--- a/src/HOL/NthRoot.thy Tue Jul 12 21:53:56 2016 +0200
+++ b/src/HOL/NthRoot.thy Tue Jul 12 22:54:37 2016 +0200
@@ -1,70 +1,87 @@
-(* Title : NthRoot.thy
- Author : Jacques D. Fleuriot
- Copyright : 1998 University of Cambridge
- Conversion to Isar and new proofs by Lawrence C Paulson, 2004
+(* Title: HOL/NthRoot.thy
+ Author: Jacques D. Fleuriot, 1998
+ Author: Lawrence C Paulson, 2004
*)
section \<open>Nth Roots of Real Numbers\<close>
theory NthRoot
-imports Deriv Binomial
+ imports Deriv Binomial
begin
+
subsection \<open>Existence of Nth Root\<close>
text \<open>Existence follows from the Intermediate Value Theorem\<close>
lemma realpow_pos_nth:
+ fixes a :: real
assumes n: "0 < n"
- assumes a: "0 < a"
- shows "\<exists>r>0. r ^ n = (a::real)"
+ and a: "0 < a"
+ shows "\<exists>r>0. r ^ n = a"
proof -
have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a"
proof (rule IVT)
- show "0 ^ n \<le> a" using n a by (simp add: power_0_left)
- show "0 \<le> max 1 a" by simp
- from n have n1: "1 \<le> n" by simp
- have "a \<le> max 1 a ^ 1" by simp
+ show "0 ^ n \<le> a"
+ using n a by (simp add: power_0_left)
+ show "0 \<le> max 1 a"
+ by simp
+ from n have n1: "1 \<le> n"
+ by simp
+ have "a \<le> max 1 a ^ 1"
+ by simp
also have "max 1 a ^ 1 \<le> max 1 a ^ n"
- using n1 by (rule power_increasing, simp)
+ using n1 by (rule power_increasing) simp
finally show "a \<le> max 1 a ^ n" .
show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r"
by simp
qed
- then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast
- with n a have "r \<noteq> 0" by (auto simp add: power_0_left)
- with r have "0 < r \<and> r ^ n = a" by simp
- thus ?thesis ..
+ then obtain r where r: "0 \<le> r \<and> r ^ n = a"
+ by fast
+ with n a have "r \<noteq> 0"
+ by (auto simp add: power_0_left)
+ with r have "0 < r \<and> r ^ n = a"
+ by simp
+ then show ?thesis ..
qed
(* Used by Integration/RealRandVar.thy in AFP *)
lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a"
-by (blast intro: realpow_pos_nth)
+ by (blast intro: realpow_pos_nth)
-text \<open>Uniqueness of nth positive root\<close>
+text \<open>Uniqueness of nth positive root.\<close>
+lemma realpow_pos_nth_unique: "0 < n \<Longrightarrow> 0 < a \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = a" for a :: real
+ by (auto intro!: realpow_pos_nth simp: power_eq_iff_eq_base)
-lemma realpow_pos_nth_unique: "\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)"
- by (auto intro!: realpow_pos_nth simp: power_eq_iff_eq_base)
subsection \<open>Nth Root\<close>
-text \<open>We define roots of negative reals such that
- @{term "root n (- x) = - root n x"}. This allows
- us to omit side conditions from many theorems.\<close>
+text \<open>
+ We define roots of negative reals such that \<open>root n (- x) = - root n x\<close>.
+ This allows us to omit side conditions from many theorems.
+\<close>
-lemma inj_sgn_power: assumes "0 < n" shows "inj (\<lambda>y. sgn y * \<bar>y\<bar>^n :: real)" (is "inj ?f")
+lemma inj_sgn_power:
+ assumes "0 < n"
+ shows "inj (\<lambda>y. sgn y * \<bar>y\<bar>^n :: real)"
+ (is "inj ?f")
proof (rule injI)
- have x: "\<And>a b :: real. (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b) \<Longrightarrow> a \<noteq> b" by auto
- fix x y assume "?f x = ?f y" with power_eq_iff_eq_base[of n "\<bar>x\<bar>" "\<bar>y\<bar>"] \<open>0<n\<close> show "x = y"
+ have x: "(0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b) \<Longrightarrow> a \<noteq> b" for a b :: real
+ by auto
+ fix x y
+ assume "?f x = ?f y"
+ with power_eq_iff_eq_base[of n "\<bar>x\<bar>" "\<bar>y\<bar>"] \<open>0 < n\<close> show "x = y"
by (cases rule: linorder_cases[of 0 x, case_product linorder_cases[of 0 y]])
(simp_all add: x)
qed
-lemma sgn_power_injE: "sgn a * \<bar>a\<bar> ^ n = x \<Longrightarrow> x = sgn b * \<bar>b\<bar> ^ n \<Longrightarrow> 0 < n \<Longrightarrow> a = (b::real)"
+lemma sgn_power_injE:
+ "sgn a * \<bar>a\<bar> ^ n = x \<Longrightarrow> x = sgn b * \<bar>b\<bar> ^ n \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
+ for a b :: real
using inj_sgn_power[THEN injD, of n a b] by simp
-definition root :: "nat \<Rightarrow> real \<Rightarrow> real" where
- "root n x = (if n = 0 then 0 else the_inv (\<lambda>y. sgn y * \<bar>y\<bar>^n) x)"
+definition root :: "nat \<Rightarrow> real \<Rightarrow> real"
+ where "root n x = (if n = 0 then 0 else the_inv (\<lambda>y. sgn y * \<bar>y\<bar>^n) x)"
lemma root_0 [simp]: "root 0 x = 0"
by (simp add: root_def)
@@ -73,16 +90,24 @@
using the_inv_f_f[OF inj_sgn_power] by (simp add: root_def)
lemma sgn_power_root:
- assumes "0 < n" shows "sgn (root n x) * \<bar>(root n x)\<bar>^n = x" (is "?f (root n x) = x")
-proof cases
- assume "x \<noteq> 0"
- with realpow_pos_nth[OF \<open>0 < n\<close>, of "\<bar>x\<bar>"] obtain r where "0 < r" "r ^ n = \<bar>x\<bar>" by auto
+ assumes "0 < n"
+ shows "sgn (root n x) * \<bar>(root n x)\<bar>^n = x"
+ (is "?f (root n x) = x")
+proof (cases "x = 0")
+ case True
+ with assms root_sgn_power[of n 0] show ?thesis
+ by simp
+next
+ case False
+ with realpow_pos_nth[OF \<open>0 < n\<close>, of "\<bar>x\<bar>"]
+ obtain r where "0 < r" "r ^ n = \<bar>x\<bar>"
+ by auto
with \<open>x \<noteq> 0\<close> have S: "x \<in> range ?f"
by (intro image_eqI[of _ _ "sgn x * r"])
(auto simp: abs_mult sgn_mult power_mult_distrib abs_sgn_eq mult_sgn_abs)
from \<open>0 < n\<close> f_the_inv_into_f[OF inj_sgn_power[OF \<open>0 < n\<close>] this] show ?thesis
by (simp add: root_def)
-qed (insert \<open>0 < n\<close> root_sgn_power[of n 0], simp)
+qed
lemma split_root: "P (root n x) \<longleftrightarrow> (n = 0 \<longrightarrow> P 0) \<and> (0 < n \<longrightarrow> (\<forall>y. sgn y * \<bar>y\<bar>^n = x \<longrightarrow> P y))"
apply (cases "n = 0")
@@ -96,72 +121,74 @@
lemma real_root_minus: "root n (- x) = - root n x"
by (clarsimp split: split_root elim!: sgn_power_injE simp: sgn_minus)
-lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"
+lemma real_root_less_mono: "0 < n \<Longrightarrow> x < y \<Longrightarrow> root n x < root n y"
proof (clarsimp split: split_root)
- have x: "\<And>a b :: real. (0 < b \<and> a < 0) \<Longrightarrow> \<not> a > b" by auto
- fix a b :: real assume "0 < n" "sgn a * \<bar>a\<bar> ^ n < sgn b * \<bar>b\<bar> ^ n" then show "a < b"
- using power_less_imp_less_base[of a n b] power_less_imp_less_base[of "-b" n "-a"]
- by (simp add: sgn_real_def x [of "a ^ n" "- ((- b) ^ n)"] split: if_split_asm)
+ have *: "0 < b \<Longrightarrow> a < 0 \<Longrightarrow> \<not> a > b" for a b :: real
+ by auto
+ fix a b :: real
+ assume "0 < n" "sgn a * \<bar>a\<bar> ^ n < sgn b * \<bar>b\<bar> ^ n"
+ then show "a < b"
+ using power_less_imp_less_base[of a n b]
+ power_less_imp_less_base[of "- b" n "- a"]
+ by (simp add: sgn_real_def * [of "a ^ n" "- ((- b) ^ n)"]
+ split: if_split_asm)
qed
-lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x"
+lemma real_root_gt_zero: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> 0 < root n x"
using real_root_less_mono[of n 0 x] by simp
lemma real_root_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> root n x"
- using real_root_gt_zero[of n x] by (cases "n = 0") (auto simp add: le_less)
+ using real_root_gt_zero[of n x]
+ by (cases "n = 0") (auto simp add: le_less)
-lemma real_root_pow_pos: (* TODO: rename *)
- "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x"
+lemma real_root_pow_pos: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x ^ n = x" (* TODO: rename *)
using sgn_power_root[of n x] real_root_gt_zero[of n x] by simp
-lemma real_root_pow_pos2 [simp]: (* TODO: rename *)
- "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x"
-by (auto simp add: order_le_less real_root_pow_pos)
+lemma real_root_pow_pos2 [simp]: "0 < n \<Longrightarrow> 0 \<le> x \<Longrightarrow> root n x ^ n = x" (* TODO: rename *)
+ by (auto simp add: order_le_less real_root_pow_pos)
lemma sgn_root: "0 < n \<Longrightarrow> sgn (root n x) = sgn x"
by (auto split: split_root simp: sgn_real_def)
lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x"
- using sgn_power_root[of n x] by (simp add: odd_pos sgn_real_def split: if_split_asm)
+ using sgn_power_root[of n x]
+ by (simp add: odd_pos sgn_real_def split: if_split_asm)
-lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x"
+lemma real_root_power_cancel: "0 < n \<Longrightarrow> 0 \<le> x \<Longrightarrow> root n (x ^ n) = x"
using root_sgn_power[of n x] by (auto simp add: le_less power_0_left)
lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x"
- using root_sgn_power[of n x] by (simp add: odd_pos sgn_real_def power_0_left split: if_split_asm)
+ using root_sgn_power[of n x]
+ by (simp add: odd_pos sgn_real_def power_0_left split: if_split_asm)
-lemma real_root_pos_unique: "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
+lemma real_root_pos_unique: "0 < n \<Longrightarrow> 0 \<le> y \<Longrightarrow> y ^ n = x \<Longrightarrow> root n x = y"
using root_sgn_power[of n y] by (auto simp add: le_less power_0_left)
-lemma odd_real_root_unique:
- "\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
-by (erule subst, rule odd_real_root_power_cancel)
+lemma odd_real_root_unique: "odd n \<Longrightarrow> y ^ n = x \<Longrightarrow> root n x = y"
+ by (erule subst, rule odd_real_root_power_cancel)
lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1"
-by (simp add: real_root_pos_unique)
+ by (simp add: real_root_pos_unique)
-text \<open>Root function is strictly monotonic, hence injective\<close>
+text \<open>Root function is strictly monotonic, hence injective.\<close>
-lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y"
+lemma real_root_le_mono: "0 < n \<Longrightarrow> x \<le> y \<Longrightarrow> root n x \<le> root n y"
by (auto simp add: order_le_less real_root_less_mono)
-lemma real_root_less_iff [simp]:
- "0 < n \<Longrightarrow> (root n x < root n y) = (x < y)"
-apply (cases "x < y")
-apply (simp add: real_root_less_mono)
-apply (simp add: linorder_not_less real_root_le_mono)
-done
+lemma real_root_less_iff [simp]: "0 < n \<Longrightarrow> root n x < root n y \<longleftrightarrow> x < y"
+ apply (cases "x < y")
+ apply (simp add: real_root_less_mono)
+ apply (simp add: linorder_not_less real_root_le_mono)
+ done
-lemma real_root_le_iff [simp]:
- "0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)"
-apply (cases "x \<le> y")
-apply (simp add: real_root_le_mono)
-apply (simp add: linorder_not_le real_root_less_mono)
-done
+lemma real_root_le_iff [simp]: "0 < n \<Longrightarrow> root n x \<le> root n y \<longleftrightarrow> x \<le> y"
+ apply (cases "x \<le> y")
+ apply (simp add: real_root_le_mono)
+ apply (simp add: linorder_not_le real_root_less_mono)
+ done
-lemma real_root_eq_iff [simp]:
- "0 < n \<Longrightarrow> (root n x = root n y) = (x = y)"
-by (simp add: order_eq_iff)
+lemma real_root_eq_iff [simp]: "0 < n \<Longrightarrow> root n x = root n y \<longleftrightarrow> x = y"
+ by (simp add: order_eq_iff)
lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified]
lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified]
@@ -169,28 +196,31 @@
lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified]
lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified]
-lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> (1 < root n y) = (1 < y)"
-by (insert real_root_less_iff [where x=1], simp)
+lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> 1 < root n y \<longleftrightarrow> 1 < y"
+ using real_root_less_iff [where x=1] by simp
-lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> (root n x < 1) = (x < 1)"
-by (insert real_root_less_iff [where y=1], simp)
+lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> root n x < 1 \<longleftrightarrow> x < 1"
+ using real_root_less_iff [where y=1] by simp
+
+lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> 1 \<le> root n y \<longleftrightarrow> 1 \<le> y"
+ using real_root_le_iff [where x=1] by simp
-lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> (1 \<le> root n y) = (1 \<le> y)"
-by (insert real_root_le_iff [where x=1], simp)
+lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> root n x \<le> 1 \<longleftrightarrow> x \<le> 1"
+ using real_root_le_iff [where y=1] by simp
-lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> (root n x \<le> 1) = (x \<le> 1)"
-by (insert real_root_le_iff [where y=1], simp)
+lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> root n x = 1 \<longleftrightarrow> x = 1"
+ using real_root_eq_iff [where y=1] by simp
-lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> (root n x = 1) = (x = 1)"
-by (insert real_root_eq_iff [where y=1], simp)
-text \<open>Roots of multiplication and division\<close>
+text \<open>Roots of multiplication and division.\<close>
lemma real_root_mult: "root n (x * y) = root n x * root n y"
- by (auto split: split_root elim!: sgn_power_injE simp: sgn_mult abs_mult power_mult_distrib)
+ by (auto split: split_root elim!: sgn_power_injE
+ simp: sgn_mult abs_mult power_mult_distrib)
lemma real_root_inverse: "root n (inverse x) = inverse (root n x)"
- by (auto split: split_root elim!: sgn_power_injE simp: inverse_sgn power_inverse)
+ by (auto split: split_root elim!: sgn_power_injE
+ simp: inverse_sgn power_inverse)
lemma real_root_divide: "root n (x / y) = root n x / root n y"
by (simp add: divide_inverse real_root_mult real_root_inverse)
@@ -201,124 +231,138 @@
lemma real_root_power: "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k"
by (induct k) (simp_all add: real_root_mult)
-text \<open>Roots of roots\<close>
+
+text \<open>Roots of roots.\<close>
lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x"
-by (simp add: odd_real_root_unique)
+ by (simp add: odd_real_root_unique)
lemma real_root_mult_exp: "root (m * n) x = root m (root n x)"
by (auto split: split_root elim!: sgn_power_injE
- simp: sgn_zero_iff sgn_mult power_mult[symmetric] abs_mult power_mult_distrib abs_sgn_eq)
+ simp: sgn_zero_iff sgn_mult power_mult[symmetric]
+ abs_mult power_mult_distrib abs_sgn_eq)
lemma real_root_commute: "root m (root n x) = root n (root m x)"
by (simp add: real_root_mult_exp [symmetric] mult.commute)
-text \<open>Monotonicity in first argument\<close>
+
+text \<open>Monotonicity in first argument.\<close>
-lemma real_root_strict_decreasing:
- "\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x"
-apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp)
-apply (simp add: real_root_commute power_strict_increasing
- del: real_root_pow_pos2)
-done
+lemma real_root_strict_decreasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 1 < x \<Longrightarrow> root N x < root n x"
+ apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N")
+ apply simp
+ apply (simp add: real_root_commute power_strict_increasing del: real_root_pow_pos2)
+ done
-lemma real_root_strict_increasing:
- "\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x"
-apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp)
-apply (simp add: real_root_commute power_strict_decreasing
- del: real_root_pow_pos2)
-done
+lemma real_root_strict_increasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 0 < x \<Longrightarrow> x < 1 \<Longrightarrow> root n x < root N x"
+ apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n")
+ apply simp
+ apply (simp add: real_root_commute power_strict_decreasing del: real_root_pow_pos2)
+ done
-lemma real_root_decreasing:
- "\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x"
-by (auto simp add: order_le_less real_root_strict_decreasing)
+lemma real_root_decreasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 1 \<le> x \<Longrightarrow> root N x \<le> root n x"
+ by (auto simp add: order_le_less real_root_strict_decreasing)
-lemma real_root_increasing:
- "\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x"
-by (auto simp add: order_le_less real_root_strict_increasing)
+lemma real_root_increasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> root n x \<le> root N x"
+ by (auto simp add: order_le_less real_root_strict_increasing)
-text \<open>Continuity and derivatives\<close>
+
+text \<open>Continuity and derivatives.\<close>
lemma isCont_real_root: "isCont (root n) x"
-proof cases
- assume n: "0 < n"
+proof (cases "n > 0")
+ case True
let ?f = "\<lambda>y::real. sgn y * \<bar>y\<bar>^n"
have "continuous_on ({0..} \<union> {.. 0}) (\<lambda>x. if 0 < x then x ^ n else - ((-x) ^ n) :: real)"
- using n by (intro continuous_on_If continuous_intros) auto
+ using True by (intro continuous_on_If continuous_intros) auto
then have "continuous_on UNIV ?f"
- by (rule continuous_on_cong[THEN iffD1, rotated 2]) (auto simp: not_less le_less n)
- then have [simp]: "\<And>x. isCont ?f x"
+ by (rule continuous_on_cong[THEN iffD1, rotated 2]) (auto simp: not_less le_less True)
+ then have [simp]: "isCont ?f x" for x
by (simp add: continuous_on_eq_continuous_at)
-
have "isCont (root n) (?f (root n x))"
- by (rule isCont_inverse_function [where f="?f" and d=1]) (auto simp: root_sgn_power n)
+ by (rule isCont_inverse_function [where f="?f" and d=1]) (auto simp: root_sgn_power True)
then show ?thesis
- by (simp add: sgn_power_root n)
-qed (simp add: root_def[abs_def])
+ by (simp add: sgn_power_root True)
+next
+ case False
+ then show ?thesis
+ by (simp add: root_def[abs_def])
+qed
-lemma tendsto_real_root[tendsto_intros]:
+lemma tendsto_real_root [tendsto_intros]:
"(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) \<longlongrightarrow> root n x) F"
using isCont_tendsto_compose[OF isCont_real_root, of f x F] .
-lemma continuous_real_root[continuous_intros]:
+lemma continuous_real_root [continuous_intros]:
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. root n (f x))"
unfolding continuous_def by (rule tendsto_real_root)
-lemma continuous_on_real_root[continuous_intros]:
+lemma continuous_on_real_root [continuous_intros]:
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))"
unfolding continuous_on_def by (auto intro: tendsto_real_root)
lemma DERIV_real_root:
assumes n: "0 < n"
- assumes x: "0 < x"
+ and x: "0 < x"
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
proof (rule DERIV_inverse_function)
- show "0 < x" using x .
- show "x < x + 1" by simp
+ show "0 < x"
+ using x .
+ show "x < x + 1"
+ by simp
show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
using n by simp
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
by (rule DERIV_pow)
show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
using n x by simp
-qed (rule isCont_real_root)
+ show "isCont (root n) x"
+ by (rule isCont_real_root)
+qed
lemma DERIV_odd_real_root:
assumes n: "odd n"
- assumes x: "x \<noteq> 0"
+ and x: "x \<noteq> 0"
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
proof (rule DERIV_inverse_function)
- show "x - 1 < x" by simp
- show "x < x + 1" by simp
+ show "x - 1 < x"
+ by simp
+ show "x < x + 1"
+ by simp
show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
using n by (simp add: odd_real_root_pow)
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
by (rule DERIV_pow)
show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
using odd_pos [OF n] x by simp
-qed (rule isCont_real_root)
+ show "isCont (root n) x"
+ by (rule isCont_real_root)
+qed
lemma DERIV_even_real_root:
- assumes n: "0 < n" and "even n"
- assumes x: "x < 0"
+ assumes n: "0 < n"
+ and "even n"
+ and x: "x < 0"
shows "DERIV (root n) x :> inverse (- real n * root n x ^ (n - Suc 0))"
proof (rule DERIV_inverse_function)
- show "x - 1 < x" by simp
- show "x < 0" using x .
-next
+ show "x - 1 < x"
+ by simp
+ show "x < 0"
+ using x .
show "\<forall>y. x - 1 < y \<and> y < 0 \<longrightarrow> - (root n y ^ n) = y"
proof (rule allI, rule impI, erule conjE)
fix y assume "x - 1 < y" and "y < 0"
- hence "root n (-y) ^ n = -y" using \<open>0 < n\<close> by simp
+ then have "root n (-y) ^ n = -y" using \<open>0 < n\<close> by simp
with real_root_minus and \<open>even n\<close>
show "- (root n y ^ n) = y" by simp
qed
-next
show "DERIV (\<lambda>x. - (x ^ n)) (root n x) :> - real n * root n x ^ (n - Suc 0)"
by (auto intro!: derivative_eq_intros)
show "- real n * root n x ^ (n - Suc 0) \<noteq> 0"
using n x by simp
-qed (rule isCont_real_root)
+ show "isCont (root n) x"
+ by (rule isCont_real_root)
+qed
lemma DERIV_real_root_generic:
assumes "0 < n" and "x \<noteq> 0"
@@ -326,84 +370,87 @@
and "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"
and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
shows "DERIV (root n) x :> D"
-using assms by (cases "even n", cases "0 < x",
- auto intro: DERIV_real_root[THEN DERIV_cong]
+ using assms
+ by (cases "even n", cases "0 < x",
+ auto intro: DERIV_real_root[THEN DERIV_cong]
DERIV_odd_real_root[THEN DERIV_cong]
DERIV_even_real_root[THEN DERIV_cong])
+
subsection \<open>Square Root\<close>
-definition sqrt :: "real \<Rightarrow> real" where
- "sqrt = root 2"
+definition sqrt :: "real \<Rightarrow> real"
+ where "sqrt = root 2"
-lemma pos2: "0 < (2::nat)" by simp
+lemma pos2: "0 < (2::nat)"
+ by simp
-lemma real_sqrt_unique: "\<lbrakk>y\<^sup>2 = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y"
-unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])
+lemma real_sqrt_unique: "y\<^sup>2 = x \<Longrightarrow> 0 \<le> y \<Longrightarrow> sqrt x = y"
+ unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])
lemma real_sqrt_abs [simp]: "sqrt (x\<^sup>2) = \<bar>x\<bar>"
-apply (rule real_sqrt_unique)
-apply (rule power2_abs)
-apply (rule abs_ge_zero)
-done
+ apply (rule real_sqrt_unique)
+ apply (rule power2_abs)
+ apply (rule abs_ge_zero)
+ done
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<^sup>2 = x"
-unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
+ unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<^sup>2 = x) = (0 \<le> x)"
-apply (rule iffI)
-apply (erule subst)
-apply (rule zero_le_power2)
-apply (erule real_sqrt_pow2)
-done
+ apply (rule iffI)
+ apply (erule subst)
+ apply (rule zero_le_power2)
+ apply (erule real_sqrt_pow2)
+ done
lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
-unfolding sqrt_def by (rule real_root_zero)
+ unfolding sqrt_def by (rule real_root_zero)
lemma real_sqrt_one [simp]: "sqrt 1 = 1"
-unfolding sqrt_def by (rule real_root_one [OF pos2])
+ unfolding sqrt_def by (rule real_root_one [OF pos2])
lemma real_sqrt_four [simp]: "sqrt 4 = 2"
using real_sqrt_abs[of 2] by simp
lemma real_sqrt_minus: "sqrt (- x) = - sqrt x"
-unfolding sqrt_def by (rule real_root_minus)
+ unfolding sqrt_def by (rule real_root_minus)
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"
-unfolding sqrt_def by (rule real_root_mult)
+ unfolding sqrt_def by (rule real_root_mult)
lemma real_sqrt_mult_self[simp]: "sqrt a * sqrt a = \<bar>a\<bar>"
using real_sqrt_abs[of a] unfolding power2_eq_square real_sqrt_mult .
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"
-unfolding sqrt_def by (rule real_root_inverse)
+ unfolding sqrt_def by (rule real_root_inverse)
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"
-unfolding sqrt_def by (rule real_root_divide)
+ unfolding sqrt_def by (rule real_root_divide)
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k"
-unfolding sqrt_def by (rule real_root_power [OF pos2])
+ unfolding sqrt_def by (rule real_root_power [OF pos2])
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x"
-unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])
+ unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x"
-unfolding sqrt_def by (rule real_root_ge_zero)
+ unfolding sqrt_def by (rule real_root_ge_zero)
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y"
-unfolding sqrt_def by (rule real_root_less_mono [OF pos2])
+ unfolding sqrt_def by (rule real_root_less_mono [OF pos2])
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y"
-unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
+ unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
-unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
+ unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)"
-unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
+ unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
-unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
+ unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
lemma real_less_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y\<^sup>2 \<Longrightarrow> sqrt x < y"
using real_sqrt_less_iff[of x "y\<^sup>2"] by simp
@@ -417,7 +464,7 @@
lemma real_less_rsqrt: "x\<^sup>2 < y \<Longrightarrow> x < sqrt y"
using real_sqrt_less_mono[of "x\<^sup>2" y] by simp
-lemma sqrt_le_D: "sqrt x \<le> y \<Longrightarrow> x \<le> y^2"
+lemma sqrt_le_D: "sqrt x \<le> y \<Longrightarrow> x \<le> y\<^sup>2"
by (meson not_le real_less_rsqrt)
lemma sqrt_even_pow2:
@@ -446,73 +493,75 @@
lemma sqrt_add_le_add_sqrt:
assumes "0 \<le> x" "0 \<le> y"
shows "sqrt (x + y) \<le> sqrt x + sqrt y"
-by (rule power2_le_imp_le) (simp_all add: power2_sum assms)
+ by (rule power2_le_imp_le) (simp_all add: power2_sum assms)
lemma isCont_real_sqrt: "isCont sqrt x"
-unfolding sqrt_def by (rule isCont_real_root)
+ unfolding sqrt_def by (rule isCont_real_root)
-lemma tendsto_real_sqrt[tendsto_intros]:
+lemma tendsto_real_sqrt [tendsto_intros]:
"(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) \<longlongrightarrow> sqrt x) F"
unfolding sqrt_def by (rule tendsto_real_root)
-lemma continuous_real_sqrt[continuous_intros]:
+lemma continuous_real_sqrt [continuous_intros]:
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))"
unfolding sqrt_def by (rule continuous_real_root)
-lemma continuous_on_real_sqrt[continuous_intros]:
+lemma continuous_on_real_sqrt [continuous_intros]:
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))"
unfolding sqrt_def by (rule continuous_on_real_root)
lemma DERIV_real_sqrt_generic:
assumes "x \<noteq> 0"
- assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2"
- assumes "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2"
+ and "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2"
+ and "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2"
shows "DERIV sqrt x :> D"
using assms unfolding sqrt_def
by (auto intro!: DERIV_real_root_generic)
-lemma DERIV_real_sqrt:
- "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2"
+lemma DERIV_real_sqrt: "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2"
using DERIV_real_sqrt_generic by simp
declare
DERIV_real_sqrt_generic[THEN DERIV_chain2, derivative_intros]
DERIV_real_root_generic[THEN DERIV_chain2, derivative_intros]
-lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
-apply auto
-apply (cut_tac x = x and y = 0 in linorder_less_linear)
-apply (simp add: zero_less_mult_iff)
-done
+lemma not_real_square_gt_zero [simp]: "\<not> 0 < x * x \<longleftrightarrow> x = 0" for x :: real
+ apply auto
+ apply (cut_tac x = x and y = 0 in linorder_less_linear)
+ apply (simp add: zero_less_mult_iff)
+ done
-lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
-apply (subst power2_eq_square [symmetric])
-apply (rule real_sqrt_abs)
-done
+lemma real_sqrt_abs2 [simp]: "sqrt (x * x) = \<bar>x\<bar>"
+ apply (subst power2_eq_square [symmetric])
+ apply (rule real_sqrt_abs)
+ done
-lemma real_inv_sqrt_pow2: "0 < x ==> (inverse (sqrt x))\<^sup>2 = inverse x"
-by (simp add: power_inverse)
+lemma real_inv_sqrt_pow2: "0 < x \<Longrightarrow> (inverse (sqrt x))\<^sup>2 = inverse x"
+ by (simp add: power_inverse)
-lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
-by simp
+lemma real_sqrt_eq_zero_cancel: "0 \<le> x \<Longrightarrow> sqrt x = 0 \<Longrightarrow> x = 0"
+ by simp
-lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
-by simp
+lemma real_sqrt_ge_one: "1 \<le> x \<Longrightarrow> 1 \<le> sqrt x"
+ by simp
lemma sqrt_divide_self_eq:
assumes nneg: "0 \<le> x"
shows "sqrt x / x = inverse (sqrt x)"
-proof cases
- assume "x=0" thus ?thesis by simp
+proof (cases "x = 0")
+ case True
+ then show ?thesis by simp
next
- assume nz: "x\<noteq>0"
- hence pos: "0<x" using nneg by arith
+ case False
+ then have pos: "0 < x"
+ using nneg by arith
show ?thesis
- proof (rule right_inverse_eq [THEN iffD1, THEN sym])
- show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz)
+ proof (rule right_inverse_eq [THEN iffD1, symmetric])
+ show "sqrt x / x \<noteq> 0"
+ by (simp add: divide_inverse nneg False)
show "inverse (sqrt x) / (sqrt x / x) = 1"
by (simp add: divide_inverse mult.assoc [symmetric]
- power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz)
+ power2_eq_square [symmetric] real_inv_sqrt_pow2 pos False)
qed
qed
@@ -523,44 +572,44 @@
apply (simp add: field_simps)
done
-lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
-apply (simp add: divide_inverse)
-apply (case_tac "r=0")
-apply (auto simp add: ac_simps)
-done
+lemma real_divide_square_eq [simp]: "(r * a) / (r * r) = a / r" for a r :: real
+ apply (simp add: divide_inverse)
+ apply (case_tac "r = 0")
+ apply (auto simp add: ac_simps)
+ done
-lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
-by (simp add: divide_less_eq)
+lemma lemma_real_divide_sqrt_less: "0 < u \<Longrightarrow> u / sqrt 2 < u"
+ by (simp add: divide_less_eq)
-lemma four_x_squared:
- fixes x::real
- shows "4 * x\<^sup>2 = (2 * x)\<^sup>2"
-by (simp add: power2_eq_square)
+lemma four_x_squared: "4 * x\<^sup>2 = (2 * x)\<^sup>2" for x :: real
+ by (simp add: power2_eq_square)
lemma sqrt_at_top: "LIM x at_top. sqrt x :: real :> at_top"
by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="power2"])
(auto intro: eventually_gt_at_top)
+
subsection \<open>Square Root of Sum of Squares\<close>
-lemma sum_squares_bound:
- fixes x:: "'a::linordered_field"
- shows "2*x*y \<le> x^2 + y^2"
+lemma sum_squares_bound: "2 * x * y \<le> x\<^sup>2 + y\<^sup>2" for x y :: "'a::linordered_field"
proof -
- have "(x-y)^2 = x*x - 2*x*y + y*y"
+ have "(x - y)\<^sup>2 = x * x - 2 * x * y + y * y"
by algebra
- then have "0 \<le> x^2 - 2*x*y + y^2"
+ then have "0 \<le> x\<^sup>2 - 2 * x * y + y\<^sup>2"
by (metis sum_power2_ge_zero zero_le_double_add_iff_zero_le_single_add power2_eq_square)
then show ?thesis
by arith
qed
lemma arith_geo_mean:
- fixes u:: "'a::linordered_field" assumes "u\<^sup>2 = x*y" "x\<ge>0" "y\<ge>0" shows "u \<le> (x + y)/2"
- apply (rule power2_le_imp_le)
- using sum_squares_bound assms
- apply (auto simp: zero_le_mult_iff)
- by (auto simp: algebra_simps power2_eq_square)
+ fixes u :: "'a::linordered_field"
+ assumes "u\<^sup>2 = x * y" "x \<ge> 0" "y \<ge> 0"
+ shows "u \<le> (x + y)/2"
+ apply (rule power2_le_imp_le)
+ using sum_squares_bound assms
+ apply (auto simp: zero_le_mult_iff)
+ apply (auto simp: algebra_simps power2_eq_square)
+ done
lemma arith_geo_mean_sqrt:
fixes x::real assumes "x\<ge>0" "y\<ge>0" shows "sqrt(x*y) \<le> (x + y)/2"
@@ -569,75 +618,77 @@
apply (auto simp: zero_le_mult_iff)
done
-lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
- "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2)*(xa\<^sup>2 + ya\<^sup>2))"
+lemma real_sqrt_sum_squares_mult_ge_zero [simp]: "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2)*(xa\<^sup>2 + ya\<^sup>2))"
by (metis real_sqrt_ge_0_iff split_mult_pos_le sum_power2_ge_zero)
lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
- "(sqrt ((x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)))\<^sup>2 = (x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)"
+ "(sqrt ((x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)))\<^sup>2 = (x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)"
by (simp add: zero_le_mult_iff)
lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<^sup>2 + y\<^sup>2) = x \<Longrightarrow> y = 0"
-by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp)
+ by (drule arg_cong [where f = "\<lambda>x. x\<^sup>2"]) simp
lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<^sup>2 + y\<^sup>2) = y \<Longrightarrow> x = 0"
-by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp)
+ by (drule arg_cong [where f = "\<lambda>x. x\<^sup>2"]) simp
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
-by (rule power2_le_imp_le, simp_all)
+ by (rule power2_le_imp_le) simp_all
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
-by (rule power2_le_imp_le, simp_all)
+ by (rule power2_le_imp_le) simp_all
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
-by (rule power2_le_imp_le, simp_all)
+ by (rule power2_le_imp_le) simp_all
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
-by (rule power2_le_imp_le, simp_all)
+ by (rule power2_le_imp_le) simp_all
lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
-by (simp add: power2_eq_square [symmetric])
+ by (simp add: power2_eq_square [symmetric])
lemma real_sqrt_sum_squares_triangle_ineq:
"sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
-apply (rule power2_le_imp_le, simp)
-apply (simp add: power2_sum)
-apply (simp only: mult.assoc distrib_left [symmetric])
-apply (rule mult_left_mono)
-apply (rule power2_le_imp_le)
-apply (simp add: power2_sum power_mult_distrib)
-apply (simp add: ring_distribs)
-apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)", simp)
-apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
-apply (rule zero_le_power2)
-apply (simp add: power2_diff power_mult_distrib)
-apply (simp)
-apply simp
-apply (simp add: add_increasing)
-done
+ apply (rule power2_le_imp_le)
+ apply simp
+ apply (simp add: power2_sum)
+ apply (simp only: mult.assoc distrib_left [symmetric])
+ apply (rule mult_left_mono)
+ apply (rule power2_le_imp_le)
+ apply (simp add: power2_sum power_mult_distrib)
+ apply (simp add: ring_distribs)
+ apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)")
+ apply simp
+ apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
+ apply (rule zero_le_power2)
+ apply (simp add: power2_diff power_mult_distrib)
+ apply simp
+ apply simp
+ apply (simp add: add_increasing)
+ done
-lemma real_sqrt_sum_squares_less:
- "\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
-apply (rule power2_less_imp_less, simp)
-apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
-apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
-apply (simp add: power_divide)
-apply (drule order_le_less_trans [OF abs_ge_zero])
-apply (simp add: zero_less_divide_iff)
-done
+lemma real_sqrt_sum_squares_less: "\<bar>x\<bar> < u / sqrt 2 \<Longrightarrow> \<bar>y\<bar> < u / sqrt 2 \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
+ apply (rule power2_less_imp_less)
+ apply simp
+ apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
+ apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
+ apply (simp add: power_divide)
+ apply (drule order_le_less_trans [OF abs_ge_zero])
+ apply (simp add: zero_less_divide_iff)
+ done
lemma sqrt2_less_2: "sqrt 2 < (2::real)"
- by (metis not_less not_less_iff_gr_or_eq numeral_less_iff real_sqrt_four real_sqrt_le_iff semiring_norm(75) semiring_norm(78) semiring_norm(85))
+ by (metis not_less not_less_iff_gr_or_eq numeral_less_iff real_sqrt_four
+ real_sqrt_le_iff semiring_norm(75) semiring_norm(78) semiring_norm(85))
-text\<open>Needed for the infinitely close relation over the nonstandard
- complex numbers\<close>
+text \<open>Needed for the infinitely close relation over the nonstandard complex numbers.\<close>
lemma lemma_sqrt_hcomplex_capprox:
- "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<^sup>2 + y\<^sup>2) < u"
+ "0 < u \<Longrightarrow> x < u/2 \<Longrightarrow> y < u/2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
apply (rule real_sqrt_sum_squares_less)
apply (auto simp add: abs_if field_simps)
apply (rule le_less_trans [where y = "x*2"])
- using less_eq_real_def sqrt2_less_2 apply force
+ using less_eq_real_def sqrt2_less_2
+ apply force
apply assumption
apply (rule le_less_trans [where y = "y*2"])
using less_eq_real_def sqrt2_less_2 mult_le_cancel_left
@@ -652,13 +703,15 @@
show "(\<lambda>x. sqrt (2 / x)) \<longlonglongrightarrow> sqrt 0"
by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
(simp_all add: at_infinity_eq_at_top_bot)
- { fix n :: nat assume "2 < n"
- have "1 + (real (n - 1) * n) / 2 * x n^2 = 1 + of_nat (n choose 2) * x n^2"
+ have "x n \<le> sqrt (2 / real n)" if "2 < n" for n :: nat
+ proof -
+ have "1 + (real (n - 1) * n) / 2 * (x n)\<^sup>2 = 1 + of_nat (n choose 2) * (x n)\<^sup>2"
by (auto simp add: choose_two of_nat_div mod_eq_0_iff_dvd)
also have "\<dots> \<le> (\<Sum>k\<in>{0, 2}. of_nat (n choose k) * x n^k)"
by (simp add: x_def)
also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
- using \<open>2 < n\<close> by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
+ using \<open>2 < n\<close>
+ by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
also have "\<dots> = (x n + 1) ^ n"
by (simp add: binomial_ring)
also have "\<dots> = n"
@@ -667,8 +720,9 @@
by simp
then have "(x n)\<^sup>2 \<le> 2 / real n"
using \<open>2 < n\<close> unfolding mult_le_cancel_left by (simp add: field_simps)
- from real_sqrt_le_mono[OF this] have "x n \<le> sqrt (2 / real n)"
- by simp }
+ from real_sqrt_le_mono[OF this] show ?thesis
+ by simp
+ qed
then show "eventually (\<lambda>n. x n \<le> sqrt (2 / real n)) sequentially"
by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
show "eventually (\<lambda>n. sqrt 0 \<le> x n) sequentially"
@@ -682,47 +736,53 @@
assumes "0 < c"
shows "(\<lambda>n. root n c) \<longlonglongrightarrow> 1"
proof -
- { fix c :: real assume "1 \<le> c"
+ have ge_1: "(\<lambda>n. root n c) \<longlonglongrightarrow> 1" if "1 \<le> c" for c :: real
+ proof -
define x where "x n = root n c - 1" for n
have "x \<longlonglongrightarrow> 0"
proof (rule tendsto_sandwich[OF _ _ tendsto_const])
show "(\<lambda>n. c / n) \<longlonglongrightarrow> 0"
by (intro tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
- (simp_all add: at_infinity_eq_at_top_bot)
- { fix n :: nat assume "1 < n"
+ (simp_all add: at_infinity_eq_at_top_bot)
+ have "x n \<le> c / n" if "1 < n" for n :: nat
+ proof -
have "1 + x n * n = 1 + of_nat (n choose 1) * x n^1"
by (simp add: choose_one)
also have "\<dots> \<le> (\<Sum>k\<in>{0, 1}. of_nat (n choose k) * x n^k)"
by (simp add: x_def)
also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
- using \<open>1 < n\<close> \<open>1 \<le> c\<close> by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
+ using \<open>1 < n\<close> \<open>1 \<le> c\<close>
+ by (intro setsum_mono2)
+ (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
also have "\<dots> = (x n + 1) ^ n"
by (simp add: binomial_ring)
also have "\<dots> = c"
using \<open>1 < n\<close> \<open>1 \<le> c\<close> by (simp add: x_def)
- finally have "x n \<le> c / n"
- using \<open>1 \<le> c\<close> \<open>1 < n\<close> by (simp add: field_simps) }
+ finally show ?thesis
+ using \<open>1 \<le> c\<close> \<open>1 < n\<close> by (simp add: field_simps)
+ qed
then show "eventually (\<lambda>n. x n \<le> c / n) sequentially"
by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
show "eventually (\<lambda>n. 0 \<le> x n) sequentially"
- using \<open>1 \<le> c\<close> by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
+ using \<open>1 \<le> c\<close>
+ by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
qed
- from tendsto_add[OF this tendsto_const[of 1]] have "(\<lambda>n. root n c) \<longlonglongrightarrow> 1"
- by (simp add: x_def) }
- note ge_1 = this
-
+ from tendsto_add[OF this tendsto_const[of 1]] show ?thesis
+ by (simp add: x_def)
+ qed
show ?thesis
- proof cases
- assume "1 \<le> c" with ge_1 show ?thesis by blast
+ proof (cases "1 \<le> c")
+ case True
+ with ge_1 show ?thesis by blast
next
- assume "\<not> 1 \<le> c"
+ case False
with \<open>0 < c\<close> have "1 \<le> 1 / c"
by simp
then have "(\<lambda>n. 1 / root n (1 / c)) \<longlonglongrightarrow> 1 / 1"
by (intro tendsto_divide tendsto_const ge_1 \<open>1 \<le> 1 / c\<close> one_neq_zero)
then show ?thesis
by (rule filterlim_cong[THEN iffD1, rotated 3])
- (auto intro!: exI[of _ 1] simp: eventually_sequentially real_root_divide)
+ (auto intro!: exI[of _ 1] simp: eventually_sequentially real_root_divide)
qed
qed
--- a/src/HOL/Transcendental.thy Tue Jul 12 21:53:56 2016 +0200
+++ b/src/HOL/Transcendental.thy Tue Jul 12 22:54:37 2016 +0200
@@ -12,10 +12,10 @@
text \<open>A fact theorem on reals.\<close>
-lemma square_fact_le_2_fact:
- shows "fact n * fact n \<le> (fact (2 * n) :: real)"
+lemma square_fact_le_2_fact: "fact n * fact n \<le> (fact (2 * n) :: real)"
proof (induct n)
- case 0 then show ?case by simp
+ case 0
+ then show ?case by simp
next
case (Suc n)
have "(fact (Suc n)) * (fact (Suc n)) = of_nat (Suc n) * of_nat (Suc n) * (fact n * fact n :: real)"
@@ -28,7 +28,6 @@
finally show ?case .
qed
-
lemma fact_in_Reals: "fact n \<in> \<real>"
by (induction n) auto
@@ -38,33 +37,33 @@
lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)"
by (simp add: pochhammer_setprod)
-lemma norm_fact [simp]:
- "norm (fact n :: 'a :: {real_normed_algebra_1}) = fact n"
+lemma norm_fact [simp]: "norm (fact n :: 'a::real_normed_algebra_1) = fact n"
proof -
- have "(fact n :: 'a) = of_real (fact n)" by simp
- also have "norm \<dots> = fact n" by (subst norm_of_real) simp
+ have "(fact n :: 'a) = of_real (fact n)"
+ by simp
+ also have "norm \<dots> = fact n"
+ by (subst norm_of_real) simp
finally show ?thesis .
qed
lemma root_test_convergence:
fixes f :: "nat \<Rightarrow> 'a::banach"
assumes f: "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> x" \<comment> "could be weakened to lim sup"
- assumes "x < 1"
+ and "x < 1"
shows "summable f"
proof -
have "0 \<le> x"
by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
from \<open>x < 1\<close> obtain z where z: "x < z" "z < 1"
by (metis dense)
- from f \<open>x < z\<close>
- have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
+ from f \<open>x < z\<close> have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
by (rule order_tendstoD)
then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
using eventually_ge_at_top
proof eventually_elim
- fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
- from power_strict_mono[OF less, of n] n
- show "norm (f n) \<le> z ^ n"
+ fix n
+ assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
+ from power_strict_mono[OF less, of n] n show "norm (f n) \<le> z ^ n"
by simp
qed
then show "summable f"
@@ -72,30 +71,30 @@
using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _ summable_geometric])
qed
+
subsection \<open>Properties of Power Series\<close>
-lemma powser_zero [simp]:
- fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra_1"
- shows "(\<Sum>n. f n * 0 ^ n) = f 0"
+lemma powser_zero [simp]: "(\<Sum>n. f n * 0 ^ n) = f 0"
+ for f :: "nat \<Rightarrow> 'a::real_normed_algebra_1"
proof -
have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
- thus ?thesis unfolding One_nat_def by simp
+ then show ?thesis
+ by (simp add: One_nat_def)
qed
-lemma powser_sums_zero:
- fixes a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
- shows "(\<lambda>n. a n * 0^n) sums a 0"
- using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"]
- by simp
-
-lemma powser_sums_zero_iff [simp]:
- fixes a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
- shows "(\<lambda>n. a n * 0^n) sums x \<longleftrightarrow> a 0 = x"
-using powser_sums_zero sums_unique2 by blast
-
-text\<open>Power series has a circle or radius of convergence: if it sums for @{term
- x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.\<close>
+lemma powser_sums_zero: "(\<lambda>n. a n * 0^n) sums a 0"
+ for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
+ using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"]
+ by simp
+
+lemma powser_sums_zero_iff [simp]: "(\<lambda>n. a n * 0^n) sums x \<longleftrightarrow> a 0 = x"
+ for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
+ using powser_sums_zero sums_unique2 by blast
+
+text \<open>
+ Power series has a circle or radius of convergence: if it sums for \<open>x\<close>,
+ then it sums absolutely for \<open>z\<close> with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.\<close>
lemma powser_insidea:
fixes x z :: "'a::real_normed_div_algebra"