author  huffman 
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child 23122  3d853d6f2f7d 
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 SEQ 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 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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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_pos: "odd (n::nat) \<Longrightarrow> 0 < n" 
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by (cases n, simp_all) 

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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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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  122 
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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152 
apply (simp add: real_root_less_mono_lemma) 
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153 
done 
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154 

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155 
lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y" 
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156 
by (auto simp add: order_le_less real_root_less_mono) 
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157 

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158 
lemma real_root_less_iff [simp]: 
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159 
"0 < n \<Longrightarrow> (root n x < root n y) = (x < y)" 
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160 
apply (cases "x < y") 
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161 
apply (simp add: real_root_less_mono) 
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162 
apply (simp add: linorder_not_less real_root_le_mono) 
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163 
done 
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164 

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165 
lemma real_root_le_iff [simp]: 
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166 
"0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)" 
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167 
apply (cases "x \<le> y") 
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168 
apply (simp add: real_root_le_mono) 
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169 
apply (simp add: linorder_not_le real_root_less_mono) 
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170 
done 
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171 

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172 
lemma real_root_eq_iff [simp]: 
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173 
"0 < n \<Longrightarrow> (root n x = root n y) = (x = y)" 
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174 
by (simp add: order_eq_iff) 
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175 

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176 
lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified] 
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177 
lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified] 
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178 
lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified] 
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179 
lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified] 
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180 
lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified] 
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181 

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182 
text {* Roots of multiplication and division *} 
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183 

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184 
lemma real_root_mult_lemma: 
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185 
"\<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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186 
by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib) 
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187 

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188 
lemma real_root_inverse_lemma: 
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189 
"\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (inverse x) = inverse (root n x)" 
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190 
by (simp add: real_root_pos_unique power_inverse [symmetric]) 
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191 

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192 
lemma real_root_mult: 
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193 
assumes n: "0 < n" 
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194 
shows "root n (x * y) = root n x * root n y" 
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195 
proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases) 
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196 
assume "0 \<le> x" and "0 \<le> y" 
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197 
thus ?thesis by (rule real_root_mult_lemma [OF n]) 
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198 
next 
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199 
assume "0 \<le> x" and "y \<le> 0" 
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200 
hence "0 \<le> x" and "0 \<le>  y" by simp_all 
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201 
hence "root n (x *  y) = root n x * root n ( y)" 
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202 
by (rule real_root_mult_lemma [OF n]) 
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203 
thus ?thesis by (simp add: real_root_minus [OF n]) 
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204 
next 
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205 
assume "x \<le> 0" and "0 \<le> y" 
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206 
hence "0 \<le>  x" and "0 \<le> y" by simp_all 
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207 
hence "root n ( x * y) = root n ( x) * root n y" 
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208 
by (rule real_root_mult_lemma [OF n]) 
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209 
thus ?thesis by (simp add: real_root_minus [OF n]) 
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210 
next 
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211 
assume "x \<le> 0" and "y \<le> 0" 
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212 
hence "0 \<le>  x" and "0 \<le>  y" by simp_all 
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213 
hence "root n ( x *  y) = root n ( x) * root n ( y)" 
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214 
by (rule real_root_mult_lemma [OF n]) 
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215 
thus ?thesis by (simp add: real_root_minus [OF n]) 
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216 
qed 
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217 

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218 
lemma real_root_inverse: 
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219 
assumes n: "0 < n" 
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220 
shows "root n (inverse x) = inverse (root n x)" 
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221 
proof (rule linorder_le_cases) 
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222 
assume "0 \<le> x" 
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223 
thus ?thesis by (rule real_root_inverse_lemma [OF n]) 
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224 
next 
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225 
assume "x \<le> 0" 
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226 
hence "0 \<le>  x" by simp 
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227 
hence "root n (inverse ( x)) = inverse (root n ( x))" 
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228 
by (rule real_root_inverse_lemma [OF n]) 
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229 
thus ?thesis by (simp add: real_root_minus [OF n]) 
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230 
qed 
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231 

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232 
lemma real_root_divide: 
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233 
"0 < n \<Longrightarrow> root n (x / y) = root n x / root n y" 
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234 
by (simp add: divide_inverse real_root_mult real_root_inverse) 
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235 

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236 
lemma real_root_power: 
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237 
"0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k" 
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238 
by (induct k, simp_all add: real_root_mult) 
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239 

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240 
lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>" 
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241 
by (simp add: abs_if real_root_minus) 
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242 

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243 
text {* Continuity and derivatives *} 
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244 

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245 
lemma isCont_root_pos: 
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246 
assumes n: "0 < n" 
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247 
assumes x: "0 < x" 
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248 
shows "isCont (root n) x" 
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249 
proof  
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250 
have "isCont (root n) (root n x ^ n)" 
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251 
proof (rule isCont_inverse_function [where f="\<lambda>a. a ^ n"]) 
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252 
show "0 < root n x" using n x by simp 
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253 
show "\<forall>z. \<bar>z  root n x\<bar> \<le> root n x \<longrightarrow> root n (z ^ n) = z" 
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254 
by (simp add: abs_le_iff real_root_power_cancel n) 
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255 
show "\<forall>z. \<bar>z  root n x\<bar> \<le> root n x \<longrightarrow> isCont (\<lambda>a. a ^ n) z" 
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256 
by (simp add: isCont_power) 
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257 
qed 
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258 
thus ?thesis using n x by simp 
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259 
qed 
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260 

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261 
lemma isCont_root_neg: 
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262 
"\<lbrakk>0 < n; x < 0\<rbrakk> \<Longrightarrow> isCont (root n) x" 
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263 
apply (subgoal_tac "isCont (\<lambda>x.  root n ( x)) x") 
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264 
apply (simp add: real_root_minus) 
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265 
apply (rule isCont_o2 [OF isCont_minus [OF isCont_ident]]) 
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266 
apply (simp add: isCont_minus isCont_root_pos) 
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267 
done 
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268 

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269 
lemma isCont_root_zero: 
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"0 < n \<Longrightarrow> isCont (root n) 0" 
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unfolding isCont_def 
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apply (rule LIM_I) 
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apply (rule_tac x="r ^ n" in exI, safe) 
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apply (simp add: zero_less_power) 
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apply (simp add: real_root_abs [symmetric]) 
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apply (rule_tac n="n" in power_less_imp_less_base, simp_all) 
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done 
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278 

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lemma isCont_real_root: "0 < n \<Longrightarrow> isCont (root n) x" 
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apply (rule_tac x=x and y=0 in linorder_cases) 
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apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero) 
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done 
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283 

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lemma DERIV_real_root: 
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assumes n: "0 < n" 
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assumes x: "0 < x" 
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shows "DERIV (root n) x :> inverse (real n * root n x ^ (n  Suc 0))" 
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proof (rule DERIV_inverse_function) 
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show "0 < x" using x . 
290 
show "x < x + 1" by simp 

291 
show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" 

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using n by simp 
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show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n  Suc 0)" 
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294 
by (rule DERIV_pow) 
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show "real n * root n x ^ (n  Suc 0) \<noteq> 0" 
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296 
using n x by simp 
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show "isCont (root n) x" 
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298 
by (rule isCont_real_root) 
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qed 
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300 

23046  301 
lemma DERIV_odd_real_root: 
302 
assumes n: "odd n" 

303 
assumes x: "x \<noteq> 0" 

304 
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n  Suc 0))" 

305 
proof (rule DERIV_inverse_function) 

306 
show "x  1 < x" by simp 

307 
show "x < x + 1" by simp 

308 
show "\<forall>y. x  1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" 

309 
using n by (simp add: odd_real_root_pow) 

310 
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n  Suc 0)" 

311 
by (rule DERIV_pow) 

312 
show "real n * root n x ^ (n  Suc 0) \<noteq> 0" 

313 
using odd_pos [OF n] x by simp 

314 
show "isCont (root n) x" 

315 
using odd_pos [OF n] by (rule isCont_real_root) 

316 
qed 

317 

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318 
subsection {* Square Root *} 
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definition 
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sqrt :: "real \<Rightarrow> real" where 
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"sqrt = root 2" 
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323 

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lemma pos2: "0 < (2::nat)" by simp 
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325 

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lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y" 
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unfolding sqrt_def by (rule real_root_pos_unique [OF pos2]) 
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328 

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lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>" 
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330 
apply (rule real_sqrt_unique) 
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331 
apply (rule power2_abs) 
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332 
apply (rule abs_ge_zero) 
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333 
done 
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334 

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lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x" 
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336 
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2]) 
22856  337 

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lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" 
22856  339 
apply (rule iffI) 
340 
apply (erule subst) 

341 
apply (rule zero_le_power2) 

342 
apply (erule real_sqrt_pow2) 

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343 
done 
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344 

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lemma real_sqrt_zero [simp]: "sqrt 0 = 0" 
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346 
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347 

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lemma real_sqrt_one [simp]: "sqrt 1 = 1" 
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349 
unfolding sqrt_def by (rule real_root_one [OF pos2]) 
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350 

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lemma real_sqrt_minus: "sqrt ( x) =  sqrt x" 
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352 
unfolding sqrt_def by (rule real_root_minus [OF pos2]) 
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353 

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354 
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y" 
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355 
unfolding sqrt_def by (rule real_root_mult [OF pos2]) 
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356 

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357 
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)" 
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358 
unfolding sqrt_def by (rule real_root_inverse [OF pos2]) 
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359 

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lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y" 
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361 
unfolding sqrt_def by (rule real_root_divide [OF pos2]) 
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362 

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363 
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k" 
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364 
unfolding sqrt_def by (rule real_root_power [OF pos2]) 
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365 

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366 
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x" 
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367 
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2]) 
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368 

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369 
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x" 
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370 
unfolding sqrt_def by (rule real_root_ge_zero [OF pos2]) 
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371 

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372 
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y" 
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373 
unfolding sqrt_def by (rule real_root_less_mono [OF pos2]) 
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374 

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375 
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y" 
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376 
unfolding sqrt_def by (rule real_root_le_mono [OF pos2]) 
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377 

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378 
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)" 
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379 
unfolding sqrt_def by (rule real_root_less_iff [OF pos2]) 
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380 

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381 
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)" 
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382 
unfolding sqrt_def by (rule real_root_le_iff [OF pos2]) 
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383 

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384 
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)" 
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385 
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2]) 
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386 

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387 
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified] 
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388 
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified] 
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389 
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified] 
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390 
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified] 
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391 
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified] 
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392 

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393 
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified] 
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394 
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified] 
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395 
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified] 
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396 
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified] 
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397 
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified] 
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398 

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399 
lemma isCont_real_sqrt: "isCont sqrt x" 
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400 
unfolding sqrt_def by (rule isCont_real_root [OF pos2]) 
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401 

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402 
lemma DERIV_real_sqrt: 
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403 
"0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2" 
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404 
unfolding sqrt_def by (rule DERIV_real_root [OF pos2, simplified]) 
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405 

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406 
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" 
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407 
apply auto 
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changeset

408 
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

409 
apply (simp add: zero_less_mult_iff) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

410 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

411 

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

412 
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" 
22856  413 
apply (subst power2_eq_square [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

414 
apply (rule real_sqrt_abs) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

415 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

416 

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

417 
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

418 
by simp (* TODO: delete *) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

419 

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

420 
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

421 
by simp (* TODO: delete *) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

422 

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

423 
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" 
22856  424 
by (simp add: power_inverse [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

425 

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

426 
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

427 
by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

428 

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

429 
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

430 
by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

431 

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

432 
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

433 
by simp 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

434 

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

435 
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

436 
by simp 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

437 

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

438 
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

439 
by simp 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

440 

22443  441 
lemma sqrt_divide_self_eq: 
442 
assumes nneg: "0 \<le> x" 

443 
shows "sqrt x / x = inverse (sqrt x)" 

444 
proof cases 

445 
assume "x=0" thus ?thesis by simp 

446 
next 

447 
assume nz: "x\<noteq>0" 

448 
hence pos: "0<x" using nneg by arith 

449 
show ?thesis 

450 
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

451 
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 

452 
show "inverse (sqrt x) / (sqrt x / x) = 1" 

453 
by (simp add: divide_inverse mult_assoc [symmetric] 

454 
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 

455 
qed 

456 
qed 

457 

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

458 
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

459 
apply (simp add: divide_inverse) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

460 
apply (case_tac "r=0") 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

461 
apply (auto simp add: mult_ac) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

462 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

463 

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

464 
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

465 
by (simp add: divide_less_eq mult_compare_simps) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

466 

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

467 
lemma four_x_squared: 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

468 
fixes x::real 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

469 
shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

470 
by (simp add: power2_eq_square) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

471 

22856  472 
subsection {* Square Root of Sum of Squares *} 
473 

474 
lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)" 

22968  475 
by (rule real_sqrt_ge_zero [OF sum_squares_ge_zero]) 
22856  476 

477 
lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 

22961  478 
by simp 
22856  479 

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

480 
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

481 

22856  482 
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: 
483 
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" 

484 
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) 

485 

486 
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: 

487 
"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

488 
by (auto simp add: zero_le_mult_iff) 
22856  489 

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

490 
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

491 
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

492 

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

493 
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

494 
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

495 

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

496 
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
22856  497 
by (rule power2_le_imp_le, simp_all) 
498 

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

499 
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

500 
by (rule power2_le_imp_le, simp_all) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

501 

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

502 
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
22856  503 
by (rule power2_le_imp_le, simp_all) 
504 

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

505 
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

506 
by (rule power2_le_imp_le, simp_all) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

507 

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

508 
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

509 
by (simp add: power2_eq_square [symmetric]) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

510 

22858  511 
lemma power2_sum: 
512 
fixes x y :: "'a::{number_ring,recpower}" 

513 
shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y" 

514 
by (simp add: left_distrib right_distrib power2_eq_square) 

515 

516 
lemma power2_diff: 

517 
fixes x y :: "'a::{number_ring,recpower}" 

518 
shows "(x  y)\<twosuperior> = x\<twosuperior> + y\<twosuperior>  2 * x * y" 

519 
by (simp add: left_diff_distrib right_diff_distrib power2_eq_square) 

520 

521 
lemma real_sqrt_sum_squares_triangle_ineq: 

522 
"sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)" 

523 
apply (rule power2_le_imp_le, simp) 

524 
apply (simp add: power2_sum) 

525 
apply (simp only: mult_assoc right_distrib [symmetric]) 

526 
apply (rule mult_left_mono) 

527 
apply (rule power2_le_imp_le) 

528 
apply (simp add: power2_sum power_mult_distrib) 

529 
apply (simp add: ring_distrib) 

530 
apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior>  2 * (a * c) * (b * d)", simp) 

531 
apply (rule_tac b="(a * d  b * c)\<twosuperior>" in ord_le_eq_trans) 

532 
apply (rule zero_le_power2) 

533 
apply (simp add: power2_diff power_mult_distrib) 

534 
apply (simp add: mult_nonneg_nonneg) 

535 
apply simp 

536 
apply (simp add: add_increasing) 

537 
done 

538 

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

539 
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

540 
complex numbers*} 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

541 
lemma lemma_sqrt_hcomplex_capprox: 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

542 
"[ 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

543 
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

544 
apply (erule_tac [2] lemma_real_divide_sqrt_less) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

545 
apply (rule power2_le_imp_le) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

546 
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

547 
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

548 
apply (rule add_mono) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

549 
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

550 
done 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

551 

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

552 
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

553 
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

554 
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

555 
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

556 
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

557 
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

558 
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

559 

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

560 
(* 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

561 
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

562 
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

563 

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

564 
(* FIXME: the stronger version of real_root_less_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

565 
breaks CauchysMeanTheorem.list_gmean_gt_iff 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

566 

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

567 
declare real_root_less_iff [simp del] 
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

568 
lemma real_root_less_iff_nonneg [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

569 
"\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> (root n x < root n 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

570 
by (rule real_root_less_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

571 

14324  572 
end 