author  huffman 
Tue, 08 May 2007 04:56:28 +0200  
changeset 22858  5ca5d1cce388 
parent 22856  eb0e0582092a 
child 22943  0b928312ab94 
permissions  rwrr 
12196  1 
(* Title : NthRoot.thy 
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Author : Jacques D. Fleuriot 

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Copyright : 1998 University of Cambridge 

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Conversion to Isar and new proofs by Lawrence C Paulson, 2004 
12196  5 
*) 
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header{*Existence of Nth Root*} 
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15131  9 
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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14324  67 

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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]) 

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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) 
14324  95 
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) 
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apply (drule_tac x = r in realpow_less) 

183 
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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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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"[ 0 \<le> x; x < y ] ==> root(Suc n) x < root(Suc n) y" 
22856  244 
apply (subgoal_tac "0 \<le> y") 
245 
apply (rule_tac n="Suc n" in power_less_imp_less_base) 

246 
apply (simp only: real_root_pow_pos2) 

247 
apply (erule real_root_pos_pos_le) 

248 
apply (erule order_trans) 

249 
apply (erule order_less_imp_le) 

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250 
done 
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251 

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252 
lemma real_root_le_mono: 
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"[ 0 \<le> x; x \<le> y ] ==> root(Suc n) x \<le> root(Suc n) y" 
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254 
apply (drule_tac y = y in order_le_imp_less_or_eq) 
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255 
apply (auto dest: real_root_less_mono intro: order_less_imp_le) 
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256 
done 
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257 

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258 
lemma real_root_less_iff [simp]: 
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259 
"[ 0 \<le> x; 0 \<le> y ] ==> (root(Suc n) x < root(Suc n) y) = (x < y)" 
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260 
apply (auto intro: real_root_less_mono) 
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261 
apply (rule ccontr, drule linorder_not_less [THEN iffD1]) 
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262 
apply (drule_tac x = y and n = n in real_root_le_mono, auto) 
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263 
done 
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264 

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265 
lemma real_root_le_iff [simp]: 
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"[ 0 \<le> x; 0 \<le> y ] ==> (root(Suc n) x \<le> root(Suc n) y) = (x \<le> y)" 
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267 
apply (auto intro: real_root_le_mono) 
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268 
apply (simp (no_asm) add: linorder_not_less [symmetric]) 
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269 
apply auto 
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270 
apply (drule_tac x = y and n = n in real_root_less_mono, auto) 
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271 
done 
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272 

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273 
lemma real_root_eq_iff [simp]: 
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274 
"[ 0 \<le> x; 0 \<le> y ] ==> (root(Suc n) x = root(Suc n) y) = (x = y)" 
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275 
apply (auto intro!: order_antisym [where 'a = real]) 
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276 
apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1]) 
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277 
apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto) 
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278 
done 
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279 

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280 
lemma real_root_pos_unique: 
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281 
"[ 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x ] ==> root (Suc n) x = y" 
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282 
by (auto dest: real_root_pos2 simp del: realpow_Suc) 
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283 

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284 
lemma real_root_mult: 
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285 
"[ 0 \<le> x; 0 \<le> y ] 
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286 
==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y" 
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287 
apply (rule real_root_pos_unique) 
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288 
apply (auto intro!: real_root_pos_pos_le 
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289 
simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2 
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290 
simp del: realpow_Suc) 
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291 
done 
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292 

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293 
lemma real_root_inverse: 
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294 
"0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))" 
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295 
apply (rule real_root_pos_unique) 
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296 
apply (auto intro: real_root_pos_pos_le 
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297 
simp add: power_inverse [symmetric] real_root_pow_pos2 
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298 
simp del: realpow_Suc) 
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299 
done 
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300 

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301 
lemma real_root_divide: 
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302 
"[ 0 \<le> x; 0 \<le> y ] 
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303 
==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)" 
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304 
apply (simp add: divide_inverse) 
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305 
apply (auto simp add: real_root_mult real_root_inverse) 
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306 
done 
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307 

20687
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308 

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309 
subsection{*Square Root*} 
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310 

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311 
text{*needed because 2 is a binary numeral!*} 
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312 
lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))" 
22856  313 
by (simp only: numeral_2_eq_2) 
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314 

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315 
lemma real_sqrt_zero [simp]: "sqrt 0 = 0" 
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316 
by (simp add: sqrt_def) 
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317 

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318 
lemma real_sqrt_one [simp]: "sqrt 1 = 1" 
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319 
by (simp add: sqrt_def) 
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320 

22856  321 
lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x" 
322 
unfolding sqrt_def numeral_2_eq_2 

323 
by (rule real_root_pow_pos2) 

324 

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325 
lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" 
22856  326 
apply (rule iffI) 
327 
apply (erule subst) 

328 
apply (rule zero_le_power2) 

329 
apply (erule real_sqrt_pow2) 

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330 
done 
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331 

22856  332 
lemma real_sqrt_abs_abs [simp]: "(sqrt \<bar>x\<bar>)\<twosuperior> = \<bar>x\<bar>" (* TODO: delete *) 
333 
by simp 

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334 

22856  335 
lemma sqrt_eqI: "\<lbrakk>r\<twosuperior> = a; 0 \<le> r\<rbrakk> \<Longrightarrow> sqrt a = r" 
336 
unfolding sqrt_def numeral_2_eq_2 

337 
by (erule subst, erule real_root_pos2) 

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338 

22856  339 
lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>" 
340 
apply (rule sqrt_eqI) 

341 
apply (rule power2_abs) 

342 
apply (rule abs_ge_zero) 

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343 
done 
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344 

22856  345 
lemma real_pow_sqrt_eq_sqrt_pow: (* TODO: delete *) 
346 
"0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)" 

347 
by simp 

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348 

22856  349 
lemma real_pow_sqrt_eq_sqrt_abs_pow2: (* TODO: delete *) 
350 
"0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 

351 
by simp 

352 

353 
lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>" (* TODO: delete *) 

354 
by simp 

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355 

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356 
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" 
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357 
apply auto 
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358 
apply (cut_tac x = x and y = 0 in linorder_less_linear) 
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359 
apply (simp add: zero_less_mult_iff) 
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360 
done 
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361 

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362 
lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)" 
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363 
by (simp add: sqrt_def real_root_gt_zero) 
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364 

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365 
lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)" 
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366 
by (auto intro: real_sqrt_gt_zero simp add: order_le_less) 
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367 

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368 

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369 
(*we need to prove something like this: 
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370 
lemma "[r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r] ==> root n a = r" 
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371 
apply (case_tac n, simp) 
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372 
apply (simp add: root_def) 
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373 
apply (rule someI2 [of _ r], safe) 
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374 
apply (auto simp del: realpow_Suc dest: power_inject_base) 
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375 
*) 
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376 

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377 
lemma real_sqrt_mult_distrib: 
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378 
"[ 0 \<le> x; 0 \<le> y ] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" 
22856  379 
unfolding sqrt_def numeral_2_eq_2 
380 
by (rule real_root_mult) 

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381 

22856  382 
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult_distrib 
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383 

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384 
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" 
22856  385 
apply (subst power2_eq_square [symmetric]) 
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changeset

386 
apply (rule real_sqrt_abs) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

387 
done 
fedb901be392
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huffman
parents:
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diff
changeset

388 

fedb901be392
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parents:
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changeset

389 
lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>" 
fedb901be392
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parents:
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changeset

390 
by simp 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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changeset

391 

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

392 
lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0" 
fedb901be392
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changeset

393 
apply (frule real_sqrt_pow2_gt_zero) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

394 
apply (auto simp add: numeral_2_eq_2) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

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

396 

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

397 
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" 
22856  398 
by (simp add: power_inverse [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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changeset

399 

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

400 
lemma real_sqrt_eq_zero_cancel: "[ 0 \<le> x; sqrt(x) = 0] ==> x = 0" 
fedb901be392
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changeset

401 
apply (drule real_le_imp_less_or_eq) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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changeset

402 
apply (auto dest: real_sqrt_not_eq_zero) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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diff
changeset

403 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

404 

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

405 
lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))" 
fedb901be392
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changeset

406 
by (auto simp add: real_sqrt_eq_zero_cancel) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset

407 

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

408 
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" 
22856  409 
apply (rule power2_le_imp_le, simp) 
410 
apply (simp add: real_sqrt_ge_zero) 

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

411 
done 
fedb901be392
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changeset

412 

22443  413 
lemma sqrt_divide_self_eq: 
414 
assumes nneg: "0 \<le> x" 

415 
shows "sqrt x / x = inverse (sqrt x)" 

416 
proof cases 

417 
assume "x=0" thus ?thesis by simp 

418 
next 

419 
assume nz: "x\<noteq>0" 

420 
hence pos: "0<x" using nneg by arith 

421 
show ?thesis 

422 
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

423 
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 

424 
show "inverse (sqrt x) / (sqrt x / x) = 1" 

425 
by (simp add: divide_inverse mult_assoc [symmetric] 

426 
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 

427 
qed 

428 
qed 

429 

22721
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huffman
parents:
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diff
changeset

430 

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

431 
lemma real_sqrt_less_mono: "[ 0 \<le> x; x < y ] ==> sqrt(x) < sqrt(y)" 
d9be18bd7a28
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parents:
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changeset

432 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

433 

d9be18bd7a28
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parents:
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changeset

434 
lemma real_sqrt_le_mono: "[ 0 \<le> x; x \<le> y ] ==> sqrt(x) \<le> sqrt(y)" 
d9be18bd7a28
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huffman
parents:
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changeset

435 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

436 

d9be18bd7a28
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parents:
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changeset

437 
lemma real_sqrt_less_iff [simp]: 
d9be18bd7a28
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changeset

438 
"[ 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:
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diff
changeset

439 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

440 

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

441 
lemma real_sqrt_le_iff [simp]: 
d9be18bd7a28
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changeset

442 
"[ 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:
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changeset

443 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

444 

d9be18bd7a28
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changeset

445 
lemma real_sqrt_eq_iff [simp]: 
d9be18bd7a28
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changeset

446 
"[ 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:
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diff
changeset

447 
by (simp add: sqrt_def) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

448 

d9be18bd7a28
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huffman
parents:
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changeset

449 
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r" 
d9be18bd7a28
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huffman
parents:
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changeset

450 
apply (simp add: divide_inverse) 
d9be18bd7a28
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parents:
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changeset

451 
apply (case_tac "r=0") 
d9be18bd7a28
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huffman
parents:
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changeset

452 
apply (auto simp add: mult_ac) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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changeset

453 
done 
d9be18bd7a28
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huffman
parents:
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changeset

454 

22856  455 
subsection {* Square Root of Sum of Squares *} 
456 

457 
lemma "(sqrt (x\<twosuperior> + y\<twosuperior>))\<twosuperior> = x\<twosuperior> + y\<twosuperior>" 

458 
by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]]) 

459 

460 
lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)" 

461 
by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 

462 

463 
lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 

464 
by (auto intro!: real_sqrt_ge_zero) 

465 

466 
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: 

467 
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" 

468 
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) 

469 

470 
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: 

471 
"sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" 

472 
by (auto simp add: zero_le_mult_iff simp del: realpow_Suc) 

473 

474 
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)" 

475 
by (rule power2_le_imp_le, simp_all) 

476 

477 
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(x\<twosuperior> + y\<twosuperior>)" 

478 
by (rule power2_le_imp_le, simp_all) 

479 

480 
lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)" 

481 
apply (subst real_sqrt_one [symmetric]) 

482 
apply (rule real_sqrt_less_iff, auto) 

483 
done 

484 

485 
lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)" 

486 
apply (subst real_sqrt_one [symmetric]) 

487 
apply (rule real_sqrt_eq_iff, auto) 

488 
done 

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

489 

22858  490 
lemma power2_sum: 
491 
fixes x y :: "'a::{number_ring,recpower}" 

492 
shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y" 

493 
by (simp add: left_distrib right_distrib power2_eq_square) 

494 

495 
lemma power2_diff: 

496 
fixes x y :: "'a::{number_ring,recpower}" 

497 
shows "(x  y)\<twosuperior> = x\<twosuperior> + y\<twosuperior>  2 * x * y" 

498 
by (simp add: left_diff_distrib right_diff_distrib power2_eq_square) 

499 

500 
lemma real_sqrt_sum_squares_triangle_ineq: 

501 
"sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)" 

502 
apply (rule power2_le_imp_le, simp) 

503 
apply (simp add: power2_sum) 

504 
apply (simp only: mult_assoc right_distrib [symmetric]) 

505 
apply (rule mult_left_mono) 

506 
apply (rule power2_le_imp_le) 

507 
apply (simp add: power2_sum power_mult_distrib) 

508 
apply (simp add: ring_distrib) 

509 
apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior>  2 * (a * c) * (b * d)", simp) 

510 
apply (rule_tac b="(a * d  b * c)\<twosuperior>" in ord_le_eq_trans) 

511 
apply (rule zero_le_power2) 

512 
apply (simp add: power2_diff power_mult_distrib) 

513 
apply (simp add: mult_nonneg_nonneg) 

514 
apply simp 

515 
apply (simp add: add_increasing) 

516 
done 

517 

14324  518 
end 