--- a/src/HOL/Hyperreal/NthRoot.thy Fri May 18 16:13:07 2007 +0200
+++ b/src/HOL/Hyperreal/NthRoot.thy Fri May 18 17:35:07 2007 +0200
@@ -7,173 +7,42 @@
header {* Nth Roots of Real Numbers *}
theory NthRoot
-imports SEQ Parity
+imports SEQ Parity Deriv
begin
subsection {* Existence of Nth Root *}
-text {*
- Various lemmas needed for this result. We follow the proof given by
- John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis
- Webnotes available at \url{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] linorder_not_le [THEN iffD1])
-apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp)
-apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc)
-done
-
-text{*Used only just below*}
-lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n"
-by (insert power_increasing [of 1 n r], simp)
-
-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] linorder_not_le [THEN iffD1])
-apply (rule_tac [2] x = 1 in exI)
-apply (rule_tac [!] setleI [THEN isUbI], safe)
-apply (simp_all (no_asm))
-apply (rule_tac [!] ccontr)
-apply (drule_tac [!] linorder_not_le [THEN iffD1])
-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, assumption, simp)
-apply (drule_tac n = n in zero_less_one [THEN realpow_less], 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"
-by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)
-
-
-subsubsection {* 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: linorder_not_less [symmetric])
-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: order_less_asym simp add: linorder_not_less [symmetric]
- iff: real_0_less_add_iff) --{*legacy iff rule!*}
-apply (simp add: linorder_not_less)
-apply (rule order_less_trans [of _ 0])
-apply (auto intro: lemma_nth_realpow_isLub_gt_zero)
-done
+text {* Existence follows from the Intermediate Value Theorem *}
-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)
-done
-
-subsubsection {* 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, assumption)
-apply (drule order_less_le_trans, auto)
-done
-
-lemma not_isUb_less_ex:
- "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
-apply (rule ccontr, erule contrapos_np)
-apply (rule setleI [THEN isUbI])
-apply (auto simp add: linorder_not_less [symmetric])
-done
-
-lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r"
-apply (simp (no_asm) add: right_distrib)
-apply (rule add_less_cancel_left [of "-r", THEN iffD1])
-apply (auto intro: mult_pos_pos
- simp add: add_assoc [symmetric] neg_less_0_iff_less)
-done
-
-lemma real_of_nat_inverse_le_iff:
- "(inverse (real(Suc n)) \<le> r) = (1 \<le> r * real(Suc n))"
-by (simp add: inverse_eq_divide pos_divide_le_eq)
-
-lemma real_mult_add_one_minus_ge_zero:
- "0 < r ==> 0 <= r*(1 + -inverse(real (Suc n)))"
-by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff)
+lemma realpow_pos_nth:
+ assumes n: "0 < n"
+ assumes a: "0 < a"
+ shows "\<exists>r>0. r ^ n = (a::real)"
+proof -
+ have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a"
+ proof (rule IVT)
+ show "0 ^ n \<le> a" using n a by (simp add: power_0_left)
+ show "0 \<le> max 1 a" by simp
+ from n have n1: "1 \<le> n" by simp
+ have "a \<le> max 1 a ^ 1" by simp
+ also have "max 1 a ^ 1 \<le> max 1 a ^ n"
+ using n1 by (rule power_increasing, simp)
+ finally show "a \<le> max 1 a ^ n" .
+ show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r"
+ by (simp add: isCont_power isCont_Id)
+ qed
+ then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast
+ with n a have "r \<noteq> 0" by (auto simp add: power_0_left)
+ with r have "0 < r \<and> r ^ n = a" by simp
+ thus ?thesis ..
+qed
-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 (Suc 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, safe)
-apply (drule_tac n = k in
- lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+)
-apply (blast intro: order_trans order_less_imp_le power_mono)
-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)
-done
+text {* Uniqueness of nth positive root *}
-text{*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 order_antisym)
-done
-
-text {* 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 order_antisym lemma_nth_realpow_isLub_gt_zero)
-done
-
-lemma realpow_pos_nth2: "(0::real) < a ==> \<exists>r. 0 < r & r ^ Suc n = a"
-by (blast intro: realpow_pos_nth)
-
-text {* uniqueness of nth positive root *}
lemma realpow_pos_nth_unique:
- "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
+ "\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)"
apply (auto intro!: realpow_pos_nth)
-apply (cut_tac x = r and y = y in linorder_less_linear, auto)
-apply (drule_tac x = r in realpow_less)
-apply (drule_tac [4] x = y in realpow_less, auto)
+apply (rule_tac n=n in power_eq_imp_eq_base, simp_all)
done
subsection {* Nth Root *}
@@ -341,7 +210,6 @@
"0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k"
by (induct k, simp_all add: real_root_mult)
-
subsection {* Square Root *}
definition