author  huffman 
Wed, 14 Mar 2007 21:52:26 +0100  
changeset 22443  346729a55460 
parent 21865  55cc354fd2d9 
child 22630  2a9b64b26612 
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{*Existence of Nth Root*} 
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theory NthRoot 
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imports SEQ Parity 
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begin 
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definition 
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root :: "[nat, real] \<Rightarrow> real" where 
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"root n x = (THE u. (0 < x \<longrightarrow> 0 < u) \<and> (u ^ n = x))" 
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definition 
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sqrt :: "real \<Rightarrow> real" where 
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"sqrt x = root 2 x" 
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text {* 
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Various lemmas needed for this result. We follow the proof given by 

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John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis 

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Webnotes available at \url{http://www.math.unl.edu/~webnotes}. 

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Lemmas about sequences of reals are used to reach the result. 

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*} 

14324  29 

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lemma lemma_nth_realpow_non_empty: 

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"[ (0::real) < a; 0 < n ] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}" 

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apply (case_tac "1 <= a") 

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apply (rule_tac x = 1 in exI) 
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apply (drule_tac [2] linorder_not_le [THEN iffD1]) 
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apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp) 
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apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc) 
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done 
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text{*Used only just below*} 
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lemma realpow_ge_self2: "[ (1::real) \<le> r; 0 < n ] ==> r \<le> r ^ n" 
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by (insert power_increasing [of 1 n r], simp) 
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lemma lemma_nth_realpow_isUb_ex: 
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"[ (0::real) < a; 0 < n ] 

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==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u" 

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apply (case_tac "1 <= a") 

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apply (rule_tac x = a in exI) 
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apply (drule_tac [2] linorder_not_le [THEN iffD1]) 
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apply (rule_tac [2] x = 1 in exI) 
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apply (rule_tac [!] setleI [THEN isUbI], safe) 

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apply (simp_all (no_asm)) 
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apply (rule_tac [!] ccontr) 

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apply (drule_tac [!] linorder_not_le [THEN iffD1]) 
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apply (drule realpow_ge_self2, assumption) 
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apply (drule_tac n = n in realpow_less) 

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apply (assumption+) 
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apply (drule real_le_trans, assumption) 
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apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp) 

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apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto) 

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done 
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lemma nth_realpow_isLub_ex: 

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"[ (0::real) < a; 0 < n ] 

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==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u" 

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by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete) 
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subsection{*First Half  Lemmas First*} 

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lemma lemma_nth_realpow_seq: 

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"isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u 

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==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}" 

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apply (safe, drule isLubD2, blast) 
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apply (simp add: linorder_not_less [symmetric]) 
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done 
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lemma lemma_nth_realpow_isLub_gt_zero: 

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"[ isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; 

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0 < a; 0 < n ] ==> 0 < u" 

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apply (drule lemma_nth_realpow_non_empty, auto) 
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apply (drule_tac y = s in isLub_isUb [THEN isUbD]) 

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apply (auto intro: order_less_le_trans) 
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done 

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lemma lemma_nth_realpow_isLub_ge: 

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"[ isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; 

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0 < a; 0 < n ] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n" 

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apply safe 
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apply (frule lemma_nth_realpow_seq, safe) 

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apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric] 
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iff: real_0_less_add_iff) {*legacy iff rule!*} 
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apply (simp add: linorder_not_less) 
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apply (rule order_less_trans [of _ 0]) 
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apply (auto intro: lemma_nth_realpow_isLub_gt_zero) 
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done 
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text{*First result we want*} 

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lemma realpow_nth_ge: 

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"[ (0::real) < a; 0 < n; 

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isLub (UNIV::real set) 

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{x. x ^ n <= a & 0 < x} u ] ==> a <= u ^ n" 

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apply (frule lemma_nth_realpow_isLub_ge, safe) 
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apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const]) 
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apply (auto simp add: real_of_nat_def) 
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done 
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subsection{*Second Half*} 

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lemma less_isLub_not_isUb: 

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"[ isLub (UNIV::real set) S u; x < u ] 

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==> ~ isUb (UNIV::real set) S x" 

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apply safe 
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apply (drule isLub_le_isUb, assumption) 

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apply (drule order_less_le_trans, auto) 

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done 
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lemma not_isUb_less_ex: 

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"~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x" 

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apply (rule ccontr, erule contrapos_np) 
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apply (rule setleI [THEN isUbI]) 
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apply (auto simp add: linorder_not_less [symmetric]) 
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done 
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lemma real_mult_less_self: "0 < r ==> r * (1 + inverse(real (Suc n))) < r" 
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apply (simp (no_asm) add: right_distrib) 
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apply (rule add_less_cancel_left [of "r", THEN iffD1]) 

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apply (auto intro: mult_pos_pos 
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simp add: add_assoc [symmetric] neg_less_0_iff_less) 
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done 
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lemma real_mult_add_one_minus_ge_zero: 

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"0 < r ==> 0 <= r*(1 + inverse(real (Suc n)))" 

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by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff) 
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lemma lemma_nth_realpow_isLub_le: 
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"[ isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; 

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0 < a; 0 < n ] ==> ALL k. (u*(1 + inverse(real (Suc k)))) ^ n <= a" 
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apply safe 
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apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex]) 
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apply (rule_tac n = k in real_mult_less_self) 
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apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe) 

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apply (drule_tac n = k in 

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lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+) 

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apply (blast intro: order_trans order_less_imp_le power_mono) 
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done 
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text{*Second result we want*} 

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lemma realpow_nth_le: 

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"[ (0::real) < a; 0 < n; 

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isLub (UNIV::real set) 

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{x. x ^ n <= a & 0 < x} u ] ==> u ^ n <= a" 

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apply (frule lemma_nth_realpow_isLub_le, safe) 
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apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult 
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[THEN LIMSEQ_pow, THEN LIMSEQ_le_const2]) 
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apply (auto simp add: real_of_nat_def) 
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done 
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text{*The theorem at last!*} 
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lemma realpow_nth: "[ (0::real) < a; 0 < n ] ==> \<exists>r. r ^ n = a" 
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apply (frule nth_realpow_isLub_ex, auto) 
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apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym) 

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done 
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(* positive only *) 

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lemma realpow_pos_nth: "[ (0::real) < a; 0 < n ] ==> \<exists>r. 0 < r & r ^ n = a" 

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apply (frule nth_realpow_isLub_ex, auto) 
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apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero) 

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done 
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lemma realpow_pos_nth2: "(0::real) < a ==> \<exists>r. 0 < r & r ^ Suc n = a" 

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by (blast intro: realpow_pos_nth) 
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(* uniqueness of nth positive root *) 

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lemma realpow_pos_nth_unique: 

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"[ (0::real) < a; 0 < n ] ==> EX! r. 0 < r & r ^ n = a" 

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apply (auto intro!: realpow_pos_nth) 

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apply (cut_tac x = r and y = y in linorder_less_linear, auto) 
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apply (drule_tac x = r in realpow_less) 

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apply (drule_tac [4] x = y in realpow_less, auto) 

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done 
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subsection {* Nth Root *} 
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lemma real_root_zero [simp]: "root (Suc n) 0 = 0" 
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apply (simp add: root_def) 
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apply (safe intro!: the_equality power_0_Suc elim!: realpow_zero_zero) 
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done 
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lemma real_root_pow_pos: 
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"0 < x ==> (root (Suc n) x) ^ (Suc n) = x" 
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apply (simp add: root_def del: realpow_Suc) 
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apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp) 
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apply (erule theI' [THEN conjunct2]) 
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done 
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lemma real_root_pow_pos2: "0 \<le> x ==> (root (Suc n) x) ^ (Suc n) = x" 
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by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos) 
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lemma real_root_pos: 
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"0 < x ==> root(Suc n) (x ^ (Suc n)) = x" 
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apply (simp add: root_def) 
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apply (rule the_equality) 
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apply (frule_tac [2] n = n in zero_less_power) 
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apply (auto simp add: zero_less_mult_iff) 
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apply (rule_tac x = u and y = x in linorder_cases) 
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apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less]) 
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apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less]) 
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apply (auto) 
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done 
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lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x" 
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by (auto dest!: real_le_imp_less_or_eq real_root_pos) 
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lemma real_root_gt_zero: 
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"0 < x ==> 0 < root (Suc n) x" 
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apply (simp add: root_def del: realpow_Suc) 
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apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp) 
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apply (erule theI' [THEN conjunct1]) 
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done 
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lemma real_root_pos_pos: 
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"0 < x ==> 0 \<le> root(Suc n) x" 
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by (rule real_root_gt_zero [THEN order_less_imp_le]) 
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lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x" 
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by (auto simp add: order_le_less real_root_gt_zero) 
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lemma real_root_one [simp]: "root (Suc n) 1 = 1" 
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apply (simp add: root_def) 
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apply (rule the_equality, auto) 
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apply (rule ccontr) 
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apply (rule_tac x = u and y = 1 in linorder_cases) 
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apply (drule_tac n = n in realpow_Suc_less_one) 
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apply (drule_tac [4] n = n in power_gt1_lemma) 
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apply (auto) 
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done 
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237 

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238 

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subsection{*Square Root*} 
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text{*needed because 2 is a binary numeral!*} 
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lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))" 
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by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 
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add: nat_numeral_0_eq_0 [symmetric]) 
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245 

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lemma real_sqrt_zero [simp]: "sqrt 0 = 0" 
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by (simp add: sqrt_def) 
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248 

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lemma real_sqrt_one [simp]: "sqrt 1 = 1" 
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by (simp add: sqrt_def) 
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251 

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lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" 
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apply (simp add: sqrt_def) 
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apply (rule iffI) 
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255 
apply (cut_tac r = "root 2 x" in realpow_two_le) 
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apply (simp add: numeral_2_eq_2) 
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apply (subst numeral_2_eq_2) 
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apply (erule real_root_pow_pos2) 
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259 
done 
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lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>" 
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262 
by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]]) 
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263 

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lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x" 
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by (simp) 
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266 

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lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>" 
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by (rule real_sqrt_pow2_iff [THEN iffD2], arith) 
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lemma real_pow_sqrt_eq_sqrt_pow: 
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"0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)" 
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apply (simp add: sqrt_def) 
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apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2) 
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done 
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275 

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lemma real_pow_sqrt_eq_sqrt_abs_pow2: 
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"0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
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by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric]) 
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lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>" 
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apply (rule real_sqrt_abs_abs [THEN subst]) 
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apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst]) 
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apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric]) 
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apply (assumption, arith) 
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done 
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286 

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lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" 
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apply auto 
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apply (cut_tac x = x and y = 0 in linorder_less_linear) 
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apply (simp add: zero_less_mult_iff) 
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291 
done 
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292 

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lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)" 
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by (simp add: sqrt_def real_root_gt_zero) 
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295 

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lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)" 
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by (auto intro: real_sqrt_gt_zero simp add: order_le_less) 
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298 

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lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)" 
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by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 
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301 

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(*we need to prove something like this: 
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lemma "[r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r] ==> root n a = r" 
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305 
apply (case_tac n, simp) 
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306 
apply (simp add: root_def) 
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apply (rule someI2 [of _ r], safe) 
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apply (auto simp del: realpow_Suc dest: power_inject_base) 
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309 
*) 
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310 

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lemma sqrt_eqI: "[r\<twosuperior> = a; 0 \<le> r] ==> sqrt a = r" 
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312 
apply (erule subst) 
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apply (simp add: sqrt_def numeral_2_eq_2 del: realpow_Suc) 
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314 
apply (erule real_root_pos2) 
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315 
done 
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316 

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lemma real_sqrt_mult_distrib: 
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"[ 0 \<le> x; 0 \<le> y ] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" 
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319 
apply (rule sqrt_eqI) 
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320 
apply (simp add: power_mult_distrib) 
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apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
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322 
done 
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323 

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lemma real_sqrt_mult_distrib2: 
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"[0\<le>x; 0\<le>y ] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" 
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by (auto intro: real_sqrt_mult_distrib simp add: order_le_less) 
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327 

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lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
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329 
by (auto intro!: real_sqrt_ge_zero) 
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330 

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331 
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: 
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332 
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" 
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333 
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) 
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334 

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lemma real_sqrt_sum_squares_mult_squared_eq [simp]: 
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336 
"sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" 
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by (auto simp add: zero_le_mult_iff simp del: realpow_Suc) 
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338 

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lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>" 
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340 
apply (rule abs_realpow_two [THEN subst]) 
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341 
apply (rule real_sqrt_abs_abs [THEN subst]) 
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342 
apply (subst real_pow_sqrt_eq_sqrt_pow) 
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343 
apply (auto simp add: numeral_2_eq_2) 
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344 
done 
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345 

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lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" 
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347 
apply (rule realpow_two [THEN subst]) 
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348 
apply (subst numeral_2_eq_2 [symmetric]) 
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349 
apply (rule real_sqrt_abs) 
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350 
done 
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351 

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352 
lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>" 
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353 
by simp 
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354 

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355 
lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0" 
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356 
apply (frule real_sqrt_pow2_gt_zero) 
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357 
apply (auto simp add: numeral_2_eq_2) 
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358 
done 
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359 

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360 
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" 
20898  361 
by (cut_tac n = 2 and a = "sqrt x" in power_inverse [symmetric], auto) 
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362 

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363 
lemma real_sqrt_eq_zero_cancel: "[ 0 \<le> x; sqrt(x) = 0] ==> x = 0" 
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364 
apply (drule real_le_imp_less_or_eq) 
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365 
apply (auto dest: real_sqrt_not_eq_zero) 
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366 
done 
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367 

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368 
lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))" 
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by (auto simp add: real_sqrt_eq_zero_cancel) 
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370 

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lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)" 
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apply (subgoal_tac "x \<le> 0  0 \<le> x", safe) 
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apply (rule real_le_trans) 
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apply (auto simp del: realpow_Suc) 
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apply (rule_tac n = 1 in realpow_increasing) 
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apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc) 
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done 
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378 

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lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)" 
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apply (simp (no_asm) add: real_add_commute del: realpow_Suc) 
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done 
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382 

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lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" 
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384 
apply (rule_tac n = 1 in realpow_increasing) 
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apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp 
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del: realpow_Suc) 
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done 
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388 

22443  389 
lemma sqrt_divide_self_eq: 
390 
assumes nneg: "0 \<le> x" 

391 
shows "sqrt x / x = inverse (sqrt x)" 

392 
proof cases 

393 
assume "x=0" thus ?thesis by simp 

394 
next 

395 
assume nz: "x\<noteq>0" 

396 
hence pos: "0<x" using nneg by arith 

397 
show ?thesis 

398 
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

399 
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 

400 
show "inverse (sqrt x) / (sqrt x / x) = 1" 

401 
by (simp add: divide_inverse mult_assoc [symmetric] 

402 
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 

403 
qed 

404 
qed 

405 

14324  406 
end 