author | haftmann |
Fri, 09 Mar 2007 08:45:50 +0100 | |
changeset 22422 | ee19cdb07528 |
parent 21865 | 55cc354fd2d9 |
child 22443 | 346729a55460 |
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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||
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header{*Existence of Nth Root*} |
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||
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theory NthRoot |
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imports SEQ Parity |
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begin |
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|
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definition |
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root :: "[nat, real] \<Rightarrow> real" where |
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"root n x = (THE u. (0 < x \<longrightarrow> 0 < u) \<and> (u ^ n = x))" |
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|
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definition |
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sqrt :: "real \<Rightarrow> real" where |
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"sqrt x = root 2 x" |
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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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|
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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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||
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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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subsection {* Nth Root *} |
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lemma real_root_zero [simp]: "root (Suc n) 0 = 0" |
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apply (simp add: root_def) |
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apply (safe intro!: the_equality power_0_Suc elim!: realpow_zero_zero) |
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done |
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lemma real_root_pow_pos: |
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"0 < x ==> (root (Suc n) x) ^ (Suc n) = x" |
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apply (simp add: root_def del: realpow_Suc) |
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apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp) |
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apply (erule theI' [THEN conjunct2]) |
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done |
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|
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lemma real_root_pow_pos2: "0 \<le> x ==> (root (Suc n) x) ^ (Suc n) = x" |
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by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos) |
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lemma real_root_pos: |
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"0 < x ==> root(Suc n) (x ^ (Suc n)) = x" |
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apply (simp add: root_def) |
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apply (rule the_equality) |
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apply (frule_tac [2] n = n in zero_less_power) |
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apply (auto simp add: zero_less_mult_iff) |
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apply (rule_tac x = u and y = x in linorder_cases) |
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apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less]) |
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apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less]) |
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apply (auto) |
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done |
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|
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lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x" |
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by (auto dest!: real_le_imp_less_or_eq real_root_pos) |
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|
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lemma real_root_gt_zero: |
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"0 < x ==> 0 < root (Suc n) x" |
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apply (simp add: root_def del: realpow_Suc) |
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apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp) |
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apply (erule theI' [THEN conjunct1]) |
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done |
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220 |
|
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221 |
lemma real_root_pos_pos: |
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"0 < x ==> 0 \<le> root(Suc n) x" |
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223 |
by (rule real_root_gt_zero [THEN order_less_imp_le]) |
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224 |
|
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225 |
lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x" |
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226 |
by (auto simp add: order_le_less real_root_gt_zero) |
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227 |
|
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228 |
lemma real_root_one [simp]: "root (Suc n) 1 = 1" |
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229 |
apply (simp add: root_def) |
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230 |
apply (rule the_equality, auto) |
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231 |
apply (rule ccontr) |
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232 |
apply (rule_tac x = u and y = 1 in linorder_cases) |
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233 |
apply (drule_tac n = n in realpow_Suc_less_one) |
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234 |
apply (drule_tac [4] n = n in power_gt1_lemma) |
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235 |
apply (auto) |
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236 |
done |
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237 |
|
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238 |
|
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239 |
subsection{*Square Root*} |
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240 |
|
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241 |
text{*needed because 2 is a binary numeral!*} |
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242 |
lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))" |
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243 |
by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 |
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244 |
add: nat_numeral_0_eq_0 [symmetric]) |
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245 |
|
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246 |
lemma real_sqrt_zero [simp]: "sqrt 0 = 0" |
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247 |
by (simp add: sqrt_def) |
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248 |
|
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249 |
lemma real_sqrt_one [simp]: "sqrt 1 = 1" |
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250 |
by (simp add: sqrt_def) |
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251 |
|
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252 |
lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" |
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253 |
apply (simp add: sqrt_def) |
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254 |
apply (rule iffI) |
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255 |
apply (cut_tac r = "root 2 x" in realpow_two_le) |
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256 |
apply (simp add: numeral_2_eq_2) |
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257 |
apply (subst numeral_2_eq_2) |
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258 |
apply (erule real_root_pow_pos2) |
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259 |
done |
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260 |
|
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261 |
lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>" |
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262 |
by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]]) |
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263 |
|
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264 |
lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x" |
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265 |
by (simp) |
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266 |
|
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267 |
lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>" |
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268 |
by (rule real_sqrt_pow2_iff [THEN iffD2], arith) |
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269 |
|
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270 |
lemma real_pow_sqrt_eq_sqrt_pow: |
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271 |
"0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)" |
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272 |
apply (simp add: sqrt_def) |
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273 |
apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2) |
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274 |
done |
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275 |
|
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276 |
lemma real_pow_sqrt_eq_sqrt_abs_pow2: |
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277 |
"0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" |
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278 |
by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric]) |
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279 |
|
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280 |
lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>" |
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281 |
apply (rule real_sqrt_abs_abs [THEN subst]) |
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282 |
apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst]) |
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283 |
apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric]) |
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284 |
apply (assumption, arith) |
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285 |
done |
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286 |
|
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287 |
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" |
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288 |
apply auto |
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289 |
apply (cut_tac x = x and y = 0 in linorder_less_linear) |
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290 |
apply (simp add: zero_less_mult_iff) |
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291 |
done |
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292 |
|
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293 |
lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)" |
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294 |
by (simp add: sqrt_def real_root_gt_zero) |
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295 |
|
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296 |
lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)" |
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297 |
by (auto intro: real_sqrt_gt_zero simp add: order_le_less) |
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298 |
|
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299 |
lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)" |
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|
300 |
by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) |
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|
301 |
|
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|
302 |
|
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|
303 |
(*we need to prove something like this: |
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|
304 |
lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r" |
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|
305 |
apply (case_tac n, simp) |
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|
306 |
apply (simp add: root_def) |
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|
307 |
apply (rule someI2 [of _ r], safe) |
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|
308 |
apply (auto simp del: realpow_Suc dest: power_inject_base) |
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|
309 |
*) |
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|
310 |
|
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|
311 |
lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r" |
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|
312 |
apply (erule subst) |
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|
313 |
apply (simp add: sqrt_def numeral_2_eq_2 del: realpow_Suc) |
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|
314 |
apply (erule real_root_pos2) |
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|
315 |
done |
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|
316 |
|
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|
317 |
lemma real_sqrt_mult_distrib: |
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|
318 |
"[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" |
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|
319 |
apply (rule sqrt_eqI) |
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|
320 |
apply (simp add: power_mult_distrib) |
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|
321 |
apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) |
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|
322 |
done |
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changeset
|
323 |
|
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|
324 |
lemma real_sqrt_mult_distrib2: |
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|
325 |
"[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" |
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|
326 |
by (auto intro: real_sqrt_mult_distrib simp add: order_le_less) |
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|
327 |
|
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|
328 |
lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
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|
329 |
by (auto intro!: real_sqrt_ge_zero) |
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|
330 |
|
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|
331 |
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: |
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|
332 |
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" |
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|
333 |
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) |
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changeset
|
334 |
|
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|
335 |
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: |
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|
336 |
"sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" |
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|
337 |
by (auto simp add: zero_le_mult_iff simp del: realpow_Suc) |
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|
338 |
|
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|
339 |
lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>" |
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|
340 |
apply (rule abs_realpow_two [THEN subst]) |
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|
341 |
apply (rule real_sqrt_abs_abs [THEN subst]) |
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parents:
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changeset
|
342 |
apply (subst real_pow_sqrt_eq_sqrt_pow) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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changeset
|
343 |
apply (auto simp add: numeral_2_eq_2) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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changeset
|
344 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
345 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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changeset
|
346 |
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" |
fedb901be392
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parents:
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changeset
|
347 |
apply (rule realpow_two [THEN subst]) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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changeset
|
348 |
apply (subst numeral_2_eq_2 [symmetric]) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
349 |
apply (rule real_sqrt_abs) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
350 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
351 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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changeset
|
352 |
lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>" |
fedb901be392
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parents:
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changeset
|
353 |
by simp |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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changeset
|
354 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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changeset
|
355 |
lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0" |
fedb901be392
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parents:
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changeset
|
356 |
apply (frule real_sqrt_pow2_gt_zero) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
357 |
apply (auto simp add: numeral_2_eq_2) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
358 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
359 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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diff
changeset
|
360 |
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" |
20898 | 361 |
by (cut_tac n = 2 and a = "sqrt x" in power_inverse [symmetric], auto) |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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diff
changeset
|
362 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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diff
changeset
|
363 |
lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0" |
fedb901be392
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parents:
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changeset
|
364 |
apply (drule real_le_imp_less_or_eq) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
365 |
apply (auto dest: real_sqrt_not_eq_zero) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
366 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
367 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
368 |
lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))" |
fedb901be392
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parents:
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changeset
|
369 |
by (auto simp add: real_sqrt_eq_zero_cancel) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
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parents:
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diff
changeset
|
370 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
371 |
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)" |
fedb901be392
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parents:
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changeset
|
372 |
apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
373 |
apply (rule real_le_trans) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
374 |
apply (auto simp del: realpow_Suc) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
375 |
apply (rule_tac n = 1 in realpow_increasing) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
376 |
apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
377 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
378 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
379 |
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)" |
fedb901be392
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huffman
parents:
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diff
changeset
|
380 |
apply (simp (no_asm) add: real_add_commute del: realpow_Suc) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
381 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
382 |
|
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
383 |
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
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diff
changeset
|
384 |
apply (rule_tac n = 1 in realpow_increasing) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
385 |
apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
386 |
del: realpow_Suc) |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
387 |
done |
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
388 |
|
14324 | 389 |
end |