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--
```     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
```