author  webertj 
Fri, 19 Oct 2012 15:12:52 +0200  
changeset 49962  a8cc904a6820 
parent 49753  a344f1a21211 
child 51478  270b21f3ae0a 
permissions  rwrr 
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(* 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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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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subsection {* Existence of Nth Root *} 
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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 
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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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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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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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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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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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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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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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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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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) 

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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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apply (erule real_root_power_cancel [OF odd_pos], simp) 

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done 

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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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23046  119 
lemma odd_real_root_unique: 
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"\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" 

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by (erule subst, rule odd_real_root_power_cancel) 

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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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text {* Root function is strictly monotonic, hence injective *} 
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lemma real_root_less_mono_lemma: 
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"\<lbrakk>0 < n; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" 
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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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apply (erule power_less_imp_less_base) 
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apply (erule (1) real_root_ge_zero) 
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apply simp 
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apply simp 
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done 
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lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" 
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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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151 

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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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154 

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155 
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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157 
apply (cases "x < y") 
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158 
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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161 

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162 
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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164 
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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167 
done 
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168 

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169 
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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171 
by (simp add: order_eq_iff) 
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172 

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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) 

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) 

202 

203 
lemma real_root_mult_exp: 

204 
"\<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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"\<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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"\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (inverse x) = inverse (root n x)" 
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247 
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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284 
hence "root n (inverse ( x)) = inverse (root n ( x))" 
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285 
by (rule real_root_inverse_lemma [OF n]) 
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286 
thus ?thesis by (simp add: real_root_minus [OF n]) 
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287 
qed 
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288 

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289 
lemma real_root_divide: 
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290 
"0 < n \<Longrightarrow> root n (x / y) = root n x / root n y" 
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291 
by (simp add: divide_inverse real_root_mult real_root_inverse) 
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292 

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293 
lemma real_root_power: 
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294 
"0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k" 
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295 
by (induct k, simp_all add: real_root_mult) 
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296 

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297 
lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>" 
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298 
by (simp add: abs_if real_root_minus) 
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299 

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300 
text {* Continuity and derivatives *} 
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301 

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302 
lemma isCont_root_pos: 
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303 
assumes n: "0 < n" 
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304 
assumes x: "0 < x" 
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305 
shows "isCont (root n) x" 
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306 
proof  
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307 
have "isCont (root n) (root n x ^ n)" 
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308 
proof (rule isCont_inverse_function [where f="\<lambda>a. a ^ n"]) 
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309 
show "0 < root n x" using n x by simp 
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310 
show "\<forall>z. \<bar>z  root n x\<bar> \<le> root n x \<longrightarrow> root n (z ^ n) = z" 
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311 
by (simp add: abs_le_iff real_root_power_cancel n) 
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312 
show "\<forall>z. \<bar>z  root n x\<bar> \<le> root n x \<longrightarrow> isCont (\<lambda>a. a ^ n) z" 
44289  313 
by simp 
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314 
qed 
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315 
thus ?thesis using n x by simp 
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316 
qed 
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317 

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318 
lemma isCont_root_neg: 
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319 
"\<lbrakk>0 < n; x < 0\<rbrakk> \<Longrightarrow> isCont (root n) x" 
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320 
apply (subgoal_tac "isCont (\<lambda>x.  root n ( x)) x") 
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321 
apply (simp add: real_root_minus) 
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322 
apply (rule isCont_o2 [OF isCont_minus [OF isCont_ident]]) 
44289  323 
apply (simp add: isCont_root_pos) 
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324 
done 
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325 

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326 
lemma isCont_root_zero: 
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327 
"0 < n \<Longrightarrow> isCont (root n) 0" 
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328 
unfolding isCont_def 
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329 
apply (rule LIM_I) 
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330 
apply (rule_tac x="r ^ n" in exI, safe) 
25875  331 
apply (simp) 
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332 
apply (simp add: real_root_abs [symmetric]) 
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333 
apply (rule_tac n="n" in power_less_imp_less_base, simp_all) 
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334 
done 
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335 

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336 
lemma isCont_real_root: "0 < n \<Longrightarrow> isCont (root n) x" 
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337 
apply (rule_tac x=x and y=0 in linorder_cases) 
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338 
apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero) 
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339 
done 
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340 

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341 
lemma DERIV_real_root: 
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342 
assumes n: "0 < n" 
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343 
assumes x: "0 < x" 
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344 
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n  Suc 0))" 
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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" 

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349 
using n by simp 
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350 
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n  Suc 0)" 
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351 
by (rule DERIV_pow) 
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352 
show "real n * root n x ^ (n  Suc 0) \<noteq> 0" 
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353 
using n x by simp 
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354 
show "isCont (root n) x" 
23441  355 
using n by (rule isCont_real_root) 
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356 
qed 
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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 

31880  375 
lemma DERIV_even_real_root: 
376 
assumes n: "0 < n" and "even n" 

377 
assumes x: "x < 0" 

378 
shows "DERIV (root n) x :> inverse ( real n * root n x ^ (n  Suc 0))" 

379 
proof (rule DERIV_inverse_function) 

380 
show "x  1 < x" by simp 

381 
show "x < 0" using x . 

382 
next 

383 
show "\<forall>y. x  1 < y \<and> y < 0 \<longrightarrow>  (root n y ^ n) = y" 

384 
proof (rule allI, rule impI, erule conjE) 

385 
fix y assume "x  1 < y" and "y < 0" 

386 
hence "root n (y) ^ n = y" using `0 < n` by simp 

387 
with real_root_minus[OF `0 < n`] and `even n` 

388 
show " (root n y ^ n) = y" by simp 

389 
qed 

390 
next 

391 
show "DERIV (\<lambda>x.  (x ^ n)) (root n x) :>  real n * root n x ^ (n  Suc 0)" 

392 
by (auto intro!: DERIV_intros) 

393 
show " real n * root n x ^ (n  Suc 0) \<noteq> 0" 

394 
using n x by simp 

395 
show "isCont (root n) x" 

396 
using n by (rule isCont_real_root) 

397 
qed 

398 

399 
lemma DERIV_real_root_generic: 

400 
assumes "0 < n" and "x \<noteq> 0" 

49753  401 
and "\<lbrakk> even n ; 0 < x \<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n  Suc 0))" 
402 
and "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D =  inverse (real n * root n x ^ (n  Suc 0))" 

403 
and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n  Suc 0))" 

31880  404 
shows "DERIV (root n) x :> D" 
405 
using assms by (cases "even n", cases "0 < x", 

406 
auto intro: DERIV_real_root[THEN DERIV_cong] 

407 
DERIV_odd_real_root[THEN DERIV_cong] 

408 
DERIV_even_real_root[THEN DERIV_cong]) 

409 

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410 
subsection {* Square Root *} 
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411 

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412 
definition 
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413 
sqrt :: "real \<Rightarrow> real" where 
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414 
"sqrt = root 2" 
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415 

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416 
lemma pos2: "0 < (2::nat)" by simp 
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417 

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418 
lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y" 
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419 
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2]) 
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420 

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421 
lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>" 
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422 
apply (rule real_sqrt_unique) 
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423 
apply (rule power2_abs) 
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424 
apply (rule abs_ge_zero) 
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425 
done 
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426 

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427 
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x" 
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428 
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2]) 
22856  429 

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430 
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" 
22856  431 
apply (rule iffI) 
432 
apply (erule subst) 

433 
apply (rule zero_le_power2) 

434 
apply (erule real_sqrt_pow2) 

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435 
done 
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436 

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437 
lemma real_sqrt_zero [simp]: "sqrt 0 = 0" 
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438 
unfolding sqrt_def by (rule real_root_zero) 
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439 

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440 
lemma real_sqrt_one [simp]: "sqrt 1 = 1" 
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441 
unfolding sqrt_def by (rule real_root_one [OF pos2]) 
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442 

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443 
lemma real_sqrt_minus: "sqrt ( x) =  sqrt x" 
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444 
unfolding sqrt_def by (rule real_root_minus [OF pos2]) 
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445 

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446 
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y" 
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447 
unfolding sqrt_def by (rule real_root_mult [OF pos2]) 
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448 

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449 
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)" 
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450 
unfolding sqrt_def by (rule real_root_inverse [OF pos2]) 
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451 

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452 
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y" 
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453 
unfolding sqrt_def by (rule real_root_divide [OF pos2]) 
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454 

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455 
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k" 
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456 
unfolding sqrt_def by (rule real_root_power [OF pos2]) 
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457 

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458 
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x" 
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459 
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2]) 
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460 

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461 
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x" 
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462 
unfolding sqrt_def by (rule real_root_ge_zero [OF pos2]) 
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463 

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464 
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y" 
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465 
unfolding sqrt_def by (rule real_root_less_mono [OF pos2]) 
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466 

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467 
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y" 
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468 
unfolding sqrt_def by (rule real_root_le_mono [OF pos2]) 
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469 

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470 
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)" 
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471 
unfolding sqrt_def by (rule real_root_less_iff [OF pos2]) 
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472 

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473 
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)" 
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474 
unfolding sqrt_def by (rule real_root_le_iff [OF pos2]) 
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475 

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476 
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)" 
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parents:
22943
diff
changeset

477 
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

478 

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

479 
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

480 
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

481 
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

482 
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

483 
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

484 

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

485 
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

486 
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

487 
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

488 
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

489 
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

490 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

491 
lemma isCont_real_sqrt: "isCont sqrt x" 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

492 
unfolding sqrt_def by (rule isCont_real_root [OF pos2]) 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

493 

31880  494 
lemma DERIV_real_sqrt_generic: 
495 
assumes "x \<noteq> 0" 

496 
assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2" 

497 
assumes "x < 0 \<Longrightarrow> D =  inverse (sqrt x) / 2" 

498 
shows "DERIV sqrt x :> D" 

499 
using assms unfolding sqrt_def 

500 
by (auto intro!: DERIV_real_root_generic) 

501 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

502 
lemma DERIV_real_sqrt: 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

503 
"0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2" 
31880  504 
using DERIV_real_sqrt_generic by simp 
505 

506 
declare 

507 
DERIV_real_sqrt_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] 

508 
DERIV_real_root_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

509 

20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

510 
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

511 
apply auto 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

512 
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

513 
apply (simp add: zero_less_mult_iff) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

514 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

515 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

516 
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" 
22856  517 
apply (subst power2_eq_square [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

518 
apply (rule real_sqrt_abs) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

519 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

520 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

521 
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" 
22856  522 
by (simp add: power_inverse [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

523 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

524 
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

525 
by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

526 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

527 
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

528 
by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

529 

22443  530 
lemma sqrt_divide_self_eq: 
531 
assumes nneg: "0 \<le> x" 

532 
shows "sqrt x / x = inverse (sqrt x)" 

533 
proof cases 

534 
assume "x=0" thus ?thesis by simp 

535 
next 

536 
assume nz: "x\<noteq>0" 

537 
hence pos: "0<x" using nneg by arith 

538 
show ?thesis 

539 
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

540 
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 

541 
show "inverse (sqrt x) / (sqrt x / x) = 1" 

542 
by (simp add: divide_inverse mult_assoc [symmetric] 

543 
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 

544 
qed 

545 
qed 

546 

22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

547 
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

548 
apply (simp add: divide_inverse) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

549 
apply (case_tac "r=0") 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

550 
apply (auto simp add: mult_ac) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

551 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

552 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

553 
lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u" 
35216  554 
by (simp add: divide_less_eq) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

555 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

556 
lemma four_x_squared: 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

557 
fixes x::real 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

558 
shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

559 
by (simp add: power2_eq_square) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

560 

22856  561 
subsection {* Square Root of Sum of Squares *} 
562 

44320  563 
lemma real_sqrt_sum_squares_ge_zero: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
564 
by simp (* TODO: delete *) 

22856  565 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

566 
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

567 

22856  568 
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: 
569 
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" 

44320  570 
by (simp add: zero_le_mult_iff) 
22856  571 

572 
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: 

573 
"sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" 

44320  574 
by (simp add: zero_le_mult_iff) 
22856  575 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

576 
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

577 
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

578 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

579 
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

580 
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

581 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

582 
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
22856  583 
by (rule power2_le_imp_le, simp_all) 
584 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

585 
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

586 
by (rule power2_le_imp_le, simp_all) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

587 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

588 
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
22856  589 
by (rule power2_le_imp_le, simp_all) 
590 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

591 
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

592 
by (rule power2_le_imp_le, simp_all) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

593 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

594 
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

595 
by (simp add: power2_eq_square [symmetric]) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

596 

22858  597 
lemma real_sqrt_sum_squares_triangle_ineq: 
598 
"sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)" 

599 
apply (rule power2_le_imp_le, simp) 

600 
apply (simp add: power2_sum) 

49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49753
diff
changeset

601 
apply (simp only: mult_assoc distrib_left [symmetric]) 
22858  602 
apply (rule mult_left_mono) 
603 
apply (rule power2_le_imp_le) 

604 
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

605 
apply (simp add: ring_distribs) 
22858  606 
apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior>  2 * (a * c) * (b * d)", simp) 
607 
apply (rule_tac b="(a * d  b * c)\<twosuperior>" in ord_le_eq_trans) 

608 
apply (rule zero_le_power2) 

609 
apply (simp add: power2_diff power_mult_distrib) 

610 
apply (simp add: mult_nonneg_nonneg) 

611 
apply simp 

612 
apply (simp add: add_increasing) 

613 
done 

614 

23122  615 
lemma real_sqrt_sum_squares_less: 
616 
"\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) < u" 

617 
apply (rule power2_less_imp_less, simp) 

618 
apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) 

619 
apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) 

620 
apply (simp add: power_divide) 

621 
apply (drule order_le_less_trans [OF abs_ge_zero]) 

622 
apply (simp add: zero_less_divide_iff) 

623 
done 

624 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

625 
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

626 
complex numbers*} 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

627 
lemma lemma_sqrt_hcomplex_capprox: 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

628 
"[ 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

629 
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

630 
apply (erule_tac [2] lemma_real_divide_sqrt_less) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

631 
apply (rule power2_le_imp_le) 
44349
f057535311c5
remove redundant lemma real_0_le_divide_iff in favor or zero_le_divide_iff
huffman
parents:
44320
diff
changeset

632 
apply (auto simp add: zero_le_divide_iff power_divide) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

633 
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

634 
apply (rule add_mono) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
28952
diff
changeset

635 
apply (auto simp add: four_x_squared intro: power_mono) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

636 
done 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

637 

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

638 
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

639 
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

640 
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

641 
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

642 
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

643 
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

644 
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

645 

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

646 
(* 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

647 
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

648 
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

649 

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end 