src/HOL/Hyperreal/NthRoot.thy
changeset 14324 c9c6832f9b22
parent 14268 5cf13e80be0e
child 14325 94ac3895822f
--- a/src/HOL/Hyperreal/NthRoot.thy	Tue Dec 23 16:53:33 2003 +0100
+++ b/src/HOL/Hyperreal/NthRoot.thy	Tue Dec 23 17:41:52 2003 +0100
@@ -5,4 +5,171 @@
                    http://www.math.unl.edu/~webnotes
 *)
 
-NthRoot = SEQ + HSeries
+header{*Existence of Nth Root*}
+
+theory NthRoot = SEQ + HSeries:
+
+text{*Various lemmas needed for this result. We follow the proof
+   given by John Lindsay Orr (jorr@math.unl.edu) in his Analysis
+   Webnotes available on the www at http://www.math.unl.edu/~webnotes
+   Lemmas about sequences of reals are used to reach the result.*}
+
+lemma lemma_nth_realpow_non_empty:
+     "[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"
+apply (case_tac "1 <= a")
+apply (rule_tac x = "1" in exI)
+apply (drule_tac [2] not_real_leE)
+apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc])
+apply (auto intro!: realpow_Suc_le_self simp add: real_zero_less_one)
+done
+
+lemma lemma_nth_realpow_isUb_ex:
+     "[| (0::real) < a; 0 < n |]  
+      ==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
+apply (case_tac "1 <= a")
+apply (rule_tac x = "a" in exI)
+apply (drule_tac [2] not_real_leE)
+apply (rule_tac [2] x = "1" in exI)
+apply (rule_tac [!] setleI [THEN isUbI])
+apply safe
+apply (simp_all (no_asm))
+apply (rule_tac [!] ccontr)
+apply (drule_tac [!] not_real_leE)
+apply (drule realpow_ge_self2 , assumption)
+apply (drule_tac n = "n" in realpow_less)
+apply (assumption+)
+apply (drule real_le_trans , assumption)
+apply (drule_tac y = "y ^ n" in order_less_le_trans)
+apply (assumption , erule real_less_irrefl)
+apply (drule_tac n = "n" in real_zero_less_one [THEN realpow_less])
+apply auto
+done
+
+lemma nth_realpow_isLub_ex:
+     "[| (0::real) < a; 0 < n |]  
+      ==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
+apply (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)
+done
+ 
+subsection{*First Half -- Lemmas First*}
+
+lemma lemma_nth_realpow_seq:
+     "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u  
+           ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"
+apply (safe , drule isLubD2 , blast)
+apply (simp add: real_le_def)
+done
+
+lemma lemma_nth_realpow_isLub_gt_zero:
+     "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
+         0 < a; 0 < n |] ==> 0 < u"
+apply (drule lemma_nth_realpow_non_empty , auto)
+apply (drule_tac y = "s" in isLub_isUb [THEN isUbD])
+apply (auto intro: order_less_le_trans)
+done
+
+lemma lemma_nth_realpow_isLub_ge:
+     "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
+         0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"
+apply (safe)
+apply (frule lemma_nth_realpow_seq , safe)
+apply (auto elim: real_less_asym simp add: real_le_def)
+apply (simp add: real_le_def [symmetric])
+apply (rule order_less_trans [of _ 0])
+apply (auto intro: real_inv_real_of_posnat_gt_zero lemma_nth_realpow_isLub_gt_zero)
+done
+
+text{*First result we want*}
+lemma realpow_nth_ge:
+     "[| (0::real) < a; 0 < n;  
+     isLub (UNIV::real set)  
+     {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"
+apply (frule lemma_nth_realpow_isLub_ge , safe)
+apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])
+apply (auto simp add: real_of_nat_def real_of_posnat_Suc)
+done
+
+subsection{*Second Half*}
+
+lemma less_isLub_not_isUb:
+     "[| isLub (UNIV::real set) S u; x < u |]  
+           ==> ~ isUb (UNIV::real set) S x"
+apply (safe)
+apply (drule isLub_le_isUb)
+apply assumption
+apply (drule order_less_le_trans)
+apply (auto simp add: real_less_not_refl)
+done
+
+lemma not_isUb_less_ex:
+     "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
+apply (rule ccontr , erule swap)
+apply (rule setleI [THEN isUbI])
+apply (auto simp add: real_le_def)
+done
+
+lemma lemma_nth_realpow_isLub_le:
+     "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
+       0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real_of_posnat k))) ^ n <= a"
+apply (safe)
+apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
+apply (rule_tac n = "k" in real_mult_less_self)
+apply (blast intro: lemma_nth_realpow_isLub_gt_zero)
+apply (safe)
+apply (drule_tac n = "k" in lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero])
+apply (drule_tac [3] conjI [THEN realpow_le2])
+apply (rule_tac [3] order_less_imp_le) 
+apply (auto intro: order_trans)
+done
+
+text{*Second result we want*}
+lemma realpow_nth_le:
+     "[| (0::real) < a; 0 < n;  
+     isLub (UNIV::real set)  
+     {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
+apply (frule lemma_nth_realpow_isLub_le , safe)
+apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
+apply (auto simp add: real_of_nat_def real_of_posnat_Suc)
+done
+
+(*----------- The theorem at last! -----------*)
+lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
+apply (frule nth_realpow_isLub_ex , auto)
+apply (auto intro: realpow_nth_le realpow_nth_ge real_le_anti_sym)
+done
+
+(* positive only *)
+lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
+apply (frule nth_realpow_isLub_ex , auto)
+apply (auto intro: realpow_nth_le realpow_nth_ge real_le_anti_sym lemma_nth_realpow_isLub_gt_zero)
+done
+
+lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
+apply (blast intro: realpow_pos_nth)
+done
+
+(* uniqueness of nth positive root *)
+lemma realpow_pos_nth_unique:
+     "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
+apply (auto intro!: realpow_pos_nth)
+apply (cut_tac x = "r" and y = "y" in linorder_less_linear)
+apply auto
+apply (drule_tac x = "r" in realpow_less)
+apply (drule_tac [4] x = "y" in realpow_less)
+apply (auto simp add: real_less_not_refl)
+done
+
+ML
+{*
+val nth_realpow_isLub_ex = thm"nth_realpow_isLub_ex";
+val realpow_nth_ge = thm"realpow_nth_ge";
+val less_isLub_not_isUb = thm"less_isLub_not_isUb";
+val not_isUb_less_ex = thm"not_isUb_less_ex";
+val realpow_nth_le = thm"realpow_nth_le";
+val realpow_nth = thm"realpow_nth";
+val realpow_pos_nth = thm"realpow_pos_nth";
+val realpow_pos_nth2 = thm"realpow_pos_nth2";
+val realpow_pos_nth_unique = thm"realpow_pos_nth_unique";
+*}
+
+end