src/HOL/Hyperreal/NthRoot.thy
author nipkow
Mon Aug 16 14:22:27 2004 +0200 (2004-08-16)
changeset 15131 c69542757a4d
parent 15085 5693a977a767
child 15140 322485b816ac
permissions -rw-r--r--
New theory header syntax.
     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 import SEQ HSeries
    11 begin
    12 
    13 text {*
    14   Various lemmas needed for this result. We follow the proof given by
    15   John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis
    16   Webnotes available at \url{http://www.math.unl.edu/~webnotes}.
    17 
    18   Lemmas about sequences of reals are used to reach the result.
    19 *}
    20 
    21 lemma lemma_nth_realpow_non_empty:
    22      "[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"
    23 apply (case_tac "1 <= a")
    24 apply (rule_tac x = 1 in exI)
    25 apply (drule_tac [2] linorder_not_le [THEN iffD1])
    26 apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp) 
    27 apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc)
    28 done
    29 
    30 text{*Used only just below*}
    31 lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n"
    32 by (insert power_increasing [of 1 n r], simp)
    33 
    34 lemma lemma_nth_realpow_isUb_ex:
    35      "[| (0::real) < a; 0 < n |]  
    36       ==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
    37 apply (case_tac "1 <= a")
    38 apply (rule_tac x = a in exI)
    39 apply (drule_tac [2] linorder_not_le [THEN iffD1])
    40 apply (rule_tac [2] x = 1 in exI)
    41 apply (rule_tac [!] setleI [THEN isUbI], safe)
    42 apply (simp_all (no_asm))
    43 apply (rule_tac [!] ccontr)
    44 apply (drule_tac [!] linorder_not_le [THEN iffD1])
    45 apply (drule realpow_ge_self2, assumption)
    46 apply (drule_tac n = n in realpow_less)
    47 apply (assumption+)
    48 apply (drule real_le_trans, assumption)
    49 apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp) 
    50 apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto)
    51 done
    52 
    53 lemma nth_realpow_isLub_ex:
    54      "[| (0::real) < a; 0 < n |]  
    55       ==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
    56 by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)
    57 
    58  
    59 subsection{*First Half -- Lemmas First*}
    60 
    61 lemma lemma_nth_realpow_seq:
    62      "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u  
    63            ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"
    64 apply (safe, drule isLubD2, blast)
    65 apply (simp add: linorder_not_less [symmetric])
    66 done
    67 
    68 lemma lemma_nth_realpow_isLub_gt_zero:
    69      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
    70          0 < a; 0 < n |] ==> 0 < u"
    71 apply (drule lemma_nth_realpow_non_empty, auto)
    72 apply (drule_tac y = s in isLub_isUb [THEN isUbD])
    73 apply (auto intro: order_less_le_trans)
    74 done
    75 
    76 lemma lemma_nth_realpow_isLub_ge:
    77      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
    78          0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"
    79 apply safe
    80 apply (frule lemma_nth_realpow_seq, safe)
    81 apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric]
    82             iff: real_0_less_add_iff) --{*legacy iff rule!*}
    83 apply (simp add: linorder_not_less)
    84 apply (rule order_less_trans [of _ 0])
    85 apply (auto intro: lemma_nth_realpow_isLub_gt_zero)
    86 done
    87 
    88 text{*First result we want*}
    89 lemma realpow_nth_ge:
    90      "[| (0::real) < a; 0 < n;  
    91      isLub (UNIV::real set)  
    92      {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"
    93 apply (frule lemma_nth_realpow_isLub_ge, safe)
    94 apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])
    95 apply (auto simp add: real_of_nat_def)
    96 done
    97 
    98 subsection{*Second Half*}
    99 
   100 lemma less_isLub_not_isUb:
   101      "[| isLub (UNIV::real set) S u; x < u |]  
   102            ==> ~ isUb (UNIV::real set) S x"
   103 apply safe
   104 apply (drule isLub_le_isUb, assumption)
   105 apply (drule order_less_le_trans, auto)
   106 done
   107 
   108 lemma not_isUb_less_ex:
   109      "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
   110 apply (rule ccontr, erule swap)
   111 apply (rule setleI [THEN isUbI])
   112 apply (auto simp add: linorder_not_less [symmetric])
   113 done
   114 
   115 lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r"
   116 apply (simp (no_asm) add: right_distrib)
   117 apply (rule add_less_cancel_left [of "-r", THEN iffD1])
   118 apply (auto intro: mult_pos
   119             simp add: add_assoc [symmetric] neg_less_0_iff_less)
   120 done
   121 
   122 lemma real_mult_add_one_minus_ge_zero:
   123      "0 < r ==>  0 <= r*(1 + -inverse(real (Suc n)))"
   124 by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff)
   125 
   126 lemma lemma_nth_realpow_isLub_le:
   127      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
   128        0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a"
   129 apply safe
   130 apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
   131 apply (rule_tac n = k in real_mult_less_self)
   132 apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe)
   133 apply (drule_tac n = k in
   134         lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+)
   135 apply (blast intro: order_trans order_less_imp_le power_mono) 
   136 done
   137 
   138 text{*Second result we want*}
   139 lemma realpow_nth_le:
   140      "[| (0::real) < a; 0 < n;  
   141      isLub (UNIV::real set)  
   142      {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
   143 apply (frule lemma_nth_realpow_isLub_le, safe)
   144 apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult
   145                 [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
   146 apply (auto simp add: real_of_nat_def)
   147 done
   148 
   149 text{*The theorem at last!*}
   150 lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
   151 apply (frule nth_realpow_isLub_ex, auto)
   152 apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym)
   153 done
   154 
   155 (* positive only *)
   156 lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
   157 apply (frule nth_realpow_isLub_ex, auto)
   158 apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero)
   159 done
   160 
   161 lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
   162 by (blast intro: realpow_pos_nth)
   163 
   164 (* uniqueness of nth positive root *)
   165 lemma realpow_pos_nth_unique:
   166      "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
   167 apply (auto intro!: realpow_pos_nth)
   168 apply (cut_tac x = r and y = y in linorder_less_linear, auto)
   169 apply (drule_tac x = r in realpow_less)
   170 apply (drule_tac [4] x = y in realpow_less, auto)
   171 done
   172 
   173 ML
   174 {*
   175 val nth_realpow_isLub_ex = thm"nth_realpow_isLub_ex";
   176 val realpow_nth_ge = thm"realpow_nth_ge";
   177 val less_isLub_not_isUb = thm"less_isLub_not_isUb";
   178 val not_isUb_less_ex = thm"not_isUb_less_ex";
   179 val realpow_nth_le = thm"realpow_nth_le";
   180 val realpow_nth = thm"realpow_nth";
   181 val realpow_pos_nth = thm"realpow_pos_nth";
   182 val realpow_pos_nth2 = thm"realpow_pos_nth2";
   183 val realpow_pos_nth_unique = thm"realpow_pos_nth_unique";
   184 *}
   185 
   186 end