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 *)
10 theory NthRoot = SEQ + HSeries:
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.*}
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
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
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
54 subsection{*First Half -- Lemmas First*}
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)
61 done
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
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
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
92 subsection{*Second Half*}
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
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
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
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
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
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
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
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
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 *}
175 end