src/HOL/Hyperreal/NthRoot.thy
 author huffman Wed Mar 14 21:52:26 2007 +0100 (2007-03-14) changeset 22443 346729a55460 parent 21865 55cc354fd2d9 child 22630 2a9b64b26612 permissions -rw-r--r--
move sqrt_divide_self_eq to NthRoot.thy
```     1 (*  Title       : NthRoot.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
```
```     5 *)
```
```     6
```
```     7 header{*Existence of Nth Root*}
```
```     8
```
```     9 theory NthRoot
```
```    10 imports SEQ Parity
```
```    11 begin
```
```    12
```
```    13 definition
```
```    14   root :: "[nat, real] \<Rightarrow> real" where
```
```    15   "root n x = (THE u. (0 < x \<longrightarrow> 0 < u) \<and> (u ^ n = x))"
```
```    16
```
```    17 definition
```
```    18   sqrt :: "real \<Rightarrow> real" where
```
```    19   "sqrt x = root 2 x"
```
```    20
```
```    21
```
```    22 text {*
```
```    23   Various lemmas needed for this result. We follow the proof given by
```
```    24   John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis
```
```    25   Webnotes available at \url{http://www.math.unl.edu/~webnotes}.
```
```    26
```
```    27   Lemmas about sequences of reals are used to reach the result.
```
```    28 *}
```
```    29
```
```    30 lemma lemma_nth_realpow_non_empty:
```
```    31      "[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"
```
```    32 apply (case_tac "1 <= a")
```
```    33 apply (rule_tac x = 1 in exI)
```
```    34 apply (drule_tac [2] linorder_not_le [THEN iffD1])
```
```    35 apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp)
```
```    36 apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc)
```
```    37 done
```
```    38
```
```    39 text{*Used only just below*}
```
```    40 lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n"
```
```    41 by (insert power_increasing [of 1 n r], simp)
```
```    42
```
```    43 lemma lemma_nth_realpow_isUb_ex:
```
```    44      "[| (0::real) < a; 0 < n |]
```
```    45       ==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
```
```    46 apply (case_tac "1 <= a")
```
```    47 apply (rule_tac x = a in exI)
```
```    48 apply (drule_tac [2] linorder_not_le [THEN iffD1])
```
```    49 apply (rule_tac [2] x = 1 in exI)
```
```    50 apply (rule_tac [!] setleI [THEN isUbI], safe)
```
```    51 apply (simp_all (no_asm))
```
```    52 apply (rule_tac [!] ccontr)
```
```    53 apply (drule_tac [!] linorder_not_le [THEN iffD1])
```
```    54 apply (drule realpow_ge_self2, assumption)
```
```    55 apply (drule_tac n = n in realpow_less)
```
```    56 apply (assumption+)
```
```    57 apply (drule real_le_trans, assumption)
```
```    58 apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp)
```
```    59 apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto)
```
```    60 done
```
```    61
```
```    62 lemma nth_realpow_isLub_ex:
```
```    63      "[| (0::real) < a; 0 < n |]
```
```    64       ==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
```
```    65 by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)
```
```    66
```
```    67
```
```    68 subsection{*First Half -- Lemmas First*}
```
```    69
```
```    70 lemma lemma_nth_realpow_seq:
```
```    71      "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u
```
```    72            ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"
```
```    73 apply (safe, drule isLubD2, blast)
```
```    74 apply (simp add: linorder_not_less [symmetric])
```
```    75 done
```
```    76
```
```    77 lemma lemma_nth_realpow_isLub_gt_zero:
```
```    78      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;
```
```    79          0 < a; 0 < n |] ==> 0 < u"
```
```    80 apply (drule lemma_nth_realpow_non_empty, auto)
```
```    81 apply (drule_tac y = s in isLub_isUb [THEN isUbD])
```
```    82 apply (auto intro: order_less_le_trans)
```
```    83 done
```
```    84
```
```    85 lemma lemma_nth_realpow_isLub_ge:
```
```    86      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;
```
```    87          0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"
```
```    88 apply safe
```
```    89 apply (frule lemma_nth_realpow_seq, safe)
```
```    90 apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric]
```
```    91             iff: real_0_less_add_iff) --{*legacy iff rule!*}
```
```    92 apply (simp add: linorder_not_less)
```
```    93 apply (rule order_less_trans [of _ 0])
```
```    94 apply (auto intro: lemma_nth_realpow_isLub_gt_zero)
```
```    95 done
```
```    96
```
```    97 text{*First result we want*}
```
```    98 lemma realpow_nth_ge:
```
```    99      "[| (0::real) < a; 0 < n;
```
```   100      isLub (UNIV::real set)
```
```   101      {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"
```
```   102 apply (frule lemma_nth_realpow_isLub_ge, safe)
```
```   103 apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])
```
```   104 apply (auto simp add: real_of_nat_def)
```
```   105 done
```
```   106
```
```   107 subsection{*Second Half*}
```
```   108
```
```   109 lemma less_isLub_not_isUb:
```
```   110      "[| isLub (UNIV::real set) S u; x < u |]
```
```   111            ==> ~ isUb (UNIV::real set) S x"
```
```   112 apply safe
```
```   113 apply (drule isLub_le_isUb, assumption)
```
```   114 apply (drule order_less_le_trans, auto)
```
```   115 done
```
```   116
```
```   117 lemma not_isUb_less_ex:
```
```   118      "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
```
```   119 apply (rule ccontr, erule contrapos_np)
```
```   120 apply (rule setleI [THEN isUbI])
```
```   121 apply (auto simp add: linorder_not_less [symmetric])
```
```   122 done
```
```   123
```
```   124 lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r"
```
```   125 apply (simp (no_asm) add: right_distrib)
```
```   126 apply (rule add_less_cancel_left [of "-r", THEN iffD1])
```
```   127 apply (auto intro: mult_pos_pos
```
```   128             simp add: add_assoc [symmetric] neg_less_0_iff_less)
```
```   129 done
```
```   130
```
```   131 lemma real_mult_add_one_minus_ge_zero:
```
```   132      "0 < r ==>  0 <= r*(1 + -inverse(real (Suc n)))"
```
```   133 by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff)
```
```   134
```
```   135 lemma lemma_nth_realpow_isLub_le:
```
```   136      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;
```
```   137        0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a"
```
```   138 apply safe
```
```   139 apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
```
```   140 apply (rule_tac n = k in real_mult_less_self)
```
```   141 apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe)
```
```   142 apply (drule_tac n = k in
```
```   143         lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+)
```
```   144 apply (blast intro: order_trans order_less_imp_le power_mono)
```
```   145 done
```
```   146
```
```   147 text{*Second result we want*}
```
```   148 lemma realpow_nth_le:
```
```   149      "[| (0::real) < a; 0 < n;
```
```   150      isLub (UNIV::real set)
```
```   151      {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
```
```   152 apply (frule lemma_nth_realpow_isLub_le, safe)
```
```   153 apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult
```
```   154                 [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
```
```   155 apply (auto simp add: real_of_nat_def)
```
```   156 done
```
```   157
```
```   158 text{*The theorem at last!*}
```
```   159 lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
```
```   160 apply (frule nth_realpow_isLub_ex, auto)
```
```   161 apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym)
```
```   162 done
```
```   163
```
```   164 (* positive only *)
```
```   165 lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
```
```   166 apply (frule nth_realpow_isLub_ex, auto)
```
```   167 apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero)
```
```   168 done
```
```   169
```
```   170 lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
```
```   171 by (blast intro: realpow_pos_nth)
```
```   172
```
```   173 (* uniqueness of nth positive root *)
```
```   174 lemma realpow_pos_nth_unique:
```
```   175      "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
```
```   176 apply (auto intro!: realpow_pos_nth)
```
```   177 apply (cut_tac x = r and y = y in linorder_less_linear, auto)
```
```   178 apply (drule_tac x = r in realpow_less)
```
```   179 apply (drule_tac [4] x = y in realpow_less, auto)
```
```   180 done
```
```   181
```
```   182 subsection {* Nth Root *}
```
```   183
```
```   184 lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
```
```   185 apply (simp add: root_def)
```
```   186 apply (safe intro!: the_equality power_0_Suc elim!: realpow_zero_zero)
```
```   187 done
```
```   188
```
```   189 lemma real_root_pow_pos:
```
```   190      "0 < x ==> (root (Suc n) x) ^ (Suc n) = x"
```
```   191 apply (simp add: root_def del: realpow_Suc)
```
```   192 apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp)
```
```   193 apply (erule theI' [THEN conjunct2])
```
```   194 done
```
```   195
```
```   196 lemma real_root_pow_pos2: "0 \<le> x ==> (root (Suc n) x) ^ (Suc n) = x"
```
```   197 by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos)
```
```   198
```
```   199 lemma real_root_pos:
```
```   200      "0 < x ==> root(Suc n) (x ^ (Suc n)) = x"
```
```   201 apply (simp add: root_def)
```
```   202 apply (rule the_equality)
```
```   203 apply (frule_tac [2] n = n in zero_less_power)
```
```   204 apply (auto simp add: zero_less_mult_iff)
```
```   205 apply (rule_tac x = u and y = x in linorder_cases)
```
```   206 apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less])
```
```   207 apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less])
```
```   208 apply (auto)
```
```   209 done
```
```   210
```
```   211 lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x"
```
```   212 by (auto dest!: real_le_imp_less_or_eq real_root_pos)
```
```   213
```
```   214 lemma real_root_gt_zero:
```
```   215      "0 < x ==> 0 < root (Suc n) x"
```
```   216 apply (simp add: root_def del: realpow_Suc)
```
```   217 apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp)
```
```   218 apply (erule theI' [THEN conjunct1])
```
```   219 done
```
```   220
```
```   221 lemma real_root_pos_pos:
```
```   222      "0 < x ==> 0 \<le> root(Suc n) x"
```
```   223 by (rule real_root_gt_zero [THEN order_less_imp_le])
```
```   224
```
```   225 lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x"
```
```   226 by (auto simp add: order_le_less real_root_gt_zero)
```
```   227
```
```   228 lemma real_root_one [simp]: "root (Suc n) 1 = 1"
```
```   229 apply (simp add: root_def)
```
```   230 apply (rule the_equality, auto)
```
```   231 apply (rule ccontr)
```
```   232 apply (rule_tac x = u and y = 1 in linorder_cases)
```
```   233 apply (drule_tac n = n in realpow_Suc_less_one)
```
```   234 apply (drule_tac [4] n = n in power_gt1_lemma)
```
```   235 apply (auto)
```
```   236 done
```
```   237
```
```   238
```
```   239 subsection{*Square Root*}
```
```   240
```
```   241 text{*needed because 2 is a binary numeral!*}
```
```   242 lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"
```
```   243 by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1
```
```   244          add: nat_numeral_0_eq_0 [symmetric])
```
```   245
```
```   246 lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
```
```   247 by (simp add: sqrt_def)
```
```   248
```
```   249 lemma real_sqrt_one [simp]: "sqrt 1 = 1"
```
```   250 by (simp add: sqrt_def)
```
```   251
```
```   252 lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
```
```   253 apply (simp add: sqrt_def)
```
```   254 apply (rule iffI)
```
```   255  apply (cut_tac r = "root 2 x" in realpow_two_le)
```
```   256  apply (simp add: numeral_2_eq_2)
```
```   257 apply (subst numeral_2_eq_2)
```
```   258 apply (erule real_root_pow_pos2)
```
```   259 done
```
```   260
```
```   261 lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>"
```
```   262 by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])
```
```   263
```
```   264 lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x"
```
```   265 by (simp)
```
```   266
```
```   267 lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>"
```
```   268 by (rule real_sqrt_pow2_iff [THEN iffD2], arith)
```
```   269
```
```   270 lemma real_pow_sqrt_eq_sqrt_pow:
```
```   271       "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)"
```
```   272 apply (simp add: sqrt_def)
```
```   273 apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2)
```
```   274 done
```
```   275
```
```   276 lemma real_pow_sqrt_eq_sqrt_abs_pow2:
```
```   277      "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)"
```
```   278 by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])
```
```   279
```
```   280 lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>"
```
```   281 apply (rule real_sqrt_abs_abs [THEN subst])
```
```   282 apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst])
```
```   283 apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric])
```
```   284 apply (assumption, arith)
```
```   285 done
```
```   286
```
```   287 lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
```
```   288 apply auto
```
```   289 apply (cut_tac x = x and y = 0 in linorder_less_linear)
```
```   290 apply (simp add: zero_less_mult_iff)
```
```   291 done
```
```   292
```
```   293 lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"
```
```   294 by (simp add: sqrt_def real_root_gt_zero)
```
```   295
```
```   296 lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
```
```   297 by (auto intro: real_sqrt_gt_zero simp add: order_le_less)
```
```   298
```
```   299 lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
```
```   300 by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero])
```
```   301
```
```   302
```
```   303 (*we need to prove something like this:
```
```   304 lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r"
```
```   305 apply (case_tac n, simp)
```
```   306 apply (simp add: root_def)
```
```   307 apply (rule someI2 [of _ r], safe)
```
```   308 apply (auto simp del: realpow_Suc dest: power_inject_base)
```
```   309 *)
```
```   310
```
```   311 lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r"
```
```   312 apply (erule subst)
```
```   313 apply (simp add: sqrt_def numeral_2_eq_2 del: realpow_Suc)
```
```   314 apply (erule real_root_pos2)
```
```   315 done
```
```   316
```
```   317 lemma real_sqrt_mult_distrib:
```
```   318      "[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)"
```
```   319 apply (rule sqrt_eqI)
```
```   320 apply (simp add: power_mult_distrib)
```
```   321 apply (simp add: zero_le_mult_iff real_sqrt_ge_zero)
```
```   322 done
```
```   323
```
```   324 lemma real_sqrt_mult_distrib2:
```
```   325      "[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) =  sqrt(x) * sqrt(y)"
```
```   326 by (auto intro: real_sqrt_mult_distrib simp add: order_le_less)
```
```   327
```
```   328 lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
```
```   329 by (auto intro!: real_sqrt_ge_zero)
```
```   330
```
```   331 lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
```
```   332      "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
```
```   333 by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
```
```   334
```
```   335 lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
```
```   336      "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
```
```   337 by (auto simp add: zero_le_mult_iff simp del: realpow_Suc)
```
```   338
```
```   339 lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>"
```
```   340 apply (rule abs_realpow_two [THEN subst])
```
```   341 apply (rule real_sqrt_abs_abs [THEN subst])
```
```   342 apply (subst real_pow_sqrt_eq_sqrt_pow)
```
```   343 apply (auto simp add: numeral_2_eq_2)
```
```   344 done
```
```   345
```
```   346 lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
```
```   347 apply (rule realpow_two [THEN subst])
```
```   348 apply (subst numeral_2_eq_2 [symmetric])
```
```   349 apply (rule real_sqrt_abs)
```
```   350 done
```
```   351
```
```   352 lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
```
```   353 by simp
```
```   354
```
```   355 lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
```
```   356 apply (frule real_sqrt_pow2_gt_zero)
```
```   357 apply (auto simp add: numeral_2_eq_2)
```
```   358 done
```
```   359
```
```   360 lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
```
```   361 by (cut_tac n = 2 and a = "sqrt x" in power_inverse [symmetric], auto)
```
```   362
```
```   363 lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
```
```   364 apply (drule real_le_imp_less_or_eq)
```
```   365 apply (auto dest: real_sqrt_not_eq_zero)
```
```   366 done
```
```   367
```
```   368 lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))"
```
```   369 by (auto simp add: real_sqrt_eq_zero_cancel)
```
```   370
```
```   371 lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
```
```   372 apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe)
```
```   373 apply (rule real_le_trans)
```
```   374 apply (auto simp del: realpow_Suc)
```
```   375 apply (rule_tac n = 1 in realpow_increasing)
```
```   376 apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc)
```
```   377 done
```
```   378
```
```   379 lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)"
```
```   380 apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
```
```   381 done
```
```   382
```
```   383 lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
```
```   384 apply (rule_tac n = 1 in realpow_increasing)
```
```   385 apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp
```
```   386             del: realpow_Suc)
```
```   387 done
```
```   388
```
```   389 lemma sqrt_divide_self_eq:
```
```   390   assumes nneg: "0 \<le> x"
```
```   391   shows "sqrt x / x = inverse (sqrt x)"
```
```   392 proof cases
```
```   393   assume "x=0" thus ?thesis by simp
```
```   394 next
```
```   395   assume nz: "x\<noteq>0"
```
```   396   hence pos: "0<x" using nneg by arith
```
```   397   show ?thesis
```
```   398   proof (rule right_inverse_eq [THEN iffD1, THEN sym])
```
```   399     show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz)
```
```   400     show "inverse (sqrt x) / (sqrt x / x) = 1"
```
```   401       by (simp add: divide_inverse mult_assoc [symmetric]
```
```   402                   power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz)
```
```   403   qed
```
```   404 qed
```
```   405
```
```   406 end
```