
paulson@12196: (*  Title       : NthRoot.thy
paulson@12196:     Author      : Jacques D. Fleuriot
paulson@12196:     Copyright   : 1998  University of Cambridge
paulson@12196:     Description : Existence of nth root. Adapted from
paulson@12196:                    http://www.math.unl.edu/~webnotes
paulson@12196: *)
paulson@12196: 
paulson@14324: header{*Existence of Nth Root*}
paulson@14324: 
paulson@14324: theory NthRoot = SEQ + HSeries:
paulson@14324: 
paulson@14324: text{*Various lemmas needed for this result. We follow the proof
paulson@14324:    given by John Lindsay Orr (jorr@math.unl.edu) in his Analysis
paulson@14324:    Webnotes available on the www at http://www.math.unl.edu/~webnotes
paulson@14324:    Lemmas about sequences of reals are used to reach the result.*}
paulson@14324: 
paulson@14324: lemma lemma_nth_realpow_non_empty:
paulson@14324:      "[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"
paulson@14324: apply (case_tac "1 <= a")
paulson@14324: apply (rule_tac x = "1" in exI)
paulson@14334: apply (drule_tac [2] linorder_not_le [THEN iffD1])
paulson@14324: apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc])
paulson@14348: apply (simp add: ); 
paulson@14348: apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc)
paulson@14324: done
paulson@14324: 
paulson@14348: text{*Used only just below*}
paulson@14348: lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n"
paulson@14348: by (insert power_increasing [of 1 n r], simp)
paulson@14348: 
paulson@14324: lemma lemma_nth_realpow_isUb_ex:
paulson@14324:      "[| (0::real) < a; 0 < n |]  
paulson@14324:       ==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
paulson@14324: apply (case_tac "1 <= a")
paulson@14324: apply (rule_tac x = "a" in exI)
paulson@14334: apply (drule_tac [2] linorder_not_le [THEN iffD1])
paulson@14324: apply (rule_tac [2] x = "1" in exI)
paulson@14324: apply (rule_tac [!] setleI [THEN isUbI])
paulson@14324: apply safe
paulson@14324: apply (simp_all (no_asm))
paulson@14324: apply (rule_tac [!] ccontr)
paulson@14334: apply (drule_tac [!] linorder_not_le [THEN iffD1])
paulson@14324: apply (drule realpow_ge_self2 , assumption)
paulson@14324: apply (drule_tac n = "n" in realpow_less)
paulson@14324: apply (assumption+)
paulson@14324: apply (drule real_le_trans , assumption)
paulson@14324: apply (drule_tac y = "y ^ n" in order_less_le_trans)
paulson@14365: apply (assumption)
paulson@14365: apply (simp); 
paulson@14334: apply (drule_tac n = "n" in zero_less_one [THEN realpow_less])
paulson@14324: apply auto
paulson@14324: done
paulson@14324: 
paulson@14324: lemma nth_realpow_isLub_ex:
paulson@14324:      "[| (0::real) < a; 0 < n |]  
paulson@14324:       ==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
paulson@14365: by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)
paulson@14365: 
paulson@14324:  
paulson@14324: subsection{*First Half -- Lemmas First*}
paulson@14324: 
paulson@14324: lemma lemma_nth_realpow_seq:
paulson@14324:      "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u  
paulson@14324:            ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"
paulson@14324: apply (safe , drule isLubD2 , blast)
paulson@14365: apply (simp add: linorder_not_less [symmetric])
paulson@14324: done
paulson@14324: 
paulson@14324: lemma lemma_nth_realpow_isLub_gt_zero:
paulson@14324:      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
paulson@14324:          0 < a; 0 < n |] ==> 0 < u"
paulson@14324: apply (drule lemma_nth_realpow_non_empty , auto)
paulson@14324: apply (drule_tac y = "s" in isLub_isUb [THEN isUbD])
paulson@14324: apply (auto intro: order_less_le_trans)
paulson@14324: done
paulson@14324: 
paulson@14324: lemma lemma_nth_realpow_isLub_ge:
paulson@14324:      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
paulson@14324:          0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"
paulson@14324: apply (safe)
paulson@14324: apply (frule lemma_nth_realpow_seq , safe)
paulson@14365: apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric])
paulson@14365: apply (simp add: linorder_not_less)
paulson@14324: apply (rule order_less_trans [of _ 0])
paulson@14325: apply (auto intro: lemma_nth_realpow_isLub_gt_zero)
paulson@14324: done
paulson@14324: 
paulson@14324: text{*First result we want*}
paulson@14324: lemma realpow_nth_ge:
paulson@14324:      "[| (0::real) < a; 0 < n;  
paulson@14324:      isLub (UNIV::real set)  
paulson@14324:      {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"
paulson@14324: apply (frule lemma_nth_realpow_isLub_ge , safe)
paulson@14324: apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])
paulson@14334: apply (auto simp add: real_of_nat_def)
paulson@14324: done
paulson@14324: 
paulson@14324: subsection{*Second Half*}
paulson@14324: 
paulson@14324: lemma less_isLub_not_isUb:
paulson@14324:      "[| isLub (UNIV::real set) S u; x < u |]  
paulson@14324:            ==> ~ isUb (UNIV::real set) S x"
paulson@14324: apply (safe)
paulson@14324: apply (drule isLub_le_isUb)
paulson@14324: apply assumption
paulson@14324: apply (drule order_less_le_trans)
paulson@14365: apply (auto)
paulson@14324: done
paulson@14324: 
paulson@14324: lemma not_isUb_less_ex:
paulson@14324:      "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
paulson@14324: apply (rule ccontr , erule swap)
paulson@14324: apply (rule setleI [THEN isUbI])
paulson@14365: apply (auto simp add: linorder_not_less [symmetric])
paulson@14324: done
paulson@14324: 
paulson@14325: lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r"
paulson@14334: apply (simp (no_asm) add: right_distrib)
paulson@14334: apply (rule add_less_cancel_left [of "-r", THEN iffD1])
paulson@14334: apply (auto intro: mult_pos
paulson@14334:             simp add: add_assoc [symmetric] neg_less_0_iff_less)
paulson@14325: done
paulson@14325: 
paulson@14325: lemma real_mult_add_one_minus_ge_zero:
paulson@14325:      "0 < r ==>  0 <= r*(1 + -inverse(real (Suc n)))"
nipkow@14355: apply (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff)
paulson@14325: done
paulson@14325: 
paulson@14324: lemma lemma_nth_realpow_isLub_le:
paulson@14324:      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
paulson@14325:        0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a"
paulson@14324: apply (safe)
paulson@14324: apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
paulson@14324: apply (rule_tac n = "k" in real_mult_less_self)
paulson@14324: apply (blast intro: lemma_nth_realpow_isLub_gt_zero)
paulson@14324: apply (safe)
paulson@14348: apply (drule_tac n = "k" in
paulson@14348:         lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero])
paulson@14348: apply assumption+
paulson@14348: apply (blast intro: order_trans order_less_imp_le power_mono) 
paulson@14324: done
paulson@14324: 
paulson@14324: text{*Second result we want*}
paulson@14324: lemma realpow_nth_le:
paulson@14324:      "[| (0::real) < a; 0 < n;  
paulson@14324:      isLub (UNIV::real set)  
paulson@14324:      {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
paulson@14324: apply (frule lemma_nth_realpow_isLub_le , safe)
paulson@14348: apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult
paulson@14348:                 [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
paulson@14334: apply (auto simp add: real_of_nat_def)
paulson@14324: done
paulson@14324: 
paulson@14348: text{*The theorem at last!*}
paulson@14324: lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
paulson@14324: apply (frule nth_realpow_isLub_ex , auto)
paulson@14324: apply (auto intro: realpow_nth_le realpow_nth_ge real_le_anti_sym)
paulson@14324: done
paulson@14324: 
paulson@14324: (* positive only *)
paulson@14324: lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
paulson@14324: apply (frule nth_realpow_isLub_ex , auto)
paulson@14324: apply (auto intro: realpow_nth_le realpow_nth_ge real_le_anti_sym lemma_nth_realpow_isLub_gt_zero)
paulson@14324: done
paulson@14324: 
paulson@14324: lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
paulson@14324: apply (blast intro: realpow_pos_nth)
paulson@14324: done
paulson@14324: 
paulson@14324: (* uniqueness of nth positive root *)
paulson@14324: lemma realpow_pos_nth_unique:
paulson@14324:      "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
paulson@14324: apply (auto intro!: realpow_pos_nth)
paulson@14324: apply (cut_tac x = "r" and y = "y" in linorder_less_linear)
paulson@14324: apply auto
paulson@14324: apply (drule_tac x = "r" in realpow_less)
paulson@14324: apply (drule_tac [4] x = "y" in realpow_less)
paulson@14365: apply (auto)
paulson@14324: done
paulson@14324: 
paulson@14324: ML
paulson@14324: {*
paulson@14324: val nth_realpow_isLub_ex = thm"nth_realpow_isLub_ex";
paulson@14324: val realpow_nth_ge = thm"realpow_nth_ge";
paulson@14324: val less_isLub_not_isUb = thm"less_isLub_not_isUb";
paulson@14324: val not_isUb_less_ex = thm"not_isUb_less_ex";
paulson@14324: val realpow_nth_le = thm"realpow_nth_le";
paulson@14324: val realpow_nth = thm"realpow_nth";
paulson@14324: val realpow_pos_nth = thm"realpow_pos_nth";
paulson@14324: val realpow_pos_nth2 = thm"realpow_pos_nth2";
paulson@14324: val realpow_pos_nth_unique = thm"realpow_pos_nth_unique";
paulson@14324: *}
paulson@14324: 
paulson@14324: end




