1 ⟹ ⌊x⌋ = n" by (fact floor_unique) lemma floor_eq3: "real n < x ⟹ x < real (Suc n) ⟹ nat ⌊x⌋ = n" by linarith lemma floor_eq4: "real n ≤ x ⟹ x < real (Suc n) ⟹ nat ⌊x⌋ = n" by linarith lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 ≤ real_of_int ⌊r⌋" by linarith lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int ⌊r⌋" by linarith lemma real_of_int_floor_add_one_ge [simp]: "r ≤ real_of_int ⌊r⌋ + 1" by linarith lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int ⌊r⌋ + 1" by linarith lemma floor_divide_real_eq_div: assumes "0 ≤ b" shows "⌊a / real_of_int b⌋ = ⌊a⌋ div b" proof (cases "b = 0") case True then show ?thesis by simp next case False with assms have b: "b > 0" by simp have "j = i div b" if "real_of_int i ≤ a" "a < 1 + real_of_int i" "real_of_int j * real_of_int b ≤ a" "a < real_of_int b + real_of_int j * real_of_int b" for i j :: int proof - from that have "i < b + j * b" by (metis le_less_trans of_int_add of_int_less_iff of_int_mult) moreover have "j * b < 1 + i" proof - have "real_of_int (j * b) < real_of_int i + 1" using ‹a < 1 + real_of_int i› ‹real_of_int j * real_of_int b ≤ a› by force then show "j * b < 1 + i" by linarith qed ultimately have "(j - i div b) * b ≤ i mod b" "i mod b < ((j - i div b) + 1) * b" by (auto simp: field_simps) then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b" using pos_mod_bound [OF b, of i] pos_mod_sign [OF