src/HOL/Hyperreal/NthRoot.thy
 author nipkow Wed Aug 18 11:09:40 2004 +0200 (2004-08-18) changeset 15140 322485b816ac parent 15131 c69542757a4d child 16775 c1b87ef4a1c3 permissions -rw-r--r--
import -> imports
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 *)
9 theory NthRoot
10 imports SEQ HSeries
11 begin
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}.
18   Lemmas about sequences of reals are used to reach the result.
19 *}
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
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)
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
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)
59 subsection{*First Half -- Lemmas First*}
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
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
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!*}
84 apply (rule order_less_trans [of _ 0])
85 apply (auto intro: lemma_nth_realpow_isLub_gt_zero)
86 done
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
98 subsection{*Second Half*}
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
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
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
120 done
123      "0 < r ==>  0 <= r*(1 + -inverse(real (Suc n)))"
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
135 apply (blast intro: order_trans order_less_imp_le power_mono)
136 done
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)
145                 [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
146 apply (auto simp add: real_of_nat_def)
147 done
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
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
161 lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
162 by (blast intro: realpow_pos_nth)
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
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 *}
186 end