src/HOL/Hyperreal/NthRoot.thy
author huffman
Mon May 14 18:03:25 2007 +0200 (2007-05-14)
changeset 22968 7134874437ac
parent 22961 e499ded5d0fc
child 23009 01c295dd4a36
permissions -rw-r--r--
tuned
     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 {* Nth Roots of Real Numbers *}
     8 
     9 theory NthRoot
    10 imports SEQ Parity
    11 begin
    12 
    13 subsection {* Existence of Nth Root *}
    14 
    15 text {*
    16   Various lemmas needed for this result. We follow the proof given by
    17   John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis
    18   Webnotes available at \url{http://www.math.unl.edu/~webnotes}.
    19 
    20   Lemmas about sequences of reals are used to reach the result.
    21 *}
    22 
    23 lemma lemma_nth_realpow_non_empty:
    24      "[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"
    25 apply (case_tac "1 <= a")
    26 apply (rule_tac x = 1 in exI)
    27 apply (drule_tac [2] linorder_not_le [THEN iffD1])
    28 apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp) 
    29 apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc)
    30 done
    31 
    32 text{*Used only just below*}
    33 lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n"
    34 by (insert power_increasing [of 1 n r], simp)
    35 
    36 lemma lemma_nth_realpow_isUb_ex:
    37      "[| (0::real) < a; 0 < n |]  
    38       ==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
    39 apply (case_tac "1 <= a")
    40 apply (rule_tac x = a in exI)
    41 apply (drule_tac [2] linorder_not_le [THEN iffD1])
    42 apply (rule_tac [2] x = 1 in exI)
    43 apply (rule_tac [!] setleI [THEN isUbI], safe)
    44 apply (simp_all (no_asm))
    45 apply (rule_tac [!] ccontr)
    46 apply (drule_tac [!] linorder_not_le [THEN iffD1])
    47 apply (drule realpow_ge_self2, assumption)
    48 apply (drule_tac n = n in realpow_less)
    49 apply (assumption+)
    50 apply (drule real_le_trans, assumption)
    51 apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp) 
    52 apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto)
    53 done
    54 
    55 lemma nth_realpow_isLub_ex:
    56      "[| (0::real) < a; 0 < n |]  
    57       ==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
    58 by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)
    59 
    60  
    61 subsubsection {* First Half -- Lemmas First *}
    62 
    63 lemma lemma_nth_realpow_seq:
    64      "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u  
    65            ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"
    66 apply (safe, drule isLubD2, blast)
    67 apply (simp add: linorder_not_less [symmetric])
    68 done
    69 
    70 lemma lemma_nth_realpow_isLub_gt_zero:
    71      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
    72          0 < a; 0 < n |] ==> 0 < u"
    73 apply (drule lemma_nth_realpow_non_empty, auto)
    74 apply (drule_tac y = s in isLub_isUb [THEN isUbD])
    75 apply (auto intro: order_less_le_trans)
    76 done
    77 
    78 lemma lemma_nth_realpow_isLub_ge:
    79      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
    80          0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"
    81 apply safe
    82 apply (frule lemma_nth_realpow_seq, safe)
    83 apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric]
    84             iff: real_0_less_add_iff) --{*legacy iff rule!*}
    85 apply (simp add: linorder_not_less)
    86 apply (rule order_less_trans [of _ 0])
    87 apply (auto intro: lemma_nth_realpow_isLub_gt_zero)
    88 done
    89 
    90 text{*First result we want*}
    91 lemma realpow_nth_ge:
    92      "[| (0::real) < a; 0 < n;  
    93      isLub (UNIV::real set)  
    94      {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"
    95 apply (frule lemma_nth_realpow_isLub_ge, safe)
    96 apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])
    97 apply (auto simp add: real_of_nat_def)
    98 done
    99 
   100 subsubsection {* Second Half *}
   101 
   102 lemma less_isLub_not_isUb:
   103      "[| isLub (UNIV::real set) S u; x < u |]  
   104            ==> ~ isUb (UNIV::real set) S x"
   105 apply safe
   106 apply (drule isLub_le_isUb, assumption)
   107 apply (drule order_less_le_trans, auto)
   108 done
   109 
   110 lemma not_isUb_less_ex:
   111      "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
   112 apply (rule ccontr, erule contrapos_np)
   113 apply (rule setleI [THEN isUbI])
   114 apply (auto simp add: linorder_not_less [symmetric])
   115 done
   116 
   117 lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r"
   118 apply (simp (no_asm) add: right_distrib)
   119 apply (rule add_less_cancel_left [of "-r", THEN iffD1])
   120 apply (auto intro: mult_pos_pos
   121             simp add: add_assoc [symmetric] neg_less_0_iff_less)
   122 done
   123 
   124 lemma real_of_nat_inverse_le_iff:
   125   "(inverse (real(Suc n)) \<le> r) = (1 \<le> r * real(Suc n))"
   126 by (simp add: inverse_eq_divide pos_divide_le_eq)
   127 
   128 lemma real_mult_add_one_minus_ge_zero:
   129      "0 < r ==>  0 <= r*(1 + -inverse(real (Suc n)))"
   130 by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff)
   131 
   132 lemma lemma_nth_realpow_isLub_le:
   133      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
   134        0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a"
   135 apply safe
   136 apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
   137 apply (rule_tac n = k in real_mult_less_self)
   138 apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe)
   139 apply (drule_tac n = k in
   140         lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+)
   141 apply (blast intro: order_trans order_less_imp_le power_mono) 
   142 done
   143 
   144 text{*Second result we want*}
   145 lemma realpow_nth_le:
   146      "[| (0::real) < a; 0 < n;  
   147      isLub (UNIV::real set)  
   148      {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
   149 apply (frule lemma_nth_realpow_isLub_le, safe)
   150 apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult
   151                 [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
   152 apply (auto simp add: real_of_nat_def)
   153 done
   154 
   155 text{*The theorem at last!*}
   156 lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
   157 apply (frule nth_realpow_isLub_ex, auto)
   158 apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym)
   159 done
   160 
   161 text {* positive only *}
   162 lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
   163 apply (frule nth_realpow_isLub_ex, auto)
   164 apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero)
   165 done
   166 
   167 lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
   168 by (blast intro: realpow_pos_nth)
   169 
   170 text {* uniqueness of nth positive root *}
   171 lemma realpow_pos_nth_unique:
   172      "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
   173 apply (auto intro!: realpow_pos_nth)
   174 apply (cut_tac x = r and y = y in linorder_less_linear, auto)
   175 apply (drule_tac x = r in realpow_less)
   176 apply (drule_tac [4] x = y in realpow_less, auto)
   177 done
   178 
   179 subsection {* Nth Root *}
   180 
   181 text {* We define roots of negative reals such that
   182   @{term "root n (- x) = - root n x"}. This allows
   183   us to omit side conditions from many theorems. *}
   184 
   185 definition
   186   root :: "[nat, real] \<Rightarrow> real" where
   187   "root n x = (if 0 < x then (THE u. 0 < u \<and> u ^ n = x) else
   188                if x < 0 then - (THE u. 0 < u \<and> u ^ n = - x) else 0)"
   189 
   190 lemma real_root_zero [simp]: "root n 0 = 0"
   191 unfolding root_def by simp
   192 
   193 lemma real_root_minus: "0 < n \<Longrightarrow> root n (- x) = - root n x"
   194 unfolding root_def by simp
   195 
   196 lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x"
   197 apply (simp add: root_def)
   198 apply (drule (1) realpow_pos_nth_unique)
   199 apply (erule theI' [THEN conjunct1])
   200 done
   201 
   202 lemma real_root_pow_pos: (* TODO: rename *)
   203   "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x"
   204 apply (simp add: root_def)
   205 apply (drule (1) realpow_pos_nth_unique)
   206 apply (erule theI' [THEN conjunct2])
   207 done
   208 
   209 lemma real_root_pow_pos2 [simp]: (* TODO: rename *)
   210   "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x"
   211 by (auto simp add: order_le_less real_root_pow_pos)
   212 
   213 lemma real_root_ge_zero: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> 0 \<le> root n x"
   214 by (auto simp add: order_le_less real_root_gt_zero)
   215 
   216 lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x"
   217 apply (subgoal_tac "0 \<le> x ^ n")
   218 apply (subgoal_tac "0 \<le> root n (x ^ n)")
   219 apply (subgoal_tac "root n (x ^ n) ^ n = x ^ n")
   220 apply (erule (3) power_eq_imp_eq_base)
   221 apply (erule (1) real_root_pow_pos2)
   222 apply (erule (1) real_root_ge_zero)
   223 apply (erule zero_le_power)
   224 done
   225 
   226 lemma real_root_pos_unique:
   227   "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
   228 by (erule subst, rule real_root_power_cancel)
   229 
   230 lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1"
   231 by (simp add: real_root_pos_unique)
   232 
   233 text {* Root function is strictly monotonic, hence injective *}
   234 
   235 lemma real_root_less_mono_lemma:
   236   "\<lbrakk>0 < n; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"
   237 apply (subgoal_tac "0 \<le> y")
   238 apply (subgoal_tac "root n x ^ n < root n y ^ n")
   239 apply (erule power_less_imp_less_base)
   240 apply (erule (1) real_root_ge_zero)
   241 apply simp
   242 apply simp
   243 done
   244 
   245 lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"
   246 apply (cases "0 \<le> x")
   247 apply (erule (2) real_root_less_mono_lemma)
   248 apply (cases "0 \<le> y")
   249 apply (rule_tac y=0 in order_less_le_trans)
   250 apply (subgoal_tac "0 < root n (- x)")
   251 apply (simp add: real_root_minus)
   252 apply (simp add: real_root_gt_zero)
   253 apply (simp add: real_root_ge_zero)
   254 apply (subgoal_tac "root n (- y) < root n (- x)")
   255 apply (simp add: real_root_minus)
   256 apply (simp add: real_root_less_mono_lemma)
   257 done
   258 
   259 lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y"
   260 by (auto simp add: order_le_less real_root_less_mono)
   261 
   262 lemma real_root_less_iff [simp]:
   263   "0 < n \<Longrightarrow> (root n x < root n y) = (x < y)"
   264 apply (cases "x < y")
   265 apply (simp add: real_root_less_mono)
   266 apply (simp add: linorder_not_less real_root_le_mono)
   267 done
   268 
   269 lemma real_root_le_iff [simp]:
   270   "0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)"
   271 apply (cases "x \<le> y")
   272 apply (simp add: real_root_le_mono)
   273 apply (simp add: linorder_not_le real_root_less_mono)
   274 done
   275 
   276 lemma real_root_eq_iff [simp]:
   277   "0 < n \<Longrightarrow> (root n x = root n y) = (x = y)"
   278 by (simp add: order_eq_iff)
   279 
   280 lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified]
   281 lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified]
   282 lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified]
   283 lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified]
   284 lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified]
   285 
   286 text {* Roots of multiplication and division *}
   287 
   288 lemma real_root_mult_lemma:
   289   "\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> root n (x * y) = root n x * root n y"
   290 by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib)
   291 
   292 lemma real_root_inverse_lemma:
   293   "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (inverse x) = inverse (root n x)"
   294 by (simp add: real_root_pos_unique power_inverse [symmetric])
   295 
   296 lemma real_root_mult:
   297   assumes n: "0 < n"
   298   shows "root n (x * y) = root n x * root n y"
   299 proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases)
   300   assume "0 \<le> x" and "0 \<le> y"
   301   thus ?thesis by (rule real_root_mult_lemma [OF n])
   302 next
   303   assume "0 \<le> x" and "y \<le> 0"
   304   hence "0 \<le> x" and "0 \<le> - y" by simp_all
   305   hence "root n (x * - y) = root n x * root n (- y)"
   306     by (rule real_root_mult_lemma [OF n])
   307   thus ?thesis by (simp add: real_root_minus [OF n])
   308 next
   309   assume "x \<le> 0" and "0 \<le> y"
   310   hence "0 \<le> - x" and "0 \<le> y" by simp_all
   311   hence "root n (- x * y) = root n (- x) * root n y"
   312     by (rule real_root_mult_lemma [OF n])
   313   thus ?thesis by (simp add: real_root_minus [OF n])
   314 next
   315   assume "x \<le> 0" and "y \<le> 0"
   316   hence "0 \<le> - x" and "0 \<le> - y" by simp_all
   317   hence "root n (- x * - y) = root n (- x) * root n (- y)"
   318     by (rule real_root_mult_lemma [OF n])
   319   thus ?thesis by (simp add: real_root_minus [OF n])
   320 qed
   321 
   322 lemma real_root_inverse:
   323   assumes n: "0 < n"
   324   shows "root n (inverse x) = inverse (root n x)"
   325 proof (rule linorder_le_cases)
   326   assume "0 \<le> x"
   327   thus ?thesis by (rule real_root_inverse_lemma [OF n])
   328 next
   329   assume "x \<le> 0"
   330   hence "0 \<le> - x" by simp
   331   hence "root n (inverse (- x)) = inverse (root n (- x))"
   332     by (rule real_root_inverse_lemma [OF n])
   333   thus ?thesis by (simp add: real_root_minus [OF n])
   334 qed
   335 
   336 lemma real_root_divide:
   337   "0 < n \<Longrightarrow> root n (x / y) = root n x / root n y"
   338 by (simp add: divide_inverse real_root_mult real_root_inverse)
   339 
   340 lemma real_root_power:
   341   "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k"
   342 by (induct k, simp_all add: real_root_mult)
   343 
   344 
   345 subsection {* Square Root *}
   346 
   347 definition
   348   sqrt :: "real \<Rightarrow> real" where
   349   "sqrt = root 2"
   350 
   351 lemma pos2: "0 < (2::nat)" by simp
   352 
   353 lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y"
   354 unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])
   355 
   356 lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>"
   357 apply (rule real_sqrt_unique)
   358 apply (rule power2_abs)
   359 apply (rule abs_ge_zero)
   360 done
   361 
   362 lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x"
   363 unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
   364 
   365 lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
   366 apply (rule iffI)
   367 apply (erule subst)
   368 apply (rule zero_le_power2)
   369 apply (erule real_sqrt_pow2)
   370 done
   371 
   372 lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
   373 unfolding sqrt_def by (rule real_root_zero)
   374 
   375 lemma real_sqrt_one [simp]: "sqrt 1 = 1"
   376 unfolding sqrt_def by (rule real_root_one [OF pos2])
   377 
   378 lemma real_sqrt_minus: "sqrt (- x) = - sqrt x"
   379 unfolding sqrt_def by (rule real_root_minus [OF pos2])
   380 
   381 lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"
   382 unfolding sqrt_def by (rule real_root_mult [OF pos2])
   383 
   384 lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"
   385 unfolding sqrt_def by (rule real_root_inverse [OF pos2])
   386 
   387 lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"
   388 unfolding sqrt_def by (rule real_root_divide [OF pos2])
   389 
   390 lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k"
   391 unfolding sqrt_def by (rule real_root_power [OF pos2])
   392 
   393 lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x"
   394 unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])
   395 
   396 lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x"
   397 unfolding sqrt_def by (rule real_root_ge_zero [OF pos2])
   398 
   399 lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y"
   400 unfolding sqrt_def by (rule real_root_less_mono [OF pos2])
   401 
   402 lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y"
   403 unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
   404 
   405 lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
   406 unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
   407 
   408 lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)"
   409 unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
   410 
   411 lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
   412 unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
   413 
   414 lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified]
   415 lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified]
   416 lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified]
   417 lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified]
   418 lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified]
   419 
   420 lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified]
   421 lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified]
   422 lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified]
   423 lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified]
   424 lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified]
   425 
   426 lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
   427 apply auto
   428 apply (cut_tac x = x and y = 0 in linorder_less_linear)
   429 apply (simp add: zero_less_mult_iff)
   430 done
   431 
   432 lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
   433 apply (subst power2_eq_square [symmetric])
   434 apply (rule real_sqrt_abs)
   435 done
   436 
   437 lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
   438 by simp (* TODO: delete *)
   439 
   440 lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
   441 by simp (* TODO: delete *)
   442 
   443 lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
   444 by (simp add: power_inverse [symmetric])
   445 
   446 lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
   447 by simp
   448 
   449 lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
   450 by simp
   451 
   452 lemma sqrt_divide_self_eq:
   453   assumes nneg: "0 \<le> x"
   454   shows "sqrt x / x = inverse (sqrt x)"
   455 proof cases
   456   assume "x=0" thus ?thesis by simp
   457 next
   458   assume nz: "x\<noteq>0" 
   459   hence pos: "0<x" using nneg by arith
   460   show ?thesis
   461   proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
   462     show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 
   463     show "inverse (sqrt x) / (sqrt x / x) = 1"
   464       by (simp add: divide_inverse mult_assoc [symmetric] 
   465                   power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
   466   qed
   467 qed
   468 
   469 lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
   470 apply (simp add: divide_inverse)
   471 apply (case_tac "r=0")
   472 apply (auto simp add: mult_ac)
   473 done
   474 
   475 subsection {* Square Root of Sum of Squares *}
   476 
   477 lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
   478 by (rule real_sqrt_ge_zero [OF sum_squares_ge_zero])
   479 
   480 lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
   481 by simp
   482 
   483 lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
   484      "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
   485 by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
   486 
   487 lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
   488      "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
   489 by (auto simp add: zero_le_mult_iff)
   490 
   491 lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
   492 by (rule power2_le_imp_le, simp_all)
   493 
   494 lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
   495 by (rule power2_le_imp_le, simp_all)
   496 
   497 lemma power2_sum:
   498   fixes x y :: "'a::{number_ring,recpower}"
   499   shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"
   500 by (simp add: left_distrib right_distrib power2_eq_square)
   501 
   502 lemma power2_diff:
   503   fixes x y :: "'a::{number_ring,recpower}"
   504   shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"
   505 by (simp add: left_diff_distrib right_diff_distrib power2_eq_square)
   506 
   507 lemma real_sqrt_sum_squares_triangle_ineq:
   508   "sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)"
   509 apply (rule power2_le_imp_le, simp)
   510 apply (simp add: power2_sum)
   511 apply (simp only: mult_assoc right_distrib [symmetric])
   512 apply (rule mult_left_mono)
   513 apply (rule power2_le_imp_le)
   514 apply (simp add: power2_sum power_mult_distrib)
   515 apply (simp add: ring_distrib)
   516 apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior> - 2 * (a * c) * (b * d)", simp)
   517 apply (rule_tac b="(a * d - b * c)\<twosuperior>" in ord_le_eq_trans)
   518 apply (rule zero_le_power2)
   519 apply (simp add: power2_diff power_mult_distrib)
   520 apply (simp add: mult_nonneg_nonneg)
   521 apply simp
   522 apply (simp add: add_increasing)
   523 done
   524 
   525 text "Legacy theorem names:"
   526 lemmas real_root_pos2 = real_root_power_cancel
   527 lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
   528 lemmas real_root_pos_pos_le = real_root_ge_zero
   529 lemmas real_sqrt_mult_distrib = real_sqrt_mult
   530 lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
   531 lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff
   532 
   533 (* needed for CauchysMeanTheorem.het_base from AFP *)
   534 lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x"
   535 by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le])
   536 
   537 (* FIXME: the stronger version of real_root_less_iff
   538  breaks CauchysMeanTheorem.list_gmean_gt_iff from AFP. *)
   539 
   540 declare real_root_less_iff [simp del]
   541 lemma real_root_less_iff_nonneg [simp]:
   542   "\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> (root n x < root n y) = (x < y)"
   543 by (rule real_root_less_iff)
   544 
   545 end