src/HOL/Hyperreal/NthRoot.thy
author paulson
Tue Dec 23 17:41:52 2003 +0100 (2003-12-23)
changeset 14324 c9c6832f9b22
parent 14268 5cf13e80be0e
child 14325 94ac3895822f
permissions -rw-r--r--
converting Hyperreal/NthRoot to Isar
     1 (*  Title       : NthRoot.thy
     2     Author      : Jacques D. Fleuriot
     3     Copyright   : 1998  University of Cambridge
     4     Description : Existence of nth root. Adapted from
     5                    http://www.math.unl.edu/~webnotes
     6 *)
     7 
     8 header{*Existence of Nth Root*}
     9 
    10 theory NthRoot = SEQ + HSeries:
    11 
    12 text{*Various lemmas needed for this result. We follow the proof
    13    given by John Lindsay Orr (jorr@math.unl.edu) in his Analysis
    14    Webnotes available on the www at http://www.math.unl.edu/~webnotes
    15    Lemmas about sequences of reals are used to reach the result.*}
    16 
    17 lemma lemma_nth_realpow_non_empty:
    18      "[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"
    19 apply (case_tac "1 <= a")
    20 apply (rule_tac x = "1" in exI)
    21 apply (drule_tac [2] not_real_leE)
    22 apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc])
    23 apply (auto intro!: realpow_Suc_le_self simp add: real_zero_less_one)
    24 done
    25 
    26 lemma lemma_nth_realpow_isUb_ex:
    27      "[| (0::real) < a; 0 < n |]  
    28       ==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
    29 apply (case_tac "1 <= a")
    30 apply (rule_tac x = "a" in exI)
    31 apply (drule_tac [2] not_real_leE)
    32 apply (rule_tac [2] x = "1" in exI)
    33 apply (rule_tac [!] setleI [THEN isUbI])
    34 apply safe
    35 apply (simp_all (no_asm))
    36 apply (rule_tac [!] ccontr)
    37 apply (drule_tac [!] not_real_leE)
    38 apply (drule realpow_ge_self2 , assumption)
    39 apply (drule_tac n = "n" in realpow_less)
    40 apply (assumption+)
    41 apply (drule real_le_trans , assumption)
    42 apply (drule_tac y = "y ^ n" in order_less_le_trans)
    43 apply (assumption , erule real_less_irrefl)
    44 apply (drule_tac n = "n" in real_zero_less_one [THEN realpow_less])
    45 apply auto
    46 done
    47 
    48 lemma nth_realpow_isLub_ex:
    49      "[| (0::real) < a; 0 < n |]  
    50       ==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
    51 apply (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)
    52 done
    53  
    54 subsection{*First Half -- Lemmas First*}
    55 
    56 lemma lemma_nth_realpow_seq:
    57      "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u  
    58            ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"
    59 apply (safe , drule isLubD2 , blast)
    60 apply (simp add: real_le_def)
    61 done
    62 
    63 lemma lemma_nth_realpow_isLub_gt_zero:
    64      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
    65          0 < a; 0 < n |] ==> 0 < u"
    66 apply (drule lemma_nth_realpow_non_empty , auto)
    67 apply (drule_tac y = "s" in isLub_isUb [THEN isUbD])
    68 apply (auto intro: order_less_le_trans)
    69 done
    70 
    71 lemma lemma_nth_realpow_isLub_ge:
    72      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
    73          0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"
    74 apply (safe)
    75 apply (frule lemma_nth_realpow_seq , safe)
    76 apply (auto elim: real_less_asym simp add: real_le_def)
    77 apply (simp add: real_le_def [symmetric])
    78 apply (rule order_less_trans [of _ 0])
    79 apply (auto intro: real_inv_real_of_posnat_gt_zero lemma_nth_realpow_isLub_gt_zero)
    80 done
    81 
    82 text{*First result we want*}
    83 lemma realpow_nth_ge:
    84      "[| (0::real) < a; 0 < n;  
    85      isLub (UNIV::real set)  
    86      {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"
    87 apply (frule lemma_nth_realpow_isLub_ge , safe)
    88 apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])
    89 apply (auto simp add: real_of_nat_def real_of_posnat_Suc)
    90 done
    91 
    92 subsection{*Second Half*}
    93 
    94 lemma less_isLub_not_isUb:
    95      "[| isLub (UNIV::real set) S u; x < u |]  
    96            ==> ~ isUb (UNIV::real set) S x"
    97 apply (safe)
    98 apply (drule isLub_le_isUb)
    99 apply assumption
   100 apply (drule order_less_le_trans)
   101 apply (auto simp add: real_less_not_refl)
   102 done
   103 
   104 lemma not_isUb_less_ex:
   105      "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
   106 apply (rule ccontr , erule swap)
   107 apply (rule setleI [THEN isUbI])
   108 apply (auto simp add: real_le_def)
   109 done
   110 
   111 lemma lemma_nth_realpow_isLub_le:
   112      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
   113        0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real_of_posnat k))) ^ n <= a"
   114 apply (safe)
   115 apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
   116 apply (rule_tac n = "k" in real_mult_less_self)
   117 apply (blast intro: lemma_nth_realpow_isLub_gt_zero)
   118 apply (safe)
   119 apply (drule_tac n = "k" in lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero])
   120 apply (drule_tac [3] conjI [THEN realpow_le2])
   121 apply (rule_tac [3] order_less_imp_le) 
   122 apply (auto intro: order_trans)
   123 done
   124 
   125 text{*Second result we want*}
   126 lemma realpow_nth_le:
   127      "[| (0::real) < a; 0 < n;  
   128      isLub (UNIV::real set)  
   129      {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
   130 apply (frule lemma_nth_realpow_isLub_le , safe)
   131 apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
   132 apply (auto simp add: real_of_nat_def real_of_posnat_Suc)
   133 done
   134 
   135 (*----------- The theorem at last! -----------*)
   136 lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
   137 apply (frule nth_realpow_isLub_ex , auto)
   138 apply (auto intro: realpow_nth_le realpow_nth_ge real_le_anti_sym)
   139 done
   140 
   141 (* positive only *)
   142 lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
   143 apply (frule nth_realpow_isLub_ex , auto)
   144 apply (auto intro: realpow_nth_le realpow_nth_ge real_le_anti_sym lemma_nth_realpow_isLub_gt_zero)
   145 done
   146 
   147 lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
   148 apply (blast intro: realpow_pos_nth)
   149 done
   150 
   151 (* uniqueness of nth positive root *)
   152 lemma realpow_pos_nth_unique:
   153      "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
   154 apply (auto intro!: realpow_pos_nth)
   155 apply (cut_tac x = "r" and y = "y" in linorder_less_linear)
   156 apply auto
   157 apply (drule_tac x = "r" in realpow_less)
   158 apply (drule_tac [4] x = "y" in realpow_less)
   159 apply (auto simp add: real_less_not_refl)
   160 done
   161 
   162 ML
   163 {*
   164 val nth_realpow_isLub_ex = thm"nth_realpow_isLub_ex";
   165 val realpow_nth_ge = thm"realpow_nth_ge";
   166 val less_isLub_not_isUb = thm"less_isLub_not_isUb";
   167 val not_isUb_less_ex = thm"not_isUb_less_ex";
   168 val realpow_nth_le = thm"realpow_nth_le";
   169 val realpow_nth = thm"realpow_nth";
   170 val realpow_pos_nth = thm"realpow_pos_nth";
   171 val realpow_pos_nth2 = thm"realpow_pos_nth2";
   172 val realpow_pos_nth_unique = thm"realpow_pos_nth_unique";
   173 *}
   174 
   175 end