author  huffman 
Tue, 17 Apr 2007 00:55:00 +0200  
changeset 22721  d9be18bd7a28 
parent 22630  2a9b64b26612 
child 22856  eb0e0582092a 
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 
12196  5 
*) 
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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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14767  22 
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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27 
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) 

14324  60 
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" 

14477  80 
apply (drule lemma_nth_realpow_non_empty, auto) 
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apply (drule_tac y = s in isLub_isUb [THEN isUbD]) 

14324  82 
apply (auto intro: order_less_le_trans) 
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done 

84 

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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*} 

108 

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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_of_nat_inverse_le_iff: 
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"(inverse (real(Suc n)) \<le> r) = (1 \<le> r * real(Suc n))" 
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by (simp add: inverse_eq_divide pos_divide_le_eq) 
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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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14324  139 
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: 

153 
"[ (0::real) < a; 0 < n; 

154 
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) 

14324  166 
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) 

14324  172 
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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177 
(* 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" 

180 
apply (auto intro!: realpow_pos_nth) 

14477  181 
apply (cut_tac x = r and y = y in linorder_less_linear, auto) 
182 
apply (drule_tac x = r in realpow_less) 

183 
apply (drule_tac [4] x = y in realpow_less, auto) 

14324  184 
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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235 
apply (rule ccontr) 
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236 
apply (rule_tac x = u and y = 1 in linorder_cases) 
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237 
apply (drule_tac n = n in realpow_Suc_less_one) 
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238 
apply (drule_tac [4] n = n in power_gt1_lemma) 
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239 
apply (auto) 
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240 
done 
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241 

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242 
lemma real_root_less_mono: 
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243 
"[ 0 \<le> x; x < y ] ==> root(Suc n) x < root(Suc n) y" 
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244 
apply (frule order_le_less_trans, assumption) 
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245 
apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst]) 
d9be18bd7a28
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246 
apply (rotate_tac 1, assumption) 
d9be18bd7a28
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247 
apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst]) 
d9be18bd7a28
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248 
apply (rotate_tac 3, assumption) 
d9be18bd7a28
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249 
apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le) 
d9be18bd7a28
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250 
apply (frule_tac n = n in real_root_pos_pos_le) 
d9be18bd7a28
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251 
apply (frule_tac n = n in real_root_pos_pos) 
d9be18bd7a28
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252 
apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing) 
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253 
apply (assumption, assumption) 
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254 
apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq) 
d9be18bd7a28
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255 
apply auto 
d9be18bd7a28
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256 
apply (drule_tac f = "%x. x ^ (Suc n)" in arg_cong) 
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257 
apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc) 
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258 
done 
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259 

d9be18bd7a28
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260 
lemma real_root_le_mono: 
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261 
"[ 0 \<le> x; x \<le> y ] ==> root(Suc n) x \<le> root(Suc n) y" 
d9be18bd7a28
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262 
apply (drule_tac y = y in order_le_imp_less_or_eq) 
d9be18bd7a28
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263 
apply (auto dest: real_root_less_mono intro: order_less_imp_le) 
d9be18bd7a28
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264 
done 
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265 

d9be18bd7a28
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266 
lemma real_root_less_iff [simp]: 
d9be18bd7a28
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267 
"[ 0 \<le> x; 0 \<le> y ] ==> (root(Suc n) x < root(Suc n) y) = (x < y)" 
d9be18bd7a28
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268 
apply (auto intro: real_root_less_mono) 
d9be18bd7a28
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269 
apply (rule ccontr, drule linorder_not_less [THEN iffD1]) 
d9be18bd7a28
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270 
apply (drule_tac x = y and n = n in real_root_le_mono, auto) 
d9be18bd7a28
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271 
done 
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272 

d9be18bd7a28
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273 
lemma real_root_le_iff [simp]: 
d9be18bd7a28
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274 
"[ 0 \<le> x; 0 \<le> y ] ==> (root(Suc n) x \<le> root(Suc n) y) = (x \<le> y)" 
d9be18bd7a28
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275 
apply (auto intro: real_root_le_mono) 
d9be18bd7a28
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276 
apply (simp (no_asm) add: linorder_not_less [symmetric]) 
d9be18bd7a28
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277 
apply auto 
d9be18bd7a28
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278 
apply (drule_tac x = y and n = n in real_root_less_mono, auto) 
d9be18bd7a28
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279 
done 
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280 

d9be18bd7a28
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281 
lemma real_root_eq_iff [simp]: 
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282 
"[ 0 \<le> x; 0 \<le> y ] ==> (root(Suc n) x = root(Suc n) y) = (x = y)" 
d9be18bd7a28
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283 
apply (auto intro!: order_antisym [where 'a = real]) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
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284 
apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1]) 
d9be18bd7a28
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285 
apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto) 
d9be18bd7a28
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286 
done 
d9be18bd7a28
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287 

d9be18bd7a28
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288 
lemma real_root_pos_unique: 
d9be18bd7a28
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289 
"[ 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x ] ==> root (Suc n) x = y" 
d9be18bd7a28
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290 
by (auto dest: real_root_pos2 simp del: realpow_Suc) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
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291 

d9be18bd7a28
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292 
lemma real_root_mult: 
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293 
"[ 0 \<le> x; 0 \<le> y ] 
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294 
==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y" 
d9be18bd7a28
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295 
apply (rule real_root_pos_unique) 
d9be18bd7a28
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296 
apply (auto intro!: real_root_pos_pos_le 
d9be18bd7a28
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297 
simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2 
d9be18bd7a28
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298 
simp del: realpow_Suc) 
d9be18bd7a28
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299 
done 
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300 

d9be18bd7a28
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301 
lemma real_root_inverse: 
d9be18bd7a28
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302 
"0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))" 
d9be18bd7a28
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303 
apply (rule real_root_pos_unique) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
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304 
apply (auto intro: real_root_pos_pos_le 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
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305 
simp add: power_inverse [symmetric] real_root_pow_pos2 
d9be18bd7a28
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306 
simp del: realpow_Suc) 
d9be18bd7a28
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307 
done 
d9be18bd7a28
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changeset

308 

d9be18bd7a28
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309 
lemma real_root_divide: 
d9be18bd7a28
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310 
"[ 0 \<le> x; 0 \<le> y ] 
d9be18bd7a28
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311 
==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)" 
d9be18bd7a28
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312 
apply (simp add: divide_inverse) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
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313 
apply (auto simp add: real_root_mult real_root_inverse) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
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314 
done 
d9be18bd7a28
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315 

20687
fedb901be392
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316 

fedb901be392
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317 
subsection{*Square Root*} 
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318 

fedb901be392
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319 
text{*needed because 2 is a binary numeral!*} 
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320 
lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))" 
fedb901be392
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321 
by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 
fedb901be392
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322 
add: nat_numeral_0_eq_0 [symmetric]) 
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changeset

323 

fedb901be392
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324 
lemma real_sqrt_zero [simp]: "sqrt 0 = 0" 
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325 
by (simp add: sqrt_def) 
fedb901be392
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changeset

326 

fedb901be392
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327 
lemma real_sqrt_one [simp]: "sqrt 1 = 1" 
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328 
by (simp add: sqrt_def) 
fedb901be392
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changeset

329 

fedb901be392
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changeset

330 
lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" 
fedb901be392
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331 
apply (simp add: sqrt_def) 
fedb901be392
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changeset

332 
apply (rule iffI) 
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changeset

333 
apply (cut_tac r = "root 2 x" in realpow_two_le) 
fedb901be392
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huffman
parents:
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diff
changeset

334 
apply (simp add: numeral_2_eq_2) 
fedb901be392
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huffman
parents:
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diff
changeset

335 
apply (subst numeral_2_eq_2) 
fedb901be392
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changeset

336 
apply (erule real_root_pow_pos2) 
fedb901be392
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337 
done 
fedb901be392
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changeset

338 

fedb901be392
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339 
lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>" 
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340 
by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]]) 
fedb901be392
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changeset

341 

fedb901be392
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342 
lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x" 
fedb901be392
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343 
by (simp) 
fedb901be392
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changeset

344 

fedb901be392
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345 
lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>" 
fedb901be392
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346 
by (rule real_sqrt_pow2_iff [THEN iffD2], arith) 
fedb901be392
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changeset

347 

fedb901be392
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348 
lemma real_pow_sqrt_eq_sqrt_pow: 
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349 
"0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)" 
fedb901be392
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350 
apply (simp add: sqrt_def) 
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351 
apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2) 
fedb901be392
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352 
done 
fedb901be392
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parents:
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changeset

353 

fedb901be392
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parents:
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354 
lemma real_pow_sqrt_eq_sqrt_abs_pow2: 
fedb901be392
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355 
"0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

356 
by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

357 

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

358 
lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

359 
apply (rule real_sqrt_abs_abs [THEN subst]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

360 
apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

361 
apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

362 
apply (assumption, arith) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

363 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

364 

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

365 
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:
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diff
changeset

366 
apply auto 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

367 
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

368 
apply (simp add: zero_less_mult_iff) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

369 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

370 

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

371 
lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

372 
by (simp add: sqrt_def real_root_gt_zero) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

373 

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

374 
lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

375 
by (auto intro: real_sqrt_gt_zero simp add: order_le_less) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

376 

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

377 
lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

378 
by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

379 

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

380 

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

381 
(*we need to prove something like this: 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

382 
lemma "[r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r] ==> root n a = r" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

383 
apply (case_tac n, simp) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

384 
apply (simp add: root_def) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

385 
apply (rule someI2 [of _ r], safe) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

386 
apply (auto simp del: realpow_Suc dest: power_inject_base) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

387 
*) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

388 

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

389 
lemma sqrt_eqI: "[r\<twosuperior> = a; 0 \<le> r] ==> sqrt a = r" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

390 
apply (erule subst) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

391 
apply (simp add: sqrt_def numeral_2_eq_2 del: realpow_Suc) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

392 
apply (erule real_root_pos2) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

393 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

394 

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

395 
lemma real_sqrt_mult_distrib: 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

396 
"[ 0 \<le> x; 0 \<le> y ] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

397 
apply (rule sqrt_eqI) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

398 
apply (simp add: power_mult_distrib) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

399 
apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

400 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

401 

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

402 
lemma real_sqrt_mult_distrib2: 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

403 
"[0\<le>x; 0\<le>y ] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

404 
by (auto intro: real_sqrt_mult_distrib simp add: order_le_less) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

405 

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

406 
lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

407 
by (auto intro!: real_sqrt_ge_zero) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

408 

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

409 
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

410 
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

411 
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

412 

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

413 
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

414 
"sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

415 
by (auto simp add: zero_le_mult_iff simp del: realpow_Suc) 
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_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

418 
apply (rule abs_realpow_two [THEN subst]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

419 
apply (rule real_sqrt_abs_abs [THEN subst]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

420 
apply (subst real_pow_sqrt_eq_sqrt_pow) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

421 
apply (auto simp add: numeral_2_eq_2) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

422 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

423 

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

424 
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

425 
apply (rule realpow_two [THEN subst]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

426 
apply (subst numeral_2_eq_2 [symmetric]) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

427 
apply (rule real_sqrt_abs) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

428 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

429 

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

430 
lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

431 
by simp 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

432 

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

433 
lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

434 
apply (frule real_sqrt_pow2_gt_zero) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

435 
apply (auto simp add: numeral_2_eq_2) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

436 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

437 

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

438 
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" 
20898  439 
by (cut_tac n = 2 and a = "sqrt x" in power_inverse [symmetric], auto) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

440 

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

441 
lemma real_sqrt_eq_zero_cancel: "[ 0 \<le> x; sqrt(x) = 0] ==> x = 0" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

442 
apply (drule real_le_imp_less_or_eq) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

443 
apply (auto dest: real_sqrt_not_eq_zero) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

444 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

445 

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

446 
lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

447 
by (auto simp add: real_sqrt_eq_zero_cancel) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

448 

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

449 
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

450 
apply (subgoal_tac "x \<le> 0  0 \<le> x", safe) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

451 
apply (rule real_le_trans) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

452 
apply (auto simp del: realpow_Suc) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

453 
apply (rule_tac n = 1 in realpow_increasing) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

454 
apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

455 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

456 

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

457 
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

458 
apply (simp (no_asm) add: real_add_commute del: realpow_Suc) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

459 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

460 

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

461 
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

462 
apply (rule_tac n = 1 in realpow_increasing) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

463 
apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

464 
del: realpow_Suc) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

465 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

466 

22443  467 
lemma sqrt_divide_self_eq: 
468 
assumes nneg: "0 \<le> x" 

469 
shows "sqrt x / x = inverse (sqrt x)" 

470 
proof cases 

471 
assume "x=0" thus ?thesis by simp 

472 
next 

473 
assume nz: "x\<noteq>0" 

474 
hence pos: "0<x" using nneg by arith 

475 
show ?thesis 

476 
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

477 
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 

478 
show "inverse (sqrt x) / (sqrt x / x) = 1" 

479 
by (simp add: divide_inverse mult_assoc [symmetric] 

480 
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 

481 
qed 

482 
qed 

483 

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

484 

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

485 
lemma real_sqrt_less_mono: "[ 0 \<le> x; x < y ] ==> sqrt(x) < sqrt(y)" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

486 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

487 

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

488 
lemma real_sqrt_le_mono: "[ 0 \<le> x; x \<le> y ] ==> sqrt(x) \<le> sqrt(y)" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

489 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

490 

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

491 
lemma real_sqrt_less_iff [simp]: 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

492 
"[ 0 \<le> x; 0 \<le> y ] ==> (sqrt(x) < sqrt(y)) = (x < y)" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

493 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

494 

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

495 
lemma real_sqrt_le_iff [simp]: 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

496 
"[ 0 \<le> x; 0 \<le> y ] ==> (sqrt(x) \<le> sqrt(y)) = (x \<le> y)" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

497 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

498 

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

499 
lemma real_sqrt_eq_iff [simp]: 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

500 
"[ 0 \<le> x; 0 \<le> y ] ==> (sqrt(x) = sqrt(y)) = (x = y)" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

501 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

502 

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

503 
lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

504 
apply (rule real_sqrt_one [THEN subst], safe) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

505 
apply (rule_tac [2] real_sqrt_less_mono) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

506 
apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

507 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

508 

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

509 
lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

510 
apply (rule real_sqrt_one [THEN subst], safe) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

511 
apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

512 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

513 

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

514 
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

515 
apply (simp add: divide_inverse) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

516 
apply (case_tac "r=0") 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

517 
apply (auto simp add: mult_ac) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

518 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

519 

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

520 

14324  521 
end 