author | wenzelm |
Tue, 03 Mar 2009 14:07:43 +0100 | |
changeset 30211 | 556d1810cdad |
parent 28952 | 15a4b2cf8c34 |
child 30273 | ecd6f0ca62ea |
permissions | -rw-r--r-- |
12196 | 1 |
(* Title : NthRoot.thy |
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Author : Jacques D. Fleuriot |
|
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Copyright : 1998 University of Cambridge |
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Conversion to Isar and new proofs by Lawrence C Paulson, 2004 |
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*) |
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||
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header {* Nth Roots of Real Numbers *} |
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theory NthRoot |
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imports Parity Deriv |
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begin |
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|
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subsection {* Existence of Nth Root *} |
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|
23009
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text {* Existence follows from the Intermediate Value Theorem *} |
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lemma realpow_pos_nth: |
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assumes n: "0 < n" |
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assumes a: "0 < a" |
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shows "\<exists>r>0. r ^ n = (a::real)" |
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proof - |
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have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a" |
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proof (rule IVT) |
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show "0 ^ n \<le> a" using n a by (simp add: power_0_left) |
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show "0 \<le> max 1 a" by simp |
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from n have n1: "1 \<le> n" by simp |
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have "a \<le> max 1 a ^ 1" by simp |
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also have "max 1 a ^ 1 \<le> max 1 a ^ n" |
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using n1 by (rule power_increasing, simp) |
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finally show "a \<le> max 1 a ^ n" . |
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show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r" |
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by (simp add: isCont_power) |
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qed |
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then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast |
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with n a have "r \<noteq> 0" by (auto simp add: power_0_left) |
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with r have "0 < r \<and> r ^ n = a" by simp |
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thus ?thesis .. |
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qed |
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(* Used by Integration/RealRandVar.thy in AFP *) |
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lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a" |
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by (blast intro: realpow_pos_nth) |
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text {* Uniqueness of nth positive root *} |
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lemma realpow_pos_nth_unique: |
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"\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)" |
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apply (auto intro!: realpow_pos_nth) |
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apply (rule_tac n=n in power_eq_imp_eq_base, simp_all) |
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done |
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||
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subsection {* Nth Root *} |
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text {* We define roots of negative reals such that |
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@{term "root n (- x) = - root n x"}. This allows |
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us to omit side conditions from many theorems. *} |
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definition |
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root :: "[nat, real] \<Rightarrow> real" where |
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"root n x = (if 0 < x then (THE u. 0 < u \<and> u ^ n = x) else |
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if x < 0 then - (THE u. 0 < u \<and> u ^ n = - x) else 0)" |
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lemma real_root_zero [simp]: "root n 0 = 0" |
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unfolding root_def by simp |
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|
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lemma real_root_minus: "0 < n \<Longrightarrow> root n (- x) = - root n x" |
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unfolding root_def by simp |
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|
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lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x" |
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apply (simp add: root_def) |
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apply (drule (1) realpow_pos_nth_unique) |
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apply (erule theI' [THEN conjunct1]) |
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done |
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lemma real_root_pow_pos: (* TODO: rename *) |
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"\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x" |
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apply (simp add: root_def) |
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apply (drule (1) realpow_pos_nth_unique) |
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apply (erule theI' [THEN conjunct2]) |
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done |
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|
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lemma real_root_pow_pos2 [simp]: (* TODO: rename *) |
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"\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x" |
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by (auto simp add: order_le_less real_root_pow_pos) |
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lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x" |
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apply (rule_tac x=0 and y=x in linorder_le_cases) |
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apply (erule (1) real_root_pow_pos2 [OF odd_pos]) |
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apply (subgoal_tac "root n (- x) ^ n = - x") |
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apply (simp add: real_root_minus odd_pos) |
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apply (simp add: odd_pos) |
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done |
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||
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lemma real_root_ge_zero: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> 0 \<le> root n x" |
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by (auto simp add: order_le_less real_root_gt_zero) |
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|
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lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x" |
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apply (subgoal_tac "0 \<le> x ^ n") |
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apply (subgoal_tac "0 \<le> root n (x ^ n)") |
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apply (subgoal_tac "root n (x ^ n) ^ n = x ^ n") |
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apply (erule (3) power_eq_imp_eq_base) |
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102 |
apply (erule (1) real_root_pow_pos2) |
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apply (erule (1) real_root_ge_zero) |
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apply (erule zero_le_power) |
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105 |
done |
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|
23046 | 107 |
lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x" |
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apply (rule_tac x=0 and y=x in linorder_le_cases) |
|
109 |
apply (erule (1) real_root_power_cancel [OF odd_pos]) |
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apply (subgoal_tac "root n ((- x) ^ n) = - x") |
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apply (simp add: real_root_minus odd_pos) |
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112 |
apply (erule real_root_power_cancel [OF odd_pos], simp) |
|
113 |
done |
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114 |
||
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lemma real_root_pos_unique: |
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"\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" |
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by (erule subst, rule real_root_power_cancel) |
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118 |
|
23046 | 119 |
lemma odd_real_root_unique: |
120 |
"\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" |
|
121 |
by (erule subst, rule odd_real_root_power_cancel) |
|
122 |
||
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lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1" |
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by (simp add: real_root_pos_unique) |
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|
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126 |
text {* Root function is strictly monotonic, hence injective *} |
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127 |
|
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128 |
lemma real_root_less_mono_lemma: |
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129 |
"\<lbrakk>0 < n; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" |
22856 | 130 |
apply (subgoal_tac "0 \<le> y") |
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apply (subgoal_tac "root n x ^ n < root n y ^ n") |
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132 |
apply (erule power_less_imp_less_base) |
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133 |
apply (erule (1) real_root_ge_zero) |
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134 |
apply simp |
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135 |
apply simp |
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136 |
done |
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137 |
|
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138 |
lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" |
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139 |
apply (cases "0 \<le> x") |
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apply (erule (2) real_root_less_mono_lemma) |
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apply (cases "0 \<le> y") |
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apply (rule_tac y=0 in order_less_le_trans) |
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apply (subgoal_tac "0 < root n (- x)") |
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apply (simp add: real_root_minus) |
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apply (simp add: real_root_gt_zero) |
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apply (simp add: real_root_ge_zero) |
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apply (subgoal_tac "root n (- y) < root n (- x)") |
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apply (simp add: real_root_minus) |
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apply (simp add: real_root_less_mono_lemma) |
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done |
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|
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lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y" |
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by (auto simp add: order_le_less real_root_less_mono) |
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|
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lemma real_root_less_iff [simp]: |
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"0 < n \<Longrightarrow> (root n x < root n y) = (x < y)" |
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apply (cases "x < y") |
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apply (simp add: real_root_less_mono) |
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apply (simp add: linorder_not_less real_root_le_mono) |
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done |
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|
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lemma real_root_le_iff [simp]: |
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"0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)" |
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apply (cases "x \<le> y") |
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apply (simp add: real_root_le_mono) |
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apply (simp add: linorder_not_le real_root_less_mono) |
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done |
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|
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lemma real_root_eq_iff [simp]: |
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"0 < n \<Longrightarrow> (root n x = root n y) = (x = y)" |
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by (simp add: order_eq_iff) |
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|
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lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified] |
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lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified] |
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lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified] |
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lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified] |
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lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified] |
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|
23257 | 179 |
lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> (1 < root n y) = (1 < y)" |
180 |
by (insert real_root_less_iff [where x=1], simp) |
|
181 |
||
182 |
lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> (root n x < 1) = (x < 1)" |
|
183 |
by (insert real_root_less_iff [where y=1], simp) |
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184 |
||
185 |
lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> (1 \<le> root n y) = (1 \<le> y)" |
|
186 |
by (insert real_root_le_iff [where x=1], simp) |
|
187 |
||
188 |
lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> (root n x \<le> 1) = (x \<le> 1)" |
|
189 |
by (insert real_root_le_iff [where y=1], simp) |
|
190 |
||
191 |
lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> (root n x = 1) = (x = 1)" |
|
192 |
by (insert real_root_eq_iff [where y=1], simp) |
|
193 |
||
194 |
text {* Roots of roots *} |
|
195 |
||
196 |
lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x" |
|
197 |
by (simp add: odd_real_root_unique) |
|
198 |
||
199 |
lemma real_root_pos_mult_exp: |
|
200 |
"\<lbrakk>0 < m; 0 < n; 0 < x\<rbrakk> \<Longrightarrow> root (m * n) x = root m (root n x)" |
|
201 |
by (rule real_root_pos_unique, simp_all add: power_mult) |
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202 |
||
203 |
lemma real_root_mult_exp: |
|
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"\<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> root (m * n) x = root m (root n x)" |
|
205 |
apply (rule linorder_cases [where x=x and y=0]) |
|
206 |
apply (subgoal_tac "root (m * n) (- x) = root m (root n (- x))") |
|
207 |
apply (simp add: real_root_minus) |
|
208 |
apply (simp_all add: real_root_pos_mult_exp) |
|
209 |
done |
|
210 |
||
211 |
lemma real_root_commute: |
|
212 |
"\<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> root m (root n x) = root n (root m x)" |
|
213 |
by (simp add: real_root_mult_exp [symmetric] mult_commute) |
|
214 |
||
215 |
text {* Monotonicity in first argument *} |
|
216 |
||
217 |
lemma real_root_strict_decreasing: |
|
218 |
"\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x" |
|
219 |
apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp) |
|
220 |
apply (simp add: real_root_commute power_strict_increasing |
|
221 |
del: real_root_pow_pos2) |
|
222 |
done |
|
223 |
||
224 |
lemma real_root_strict_increasing: |
|
225 |
"\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x" |
|
226 |
apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp) |
|
227 |
apply (simp add: real_root_commute power_strict_decreasing |
|
228 |
del: real_root_pow_pos2) |
|
229 |
done |
|
230 |
||
231 |
lemma real_root_decreasing: |
|
232 |
"\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x" |
|
233 |
by (auto simp add: order_le_less real_root_strict_decreasing) |
|
234 |
||
235 |
lemma real_root_increasing: |
|
236 |
"\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x" |
|
237 |
by (auto simp add: order_le_less real_root_strict_increasing) |
|
238 |
||
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text {* Roots of multiplication and division *} |
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240 |
|
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241 |
lemma real_root_mult_lemma: |
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242 |
"\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> root n (x * y) = root n x * root n y" |
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243 |
by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib) |
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244 |
|
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245 |
lemma real_root_inverse_lemma: |
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246 |
"\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (inverse x) = inverse (root n x)" |
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by (simp add: real_root_pos_unique power_inverse [symmetric]) |
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248 |
|
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249 |
lemma real_root_mult: |
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250 |
assumes n: "0 < n" |
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251 |
shows "root n (x * y) = root n x * root n y" |
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252 |
proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases) |
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253 |
assume "0 \<le> x" and "0 \<le> y" |
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254 |
thus ?thesis by (rule real_root_mult_lemma [OF n]) |
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255 |
next |
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256 |
assume "0 \<le> x" and "y \<le> 0" |
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257 |
hence "0 \<le> x" and "0 \<le> - y" by simp_all |
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258 |
hence "root n (x * - y) = root n x * root n (- y)" |
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259 |
by (rule real_root_mult_lemma [OF n]) |
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260 |
thus ?thesis by (simp add: real_root_minus [OF n]) |
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261 |
next |
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262 |
assume "x \<le> 0" and "0 \<le> y" |
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263 |
hence "0 \<le> - x" and "0 \<le> y" by simp_all |
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264 |
hence "root n (- x * y) = root n (- x) * root n y" |
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265 |
by (rule real_root_mult_lemma [OF n]) |
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266 |
thus ?thesis by (simp add: real_root_minus [OF n]) |
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267 |
next |
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268 |
assume "x \<le> 0" and "y \<le> 0" |
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269 |
hence "0 \<le> - x" and "0 \<le> - y" by simp_all |
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270 |
hence "root n (- x * - y) = root n (- x) * root n (- y)" |
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271 |
by (rule real_root_mult_lemma [OF n]) |
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272 |
thus ?thesis by (simp add: real_root_minus [OF n]) |
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273 |
qed |
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274 |
|
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275 |
lemma real_root_inverse: |
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276 |
assumes n: "0 < n" |
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277 |
shows "root n (inverse x) = inverse (root n x)" |
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278 |
proof (rule linorder_le_cases) |
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279 |
assume "0 \<le> x" |
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280 |
thus ?thesis by (rule real_root_inverse_lemma [OF n]) |
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281 |
next |
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282 |
assume "x \<le> 0" |
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283 |
hence "0 \<le> - x" by simp |
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changeset
|
284 |
hence "root n (inverse (- x)) = inverse (root n (- x))" |
617140080e6a
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huffman
parents:
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diff
changeset
|
285 |
by (rule real_root_inverse_lemma [OF n]) |
617140080e6a
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huffman
parents:
22943
diff
changeset
|
286 |
thus ?thesis by (simp add: real_root_minus [OF n]) |
617140080e6a
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huffman
parents:
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diff
changeset
|
287 |
qed |
22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset
|
288 |
|
22956
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huffman
parents:
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changeset
|
289 |
lemma real_root_divide: |
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huffman
parents:
22943
diff
changeset
|
290 |
"0 < n \<Longrightarrow> root n (x / y) = root n x / root n y" |
617140080e6a
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huffman
parents:
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diff
changeset
|
291 |
by (simp add: divide_inverse real_root_mult real_root_inverse) |
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huffman
parents:
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diff
changeset
|
292 |
|
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huffman
parents:
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diff
changeset
|
293 |
lemma real_root_power: |
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huffman
parents:
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diff
changeset
|
294 |
"0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k" |
617140080e6a
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huffman
parents:
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diff
changeset
|
295 |
by (induct k, simp_all add: real_root_mult) |
22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset
|
296 |
|
23042
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huffman
parents:
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diff
changeset
|
297 |
lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>" |
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huffman
parents:
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diff
changeset
|
298 |
by (simp add: abs_if real_root_minus) |
492514b39956
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huffman
parents:
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diff
changeset
|
299 |
|
492514b39956
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huffman
parents:
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diff
changeset
|
300 |
text {* Continuity and derivatives *} |
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huffman
parents:
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diff
changeset
|
301 |
|
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huffman
parents:
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diff
changeset
|
302 |
lemma isCont_root_pos: |
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huffman
parents:
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diff
changeset
|
303 |
assumes n: "0 < n" |
492514b39956
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huffman
parents:
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diff
changeset
|
304 |
assumes x: "0 < x" |
492514b39956
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huffman
parents:
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diff
changeset
|
305 |
shows "isCont (root n) x" |
492514b39956
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huffman
parents:
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diff
changeset
|
306 |
proof - |
492514b39956
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huffman
parents:
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diff
changeset
|
307 |
have "isCont (root n) (root n x ^ n)" |
492514b39956
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huffman
parents:
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diff
changeset
|
308 |
proof (rule isCont_inverse_function [where f="\<lambda>a. a ^ n"]) |
492514b39956
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huffman
parents:
23009
diff
changeset
|
309 |
show "0 < root n x" using n x by simp |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
310 |
show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> root n (z ^ n) = z" |
492514b39956
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huffman
parents:
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diff
changeset
|
311 |
by (simp add: abs_le_iff real_root_power_cancel n) |
492514b39956
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huffman
parents:
23009
diff
changeset
|
312 |
show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> isCont (\<lambda>a. a ^ n) z" |
23069
cdfff0241c12
rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents:
23049
diff
changeset
|
313 |
by (simp add: isCont_power) |
23042
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huffman
parents:
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changeset
|
314 |
qed |
492514b39956
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huffman
parents:
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diff
changeset
|
315 |
thus ?thesis using n x by simp |
492514b39956
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huffman
parents:
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diff
changeset
|
316 |
qed |
492514b39956
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huffman
parents:
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diff
changeset
|
317 |
|
492514b39956
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huffman
parents:
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diff
changeset
|
318 |
lemma isCont_root_neg: |
492514b39956
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huffman
parents:
23009
diff
changeset
|
319 |
"\<lbrakk>0 < n; x < 0\<rbrakk> \<Longrightarrow> isCont (root n) x" |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
320 |
apply (subgoal_tac "isCont (\<lambda>x. - root n (- x)) x") |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
321 |
apply (simp add: real_root_minus) |
23069
cdfff0241c12
rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents:
23049
diff
changeset
|
322 |
apply (rule isCont_o2 [OF isCont_minus [OF isCont_ident]]) |
23042
492514b39956
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huffman
parents:
23009
diff
changeset
|
323 |
apply (simp add: isCont_minus isCont_root_pos) |
492514b39956
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huffman
parents:
23009
diff
changeset
|
324 |
done |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
325 |
|
492514b39956
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huffman
parents:
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diff
changeset
|
326 |
lemma isCont_root_zero: |
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huffman
parents:
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diff
changeset
|
327 |
"0 < n \<Longrightarrow> isCont (root n) 0" |
492514b39956
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huffman
parents:
23009
diff
changeset
|
328 |
unfolding isCont_def |
492514b39956
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huffman
parents:
23009
diff
changeset
|
329 |
apply (rule LIM_I) |
492514b39956
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huffman
parents:
23009
diff
changeset
|
330 |
apply (rule_tac x="r ^ n" in exI, safe) |
25875 | 331 |
apply (simp) |
23042
492514b39956
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huffman
parents:
23009
diff
changeset
|
332 |
apply (simp add: real_root_abs [symmetric]) |
492514b39956
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huffman
parents:
23009
diff
changeset
|
333 |
apply (rule_tac n="n" in power_less_imp_less_base, simp_all) |
492514b39956
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huffman
parents:
23009
diff
changeset
|
334 |
done |
492514b39956
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huffman
parents:
23009
diff
changeset
|
335 |
|
492514b39956
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huffman
parents:
23009
diff
changeset
|
336 |
lemma isCont_real_root: "0 < n \<Longrightarrow> isCont (root n) x" |
492514b39956
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huffman
parents:
23009
diff
changeset
|
337 |
apply (rule_tac x=x and y=0 in linorder_cases) |
492514b39956
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huffman
parents:
23009
diff
changeset
|
338 |
apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero) |
492514b39956
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huffman
parents:
23009
diff
changeset
|
339 |
done |
492514b39956
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huffman
parents:
23009
diff
changeset
|
340 |
|
492514b39956
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huffman
parents:
23009
diff
changeset
|
341 |
lemma DERIV_real_root: |
492514b39956
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huffman
parents:
23009
diff
changeset
|
342 |
assumes n: "0 < n" |
492514b39956
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huffman
parents:
23009
diff
changeset
|
343 |
assumes x: "0 < x" |
492514b39956
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huffman
parents:
23009
diff
changeset
|
344 |
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))" |
492514b39956
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huffman
parents:
23009
diff
changeset
|
345 |
proof (rule DERIV_inverse_function) |
23044 | 346 |
show "0 < x" using x . |
347 |
show "x < x + 1" by simp |
|
348 |
show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" |
|
23042
492514b39956
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huffman
parents:
23009
diff
changeset
|
349 |
using n by simp |
492514b39956
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huffman
parents:
23009
diff
changeset
|
350 |
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)" |
492514b39956
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huffman
parents:
23009
diff
changeset
|
351 |
by (rule DERIV_pow) |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
352 |
show "real n * root n x ^ (n - Suc 0) \<noteq> 0" |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
353 |
using n x by simp |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
354 |
show "isCont (root n) x" |
23441 | 355 |
using n by (rule isCont_real_root) |
23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
356 |
qed |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
357 |
|
23046 | 358 |
lemma DERIV_odd_real_root: |
359 |
assumes n: "odd n" |
|
360 |
assumes x: "x \<noteq> 0" |
|
361 |
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))" |
|
362 |
proof (rule DERIV_inverse_function) |
|
363 |
show "x - 1 < x" by simp |
|
364 |
show "x < x + 1" by simp |
|
365 |
show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" |
|
366 |
using n by (simp add: odd_real_root_pow) |
|
367 |
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)" |
|
368 |
by (rule DERIV_pow) |
|
369 |
show "real n * root n x ^ (n - Suc 0) \<noteq> 0" |
|
370 |
using odd_pos [OF n] x by simp |
|
371 |
show "isCont (root n) x" |
|
372 |
using odd_pos [OF n] by (rule isCont_real_root) |
|
373 |
qed |
|
374 |
||
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
375 |
subsection {* Square Root *} |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
376 |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
377 |
definition |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
378 |
sqrt :: "real \<Rightarrow> real" where |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
379 |
"sqrt = root 2" |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
380 |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
381 |
lemma pos2: "0 < (2::nat)" by simp |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
382 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
383 |
lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
384 |
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2]) |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
385 |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
386 |
lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
387 |
apply (rule real_sqrt_unique) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
388 |
apply (rule power2_abs) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
389 |
apply (rule abs_ge_zero) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
390 |
done |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
391 |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
392 |
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
393 |
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2]) |
22856 | 394 |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
395 |
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" |
22856 | 396 |
apply (rule iffI) |
397 |
apply (erule subst) |
|
398 |
apply (rule zero_le_power2) |
|
399 |
apply (erule real_sqrt_pow2) |
|
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
400 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
401 |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
402 |
lemma real_sqrt_zero [simp]: "sqrt 0 = 0" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
403 |
unfolding sqrt_def by (rule real_root_zero) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
404 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
405 |
lemma real_sqrt_one [simp]: "sqrt 1 = 1" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
406 |
unfolding sqrt_def by (rule real_root_one [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
407 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
408 |
lemma real_sqrt_minus: "sqrt (- x) = - sqrt x" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
409 |
unfolding sqrt_def by (rule real_root_minus [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
410 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
411 |
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
412 |
unfolding sqrt_def by (rule real_root_mult [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
413 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
414 |
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
415 |
unfolding sqrt_def by (rule real_root_inverse [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
416 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
417 |
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
418 |
unfolding sqrt_def by (rule real_root_divide [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
419 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
420 |
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
421 |
unfolding sqrt_def by (rule real_root_power [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
422 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
423 |
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
424 |
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
425 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
426 |
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
427 |
unfolding sqrt_def by (rule real_root_ge_zero [OF pos2]) |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
428 |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
429 |
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
430 |
unfolding sqrt_def by (rule real_root_less_mono [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
431 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
432 |
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
433 |
unfolding sqrt_def by (rule real_root_le_mono [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
434 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
435 |
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
436 |
unfolding sqrt_def by (rule real_root_less_iff [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
437 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
438 |
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
439 |
unfolding sqrt_def by (rule real_root_le_iff [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
440 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
441 |
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
442 |
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
443 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
444 |
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
445 |
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
446 |
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
447 |
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
448 |
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
449 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
450 |
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
451 |
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
452 |
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
453 |
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
454 |
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified] |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
455 |
|
23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
456 |
lemma isCont_real_sqrt: "isCont sqrt x" |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
457 |
unfolding sqrt_def by (rule isCont_real_root [OF pos2]) |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
458 |
|
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
459 |
lemma DERIV_real_sqrt: |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
460 |
"0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2" |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
461 |
unfolding sqrt_def by (rule DERIV_real_root [OF pos2, simplified]) |
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset
|
462 |
|
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
463 |
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
464 |
apply auto |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
465 |
apply (cut_tac x = x and y = 0 in linorder_less_linear) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
466 |
apply (simp add: zero_less_mult_iff) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
467 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
468 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
469 |
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" |
22856 | 470 |
apply (subst power2_eq_square [symmetric]) |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
471 |
apply (rule real_sqrt_abs) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
472 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
473 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
474 |
lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>" |
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
475 |
by simp (* TODO: delete *) |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
476 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
477 |
lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0" |
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
478 |
by simp (* TODO: delete *) |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
479 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
480 |
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" |
22856 | 481 |
by (simp add: power_inverse [symmetric]) |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
482 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
483 |
lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0" |
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
484 |
by simp |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
485 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
486 |
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" |
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
487 |
by simp |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
488 |
|
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
489 |
lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
490 |
by simp |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
491 |
|
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
492 |
lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
493 |
by simp |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
494 |
|
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
495 |
lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
496 |
by simp |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
497 |
|
22443 | 498 |
lemma sqrt_divide_self_eq: |
499 |
assumes nneg: "0 \<le> x" |
|
500 |
shows "sqrt x / x = inverse (sqrt x)" |
|
501 |
proof cases |
|
502 |
assume "x=0" thus ?thesis by simp |
|
503 |
next |
|
504 |
assume nz: "x\<noteq>0" |
|
505 |
hence pos: "0<x" using nneg by arith |
|
506 |
show ?thesis |
|
507 |
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) |
|
508 |
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) |
|
509 |
show "inverse (sqrt x) / (sqrt x / x) = 1" |
|
510 |
by (simp add: divide_inverse mult_assoc [symmetric] |
|
511 |
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) |
|
512 |
qed |
|
513 |
qed |
|
514 |
||
22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset
|
515 |
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r" |
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset
|
516 |
apply (simp add: divide_inverse) |
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset
|
517 |
apply (case_tac "r=0") |
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset
|
518 |
apply (auto simp add: mult_ac) |
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset
|
519 |
done |
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset
|
520 |
|
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
521 |
lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
522 |
by (simp add: divide_less_eq mult_compare_simps) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
523 |
|
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
524 |
lemma four_x_squared: |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
525 |
fixes x::real |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
526 |
shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
527 |
by (simp add: power2_eq_square) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
528 |
|
22856 | 529 |
subsection {* Square Root of Sum of Squares *} |
530 |
||
531 |
lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)" |
|
22968 | 532 |
by (rule real_sqrt_ge_zero [OF sum_squares_ge_zero]) |
22856 | 533 |
|
534 |
lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
|
22961 | 535 |
by simp |
22856 | 536 |
|
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
537 |
declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp] |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
538 |
|
22856 | 539 |
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: |
540 |
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" |
|
541 |
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) |
|
542 |
||
543 |
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: |
|
544 |
"sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
545 |
by (auto simp add: zero_le_mult_iff) |
22856 | 546 |
|
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
547 |
lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<twosuperior> + y\<twosuperior>) = x \<Longrightarrow> y = 0" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
548 |
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
549 |
|
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
550 |
lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<twosuperior> + y\<twosuperior>) = y \<Longrightarrow> x = 0" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
551 |
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
552 |
|
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
553 |
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
22856 | 554 |
by (rule power2_le_imp_le, simp_all) |
555 |
||
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
556 |
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
557 |
by (rule power2_le_imp_le, simp_all) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
558 |
|
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
559 |
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
22856 | 560 |
by (rule power2_le_imp_le, simp_all) |
561 |
||
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
562 |
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
563 |
by (rule power2_le_imp_le, simp_all) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
564 |
|
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
565 |
lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
566 |
by (simp add: power2_eq_square [symmetric]) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
567 |
|
22858 | 568 |
lemma power2_sum: |
569 |
fixes x y :: "'a::{number_ring,recpower}" |
|
570 |
shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23475
diff
changeset
|
571 |
by (simp add: ring_distribs power2_eq_square) |
22858 | 572 |
|
573 |
lemma power2_diff: |
|
574 |
fixes x y :: "'a::{number_ring,recpower}" |
|
575 |
shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23475
diff
changeset
|
576 |
by (simp add: ring_distribs power2_eq_square) |
22858 | 577 |
|
578 |
lemma real_sqrt_sum_squares_triangle_ineq: |
|
579 |
"sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)" |
|
580 |
apply (rule power2_le_imp_le, simp) |
|
581 |
apply (simp add: power2_sum) |
|
582 |
apply (simp only: mult_assoc right_distrib [symmetric]) |
|
583 |
apply (rule mult_left_mono) |
|
584 |
apply (rule power2_le_imp_le) |
|
585 |
apply (simp add: power2_sum power_mult_distrib) |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23475
diff
changeset
|
586 |
apply (simp add: ring_distribs) |
22858 | 587 |
apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior> - 2 * (a * c) * (b * d)", simp) |
588 |
apply (rule_tac b="(a * d - b * c)\<twosuperior>" in ord_le_eq_trans) |
|
589 |
apply (rule zero_le_power2) |
|
590 |
apply (simp add: power2_diff power_mult_distrib) |
|
591 |
apply (simp add: mult_nonneg_nonneg) |
|
592 |
apply simp |
|
593 |
apply (simp add: add_increasing) |
|
594 |
done |
|
595 |
||
23122 | 596 |
lemma real_sqrt_sum_squares_less: |
597 |
"\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) < u" |
|
598 |
apply (rule power2_less_imp_less, simp) |
|
599 |
apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) |
|
600 |
apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) |
|
601 |
apply (simp add: power_divide) |
|
602 |
apply (drule order_le_less_trans [OF abs_ge_zero]) |
|
603 |
apply (simp add: zero_less_divide_iff) |
|
604 |
done |
|
605 |
||
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
606 |
text{*Needed for the infinitely close relation over the nonstandard |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
607 |
complex numbers*} |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
608 |
lemma lemma_sqrt_hcomplex_capprox: |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
609 |
"[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u" |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
610 |
apply (rule_tac y = "u/sqrt 2" in order_le_less_trans) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
611 |
apply (erule_tac [2] lemma_real_divide_sqrt_less) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
612 |
apply (rule power2_le_imp_le) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
613 |
apply (auto simp add: real_0_le_divide_iff power_divide) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
614 |
apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst]) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
615 |
apply (rule add_mono) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
616 |
apply (auto simp add: four_x_squared simp del: realpow_Suc intro: power_mono) |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
617 |
done |
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset
|
618 |
|
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
619 |
text "Legacy theorem names:" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
620 |
lemmas real_root_pos2 = real_root_power_cancel |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
621 |
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
622 |
lemmas real_root_pos_pos_le = real_root_ge_zero |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
623 |
lemmas real_sqrt_mult_distrib = real_sqrt_mult |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
624 |
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
625 |
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
626 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
627 |
(* needed for CauchysMeanTheorem.het_base from AFP *) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
628 |
lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x" |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
629 |
by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
630 |
|
14324 | 631 |
end |