author | wenzelm |
Thu, 05 Jan 2006 22:29:55 +0100 | |
changeset 18585 | 5d379fe2eb74 |
parent 16775 | c1b87ef4a1c3 |
child 19765 | dfe940911617 |
permissions | -rw-r--r-- |
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(* Title : NthRoot.thy |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Conversion to Isar and new proofs by Lawrence C Paulson, 2004 |
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*) |
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header{*Existence of Nth Root*} |
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theory NthRoot |
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imports SEQ HSeries |
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begin |
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text {* |
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Various lemmas needed for this result. We follow the proof given by |
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John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis |
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Webnotes available at \url{http://www.math.unl.edu/~webnotes}. |
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Lemmas about sequences of reals are used to reach the result. |
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*} |
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lemma lemma_nth_realpow_non_empty: |
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"[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}" |
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apply (case_tac "1 <= a") |
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apply (rule_tac x = 1 in exI) |
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apply (drule_tac [2] linorder_not_le [THEN iffD1]) |
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apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp) |
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apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc) |
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done |
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text{*Used only just below*} |
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lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n" |
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by (insert power_increasing [of 1 n r], simp) |
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lemma lemma_nth_realpow_isUb_ex: |
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"[| (0::real) < a; 0 < n |] |
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==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u" |
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apply (case_tac "1 <= a") |
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apply (rule_tac x = a in exI) |
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apply (drule_tac [2] linorder_not_le [THEN iffD1]) |
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apply (rule_tac [2] x = 1 in exI) |
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apply (rule_tac [!] setleI [THEN isUbI], safe) |
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apply (simp_all (no_asm)) |
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apply (rule_tac [!] ccontr) |
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apply (drule_tac [!] linorder_not_le [THEN iffD1]) |
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apply (drule realpow_ge_self2, assumption) |
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apply (drule_tac n = n in realpow_less) |
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apply (assumption+) |
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apply (drule real_le_trans, assumption) |
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apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp) |
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apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto) |
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done |
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lemma nth_realpow_isLub_ex: |
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"[| (0::real) < a; 0 < n |] |
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==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u" |
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by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete) |
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subsection{*First Half -- Lemmas First*} |
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lemma lemma_nth_realpow_seq: |
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"isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u |
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==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}" |
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apply (safe, drule isLubD2, blast) |
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apply (simp add: linorder_not_less [symmetric]) |
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done |
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lemma lemma_nth_realpow_isLub_gt_zero: |
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"[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; |
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0 < a; 0 < n |] ==> 0 < u" |
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apply (drule lemma_nth_realpow_non_empty, auto) |
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apply (drule_tac y = s in isLub_isUb [THEN isUbD]) |
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apply (auto intro: order_less_le_trans) |
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done |
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lemma lemma_nth_realpow_isLub_ge: |
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"[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; |
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0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n" |
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apply safe |
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apply (frule lemma_nth_realpow_seq, safe) |
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apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric] |
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iff: real_0_less_add_iff) --{*legacy iff rule!*} |
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apply (simp add: linorder_not_less) |
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apply (rule order_less_trans [of _ 0]) |
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apply (auto intro: lemma_nth_realpow_isLub_gt_zero) |
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done |
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text{*First result we want*} |
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lemma realpow_nth_ge: |
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"[| (0::real) < a; 0 < n; |
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isLub (UNIV::real set) |
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{x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n" |
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apply (frule lemma_nth_realpow_isLub_ge, safe) |
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apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const]) |
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apply (auto simp add: real_of_nat_def) |
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done |
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subsection{*Second Half*} |
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lemma less_isLub_not_isUb: |
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"[| isLub (UNIV::real set) S u; x < u |] |
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==> ~ isUb (UNIV::real set) S x" |
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apply safe |
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apply (drule isLub_le_isUb, assumption) |
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apply (drule order_less_le_trans, auto) |
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done |
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lemma not_isUb_less_ex: |
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"~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x" |
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apply (rule ccontr, erule contrapos_np) |
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apply (rule setleI [THEN isUbI]) |
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apply (auto simp add: linorder_not_less [symmetric]) |
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done |
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lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r" |
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apply (simp (no_asm) add: right_distrib) |
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apply (rule add_less_cancel_left [of "-r", THEN iffD1]) |
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apply (auto intro: mult_pos_pos |
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simp add: add_assoc [symmetric] neg_less_0_iff_less) |
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done |
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lemma real_mult_add_one_minus_ge_zero: |
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"0 < r ==> 0 <= r*(1 + -inverse(real (Suc n)))" |
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by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff) |
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lemma lemma_nth_realpow_isLub_le: |
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"[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; |
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0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a" |
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apply safe |
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apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex]) |
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apply (rule_tac n = k in real_mult_less_self) |
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apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe) |
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apply (drule_tac n = k in |
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lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+) |
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apply (blast intro: order_trans order_less_imp_le power_mono) |
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done |
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text{*Second result we want*} |
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lemma realpow_nth_le: |
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"[| (0::real) < a; 0 < n; |
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isLub (UNIV::real set) |
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{x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a" |
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apply (frule lemma_nth_realpow_isLub_le, safe) |
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apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult |
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[THEN LIMSEQ_pow, THEN LIMSEQ_le_const2]) |
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apply (auto simp add: real_of_nat_def) |
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done |
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text{*The theorem at last!*} |
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lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a" |
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apply (frule nth_realpow_isLub_ex, auto) |
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apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym) |
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done |
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(* positive only *) |
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lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a" |
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apply (frule nth_realpow_isLub_ex, auto) |
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apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero) |
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done |
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lemma realpow_pos_nth2: "(0::real) < a ==> \<exists>r. 0 < r & r ^ Suc n = a" |
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by (blast intro: realpow_pos_nth) |
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(* uniqueness of nth positive root *) |
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lemma realpow_pos_nth_unique: |
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"[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a" |
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apply (auto intro!: realpow_pos_nth) |
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apply (cut_tac x = r and y = y in linorder_less_linear, auto) |
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apply (drule_tac x = r in realpow_less) |
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apply (drule_tac [4] x = y in realpow_less, auto) |
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done |
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ML |
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{* |
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val nth_realpow_isLub_ex = thm"nth_realpow_isLub_ex"; |
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val realpow_nth_ge = thm"realpow_nth_ge"; |
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val less_isLub_not_isUb = thm"less_isLub_not_isUb"; |
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val not_isUb_less_ex = thm"not_isUb_less_ex"; |
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val realpow_nth_le = thm"realpow_nth_le"; |
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val realpow_nth = thm"realpow_nth"; |
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val realpow_pos_nth = thm"realpow_pos_nth"; |
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val realpow_pos_nth2 = thm"realpow_pos_nth2"; |
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val realpow_pos_nth_unique = thm"realpow_pos_nth_unique"; |
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*} |
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end |