src/HOL/Hyperreal/NthRoot.thy
author huffman
Tue Apr 17 00:55:00 2007 +0200 (2007-04-17)
changeset 22721 d9be18bd7a28
parent 22630 2a9b64b26612
child 22856 eb0e0582092a
permissions -rw-r--r--
moved root and sqrt lemmas from Transcendental.thy 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_of_nat_inverse_le_iff:
   132   "(inverse (real(Suc n)) \<le> r) = (1 \<le> r * real(Suc n))"
   133 by (simp add: inverse_eq_divide pos_divide_le_eq)
   134 
   135 lemma real_mult_add_one_minus_ge_zero:
   136      "0 < r ==>  0 <= r*(1 + -inverse(real (Suc n)))"
   137 by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff)
   138 
   139 lemma lemma_nth_realpow_isLub_le:
   140      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
   141        0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a"
   142 apply safe
   143 apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
   144 apply (rule_tac n = k in real_mult_less_self)
   145 apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe)
   146 apply (drule_tac n = k in
   147         lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+)
   148 apply (blast intro: order_trans order_less_imp_le power_mono) 
   149 done
   150 
   151 text{*Second result we want*}
   152 lemma realpow_nth_le:
   153      "[| (0::real) < a; 0 < n;  
   154      isLub (UNIV::real set)  
   155      {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
   156 apply (frule lemma_nth_realpow_isLub_le, safe)
   157 apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult
   158                 [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
   159 apply (auto simp add: real_of_nat_def)
   160 done
   161 
   162 text{*The theorem at last!*}
   163 lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
   164 apply (frule nth_realpow_isLub_ex, auto)
   165 apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym)
   166 done
   167 
   168 (* positive only *)
   169 lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
   170 apply (frule nth_realpow_isLub_ex, auto)
   171 apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero)
   172 done
   173 
   174 lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
   175 by (blast intro: realpow_pos_nth)
   176 
   177 (* uniqueness of nth positive root *)
   178 lemma realpow_pos_nth_unique:
   179      "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
   180 apply (auto intro!: realpow_pos_nth)
   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)
   184 done
   185 
   186 subsection {* Nth Root *}
   187 
   188 lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
   189 apply (simp add: root_def)
   190 apply (safe intro!: the_equality power_0_Suc elim!: realpow_zero_zero)
   191 done
   192 
   193 lemma real_root_pow_pos: 
   194      "0 < x ==> (root (Suc n) x) ^ (Suc n) = x"
   195 apply (simp add: root_def del: realpow_Suc)
   196 apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp)
   197 apply (erule theI' [THEN conjunct2])
   198 done
   199 
   200 lemma real_root_pow_pos2: "0 \<le> x ==> (root (Suc n) x) ^ (Suc n) = x"
   201 by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos)
   202 
   203 lemma real_root_pos: 
   204      "0 < x ==> root(Suc n) (x ^ (Suc n)) = x"
   205 apply (simp add: root_def)
   206 apply (rule the_equality)
   207 apply (frule_tac [2] n = n in zero_less_power)
   208 apply (auto simp add: zero_less_mult_iff)
   209 apply (rule_tac x = u and y = x in linorder_cases)
   210 apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less])
   211 apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less])
   212 apply (auto)
   213 done
   214 
   215 lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x"
   216 by (auto dest!: real_le_imp_less_or_eq real_root_pos)
   217 
   218 lemma real_root_gt_zero:
   219      "0 < x ==> 0 < root (Suc n) x"
   220 apply (simp add: root_def del: realpow_Suc)
   221 apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp)
   222 apply (erule theI' [THEN conjunct1])
   223 done
   224 
   225 lemma real_root_pos_pos: 
   226      "0 < x ==> 0 \<le> root(Suc n) x"
   227 by (rule real_root_gt_zero [THEN order_less_imp_le])
   228 
   229 lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x"
   230 by (auto simp add: order_le_less real_root_gt_zero)
   231 
   232 lemma real_root_one [simp]: "root (Suc n) 1 = 1"
   233 apply (simp add: root_def)
   234 apply (rule the_equality, auto)
   235 apply (rule ccontr)
   236 apply (rule_tac x = u and y = 1 in linorder_cases)
   237 apply (drule_tac n = n in realpow_Suc_less_one)
   238 apply (drule_tac [4] n = n in power_gt1_lemma)
   239 apply (auto)
   240 done
   241 
   242 lemma real_root_less_mono:
   243      "[| 0 \<le> x; x < y |] ==> root(Suc n) x < root(Suc n) y"
   244 apply (frule order_le_less_trans, assumption)
   245 apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst])
   246 apply (rotate_tac 1, assumption)
   247 apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst])
   248 apply (rotate_tac 3, assumption)
   249 apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le)
   250 apply (frule_tac n = n in real_root_pos_pos_le)
   251 apply (frule_tac n = n in real_root_pos_pos)
   252 apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing)
   253 apply (assumption, assumption)
   254 apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq)
   255 apply auto
   256 apply (drule_tac f = "%x. x ^ (Suc n)" in arg_cong)
   257 apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc)
   258 done
   259 
   260 lemma real_root_le_mono:
   261      "[| 0 \<le> x; x \<le> y |] ==> root(Suc n) x \<le> root(Suc n) y"
   262 apply (drule_tac y = y in order_le_imp_less_or_eq)
   263 apply (auto dest: real_root_less_mono intro: order_less_imp_le)
   264 done
   265 
   266 lemma real_root_less_iff [simp]:
   267      "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)"
   268 apply (auto intro: real_root_less_mono)
   269 apply (rule ccontr, drule linorder_not_less [THEN iffD1])
   270 apply (drule_tac x = y and n = n in real_root_le_mono, auto)
   271 done
   272 
   273 lemma real_root_le_iff [simp]:
   274      "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x \<le> root(Suc n) y) = (x \<le> y)"
   275 apply (auto intro: real_root_le_mono)
   276 apply (simp (no_asm) add: linorder_not_less [symmetric])
   277 apply auto
   278 apply (drule_tac x = y and n = n in real_root_less_mono, auto)
   279 done
   280 
   281 lemma real_root_eq_iff [simp]:
   282      "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)"
   283 apply (auto intro!: order_antisym [where 'a = real])
   284 apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1])
   285 apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto)
   286 done
   287 
   288 lemma real_root_pos_unique:
   289      "[| 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x |] ==> root (Suc n) x = y"
   290 by (auto dest: real_root_pos2 simp del: realpow_Suc)
   291 
   292 lemma real_root_mult:
   293      "[| 0 \<le> x; 0 \<le> y |] 
   294       ==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y"
   295 apply (rule real_root_pos_unique)
   296 apply (auto intro!: real_root_pos_pos_le 
   297             simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2 
   298             simp del: realpow_Suc)
   299 done
   300 
   301 lemma real_root_inverse:
   302      "0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))"
   303 apply (rule real_root_pos_unique)
   304 apply (auto intro: real_root_pos_pos_le 
   305             simp add: power_inverse [symmetric] real_root_pow_pos2 
   306             simp del: realpow_Suc)
   307 done
   308 
   309 lemma real_root_divide: 
   310      "[| 0 \<le> x; 0 \<le> y |]  
   311       ==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)"
   312 apply (simp add: divide_inverse)
   313 apply (auto simp add: real_root_mult real_root_inverse)
   314 done
   315 
   316 
   317 subsection{*Square Root*}
   318 
   319 text{*needed because 2 is a binary numeral!*}
   320 lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"
   321 by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 
   322          add: nat_numeral_0_eq_0 [symmetric])
   323 
   324 lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
   325 by (simp add: sqrt_def)
   326 
   327 lemma real_sqrt_one [simp]: "sqrt 1 = 1"
   328 by (simp add: sqrt_def)
   329 
   330 lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
   331 apply (simp add: sqrt_def)
   332 apply (rule iffI) 
   333  apply (cut_tac r = "root 2 x" in realpow_two_le)
   334  apply (simp add: numeral_2_eq_2)
   335 apply (subst numeral_2_eq_2)
   336 apply (erule real_root_pow_pos2)
   337 done
   338 
   339 lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>"
   340 by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])
   341 
   342 lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x"
   343 by (simp)
   344 
   345 lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>"
   346 by (rule real_sqrt_pow2_iff [THEN iffD2], arith)
   347 
   348 lemma real_pow_sqrt_eq_sqrt_pow: 
   349       "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)"
   350 apply (simp add: sqrt_def)
   351 apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2)
   352 done
   353 
   354 lemma real_pow_sqrt_eq_sqrt_abs_pow2:
   355      "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
   356 by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])
   357 
   358 lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>"
   359 apply (rule real_sqrt_abs_abs [THEN subst])
   360 apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst])
   361 apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric])
   362 apply (assumption, arith)
   363 done
   364 
   365 lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
   366 apply auto
   367 apply (cut_tac x = x and y = 0 in linorder_less_linear)
   368 apply (simp add: zero_less_mult_iff)
   369 done
   370 
   371 lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"
   372 by (simp add: sqrt_def real_root_gt_zero)
   373 
   374 lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
   375 by (auto intro: real_sqrt_gt_zero simp add: order_le_less)
   376 
   377 lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
   378 by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 
   379 
   380 
   381 (*we need to prove something like this:
   382 lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r"
   383 apply (case_tac n, simp) 
   384 apply (simp add: root_def) 
   385 apply (rule someI2 [of _ r], safe)
   386 apply (auto simp del: realpow_Suc dest: power_inject_base)
   387 *)
   388 
   389 lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r"
   390 apply (erule subst)
   391 apply (simp add: sqrt_def numeral_2_eq_2 del: realpow_Suc)
   392 apply (erule real_root_pos2)
   393 done
   394 
   395 lemma real_sqrt_mult_distrib: 
   396      "[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)"
   397 apply (rule sqrt_eqI)
   398 apply (simp add: power_mult_distrib)  
   399 apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
   400 done
   401 
   402 lemma real_sqrt_mult_distrib2:
   403      "[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) =  sqrt(x) * sqrt(y)"
   404 by (auto intro: real_sqrt_mult_distrib simp add: order_le_less)
   405 
   406 lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
   407 by (auto intro!: real_sqrt_ge_zero)
   408 
   409 lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
   410      "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
   411 by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
   412 
   413 lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
   414      "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
   415 by (auto simp add: zero_le_mult_iff simp del: realpow_Suc)
   416 
   417 lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>"
   418 apply (rule abs_realpow_two [THEN subst])
   419 apply (rule real_sqrt_abs_abs [THEN subst])
   420 apply (subst real_pow_sqrt_eq_sqrt_pow)
   421 apply (auto simp add: numeral_2_eq_2)
   422 done
   423 
   424 lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
   425 apply (rule realpow_two [THEN subst])
   426 apply (subst numeral_2_eq_2 [symmetric])
   427 apply (rule real_sqrt_abs)
   428 done
   429 
   430 lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
   431 by simp
   432 
   433 lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
   434 apply (frule real_sqrt_pow2_gt_zero)
   435 apply (auto simp add: numeral_2_eq_2)
   436 done
   437 
   438 lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
   439 by (cut_tac n = 2 and a = "sqrt x" in power_inverse [symmetric], auto)
   440 
   441 lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
   442 apply (drule real_le_imp_less_or_eq)
   443 apply (auto dest: real_sqrt_not_eq_zero)
   444 done
   445 
   446 lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))"
   447 by (auto simp add: real_sqrt_eq_zero_cancel)
   448 
   449 lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
   450 apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe)
   451 apply (rule real_le_trans)
   452 apply (auto simp del: realpow_Suc)
   453 apply (rule_tac n = 1 in realpow_increasing)
   454 apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc)
   455 done
   456 
   457 lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)"
   458 apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
   459 done
   460 
   461 lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
   462 apply (rule_tac n = 1 in realpow_increasing)
   463 apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp 
   464             del: realpow_Suc)
   465 done
   466 
   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 
   484 
   485 lemma real_sqrt_less_mono: "[| 0 \<le> x; x < y |] ==> sqrt(x) < sqrt(y)"
   486 by (simp add: sqrt_def)
   487 
   488 lemma real_sqrt_le_mono: "[| 0 \<le> x; x \<le> y |] ==> sqrt(x) \<le> sqrt(y)"
   489 by (simp add: sqrt_def)
   490 
   491 lemma real_sqrt_less_iff [simp]:
   492      "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) < sqrt(y)) = (x < y)"
   493 by (simp add: sqrt_def)
   494 
   495 lemma real_sqrt_le_iff [simp]:
   496      "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) \<le> sqrt(y)) = (x \<le> y)"
   497 by (simp add: sqrt_def)
   498 
   499 lemma real_sqrt_eq_iff [simp]:
   500      "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) = sqrt(y)) = (x = y)"
   501 by (simp add: sqrt_def)
   502 
   503 lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)"
   504 apply (rule real_sqrt_one [THEN subst], safe)
   505 apply (rule_tac [2] real_sqrt_less_mono)
   506 apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto)
   507 done
   508 
   509 lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)"
   510 apply (rule real_sqrt_one [THEN subst], safe)
   511 apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto)
   512 done
   513 
   514 lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
   515 apply (simp add: divide_inverse)
   516 apply (case_tac "r=0")
   517 apply (auto simp add: mult_ac)
   518 done
   519 
   520 
   521 end